-
For a general
Nf -flavor Nambu-Jona-Lasinio model, the Lagrangian density is given by [39]LNJL=ˉψ(iγμ∂μ−ˆmc)ψ+LS+LKMT,LS=GsN2f−1∑α=0[(ˉψλαψ)2+(ˉψiγ5λαψ)2],LKMT=−K[detˉψ(1+γ5)ψ+detˉψ(1−γ5)ψ],
(1) where
λα (α=0,1,⋯,N2f−1) is theNf -flavor Gell-Mann matrix withλ0=√2/Nf , andˆmc=diag(mu,md,ms,⋯) is the current quark mass matrix. In the special casemu=md=ms=⋯=0 andK=0 ,LNJL is invariant under the group transformationSUC(Nc)⊗SUV(Nf)⊗SUA(Nf)⊗ UB(1)⊗UA(1) .LKMT is the so-called Kobayashi-Maskawa-t'Hooft term withK<0 , designed to break theUA(1) symmetry. For the three-flavor case (Nf=3 ),LKMT contains six-fermion interactions and can efficiently describe the mass splitting betweenη andη′ . In this work, we consider the two-flavor case, whereLKMT contains only four-fermion interactions, like the mesonic interaction termLS . The Lagrangian density of the general two-flavor NJL model is given byLNJL=ˉψ(iγμ∂μ−m0)ψ+G[(ˉψψ)2+(ˉψiγ5τψ)2]+G′[(ˉψτψ)2+(ˉψiγ5ψ)2],
(2) where
G=Gs−K,G′=Gs+K , and we assumemu= md=m0 . Since the masses of scalar-isovector and pseudoscalcar-isoscalar mesons in the two-flavor case are much larger than the sigma meson and pions, we consider the maximal axial symmetry breaking case|K|=Gs , which leads to the minimal NJL modelLNJL=ˉψ(iγμ∂μ−m0)ψ+G[(ˉψψ)2+(ˉψiγ5τψ)2].
(3) In this work, we study this minimum NJL model for the sake of simplicity.
In the functional path integral formalism, the partition function of the NJL model can be written as
ZNJL=∫[dψ][dˉψ]exp{i∫d4xLNJL}.
(4) Introducing two auxiliary fields
σ andπ , which satisfy equations of motionσ=−2Gˉψψ,π=−2Gˉψiγ5τψ , and applying the Hubbard-Stratonovich transformation, we obtainZNJL=∫[dψ][dˉψ][dσ][dπ]exp{iS[ψ,ˉψ,σ,π]},
(5) where the action reads
S[ψ,ˉψ,σ,π]=−∫d4xσ2+π24G+∫d4x∫d4x′ˉψ(x)×G−1(x,x′)ψ(x′),G−1(x,x′)=[iγμ∂μ−m0−(σ+iγ5τ⋅π)]δ(x−x′).
(6) Subsequently, we integrate out the quark field and obtain
ZNJL=∫[dσ][dπ]exp{iSeff[σ,π]},Seff[σ,π]=−14G∫d4x(σ2+π2)−iTrlnG−1(x,x′).
(7) The partition function cannot be evaluated precisely. We assume that the sigma field acquires a non-vanishing expectation value
⟨σ(x)⟩=υ and set⟨(x)⟩=0 , which characterizes the dynamical chiral symmetry breaking (DCSB). Then, the auxiliary fields can be expanded around their expectation values. After performing the field shifts,σ(x)→υ+σ(x) andπ(x)→0+π(x) , we expand the effective actionSeff[σ,π] in powers of the fluctuationsσ(x) andπ(x) . We haveSeff[σ,π]=S(0)eff+S(1)eff[σ,π]+S(2)eff[σ,π]+⋯.
(8) The mean-field part
S(0)eff=Seff[υ,0] can be evaluated asS(0)effV4=υ24G−2NcNf∫d3k(2π)3Ek,
(9) where
Ek=√k2+M2 with the effective quark massM=m0+υ . Since the NJL model is not renormalizable, we employ a hard cutoffΛ to regularize the integral over the quark momentumk (|k|<Λ ). The condensateυ should be determined by minimizingS(0)eff , i.e.,∂S(0)eff/∂υ=0 , which gives rise to the gap equationM−m0=4GNcNfM∫d3k(2π)31Ek.
(10) In the chiral limit
m0=0 , we find that ifG>π2/(NcNfΛ2) [39,40], the sigma field acquires a nonvanishing expectation valueυ≠0 , and hence the DCSB occurs.The gap Eq. (10) ensures that the linear term
S(1)eff[σ,π] vanishes. The mesons in the NJL model are regarded as collective excitations, which are characterized by the Gaussian fluctuation termS(2)eff[σ,π] . Using the derivative expansionTrln(1−GΣ)=−∞∑n=11nTr(GΣ)n,
(11) with
G=(γμKμ−M)−1 being the mean-field quark propagator andΣ=σ+iγ5τ⋅π , we obtainS(2)eff[σ,π]=−12∫d4Q(2π)4[D−1σ(Q)σ(Q)σ(−Q)+D−1π(Q)π(Q)⋅π(−Q)],D−1σ,π(Q)=12G−Πσ,π(Q).
(12) Here the polarization functions
Πσ,π(Q) are given byΠσ,π(Q)=4iNcNf∫d4K(2π)41K2−M2−2iNcNf(Q2−ε2σ,π)I(Q2),I(Q2)=∫d4K(2π)41[(K+Q/2)2−M2][(K−Q/2)2−M2],
(13) with
εσ=2M andεπ=0 .The masses of the mesons are determined by the pole of their propagators, i.e.,
D−1σ,π(Q2=m2σ,π)=0 . We obtainm2σ,π=−m0M14iGNcNfI(m2σ,π)+ε2σ,π.
(14) The function
I(Q2) changes very slowly withQ2 . Therefore, we can approximateI(m2σ,π)≈I(0) . The meson masses are given bym2π≈−m0M14iGNcNfI(0), m2σ≈m2π+4M2.
(15) Near the poles, the meson propagators can be efficiently approximated as
Dσ,π(Q)≃g2σqq,πqqQ2−m2σ,π,
(16) where the meson-quark couplings are given by
g−2σqq,πqq≡∂Πσ,π∂Q2|Q2=m2σ,π≈−2iNcNfI(0).
(17) To determine the model parameters, i.e., the current quark mass
m0 , the coupling constant G, and the cutoffΛ , we need to derive the pion decay constantfπ in the NJL model. This can be obtained by calculating the matrix element of the vacuum to one-pion axial-vector current transition. We haveiQμfπδij=−Tr∫d4K(2π)4[iγμγ5τi2iG(K+Q/2)×igπqqγ5τjiG(K−Q/2)]=2NcNfgπqqMQμI(Q2)δij.
(18) Using Eq. (17), we obtain
f2π≈−2iNcNfM2I(0).
(19) Applying the result
M=−2G⟨ˉψψ⟩0+m0 , we recover the Gell-Mann-Oakes-Renner relationm2πf2π≈−m0⟨ˉψψ⟩0.
(20) The model parameters can be fixed by matching the pion mass
mπ , the pion decay constantfπ , and the chiral condensate⟨ˉψψ⟩0 . For the physical case, we choosem0=5 MeV,G=4.93GeV−2 , andΛ=653MeV , which yieldsmπ=134MeV ,fπ=93MeV , and⟨ˉuu⟩0= −(250MeV)3 . In the chiral limit,m0=0 , we useG=5.01GeV−2 , andΛ=650MeV . -
The partition function of the NJL model at finite temperature T can be given by the imaginary time formalism,
ZNJL=∫[dψ][dˉψ]exp{∫dx[LNJL+ˉψˆμγ0ψ]}.
(21) Here and in the following,
x=(τ,r) withτ being the imaginary time. We use the notation∫dx≡∫β0dτ∫d3r withβ=1/T . The chemical potential matrixˆμ is diagonal in flavor space,ˆμ=diag(μu,μd) . A useful parameterization of the chemical potentials is given byμu=13μB+12μI,μd=13μB−12μI,
(22) corresponding to introducing two conserved charges, the baryon number and the third component of the isospin. In this work, we consider the case
μI=0 for the sake of simplicity. We therefore setμu=μd≡μ . Our theory can be easily generalized to nonzero isospin chemical potential,μI≠0 . A large isospin chemical potential leads to the Bose-Einstein condensation of charged pions and the BEC-BCS crossover [86-96].Introducing two auxiliary fields
σ andπ , which satisfy equations of motionσ=−2Gˉψψ,π=−2Gˉψiγ5τψ , and applying the Hubbard-Strotonovich transformation, we obtainZNJL=∫[dψ][dˉψ][dσ][dπ]exp{−S[ψ,ˉψ,σ,π]},
(23) where the action is given by
S[ψ,ˉψ,σ,π]=∫dxσ2(x)+π2(x)4G−∫dx∫dx′ˉψ(x)G−1(x,x′)ψ(x′),
(24) with the inverse of the fermion Green's function
G−1(x,x′)={γ0(−∂τ+μ)+iγ⋅∇−m0−[σ(x)+iγ5τ⋅π(x)]}δ(x−x′).
(25) We integrate out the quark field and obtain
ZNJL=∫[dσ][dπ]exp{−Seff[σ,π]},Seff[σ,π]=∫dxσ2(x)+π2(x)4G−TrlnG−1(x,x′).
(26) At low temperature, we expect that the DCSB persists and we set
⟨σ(x)⟩=υ and⟨π(x)⟩=0 . Applying again the field shiftsσ(x)→υ+σ(x) andπ(x)→0+π(x) , we expand the effective actionSeff[σ,π] in powers of the fluctuationsσ(x) andπ(x) and obtainSeff[σ,π]=S(0)eff+S(1)eff[σ,π]+S(2)eff[σ,π]+⋯.
(27) In this work, we neglect mesonic fluctuations of order higher than the Gaussian. The linear term
S(1)eff[σ,π] can be shown to vanish. The partition function in this Gaussian approximation is given byZNJL≈exp{−S(0)eff}∫[dσ][dπ]exp{−S(2)eff[σ,π]}.
(28) Evidently, the advantage of this Gaussian approximation is that we can complete the path integral over the fluctuation fields
σ(x) andπ(x) . The thermodynamic potentialΩ=−lnZNJL/(βV) is given byΩ≈ΩMF+ΩFL,
(29) where the mean-field contribution reads
ΩMF=1βVS(0)eff,
(30) and the meson-fluctuation contribution is given by
ΩFL=−1βVln[∫[dσ][dπ]exp{−S(2)eff[σ,π]}].
(31) -
At finite temperature, the mean-field part
S(0)eff= Seff[υ,0] is given byS(0)eff=βVυ24G−∑n∑klndet[G−1(ikn,k)],
(32) where
G−1(ikn,k)=(ikn+μ)γ0−γ⋅k−M
(33) is the inverse of the mean-field quark Green's function in momentum space, with
kn=(2n+1)πT (n∈Z ) being the fermion Matsubara frequency andM=m0+υ as the effective quark mass. The mean-field thermodynamic potential can be evaluated asΩMF=υ24G−2NcNf∫d3k(2π)3{Ek+1βln[1+e−β(Ek−μ)]+1βln[1+e−β(Ek+μ)]},
(34) where
Ek=√k2+M2 . As in the zero temperature case, we also regularize the integral over the quark momentumk via a hard cutoffΛ (|k|<Λ ). The chiral condensateυ is determined by minimizingS(0)eff , i.e.,∂S(0)eff/∂υ=0 , leading to the gap equationM−m0=4GNcNfM∫d3k(2π)31−f(Ek−μ)−f(Ek+μ)Ek.
(35) Here,
f(E)=1/(1+eβE) is the Fermi-Dirac distribution. If the phase transition is of first order, the gap equation has multiple solutions. In this case, we compare their grand potentials and find the physical solution ofυ .Figure 1 shows the effective quark mass M as a function of T for various values of the chemical potential
μ in the chiral limit (m0=0 ). Figure 2 shows the well-known phase diagram of the NJL model in the T-μ plane. At small chemical potential, the chiral phase transition is of second order. It becomes of first order at largeμ . Hence, a tricritical point appears. For physical current quark mass, the second-order phase transition turns into a crossover, and the tricritical point becomes a critical endpoint. -
Here, we include the mesonic degrees of freedom. To this end, we consider the excitations corresponding to the fluctuation fields
σ(x) andπ(x) . It is convenient to work in the momentum space by defining the Fourier transformationϕm(x)=∑Qϕm(Q)e−iqlτ+iq⋅r, m=0,1,2,3,
(36) where
ϕ0=σ andϕi=πi (i=1,2,3 ). HereQ≡(iql,q) withql=2lπT (l∈Z ) as the boson Matsubara frequency. The notation∑Q=∑l∫d3q(2π)3 is used throughout. In the momentum space, the inverse of the quark Green's functionG−1 readsG−1(K,K′)=G−1(K)δK,K′−ΣFL(K,K′),
(37) where
K=(ikn,k) andΣFL(K,K′)=3∑m=0Γmϕm(K−K′).
(38) Here we have defined
Γ0=1 andΓi=iγ5τi (i=1,2,3 ). Applying the derivative expansion, we obtainS(2)eff[σ,π]=βV23∑m,n=0∑Qϕm(−Q)[D−1(Q)]mnϕn(Q),
(39) where
[D−1(Q)]mn=δmn2G+Πmn(Q)
(40) is the inverse of the meson Green's function. The polarization function
Πmn(Q) is defined asΠmn(Q)=1βV∑KTr[G(K)ΓmG(K+Q)Γn].
(41) The notation
∑K=∑n∫d3k(2π)3 will be used throughout. Since we consider the caseμI=0 , the off-diagonal components vanishes, i.e.,Πmn(Q)=δmnΠm(Q) . It is also evident thatΠm(−Q)=Πm(Q) .The meson polarization functions
Πm(Q) can be evaluated asΠ0(iql,q)=NcNf∫d3k(2π)3[(1−f(E+k)−f(E−k+q)iql−Ek−Ek+q−1−f(E−k)−f(E+k+q)iql+Ek+Ek+q)(1+k⋅(k+q)−M2EkEk+q)+(f(E−k)−f(E−k+q)iql+Ek−Ek+q−f(E+k)−f(E+k+q)iql−Ek+Ek+q)×(1−k⋅(k+q)−M2EkEk+q)]
(42) for
m=0 , andΠm(iql,q)=NcNf∫d3k(2π)3[(1−f(E+k)−f(E−k+q)iql−Ek−Ek+q−1−f(E−k)−f(E+k+q)iql+Ek+Ek+q)(1+k⋅(k+q)+M2EkEk+q)+(f(E−k)−f(E−k+q)iql+Ek−Ek+q−f(E+k)−f(E+k+q)iql−Ek+Ek+q)×(1−k⋅(k+q)+M2EkEk+q)]
(43) for
m=1,2,3 . Here, we definedE±k=Ek±μ for convenience. In the chiral limit, we can show that from the gap equation,1/(2G)+Πm(0,0)=0 (m=1,2,3 ) in the chiral symmetry broken phaseM≠0 , which manifests the fact that the pions are Goldstone bosons in this phase. In the chiral symmetry restored phase,M=0 , we obtainΠ0(Q)=Π1(Q)=Π2(Q)=Π3(Q) , which indicates that the sigma meson and the pions become degenerate.In the Gaussian approximation, the path integral over
ϕm can be completed. The mesonic contribution to the thermodynamic potential can be evaluated asΩFL=12βV∑Qlndet[D−1(Q)]=123∑m=01β∑l∫d3q(2π)3ln[12G+Πm(iql,q)]eiql0+.
(44) We can convert the summation over the boson Matsubara frequency to an contour integration and obtain [74]
ΩFL=−3∑m=0∫d3q(2π)3∫∞0dω2πi[ω2+1βln(1−e−βω)]×ddωln[1+2GΠm(ω+i0+,q)1+2GΠm(ω−i0+,q)].
(45) This result could be related to the Bethe-Uhlenbeck expression, i.e., the second virial contribution in terms of the two-body scattering phase shift [74]. We note that
1+2GΠm(ω+i0+,q) is proportional to the T-matrix for the quark-antiquark scattering in them -channel, with total energyω and momentumq . The scattering matrix element can be written in the Jost representation asSm(ω,q)=1+2GΠm(ω−i0+,q)1+2GΠm(ω+i0+,q).
(46) The S-matrix element may has poles corresponding to mesonic bound states. Above the threshold for elastic scattering, it can be represented by a scattering phase shift as
Sm(ω,q)=e2iϕm(ω,q).
(47) Combining a possible pole term and the scattering contribution, we have [74]
ΩFL=3∑m=0∫d3q(2π)3∫∞0dω[ω2+1βln(1−e−βω)]×[δ(ω−εm(q))+1π∂ϕm(ω,q)∂ω],
(48) where the mesonic pole energy can be given by
εm(q)=√q2+m2m , with the in-medium meson massmm . At low temperature, the above expression explicitly recovers the fact that thermodynamic quantities are dominated by the lightest mesonic excitations, i.e., the pions [75]. In the chiral limit, the pressure of the system at low temperature can be well given by the pressure of a gas of noninteracting massless pions,p=π2T4/30 . -
We now start to study the linear response of the hot and dense matter in the NJL model, based on the description of the equilibrium thermodynamics in the last section. In this section, we introduce a generic theoretical framework to compute the following imaginary-time-ordered current-current correlation function
Πμν(τ−τ′,r−r′)=−⟨Tτ[Jμ(τ,r)Jν(τ′,r′)]⟩c,
(49) where
Jμ(τ,r) can be any current operator. The notation⟨⋯⟩c denotes the connected piece of the correlation function. For a pure fermionic field theory with a Lagrangian densityL[ψ,ˉψ]=ˉψ(iγμ∂μ−m0)ψ+Lint[ψ,ˉψ],
(50) the current operator is given by
Jμ=ˉψΓμψ,
(51) where
Γμ=γμˆX withˆX depicting any Hermitian matrix in the spin, flavor, and color spaces. For instance, the electromagnetic current is defined byˆX=diag(2e/3,−e/3) in the flavor space with e being the elementary electric charge andˆX=γ5 in the spin space gives the axial vector current.Parallel to the path integral approach to the equilibrium thermodynamics, we introduce a path integral formalism for the linear response. In this formalism, we introduce an external source term to compute the correlation function
Πμν(τ,r) . The external source physically represents an external perturbation applied to the system. The external source here is actually an external gauge fieldAμ(τ,r) which couples to the currentJμ(τ,r) . We still usex=(τ,r) for convenience. The partition function with the external source is given byZ[A]=∫[dψ][dˉψ]exp{−S[ψ,ˉψ;A]},
(52) where the action reads
S[ψ,ˉψ;A]=∫dx {−L[ψ,ˉψ]−μˉψγ0ψ+Aμ(x)ˉψΓμψ}.
(53) It is convenient to use the generating functional
W[A] defined asZ[A]=exp{−W[A]}.
(54) If the the generating functional can be computed exactly, the correlation function is given by
Πμν(τ−τ′,r−r′)=δ2W[A]δAμ(τ,r)δAν(τ′,r′)|A=0.
(55) In practice, we need to evaluate the generating functional in some approximations. It is convenient to work in the momentum space by making the Fourier transform
Aμ(x)=∑QAμ(Q)e−iqlτ+iq⋅r.
(56) To evaluate the correlation function, we expand the generating functional
W[A] in powers ofAμ(Q) . The expansion can be formally given byW[A]=W(0)+W(1)[A]+W(2)[A]+⋯,
(57) where
W(n) is the nth-order expansion inAμ(Q) . The zeroth-order contributionW(0) recovers the equilibrium grand potentialΩ with a vanishing external source,W(0)=βVΩ.
(58) The first-order contribution
W(1)[A] provides nothing but the thermodynamic relation for the charge densitynX=⟨J0⟩ . We havenX=−∂Ω/∂μX , where the chemical potential is defined asμX=A0(Q=0) . Hence, we haveW(1)[A]βV=−nXA0(Q=0).
(59) The second-order contribution
W(2)[A] characterizes the linear response. It can be formally given byW(2)[A]βV=12∑QΠμν(Q)Aμ(−Q)Aν(Q).
(60) Here
Πμν(Q) is just the correlation function in the momentum space. The static and long-wavelength limit of its00 -component,Π00(Q=0) , is related to to the number susceptibility, i.e.,limq→0Π00(iql=0,q)=∂2Ω(T,μX)∂μ2X.
(61) For instance, for the vector current with
ˆX=1 ,Π00(Q=0) is proportional to the baryon number susceptibility. The above discussions are precise if the generating functionalW[A] or its second-order expansionW(2)[A] can be computed exactly.Followingly, we turn to the NJL model. The partition function of the NJL model with the external source is given by
ZNJL[A]=∫[dψ][dˉψ]exp{−S[ψ,ˉψ;A]},
(62) with the action
S[ψ,ˉψ;A]=∫dx {−LNJL[ψ,ˉψ]−μˉψγ0ψ+Aμ(x)ˉψΓμψ}.
(63) Again, introducing two auxiliary fields
σ andπ , which satisfy equations of motionσ=−2Gˉψψ,π= −2Gˉψiγ5τψ , and applying the Hubbard-Strotonovich transformation, we obtainZNJL[A]=∫[dψ][dˉψ][dσ][dπ]exp{−S[ψ,ˉψ,σ,π;A]},
(64) where the action now reads
S[ψ,ˉψ,σ,π;A]=∫dxσ2(x)+π2(x)4G−∫dx∫dx′ˉψ(x)G−1A(x,x′)ψ(x′),
(65) with the inverse of the fermion Green's function
G−1A(x,x′)={γ0(−∂τ+μ)+iγ⋅∇−m0−[σ(x)+iγ5τ⋅π(x)]−ΓμAμ(x)}δ(x−x′).
(66) Integrating out the quark field yields
ZNJL[A]=∫[dσ][dπ]exp{−Seff[σ,π;A]},Seff[σ,π;A]=∫dxσ2(x)+π2(x)4G−TrlnG−1A(x,x′).
(67) The treatment of the the expectation values of the meson fields
σ(x) andπ(x) , or their classical fieldsσcl(x) andπcl(x) , becomes nontrivial. In the absence of the external source, we chooseσcl(x)=υ andπcl(x)=0 , which are static and homogeneous. However, in the absence of the external source, they are generally no longer static and homogeneous. Again, we apply the field shifts,σ(x)→ σcl(x)+σ(x) andπ(x)→πcl(x)+π(x) , and expand the effective actionSeff[σ,π] in powers of the fluctuationsσ(x) andπ(x) . We obtainSeff[σ,π;A]=S(0)eff[A]+S(1)eff[σ,π;A]+S(2)eff[σ,π;A]+⋯.
(68) Parallel to the case without external source, we neglect the mesonic fluctuations of order higher than the Gaussian. The linear term
S(1)eff[σ,π;A] can be shown to vanish once the classical fieldsσcl(x) andπcl(x) are determined by minimizingS(0)eff[A] . The partition function in the Gaussian approximation is given byZNJL≈exp{−S(0)eff[A]}∫[dσ][dπ]exp{−S(2)eff[σ,π;A]}.
(69) Therefore, in this Gaussian approximation, the generating functional
WNJL[A] includes both the mean-field (MF) and the meson-fluctuation (FL) contributions. We haveWNJL[A]=WMF[A]+WFL[A],
(70) where
WMF[A]=S(0)eff[A],WFL[A]=−ln[∫[dσ][dπ]exp{−S(2)eff[σ,π;A]}].
(71) In the path integral, we can treat the equilibrium thermodynamics and the linear response at the same footing. The mean-field and the meson-fluctuation contributions to the generating functional can be expanded in powers of the external source as
WMF[A]=W(0)MF+W(1)MF[A]+W(2)MF[A]+⋯,WFL[A]=W(0)FL+W(1)FL[A]+W(2)FL[A]+⋯.
(72) The zeroth-order contributions recover the equilibrium thermodynamic potentials, i.e.,
W(0)MF=βVΩMF andW(0)FL=βVΩFL .So far, the dependence on the classical fields
σcl(x) andπcl(x) is not explicitly shown. They are not independent quantities and should be determined as functionals of the external source via some gap equations. We writeWNJL[A]=WMF[A;σcl,πcl]+WFL[A;σcl,πcl].
(73) Parallel to the theory of the equilibrium thermodynamics, we require that the classical fields are determined by minimizing the mean-field part of the generating functional, i.e.,
δWMF[A;σcl,πcl]δσcl(x)=0,δWMF[A;σcl,πcl]δπcl(x)=0.
(74) Once this extreme condition is imposed, we can show that the linear term
S(1)eff[σ,π;A] vanishes exactly. Moreover, it is also necessary to maintain the Goldstone's theorem. Solving the extreme condition formally, we haveσcl(x)=Fσ[A], πcl(x)=Fπ[A].
(75) Substituting these solutions into the generating functional, we finally eliminate the dependence on the classical fields.
In the following sections, we will study the response functions in the mean-field approximation (
Seff≃S(0)eff, WNJL≃WMF ) and in the Gaussian-fluctuation approximation (Seff≃S(0)eff+S(2)eff,WNJL≃WMF+WFL ). Here, we have truncated the mesonic fluctuations up to the quadratic order, since higher-order contributions cannot be analytically treated. The mean-field truncation, corresponding to the random phase approximation of the linear response, is obviously self-consistent, as has been verified in numerous studies of the many-body theory. The Gaussian-fluctuation truncation takes into account the contribution from the collective modes (mesons). The contributions higher than the Gaussian may correspond to the interaction between mesons, which are assumed to be weak and therefore can be neglected. However, one can show that the Gaussian-fluctuation approximation also preserves the Ward-Takahashi identity and hence the conservation laws [97-99]. In the context of the electromagnetic response of superconductors, such a Gaussian-fluctuation approximation leads to a gauge invariant linear response theory [97-99]. -
We first present the linear response in the mean-field approximation, i.e.,
WNJL[A]≃WMF[A] . We will see that the response functions in this approximation recovers the famous random phase approximation (RPA) developed in early condensed matter theory. Since we are interested in the response to an infinitesimal external source, we expect that the induced perturbations to the classical fields are also infinitesimal. Therefore, we haveσcl(x)=υ+η0(x), πcl(x)=0+η(x),
(76) where the static and uniform part
υ is the chiral condensate with vanishing external source. The generating functional in the mean-field approximation is given byWMF[A;σcl,πcl]=∫dxσ2cl(x)+π2cl(x)4G−Trln[G−1A(x,x′)].
(77) Here
G−1A is the inverse of the fermion Green's function in the mean-field approximation with external source. It can be expressed asG−1A(x,x′)=G−1(x,x′)−ΣA(x,x′),
(78) where the two terms are defined as
G−1(x,x′)=[γ0(−∂τ+μ)+iγ⋅∇−M]δ(x−x′),
ΣA(x,x′)=[3∑m=0Γmηm(x)+ΓμAμ(x)]δ(x−x′).
(79) Here
M=m0+υ is the effective quark mass as we have defined in the absence of the external source.Now we turn to the momentum space via the Fourier transform
ηm(x)=∑Qηm(Q)e−iqlτ+iq⋅r.
(80) In the momentum space, the inverse of the fermion Green's function is given by
G−1A(K,K′)=G−1(K)δK,K′−ΣA(K,K′),
(81) where
G−1(K) is given by (33) andΣA(K,K′)=3∑m=0Γmηm(K−K′)+ΓμAμ(K−K′).
(82) Using the derivative expansion, we can expand the generating functional in powers of the external source as well as the induced perturbations
ηm . We haveWMF[A;η]=W(0)MF+W(1)MF[A;η]+W(2)MF[A;η]+⋯,
(83) where it is obvious that
W(0)MF=βVΩMF . Note that the induced perturbation should be finally eliminated via the gap equation (74).The linear term
W(1)MF[A;η] can be evaluated asW(1)MF[A;η]βV=[υ2G+1βV∑KTrG(K)]η0(0)+1βV∑KTr[G(K)iγ5τ]⋅η(0)+1βV∑KTr[G(K)Γμ]Aμ(0).
(84) It is related only to the
Q=0 component of the external source and the induced perturbations. The explicit form ofG(K) can be evaluated asG(K)=1ikn−EkΛ+(k)γ0+1ikn+EkΛ−(k)γ0,
(85) where the the energy projectors
Λ±(k) are given byΛ±(k)=12[1±γ0(γ⋅k+M)Ek].
(86) Using the gap equation (35), we can show that the only nonvanishing part is related to the number density, i.e.,
W(1)MF[A;η]βV=−(nX)MFA0(0),
(87) where the number density is given by
(nX)MF=1βV∑KTr[G(K)Γ0].
(88) It is evident that
(nX)MF=−∂ΩMF/∂μX with the chemical potentialμX=A0(0) .The linear response is characterized by the quadratic term
W(2)MF[A;η] . By making use of the derivative expansion and completing the trace in the momentum space, we obtainW(2)MF[A;η]=βV4G3∑m=0∑Qηm(−Q)ηm(Q)+12∑K∑K′Tr[G(K)ΣA(K,K′)G(K′)ΣA(K′,K)].
(89) Defining
Q=K′−K , we obtainW(2)MF[A;η]βV=12∑QΠμνb(Q)Aμ(−Q)Aν(Q)+123∑m=0∑Q[12G+Πm(Q)]ηm(−Q)ηm(Q)+3∑m=0∑QCμm(Q)Aμ(−Q)ηm(Q).
(90) Here the the bare response function
Πμνb(Q) is defined asΠμνb(Q)=1βV∑KTr[G(K)ΓμG(K+Q)Γν]
(91) and the coupling function
Cμm(Q) is given byCμm(Q)=1βV∑KTr[G(K)ΓμG(K+Q)Γm].
(92) The meson polarization functions
Πm(Q) are given in Sec. 3.The final task is to eliminate the induced perturbations. For the purpose of linear response, the induced perturbations
ηm(Q) can be determined byδW(2)MF[A;η]δηm(Q)=0.
(93) Using the explicit form of
W(2)MF[A;η] , we obtainηm(Q)=−Cμm(−Q)Aμ(Q)12G+Πm(Q)+O(A2),ηm(−Q)=−Cμm(Q)Aμ(−Q)12G+Πm(Q)+O(A2),
(94) where we have applied the fact that
Πm(−Q)=Πm(Q) . Using the above results to eliminate the induced perturbations, we finally obtainW(2)MF[A]βV=12∑QΠμνMF(Q)Aμ(−Q)Aν(Q),
(95) where the full response function in the mean-field theory reads
ΠμνMF(Q)=Πμνb(Q)−3∑m=0Cμm(Q)Cνm(−Q)12G+Πm(Q).
(96) This result recovers nothing but the quasi-particle random phase approximation widely used in condensed matter theory [81]. We note that in addition to the pure quasi-particle contribution
Πμνb(Q) , the linear response can couple to the collective mesonic modes onceCμm(Q)≠0 . Hence, the response function reveals meson properties and also possibly phase transitions.In the chiral limit (
m0=0 ), we can show thatCμm(Q)=0 in the chiral symmetry restored phase (T>Tc ). In this case, the quasi-particle random phase approximation just describes the linear response of a hot and dense gas of non-interacting quarks. This is obviously inadequate. We will discuss the linear response theory beyond the quasi-particle random phase approximation in Sec. 7. -
As an application of the mean-field theory or the random phase approximation, we study the linear responses to some density perturbations. To be specific, we consider the following
ˆX operators: (1)ˆX=1 , corresponding to the vector current; (2)ˆX=τ3 , corresponding to the isospin vector current; (3)ˆX=γ5 , corresponding to the axial vector current; (4)ˆX=τ3γ5 , corresponding to the isospin axial vector current. The0 -component of the currentJμ is related to the baryon density, isospin density, axial baryon density, and axial isospin density, respectively. The density response functionχ(iql,q) is given by the00 -component of the response functionΠμν(Q) . In the mean-field theory, it is given byχ(iql,q)=Π00MF(Q)=Π00b(Q)−3∑m=0C0m(Q)C0m(−Q)12G+Πm(Q).
(97) In practice, we define the dynamic structure factor
S(ω,q) , which is related to the density response functionχ(iql,q) via the fluctuation-dissipation theorem. It is defined asS(ω,q)=−1π11−e−βωImχ(ω+iϵ,q).
(98) In the following, we are interested in the long-wavelength limit
q=0 and focus on the pure dynamical effect. -
For the vector current
ˆX=1 , the bare response function is given byΠ00b(Q)=1βV∑KTr[G(K)γ0G(K+Q)γ0].
(99) At
q=0 , we can show thatΠ00b(iql,q=0) vanishes. The coupling function is given byC0m(Q)=1βV∑KTr[G(K)γ0G(K+Q)Γm].
(100) At
q=0 , we can show thatC0m(iql,q=0) vanish for allm=0,1,2,3 . Therefore, for the baryon density response, the dynamic structure factor vanishes atq=0 , i.e.,S(ω,q=0)=0.
(101) -
For the isospin vector current
ˆX=τ3 , the bare response function is given byΠ00b(Q)=1βV∑KTr[G(K)γ0τ3G(K+Q)γ0τ3].
(102) At
q=0 , we can show thatΠ00b(iql,q=0) vanishes. The coupling function is given byC0m(Q)=1βV∑KTr[G(K)γ0τ3G(K+Q)Γm].
(103) At
q=0 , we can show thatC0m(iql,q=0) vanish for allm=0,1,2,3 . Therefore, for the isospin density response, the dynamic structure factor also vanishes atq=0 , i.e.,S(ω,q=0)=0.
(104) -
For the axial vector current
ˆX=γ5 , the bare response function is given byΠ00b(Q)=1βV∑KTr[G(K)γ0γ5G(K+Q)γ0γ5].
(105) Completing the trace and the Matsubara sum, we obtain
Π00b(iql,q=0)=2NcNf∫d3k(2π)3M2E2k(1iql−2Ek−1iql+2Ek)×[1−f(Ek−μ)−f(Ek+μ)].
(106) The coupling function is given by
C0m(Q)=1βV∑KTr[G(K)γ0γ5G(K+Q)Γm].
(107) At
q=0 , we can show thatC0m(iql,q=0) vanish for allm=0,1,2,3 . Therefore, the axial baryon density response has a nonzero dynamical structure factor atq=0 . It does not couple to the mesonic modes and is given byχ(iql,q=0)=Π00b(iql,q=0) . The dynamical structure factor readsS(ω,q=0)=NcNfM22π2√ω2−4M2ωΘ(|ω|−2M)1−e−βω×[1−1eβ(12ω−μ)+1−1eβ(12ω+μ)+1].
(108) We note that a similar result was also obtained in Ref. [100]. It is evident that the dynamical structure factor for the axial baryon density response is a direct reflection of the quark mass gap.
S(ω,q=0) is nonzero only for|ω| larger than two times the quark mass gap. Figure 3 shows the dynamical structure factorS(ω,q=0) for various values of the temperature. With increasing temperature, the thresholdωth=2M becomes smaller, and finallyωth→0 in the high T limit. -
For the isospin axial vector current
ˆX=τ3γ5 , the bare response function is given byΠ00b(Q)=1βV∑KTr[G(K)γ0τ3γ5G(K+Q)γ0τ3γ5].
(109) Completing the trace and the Matsubara sum, we obtain
Π00b(iql,q=0)=2NcNf∫d3k(2π)3M2E2k(1iql−2Ek−1iql+2Ek)×[1−f(Ek−μ)−f(Ek+μ)].
(110) The coupling function is given by
C0m(Q)=1βV∑KTr[G(K)γ0τ3γ5G(K+Q)Γm].
(111) At
q=0 , we can show thatC0m(iql,q=0) vanish form=0,1,2 . The nonzero couplingC03(iql,q=0) is given byC03(iql,q=0)=2iNcNf∫d3k(2π)3MEk(1iql−2Ek+1iql+2Ek)×[1−f(Ek−μ)−f(Ek+μ)].
(112) Thus, the axial isospin density response couples to the neutral pion mode
π0 .The full response function reads
χ(iql,q=0)=Π00b(iql,q=0)−C03(iql,q=0)C03(−iql,q=0)12G+Π3(iql,q=0).
(113) Here
Π3(iql,q=0) is given byΠ3(iql,q=0)=2NcNf∫d3k(2π)3(1iql−2Ek−1iql+2Ek)×[1−f(Ek−μ)−f(Ek+μ)].
(114) To evaluate the dynamical structure factor, we make use of the following results,
ImΠ00b(ω+iϵ,q=0)=−NcNfM22π√ω2−4M2ωΘ(|ω|−2M)[1−1eβ(12ω−μ)+1−1eβ(12ω+μ)+1],ReΠ3(ω+iϵ,q=0)=2NcNfP∫d3k(2π)3(1ω−2Ek−1ω+2Ek)[1−f(Ek−μ)−f(Ek+μ)],ImΠ3(ω+iϵ,q=0)=−NcNfω√ω2−4M28πΘ(|ω|−2M)[1−1eβ(12ω−μ)+1−1eβ(12ω+μ)+1],ReC03(ω+iϵ,q=0)=NcNfM√ω2−4M24πΘ(|ω|−2M)sgn(ω)[1−1eβ(12ω−μ)+1−1eβ(12ω+μ)+1],ImC03(ω+iϵ,q=0)=2NcNfP∫d3k(2π)3MEk(1ω−2Ek+1ω+2Ek)[1−f(Ek−μ)−f(Ek+μ)].
(115) Here
P denotes the principal value. The imaginary part ofχ(ω+iϵ,q=0) can be expressed asImχ(ω+iϵ)=ImΠ3(ω+iϵ)[12G+ReΠ3(ω+iϵ)]2+[ImΠ3(ω+iϵ)]2×[ImC03(ω+iϵ)−2Mω(12G+ReΠ3(ω+iϵ))]2.
(116) Here, we have suppressed the condition
q=0 for convenience. Therefore, we expect that at low temperature, the dynamical structure factor for the axial isospin density response reveals a pole plus continuum structure. For|ω|>2M ,ImΠ3(ω+iϵ) is nonzero, and hence the dynamical structure factor shows a continuum. For|ω|<2M ,ImΠ3(ω+iϵ) vanishes and thus the dynamical structure factor is simply proportional to a delta function. We haveImχ(ω+iϵ)=π[ImC03(ω+iϵ)]2δ(12G+ReΠ3(ω+iϵ)).
(117) It is evident that the pole is located at the pion mass. In the chiral limit, this pole is located exactly at
ω=0 forT<Tc , and it disappears forT>Tc . For physical current quark mass, there is a Mott transition temperatureT=Tm determined by the equationmπ(T)=2M(T) . Figure 4 shows the dynamical structure factorS(ω,q=0) for temperatures below and aboveTm . ForT<Tm , the pion is a bound state and henceS(ω,q=0) shows a pole plus continuum structure. Above the Mott transition temperature, the pole disappears andS(ω,q=0) shows only a continuum. The threshold of the continuum is also located atωth=2M . -
In the chiral limit (
m0=0 ), the quarks become massless (M=0 ) above the chiral phase transition temperature. In this case, we can show thatCμm(Q)=0 in the chiral symmetry restored phase. Therefore, the quasi-particle random phase approximation simply describes the linear response of a system of non-interacting massless quarks. However, it is generally expected that mesonic fluctuations play an important role above and near the chiral phase transition, indicating that the random phase approximation is inadequate for such a strongly interacting system. In this part, we consider a linear response theory beyond the random phase approximation. To this end, we recall that the generating functional in the Gaussian approximation can be expressed asWNJL[A]=WMF[A]+WFL[A].
(118) In the previous random phase approximation, the meson-fluctuation contribution
WFL[A] is neglected. We expect that this part becomes rather important near and above the chiral phase transition, where the quarks become massless and the mesonic degrees of freedom are still important. This is a general feature of a strongly interacting fermionic system. In strong-coupling superconductors, the pair fluctuation has an important contribution to the transport properties above and near the superconducting transition temperature [84,85].We consider the contribution from mesonic fluctuations. To derive the generating functional
WFL[A] , we first note thatG−1A(x,x′)=G−1A(x,x′)−ΣFL(x,x′),
(119) where
ΣFL includes mesonic fluctuation fields,ΣFL(x,x′)=∑3m=0Γmϕm(x)δ(x−x′),
(120) and
G−1A(x,x′) is the mean-field quark Green's function with the external source,G−1A(x,x′)=G−1(x,x′)−ΣA(x,x′),
(121) with
G−1(x,x′) andΣA(x,x′) given in Eq. (79). Converting to the momentum space, we haveG−1A(K,K′)=G−1A(K,K′)−ΣFL(K,K′)ΣFL(K,K′)=3∑m=0Γmϕm(K−K′).
(122) Starting from Eqs. (67) and (68) and applying the derivative expansion, we obtain
S(2)eff[σ,π]=βV23∑m,n=0∑Q,Q′ϕm(−Q)[D−1A(Q,Q′)]ϕn(Q′),
(123) where
[D−1A(Q,Q′)]mn=δmn2GδQ,Q′+1βV∑K,K′Tr[GA(K,K′−Q)ΓmGA(K′,K+Q′)Γn].
(124) Again, the path integral over the fluctuation fields
ϕm can be calculated, and we obtainWFL[A]=12Trln[D−1A(Q,Q′)].
(125) The trace here is also taken in the momentum space.
The next step is to expand
WFL[A] in powers of the external source and induced perturbations. To this end, we first expand the inverse meson propagatorD−1A(Q,Q′) in powers ofAμ andηm . The expansion takes the formD−1A(Q,Q′)=D−1(Q)δQ,Q′+Σ(1)(Q,Q′)+Σ(2)(Q,Q′)+⋯.
(126) Here,
D(Q) is the meson propagator evaluated in Sec. 3, andΣ(n) denotes the nth-order expansion inAμ andηm . In practice, we only need to evaluate the expansion up to the second order, since the higher order contributions are irrelevant to the linear response. LikeD−1 ,Σ(1) andΣ(2) are4×4 matrices in the space spanned bym=0,1,2,3 .To obtain
Σ(1) andΣ(2) , we note that in the momentum space, the inverse of the mean-field quark Green's function with external source,G−1A , is given byG−1A(K,K′)=G−1(K)δK,K′−ΣA(K,K′),
(127) where
ΣA(K,K′)=3∑m=0Γmηm(K−K′)+ΓμAμ(K−K′).
(128) For convenience, here we express
ΣA in a more compact formΣA(K,K′)=7∑i=0˜ΓiΦi(K−K′),
(129) where
˜Γ is a compact notation of(Γm,Γμ) andΦ is a compact form of(ηm,Aμ) . Herei=0,1,2,3 still stands forηm withm=0,1,2,3 , andi=4,5,6,7 stands forAμ withμ=0,1,2,3 . Applying the Taylor expansion for matrix functions, we obtainGA=G+GΣAG+GΣAGΣAG+⋯.
(130) This compact form of the Taylor expansion should be understood in all spaces. In the momentum space, we have explicitly
GA(K,K′)=G(K,K′)+∑K1,K2G(K,K1)ΣA(K1,K2)G(K2,K′)+∑K1,K2,K3,K4G(K,K1)ΣA(K1,K2)G(K2,K3)×ΣA(K3,K4)G(K4,K′)+⋯.
(131) According to the fact that
G(K,K′)=G(K)δK,K′ andΣA(K,K′)=ΣA(K−K′) , this can be simplified toGA(K,K′)=G(K)δK,K′+G(K)ΣA(K−K′)G(K′)+∑K′′G(K)ΣA(K−K′′)G(K′′)ΣA(K′′−K′)×G(K′)+⋯.
(132) The explicit form of
Σ(1) andΣ(2) can be derived by using the above expansion forGA .Σ(1) is composed of one zeroth-order and one first-order contributions ofGA . It is explicitly given byΣ(1)mn(Q,Q′)=7∑i=0[X(1)]imn(Q,Q′)Φi(Q−Q′),
(133) where the coefficients are given by
[X(1)]imn(Q,Q′)=1βV∑KTr[G(K)ΓmG(K+Q)˜ΓiG(K+Q′)Γn]+1βV∑KTr[G(K)˜ΓiG(K+Q′−Q)×ΓmG(K+Q′)Γn].
(134) Σ(2) includes two types of contributions. We haveΣ(2)=Σ(2a)+Σ(2b).
(135) Σ(2a) is composed of one zeroth-order and one second-order contributions ofGA . It is given byΣ(2a)mn(Q,Q′)=1βV7∑i,j=0∑K,K′[X(2a)]ijmn(Q,Q′;K,K′)Φi(Q1)Φj(Q2)+1βV7∑i,j=0∑K,K′[Y(2a)]ijmn(Q,Q′;K,K′)×Φi(Q3)Φj(Q4).
(136) Here the momenta
Q1,Q2,Q3 , andQ4 are defined asQ1=K−K′+Q, Q2=K′−K−Q′,Q3=K−K′, Q4=K′−K+Q−Q′.
(137) The expansion coefficients are given by
[X(2a)]ijmn(Q,Q′;K,K′)=Tr[G(K)ΓmG(K+Q)˜ΓiG(K′)טΓjG(K+Q′)Γn],[Y(2a)]ijmn(Q,Q′;K,K′)=Tr[G(K)˜ΓiG(K′)˜ΓjG(K+Q′−Q)×ΓmG(K+Q′)Γn].
(138) Σ(2b) is composed of two second-order contributions ofGA . It readsΣ(2b)mn(Q,Q′)=1βV7∑i,j=0∑K,K′[X(2b)]ijmn×(Q,Q′;K,K′)Φi(Q1)Φj(Q2),
(139) where the expansion coefficient is given by
[X(2b)]ijmn(Q,Q′;K,K′)=Tr[G(K)˜ΓiG(K′−Q)×ΓmG(K′)˜ΓjG(K+Q′)Γn].
(140) -
Now we express the meson-fluctuation contribution to the generating functional as
WFL[A;η]=12Trln[D−1(Q)δQ,Q′+Σ(1)(Q,Q′)+Σ(2a)(Q,Q′)+Σ(2b)(Q,Q′)+⋯].
(141) Here we start to demonstrate the explicit dependence on induced perturbations. Applying the trick of derivative expansion, we can expand
WFL in powers of the external source as well as the induced perturbations. We haveWFL[A;η]=W(0)FL+W(1)FL[A;η]+W(2)FL[A;η]+⋯.
(142) It is evident that
W(0)FL=βVΩFL and hence the present linear response theory including the meson-fluctuation contribution is parallel to the meson-fluctuation theory of the equilibrium thermodynamics. The first-order expansion is given byW(1)FL[A;η]=12∑QTr4D[D(Q)Σ(1)(Q,Q)].
(143) Here, the trace
Tr4D is now taken only in the four-dimensional space spanned bym,n=0,1,2,3 . The second-order expansion can be expressed asW(2)FL[A;η]=W(AL)FL+W(SE)FL+W(MT)FL,
(144) where the three kinds of contributions are given by
W(AL)FL[A;η]=−14∑Q∑Q′Tr4D[D(Q)Σ(1)(Q,Q′)×D(Q′)Σ(1)(Q′,Q)],W(SE)FL[A;η]=12∑QTr4D[D(Q)Σ(2a)(Q,Q)],W(MT)FL[A;η]=12∑QTr4D[D(Q)Σ(2b)(Q,Q)],
(145) which correspond diagrammatically to the Aslamazov-Lakin (AL), self-energy (SE) or density-of-state, and Maki-Thompson (MT) contributions.
To obtain the response functions, we need to eliminate the induced perturbations
ηm(Q) . Noting that the present theory of linear response is a natural generalization of the meson-fluctuation theory of the equilibrium thermodynamics, where the order parameter is determined at the mean-field level, we determine the induced perturbationsηm(Q) still by minimizing the mean-field generation functional, i.e.,δW(2)MF[A;η]δηm(Q)=0,
(146) which leads to
ηm(Q)=−Cμm(−Q)12G+Πm(Q)Aμ(Q)+O(A2),ηm(−Q)=−Cμm(Q)12G+Πm(Q)Aμ(−Q)+O(A2).
(147) Later, we will show that the use of the above relations is also crucial to recover the correct number susceptibility in the static and long-wavelength limit.
-
Unlike the mean-field theory or random phase approximation, the first-order contribution, Eq. (143), becomes highly nontrivial. It can be expressed as
W(1)FL[A;η]=βV7∑i=1CiΦi(0),
(148) where the coefficients read
Ci=12βV∑QTr4D{D(Q)[X(1)]i(Q,Q)}.
(149) Using the explicit expression of
X(1) , we can show that possible nonvanishing coefficients areC0=∂ΩFL(M,μX)∂M, C4=∂ΩFL(M,μX)∂μX.
(150) Since we consider only nonzero baryon chemical potential, here the effective chemical potential
μX=A0(0) is nonvanishing only for the vector current case(ˆX=1) . Thus,C4 is nonvanishing only for the caseˆX=1 , whereμX corresponds to the quark chemical potentialμ . The fact thatC0≠0 indicates that the first-order contributionW(1)FL[A;η] cannot be simply neglected, since it does contribute to the linear response. To understand this, we note that when eliminating the induced perturbationη0(0) , Eq. (147) is not adequate. Actually, the contributions of the orderO(A2) in Eq. (147) become important. To obtain these contributions, we should expand the mean-field generating functionalWMF[A;η] up to the third order in A andη . We haveW(3)MF[A;η]=13∑K∑K′∑K′′Tr[G(K)ΣA(K,K′)G(K′)×ΣA(K′,K′′)G(K′′)ΣA(K′′,K)].
(151) By defining
K′=K+Q andK′′=K+Q′ , we obtainW(3)MF[A;η]βV=137∑i,j,k=0∑Q∑Q′Fijk(Q,Q′)×Φi(−Q)Φj(Q−Q′)Φk(Q′),
(152) where the function
Fijk(Q,Q′) is defined asFijk(Q,Q′)=1βV∑KTr[G(K)˜ΓiG(K+Q)˜ΓjG(K+Q′)˜Γk].
(153) Using the extreme condition
δWMF[A;η]δη0(Q)=0
(154) with
WMF=W(0)MF+W(1)MF+W(2)MF+W(3)MF+⋯ , we obtainη0(0)=R1A0(0)+127∑i,j=0∑QUij(Q)Φi(−Q)Φj(Q)+⋯,
(155) where the coefficients
R1 andUij(Q) are given byR1=−limQ→0C00(−Q)12G+Π0(Q),Uij(Q)=23limQ′→0F0ij(−Q′,Q)+Fi0j(Q+Q′,Q)+Fij0(Q,Q′)12G+Π0(Q).
(156) Here the the static and long-wavelength limit of an arbitrary function
A(Q) should be understood aslimQ→0 A(Q)=limq→0A(iql=0,q) . For the purpose of linear response, we apply Eq. (147) and obtainη0(0)=R1A0(0)+12∑QRμν2(Q)Aμ(−Q)Aν(Q)+O(A3).
(157) Here, the explicit form of the function
Rμν2(Q) is not shown. It is evident thatR1=∂M(μX)∂μX, limQ→0R002(Q)=∂2M(μX)∂μ2X.
(158) Substituting the expansion (155) into Eq. (148), we eliminate the induced perturbations and obtain
W(1)FL[A]βV=−(nX)FLA0(0)+12∑QΠμνOP(Q)Aμ(−Q)Aν(Q)+⋯,
(159) where
(nX)FL is the fluctuation contribution to the charge density,(nX)FL=−∂ΩFL(M,μX)∂μX−∂ΩFL(M,μX)∂M∂M(μX)∂μX.
(160) The first-order term
W(1)FL thus yields a nontrivial contribution to the response function, which is given byΠμνOP(Q)=C0Rμν2(Q).
(161) It is evident that this contribution is due to the non-vanishing chiral condensate. In the chiral limit, this contribution vanishes above the phase transition temperature, where
C0=0 . Therefore, we denote it as the order parameter induced (OP) contribution, which can be expressed asW(OP)FL[A]=βV2∑QΠμνOP(Q)Aμ(−Q)Aν(Q).
(162) -
The Aslamazov-Lakin contribution is given by
W(AL)FL[A;η]=−14∑Q∑Q′Tr4D[D(Q)Σ(1)(Q,Q′)×D(Q′)Σ(1)(Q′,Q)].
(163) After some manipulation, it can be expressed as
W(AL)FL[A;η]=βV27∑i,j=0∑QΞALij(Q)Φi(−Q)Φj(Q),
(164) where the function
ΞALij(Q) is given byΞALij(Q)=−121βV∑PTr4D{D(P)[X(1)]i(P,P+Q)×D(P+Q)[X(1)]j(P+Q,P)}.
(165) Here the matrices
[X(1)]i(P,P+Q) and[X(1)]j(P+Q,P) are defined as[X(1)]imn(P,P+Q)=1βV∑KTr[G(K)ΓmG(K+P)טΓiG(K+P+Q)Γn]+1βV∑KTr[G(K)˜ΓiG(K+Q)×ΓmG(K+P+Q)Γn],[X(1)]jmn(P+Q,P)=1βV∑KTr[G(K)ΓmG(K+P+Q)טΓjG(K+P)Γn]+1βV∑KTr[G(K)˜ΓjG(K−Q)×ΓmG(K+P)Γn].
(166) We finally use Eq. (147) to eliminate the induced perturbations and obtain the AL contribution
W(AL)FL[A]=βV2∑QΠμνAL(Q)Aμ(−Q)Aν(Q),
(167) where
ΠμνAL(Q) is the AL contribution to the response function. -
The self-energy or density-of-state contribution is given by
W(SE)FL[A;η]=12∑QTr4D[D(Q)Σ(2a)(Q,Q)],
(168) After some manipulations, it can be expressed as
W(SE)FL[A;η]=βV28∑i,j=1∑QΞSEij(Q)Φi(−Q)Φj(Q),
(169) where the function
ΞSEij(Q) is given byΞSEij(Q)=1βV∑PTr4D[D(P)Yij(P,Q)]+1βV∑PTr4D[D(P)Zij(P,Q)].
(170) Here the matrices
Yij(P,Q) andZij(P,Q) are defined asYijmn(P,Q)=1βV∑KTr[G(K−P)ΓmG(K)טΓiG(K+Q)˜ΓjG(K)Γn],Zijmn(P,Q)=1βV∑KTr[G(K)˜ΓiG(K+Q)טΓjG(K)ΓmG(K+P)Γn].
(171) We finally use Eq. (147) to eliminate the induced perturbations and obtain the SE contribution
W(SE)FL[A]=βV2∑QΠμνSE(Q)Aμ(−Q)Aν(Q),
(172) where
ΠμνSE(Q) is the SE contribution to the response function. -
The Maki-Thompson contribution is given by
W(MT)FL[A;η]=12∑QTr4D[D(Q)Σ(2b)(Q,Q)].
(173) After some manipulation, it can be expressed as
W(MT)FL[A;η]=128∑i,j=1∑QΞMTij(Q)Φi(−Q)Φj(Q),
(174) where the function
ΞMTij(Q) is given byΞMTij(Q)=1βV∑PTr4D[D(P)Wij(P,Q)].
(175) Here the matrix
Wij(P,Q) is defined asWij(P,Q)=1βV∑KTr[G(K)˜ΓiG(K+Q)×ΓmG(K+P+Q)˜ΓjG(K+P)Γn].
(176) We finally use Eq. (147) to eliminate the induced perturbations and obtain the MT contribution
W(MT)FL[A]=βV2∑QΠμνMT(Q)Aμ(−Q)Aν(Q),
(177) where
ΠμνMT(Q) is the MT contribution to the response function.Combining all contributions, the meson-fluctuation contribution to the linear response is given by
W(2)FL[A]=12∑QΠμνFL(Q)Aμ(−Q)Aν(Q),
(178) where
ΠμνFL(Q) is a summation of all the above contributions,ΠμνFL(Q)=ΠμνOP(Q)+ΠμνAL(Q)+ΠμνSE(Q)+ΠμνMT(Q).
(179) Summarizing the mean-field and the meson-fluctuation contributions, we have
W(2)NJL[A]=12∑QΠμν(Q)Aμ(−Q)Aν(Q),
(180) where the full response function within the present theory is given by
Πμν(Q)=ΠμνMF(Q)+ΠμνFL(Q).
(181) -
Now, we verify the static and long-wavelength limit of the above linear response theory. In this limit, it is obvious that the density response function
Π00(Q→0) should recover the charge susceptibilityκX associated with the channel X, i.e.,Π00(Q→0)=−κX=∂2Ω(T,μX)∂μ2X.
(182) In condensed matter theory, this is the so-called compressibility sum rule [75]. Here, we emphasize that the correct static and long-wavelength limit of an arbitrary function
A(Q) should be understood asA(Q→0)=limq→0A(iql=0,q).
(183) In the mean-field theory, the thermodynamic potential is given by
ΩMF(μX,M) , where the dependence on the temperature is not explicitly shown. Note that the effective quark mass M is also an implicit function ofμX ,M=M(μX) , which should be determined by the mean-field gap equation∂ΩMF(μX,M)∂M=0.
(184) The charge susceptibility can be evaluated as
(κX)MF=−∂2ΩMF(μX,M)∂μ2X−∂2ΩMF(μX,M)∂μX∂M∂M(μX)∂μX.
(185) The quantity
∂M/∂μX can be deduced from the gap equation. We have∂2ΩMF(μX,M)∂μX∂M+∂2ΩMF(μX,M)∂M2∂M(μX)∂μX=0,
(186) which leads to
∂M(μX)∂μX=−∂2ΩMF(μX,M)∂μX∂M[∂2ΩMF(μX,M)∂M2]−1.
(187) Hence we obtain
(κX)MF=−∂2ΩMF(μX,M)∂μ2X+[∂2ΩMF(μX,M)∂μX∂M]2[∂2ΩMF(μX,M)∂M2]−1.
(188) In contrast, from the linear response theory, we have
Π00MF(Q)=Π00b(Q)−3∑m=0C0m(Q)C0m(−Q)12G+Πm(Q).
(189) In the static and long-wavelength limit
Q→0 , we haveC0m(Q)→0 form=1,2,3 . Thus, we obtainΠ00MF(Q→0)=Π00b(Q→0)−[C00(Q→0)]212G+Π0(Q→0).
(190) Using the explicit form of the above functions, we can show that
Π00b(Q→0)=∂2ΩMF(μX,M)∂μ2X,C00(Q→0)=∂2ΩMF(μX,M)∂μX∂M,12G+Π0(Q→0)=∂2ΩMF(μX,M)∂M2.
(191) Thus, the compressibility sum rule is satisfied in the mean-field theory, i.e.,
Π00MF(Q→0)=−(κX)MF.
(192) When the meson fluctuations are taken into account, we have
κX=(κX)MF+(κX)FL,
(193) where the meson-fluctuation contribution can be evaluated as
(κX)FL=−∂2ΩFL(μX,M)∂μ2X−2∂2ΩFL(μX,M)∂μX∂M∂M(μX)∂μX−∂2ΩFL(μX,M)∂M2[∂M(μX)∂μX]2−∂ΩFL(μX,M)∂M∂2M(μX)∂μ2X.
(194) We note that the effective quark mass
M(μX) is still determined by the mean-field gap equation. On the other hand, the meson-fluctuation contribution to the density response function can be decomposed asΠ00FL(Q)=Π00OP(Q)+Π00AL(Q)+Π00SE(Q)+Π00MT(Q).
(195) We can show that the first three terms in
(κX)FL are related to the sum of AL, SE, and MT contributions in theQ→0 limit,limQ→0[Π00AL(Q)+Π00SE(Q)+Π00MT(Q)]=∂2ΩFL(μX,M)∂μ2X+2∂2ΩFL(μX,M)∂μX∂M∂M(μX)∂μX+∂2ΩFL(μX,M)∂M2[∂M(μX)∂μX]2.
(196) To prove this, we recall that in the presence of only
A0(Q) , the induced perturbations are given byηm(Q)=−C0m(−Q)A0(Q)12G+Πm(Q)+O(A2),ηm(−Q)=−C0m(Q)A0(−Q)12G+Πm(Q)+O(A2).
(197) In the limit
Q→0 , onlyη0 survives and hencelimQ→0η0(Q)A0(Q)=−limQ→0C00(−Q)12G+Π0(Q)=∂M(μX)∂μX.
(198) The order parameter induced contribution,
Π00OP(Q) , is related to the last term in(κX)FL . We have\begin{array}{l} \Pi_{\rm OP}^{00}(Q\rightarrow0) = {\cal C}_0{\cal R}_2^{00}(Q\rightarrow 0). \end{array}
(199) Using the fact that
\begin{split} {\cal C}_0 = \frac{\partial\Omega_{\rm FL}(\mu_{X},M)}{\partial M}, \ \ \ \ \ \ \lim_{Q\rightarrow0}{\cal R}_2^{00}(Q) = \frac{\partial^2 M(\mu_{X})}{\partial \mu_{X}^2}, \end{split}
(200) we find that the the last term in
(\kappa_{X})_{\rm FL} is exactly given by the OP contribution. We can further understand this result by working out the explicit form\begin{split} \frac{\partial^2 M(\mu_{X})}{\partial \mu_{X}^2} =& - \left\{\frac{\partial^3\Omega_{\rm MF}(\mu_{X},M)}{\partial \mu_{X}^2\partial M}+2\frac{\partial^3\Omega_{\rm MF}(\mu_{X},M)}{\partial \mu_{X}\partial M^2} \frac{\partial M(\mu_{X})}{\partial\mu_{X}}\right.\\&\left.+\frac{\partial^3\Omega_{\rm MF}(\mu_{X},M)}{\partial M^3}\left[\frac{\partial M(\mu_{X})}{\partial\mu_{X}}\right]^2\right\}\\& \times\left[\frac{\partial^2\Omega_{\rm MF}(\mu_{X},M)}{\partial M^2}\right]^{-1}. \end{split}
(201) In summary, we have shown that the compressibility sum rule is exactly satisfied in the linear response theory including the meson fluctuations. The order parameter induced contribution is rather crucial to recover the correct static and long-wavelength limit.
-
One special case we are interested in is the chiral symmetry restored phase (
T>T_c ) in the chiral limit (m_0 = 0 ). In this case, we have{\cal C}_0 = 0 , and hence the order parameter induced contribution vanishes. Also, we haveC_{m}^\mu(Q) = 0 , indicating that we do not need to consider the induced perturbations\eta_{m}(Q) . In this case, the formalism becomes rather simple and we can identify various contributions diagrammatically.In the chiral symmetry restored phase, the sigma meson and pions become degenerate. We have
\begin{array}{l} [{ D}(Q)]_{mn} = {\cal D}(Q)\delta_{mn} , \end{array}
(202) where the propagator of the mesonic modes above
T_c is given by\begin{split} {\cal D}^{-1}(Q) =& \frac{1}{2G}+N_cN_f \int{{\rm d}^3{ k}\over (2\pi)^3}\Bigg[\left(\frac{1-f(E_{ k}^+)-f(E_{{ k}+{ q}}^-)}{iq_l-E_{ k}-E_{{ k}+{ q}}}\right.\\&\left.-\frac{1-f(E_{ k}^-)-f(E_{{ k}+{ q}}^+)}{iq_l+E_{ k}+E_{{ k}+{ q}}}\right) \left(1+\frac{{ k}\cdot ({ k+ q})}{E_{ k} E_{{ k}+{ q}}}\right)\\ &+\left(\frac{f(E_{ k}^-)-f(E_{{ k}+{ q}}^-)}{iq_l+E_{ k}-E_{{ k}+{ q}}}-\frac{f(E_{ k}^+)-f(E_{{ k}+{ q}}^+)}{iq_l-E_{ k}+E_{{ k}+{ q}}}\right)\\&\times \left(1-\frac{{ k}\cdot ({ k+ q})}{E_{ k} E_{{ k}+{ q}}}\right)\Bigg]. \end{split}
(203) Here
E_{ k} = |{ k}| forT>T_c . Due to the degeneracy of the sigma meson and pions, various contributions to the linear response aboveT_c become simple. -
Above
T_c , the Aslamazov-Lakin contribution is given by\begin{split} {\cal W}_{\rm FL}^{({\rm AL})} = \frac{\beta V}{2}\sum\limits_{Q} \Pi_{\rm AL}^{\mu\nu}(Q)A_{\mu}(-Q)A_{\nu}(Q), \end{split}
(204) where the AL response function
\Pi_{\rm AL}^{\mu\nu}(Q) is given by\begin{split} \Pi_{\rm AL}^{\mu\nu}(Q) = -\frac{2}{\beta V}\sum\limits_{P}\left[{\cal D}(P){\cal D}(P+Q){\cal X}^\mu(P,Q){\cal X}^\nu(-P,-Q)\right]. \end{split}
(205) The function
{\cal X}^\mu(P,Q) here is defined as\begin{split} {\cal X}^\mu(P,Q) =& \frac{1}{\beta V}\sum\limits_K{\rm Tr} \left[{\cal G}(K)\Gamma^{\mu}{\cal G}(K+Q){\cal G}(K-P)\right]\\ &+\frac{1}{\beta V}\sum_K{\rm Tr}\left[{\cal G}(K-Q)\Gamma^{\mu}{\cal G}(K){\cal G}(K+P)\right]. \end{split}
(206) Here, the quark propagator
{\cal G}(K) is given in Eq. (85) withM = 0 . The AL contribution can be diagrammatically demonstrated in Fig. 5. -
Above
T_c , the self-energy or density-of-state contribution is given by\begin{split} {\cal W}_{\rm FL}^{({\rm SE})} = \frac{\beta V}{2}\sum\limits_{Q} \Pi_{\rm SE}^{\mu\nu}(Q)A_{\mu}(-Q)A_{\nu}(Q), \end{split}
(207) where the SE response function
\Pi_{\rm SE}^{\mu\nu}(Q) is given by\begin{split} \Pi_{\rm SE}^{\mu\nu}(Q) = \frac{8}{\beta V}\sum\limits_{P}\left[{\cal D}(P){\cal Y}^{\mu\nu}(P,Q)\right]. \end{split}
(208) Here, the function
{\cal Y}^{\mu\nu}(P,Q) is explicitly given by\begin{split} {\cal Y}^{\mu\nu}(P,Q) = \frac{1}{\beta V}\sum\limits_{K}{\rm Tr}\left[{\cal G}(K)\Gamma^{\mu}{\cal G}(K+Q)\Gamma^{\nu}{\cal G}(K){\cal G}(K+P)\right]. \end{split}
(209) Note that
\Pi_{\rm SE}^{\mu\nu}(Q) can also be written as\begin{split} \Pi_{\rm SE}^{\mu\nu}(Q) = \frac{2}{\beta V}\sum\limits_{K}{\rm Tr}\left[\Gamma^{\mu}{\cal G}(K+Q)\Gamma^{\nu}{\cal G}(K)\Sigma_{q}(K){\cal G}(K)\right], \end{split}
(210) where
\Sigma_{q} is the quark self-energy,\begin{split} \Sigma_{q}(K) = \frac{4}{\beta V}\sum\limits_{P}\left[{\cal D}(P){\cal G}(K+P)\right]. \end{split}
(211) The SE contribution is diagrammatically illustrated in Fig. 6.
Figure 6. Diagrammatic representation of the self-energy contribution. Notations are the same as in Fig. 5.
-
Above
T_c , the Maki-Thompson contribution is given by\begin{split} {\cal W}_{\rm FL}^{({\rm MT})} = \frac{\beta V}{2}\sum\limits_{Q} \Pi_{\rm MT}^{\mu\nu}(Q)A_{\mu}(-Q)A_{\nu}(Q), \end{split}
(212) where the MT response function
\Pi_{\rm MT}^{\mu\nu}(Q) is given by\begin{split} \Pi_{\rm MT}^{\mu\nu}(Q) = \frac{4}{\beta V}\sum\limits_{P}\left[{\cal D}(P){\cal W}^{\mu\nu}(P,Q)\right]. \end{split}
(213) Here, the function
{\cal W}^{\mu\nu}(P,Q) is explicitly given by\begin{split} {\cal W}^{\mu\nu}(P,Q) =& \frac{1}{\beta V}\sum\limits_{K}{\rm Tr} \left[{\cal G}(K)\Gamma^{\mu}{\cal G}(K+Q)\right.\\&\times\left.{\cal G}(K+P+Q)\Gamma^{\nu}{\cal G}(K+P)\right]. \end{split}
(214) The MT contribution is diagrammatically illustrated in Fig. 7.
Figure 7. Diagrammatic representation of the Maki-Thompson contribution. Notations are the same as in Fig. 5.
In summary, the AL, SE, and MT contributions can be diagrammatically identified in the chiral symmetry restored phase. These contributions include the propagator of the degenerate mesonic modes,
{\cal {D}}(Q) . Near the chiral phase transition temperature, these mesonic modes are soft modes and are nearly massless. Therefore, we expect that these mesonic modes have a significant effect on the transport properties near the transition.
Response functions of hot and dense matter in the Nambu-Jona-Lasino model
- Received Date: 2019-04-10
- Accepted Date: 2019-06-09
- Available Online: 2019-09-01
Abstract: We investigate current-current correlation functions, or the so-called response functions of a two-flavor Nambu-Jona-Lasino model at finite temperature and density. The linear response is investigated introducing the conjugated gauge fields as external sources within the functional path integral approach. The response functions can be obtained by expanding the generational functional in powers of the external sources. We derive the response functions parallel to two well-established approximations for equilibrium thermodynamics, namely mean-field theory and a beyond-mean-field theory, taking into account mesonic contributions. Response functions based on the mean-field theory recover the so-called quasiparticle random phase approximation. We calculate the dynamical structure factors for the density responses in various channels within the random phase approximation, showing that the dynamical structure factors in the baryon axial vector and isospin axial vector channels can be used to reveal the quark mass gap and the Mott dissociation of mesons, respectively. Noting that the mesonic contributions are not taken into account in the random phase approximation, we also derive the response functions parallel to the beyond-mean-field theory. We show that the mesonic fluctuations naturally give rise to three kinds of famous diagrammatic contributions: the Aslamazov-Lakin contribution, the self-energy or density-of-state contribution, and the Maki-Thompson contribution. Unlike the equilibrium case, in evaluating the fluctuation contributions, we need to carefully treat the linear terms in external sources and the induced perturbations. In the chiral symmetry breaking phase, we find an additional chiral order parameter induced contribution, which ensures that the temporal component of the response functions in the static and long-wavelength limit recovers the correct charge susceptibility defined using the equilibrium thermodynamic quantities. These contributions from mesonic fluctuations are expected to have significant effects on the transport properties of hot and dense matter around the chiral phase transition or crossover, where the mesonic degrees of freedom are still important.