Talk given at the Workshop in Catania University

Vector meson condensation within NJL model
Marco Frasca
Quantum Chromodynamics in strong magnetic ﬁelds, Catania,
September 23, 2013

Plan of the Talk 2 / 36
Plan of the talk
Low-energy QCD
Scalar ﬁeld theory
Yang-Mills theory
Yang-Mills Green function
QCD in the infrared limit
ρ meson condensation
Emerging of ρ condensation
NJL and magnetic ﬁeld
Mass of the ρ in NJL model
Quantum phase transition in QCD
Conclusions

Low-energy QCD Scalar field 3 / 36
Scalar field
A classical field theory for a massless scalar field is given by
φ+λφ3=j
The homogeneous equation can be exactly solved by [M. Frasca, J.
Nonlin. Math. Phys. 18, 291 (2011)]
Exact solution
φ=µ( 2
λ )
1
4 sn(p·x+θ,i) p2=µ2 λ
2
being sn an elliptic Jacobi function and µ and θ two constants. This
solution holds provided the given dispersion relation holds and
represents a free massive solution notwithstanding we started from a
massless theory.
Mass arises from the nonlinearities when λ is taken to be finite.

Scalar ﬁeld
When there is a current we ask for a solution in the limit λ→∞ as our
aim is to understand a strong coupling limit. We consider φ = φ[j]
and put
φ[j]=φ0(x)+ d4x
δφ[j]
δj(x ) j=0
j(x )+ d4x d4x
δ2φ[j]
δj(x )δj(x )
j=0
j(x )j(x )+...
This is a current expansion where ∆(x−x )=
δφ[j]
δj(x ) j=0
is the Green
function
One can prove that this is indeed so provided
Green function equation
∆(x)+3λ[φ0(x)]2∆(x)=δ4(x).
This is an exactly solvable linear equation.

Scalar ﬁeld
The exact solution of the equation
∆(x)+λ[φ0(x)]2∆(x)=δ4(x)
it is straightforwardly obtained and its Fourier transformed
counterpart is [M. Frasca, J. Nonlin. Math. Phys. 18, 291 (2011)]
∆(p)= ∞
n=0(2n+1)2 π3
4K3(i)
e
−(n+ 1
2 )π
1+e−(2n+1)π
. 1
p2−m2
n+i
provided the phases of the exact solutions are ﬁxed to θm=(4m+1)K(i)
being K(i)=1.311028777....
The poles are determined by
mn=(2n+1) π
2K(i) (λ
2 )
1
4 µ.

Scalar field
We can formulate a quantum field theory for the scalar field starting
from the generating functional
Z[j]= [dφ] exp[i d4x(1
2
(∂φ)2−λ
4
φ4+jφ)].
We can rescale the space-time variable as x →
√
λx and rewrite the
functional as
Z[j]= [dφ] exp[i 1
λ
d4x(1
2
(∂φ)2−1
4
φ4+ 1
λ
jφ)].
Then we can seek for a solution series as φ= ∞
n=0 λ−nφn and rescale the
current j → j/λ being this arbitrary.
We work on-shell given the equation for the exact solution to hold
φ0+φ3
0=0
This is completely consistent with our preceding formulation [M.
Frasca, Phys. Rev. D 73, 027701 (2006)] but now all is fully
covariant.

Scalar field
Using the current expansion that holds at strong coupling λ→∞ one
has
φ[j]=φ0+ d4x ∆(x−x )j(x )+...
it is not difficult to write the generating functional at the leading
order in a Gaussian form
Z0[j]=exp[ i
2
d4x d4x j(x )∆(x −x )j(x )].
This conclusion is really important: It says that there exists an infinite
set of scalar field theories in d=3+1 that are trivial in the infrared
limit and that obtain a mass gap even if are massless.
This functional describes a set of free particles with a mass spectrum
mn=(2n+1) π
2K(i) (λ
2 )
1
4 µ
that are the poles of the propagator, the same as in classical theory.

Low-energy QCD Yang-Mills field 8 / 36
Yang-Mills field
A classical field theory for the Yang-Mills field is given by
∂µ∂µAa
ν −(1− 1
ξ )∂ν (∂µAa
µ)+gf abc Abµ(∂µAc
ν −∂ν Ac
µ)+gf abc ∂µ(Ab
µAc
ν )+g2f abc f cde AbµAd
µAe
ν =−ja
ν .
For the homogeneous equations, we want to study it in the formal
limit g → ∞. We note that a class of exact solutions exists if we take
the potential Aa
µ just depending on time, after a proper selection of
the components [see Smilga (2001)]. These solutions are the
same of the scalar field when spatial coordinates are set to zero
(rest frame).
Differently from the scalar field, we cannot just boost away these
solutions to get a general solution to Yang-Mills equations due to
gauge symmetry. But we can try to find a set of similar solutions
with the proviso of a gauge choice.
This kind of solutions will permit us to prove that a set of them
exists supporting a trivial infrared fixed point to build up a
quantum field theory.

Yang-Mills field
Exactly as in the case of the scalar field we assume the following
solution to our field equations
Aa
µ[j]=Aa
µ[0]+ d4x Dab
µν (x−x )jbν (x )+...
Also in this case this boils down to an expansion in powers of the
currents as already guessed in the ’80 [R. T. Cahill and C. D.
Roberts, Phys. Rev. D 32, 2419 (1985)].
This implies that the corresponding quantum theory, in a very strong
coupling limit, takes a Gaussian form and is trivial.
The crucial point, as already pointed out in the eighties [T. Goldman
and R. W. Haymaker, Phys. Rev. D 24, 724 (1981), T. Cahill and C.
D. Roberts (1985)], is the determination of the gluon propagator
in the low-energy limit.

Yang-Mills field
The question to ask is: Does a set of classical solutions exist for
Yang-Mills equations supporting a trivial infrared fixed point for the
corresponding quantum theory?
The answer is yes! These solutions are instantons in the form
Aa
µ = ηa
µφ with ηa
µ a set of constants and φ a scalar field.
By direct substitution into Yang-Mills equations one recovers the
equation for φ that is
∂µ∂µφ− 1
N2−1
1− 1
ξ
(ηa·∂)2φ+Ng2φ3=−jφ
being jφ=ηa
µjµa.
In the Landau gauge (Lorenz gauge classically) this equation is exactly
that of the scalar field given above and we can solve it accordingly.
So, a set of solutions of the Yang-Mills equations exists
supporting a trivial infrared fixed point. Our aim is to study the
theory in this case.

Yang-Mills-Green function
The instanton solutions given above permit us to write down
immediately the propagator for the Yang-Mills equations in the
Landau gauge for SU(N) being exactly the same given for the scalar
field:
Gluon propagator in the infrared limit
∆ab
µν (p)=δab ηµν −
pµpν
p2
∞
n=0
Bn
p2−m2
n+i
+O 1√
Ng
being
Bn=(2n+1)2 π3
4K3(i)
e
−(n+ 1
2 )π
1+e−(2n+1)π
and
mn=(2n+1) π
2K(i)
Ng2
2
1
4
Λ
This is the propagator of a massive free field theory due to the
Källen-Lehman representation.

Yang-Mills field
We can now take the form of the propagator given above, e.g. in the
Landau gauge, as
Dab
µν (p)=δab ηµν −
pµpν
p2
∞
n=0
Bn
p2−m2
n+i
+O 1√
Ng
to do a formulation of Yang-Mills theory in the infrared limit.
Then, the next step is to use the approximation that holds in a strong
coupling limit
Aa
µ[j]=Aa
µ[0]+ d4x Dab
µν (x−x )jbν (x )+O(j2)
and we note that, in this approximation, the ghost field just decouples
and becomes free and one finally has at the leading order
Z0[j]=N exp[ i
2
d4x d4x jaµ(x )Dab
µν (x −x )jbν (x )].
Yang-Mills theories exist that have an infrared trivial fixed point in
the limit of the coupling going to infinity and we expect the running
coupling to go to zero lowering energies. So, the leading order
propagator cannot confine.

Yang-Mills ﬁeld
A direct comparison of our results with lattice data gives the following
ﬁgure:
that is strikingly good below 1 GeV (ref. A. Cucchieri, T. Mendes, PoS
LAT2007, 297 (2007)).

Low-energy QCD QCD at infrared limit 14 / 36
QCD at infrared limit
When use is made of the trivial infrared fixed point, QCD action can
be written down quite easily.
Indeed, we will have for the gluon field
Sgf =1
2
d4x d4x jµa(x )Dab
µν (x −x )jνb(x )+O 1√
Ng
+O(j2
)
and for the quark fields
Sq= q d4x¯q(x) i /∂−mq−gγµ λa
2
d4x Dab
µν (x−x )jνb(x )
−g2γµ λa
2
d4x Dab
µν (x−x ) q ¯q (x ) λb
2
γν q (x )+O 1√
Ng
+O(j2
) q(x)
We recognize here an explicit Yukawa interaction and a
Nambu-Jona-Lasinio non-local term. Already at this stage we are able
to recognize that NJL is the proper low-energy limit for QCD.

Now we operate the Smilga’s choice ηa
µηb
ν = δab(ηµν − pµpν/p2) for
the Landau gauge.
We are left with the infrared limit for QCD using conservation of
currents
Sgf =1
2
d4x d4x ja
µ(x )∆(x −x )jµa(x )+O 1√
Ng
+O(j2
)
Sq= q d4x¯q(x) i /∂−mq−gγµ λa
2
d4x ∆(x−x )ja
µ(x )
−g2γµ λa
2
d4x ∆(x−x ) q ¯q (x ) λa
2
γµq (x )+O 1√
Ng
+O(j2
) q(x)
We want to give explicitly the contributions from gluon resonances.
In order to do this, we introduce the bosonic currents ja
µ(x) = ηa
µj(x)
with the current j(x) that of the gluonic excitations.

Using the relation ηa
µηµa=3(N2
c −1) we get in the end
Sgf =3
2
(N2
c −1) d4x d4x j(x )∆(x −x )j(x )+O 1√
Ng
+O(j2
)
Sq= q d4x¯q(x) i /∂−mq−gηa
µγµ λa
2
d4x ∆(x−x )j(x )
−g2γµ λa
2
d4x ∆(x−x ) q ¯q (x ) λa
2
γµq (x )+O 1√
Ng
+O(j2
) q(x)
Now, we recognize that the gluon propagator is just a sum of Yukawa
propagators weighted by exponential damping terms. So, we
introduce the σ ﬁeld and truncate at the lowest excitation.
This means the we can write the bosonic currents contribution as
coming from a single bosonic ﬁeld written down as
σ(x)=
√
3(N2
c −1)/B0 d4x ∆(x−x )j(x ).

So, low-energy QCD yields a non-local NJL model as given in [M.
Frasca, Phys. Rev. C 84, 055208 (2011)]
Sσ= d4x[1
2
(∂σ)2−1
2
m2
0σ2
]
Sq= q d4x¯q(x) i /∂−mq−g
B0
3(N2
c −1)
ηa
µγµ λa
2
σ(x)
−g2γµ λa
2
d4x ∆(x−x ) q ¯q (x ) λa
2
γµq (x )+O 1√
Ng
+O(j3
) q(x)
Now, we obtain directly from QCD (2G(0) = G is the standard NJL
coupling)
G(p)=−1
2
g2 ∞
n=0
Bn
p2−(2n+1)2(π/2K(i))2σ+i
=G
2
C(p)
with C(0) = 1 ﬁxing in this way the value of G using the gluon
propagator. This yields an almost perfect agreement with the case of
an instanton liquid (see ref. in this page).

Nambu-Jona-Lasinio model from QCD at low-energies
SNJL =
q
d4
x¯q(x) i /∂ − mq q(x)
+ d4
x G(x − x )
q q
¯q(x)
λa
2
γµ
¯q (x )
λa
2
γµq (x )q(x)
+ d4
x
1
2
(∂σ)2
−
1
2
m2
0σ2
.
We need to:
1 Turn on the electromagnetic interaction.
2 Fierz transform it.
3 Bosonize it.

ρ meson condensation Emerging of ρ condensation 19 / 36
A rather elementary argument [Chernodub, Phys. Rev. D 82, 085011
(2010)] starts by simply observing that a relativistic particle in a
magnetic ﬁeld has
ε2
n,sz
(pz) = p2
z + (2n − 2sz + 1)eBext + m2
.
n is a non-negative integer and sz the spin projection.
For the π (spin 0 mass about 139 MeV) and ρ meson (spin 1 and
mass about 770 MeV) happens that
m2
π± (Bext) = m2
π± + eBext , m2
ρ± (Bext) = m2
ρ± − eBext .
the π becomes heavier while the ρ becomes lighter possibly hindering
the decay ρ → ππ.
But, more important, from this simple argument a critical magnetic
ﬁeld seems to exist that, once overcame, the ρ change into a
tachyonic mode: A condensate forms
Bc =
m2
ρ
e
≈ 1016
Tesla .

A more substantial argument [Chernodub, ibid.] is based on the
vector meson Lagrangian [Djukanovic et al. Phys. Rev. Lett. 95,
012001 (2005)]
L = −
1
4
FµνFµν
−
1
2
(Dµρν − Dνρµ)†
(Dµ
ρν
− Dν
ρµ
) + m2
ρ ρ†
µρµ
−
1
4
ρ(0)
µν ρ(0)µν
+
m2
ρ
2
ρ(0)
µ ρ(0)µ
+
e
2gs
Fµν
ρ(0)
µν .
For a ρ meson the gyromagnetic ratio is anomalously large being 2.
This Lagrangian describes a non-minimal coupling between vector
mesons and electromagnetic ﬁeld with gs ≈ 5.88.
High gs and so high g is the reason for condensation. If we consider
the homogeneous case for a moment we get
0(ρ)=1
2
B2
ext +2(m2
ρ−eBext ) |ρ|2+2g2
s |ρ|4 .
being ρ− ≡
√
2ρ, ρ+ = 0 and this permits to draw an identical
conclusion as in the aforementioned simplest argument.

Condensation of charged particles implies a superconductor state.
One needs to show that NJL model should also confirm ρ
condensation. A first computation was given in Chernodub, Phys.
Rev. Lett. 106, 142003 (2011).
A first computation on lattice confirmed ρ condensation [Braguta et
al., Phys. Lett. B718 (2012) 667].
The idea of spin-1 particles that condensate apparently clashes with
Vafa-Witten and Elitzur theorems [Hidaka and Yamamoto, Phys. Rev.
D 87, 094502 (2013)]. Hidaka and Yamamoto also performed lattice
computations disclaiming ρ condensation. Chernodub responses are
Phys.Rev.D 86, 107703 (2012), arXiv:1309.4071 [hep-ph].
NJL model evades Vafa-Witten theorem and condensation could be
seen there. But we just proved that a non-local NJL model is the
low-energy limit of QCD so, condensation in NJL is condensation in
QCD.

ρ meson condensation NJL and magnetic field 22 / 36
Turning on the electromagnetic field
Nambu-Jona-Lasinio model in an electromagnetic field
Introduce the gauge covariant derivative
Dµ = ∂µ − ieqAµ
and then,
SNJL =
q
d4
x¯q(x) i /∂ + eq /A − mq q(x)
+ d4
x G(x − x )
q q
¯q(x)
λa
2
γµ
¯q (x )
λa
2
γµq (x )q(x)
+ d4
x
1
2
(∂σ)2
−
1
2
m2
0σ2

Fierz transforming
Nambu-Jona-Lasinio model after Fierz transformation
We take the local and two-ﬂavor approximations that are enough for our
aims. So,
SNJL =
q=u,d
d4
x¯q(x) i /∂ + eq /A − mq q(x)+ d4
x
1
2
(∂σ)2
−
1
2
m2
0σ2
+
G
2
q=u,d
d4
x (¯q(x)q(x))2
+ (¯q(x)iγ5
τq(x))2
−
G
4
q=u,d
d4
x (¯q(x)γµ
τaq(x))2
+ (¯q(x)γ5
γµ
τaq(x))2
from which we get GS = G and GV = GS /2 = G/2.

Bosonization
Nambu-Jona-Lasinio model after bosonization
Introducing bosonic degrees of freedom
SB = d4
x

1
2
(∂σ)2
−
1
2
m2
0σ2
+
q=u,d
¯q i ˆD − mq q
−
1
2G
σ2
+ π · π +
1
G
(Va · Va + Aa · Aa)
being i ˆD = i /D − mq + /ˆV + γ5 /ˆA − (σ + iγ5π · τ). After integration of the
fermionic degrees of freedom yields
SB = −iTrln(i ˆD − mq) + d4x[− 1
2G (σ2+π·π)+ 1
G
(Va·Va+Aa·Aa)].
The fermionic determinant is singular and needs regularization.

One-loop potential
One-loop potential is easily obtained by
V1L = −iTrlnS−1
q (x, x )
The fermionic propagator is obtained through Ritus-Leung-Wang
technique
Sq(x,y)= ∞
k=0
dpt dpy dpz
(2π)4 EP (x)Λk
i
/P−mf
¯EP (y).
This yields (βk = 2 − δk0 degeneracy of Landau levels)
V1L = −NcT
q=u,d
|eqB|
2π
∞
k=0
βk
∞
n=−∞
dpz
2π
×
ln((2n + 1)2
π2
T2
+ p2
z + 2k|eqB| + v2
).

Moving to finite temperature and magnetic field
Moving rules
From the fermionic determinant and the one-loop potential we learn the
rules to move from a standard NJL model to finite temperature and
magnetic field:
1 Change euclidean integrals to
d4pE
(2π)4
→ T
q
|eqB|
2π
k
βk
∞
n=−∞
dpz
2π
2 Change p0 → (2n + 1)πT
3 Change the third component of momenta to
p2
z → p2
z + 2k|eqB|
4 Finally, regularize.

ρ meson condensation Mass of the ρ in NJL model 27 / 36
ρ mass and NJL model
ρ mass in NJL model comes out as the pole of the corresponding
propagator [Ebert, Reinhardt and Volkov, Prog. Part. Nucl. Phys. 33,
1 (1994); Bernard, Meissner and Osipov, Phys. Lett. B 324, 201
(1994)].
This result is well-known in literature and is
m2
ρ =
3
8GV NcJ2(0)
being in our case GV = G/2.
J2(0) is the following integral
J2(0) = −iNf
d4p
(2π)4
1
(p2 − v2 + i )2
in need of regularization being divergent.
Various techniques of regularization (e.g. Pauli-Villars) provide the
value
J2(0) =
Nf
16π2
ln
Λ2
v2
.

ρ mass and NJL model
One can yield a numerical evaluation of the ρ mass from the above
formula starting from the values in [Frasca, AIP Conf. Proc. 1492,
177 (2012); Nucl. Phys. Proc. Suppl. 234, 329 (2013)].
Λ si NJL cut-off while v is the effective quark mass. At this level of
approximation one gets 670 MeV that is rather good. Further
corrections can be added [Klevansky and Lemmer, hep-ph/9707206
(unpublished)].
Our aim is to generalize the equation for the mass to finite
temperature and magnetic field. This will be accomplished through
the moving rules introduced above.

ρ mass at finite T and B
We need to move the integral J2(0) to the case of finite temperature
and magnetic field.
This is immediately done through the moving rules introduced in the
NJL model yielding
NJL integral for the ρ mass
J2(B, T) = T
q
|eqB|
2π
k
βk
∞
n=−∞
dpz
2π
×
1
[(2n + 1)2π2T2 + p2
z + 2k|eqB| + v2]2
We are interested in the limit of decreasing T (T → 0).

ρ meson condensation Quantum phase transition in QCD 30 / 36
Strategy
A quantum phase transition occurs when some other parameter is
varied rather than the temperature (taken to be 0). In our case this
parameter is the magnetic field.
We have to show that the integral J2 changes sign crossing 0, at
T=0, for a critical magnetic field.
We observe that such a zero in J2 is just signaling that non-local
effects come into play at the critical magnetic field.

Computation
Firstly, we perform the Matsubara sum that gives
∞
n=−∞
1
[(2n+1)2π2T2+p2
z +2k|eqB|+v2]2 = 1
4T
cosh
ωk (pz )
2T
sinh
ωk (pz )
2T
−
ωk (pz )
2T
ω3
k
(pz ) cosh2 ωk (pz )
2T
.
At small T yields
∞
n=−∞
1
[(2n+1)2π2T2+p2
z +2k|eqB|+v2]2 ≈ 1
4T
1−
2ωk (pz )
T
e
−
ωk (pz )
T
ω3
k
(pz )
.
being ωk (pz )=
√
p2
z +2k|eqB|+v2.

Computation
Integrating on pz one gets
J2(B, T) ≈
q
|eqB|
16π2
k
βk




1
2k|eqB| + v2
−
2
T
F
√
2k|eqB|+v2
T
2k|eqB| + v2




being F(a) = π − 2a K0(a) + π
2 (K0(a)L1(a) + K1(a)L0(a)) where
Kn are Macdonald functions and Ln modiﬁed Struve functions.
We are in need to regularize the harmonic series and this is done in a
standard way as
∞
k=0
βk
1
2k|eqB| + v2
= −
1
v2
−
1
|eqB|
ψ
v2
2|eqB|
being ψ(a) the polygamma function.

Recovering the ρ mass
ρ mass at B = 0 can ber ecovered if we put an ultraviolet cut-oﬀ on
Landau levels, N, and expand the polygamma function [Ruggieri,
Tachibana and Greco,arXiv:1305.0137 [hep-ph]].
The harmonic sum takes the form
N
k=0
βk
1
2k|eqB| + v2
≈
1
|eqB|
ln
Λ2
v2
+
1
3
|eqB|
v4
+ . . . .
Notice that the corrections due to the temperature goes to 0
exponentially for T → 0.
So, we have recovered the J2 integral for B ≈ 0
J2(B, 0)
B→0
=
1
8π2
ln
Λ2
v2
+
q=u,d
1
3
|eqB|2
v4

High magnetic field limit
We can collect all the results given above to get
J2(B,T)≈ 1
16π2 q





−
|eqB|
v2 −ψ v2
2|eqB|
−
2|eqB|
T
∞
k=0
F


√
2k|eqB|+v2
T


√
2k|eqB|+v2





.
In the limit T → 0, the contributions due to the temperature are
exponentially damped and one should ask if the remaining function
can change sign and for what value of the magnetic field this happens.
Indeed, this is so yielding the conclusion
Critical magnetic field
|eBc|
v2
= 0.9787896 . . .
This shows that in the NJL at increasing magnetic field, with the
temperature going to zero, there is a quantum phase transition.

Conclusions 35 / 36
Conclusions
We derived the Nambu-Jona-Lasinio model from QCD with all the
parameters properly fixed.
The NJL model has been extended to the case of interaction with a
magnetic field.
Computations have been performed providing the dependence of the
ρ meson mass from the magnetic field and the temperature.
It has been shown that a quantum phase transition happens
increasing the magnetic field to a critical value for the ρ particle.
This yields strong support to the hypothesis by Chernodub of a
superconductive state for such mesons.
Enlightening discussions with Marco Ruggieri are gratefully ac-
knowledged.

Conclusions 36 / 36
Thanks a lot for your attention!

Talk given at the Workshop in Catania University

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Talk given at the Workshop in Catania University