aa r X i v : . [ nu c l - t h ] S e p Magnetic field effect on pion superfluid
Shijun Mao ∗ School of Physics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China (Dated: September 25, 2020)Magnetic field effect on pion superfluid phase transition is investigated in frame of a Pauli-Villarsregularized NJL model. Instead of directly dealing with charged pion condensate, we apply theGoldstone’s theorem (massless Goldstone boson π + ) to determine the onset of pion superfluid phase,and obtain the phase diagram in magnetic field, temperature, isospin and baryon chemical potentialspace. At weak magnetic field, it is analytically proved that the critical isospin chemical potential ofpion superfluid phase transition is equal to the mass of π + meson in magnetic field. The pion super-fluid phase is retarded to higher isospin chemical potential, and can survive at higher temperatureand higher baryon chemical potential under external magnetic field. PACS numbers: 12.38.-t, 25.75.Nq, 14.80.Mz
The study of QCD at finite isospin density and thecorresponding pion superfluid phase attracts much at-tention due to its relation to the investigation of com-pact stars, isospin asymmetric nuclear matter, andheavy-ion collisions at intermediate energies. On nu-merical side, while there is not yet precise lattice re-sults at finite baryon density due to the Fermion signproblem, it is in principle no problem to do latticesimulation at finite isospin density [1–3]. On analyt-ical side, effective models such as Nambu–Jona-Lasiniomodel (NJL), linear sigma model and chiral perturba-tion theory have been widely used to investigate pionsuperfluid phase structure [4–26]. There are two equiv-alent criteria for the critical point of pion superfluidphase transition, the non-vanishing charged pion con-densate and the massless π + meson, which correspondto the spontaneous breaking of isospin symmetry andthe Goldstone boson, respectively, guaranteed by theGoldstone’s theorem [27, 28]. With vanishing tempera-ture, the critical isospin chemical potential µ cI is thepion mass in vacuum m π . When µ I > m π , the u quark and ¯ d quark form coherent pairs and condensate,and the system enters the pion superfluid phase [1–26]. At hadron level, in the normal phase ( µ I < m π )without charged pion condensate, different pion modesexplicitly show the mass splitting according to theirisospin, with m π ± = m π ∓ µ I and m π = m π . As µ I = µ cI = m π , the excitation of π + meson is free withzero momentum, which indicates the onset of pion su-perfluid phase [2, 12, 15–17, 24]. Inside the pion super-fluid phase ( µ I ≥ m π ), π + meson keeps massless as theGoldstone mode [2, 12, 15–17, 24].Recently, the magnetic properties of QCD matter be-come important. For instance, a certain class of neu-tron stars (magnetars) exhibits intense magnetic fieldsof strengths up to 10 − Gauss at the star surfaceand the field is expected to become stronger towardsthe star center, about 10 Gauss [29, 30]. However,the magnetic field effect on the pion superfluid is stillan open question. The difficulty lies in the fact thatthe pion superfluid is a phase with charged pion con-densate. It breaks both the isospin symmetry in theflavor space and the translational invariance in the co- ∗ Electronic address: [email protected] ordinate space, and thus the Fourier transformationbetween coordinate and momentum spaces is not assimple as for neutral condensate or without magneticfield. LQCD simulations exhibit a sign problem at fi-nite isospin chemical potential and magnetic field. Byusing a Taylor expansion in the magnetic field, it isreported that at vanishing temperature, the onset ofpion condensate shifts to larger isospin chemical poten-tial under magnetic fields [31], which is qualitativelyconsistent with the enhancement of the charged pionmass with growing magnetic fields [32]. In the study ofeffective models, people also focus on the charged pioncondensate but the interaction between the chargedpion condensate and the magnetic field is simply ne-glected in Ref. [33, 34] or taken into account by theGinzburg-Landau approach assuming a tiny condensatein Ref. [35].In this paper, we will study the pion superfluid phasetransition at finite magnetic field, temperature, isospinand baryon chemical potential in frame of a Pauli-Villars regularized NJL model, which is inspired by theBardeen-Cooper-Shrieffer (BCS) theory and describesremarkablely well the quark pairing mechanisms andhadron mass spectra [36–41]. Instead of directly dealingwith charged pion condensate, we investigate the mag-netic field effect on pion superfluid through its Gold-stone mode π + , determining the critical point of pionsuperfluid phase transition by the massless π + meson.Seriously taking into account the breaking of transla-tional invariance for charged particles, the pion prop-agators in terms of quark bubbles are analytically de-rived, and pion masses are solved. At weak magneticfield and vanishing temperature and baryon chemicalpotential, we analytically prove that the critical isospinchemical potential of pion superfluid phase transition isequal to the π + mass in magnetic field, the same as thevanishing magnetic field case [2, 12, 15–17, 24]. Un-der external magnetic field, the pion superfluid phaseis shifted to higher isospin chemical potential, and cansurvive at higher temperature and higher baryon chem-ical potential.The two-flavor NJL model is defined through the La-grangian density in terms of quark fields ψ [36–41] L = ¯ ψ ( iγ ν D ν − m + γ µ ) ψ + G h(cid:0) ¯ ψψ (cid:1) + (cid:0) ¯ ψiγ ~τ ψ (cid:1) i . (1)Here the covariant derivative D ν = ∂ ν + iQA ν cou-ples quarks with electric charge Q = diag ( Q u , Q d ) = diag (2 e/ , − e/
3) to the external magnetic field B =(0 , , B ) in z -direction through the potential A ν =(0 , , Bx , µ = diag ( µ u , µ d ) = diag ( µ B / µ I / , µ B / − µ I /
2) is amatrix in the flavor space, with µ u and µ d being the u - and d -quark chemical potentials and µ B and µ I be-ing the baryon and isospin chemical potentials. G isthe coupling constant in scalar and pseudo-scalar chan-nels. At finite isospin chemical potential and magneticfield, the isospin symmetry SU (2) I is broken down to U (1) I symmetry, and the chiral symmetry SU (2) A isbroken down to U (1) A symmetry. With the sponta-neous breaking of chiral U (1) A symmetry and isospin U (1) I symmetry, the Goldstone mode reads π mesonand π + meson, respectively. m is the current quarkmass characterizing the explicit chiral symmetry break-ing.Corresponding to the symmetries and their sponta-neous breaking, we have two order parameters, neu-tral chiral condensate h ¯ ψψ i for chiral restoration phasetransition and charged pion condensate h ¯ ψγ τ ψ i forpion superfluid phase transition. Under magneticfields, the charged pion condensate breaks both theisospin symmetry in the flavor space and the trans-lational invariance in the coordinate space, and thusthe Fourier transformation between coordinate and mo-mentum spaces is not as simple as for neutral conden-sate or without magnetic field. In our current work,to avoid the complication and difficulty of dealing withcharged pion condensate under magnetic field, we willstart from the normal phase only with neutral chiralcondensate and determine the critical point of pion su-perfluid phase transition by the appearance of Gold-stone boson, massless π + meson. Physically, it is equiv-alent to define the phase transition by the order param-eter (charged pion condensate) and Goldstone mode(massless π + meson), as guaranteed by the Goldstone’stheorem [2, 12, 27, 28].In mean field approximation, the chiral condensate h ¯ ψψ i or the dynamical quark mass m q = m − G h ¯ ψψ i is controlled by the gap equation [42–48], m = m q (1 − GJ ) , (2) J = 3 X f,n α n | Q f B | π Z dp π E f (3) × [1 − f ( E f + µ f ) − f ( E f − µ f )] , with the summation over all flavors and Landau en-ergy levels, spin factor α n = 2 − δ n , quark energy E f = q p + 2 n | Q f B | + m q , and Fermi-Dirac distribu-tion function f ( x ) = 1 / ( e x/T + 1).As quantum fluctuations above the mean field,mesons are constructed through quark bubble summa-tions in the frame of random phase approximation [37–41]. Taking into account of the interaction betweencharged mesons and magnetic fields, and generalizingour derivations in Ref. [48] to finite quark chemical po-tential, the meson propagator D M can be expressed interms of the meson polarization function Π M with con- served Ritus momentum ¯ k , D M (¯ k ) = G − G Π M (¯ k ) . (4)The meson pole mass m M is defined through the poleof the propagator at zero momentum,1 − G Π M ( k = m M ) = 0 . (5)Based on the Goldstone’s theorem for the sponta-neous breaking of isospin symmetry, massless Gold-stone mode π + exists in the pion superfluid phase.Therefore, the critical isospin chemical potential µ cπI for pion superfluid can be identified by the condition m π + ( B, T, µ B , µ cπI ) = 0 . (6)For the π + meson, we haveΠ π + ( k ) = J + J ( k ) , (7) J ( k ) = X n,n ′ Z dp π j n,n ′ ( k )4 E n E n ′ (8) × (cid:2) f ( − E n ′ − µ u ) − f ( E n − µ d ) k + µ I + E n ′ + E n + f ( E n ′ − µ u ) − f ( − E n − µ d ) k + µ I − E n ′ − E n (cid:3) ,j n,n ′ ( k ) = (cid:2) ( k + µ I ) / − n ′ | Q u B | − n | Q d B | (cid:3) j + n,n ′ − p n ′ | Q u B | n | Q d B | j − n,n ′ , (9)with the u -quark energy E n ′ = q p + 2 n ′ | Q u B | + m q and d -quark energy E n = q p + 2 n | Q d B | + m q . Thecoefficients j ± n,n ′ are detailed derived in our previouswork [48]. Note that the lowest-Landau-level term with n = n ′ = 0 do not contribute to the polarization func-tion with j ± , = 0. Because the spins of u and ¯ d quarksat the lowest Landau level are aligned parallel to themagnetic field, but π + meson has spin zero. This leadsto the heavy π + mass in magnetic field [48] and thusdelays the pion superfluid in magnetic field (see thediscussions of Fig.1).Because of the four-fermion interaction, the NJLmodel is not a renormalizable theory and needs reg-ularization. The magnetic field does not cause extraultraviolet divergence but introduces discrete Landaulevels and anisotropy in momentum space. To guaran-tee the law of causality in anisotropic systems, we takeinto account the gauge invariant Pauli-Villars regular-ization scheme [47, 48]. The three parameters in theNJL model, namely the current quark mass m = 5MeV, the coupling constant G = 3 .
44 GeV − and thePauli-Villars mass parameter Λ = 1127 MeV are fixedby fitting the chiral condensate h ¯ ψψ i = − (250 MeV) ,pion mass m π = 134 MeV and pion decay constant f π = 93 MeV in vacuum with T = µ B = µ I = 0 and B = 0.In Fig.1, we plot the critical isospin chemical poten-tial µ cπI (black and red solid lines) for pion superfluidphase transition as a function of magnetic field at T = µ B = 0, which is determined by the condition of mass-less Goldstone boson m π + ( B, T = µ B = 0 , µ cπI ) = 0. eB (cid:144) m Π H M e V L Μ I cc Μ I c Π M Π + FIG. 1: Critical isospin chemical potential µ cπI (black andred solid lines) for pion superfluid phase transition, and µ ccI (green dotted line) for chiral restoration phase transition asa function of magnetic field at T = µ B = 0. π + mass inmagnetic field M π + = m π + ( B, T = µ B = µ I = 0) is plottedin blue dashed line for reference. µ cπI increases with magnetic field, which is qualitativelyconsistent with the conclusion of LQCD [31] and modelcalculations [35], and this means that magnetic fielddelays/disfavors the pion superfluid phase transition atfinite isospin chemical potential. Physically, it can beunderstood in this way. Locating both the two con-stituent quarks at the lowest Landau level are forbid-den for charged pions due to its zero spin. Accordingto the quark energy E f = q p + 2 n | Q f B | + m q , dif-ferent electric charges of u and d quarks indicate dif-ferent effective quark mass q n | Q f B | + m q with finitemagnetic field and zero momentum p = 0. This massdifference plays the role of effective Fermi surface mis-match when u quark and ¯ d quark form cooper pairs.The larger the magnetic field (mass difference) is, themore difficult to form pion superfluid becomes, and thisleads to the increasing µ cπI in magnetic field.Critical isospin chemical potential µ ccI for chiralrestoration phase transition, see green dotted line inFig.1, is determined by the dynamical quark mass. Atfinite magnetic field, chiral restoration is a first orderphase transition, and the quark mass jumps from alarge value to a small value. It is noticeable that µ ccI and µ cπI are different from each other, except for onepoint at eB = 4 . m π , with µ ccI > µ cπI at eB < . m π and µ ccI < µ cπI at eB > . m π .The critical isospin chemical potential µ cπI is sepa-rated into two parts, denoted by the connecting pointof red and black solid lines at eB = 4 . m π in Fig.1.For eB < . m π , we observe that the critical isospinchemical potential is equal to the π + mass in magneticfield, with µ cπI = M π + = m π + ( B, T = µ B = µ I = 0),as shown by the overlap between the black solid lineand blue dashed line in Fig.1. This conclusion can beanalytically proved, similar as the case without mag-netic field [12]. At T = 0, the Fermi-Dirac distribution f ( x ) becomes a Heaviside step function θ ( − x ). Withfixed magnetic field, we solve a constant quark mass m q ( B, T = µ B = 0 , µ I ) = m q ( B, T = µ B = µ I = 0)from gap equation (2), before the chiral restoration hap-pens. And by straightforward comparison of gap equa-tion (2) and pole equation (5), a linearly decreasing π + mass is obtained m π + ( B, T = µ B = 0 , µ I ) = M π + − µ I .Applying the Goldstone’s theorem, the critical isospinchemical potential µ cπI for pion superfluid is determinedby the condition m π + ( B, T = µ B = 0 , µ cπI ) = 0. There-fore, we solve µ cπI = M π + . At eB = 4 . m π , both thepion superfluid phase transition and the chiral restora-tion phase transition happen at the same critical isospinchemical potential µ cπI = µ ccI . Since chiral restoration isa first order phase transition, associated with the quarkmass jump. It leads to the discontinuous µ cπI for pionsuperfluid phase transition, as shown by the differentslope of black and red lines around eB = 4 . m π . For eB > . m π , no such analytical derivations are avail-able and we should rely on the numerical calculations.The critical isospin chemical potential µ cπI is deviatedfrom M π + , although they both increase in magneticfields. With stronger magnetic field, the deviation be-comes larger. Μ I H MeV L T H M e V L eB = = m Π normal phasepion superfluid FIG. 2: Pion superfluid phase diagram in µ I − T plane with µ B = 0 and fixed magnetic field. The black dashed line isfor eB/m π = 0, and red solid line for eB/m π = 5. We now turn on the temperature effect and depictthe pion superfluid phase diagram in µ I − T plane with µ B = 0 and fixed magnetic field eB/m π = 0 (blackdashed line) and eB/m π = 5 (red solid line) in Fig.2.The phase transition line determined by the massless π + meson divides the µ I − T plane into two regions.The pion superfluid phase is located in high isospinchemical and low temperature region, and the quarksare in normal phase for low isospin chemical potentialand/or high temperature region. With increasing tem-perature, the quark thermal motion becomes strong. Itprohibits the quark pairing and leads to the phase tran-sition from pion superfluid phase to normal phase. Thecritical temperature increases with isospin chemical po-tential. Comparing with vanishing magnetic field case,the pion superfluid phase is retarded to higher isospinchemical potential, and it survives in higher tempera-ture under finite magnetic field.Fig.3 is the phase diagram in µ I − µ B plane with T = 0 and fixed magnetic field. The black dashed lineis for eB/m π = 0, and red solid line for eB/m π = 5.Pion superfluid phase locates in high isospin chemicalpotential and low baryon chemical potential region. Inthe low isospin chemical potential and/or high baryonchemical potential region, quarks are in normal phase.At zero baryon chemical potential, the u quark and ¯ d quark form coherent pairs and condensate on a uniform Μ I H MeV L Μ B H M e V L eB = = m Π normal phasepion superfluid FIG. 3: Pion superfluid phase diagram in µ I − µ B planewith T = 0 and fixed magnetic field. The black dashed lineis for eB/m π = 0, and red solid line for eB/m π = 5. Fermi surface, as µ I > µ cπI . When the baryon chem-ical potential is switched on, there appears a Fermisurface mismatch between the u quark and ¯ d quark,and it causes the phase transition from pion superfluidphase to normal phase. The critical baryon chemi-cal potential increases with isospin chemical potential.With stronger magnetic field, the pion superfluid phasehappens at higher isospin chemical potential and sur-vives at higher baryon chemical potential. It shouldbe mentioned that even in large baryon chemical po- tential case, we still neglect the color superconductorphase. The competition between color superconductorand pion superfluid in µ I − µ B plane will be studiedelsewhere.Magnetic field effect on pion superfluid phase tran-sition is studied in frame of a Pauli-Villars regularizedNJL model. Instead of directly dealing with chargedpion condensate, we apply the Goldstone’s theorem(massless Goldstone boson π + ) to determine the onsetof pion superfluid phase. Seriously taking into accountthe breaking of translational invariance, the chargedpion propagator is constructed at finite magnetic field,temperature and chemical potential, and the π + massand pion superfluid phase diagram are obtained. Atweak magnetic field and vanishing temperature andbaryon chemical potential, it is analytically provedthat the critical isospin chemical potential µ cπI is equalto the π + mass in magnetic field, µ cπI = M π + . Underexternal magnetic field, the pion superfluid phase isretarded to higher isospin chemical potential, andcan survive at higher temperature and higher baryonchemical potential. Acknowledgement:
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