Part VIII - The Standard Model
5 likes3,316 views
The document summarizes key aspects of the Standard Model of particle physics. It describes how the Standard Model accounts for fundamental particles like quarks and leptons that interact via four fundamental forces - gravitation, electromagnetism, weak force, and strong force. These interactions are mediated by exchange of spin-1/2 bosons. The Standard Model has been very successful in explaining experimental observations, but questions remain like incorporating gravity and the origin of particle masses.
![Each of the quarks in the first Table can exist in three forms distinguished by the so-
called color quantum number associated with the strong interaction coupling (i.e., an
interaction is where the point particles or their constituents actually mix together via a
potential – highlighted as in Figures) which can take the values red, blue, or green.
9
2017
MRT
)]q()q([3 NNB −=
There are no known interactions that mix quarks with leptons and hence the total
quark number (i.e., the number of quarks minus the number of antiquarks, N(q)−N(q))
is conserved. This is referred to as the baryon number conservation since:
This rule is crucial to the stability of protons and hence of matter itself.
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-9-320.jpg)
![Because of this vacuum polarization screening effect, the electromagnetic coupling
varies with distance, r (see Figure), so that the fine structure constant α is not really a
constant at all but varies with r. It turns out that the e+e− loop gives:
17
A negative charge, represented by a ‘point’
particle, which is ‘screened’ by vacuum
polarization (i.e., the creation of e+e− pairs)
[how organized the spread of polarization is
is a matter of speculation] making the whole
QED-picture-view of an electron as ‘fuzzy’.
2017
MRT
In QCD it is found that similarly (i.e., for r>>hc/ΛC):
1
o
ln
π2
)(
−
Λ
≅
r
cb
r
C
s
h
α
1
e
1ln
π3
1)(
−
+−≅
crm
r
hα
αα
where α is the value measured at large r (i.e., r>>h/mec, the electron’s Compton
wavelength). Evidently, α is predicted to become very large at very short distances (i.e.,
as r→h/[mecexp(3π/α)]≅10−300 m), by which point perturbation theory must break down.
Our equation above still indicates that QED alone cannot be correct at high energies!
where bo is a constant (see Renormalization chapter), is positive
because the self-coupling of the gluons results in antiscreening.
Hence αs →0 as r→0, which gives rise to the so-called asymp-
totic freedom (i.e., the fact that quarks and gluons inside a
hadron behave like free particles when very close together). On
the other hand, αs apparently diverges as r→RC ≡hc/ΛC, where
ΛC is the hadronic energy scale. This divergence simply heralds
the breakdown of perturbation theory, of course, but nonetheless
it leads us to expect that the strength of the force between
quarks will increase if they are pulled apart. As a result, quarks
and gluons are confined inside hadrons which size is of order RC.
+
−
+
−
+
−
+
−
+
−
+ −+− −−−−
+
−
+
−
+
−](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-17-320.jpg)
![As we have just seen, the strength of the color coupling, αs, is not constant but
becomes stronger as the separation between the quarks increases (i.e., αs(r)≅1/[(bo/2π)
⋅ln(hc/ΛC r)]). So, unlike electrons in atoms, quarks are permanently bound together, and
confined within hadrons. It is only color-neutral combinations of quarks that can have a
vanishing color coupling and hence occur as free particles. There are two different ways
in which these colorless hadrons can be made, given the three different colors of
quarks qk (k = R for red,G for green, andB for blue):
19
2017
MRT
∑=+−=
kji
kji
kji 321
B
3
R
2
G
1
B
3
G
2
R
1 qqq
6
1
)qqqqqq(
6
1
B εK
The Hadrons
where εijk is the anti-symmetric permutation tensor, or;
∑=++=
k
kk
21
B
2
B
1
G
2
G
1
R
2
R
1 qq
3
1
)qqqqqq(
3
1
M
These hadrons are color-neutral in a somewhat analogous way to that whereby
one can produce white either by an equal mixture of the three primary colors (red,
green, and blue) like B above, or by mixing green (say) with its complementary
color, magenta (=red+blue), like M above.
2) We can take a quark q1 of one color and an antiquark q2 which has the opposite
(complementary) color, and thereby make a meson:
1) We can take an admixture of three quarks, q1, q2, and q3, say, which have different
colors, and thereby obtain a baryon:
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-19-320.jpg)
![In particles made of heavy quarks (i.e., mqc2 >>ΛC), such as the ψ(cc) charmonium
states, the ϒ(bb) bottomonium states (N.B., the high mass of the top quark, toponium tt,
does not exist since the top quark decays through the electroweak interaction before a
bound state can form), the velocities of the quarks are very much less than the speed of
light, c, and relativistic effects should not be too important. In such cases it is reasonably
satisfactory to treat the particles as if they were made of just their constituent valence
quarks, with an interaction potential between the q and q that takes the form:
25
2017
MRT
r
r
rV s )(
3
4
)(
α
−=
where the factor of 4/3 comes from the sum over all possible colors of quarks, and over
the colors of the gluons that can be exchanged.
Since αs varies with r in αs(r)≅1/[(bo/2π)ln(hc/ΛC r)], this can be rewritten:
Λ
−≈
r
c
rb
rV
C
h
ln
π2
3
4
)(
o
or, to represent better the behavior at large r (i.e., where αs(r) ≅1/[(bo/2π)ln(hc/ΛC r)] is
invalid):
rT
r
rV s
o
3
4
)( +−≈
α
where To is called the ‘string tension’.
_
_
_
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-25-320.jpg)
![Another important reason for this confidence in QCD, and indeed in the whole
SU(3)⊗SU(2)⊗U(1) Standard Model of the strong, weak, and electromagnetic properties
of matter, is that it correctly predicts the outcome of a great variety of scattering
experiments: electron-positron scattering (e.g., e+e−), deep inelastic scattering (e.g., e−p),
and hadron scattering (e.g., pp), to name a few.
27
In a two-body scattering process such as (see Figure - Left):
dcba +→+
Scattering
According to quantum theory a high-energy beam of energy E=p/c has an associated
wavelength λ=hc/E (i.e., λ∝1/E), which determines the shortest distance which can be
resolved. Hence, to probe short distances we require very high energies.
each particle has four-vector momentum matrix:
T
][ pcEp =µ
where µ =0,1,2,3 corresponding to the timelike, 0, and the
three spacelike components, 1,2,3, respectively, E being its
energy and p its momentum, in a given Lorentz frame. We will
employ the usual Minkowski space-time metric, ηµν , with sig-
nature (+,−,−,−), so that the Lorentz-invariant scalar product:
2
o
2
23
0
2
)()( cm
c
E
ppppppp =−
=≡≡⋅≡ ∑∑=
p
µν
ν
µν
µ
µ
µ
µ
η
where mo is the is the particle’s rest mass (moc being the
reference or standard momentum kµ ). It is usually more
convenient to work in units where c≡ 1 so this mass-shell
condition above becomes p2 =m2 with mo≡m from now on.
_
(Left) The scattering process a +b→ c +d
with the four-momenta labelled. (Right)
This same process in the center-of-mass
(CM) system. Here θ is the scattering
angle of particle c with respect to the
beam direction and d Ω is the element of
solid angle into which c is scattered.
2017
MRT](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-27-320.jpg)
![The scattering process can be described by three Lorentz-invariant Mandelstam
variables:
28
2017
MRT
22
22
22
)()(
)()(
)()(
bcda
bdca
dcba
ppppu
ppppt
pppps
−=−≡
−=−≡
+=+≡
which can readily be shown to satisfy:
2222
dcba mmmmuts +++=++
so only two of the s, t, or u are independent.
],[],[ pp −== bbaa EpEp and
represents the square of the total center-of-mass energy, ECM ≡Ea +Eb, while the center-
of-mass frame scattering angle θ is given by:
θcos222 2222
cacacacaca EEmmppmmt pp+−+=⋅−+=
At high energies, where all the masses are negligible, so that |pa|~Ea, &c., we find:
)cos1(
2
)cos1(
2
θθ +−≈−−≈
s
u
s
t and
In the center-of-mass system of the scattering process (see previous Figure - Right):
22
],[ CMEEEs ba =+= 0
and:](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-28-320.jpg)
![For the purpose of keeping the relativistic covariance of the physics more apparent, it
is convenient to replace L by the Lagrangian density function L so that:
31
2017
MRT
∫= x3
dL L
and thus the action S above can also be rewritten covariantly as:
∫∫∫ === xddtdtdLS 43
LL x
since d4x=dtd3x where x is the space-time four-vector with components xµ =[t,x].
In relativistic field theory, we replace qi by the field φ(xµ), so the index i is replaced by
the space-time coordinates x≡xµ, and the Lagrangian is a Lorentz scalar function of
φ(x). Since each ∂/∂x can be associated with a similar term involving ∇∇∇∇, we can make up
the covariant derivative (and its equivalent shorthand notation):
0
)()()(
3
0
3
1
=
∂
∂
−
∂∂
∂
∂=
∂
∂
−
∂∂
∂
∂
∂
+
∂∂
∂
∂
∂
∑∑ ==
φφφφφ µ µ
µ
LLLLL
i i
i
t xt
So, for a given choice of Lagrangian L(φ,∂µφ), this set of space-time Euler-Lagrange
equations can be used to deduce the equations that will be satisfied by the field φ(x).
and so the Euler-Lagrange equations become:
φ
φ
µµ
∂≡
∂
∂
x](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-31-320.jpg)
![Thus, for a free scalar (i.e.,spin-0) particle of (rest) mass m the Lagrangian density is:
32
2017
MRT
22
2
1
))((
2
1
φφφ
µ
µ
µ m−∂∂= ∑L
and so the differentials are:
φφ
φφ
φφφ
φφ
µ
µ
µ
µ
µµ
222
2
1
0)(0))((
2
1
)()(
mm −=
∂
∂
−=
∂
∂
∂=−
∂∂
∂∂
∂
=
∂∂
∂
∑
LL
and
which, when substituted into the space-time Euler-Lagrange equations, gives the Klein-
Gordon equation:
0)][(
)(
222
=+=+
∂∂=+∂∂=
∂
∂
−
∂∂
∂
∂ ∑∑∑ φφφφφφ
φφ µ
µ
µ
µ
µ
µ
µ µ
µ mmm
LL
where we have introduced the d’Alembertian operator:
2
2
2
∇−
∂
∂
=∂∂= ∑ tµ
µ
µ
In view of the usual quantum-mechanical operator replacements (h≡1) E→i∂/∂t and p→
−i∇∇∇∇ which, in four-vector notation, given [E,p]→[i∂/∂t,−i∇∇∇∇]=i[∂/∂t,−∇∇∇∇], gives the four-
momentum as:
µµ
∂−→ ip
Now, φ +m2φ =0 simply ensures that the field obeys the mass-shell condition p2=m2.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-32-320.jpg)
![The flow of particles is represented in space-time by the current four-vector:
33
2017
MRT
*)*( φφφφ µµµ
∂−∂= iJ
],[ Jρµ
=J
where ρ is the particle density and J is the flux current. It is a conserved quantity:
0=∂=•+
∂
∂
∑µ
µ
µ
ρ
J
t
J∇∇∇∇
For example, the current for the scalar boson field φ is:
In order to discuss vector (i.e., spin-h) particles, we need to consider properties of the
classical electromagnetic field, which are summarized in Maxwell’s equations:
0
B
EB
J
E
BE
=
∂
∂
=•
=
∂
∂
=•
t
t
++++××××∇∇∇∇∇∇∇∇
−−−−××××∇∇∇∇∇∇∇∇
and
,,
0
ρ
where E and B are the electric and magnetic field strengths, respectively. Here ρ is the
charge density and J is the charge-current density (which, unlike Jµ =[ρ,J] includes a
factor of e). As usual in particle physics, we use the Heaviside-Lorentz units where εo ≡1.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-33-320.jpg)
![It is convenient to introduce the four-vector potential:
34
2017
MRT
µννµµν AAF ∂−∂=
],[ Aφµ
=A
so that:
φ∇∇∇∇−−−−××××∇∇∇∇
t∂
∂
−==
A
EAB and
Then, since ∇∇∇∇•∇∇∇∇××××A=0 and ∇∇∇∇××××∇∇∇∇φ =0, the last two Maxwell equations (i.e., ∇∇∇∇•B=0 and
∇∇∇∇××××E++++∂B/∂t=0) are satisfied automatically, while the first two (i.e., ∇∇∇∇•E=ρ and ∇∇∇∇××××B−−−−
∂E/∂t=J) can be re-expressed in terms of the field-strength tensor:
in the form:
νµ
µ
µ
νν
µ
µν
µ JAAF =
∂∂−=∂ ∑∑
where Jµ =[ρ,J] is the charge-current four-vector that satisfies Σµ∂µ Jµ =0.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-34-320.jpg)
![For a free spin-h particle of (rest) mass M, we take:
37
2017
MRT
instead of L =−¼Σµν Fµν Fµν−Σµ Jµ Aµ above with Jµ =0. This leads, using the space-time
Euler-Lagrange equations again, to the Proca equation:
∑∑ +−=
µ
µ
µ
µν
µν
µν AAMFF 2
2
1
4
1
L
02
=+∂∑ ν
µ
µν
µ AMF
If we take the divergence of this equation (i.e., operate on it with ∂ν), we find Σν ∂ν Aν =0
(for M2 ≠0). This is not a necessary condition (i.e., not a choice of gauge). The Proca
equation then becomes:
0)( 2
=+ ν
AM
which leads again to the mass-shell condition p2 =M2≠0.
Polarization vectors of a massive spin-h particle automatically satisfy Σµ qµε µ =0, but as
there is no gauge invariance there are three degrees of freedom, corresponding to
helicities ±1 and 0. For example, a particle with momentum q along the z-axis has ε (r=±1)
=m(1/√2)[0,1,±i,0] and ε (r=0)=(1/M)[|q|,0,0,E]. They satisfy the completeness relation:
2
)()(
*][
M
qq
r
rr νµ
µννµ ηεε +−=∑
Ok,from now on we set h≡1 so as to describe spin-h particles will be spin-1 particles.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-37-320.jpg)
![In order that the equation obtained by operating with (iΣµγ µ∂µ +m) on (iΣµγ µ∂µ −m)ψ =0
should be the Klein-GordonequationΣµ∂µ ∂µφ +m2φ =0(so as to guarantee that E2 =p2 +m2),
we require (i.e., the Clifford algebra):
39
2017
MRT
µνµννµνµ
ηγγγγγγ 2},{ =+≡
called an anticommutator (to contrast with the commutator [σµ,σ ν ]≡σµσν −σνσµ). All of
the spin properties of the theory follow from this equation for the γ matrices and
particular representations are not necessary. However, it is often much easier to work
with a suitable chosen representation. We shall take the γ matrices to be unitary, so
from the anticommutator {γ µ,γν} above we find (Exercise):
kk
γγγγ −== †0†0
and
These equations can be summarized by the relation:
00†
γγγγ µ k
=
The ‘standard’ (or Pauli-Dirac) representation is given by:
−
=
=
0
0
0
00
k
kk
I
I
σ
σ
γγ and
where I is the 2×2 unit matrix and the σ k are the usual 2×2 Pauli matrices:
−
=
−
=
=
10
01
0
0
01
10
321 σσσ and,
i
i](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-39-320.jpg)
![Bilinear quantities with simple Lorentz transformation properties can be built from ψ
and ψ . It can be shown (Exercise) that:
42
2017
MRT
ψψ
ψγγψ
ψγψ
ψγψ
ψψ
µν
µ
µ
Σ
5
5
transform as a scalar, pseudoscalar, vector, axial-vector and (antisymmetric) tensor,
respectively, where the tensor is the commutator of the γ µ (i.e., that set of 4×4 matrices):
],[
2
)(
2
νµµννµµν
γγγγγγ
ii
≡−≡Σ
The Lagrangian which, using the space-time Euler-Lagrange equations, leads to the
free Dirac equation (iΣµγ µ∂µ −m)ψ =0 is:
ψψψψ mi −∂/=L
or, using the Feynman slash notation (i.e., a≡Σµγ µaµ):
ψψψγψ
µ
µ
µ
mi −∂= ∑L
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-42-320.jpg)
![For a massive spin-1 particle, the Proca equation leads to the propagator:
50
2017
MRT
That the numerator corresponds to the sum over spin states can be checked by
comparing with Σr[εµ
(r)]*εν
(r) =−ηµν +qµ qν /M2.
+−
− 222
M
qq
Mq
i νµ
µνη
Finally, the propagator for a spin-½ particle will be the inverse of the Dirac operator
(iΣµγ µ∂µ −m)ψ =0 :
Again the numerator contains the (completeness) sum over spins:
2222
)()(
)(
)(
)( mq
mqi
mq
mqi
mq
mq
mq
i
mq
i
−
+/≡
−
+⋅
=
+⋅
+⋅
−⋅
=
−⋅
γ
γ
γ
γγ
∑=
=+/
2,1
)()(
)()(
s
ss
ququmq
where ψ =u(q)exp(−iq⋅x) and u ≡u†γ 0, and where a relativistically invariant normalization
of the spinors is used:
sr
sr
muu δ2)()(
=
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-50-320.jpg)
![It is useful to generalize the idea of global invariance. Thus, the isotopic spin
invariance of nuclear physics under p↔n, or of the weak interaction can be represented
as an invariance of the system under transformations within an isospin multiplet; for
example, within the quark doublet ψ =[u d]T. The most general such transformation is:
54
2017
MRT
ψψψ
ατ∑ =
≡→
3
12e k kk
i
U
where the τk are the 2×2 Pauli isospin matrices (like the σk ):
−
=
−
=
=
10
01
0
0
01
10
321 τττ and,
i
i
and the αk (k=1,2,3) are the phase rotation parameters in the three orthogonal directions
in isospin space. The requirement that U is unitary requires the τk to be unitary and,
since Tr(τk)=0 for all k, the U are unimodular in that:
1ee)det(
3
1
Tr
2)(lnTr
==≡
∑ =k kk
i
U
U
ατ
The Lie group of all unitary 2×2 matrices with unit determinant is called SU(2). The
addition of the unit matrix I to the τk above would give the U(2) group of all unitary 2×2
matrices, but I would simply produce a change of the phase of u and d by the same
amount. Such a U(1) transformation is just like φ →exp(iα)φ and corresponds to the
conservation of quark or baryon number and has nothing to do with isospin itself.
Locally, U(2) is isomorphic to SU(2)⊗U(1).](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-54-320.jpg)
![The Pauli matrices τk are called generators of the isospin transformations, and their
commutation relations:
55
2017
MRT
∑=
=
3
1
2],[
k
kjkiji i τεττ
(where εijk is the permutation tensor) are called the Lie algebra of the group, the εijk
being the structure constants. The doublet ψ =[u d]T, which has the same dimension as
the generators τk , is called the fundamental representation of the group. Since the τk do
not commute, neither in general will two transformations like ψ →exp[(i/2)Σkτkαk]ψ (i.e.,
U2U1 ≠U1U2), and so this group is non-Abelian. These transformations are still unitary:
Noether’s theorem tells us that if this is a good symmetry then any component of
isospin will be a conserved quantity. However, as the τk do not commute, only one
component is measurable at a time, and by convention this is taken to be the third
component (which thus has the diagonal matrix in the basis τ1, τ2, and τ3 above).
Hence, the isospin 3-axis component, T3, is (in units of h):
is conserved if ψ →Uψ ≡ exp[(i/2)Σkτkαk]ψ is a symmetry of the Lagrangian.
33
2
1
τ=T
where U† is the Hermitian adjoint matrix of U.
1== ††
UUUU](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-55-320.jpg)
![Similarly, the strong interaction is invariant under permutations of the colors of the
quarks, so that if we write the quark wave function as a fundamental color triplet repre-
sentation ψ =[R G B]T we will have invariance under SU(3) transformations of the form:
56
2017
MRT
ψψψ
αλ∑ =
≡→
8
12e a aa
i
U
where U are unitary unimodular 3×3 matrices, the αa (a=1,2,…,8) are the eight phase
angles, and the λa (Gell-Mann) matrices are eight independent traceless, Hermitian 3×3
matrices which generate the group. They are the equivalent of the 2×2 Pauli matrices for
SU(2) and the conventional choice is given in the Table on the next slide.
)
(2
2],[
88776655
44332211
8
1
λλλλ
λλλλ
λλλ
abababab
abababab
c
cabcba
ffff
ffffi
fi
++++
+++=
= ∑=
where the structure constants fabc are also given in the Table on the next slide, where
you will notice that the only diagonal matrices are λ3 and λ8. For example:
The λa matrices satisfy the Lie algebra:
)(22],[ 83787377637653754374337323721371
8
1
3773 λλλλλλλλλλλ ffffffffifi
c
cc +++++++== ∑=
where [λ3,λ7]≡λ3λ7 −λ7λ3.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-56-320.jpg)
![59
2017
MRT
Scalar particle propagator (momentum p) 22
1
Mp
i
−
Fermion propagator (momentum q) 22
mq
mq
i
−
+⋅γ
Massless vector propagator (momentum k) 2
k
i
µνη
Fermion-photon vertex (charge e) µ
γi−
Scalar boson-photon vertices (charge e)
µ
)( ppi ′+−
p
q
e
p
p′
Three-gluon vertex (strong coupling gs)
])()(
)[(
1332
21
µλννλµ
µνλ
ηη
η
kkkk
kkfabc
−+−+
−−
Four-gluon vertex (strong coupling gs)
Quark-gluon vertex (strong coupling gs)
µα
γλ ji
i
2
−
Massive vector propagator (momentum p) 22
2
/
Mp
Mpp
i
−
− νµµνη
gs
α
j
i
µν
ηi2
e
gs
b
k2, fν
c
k3, fλ
a
k1, fµ
gs
2ρ, d
µ, a
λ, c
ν, b
)](
)(
)([
ρνµλρλµν
λρµνλνµρ
νλµρνρµλ
ηηηη
ηηηη
ηηηη
−+
−+
−−
ebcade
edbace
ecdabe
ff
ff
ffi
e2
As as general rule
(due to limited
space available at
times) Feyman
diagrams are
draws as:
or:
but should be
viewed as:
or:
or
k
p
e
gs
e
e2
gs
2
gs
2](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-59-320.jpg)
![A local gauge transformation is one in which:
61
2017
MRT
)(e)()(e)( )()(
xxxx xixi
ψψψψ χχ −
→→ and
so that, in contrast to the global transformation φ →exp(iχ)φ, the phase change, χ(x),
can be different at every space-time point x=xµ.This is obviously not an invariance of the
free-particle Lagrangian L1 = L =iΣµψγ µ∂µψ −mψ ψ , since under ψ (x)→exp[iχ(x)]ψ (x):
∑∑∑ ∂−=∂−=−∂→ −−
µ
µ
µ
µ
µ
µχχ
µ
χ
µ
µχ
χχψγψψψψγψ Jxxxmxxi xixixixi
11
)()()()(
1 )()](e[)](e[)](e[])(e[ LLL
Only if ∂µχ=0 (i.e., if χ is independent of x), is L1 unchanged because of the derivative
involved in the energy-momentum term. However, if in L =iΣµψγ µ∂µψ −mψ ψ we make
the replacement (c.f., i∂µ →i∂µ −eAµ):
)(xAeiD µµµµ +∂≡→∂
where Aµ (x) is some vector field, we get instead:
χµµµ ∂−→
e
AA
1
which precisely cancels the additional and unwanted term in the development L1
above. Dµ defined in ∂µ →Dµ ≡∂µ +ieAµ(x) is called the ‘covariant derivative’.
which is invariant under ψ (x)→exp[iχ(x)]ψ (x) and ψ (x)→exp[−iχ(x)]ψ (x) provided that
at the same time we make the replacement:
∑−=
µ
µ
µ
AJe1LL
_ _
_ _
__](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-61-320.jpg)
![Apart from the fact that it does not include the energy of the electromagnetic field, L =
L1 −eΣµ JµAµ is just the same as L =iΣµψ γ µ∂µψ −mψψ −eΣµ JµAµ −¼Σµν Fµν Fµν, and Aµ →
Aµ −(1/e)∂µχ is just a gauge transformation of the type Aµ→Aµ +∂µ χ that we saw earlier,
which we know leaves the physical E and B fields, and hence the Lorentz force,
unaltered. Thus, if we choose to identify the ‘gauge field’ Aµ with the electromagnetic
potential and e with the charge of the fermion, we have essentially deduced the
electromagnetic properties of a charged fermion just by requiring the gauge invariance
of its Lagrangian. To make the identification complete we must add the electromagnetic
field energy term LVector =−¼Σµν Fµν Fµν (with Fµν =∂µ Aν − ∂ν Aµ), which is of course gauge-
invariant by itself; indeed, it is essentially the only simple gauge-invariant quantity we
can construct involving ∂µ Aν.
62
2017
MRT
Hence, the gauge invariance of the electromagnetic potentials, which in classical
electromagnetism seems to be just a nuisance reflecting the fact that the four-vector Aµ
has too many degrees of freedom, is seen to play a crucial role in the quantum theory of
charged particles. It is needed to compensate for the phase freedom of the particle’s
wave function ψ (x)→exp[iχ(x)]ψ (x) (and also its adjoint ψ (x)→exp[−iχ(x)]ψ (x)). We
have already noted that the term ½ M2Σµ Aµ Aµ in the free spin-1 particle of mass M
Lagrangian L =−¼Σµν Fµν Fµν+½ M2Σµ Aµ Aµ (this Lagrangian leads to the Proca equation
Σµ ∂µ Fµν+ M2 Aν =0) and so the required gauge invariance is a property only of massless
vector fields.
The extra significance of the potential Aµ in quantum theory has been made more
explicit in the work of Aharonov and Bohm (c.f., P.D.B. Collins, et al., P. 49ff).
_ _
_ _](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-62-320.jpg)
![In light of this success it seems worthwhile to explore the consequences of turning non-
Abelian global symmetries such as isospin SU(2) or color SU(3) into local gauge
symmetries. Thus, if the fundamental doublet ψ =[u d]T is transformed as:
63
2017
MRT
Yang-Mills Gauge Theories
where now the αk(x) are functions of the space-time coordinates x=xµ, the Lagrangian
density describing this doublet will only be invariant under this local SU(2) symmetry if
we make the replacements:
→
=
∑ =
d
u
e
d
u
3
1
)(
2 k kk x
i
ατ
ψ
∑+∂≡→∂
k
k
k xWg
i
D )(
2
µµµµ τ
where g is some (arbitrary) coupling strength and Wk
µ(x) are three independent gauge
field potentials acting in different directions in isospin space. They form the adjoint
representation of the group, since their transformation properties are the same as those
of the τk generators.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-63-320.jpg)
![The requirement of gauge invariance is that the four-momentum term Σµψγ µDµψ
should be unchanged provided that we make the transformation Wµ→Wµ+δ Wµ in
analogy with Aµ→Aµ−(1/e)∂µχ. For infinitesimal αk(x), we can consider the change:
65
2017
MRT
)()(
2
1)( xx
i
x ψψ
•+→ ααααττττ
so that:
∑∑
•
+
+•
+∂
•
+→
µ
µµµ
µ
µ
µ
µ
ψδγψψγψ ααααττττττττααααττττ
2
1)(
22
1
i
g
ii
D WW
which is unchanged provided:
)])(())([(
2
ααααττττττττττττααααττττααααττττττττ ••−••+∂•−=• µµµµδ WWW g
i
g
The term in the square brackets is:
∑ ∑∑∑∑∑∑
=−=−
ji k
kjkiji
ji
ijjiji
i
ii
j
jj
j
jj
i
ii iWWWW τεατττταατττατ 2)(
from [τi,τj]=2iΣkεijkτk, and so we need:
∑−∂−=
ji
j
ijkik
k
xWx
g
W )()(
1
µµµ αεαδ
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-65-320.jpg)
![We can make a local SU(3) gauge transformation of the fundamental fermion color
triplet ψ =[R G B]T of the form:
68
2017
MRT
Quantum Chromodynamics (QCD)
)(e)(
8
1
)(
2 xx a aa x
i
ψψ
αλ∑ =
→
and, completely in parallel with the previous chapter, the kinetic energy term will
preserve gauge invariance if (c.f., ∂µ →Dµ ≡∂µ + (ig/2)Σkτk Wk
µ):
∑+∂≡→∂
a
a
as xGg
i
D )(
2
µµµµ λ
where Ga
µ(x) are eight gluon field potentials (N.B., a=1,2,…,8), provided that these
potentials transform as (c.f., Wi
µν= ∂µ Wi
ν −∂ν Wi
µ −gΣjk εijk Wj
µWk
ν):
∑−∂−→
ab
c
babca
s
aa
xGxfx
g
xGxG )()()(
1
)()( µµµµ λλ
from [λa ,λb]=2iΣc fabcλc. To ensure the gauge invariance of the gluon field-strength
tensor, it is defined as (c.f., Wi
µν= ∂µ Wi
ν −∂ν Wi
µ −gΣjk εijk Wj
µWk
ν ):
∑−∂−∂≡
bc
cb
abcs
aaa
GGfgGGG νµµννµµν](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-68-320.jpg)
![Since αo is not a measurable quantity we can reparametrize this result, and thereby
renormalize the charge, by defining that α ≡α(q2 =0) has the value measured in very
low-energy scattering experiments (q2 <<me
2), α ≅1/137. Then:
72
2017
MRT
21472
e
300π32
e
2
)GeV10(10e ≈≈→ mmq α
−
−
≈
2
e
22
e
2
eff
ln
3π
1
)(
m
qm
q
α
α
α
gives the leading logarithmic variation of the coupling as |q2| is increased from zero. As
we have already noted from its Fourier transform α(r)≅α/[1−(α/3π)ln(1+h/merc)],αeff (q2)
above has the consequence that αeff (q2)→∞ as:
This is the (Landau energy)2. In practice, of course, there are several charged leptons
and quarks that contribute to the vacuum polarization as in the previous Figure, but still
QED seems to require some modification as one approaches the unimaginably high
energy given by the (Landau energy)2. This is not very surprising, because gravitation
must change things as |q|→EP, the Planck scale. However, perturbation theory with α =
O(10−2) should still be satisfactory at all practically attainable energies.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-72-320.jpg)
![Hence the coupling αo and mass mo, the parameters that appear in the original
Lagrangian L =iΣµψ γ µ∂µψ −mψψ −eΣµ JµAµ −¼Σµν Fµν Fµν, are not observable quantities.
Infinities seem to arise in the ‘observed’ quantities α and m, but this only means that αo
and mo are not in fact finite. The solution to the problem is to reparametrize the
expressions for truly observable quantities, such as scattering cross sections for
example, in terms of the finite parameters α and m so that:
74
])()(1[
])()(1[
2
ouv2ouv1o
2
ouv2ouv1o
L
L
+Λ+Λ+=
+Λ+Λ+=
αα
αααα
ggmm
ff
where the coefficients fi and gi involve the ultraviolet cut-off and diverge as Λuv
2 →∞.
The cross section for electron-muon
elastic scattering.
2017
MRT
Then if we calculate some cross section, such as electron-
muon scattering (see Figure), we find that the result can be
written in the form:
])([),,( ouv1o
2
ouvoo L+Λ+=Λ= ασσαασ mf
We now eliminate αo and mo in favor of α and m by inverting α=
αo[1 + f1(Λuv)αo + f2(Λuv)αo
2 +…] and m =mo[1 + g1(Λuv)αo
+g2(Λuv)αo
2 +…] above, all the divergent terms cancel. The Λuv
dependence of the coefficients σi (Λuv) is cancelled by that of
the coefficients fi and gi, and we end up with:
)(),( 1o
2
L++== ασσαασ mf
which is free of divergence. Since all we have done is change
variables, this new result is the same as the previous one (i.e.,
σ = f vs f ), but it is now expressed in terms of finite parameters.
+ + K+
2
=σ
e
µ
_ _
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-74-320.jpg)
![To make the Green’s function propagators finite, we need to re-scale the fields ψ and
Aµ similarly (i.e., ‘wavefunction renormalization’). It turns out that QED is a renormaliza-
ble theory in the sense that once the divergences in the coupling e, the mass m, and the
normalization of the fields ψ and Aµ have been rescaled in this way, all the physical quan-
tities are finite. Hence, if the Lagrangian L =iΣµψ γ µ∂µψ −mψψ −eΣµ JµAµ −¼Σµν Fµν Fµν
is taken to be written in terms of the bare quantities ψo, mo, Aµ
o, and eo:
75
2017
MRT
∑∑∑ −−
−∂=
µν
µν
µν
µ
µ
µ
µ
µ
µ
ψγψψγψ o
oo
oooooo
4
1
FFAemiL
these bare quantities can be related to the physical ones by the renormalization constant
Zi as:
e
ZZ
Z
eAZAmZmZ m
32
1
o3
o
o2o ==== and,, µµψψ
where the Zi are functions of Λuv, and so:
L+Λ+Λ+=Λ )()(1)( uv2
2
uv1uv iii ffZ αα
where the functions fin(Λuv) contain the divergences as Λuv→∞, like αeff (r)≅αo/{1−
(αo/3π)ln[(m2−q2)/Λuv
2)]}. These infinities are then absorbed into the definition of the bare
quantities ψo, mo, Aµ
o, and eo above so that the physical quantities are finite. That this can
be done while keeping the same form for the Lagrangian L(ψo,mo,Aµ
o,eo) above as
the original L(ψ,m,Aµ ,e) shows that QED is a renormalizable theory.
__](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-75-320.jpg)
![We saw that QCD has all the essential ingredients required for the theory of strong inter-
actions, namely, asymptotic freedom and the possibility of accounting for color confi-
nement. So, we write the Lagrangian of the Quantum Chromodynamics (QCD) chapter:
79
2017
MRT
a
a
a
GG µνµν
λ
∑=
=
8
1
2
Strong Interactions and Chiral Symmetry
∑∑∑∑∑∑∑ −−=
µν
µν
µν
µ
µ
µ
γ
a
a
a
k
kk
lk
llkk GGmDi
4
1
qqq][q
q
q
q
QCDL
where now we have LQCD expressed as a function of the quark field qk of flavor q=u,d,
s, c, b, t and color k=1,2,3 (or R, B, and G for red, blue, and green). We will suppress
the color indices (i.e., k and l) and all summations signs and rewrite LQCD in the form:
)(Tr
2
1
qqqqQCD
µν
µνµ
µ
γ GGMDi −−=L
where q is the column vector [u d s c b t]T, q is the row vector [u d s c b t], and M is
the diagonal mass matrix in flavor space with eigenvalues mu, md, ms, mc, mb, and mt.
Also, we have introduced:
and used Tr(λaλb)=2δ ab, where λa (with a=1,2,…,8, which correspond to the various
gluon types) are the SU(3) matrices. The last Lagrangian is completely determined
by the requirements of renormalizability and color gauge invariance, except for the
number of quark flavors, Nf , and their masses mq.
_ _ _ _ _ _ _](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-79-320.jpg)
![The QCD Lagrangian possesses a high degree of symmetry, most notable of these
being that if mu =md then the Lagrangian is invariant under the isospin or SU(2) flavor
transformation ψ =[u d]T→ ψ =Uψ =exp[(i/2)Σkτk αk(x)][u d]T:
80
2017
MRT
qq U→
where q=[u d]T only.
To display the full symmetry of LQCD in the limit mq →0 we rewrite it in terms of the left-
and right-handed quark fields:
q)1(
2
1
qq)1(
2
1
q 55
γγ +=−= RL and
where γ 5 =iγ 0γ 1γ 2γ 3 or off-diagonal I2×2. Using these ‘chiral’ fields, LQCD becomes:
−−−+= ∑µν
µν
µνµ
µ
µ
µ
γγ GGMMDiDi LRRLRRLL Tr
2
1
qqqqqqqqQCDL
In the absence of the quark mass matrix (i.e., for M=0), we see that this Lagrangian is
invariant under two separate groups of unitary transformations:
RRRLLL UU qqqq →→ and
that is, under independent rotations of qL and qR in the space of quark flavors. We
thus have a U(Nf )L⊗U(Nf )R flavor symmetry and this symmetry should be realized
for those flavors with mass mq much less than the hadronic mass scale ΛC. It should
thus be good for u and d quarks, but more approximate for s quarks, so we can then
say that the QCD Lagrangian has an approximate U(3)L⊗U(3)R ‘chiral’ symmetry.
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-80-320.jpg)
![The standard theory of electroweak interactions is based on the gauge group
SU(2)⊗U(1) and is known as the Glashow-Weinberg-Salam model. Sheldon Glashow,
Abdus Salam, and Steven Weinberg were awarded the 1979 Nobel Prize in Physics for
their contributions to the unification of the weak and electromagnetic interaction between
elementary particles. Glashow (1961) originally unified the weak and electromagnetic
interactions using the SU(2)⊗U(1) gauge group, and Weinberg (1967) and Salam (1968)
showed how the weak gauge bosons could acquire their mass without spoiling the re-
normalizability. Major experimental support for the model came with observation of weak
neutral currents in 1973, followed by the discovery of the weak gauge boson themselves
(W± and Z) in 1983 and, of course, the discovery of the Higgs boson (H0) in 2014.
90
2017
MRT
µµµµµ B
Y
giWTgiD
k
k
k
2
3
1
′++∂≡→∂ ∑=
The SU(2)⊗U(1) Gauge Theory
The gauge group SU(2)⊗U(1) has four vector fields, three associated with the adjoint
representation of SU(2), which we denote Wk
µ, and one with U(1) denoted by Bµ . The
Lagrangian is made gauge-invariant by replacing ∂µ in the fermion kinetic energy term by
the covariant derivative Dµ:
where g, g′ and Tk, Y/2 are the SU(2), U(1) couplings and group generators, respectively.
The Tk satisfy the SU(2) algebra (c.f., [τi,τj]=2iΣkεijkτk):
∑=
k
kjkiji TiTT ε],[](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-90-320.jpg)
![We now come to the crucial problem of mass generation. The previous Lagrangian (i.e.,
L=Σf { fLΣµγ µ[i∂µ−gΣi(τi /2)Wi
µ −g′(Y/2)Bµ]fL + fRΣµγ µ[i∂µ −g′(Y/2)Bµ]fR}−¼ΣµνΣiWi
µνWi
µν
−¼Σµν Bµν Bµν) describes massless gauge bosons interacting with massless fermions. A
gauge boson mass term is not gauge-invariant, and a Dirac mass term:
94
2017
MRT
)()1(
2
1
)1(
2
1 55
RLLRmmm ψψψψψγγψψψ +−=
++−−=−
SSB in the Electroweak Model
is also excluded because ψL is a member of an SU(2) doublet while ψR is a single, so
that this term too manifestly breaks gauge invariance. We appear to have reached an
impasse! How can we generate gauge-boson and fermion masses without destroying
the renormalizability of the theory, which depends so critically on the gauge symmetry of
the interaction?
The answer is the Higgs mechanism that we discussed earlier. We introduce
elementary scalar (Higgs) fields, φ, which couple gauge-invariantly to the gauge bosons
through the covariant derivative (c.f., ∂µ →Dµ ≡∂µ +igΣiTi Wi
µ +ig′(Y/2)Bµ):
2
3
1
2†
2
φφφφ µµµµ
µ
µ
′++∂→∂≡∂∂ ∑=
B
Y
giWTgi
k
k
k
and to the fermions through so-called Yukawa couplings of the form:
)]()[( †
LRRLYG ψφψψφψ +−
_ _](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-94-320.jpg)
![Clearly, we require a φ field that is an SU(2) doublet if the Yukawa coupling above is to
be gauge-invariant. We take this Higgs doublet to be:
95
2017
MRT
+
+
≡
=
+
)(
2
1
)(
2
1
43
21
0
φφ
φφ
φ
φ
φ
i
i
with φi real, while the Hermitian conjugate doublet, φ†, describes the antiparticles φ− and
φ0. The charge assignments of the components of φ follow from the Yukawa coupling.
This term is only SU(2)⊗U(1) gauge-invariant if φ is a doublet (T=½) with Y=½.
Besides the Yukawa terms, the Lagrangian can also contain a self-interaction between
the Higgs fields. The most general SU(2)-invariant and renormalizable form is:
)(φVH −=L
where λ must be positive for V(φ) to be bounded from below (c.f., L =½ΣiΣµ∂µφi ∂µφi −
[−½µ2Σi(φiφi)+¼λΣi(φiφi)2]). An ordinary scalar mass term in LH would have the form
−M2φ†φ, but for spontaneous symmetry breaking we require the coefficient of φ†φ to be
positive. Indeed, with µ2 and λ positive the Higgs potential V(φ) is at its minimum when:
λ
µ
φφφφφφ
2
2
4
2
3
2
2
2
1
†
2
1
)(
2
1
=+++≡
2††2
)()()( φφλφφµφ +−=V
with the Higgs Potential:
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-95-320.jpg)
![In perturbative field theory, we expand φ about some particular minimum of V(φ). We
choose the minimum that has the vacuum expectation value:
96
2017
MRT
( )
λ
µ
φφ
2
3 004,2,1000 ≡≡== vii and
The particle quanta of the theory corresponds to quantum fluctuations of φ3(x) about the
value φ3 =v, rather than φ3(x) itself, that is, to:
vxxH −≡ )()( 3φ
It is therefore desirable to re-express the Lagrangian in terms of H rather than φ3. We
then find that |∂µφ|2 above and the Yukawa coupling −GY [(ψLφ)ψR +ψR(φ†ψL)] contain
boson and fermion mass terms of the form:
ψψ
µ
µ
µ )()( 2
vGWWvg Yand∑
So, somewhat surprisingly, all the weak gauge boson and fermion masses can be
generated by introducing just one complex SU(2)L doublet of Higgs fields!
_ _](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-96-320.jpg)
![By choosing the nonvanishing expectation value to be that of the neutral field φ3, we
ensure that the vacuum is invariant under U(1)EM of QED, and that the photon remains
massless. Then φ =[φ+ φ0]T gives:
97
2017
MRT
=
≡
+
v
0
2
1
00 0
φ
φ
φ
Now, because:
00 ≠≠ φφτ Yi and
both SU(2) and U(1)Y are spontaneously broken, but:
0
0
00
01
2
1
22
3
=
=
+=
v
Y
Q φ
τ
φ
and hence the vacuum remains invariant under U(1)EM gauge transformation. We
therefore expect three massive gauge bosons and one massless gauge boson.
In summary, both Goldstone and Higgs phenomena generalize to non-Abelian
symmetries. In the case of global symmetries, for every generator of a broken global
symmetry, there is a massless particle. For local symmetries, each broken generator
gives rise to a massive gaige boson.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-97-320.jpg)
![The masses of the gauge bosons can be found by substituting the expectation value 〈φ〉
=(1/√2)[0 v]T into |∂µφ|2. The relevant term in the Lagrangian is:
98
2017
MRT
∑
∑
−+
++′+′−=
′+−+
−′+
=
′+
µ
µ
µµµµµ
µµµµ
µµµµ
µµ φ
τ
WWgvBgWgBgWgv
vBgWgWiWg
WiWgBgWg
B
Y
giWgi
k
kk
2
23232
2
321
213
2
2
1
)(0)(
8
1
0
)(
)(
8
1
22
Gauge Boson Masses
where:
)(
2
1 21
WiWW m≡±
The mass matrix of the neutral fields is off-diagonal in the {W3,B} basis. As expected,
one of the mass eigenvalues is zero, and we have displayed this in |(…)〈φ〉|2 above with
the orthogonal combination of fields to that in the first term. The normalized neutral mass
eigenstates are thus:
where we have introduced θw, the Weinberg or weak mixing angle, defined by:
2222
sincos
gg
g
gg
g
ww
′+
′
=
′+
= θθ and
wwww BW
gg
BgWg
ABW
gg
BgWg
Z θθθθ µµ
µµ
µµµ
µµ
µ cossinsincos 3
22
3
3
22
3
+≡
′+
+′
=−≡
′+
′−
= &](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-98-320.jpg)
![By comparing |[igΣk(τk /2)Wk
µ +ig′(Y/2)Bµ]〈φ〉 |2 with the mass terms we would find in
the Lagrangian of the physical W±
µ, Zµ and photon Aµ fields, namely:
99
2017
MRT
∑∑∑ ++−+
µ
µ
µ
µ
µ
µ
µ
µ
µ AAMZZMWWM 2
γ
2
Z
2
W
2
1
2
1
we see that:
0
2
1
2
1
γ
22
ZW =′+== MggvMgvM and,
and so:
w
gg
g
ggv
gv
M
M
θcos
2
1
2
1
2222Z
W
=
′+
=
′+
=
The inequality MZ ≠MW is due to the mixing between the W3
µ and Bµ fields.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-99-320.jpg)
![We can rewrite the fermion-gauge boson electroweak interaction terms in the SU(2)⊗U(1)
invariant Lagrangian density (c.f., L =Σf { fLΣµγ µ[i∂µ−gΣi(τi /2)Wi
µ −g′(Y/2)Bµ]fL +
fRΣµγ µ[i∂µ −g′(Y/2)Bµ]fR}−¼ΣµνΣiWi
µνWi
µν −¼Σµν Bµν Bµν) in the form:
100
2017
MRT
∑∑∑ ′−−=
µ
µ
µ
µ
µ
µ
B
J
gWJg Y
i
i
i
2
EWL
where:
ψγψψ
τ
γψ µµµµ
YJJ YL
i
Li == and
2
Then using W± ≡(1/√2)(W1 miW2), and Zµ≡W3
µcosθw −Bµsinθw and Aµ ≡W3
µsinθw +Bµcosθw,
to express LEW in terms of the physical fields W ±, Z, and A, we obtain:
∑∑∑ ′+−
′+
′
−+−= −+
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
ZJggAJ
gg
gg
WJWJ
g
nccccc
22
EM
22
†
EW )(
2
L
In this way we can identify the physical currents as linear combinations of the SU(2) and
U(1) currents Ji and JY. Thus:
LL iJiJJ ψττγψ µµµµ
)(
2
1
)(
2
1
2121cc +=+≡
is seen to be the weak charged-current of Jµ=νeγ µ½(1−γ 5)e+νµγ µ½(1−γ 5)µ+
uγ µ½(1−γ 5)d′+cγ µ½(1−γ 5)s′ (c.f., Weak Interactions chapter) which couples to
the W+ boson.
Gauge Boson Mixing and Coupling
_
_
_ _
_ _](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-100-320.jpg)
![To obtain all the interactions and masses generated by the Higgs mechanism, we
need only substitute (as in H(x)≡φ3(x)−v):
104
2017
MRT
+
=
)(
0
2
1
)(
xHv
xφ
into the Lagrangian for the Higgs sector, which is the sum of the prior results: |∂µφ|2,
−GY[(ψLφ)ψR +ψR(φ†ψL)], and LH =V(φ) (with Higgs potential V(φ)=−µ2(φ†φ)+λ(φ†φ)2),
obtained earlier. We then find that of the four scalar fields φi(x) of φ =[φ+ φ0]T (where φ+≡
(1/√2)(φ1+iφ2) and φ0≡(1/√2)(φ3+iφ4)), the only one that remains is H(x). The other three
fields are spurious and we can remove all trace of them from the Lagrangian. To see
this, we write φ(x) in terms of H(x) and three new fields θk(x) (k=1,2,3), defined by:
+
= •
)(
0
e
2
1
)( )(
xHv
x vxi θθθθττττ
φ
θk and H fully parametrize all possible deviations from the vacuum. Given this form, we
can use gauge freedom to set θk =0. This choice is known as the ‘unitary’ gauge, as only
fields that correspond to physical particles appear in the Lagrangian. However, we
cannot have just lost three degrees of freedom as a result of spontaneous breaking of
symmetry and translating the field variables. What has happened is that in generating
masses for the three weak bosons we have increased their polarization degrees of free-
dom from 2 to 3. They can now have longitudinal polarization too. The phases θk of
three of the Higgs fields have been surrendered to make the gauge fields massive!
_ _](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-104-320.jpg)
![So, the W±, Z bosons, φ± and (i/√2)(φ0 −φ0), out of the four in the original complex
doublet. In the minimal model, the values of the masses are predicted in terms of the
couplings. From MW=½vg, MZ=½v√(g2 +g′2), MW/MZ =cosθw, and e=gg′/√(g2 +g′2) we
have:
105
2017
MRT
GeV90
sin
GeV80
sin
GeV3.37
sin2
1
2
1 W
ZW ≅=≅===
www
M
M
ev
gvM
θθθ
and
where we have used the experimentally determined value of sinθw≅0.23. These
predictions are in excellent agreement with the masses of the W± and Z bosons that
were subsequently discovered.
The fourth parameter of the model, λ (or alternatively µ2 ≡λv2) controls the form of the
potential V(φ) and determines the mass of the Higgs particle associated with the
remaining field H(x). On substituting φ(x)=(1/√2)[0 v+H(x)]T into the Higgs potential V(φ)
=−µ2(φ†φ)+λ(φ†φ)2, we find:
L+−=+−+= 2242
2
)2(
2
1
)(
4
)(
2
HvHvHv
v
H λ
λλ
L
where the higher-order terms represent the self-couplings of the Higgs boson, H. From
this last equation, we conclude:
222
H 22 µλ == vM
The mass of the Higgs is not predicted,since neither µ2 not λ is determined,only their
ratio v2. However, the Higgs couplings to other bosons are completely determined!
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-105-320.jpg)
![The Higgs-fermion couplings give masses to the fermions. We begin by considering the
Yukawa term −GY[(ψLφ)ψR +ψR(φ†ψL)] for the electron doublet:
106
2017
MRT
+
−= −
+
L
RRLY G
e
][ee]e[ e0
0ee
e ν
φφ
φ
φ
νL
Fermion Masses and Couplings
When we spontaneously break the symmetry and substitute φ(x)=(1/√2)[0 v+H(x)]T, this
term becomes:
)Hee()ee()eeee)((
2
e
e
ee
v
m
mHv
G
LRRLY −−≡++−=L
revealing that the electron’s mass and coupling are:
W
eee
e
2
)eeH(
2 M
mg
v
m
g
vG
m === and
Since Ge is arbitrary, the actual mass of the electron is not predicted, but its Higgs
coupling is specified and, being proportional to me/MW, is very small.
W
qq
2
)qqH(
M
mg
v
m
g ==
On the other hand, the Higgs couplings to the quarks are:
_ _](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-106-320.jpg)
![The quark masses (and couplings) are generated in an analogous manner. However,
φ(x)=(1/√2)[0 v+H(x)]T gives a mass only to the lower member of the fermion doublet,
and to generate a mass for the upper member we must construct from φ a new Higgs
doublet with a neutral upper member:
107
2017
MRT
+
→
−
== − 0
)(
2
1
*
0
2
xHv
ic
breaking
φ
φ
φτφ
Owing to the special properties of SU(2), φc transforms identically to φ, but has opposite
weak hypercharge, Y(φc)=−1. The most general SU(2)⊗U(1) invariant Yukawa terms for
the [u d] quark doublet are then:
conjugateHermitianu]du[d]du[
0
u0d
u
+
−
−
−= −
+
RLRLY GG
φ
φ
φ
φ
L
which, on substitution of φ(x)=(1/√2)[0 v+H(x)]T and φc =(1/√2)[v+H(x) 0]T above,
reveals that the mass and qqH coupling terms are:
++−=
v
H
mmY 1)uudd( ud
u
L
where mq =(1/√2)Gqv.
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-107-320.jpg)
![To date, there are no confirmed experimental results that contradicts the Standard Model
of Particle Physics. Why, then, should we wish to go beyond it? Why are we convinced
that it cannot be the whole truth?
108
2017
MRT
In view of this fact, it is perhaps reassuring that the Standard
Model itself, although experimentally well confirmed, has
several unsatisfactory, or at least, unnatural features. The
gauge group has three factors (i.e., SU(3)⊗SU(2)⊗U(1)) and
hence has three independent coupling constants. It offers a
glimpse of unification in the breakdown of SU(2)⊗U(1)Y →U(1)Q,
but this does not take us very far. Also, it is strange that one of
these factors (i.e., SU(2)) distinguishes between left (L) and
right (R) handed states. As a consequence, the Standard Model
includes 45 massive fermion fields (see Figure) arranged in
left-handed SU(2)L doublets and right-handed SU(2)R singlets.
Since parity is maximally violated by the weak interaction, there
is no right-handed neutrinos and only left-handed fermions (and
right handed anti-fermions) are sensitive to the weak
interaction. The primes on down-type quarks [with color
charge red (R), green (G) or blue (B)] and neutrinos
correspond to gauge eigenstates.
Why Go Beyond the Standard Model?
Undoubtedly, the most compelling reason for our dissatisfaction is that the Standard
Model does not include gravity. Attempts to quantize general relativity results in a
nonrenormalizable field theory. Such a theory may give the correct results in the lowest-
order (i.e., at the classical level) but it but it does not permit a proper quantum
calculation of any experimental quantity.
111
21
τ
21
µ
21
e
31
R,
32
B,
31
B,
32
B,
31
B,
32
B,
61
B,
61
B,
61
B,
31
G,
32
G,
31
G,
32
G,
31
G,
32
G,
61
G,
61
G,
61
G,
31
R,
32
R,
31
R,
32
R,
31
R,
32
R,
61
R,
61
R,
61
R,
τµe
τµe
b
t
s
c
d
u
b
t
s
c
d
u
b
t
s
c
d
u
b
t
s
c
d
u
b
t
s
c
d
u
b
t
s
c
d
u
−−−
−−−
−−−
−−−
−−−
′
′
′
′
′
′
′
′
′
′
′
′
RRR
LLL
R
R
R
R
R
R
LLL
R
R
R
R
R
R
LLL
R
R
R
R
R
R
LLL
ννν
Standard Model fields with their associated
charges (Y top right, chirality (bottom right
of each doublet or singlet and color k =R,G,
B ,bottom right of each doublet or singlet).](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-108-320.jpg)
![The Higgs spontaneous symmetry breakdown mechanism, which is crucial to the
success of the Standard Model, requires an inelegant and arbitrary addition to the
Lagrangian – if we didn’t need it, then why have a Higgs coupling to start off with?
Perhaps even worse is the fact that the theory offers no explanation for family
replication. The old question of who ordered the muon, has changed into why are there
three families, but it still remains unanswered. Related to this is our complete ignorance
as to the origin of the parameters in the mass matrix, all of which seem quite arbitrary.
109
2017
MRT
• Three coupling parameters (i.e., the constants g1 =α, g2 =g, and g3 =gs);
• The two Higgs parameters (i.e., MH and λ);
• The nine fermion (i.e., quarks and leptons) masses (i.e., me , mu , md ; mµ , ms , mc , and
mτ , mb , mt );
• The three mixing angles (i.e., θc, cosθw, and sinθw);
• One phase angle in the quark-mixing Cabibbo-Kobayashi-Maskawa [V] (CKM) matrix
(i.e.,δ) who QCD Lagrangian) requirement of a single true vacuum.
There are even more parameters now with three neutrinos having finite masses and
mixing.
Hence, any model that might explain or relate some of the above parameters is worth
considering, and it is to such models that we now turn. We will begin with the unification
of the Standard Models itself.
Indeed, the number of free parameters in the Standard Model totals 19; namely:](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-109-320.jpg)
![One of the most outstanding puzzles of the Standard Model is the structure of fermion
masses and mixing angles. The masses of quarks and leptons show a hierarchical
structure (see Table – all in GeV) suggesting the possibility of some underlying pattern.
110
2017
MRT
Experimental measurements yield the following approximate structure for the Cabibbo-
Kobayashi-Maskawa (CKM) matrix:
=
999.004.0008.0
04.0973.022.0
003.022.0974.0
][V
Again, this displays an interesting structure. It is close to the identity matrix, with
small off-diagonal mixing entries, except for the Cabibbo 1-2 entry, which is
somewhat larger.
First Generation* Second Generation Third Generation
U-quarks u c t
2×10−3 2×10−1 173
D-quarks d s b
4×10−3 1×10−1 3
leptons e µ τ
0.51×10−3 1.05×10−1 1.7
* Here the First Generation is labeled by i = 1 so that U = Ui and D = Di or U1 = u and D1 = d, &c. Ibid for i = 2 and i = 3.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-110-320.jpg)
![In summary, the hypercharges of the Standard Model in the Table below are related to
their usual electric charges by QEM =Y+T3, where T3 =diag[½,−½] is an SU(2)L generator.
111
2017
MRT
SM QUDLEs* SU(3)C SU(2)L U(1)Y
UL
i =[Ui,Di]L 3 2 1/6
UR
i 3 1 −2/3
DR
i 3 1 +1/3
Li =[νi,Ei]L 1 2 −1/3
ER
i 1 1 +1
H=[H−,H0] 1 2 −1/2
* Gauge quantum number of the Standard Model of quarks, letons and the Higgs scalars.
_
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-111-320.jpg)
![But first we have to ensure that the generator of U(1)Y transformations has the same
normalization as the other generators of G. We require that all generators TI of G should
satisfy:
116
2017
MRT
JIJI cTT δ=)(Tr
for arbitrary I, J indices and where the trace is taken over any representation of G and c
is an irrelevant (representation-dependent) constant. In the Standard Model the
normalization of the U(1)Y generator was arbitrary, but now it must satisfy this last
equation. We can compare Y/2 with, for example, T3 of weak isospin. If we assume that
the 15 fermions of (1,2)⊕(3,2)⊕(1,1)⊕(3,1)⊕(3,1) above completely fill a (possibly
reducible) representation of G, then the eigenvalues of the Table on Slide 89 give:
3
5
])())[(31(
])(3)(3)2()(6)1(2[
)(Tr
)(Tr
2
2
12
2
1
2
3
22
3
422
3
12
4
1
2
3
2
4
1
=
−++
+−+++−
=
T
Y
Upon embedding U(1)Y in G we must therefore renormalize the coupling g′ defined by
the electroweak Lagrangian LEW=−gΣµΣi Ji
µWi
µ−g′Σµ(JY
µ/2)Bµ accordingly, by taking:
2
1
2
5
3
gg =′
and hence from the definition of cosθw and sinθw above and g3 =g2 =g1 we have:
375.0
8
3
sin 2
15
32
2
2
15
3
22
2
2
2
==
+
=
′+
′
=
gg
g
gg
g
wθ
_ _](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-116-320.jpg)
![Is the unification shown in the previous Figure reasonable? That is, can we find a
grand unified scale MX at which the αk are equal, such that at present energies we
obtain the values of αk actually measured? To check whether this is possible we use the
known values of the strong and electromagnetic couplings:
119
2017
MRT
15.010.0)()( 2
W
2
W3 −=≡ MM sαα
and:
from a relation obtain from calculating the running of the coupling α in the electroweak
sector (i.e., α(MW)/α(me)≅{1−[α(me)/3π][ΣFQF
2ln(MW
2/mF
2)]}≅1.073, where F indicates
the fermion family – not its field f ), together with the evolution equations (i.e., 1/αGUT
above), to solve the matching condition:
128
1
cos
5
3
sin)( 2
1
2
2
2
W =
== wwM θαθαα
)()()( 2
X1
2
X2
2
X3 MMM ααα ==
for MX and sin2θw(MW
2). The solution gives:
.220-20.0~sinGeV10-10~ 21414
X wM θand
and αk(MW
2)~1/40. For energies well aboveMX, G would be a good symmetry with only
this one coupling. The additional X gauge boson loos increase the value of bk in 1/αGUT
above and the evolution is indicated by the dashed line (− − −) in the previous Figure.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-119-320.jpg)
![The prediction sin2θw =0.20-0.22 (in contrast to sin2θw =3/8) is in excellent agreement
with observations and is one of the successes of the grand unification idea. The result is
relatively insensitive to the details of the calculation. A more careful analysis has been
done allowing for mass effects and including the two-loop contributions of the β-
functions so we can write:
120
∑+≡
l
lklkkk
k
b
d
d
ααααβ
µ
α
µ 22
)(
In the minimal SU(5) grand unified theory the more detailed predictions are:
Λ
−±==
Λ
−×= ±
2.0
ln006.00.0070.216GeV)20(sinGeV
2.0
101.3 MS22
03.1
MS3.014
X µθwM and
The largest uncertainty is in the input value of α3 in α3(MW
2)=αs(MW
2)=0.10-0.15 above,
or ΛC. Increasing ΛC increases α3 (c.f., α3(q2)=1/[(bo/4π)ln(Q2/ΛC
2)]) and so the couplings
αk do not become equal until larger MX. This dependence is shown explicitly in MX and
sin2θw above in terms of ΛC (in GeV) defined in the MS renormalization scheme. The fact
that MX is of the same order as the lower bound given in MX >1015-1015 GeV is
remarkable and gives credence to the idea of grand unified theories (GUTs). The
couplings apparently imply a symmetry-breaking scale that is large enough to inhibit
proton decay sufficiently.
~
__
2017
MRT](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-120-320.jpg)
![Another consequence of grand unification is that the masses of the fermions should be
related. Indeed, many models with economical Higgs structure predict that:
121
bτsµde mmmmmm === and,
at the unification scale MX. At first sight, these equalities appear just as disastrous as the
coupling equalities g3=g2 =g1 or α3 =α2 =α/sin2θw (with sin2θw =3/8). However, the masses
also depend on the renormalization scale and to see whether such relations are
reasonable we must use the renormalization group equations to continue down to
present energies. In the one-loop approximation (i.e., m(Q2)=m(µ2)[α(Q2)/α(µ2)]γ o/bo)they
give:
b
M
Mmm
f
)(
)(
)()(
X
Xff
γ
α
µα
µ
=
and so at scale µ:
13q 1
X1
1
4
X3
3
3
1
)(
X
q
)(
)(
)(
)(
)(
)(
)(
)(
bb
k
b
k
k
MMMm
m k
kk
=
= ∏=
−
α
µα
α
µα
α
µα
µ
µ
γγ l
l
where the bk are given in b3 =11−(4/3)Ng, b2 =22/3−(4/3)Ng −(1/6)NH, and b1 = −(4/3)Ng −
(1/10)NH and the γk by γ (α)=−(γo/2π)α and γo =(3/2)[(N2 −1)/N]. There is no k=2
contribution since quarks and leptons have the same SU(2) interactions (i.e., γl
2 =γq
2).
Also, γl
3 =0 because gluons do not couple to leptons. 2017
MRT](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-121-320.jpg)
![The possible groups of rank 4 are:
125
2017
MRT
)(SU)(SU)(SU 335 ⊗and
We can exclude the latter because one factor must be SU(3) of color, but the leptons
do not carry color and must therefore lie in different representations to the quarks.
However, since the sum of the charges of the quarks and leptons within any given
multiplet of G must be zero, we require that the trace of the charge operator Q be zero
(i.e., TrQ=0) and the latter gives TrQ≠0 either for the known quarks or for the leptons.
This leaves only SU(5). Can the 15 left-handed fermion of (1,2)⊕(3,2)⊕(1,1)⊕(3,1)⊕(3,1)
above be accommodated in representations of SU(5)? Remarkably, they can, but not in
a single irreducible representation. Rather they fill two representations: a 5 and a 10
dimensional representation, with the following SU(3), SU(2) content:
44444 844444 76444 8444 76 105
1113232113 ),(),(),(),(),( ⊕⊕⊕⊕
L]e[ e
−
ν⊕ Lii ]du[⊕L
c
i )(d ⊕ L
c
i )(u ⊕
c
Le
where for each multiplet ΣQ=0, provided:
−=−= eud
3
1
2
1
QQQ
Thus, it is predicted, correctly, that quarks have ⅓-integral charge because they
come in three colors.
_
_ _](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-125-320.jpg)
![The SU(3), SU(2) decomposition of the fundamental five-dimensional representation of
SU(5) is:
128
2017
MRT
Recalling that the SU(3) and SU(2) product representations decompose into the
symmetric and antisymmetric representations:
Grand Unified SU(5)
),(),( 21135 ⊕=
ASAS 13223633 ⊕=⊗⊕=⊗ and
it can be checked that the SU(5) product representation decomposes as:
AS
AS
1015
1123133123162113211355
⊕=
⊕⊕⊕⊕⊕=⊕⊗⊕=⊗ )],(),(),[()],(),(),[()],(),[()],(),[(](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-128-320.jpg)
![The family of left-handed fermions (3,1)⊕(1,2)⊕(3,2)⊕(3,1)⊕(1,1) above therefore fits
neatly into a 5 and a 10 of SU(5) with the assignment:
129
2017
MRT
It is often convenient to use the 5 of right-handed fields instead of the 5 of left-handed
fields. We have identified color SU(3) with the first three indices of SU(5) and SU(2) with
the remaining two. To conform with our previous choice of [ν,e−]L as the SU(2) doublet,
we take the conjugate to be [e−,−ν]L (c.f., φc =iτ2φ*=[φ 0,−φ −]T of the Fermion Masses
and Couplings chapter). We shall return to the assignment of the u, d, uc, ec or the dc to
the 5 and of the uc to the 10 are necessary to satisfy the hypercharge (charge)
requirement TrY=0 (TrQ=0) in a given multiplet. It would be violated if dc ↔uc.
55 =
−
==
−
=
+−
R
c
R
L
c
c
c
L
ν
ψ
ν
ψ
e
d
d
d
e
d
d
d
3
2
1
3
2
1
or
_ _
_
_
_
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-129-320.jpg)
![The SU(5) Lagrangian contains a gauge-invariant interaction term involving the
multiplet (c.f., previous ‘ψL or ψR
c ’ equation) of the form:
130
2017
MRT
L+
+∂= ∑ ∑= =
3
0
24
1
2µ
µµ
µ ψ
λ
γψ c
R
I
I
Ic
R AgiiL
There are 52 −1=24 traceless, Hermitian matrix generators of SU(5), λI /2, chosen as in
SU(3) or SU(2) (c.f., Table on Slide 57). They satisfy:
∑=
==
24
1
2],[2)(Tr
K
KKJIJIJIJI ci λλλδλλ and
where cIJK are the structure constants of SU(5). A convenient representation is given in
the Table.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-130-320.jpg)
![Thus there are 24 gauge bosons, Aµ
I, which lie in the adjoint representation of SU(5),
that is, in the nonsinglet piece of the product 5⊗5=24⊕1. The SU(3), SU(2)
decomposition of the 24-dimensional representation is:
132
2017
MRT
where the color labels of the particles have been suppressed. So, in addition to the 12
gauge bosons of the Standard Model there are 12 new bosons (X, Y, and their
antiparticles X, Y) that lie in the fundamental representations of both SU(3) and SU(2).
On account of the summation in L above it is convenient to introduce a 5×5 traceless
matrix of the gauge fields Aµ (N.B., g1, …, g8 and H component order is speculative):
),(),(),(),(),( 322311311824 ⊕⊕⊕⊕=
Gluons, g
Higgs, H
43421
γZ,W 0±
X, YWi B X, Y
_ _
µ
µ
µµ λλ
B
YYY
XXX
YX
YX
YX
W
Y
X
YXg
AA
i
i
i
iia
I
I
I
2
γW
WZ
ggg
gHg
ggg
H,
2
1
2
24
321
0
321
33876
2254
11321
24
1
+
=
==
+
−=
∑
where λ24 is the diagonal matrix given in the previous Table and W and B are the
SU(2) and U(1) gauge fields as in L=Σf { fLΣµγ µ[i∂µ−gΣi(τi /2)Wi
µ −g′(Y/2)Bµ ]fL +
fRΣµγ µ[i∂µ −g′(Y/2)Bµ] fR}−¼ΣµνΣiWi
µνWi
µν −¼Σµν Bµν Bµν.
_ _
_
_
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-132-320.jpg)
![As we have anticipated, the Lagrangian L =iψR
c Σµγµ[∂µ−igΣI(λI /2)Aµ
I ]ψR
c describes
not only standard-model transitions but also lepton-quark transitions mediated by the
new superheavy gauge bosons X and Y. Examples of such transitions are shown in the
Figure. It is clear that these bosons must have charge QX =4/3 and QY =1/3. As the 10 of
5⊗5=15S⊕10A above is the antisymmetric part of this 5⊗5 its components have the form:
133
Lepton-quark transitions in SU(5).
2017
MRT
This indicates how the u, d, uc, ec fields are to be assigned to
the 10. We put [using the Langacker (1981) sign convention]:
L
cc
cc
cc
NM
−
−−−
−−−
−−−
=
+
+
0eddd
e0uuu
du0uu
duu0u
duuu0
2
1
321
321
3312
2213
1123
χ
g
e−
d
X
g
ν
d
Y
_
where φM with M=1,…,5 transforms as the fundamental 5-dimensional representation of
SU(5).
)(
2
1
MNNMNM φφφφχ −=
where (χ4k , χ5k) have been identified with (uk ,dk)L and transform
as (3,2) of SU(3), SU(2), whereas:
∑=
=
3
1
u
2
1
k
c
kLkjiji εχ
with color labels i, j,k=1,2,3, transforms as a 3 of color.
_
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-133-320.jpg)
![The covariant derivative for a fundamental 5, φM, is:
134
2017
MRT
∑=
•+∂=
5
1
2
1
)(
N
N
NM
MM giD φφφ µµµ Aλλλλ
as in L =iψR
c Σµγµ[∂µ−igΣI(λI /2)Aµ
I ]ψR
c, and so for the antisymmetric 5⊗5 representation
of χMN =(1/√2)(φM φN −φN φM) we obtain:
∑∑ ==
•+
•+∂=
5
1
5
1
2
1
2
1
)(
P
PM
QPQ
NQ
QM
PMPM gigiD χχχχ µµµµ AA λλλλλλλλ
Inserting this covariant derivative into the kinetic-energy term iTr(χ Σµγ µDµχ), we obtain
the gauge interactions of the 10 multiplet:
∑∑
•−=
µ
µ
µ
χγχ
N
PN
NM
PMg Aλλλλ
2
1
)(2nInteractioL
The factor 2 occurs because the two gauge terms in the above for (Dµχ)MP give identical
contributions, which in turn follows on using χMN =−χNM (twice). If we also include the
gauge-interaction term of the 5 multiplet ψR
c given in L =iψR
c Σµγµ[∂µ−igΣI(λI /2)Aµ
I ]ψR
c,
we obtain:
∑∑∑∑
•−
•−=
µ
µ
µ
µ
µ
µ
ψγψχγχ
N
N
c
R
NM
M
c
R
N
PN
NM
PM )(
2
1
)(
2
1
)(2nInteractio AA λλλλλλλλL
_
_
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-134-320.jpg)
![This last LInteraction Lagrangian describes all the SU(5) gauge interactions. For example,
the SU(3) color gauge theory of the Standard Model is obtained by taking M,N,P=1,2,3,
while the Glashow-Weinberg-Salam SU(2)⊗U(1) is contained in M, N=4,5. On the other
hand, taking M=1,2,3 with N=4,5 in the second term in L Interaction above gives new
transitions mediated by the superheavy gauge group (X,Y). This results in lepton-quark
transitions:
135
2017
MRT
RR d],e[ →+
ν
Similarly, the first term in LInteraction gives the transitions:
LLL u]d,u[]u,d[e →→+
and
and the inverse transitions follow from M↔N. Because they can produce such
transitions, the X and Y bosons are sometimes called leptoquarks. They mediate proton
or neutron decay. For example, the Figure shows diagrams for p→e+uu, where the uu
form a neutral meson, π0, ρ0, &c., and there are similar diagrams for p→e+dd & p→ν du.
Although the above transitions violate both baryon and lepton number conservation, they
do not change the value of B−L. The X and Y bosons carry B−L of ⅔ and B−L is
conserved in SU(5).
Diagrams for the proton decay process p→e+uu. ‘ p{’ and ‘ }π0,ρ0,…’ are the same for all three diagrams.
Y
p u
d
u
u
e−
u
_
d
u
u
u
e−
u
_X
Y
d
u
u
u
e−
u
_
π0,ρ0,…
_ _
_ _
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-135-320.jpg)
![Suppose, for a moment, that Φ is the only Higgs representation. It is convenient to
reassemble the 24 Higgs fields in the form of a 5×5 traceless matrix:
137
2017
MRT
)())((Tr
2
1 †
Φ−
ΦΦ= ∑Φ VDD
µ
µ
µL
The most general form of the scalar self-coupling potential is:
∑=
Φ=Φ
24
1 2I
I
Iλ
just as we did for the gauge fields in Aµ =ΣI(λI /2)Aµ
I. The Higgs contribution to the
Lagrangian is then (c.f., L =½Σi Σµ ∂µφi∂µφi −[−½λ2Σi (φiφi)+¼λΣi (φiφi)2]):
)(Tr
2
1
])[(Tr
4
1
)(Tr
2
1
)( 42222
Φ+Φ+Φ−=Φ baMV
apart from an inessential cubic term which has been dropped by imposing the discrete
symmetry under Φ→−Φ.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-137-320.jpg)
![To determine the resulting masses of the gauge bosons, we follow the procedure of
the Gauge Boson Mixing and Coupling chapter. First, we obtain the covariant derivative
for the 24 adjoint field Φ. For a fundamental 5 representation the derivative is given in
(Dµ φ)M = ∂µφM +igΣN [(λλλλ/2)•Aµ)MN φN], but for a 24 it is more complicated. We have:
139
2017
MRT
where the 24-dimensional matrices, TI, represent the generators in the adjoint
representation, and may be evaluated using the general group-theoretic result for the
adjoint representation:
∑ Φ+Φ∂=Φ
JI
JJKIIKK ATgiD )()( µµµ
KJIKJI ciT =)(
the cIJK being the structure constants of the group (c.f., Tr (λI λJ)=2δIJ and also [λI ,λJ]=
2iΣK cIJK λK). Using Φ=ΣI(λI /√2)ΦI, we may rewrite (Dµ Φ)K as:
)(
2
µµµµµµ λ
AAgiAcigiD
KJI
JI
K
KJI Φ−Φ+Φ∂=Φ+Φ∂=Φ ∑
where the last equality follows from the commutation relations of the λ matrices, [λI ,λJ]=
2iΣK cIJK λK and Aµ =ΣI(λI /2)Aµ
I.The next step is to substitute the covariant derivative DµΦ
above into the Lagrangian LΦ =½Tr [(DµΦ)(DµΦ†)]−V(Φ) and to replace Φ by its vacuum
expectation value 〈0|Φ|0〉=vo[::] above.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-139-320.jpg)
![The gauge boson masses can then be extracted from the term:
140
2017
MRT
where here Aµ
I denotes the linear combinations of the original Aµ
I that diagonalizes the
mass matrix. Since 〈Φ〉 of 〈0|Φ|0〉=vo[::] above is proportional to the unit matrix in the
SU(3) and SU(2) subspaces, these components commute with Aµ and the standard-
model gauge bosons remains massless. A mass is only generated for the X and Y
bosons. Substituting 〈0|Φ|0〉=vo[::] into the Lmass Lagrangian above, we obtain:
∑∑=Φ−Φ=
µ
µ
µ
I
III AAMAA
g 22
2
mass
2
1
])[(Tr
2
L
2
o
22
Y
2
X
8
85
vgMM ==
The number of Higgs fields that remain massless (i.e., Goldstone bosons) is equal to
the number of generators for which the symmetry is broken (c.f., Spontaneous
Symmetry Breaking (SSB) chapter). Thus, for the SU(5)→SU(3)⊗SU(2)⊗U(1) symmetry
breaking, we have 24−8−3−1=12 broken generators, and hence 12 massless Higgs
fields that are eaten up by the X and Y bosons as they acquire mass. They become the
longitudinal polarization states of the massive X and Y bosons. The 12 remaining Higgs
fields acquire masses of order vo ~MX from vo =2M2/(15a−7b), but since they do not
couple to fermions they are of no further interest.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-140-320.jpg)
![The second stage of breaking of SU(5) to give the observed physics is the electroweak
symmetry breaking of the Standard Model. In the minimal SU(5) theory, this is
accompanied by a complex 5-dimensional Higgs multiplet H:
141
2017
MRT
where (φ +,φ 0) is the analog of the conventional doublet 〈Φ〉=〈0|[φ+ φ 0]T|0〉= (1/√2)[0 v]T
and where the nonvanishing vacuum expectation value arises from a Higgs potential:
=
=
+
v
HH
H
H
H
0
0
0
0
2
1
00
0
3
2
1
with
φ
φ
2††2
)()()( HHHHHV λµ +−=
with µ2 >0 and λ>0 in the usual way (c.f., V(φ)=−µ2(φ†φ)+λ(φ†φ)2). It can then be found:
λµ 22
v=
and that the W± and Z0 bosons acquire masses:
2222
Z
2
W
4
1
cos vgMM w == θ
as found in the Gauge Boson Masses chapter. However, the color triplet of Higgs
fields, HM with M=1,2,3 of H=[:] and 〈0|H|0〉=(1/√2)[::] above remains massless.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-141-320.jpg)
![In the Standard Model, the fermion masses were generated by symmetry breaking
through Yukawa couplings to the Higgs field (c.f., Fermion Masses chapter). Is a similar
mechanism possible in SU(5)? The left-handed fermions lie in ψL(5) and χ(10) multiplets
and so the Higgs representations that can be responsible for these masses must occur
in the following decompositions:
142
2017
MRT
Fortunately, these products do not contain a 24 because a coupling to the Φ(24) Higgs of
Φ=ΣI(λI /√2)ΦI would normally generate fermion masses of order MX. However the 5 of
Higgs, H of H=[:] above looks promising. From 5⊗10=5⊕45 and 10⊗10=5⊕45⊕50 we
see that:
504551010455105151055 ⊕⊕=⊗⊕=⊗⊕=⊗ and,,
Fermion Masses Again
LL ⊕=⊗⊗⊕=⊗⊗ 15101015105 and
and so we can construct two SU(5)-invariant Yukawa coupling terms:
where the ε arises because the relevant coupling in 10⊗10=5⊕45⊕50 is totally
antisymmetric.
h.c.)()(
4
1
)()( †
++= ∑∑ pnmlk
pnmLlk
c
RpnmlkU
lk
llkLk
c
RDY HGHG χχεχψL
_ _ __
_ __
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-142-320.jpg)
![Substituting the vacuum expectation value vδi5/√5 of 〈0|H|0〉=(1/√2)[::] for Hl, we
obtain:
143
2017
MRT
at the unification mass scale (c.f., me =md, mµ =ms, and mτ =mb at the unification scale MX),
while mu =(1/√2)vGU. Similarly, for other generations mµ =ms and mτ =mb, but presumably
larger couplings GD, GU.
DGvmm
2
1
de ==
In summary, at the first stage of symmetry breaking a Φ(24) of Higgs generates the
masses of the superheavy X and Y bosons, while at the second stage an H(5) of Higgs
generates both the masses of the W, Z bosons and of the fermions. The vacuum
expectation values of the Higgs fields are vo ~1014 GeV and v~102 GeV, respectively. This
attractive model has a fundamental flaw, however, which goes under the name of
hierarchy problem.
In minimal SU(5) we therefore have:
∑∑
∑∑
−
+−=
++=
k
kk
U
k
kkD
kji
kLji
c
Rkji
U
k
kLk
c
R
D
Y
Gv
Gv
GvGv
uu
2
ddee
2
1
h.c.)()(
2
)()(
2
)( 45mass ψχεχψL
where i, j,k=1,2,3 are the color labels and we have used χij=(1/√2)Σkεijk uL
c
k.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-143-320.jpg)
![We have already noticed that the 5 of Higgs (i.e., H=[:] above), contains a massless
Higgs color triplet Hk . In fact, in the minimal SU(5) model there are two Higgs color
triplets of charge −⅓, namely, Φk5 and Hk , together with their antiparticles. One linear
combination, which is dominantly Φk5, is eaten by the Y boson to generate its mass,
while the other, which is predominantly Hk , remains massless and through B-violating
process such as:
144
2017
MRT
would allow protons to decay extremely rapidly!
ueHud +
→→ k
Hierarchy Problem
A second problem is that, although it is attractive to have two separate terms in the
Higgs potential V(Φ)+V(H) each with its own minimum, such a theory is non renorma-
lizable. There must be gauge-invariant quartic terms involving both Φ and H of the form:
HHHHHV 2†2†
)(Tr),( Φ+Φ=Φ βα
Even if we artificially try to omit these cross-coupling terms from the potential, they will
be generated automatically by radiative corrections. The combined potential V(Φ)+V(H)
+V(Φ,H) can be arranged to have a single minimum at a point where the expectation
values of Φ and H are given by 〈0|Φ|0〉=vo[::] and 〈0|H|0〉=(1/√2)[::], respectively, but
there are then extra terms in the equations vo =2M2/(15a−7b) and µ2 =2v2λ, that
determine vo and v:
2
o
2222
o
2
2
3
5
2
3
10
3
)715(
2
1
vvvvbaM
−++≅
+++≅ βεβαλµβα and
_](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-144-320.jpg)
![Strictly speaking, the vacuum expectation value of Φ can no longer be exactly of the
form 〈0|Φ|0〉=vo[::], since it acquires a tiny SU(2) isotriplet breaking part ε:
145
2017
MRT
withε ≅ (3β/20b)(v/vo)2.However,since v<<vo, conditionM2≅½(15a+7b)vo
2+[α+(3/10)β ]v2
above is little changed from its original form. But µ2≅λv2+(3/2)[5α +(3/2)β −εβ]vo
2 is
grossly different. The extra vo
2 terms now imply that the natural size of v is of order vo.
The problem is that the mixing terms (e.g., Tr (Φ2)H†H) generate contributions to the
mass of H of order vo. This is fine for the color triplet components Hk, and would
suppress Higgs-mediated proton decay to an acceptable level and so cure our first
problem. However, for the standard-model Higgs φ it is a disaster.
+−
−−
=Φ
22
3
22
3
o
0000
0000
00100
00010
00001
00
ε
ε
v](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-145-320.jpg)
![To obtain the desired result:
146
2017
MRT
Even if such a cancellation were arranged, it would be upset by radiative corrections. At
first order they would generate a mass ≈gvo
2 for H5, and we would have to retune the
parameters to cancel this, too, and so on to higher and higher orders.
12
X
W
o
10~~ −
M
M
v
v
we have to choose the Higgs parameters with an astonishing accuracy if we are to
ensure that in µ2≅λv2+(3/2)[5α +(3/2)β −εβ]vo
2:
2
o
242
o
2
)10(
2
3
5
2
3
vOv −
=
+− βαµ
This necessity for fine tuning is a disease of all grand unified theories and is known as
the (gauge) hierarchy problem. It is difficult to sustain naturally two vastly different
scales of symmetry breaking, since radiative corrections mix the scales and, without fine
tuning, will always equalize them.](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-146-320.jpg)
![Loop diagrams like those in the Figure below give quadratically divergence
renormalization corrections to the bare Higgs mass of the form:
148
2017
MRT
2
2
2
24
4
2
H
π8
~
1
)π2(
Λ∝∆ ∫
Λ g
k
kd
gM
where Λ is the (ultra-violet) cutoff and g represents the coupling (i.e., ). Even after this
momentum-independent contribution has been subtracted, there is still the usual
logarithmic momentum-dependent contribution to the mass:
Renormalization correction of the Higgs boson mass due to (Left) the λφ 4 term, (Middle) gauge boson
loops, and (Right) fermion loops.
H H H
µ
q
qM ln~)(H∆
just like that of the fermion masses (c.f., m(q2)=mo{1+(3α/4π)ln[(mo
2−q2)/Λuv]+…).
g g g g g](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-148-320.jpg)
![If we try to explain the Higgs couplings by embedding the Standard Model in some
larger GUT (e.g., SU(5), as discussed in the previous chapters) the breakdown:
149
2017
MRT
if MX ~1014 GeV (c.f., µ2 −(3/2)[5α +(3/2)β]vo
2=O(10−24)vo
2 ).
EM)()()()()()(
HX
131235 USUUSUSUSU CMYLCM
⊗ →⊗⊗ →
requires two types of particles (i.e., Φs with MΦ =O(MX) in addition to the usual Hs with
MH =O(MW)). In order to keep the Hs light while the are heavy, we must ensure
cancellation of the divergence ∆ MH
2 above to an accuracy:
24
2
X
W
2
H
10~~~ −
Φ
M
M
M
M](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-149-320.jpg)
![2017
MRT
D. Perkins, Introduction to High Energy Physics, 4-th Edition, Cambridge, 2000.
University of Oxford, England
This highly regarded textbook for advanced undergraduates provides a comprehensive introduction to modern particle physics.
Coverage emphasizes the balance between experiment and theory. It places stress on the phenomenological approach and basic
theoretical concepts rather than rigorous mathematical detail. Donald Perkins also details recent developments in elementary particle
physics, as well as its connections with cosmology and astrophysics. A number of key experiments are also identified along with a
description of how they have influenced the field. Perkins presents most of the material in the context of the Standard Model of
quarks and leptons. He also fully explores the shortcomings of this model and new physics beyond its compass (such as
supersymmetry, neutrino mass and oscillations, GUTs and superstrings) […].
P.D.B. Collins, A.D. Martin, E.J. Squires, Particle Physics and Cosmology, Wiley, 1989.
University of Durham, England
This readable introduction to particle physics and cosmology discusses the interaction of these two fundamental branches of physics
and considers recent advances beyond the Standard Models. Eight chapters comprise a brief introduction to the gauge theories of
the strong and the electroweak interactions, the so-called grand unified theories, and general relativity. Ten more chapters address
recent concepts such as composite fermions and bosons, supersymmetry, quantum gravity, supergravity, and strings theories, and
relate them to modern cosmology and experimental astronomy.
M. Kaku, Quantum Field Theory – A Modern Introduction, Oxford University Press, 1993
City College of the CUNY
The rise of quantum electrodynamics (QED) made possible a number of excellent textbooks on quantum field theory in the 1960s.
However, the rise of quantum chromodynamics (QCD) and the Standard Model has made it urgent to have a fully modern textbook
for the 1990s and beyond. Building on the foundation of QED, Quantum Field Theory: A Modern Introduction presents a clear and
comprehensive discussion of the gauge revolution and the theoretical and experimental evidence which makes the Standard Model
the leading theory of subatomic phenomena. The book is divided into three parts: Part I, Fields and Renormalization, lays a solid
foundation by presenting canonical quantization, Feynman rules and scattering matrices, and renormalization theory. Part II, Gauge
Theory and the Standard Model, focuses on the Standard Model and discusses path integrals, gauge theory, spontaneous symmetry
breaking, the renormalization group, and BPHZ quantization. Part III, Non-perturbative Methods and Unification, discusses more
advanced methods which now form an essential part of field theory, such as critical phenomena, lattice gauge theory, instantons,
supersymmetry, quantum gravity, supergravity, and superstrings.
S. Weinberg, The Quantum Theory of Fields, Volume II, Cambridge University Press, 1996.
Josey Regental Chair in Science at the University of Texas at Austin
In this second volume of The Quantum Theory of Fields, Nobel Laureate Steven Weinberg continues his masterly exposition of
quantum theory. Volume 2 (of 3) provides an up-to-date and self-contained account of the methods of quantum field theory, and
how they have led to an understanding of the weak, strong, and electromagnetic interactions of the elementary particles. The
presentation of modern mathematical methods is throughout interwoven with accounts of the problems of elementary particle
physics and condensed matter physics to which they have been applied.
156
References / Study Guide](https://image.slidesharecdn.com/linked-inslides-sm-150116221208-conversion-gate01/85/Part-VIII-The-Standard-Model-156-320.jpg)
Part VIII - The Standard Model
- 1. PART VIII – THE STANDARD MODEL From First Principles March 2017 – R3.1 Maurice R. TREMBLAY
- 2. Forward 2 2017 MRT Quantum field theory has emerged as the most successful physical framework describing the subatomic world. Both its computational power and its conceptual scope are remarkable. Its predictions for the interactions between electrons and photons have proved to be correct to within one part in 108 (i.e., a billion). Furthermore, it can adequately explain the interactions of three of the four known fundamental forces in the universe. The success of quantum field theory as a theory of subatomic forces is today embodied in what is called the Standard Model. In fact, at present, there is no known experimental deviation from the Standard Model (excluding gravity). To be specific, the Standard Model of particle physics is a partially unified quantum gauge field theory for the electromagnetic and weak interactions, which exhibits a broken SU(2)L⊗U(1)EM symmetry, together with the SU(3)C symmetric quantum chromodynamics for the strong interaction. As such, it seems to give a completely satisfactory account of the inter- actions of the fundamental particles, which are the quarks and leptons. Unfortunately, their gravitational interactions appear to be entirely in accord with classical general relativity, but so far no consistent quantized version of this theory has been devised or even tested. All of the quantum field theories that have been successful in describing and testing the fundamental interactions of nature are gauge theories, that is to say that they are invariant under gauge (i.e., phase) transformations of field potentials. This property has long been recognized in classical electromagnetism and so was built into quantum electrodynamics(QED) from the start. It also turned out to be the key to the development of quantum chromodynamics(QCD) for the strong color interaction,and, albeit with symmetry breaking, to the formulation of the unified electroweak theory.
- 3. The next plausible step beyond the Standard Model may be the Grand Unified Theories (GUT), which are based on gauging a single Lie group, such as SU(5) or O(10). The following Chart shows how gauge theories based on Lie groups have united the funda- mental forces of nature: GUT mean to unite the Strong Force with the Electroweak Force. 3 2017 MRT As with my other work, nothing of this is new or even developed first hand but the content (or maybe its clarity and the way it is organized) is original in the fact that it displays an abridged yet concise and straightforward mathematical development of the Standard Model as I understand it and wish it to be presented to the layman. Now, as a matter of convention,I have included the setting h≡c≡1 in most of the equations and ancillary theoretical discussions (N.B., these units have been reinstated wherever one might be confronted with an observational fact that needs to be made), but, as the astute reader will surely notice, I will forgo as much as I can the summation convention and display the summation signs so as to highlight the rich notation that they do convey. According to GUT, the energy scale at which the unification of all three particle forces takes is enormously large, above 1015 GeV, just below the Planck energy (i.e., EP ≅ 2.43×1018 GeV). Near the instant of the Big Bang, where such energies were found, the theory predicts that all three particle forces were unified by one GUT symmetry. In this picture, as the universe rapidly cooled down, the original GUT symmetry was broken down successively into the present-day symmetries of the Standard Model. Electricity Magnetism Weak Force Strong Force Gravity U(1) SU(2)⊗U(1) SU(5),O(10)? Weinberg-Salam (Electroweak) SU(2) SU(3) GUT Superstrings? The Standard Model SU(3)⊗SU(2)⊗U(1) GL(4,R),O(3,1)? OSp(N/4)? Maxwell (Electromagnetism) General Relativity
- 4. Gauge Boson Mixing and Coupling Fermion Masses and Couplings Why Go Beyond the Standard Model? Grand Unified Theories General Consequences of Grand Unification Possible Choices of the Grand Unified Group Grand Unified SU(5) Spontaneous Symmetry Breaking in SU(5) Fermion Masses Again Hierarchy Problem Higgs Scalars and the Hierarchy Problem Appendix – Useful Figures References Contents 2017 MRT PART VIII – THE STANDARD MODEL Introduction The Particles The Forces The Hadrons Scattering Field Equations Fermions Particle Propagators Noether’s Theorem and Global Invariance Local Gauge Invariance in QED Yang-Mills Gauge Theories Quantum Chromodynamics (QCD) Renormalization Strong Interactions and Chiral Symmetry Spontaneous Symmetry Breaking (SSB) Weak Interactions The SU(2)⊗U(1) Gauge Theory SSB in the Electroweak Model Gauge Boson Masses “Gravity is separate, because if we were only interested in the physics of individual particles, we wouldn’t know about it at all. It is only because we have some experience with huge collections of particles put together into planets and stars that we know about gravity.” H. Georgi, Lie Algebras in Particle Physics – Preface to Chapter 18 (Unified Theories and SU(5)), 1999. 4
- 5. Introduction By the end of these slides, you will be able to read and understand the key ideas of the Standard Model as described in Wikipedia: https://en.wikipedia.org/wiki/Standard_Model. Theoretical aspects - Construction of the Standard Model Lagrangian 2017 MRT 5
- 6. Fundamental forces 6 2017 MRT Gravitation Weak Electromagnetic Strong (Electroweak) γW+ W− Z0GravitonG(Theoretical) Gluons g1…g8 Mesons ud Hadrons Color charge Quarks, Gluons Flavor Electric charge QEM Quarks, Leptons Quarks Hadrons Quarks
- 7. According to the Standard Model of elementary particle physics, the universe is made of a set of fundamental spin-h/2 fermions, the leptons and quarks (see Table). The Particles 7 2017 MRT Quark Up u ⅔ 0.005 Down d −⅓ 0.01 Charm c ⅔ 1.5 Strange s −⅓ 0.2 Top (disc. 1994) t ⅔ 170 Bottom (disc. 1977) b −⅓ 4.7 Lepton Symbol Charge (e) Mass (GeV/c2)* Electron e− −1 0.000511 e-Neutrino νe 0 <7×10−9 Muon µ− −1 0.106 µ-Neutrino νµ 0 <0.0003 Tau τ− −1 1.7771 τ-Neutrino ντ 0 <0.03 * Masses given in units of 1Giga-eV/c2=1.782662×10−27 kg,with 1eV=1.602×10−19J and c is the velocity of light (in vacuum). Fermions interact through the exchange of spin-h bosons in a way that is precisely de- termined by local gauge invariance and through gravitation, and also through the ex- change of some spin-0 Higgs particles, which play a crucial role in generating mass.
- 8. These fermions may be divided into three generations (or families) (see Table). 8 First Generation Second Generation Third Generation Leptons e−, νe µ−, νµ τ−, ντ Quarks u, d c, s t, b Each generation contains two flavors of Quark, which enjoy strong interactions, and two leptons, which do not. 2017 MRT Each particle has an associated antiparticle with the same mass but opposite quantum numbers, so there are antileptons like e+ (with L=−1) and antiquarks like d(with B=−⅓). The leptons carry unit lepton number L=+1, but zero baryon number, while the quarks have one-third (fractional)baryon number B=+⅓, but zero lepton number L=0. The net lepton number and net baryon number appear to be conserved in all the interactions, as of course is the net electric charge. The leptons group naturally into pairs because in all processes the total number of particles of each generation, for example: )()e()()e( eee νν NNNNL −−+≡ +− appears to be conserved (and similarly for (µ−,νµ) and (τ−,ντ)). These rules are often referred to as conservation of electron number Le, muon number Lµ, and tau number Lτ. The masses, me, mµ, and mτ, of the charged leptons e−, µ−, τ− exhibit no discernable pattern, so a fourth (or further generations) is considered to be unlikely. _
- 9. Each of the quarks in the first Table can exist in three forms distinguished by the so- called color quantum number associated with the strong interaction coupling (i.e., an interaction is where the point particles or their constituents actually mix together via a potential – highlighted as in Figures) which can take the values red, blue, or green. 9 2017 MRT )]q()q([3 NNB −= There are no known interactions that mix quarks with leptons and hence the total quark number (i.e., the number of quarks minus the number of antiquarks, N(q)−N(q)) is conserved. This is referred to as the baryon number conservation since: This rule is crucial to the stability of protons and hence of matter itself. _
- 10. In the Standard Model these fundamental particles undergo four known types of gauge interaction – gravitation, electromagnetism, and the weak and strong nuclear forces – and also interact with Higgs bosons. 10 rr mm GrV N 1 )( 21 ∝−= The Forces According to Newton’s nonrelativistic theory of gravitation, the potential energy between two point-particles of mass m1 and m2 that are separated by a distance r is: where GN is Newton’s gravitational constant (see Table). The potential acts only over r−1. Constant Symbol Value Newton’s gravitational constant GN 6.67259×10−11 m3 kg−1 s−2 Velocity of light (vacuum) c 299,792,458 m s−1 Planck’s constant (Dirac’s h-bar) h≡h/2π 1.05457×10−3 J s Conversion constant hc 0.19733 GeV fm Fine structure constant α≡e2/4πεohc (137.03599)−1 Fermi constant GF/(hc)3 1.16637×10−5 GeV−2 2017 MRT A dimensionless measure of the strength of the gravitational coupling is given by GN m2/hc and if we insert a typical mass, such as that of the proton, mp, into the above equation we find that GN mp 2/hc≅10−40 is so extraordinarily small that gravity can safely be neglected in most practical aspects of particle physics.
- 11. According to general relativity, gravity really couples to the total energy, E≡mc2, not just the rest mass, mo, and if we write GN m2/hc as GN E2/hc5 we find that the coupling is unity for E=EP ≡MP c2, where MP =√(hc/GN)=1.2×1019 GeV/c2, the so-called Planck mass. Hence, gravity certainly cannot be neglected if we want to explore what may happen at such very high energies, EP =pc (with the momentum p defined by de Broglie as p=h/λ= 2πh/λ where we can also identify D=λ/2π as lP at that energy scale), that is if we want to probe to distances of r≅lP, the Planck length, defined by lP=hc/EP=√(hGN /c3)=1.6×10−35 m. 11 Now, according to quantum field theory, the gravitational force is carried by the particle ‘quantum’ of gravitational radiation called the graviton, G (see Figure). Then GN m2/hc gives a measure of the probability of a graviton being exchanged, which is clearly very small unless energies approaching EP or distances approaching lP are encountered. 2017 MRT Since the gravitational force is of infinite range, these gravitons must be massless, and because in general relativity the quantum field represents fluctuations of the rank-2 metric tensor of space- time, gµν , it must be spin-2h. Technically, the fact that gravity is found to have a long range automatically means that the interaction energy depends on separation as 1/r. The graviton must have a mass mG=0 so that the force proportional to 1/r2 results from an interaction and its spin cannot be h/2 since their could be no interference between the amplitudes of the single exchange, and no exchange nor can it have spin-h because one consequence of spin-h is that likes repel, and unlike attract as is the case in electromagnetism. Spin-0 is also out of the question due to the gravitational behavior of the binding energies. The gravitational interaction of two masses, m1 and m2, represented (Left) classically by force of gravity and (Right) quantum mechanically by virtual graviton exchange (x-t plane). The factor κ =√(8πGN)/c2 is the gravitational coupling. 2m 1m 2m 1m G ct x κ ≅2×106s⋅(kg⋅m)−½ κ
- 12. Of more immediate concern is the electromagnetic interaction. According to Coulomb’s law, the interaction potential between particles with charges Q1 and Q2, respectively, separated by a distance r is: 12 The electromagnetic interaction between a positron e+ and an electron e−, represented (Left) classically by lines of force and (Right) quantum mechanically by virtual photon exchange with coupling strengths ±e. 2017 MRT Since all particle charges appear to be simple multiples of the electron’s charge (−e where e=1.60217×10−19 C (or A⋅s) since 1 C =1 A⋅s) a convenient dimensionless measure of the electro- magnetic coupling is α≡e2/4πεohc=e2/4π≅(137)−1≅0.007 if we adopt the Heaviside-Lorentz units (i.e., εo ≡1) and, as is common in particle physics, set h ≡c≡ 1. In these units, e= √(4πα)≅0.303 and is dimensionless. Note that the photon has no mass, mγ = 0. − e + e rr QQ rV 1 επ4 1 )( 21 o ∝= where εo is the permittivity of the vacuum, whose value depends on the units adopted for the charge. In MKS units, εo =8.854×10−12 Fm−1 (A2 s4 m−3 kg−1) since 1 F=1 s4A2 m−2 kg−1. The quantum of the electromagnetic field is the photon, γ, and in quantum field theory it carries the electromagnetic force (see Figure). Since this force is also of infinite range, the photon must be massless, mγ = 0, and since it represents the U(1) gauge- invariant electromagnetic potential Aµ (a tensor of rank-1) it must have spin-h. It is this gauge-invariance property that ensures charge conservation and makes quantum electrodynamics (QED) a renormalizable theory (i.e., one that has only a finite number of divergences which, once subtracted away by absorption into the ‘bare’ parameters, leave a finite and sensible theory). − e + e γ +e ≅1.6×10−19 A⋅s −e
- 13. The weak interaction, which cause β-decays (e.g., such as n→pe−νe or muon decay µ− →e−νeνµ), is of very short range. In fact, originally it was thought to be point-like (see Figure - Top Left) with a strength given by the Fermi (circa 1932) weak coupling constant GF (see previous Table). It is a universal interaction in the sense that all quarks and leptons have the same overall weak coupling strength. The dimensionless coupling for a particle having the typical hadronic mass mp is thus GF mp 2c4/(hc)3 ≅1×10−5 which, when compared to α≡e2/4πεohc≅7×10−3, explains why this is called the weak interaction. 13 (Top Left) The β-decay n → pe−νe by Fermi’s point-like interaction of strength GF. (Top Right) The same process mediated by virtual W− exchange, with coupling strength g. (Bottom) Another weak interaction process νµe− scattering mediated by virtual Z0 exchange. 2017 MRT According to the Glashow-Weinberg-Salam theory the weak force is in fact carried by very massive, spin-h, vector bosons W+, W−, and Z0 (see Figure) which generate an approximately SU(2) isospin-invariant weak interaction. The apparent weakness of the interaction is due, not to the smallness of the coupling 1×10−5 would seem to imply, but to the improbability of these very massive virtual particle being emitted. In fact, the weak interaction coupling g is comparable to e above, with GF ~ g2/MW 2 ~ e2/MW 2. More precisely, the couplings are related by: where sin2θw =0.222±0.011(1994),θw being the Weinberg weak mixing angle between the electromagnetic and weak interactions. Given our knowledge of α, GF , and sin2θw , our equation above can be used to deduce that MW ≅80 GeV/c2 and hence that the range of the weak interaction is ~ h/MWc ≅ 2.5×10−18 m ≅0.003 fm (with 1 fm =10−15 m). − e n p FG eν − e n p eν W− g g Z0 g g − e− e µν µν w F cMcM g c G θ α 242 W 2 W 2 3 sin2 π 24)( == hh ⇒ _ _ _
- 14. Parity is not conserved in weak interactions because these W bosons couple only to left-handed (L) chiral projections of the quark and lepton fields. This means that the Ws couple to relativistic particles (i.e., with E>>mc2) only if they are spinning left-handedly about their direction of motion (see Figure). The SU(2)L isospin symmetric weak coupling of left-handed fermions is often called quantum flavor dynamics (QFD). 14 Particles traveling along an arbitrary positive z-direction with (Left) left-handed (anticlockwise) or (Right) right-handed (clockwise) spins along their directions of motion. Try it with your left / right hands. The particles’ spin is represented by the hashed arrow ( ⋅⋅⋅⋅ for tip and ×××× for base). 2017 MRT It turns out that if we start with a renormalizable SU(2) invariant gauge theory, which necessarily has massive W and Z fields, but then spontaneously break the symmetry by adding a Higgs scalar boson that has nonvanishing vacuum expectation value, W and Z (and in fact that quarks and leptons too) acquire finite masses, yet the renormalizability is retained. As a result of mixing with the proton (with mixing angle θw) the Higgs boson is not predicted, but is expected to be of the same order as MW. A chiral phenomenon is one that is not identical to its mirror image. The spin of a particle may be used to define a handedness, or helicity, for that particle which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity. Invariance under parity by a Dirac fermion is called chiral symmetry. An experiment on the weak decay of Cobalt-60 (i.e., 60Co) nuclei carried out by Chien-Shiung Wu and collaborators in 1957 demonstrated that parity is not a symmetry of the universe. The helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion (see Figure). L.H. Nature has favored ×××× ⋅⋅⋅⋅ z R.H. ×××× ×××× z
- 15. Finally, we come to the strong force that binds quarks together to form hadrons, and hadrons into complex nuclei. This force is associated with the color coupling, which can take one of three possible values: red (R), green (G) or blue (B). The strong force is invariant under transformations among these three colors. Indeed, the nature of the strong force can be deduced simply by demanding that it should obey an exact local SU(3)C color symmetry (i.e., the C subscript), analogous to the U(1) invariance of electromagnetism. The massless quanta of this quantum chromodynamics (QCD) color field are called gluons, g (e.g., see Figure - Left), and because of the SU(3)C invariance it turns out that there must be 32 −1=8 different color combinations of gluons. And since the gluons carry color they can also couple to each other (e.g., see Figure - Right). 15 (Left) A quark-antiquark interaction by virtual gluon exchange, with strong coupling strength gs. (Right) The coupling of two gluons by gluon exchange, also of strength gs. 2017 MRT g qq q q gs gs g g g g g gs gs The strength of the strong coupling, gs, gives the probability of a quark or gluon emitting a gluon as in the Figure, and it is convenient to introduce: π4 2 s s g ≡α
- 16. Before we can comment on the magnitude of the αs coupling, we have to take note of the fact that ‘vacuum polarization’ (i.e., the creation of virtual particle-antiparticle pairs) greatly complicates the description of relativistic quantum states. The uncertainty principle permits the existence of such pairs of particles (e.g., the best known being the positron e+ and electron e− pairs), each of mass me, for time periods T less than h/2mec2 (see Figure). 16 Virtual e+e− creation in the electromagnetic photon field, with coupling strength √α = e/√(4πεohc). 2017 MRTα α − e + e If the coupling that determines the probability of such pair creation is small (e.g., like the fine structure constant, α =e2/4πεohc, in QED), this effect can be treated as a small perturbation. This is the basis of perturbative quantum field theory, in which physical quantities are represented as power series in α. It successfully accounts for such quantities as the anomalous magnetic moments of the electron and muon, and the Lamb shift in atoms, for example, which are a direct result of vacuum polarization.
- 17. Because of this vacuum polarization screening effect, the electromagnetic coupling varies with distance, r (see Figure), so that the fine structure constant α is not really a constant at all but varies with r. It turns out that the e+e− loop gives: 17 A negative charge, represented by a ‘point’ particle, which is ‘screened’ by vacuum polarization (i.e., the creation of e+e− pairs) [how organized the spread of polarization is is a matter of speculation] making the whole QED-picture-view of an electron as ‘fuzzy’. 2017 MRT In QCD it is found that similarly (i.e., for r>>hc/ΛC): 1 o ln π2 )( − Λ ≅ r cb r C s h α 1 e 1ln π3 1)( − +−≅ crm r hα αα where α is the value measured at large r (i.e., r>>h/mec, the electron’s Compton wavelength). Evidently, α is predicted to become very large at very short distances (i.e., as r→h/[mecexp(3π/α)]≅10−300 m), by which point perturbation theory must break down. Our equation above still indicates that QED alone cannot be correct at high energies! where bo is a constant (see Renormalization chapter), is positive because the self-coupling of the gluons results in antiscreening. Hence αs →0 as r→0, which gives rise to the so-called asymp- totic freedom (i.e., the fact that quarks and gluons inside a hadron behave like free particles when very close together). On the other hand, αs apparently diverges as r→RC ≡hc/ΛC, where ΛC is the hadronic energy scale. This divergence simply heralds the breakdown of perturbation theory, of course, but nonetheless it leads us to expect that the strength of the force between quarks will increase if they are pulled apart. As a result, quarks and gluons are confined inside hadrons which size is of order RC. + − + − + − + − + − + −+− −−−− + − + − + −
- 18. The crucial parameter characterizing the strong interaction is the hadronic energy scale ΛC. It turns out to be: 18 2017 MRT GeV3.02.0 −≅ΛC Not surprisingly, it is of the same order of magnitude as the masses of the lightest hadrons. In summary, in order to account for all the known types of interactions, we need the spin-2h graviton and the 12 spin-h gauge bosons listed in the Table. The Standard Model of particle physics is thus gravitation, together with the above SU(3)C⊗SU(2)L⊗U(1)Y gauge-invariant strong and electroweak interactions. After the spontaneous breaking of the symmetry as a result of the Higgs coupling, we are left with SU(2)L⊗U(1)EM as exact gauge symmetries, and the 8 gluons g1…g8 and the photon γ as massless particles. Constant Symbol Spin (h) Mass (GeV/c2) Charge (e) Graviton† G 2 0 0 Photon γ 1 0 0 Charged weak bosons W± 1 81.5 ±1 Neutral weak boson Z0 1 92.5 0 Gluons* g1…g8 1 0 0 Higgs H 0 126 0 * <0.0002 eV/c2 (Experimental limit). †If I were to speculate based on a discernable pattern, since the Higgs generates the mass of both weak bosons would it be conceivable that the photon and the gluons generate the graviton in some theoretically yet-to-be-formulated model? TBD…
- 19. As we have just seen, the strength of the color coupling, αs, is not constant but becomes stronger as the separation between the quarks increases (i.e., αs(r)≅1/[(bo/2π) ⋅ln(hc/ΛC r)]). So, unlike electrons in atoms, quarks are permanently bound together, and confined within hadrons. It is only color-neutral combinations of quarks that can have a vanishing color coupling and hence occur as free particles. There are two different ways in which these colorless hadrons can be made, given the three different colors of quarks qk (k = R for red,G for green, andB for blue): 19 2017 MRT ∑=+−= kji kji kji 321 B 3 R 2 G 1 B 3 G 2 R 1 qqq 6 1 )qqqqqq( 6 1 B εK The Hadrons where εijk is the anti-symmetric permutation tensor, or; ∑=++= k kk 21 B 2 B 1 G 2 G 1 R 2 R 1 qq 3 1 )qqqqqq( 3 1 M These hadrons are color-neutral in a somewhat analogous way to that whereby one can produce white either by an equal mixture of the three primary colors (red, green, and blue) like B above, or by mixing green (say) with its complementary color, magenta (=red+blue), like M above. 2) We can take a quark q1 of one color and an antiquark q2 which has the opposite (complementary) color, and thereby make a meson: 1) We can take an admixture of three quarks, q1, q2, and q3, say, which have different colors, and thereby obtain a baryon: _
- 20. The lightest baryons (B=1) and mesons (B=0) are listed in the following Table. 20 2017 MRT Baryons Spin-h/2 Spin-3h/2 Quark Content Particle Mass (GeV/c2) Particle Mass (GeV/c2) uuu ∆++ 1.232 uud p 0.9383 ∆+ 1.232 udd n 0.9396 ∆0 1.232 ddd ∆− 1.232 uus Σ+ 1.1894 Σ+ (1385) 1.3828 uds Σ0 1.1925 Σ0 (1385) 1.3837 Λ(*) 1.1156 dds Σ− 1.1973 Σ− (1385) 1.3872 uss Ξ0 1.3149 Ξ0 (1530) 1.5318 dss Ξ− 1.3213 Ξ− (1530) 1.5350 sss Ω− 1.6724 Mesons Spin-0 Spin-h Quark Content Particle Mass (GeV/c2) Particle Mass (GeV/c2) ud, du π± 0.13957 ρ± 0.770 (uu –dd)/√2 π0 0.13497 ρ0 0.770 us, su K± 0.49365 K*± 0.8921 ds,sd K0,K0 0.49767 K*0, K*0 0.8921 (uu –dd)/√2 η 0.5488 ω 0.782 ss η′ 0.9575 φ 1.0194 cd,dc D± 1.8693 D*± 2.0101 cu, uc D0,D0 1.8645 D*0, D*0 2.0072 cs,sc Ds ± 1.969 Ds*± 2.113 cc ηc 2.980 ψ 3.0969 ub, bu B± 5.278 B*± 5.325 db,bd B0,B0 5.279 B*0, B*0 5.325 sb,bs Bs 0,Bs 0 5.366 Bs*0, Bs*0 5.415 bb ηb 9.388 ϒ 9.4603 * Heavier baryons, such as Λc = udc, can be made by substituting the heavier c, b, or t quarks for any of those shown.
- 21. Baryons such as the proton p=uud and neutron n=udd come in states where the 3 quark composing them have their spins oriented ↑↑↓ (i.e., up-up-down) so that the pro- ton and neutron both have spin-h/2. For other particles, the spins can be ↑↑↑ and so there are heavier spin-3h/2 states made of the same set of quarks (i.e., ∆+ and ∆0, respecti-vely) that are unstable and decay (e.g., ∆+ →pπ0, where π0 is a neutral pion). Mesons with B=0 consist of a quark (B=⅓) together with an antiquark (B=−⅓) (e.g., the spin-0 π+=ud is an ↑↓ spin state, while the spin-h ρ+ is made of the same quarks but in the ↑↑ spin state). 21 2017 MRT 2 c mm CΛ +≈ currenttconstituen Since quarks are always confined within hadrons, it is not possible to measure their masses directly. The values quoted in the previous Table are so-called current quark masses deduced from what is called current algebra. It is important to notice that the light hadrons (i.e., those made of only u, d, and s quarks) are very different from most of the composite states found in physics because their masses are much lighter than the sum of the individual current algebra quark masses. The mass of a hadron can be thought of as being made up of the sum of the kinetic energies of the quarks, which are of order ΛC (i.e., the hadronic energy scale), together with their current (algebra) masses. It is sometimes useful to introduce a constituent quark mass of magnitude: _ so that, in terms of constituent masses, mp ≅2mu +md for example. Although the current masses of the u and d are slightly different, their constituent masses are almost identical (both ~ΛC /c2), which is why the proton and neutron masses are almost equal, mp ≅mn.
- 22. The resulting symmetry under the interchange of u and d quarks produces the SU(2) isospin symmetry of nuclear physics. The s quark is not all that much heavier, so there is also an approximate SU(3) flavor symmetry among those hadrons that can be made out of just u, d, and s quarks. The c, b, and t quarks are much heavier than ΛC, however, so there is not much difference between their current (algebra) and constituent masses. 22 Because the heavier quarks undergo weak decays (e.g., s→ue−νe, c→sµ+νµ, b→cud, &c.), the proton is the only stable baryon. The neutron is nearly stable, with a lifetime τ ~ 15 minutes for its decay, n→pe−νe (i.e., in terms of quarks, d→ue−νe), because mn ≅mp, and of course it is stable when inside a nucleus if its extra binding energy exceeds (mn −mp −me)c2. In these decays the total number of quarks, and hence the baryon number B, is conserved. Indeed, B is conserved in all interactions, a fact that is crucial to the stability of atomic matter. Quark diagrams for the dominant decays of the lightest mesons, π+ and π0. 2017 MRT All of the mesons are unstable because even the lightest, the pion, can decay, viz: as a result of quark-antiquark annihilation (see Figure). γγπµπµπ 0 µµ →→→ −−++ and, νν + π µν + µu d W+ )d(u 0 π )d(u γ γ _ _ __
- 23. Any attempt to knock a quark out of a hadron, for example by hitting it with an electron (see Figure - Left), or spontaneous hadronic decay processes (see Figure - Right), involve a stretching of the color lines of force. This results in the creation of quark- antiquark qq pairs in the vacuum, so the lines of force get shortened, but they still always begin on a q and end on a q. Hence, a quark that is knocked out of one hadron necessarily ends up inside another. The forces between quarks and gluons increase with distance, so they are confined within hadrons whose radii: 23 (Left) Deep inelastic electron-proton scattering in which a quark is knocked out of the proton by a virtual photon γ. This is followed by the creation of qq pairs in the stretched color field and hence the production of hadrons. (Right) The decay of the unstable ρ+ meson into a π+π0 state through dd creation. C C c RR Λ ≡≈ h 2017 MRT and typically about 10−15 m (1 fm), because the strong color interaction energy scale is approximately ΛC ~ 0.2-0.3 GeV. e p q q q N e γ d d d u u d + ρ 0 π + π g g _ _ _ _
- 24. Of course, even though hadrons are color-neutral they can still interact with each other by the exchange of gluons and quarks (see Figure - proton-proton scattering with p=uud). The force results from the polarization of the color charge. 24 2017 MRT Proton-proton scattering by (Left) gluon exchange, (Middle Left) quark exchange, (Middle Right) π0 pion exchange (i.e., as a result of the bonding of the exchanged quarks in Middle Left), and (Right) ‘Pomeron’ exchange (i.e., interacting gluons). If quarks are exchanged over distances >RC, they may bind together to form mesons (see Figure - Middle Right) which is why the long-range part of the interaction was originally identified by Yukawa as meson exchange. The gluonic force may similarly be identified with the ‘Pomeron’ effect (i.e., interacting gluons), which was introduced in the late 1950s to account for elastic and diffractive scattering processes in which there is no exchange of flavor quantum numbers. Gluons carry color and hence couple to each other (see Figure - Right), whereas photons of course do not carry charge and so cannot couple together directly. g g 0 πq p p p p ‘Pomeron’ exchange
- 25. In particles made of heavy quarks (i.e., mqc2 >>ΛC), such as the ψ(cc) charmonium states, the ϒ(bb) bottomonium states (N.B., the high mass of the top quark, toponium tt, does not exist since the top quark decays through the electroweak interaction before a bound state can form), the velocities of the quarks are very much less than the speed of light, c, and relativistic effects should not be too important. In such cases it is reasonably satisfactory to treat the particles as if they were made of just their constituent valence quarks, with an interaction potential between the q and q that takes the form: 25 2017 MRT r r rV s )( 3 4 )( α −= where the factor of 4/3 comes from the sum over all possible colors of quarks, and over the colors of the gluons that can be exchanged. Since αs varies with r in αs(r)≅1/[(bo/2π)ln(hc/ΛC r)], this can be rewritten: Λ −≈ r c rb rV C h ln π2 3 4 )( o or, to represent better the behavior at large r (i.e., where αs(r) ≅1/[(bo/2π)ln(hc/ΛC r)] is invalid): rT r rV s o 3 4 )( +−≈ α where To is called the ‘string tension’. _ _ _ _
- 26. This arises because gluons interact with each other and so produce a cigar- or string- shaped distribution of color lines of force (see Figure), quite different from Coulomb’s law. Since the energy density in the ‘string’ is approximately constant, the energy of the string increases with r and gives rise to a potential V(r)~To r at large r. It therefore needs an infinite amount of energy to drag the quarks apart (unless new qq pairs are created), which is presumably the explanation for the confinement of quarks in hadrons. 26 The long-ranged interaction in QED, in which the field’s energy density represen- ted by the separation of the lines of force ~r−2, contrasted with the QCD interaction of a quark and an antiquark, in which the energy density is constant inside the color ‘cigar’ or ‘string’. 2017 MRT − e + e r QED QCD q q _
- 27. Another important reason for this confidence in QCD, and indeed in the whole SU(3)⊗SU(2)⊗U(1) Standard Model of the strong, weak, and electromagnetic properties of matter, is that it correctly predicts the outcome of a great variety of scattering experiments: electron-positron scattering (e.g., e+e−), deep inelastic scattering (e.g., e−p), and hadron scattering (e.g., pp), to name a few. 27 In a two-body scattering process such as (see Figure - Left): dcba +→+ Scattering According to quantum theory a high-energy beam of energy E=p/c has an associated wavelength λ=hc/E (i.e., λ∝1/E), which determines the shortest distance which can be resolved. Hence, to probe short distances we require very high energies. each particle has four-vector momentum matrix: T ][ pcEp =µ where µ =0,1,2,3 corresponding to the timelike, 0, and the three spacelike components, 1,2,3, respectively, E being its energy and p its momentum, in a given Lorentz frame. We will employ the usual Minkowski space-time metric, ηµν , with sig- nature (+,−,−,−), so that the Lorentz-invariant scalar product: 2 o 2 23 0 2 )()( cm c E ppppppp =− =≡≡⋅≡ ∑∑= p µν ν µν µ µ µ µ η where mo is the is the particle’s rest mass (moc being the reference or standard momentum kµ ). It is usually more convenient to work in units where c≡ 1 so this mass-shell condition above becomes p2 =m2 with mo≡m from now on. _ (Left) The scattering process a +b→ c +d with the four-momenta labelled. (Right) This same process in the center-of-mass (CM) system. Here θ is the scattering angle of particle c with respect to the beam direction and d Ω is the element of solid angle into which c is scattered. 2017 MRT
- 28. The scattering process can be described by three Lorentz-invariant Mandelstam variables: 28 2017 MRT 22 22 22 )()( )()( )()( bcda bdca dcba ppppu ppppt pppps −=−≡ −=−≡ +=+≡ which can readily be shown to satisfy: 2222 dcba mmmmuts +++=++ so only two of the s, t, or u are independent. ],[],[ pp −== bbaa EpEp and represents the square of the total center-of-mass energy, ECM ≡Ea +Eb, while the center- of-mass frame scattering angle θ is given by: θcos222 2222 cacacacaca EEmmppmmt pp+−+=⋅−+= At high energies, where all the masses are negligible, so that |pa|~Ea, &c., we find: )cos1( 2 )cos1( 2 θθ +−≈−−≈ s u s t and In the center-of-mass system of the scattering process (see previous Figure - Right): 22 ],[ CMEEEs ba =+= 0 and:
- 29. The differential cross section, dσ, which is the probability per unit incident flux of particle c being scattered into a given element of solid angle dΩCM =dϕ d(cosθ) (see previous Figure - Right), is given by: 29 2017 MRT 2 2 π64 M a c sd d p p = ΩCM σ or from t~2|pa||pc|cosθ above (N.B., s=ECM 2=(Ea +Eb)2): ∫ Ω=+→Γ d m cba a 2 22 π32 )( M p 2 2 2 2 6π1 11 π64 1 MM sstd d a ≈= p σ Similarly, the width for the decay a→b+c is: where p is the momentum of particle b or c in the rest frame of a and the integration is over all directions of p. 2 2 π64 1 M sd d ≈ ΩCM σ where M is the scattering amplitude. We have to integrate this over the scattering angle θ to get the actual scattering cross-section, σ = ∫(dσ /dΩCM)dΩCM =∫(dσ /dΩCM)sinθdθdϕ. Now, if we neglect the masses:
- 30. In classical mechanics, the equations of motion of a dynamical system can be derived from the Lagrangian function L(qi,qi) (N.B., the dot ‘⋅⋅⋅⋅’ over the generalized coordinate q is shorthand for q=∂q/∂t) with qi being the generalized coordinates of the system, which is defined to be: 30 2017 MRT )()(),( iiii qVqTqqL −= && Field Equations ⋅⋅⋅⋅ where T is the kinetic energy and V is the potential energy. The action S involved in the motion of the system from one configuration at time t1 to another at t2 is given by: ∫= 2 1 t t tdLS and, according to the principle of least action, the path actually taken by the system will be the one that minimizes S. It is readily shown that the condition for S to be a minimum is that L should obey the Euler-Lagrange equations: 0= ∂ ∂ − ∂ ∂ ii q L q L td d & These equations of motion are equivalent to Newton’s laws of motion. Thus, for a qi →xi = x example, by using T=½mx2 in L(x,x)=T(x)−V(x) above one can deduce from the Euler- Lagrange equations that a particle’smotion will obeyNewton’s second law if expressed as: where F(x) is the force at point x. ⋅⋅⋅⋅ )()(2 2 xFx x ≡−= V td d m ∇∇∇∇ ⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅⋅⋅
- 31. For the purpose of keeping the relativistic covariance of the physics more apparent, it is convenient to replace L by the Lagrangian density function L so that: 31 2017 MRT ∫= x3 dL L and thus the action S above can also be rewritten covariantly as: ∫∫∫ === xddtdtdLS 43 LL x since d4x=dtd3x where x is the space-time four-vector with components xµ =[t,x]. In relativistic field theory, we replace qi by the field φ(xµ), so the index i is replaced by the space-time coordinates x≡xµ, and the Lagrangian is a Lorentz scalar function of φ(x). Since each ∂/∂x can be associated with a similar term involving ∇∇∇∇, we can make up the covariant derivative (and its equivalent shorthand notation): 0 )()()( 3 0 3 1 = ∂ ∂ − ∂∂ ∂ ∂= ∂ ∂ − ∂∂ ∂ ∂ ∂ + ∂∂ ∂ ∂ ∂ ∑∑ == φφφφφ µ µ µ LLLLL i i i t xt So, for a given choice of Lagrangian L(φ,∂µφ), this set of space-time Euler-Lagrange equations can be used to deduce the equations that will be satisfied by the field φ(x). and so the Euler-Lagrange equations become: φ φ µµ ∂≡ ∂ ∂ x
- 32. Thus, for a free scalar (i.e.,spin-0) particle of (rest) mass m the Lagrangian density is: 32 2017 MRT 22 2 1 ))(( 2 1 φφφ µ µ µ m−∂∂= ∑L and so the differentials are: φφ φφ φφφ φφ µ µ µ µ µµ 222 2 1 0)(0))(( 2 1 )()( mm −= ∂ ∂ −= ∂ ∂ ∂=− ∂∂ ∂∂ ∂ = ∂∂ ∂ ∑ LL and which, when substituted into the space-time Euler-Lagrange equations, gives the Klein- Gordon equation: 0)][( )( 222 =+=+ ∂∂=+∂∂= ∂ ∂ − ∂∂ ∂ ∂ ∑∑∑ φφφφφφ φφ µ µ µ µ µ µ µ µ µ mmm LL where we have introduced the d’Alembertian operator: 2 2 2 ∇− ∂ ∂ =∂∂= ∑ tµ µ µ In view of the usual quantum-mechanical operator replacements (h≡1) E→i∂/∂t and p→ −i∇∇∇∇ which, in four-vector notation, given [E,p]→[i∂/∂t,−i∇∇∇∇]=i[∂/∂t,−∇∇∇∇], gives the four- momentum as: µµ ∂−→ ip Now, φ +m2φ =0 simply ensures that the field obeys the mass-shell condition p2=m2.
- 33. The flow of particles is represented in space-time by the current four-vector: 33 2017 MRT *)*( φφφφ µµµ ∂−∂= iJ ],[ Jρµ =J where ρ is the particle density and J is the flux current. It is a conserved quantity: 0=∂=•+ ∂ ∂ ∑µ µ µ ρ J t J∇∇∇∇ For example, the current for the scalar boson field φ is: In order to discuss vector (i.e., spin-h) particles, we need to consider properties of the classical electromagnetic field, which are summarized in Maxwell’s equations: 0 B EB J E BE = ∂ ∂ =• = ∂ ∂ =• t t ++++××××∇∇∇∇∇∇∇∇ −−−−××××∇∇∇∇∇∇∇∇ and ,, 0 ρ where E and B are the electric and magnetic field strengths, respectively. Here ρ is the charge density and J is the charge-current density (which, unlike Jµ =[ρ,J] includes a factor of e). As usual in particle physics, we use the Heaviside-Lorentz units where εo ≡1.
- 34. It is convenient to introduce the four-vector potential: 34 2017 MRT µννµµν AAF ∂−∂= ],[ Aφµ =A so that: φ∇∇∇∇−−−−××××∇∇∇∇ t∂ ∂ −== A EAB and Then, since ∇∇∇∇•∇∇∇∇××××A=0 and ∇∇∇∇××××∇∇∇∇φ =0, the last two Maxwell equations (i.e., ∇∇∇∇•B=0 and ∇∇∇∇××××E++++∂B/∂t=0) are satisfied automatically, while the first two (i.e., ∇∇∇∇•E=ρ and ∇∇∇∇××××B−−−− ∂E/∂t=J) can be re-expressed in terms of the field-strength tensor: in the form: νµ µ µ νν µ µν µ JAAF = ∂∂−=∂ ∑∑ where Jµ =[ρ,J] is the charge-current four-vector that satisfies Σµ∂µ Jµ =0.
- 35. The Maxwell equations can be derived from the Lagrangian (density): 35 2017 MRT χχ φφ χ µµµ ∂+→ ∂ ∂ +→ → AA t ∇∇∇∇−−−−AA ∑∑ −−= µ µ µ µν µν µν AJFF 4 1 L By substituting this Lagrangian into the space-time Euler-Lagrange equations and minimizing the action with respect to each component Aµ, we again obtain Σµ ∂µ Fµν =Jν. In quantum field theory, Aµ can be regarded as the wave function of the photon, and the expression Σµ ∂µ Fµν =Jν is then the wave equation that the photon has to satisfy. However, the B=∇∇∇∇××××A and E=−∂A/∂t−−−−∇∇∇∇φ equations do not specify Aµ uniquely in terms of the physical E and B fields, since under a so-called ‘gauge’ transformation of the form: where χ can be any scalar function of the space-time coordinate x, the fields B, E, and hence Fµν are all completely unchanged, as may readily be checked by direct substitu- tion in B=∇∇∇∇××××A, E=−∂A/∂t−−−−∇∇∇∇φ, and Fµν =∂µ Aν −∂ν Aµ (Exercise). It is often useful to make use of this gauge freedom and fix the gauge so that Aµ satisfies the Lorentz condition: in which case Σµ ∂µ Fµν = Aν −∂ν (Σµ ∂µ Aµ)= Jν reduces to Aµ =Jµ. Redundancy! 0=∂∑µ µ µ A
- 36. For non-interacting photons, Jµ =0, and Aµ =Jµ is simply the requirement that the mass-shell condition p2 =m2=0 be satisfied (i.e., we will use the conventional q2 =0 for a massless photon of four-momentum q). The plane-wave solutions have the form: 36 2017 MRT xqi qA ⋅− = e)(µµ ε with q2 =0, where ε µ is the polarization vector of the photon (N.B., we now use the four- vector shorthand q⋅x ≡Σµ qµ xµ). The Lorentz condition Σµ ∂µ Aµ=0 requires that: 0=∑µ µ µεq which reduces the apparent four degrees of freedom to three. However, one of these three is spurious (i.e., a dummy one) because the Lorentz condition does not completely fix the gauge. Transformations like Aµ→Aµ +∂µχ are still allowed provided χ =0; for example χ=iaexp(−iq⋅x). On substituting this, together with Aµ=ε µ(q)exp(−iq⋅x), into Aµ →Aµ +∂µ χ one sees that no physical quantity is changed by the replacement εµ→εµ +aqµ so two polarization vectors that differ by some multiple of the four-momentum describe the same photon. We may use this freedom to set ε0≡0, whereupon the Lorentz condi- tion Σµ ∂µ Aµ=0 becomes q•εεεε=0 and the photon has only transverse polarization. This (non-covariant) choice of gauge is known as the Coulomb gauge. For a photon travelling along the z-axis, the two independent polarization vectors can be taken to be εεεεR,L = m(1/√2)(êx ± iêy) where êk are unit vectors along the k-axis (k=1,2,3). These describe a photon with spin projection along the direction of motion (i.e., helicities) of ±1, respec- tively (as can be easily verified from their behavior under rotations about the z-axis).
- 37. For a free spin-h particle of (rest) mass M, we take: 37 2017 MRT instead of L =−¼Σµν Fµν Fµν−Σµ Jµ Aµ above with Jµ =0. This leads, using the space-time Euler-Lagrange equations again, to the Proca equation: ∑∑ +−= µ µ µ µν µν µν AAMFF 2 2 1 4 1 L 02 =+∂∑ ν µ µν µ AMF If we take the divergence of this equation (i.e., operate on it with ∂ν), we find Σν ∂ν Aν =0 (for M2 ≠0). This is not a necessary condition (i.e., not a choice of gauge). The Proca equation then becomes: 0)( 2 =+ ν AM which leads again to the mass-shell condition p2 =M2≠0. Polarization vectors of a massive spin-h particle automatically satisfy Σµ qµε µ =0, but as there is no gauge invariance there are three degrees of freedom, corresponding to helicities ±1 and 0. For example, a particle with momentum q along the z-axis has ε (r=±1) =m(1/√2)[0,1,±i,0] and ε (r=0)=(1/M)[|q|,0,0,E]. They satisfy the completeness relation: 2 )()( *][ M qq r rr νµ µννµ ηεε +−=∑ Ok,from now on we set h≡1 so as to describe spin-h particles will be spin-1 particles.
- 38. We now turn to spin-½ particles (with h≡1) which obey Fermi-Dirac statistics (N.B., as opposed to spin-0, spin-1, and spin-2 particles which obey Bose-Einstein statistics), for which the situation is more complicated. Fermions are described by four-component spinor fields: 38 2017 MRT = 4 3 2 1 ψ ψ ψ ψ ψ Fermions For free particles of (rest) mass m (with h≡1), the spinor ψ (x) satisfies the Dirac equation: 00o = −∂⇔= −∂ ∑∑ ψγψγ µ µ µ µ µ µ micmi h where the γ µ are a set of 4×4 matrices. A more compact Feynman slash notation, in which: is often used. With this notation, the Dirac equation above becomes (Dirac 1928): 0)( =−∂/ ψmi aa /≡∑µ µ µ γ
- 39. In order that the equation obtained by operating with (iΣµγ µ∂µ +m) on (iΣµγ µ∂µ −m)ψ =0 should be the Klein-GordonequationΣµ∂µ ∂µφ +m2φ =0(so as to guarantee that E2 =p2 +m2), we require (i.e., the Clifford algebra): 39 2017 MRT µνµννµνµ ηγγγγγγ 2},{ =+≡ called an anticommutator (to contrast with the commutator [σµ,σ ν ]≡σµσν −σνσµ). All of the spin properties of the theory follow from this equation for the γ matrices and particular representations are not necessary. However, it is often much easier to work with a suitable chosen representation. We shall take the γ matrices to be unitary, so from the anticommutator {γ µ,γν} above we find (Exercise): kk γγγγ −== †0†0 and These equations can be summarized by the relation: 00† γγγγ µ k = The ‘standard’ (or Pauli-Dirac) representation is given by: − = = 0 0 0 00 k kk I I σ σ γγ and where I is the 2×2 unit matrix and the σ k are the usual 2×2 Pauli matrices: − = − = = 10 01 0 0 01 10 321 σσσ and, i i
- 40. It is convenient to define a fifth matrix, usually denoted by γ 5, as: 40 2017 MRT 32105 γγγγγ i≡ Clearly from {γ µ,γ ν}: 50123†0†1†2†3†5 γγγγγγγγγγ ==−= ii We also find that (Exercise): 0},{1)( 525 == µ γγγ and In our standard representation: = 0 05 I I γ and (Exercise): kk γγγγ −== †0†0 and
- 41. 41 2017 MRT 0=−∂− ∑ ψγψ µ µ µ mi The Hermitian conjugate of the Dirac equation iΣµγ µ∂µψ −mψ =0 is: 0††† =−∂− ∑ ψγψ µ µ µ mi which, on multiplying by γ 0 from the right: 00†0†† =−∂− ∑ γψγγψ µ µ µ mi and using γ µ† =γ 0γ µγ 0 and the fact that (γ 0)2 =1, becomes: where we have introduced the adjoint (row) spinor: 0† γψψ ≡ On multiplying iΣµγ µ∂µψ −mψ =0 from the left by ψ and −iΣµ ∂µψγ µ −mψ =0 from the right by ψ and subtracting, we deduce that: ψγψ µµ ≡J is the probability flux or current that satisfies the conservation equation Σµ∂µ Jµ =0. The probability density: is positive definite. This was part of the original motivation for the introduction of Dirac’s equation (as the story goes, he was staring into a fireplace at Cambridge). ψψψγψρ †00 ==≡ J _ _ _
- 42. Bilinear quantities with simple Lorentz transformation properties can be built from ψ and ψ . It can be shown (Exercise) that: 42 2017 MRT ψψ ψγγψ ψγψ ψγψ ψψ µν µ µ Σ 5 5 transform as a scalar, pseudoscalar, vector, axial-vector and (antisymmetric) tensor, respectively, where the tensor is the commutator of the γ µ (i.e., that set of 4×4 matrices): ],[ 2 )( 2 νµµννµµν γγγγγγ ii ≡−≡Σ The Lagrangian which, using the space-time Euler-Lagrange equations, leads to the free Dirac equation (iΣµγ µ∂µ −m)ψ =0 is: ψψψψ mi −∂/=L or, using the Feynman slash notation (i.e., a≡Σµγ µaµ): ψψψγψ µ µ µ mi −∂= ∑L _
- 43. We must next consider the coupling between spin-½ and spin-1 particles. 43 2017 MRT 10† − −== Cψcc T γψψ 0)( =−−∂∑ ψψγ µ µµ µ mAei The corresponding equation written for an antiparticle (i.e., with e→−e) is satisfied by: *0ψγψψ TT CCc =≡ with ψ =ψ †γ 0 and where c, the notation for a charge-conjugation matrix, satisfies: T µµ γγ −=− CC 1 So, if ψ is regarded as a field operator that annihilates a particle or creates an antiparticle, then ψ c is the charge-conjugation field, which annihilates an antiparticle or creates a particle. From ψ c=Cψ T= Cγ 0 Tψ * above we find that: In classical electromagnetism, the force experienced by a particle of charge e in an electromagnetic field is the Lorentz force F=e(E++++v××××B), which is obtained if in the classical Lagrangian we make the replacement pµ →pµ−eAµ, where Aµ is the vector potential. Similarly, in quantum theory we make the replacement i∂µ →i∂µ −eAµ. Hence, the Dirac equation for a particle in an electromagnetic field is: _ _
- 44. Some useful bilinear identities that follow from these results are: 44 2017 MRT − − == 0 0 01 10 01 10 02 γγiC 12212 1 121 )()( ψψηψηψψψψψ Γ=Γ−=Γ−=Γ − TTTTT CCcc where by using C −1γµ C=−γµ T we find that η=(1,1,−1,1,−1) for Γ={1,γ 5,γ µ,γ µγ 5,Σµν}, respectively (N.B., in transposing, add a ‘−’ sign since fermion fields anticommute). One consequence of the above results is that the current J µ=ψ γ µψ satisfies ψγµψ =−ψ cγµψ c while ψ cψ c =ψγµψ , which is just what one would expect to be the result of the charge- conjugation operation. In the standard representation (N.B., differs from PART IV’s Dirac basis) of γ 0 and γ k : More generally, in all representations of interest, we have: CCCC −=== −1†T − − = − − = = = 0 0 0 0 0 0 0 0 10 01 10 01 0 0 0 0 01 10 01 10 10 01 10 01 3210 γγγγ and,, i i i i a possible choice for C that satisfies C−1γµ C=−γµ T is: _ _ _ _ _
- 45. We shall later meet two important special types of fermion field. The first is the chiral or Weyl fermion defined by: 45 2017 MRT RLRL ,, 5 ψψγ m= where the suffixes L, R are used to indicate that these eigenstates of γ 5 correspond to left-, right-handed chiralities (i.e., which in the zero-mass or infinite-momentum limit correspond to helicities m½), respectively. We can project out the left- or right-handed parts of a general spinor ψ with the projection operators ½(1mγ 5): ψγψ )1( 2 1 5 , m=RL From this last relation we can form the conjugate of ψL,R : )1( 2 1 )1( 2 1 50†5† , γψγγψψ mm ==RL The charge of sign in front of the γ 5 occurs because it anticommutes with the γ 0. We can therefore write: RLLR ψψψψψ γγγγ ψψψ += − + + − + + = 2 1 2 1 2 1 2 1 5555 since (1+γ 5)(1−γ 5)=0. Similarly, we find: LLRR ψγψψγψψγψ µµµ += So the scalar term ψψ mixes R, L fermions whereas the vector term ψ γ µψ does not. __
- 46. The other special case is the Majorana fermion, which by definition is its own charge conjugate: 46 2017 MRT ψψγψψ == *0 T or Cc Obviously, a Majorana spinor must have zero for its charge (and for other additive quantum numbers like lepton number). Some important results for Majorana spinors follow from our previously derived useful bilinear identities. For example, for spinors χ and ψ: ψχχψχγψχψ === † 0 †† )( A given spinor cannot satisfy both Majorana and Weyl conditions. To see this, suppose that, for example, ψ is a right-handed Weyl spinor and hence satisfies: ψψγ =5 Then we find: **)( 05055 ψγγψγγψγ TTT CCc == By using γ 5= iγ 0γ 1γ 2γ 3 and C−1γµ C=−γµ T repeatedly, which gives: cc C ψψγγψγ −=−= ** 5 05 T using {γ 5,γ µ}=0, γ 5†=−iγ 3†γ 2†γ 1†γ 0†=iγ 3γ 2γ 1γ 0, and γ 5ψ =ψ above. Thus, a right- handed Weyl spinor ψ becomes a left-handed spinor ψ c upon charge conjugation. It follows therefore that a Weyl ψ cannot be identical to ψ c and so cannot satisfy the Majorana condition ψ c=ψ above.
- 47. Finally, we mention particles of spin-3/2, which are described by so-called Rarita- Schwinger fields, ψµ, where µ is a vector index, and, for each value of µ, ψµ is a four- component spinor. For a particle of (rest) mass m, the Lagrangian: 47 2017 MRT 0)( =−∂/ µψψ mi ∑∑ +∂−= µ µ µ µνρσ σρνµ µνρσ ψψψγγψε m5 2 1 L leads to the equation of motion: 05 =−∂− ∑ µ νρσ σρν µνρσ ψψγγε m If m≠0, the divergence of this equation, and it scalar product with γµ, lead to the two constraints: The 2×2 condition Σµ∂µψµ =Σµγ µψµ =0 above reduce the original 4×4 degrees of freedom of ψµ to eight, which corresponds to the four possible helicity states (±3/2, ±1/2) of the particle and its antiparticle. If m=0, the constraint Σµ∂µψµ =Σµγ µψµ =0 above do not follow from the equation of motion but are a choice of gauge. Gauge invariance eliminates two more degrees of freedom, leaving only the ±3/2 helicity states for a massless spin-3/2 particle. which allows us to write the equation of motion in the simpler Dirac-like form: 0==∂ ∑∑ µ µ µ µ µ µ ψγψ
- 48. Exact solutions of relativistic quantum field theories are not known, so it is usually necessary to use perturbation methods in which the amplitude for the process of interest is expressed as a power series in the coupling constant. The various terms in the expansion are given by Feynman diagrams that can be evaluated by a set of Feynman rules. In particular, the internal lines of a diagram are the propagators that represent the motion of virtual intermediate state particles from one vertex to another. These propagators are Green’s functions in the sense that they correspond to the inverse of the operator that appears in the wave equation for the particle. Thus, for a scalar field with source term ρ, the wave equation is (c.f., Σµ ∂µ ∂µφ +m2φ= φ +m2φ = 0 with m→M): 48 2017 MRT Particle Propagators ρφφ =−≡+− )()( 222 MqM and the particle propagator is: εiMq i +− 22 The conventional factor i in the numerator is useful to obtain a simple and consistent set of Feynman rules, and the +iε, where ε is a positive infinitesimal constant, is needed to give the correct definition of the propagator in the region of the singularity at q2 =M2. The propagators of particles that have spin are also given by i/(q2 −M2 +iε), but in addition we must include a (completeness) sum over all the possible intermediate spin states |σ 〉, so instead we take: ∑+− s ss iMq i ε22
- 49. For a massless vector field this does not specify the propagator uniquely, because of gauge invariance. In a Lorentz gauge the wave equation Σµ ∂µ Fµν = Aν −Σµ ∂ν ∂µ Aµ = Jν can be rewritten in momentum space as: 49 2017 MRT ν µ µ µννµ ξη JAqqq =−− ∑ )( 2 where ξ is an arbitrary parameter. The term containing ξ does not contribute because of Σµ ∂µ Aµ =0. It is easy to verify (Exercise) that: ν λ µ λµµλµνµν δ ξ ξη ξη = − +−−− ∑ 42 2 1 )( q qq q qqq so we take: as the propagator. In the last propagator, the final term (i.e., with coefficient ξ/(ξ−1)) does not contribute if the particle is coupled only to conserved currents for which Σµqµ Jµ =0 and so it is usual to choose ξ=0, which gives the propagator in the Feynman gauge. The numerator of the above propagator contains the (completeness) sum over the four spin states of a virtual photon (q2 ≠0), µ =0,1,2,3. For a real photon (q2 =0) the contribu- tions of the longitudinal and timelike spin states cancel each other, leaving only two transverse spins allowed by Σµ qµε µ =0. − −− 22 1 q qq q i λµ µλ ξ ξ η
- 50. For a massive spin-1 particle, the Proca equation leads to the propagator: 50 2017 MRT That the numerator corresponds to the sum over spin states can be checked by comparing with Σr[εµ (r)]*εν (r) =−ηµν +qµ qν /M2. +− − 222 M qq Mq i νµ µνη Finally, the propagator for a spin-½ particle will be the inverse of the Dirac operator (iΣµγ µ∂µ −m)ψ =0 : Again the numerator contains the (completeness) sum over spins: 2222 )()( )( )( )( mq mqi mq mqi mq mq mq i mq i − +/≡ − +⋅ = +⋅ +⋅ −⋅ = −⋅ γ γ γ γγ ∑= =+/ 2,1 )()( )()( s ss ququmq where ψ =u(q)exp(−iq⋅x) and u ≡u†γ 0, and where a relativistically invariant normalization of the spinors is used: sr sr muu δ2)()( = _
- 51. In quantum theory, although the relative phases of wave functions are of crucial importance in determining interference effects and the like, the absolute phase of a wave function is unmeasurable and arbitrary. It is not surprising, therefore, that the Lagrangians of the previous chapter such as L =−½Σµ(∂µφ)(∂µφ) −½m2φ 2 are unchanged by phase transformations of the form: 51 2017 MRT )(*e)(*)(e)( xxxx ii φφφφ αα − →→ and Noether’s Theorem and Global Invariance where φ* is the complex-conjugate (i.e., i→−i) of φ. Indeed, at first sight this seems an entirely trivial observation. However, according to Noether’s theorem: This can be readily be demonstrated by considering infinitesimal values of α such that: ***)1(*e*)1(e φδφφαφφφδφφαφφ αα +≡−≈→+≡+≈→ − ii ii and so the change in the fields is δφ=iαφ and δφ*=−iαφ*. The change in the Lagrangian resulting from this replacement is: An invariance necessarily implies the existence of a conserved current associated with the particle. ∑∑∑ ∑∑ →∂+ →+ ∂∂ ∂ ∂+ ∂∂ ∂ ∂− ∂ ∂ = ∂ ∂∂ ∂ + ∂ ∂ +∂ ∂∂ ∂ + ∂ ∂ = µ µ µ µ µ µ µ µ µ µ µµ µ µ φφφδφφφδ φ φδ φφ φδ φ φδ φ φδ φ φδ φ δ *** )()( *)( *)( * * )( )( LLL LLLL L
- 52. The first term vanishes by virtue of the Euler-Lagrange equations: 52 2017 MRT 0 )( = ∂ ∂ − ∂∂ ∂ ∂∑ φφµ µ µ LL as does the corresponding term when φ* replaces φ, and so we end up with: ∑∑ ∂−∂∂= ∂∂ ∂ + ∂∂ ∂ ∂= µ µµ µ µ µµ µ φφφφαφδ φ φδ φ δ )**(* *)()( i LL L from L =−½Σµ(∂µφ)(∂µφ) −½m2φ 2. However, we have already noted that the Lagrangian is in fact unchanged and so δ L =0, and hence the particle current for a complex field φ: *)*( φφφφ µµµ ∂−∂= iJ introduced earlier for the scalar boson field φ must be a conserved quantity (i.e., it must obey Σµ ∂µ Jµ=0). This implies the conservation of probability and hence of the charge or any other similar additive quantity that can be associated with the complex field φ. Note that under φ →φ* the sign of the current is reversed, and so if φ has charge e (say) then φ* has charge −e (i.e., φ* represents the antiparticle of φ). In the same way, the invariance of the Dirac Lagrangian L =iΣµψγ µ∂µψ −mψ ψ under ψ →exp(iα)ψ and ψ →exp(−iα)ψ leads to the conserved current for a fermionic field ψ : ψγψ µµ =J _ _ _ _
- 53. The transformation φ →exp(iα)φ (and its conjugate φ* →exp(−iα)φ*) is often referred to as a global transformation in the sense that φ(x) has its phase changed by the same amount, α, globally for all values of x. It is also clearly a unitary transformation (i.e., one which preserves the normalization of φ), in that: 53 2017 MRT φφφφφφφφφφ αααα *e*e*ee** 0)( ===→ +−− iiii since exp(0)≡1. If we make two such unitary transformations, say: φφφφφφ αα 21 ee 21 ii UU ≡→≡→ and then obviously: φφφφφ αααα 12 )()( 21 1221 ee UUUU iiii ===→ ++ and so these transformations commute with each other (i.e., they are Abelian unitary transformations). The set of all such transformations is given by varying α within the range 0≤α<2π. It is generally referred to as the group U(1) of all the unitary transformations which depend on a single parameter α. Clearly dU/dα =iU, and groups that are differentiable with respect to the group parameters in this way are called Lie groups.
- 54. It is useful to generalize the idea of global invariance. Thus, the isotopic spin invariance of nuclear physics under p↔n, or of the weak interaction can be represented as an invariance of the system under transformations within an isospin multiplet; for example, within the quark doublet ψ =[u d]T. The most general such transformation is: 54 2017 MRT ψψψ ατ∑ = ≡→ 3 12e k kk i U where the τk are the 2×2 Pauli isospin matrices (like the σk ): − = − = = 10 01 0 0 01 10 321 τττ and, i i and the αk (k=1,2,3) are the phase rotation parameters in the three orthogonal directions in isospin space. The requirement that U is unitary requires the τk to be unitary and, since Tr(τk)=0 for all k, the U are unimodular in that: 1ee)det( 3 1 Tr 2)(lnTr ==≡ ∑ =k kk i U U ατ The Lie group of all unitary 2×2 matrices with unit determinant is called SU(2). The addition of the unit matrix I to the τk above would give the U(2) group of all unitary 2×2 matrices, but I would simply produce a change of the phase of u and d by the same amount. Such a U(1) transformation is just like φ →exp(iα)φ and corresponds to the conservation of quark or baryon number and has nothing to do with isospin itself. Locally, U(2) is isomorphic to SU(2)⊗U(1).
- 55. The Pauli matrices τk are called generators of the isospin transformations, and their commutation relations: 55 2017 MRT ∑= = 3 1 2],[ k kjkiji i τεττ (where εijk is the permutation tensor) are called the Lie algebra of the group, the εijk being the structure constants. The doublet ψ =[u d]T, which has the same dimension as the generators τk , is called the fundamental representation of the group. Since the τk do not commute, neither in general will two transformations like ψ →exp[(i/2)Σkτkαk]ψ (i.e., U2U1 ≠U1U2), and so this group is non-Abelian. These transformations are still unitary: Noether’s theorem tells us that if this is a good symmetry then any component of isospin will be a conserved quantity. However, as the τk do not commute, only one component is measurable at a time, and by convention this is taken to be the third component (which thus has the diagonal matrix in the basis τ1, τ2, and τ3 above). Hence, the isospin 3-axis component, T3, is (in units of h): is conserved if ψ →Uψ ≡ exp[(i/2)Σkτkαk]ψ is a symmetry of the Lagrangian. 33 2 1 τ=T where U† is the Hermitian adjoint matrix of U. 1== †† UUUU
- 56. Similarly, the strong interaction is invariant under permutations of the colors of the quarks, so that if we write the quark wave function as a fundamental color triplet repre- sentation ψ =[R G B]T we will have invariance under SU(3) transformations of the form: 56 2017 MRT ψψψ αλ∑ = ≡→ 8 12e a aa i U where U are unitary unimodular 3×3 matrices, the αa (a=1,2,…,8) are the eight phase angles, and the λa (Gell-Mann) matrices are eight independent traceless, Hermitian 3×3 matrices which generate the group. They are the equivalent of the 2×2 Pauli matrices for SU(2) and the conventional choice is given in the Table on the next slide. ) (2 2],[ 88776655 44332211 8 1 λλλλ λλλλ λλλ abababab abababab c cabcba ffff ffffi fi ++++ +++= = ∑= where the structure constants fabc are also given in the Table on the next slide, where you will notice that the only diagonal matrices are λ3 and λ8. For example: The λa matrices satisfy the Lie algebra: )(22],[ 83787377637653754374337323721371 8 1 3773 λλλλλλλλλλλ ffffffffifi c cc +++++++== ∑= where [λ3,λ7]≡λ3λ7 −λ7λ3.
- 57. 57 2017 MRT The 3×3 traceless Hermitian λa matrices of SU(3) are: The nonvanishing totally antisymmetric structure constants are: and all other fabc (a,b, c=1,2,…,8) are either related to these by antisymmetry (e.g., f156 =− f165) or they vanish. The trace (the sum of the elements on the main diagonal) of the product of two λ matrices is: ,and ,,,, ,,, − = −= = − = = ≡ −= ≡ − = ≡ = 200 010 001 3 1 00 00 000 010 100 000 00 000 00 001 000 100 00 0 000 010 001 00 0 000 00 00 00 0 000 001 010 8 7654 3 3 2 2 1 1 λ λλλλ τ λ τ λ τ λ i i i i i i 2 3 2 1 1 678458376345257246165147123 ========= fffffffff and, abba δλλ 2)(Tr = where the Kronecker delta δab is equal to 1 for a =b and 0 for a ≠b.
- 58. Quantum electrodynamic (QED) is the quantum theory of the interactions of charged particles. In classical electromagnetism, the force experienced by a particle of charge e in electromagnetic fields is the Lorentz force: 58 2017 MRT )( BvEF ××××++++e= Local Gauge Invariance in QED which is obtained if in the classical Lagrangian for the particle we make the replacement: µµµ Aepp −→ Correspondingly, in quantum theory we make the replacement: µµµ Aeii −∂→∂ and so, for example, the Lagrangian describing a charged spin-½ particle in an electromagnetic field is (from L =iΣµψγ µ∂µψ −mψ ψ and L =−¼Σµν Fµν Fµν−Σµ Jµ Aµ ): ∑∑∑ −− −∂= µν µν µν µ µ µ µ µ µ ψγψ FFAJemi 4 1 L where Jµ is just the current obtain earlier as Jµ ≡ ψγ µψ. The first term gives rise to the fermion’s propagator i(γ ⋅q+m)/(q2 −m2), the second to the fermion-photon vertex coup- ling, while the last term produces the photon propagator −(i/q2)ηµν as in the Feynman Rules of the Table on the next slide which can be represented schematically as: =L + + e _ _ _
- 59. 59 2017 MRT Scalar particle propagator (momentum p) 22 1 Mp i − Fermion propagator (momentum q) 22 mq mq i − +⋅γ Massless vector propagator (momentum k) 2 k i µνη Fermion-photon vertex (charge e) µ γi− Scalar boson-photon vertices (charge e) µ )( ppi ′+− p q e p p′ Three-gluon vertex (strong coupling gs) ])()( )[( 1332 21 µλννλµ µνλ ηη η kkkk kkfabc −+−+ −− Four-gluon vertex (strong coupling gs) Quark-gluon vertex (strong coupling gs) µα γλ ji i 2 − Massive vector propagator (momentum p) 22 2 / Mp Mpp i − − νµµνη gs α j i µν ηi2 e gs b k2, fν c k3, fλ a k1, fµ gs 2ρ, d µ, a λ, c ν, b )]( )( )([ ρνµλρλµν λρµνλνµρ νλµρνρµλ ηηηη ηηηη ηηηη −+ −+ −− ebcade edbace ecdabe ff ff ffi e2 As as general rule (due to limited space available at times) Feyman diagrams are draws as: or: but should be viewed as: or: or k p e gs e e2 gs 2 gs 2
- 60. This form of coupling: 60 2017 MRT ∑∑ −=− µ µ µ µ µ µ ψγψ AeAJe is often referred to as the ‘minimal coupling’ of a spin-½ particle to the electromagnetic potential because it contains just the change and the point-like Dirac magnetic moment, but no anomalous magnetic moment or other momentum-dependent terms of the type one obtains for composite spin-½ systems (e.g., proton). It is remarkable that the form of L =iΣµψ γ µ∂µ ψ −mψψ −eΣµψ γ µAµψ −¼Σµν Fµν Fµν, which successfully describes the electromagnetic properties of elementary fermions, can be deduced simply by demanding that the Lagrangian must be invariant under local phase transformations, which for historical reasons are called ‘gauge transformations’. _ _ _
- 61. A local gauge transformation is one in which: 61 2017 MRT )(e)()(e)( )()( xxxx xixi ψψψψ χχ − →→ and so that, in contrast to the global transformation φ →exp(iχ)φ, the phase change, χ(x), can be different at every space-time point x=xµ.This is obviously not an invariance of the free-particle Lagrangian L1 = L =iΣµψγ µ∂µψ −mψ ψ , since under ψ (x)→exp[iχ(x)]ψ (x): ∑∑∑ ∂−=∂−=−∂→ −− µ µ µ µ µ µχχ µ χ µ µχ χχψγψψψψγψ Jxxxmxxi xixixixi 11 )()()()( 1 )()](e[)](e[)](e[])(e[ LLL Only if ∂µχ=0 (i.e., if χ is independent of x), is L1 unchanged because of the derivative involved in the energy-momentum term. However, if in L =iΣµψγ µ∂µψ −mψ ψ we make the replacement (c.f., i∂µ →i∂µ −eAµ): )(xAeiD µµµµ +∂≡→∂ where Aµ (x) is some vector field, we get instead: χµµµ ∂−→ e AA 1 which precisely cancels the additional and unwanted term in the development L1 above. Dµ defined in ∂µ →Dµ ≡∂µ +ieAµ(x) is called the ‘covariant derivative’. which is invariant under ψ (x)→exp[iχ(x)]ψ (x) and ψ (x)→exp[−iχ(x)]ψ (x) provided that at the same time we make the replacement: ∑−= µ µ µ AJe1LL _ _ _ _ __
- 62. Apart from the fact that it does not include the energy of the electromagnetic field, L = L1 −eΣµ JµAµ is just the same as L =iΣµψ γ µ∂µψ −mψψ −eΣµ JµAµ −¼Σµν Fµν Fµν, and Aµ → Aµ −(1/e)∂µχ is just a gauge transformation of the type Aµ→Aµ +∂µ χ that we saw earlier, which we know leaves the physical E and B fields, and hence the Lorentz force, unaltered. Thus, if we choose to identify the ‘gauge field’ Aµ with the electromagnetic potential and e with the charge of the fermion, we have essentially deduced the electromagnetic properties of a charged fermion just by requiring the gauge invariance of its Lagrangian. To make the identification complete we must add the electromagnetic field energy term LVector =−¼Σµν Fµν Fµν (with Fµν =∂µ Aν − ∂ν Aµ), which is of course gauge- invariant by itself; indeed, it is essentially the only simple gauge-invariant quantity we can construct involving ∂µ Aν. 62 2017 MRT Hence, the gauge invariance of the electromagnetic potentials, which in classical electromagnetism seems to be just a nuisance reflecting the fact that the four-vector Aµ has too many degrees of freedom, is seen to play a crucial role in the quantum theory of charged particles. It is needed to compensate for the phase freedom of the particle’s wave function ψ (x)→exp[iχ(x)]ψ (x) (and also its adjoint ψ (x)→exp[−iχ(x)]ψ (x)). We have already noted that the term ½ M2Σµ Aµ Aµ in the free spin-1 particle of mass M Lagrangian L =−¼Σµν Fµν Fµν+½ M2Σµ Aµ Aµ (this Lagrangian leads to the Proca equation Σµ ∂µ Fµν+ M2 Aν =0) and so the required gauge invariance is a property only of massless vector fields. The extra significance of the potential Aµ in quantum theory has been made more explicit in the work of Aharonov and Bohm (c.f., P.D.B. Collins, et al., P. 49ff). _ _ _ _
- 63. In light of this success it seems worthwhile to explore the consequences of turning non- Abelian global symmetries such as isospin SU(2) or color SU(3) into local gauge symmetries. Thus, if the fundamental doublet ψ =[u d]T is transformed as: 63 2017 MRT Yang-Mills Gauge Theories where now the αk(x) are functions of the space-time coordinates x=xµ, the Lagrangian density describing this doublet will only be invariant under this local SU(2) symmetry if we make the replacements: → = ∑ = d u e d u 3 1 )( 2 k kk x i ατ ψ ∑+∂≡→∂ k k k xWg i D )( 2 µµµµ τ where g is some (arbitrary) coupling strength and Wk µ(x) are three independent gauge field potentials acting in different directions in isospin space. They form the adjoint representation of the group, since their transformation properties are the same as those of the τk generators.
- 64. In fact: 64 2017 MRT − = −+ − = − + − + = − + − + =++= •≡ + − ∑ 3 3 321 213 3 3 2 2 1 1 3213 3 2 2 1 1 2 2 0 0 0 0 0 0 10 01 0 0 01 10 µµ µµ µµµ µµµ µ µ µ µ µ µ µµµµµµ µµ τττ τ WW WW WWiW WiWW W W Wi Wi W W WW i i WWWW W k k k Wττττ where Wk ≡W is a vector in isospin space, and where W± ≡(1/√2)(W1 miW2) can be identified with the charged gauge boson since the isospin step-up and step-down (i.e., also called ladder) operators ½(τ1 ±iτ2) change d↔u accompanied by the absorption of a charged W± boson. The interaction term in the Lagrangian is then: ++−−=•− ∑∑∑∑∑ −+ ud2du2dduu 2 1 2 1 33 µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ γγγγψγψ WWWWgg Wττττ which gives identical charged and neutral weak current couplings apart from the SU(2) Clebsch-Gordan coefficient √2.
- 65. The requirement of gauge invariance is that the four-momentum term Σµψγ µDµψ should be unchanged provided that we make the transformation Wµ→Wµ+δ Wµ in analogy with Aµ→Aµ−(1/e)∂µχ. For infinitesimal αk(x), we can consider the change: 65 2017 MRT )()( 2 1)( xx i x ψψ •+→ ααααττττ so that: ∑∑ • + +• +∂ • +→ µ µµµ µ µ µ µ ψδγψψγψ ααααττττττττααααττττ 2 1)( 22 1 i g ii D WW which is unchanged provided: )])(())([( 2 ααααττττττττττττααααττττααααττττττττ ••−••+∂•−=• µµµµδ WWW g i g The term in the square brackets is: ∑ ∑∑∑∑∑∑ =−=− ji k kjkiji ji ijjiji i ii j jj j jj i ii iWWWW τεατττταατττατ 2)( from [τi,τj]=2iΣkεijkτk, and so we need: ∑−∂−= ji j ijkik k xWx g W )()( 1 µµµ αεαδ _
- 66. This is similar to QED’s Aµ →Aµ−(1/e)∂µχ except for the final term, which reflects the non-Abelian nature of the W fields (N.B., Wk ≡W is a vector in isospin space). The energy of these fields can be written like LVector=−¼Σµν Fµν Fµν as: 66 2017 MRT ∑ ∑= = −= 3 0, 3 1 4 1 νµ µν µν i i i W WWL but because the fields do not commute this is only gauge-invariant if instead of Σµ ∂µ Fµν = Jν we define the field strength Wi µν to be: ∑−∂−∂≡ jk kj jki iii WWgWWW νµµννµµν ε Hence, the full Lagrangian of this SU(2) gauge-invariant theory is: ∑∑∑∑∑ −− −∂= µν µν µν µ µ µ µ µ µ ψτγψψγψ i i i i i i WWWgmi 4 1 )( 2 1 L The first term is just the four-momentum of the fermions, the second is the interaction of the fermion isospin current with the W fields and the last term gives the kinetic energy of the W fields as in LW = −¼Σµν Σi Wi µνWi µν above. Hence the gauge transformation must be: ∑−∂−→ ji j ijkik kk xWxx g xWxW )()()( 1 )()( µµµµ αεα
- 67. However, when Wi µν= ∂µ Wi ν −∂ν Wi µ −gΣjk εijk Wj µWk ν is substituted into the Lagrangian LW = −¼Σµν Σi Wi µνWi µν , there is also a term of order g coupling three W fields together and a term of order g2 coupling four W fields. These self-interactions of the W fields are typical of non-Abelian theories. 67 2017 MRT Unfortunately, this theory is of no use for the weak interaction because it gives identical couplings to right- and left-handed fermions and so conserves parity and, even more serious, in order to preserve the gauge invariance it is essential to have massless W bosons, which would give rise to a weak interaction of infinite range like electromagnetism. Only if the gauge symmetry is broken by the inclusion of mass terms like ½ M2Σµ Aµ Aµ it is possible to achieve agreement with experiment. A way of doing this without destroying the beneficial features of gauge theories will be discussed in the Electroweak Interactions chapter. Instead, we go straight on to color SU(3), a non- Abelian symmetry that is indeed unbroken! This can be represented schematically as: =L + + g g + + g2
- 68. We can make a local SU(3) gauge transformation of the fundamental fermion color triplet ψ =[R G B]T of the form: 68 2017 MRT Quantum Chromodynamics (QCD) )(e)( 8 1 )( 2 xx a aa x i ψψ αλ∑ = → and, completely in parallel with the previous chapter, the kinetic energy term will preserve gauge invariance if (c.f., ∂µ →Dµ ≡∂µ + (ig/2)Σkτk Wk µ): ∑+∂≡→∂ a a as xGg i D )( 2 µµµµ λ where Ga µ(x) are eight gluon field potentials (N.B., a=1,2,…,8), provided that these potentials transform as (c.f., Wi µν= ∂µ Wi ν −∂ν Wi µ −gΣjk εijk Wj µWk ν): ∑−∂−→ ab c babca s aa xGxfx g xGxG )()()( 1 )()( µµµµ λλ from [λa ,λb]=2iΣc fabcλc. To ensure the gauge invariance of the gluon field-strength tensor, it is defined as (c.f., Wi µν= ∂µ Wi ν −∂ν Wi µ −gΣjk εijk Wj µWk ν ): ∑−∂−∂≡ bc cb abcs aaa GGfgGGG νµµννµµν
- 69. Hence, the full Lagrangian of the theory is (c.f., the Lagrangian of the SU(2) gauge- invariant theory L=iψ (Σµγ µ∂µ −m)ψ −(g/2)Σµ Σi (ψγ µτi ψ)Wi µ −¼Σµν Σi Wi µνWi µν ): 69 2017 MRT ∑∑∑∑∑ −− −∂= µν µν µν µ µ µ µ µ µ ψλγψψγψ a a a a a as GGGgmi 4 1 )( 2 1 qL This can be represented schematically as: The Feynman rules corresponding to this Lagrangian are summarized in the previous Table. Again, the last term involves not just the gluon propagator but also cubic (3-rd order) and quartic (4-th order) self-couplings of order gs and gs 2 respectively from Ga µν= ∂µGa ν −∂ν Ga µ −gsΣbc fabcGb µGc ν . They arise because gluons both carry color and couple to color, and hence couple to each other. By contrast, in QED the photons couple to the charge but do not carry charge themselves, and hence cannot couple directly to each other. +=L + gs gs + + gs 2 g q _ _
- 70. A major obstacle to the application of quantum field theories is that naively they predict that all physical observable quantities such as charge, mass, &c., are infinite. To understand why, let us examine, for example, the electromagnetic coupling in QED, which involves the photon propagator in the Feynman gauge (see Figure (a)): 70 2017 MRT 2 q i µνη − Renormalization But if we also consider the lowest-order vacuum polarization correction, involving e+e− loop as in Figure (b), we get an additional contribution: (a) The electron-photon bare coupling eo and (b) the electron loop diagrams that modify the photon propagator, leading to a renormalization of the charge coupling. − −− +/−/ − +/ −− ∫ ∞ 20 2 e 2 e o2 e 2 e o4 4 2 )( )()( Tr )π2( q i mqk mqki ei mk mki ei kd q i νννµµµ η γγ η (a) (b) + +eo q eo q eo eo qk kq − ν νµ µ where eo is the ‘bare’ charge coupling at each vertex and me is the (rest) mass of the electron. The integral appears to diverge like ∫(1/k2)d4k. However, when the γ matrices are multiplied out it is found to behave only like ∫(1/k4)d4k, but this still diverges logarithmically.
- 71. If we decide to impose an ‘ultraviolet’ cutoff at k2 =Λuv 2 we find that the sum of the last two results (i.e., photon propagator and additional contribution) is of the form: 71 2017 MRT + Λ − +− )(ln 12π 1 2 2 uv 22 e 2 2 o 2 qF qme q i νµη where F(q2) is a finite function of q2 that vanishes as q2 →0. Hence, with the additional contribution integral, the effective charge we measure at low |q2|<<me 2 is not eo but the renormalized charge given by: Λ −= 2 e 2 uv 2 2 o2 o 2 eff ln 12π 1 m e ee which becomes infinitely different from eo as Λuv 2 →∞. However, this is only the lowest- order correction, and if we take just the leading logarithm of each term in the full series in the previous Figure, we find that the effective coupling has the form: Λ − − ≈ + Λ − + Λ − +≈ 2 uv 22 eo o 2 2 uv 22 eo 2 uv 22 eo o 2 eff ln 3π 1 ln 3π ln 3π 1)( qm qmqm q α ααα αα L where we have introduced αeff ≡eeff 2/4π and αo ≡eo 2/4π.
- 72. Since αo is not a measurable quantity we can reparametrize this result, and thereby renormalize the charge, by defining that α ≡α(q2 =0) has the value measured in very low-energy scattering experiments (q2 <<me 2), α ≅1/137. Then: 72 2017 MRT 21472 e 300π32 e 2 )GeV10(10e ≈≈→ mmq α − − ≈ 2 e 22 e 2 eff ln 3π 1 )( m qm q α α α gives the leading logarithmic variation of the coupling as |q2| is increased from zero. As we have already noted from its Fourier transform α(r)≅α/[1−(α/3π)ln(1+h/merc)],αeff (q2) above has the consequence that αeff (q2)→∞ as: This is the (Landau energy)2. In practice, of course, there are several charged leptons and quarks that contribute to the vacuum polarization as in the previous Figure, but still QED seems to require some modification as one approaches the unimaginably high energy given by the (Landau energy)2. This is not very surprising, because gravitation must change things as |q|→EP, the Planck scale. However, perturbation theory with α = O(10−2) should still be satisfactory at all practically attainable energies.
- 73. Similarly, we find that the fermion propagator of Figure (1a) obtains divergent modifica- tions from the graphs such as Figure (1b) that produce a change in its mass of the form: 73 2017 MRT + Λ − +≈ L2 uv 22 o o 2 ln π4 3 1)( qm mqm α which again can be incorporated by renormalizing to the physical mass m measured at q2 <<m2. The vertices of Figure (2a) get renormalized by diagrams like Figure (2b) & (2c). (1a) The fermion propagator and (1b) the radiative corrections that renormalize its mass. + + (1a) (1b) (2a) The electron-photon vertex (upper) and the four-photon coupling (lower) and (2b), (2c), … some of the radiative corrections that renormalize the electron and photon wave functions. + (2a) (2b) + (2c) +
- 74. Hence the coupling αo and mass mo, the parameters that appear in the original Lagrangian L =iΣµψ γ µ∂µψ −mψψ −eΣµ JµAµ −¼Σµν Fµν Fµν, are not observable quantities. Infinities seem to arise in the ‘observed’ quantities α and m, but this only means that αo and mo are not in fact finite. The solution to the problem is to reparametrize the expressions for truly observable quantities, such as scattering cross sections for example, in terms of the finite parameters α and m so that: 74 ])()(1[ ])()(1[ 2 ouv2ouv1o 2 ouv2ouv1o L L +Λ+Λ+= +Λ+Λ+= αα αααα ggmm ff where the coefficients fi and gi involve the ultraviolet cut-off and diverge as Λuv 2 →∞. The cross section for electron-muon elastic scattering. 2017 MRT Then if we calculate some cross section, such as electron- muon scattering (see Figure), we find that the result can be written in the form: ])([),,( ouv1o 2 ouvoo L+Λ+=Λ= ασσαασ mf We now eliminate αo and mo in favor of α and m by inverting α= αo[1 + f1(Λuv)αo + f2(Λuv)αo 2 +…] and m =mo[1 + g1(Λuv)αo +g2(Λuv)αo 2 +…] above, all the divergent terms cancel. The Λuv dependence of the coefficients σi (Λuv) is cancelled by that of the coefficients fi and gi, and we end up with: )(),( 1o 2 L++== ασσαασ mf which is free of divergence. Since all we have done is change variables, this new result is the same as the previous one (i.e., σ = f vs f ), but it is now expressed in terms of finite parameters. + + K+ 2 =σ e µ _ _ _
- 75. To make the Green’s function propagators finite, we need to re-scale the fields ψ and Aµ similarly (i.e., ‘wavefunction renormalization’). It turns out that QED is a renormaliza- ble theory in the sense that once the divergences in the coupling e, the mass m, and the normalization of the fields ψ and Aµ have been rescaled in this way, all the physical quan- tities are finite. Hence, if the Lagrangian L =iΣµψ γ µ∂µψ −mψψ −eΣµ JµAµ −¼Σµν Fµν Fµν is taken to be written in terms of the bare quantities ψo, mo, Aµ o, and eo: 75 2017 MRT ∑∑∑ −− −∂= µν µν µν µ µ µ µ µ µ ψγψψγψ o oo oooooo 4 1 FFAemiL these bare quantities can be related to the physical ones by the renormalization constant Zi as: e ZZ Z eAZAmZmZ m 32 1 o3 o o2o ==== and,, µµψψ where the Zi are functions of Λuv, and so: L+Λ+Λ+=Λ )()(1)( uv2 2 uv1uv iii ffZ αα where the functions fin(Λuv) contain the divergences as Λuv→∞, like αeff (r)≅αo/{1− (αo/3π)ln[(m2−q2)/Λuv 2)]}. These infinities are then absorbed into the definition of the bare quantities ψo, mo, Aµ o, and eo above so that the physical quantities are finite. That this can be done while keeping the same form for the Lagrangian L(ψo,mo,Aµ o,eo) above as the original L(ψ,m,Aµ ,e) shows that QED is a renormalizable theory. __
- 76. Similarly, in QCD the masses and couplings will get renormalized. However, the form of the coupling-constant renormalization is quite different from that in QED because of the self-coupling of the gluons. The lowest-order quark-gluon coupling (see Figure) is corrected by the higher-order terms and so the effective coupling is: 76 2017 MRT + −+ −−=≡ L 2 2 2 o o 2 2 o o o 2 eff2 ln 4π ln 4π 1 4π )( µ α µ α αα qbqbg q ss s s s where µ2 is the arbitrary renormalization point (i.e., the value of −q2 at which αs =αs o, the measured value). Hence: (a) The quark-gluon vertex, (b) the quark loop, and (c) the gluon loop, diagrams, which modify the gluon propagator and hence renormalize the color coupling. Λ = −+ ≈ 2 2 o o 2 2 o o o 2 ln 4π 1 ln 4π 1 )( C ss s s Qbqb q α µ α α α where we have introduced Q2 ≡–q2 and ΛC 2=µ2exp(−4π/αs obo) is the position of the so- called ‘Landau pole’, since αs o(Q2)→∞ as Q2 →ΛC 2. + gs 2 λ gs 2 λ +gs
- 77. Now, it is found that: 77 2017 MRT fc NNb 3 2 3 11 o −= where Nc =3 is the number of colors, and Nf is the number of flavors of quark. This first term arises from the gluon loops because: abc N de bdeade Nff c δ=∑ − = 1 1 2 while the Nf term is the same as the electron loop in QED with e→gs /2. As long as Nf < 11Nc /2=33/2, the sign of bo is positive and hence the sign of variation of as with q2 is opposite to that in QED, which has only the negative Nf term. Hence, we see from the previous αs o(q2) equation that when Q2 →∞, αs →0. This means that the effective coupling vanishes and we obtain the so-called ‘asymptotic freedom’. However, in ‘hard’ large-momentum-transfer processes, the quarks and gluons inside a hadron are predicted to behave as if they were free, in agreement with observation. So, on the other hand, for Q2 →ΛC 2, αs →∞, and so the perturbation series breaks down. It is this effect that is believed to account for the confinement of quarks and gluons inside hadrons within a radius R~hc/ΛC≅1 fm (i.e., ΛC≅0.2 GeV). It is only because QED and QCD involve massless vector particles and dimensionless couplings, α and αs, that they can be renormalized in this way!
- 78. Theories with massive vector boson are not generally renormalizable. They are renormalizable, however, if the bosons acquire a mass as a result of a spontaneous breaking of gauge symmetry, as will be discussed in the Electroweak Interactions chapter, or if the boson couples only through conserved currents that satisfy Σµ∂µ Jµ =0 (i.e., Σµqµ Jµ =0), in which case we get (c.f., i(−ηµν +qµqν /M2)/(q2 −M2) in the Rules): 78 2017 MRT † 22 2 µ νµ µν µ η J Mq M qq J − +− and the contribution of the second (i.e., qµqν /M2), dangerous term in the propagator numerator will vanish and so will not affect the asymptotic behavior. It is evident that theories involving particles with spin greater than 1 (e.g., gravity – as it is theoretically propagated by the graviton which has spin-2), will not generally be renormalizable either. One last thing. The masses mq in the QCD Lagrangian are referred to as the ‘current’ quark masses; they are the parameters that specify the chiral symmetry breaking. In QCD, a bare quark is surrounded by a cloud of gluons and quark-antiquark qq pairs, and the energy (the mass) of the cloud contained in a sphere of radius r decreases as the renormalization scale increases (i.e., µ ~1/r). _
- 79. We saw that QCD has all the essential ingredients required for the theory of strong inter- actions, namely, asymptotic freedom and the possibility of accounting for color confi- nement. So, we write the Lagrangian of the Quantum Chromodynamics (QCD) chapter: 79 2017 MRT a a a GG µνµν λ ∑= = 8 1 2 Strong Interactions and Chiral Symmetry ∑∑∑∑∑∑∑ −−= µν µν µν µ µ µ γ a a a k kk lk llkk GGmDi 4 1 qqq][q q q q QCDL where now we have LQCD expressed as a function of the quark field qk of flavor q=u,d, s, c, b, t and color k=1,2,3 (or R, B, and G for red, blue, and green). We will suppress the color indices (i.e., k and l) and all summations signs and rewrite LQCD in the form: )(Tr 2 1 qqqqQCD µν µνµ µ γ GGMDi −−=L where q is the column vector [u d s c b t]T, q is the row vector [u d s c b t], and M is the diagonal mass matrix in flavor space with eigenvalues mu, md, ms, mc, mb, and mt. Also, we have introduced: and used Tr(λaλb)=2δ ab, where λa (with a=1,2,…,8, which correspond to the various gluon types) are the SU(3) matrices. The last Lagrangian is completely determined by the requirements of renormalizability and color gauge invariance, except for the number of quark flavors, Nf , and their masses mq. _ _ _ _ _ _ _
- 80. The QCD Lagrangian possesses a high degree of symmetry, most notable of these being that if mu =md then the Lagrangian is invariant under the isospin or SU(2) flavor transformation ψ =[u d]T→ ψ =Uψ =exp[(i/2)Σkτk αk(x)][u d]T: 80 2017 MRT qq U→ where q=[u d]T only. To display the full symmetry of LQCD in the limit mq →0 we rewrite it in terms of the left- and right-handed quark fields: q)1( 2 1 qq)1( 2 1 q 55 γγ +=−= RL and where γ 5 =iγ 0γ 1γ 2γ 3 or off-diagonal I2×2. Using these ‘chiral’ fields, LQCD becomes: −−−+= ∑µν µν µνµ µ µ µ γγ GGMMDiDi LRRLRRLL Tr 2 1 qqqqqqqqQCDL In the absence of the quark mass matrix (i.e., for M=0), we see that this Lagrangian is invariant under two separate groups of unitary transformations: RRRLLL UU qqqq →→ and that is, under independent rotations of qL and qR in the space of quark flavors. We thus have a U(Nf )L⊗U(Nf )R flavor symmetry and this symmetry should be realized for those flavors with mass mq much less than the hadronic mass scale ΛC. It should thus be good for u and d quarks, but more approximate for s quarks, so we can then say that the QCD Lagrangian has an approximate U(3)L⊗U(3)R ‘chiral’ symmetry. _
- 81. 1019 GeV 1 GeV 1 MeV 1 eV 18 20 14 10 6 2 −2 −6 −10 −14 log10M(GeV) ← kT (universe) MGUT? ← MP mγ <10−15 eV ← Solar / Atmospheric anomalies νe ← νµ ← ντ ← e µ τ LeptonsQuarks Bosons Z, W s t b c u,d H Adapted from Fig. 1.7 of D.H. Perkins - Introduction to High Energy Physics For U(3)L we can construct nine Noether currents plus another nine for U(3)R. It is convenient to form the vector and axial-vector combinations V=R+L and A=R−L, respectively. Using the subscript 5 of γ 5 =iγ 0γ 1γ 2γ 3 to distinguish the latter, we have: 81 qqq)2(q:)()( qqq)2(q:)()( 5555 γγλγγ γλγ µµµµ µµµµ ==⊗ ==⊗ JJUSU JJUSU aa AA Baa VV and and 13 13 where λa (with a=1,2,…,8) are the SU(3) matrices in u,d,s flavor space. In the chiral symmetry limit, mq →0, each of the currents is expected to be conserved (i.e.,Σµ∂µ Jµ =0). 2017 MRT The SU(3)V ⊗SU(3)A symmetries are realized in the normal way and correspond to SU(3) flavor invariance and baryon number conservation, respectively. The observed hadron states form SU(3) flavor multiplets approximately degenerate in mass (see Figure). Since the quark masses are not in fact degene- rate, the symmetry is broken. However, although they are not equal, mu and md are small compared to the hadronic mass scale ΛC. It is for this reason that SU(2) or isospin symmetry is so well satisfied. Interacting quarks (in hadrons) always have energies of at least the order of ΛC and it makes little difference weather mu or md are a few MeV or zero. Because ms is larger, the SU(3) flavor symmetry is much more approximate. However, all the baryon representations of chiral symmetry are either massless or form parity doublets, which is not even an approximate property of the observed hadron spectrum (e.g., there is no particle with spin-parity ½− which is approximately degenerate in mass with the proton, which has spin-parity ½+).
- 82. Because of the lack of observed + and − parity in hadronic experiments, something must break chiral symmetry! In fact, what happen is a special case of what is called Spontaneous Symmetry Breaking (SSB) in which, although the Lagrangian of the theory is invariant under some symmetry group, the vacuum of the theory (N.B., and the observed physics) is not! The symmetry is then to be realized in the ‘Nambu-Goldstone’ mode… 82 2017 MRT )( 2 1 φφφ µ µ µ V−∂∂= ∑L Spontaneous Symmetry Breaking (SSB) To understand what is involved, we need to recall how we obtain the physical consequences of a field-theory Lagrangian. Suppose, for example, we consider a real scalar field with Lagrangian: and consequent classical equation of motion: 0= ∂ ∂ + φ φ V Free particle states are the solutions of this equation with only a quadratic term φ in the potential V(φ).
- 83. 83 2017 MRT 0 )( = ∂ ∂ φ φV Earlier, we had for the Lagrangian density of a free scalar spin-0 particle of mass m: 22 2 1 2 1 φφφ µ µ µ m−∂∂= ∑L By comparing our previous Lagrangian with the one above, we see that the coefficient of this term specifies the mass m of the particle (i.e., V(φ)=½m2φ2). The vacuum state, which by definition is the state in which there are no particles, occurs when: which in this case is when φ =0 as one would expect. Higher-order terms in V(φ) correspond to the interactions between these particles. Now the differential equation φ +∂V/∂φ =0 also has solutions φ =constant at any value of φ for which ∂V/∂φ =0 above holds. The ‘no particles’ or ‘vacuum’ state will then be one in which the expectation value of φ takes one of these constant values. These are several possibilities. It could be that ∂V/∂φ =0 has only one solution. In order for the energy to be bounded from below, this solution must be at the minimum of the potential, and it will then correspond to the unique vacuum of the theory. On the other hand, there could be several solutions of this equation. The maxima of the potential are unstable, but all the minima can be regarded as possible vacua (i.e., as possible no-particle states of the theory). If there is more than one such minimum then the lowest would be the ultimate vacuum state of this world. In some cases, there may be several such minima that have the same value for the potential (i.e., the vacuum may be degenerate!)
- 84. We shall show how this leads to the spontaneous breakdown of the symmetry of the Lagrangian by examining a particular example. Consider a theory with N real scalar fields, φi, and suppose the Lagrangian has the form: 84 +−−∂∂=−= ∑∑∑∑ === = N i ii N i ii N i iiVT 1 2 1 2 1 3 0 )( 4 1 )( 2 1 2 1 φφλφφµφφ µ µ µL The first term, the kinetic energy, is invariant under rotations of the φi space (i.e., under the O(N) group). Since the potential energy is a function only of the ‘length’ Σi(φiφi), it is similarly invariant. Thus we have a theory that possesses global O(N) invariance. A normal mass term in the Lagrangian above has negative φ2 and the potential takes the form of the upper curve in the Figure (N.B., λ must be positive so that the energy is bounded from below). The potential V=−½µ2Σi(φiφi)+¼λΣi(φiφi)2, with λ>0, and with the quadratic term having different signs. For µ2 >0 there is a minimum at |Σiφiφi |½ =v for the Higgs potential curve. 2017 MRT Now, the equation: 0)( 4 1 )( 2 1 22 = +− ∂ ∂ = ∂ ∂ ∑∑ i ii i ii ii V φφλφφµ φφ |Σiφiφi |½ V µ2 >0 µ2 <0 v yields (N.B., ∂(φiφi)/∂φi =2φj and ∂(φiφi)2/∂φi = 4(φiφi)φj, for all j): 0)(2 = +− ∑ j i iij φφφλφµ which have a unique solution: 0=jφ This then corresponds to the unique vacuum of the theory. V(φ)
- 85. On the other hand, if µ2 is positive (i.e., µ2 >0), the potential, which is often referred to as the Higgs potential (N.B., the Higgs potential will soon be seen to be represented by V(φ)=−µ2(φ†φ)+λ(φ†φ)2) is shown by the lower curve in the previous Figure and thus the solution φi =0 corresponds to a maximum of V(φ). The minimum now occurs at: 85 2017 MRT 2 2 2 )(0)( v i ii i ii ≡=⇒=+− ∑∑ λ µ φφφφλµ which gives the vacuum of the theory in this case. We see that Σi(φiφi)=µ2/λ≡v2 above only fixes the length of the φ vector in φi space, and says nothing about its direction. Thus, the vacuum state is infinitely degenerate; any direction gives a vacuum state of the same energy! This degeneracy is due to the O(N) invariance of the original Lagrangian. However, although Σi(φiφi)=µ2/λ≡v2 is invariant under O(N) rotations, any particular solution corresponds to a vector pointing in some given direction and therefore no longer has the O(N) invariance. This is the origin of spontaneous symmetry breaking!
- 86. At this stage it is convenient to assume that the actual ground state corresponds to a particular solution of Σi(φiφi)=µ2/λ≡v2. By suitable choice of the axes in φi space we can arrange that this vacuum is: 86 2017 MRT vNi ≡== λ µ φφ 2 0 and with i=1,2,…,N−1. Perturbation theory involves an expansion of L in φ around the maximum of the potential. We therefore write the Lagrangian in terms of a new field η(x) defined by: vxx N −≡ )()( φη but keeping the original φi(x), so: +−∂∂+∂∂= ∑∑∑ fieldsthein higherandcubicterms22 2 1 2 1 ηµηηφφ µ µ µ µ µ µ i iiL Thus, we have (N−1) massless scalar fields φi, with a global O(N−1) symmetry, and a single scalar Higgs field η of mass m with: 02 22 >= µm Recall that a mass term in a Lagrangian, like the one for a free scalar spin-0 particle of mass m, has the form −½m2η2. There are also complicated interactions between these fields arising from the neglected higher-order terms in the above Lagrangian.
- 87. The particles associated with the massless fields are referred to a Goldstone bosons and their existence is a general feature of the spontaneous breakdown of a global symmetry. The number of such Goldstone bosons is always equal to the difference between the order (i.e., number of generators) of the original symmetry group and the order of the surviving symmetry group. They can be understood physically as being excitations along the symmetry directions in which the potential is unchanged. In the above example, the original group is O(N), with ½N(N−1) generators, and the final group is O(N−1) with ½(N−1)(N−2) generators, so there are (N−1) massless particles, as can be seen explicitly in L =½ΣiΣµ∂µφi ∂µφi +½Σµ∂µη∂µη−µ2η 2 +O(n). 87 2017 MRT In closing this Chapter we offer a few words about the chiral symmetry problem. Because the approximate SU(3)L⊗SU(3)R symmetry is not seen in nature, it appears that is must be spontaneously broken down to the SU(3)V that is observed. This assumption that chiral symmetry is spontaneously broken makes a testable prediction. The arguments of the last few slides show that the breakdown of SU(3)L⊗SU(3)R to SU(3)V requires the existence of eight pseudoscalar Goldstone bosons, one for each of the broken generators. These bosons would be massless in the limit of zero quark masses. The known pseudoscalar octet (π,K,η) are obvious candidates, and support for this identification comes from the fact that the pion, which is the Goldstone isotriplet boson associated with the breaking of the SU(2)L⊗SU(2)R symmetry, is far the lightest of the mesons. The identification of the pion as a Goldstone boson leads to other interesting, and verifiable, results (c.f., P.D.B. Collins, et al., P. 75ff).
- 88. Some examples of weak interactions, and their descriptions in terms of quarks and leptons, are: 88 2017 MRT µ µ eµ e µus:K µdu:π eµ:µ ued: ν ν νν ν ++ ++ +++ − →− →− →− →− decay decay decay decayneutron β Weak Interactions which are pictured in the Figure. In contrast to the strong and electromagnetic inter- actions these weak decays are distinctive because they involve a change of the particle type, with transitions such as u↔d and/or e− ↔νe. Weak decays pictured in terms of (Top) the effective 4-fermion interaction and (Bottom) W-exchange. decayneutron −β decay−+ µ decay−+ π decay−+ K du dn p − e eν n p − e eν W− u u d + e+ µ W+ µν eν eν + e+ µ µν d W+ + π + µ µν u + π µν + µ cosθc cosθc s W+ + K + µ µν u + K µν + µ sinθc sinθc
- 89. Apart from the very small CP (charge-parity) violations seen in neutral K decays, all such weak processes were successfully described by a current-current effective interaction of the form: 89 2017 MRT ∑= −= 3 0 † 2 4 µ µ µ JJ GF WL with the currents: s)1( 2 1 cd)1( 2 1 uµ)1( 2 1 e)1( 2 1 555 µ 5 e ′−+′−+−+−= γγγγγγνγγν µµµµµ J where particle names are used to denote Dirac spinors. The d and s quarks occur in the weak current in the ‘rotated’ form: cccc θθθθ cosscosdssinscosdd +−≡′+≡′ and where θc , the Cabibbo angle, was introduced in the u↔d′ term to allow for the strangeness-changing weak interactions (e.g., K+ →µ+νµ) with amplitudes suppressed by an amount sinθc/cosθc compared to strangeness-conserving interactions (e.g., π+ →µ+νµ) as shown in the previous Figure. Data gives the sine of the Cabibbo angle as sinθc≅0.22. Unlike QED and QCD, the weak interactions are chiral (i.e., they do not treat the left- and right-handed components of the fermions equally) and hence do not conserve parity (P). Also, they are not invariant under particle-antiparticle conjugation (C) because left-handed fermions appear in the current Jµ above but not left-handed antifermions.
- 90. The standard theory of electroweak interactions is based on the gauge group SU(2)⊗U(1) and is known as the Glashow-Weinberg-Salam model. Sheldon Glashow, Abdus Salam, and Steven Weinberg were awarded the 1979 Nobel Prize in Physics for their contributions to the unification of the weak and electromagnetic interaction between elementary particles. Glashow (1961) originally unified the weak and electromagnetic interactions using the SU(2)⊗U(1) gauge group, and Weinberg (1967) and Salam (1968) showed how the weak gauge bosons could acquire their mass without spoiling the re- normalizability. Major experimental support for the model came with observation of weak neutral currents in 1973, followed by the discovery of the weak gauge boson themselves (W± and Z) in 1983 and, of course, the discovery of the Higgs boson (H0) in 2014. 90 2017 MRT µµµµµ B Y giWTgiD k k k 2 3 1 ′++∂≡→∂ ∑= The SU(2)⊗U(1) Gauge Theory The gauge group SU(2)⊗U(1) has four vector fields, three associated with the adjoint representation of SU(2), which we denote Wk µ, and one with U(1) denoted by Bµ . The Lagrangian is made gauge-invariant by replacing ∂µ in the fermion kinetic energy term by the covariant derivative Dµ: where g, g′ and Tk, Y/2 are the SU(2), U(1) couplings and group generators, respectively. The Tk satisfy the SU(2) algebra (c.f., [τi,τj]=2iΣkεijkτk): ∑= k kjkiji TiTT ε],[
- 91. We need to specify the action of these generators on the fermion fields. Parity violation is incorporated by assigning the left- and right-handed components of the fermions to different group representations. Motivated by the pervious current Jµ, all the left-handed fermions are taken to transform as doublets under SU(2)L, while the right-handed fermions are singlets. For example, in the first generation of leptons and quarks we have the SU(2)L multiplets: 91 2017 MRT RR L R L du d u e e ee ′ ′ − − and,,, ν and so SU(2)L generators act as follows: 0 2 1 == RkLkLk TT ψψτψ and where the τk are the 2×2 Pauli matrices. Identical assignments are made for the other generations of fermions like νµ, µ−, c, and s′.
- 92. Since the weak interaction involves charged W± bosons it must be related to electro- magnetism, and to incorporate QED in the model we have to identify some linear combination of the weak generators with the electric charge operator Q. Clearly T3 is closely related to Q because adjacent members of an isospin multiplet are eigenstates of T3 with eigenvalues that differ by one unit of charge (in units of e). We may therefore write: 92 2017 MRT )(2 2 33 TQY Y TQ −=⇔+= and use this relation to specify the eigenvalues of the U(1) generator, Y/2 (N.B., the factor ½ being purely conventional). Tk and Y are referred to as the weak isospin and weak hypercharge generators, respectively, and their eigenvalues for the fermion fields are listed in the Table. Lepton T T3 Q Y Quark T T3 Q Y νe ½ ½ 0 −1 uL ½ ½ ⅔ ⅓ e− L ½ −½ −1 −1 dL ½ −½ −⅓ ⅓ uR 0 0 ⅔ e− R 0 0 −1 −2 dR 0 0 −⅓ −⅔ 3 4 The group structure permits an arbitrary hypercharge assignment for each left-handed doublet and each right-handed singlet, and so we have chosen Y to give the correct electrical charges according to Q=T3 +Y/2 above. Evidently, charge quantization must be put in by hand in this SU(2)⊗U(1) theory.
- 93. With the inclusion of the gauge boson kinetic energy terms, the SU(2)⊗U(1) invariant Lagrangian density takes the form: 93 2017 MRT ∑∑∑ ∑ ∑∑ ∑ −− ′−∂+ ′−−∂= µν µν µν µν µν µν µ µµ µ µ µµµ µ γ τ γ BBWW fB Y giffB Y gWgif i i i f RRL i ii L 4 1 4 1 222 L where the Σf sum is over left- and right-handed fermion fields (i.e., fL and fR, respectively). The quantities in round brackets are the covariant derivatives ∂µ →Dµ ≡∂µ + igΣiTi Wi µ +ig′(Y/2)Bµ (or rather iDµ ), which introduce the fermion-fermion-gauge boson couplings. The field strength tensors of SU(2) and U(1) gauge fields are given by: ∑−∂−∂= kj kj jki iii WWgWWW νµµννµµν ε and: The term bilinear in Wµν in the Lagrangian L above generates the trilinear and quadrilinear self-couplings of the Wµ fields that are a characteristic of non-Abelian gauge theories. µννµµν BBB ∂−∂=
- 94. We now come to the crucial problem of mass generation. The previous Lagrangian (i.e., L=Σf { fLΣµγ µ[i∂µ−gΣi(τi /2)Wi µ −g′(Y/2)Bµ]fL + fRΣµγ µ[i∂µ −g′(Y/2)Bµ]fR}−¼ΣµνΣiWi µνWi µν −¼Σµν Bµν Bµν) describes massless gauge bosons interacting with massless fermions. A gauge boson mass term is not gauge-invariant, and a Dirac mass term: 94 2017 MRT )()1( 2 1 )1( 2 1 55 RLLRmmm ψψψψψγγψψψ +−= ++−−=− SSB in the Electroweak Model is also excluded because ψL is a member of an SU(2) doublet while ψR is a single, so that this term too manifestly breaks gauge invariance. We appear to have reached an impasse! How can we generate gauge-boson and fermion masses without destroying the renormalizability of the theory, which depends so critically on the gauge symmetry of the interaction? The answer is the Higgs mechanism that we discussed earlier. We introduce elementary scalar (Higgs) fields, φ, which couple gauge-invariantly to the gauge bosons through the covariant derivative (c.f., ∂µ →Dµ ≡∂µ +igΣiTi Wi µ +ig′(Y/2)Bµ): 2 3 1 2† 2 φφφφ µµµµ µ µ ′++∂→∂≡∂∂ ∑= B Y giWTgi k k k and to the fermions through so-called Yukawa couplings of the form: )]()[( † LRRLYG ψφψψφψ +− _ _
- 95. Clearly, we require a φ field that is an SU(2) doublet if the Yukawa coupling above is to be gauge-invariant. We take this Higgs doublet to be: 95 2017 MRT + + ≡ = + )( 2 1 )( 2 1 43 21 0 φφ φφ φ φ φ i i with φi real, while the Hermitian conjugate doublet, φ†, describes the antiparticles φ− and φ0. The charge assignments of the components of φ follow from the Yukawa coupling. This term is only SU(2)⊗U(1) gauge-invariant if φ is a doublet (T=½) with Y=½. Besides the Yukawa terms, the Lagrangian can also contain a self-interaction between the Higgs fields. The most general SU(2)-invariant and renormalizable form is: )(φVH −=L where λ must be positive for V(φ) to be bounded from below (c.f., L =½ΣiΣµ∂µφi ∂µφi − [−½µ2Σi(φiφi)+¼λΣi(φiφi)2]). An ordinary scalar mass term in LH would have the form −M2φ†φ, but for spontaneous symmetry breaking we require the coefficient of φ†φ to be positive. Indeed, with µ2 and λ positive the Higgs potential V(φ) is at its minimum when: λ µ φφφφφφ 2 2 4 2 3 2 2 2 1 † 2 1 )( 2 1 =+++≡ 2††2 )()()( φφλφφµφ +−=V with the Higgs Potential: _
- 96. In perturbative field theory, we expand φ about some particular minimum of V(φ). We choose the minimum that has the vacuum expectation value: 96 2017 MRT ( ) λ µ φφ 2 3 004,2,1000 ≡≡== vii and The particle quanta of the theory corresponds to quantum fluctuations of φ3(x) about the value φ3 =v, rather than φ3(x) itself, that is, to: vxxH −≡ )()( 3φ It is therefore desirable to re-express the Lagrangian in terms of H rather than φ3. We then find that |∂µφ|2 above and the Yukawa coupling −GY [(ψLφ)ψR +ψR(φ†ψL)] contain boson and fermion mass terms of the form: ψψ µ µ µ )()( 2 vGWWvg Yand∑ So, somewhat surprisingly, all the weak gauge boson and fermion masses can be generated by introducing just one complex SU(2)L doublet of Higgs fields! _ _
- 97. By choosing the nonvanishing expectation value to be that of the neutral field φ3, we ensure that the vacuum is invariant under U(1)EM of QED, and that the photon remains massless. Then φ =[φ+ φ0]T gives: 97 2017 MRT = ≡ + v 0 2 1 00 0 φ φ φ Now, because: 00 ≠≠ φφτ Yi and both SU(2) and U(1)Y are spontaneously broken, but: 0 0 00 01 2 1 22 3 = = += v Y Q φ τ φ and hence the vacuum remains invariant under U(1)EM gauge transformation. We therefore expect three massive gauge bosons and one massless gauge boson. In summary, both Goldstone and Higgs phenomena generalize to non-Abelian symmetries. In the case of global symmetries, for every generator of a broken global symmetry, there is a massless particle. For local symmetries, each broken generator gives rise to a massive gaige boson.
- 98. The masses of the gauge bosons can be found by substituting the expectation value 〈φ〉 =(1/√2)[0 v]T into |∂µφ|2. The relevant term in the Lagrangian is: 98 2017 MRT ∑ ∑ −+ ++′+′−= ′+−+ −′+ = ′+ µ µ µµµµµ µµµµ µµµµ µµ φ τ WWgvBgWgBgWgv vBgWgWiWg WiWgBgWg B Y giWgi k kk 2 23232 2 321 213 2 2 1 )(0)( 8 1 0 )( )( 8 1 22 Gauge Boson Masses where: )( 2 1 21 WiWW m≡± The mass matrix of the neutral fields is off-diagonal in the {W3,B} basis. As expected, one of the mass eigenvalues is zero, and we have displayed this in |(…)〈φ〉|2 above with the orthogonal combination of fields to that in the first term. The normalized neutral mass eigenstates are thus: where we have introduced θw, the Weinberg or weak mixing angle, defined by: 2222 sincos gg g gg g ww ′+ ′ = ′+ = θθ and wwww BW gg BgWg ABW gg BgWg Z θθθθ µµ µµ µµµ µµ µ cossinsincos 3 22 3 3 22 3 +≡ ′+ +′ =−≡ ′+ ′− = &
- 99. By comparing |[igΣk(τk /2)Wk µ +ig′(Y/2)Bµ]〈φ〉 |2 with the mass terms we would find in the Lagrangian of the physical W± µ, Zµ and photon Aµ fields, namely: 99 2017 MRT ∑∑∑ ++−+ µ µ µ µ µ µ µ µ µ AAMZZMWWM 2 γ 2 Z 2 W 2 1 2 1 we see that: 0 2 1 2 1 γ 22 ZW =′+== MggvMgvM and, and so: w gg g ggv gv M M θcos 2 1 2 1 2222Z W = ′+ = ′+ = The inequality MZ ≠MW is due to the mixing between the W3 µ and Bµ fields.
- 100. We can rewrite the fermion-gauge boson electroweak interaction terms in the SU(2)⊗U(1) invariant Lagrangian density (c.f., L =Σf { fLΣµγ µ[i∂µ−gΣi(τi /2)Wi µ −g′(Y/2)Bµ]fL + fRΣµγ µ[i∂µ −g′(Y/2)Bµ]fR}−¼ΣµνΣiWi µνWi µν −¼Σµν Bµν Bµν) in the form: 100 2017 MRT ∑∑∑ ′−−= µ µ µ µ µ µ B J gWJg Y i i i 2 EWL where: ψγψψ τ γψ µµµµ YJJ YL i Li == and 2 Then using W± ≡(1/√2)(W1 miW2), and Zµ≡W3 µcosθw −Bµsinθw and Aµ ≡W3 µsinθw +Bµcosθw, to express LEW in terms of the physical fields W ±, Z, and A, we obtain: ∑∑∑ ′+− ′+ ′ −+−= −+ µ µ µ µ µ µ µ µ µ µ µ ZJggAJ gg gg WJWJ g nccccc 22 EM 22 † EW )( 2 L In this way we can identify the physical currents as linear combinations of the SU(2) and U(1) currents Ji and JY. Thus: LL iJiJJ ψττγψ µµµµ )( 2 1 )( 2 1 2121cc +=+≡ is seen to be the weak charged-current of Jµ=νeγ µ½(1−γ 5)e+νµγ µ½(1−γ 5)µ+ uγ µ½(1−γ 5)d′+cγ µ½(1−γ 5)s′ (c.f., Weak Interactions chapter) which couples to the W+ boson. Gauge Boson Mixing and Coupling _ _ _ _ _ _
- 101. The coupling g is therefore related to the Fermi coupling GF by g2/2MW 2 =4GF /√2, and hence, by using MW=½vg for MW, we can determine the vacuum expectation value of the Higgs field: 101 2017 MRT GeV246)2( 2 21W === − FG g M v using GF =1.16637×10−5 GeV−2 ≅10−5/mp 2 where mp is the proton mass. ψγψ µµµµ QJJJ Y =+≡ 2 1 3EM and so by construction is just the usual electromagnetic current (c.f., Q=T3 +Y/2). Hence, the electromagnetic charge is: from cosθw =g/√(g2 +g′2) and sinθw =g′/√(g2+g′2). ww gg gg gg e θθ cossin 22 ′== ′+ ′ = The current in the second term in LEW is:
- 102. Finally, we identify the weak neutral-current to the Z boson in LEW as (N.B., unlike the charged current, the neutral current couples to both right- and left-handed fermions): 102 2017 MRT ψγγψψθγγψθ µµµµµ )( 2 1 sin)1( 2 1 sin 52 3 5 EM 2 3nc AVww ccQTJJJ −≡ −−=−≡ where: ff Awf ff V TcQTc 3 2 3 sin2 ≡−≡ andθ the values of which are listed for the various fermions in the Table (with sin2θw =0.222). f Qf cf A cf V νe,νµ,… 0 ½ ½ e−,µ−,… −1 −½ −½+2sin2θw ≅−0.04 u,c,… ⅔ ½ ½−(4/3)sin2θw ≅0.19 d,s,… −⅓ −½ −½+⅔sin2θw ≅−0.35 The coupling in LEW : w g gg θcos 22 =′+ contains an extra cosθw on account of the mass difference between the Z and W bosons (c.f., MW/MZ =cosθw).
- 103. It is customary to introduce the parameter: 103 2017 MRT ∑ ∑ −+ = i ii i iiii Yv YTTv 22 22 2 1 4 1 )1( ρ wM M θ ρ 22 Z 2 W cos ≡ which specifies the relative strength of the neutral- and charged-current weak interactions. The Weinberg-Salam electroweak model with a single Higgs doublet has ρ =1, which is in excellent agreement with experiment. The Higgs sector could be much more complicated.If there are several representations (i=1,2,…,N) of Higgs scalars whose neutral members acquire expectation values vi, then: where Ti and Yi are, respectively, the weak isospin and hypercharge or representation i.
- 104. To obtain all the interactions and masses generated by the Higgs mechanism, we need only substitute (as in H(x)≡φ3(x)−v): 104 2017 MRT + = )( 0 2 1 )( xHv xφ into the Lagrangian for the Higgs sector, which is the sum of the prior results: |∂µφ|2, −GY[(ψLφ)ψR +ψR(φ†ψL)], and LH =V(φ) (with Higgs potential V(φ)=−µ2(φ†φ)+λ(φ†φ)2), obtained earlier. We then find that of the four scalar fields φi(x) of φ =[φ+ φ0]T (where φ+≡ (1/√2)(φ1+iφ2) and φ0≡(1/√2)(φ3+iφ4)), the only one that remains is H(x). The other three fields are spurious and we can remove all trace of them from the Lagrangian. To see this, we write φ(x) in terms of H(x) and three new fields θk(x) (k=1,2,3), defined by: + = • )( 0 e 2 1 )( )( xHv x vxi θθθθττττ φ θk and H fully parametrize all possible deviations from the vacuum. Given this form, we can use gauge freedom to set θk =0. This choice is known as the ‘unitary’ gauge, as only fields that correspond to physical particles appear in the Lagrangian. However, we cannot have just lost three degrees of freedom as a result of spontaneous breaking of symmetry and translating the field variables. What has happened is that in generating masses for the three weak bosons we have increased their polarization degrees of free- dom from 2 to 3. They can now have longitudinal polarization too. The phases θk of three of the Higgs fields have been surrendered to make the gauge fields massive! _ _
- 105. So, the W±, Z bosons, φ± and (i/√2)(φ0 −φ0), out of the four in the original complex doublet. In the minimal model, the values of the masses are predicted in terms of the couplings. From MW=½vg, MZ=½v√(g2 +g′2), MW/MZ =cosθw, and e=gg′/√(g2 +g′2) we have: 105 2017 MRT GeV90 sin GeV80 sin GeV3.37 sin2 1 2 1 W ZW ≅=≅=== www M M ev gvM θθθ and where we have used the experimentally determined value of sinθw≅0.23. These predictions are in excellent agreement with the masses of the W± and Z bosons that were subsequently discovered. The fourth parameter of the model, λ (or alternatively µ2 ≡λv2) controls the form of the potential V(φ) and determines the mass of the Higgs particle associated with the remaining field H(x). On substituting φ(x)=(1/√2)[0 v+H(x)]T into the Higgs potential V(φ) =−µ2(φ†φ)+λ(φ†φ)2, we find: L+−=+−+= 2242 2 )2( 2 1 )( 4 )( 2 HvHvHv v H λ λλ L where the higher-order terms represent the self-couplings of the Higgs boson, H. From this last equation, we conclude: 222 H 22 µλ == vM The mass of the Higgs is not predicted,since neither µ2 not λ is determined,only their ratio v2. However, the Higgs couplings to other bosons are completely determined! _
- 106. The Higgs-fermion couplings give masses to the fermions. We begin by considering the Yukawa term −GY[(ψLφ)ψR +ψR(φ†ψL)] for the electron doublet: 106 2017 MRT + −= − + L RRLY G e ][ee]e[ e0 0ee e ν φφ φ φ νL Fermion Masses and Couplings When we spontaneously break the symmetry and substitute φ(x)=(1/√2)[0 v+H(x)]T, this term becomes: )Hee()ee()eeee)(( 2 e e ee v m mHv G LRRLY −−≡++−=L revealing that the electron’s mass and coupling are: W eee e 2 )eeH( 2 M mg v m g vG m === and Since Ge is arbitrary, the actual mass of the electron is not predicted, but its Higgs coupling is specified and, being proportional to me/MW, is very small. W qq 2 )qqH( M mg v m g == On the other hand, the Higgs couplings to the quarks are: _ _
- 107. The quark masses (and couplings) are generated in an analogous manner. However, φ(x)=(1/√2)[0 v+H(x)]T gives a mass only to the lower member of the fermion doublet, and to generate a mass for the upper member we must construct from φ a new Higgs doublet with a neutral upper member: 107 2017 MRT + → − == − 0 )( 2 1 * 0 2 xHv ic breaking φ φ φτφ Owing to the special properties of SU(2), φc transforms identically to φ, but has opposite weak hypercharge, Y(φc)=−1. The most general SU(2)⊗U(1) invariant Yukawa terms for the [u d] quark doublet are then: conjugateHermitianu]du[d]du[ 0 u0d u + − − −= − + RLRLY GG φ φ φ φ L which, on substitution of φ(x)=(1/√2)[0 v+H(x)]T and φc =(1/√2)[v+H(x) 0]T above, reveals that the mass and qqH coupling terms are: ++−= v H mmY 1)uudd( ud u L where mq =(1/√2)Gqv. _
- 108. To date, there are no confirmed experimental results that contradicts the Standard Model of Particle Physics. Why, then, should we wish to go beyond it? Why are we convinced that it cannot be the whole truth? 108 2017 MRT In view of this fact, it is perhaps reassuring that the Standard Model itself, although experimentally well confirmed, has several unsatisfactory, or at least, unnatural features. The gauge group has three factors (i.e., SU(3)⊗SU(2)⊗U(1)) and hence has three independent coupling constants. It offers a glimpse of unification in the breakdown of SU(2)⊗U(1)Y →U(1)Q, but this does not take us very far. Also, it is strange that one of these factors (i.e., SU(2)) distinguishes between left (L) and right (R) handed states. As a consequence, the Standard Model includes 45 massive fermion fields (see Figure) arranged in left-handed SU(2)L doublets and right-handed SU(2)R singlets. Since parity is maximally violated by the weak interaction, there is no right-handed neutrinos and only left-handed fermions (and right handed anti-fermions) are sensitive to the weak interaction. The primes on down-type quarks [with color charge red (R), green (G) or blue (B)] and neutrinos correspond to gauge eigenstates. Why Go Beyond the Standard Model? Undoubtedly, the most compelling reason for our dissatisfaction is that the Standard Model does not include gravity. Attempts to quantize general relativity results in a nonrenormalizable field theory. Such a theory may give the correct results in the lowest- order (i.e., at the classical level) but it but it does not permit a proper quantum calculation of any experimental quantity. 111 21 τ 21 µ 21 e 31 R, 32 B, 31 B, 32 B, 31 B, 32 B, 61 B, 61 B, 61 B, 31 G, 32 G, 31 G, 32 G, 31 G, 32 G, 61 G, 61 G, 61 G, 31 R, 32 R, 31 R, 32 R, 31 R, 32 R, 61 R, 61 R, 61 R, τµe τµe b t s c d u b t s c d u b t s c d u b t s c d u b t s c d u b t s c d u −−− −−− −−− −−− −−− ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ RRR LLL R R R R R R LLL R R R R R R LLL R R R R R R LLL ννν Standard Model fields with their associated charges (Y top right, chirality (bottom right of each doublet or singlet and color k =R,G, B ,bottom right of each doublet or singlet).
- 109. The Higgs spontaneous symmetry breakdown mechanism, which is crucial to the success of the Standard Model, requires an inelegant and arbitrary addition to the Lagrangian – if we didn’t need it, then why have a Higgs coupling to start off with? Perhaps even worse is the fact that the theory offers no explanation for family replication. The old question of who ordered the muon, has changed into why are there three families, but it still remains unanswered. Related to this is our complete ignorance as to the origin of the parameters in the mass matrix, all of which seem quite arbitrary. 109 2017 MRT • Three coupling parameters (i.e., the constants g1 =α, g2 =g, and g3 =gs); • The two Higgs parameters (i.e., MH and λ); • The nine fermion (i.e., quarks and leptons) masses (i.e., me , mu , md ; mµ , ms , mc , and mτ , mb , mt ); • The three mixing angles (i.e., θc, cosθw, and sinθw); • One phase angle in the quark-mixing Cabibbo-Kobayashi-Maskawa [V] (CKM) matrix (i.e.,δ) who QCD Lagrangian) requirement of a single true vacuum. There are even more parameters now with three neutrinos having finite masses and mixing. Hence, any model that might explain or relate some of the above parameters is worth considering, and it is to such models that we now turn. We will begin with the unification of the Standard Models itself. Indeed, the number of free parameters in the Standard Model totals 19; namely:
- 110. One of the most outstanding puzzles of the Standard Model is the structure of fermion masses and mixing angles. The masses of quarks and leptons show a hierarchical structure (see Table – all in GeV) suggesting the possibility of some underlying pattern. 110 2017 MRT Experimental measurements yield the following approximate structure for the Cabibbo- Kobayashi-Maskawa (CKM) matrix: = 999.004.0008.0 04.0973.022.0 003.022.0974.0 ][V Again, this displays an interesting structure. It is close to the identity matrix, with small off-diagonal mixing entries, except for the Cabibbo 1-2 entry, which is somewhat larger. First Generation* Second Generation Third Generation U-quarks u c t 2×10−3 2×10−1 173 D-quarks d s b 4×10−3 1×10−1 3 leptons e µ τ 0.51×10−3 1.05×10−1 1.7 * Here the First Generation is labeled by i = 1 so that U = Ui and D = Di or U1 = u and D1 = d, &c. Ibid for i = 2 and i = 3.
- 111. In summary, the hypercharges of the Standard Model in the Table below are related to their usual electric charges by QEM =Y+T3, where T3 =diag[½,−½] is an SU(2)L generator. 111 2017 MRT SM QUDLEs* SU(3)C SU(2)L U(1)Y UL i =[Ui,Di]L 3 2 1/6 UR i 3 1 −2/3 DR i 3 1 +1/3 Li =[νi,Ei]L 1 2 −1/3 ER i 1 1 +1 H=[H−,H0] 1 2 −1/2 * Gauge quantum number of the Standard Model of quarks, letons and the Higgs scalars. _ _
- 112. The Standard Model is based on the SU(3)C⊗SU(2)L⊗U(1)Y gauge group, which is spontaneously broken to SU(3)C⊗U(1)Q at a scale of order MW. Despite its ability to describe all available data, and its success in predicting new phenomena, we have explained in the previous chapter why it can not be regarded as the final theory. This Standard Model is not really a unified theory at all, because there are three different gauge interactions (i.e., gs, g, and g′), each with its own coupling strength. Moreover, owing to the presence of the Abelian U(1) group, the quantization of electric charge is not explained, and the conserved equality of the charges: 112 2017 MRT )p()e( QQ −=− which must be good to at least 1 part in 1020 to account for the observed electrical neutrality of bulk matter, remains a mystery. Grand Unified Theories The essential idea of grand unification is to try to embed SU(3)C⊗SU(2)L⊗U(1)Y in some simple gauge group G that has just a single coupling g, and to suppose that at high energies, above some unification scale MX, all phenomena satisfy the symmetry of G (c.f., Georgi and Glashow (1974) - http://pcbat1.mi.infn.it/~battist/astrop/su5.pdf). The different couplings gs, g, and g′ observed at low energies would then arise because the unified group G is spontaneously broken, first at the mass scale MX then followed by the electroweak breaking in the region of MW, as indicated by: G SU(3)C⊗SU(2)L⊗U(1)Y SU(3)C⊗U(1)Q MWMX energy
- 113. It is possible to find a unified gauge group G whose representations can accommodate all of the observed particles? The known fermions come in families containing 15 members, and each family decomposes into the SU(3)C⊗SU(2)L representation: 113 2017 MRT ),(),(),(),(),( 1313112321 ⊕⊕⊕⊕ where k=1,2,3 labels the color index. We have replaced the right-handed fermions by their charge-conjugate partners (e.g., eL c =eL +). This is necessary because gauge interactions conserve chirality: L − e eν Lk k d u c Le L c k )u( L c k )d(⊕⊕⊕⊕ ∑∑∑ += µ µ µ µ µ µ µ µ µ ψγψψγψψγψ RRLL AAA and so right- and left-handed fermions cannot be put in the same irreducible representation.
- 114. So, rather than regard ψL and ψR as two independent fundamental fields, we instead choose ψL and ψL c. The fields ψL and ψL c annihilate left-handed particles and antiparticles, respectively (or create right-handed antiparticles and particles). The relationship between ψR and ψL c is: 114 2017 MRT * 0)( RRLL c L c L CCPCCPP ψγψψψψψ TTTT ====≡ where PR,L=½(1±γ 5) are the right/left-handed projection operators, T stands for the transpose, and C is the charge-conjugate matrix. Thus, the complex-conjugate of a field that annihilates right-handed particles is in essence a field that annihilates left-handed antiparticles. C is the matrix that matches the components of ψR c to those of ψL c. It follows from the above equation that: )(1 0 †† 0 †* 0 † CCC RRR c L c L TTT ψψγγψγψψ =−==≡ − where we have used C−1γµ C=−γµ T and CT =C†=C−1=−C. We seek a unifying group G with representations that can accommodate the family of 15 left-handed fermions of (1,2)⊕(3,2)⊕(1,1)⊕(3,1)⊕(3,1) above in a neat and simple way. _ _
- 115. In addition to the 12 known gauge bosons of the Standard Model (the eight gluons g1…g8, W±, Z, and γ ), in a grand unified theory there must be gauge bosons, X, which link the quarks and leptons that lie within the same multiplet of G. These X bosons will mediate new interactions that violate baryon number (B) and lepton number (L) conservation. Evidently, these new interactions must be sufficiently weak to have eluded detection so far, which means that the X bosons must be very massive. For example, the B-violating interactions would make the proton unstable. Its decay amplitude will be of order (mp/MX)2, so, on dimensional grounds, we expect the proton lifetime: 115 2017 MRT A 2014 result, with 260 kT·yr of data searching for decay to K-mesons, set a lower limit of τp >5.9×1033 yr, close to a supersymmetry (SUSY) prediction of near 1034 yr. The proton thus appears to be stable which implies that MX >1015-1015 GeV. So grand unification, it it exists, must be a very high-energy phenomenon. Nevertheless, MX can still be much smaller than the Planck mass, ~1019 GeV, the scale at which gravitational interactions become strong. Hence, it is possible to discuss grand unification without including gravity. General Consequences of Grand Unification ~ 5 p 4 X 22p 1 ~ m M c h α τ ~ If G is to be a good symmetry at super-high energies, we expect the Standard Model couplings to become equal g3=g2 =g1 where g3≡gs, g2 ≡g, and g1 ≡g′ denote SU(3)C, SU(2)L, and U(1)Y coupling, respectively. The equality g2 =g1 is usually expressed in terms of sin2θw using cosθw =g/√(g2 +g′2) and sinθw =g′/√(g2 +g′2).
- 116. But first we have to ensure that the generator of U(1)Y transformations has the same normalization as the other generators of G. We require that all generators TI of G should satisfy: 116 2017 MRT JIJI cTT δ=)(Tr for arbitrary I, J indices and where the trace is taken over any representation of G and c is an irrelevant (representation-dependent) constant. In the Standard Model the normalization of the U(1)Y generator was arbitrary, but now it must satisfy this last equation. We can compare Y/2 with, for example, T3 of weak isospin. If we assume that the 15 fermions of (1,2)⊕(3,2)⊕(1,1)⊕(3,1)⊕(3,1) above completely fill a (possibly reducible) representation of G, then the eigenvalues of the Table on Slide 89 give: 3 5 ])())[(31( ])(3)(3)2()(6)1(2[ )(Tr )(Tr 2 2 12 2 1 2 3 22 3 422 3 12 4 1 2 3 2 4 1 = −++ +−+++− = T Y Upon embedding U(1)Y in G we must therefore renormalize the coupling g′ defined by the electroweak Lagrangian LEW=−gΣµΣi Ji µWi µ−g′Σµ(JY µ/2)Bµ accordingly, by taking: 2 1 2 5 3 gg =′ and hence from the definition of cosθw and sinθw above and g3 =g2 =g1 we have: 375.0 8 3 sin 2 15 32 2 2 15 3 22 2 2 2 == + = ′+ ′ = gg g gg g wθ _ _
- 117. The predictions that at the super-heavy scale MX: 117 8 3 sin sin 2 223 === w w θ θ α αα and where αk =gk 2/4π, are so different from the values observed at low energies that grand unification appears to be ruled out immediately. However, it must be remembered that the couplings are scale-dependent and so we must use renormalization group equations to continue these relations from MX down to the energies at which the αi have been measured. 2017 MRT HgHgg NNbNNbNb 10 1 3 4 6 1 3 4 3 22 3 4 11 123 −−=−−=−= and, +=≡ 2 2 X 22 XGUT ln π4)( 1 )( 11 µµααα Mb M k kk with: (here bk =(bo)k of bo=(11/3)N−(2/3)Nf with k=3, 2, 1 for SU(i)), where Ng is the number of families (or generations) of fermions and NH is the number of Higgs doublets in the electroweak sector. In the one-loop approximation, the solution 1/α(Q2)≈1/α(µ2)+ (bo/4π)ln(Q2/µ2) to the renormalization group equations gives:
- 118. The individual terms in these last three equations for b3, b2, and b1 correspond, respectively, to the contributions to bk from the gauge boson, fermion and Higgs loops, like shown in the Figure on Slide 73. The behavior of the equation for 1/αGUT above with Ng =3 and NH =3 is shown in the Figure and reflects the property that b3 >b2 >0 and b1 <0. 118 2017 MRT The variation of the effective coupling constants, αk(µ), with the energy scale µ. 0 10 20 30 40 50 60 70 2 WM 0 1010 1020 1030 1040 µ2 (GeV2) kα 1 2 1 α 1 1 8 3 α 3 1 α 2 XM GUT 1 α
- 119. Is the unification shown in the previous Figure reasonable? That is, can we find a grand unified scale MX at which the αk are equal, such that at present energies we obtain the values of αk actually measured? To check whether this is possible we use the known values of the strong and electromagnetic couplings: 119 2017 MRT 15.010.0)()( 2 W 2 W3 −=≡ MM sαα and: from a relation obtain from calculating the running of the coupling α in the electroweak sector (i.e., α(MW)/α(me)≅{1−[α(me)/3π][ΣFQF 2ln(MW 2/mF 2)]}≅1.073, where F indicates the fermion family – not its field f ), together with the evolution equations (i.e., 1/αGUT above), to solve the matching condition: 128 1 cos 5 3 sin)( 2 1 2 2 2 W = == wwM θαθαα )()()( 2 X1 2 X2 2 X3 MMM ααα == for MX and sin2θw(MW 2). The solution gives: .220-20.0~sinGeV10-10~ 21414 X wM θand and αk(MW 2)~1/40. For energies well aboveMX, G would be a good symmetry with only this one coupling. The additional X gauge boson loos increase the value of bk in 1/αGUT above and the evolution is indicated by the dashed line (− − −) in the previous Figure.
- 120. The prediction sin2θw =0.20-0.22 (in contrast to sin2θw =3/8) is in excellent agreement with observations and is one of the successes of the grand unification idea. The result is relatively insensitive to the details of the calculation. A more careful analysis has been done allowing for mass effects and including the two-loop contributions of the β- functions so we can write: 120 ∑+≡ l lklkkk k b d d ααααβ µ α µ 22 )( In the minimal SU(5) grand unified theory the more detailed predictions are: Λ −±== Λ −×= ± 2.0 ln006.00.0070.216GeV)20(sinGeV 2.0 101.3 MS22 03.1 MS3.014 X µθwM and The largest uncertainty is in the input value of α3 in α3(MW 2)=αs(MW 2)=0.10-0.15 above, or ΛC. Increasing ΛC increases α3 (c.f., α3(q2)=1/[(bo/4π)ln(Q2/ΛC 2)]) and so the couplings αk do not become equal until larger MX. This dependence is shown explicitly in MX and sin2θw above in terms of ΛC (in GeV) defined in the MS renormalization scheme. The fact that MX is of the same order as the lower bound given in MX >1015-1015 GeV is remarkable and gives credence to the idea of grand unified theories (GUTs). The couplings apparently imply a symmetry-breaking scale that is large enough to inhibit proton decay sufficiently. ~ __ 2017 MRT
- 121. Another consequence of grand unification is that the masses of the fermions should be related. Indeed, many models with economical Higgs structure predict that: 121 bτsµde mmmmmm === and, at the unification scale MX. At first sight, these equalities appear just as disastrous as the coupling equalities g3=g2 =g1 or α3 =α2 =α/sin2θw (with sin2θw =3/8). However, the masses also depend on the renormalization scale and to see whether such relations are reasonable we must use the renormalization group equations to continue down to present energies. In the one-loop approximation (i.e., m(Q2)=m(µ2)[α(Q2)/α(µ2)]γ o/bo)they give: b M Mmm f )( )( )()( X Xff γ α µα µ = and so at scale µ: 13q 1 X1 1 4 X3 3 3 1 )( X q )( )( )( )( )( )( )( )( bb k b k k MMMm m k kk = = ∏= − α µα α µα α µα µ µ γγ l l where the bk are given in b3 =11−(4/3)Ng, b2 =22/3−(4/3)Ng −(1/6)NH, and b1 = −(4/3)Ng − (1/10)NH and the γk by γ (α)=−(γo/2π)α and γo =(3/2)[(N2 −1)/N]. There is no k=2 contribution since quarks and leptons have the same SU(2) interactions (i.e., γl 2 =γq 2). Also, γl 3 =0 because gluons do not couple to leptons. 2017 MRT
- 122. If we evaluate mq(µ)/ml (µ) above at µ=10 GeV, the bb threshold, then we find: 122 3 τ b ≅ m m which is in excellent agreement with the observed masses. We cannot predict the corresponding ratios for the first two families, since mq(µ)/ml(µ) above is not reliable for µ values α3(µ)~O(1). However, the prediction for the ratio: 200 )( )( e µ d ≅= m m m ms µ µ should be essentially independent of renormalization effects and can therefore be compared with the current-algebra prediction for the quark masses given by ms /md ≅20 at large µ. This order of magnitude discrepancy poses a problem for grand unified theories with minimal Higgs structure. 2017 MRT
- 123. Any candidate for the grand unified group G must satisfy the following requirements: 1. Since G contains the Standard Model, SU(3)⊗SU(2)⊗U(1), it must have rank of at least 4 (i.e., the rank of a group is the maximum number of generators that can be diagonalized simultaneously – that can have simultaneous eigenvalues). As there are two diagonal generators for SU(3), one for SU(2), and one for U(1), G must have rank ≥4; 2. G must have complex representation (e.g., SU(3) in which the 3 transforms differently to the adjoint 3≡3* representation). This is because parity violation requires that left- and right-handed fermions must belong to different representations of the gauge group. Since ψ and ψ c have opposite helicities, and lie in are different. A consequence of this is that Dirac mass terms mψRψL, which are not invariant, are forbidden by the symmetry, which is probably the reason why there are light fermions (i.e., mf << MX) in nature; 3. G should have a single gauge coupling so all the interactions are truly unified. It should therefore be a ‘simple’ group (or the product of identical ‘simple’ groups whose couplings are required to be equal by some discrete symmetry); 4. The known fermions should fit economically into representations of G, and, since the unified gauge theory should be renormalizable, it must be free of anomalies. 123 2017 MRT Possible Choices of the Grand Unified Group _ _
- 124. An exhaustive list of ‘simple’ Lie groups is give in the Table. It is apparent that the above requirements severely limit the candidates for G. 124 2017 MRT Cartan Name Classical Name Rank* Order* Complex Representation An (N≥1) SU(N+1) N N(N+1) N≥2 Bn (N≥2) SO(2N+1) N N(2N+1) No Cn (N≥3) Sp(2N) N N(2N+1) No Dn (N≥4) SO(2N) N N(2N−1) N=5,7,9, … G2 G2 2 14 No F2 F4 4 52 No E6 E6 6 78 Yes E7 E7 7 133 No E8 E8 8 248 No * The order is the number of generators of the group and its rank is the maximum number that are simultaneously diagonalizable. In terms of the fundamental representation, SU(N) is the set of N×N complex, unitary matrices with unit determinant, SO(N) is the set of real orthogonal matrices with unit determinant, and Sp(2N) are real symplectic 2N×2N matrices that leave invariant the skew-symmetric matrix M with Mi,i−1 =−Mi−1,i and all other components zero. The final five entries are know as the exceptional groups, SO(N) with N =3,4,5,6 are equivalent locally to SU(2), SU(2)⊗SU(2), Sp(4), and SU(4), respectively. For example, for N= 4: A4 , SU(5), rank 4, Order 4(4+1)=20, and allows for a complex representation; B4 , SO(9), rank 4, order 4(2⋅4+1)=36, but does not allow for a complex representation; C4 , Sp(8), rank 4, order 4(2⋅4+1)=36, but does not allow for a complex representation; D4 , SO(8), rank 4, order 4(2⋅4−1)=28, but does not allow for a complex representation (since N=4); G2, F2, E6, E7, and E8 do not apply.
- 125. The possible groups of rank 4 are: 125 2017 MRT )(SU)(SU)(SU 335 ⊗and We can exclude the latter because one factor must be SU(3) of color, but the leptons do not carry color and must therefore lie in different representations to the quarks. However, since the sum of the charges of the quarks and leptons within any given multiplet of G must be zero, we require that the trace of the charge operator Q be zero (i.e., TrQ=0) and the latter gives TrQ≠0 either for the known quarks or for the leptons. This leaves only SU(5). Can the 15 left-handed fermion of (1,2)⊕(3,2)⊕(1,1)⊕(3,1)⊕(3,1) above be accommodated in representations of SU(5)? Remarkably, they can, but not in a single irreducible representation. Rather they fill two representations: a 5 and a 10 dimensional representation, with the following SU(3), SU(2) content: 44444 844444 76444 8444 76 105 1113232113 ),(),(),(),(),( ⊕⊕⊕⊕ L]e[ e − ν⊕ Lii ]du[⊕L c i )(d ⊕ L c i )(u ⊕ c Le where for each multiplet ΣQ=0, provided: −=−= eud 3 1 2 1 QQQ Thus, it is predicted, correctly, that quarks have ⅓-integral charge because they come in three colors. _ _ _
- 126. Next we should see whether there are any candidates for G with rank 5. From the previous Table we find that the only possibilities that contain an SU(3) color subgroup and that have complex representations are: 126 2017 MRT We must exclude SU(6) since, although it has a 15-dimensional representation, its SU(3)⊗SU(2) decomposition does not match that of (1,2)⊕(3,2)⊕(1,1)⊕(3,1)⊕(3,1) above for a single fermion family. On the other hand, SO(10) contains a 16-dimensional irreducible representation. Is it suitable? SO(10) contains SU(5) as a subgroup and the representation decomposes into: )(SO)(SU 106 and ⊕⊕= 51016 where the 10 and 5 are preciselythe SU(5) representationof (3,1)⊕(1,2)⊕(3,2)⊕(3,1)⊕(1,1) above. The interpretation for the extra SU(5) singlet (i.e., 1) is to postulate the existence of a right-handed neutrino, νR, and hence a νL c. 1_ _ _ _ _
- 127. In the minimal SU(5) grand unified model the neutrino has to be massless. So the observed maximal parity violation is a law of nature but it is not obvious why this should be so. It is perhaps more natural to assume that G possesses left-right symmetry and that the parity violation we observe at low energies is a result of symmetry breaking. For this to be possible G must contain SU(2)L⊗SU(2)R subgroups, not just SU(2)L. The simplest such G is precisely SO(10). The breaking of SO(10) down to SU(3)⊗SU(2)⊗U(1) is much more complicated than SU(5) breaking, and nature could choose one of the alternative chains: 127 2017 MRT and each stage of the symmetry breaking could occur at a different mass scale. A nice feature of the parity-conserving SO(10) symmetry is that the fermions all appear in a single representation and it is automatically free of anomalies. Hence, this group also satisfies all our above requirements (1. to 4.). SU(4)⊗SU(2)L⊗SU(2)R SO(10) SU(4)⊗SU(2)⊗U(1) SU(5) SU(3)⊗SU(2)⊗U(1) If we consider even higher-rank groups, the only ones that are automatically anomaly- free, and that admit complex representations, are SO(2N+2) with N≥2 and the exceptional rank-6 group E6. We conclude that the only natural candidates for the grand unification group G are the gauge groups SU(5) (≡E4), SO(10) (≡E5), and E6, of rank 4, 5, 6, respectively. SU(5) if frequently called the Georgi-Glashow model after its original proposers and it is the simplest of the grand unified theory.
- 128. The SU(3), SU(2) decomposition of the fundamental five-dimensional representation of SU(5) is: 128 2017 MRT Recalling that the SU(3) and SU(2) product representations decompose into the symmetric and antisymmetric representations: Grand Unified SU(5) ),(),( 21135 ⊕= ASAS 13223633 ⊕=⊗⊕=⊗ and it can be checked that the SU(5) product representation decomposes as: AS AS 1015 1123133123162113211355 ⊕= ⊕⊕⊕⊕⊕=⊕⊗⊕=⊗ )],(),(),[()],(),(),[()],(),[()],(),[(
- 129. The family of left-handed fermions (3,1)⊕(1,2)⊕(3,2)⊕(3,1)⊕(1,1) above therefore fits neatly into a 5 and a 10 of SU(5) with the assignment: 129 2017 MRT It is often convenient to use the 5 of right-handed fields instead of the 5 of left-handed fields. We have identified color SU(3) with the first three indices of SU(5) and SU(2) with the remaining two. To conform with our previous choice of [ν,e−]L as the SU(2) doublet, we take the conjugate to be [e−,−ν]L (c.f., φc =iτ2φ*=[φ 0,−φ −]T of the Fermion Masses and Couplings chapter). We shall return to the assignment of the u, d, uc, ec or the dc to the 5 and of the uc to the 10 are necessary to satisfy the hypercharge (charge) requirement TrY=0 (TrQ=0) in a given multiplet. It would be violated if dc ↔uc. 55 = − == − = +− R c R L c c c L ν ψ ν ψ e d d d e d d d 3 2 1 3 2 1 or _ _ _ _ _ _
- 130. The SU(5) Lagrangian contains a gauge-invariant interaction term involving the multiplet (c.f., previous ‘ψL or ψR c ’ equation) of the form: 130 2017 MRT L+ +∂= ∑ ∑= = 3 0 24 1 2µ µµ µ ψ λ γψ c R I I Ic R AgiiL There are 52 −1=24 traceless, Hermitian matrix generators of SU(5), λI /2, chosen as in SU(3) or SU(2) (c.f., Table on Slide 57). They satisfy: ∑= == 24 1 2],[2)(Tr K KKJIJIJIJI ci λλλδλλ and where cIJK are the structure constants of SU(5). A convenient representation is given in the Table.
- 131. The 5×5 traceless Hermitian λI matrices of SU(5) are: and λ12, λ13, …, and λ20 are obtained by continuing to put 1 and mi in the same pattern in the off diagonal blocks. Lastly, for λ21, λ22, λ23, and λ20: c&,,, = − = = = 0 0 0 0 0 0 00 0 000 010 00 01 00 000 00 00 00 0 000 001 00 00 01 11109 λλλ λ λ i i a a with i =1,2,3 and τi are the Pauli isospin matrices. − − − = =+ 30 03 200 020 002 15 1 0 0 0 00 2420 λ τ λ and i i with a=1, …,8, where λa are the SU(3) λ matrices of Table on Slide 55. Then, we expand for I=9, …,20:
- 132. Thus there are 24 gauge bosons, Aµ I, which lie in the adjoint representation of SU(5), that is, in the nonsinglet piece of the product 5⊗5=24⊕1. The SU(3), SU(2) decomposition of the 24-dimensional representation is: 132 2017 MRT where the color labels of the particles have been suppressed. So, in addition to the 12 gauge bosons of the Standard Model there are 12 new bosons (X, Y, and their antiparticles X, Y) that lie in the fundamental representations of both SU(3) and SU(2). On account of the summation in L above it is convenient to introduce a 5×5 traceless matrix of the gauge fields Aµ (N.B., g1, …, g8 and H component order is speculative): ),(),(),(),(),( 322311311824 ⊕⊕⊕⊕= Gluons, g Higgs, H 43421 γZ,W 0± X, YWi B X, Y _ _ µ µ µµ λλ B YYY XXX YX YX YX W Y X YXg AA i i i iia I I I 2 γW WZ ggg gHg ggg H, 2 1 2 24 321 0 321 33876 2254 11321 24 1 + = == + −= ∑ where λ24 is the diagonal matrix given in the previous Table and W and B are the SU(2) and U(1) gauge fields as in L=Σf { fLΣµγ µ[i∂µ−gΣi(τi /2)Wi µ −g′(Y/2)Bµ ]fL + fRΣµγ µ[i∂µ −g′(Y/2)Bµ] fR}−¼ΣµνΣiWi µνWi µν −¼Σµν Bµν Bµν. _ _ _ _ _
- 133. As we have anticipated, the Lagrangian L =iψR c Σµγµ[∂µ−igΣI(λI /2)Aµ I ]ψR c describes not only standard-model transitions but also lepton-quark transitions mediated by the new superheavy gauge bosons X and Y. Examples of such transitions are shown in the Figure. It is clear that these bosons must have charge QX =4/3 and QY =1/3. As the 10 of 5⊗5=15S⊕10A above is the antisymmetric part of this 5⊗5 its components have the form: 133 Lepton-quark transitions in SU(5). 2017 MRT This indicates how the u, d, uc, ec fields are to be assigned to the 10. We put [using the Langacker (1981) sign convention]: L cc cc cc NM − −−− −−− −−− = + + 0eddd e0uuu du0uu duu0u duuu0 2 1 321 321 3312 2213 1123 χ g e− d X g ν d Y _ where φM with M=1,…,5 transforms as the fundamental 5-dimensional representation of SU(5). )( 2 1 MNNMNM φφφφχ −= where (χ4k , χ5k) have been identified with (uk ,dk)L and transform as (3,2) of SU(3), SU(2), whereas: ∑= = 3 1 u 2 1 k c kLkjiji εχ with color labels i, j,k=1,2,3, transforms as a 3 of color. _ _
- 134. The covariant derivative for a fundamental 5, φM, is: 134 2017 MRT ∑= •+∂= 5 1 2 1 )( N N NM MM giD φφφ µµµ Aλλλλ as in L =iψR c Σµγµ[∂µ−igΣI(λI /2)Aµ I ]ψR c, and so for the antisymmetric 5⊗5 representation of χMN =(1/√2)(φM φN −φN φM) we obtain: ∑∑ == •+ •+∂= 5 1 5 1 2 1 2 1 )( P PM QPQ NQ QM PMPM gigiD χχχχ µµµµ AA λλλλλλλλ Inserting this covariant derivative into the kinetic-energy term iTr(χ Σµγ µDµχ), we obtain the gauge interactions of the 10 multiplet: ∑∑ •−= µ µ µ χγχ N PN NM PMg Aλλλλ 2 1 )(2nInteractioL The factor 2 occurs because the two gauge terms in the above for (Dµχ)MP give identical contributions, which in turn follows on using χMN =−χNM (twice). If we also include the gauge-interaction term of the 5 multiplet ψR c given in L =iψR c Σµγµ[∂µ−igΣI(λI /2)Aµ I ]ψR c, we obtain: ∑∑∑∑ •− •−= µ µ µ µ µ µ ψγψχγχ N N c R NM M c R N PN NM PM )( 2 1 )( 2 1 )(2nInteractio AA λλλλλλλλL _ _ _
- 135. This last LInteraction Lagrangian describes all the SU(5) gauge interactions. For example, the SU(3) color gauge theory of the Standard Model is obtained by taking M,N,P=1,2,3, while the Glashow-Weinberg-Salam SU(2)⊗U(1) is contained in M, N=4,5. On the other hand, taking M=1,2,3 with N=4,5 in the second term in L Interaction above gives new transitions mediated by the superheavy gauge group (X,Y). This results in lepton-quark transitions: 135 2017 MRT RR d],e[ →+ ν Similarly, the first term in LInteraction gives the transitions: LLL u]d,u[]u,d[e →→+ and and the inverse transitions follow from M↔N. Because they can produce such transitions, the X and Y bosons are sometimes called leptoquarks. They mediate proton or neutron decay. For example, the Figure shows diagrams for p→e+uu, where the uu form a neutral meson, π0, ρ0, &c., and there are similar diagrams for p→e+dd & p→ν du. Although the above transitions violate both baryon and lepton number conservation, they do not change the value of B−L. The X and Y bosons carry B−L of ⅔ and B−L is conserved in SU(5). Diagrams for the proton decay process p→e+uu. ‘ p{’ and ‘ }π0,ρ0,…’ are the same for all three diagrams. Y p u d u u e− u _ d u u u e− u _X Y d u u u e− u _ π0,ρ0,… _ _ _ _ _
- 136. The symmetry must be broken spontaneously at two very different scales: first at ~1014 GeV to generate masses for the superheavy gauge bosons X, Y, and secondly at ~100 GeV to generate masses for the W±, Z0. In the minimal SU(5) model this is accompanied by two multiplets of Higgs scalar fields (i.e., a real adjoint 24-dimensional representation with indices I,J,K=1,…,24 ), ΦI, and a complex 5-dimensional representation (with indices M,N=1,…,5), HI =(HM ,φ). HM is a color triplet and φ, which is a color singlet, is the usual SU(2) Higgs doublet of the Standard Model). Hence: 136 2017 MRT where 〈…〉 denotes the vaccum expectation value of the field. It is assumed that there are no new bosons of fermions in the enormous, and unexplored, mass range between MZ and MX. This supposedly uninteresting region is known as the desert. Spontaneous Symmetry Breaking in SU(5) )(U)(SU)(U)(SU)(SU)(SU 131235 ⊗⊗⊗ 24 〈Φ〉∼1014GeV 5 〈H〉∼102GeV
- 137. Suppose, for a moment, that Φ is the only Higgs representation. It is convenient to reassemble the 24 Higgs fields in the form of a 5×5 traceless matrix: 137 2017 MRT )())((Tr 2 1 † Φ− ΦΦ= ∑Φ VDD µ µ µL The most general form of the scalar self-coupling potential is: ∑= Φ=Φ 24 1 2I I Iλ just as we did for the gauge fields in Aµ =ΣI(λI /2)Aµ I. The Higgs contribution to the Lagrangian is then (c.f., L =½Σi Σµ ∂µφi∂µφi −[−½λ2Σi (φiφi)+¼λΣi (φiφi)2]): )(Tr 2 1 ])[(Tr 4 1 )(Tr 2 1 )( 42222 Φ+Φ+Φ−=Φ baMV apart from an inessential cubic term which has been dropped by imposing the discrete symmetry under Φ→−Φ.
- 138. If M2 >0, spontaneous symmetry breaking occurs as described in the Spontaneous Symmetry Breaking (SSB) chapter. It is also necessary for the quartic terms in V(Φ) above to be positive to ensure that this potential is bounded from below, which requires that 15a+7b>0. If b>0 it can be shown (c.f., L.-F. Li, 1974) that when one diagonalizes the matrix Φ and impose the condition ∂V/∂Φ=0, Φ acquires a vacuum expectation value of the desired SU(3)⊗SU(2)⊗U(1) invariant form: 138 2017 MRT − − =Φ 2 3 2 3 o 0000 0000 00100 00010 00001 00 v where vo is a constant. Substituting this vacuum expectation value of 〈0|Φ|0〉 into V(Φ) above gives: ++−=Φ 4 )715( 4 15 )( 2 o22 o v baMvV which has a minimum when: ba M v 715 2 2 2 o + =
- 139. To determine the resulting masses of the gauge bosons, we follow the procedure of the Gauge Boson Mixing and Coupling chapter. First, we obtain the covariant derivative for the 24 adjoint field Φ. For a fundamental 5 representation the derivative is given in (Dµ φ)M = ∂µφM +igΣN [(λλλλ/2)•Aµ)MN φN], but for a 24 it is more complicated. We have: 139 2017 MRT where the 24-dimensional matrices, TI, represent the generators in the adjoint representation, and may be evaluated using the general group-theoretic result for the adjoint representation: ∑ Φ+Φ∂=Φ JI JJKIIKK ATgiD )()( µµµ KJIKJI ciT =)( the cIJK being the structure constants of the group (c.f., Tr (λI λJ)=2δIJ and also [λI ,λJ]= 2iΣK cIJK λK). Using Φ=ΣI(λI /√2)ΦI, we may rewrite (Dµ Φ)K as: )( 2 µµµµµµ λ AAgiAcigiD KJI JI K KJI Φ−Φ+Φ∂=Φ+Φ∂=Φ ∑ where the last equality follows from the commutation relations of the λ matrices, [λI ,λJ]= 2iΣK cIJK λK and Aµ =ΣI(λI /2)Aµ I.The next step is to substitute the covariant derivative DµΦ above into the Lagrangian LΦ =½Tr [(DµΦ)(DµΦ†)]−V(Φ) and to replace Φ by its vacuum expectation value 〈0|Φ|0〉=vo[::] above.
- 140. The gauge boson masses can then be extracted from the term: 140 2017 MRT where here Aµ I denotes the linear combinations of the original Aµ I that diagonalizes the mass matrix. Since 〈Φ〉 of 〈0|Φ|0〉=vo[::] above is proportional to the unit matrix in the SU(3) and SU(2) subspaces, these components commute with Aµ and the standard- model gauge bosons remains massless. A mass is only generated for the X and Y bosons. Substituting 〈0|Φ|0〉=vo[::] into the Lmass Lagrangian above, we obtain: ∑∑=Φ−Φ= µ µ µ I III AAMAA g 22 2 mass 2 1 ])[(Tr 2 L 2 o 22 Y 2 X 8 85 vgMM == The number of Higgs fields that remain massless (i.e., Goldstone bosons) is equal to the number of generators for which the symmetry is broken (c.f., Spontaneous Symmetry Breaking (SSB) chapter). Thus, for the SU(5)→SU(3)⊗SU(2)⊗U(1) symmetry breaking, we have 24−8−3−1=12 broken generators, and hence 12 massless Higgs fields that are eaten up by the X and Y bosons as they acquire mass. They become the longitudinal polarization states of the massive X and Y bosons. The 12 remaining Higgs fields acquire masses of order vo ~MX from vo =2M2/(15a−7b), but since they do not couple to fermions they are of no further interest.
- 141. The second stage of breaking of SU(5) to give the observed physics is the electroweak symmetry breaking of the Standard Model. In the minimal SU(5) theory, this is accompanied by a complex 5-dimensional Higgs multiplet H: 141 2017 MRT where (φ +,φ 0) is the analog of the conventional doublet 〈Φ〉=〈0|[φ+ φ 0]T|0〉= (1/√2)[0 v]T and where the nonvanishing vacuum expectation value arises from a Higgs potential: = = + v HH H H H 0 0 0 0 2 1 00 0 3 2 1 with φ φ 2††2 )()()( HHHHHV λµ +−= with µ2 >0 and λ>0 in the usual way (c.f., V(φ)=−µ2(φ†φ)+λ(φ†φ)2). It can then be found: λµ 22 v= and that the W± and Z0 bosons acquire masses: 2222 Z 2 W 4 1 cos vgMM w == θ as found in the Gauge Boson Masses chapter. However, the color triplet of Higgs fields, HM with M=1,2,3 of H=[:] and 〈0|H|0〉=(1/√2)[::] above remains massless.
- 142. In the Standard Model, the fermion masses were generated by symmetry breaking through Yukawa couplings to the Higgs field (c.f., Fermion Masses chapter). Is a similar mechanism possible in SU(5)? The left-handed fermions lie in ψL(5) and χ(10) multiplets and so the Higgs representations that can be responsible for these masses must occur in the following decompositions: 142 2017 MRT Fortunately, these products do not contain a 24 because a coupling to the Φ(24) Higgs of Φ=ΣI(λI /√2)ΦI would normally generate fermion masses of order MX. However the 5 of Higgs, H of H=[:] above looks promising. From 5⊗10=5⊕45 and 10⊗10=5⊕45⊕50 we see that: 504551010455105151055 ⊕⊕=⊗⊕=⊗⊕=⊗ and,, Fermion Masses Again LL ⊕=⊗⊗⊕=⊗⊗ 15101015105 and and so we can construct two SU(5)-invariant Yukawa coupling terms: where the ε arises because the relevant coupling in 10⊗10=5⊕45⊕50 is totally antisymmetric. h.c.)()( 4 1 )()( † ++= ∑∑ pnmlk pnmLlk c RpnmlkU lk llkLk c RDY HGHG χχεχψL _ _ __ _ __ _
- 143. Substituting the vacuum expectation value vδi5/√5 of 〈0|H|0〉=(1/√2)[::] for Hl, we obtain: 143 2017 MRT at the unification mass scale (c.f., me =md, mµ =ms, and mτ =mb at the unification scale MX), while mu =(1/√2)vGU. Similarly, for other generations mµ =ms and mτ =mb, but presumably larger couplings GD, GU. DGvmm 2 1 de == In summary, at the first stage of symmetry breaking a Φ(24) of Higgs generates the masses of the superheavy X and Y bosons, while at the second stage an H(5) of Higgs generates both the masses of the W, Z bosons and of the fermions. The vacuum expectation values of the Higgs fields are vo ~1014 GeV and v~102 GeV, respectively. This attractive model has a fundamental flaw, however, which goes under the name of hierarchy problem. In minimal SU(5) we therefore have: ∑∑ ∑∑ − +−= ++= k kk U k kkD kji kLji c Rkji U k kLk c R D Y Gv Gv GvGv uu 2 ddee 2 1 h.c.)()( 2 )()( 2 )( 45mass ψχεχψL where i, j,k=1,2,3 are the color labels and we have used χij=(1/√2)Σkεijk uL c k.
- 144. We have already noticed that the 5 of Higgs (i.e., H=[:] above), contains a massless Higgs color triplet Hk . In fact, in the minimal SU(5) model there are two Higgs color triplets of charge −⅓, namely, Φk5 and Hk , together with their antiparticles. One linear combination, which is dominantly Φk5, is eaten by the Y boson to generate its mass, while the other, which is predominantly Hk , remains massless and through B-violating process such as: 144 2017 MRT would allow protons to decay extremely rapidly! ueHud + →→ k Hierarchy Problem A second problem is that, although it is attractive to have two separate terms in the Higgs potential V(Φ)+V(H) each with its own minimum, such a theory is non renorma- lizable. There must be gauge-invariant quartic terms involving both Φ and H of the form: HHHHHV 2†2† )(Tr),( Φ+Φ=Φ βα Even if we artificially try to omit these cross-coupling terms from the potential, they will be generated automatically by radiative corrections. The combined potential V(Φ)+V(H) +V(Φ,H) can be arranged to have a single minimum at a point where the expectation values of Φ and H are given by 〈0|Φ|0〉=vo[::] and 〈0|H|0〉=(1/√2)[::], respectively, but there are then extra terms in the equations vo =2M2/(15a−7b) and µ2 =2v2λ, that determine vo and v: 2 o 2222 o 2 2 3 5 2 3 10 3 )715( 2 1 vvvvbaM −++≅ +++≅ βεβαλµβα and _
- 145. Strictly speaking, the vacuum expectation value of Φ can no longer be exactly of the form 〈0|Φ|0〉=vo[::], since it acquires a tiny SU(2) isotriplet breaking part ε: 145 2017 MRT withε ≅ (3β/20b)(v/vo)2.However,since v<<vo, conditionM2≅½(15a+7b)vo 2+[α+(3/10)β ]v2 above is little changed from its original form. But µ2≅λv2+(3/2)[5α +(3/2)β −εβ]vo 2 is grossly different. The extra vo 2 terms now imply that the natural size of v is of order vo. The problem is that the mixing terms (e.g., Tr (Φ2)H†H) generate contributions to the mass of H of order vo. This is fine for the color triplet components Hk, and would suppress Higgs-mediated proton decay to an acceptable level and so cure our first problem. However, for the standard-model Higgs φ it is a disaster. +− −− =Φ 22 3 22 3 o 0000 0000 00100 00010 00001 00 ε ε v
- 146. To obtain the desired result: 146 2017 MRT Even if such a cancellation were arranged, it would be upset by radiative corrections. At first order they would generate a mass ≈gvo 2 for H5, and we would have to retune the parameters to cancel this, too, and so on to higher and higher orders. 12 X W o 10~~ − M M v v we have to choose the Higgs parameters with an astonishing accuracy if we are to ensure that in µ2≅λv2+(3/2)[5α +(3/2)β −εβ]vo 2: 2 o 242 o 2 )10( 2 3 5 2 3 vOv − = +− βαµ This necessity for fine tuning is a disease of all grand unified theories and is known as the (gauge) hierarchy problem. It is difficult to sustain naturally two vastly different scales of symmetry breaking, since radiative corrections mix the scales and, without fine tuning, will always equalize them.
- 147. Although the standard SU(3)C⊗SU(2)L⊗U(1)Y theory gives a very satisfactory account of the interaction of the gauge bosons with the fundamental fermions (i.e., the quarks and leptons), the Higgs sector presents much more of a problem. As we have seen in the Spontaneous Symmetry Breaking (SSB) chapter, the higgs particle H plays the crucial role of breaking the original symmetry down to SU(3)C⊗U(1)EM, thereby giving mass to the W±, Z0 bosons. Also, through its coupling to the fermions, it is responsible for their masses and mixing angles. However, these couplings are completely arbitrary parameters and so the theory can not explain, for example, why the (fundamental) fermion (rest) masses (c.f., Table on Slide 4) vary over at least five orders of magnitude (me to mt). Furthermore, the mass of the Higgs particle itself in not predicted; it has been derived earlier as: 147 2017 MRT GeV246 2 1 21 = = FG v Higgs Scalars and the Hierarchy Problem vM λµ 22 2 H == where: but λ is unknown so if perturbation theory is to be valid for the Higgs interactions, then the coupling is λ<1 such that MH <1000 GeV. The actual value is (c.f., CERN LHC 2014): GeV0.30125.02H ±≅M based on proton-proton collisions, which is about half the value of v above.
- 148. Loop diagrams like those in the Figure below give quadratically divergence renormalization corrections to the bare Higgs mass of the form: 148 2017 MRT 2 2 2 24 4 2 H π8 ~ 1 )π2( Λ∝∆ ∫ Λ g k kd gM where Λ is the (ultra-violet) cutoff and g represents the coupling (i.e., ). Even after this momentum-independent contribution has been subtracted, there is still the usual logarithmic momentum-dependent contribution to the mass: Renormalization correction of the Higgs boson mass due to (Left) the λφ 4 term, (Middle) gauge boson loops, and (Right) fermion loops. H H H µ q qM ln~)(H∆ just like that of the fermion masses (c.f., m(q2)=mo{1+(3α/4π)ln[(mo 2−q2)/Λuv]+…). g g g g g
- 149. If we try to explain the Higgs couplings by embedding the Standard Model in some larger GUT (e.g., SU(5), as discussed in the previous chapters) the breakdown: 149 2017 MRT if MX ~1014 GeV (c.f., µ2 −(3/2)[5α +(3/2)β]vo 2=O(10−24)vo 2 ). EM)()()()()()( HX 131235 USUUSUSUSU CMYLCM ⊗ →⊗⊗ → requires two types of particles (i.e., Φs with MΦ =O(MX) in addition to the usual Hs with MH =O(MW)). In order to keep the Hs light while the are heavy, we must ensure cancellation of the divergence ∆ MH 2 above to an accuracy: 24 2 X W 2 H 10~~~ − Φ M M M M
- 150. The severity of this problem can be appreciated by considering a Higgs potential (i.e., like V(Φ,H)=αH†HTr(Φ2)+β H†Φ2H) with two very different energy scales. If we are to have one set of Higgs fields φ with associated particles H arising from a vacuum expectation value v≅102 GeV and another set Φ with vacuum expectation value vo ≅1014 GeV, then the kind of potential we need is (c.f., V(φ)=−µ2(φ†φ)+λ (φ†φ)2): 150 224 Φφg where g is the gauge coupling, through diagrams such as that in the Figure. 22 o 2 2 222 1 )()(),( vvV −Φ+−=Φ λφλφ However, since both sets of Higgs fields interact with the gauge bosons, we get (after renormalization) corrections to V(φ,Φ) above of order: Corrections to the Higgs potential V(φ,Φ) =λ1(|φ |2 −v2)2 +λ2(|Φ|2 −vo 2)2 due to couplings between the light Higgs H and heavy Φ through gauge boson exchange giving g4φ2Φ2. 2017 MRT If g4φ2Φ2 is added to V(φ,Φ) above the minimum of the potential with respect to φ is shifted from |φ |=v to: o 2 1 2 o 4 2 vvgvg αλφ ≅≅≅ Φ H Φ H g g g g for λ1 ≈1 and the vacuum expectation value of the lower Higgs field gets moved up to within order α(≈10−2) of the higher mass scale unless there are additional contributions to the potential, adjusted to an accuracy v 2/vo 2 ~10−24, that cancel away these corrections. This is the Hierarchy problem. It is evidently not possible to have a hierarchy of spontaneous symmetry breakdown at different mass scales through a succession of Higgs couplings without very fine and artificial tuning of the parameters. Supersymmetry could resolve this problem.
- 151. Appendix – Useful Figures Particle Data Group: http://pdg.lbl.gov/2014/reviews/rpp2014-rev-quark-model.pdf. Figure 15.1: SU(4) weight diagram showing the 16-plets for the (a) pseudoscalar mesons and (b) vector mesons, made of the u, d, s, and c quarks as a function of isospin T3, charm C, and hypercharge Y = B + S − C/3. The nonets of light mesons occupy the central planes to which the cc states have been added. (b) (a) T3 Y C _ 2017 MRT
- 152. Figure 15.4: SU(4) multiplets of baryons made of u, d, s, and c quarks. (a) The 20-plet with an SU(3) octet. (b) The 20-plet with an SU(3) decuplet. (b) (a) 2017 MRT
- 153. Figure: The pattern of weak isospin, T3, and weak hypercharge, YW = Q − T3, of the known elementary particles, showing electric charge, Q, along the weak mixing angle. The neutral Higgs field (circled) breaks the electroweak symmetry and interacts with other particles to give them mass. Three components of the Higgs field become part of the massive W± and Z bosons. 2017 MRT
- 154. Figure: Summary of interactions between particles described by the Standard Model. 2017 MRT
- 155. Figure: The above interactions form the basis of the standard model. Feynman diagrams in the standard model are built from these vertices. Modifications involving Higgs boson interactions and neutrino oscillations are omitted. The charge of the W bosons is dictated by the fermions they interact with; the conjugate of each listed vertex (i.e. reversing the direction of arrows) is also allowed. 2017 MRT
- 156. 2017 MRT D. Perkins, Introduction to High Energy Physics, 4-th Edition, Cambridge, 2000. University of Oxford, England This highly regarded textbook for advanced undergraduates provides a comprehensive introduction to modern particle physics. Coverage emphasizes the balance between experiment and theory. It places stress on the phenomenological approach and basic theoretical concepts rather than rigorous mathematical detail. Donald Perkins also details recent developments in elementary particle physics, as well as its connections with cosmology and astrophysics. A number of key experiments are also identified along with a description of how they have influenced the field. Perkins presents most of the material in the context of the Standard Model of quarks and leptons. He also fully explores the shortcomings of this model and new physics beyond its compass (such as supersymmetry, neutrino mass and oscillations, GUTs and superstrings) […]. P.D.B. Collins, A.D. Martin, E.J. Squires, Particle Physics and Cosmology, Wiley, 1989. University of Durham, England This readable introduction to particle physics and cosmology discusses the interaction of these two fundamental branches of physics and considers recent advances beyond the Standard Models. Eight chapters comprise a brief introduction to the gauge theories of the strong and the electroweak interactions, the so-called grand unified theories, and general relativity. Ten more chapters address recent concepts such as composite fermions and bosons, supersymmetry, quantum gravity, supergravity, and strings theories, and relate them to modern cosmology and experimental astronomy. M. Kaku, Quantum Field Theory – A Modern Introduction, Oxford University Press, 1993 City College of the CUNY The rise of quantum electrodynamics (QED) made possible a number of excellent textbooks on quantum field theory in the 1960s. However, the rise of quantum chromodynamics (QCD) and the Standard Model has made it urgent to have a fully modern textbook for the 1990s and beyond. Building on the foundation of QED, Quantum Field Theory: A Modern Introduction presents a clear and comprehensive discussion of the gauge revolution and the theoretical and experimental evidence which makes the Standard Model the leading theory of subatomic phenomena. The book is divided into three parts: Part I, Fields and Renormalization, lays a solid foundation by presenting canonical quantization, Feynman rules and scattering matrices, and renormalization theory. Part II, Gauge Theory and the Standard Model, focuses on the Standard Model and discusses path integrals, gauge theory, spontaneous symmetry breaking, the renormalization group, and BPHZ quantization. Part III, Non-perturbative Methods and Unification, discusses more advanced methods which now form an essential part of field theory, such as critical phenomena, lattice gauge theory, instantons, supersymmetry, quantum gravity, supergravity, and superstrings. S. Weinberg, The Quantum Theory of Fields, Volume II, Cambridge University Press, 1996. Josey Regental Chair in Science at the University of Texas at Austin In this second volume of The Quantum Theory of Fields, Nobel Laureate Steven Weinberg continues his masterly exposition of quantum theory. Volume 2 (of 3) provides an up-to-date and self-contained account of the methods of quantum field theory, and how they have led to an understanding of the weak, strong, and electromagnetic interactions of the elementary particles. The presentation of modern mathematical methods is throughout interwoven with accounts of the problems of elementary particle physics and condensed matter physics to which they have been applied. 156 References / Study Guide