Spontaneous parity violation in extreme conditions: an effective lagrangian analysis
aa r X i v : . [ h e p - ph ] A p r Eur. Phys. J. C manuscript No. (will be inserted by the editor)
Spontaneous parity violation in extreme conditions: aneffective lagrangian analysis
Alexander A. Andrianov a,1,2 , Vladimir A. Andrianov b,1 , Domenec Espriu c,2 V.A. Fock Department of Theoretical Physics, Saint-Petersburg State University, ul. Ulianovskaya,1, 198504 St. Petersburg,Russia Departament dEstructura i Constituents de la Mat´eria and Institut de Ci´encies del Cosmos (ICCUB) Universitat deBarcelona, Mart´ı Franqu`es, 1, E08028 Barcelona, SpainReceived: date / Accepted: date
Abstract
We investigate how large baryon densities(and possibly high temperatures) may induce sponta-neous parity violation in the composite meson sector ofvector-like gauge theory (presumably QCD or techni-QCD) . The analysis at intermediate energy scales isdone by using an extended σ -model lagrangian that in-cludes two scalar and two pseudoscalar multiplets andfulfills low-energy constraints for vector-like gauge the-ories. We elaborate on a novel mechanism of paritybreaking based on the interplay between lightest andheavier meson condensates, which therefore cannot berealized in the simplest σ model. The results are rele-vant for an idealized homogeneous and infinite nuclear(quark or techniquark) matter where the influence ofdensity can be examined with the help of a constantchemical potential. The model is able to describe sat-isfactorily the first-order phase transition to stable nu-clear matter, and predicts a second-order phase tran-sition to a state where parity is spontaneously bro-ken. We argue that the parity breaking phenomenonis quite generic when a large enough chemical potentialis present. Current quark masses are explicitly takeninto account in this work and shown not to change thegeneral conclusions. Emergent parity violation for sufficiently large valuesof the baryon chemical potential (and/or temperature)has been attracting much interest during several decades(see reviews [1]). Yet the reliable prediction of par-ity violation effects has not been done from the first a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] principles. In our work we investigate how large baryondensities (and possibly high temperatures) may inducespontaneous parity violation in the composite mesonsector of vector-like gauge theory (presumably QCD ortechni-QCD). The analysis is performed by using anextended σ -model lagrangian that includes two scalarand two pseudoscalar multiplets and fulfills low-energyconstraints. This is a model inspired by but not ex-actly equivalent to QCD as its coupling constants aretaken as empirical parameters to be measured in mesonphysics.At finite baryon density pion condensation is con-jectured in nuclear physics long ago in [2] and it seemsto be a plausible possibility which however cannot beproved in simple models describing pion-nucleon inter-actions. In this paper this long-standing idea will in factbe vindicated.In this paper we shall attempt to explore the in-teresting issue of parity breaking employing effectivelagrangian techniques, useful to explore the range ofnuclear densities where the hadron phase still persistsand quark percolation does not occur yet. Our effec-tive lagrangian is a realization of the generalized lin-ear σ model, but including the two lowest lying res-onances in each channel, those that are expected toplay a role in this issue. This seems to be the mini-mal model where the interesting possibility of paritybreaking can be realized. Namely, condensation of oneof the pseudoscalar fields can arise on the backgroundof two-component scalar condensate so that the chiralconstant background cannot be rotated away by trans-formation of two complex scalar multiplets preservingspace parity. The use of effective lagrangians is also cru-cial to understand how would parity breaking originat-ing from a finite baryon density eventually reflect inhadronic physics. A pre-QCD attempt to describe two multiplets ofscalar and pseudoscalar mesons was done in [3] with areduced set of operators and a chiral symmetry break-ing (CSB) pattern not quite compatible with QCD.We have been basically inspired by our previous workson extended quark models [4–6] where two differentschemes with linear and non-linear realization of chi-ral symmetry were adopted to incorporate heavy pionsand scalar mesons within an effective quark model withquark self-interactions. A certain resemblance can bealso found with the model [7] where two SU (3) F mul-tiplets have been associated with two-quark and four-quark meson states although we do not share the as-sumption in [7] concerning the dominance of four-quarkcomponent in radially excited mesons. The model in [8]is also of relevance in studying of vacuum for extended σ models.The present work is an extension of the preliminaryresults concisely reported in [9, 10] which includes cor-rections beyond the chiral limit, linear in the currentquark masses. We will give also the qualitative expla-nation of a possible origin of the model, provide the de-tailed proofs of several statements that were only enun-ciated in [9, 10] and derive a number of new thermo-dynamic relations for finite temperatures and chemicalpotentials.The paper is organized as follows. In section 2 thebosonization of QCD quark currents in the color-singletsector is discussed and the ingredients of the general-ized σ model are indicated. In Section 3 we introducethe σ model with two multiplets of isosinglet scalar andisotriplet pseudoscalar fields. The effective potential fortwo multiplets of scalar and pseudoscalar mesons is ob-tained and the mass-gap equations and second varia-tions at the minima are derived. In Section 4 the ex-istence of a region in the coupling constant space isproven where there are four minima of the effective po-tential playing the crucial role in realization of stablebaryon matter via a first order phase transition. In Sec-tion 5 we shall introduce the finite chemical potentialand temperature and see how they modify the effectivetheory and the vacuum state. Temperature and baryonchemical potential appear through the one-quark loopfree energy. In sections 6 we investigate the emergenceof spontaneous parity breaking (SPB) phase. The massgap equations and critical lines for the parity breakingphase transition are derived and corrections beyond thechiral limit are taken into account to the leading orderin quark masses. In Section 7 it is established that thetransition to the SPB phase is of second order. In Sec-tion 8 the kinetic terms are considered in order to de-termine the physical masses of scalar and pseudoscalarmesons and extract some physical consequences. In par- ticular, it is proven that for massive quarks only threemassless states characterize the SPB phase transition.In SPB phase the masses of four light pseudoscalarstates are obtained. We notice however that in the SPBphase strictly speaking there are no genuine scalar orpseudoscalar states as each of massive states can equallywell decay into two and three (pseudo)scalars. Section9 is devoted to a description of nuclear matter and theapproach to the condensation point of stable baryonmatter. In order to describe adequately the saturationpoint transition to stable baryon matter we supplementthe effective lagrangian with an ω meson coupling to theisosinglet quark current which influences the repulsivepart of nuclear forces[11] and thereby supports the for-mation of stable nuclear matter. We obtain a first-orderphase transition at the saturation point. In Section 10we attempt to confront the on-set of empirical constantsof two-multiplet model with meson and nuclear matterphenomenology. We summarize our findings in the con-clusions Section. In Appendix A we prove that the chi-ral collapse affecting the simplest σ models and/or theone-multiplet Nambu-Jona-Lasinio (NJL) models [12]does not occur in our two-multiplet model. In AppendixB the proposal for description of (in)compressibilities inthe mean-field approach is formulated with the help ofmatching quark and nuclear matters.The range of intermediate nuclear densities whereour effective lagrangian could be used is of high interestas they may be reached in both compact stars [13] andheavy-ion collisions [14]. Its relevance can be qualita-tively motivated by the fact that at substantially largerdensities typical distances between baryons are shrink-ing considerably and meson excitations with Comptonwave lengths much shorter than the pion wave lengthstart playing an important role. Can the spontaneousparity breaking be realized in heavy ion collisions orin neutron stars? In order to answer this question wemight appeal to lattice QCD for help and in fact thispossibility has been studied intensively for quite sometime [15–17]. However the lattice results for sufficientlylarge values of the baryon chemical are not known quan-titatively and rigorously yet.It is worth to mention some previous studies dealingwith the problem of strong interactions at zero tempera-ture and finite chemical potential: depending on a valueof nuclear density, a variety of methods are involvedfrom using meson-nucleon [1, 18] or quark-meson[13, 19]lagrangians for low-dense nuclear matter to models ofthe NJL type[20] for high-dense quark matter[21]. Al-though the issue of SPB in hadronic phase has beentouched upon in the pion-nucleon theory[1, 18, 22] andin NJL models [23] the reliability of the models used isnot quite clear for intermediate nuclear densities. The reason is discussed in the next Section: they are not richenough to explore the subtle phenomenology involved.More recently the phenomenon of parity breakingwas assumed to be present in meta-stable nuclear bub-bles with non-zero axial charge generated by nontrivialtopological charge in hot nuclear matter [24] and/orin the presence of a strong background magnetic fields[25, 26]. It was also shown [27] that the associated ax-ial chemical potential causes a distortion of the en-ergy spectrum of photons and vector particles ( ρ and ω mesons) due to a Chern-Simons term that is gener-ated. In addition scalar and pseudoscalar mesons get amomentum dependent effective mass [28]. However thisphenomenon is theoretically somewhat different in itsorigin to the previous one and it will not be discussedin the present paper. In order to elaborate an effective lagrangian for compos-ite meson states starting from QCD or QCD-like theo-ries we revisit the properties of color-singlet (quasi)localquark currents in the vacuum with spontaneously bro-ken chiral symmetry. This phenomenon emerges due toa non-zero value of quark condensate h ¯ qq i and can beassociated to the CSB scale Λ (in QCD it is presum-ably ∼ GeV ). This CSB due to quark condensationmakes the quark bilinears to be interpolating operatorsfor meson fields (in the limit of large number of colors).In particular, the scalar and pseudoscalar quark densi-ties effectively describe the creation or annihilation ofscalar and pseudoscalar mesons¯ qq ( x ) ≃ Λ ∞ X l =1 Z (1) l σ l ( x );¯ qγ τ a q ( x ) ≃ Λ ∞ X l =1 Z (1) l π al ( x ) , (1)where the normalized meson fields σ l , π al describe thefamilies of resonances with the same quantum numbersbut increasing masses (radial Regge trajectories)[29]and the set of normalization constants Z (1) l is intro-duced. The constituent quark fields are denoted as ¯ q, q and τ a , a = 1 , , h σ l i of scalar fields, h ¯ qq i ≃ Λ ∞ X l =1 Z (1) l h σ l i , (2)which represents the condition on the choice of poten-tial in a QCD motivated sigma model. In this paper werestrict ourselves with consideration of two light flavorsrelated to u, d quarks and therefore the approximatechiral symmetry of the quark sector is SU (2) L × SU (2) R .Keeping in mind confinement we have retained in(1) only the one-resonance states as leading ones whilebeing aware of that the total saturation of quark cur-rents includes, of course, also multi-resonance states.Thus we use the large- N c approach where resonancesbehave like true elementary particles with zero widthsand multi-resonant states can be neglected.Let us comment a bit more on the previous relation.On the left-hand side one sees an operator of canonicaldimension 3 whereas on the right-hand side one findsfield operators of canonical dimension 1. This drasticchange in dimensions is a consequence of CSB and itmodifies the dimensional analysis of what must be in-cluded into an effective lagrangian. More exactly, in or-der to replace the non-perturbative regime of QCD atlow and intermediate energies by a hadron effective la-grangian one has to apply this dimensional countingin the CSB phase [5] to all possible combinations ofcolor-singlet operators arising in the chiral expansionin inverse powers of the CSB scale Λ .To be specific the chiral invariant local operatorsplaying the leading role in the low-energy effective la-grangian for meson dynamics are1 Λ (cid:2) (¯ qq ) − ¯ qγ τ a q ¯ qγ τ a q (cid:3) ≃ Λ ∞ X l,m =1 Z (2) lm [ σ l σ m + π al π a,m ];1 Λ [(¯ qq ) − ¯ qγ τ a q ¯ qγ τ a q ] ≃ ∞ X l,m,n,r =1 Z (4) lmnr [ σ l σ m + π al π a,m ][( σ n σ r + π an π a,r ];1 Λ [ ∂ µ (¯ qq ) ∂ µ (¯ qq ) − ∂ µ (¯ qγ τ a q ) ∂ µ (¯ qγ τ a q )] ≃ ∞ X l,m =1 ˜ Z (2) lm [ ∂ µ σ l ∂ µ σ m + ∂ µ π al ∂ µ π a,m ] , (3)where the matrices Z (2) lm , Z (4) lmnr , ˜ Z (2) lm must be symmet-ric under transposition of indices in order to provideglobal chiral invariance. The superscript numbers indi-cate the powers of interpolating meson fields. The termsquadratic in scalar fields must trigger an instability inthe potential that leads to CSB in the effective meson theory due to condensation of scalar fields, h σ l i 6 = 0 forsome l (see. e.g. [5, 7, 12]).The above set of operators is not complete and canbe extended with the help of form factors that are poly-nomials in derivatives [5]. For example, using the sameCSB scale Λ one can add into the effective quark la-grangian the vertices built of the elements¯ q ←→ ∂ k Λ k q ( x ) ≃ Λ ∞ X l =1 Z (1) ,kl σ l ( x );¯ qγ τ a ←→ ∂ k Λ k q ( x ) ≃ Λ ∞ X l =1 Z (1) ,kl π al ( x ) , ←→ ∂ ≡
14 ( −→ ∂ µ − ←− ∂ µ )( −→ ∂ µ − ←− ∂ µ ) , (4)which may give numerically comparable contributionsfor several k [5]. σ and pseudoscalar π a fields. Spontaneous CSB emerges due to a non-zerovalue for h σ i ∼ h ¯ qq i /Λ , Λ ∼ πF π with F π beinga weak pion decay coupling constant. Current algebratechniques indicate that in order to relate this model toQCD one has to choose a real condensate for the scalardensity, with its sign opposite to current quark masses,and avoid any parity breaking due to a v.e.v. of thepseudoscalar density. The introduction of a chemicalpotential does not change the phase of the condensateand therefore does not generate any parity breaking.This is just fine because in normal conditions paritybreaking does not take place in QCD. However, if twodifferent scalar fields condense with a relative phase be-tween the two v.e.v.’s the opportunity of spontaneousparity breaking may arise.Let us consider a model with two multiplets of scalar( ˜ σ j ) and pseudoscalar ( ˜ π aj ) fields H j = ˜ σ j I + i ˆ π j , j = 1 , H j H † j = (˜ σ j +(˜ π aj ) ) I , (5)where I is an identity 2 × π j ≡ ˜ π aj τ a with τ a being a set of Pauli matrices. We shall deal with ascalar system globally symmetric respect to SU (2) L × SU (2) R rotations in the exact chiral limit and next con-sider the soft breaking of chiral symmetry by currentquark masses. We should think of these two chiral mul-tiplets as representing the two lowest-lying radial states for a given J P C . Of course one could add more multi-plets, representing higher radial and spin excitations,to obtain a better description of QCD, but the presentmodel, without being completely realistic, already pos-sesses all the necessary ingredients to study SPB. In-clusion of higher-mass states would be required at sub-stantially larger baryon densities when typical distancesbetween baryons are shrinking considerably and mesonexcitations with Compton wave lengths much shorterthan the pion wave length start playing an importantrole.Let us define the effective potential of this general-ized sigma-model. First we write the most general Her-mitian potential at zero µ , V eff = 12 tr − X j,k =1 H † j ∆ jk H k + λ ( H † H ) + λ ( H † H ) + λ H † H H † H + 12 λ ( H † H H † H + H † H H † H )+ 12 λ ( H † H + H † H ) H † H + 12 λ ( H † H + H † H ) H † H (cid:27) + O ( | H | Λ ) , (6)which contains 9 real constants ∆ jk , λ A ; A = 1 , . . . , . However this set of constants can be reduced (see sect.4).QCD bosonization rules in the large N c limit prescribe ∆ jk ∼ λ A ∼ N c . The neglected terms will be sup-pressed by inverse power of the CSB scale Λ ∼ H j to be of the order of theconstituent mass 0 . ÷ . ∆ with an operator i tr n ( H † H − H † H ) o , (7)and four more operators i tr n H † H H † H − H † H H † H o ,i tr n ( H † H − H † H ) H † H o ,i tr n ( H † H − H † H ) H † H o ; i (cid:20)(cid:16) tr n H † H o (cid:17) − (cid:16) tr n H H † o (cid:17) (cid:21) . (8)But for the scalar multiplets (5) in SU (2) L × SU (2) R representation these operators identically vanish (seebelow).There are also two operators with two disconnectedtraces which seem to complete the full set of operators (cid:16) tr n H † H o (cid:17) + (cid:16) tr n H H † o (cid:17) ;tr n H † H o tr n H H † o . (9) However for the scalar multiplets (5) they are not in-dependent and can be expressed as the linear combi-nation of operators with constants λ , λ in (6). Theproofs of above statements can be easily done with thehelp of so called chiral parameterization. Namely, onecan use the global invariance of the model to factor outthe Goldstone boson fields with the help of the chiralparameterization H ( x ) = σ ( x ) ξ ( x ); H ( x ) = ξ ( x ) (cid:16) σ ( x ) + i ˆ π ( x ) (cid:17) ξ ( x ); ξ ≡ exp (cid:18) i π a τ a F (cid:19) = cos p ( π a ) F ! + i π a τ a p ( π a ) sin p ( π a ) F ! , (10)which differs from eq. (5) in notation. The constant F is related to the bare pion decay constant and will bedefined later when the kinetic terms are normalized.This kind of parameterization preserves the parities of σ ( x ) and ˆ π to be even and odd respectively in theabsence of SPB. Then the contribution of the four ad-ditional operators (7),(8) vanishes identically, whereasthe operator (9) turns out to be a combination of op-erators with constants λ , . Finally the potential (6) isfurther simplified to V eff = − X j,k =1 σ j ∆ jk σ k − ∆ ( π a ) (11)+ λ (cid:16) ( π a ) (cid:17) + (cid:16) ( λ − λ ) σ + λ σ σ + 2 λ σ (cid:17) ( π a ) + λ σ + λ σ + ( λ + λ ) σ σ + λ σ σ + λ σ σ . The current quark mass m q corresponds to the av-erage of the external scalar sources M j ( x ) = s j ( x ) + iτ a p aj ( x ), namely, h M j ( x ) i = − d j m q and thus the rel-evant new terms beyond the chiral limit can be pro-duced with the help of the formal replacement H j → c j m q in all quadratic and quartic operators included in(6) and by adding these new terms with new constantsinto the effective potential. We will consider the twoflavor case and retain the terms softly breaking chiralsymmetry and linear in H j and m q , thereby neglect-ing terms cubic in scalar fields exploiting the non-linearequivalence transformation H j → H j + X k,l,m =1 , b jklm H k H † l H m . It corresponds to the choice of external scalar sourceslinear in H j , P j =1 , tr ( M † j H j + H † j M j ) . Thus we addtwo new terms to our effective potential (12) − m q tr h d ( H + H † ) + d ( H + H † ) i . (12) Making use of our chiral parametrization of the fields H j through the chiral field U ≡ ξ = cos | π a | F + i τ a π a | π a | sin | π a | F , (13)one derives the following extension of the effective po-tential (12) ∆V eff ( m q ) = 2 m q (cid:20) − ( d σ + d σ ) cos | π a | F + d π a π a | π a | sin | π a | F (cid:21) . (14)The effective potential (12), (14) will be used to searchfor CSB and for the derivation of meson masses.3.2 Mass-gap equations and second variations odeffective potentialLet us now investigate the possible appearance of a non-zero v.e.v.’s of pseudoscalar fields. Some time ago it wasproved in [31] that parity and vector flavor symmetrycould not undergo spontaneous symmetry breaking ina vector-like theory such as QCD at normal vacuumconditions at zero chemical potential . Finite baryondensity however may result in a breaking of parity in-variance by simply circumventing the hypothesis of thetheorem. Indeed the presence of a finite chemical po-tential leads to the appearance of a constant imaginaryzeroth-component of a vector field and the conditionsunder which the results of [31] were proven are not ful-filled anymore.Accordingly let us check the possibility of conden-sation of the neutral isospin pseudoscalar components(in order not to violate charge conservation), π a = π δ a , π a = ρδ a , (15)and for the vacuum solutions take π ± = π ± = 0. In thiscase one obtains four mass-gap equations as the pioncondensate h π i 6 = 0 becomes, in principle, possible,unlike in the chiral limit, − ∆ σ + ∆ σ ) − m q d cos π F + 4 λ σ +3 λ σ σ + 2( λ + λ ) σ σ + λ σ + ρ (cid:16) λ − λ ) σ + λ σ (cid:17) = 0 , (16) − ∆ σ + ∆ σ ) − m q d cos π F + λ σ +2( λ + λ ) σ σ + 3 λ σ σ + 4 λ σ + ρ (cid:16) λ σ + 4 λ σ (cid:17) = 0 , (17)( d σ + d σ ) sin π F + d ρ cos π F = 0 , (18) m q d sin π F + ρ (cid:16) − ∆ + ( λ − λ ) σ + λ σ σ +2 λ ( σ + ρ ) (cid:17) = 0= ρ (cid:16) − m q d ( d σ + d σ ) cos π F − ∆ + ( λ − λ ) σ + λ σ σ + 2 λ ( σ + ρ ) (cid:17) , (19)where the last equality follows from eq. (18). As well theequality ρ = 0 entails π = 0 from eq. (18) if d σ + d σ = 0 and d = 0. However, as will be seen below,the combination d σ + d σ is related to the quarkcondensate h d σ + d σ i = −h ¯ qq i > , (20)hence, this combination cannot be zero. For d = 0 onehas always h π i = 0 and the parity breaking pattern re-mains the same as for the massless case. We neglect thepossibilities h π i = F nπ, n = 0 , ± , ± , . . . , as not rel-evant for the physics studied in this paper. For d = 0both pseudoscalar v.e.v. h π i and ρ can arise simulta-neously only. To avoid spontaneous parity breaking inthen normal vacuum of QCD, it is thus sufficient toimpose,( λ − λ ) σ + λ σ σ +2 λ σ − ∆ − m q d ( d σ + d σ ) > , (21)on the mass-gap solutions σ j = h σ j i in the vicinity of aminimum of effective potential. It follows from the lastline in eq. (19). Since QCD in normal conditions doesnot lead to parity breaking, the low-energy model mustfulfill (21).For the parity-even vacuum state the necessary con-dition to have a minimum for non-zero σ j = h σ j i (forvanishing ρ ), equivalent to the condition of having CSBin QCD, can be derived from the condition to get a lo-cal maximum (or at least a saddle point) for zero σ j . Atthis point the extremum is characterized by the matrix − ∆ jk in (6). It must have at least one negative eigen-value. This happens either for Det ∆ > , tr { ∆ } > ∆ < σ j = h σ j i . The sufficient conditionsfollow from the positivity of the second variation for anon-trivial solution of the two first equations (16), (17)at ρ = 0. The matrix containing the second variationsˆ V (2) for the scalar sector is12 V (2) σ = − ∆ + 6 λ σ + 3 λ σ σ + ( λ + λ ) σ ,V (2) σ = − ∆ + 3 λ σ + 4( λ + λ ) σ σ + 3 λ σ , V (2) σ = − ∆ + ( λ + λ ) σ + 3 λ σ σ + 6 λ σ . (22) In turn, the nonzero elements of the second variationsˆ V (2) in the pseudoscalar sector are( V (2) πjk ) ab ≡ V (2) πjk δ ab ;12 V (2) π = m q d σ + d σ F , V (2) π = 2 m q d F , V (2) π = − ∆ + ( λ − λ ) σ + λ σ σ + 2 λ σ . (23)The required conditions are given by tr n ˆ V (2) o > V (2) > σ j = h σ j i . For positive matrices itmeans that V (2) σjj > V (2) πkk > . (24)The diagonalization of the matrix ( V (2) πjk ) leads to phys-ical mass states for pseudoscalar mesons π, Π which arethe mixtures of π , π . The eigenvalues of (23) even-tually give their masses squared and thereby must bepositive according to the inequality (21). The latter cor-responds to the positivity of the determinant,Det ˆ V (2) π = 4 m q d σ + d σ F (cid:16) − ∆ + ( λ − λ ) σ + λ σ σ + 2 λ σ − m q d ( d σ + d σ ) (cid:17) , (25)wherefrom it becomes evident that the inequality (21)is also a necessary condition for the absence of sponta-neous parity breaking. Indeed it follows from the pos-itivity of matrix element V (2) π that the combination h d σ + d σ i > . In fact, to the leading order in m q themasses of a lighter π and a heavier Π mesons are pro-portional to V (2) π and V (2) π , respectively (see (23)). Amore detailed analysis of the pseudoscalar meson spec-trum will be given in Subsect.8.2. The requirement tohave a positive determinant of the matrix V (2) πjk is sup-ported by (21).The two set of conditions, namely those presentedin eq. (21) and in eq. (24) represent restrictions thatthe symmetry breaking pattern of QCD imposes on itslow-energy effective realization at vanishing chemicalpotential.One can easily find the correction linear in m q tothe vacuum solution in the chiral limit h σ j i ( m q ) ≃ h σ j i (0) + 2 m q ∆ σj ; ∆ σ ≡ (cid:18) ∆ σ ∆ σ (cid:19) = (cid:16) ˆ V (2) σ (cid:17) − · d = 1Det ˆ V (2) σ d V (2) σ − d V (2) σ d V (2) σ − d V (2) σ ! ; d ≡ (cid:18) d d (cid:19) . Using these equations the corrections to the masses ofscalar and heavy pseudoscalar mesons can be derived straightforwardly. In particular, for scalar mesons thecorrections to the mass matrix are ∆V (2) σkl = 2 m q X j,m =1 , ∂ j V (2) σkl (cid:0) ˆ V (2) σ (cid:1) − jm d m h ∂ j ≡ ∂ σ j i = 2 m q X j,m =1 , ∂ k V (2) σlj (cid:0) ˆ V (2) σ (cid:1) − jm d m = 2 m q ∂ k (cid:16) ˆ V (2) σ (cid:17) · (cid:16) ˆ V (2) σ (cid:17) − · d , (26)whereas in the pseudoscalar sector ∆V (2) π = 2 m q X j,m =1 , ∂ j V (2) π (cid:0) ˆ V (2) σ (cid:1) − jm d m . (27)The latter term saturates the current quark mass cor-rection for heavy pseudoscalar meson masses. Let us investigate how many extrema the effective po-tential possesses for different values of the coupling con-stants. In this Section we take the chiral limit m q = 0for simplicity. It turns out that when the chemical po-tential and temperature are zero one can eliminate oneof the constant in the effective potential by a redefini-tion of the fields. Indeed, one can change the variable H = αH + β ˜ H , (28)using a linear transformation with real coefficients α, β (to preserve reality of ˜ σ j , π aj ). With the help of this re-definition one can diagonalize the quadratic part in (6)and make its coefficients equal ˜ ∆ = ˜ ∆ = det ˆ ∆/∆ ≡ ∆ . Then X j,k =1 tr n H † j ∆ jk H k o = ∆ tr n H † H + ˜ H † ˜ H o . (29)A further reduction of the coupling constants affects thedependence of free energy on finite chemical potentialand temperature (see below), but it can be implementedwhen both external control parameters vanish; namelywe perform the orthogonal rotation of two fields H = cos φ ˇ H + sin φ ˇ H , ˜ H = − sin φ ˇ H + cos φ ˇ H . (30)Then the coefficient in the operator ( ˇ H † ˇ H + ˇ H † ˇ H ) ˇ H † ˇ H becomes equal toˇ λ cos φ = λ − λ + λ − λ ) tan φ − λ − λ ) tan φ + 2( λ + λ − λ ) tan φ − λ tan φ ≡ P λ (tan φ ) . (31) One can always fix λ < H . Thenif λ < P λ (0) < φ ≫ P λ (tan φ ) ∼ − λ tan φ > P λ (tan φ ) = 0 has at least one (positive) real root. In the complementary region λ ≥ P λ (0) >
0. In this case one can look at tan φ = ± P λ ( ±
1) = − λ − λ ) ± λ − λ ) , (32)so that one of these combinations is negative. Againthe comparison with the asymptotics allows to concludethat there is a real root for P λ (tan φ ) = 0. Thus forany sign of λ it can be eliminated by a proper rotationof scalar fields.Let us take the basis of operators with ˇ λ = 0 .Then, after renaming the fields V eff = − ∆ (cid:16) ( σ ) + ( σ ) (cid:17) + ˇ λ (cid:16) ( π a ) (cid:17) +( π a ) (cid:16) − ∆ + (ˇ λ − ˇ λ ) σ + ˇ λ σ σ + 2ˇ λ σ (cid:17) +ˇ λ σ + ˇ λ σ + (ˇ λ + ˇ λ ) σ σ + ˇ λ σ σ . (33)This potential simplifies the mass gap equations andsecond variations in order to investigate their solutionsanalytically. The effective potential must provide thefamiliar CSB at normal conditions ( µ = T = 0). Thus inthe chiral limit there are at least two minima related bythe symmetry rotation σ , → − σ , and one maximumat the origin. This is implemented by assigning a realsinglet v.e.v. h σ i > H thereby selecting one ofthe minima.In this Section we shall assume ˇ λ = 0 in orderto determine the different vacua of the theory at zerotemperature and chemical potential.4.1 Search for the extrema of effective potentialIn the parity symmetric case the second eq. (17) reads σ ( − ∆ + (ˇ λ + ˇ λ ) σ + 32 ˇ λ σ σ + 2ˇ λ σ ) = 0 . (34)One of its solutions is σ (0)2 = 0 and directly from eq.(16)one finds σ (0)2 = 0 , ( σ (0)1 ) = ∆ λ . (35)For stable solutions ˇ λ > ∆ > σ ( m )1 , ; m = 1 , , σ = 0 . With a combination of the min minminmin saddlesaddle saddlesaddle (cid:86) (cid:86) max (1) t (2) t (3) t (0)1 (cid:86) (1) -t (2) -t (3) -t (0) 1 (cid:86)(cid:16) Fig. 1
Extrema of effective potential in the reduction basis: themaximum is placed in the square, four minima are located inthe circles and the corresponding four saddle points are depictedby the lentils. The existence of two solutions t (2) , t (3) with posi-tive values of σ j is governed by condition (38). Which one corre-sponds to the true minimum depends on the actual value of thephenomenological constants. mass-gap eqs. (16) and (34) one can decouple the equa-tion in terms of the ratio t = σ /σ , P ( t ) = t − at − bt + c = 0 , (36) a = 2 (cid:0) ˇ λ + ˇ λ − λ (cid:1) − ˇ λ , b = 3 ,c = 2 (cid:0) ˇ λ + ˇ λ − λ (cid:1) − ˇ λ , where the sign is fixed for ˇ λ < c > P (0) > , P ′ (0) < P ( t ) P ′ ( t ) = 0 −→ t − at − b = 0 ,t ± = 13 a ± r a + b, t + > , t − < . (37)All together it means that a negative solution t (1) < t (2) < t (3) , and separated by a minimum of cubicpolynomial P . Therefore the existence of two positivesolutions is regulated by the sign of P ( t + ). They existif P ( t + ) = c − a a r a b ! < c < a a + q a + b − a + q a + b . (38) Evidently it takes place for some positive a , i.e. forˇ λ + ˇ λ > λ . (39)Finally we can have at most two minima for posi-tive σ , namely, at σ (0)2 = 0 and at t (2) or t (3) which en-tails two more minima for negative σ due to symmetryunder σ j → − σ j . Then other solutions correspond tosaddle points as four minima must be separated by foursaddle points situated around the maximum, see Fig.1.This configuration is unique for potentials bounded be-low, namely, any saddle point connects two adjacentminima. Later on we will see that in order to implementa first-order phase transition to stable nuclear matterwe just need two minima for positive σ . Thereby, inthe half plane of positive σ one has to reveal four solu-tions, namely, one is σ (0)2 = 0 and three for σ = 0 whichare inevitably mark one more minimum and two saddlepoints. Thereby the condition (38) should be satisfiedin order to be able to describe the saturation point .After the appropriate roots t j are found one can useEq. (34) and find the v.e.v. of σ j , σ = ∆ (ˇ λ + ˇ λ ) + ˇ λ t j + 2ˇ λ t j > σ = t j σ . (40)The latter inequality holds for any t j ifˇ λ (ˇ λ + ˇ λ ) ≥
932 (ˇ λ ) . (41)Otherwise the existence of real σ for solutions t j needsa more subtle investigation.Let us recall that all the inequalities obtained in thisSection are referred to the field basis with fully diagonal ∆ ij = ∆δ ij and with ˇ λ = 0. However it is evident thatthe qualitative structure of extrema is independent ofthe basis choice.4.2 Selection of the minimaIn all cases the conditions of minimum come from thepositive definiteness of the matrix of second variationsof effective potential,12 V (2) σ = − ∆ + 6ˇ λ σ + (ˇ λ + ˇ λ ) σ > , (42) V (2) σ = 4(ˇ λ + ˇ λ ) σ σ + 3ˇ λ σ , V (2) σ = − ∆ + (ˇ λ + ˇ λ ) σ + 3ˇ λ σ σ + 6ˇ λ σ > , V (2) π = − ∆ + (ˇ λ − ˇ λ ) σ + ˇ λ σ σ + 2ˇ λ σ > V (2) π = V (2) π = 0 . For σ (0)2 = 0 they read(ˇ λ ± ˇ λ ) > λ , −→ ˇ λ > | ˇ λ | (43)for ∆ > c > σ ( m )2 = 0 one obtains a number of bounds onthe solution from the second variation12 V (2) σ = ( σ ) h λ −
12 ˇ λ t i > ,V (2) σ = ( σ ) h λ + ˇ λ ) t + 3ˇ λ t i , V (2) σ = ( σ ) h
32 ˇ λ t + 4ˇ λ t i > , V (2) π = ( σ ) h − λ −
12 ˇ λ t i > . (44)Evidently if ˇ λ > σ > → t >
0. The remainingbound must come from the positivity, det ˆ V (2) > N c (mean field) approach. This gives a prescription toconnect the properties of quark and nuclear matter andestimate the parameters of our model to reproduce me-son phenomenology and the bulk characteristics of nu-clear matter such as binding energy, normal nucleardensity and (in)compressibility.The meson degrees of freedom present in our modelappear after bosonization of QCD in the vacuum andthe relevant effective potential is given in Sec. 3, eq. (6).The effects of infinite homogeneous baryon matter onthe effective meson lagrangian are described by the baryonchemical potential µ , which is transmitted to the me-son lagrangian via a local quark-meson coupling (in theleading order of chiral expansion µ /Λ ). In turn, in thelarge N c limit one can neglect the temperature depen-dence due to meson collisions. The temperature T isinduced with the help of the imaginary time Matsub-ara formalism for quark Green functions[32] ω n = (2 n + 1) πβ , β = 1 kT . (45) For real physics with 3 colors this approximation tothermal properties of mesons is expected to be less pre-cise as meson loops contribute substantially to the ther-modynamic characteristics for large temperatures (firstof all a hot pion gas). Nevertheless it should be sufficientto describe qualitatively the interplay between baryondensity and temperature at the phase transition.Without loss of generality we can specify one of thecollective fields H j , namely, H as that one which haslocal coupling to quarks: this actually defines the chi-ral multiplet H . The set of coupling constants in (6)is sufficient to support this choice as well as to fix theYukawa coupling constant to unity. Accordingly, we se-lect the basis in which finite density and temperaturewere transmitted to the boson sector by means of ∆ L q = ¯ q R H q L + ¯ q L H † q R −→ ¯ Qσ Q, (46)where Q L = ξq L , Q R = ξ † q R ; ξ = exp { i ˆ π/ F } standfor constituent quarks [6]. Then for finite temperaturesand chemical potentials the free Fermi gas contributionto the generalized σ model lagrangian originates fromthe quark action in Euclidean space-time (thermal fieldtheory) [33], S q = β Z dτ Z d x q † (cid:16) ∂ − γ µ + ( H P L + H † P R ) (cid:17) q ≃ β Z dτ Z d x ¯ Q ( ∂ − γ µ + σ ) Q, (47)where P L,R ≡ (1 ± γ ). As we want to calculate theeffective potential we neglect the gradient of chiral fields ∂ µ ξ ∼ ∆V eff ( σ , µ, β ) = V eff ,Q ( σ , µ, β ) − V eff ,Q ( σ , , − N β Z d pπ n log (cid:16) − β ( E − µ )) (cid:17) + log (cid:16) − β ( E + µ )) (cid:17)o = − N β ∞ Z σ dEE q E − σ n log (cid:16) − β ( E − µ )) (cid:17) + log (cid:16) − β ( E + µ )) (cid:17)o (48)= − N ∞ Z σ dE (cid:16) E − σ (cid:17) / cosh( βµ ) + exp( − βE )cosh( βµ ) + cosh( βE ) , where E = p p + σ . The last expression for the Fermigas free energy in (48) can be obtained with the help of integration by part and an appropriate change of energyand momentum variables.Accordingly the complete effective potential is spec-ified as, V eff ( σ j , π aj ; µ, β ) = V eff ( σ j , π aj ; 0 , ∆V eff ( σ , µ, β ) . Following this recipe the quark temperature and chem-ical potential dependence can be derived for mass gapequations – the conditions for a minimum of the effec-tive potential. Namely, taking into account the choiceof variables (10) the first equation (16) is modified to − ∆σ − m q d cos π F + 4 λ σ + 3 λ σ σ +2( λ + λ ) σ σ + λ σ + ρ (cid:16) λ − λ ) σ + λ σ (cid:17) +2 N σ A ( σ , µ, β ) = 0 , (49)with notation N ≡ N c N f π . In turn A ( σ , µ, β ) = 12 N σ ∂ σ ∆V e ff ( σ , β, µ )= 2 Z ∞ σ dE q E − σ cosh( βµ ) + exp( − βE )cosh( βµ ) + cosh( βE ) . (50)When using the gap equations (17), (18), (19)and (49)one finds the value of the effective potential at its min-ima (”on shell”), V eff ( σ j , π aj ; µ, β ) (cid:12)(cid:12) σ j = h σ j i ; π a = δ a h π i ; π a = δ a ρ ≡ e V eff( µ, β ) = − ∆ (cid:16) h σ i + h σ i + ρ (cid:17) + 32 m q (cid:20) − ( d h σ i + d h σ i ) cos h π i F + d ρ sin h π i F (cid:21) + ∆V eff ( h σ i , µ, β ) − N h σ i A ( h σ i , µ, β ); (51) ∆V eff − N h σ i A = − N ∞ Z h σ i dE (4 E − h σ i ) (cid:16) E − h σ i (cid:17) / × cosh( βµ ) + exp( − βE )cosh( βµ ) + cosh( βE ) . (52)In order to derive it we split the vacuum potential (12),(14) into three pieces according to their field dimension, V eff ( σ j , π aj ; 0 ,
0) = V (2) eff + V (4) eff + ∆V (1) eff ( m q ) . (53)Next let us multiply eq. (49) by σ , eq. (17) by σ andeq. (19) by ρ and sum up. In this way the followingon-shell identity is obtained, V (4) eff ( h σ j i ) = − V (2) eff ( h σ j i ) − ∆V (1) eff ( m q )( h σ j i ) − N h σ i A ( h σ i , µ, β ) . (54) The final result (51) can be derived by insertion of thisidentity in eq. (53).Let us notice that the chosen specification of col-lective fields H j is compatible with the transforma-tion (28) and therefore one can proceed to the diag-onal quadratic part of the potential (6). However theadditional linear transformation (30) would split theconstituent mass in the quark Yukawa vertex into twofields¯ q R H q L + h. c. −→ ¯ q R (cos φ ˇ H + sin φ ˇ H ) q L + h. c.It means that a possible change of the basis used in Sec.4 to eliminate the constant λ would affect the chemicalpotential driver ∆V e ff ( σ , β, µ ) → ∆V e ff ( p (cos φ ˇ σ + sin φ ˇ σ ) + ˇ ρ , β, µ ) . (55)Thereby all the mass gap equations (16)–(19) wouldobtain new contributions depending on T and µ mak-ing the equations less tractable. In order to keep thesimplified form of the mass gap equations we prefer toretain the single scalar field in the Yukawa vertex andinclude the dependence on environment conditions inone mass-gap equation only . Correspondingly we take,in general, λ = 0.However, the qualitative results derived in the pre-vious Section on the different vacua for vanishing tem-perature and chemical potential remain obviously valid.Namely, in the CSB regime one has at most one max-imum, four minima and four saddle points at our dis-posal in order to simulate nuclear matter properties.5.2 Thermodynamic properties of the model at T = 0Thermodynamically the system is described by the pres-sure P , the energy density, ε and the entropy density S . The pressure is determined by the potential densitydifference with and without the presence of chemicalpotential, dP = − dVP ( µ, β ) ≡ e V eff (0 , − e V eff ( µ, β ) , (56)The energy density is related to the pressure, baryondensity and entropy density by ε = − P + N c µ̺ B + T S. (57)The chemical potential is defined as ∂ ̺ B ε = N c µ, (58) with the entropy and volume held fixed. The factor N c is introduced to relate the quark and baryon chemicalpotentials. Since ε is independent of µ N c ∂ µ P = ̺ B = − N c ∂ µ e V eff (59)= N f π ∞ Z h σ i dEE p E − h σ i sinh( βµ )cosh( βµ ) + cosh( βE ) , where h σ i = σ ( µ, β ) on shell.In turn the entropy is defined as S = ∂ T P = − ∂ T e V eff , T S = β∂ β e V eff . (60)with the baryon density and volume held fixed.The above equation allows to calculate the energydensity (57) in our model in terms of the effective po-tential on shell, ε = ( − µ∂ µ − β∂ β ) ˜ V eff ( µ, β ) . (61)5.3 Zero temperature and finite densityIn this subsection we consider the zero-temperature caseand study the regime of chemical potentials compara-ble with the v.e.v. σ . At zero temperature T = 0 thecontribution from µ to the effective potential is ∆V eff ( σ , µ ) = − θ ( µ − σ ) 43 N µ Z σ dE (cid:16) E − σ (cid:17) / = N θ ( µ − σ ) " µσ q µ − σ − µ µ − σ ) / − σ ln µ + p µ − σ σ . (62)The total value of the effective potential at its minimumis e V eff( µ ) = − ∆ (cid:16) h σ i + h σ i + ρ (cid:17) + 32 m q (cid:20) − ( d h σ i + d h σ i ) cos h π i F + d ρ sin h π i F (cid:21) − N µ (cid:16) µ − h σ i (cid:17) / θ (cid:16) µ − h σ i (cid:17) . (63)Higher-order terms of the chiral expansion in 1 /Λ arenot considered.Accordingly in the first mass gap equation (49) A ( σ , µ, β ) β →∞ = 2 θ ( µ − σ ) Z µσ dE q E − σ = θ ( µ − σ ) h µ q µ − σ − σ ln µ + p µ − σ σ i . (64) Then the second variation of effective potential is mod-ified in the only element12 V (2) σ = − ∆ + 6 λ σ + 3 λ σ σ + ( λ + λ ) σ + N θ ( µ − σ ) " µ q µ − σ − σ ln µ + p µ − σ σ . (65)The effective potential (62),(63) is normalized to repro-duce the baryon density for quark matter ̺ B = − N c ∂ µ ∆V eff ( σ , µ ) (cid:12)(cid:12)(cid:12) σ = h σ i = σ ( µ ) (66)= − N c d V eff( µ )d µ = N f π p F = N f π (cid:16) µ − σ ( µ ) (cid:17) / , where the quark Fermi momentum is p F = p µ − σ ( µ ). in the chiral limit m q = 0 for simplicity, where the strictinequality (21) does not hold and instead for µ ≥ µ crit ( λ − λ ) σ + λ σ σ + 2 λ (cid:16) σ + ρ (cid:17) = ∆, (67)so that Eq.(19) admits non-zero values of ρ and therebySPB arises. After substituting ∆ from (67) into the sec-ond eq. (16) one finds that λ σ + 4 λ σ σ + λ (cid:16) σ + ρ (cid:17) = 0 , (68)where we have taken into account that h σ i 6 = 0. This,together with (67) completely fixes the v.e.v.’s of thescalar fields σ , . If λ = 0 and/or λ = 0 equations(67) or (68) unambiguously determine the relation be-tween h σ i and h σ i . Otherwise if λ λ = 0 these twoequations still allow to get rid of the v.e.v. of pseu-doscalar field leading to the relation (cid:16) λ λ + λ ( λ − λ ) (cid:17) σ + (cid:16) λ λ − λ (cid:17) σ σ = − λ ∆, (69)whose solution is h σ i = A h σ i + B h σ i > A ≡ λ λ + λ ( λ − λ ) λ − λ λ ; B ≡ λ ∆λ − λ λ . (70)Thus in the parity breaking phase the relation betweenthe two scalar v.e.v’s is completely determined and, inparticular, does not depend neither on ρ nor on µ . The first mass gap equation (49) can be brought tothe form ∆ = 2 λ σ + λ σ σ + ( λ − λ )( σ + ρ )+ N A ( σ , µ, β ) , (71)if one employs eq. (68). Together with eq. (70) it allowsto find all v.e.v.’s of the scalar fields σ j , ρ as functionsof temperature and chemical potential.Let us now find the critical value of the chemicalpotential, namely the value where ρ ( µ c ) = 0, but equa-tions (67), (68), (70) hold. Combining the two equations(67), (68) λ r + 4 λ r + λ = 0; r ≡ h σ ih σ i . (72)In order for a SPB phase to exist this equation has topossess real solutions. If λ = 0 there is only one so-lution corresponding to a second order transition, butthere may exist other solutions that fall beyond theaccuracy of our low energy model (which becomes in-appropriate for small values of σ ).We stress that equations (70) and (72) contain onlythe structural constants of the potential and do not de-pend on temperature or chemical potential manifestly.Thus using the critical values r crit = r ± = − λ ± p λ − λ λ λ (73)one can immediately calculate h σ i ± ( ∆, λ j ) = s Br ± − A ; h σ i ± ( ∆, λ j ) = r ± h σ i ± , (74)where h σ i i ± are the corresponding critical values.After substituting these values into equation (71)for each critical set of h σ i i one derives the boundary ofthe parity breaking phase N A ( σ ± , µ crit , β crit ) = ∆ − λ ( h σ i ± ) − λ h σ i ± h σ i ± − ( λ − λ )( h σ i ± ) . (75)It must be positive at critical values of h σ i i ± . The rela-tion (75) defines a strip in the T − µ plane where parityis spontaneously broken. From (50) one can obtain that A > A → ∞ when
T, µ → ∞ . It means that for any nontrivial solution h σ i ± , h σ i ± the parity breakingphase boundary exists.Thus we have proved that if the phenomenon of par-ity breaking is realized for zero temperature it will takeplace in a strip including lower chemical potentials buthigher temperatures. 6.2 Mass-gap equations in SPB beyond the chiral limitLet us now examine again the possible existence of acritical point where the strict inequality (21) does nothold and for µ > µ crit ( λ − λ ) σ + λ σ σ + 2 λ (cid:16) σ + ρ (cid:17) − ∆ (76)= m q d ( d σ + d σ ) cos π F = m q d p d ρ + ( d σ + d σ ) , where the following consequence of equation(18) hasbeen used:cos π F = d σ + d σ p d ρ + ( d σ + d σ ) . (77)When combining equation (76) with (16), (17) onefinds that d (cid:16) λ σ + 4 λ σ σ + λ ( σ + ρ ) (cid:17) = 2 d (cid:16) − ∆ + 2 λ σ + λ σ σ +( λ − λ )( σ + ρ ) + N A ( σ , µ, β ) (cid:17) , (78) d (cid:16) λ σ + 4 λ σ σ + λ ( σ + ρ ) (cid:17) = 2 d (cid:16) − ∆ + ( λ − λ ) σ + λ σ σ + 2 λ ( σ + ρ ) (cid:17) , (79)where we have taken into account that h σ i 6 = 0. Thesetwo relations determine the v.e.v.’s of the scalar fields σ , . If λ = λ = 0 and/or λ = λ , λ = 0 equations(78) and (79) firmly fix the relation between h σ i and h σ i . Otherwise an appropriate combination of thesetwo equations still allows us to get rid of the v.e.v. of thepseudoscalar field . Thus in the parity breaking phasethe relation between the two scalar v.e.v’s is completelydetermined and in particular does not depend neitheron ρ nor on µ . Using equations (76), (78) and (79) onecan easily eliminate the variables ρ and σ , obtaining anequation for the variable σ /µ . The latter completesthe determination of the v.e.v.’s.We notice that in the chiral limit m q → d , d become arbitrary and therefore (78), (79)entail three independent relations coinciding with (67),(68), (71). Let us find the character of the phase transition to theSPB phase. In this Section, for brevity, we employ the We recall that in the presence of SPB the distinction betweenscalars and pseudoscalars is a nominal one.3 v.e.v.’s of variables σ j = h σ j i = σ j ( µ ) , ρ = h ρ i = ρ ( µ )as functions of the chemical potential µ on shell. Forsmall values of µ − σ ( µ ) >
0, we know that the valueof the odd parity condensate h ρ i is zero. Setting ρ = 0in equations (16), (17), (49) and using (71) and differ-entiating w.r.t. µ we get X k =1 , ˆ V (2) σjk ∂ µ σ k = − N σ q µ − σ δ j , (80)or, after inversion of the matrix of second variations, ∂ µ σ = − N σ q µ − σ V (2) σ Det ˆ V (2) σ < ,∂ µ σ = 4 N σ q µ − σ V (2) σ Det ˆ V (2) σ . (81)The possibility of SPB is controlled by the inequality(21); in order to approach a SPB phase transition whenthe chemical potential is increasing we have to diminishthe l.h.s. of inequality (21) and therefore we need tohave ∂ µ h ( λ − λ ) σ + λ σ σ + 2 λ σ − m q d ( d σ + d σ ) i < . (82)This is equivalent (using (81)) to (cid:16) λ σ + 4 λ σ + m q d ( d σ + d σ ) (cid:17) V (2) σ << (cid:16) λ − λ ) σ + λ σ + m q d d ( d σ + d σ ) (cid:17) V (2) σ . (83)This last inequality is a necessary condition that has tobe satisfied by the model at zero chemical potential forit to be potentially capable of yielding SPB. Evidently,this inequality must hold across the critical point inorder that ∂ µ ρ > , ∂ µ ( π ) > π appears spontaneously thevector SU (2) symmetry is broken to U (1) and two charged Π mesons are expected to possess zero masses as dic-tated by the Goldstone theorem. For simplicity let usconsider zero temperature. In the chiral limit the matrixof second variations in essential variables σ , σ , π hasthe rank 3, ˆ V (2) = (cid:16) V (2) mn (cid:17) ; m, n = 1 , , , where the index ’0’ is engaged for variation of the neutral pseu-doscalar field π . This matrix reads12 V (2) σ = − ∆ + 6 λ σ + 3 λ σ σ +( λ + λ ) σ + ( λ − λ ) ρ + N " µ q µ − σ − σ ln µ + p µ − σ σ ≡ V ,V (2) σ = 3 λ σ + 4( λ + λ ) σ σ +3 λ σ + λ ρ ≡ V , V (2) σ = − ∆ + ( λ + λ ) σ + 3 λ σ σ +6 λ σ + 2 λ ρ ≡ V , (84) V (2) σπ = (cid:16) λ − λ ) σ + 2 λ σ (cid:17) ρ ≡ V ρ,V (2) σπ = (cid:16) λ σ + 8 λ σ (cid:17) ρ ≡ V ρ, V (2) π = 4 λ ρ ≡ V ρ . (85)We notice that the second variation of charged pseu-doscalar fields π ± vanishes V (2) π ±∓ = 0 and thereforethese fields are massless Goldstone bosons.Now we are able to check the character of phasetransition. The qualitative behavior of the order param-eters: dynamical mass σ j ( µ ) and parity-odd condensate ρ ( µ ), is shown on Fig.2. It is justified when using consis-tently equations (17), (49) and the condition (76) in theSPB phase. Then one obtains the differential equationson functions σ j ( µ ) , ρ ( µ ), following the same strategyas for (81), ∂ µ σ = − N σ q µ − σ V V − V Det ˆ V < ,∂ µ σ = − N σ q µ − σ V V − V V Det ˆ V ,∂ µ ρ = − N σ q µ − σ V V − V V Det ˆ V . (86)The last derivative must be positive in order to generateparity breaking and this is guaranteed by the inequality(83). Let us compare the derivatives of the dynamicmass σ across the phase transition point. For µ → µ crit − i µ → µ crit + i ∂ µ σ (cid:12)(cid:12)(cid:12) µ crit + i − ∂ µ σ (cid:12)(cid:12)(cid:12) µ crit − i = − N σ q µ − σ ( V V − V Det ˆ
V − V (2) σ Det ˆ V (2) σ ) = − N σ q µ − σ (cid:0) V V (2) σ − V V (2) σ (cid:1) Det ˆ V Det ˆ V (2) σ < , (87)4
V − V (2) σ Det ˆ V (2) σ ) = − N σ q µ − σ (cid:0) V V (2) σ − V V (2) σ (cid:1) Det ˆ V Det ˆ V (2) σ < , (87)4 SPB
Jumps of derivatives 2nd order phase transition
Fig. 2
The SPB phase transition of second order: the dashedline depicts the SPB breaking phase and the solid line stands forthe v.e.v. of ”dynamical” mass. The plot is only qualitative. provided that both determinants are positive (they de-termine the spectrum of meson masses squared ) andinequality (83) holds across. Thus the dynamic massderivative is discontinuous and the phase transition isof second order.7.2 Inclusion of current quark massesFor non-vanishing current quark masses the deviationlinear in m q in the parity breaking phase affects also thepseudoscalar parameters h ρ i = ρ ( µ ) and h π i = π ( µ ).After usage of equation (77) one findsˆ ∆ ≡ ∆σ ( m q ) ∆σ ( m q ) ∆ρ ( m q ) ≃ m q ∆ ∆ ∆ ρ ; (88)˜ ∆ ≡ ∆ ∆ ∆ = (cid:16) ˆ V (cid:17) − · d d d d σ + d σ d σ + d σ p d ρ + ( d σ + d σ ) , where in order to keep the leading order all parametersmust be taken in the chiral limit. As to the v.e.v. ofneutral pion field it does not need any mass correctionsto the leading order and must be taken from the massindependent eq. (18) π = − arctan (cid:18) d ρd σ + d σ (cid:19) . (89) Accordingly, the mass corrections to the matrix of sec-ond variation ∆ ˆ V (2) , equation (85), takes the form V (2) σjl ( m q ) = V (2) σjl (0) + X m =1 , , ∂ m (cid:16) V (2) σjl (cid:17) ˆ ∆ m , (90) V (2) σ j π ( m q ) = V (2) σπj + X m =1 , , ∂ m (cid:16) V (2) σπj (cid:17) ˆ ∆ m , (91)( ∂ m ) ≡ ( ∂ σ , ∂ σ , ∂ ρ ) ; V (2) σ π = 2 m q d F sin π F , V (2) σ π = 2 m q d F sin π F , (92)12 V (2) π = m q d σ + d σ F cos π F − d ρF sin π F ! , (93) V (2) π = 2 m q d F cos π F , (94)12 V (2) π = 4 λ ρ (0) + 16 m q λ ∆ + m q d d σ + d σ cos π F , (95) V (2) π +1 π − = − m q d ρ ( π ) sin π F = 2 m q d σ + d σ ( π ) sin π F cos π F , (96) V (2) π +1 π − = V (2) π +2 π − = 2 m q d π sin π F , (97) V (2) π +2 π − = 2 m q d d σ + d σ cos π F , (98)where the r.h.s. are evaluated with the help of eqs.(16)-(18), (76) - (78) and the v.e.v’s for σ j , ρ are takenin the chiral limit. We notice that convexity aroundthis minimum implies that all diagonal elements arenon-negative. This gives positive masses for two scalarand four pseudoscalar mesons, whereas the doublet ofcharged of π mesons remains massless. The latter canbe easily checked from the vanishing determinant of thelast matrix V (2) π + j π − l in eqs. (96)-(98). Of course, quanti-tatively the mass spectrum can be obtained only afterkinetic terms are properly normalized.If the soft breaking of chiral symmetry occurs onlyin the H channel, d = 0 then it follows from eq. (18)that light pions do not condense h π i = 0 and do notmix with other states as the off-diagonal matrix ele-ments (92), (94) and (97) vanish. The second pair ofcharged pseudoscalars π ± becomes massless manifestly. σ model In this Section we examine the fluctuations around theconstant solutions of the mass-gap equations (49),(17),(18),(19), and introduce appropriate notations for thefluctuations Σ j , ˆ Π around v.e.v.’s h σ j i , h ρ i so that σ j ≡h σ j i + Σ j ,ˆ π = τ h π i + ˆ π , ˆ π = τ h ρ i + ˆ Π . These v.e.v’s h σ j i , h ρ i must be used in all previous relations for thesecond variation of the potential. In calculations of thekinetic term we retain only the terms in the chiral limitkeeping our interest to the masses of scalar and pseu-doscalar mesons at the leading order of the expansionin current quark masses, m q . Thus in the kinetic termswe take h π i ≃ H without changes in the chemical potentialdriver (62). However the rescaling of the field H ispossible at the expense of an appropriate redefinitionsof other coupling constants and this freedom can beused to fix one of the constants which appear in thekinetic term. Thus we take the general kinetic termsymmetric under SU (2) L × SU (2) R global rotations tobe L kin = 14 X j,k =1 A jk tr n ∂ µ H † j ∂ µ H k o . (99)With the chiral parameterization (10) one can separatethe bare Goldstone boson action, L kin = 12 X j,k =1 A jk ∂ µ σ j ∂ µ σ k + 14 X j,k =1 A jk σ j σ k tr n ∂ µ U † ∂ µ U o (100)+ 12 i X j =1 A j tr n σ j (cid:16) ξ † ( ∂ µ ξ ) ξ † − ∂ µ ξ ( ξ † ) ∂ µ ξ (cid:17) ˆ π − σ j ξ † ∂ µ U ξ † ∂ µ ˆ π + ∂ µ σ j ξ † ∂ µ U ξ † ˆ π o + 14 A tr n ∂ µ ˆ π ∂ µ ˆ π − ∂ µ ξξ † ˆ π ξ † ∂ µ ξ ˆ π − ( ∂ µ ξξ † ∂ µ ξξ † + ξ † ∂ µ ξξ † ∂ µ ξ )(ˆ π ) +[ ξ † , ∂ µ ξ ][ˆ π , ∂ µ ˆ π ] o . After selecting out the v.e.v. h H j i = h σ j i let us explorethe kinetic part quadratic in fields. We expand U = 1 + i ˆ π/F + · · · , ξ = 1 + i ˆ π/ F + · · · and use thenotations defined at the beginning of this Section. Thenthe quadratic part looks as follows L (2) kin = 12 X j,k =1 A jk " ∂ µ Σ j ∂ µ Σ k + 1 F h σ j ih σ k i ∂ µ π a ∂ µ π a (101)+ 1 F X j =1 A j " −h ρ i ∂ µ Σ j ∂ µ π + h σ j i ∂ µ π a ∂ µ Π a + 12 A " h ρ i F ∂ µ π ∂ µ π + ∂ µ Π a ∂ µ Π a , which shows manifestly the mixture between bare pseu-doscalar states and, in the SPB phase, also betweenscalar and pseudoscalar states.Let us define F = X j,k =1 A jk h σ j ih σ k i , ζ ≡ F X j =1 A j h σ j i . (102)8.2 Parity-symmetric phaseIn the symmetric phase h ρ i = 0 , ˆ π = ˆ Π one diagonal-izes by shifting the pion field π a = ˜ π a − ζΠ a , (103) L (2) kin,π = 12 ∂ µ ˜ π a ∂ µ ˜ π a + 12 ( A − ζ ) ∂ µ Π a ∂ µ Π a ,A − ζ = h σ i det AF > . (104)Taking into account the modification of the matrix ofsecond variations (93)-(98) after shifting (103) one findsthe masses of light and heavy pseudoscalars to the lead-ing order in current quark mass( ˜ V (2) π ) ab = ( V (2) π ) ab = δ ab m q d h σ i + d h σ i F = δ ab m π , ( ˜ V (2) π ) ab = ( V (2) π ) ab − ζ ( V (2) π ) ab = δ ab m q (cid:18) d F − ζ d h σ i + d h σ i F (cid:19) , ( ˜ V (2) π ) ab = (cid:16) ( V (2) π ) ab − ζ ( V (2) π ) ab + ζ ( V (2) π ) ab (cid:17) = δ ab (cid:16) − ∆ + ( λ − λ ) h σ i + λ h σ ih σ i +2 λ h σ i − ζm q d F + ζ m q d h σ i + d h σ i F (cid:17) = δ ab (( A − ζ )) m Π + O ( m q ) . (105)at the leading order in m q because it is assumed that m Π ≫ m π far below the P-breaking transition point SPB
P-state mixture0 c Fig. 3
Masses of pseudoscalar states in P-symmetric and in SPBphases. The light and heavy charged pseudoscalars are depictedwith solid lines, the dotted line corresponds to the neutral lightpseudoscalar and the dashed line stands for the neutral heavyone. The plot is only qualitative. in chemical potential (see Fig. 3). We notice that inthis region the off-diagonal element does not make anyinfluence. At the point of the SPB phase transition onehas to impose the condition (76) which leads to( ˜ V (2) π ) ab = δ ab m q d h σ i + d h σ i (cid:16) d − ζ d h σ i + d h σ i F (cid:17) . (106)Evidently the determinant of ( ˜ V (2) π ) vanishes and onereveals zero modes for all three pion states, one neutraland two charged. They represent the true Goldstonemodes (in the limit of exact isospin symmetry m u = m d ). At the P-breaking transition point h ρ i = 0, whentaking into account the normalization of kinetic terms(101) with the definitions (102) one finds the values ofthree massive modes m Π = 2 m q A d − A d d + A d det A ( d h σ i + d h σ i )= 2 m q ( P j,k =1 , d j [ A − ] jk d k ) P j,k =1 , d j h σ j i . (107)Thus in the chiral limit, at the phase transitionpoint one reveals six zero modes and beyond the chirallimit only three ones (see Fig.3).8.3 Masses of light states in SPB phaseIn the SPB phase the situation is more involved: pseu-doscalar states mix with scalar ones. In particular, thediagonalization of kinetic terms is different for neutraland charged pions because the vector isospin symmetry is broken: SU (2) V → U (1). Namely˜ π ± = π ± + ζΠ ± , ˜ π = π + F F + A h ρ i (cid:16) ζΠ − h ρ i F X j =1 A j Σ j (cid:17) . (108)In this way SPB induces mixing of both massless andheavy neutral pions with scalars. The (partially) diag-onalized kinetic term has the following form L (2) kin = ∂ µ ˜ π ± ∂ µ ˜ π ∓ + 12 (cid:16) A h ρ i F (cid:17) ∂ µ ˜ π ∂ µ ˜ π +( A − ζ ) ∂ µ Π ± ∂ µ Π ∓ + 12 ( A − F F + A h ρ i ζ ) ∂ µ Π ∂ µ Π + 12 X j,k =1 A jk F + h ρ i det Aδ j δ k F + A h ρ i ∂ µ Σ j ∂ µ Σ k − F h ρ i F + A h ρ i ζ∂ µ Π X j =1 A j ∂ µ Σ j . (109)We see that even in the massless pion sector the isospinbreaking SU (2) V → U (1) occurs: neutral pions becomeless stable with a larger decay constant. Another obser-vation is that in the charged meson sector the relation-ship between massless π and Π remain the same as inthe symmetric phase.Beyond the chiral limit one can derive the masses ofthe lightest pseudo-goldstone states. When h ρ i ≫ m q then in the mass matrix (90)- (98) the heavy massparts (90), (91), (95) and the light mass ones (93),(96) - (98) combine into an approximately block di-agonal form with additional off-diagonal elements (92)and (94), proportional to m q . The latter leads to fac-torization of the light pseudoscalar meson sector fromthe heavy meson one to the order of m q . Thus neglect-ing the mixture of heavy and light states one deals withthe light sector built of (93), (96) -(98) which after di-agonalizing the kinetic term by (108) projected on thelight state sector gives the light pseudoscalar masses m π = 2 m q (cid:16) A h ρ i F (cid:17) − × (cid:18) d h σ i + d h σ i F cos h π i F − d h ρ i F sin h π i F (cid:19) ,m π ± = 0 , (110) m Π ± = 2 m q cos h π i F ( A − ζ )( d h σ i + d h σ i ) × (cid:18) d − ζ d h σ i + d h σ ih π i tan h π i F (cid:19) . (111)Thus in the SPB one finds two massless chargedpseudoscalars and three light pseudoscalars with masses linear in the current quark mass (see Fig.3). Theseequations represent the generalization of the Gell-Mann-Oakes-Renner relation in the phase with broken parity.We notice that the masses of neutral and chargedpseudoscalars do not coincide in the well developed SPBphase, just realizing the spontaneous breaking of isospinsymmetry. One can also guess that the manifest break-ing of SU (2) symmetry due to different masses of u and d quarks will supply the Goldstone bosons ˜ π ± with tinymasses proportional to the difference m u − m d . When keeping in mind QCD we assume that the quarkmatter is equivalent to nuclear matter when their av-erage baryon densities coincide, at least in what re-spects meson properties. One could also think abouttechniquarks and the two multiplets of composite Higgsmesons.Thermodynamical characteristics of such a matterare the pressure, P , and the energy density, ε . The pres-sure is determined in the presence of chemical potentialby (56), defined for σ j satisfying the mass gap equa-tion. The pressure at zero nuclear density must vanish.In this case the energy and baryon densities are relatedto the pressure as follows ε = − P + N c µ̺ B ; ∂ µ P = N c ̺ B ; ∂ ̺ ε = N c µ. (112)The direct connection between energy density and pres-sure reads P = ̺ B ∂ ̺ (cid:18) ε̺ B (cid:19) . (113)Evidently the energy per baryon has an extremum whenthe pressure vanishes. Since the pressure is an increas-ing function of the density as we have seen, obviouslyvanishing at zero density, and infinite nuclear matter isstable (thus implying zero pressure) the phase diagramin the P, ̺ B plane is necessarily discontinuous with val-ues of density in the interval (0 , ̺ ) not correspondingto equilibrium states ( ̺ is nuclear matter density). Wewill see below how this is realized in our model.9.1 On the way to stable nuclear matterOur model consisting of two scalar isomultiplets is stillsomewhat too simple in one aspect. The stabilization ofnuclear matter requires not only attractive scalar forces(scalars) but also repulsive ones (vector-mediated). Con-ventionally [11], the latter ones are associated to the in-teractions mediated by the iso-singlet vector ω meson. Let us supplement our action with the free ω mesonlagrangian and its coupling to quarks ∆ L ω = − ω µν ω µν + 12 m ω ω µ ω µ − g ω ¯ qq ¯ qγ µ ω µ q, (114)with a coupling constant g ω ¯ qq ∼ O (1 / √ N c ). After bo-sonization of QCD or QCD-like theories, on symmetrygrounds, any vector field interacts with scalars in theform of commutator and therefore ω µ does not show upin the effective potential H j fields to the lowest order.However in the quark lagrangian the time component ω interplays with the chemical potential and it is ofimportance to describe the dense nuclear matter prop-erties. Let us assign a constant v.e.v. for this component g ω ¯ qq h ω i ≡ ˜ ω . Then one needs to compute the modifi-cation of the effective potential due to the replacement µ → µ + ˜ ω ≡ ˜ µ . The variable ˜ ω , and accordingly ˜ µ , isdynamical and in addition appears quadratically in themass term in (114) which reads ∆V ω = − m ω h ω i = −
12 (˜ µ − µ ) G ω ,G ω ≡ g ω ¯ qq m ω ≃ O ( 1 N c ) . (115)The term (115) supplements the effective potential (63): e V eff ,ω ( µ ) = e V eff (˜ µ ) + ∆V ω ( µ, ˜ µ ). Correspondingly theextremum condition for the variation of the variable ˜ µ involves both the scalar part of the effective potential(63) and the vector one (115) and due to (59) takes thefollowing form N c ̺ B ( µ ) = N c N f π p F (˜ µ ) = µ − ˜ µG ω , (116)from this one finds ˜ µ ( µ ) after solving the mass-gapequations (49), (17) and (19).Finally the extended effective potential at a mini-mum reads e V eff ,ω ( µ ) = − ∆ (cid:16) σ (˜ µ ) + σ (˜ µ ) + h ρ i (˜ µ ) (cid:17) − h N µ (cid:16) ˜ µ − σ (˜ µ ) (cid:17) / + G ω N (cid:16) ˜ µ − σ (˜ µ ) (cid:17) i θ (cid:16) ˜ µ − σ (˜ µ ) (cid:17) , (117)where h σ j i = σ j (˜ µ ). Let us define the v.e.v.’s of scalarfield σ in vacuum at the two minima as σ ∗ (0) < σ ♯ (0) .Let us select out the parameter subspace such that theminimum corresponding to σ ♯ (0) , σ ♯ (0) is lower thanthe another minimum at σ ∗ (0) , σ ∗ (0) . Then for parity-even matter h ρ i = 0, one seeks for the nuclear mat-ter stability at a value of chemical potential ˜ µ s with σ ∗ (0) ≤ ˜ µ s < σ ♯ (0) . The corresponding baryon matter SPB
Jumps of derivativesSaturation point s (cid:80) c (cid:80) (0) (cid:86) (0) (cid:86) ( ) (cid:86) (cid:80) ( ) s (cid:86) (cid:80) (cid:13) (cid:80) ( ) 0 (cid:85) (cid:80) (cid:122) ( ) (cid:85) (cid:80) Fig. 4
Saturation point meets spontaneous parity breaking: at µ = µ s the pressures for the two solutions σ ♯ , σ ∗ become equaland the solutions interchange realizing the 1st order phase tran-sition. At a larger chemical potential µ c the 2nd order SPB phasetransition occurs. stability condition ∆P = P ( σ ♯ (0)) − P ( σ ∗ (˜ µ s )) = 0,eq. (56), can be formulated as ∆ (cid:16) ( σ ♯ (0)) + ( σ ♯ (0)) − ( σ ∗ (˜ µ s )) − ( σ ∗ (˜ µ s )) (cid:17) = N c N f π ˜ µ s p F (˜ µ s ) + G ω N c N f π p F (˜ µ s )= N c µ s ̺ B ( µ s ) + G ω N c ̺ B ( µ s ) , (118)taking into account (117) and (115) . Herein ˜ µ s is re-lated to the physical value of µ s by (116) and it is as-sumed that parity is not violated h ρ i = 0. This relationrepresents the condition for the formation of stable sym-metric nuclear matter in result of first-order phase tran-sition [11]. It can be fulfilled by an appropriate choiceof the vector coupling constant G ω as typically the firstterm in the r.h.s. of (118) is smaller than the one on thel.h.s. The first order phase transition at the saturationpoint is illustrated on Fig.4.At finite temperatures one has to modify the ther-modynamic relations. The modification of the effec-tive potential due to ω mesons is given by (115). Thus e V eff ,ω ( µ, β ) ≡ e V eff(˜ µ, β )+ ∆V ω (˜ µ, µ ) which should hence-forth be used in all the previous thermodynamical for-mulae. The replacement µ → ˜ µ makes all expectationvalues depend rather on ˜ µ which is determined via thevariation of e V eff˜ µ − µG ω = − N c ̺ B (cid:16) β, µ, σ (cid:17) = ∂ ˜ µ e V eff (˜ µ, β ) . (119)The saturation point at µ = µ s where nuclear matterforms is characterized by the energy crossing condition for P, T = 0, ∆ (cid:16)(cid:0) σ ♯ (cid:1) + (cid:0) σ ♯ (cid:1) − (cid:0) σ ∗ (cid:1) − (cid:0) σ ∗ (cid:1) (cid:17) = N c µ s (cid:16) ̺ B ( β, µ s , σ ∗ ) − ̺ B ( β, µ s , σ ♯ ) (cid:17) + 12 T (cid:16) S ( β, µ s , σ ∗ ) − S ( β, µ s , σ ♯ ) (cid:17) + G ω N c (cid:16) ̺ B ( β, µ s , σ ∗ ) − ̺ B ( β, µ s , σ ♯ ) (cid:17) , (120)where ˜ µ s is related to the physical value of µ s by equa-tion (119) and σ ♯j ≡ σ ♯j (˜ µ s , β ); σ ∗ j ≡ σ ∗ j (˜ µ s , β ).The latter relation represents the condition for theexistence of symmetric nuclear matter. It can be alwaysfulfilled by an appropriate choice of G ω .9.2 Saturation point meets spontaneous paritybreakingLet us search for the domain of parameters in the modelproviding the realization of both stable nuclear matterand the regime of SPB. The former is associated with afirst-order phase transition and implies the existenceof two minima at zero chemical potential which arepossibly moving when the chemical potential increases.The highest, metastable minimum must start movingat chemical potentials µ smaller than the value of thedynamical mass of the lowest minimum, σ ♯ and largerthan the v.e.v. σ ∗ at the highest minimum, σ ∗ ≤ µ < σ ♯ .This metastable minimum may reach the lowest one ifthe density and omega meson effects are taken into ac-count. Then a first-order phase transition to normal nu-clear matter occurs when pressures become equal, eq.(118).In order to simplify our search we make a particularchoice of λ = 0 (not a reduction by (30)!). In this caseone of the solutions is σ (0)2 = 0 and 2 λ ( σ (0)1 ) = ∆ and it is a minimum as it follows from (22), (23) (inthe chiral limit) provided that λ > , ( λ + λ ) > λ .When σ (0)2 = 0 a higher symmetry Z × Z arises for theeffective potential in the vicinity of such a minimum asthe contribution of the vertex with λ into the secondvariation vanishes with σ (0)2 . For σ = 0 one can ob-tain eq. (36) for the ratio t = σ /σ . As it is analyzedin subsection 4.2, it has, in general, one or three realroots. For our purposes eq. (36) must have three real so-lutions: one corresponding to a minimum t (3) > t (1) < , t (2) > t (3) one finds a unique solution for σ > Let us assume the minimum with σ (0)2 = 0 to bethe higher one at zero chemical potentials σ (0)1 ≡ σ ∗ .For this choice to be realized it is sufficient to fulfill theinequality σ (0)1 ≡ σ ∗ < σ (3)1 ≡ σ ♯ . It turns out that inorder to provide it one has to satisfy the inequality2 λ t + 32 λ t + ( λ + λ − λ ) < , (121)which implies9 λ ≥ λ ( λ + λ − λ ); 0 < t (3) < − λ λ . (122)When at the critical value of µ = µ s < σ ♯ , the solu-tion with σ ∗ = 0 describes a saturation point then thefurther evolution of the meson background for higherchemical potentials is characterized by the followingequation σ ∗ ( µ ∗ )2 λ ( σ ∗ ) ( µ ) = ∆ − N A ( σ ∗ , µ ) . (123)As the last term is monotonously increasing with chem-ical potential the v.e.v of scalar field is decreasing. Nowwe approach to the P-breaking regime and employ eq.(67) at the expected phase transition point. Its solutionis σ ,c = ∆λ − λ < ( σ ) = ∆ λ . (124)Thus the feasibility of spontaneous P-breaking dependson the realization of the inequality σ ,c < σ ∗ < σ . Thecombination of the regime of nuclear matter saturationfor normal baryon density and of the P-breaking phaseat a higher baryon density is qualitatively depicted onFig.4.Let us collect the inequalities providing the requiredconvexity of the two minima and the very existence ofboth the stable nuclear matter and a parity breakingphase for higher densities (see [9, 10]) − λ λ > t (3) > max h − λ λ ; − λ λ i ,λ , , , > , λ > λ , λ > λ λ > λ ,∆ > , ( λ ± λ ) > λ , ( λ + λ ) > λ , (125)in addition to those ones derived above. For a moredefinite numerical estimation of these six constants as-sociated to QCD there is not at present enough exper-imental or phenomenological information.
10 Confronting the two-multiplet model withmeson and nuclear matter phenomenology
We assume that the quark matter is equivalent to nu-clear matter when their average baryon densities coin-cide, at least in what respects meson properties. Thusthe two-multiplet model investigated in our paper couldbe exploited to explore baryon matter properties inthe mean-field approach. The baryon matter normal-ization we will apply at the normal baryon density.The normal density of infinite nuclear matter [11] is ̺ ≃ . ÷ .
16 fm − that corresponds to the av-erage distance 1 . ÷ . ∆ ij , three normalization pa-rameters for kinetic terms A ij and six coupling con-stants for dim-4 meson self-interaction λ k . Beyond thechiral limit one has also the two vertices linear in cur-rent quark masses parameterized by d j . At last in or-der to provide the first order phase transition to stablebaryon matter one has to include the repulsive forcesgenerated by ω meson with the relevant coupling con-stant G ω in (115). All together one has 15 constantsto be found from spectral characteristics of mesons andstable baryon matter.The reparameterization (28) of the scalar field H discussed in Sec.4 allows to reduce the mass-like param-eters to only one, ∆ ij → ∆δ ij . Thus 13 independentparameters must be fixed from hadron phenomenology.The first source for determination of coupling con-stants and mass scales of the model comes from themass spectrum of two lightest multiplets of scalar isoscalarand pseudoscalar isotriplet mesons. The pseudoscalarmeson masses are known with a reasonable precisionaccording to [34]. In particular the heavy pion Π massstarts from ∼ Namely, the normal density of infinite nuclear matter[11] is ̺ ≃ . ÷ .
16 fm − that corresponds to the av-erage distance 1 . ÷ . P = 0 and µ N = ε̺ B = energy per baryon= m N − E bound = (939 − M eV = 923
M eV. (126)The quark matter chemical potential is defined as ∂ ̺ ε = µ N = N c µ . Therefore at the saturation point µ s =308 M eV . Then from (120) it can be established that ifnormal nuclear matter is formed at the chemical poten-tial µ s ≃ M eV then it can stabilized by ω mesoncondensate with G ω ∼ (10 ÷ GeV − in a qualitativeagreement to what is known from other model estima-tions [35].Evidently for a more definite numerical estimationof the entire set of 13 constants there is not at presentenough experimental or phenomenological information,although it can be shown that the tentative values as-sumed in [9] for λ ∼ . , λ ∼ , ∆ ∼ . GeV − maylead to the occurrence of SPB at about three times nor-mal nuclear densities.Still we pay hopes to collect the required number ofinputs from hadron phenomenology to falsify the real-ization of spontaneous parity breaking in dense baryonmatter or vice versa the discovery of SPB in heavy ioncollisions [27] might give the missing data to fix themodel parameters with a reasonable precision.
11 Conclusions
In this paper we followed the preliminary investiga-tions in [9, 10] and explored the issue of parity breakingin dense baryon matter employing effective lagrangiantechniques. – Our effective lagrangian is a realization of the gen-eralized linear σ model, but including the two low-est lying resonances in each channel, those that areexpected to play a role in this issue. It can be as-sociated with QCD or QCD-like technicolor modelsand includes the vertices of soft breaking of chiralsymmetry presumably generated by current quarkmasses. In this minimal model condensation of oneof the pseudoscalar fields can arise on the back-ground of two-component scalar condensate so thatthe chiral constant background cannot be rotatedaway by transformation of two complex scalar mul-tiplets preserving space parity. The use of effectivelagrangians is crucial to understand how would par-ity breaking originating from a finite baryon densityeventually reflect in hadronic physics. – We conclude that parity breaking is a realistic pos-sibility in nuclear matter at moderate densities andnon-zero current quark masses. To prove it we in-cluded a chemical potential for the quarks that cor-responds to a finite density of baryons and investi-gate the pattern of symmetry breaking in its pres-ence. The necessary and sufficient conditions (be-yond the chiral limit) were found for a phase whereparity is spontaneously broken to exist. It also ex-tends to finite temperature although for large tem-peratures the hot pion gas corrections must be takeninto account. – As a consequence of SPB a strong mixing betweenscalar and pseudoscalar states appears that trans-late spontaneous parity breaking into meson decays.The mass eigenstates will decay both in odd andeven number of ”pions” simultaneously. – At the very point of the phase transition leading toparity breaking one has six massless pion-like statesin the chiral limit and the three massless states whenthe quark masses are taken into account. After cross-ing the phase transition, in the parity broken phase,the massless charged pseudoscalar states remain asGoldstone bosons enhancing charged pion produc-tion, whereas the additional neutral pseudoscalarstate becomes massive. – As a bonus we have gotten a rather good descrip-tion of several aspects of nuclear physics; in par-ticular a good description of the physics associatedto the condensation transition where nuclear mat-ter becomes the preferred solution. The model isrich enough to provide the relevant characteristicswhile avoiding some undesirable properties of sim-pler models, such as the chiral collapse (see Ap-pendix A). Other nuclear properties such as (in)-compressibilities are well described too (see Appen-dix B).We have presented our results trying to avoid as muchas possible specific numerical values for the differentquantities and parameters. Not only is this proceduremore general but also the logical connections are bet-ter outlined. The main conclusion of our studies is thatspontaneous parity breaking seems to be a rather genericphenomenon at finite density. It would be interesting toinvestigate how this new phenomenon could modify theequation of state of neutron stars (the density of suchobjects seems to be about right for it). It is also manda-tory to investigate in detail the appearance of a SPBphase in heavy-ion collisions.
12 Acknowledgments
This work was supported by Research grant FPA2010-20807. AA and VA are partially supported by GrantRFBR 13-02-00127 and by Saint-Petersburg State Uni-versity research grant 11.38.660.2013.
Appendix A: Stable baryon matter withoutchiral collapse
A viable model of dense baryon (quark/nucleon) mat-ter must reveal the phase transition to a stable boundstate at the normal nuclear density ̺ B = ̺ for infinitehomogeneous symmetric nuclear matter at the so called“saturation point”. This phase transition is believed tobe of first order similar to the vapor condensation intoliquid: from droplets of heavy nuclei to a homogeneousnuclear liquid. However in simple quark models of theNJL type [12] this phase transition (for vanishing cur-rent quark masses) goes to the chirally symmetric phasewith zero dynamical mass (zero v.e.v. of scalar fields),so called “chiral collapse”. When it happens the typicalbaryon density is substantially larger that the normalone ̺ B,c = 2 . ̺ . For this reason these simple modelscannot be a reasonable guide to phase transitions invery dense nuclear matter.Let us examine under which conditions the satura-tion point in our model happens to be at normal nucleardensity and is not accompanied by the chiral collapsekeeping the dynamical mass different from zero. To an-alyze this problem we have to examine the pressure incold ( T = 0), dense baryon matter.We remind that in so far as our system undergoesspontaneous CSB the effective potential (33) does notreveal any minimum at the origin in variables σ j (for µ = 0) and may have either a saddle point,det h ˆ V (2) i ( σ j = 0) < h ˆ V (2) i (0) > , tr n ˆ V (2) o (0) < . One has a positive definite matrix of second variations(22), (23) of the effective potential in the vicinity of aCSB solution,det h ˆ V (2) i ( σ ♯j ) >
0; tr n ˆ V (2) o ( σ ♯j ) > . It means that V eff( σ ♯j ) < V eff(0) . These properties allow us to guess that at somevalue of chemical potential µ s < σ ♯ and smaller values of v.e.v.’s for scalar fields σ ∗ < µ s the deficit in scalarbackground energies on the left-hand side of (118) maybe exactly compensated by contributions from the nu-clear density and omega-meson repulsion on the right-hand side so that P ( σ ∗ j , µ s ) = 0 and the system under-goes a first-order phase transition to the stable quark(nuclear) matter.Let us now prove that for a large variety of couplingconstants admitting CSB (and SPB, see below) one ofthe v.e.v. σ ∗ j = 0 in the chiral limit, and the chiralcollapse is impossible. Indeed, suppose that σ ∗ j = 0 at µ ∗ < σ ♯ where the pressure vanishes then( µ ∗ ) + 8 G ω N µ ∗ ) = − N (cid:12)(cid:12)(cid:12) V eff ( σ ♯j , µ = 0) (cid:12)(cid:12)(cid:12) . (A.1)In this case the second variation matrix for the effectivepotential (22), (65) reads12 V (2) σ = − ∆ + N ( µ ∗ ) ,V (2) σ = 0 , V (2) σ = − ∆. (A.2)In order to induce SPB one takes ∆ > µ ∗ the second variation matrix is never positive definiteand one reveals either a saddle point or a maximum ata presumed saturation point whereas a maximum re-mains for vacuum values with µ ∗ < σ ♯ . As we have toguarantee the existence of stable nuclear matter withnormal baryon density we consider further on ∆ > V (2) σ to describethe behavior around extremum at the origin. Evidently,there is always a value of µ ∗ for which it becomes pos-itive and chiral collapse is inevitable. But when com-paring with our model one concludes that the reasonfor appearance of chiral collapse is not the absence ofconfinement [12] but the inefficiency of a one-channellinear sigma model in representing the complicated chi-ral dynamics in hadronic physics. Appendix B: Proposal for (in)compressibilities:matching quark and nuclear matters
The (in)compressibility in the quark matter must bedefined as K ( µ ) = ∂ ̺ B P , where the derivative is madewith the help of the function ̺ B ( µ ) given in (66). For azero pressure state (such as stable nuclear matter) thisequals K = ̺ B ∂ ̺ B (cid:18) ε̺ B (cid:19) (cid:12)(cid:12) P =0 , (B.3) which must be positive since it corresponds to a min-imum of the energy per baryon. In our model this isindeed the case K ( µ ) = N c ̺ B ∂ ̺ B µ = p F (˜ µ ) (cid:16) ˜ µ − σ ∂ ˜ µ σ (˜ µ ) (cid:17) + 9 G ω ̺ B ( µ ) ,∂ ˜ µ σ = − N σ q ˜ µ − σ V (2) σ σ det ˆ V (2) < , (B.4)where N c = 3 is assumed and σ j = σ j (˜ µ ).In order to adjust the properties of stable baryonmatter in a quark description we parameterize the (in)-compressibility as follows: K = aN c ̺ B ∂ ̺ µ (cid:12)(cid:12)(cid:12) P =0 . Let usshow that the matching of quark and nuclear matterat the saturation point is provided by the normaliza-tion factor a = 9 in meson-nucleon models and a = 1for quark-meson models. The incompressibility must bepositive (giving a minimum of the energy per baryon)when a stable nuclear matter is formed at zero pres-sure. The derivative of the dynamical mass is given in(B.4) wherefrom it follows that the derivative of thepressure is always positive. Thus if there is a solutionwith σ ∗ ≡ σ ( µ ∗ ) < µ ∗ < σ ♯ providing P = 0, thephase transition emerges to the stable nuclear matterstate.In the terms of nuclear d.o.f. one defines ε = − P + µ N ̺ B . (B.5)At the saturation point P = 0 and µ N = ε̺ B = energy per baryon = m N − E bound . (B.6)The quark matter chemical potential is defined as ∂ ̺ ε = µ N = N c µ . Let us use this point for the quark-hadronmatching ̺ B ( µ N ) = ̺ B ( µ s ) ,p F,N = ( µ N ) − ( m N ) ≃ p F,q = ( µ s ) − ( σ ∗ ) , (B.7)if we neglect the vector meson shift ˜ µ ≃ µ . However thematching of densities does not provide the equivalenceof derivatives w.r.t. chemical potentials. Namely aroundthe saturation point ∂p F ∂µ N ≃ µ N = 2 N c µ s = ∂p F ∂µ ≃ µ s , (B.8)where (subdominant) derivatives of dynamical massesare neglected for a moment. Now let us try to extendthe matching to the (in)compressibilities, providing thecorrect N c factors. For hadron matter K N ( µ s ) = 9 ̺ B (cid:16) ∂̺ B ∂µ N (cid:17) − ≃ p F µ N = 3 p F N c µ s , (B.9) whereas for quark matter, K Q ( µ s ) = N c ̺ B (cid:16) ∂̺ B ∂µ (cid:17) − ≃ N c p F µ s , (B.10)they match each other for N c = 3. One could do thingseven better. If the coefficient for hadron matter we re-define 9 → N c and we introduce the coefficient 3 /N c for quark matter then both definitions match for any N c to the leading order. At least one then could succeedin their matching around normal density.Going back to quark matter description it wouldmean that K ( µ s ) = p F (˜ µ s ) (cid:16) ˜ µ s − σ ∂ ˜ µ σ (˜ µ s ) (cid:17) + 3 N c G ω ̺ B ( µ s ) , (B.11)has a finite limit at large N c coinciding with what weget for hadron matter if one remembers that G ω ∼ /N c . References
1. A.B. Migdal, Rev. Mod. Phys. 50 (1978) 107; D.Bailin and A. Love, Phys. Rep. 107, 325 (1984);T. Ericson, W. Weise, Pions and nuclei,(ClarendonPress, Oxford, 1988); C.-H. Lee, Phys. Rep. 275(1996) 197; M. Prakash, I. Bombaci, M. Prakash,P. J. Ellis, J. M. Lattimer and R. Knorren, Phys.Rep. 280 (1997) 1.2. A.B. Migdal, Zh. Eksp. Teor. Fiz. 61 (1971) 2210[Sov. Phys. JETP 36 (1973) 1052]; R.F. Sawyer,Phys. Rev. Lett. 29 (1972) 382; D.J. Scalapino,Phys. Rev. Lett. 29 (1972) 386; G. Baym, Phys.Rev. Lett. 30 (1973) 1340; A.B. Migdal, O.A.Markin and I.N. Mishustin, Sov. Phys. JETP, 39(1974) 212.3. I. Bars and M.B. Halpern, Phys. Rev. D7 (1973)3043.4. D. Espriu, E. de Rafael and J. Taron, Nucl. Phys.B 345, 22 (1990) [Erratum-ibid. B 355, 278 (1991)].5. A.A. Andrianov and V.A. Andrianov, Int. J. Mod.Phys. A 8, 1981 (1993); Theor. Math. Phys. 94, 3(1993) [Teor. Mat. Fiz. 94, 6 (1993)]; Nucl. Phys.Proc. Suppl. 39BC (1995) 257; Int. J. Mod. Phys. A20, 1850 (2005); A.A. Andrianov, V.A. Andrianovand V.L. Yudichev, Theor. Math. Phys. 108, 1069(1996) 1069 .6. A.A. Andrianov, D. Espriu and R. Tarrach, Nucl.Phys. B533 (1998) 429; Nucl. Phys. Proc. Suppl.86 (2000) 275.7. D. Black, A.H. Fariborz, S. Moussa, S. Nasri andJ. Schechter, Phys. Rev. D 64, 014031 (2001); A.H.Fariborz, R. Jora and J. Schechter, Phys. Rev. D
72, 034001 (2005); Phys. Rev. D 77, 034006 (2008);Fariborz,A.H., Jora,R., Schechter,J., Shahid,M.N.,Phys. Rev. D 83, 034018 (2011); Fariborz,A.H.,Jora,R., Schechter,J., NaeemShahid,M. Phys. Rev.D 84, 094024 (2011); Fariborz, A.H., Jora,R.,Schechter,J., Shahid, M.N. Phys. Rev. D 84, 113004(2011).8. F. Giacosa, Eur. Phys. J. C 65, 449 (2010); S. Gal-las, F. Giacosa and D. H. Rischke, Phys. Rev. D82, 014004 (2010).9. A.A.Andrianov, D.Espriu, Phys. Lett. B 663 (2008)450.10. A.A.Andrianov, V.A. Andrianov, D.Espriu, Phys.Lett. B678 (2009) 416; Phys. Part. Nucl. 41, 896(2010); PoS QFTHEP2010, 054 (2010).11. J.D.Walecka, Theoretical Nuclear and SubnuclearPhysics, World Scientific Publishing Co. Pte.Ltd.(2004) pp.607; B.D.Serot and J.D.Walecka, Int.J. Mod. Phys. E6 (1997) 515.12. M. Buballa, Nucl.Phys. A611 (1996) 393.13. S. Pal, M. Hanauske, I. Zakout, H. Stocker and W.Greiner, Phys. Rev. C, 60, 015802 (1999) .14. P. Braun-Munzinger, K. Redlich and J. Stachel, In”Hwa, R.C. (ed.) et al.: Quark gluon plasma” 491-599; arXiv: nucl-th/0304013.15. O. Philipsen, Eur. Phys. J. ST 152 (2007) 29 .16. M.P.Lombardo, PoS CPOD2006 (2006) 003[hep-lat/0612017]17. M.A. Stephanov, PoS LAT2006 (2006) 024[hep-lat/0701002] .18. G.E. Brown, M. Rho, Phys. Rep. 363 (2002) 85; D.Toublan and J. B. Kogut, Phys. Lett. B 564, 212(2003); M. Frank, M. Buballa and M. Oertel, Phys.Lett. B 562, 221 (2003).19. D. Bailin, J. Cleymans and M.D. Scadron, Phys.Rev. D 31, 164 (1985); O. Scavenius, ´A. M´ocsy,I.N. Mishustin and D.H. Rischke, Phys. Rev. C 64,045202 (2001) .20. Bernard, V., Meissner, Ulf-G., Zahed, I., Phys. RevD36 (1987) 819; M. Asakawa and K. Yazaki, Nucl.Phys. A 504 (1989) 668; T. Hatsuda and T. Ku-nihiro, Phys. Rep., 247, 221 (1994); A. Delfino, J.Dey, M. Dey, M. Malheiro, Phys. Lett. B363, (1995)17; M. Buballa, Phys. Rept. 407 (2005) 205; D. N.Walters and S. Hands, Nucl. Phys. Proc. Suppl.140 (2005) 532; H. Abuki, R. Anglani, R. Gatto,M. Pellicoro and M. Ruggieri, Phys. Rev. D 79,034032 (2009).21. M. Alford, K. Rajagopal and F. Wilczek, Phys.Lett. B422, 247 (1998); R. Rapp, T. Schafer, E.V. Shuryak and M. Velkovsky, Phys. Rev. Lett. 81(1998) 53; M. G. Alford, A. Schmitt, K. Rajagopaland T. Schafer, arXiv: 0709.4635 [hep-ph]. 22. K. Takahashi, Phys. Rev. C 66 (2002) 025202 .23. A. Barducci, R. Casalbuoni, G. Pettini, and L.Ravagli, Phys. Rev. D 69 (2004) 096004; D.Ebert and K.G. Klimenko, J.Phys. G32 (2006)599; Eur.Phys.J. C46 (2006) 771; Phys. Rev. D80, 125013 (2009); D.Ebert, K.G. Klimenko, A.V.Tyukov and V.C. Zhukovsky, Phys. Rev. D 78,045008 (2008) .24. D. Kharzeev, R.D. Pisarski and M.H.G. Tytgat,Phys. Rev. Lett. 81 (1998) 512; D. Kharzeev andR.D. Pisarski, Phys. Rev. D 61 (2000) 111901(R);D. Kharzeev, Phys. Lett. B 633 (2006) 260; Ann.Phys. (NY) 325 (2010) 205; ; D. Kharzeev and A.Zhitnitsky , Nucl. Phys. A 797 (2007) 67 .25. D.E. Kharzeev, L.D. McLerran and H.J.Warringa,Nucl. Phys. A 803 (2008) 227; K. Fukushima, D.E. Kharzeev, H. J. Warringa, Phys. Rev. D 78,074033 (2008); Nucl. Phys. A 836 (2010) 311; K.Fukushima and K. Mameda, Phys. Rev. D 86,071501 (2012).26. Buividovich,P.V. Chernodub,M.N.Luschevskaya,E.V. Polikarpov,M.I. Phys. Rev.D 80 (2009) 054503; Nucl. Phys. B826 (2010) 313;Buividovich,P.V. Chernodub,M.N. Kharzeev,D.E.Kalaydzhyan,T. Luschevskaya,E.V. Polikar-pov,M.I. Phys. Rev. Lett. 105 (2010) 132001; V.V.Braguta, P.V. Buividovich, T. Kalaydzhyan, S.V.Kuznetsov and M.I. Polikarpov, Phys. Atom. Nucl.75, 488 (2012); P.V. Buividovich, T. Kalaydzhyan,M.I. Polikarpov, Phys. Rev. D 86, 074511 (2012).27. A.A. Andrianov, V.A. Andrianov, D. Espriu andX. Planells, PoS QFTHEP2010, 053 (2010); AIPConf.Proc. 1343 (2011) 450; PoS QFTHEP2011,025 (2011); Theor. Math. Phys. 170, 17 (2012)[Teor. Mat. Fiz. 170, 22 (2012)]; Phys. Lett. B 710,230 (2012) .28. A.A. Andrianov, D. Espriu and X. Planells, Eur.Phys. J. C 73, 2294 (2013); ibid.
C 74, 2776 (2014).29. S.S. Afonin, A.A. Andrianov, V.A. Andrianov andD.Espriu, JHEP 0404, 039 (2004).30. M. Gell-Mann and M. Levy, Nuovo Cim. 16 (1960)705 .31. D. Weingarten, Phys. Rev. Lett. 51, 1830 (1983);C. Vafa and E. Witten, Phys. Rev. Lett. 53 (1984)535; S. Nussinov, Phys. Rev. Lett. 52, 966 (1984);D. Espriu, M. Gross and J.F. Wheater, Phys. Lett.B 146, 67 (1984); for a review see, S. Nussinov andM. Lambert, Phys. Rept. 362 (2002) 193 .32. T. Matsubara, Prog. Theor. Phys. 14 (1955), 351.33. J.I. Kapusta and C. Gale, Finite-Temperature FieldTheory Principles and Applications, CambridgeUniversity Press (2006), 428 pp.4