Large N solution of generalized Gross-Neveu model with two coupling constants Christian Boehmer and Michael Thies ∗ † Institut fu¨r Theoretische Physik III, Universita¨t Erlangen-Nu¨rnberg, D-91058 Erlangen, Germany (Dated: January 7, 2010) The Gross-Neveu model in 1+1 dimensions is generalized to the case of different scalar and pseudoscalar coupling constants. This enables us to interpolate smoothly between the standard massless Gross-Neveu models with either discrete or continuous chiral symmetry. We present the solution of the generalized model in the large N limit including the vacuum, fermion-antifermion scattering and bound states, solitonic baryons with fractional baryon number and the full phase diagram at finite temperatureand chemical potential. 0 1 PACSnumbers: 11.10.-z,11.10.Kk,11.10.St 0 2 I. INTRODUCTION differences in the phase diagrams and baryon structure n come about. Moreover, we would like to explore an al- a J ThesustainedinterestinGross-Neveu(GN)modelsin ternative mechanism for breaking chiral symmetry ex- 7 1+1dimensions[1]stemstoalargeextentfromtheirchi- plicitly, different fromthe usual bare mass term. To this ralproperties. ThusthesimplestmodelwithLagrangian end,weconsideraLagrangiansimilartoEq.(3),butwith ] different (attractive) scalar and pseudoscalar couplings, h 1 -t L=ψ¯iγµ∂µψ+ 2g2(ψ¯ψ)2 (1) =ψ¯iγµ∂ ψ+ 1g2(ψ¯ψ)2+ 1G2(ψ¯iγ ψ)2. (5) p L µ 2 2 5 e (suppressing flavor indices, i.e., ψ¯ψ = N ψ¯ ψ etc.) h k=1 k k ByvaryingG2 from0tog2,wegenerateafamilyoftheo- [ has a discrete chiral Z2 symmetry P riesinterpolatingbetweenthe GNandtheNJL2 models. 3 ψ γ ψ, (2) TheideatogeneralizetheGNmodelinthisfashionisnot 5 v → new. Thus for instance, Klimenko has studied a closely 4 whereas the chiral GN model or, equivalently, the two- relatedproblemlongtimeago[9,10]. However,sincethe 1 dimensional Nambu–Jona-Lasinio model (NJL ) [2], role of inhomogeneous condensates has only been appre- 7 2 ciatedinrecentyears,thereisalmostnooverlapbetween 3 . =ψ¯iγµ∂ ψ+ 1g2(ψ¯ψ)2+ 1g2(ψ¯iγ ψ)2, (3) the present work and these earlier studies. 9 L µ 2 2 5 The methods which we shall use in our investigation 0 havebeendevelopedduringthelastfewyearsinaneffort 9 possesses a continuous chiral U(1) symmetry, toclarifythephasestructureofmasslessandmassiveGN 0 models. Asaresult,wehavenowatourdisposalawhole : ψ eiαγ5ψ. (4) v → toolbox of analytical and numerical instruments. The Xi Chiral symmetry and in particular its breakdown man- most important keywords are: the derivative expansion, asymptotic expansions, perturbation theory, Ginzburg- r ifest themselves in such diverse physical phenomena as a dynamical fermion masses, the meson spectrum, topo- Landau (GL) theory and numerical Hartree-Fock (HF) approach including the Dirac sea. This will enable us to logicaleffects inthe structure ofbaryons,andrichphase solve the generalized GN model (5) in a rather straight- diagrams at finite density and temperature with various forward fashion, although the model is far from trivial. types of homogeneous and solitonic crystal phases, see Its two limiting cases, the standard massless GN and the introductory review article [3] as well as the recent NJL models, can both be solved analytically. This is updates in [4–6]. By adding a bare mass term to the 2 unfortunately not true for the generalized model which Lagrangian, one breaks the chiral symmetry explicitly in this respect is closer to the massive NJL model [8]. and gets additional insights into the symmetry aspects 2 This paper is organized as follows. We present our of both models [7, 8]. Nevertheless, studies of models computations and results starting with mostly analyti- (1)and(3)withtheirstrikinglydifferentpropertieshave cal work and ending with purely numerical results. The remained somewhat disconnected. logic of the HF approach demands that we begin with a In the present work, we propose and solve a simple discussion of the vacuum, dynamical fermion mass and field theoretical model which interpolates continuously coupling constant renormalization in Sec. II. Sec. III is between the Lagrangians (1) and (3). Our motivation dedicatedto fermion-fermionbound states(mesons)and is to get a better understanding of how the conspicuous scattering. In Sec. IV, we solve the theory in the baryon sector as well as for low density soliton crystals in the vicinityofthechirallimit,usingakindofchiralperturba- ∗[email protected] tion theory obtained from the derivative expansion. We †[email protected] then begin our study of thermodynamics at finite tem- 2 perature and chemical potential with an investigationof IfwechoosethefollowingrelationsbetweentheUVcutoff the tricriticalbehaviornearthe chirallimit inSec.V. In Λ/2 and the bare coupling constants g2,G2, Sec. VI the microscopic GL approach underlying Sec. V π is extended to more general coupling constants, and the lnΛ = ξ , Ng2 − 1 tricritical point of the generalized GN model is deter- π mined exactly. Some technical details are deferred to lnΛ = ξ , (11) the appendix. Sec. VII is devoted to the full phase dia- NG2 − 2 gramofthe generalizedGNmodelforarbitrarycoupling (m,M)is welldefinedinthe limit Λ (dropping vac constants, chemical potential and temperature, only ac- E →∞ the irrelevant quadratic divergence) and given by cessible via a numericalrelativistic HF calculation. As a by-product, we also present information about baryons m2+M2 ξ m2 ξ M2 away from the chiral limit. The paper ends with a con- vac = ln m2+M2 1 + 1 + 2 . E 4π − 2π 2π cluding section, Sec. VIII. (12) (cid:2) (cid:0) (cid:1) (cid:3) Minimize with respect to m,M, vac E II. VACUUM, DYNAMICAL FERMION MASS, 0 = m 2ξ +ln m2+M2 , 1 RENORMALIZATION 0 = M(cid:2)2ξ2+ln(cid:0)m2+M2(cid:1)(cid:3) . (13) Consider the Lagrangianof the generalizedGN model Theseequationsonly(cid:2)admitas(cid:0)olutionwit(cid:1)h(cid:3)nonvanishing with two coupling constants in 1+1 dimensions, Eq. (5). m and M if ξ = ξ = 1ln(m2+M2). This takes us For G2 = g2, it coincides with the one from the mass- back to the N1JL 2mode−l2with its infinitely degenerate 2 less NJL2 model, Eq. (3). For G2 = 0, we recover the vacua along the chiral circle of radius √m2+M2. The massless GN model, Eq. (1). The case G2 > g2 can be other options are m=0,M =0,ξ unspecified and 2 mappedontoG2 <g2bymeansofachiralrotationabout 6 a quarter of a circle, 1 m2 ξ = lnm2, = , (14) 1 vac −2 E −4π ψ eiγ5π/4ψ. (6) → or else m=0,M =0,ξ unspecified and 1 6 Since this is a canonical transformation, we may assume 0<G2 <g2 without loss of generality. Hence the gener- ξ = 1lnM2, = M2. (15) alized GN model can serve as a continuous interpolation 2 −2 Evac − 4π between two well-studied model field theories with dis- Thevacuumenergyislowestform=0ifξ <ξ andfor tinct symmetry properties. Notice that the generalized 6 1 2 M =0ifξ >ξ . InviewoftheremarkbelowEq.(6),we Lagrangian (5) always has the discrete chiral symmetry 6 1 2 ψ γ ψ under which ψ¯ψ and ψ¯iγ ψ change sign. The may adopt the first scenario. Choosing units such that con→tinu5ouschiralsymmetryψ eiα5γ5ψisonlyrecovered m=1anddenotingξ2(>0)byξ fromnowon,wefinally at the point g2 =G2. → get the renormalization conditions (gap equations) To find the vacuum in the large N limit, we introduce π = lnΛ, homogeneous scalar and pseudoscalar condensates, Ng2 π π m = g2 ψ¯ψ , = ξ+ =ξ+lnΛ. (16) − h i NG2 Ng2 M = G2 ψ¯iγ ψ . (7) 5 − h i With the help of these relations, all physical quantities The Dirac-Hartree-Fockequation can be expressed in terms of the scale m (set equal to 1) and the dimensionless parameter ξ which serves to γ i∂ +γ0m+iγ1M ψ =Eψ (8) − 5 x interpolate between the massless NJL2 (ξ = 0) and GN then yields t(cid:0)he single particle energ(cid:1)ies (ξ =∞)models. Thisexpectationisborneoutinthefol- lowingsections, supporting ourrenormalizationmethod. E = k2+m2+M2 (9) ± and the (cutoff regularipzed) vacuum energy, III. MESON SPECTRUM AND FERMION-ANTIFERMION SCATTERING Λ/2 dk m2 M2 = k2+m2+M2+ + Evac − 2π 2Ng2 2NG2 In the large N limit, fermion-antifermion bound and Z−Λ/2 p scattering states can conveniently be derivedvia the rel- Λ2 m2+M2 m2+M2 = + ln 1 ativistic random phase approximation (RPA) [11, 12]. −8π 4π Λ2 − (cid:20) (cid:18) (cid:19) (cid:21) Since the scalarand pseudoscalarchannels decouple and m2 M2 theHFvacuumisthesameasintheGNorNJL model, + + . (10) 2 2Ng2 2NG2 thisanalysisrequiresonlyminorchangesofthestandard 3 calculationfortheNJL model. Considerfirstthebound now free of divergences. Eq. (23) is the same as in the 2 stateproblem. Thescalarchannelhasbeenspelledoutin standard GN and NJL models and gives the familiar 2 alldetailinRef.[12]whereitisshownthattheeigenvalue result for the scalar (σ) meson mass, =2. The right- M equation assumes the form handsideofEq.(24)isindependentofP andcanreadily be evaluated in the cm frame of the meson (P =0), dk 1 = 2Ng2 u¯(k)v(k P)u¯(k P)v(k) 2π − − 2 1 Z ξ = M dk E(k P,k) − 2 √k2+1( 2 4 4k2) × 2(P) E−2(k P,k) (17) 1 Z 1 M − − E − − = arctan (25) Here, P is the total momentum of the fermion- √η 1 √η 1 − − antifermionsystem,u,v arepositiveandnegativeenergy with HF spinors, and 4 E(k)= k2+1, E(k ,k)=E(k )+E(k). (18) η = (26) ′ ′ 2 M The energpy of the meson is denoted by (P) = E Solvingthetranscendentalequation(25)numerically,the √P2+ 2. An analogous computation in the pseu- pseudoscalar(π)mesonmassisfoundtorisefrom =0 M doscalar channel gives M at ξ = 0 to 2 at ξ , see Fig. 1. The first limit is as → ∞ dk expected – this is the would-be Goldstone boson of the 1 = −2NG2 2πu¯(k)iγ5v(k−P)u¯(k−P)iγ5v(k) NJL2 model. The 2nd one is surprising at first glance, Z since we are supposed to reach the GN model in this E(k P,k) − . (19) limit. The GN model does not have any pseudoscalar × 2(P) E2(k P,k) fermion-antifermioninteraction, let alone a bound state. E − − Tobetterunderstandwhatisgoingon,webrieflyturn Use of the identities tothefermion-antifermionscatteringproblem. Since the 4+P2 E2(k P,k) RPA equations have a separable kernel with one-term u¯(k)v(k P)u¯(k P)v(k)= − − − − 4E(k)E(k P) separablepotentialsinthe scalarandpseudoscalarchan- − (20) nels,thisis straightforward[13]. Theenergydependence ofthe scatteringmatrix isencodedinthe followingfunc- P2 E2(k P,k) u¯(k)iγ v(k P)u¯(k P)iγ v(k)= − − tions of the Mandelstam variable s, 5 5 − − − 4E(k)E(k P) − (21) Ng2 putstheseeigenvalueequationsintothemoreconvenient τσ = 1+Ng2 dk 1 4k2 form 2π√1+k2 s 4(1+k2)+iǫ − Ng2 dk 1 1 τ = R NG2 (27) 1 = 2 Z 2π (cid:18)E(k−P) + E(k)(cid:19) π 1+NG2 2dπk√11+k2s−44((11++kk22))+iǫ 4+P2 E2(k P,k) R − − , Upon isolating the divergent part of the integrals and × 2(P) E2(k P,k) using the renormalizationconditions, this becomes E − − NG2 dk 1 1 1 = + (s 4) 2 Z 2π (cid:18)E(k−P) E(k)(cid:19) τσ−1 = 2−π I(s) P2 E2(k P,k) ×E2(P−)−E2(k−−P,k). (22) τπ−1 = πξ + 2sπI(s) If we regularize the momentum integrals with the same 1 1 I(s) = dk (28) cutoffΛ/2asusedinthetreatmentofthevacuumenergy √1+k2s 4(1+k2)+iǫ Z − and use the renormalizationconditions Eqs.(16), we get the renormalized eigenvalue conditions where the integral I(s) can be evaluated in closed form, dk 1 1 2 s 0 = + I(s)= arctan (s<4) (29) 2π E(k P) E(k) − s(4 s) 4 s Z (cid:18) − (cid:19) − r − 4+P2 2(P) −E , (23) p × 2(P) E2(k P,k) E − − 1 √s+√s 4 2ξ dk 1 1 I(s)= ln − iπ (s>4) = + s(s 4) √s √s 4 − π 2π E(k P) E(k) − (cid:18) − − (cid:19) Z (cid:18) − (cid:19) (30) P2 2(P) τ has thepexpected pole at s = 4 corresponding to the −E , (24) σ × 2(P) E2(k P,k) marginally bound scalar meson with = 2. The pole E − − M 4 σ sates in the baryon state are x dependent, 2 γ i∂ +γ0S(x)+iγ1P(x) ψ =Eψ, (31) 5 x − π with (cid:2) (cid:3) 1.5 S = g2 ψ¯ψ , − h i M P = G2 ψ¯iγ ψ . (32) 5 − h i 1 As is well known, the HF energy can be written as the sum over single particle energies of occupied orbits and a double counting correction. Only this last part is dif- 0.5 ferent in the present case. Due to the renormalization condition (16), it depends on the parameter ξ, 0 2 4 6 8 10 = S2 + P2 = S2+P2 lnΛ+ ξ P2. (33) ξ Ed.c. 2Ng2 2NG2 2π 2π The cutoff dependent term cancels exactly the logarith- FIG. 1: Masses of σ and π mesons vs. ξ in the large N limit mic divergence in the sum over single particle energies. of the generalized GN model, obtained from Eqs. (23-26). Only the last term in Eq. (33) is different from what it was before. Consequently, we can simply take over the effective action from Ref. [16], set the confinement pa- ofτ inturncoincides withthe massofthe pseudoscalar rameter γ = 0 (vanishing bare fermion mass) and add π meson, see Eqs. (25,26). According to the 2nd line of the new contribution proportional to ξ from Eq. (33). Eq.(28), the strengthofthe pseudoscalarscatteringma- Adopting polar coordinates in field space, trix vanishes like 1/ξ for ξ . We therefore arrive ∼ → ∞ S iP =(1+λ)e2iχ, (34) atthe followingpicture: As ξ , the pseudoscalarin- − →∞ teraction vanishes, in accordance with the expected GN andworkingatthesameorderinthederivativeexpansion limit. However, since an arbitrary weak attractive in- as in[16], we then getat once the energy density ( =∂ ′ x teraction is sufficient to support a bound state in 1+1 and χIV denotes the 4th derivative of χ) dimensions, the pseudoscalar bound state pole persists, 1 1 the binding energy going to zero. As we shall see later 2π = ξ(1+λ)2sin2(2χ)+(χ)2 (χ )2+ (χ )2 ′ ′′ ′′′ on,thisdecoupledπ mesonhasnoinfluenceonanyother E − 6 30 observables of the model in the large N limit, so that it 1 (χIV)2 1 (χ )4+λ2+ 1 (λ)2+ 1λ3 ′′ ′ does not really upset our goal of interpolating between − 140 − 45 12 3 the NJL2 and GN models. 1 (λ )2 1λ(λ)2 1 λ4+ 1λ(χ )2 ′′ ′ ′′ − 120 − 6 − 12 3 1 1 1 + λ(χ )2+ λχ χIV λ2(χ )2. (35) ′′′ ′′ ′′ IV. BARYONS AND SOLITON CRYSTALS AT 15 5 − 2 SMALL ξ AND LOW DENSITY We have to vary the energy functional with respect to λ and χ and solve the Euler-Lagrange equations, then The derivative expansion is a standard technique compute baryonnumber and baryonmass. Although we to deal with quantum mechanical particles subject to shallfollowthesameprocedureasinRef.[16],theresults smooth potentials [14, 15]. In Ref. [16] it has been will be quite different, reflecting the different ways in adapted to the particular needs of the HF approach whichchiralsymmetryisbrokeninthesetwomodels. For for low dimensional fermion field theories. In effect, it simplicity, take first the case of the leading order (LO) amounts to integrating out the fermions in favor of an derivative expansion. Here, we only keep two terms in effective bosonic field theory, where the scalar and pseu- the energy density, doscalarfieldscanbeidentifiedwiththeHFpotentialsre- lated to the composite fermion operatorsψ¯ψ and ψ¯iγ5ψ. 2πE =ξsin2(2χ)+(χ′)2. (36) ForbaryonsinthemassiveNJL modelitleadstoachiral 2 Rescaling the chiral phase field and its spatial argument expansion in closed analytical form [16]. Note that this as follows, method can only handle fully occupied valence levels at present. 1 χ(x)= θ(y), y =2 ξx, (37) Since the HF equationin the problem athand has the 4 same form as in the NJL model, we can take over the p 2 we recognize the (static) sine-Gordon action (˙ =∂ ) derivationoftheeffectiveactionfromRef.[16]almostlit- y erally. The Dirac-HFequationis written asin Eqs.(7,8) 4π 1 = θ˙2 cosθ+1. (38) except that the scalar(S) and pseudoscalar(P) conden- ξ E 2 − 5 The Euler-Lagrange equation is the time-independent AsthewholewindingnumberofχresidesintheLOterm sine-Gordon equation χ ,baryonnumberisalways1/2. Thereforethe complex 0 potential S iP traces out half a turn aroundthe chiral θ¨=sinθ, (39) − circle. This is confirmed by plotting S and P, showing kink-like behavior of S like in the massless GN model, sothatthebaryoncanbeidentifiedwiththesine-Gordon see Fig. 2. The presence of a non-vanishing P signals kink that we are dealing with a new kind of solitonic baryon θ =4arctaney. (40) here which did not show up yet in any other variant of the GN model family. But unlike in the massive NJL2 model, this object has Let us now turn to periodic solutions of the Euler- baryonnumber1/2,exactlylikethekinkinthestandard Lagrange equations in the derivative expansion. They GN model (with fully occupied zero-mode), are expected to approximate systematically the ground state ofmatter atlow densities andin the vicinity ofthe χ 1 1 N = dx ′ = [χ( ) χ( )]= . (41) chiral limit ξ = 0. Since the resulting expressions are B π π ∞ − −∞ 2 rather lengthy, we only give them up to NNLO here, Z Here we have used the topological relationship between π 1 baryon number and winding number of the chiral phase χ0 = + am 4 2 [11,16]. The massofthis kink-likebaryonis foundto be 1 λ = cn2 1 M √ξ m −4 B π = = , (42) ζ 1 ζ N π 2π χ = + sncn dnZ 1 24 16 − 24κ2 where, in the 2nd step, we have made use of Eq. (25) to (cid:18) (cid:19) 13 ζ ζ 1+κ2 4 κ2 LOIninthξeasnadmdeevneointe,dhitghheeprioorndemracsaslcbuylamtiπo.ns closely fol- λ2 = 96 − 24 sn4+ 24 − 24κ2 sn2+ 9−6κ2 (cid:18) (cid:19) (cid:18) (cid:19) low Ref. [16]. We find it useful to switch from the pa- ζ sncndnZ rameter ξ to mπ by means of Eq. (25), − 24κ2 ζ3 (κ2 5)ζ2 (61+30κ2)ζ ξ ≈ 14m2π+ 214m4π + 1210m6π+ 5160m8π, (43) χ2 = (cid:18)576κ2 + 57−6κ2 + 2880κ2 59κ2 44 and to expand χ and λ into Taylor series in mπ, + − sncn 2304κ2 (cid:19) χλ ≈≈ χm02π+λ1m+2πmχ14π+λ2m+4πmχ26π+λ3m. 6πχ3, (44) − (cid:18)57ζ63κ4 + (κ228−8κ34)ζ2 + (612+88300κκ42)ζ(cid:19)dnZ The Euler-Lagrange equations corresponding to the ef- 13 + ζ + ζ2 sn3cn ζ2 sncnZ2 fective action (35) can then be solved analytically with − 1152 96 576 − 576κ2 (cid:18) (cid:19) the NNNLO results (y =mπx) ζ2 ζ + + dnsn2Z (47) χ0 = arctaney (cid:18)288κ2 96κ2(cid:19) 1 1 Here, λ = 1 −4cosh2y K 1 sinhy ζ =(1 κ2) , (48) χ = − E 1 16cosh2y E,Karecompleteellipticintegralsofκandam,sn,cn,dn 1 10cosh2y 13 λ = − (45) and Z are standard Jacobi elliptic functions with spatial 2 −96 cosh4y argument 1 sinhy(11cosh2y 26) m χ2 = −2304 cosh4y − z = κπx (49) 1 562cosh4y 3090cosh2y+2811 andelliptic modulus κ. The meandensity canbe simply λ = − 3 −5760 cosh6y inferred from the period of the crystal, sinhy (6271cosh4y+29588cosh2y 26784) m π χ = − ρ= . (50) 3 1382400 cosh6y 4κK By way of example, we show in Fig. 3 the scalar and The baryon mass becomes pseudoscalar potentials corresponding to ξ = 0.2 (as in MB = mπ 1 1 m2 + 13 m4 1193 m6 (46) Fig. 2) and the density ρ=0.05. Again the convergence N 2π − 36 π 3600 π− 705600 π seems to be very good. (cid:18) (cid:19) 6 1 us with a way of testing the results from the derivative expansion. Starting point is the divergence of the axial S current in the generalized GN model 0.5 ∂ jµ = 2(g2 G2)ψ¯ψψ¯iγ ψ µ 5 − − 5 = 2 Sψ¯iγ ψ Pψ¯ψ 5 − S,P 0 P 1 1 (cid:0) (cid:1) = 2N SP − NG2 − Ng2 (cid:18) (cid:19) 2Nξ = SP, (52) –0.5 − π where we have taken a ground state expectation value and used large N factorization. Owing to the properties –1 –4 –2 0 2 4 y j0 =j1, j1 =j0 (53) 5 5 FbDEaIqaGrssy.h.oe(n4d25:ic)nu.PSfragrepla emrStvcheaesl:adrLeOr(iSv(a)stiinvaene-dGexopprsdaeonunsdi)oo,nssc,oalξliadr=cu(0rPv.2)e,sm:pNoπtNe≈nNtiL0aO.l8s,38sfoe9er. specific for 1+1∂1dρim(xe)n=sio−ns2,NπwξeSg(ext)Pfo(rxs)t.ationary sta(5te4s) en ts Twofold integration for the baryon case then leads to a sum rule relating baryon number directly to an integral 1 over the HF potentials S,P, 2Nξ x S ρ(x) = dxS(x)P(x) (55) ′ ′ ′ 0.5 − π P 1 = 2ξ Z∞−∞dx x dxS(x)P(x) ′ ′ ′ 2 − π 0 Z−∞ Z−∞ S;P 2ξ ∞ = dxxS(x)P(x). (56) π Z−∞ –0.5 In the last step, partial integration was used. Inserting the results for S,P from the baryon, i.e., –1 S = +(1+λ)cos(2χ), –8 –6 –4 –2 0 2 4 6 8 10 P = (1+λ)sin(2χ), (57) − x withχ,λfromEqs.(45),wefindthatthesumrule(56)is FIG. 3: Soliton crystal for generalized GN model, ξ = only violated atO(m8). This is a goodindependent test 0.2,mπ 0.8389,ρ = 0.05. Dashed curves: LO (sine Gor- π ≈ ofaconsiderableamountofalgebrabehindthederivative don), solid curves: NNLO,see Eqs. (47). expansion. Since the derivative expansion is anyway expected to V. PHASE DIAGRAM NEAR THE NJL2 be most useful at low densities, we note the following TRICRITICAL POINT (ξ =0) simplification in the low density limit: for κ 1, we → can use the approximation ζ 0 and keep κ only in the We startourinvestigationof the phase diagramof the ≈ arguments of the Jacobi elliptic functions. Expressions generalized GN model by zooming in onto the tricritical (47) then reduce to periodic extensions of the baryon point at ξ = 0, i.e., of the NJL2 model. In Ref. [18] it results obtained by simply replacing wasshownthatthisregioniswellsuitedforthederivative expansion,whichhereleadstoa(microscopic)Ginzburg- 1 sn(z,κ) Landau type theory. In that work, chiral symmetry was coshy , sinhy (51) → cn(z,κ) → cn(z,κ) broken as usual by means of a bare fermion mass term. Here instead we break it by choosing two slightly dif- in Eqs. (45). ferent coupling constants in the scalar and pseudoscalar Finally, we derive a sum rule for the baryon number channels. The central quantity of interest is the grand of a single baryon, following Ref. [17]. This will equip canonical potential which differs in these two cases only 7 by the double counting correction. Since the latter is with the constant independent of temperature and chemical potential, the 16e2C situation is very similar to the one in the preceding sec- a= 6.03198, (66) tion. Once again we can take over the effective action 7ζ(3) ≈ from the literature about the massive NJL model [18]. 2 the reduced grand canonical potential density The only necessary modification is to replace the double counting correction term coming from the bare mass by 2πa the one proportional to ξ, cf. Eq. (33). For the present Ψ˜eff = ξ2 Ψeff (67) purpose,itisadvantageoustocombinetheHFpotentials S,P into one complex field φ = S iP. The result for becomes indeed independent of ξ, − thegrandcanonicalpotentialdensitytotheorderneeded here (dropping a field independent part) then becomes Ψ˜eff = ϕ˙ 2 iν(ϕϕ˙∗ ϕ˙ϕ∗)+(ν2 σ2)ϕ2+ ϕ4 | | − − − | | | | a ξ (ϕ ϕ∗)2. (68) Ψ =α φ2+α (φφ )+α φ4+ φ 2 + ( φ)2 −4 − eff 2 3 ′∗ 4 ′ | | ℑ | | | | 2π ℑ (58) The Euler-Lagrangeequation (cid:0) (cid:1) with a ϕ¨ 2iνϕ˙ +(σ2 ν2)ϕ 2ϕ2ϕ (ϕ ϕ )=0 (69) 1 − − − | | − 2 − ∗ α = [ln(4πT)+ Ψ(z)] 2 2π ℜ differsfromthecomplexnon-linearSchr¨odingerequation 1 α3 = −8π2TℑΨ(1)(z) by the term ∼ ϕ∗. This has prevented us from finding the solution in closed analytical form. Let us first de- 1 α = Ψ(2)(z) (59) termine the expected 2nd order phase boundaries. The 4 −64π3T2ℜ phase boundary between massless and massive homoge- and neousphasescaneasilybefoundbyminimizingΨ˜ with eff the ansatz ϕ=m and setting m=0 in the condition for 1 iµ z = + . (60) the non-trivial solution. The result in the new coordi- 2 2πT nates is the straight line We denote the digamma and polygamma functions as σ =ν. (70) d dn Ψ(z)= lnΓ(z), Ψ(n)(z)= Ψ(z). (61) dz dzn Nextconsiderthe phaseboundaryseparatingthe crystal phase from the chirally restored (m = 0) homogeneous In the chiral limit (ξ =0), the tricritical point is located phase. Here we use the ansatz (see Sec. IV of Ref. [18] at for the justification) eC µ =0, T =T = (62) ϕ=c cos(qu)+id sin(qu) (71) t t c 0 0 π with Euler’s constant C 0.577216. Following Ref. [18], and evaluate the spatial average of Ψ˜eff, keeping only weexpandthecoefficient≈s(59)ofthe GLeffectiveaction terms up to 2nd order in c0,d0, around the tricritical point (62), Ψ˜ = c2+2 c d + d2, (72) h effi M11 0 M12 0 0 M22 0 7 1 α2 ζ(3)e−2Cµ2 e−Cτ2 with ≈ 8π − 2 α3 ≈ 87πζ(3)e−2Cµ M11 = 12(q2+ν2−σ2) 7 α4 ζ(3)e−2C (63) M12 = −νq ≈ 32π 1 = (a+q2+ν2 σ2). (73) 22 with τ = √T T. The ξ-dependence can now be re- M 2 − c − moved as follows. Rescaling the field and the coordiante As explained in Ref. [8], the phase boundary is now de- according to fined by the conditions φ(x) = ξ1/2ϕ(u), u=ξ1/2x ∂ det =0, det =0, (74) φ′(x) = ξϕ˙(u), φ′′(x) = ξ3/2ϕ¨(u) (64) M ∂q2 M and introducing rescaled thermodynamic variables yielding the critical curve 2µ a τ a(8ν2 a) ν = , σ = (65) σ = − . (75) ξ1/2 rTcξ1/2 p 4ν 8 The wave number q obeys 5 a q = σ2+ν2 . (76) − 2 4 r The tricritical point can be identified with the point of intersection of the two critical curves (70) and (75), 3 m>0 crystal σ √a σ =ν = . (77) t t 2 2 Goingbacktotheoriginal,unscaledvariables,thistrans- lates into 1 m=0 1 T = T 1 ξ , t c − 4 (cid:18) (cid:19) √a 0 1 2 3 4 5 µ = ξ1/2. (78) ν t 4 FIG. 4: Rescaled phase diagram near the tricritical point of Notice that q vanishes at the tricritical point. We ex- the NJL2 model. Straight line: 2nd order phase boundary, pectthatathirdcriticallineendsatthetricriticalpoint, Eq.(70). Dashedcurve: 2ndorderphaseboundary,Eq.(75). namelythe1storderphaseboundaryseparatingthecrys- Solidcurve: 1storderphaseboundary,numericalcalculation. tal from the massive Fermi gas phase. It has to be de- The 3 critical curves meet at the tricritical point σt = νt = termined numerically. To this end, we insert the Fourier √a/2. Theparameter ξ hasbeen eliminated bythechoiceof series ansatz variables, see Eq. (65). ϕ= c cos[(2n+1)qu]+i d sin[(2n+1)qu] (79) n n n n X X 0.57 into Eq. (68) and minimize the effective action with re- 0.568 spect to the parameters c ,d and q. By keeping only n n wave numbers which are odd multiples of q, we restrict 0.566 ourselvesto potentials which are antiperiodic over half a 0.564 period, 0.562 ϕ(u+π/q)= ϕ(u). (80) T 0.56 − Thiskindofshapeisindeedfavoredbytheminimization, 0.558 aswasthecaseforthemasslessGNmodel. Itshowsthat 0.556 discretechiralsymmetryandtranslationalsymmetryare brokendownto a discrete combinationof the 2 transfor- 0.554 mations, namely 0.552 ψ(x) γ5ψ(x+π/q) (81) 0.55 0 0.02 0.04 0.06 0.08 0.1 → µ from which Eq. (80) for bilinears follows. In practice, we found that it is sufficient to keep c0,c1,d0,d1 in the FIG. 5: Reconstructed phase diagram of generalized GN expansion (79). Comparing the reduced grand potential modelnearthetricriticalpointofNJL2 modelforξ =0.0001, withtheonefromthe homogeneousmassivesolution,we 0.0002, 0.0004, 0.0007, 0.001, 0.002, 0.004, 0.007, 0.01, from can locate the phase boundary. The result of the calcu- left to right. All curves are obtained from the ones shown in lationis showninFig.4togetherwiththe two2ndorder Fig. 4, butν,τ values upto 50 are needed for thesmallest ≈ phaseboundariesdiscussedabove. Duetotherescalings, ξ value. this is a kind of universal phase diagram which contains all information about the actual phase diagram in the vicinity of the tricritical point at ξ =0. By undoing the rescalingwecanreconstructthephasediagramsforsmall VI. EXACT TRICRITICAL BEHAVIOR FROM ξ values in a limited region of the (µ,T) plane. This is GINZBURG-LANDAU THEORY showninFig. 5. Here one sees nicely the transitionfrom the behavior qualitatively familiar from the GN model As ξ varies from 0 to , the tricritical point of the ∞ to the one from the massless NJL model. The angle generalized GN model moves from the NJL to the GN 2 2 between the two phase boundaries delimiting the crystal tricritical point, i.e. from µ = 0,T = 0.5669 to µ = atthe tricriticalpointis consistentwith zero,just likein 0.6082,T = 0.3183. Since the HF potential φ = S iP − the standard GN model. vanishesatthetricriticalpointanditsperiodisexpected 9 to diverge, the derivative expansion should be sufficient or, equivalently, to determine the exacttricriticalbehaviorfor all ξ. As a matter of fact, this will enable us to determine analyti- 0 = Q4+ ξ α3 2+ 2α2 Q2+ α2ξ + α22 cally the location of the tricritical point as a function of 2πα4 −(cid:18)α4(cid:19) α4 ! 2πα24 α24 ξ. Wewillalsobeinterestedinthebehaviorofthephase ξ α α2 boundariesinthevicinityofthetricriticalpoint. Itturns 0 = Q2+ + 2 3 . (90) outthattheregionofvalidityoftheGLtheoryasdefined 4πα4 α4 − 2α24 in Eq.(58) shrinks rapidly with increasing ξ. One of the These two equations determine Q and the critical curve reasonsisthe factthatbothα andα vanishatthe GN 2 4 in the (µ,T) plane. The tricritical point must lie on this tricritical point, so that it would be necessary to go to curveandonthecurveα =0. Thisgivesthe conditions 2 higher orders in the derivative expansion for large ξ. To Q=0 and keepthe analyticalwork reasonablysimple, we therefore analyzethephaseboundariesonlyformoderateξ values. 2πα2 ξ = 3 , (91) We startonce againfromthe GL effective action(58). α Consider first the homogeneous phases. The constant 4 (cid:12)t (cid:12) ansatz φ=m yields where the right hand side is to(cid:12)be evaluated at the tri- (cid:12) critical point. Using Eqs. (59), we finally arrive at the Ψeff =α2m2+α4m4. (82) followingparametricrepresentationofthe dependenceof the tricritical point (µ ,T ) on ξ (parameter ν˜), t t t Minimizing with respect to m, we find either m=0 or α2 ξ = 2 ℑΨ(1)(zt) 2 z = 1 + iν˜t m=r−2α4 (α2 <0). (83) − (cid:2)ℜΨ(2)(zt)(cid:3) (cid:18) t 2 2π(cid:19) 1 We thus recover the well known result for the phase Tt = e−ℜΨ(zt) 4π boundary between massless and massive Fermi gas µ = ν˜T . (92) t t t phases, namely This result should hold exactly in the generalized GN α =0 (84) 2 model,sinceGLtheorybecomesrigorousatthetricritical point. It has the correct limits for ξ 0 (NJL ) and or, parametrically (parameter ν˜), → 2 ξ (GN), as follows immediately from the vanishing →∞ T = 1 e−ℜΨ(z) z = 1 +i ν˜ , owfeαr3ecaonvderαt4h,ereasspyemctpivtoetlyic. bMeohraevoivoerro,fby(µex,pTa)ndfoinrgξinν0t 4π 2 2π t t (cid:18) (cid:19) found in Sec. V, cf. Eq. (78). → µ = ν˜T. (85) We now determine the shape of the phase boundaries near the tricritical point for finite ξ values. To this end, Next consider the 2nd order phase boundary between wemeasurechemicalpotentialandtemperaturefromthe crystal and massless homogeneous phase. As in Sec. V, tricritical point (at fixed ξ), the ansatz µ = µ +δ, φ=c cos(Qx)+id sin(Qx) (86) t 0 0 T = T +τ. (93) t is adequate for a continuous phase transition which can be treated in perturbation theory. The spatial average We then rotate the coordinate frame in the (δ,τ) plane of the effective action, keeping only quadratic terms in such that the new axes are tangential and normalto the (c0,d0), then becomes homogeneous phase boundary α2 =0, Ψ = c2+2 c d + d2 (87) δ cosθ sinθ σ h effi M11 0 M12 0 0 M22 0 τ = sinθ −cosθ η (94) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) where with 1 M11 = 2 α2+α4Q2 sinθ = I1, cosθ = ζ (95) (cid:0)1 (cid:1) Ω Ω = α Q 12 3 M −2 We have defined 1 ξ 22 = α2+α4Q2+ . (88) ζ =2π+ν˜I , Ω= I2+ζ2, I = Ψ(1)(z ). (96) M 2(cid:18) 2π(cid:19) t 1 1 1 ℑ t q The 2nd order phase boundary is again defined by Due to the cusp, the phase boundaries lie in the region around the tricritical point where ∂ detM=0, ∂Q2 detM=0 (89) σ ε, η ε2. (97) ∼ ∼ 10 In this region, the Taylor expansion The choice α2 = a22ε2+... λ = 2a31a40−a30a41 a2 α = a +a ε+a ε2+... 40 3 30 31 32 α4 = a40+a41ε+a42ε2+... (98) χ = a30λ (107) a r 40 holds with calculable coefficients given in the appendix. then yields the simpler expression We first determine the shape of the 2nd order phase boundary from (90,91) and (98). To leading order in Ψ = (f )2 (f )2+f4+κf2 (108) ε, we find eff N ′′ − ′ with only two res(cid:2)idual parameters (cid:3) a (2a a a a ) Q2 = 30 31 40− 30 41 ε (99) 2a3 = a λ4, 40 40 N a 1 and the following condition for the phase boundary, κ = 22 . (109) a λ2 40 0=4a a a a +4a a3 4a2 a2 a2 a2 . (100) 30 31 40 41 22 40− 31 40− 30 41 Now we focus on the reduced effective action We can also determine the ratio d0/c0 of imaginary to Ψeff =ψ =(f )2 (f )2+f4+κf2. (110) real amplitudes, eff ′′ ′ − N d 2a a a a As we have not been able to solve the Euler-Lagrange 0 = 31 40− 30 41√ε. (101) c 2a a equation 0 r 30 40 Computing the 1st order phase boundary is the most fIV +f′′+2f3+κf =0 (111) complicatedtask. Letusdecomposeφintorealandimag- analytically, we minimize the reduced effective action inary parts and assume the following LO behavior in ε, with the Fourier series ansatz φ = F +iG nmax y = ε1/2x f(z)= c cos[(2n+1)qz]. (112) n F(x) = εF0(y), F′(x)=ε3/2F˙0(y) nX=0 G(x) = ε3/2G (y), G(x)=ε2G˙ (y) (102) Provided we keep only one term in the sum (n =0), 0 ′ 0 max everythingcanbeworkedoutanalyticallywiththeresult These assumptions will be justified a posteriori once we have constructed a consistent solution. We then get 1 4κ c = − , 0 6 a2 r Ψ = a F G˙ +a G F˙ + 30G2+a F˙2 ε3 1 eff − 30 0 0 30 0 0 a 0 40 0 q = . (113) (cid:18) 40 (cid:19) √2 + a F2 a F G˙ +a G F˙ +a F4 22 0 − 31 0 0 31 0 0 40 0 The (spatially averaged) reduced effective action in this +(cid:16)a G˙2+a F˙2 ε4 (103) approximationis given by 40 0 41 0 (cid:17) 1 G0 can be eliminated as follows: Vary the O(ε3) term hψeffi=−96(1−4κ)2. (114) with respect to G , find the condition 0 The 2nd orderphase boundaryis obtainedfrom ψ = a h effi G = 40F˙ . (104) 0 and assumes the simple form 0 0 −a 30 1 If we insert this relation into Eq. (103), the ε3 term dis- κ= . (115) 4 appears after a partial integration and we are left with Thehomogeneous,massivesolutionintherescaledmodel a3 2a a a a is characterized by Ψ = 40F¨2 31 40− 30 41F˙2+a F4+a F2. eff a2 0 − a 0 40 0 22 0 30 30 κ (105) q =0, c = (116) 0 Here we have set the formal expansion parameter ε = 1 −2 r since it is not needed anymore. The coefficients may be and has the reduced action simplified by rescaling, 1 F0(y)=λf(χy). (106) ψhom =−4κ2. (117)