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 theoryYang-Mills theoryYang-Mills Green functionQCD in the infrared limit

ρ meson condensation

Emerging of ρ condensationNJL and magnetic fieldMass of the ρ in NJL modelQuantum 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( 2λ)

14 sn(p·x+θ,i) p2=µ2

√λ2

being sn an elliptic Jacobi function and µ and θ two constants. Thissolution holds provided the given dispersion relation holds andrepresents a free massive solution notwithstanding we started from amassless 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 ouraim 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 transformedcounterpart 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−m2n+iε

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) π2K(i) (λ2 )

14 µ.


Scalar field

We can formulate a quantum field theory for the scalar field startingfrom the generating functional

Z [j]=∫

[dφ] exp[i∫d4x( 1

2(∂φ)2−λ

4φ4+jφ)].

We can rescale the space-time variable as x →√λx and rewrite the

functional as

Z [j]=∫

[dφ] exp[i 1λ

∫d4x( 1

2(∂φ)2− 1

4φ4+ 1

λjφ)].

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+φ30=0

This is completely consistent with our preceding formulation [M.Frasca, Phys. Rev. D 73, 027701 (2006)] but now all is fullycovariant.


Scalar field

Using the current expansion that holds at strong coupling λ→∞ onehas

φ[j]=φ0+∫d4x ′∆(x−x ′)j(x ′)+...

it is not difficult to write the generating functional at the leadingorder in a Gaussian form

Z0[j]=exp[ i2

∫d4x′d4x′′j(x′)∆(x′−x′′)j(x′′)].

This conclusion is really important: It says that there exists an infiniteset of scalar field theories in d=3+1 that are trivial in the infraredlimit 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) π2K(i) (λ2 )

14 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∂µAaν−(1− 1ξ )∂ν (∂µAaµ)+gf abcAbµ(∂µAcν−∂νA

cµ)+gf abc∂µ(AbµAcν )+g2f abc f cdeAbµAdµAeν=−jaν .

For the homogeneous equations, we want to study it in the formallimit g →∞. We note that a class of exact solutions exists if we takethe potential Aa

µ just depending on time, after a proper selection ofthe components [see Smilga (2001)]. These solutions are thesame of the scalar field when spatial coordinates are set to zero(rest frame).

Differently from the scalar field, we cannot just boost away thesesolutions to get a general solution to Yang-Mills equations due togauge symmetry. But we can try to find a set of similar solutionswith the proviso of a gauge choice.

This kind of solutions will permit us to prove that a set of themexists supporting a trivial infrared fixed point to build up aquantum field theory.


Yang-Mills field

Exactly as in the case of the scalar field we assume the followingsolution to our field equations

Aaµ[j]=Aaµ[0]+∫d4x′Dab

µν (x−x′)jbν (x′)+...

Also in this case this boils down to an expansion in powers of thecurrents 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 strongcoupling limit, takes a Gaussian form and is trivial.

The crucial point, as already pointed out in the eighties [T. Goldmanand R. W. Haymaker, Phys. Rev. D 24, 724 (1981), T. Cahill and C.D. Roberts (1985)], is the determination of the gluon propagatorin the low-energy limit.


Yang-Mills field

The question to ask is: Does a set of classical solutions exist forYang-Mills equations supporting a trivial infrared fixed point for thecorresponding quantum theory?

The answer is yes! These solutions are instantons in the formAaµ = ηaµφ with ηaµ a set of constants and φ a scalar field.

By direct substitution into Yang-Mills equations one recovers theequation for φ that is

∂µ∂µφ− 1N2−1

(1− 1

ξ

)(ηa·∂)2φ+Ng2φ3=−jφ

being jφ=ηaµjµa.

In the Landau gauge (Lorenz gauge classically) this equation is exactlythat of the scalar field given above and we can solve it accordingly.

So, a set of solutions of the Yang-Mills equations existssupporting a trivial infrared fixed point. Our aim is to study thetheory in this case.


Yang-Mills-Green function

The instanton solutions given above permit us to write downimmediately the propagator for the Yang-Mills equations in theLandau gauge for SU(N) being exactly the same given for the scalarfield:

Gluon propagator in the infrared limit

∆abµν(p)=δab

(ηµν−

pµpν

p2

)∑∞n=0

Bnp2−m2

n+iε+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) π2K(i)

(Ng2

2

) 14

Λ

This is the propagator of a massive free field theory due to theKallen-Lehman representation.


Yang-Mills field

We can now take the form of the propagator given above, e.g. in theLandau gauge, as

Dabµν(p)=δab

(ηµν−

pµpν

p2

)∑∞n=0

Bnp2−m2

n+iε+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 strongcoupling limit

Aaµ[j]=Aa

µ[0]+∫d4x ′Dab

µν(x−x ′)jbν(x ′)+O(j2)

and we note that, in this approximation, the ghost field just decouplesand becomes free and one finally has at the leading order

Z0[j]=N exp[ i2

∫d4x ′d4x ′′jaµ(x ′)Dab

µν(x ′−x ′′)jbν(x ′′)].

Yang-Mills theories exist that have an infrared trivial fixed point inthe limit of the coupling going to infinity and we expect the runningcoupling to go to zero lowering energies. So, the leading orderpropagator cannot confine.


Yang-Mills field

A direct comparison of our results with lattice data gives the followingfigure:

that is strikingly good below 1 GeV (ref. A. Cucchieri, T. Mendes, PoSLAT2007, 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 canbe written down quite easily.

Indeed, we will have for the gluon field

Sgf = 12

∫d4x ′d4x ′′

[jµa(x ′)Dab

µν(x ′−x ′′)jνb(x ′′)+O(

1√Ng

)+O(j2)

]and for the quark fields

Sq=∑

q

∫d4xq(x)

[i /∂−mq−gγµ λ

a

2

∫d4x ′Dab

µν(x−x ′)jνb(x ′)

−g2γµ λa

2

∫d4x ′Dab

µν(x−x ′)∑

q′ q′(x ′)λ

b

2γνq′(x ′)+O

(1√Ng

)+O(j2)

]q(x)

We recognize here an explicit Yukawa interaction and aNambu-Jona-Lasinio non-local term. Already at this stage we are ableto recognize that NJL is the proper low-energy limit for QCD.


QCD in the infrared limit

Now we operate the Smilga’s choice ηaµηbν = δab(ηµν − pµpν/p

2) forthe Landau gauge.

We are left with the infrared limit for QCD using conservation ofcurrents

Sgf = 12

∫d4x ′d4x ′′

[jaµ(x ′)∆(x ′−x ′′)jµa(x ′′)+O

(1√Ng

)+O(j2)


Sq=∑

q

∫d4xq(x)

[i /∂−mq−gγµ λ

a

2

∫d4x ′∆(x−x ′)jaµ(x ′)

−g2γµ λa

2

∫d4x ′∆(x−x ′)

∑q′ q′(x ′)λ

a

2γµq′(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 jaµ(x) = ηaµj(x)with the current j(x) that of the gluonic excitations.


QCD in the infrared limit

Using the relation ηaµηµa=3(N2

c−1) we get in the end

Sgf = 32

(N2c−1)

∫d4x ′d4x ′′

[j(x ′)∆(x ′−x ′′)j(x ′′)+O

(1√Ng

)+O(j2)


Sq=∑

q

∫d4xq(x)

[i /∂−mq−gηaµγµ λ

a

2

∫d4x ′∆(x−x ′)j(x ′)

−g2γµ λa

2

∫d4x ′∆(x−x ′)

∑q′ q′(x ′)λ

a

2γµq′(x ′)+O

(1√Ng

)+O(j2)

]q(x)

Now, we recognize that the gluon propagator is just a sum of Yukawapropagators weighted by exponential damping terms. So, weintroduce the σ field and truncate at the lowest excitation.

This means the we can write the bosonic currents contribution ascoming from a single bosonic field written down asσ(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

2m2

0σ2]

and for the quark fields

Sq=∑

q

∫d4xq(x)

[i /∂−mq−g

√B0

3(N2c−1)

ηaµγµ λa

2σ(x)

−g2γµ λa

2

∫d4x ′∆(x−x ′)

∑q′ q′(x ′)λ

a

2γµq′(x ′)+O

(1√Ng

)+O(j3)

]q(x)

Now, we obtain directly from QCD (2G(0) = G is the standard NJLcoupling)

G(p)=− 12g2∑∞

n=0Bn

p2−(2n+1)2(π/2K(i))2σ+iε=G

2C(p)

with C(0) = 1 fixing in this way the value of G using the gluonpropagator. This yields an almost perfect agreement with the case ofan instanton liquid (see ref. in this page).



Nambu-Jona-Lasinio model from QCD at low-energies

SNJL =∑q

∫d4xq(x)

[i /∂ −mq

]q(x)

+

∫d4x ′G(x − x ′)

∑q

∑q′

q(x)λa

2γµq′(x ′)

λa

2γµq

′(x ′)q(x)

+

∫d4x

[1

2(∂σ)2 − 1

2m2

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

Emerging of ρ condensation

A rather elementary argument [Chernodub, Phys. Rev. D 82, 085011(2010)] starts by simply observing that a relativistic particle in amagnetic field has

ε2n,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 andmass about 770 MeV) happens that

m2π±(Bext) = m2

π± + eBext , m2ρ±(Bext) = m2

ρ± − eBext .

the π becomes heavier while the ρ becomes lighter possibly hinderingthe decay ρ→ ππ.But, more important, from this simple argument a critical magneticfield seems to exist that, once overcame, the ρ change into atachyonic mode: A condensate forms

Bc =m2ρ

e≈ 1016 Tesla .



A more substantial argument [Chernodub, ibid.] is based on thevector meson Lagrangian [Djukanovic et al. Phys. Rev. Lett. 95,012001 (2005)]

L = −1

4FµνF

µν − 1

2(Dµρν − Dνρµ)†(Dµρν − Dνρµ) + m2

ρ ρ†µρ

µ

−1

4ρ(0)µν ρ

(0)µν+m2ρ

2ρ(0)µ ρ(0)µ +

e

2gsFµνρ(0)

µν .

For a ρ meson the gyromagnetic ratio is anomalously large being 2.This Lagrangian describes a non-minimal coupling between vectormesons and electromagnetic field with gs ≈ 5.88.High gs and so high g is the reason for condensation. If we considerthe homogeneous case for a moment we get

ε0(ρ)= 12B2ext+2(m2

ρ−eBext) |ρ|2+2g2s |ρ|4 .

being ρ− ≡√

2ρ, ρ+ = 0 and this permits to draw an identicalconclusion as in the aforementioned simplest argument.



Condensation of charged particles implies a superconductor state.One needs to show that NJL model should also confirm ρcondensation. A first computation was given in Chernodub, Phys.Rev. Lett. 106, 142003 (2011).

A first computation on lattice confirmed ρ condensation [Braguta etal., Phys. Lett. B718 (2012) 667].

The idea of spin-1 particles that condensate apparently clashes withVafa-Witten and Elitzur theorems [Hidaka and Yamamoto, Phys. Rev.D 87, 094502 (2013)]. Hidaka and Yamamoto also performed latticecomputations disclaiming ρ condensation. Chernodub responses arePhys.Rev.D 86, 107703 (2012), arXiv:1309.4071 [hep-ph].

NJL model evades Vafa-Witten theorem and condensation could beseen there. But we just proved that a non-local NJL model is thelow-energy limit of QCD so, condensation in NJL is condensation inQCD.

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

∫d4xq(x)

[i /∂ + eq /A−mq

]q(x)

+

∫d4x ′G(x − x ′)

∑q

∑q′

q(x)λa

2γµq′(x ′)

λa

2γµq

′(x ′)q(x)

+

∫d4x

[1

2(∂σ)2 − 1

2m2

0σ2

]


Fierz transforming

Nambu-Jona-Lasinio model after Fierz transformation

We take the local and two-flavor approximations that are enough for ouraims. So,

SNJL =∑q=u,d

∫d4xq(x)

[i /∂ + eq /A−mq

]q(x)+

∫d4x

[1

2(∂σ)2 − 1

2m2

0σ2

]

+G

2

∑q=u,d

∫d4x

[(q(x)q(x))2 + (q(x)iγ5τq(x))2

]−G

4

∑q=u,d

∫d4x

[(q(x)γµτaq(x))2 + (q(x)γ5γµτaq(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 =

∫d4x

1

2(∂σ)2 − 1

2m2

0σ2 +

∑q=u,d

q(iD −mq

)q

− 1

2G

(σ2 + π · π

)+

1

G(Va · Va + Aa · Aa)

]being iD = i /D −mq + /V + γ5 /A− (σ + iγ5π · τ ). After integration of thefermionic degrees of freedom yields

SB = −iTrln(iD −mq) +∫d4x[− 1

2G (σ2+π·π)+ 1G

(Va·Va+Aa·Aa)].

The fermionic determinant is singular and needs regularization.


One-loop potential

One-loop potential is easily obtained by

V1L = −iTrlnS−1q (x , x ′)

The fermionic propagator is obtained through Ritus-Leung-Wangtechnique

Sq(x ,y)=∑∞

k=0

∫ dpt dpy dpz

(2π)4 EP(x)Λki

/P−mfEP(y).

This yields (βk = 2− δk0 degeneracy of Landau levels)

V1L = −NcT∑q=u,d

|eqB|2π

∞∑k=0

βk

∞∑n=−∞

∫dpz2π×

ln((2n + 1)2π2T 2 + p2z + 2k |eqB|+ v2).


Moving to finite temperature and magnetic field

Moving rules

From the fermionic determinant and the one-loop potential we learn therules to move from a standard NJL model to finite temperature andmagnetic field:

1 Change euclidean integrals to∫d4pE(2π)4

→ T∑q

|eqB|2π

∑k

βk

∞∑n=−∞

∫dpz2π

2 Change p0 → (2n + 1)πT

3 Change the third component of momenta to

p2z → 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 correspondingpropagator [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

m2ρ =

3

8GVNcJ2(0)

being in our case GV = G/2.J2(0) is the following integral

J2(0) = −iNf

∫d4p

(2π)4

1

(p2 − v2 + iε)2

in need of regularization being divergent.Various techniques of regularization (e.g. Pauli-Villars) provide thevalue

J2(0) =Nf

16π2ln

(Λ2

v2

).


ρ mass and NJL model

One can yield a numerical evaluation of the ρ mass from the aboveformula 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 ofapproximation one gets 670 MeV that is rather good. Furthercorrections can be added [Klevansky and Lemmer, hep-ph/9707206(unpublished)].

Our aim is to generalize the equation for the mass to finitetemperature and magnetic field. This will be accomplished throughthe moving rules introduced above.


ρ mass at finite T and B

We need to move the integral J2(0) to the case of finite temperatureand magnetic field.

This is immediately done through the moving rules introduced in theNJL model yielding

NJL integral for the ρ mass

J2(B,T ) = T∑q

|eqB|2π

∑k

βk

∞∑n=−∞

∫dpz2π×

1

[(2n + 1)2π2T 2 + p2z + 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 isvaried rather than the temperature (taken to be 0). In our case thisparameter is the magnetic field.

We have to show that the integral J2 changes sign crossing 0, atT=0, for a critical magnetic field.

We observe that such a zero in J2 is just signaling that non-localeffects come into play at the critical magnetic field.


Computation

Firstly, we perform the Matsubara sum that gives

∑∞n=−∞

1

[(2n+1)2π2T2+p2z +2k|eqB|+v2]2

= 14T

cosh

(ωk (pz )

2T

)sinh

(ωk (pz )

2T

)−ωk (pz )

2T

ω3k

(pz ) cosh2(ωk (pz )

2T

) .

At small T yields

∑∞n=−∞

1

[(2n+1)2π2T2+p2z +2k|eqB|+v2]2

≈ 14T

1− 2ωk (pz )T

e−(ωk (pz )

T

)ω3k

(pz ).

being ωk (pz )=√

p2z+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))]

whereKn are Macdonald functions and Ln modified Struve functions.

We are in need to regularize the harmonic series and this is done in astandard way as

∞∑k=0

βk1

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 onLandau 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

βk1

2k|eqB|+ v2≈ 1

|eqB|ln

(Λ2

v2

)+

1

3

|eqB|v4

+ . . . .

Notice that the corrections due to the temperature goes to 0exponentially for T → 0.

So, we have recovered the J2 integral for B ≈ 0

J2(B, 0)B→0

=1

8π2ln

(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 )≈ 116π2

∑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 areexponentially damped and one should ask if the remaining functioncan 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 thetemperature going to zero, there is a quantum phase transition.

Conclusions 35 / 36

Conclusions

We derived the Nambu-Jona-Lasinio model from QCD with all theparameters properly fixed.

The NJL model has been extended to the case of interaction with amagnetic field.

Computations have been performed providing the dependence of theρ meson mass from the magnetic field and the temperature.

It has been shown that a quantum phase transition happensincreasing the magnetic field to a critical value for the ρ particle.

This yields strong support to the hypothesis by Chernodub of asuperconductive 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

Technology

Transcript of Talk given at the Workshop in Catania University