Generalized Beth--Uhlenbeck approach to mesons and diquarks in hot, dense quark matter
aa r X i v : . [ h e p - ph ] M a y Generalized Beth–Uhlenbeck approach to mesons and diquarksin hot, dense quark matter
D. Blaschke a,b , M. Buballa c , A. Dubinin a , G. R¨opke d , D. Zablocki a,c a Institute for Theoretical Physics, University of Wroc law, pl. M. Borna 9, 50-204 Wroc law, Poland b Laboratory for Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie ul. 6, 141980 Dubna, Russia c Institute for Nuclear Physics, Technical University of Darmstadt, Schlossgartenstr. 2, 64289 Darmstadt, Germany d Institut f¨ur Physik, Universit¨at Rostock, Universit¨atsplatz 3, 18055 Rostock, Germany
Abstract
An important first step in the program of hadronization of chiral quark models is the bosonization in mesonand diquark channels. This procedure is presented at finite temperatures and chemical potentials for theSU(2) flavor case of the NJL model with special emphasis on the mixing between scalar meson and scalardiquark modes which occurs in the 2SC color superconducting phase. The thermodynamic potential isobtained in the gaussian approximation for the meson and diquark fields and it is given the Beth-Uhlenbeckform. This allows a detailed discussion of bound state dissociation in hot, dense matter (Mott effect) interms of the in-medium scattering phase shift of two-particle correlations. It is shown for the case withoutmeson-diquark mixing that the phase shift can be separated into a continuum and a resonance part. In thelatter, the Mott transition manifests itself by a change of the phase shift at threshold by π in accordancewith Levinson’s theorem, when a bound state transforms to a resonance in the scattering continuum. Theconsequences for the contribution of pionic correlations to the pressure are discussed by evaluating the Beth-Uhlenbeck equation of state in different approximations. A similar discussion is performed for the scalardiquark channel in the normal phase. Further developments and applications of the developed approach areoutlined. Keywords: quantum chromodynamics; chiral symmetry; quark-gluon plasma; mesons; diquarks;Bethe-Salpeter equation; Beth-Uhlenbeck equation; Mott effect
1. Introduction
One of the long-standing problems of low-energy hadron physics concerns the hadronization of QCD,i.e., the transformation of QCD formulated as a gauge theory of quark and gluon fields to an effectivetheory formulated in terms of hadrons, the observable low-energy degrees of freedom. The general idea isto start from the path integral representation of the partition function (generating functional) of QCD andto “integrate out” the elementary quark and gluon degrees of freedom, leaving an effective theory in termsof collective hadronic degrees of freedom. This task may be schematically depicted as the map Z QCD = Z D ¯ q D q D A exp { A QCD [¯ q, q, A ] } −→ Z D M D ¯ B D B exp { A eff [ ¯ B, B, M ] } , (1)where A QCD [¯ q, q, A ] is the Euclidean action functional of QCD in terms of quark (¯ q, q ) and gluon ( A ) fields,while A eff [ ¯ B, B, M ] is an effective low-energy action functional in terms of baryon ( ¯
B, B ) and meson ( M )fields. Since the map (1) involves, among others, the still open problem of quark confinement, we do not aim Email addresses: [email protected] (D. Blaschke), [email protected] (M. Buballa), [email protected] (A. Dubinin), [email protected] (G. R¨opke), [email protected] (D. Zablocki)
Preprint submitted to Elsevier September 14, 2018 t a solution on the level of mathematical rigor. A slightly modified, solvable version of this program (1) doesstart from the global color model of QCD where all nonabelian aspects of the gluon sector are absorbed intononperturbative gluon n-point functions (Schwinger functions) starting with the gluon propagator (2-pointfunction) coupling to quark currents via nonperturbative vertex functions (3-point functions), see [1–4] forbasic reviews.After Fierz rearrangement of the current-current interaction [5, 6], the theory obtains the Yukawa-likeform with Dirac quark fields coupled to the spectrum of meson and diquark channels, thus generating4-fermion interactions in the relevant quark-antiquark and quark-quark bilinears [7, 8], see also [9].The hadronization proceeds further via the two-step Hubbard-Stratonovich procedure by: 1) introducingauxiliary collective fields and functional integration over them such that the 4-fermion interaction terms arereplaced by Yukawa couplings to the collective fields, and 2) integrating out the quark fields so that aneffective theory results which is formulated in meson and diquark degrees of freedom, highly nonlinear dueto the Fermion determinant, recognizable as “Tr ln” terms in the effective bosonized action.In this way one obtains a bosonized chiral quark model, but not yet the desired hadronization in termsof mesons and baryons, the physical degrees of freedom of low-energy QCD and hadronic matter. A possiblescheme for introducing baryons as quark-diquark bound states and integrating out the colored and thereforenot asymptotically observable diquark fields has been suggested by Cahill and collaborators [7, 10–12]. Itwas afterwards elaborated by Reinhardt [8] and developed further by including the solitonic aspects of afield theoretic description of the nucleon [13], see also Refs. [9, 14–16]. Introductory reviews on the lattercan be found, e.g., in [17–19].It is the aim of our study to extend the hadronization of effective chiral quark models for QCD to nonzerotemperatures and densities. The goal is to investigate the modification of thermodynamic properties ofhadronic matter evoked under extreme conditions by the onset of (partial) chiral symmetry restoration. Inthis first part of our work we investigate in detail the two-particle correlations (mesons and diquarks) in a hotand dense quark matter medium and the response of their spectral properties to medium dependent meanfields signaling chiral symmetry breaking (quark mass gap) and color superconductivity (diquark pairinggap). Of central interest is the relation of the meson and diquark spectrum to the density and temperaturedependence of these order parameters (quark mass and pairing gap), the question under which conditionsmesons and diquarks may exist as real bound states or appear only as a correlation in the continuum ofquark-antiquark and quark-quark scattering states, resp. The transition of a correlation from the discretebound state spectrum to a resonant continuum state is called Mott effect and will be a central theme for ourstudy. When it occurs at low temperatures and high densities under fulfillment of the conditions for Bosecondensation, the Mott effect serves as a mechanism for the BEC-BCS crossover [20] which recently becamevery topical when in atomic traps this transition could be studied in detail under laboratory conditions.While numerous works have recently studied the thermodynamics of quark matter on the mean-fieldlevel including the effects of the medium dependence of the order parameters, not so much is known beyondthe mean field, about hadronic correlations and their backreaction to the structure of the model QCD phasediagram and its thermodynamics. Here we will elaborate on the generalized Beth-Uhlenbeck form of theequation of state (EoS) which is systematically extended from studying mesonic correlations [21] to theinclusion of diquark degrees of freedom. To that end we will employ a Nambu–Jona-Lasinio-type quarkmodel with fourpoint interactions in mesonic (quark-antiquark) and diquark (quark-quark) channels. Weshall discuss here the importance of the interplay of the resonant states with the residual non-resonant inthe continuum of scattering states. Due to the Levinson theorem both contributions have the tendency tocompensate each other in quark matter above the Mott transition [22–24], see also [25].The most intriguing questions will occur when on the basis of this in-medium bosonized effective chiralquark model the next step of the hadronization program nucleon will be performed and diquarks will be“integrated out” in favor of baryons so that nuclear matter can be described in the model QCD phasediagram. Exploratory studies within the framework of an effective local NJL-type model for the quark-diquark interaction vertex have revealed a first glimpse at the modification of the nucleon spectral function inthe different regions of the model QCD phase diagram, including chirally restored and color superconductingphases [26].We will prepare the ground for a Beth-Uhlenbeck description of nuclear matter, to be discussed in future2ork. In particular, at zero temperature the structure of a Walecka model for nuclear matter shall emergeunder specified conditions. Earlier work in this direction [27–29] has demonstrated this possibility althoughno unified description of the nuclear-to-quark matter transition has been possible and the elucidation ofphysical mechanisms for the very transition between the hadronic and the quark matter phases of lowenergy QCD has been spared out. Our study aims at indicating directions for filling this gap by providinga detailed discussion of the Mott mechanism for the dissociation of hadronic bound states of quarks withinthe NJL model description of low-energy QCD on the example ot two-particle correlations (mesons anddiquarks).Our approach can be considered as complementary to lattice gauge theory, where Z QCD is calculatedin ab-initio Monte-Carlo simulations without further approximations other than the discretization of space-time [30, 31]. While being more rigorous, lattice calculations are often lacking a clear physical interpretationof the results. Moreover, because of the fermion sign problem, they are restricted to vanishing or smallchemical potential. In this situation, effective chiral quark model approaches, like the one employed here,can give invaluable methodological guidance to elucidate how effects like bound state dissociation in hotand dense quark matter, as seen in spectral functions, may manifest themselves in two-particle correlatorswhich are objects accessible to lattice QCD simulations.This work is organized as follows. In Sect. 2 we define our model and derive the thermodynamic potentialin mean-field approximation and its corrections to gaussian order in normal and color superconductingmatter. The corresponding meson and diquark spectra are discussed in Sect. 3. Sect. 4 is the main partof this article. Here we derive a generalized Beth-Uhlenbeck form for the thermodynamic potential of two-particle correlations in quark matter as the appropriate representation for discussing the Mott effect, thedissociation of hadronic bound states of quarks induced by the lowering of the quark continuum by thechiral symmetry restoration at finite temperatures and chemical potentials. We evaluate the temperaturedependence of the phase shifts for pion, sigma meson and scalar diquark interaction channels and obtainthe corresponding correlation contributions to the EoS. In closing Sect. 4 we give an outlook to furtherdevelopments and applications of the formalism. In Sect. 5 we summarize our main results and present theconclusions of this work.
2. The model: mean field approximation and beyond
We consider a system of quarks with N f = 2 flavor and N c = 3 color degrees of freedom at temperature T and chemical potential µ , described by the Nambu–Jona-Lasinio (NJL) type Lagrangian L = L + L S + L V + L D . (2)The free part is given by L = ¯ q (i /∂ − m + µγ ) q , (3)with the bare quark mass m . Here we have assumed isospin symmetry, i.e., equal masses and chemicalpotentials for up and down quarks. The quarks interact via local four-point vertices in the scalar-isoscalar,pseudoscalar-isovector and vector-isoscalar quark-antiquark channels, L S = G S (cid:2) (¯ qq ) + (¯ q i γ ~τ q ) (cid:3) , (4) L V = − G V (¯ qγ µ q ) , (5)augmented with a quark-quark interaction in the scalar color-antitriplet channel, L D = G D X A =2 , , (¯ q i γ τ λ A q c )(¯ q c i γ τ λ A q ) . (6)Here q c = C ¯ q T with C = i γ γ denote the charge conjugate quark fields, λ A , A = 2 , ,
7, the antisymmetricGell-Mann matrices in color space and τ i , i = 1 , ,
3, the Pauli matrices in flavor space. G S , G V and G D aredimensionful coupling constants. 3ventually, one should also couple the quarks to an effective Polyakov loop variable [32–34], in order todescribe confinement effects in a more realistic way. Although in principle straightforward, this would leadto the appearance of more complicated dispersion relations in the expressions below [35]. For simplicity, wetherefore leave this extention for a later publication.Our model then has five parameters: the bare quark mass m , the coupling constants G S , G V and G D anda cutoff parameter Λ, which is needed because the interaction is not renormalizable. While m , G S and Λare typically fitted to vacuum properties (mass and decay constant) of the pion and to the chiral condensate[36–38], the two other coupling constants are less constrained. They are sometimes related to the scalarcoupling via Fierz transformation of a color-current interaction, which yields G V = G S / G D = 3 / G S .Alternatively, G V can be fixed by fitting the mass of vector mesons, which gives higher values [39]. Anotherpossibility is to constrain G V and G D from compact star and heavy-ion phenomenology [41]. In the presentpaper, we will specify the parameters only in the context of the numerical examples discussed in Sect. 4,while most of the analytic expressions are more general. In future extensions of the model one could thentry to fix G V and G D by fitting baryon and nuclear matter properties.The bulk thermodynamic properties of the model at temperature T and chemical potential µ are encodedin the thermodynamic potential per volumeΩ( T, µ ) = − TV ln Z ( T, µ ) , (7)with the grand partion function Z which is given as a functional integral involving the above Lagrangian, Z = Z D q D ¯ q exp (cid:20)Z d x E L (cid:21) , (8)where R d x E = R β d τ R d x denotes an integration over the Euclidean four-volume, i.e., over the three-space with volume V and imaginary time τ restricted to the interval between 0 and β = 1 /T . We performa bosonization by means of Hubbard-Stratonovich transformations. To this end, we introduce the auxiliarymeson fields σ , ~π and ω µ in the scalar, pseudoscalar and vector channel, respectively, as well as the complexauxiliary scalar diquark fields ∆ A and their complex conjugate fields ∆ ∗ A . Then, after introducing Nambu-Gorkov bispinors Ψ ≡ √ (cid:18) qq c (cid:19) ¯Ψ ≡ √ (cid:0) ¯ q ¯ q c (cid:1) (9)the grand partition function takes the form Z = Z D σ D ~π D ω µ D ∆ A D ∆ ∗ A exp (cid:20)Z d x E (cid:18) − σ + ~π G S − ∆ ∗ A ∆ A G D (cid:19)(cid:21) Z D Ψ D ¯Ψ exp (cid:20)Z d x E ¯Ψ S − Ψ (cid:21) , (10)with the inverse Nambu-Gorkov propagator defined as S − ≡ (cid:18) i /∂ + µγ + γ µ ω µ − m − σ − i γ ~τ · ~π i∆ A γ τ λ A i∆ ∗ A γ τ λ A i /∂ − µγ − γ µ ω µ − m − σ − i γ ~τ T · ~π (cid:19) . (11)Since the functional integral over the bispinor fields is of gaussian type the quark degrees of freedom can beintegrated out and we are able to express the partition function in terms of collective fields only, viz., Z = Z D σ D ~π D ω µ D ∆ A D ∆ ∗ A e − R d x E (cid:26) σ ~π G S − ω µ G V + | ∆ A | G D (cid:27) + ln det ( βS − ) . (12)The determinant is to be taken over Dirac-, flavor-, color- and Nambu-Gorkov indices as well as overEuclidean 4-volume. The latter is accomplished after Fourier transforming the inverse quark propagator tothe momentum space and Matsubara representation given by S − = (cid:18) (i z n + µ ∗ ) γ − γ · ( p + ω ) − m − i γ ~τ · ~π ∆ A i γ τ λ A ∆ ∗ A i γ τ λ A (i z n − µ ∗ ) γ − γ · ( p − ω ) − m − i γ ~τ T · ~π (cid:19) (13)4ith z n = (2 n + 1) πT being fermionic Matsubara frequencies. Bold-face symbols denote space-like com-ponents of four-vectors and gamma matrices, and the transposed isospin matrices are given by ~τ T =( τ , − τ , τ ). Finally, we have introduced the combinations m = m + σ , µ ∗ = µ + ω , (14)which can be interpreted as an effective (constituent) quark mass and an effective chemical potential. To proceed further, we employ the homogenous mean-field approximation, i.e., we replace all occurringfields by homogeneous and isotropic mean fields. Then the functional integration in Eq. (12) becomestrivial, and the partition function essentially factorizes into a gaussian part and a contribution from theinverse quark propagator. Accordingly, the thermodynamic potential per volume, can be separated into acondensate part and a contribution from the quarks,Ω MF = Ω cond + Ω Q , (15)with Ω cond = σ G S + | ∆ MF | G D − ω G V (16)and Ω Q = − TV Tr ln (cid:0) βS − (cid:1) . (17)Here the functional trace symbol Tr stands for summation over Matsubara frequencies and three-momenta aswell as for the trace over internal degrees of freedom, i.e., color, flavor, Dirac and Nambu-Gorkov space. Theextra factor of arises from the artificial doubling of the degrees of freedom in Nambu-Gorkov formalism.The isotropy implies ω MF = 0, hence ω MF denotes the 0-th component only. Also, anticipating thatpseudoscalar mean fields are disfavored at nonzero bare quark masses and vanishing isospin chemical poten-tial, ~π MF vanishes as well, and therefore was dropped in Eq. (16). Moreover, we have taken the freedom toperform a global color rotation, so that in the diquark sector the only non-vanishing mean fields correspondto the A = 2 direction, meaning that only the first two quark colors (red and green) participate in thecondensate, while the third color (blue) remains unpaired.The inverse quark propagator then takes the form S − = (cid:18) (i z n + µ ∗ ) γ − γ · p − m ∆ MF i γ τ λ ∆ ∗ MF i γ τ λ (i z n − µ ∗ ) γ − γ · p − m (cid:19) . (18)The inverse propagator can be inverted to obtain the propagator S MF ≡ (cid:16) G + F − F + G − (cid:17) . (19)The resulting normal and anomalous Nambu-Gorkov components are G ± p = X s p X t p t p E ± s p p t p E ± s p p − s p ξ ± s p p i z n − t p E ± s p p P rg + 1i z n + s p ξ ± s p p P b ! Λ − s p p γ , (20) F ± p = i X s p ,t p t p E ± s p p ∆ ± MF i z n − t p E ± s p p τ λ Λ s p p γ , (21)where s p , t p = ± +MF , ∆ − MF ) ≡ (∆ ∗ MF , ∆ MF ). P rg = diag c (1 , , P b = diag c (0 , ,
1) are colorprojection operators, and Λ ± p = [1 ± γ ( γ · p + m ) /E p ] / ξ ± p = E p ± µ ∗ (22)5or the blue quarks and E ± p = q ( ξ ± p ) + | ∆ MF | (23)for the red and green quarks, where E p = p | p | + m .The mean-field values are obtained as stationary points of the thermodynamic potential, i.e., ∂ Ω MF ∂σ MF = σ MF G S − TV Tr (cid:18) S MF ∂S − ∂σ MF (cid:19) , (24)0 = ∂ Ω MF ∂ω MF = − ω MF G V − TV Tr (cid:18) S MF ∂S − ∂ω MF (cid:19) , (25)0 = ∂ Ω MF ∂ ∆ ∗ MF = ∆ MF G D − TV Tr (cid:18) S MF ∂S − ∂ ∆ ∗ MF (cid:19) , (26)0 = ∂ Ω MF ∂ ∆ MF = ∆ ∗ MF G D − TV Tr (cid:18) S MF ∂S − ∂ ∆ MF (cid:19) . (27)The derivatives of the inverse propagator basically reduce to the vertex functions, ∂S − ∂σ MF = − (cid:18) (cid:19) , ∂S − ∂ω MF = (cid:18) γ − γ (cid:19) , (28) ∂S − ∂ ∆ ∗ MF = (cid:18) γ λ τ (cid:19) , ∂S − ∂ ∆ MF = (cid:18) γ λ τ (cid:19) , (29)and after performing the trace in Nambu-Gorkov space the gap equations become σ MF = − G S TV tr (cid:0) G + p + G − p (cid:1) , (30) ω MF = − G V TV tr (cid:2)(cid:0) G + p − G − p (cid:1) γ (cid:3) , (31)∆ MF = 2 G D TV tr (cid:2) F − p i γ τ λ (cid:3) , (32)∆ ∗ MF = 2 G D TV tr (cid:2) F + p i γ τ λ (cid:3) , (33)where tr denotes the remaining functional trace. Carrying out the trace in color, flavor and Dirac space andperforming the sum over Matsubara frequencies using T X n iz n − x = n ( x ) , (34)with n ( x ) = [exp( x/T ) + 1] − being the Fermi distribution function, we finally obtain m − m = 4 N f G S m Z d p (2 π ) E p (cid:26)(cid:2) − n ( E − p ) (cid:3) ξ − p E − p + (cid:2) − n ( E + p ) (cid:3) ξ + p E + p + n ( − ξ + p ) − n ( ξ − p ) (cid:27) , (35) µ − µ ∗ = 4 N f G V Z d p (2 π ) (cid:20) [1 − n ( E + p )] ξ + p E + p − [1 − n ( E − p )] ξ − p E − p + n ( ξ − p ) − n ( ξ + p ) (cid:21) , (36)∆ MF = 4 N f G D ∆ MF Z d p (2 π ) (cid:20) − n ( E − p ) E − p + 1 − n ( E + p ) E + p (cid:21) . (37)For ∆ ∗ MF one gets just the complex conjugate of Eq. (37). Moreover, only the modulus of ∆ MF is fixed bythe gap equations, while the choice of the phase is arbitrary. In practice, ∆ MF is therefore usually chosen6o be real. However, the complex nature of the diquark field must be kept in mind when fluctuations aretaken into account.The corresponding thermodynamic potential is then readily evaluated toΩ MF = σ G S − ω G V + | ∆ MF | G D − N f Z d p (2 π ) h E + p + 2 T ln(1 + e − E + p /T ) + E − p + 2 T ln(1 + e − E − p /T )+ E p + T ln(1 + e − ξ + p /T ) + T ln(1 + e − ξ − p /T ) i . (38)In general the gap equations have more than one solution. The stable solution is then the set of selfconsistentmean fields which minimizes the thermodynamic potential. For the standard choice of attractive scalarquark-antiquark and quark-quark interactions and repulsive vector interactions, this solution correspondsto a minimum w.r.t. σ MF and ∆ MF , but to a maximum w.r.t. ω MF . The latter is just a constraint forthermodynamic consistency.The above expressions get strongly simplified in the non-superconducting (“normal”) phase, where thediquark condensates vanish, ∆ MF = 0. In this case, the remaining gap equations reduce to m − m = 4 N f N c G S Z d p (2 π ) mE p (cid:2) − n ( ξ + p ) − n ( ξ − p ) (cid:3) , (39) µ − µ ∗ = 4 N f N c G V Z d p (2 π ) (cid:2) n ( ξ − p ) − n ( ξ + p ) (cid:3) , (40)while the thermodynamic potential becomesΩ MF = σ G S − ω G V − N f N c Z d p (2 π ) h E p + T ln (cid:16) − ( E p − µ ∗ ) /T (cid:17) + T ln (cid:16) − ( E p + µ ∗ ) /T (cid:17)i . (41)The prefactors N f , N c are obtained naturally from the color and flavor traces and are a consequence ofpersisting isospin and color symmetry. In mean-field approximation the thermodynamic potential corresponds to a Fermi gas of quasi-particleswith dispersion relations which could be strongly modified compared to the non-interacting case by thevarious mean fields. On the other hand, the effects of low-lying bosonic excitations, in particular of theGoldstone bosons of the spontaneously broken symmetries, are completely missing. This could be a ratherbad approximation at low temperatures, where the excitation of the fermionic quasi-particles is stronglysuppressed by large constituent quark masses or pairing gaps, so that the Goldstone bosons are the dominantdegrees of freedom.In this section we therefore allow for fluctuations of the meson and diquark fields around their mean-field values and derive their contributions to the thermodynamic potential. In this article, we will fix theorder parameters on the mean-field level only, i.e., by the gap equations derived in the previous subsection.Eventually, one should derive the generalized gap equations, where the fluctuation corrections to the mean-field thermodynamic potential are taken into account in the minimization procedure.We slightly change our notation in the σ and ω channels and introduce shifted fields, σ → σ MF + σ , (42) ω → ω MF + ω , (43)so that, from now on, σ and ω denote only the fluctuating parts of the fields. Obviously, this is also truefor the meson fields with vanishing mean fields, i.e., pions and the space-like components of the ω . For thediquarks we write ∆ = ∆ MF + δ , ∆ ≡ δ , ∆ ≡ δ , (44)7nd analogously for the complex conjugate fields.The gaussian terms in the partition function, Eq. (12), are then to be replaced by σ + ~π → σ + 2 σ MF σ + σ + ~π , (45) ω µ → ω + 2 ω MF ω + ω µ , (46) | ∆ A | = | ∆ MF | + ∆ MF δ ∗ + ∆ ∗ MF δ + | δ A | , (47)where a sum over the indices µ and A was implied, as before.In addition, the fluctuations contribute to the partition function via the inverse propagator, which canbe written as S − = S − + Σ , (48)with Σ = (cid:18) γ µ ω µ − σ − i γ ~τ · ~π δ A i γ τ λ A δ ∗ A i γ τ λ A − γ µ ω µ − σ − i γ ~τ T · ~π (cid:19) . (49)The logarithmic term in Eq. (12) then takes the formln det (cid:0) βS − (cid:1) = Tr ln (cid:0) βS − + β Σ (cid:1) = Tr ln (cid:0) βS − (cid:1) + Tr ln(1 + S MF Σ) (50)and can be Taylor-expanded in the fluctuating fields. Here we expand it up to quadratic (gaussian) orderas to account for two-particle correlations. Higher-order correlations will be ignored and are left to a laterinvestigation. Those correlations would include baryons. We obtainln det (cid:0) βS − (cid:1) = Tr ln (cid:0) βS − (cid:1) + Tr (cid:18) S MF Σ − S MF Σ S MF Σ (cid:19) + O (cid:0) Σ (cid:1) , (51)and the partition function in gaussian approximation neglecting contributions O (cid:0) Σ (cid:1) can be written as Z Gauß = Z MF Z D σ D ~π D ω µ D δ A D δ ∗ A e A (1) + A (2) (52)with the mean-field partition function Z MF = exp ( − βV Ω MF ), and two correction terms to the Euclideanaction which are linear and quadratic in the fluctuations, viz., A (1) = − βV (cid:18) σ MF σ G S + ∆ MF δ ∗ + ∆ ∗ MF δ G D − ω MF ω G V (cid:19) + 12 Tr [ S MF Σ] , (53)and A (2) = − βV σ + ~π G S + | δ A | G D − ω µ G V ! −
14 Tr [ S MF Σ S MF Σ] . (54)The linear correction term vanishes as a result of the mean-field gap equations. To show this, we evaluatethe Nambu-Gorkov trace of the last term in Eq. (53), − Tr [ S MF Σ] = 12 tr (cid:2) G + p ( σ + i γ ~τ · ~π − γ µ ω µ ) + G − p ( σ + i γ ~τ T · ~π + γ µ ω µ ) − F + p δ A i γ τ λ A − F − p δ ∗ A i γ τ λ A (cid:3) . (55)The pion fields and the space-like ω -components drop out in the subsequent flavor trace and momentumintegral, respectively, as the normal propagator components, given in Eq. (20), respect the isospin symmetryand the isotropy of the medium. Similarly, the A = 5 , (cid:18) σ MF G S + TV tr (cid:0) G + p + G − p (cid:1)(cid:19) σ = 0 (56)12 (cid:18) − ω MF G V − TV tr (cid:2)(cid:0) G + p − G − p (cid:1) γ (cid:3)(cid:19) ω = 0 (57)12 (cid:18) ∆ MF G D − TV tr (cid:2) F − p i γ τ λ (cid:3)(cid:19) δ ∗ = 0 (58)12 (cid:18) ∆ ∗ MF G D − TV tr (cid:2) F + p i γ τ λ (cid:3)(cid:19) δ = 0 . (59)Hence, the partition function reads Z Gauß = Z MF Z D σ D ~π D ω µ D δ A D δ ∗ A e A (2) . (60)By construction, the exponent is bilinear in the fields, so that the path integrals can be carried out. To thatend, we combine all fields in a vector X = ~πω µ σδ A δ ∗ A , X † = ( ~π T , ω µ , σ, δ ∗ A , δ A ) , (61)where all vector-meson components µ and all diquark fields, A = 2 , ,
7, are implied, as it was in the partitionfunction. The trace in the exponent can then be written as12 Tr[ S MF Σ S MF Σ] = − X † Π X , (62)with the polarization matrix Π. In addition, we have the gaussian terms in front of the trace, which arediagonal in this basis. Combining both parts, we write σ + ~π G S + | δ A | G D − ω µ G V + 12 TV Tr[ S MF Σ S MF Σ] = X † ˜ S − X , (63)with a, in general non-diagonal, propagator matrix ˜ S . The partition function is then readily evaluated as Z Gauß = Z MF Z D X e − R d x E { X † ˜ S − X } = Z MF h det (cid:16) β ˜ S − (cid:17)i − / . (64)Note that for evaluating the gaussian functional integrals over the set of bosonic fields forming the com-ponents of the vector X we performed a Fourier transformation of these fields to their representation inthe space of three-momenta and Matsubara frequencies where they are normalized to be dimensionless, seechapter 2.3 of [40]. After diagonalizing ˜ S − , the partition function factorizes, Z Gauß = Z MF Y X Z X , (65)with Z X = (cid:2) det (cid:0) β S − X (cid:1)(cid:3) − d X / being the partition function related to the propagator S X of the d X − folddegenerate eigenmode X operating in the d X − dimensional subspace of the general propagator ˜ S of two-quark correlations. Accordingly, the thermodynamic potential becomes a sum of the mean-field part andthe fluctuation parts related to these modes,Ω Gauß = − TV ln Z Gauß = Ω MF + Ω (2) = Ω cond + Ω Q + X X Ω X , (66)9ith Ω X ( T, µ ) = d X TV Tr ln (cid:2) β S − (i z n , q ) (cid:3) , (67)where Tr stands for summation over 3-momenta and Matsubara frequencies of the bosonic two-particlecorrelation in the channel X.The elements of the polarization matrix Π are explicitly listed in App. A for the simplified case where thevector fields ω µ have been neglected. In the 2SC phase, i.e., for ∆ MF = 0, the σ -meson mode mixes with thediquark modes δ and δ ∗ , formally evident from the occurrence of non-diagonal elements in the polarizationmatrix. Physically, this reflects the non-conservation of baryon number in the superfluid medium.In the non-superconducting phase where ∆ MF = 0, on the other hand, baryon number is conserved andmeson, diquark and anti-diquark modes decouple,Ω Gauß = Ω cond + Ω Q + Ω M + Ω D + Ω ¯D . (68)The three diquark modes D = δ A , related to A = 2 , ,
7, are degenerate because of the persisting colorsymmetry, whereas the chemical potential causes a splitting between diquarks and anti-diquarks, Ω ¯D ( T, µ ) =Ω D ( T, − µ ). In the meson sector, the pions always decouple from the scalar and vector modes because ofparity and angular momentum conservation. On the other hand, the ω -mode can mix with the σ at nonzero µ . Neglecting again the vector fields, all elementary meson and diquark modes of the original Lagrangiandecouple in the normal phase,Ω Gauß = Ω cond + Ω Q + Ω σ + Ω π + Ω D + Ω ¯D , (69)and the correlation contributions from the composite fields are determined byΩ X ( T, µ ) = d X T X n Z d q (2 π ) ln (cid:2) β S − (i z n , q ) (cid:3) S − (i z n , q ) = 1 G X − Π X (i z n , q ) . (70)Here G X is the coupling, related to the channel X, i.e., G X = 2 G S for X = σ, π and G X = 2 G D for X = δ A , δ ∗ A , and Π X denotes the diagonal element of the polarization matrix in this channel. The correspondingdegeneracy factors d X are d σ = 1 for the sigma meson and d π = d D = d ¯ D = 3 for pions, diquarks andanti-diquarks, respectively.
3. Meson and diquark spectra and their mixing in the 2SC phase
In this section, we want to discuss the meson and diquark spectra, which are given by the poles of thepropagator matrix, i.e., by the zeroes of det ˜ S − . In the normal phase, this is simplified by the fact that thevarious meson and diquark modes separate, as mentioned above. In the 2SC phase, on the other hand, thepolarization matrix has non-diagonal elements which cause a mixing of the σ mode with the A = 2 diquarksand anti-diquarks. Therefore, we will mainly concentrate on this mixing.The relevant piece of the inverse propagator matrix has the form˜ S − ≡ S − σσ S − σδ S − σδ ∗ S − δ ∗ σ S − δ ∗ δ S − δ ∗ δ ∗ S − δ σ S − δ δ S − δ δ ∗ = G S − Π σσ − Π σδ − Π σδ ∗ − Π δ ∗ σ G D − Π δ ∗ δ − Π δ ∗ δ ∗ − Π δ σ − Π δ δ G D − Π δ δ ∗ , (71)with the polarization-matrix elements given in App. A. These matrix elements and therefore also the ele-ments of the inverse propagator matrix depend on an external three-momentum q and an external bosonic10atsubara frequency z n . In the following both functions will be analytically continued to the complex plane,replacing iz n by the complex variable z .In order to determine the eigenmodes, ˜ S − needs to be diagonalized. Thereby we will restrict ourselvesto mesons and diquarks which are at rest in the medium, q = 0. Then, as shown in App. A.2, the polarizationmatrix elements simplify dramatically. Combining them with the diagonal coupling terms, we have S − σσ ( z ) = 12 G S + 8 I σ ( z ) + 16 m | ∆ MF | I ( z ) (72) S − δ δ ( z ) = − I ( z ) (73) S − δ ∗ δ ( z ) = 14 G D − I ∆ − zI ( z ) − (cid:0) | ∆ MF | − z (cid:1) I ( z ) (74) S − σδ ( z ) = 4 m ∆ MF ( zI ( z ) + 2 I ( z )) , (75)while the remaining elements of ˜ S − follow from the symmetry relations Eqs. (A.9)-(A.13.) Here we havedropped the three-momentum argument, q = 0, for brevity. The constant I ∆ and the functions I i ( z ) are thefinite-temperature extensions of the integrals introduced in Ref. [42] and are explicitly given in App. A.2.The above expressions are general and valid in all phases. In particular, we see that the mixing termsvanish in the normal phase, where ∆ MF = 0. In the 2SC phase, where the mixing persists, it is more tedious,but straightforward to calculate the determinant of ˜ S − . The corresponding eigenmodes, i.e., the zeroes ofthe determinant must in general be determined numerically.An exception are the Goldstone modes in the 2SC phase, which can be found analytically. These modesare related to the spontaneous breaking of the color SU (3) symmetry down to SU (2). In this context weremind that the color symmetry is a global symmetry in the present model. In QCD, where it is a localgauge symmetry, the would-be Goldstone modes, which are related to the five generators of the brokensymmetry are “eaten” by five gluons, giving them a non-zero Meissner mass. Hence, naively, one wouldexpect that in the present model there are five Goldstone bosons in the 2SC phase. However, as shownin Ref. [43] there are in fact only three Goldstone bosons, with two of them having a quadratic dispersionrelation, in agreement with the Nielsen-Chadha theorem [44]. This abnormal number of Goldstone bosonscan be related to the nonzero color charge [43, 45, 46], which arises in the 2SC phase as a consequence ofthe fact that only red and green quarks are paired. (In QCD, the 2SC phase is always color neutralized bythe background gluon field [47].)In order to identify the Goldstone modes, we evaluate the matrix elements at z = 0. In this limit (andadditionally choosing ∆ MF to be a real quantity) we can make extensive use of the symmetry relationsquoted in Eqs. (A.9)- (A.13), to show that the determinant can be written asdet ˜ S − = (cid:16) S − δ δ ∗ − S − δ δ (cid:17) h S − σσ (cid:16) S − δ δ ∗ + S − δ δ (cid:17) − S − σδ i . (76)Evaluating Eqs. (73) and (74) at z = 0 and using the 2SC gap equation,1 = 8 G D I ∆ , (77)one then finds that the first term, S − δ δ ∗ − S − δ δ vanishes, thus proving the existence of a Goldstone mode.The mixing-problem gets strongly simplified if the quarks in the 2SC phase are strictly massless, whichis the case in the chiral limit for a sufficiently weak diquark coupling G D . In this case the mixing betweenthe σ and the diquark and anti-diquark vanishes, see Eq. (75), i.e., the mixing is restricted to the A = 2diquark and anti-diquark sector. The determinant of the corresponding mixing matrix is then given bydet ˜ S − , D¯D ( z ) = 4 z [( z − ) I ( z ) − I ( z ) ] , (78)which makes the Goldstone mode at z = 0 explicit. In addition, it has a second root, which can be foundby solving the selfconsistent equation z = 2∆ MF s (cid:18) I ( z )∆ MF I ( z ) (cid:19) . (79)11his solution is manifestly above the threshold for pair breaking 2∆ MF and thus unstable.The remaining Goldstone modes are found in the other color directions of the diquark sector, which donot mix. In these channels the inverse propagators read S − δ ∗ δ ( z ) = S − δ ∗ δ ( z ) = 14 G D − I ∆ − zI ( z ) + ( | ∆ MF | − z ) I ( z ) (80)and S − δ δ ∗ ( z ) = S − δ δ ∗ ( z ) = S − δ ∗ δ ( − z ). Observing that I ( z = 0) = − I ∆ / | ∆ MF | and using the gap equationagain, one finds that these inverse propagators also vanish at z = 0.A closer inspection shows that det ˜ S − yields a factor z (as evident in the chiral limit from Eq. (78)),whereas S − δ A δ ∗ A ( z ) and S − δ ∗ A δ A in the A = 5 and 7 sectors only contribute a factor z each, so that diquarksand anti-diquarks must be combined to be counted as a full Goldstone mode. This leads to a total numberof three Goldstone bosons, as already mentioned above.For completeness, the pion propagator at rest is given by S − ππ ( z ) = 12 G S + 8 I π ( z ) , (81)with the function I π ( z ) as defined in Eq. (A.33). As well known, in the chiral limit, the pions are theGoldstone bosons in the chirally broken normal phase ( m = 0), while for m = 0, they become degeneratewith the σ meson.Finally, we would like to point out that, in general, the pole energies of the eigenmodes at q = 0 shouldnot be called “masses”, although this is quite common in the literature [48, 49]. To see this, we recall thatthe propagator of a free boson with mass m X at boson-chemical potential µ X has the form S freeX = 1( z + µ X ) − q − m , (82)which, for q = 0, has poles at z = ± m X − µ X ≡ ω ± X . Thus, we should identify the mass and the chemicalpotential of the bosonic mode X from its pole energies as [50] m X = 12 ( ω +X − ω − X ) , µ X = −
12 ( ω +X + ω − X ) . (83)Of course, in the normal phase, the assignment of the chemical potentials corresponds to the net quark-number content of the boson, i.e., µ X = 0 for the mesons, µ X = 2 µ ∗ for the diquarks, and µ X = − µ ∗ forthe anti-diquarks. As a consequence, the poles of the diquark and anti-diquark propagators split, even at T = 0 and low chemical potentials (see, e.g., Ref. [42]), while their true masses stay at their vacuum valuesuntil the lowest excitation threshold is reached or a phase transition takes place.In the 2SC phase, on the other hand, where baryon number is not conserved and mixing takes place,the chemical potentials µ X related to the various modes are less clear a priori and must be determined fromEq. (83).
4. Generalized Beth-Uhlenbeck equation of state
In the present section we will formulate the thermodynamics of two-particle correlations in a form whichis known as the Beth-Uhlenbeck EoS [51, 52]. The standard Beth-Uhlenbeck formula considers the secondvirial coefficient for the EoS that contains the contribution of bound states and scattering states in thelow-density limit. In dense matter, the single-particle properties as well as the two-particle properties aremodified. This is seen in the corresponding spectral functions where the δ -like peaks describing the single-particle states and two-particle bound states (in the two-particle propagator) are shifted and broadened. Themedium modifications of these quasiparticles are given in lowest approximation by the self-energy and (in thetwo-particle case) screening and Pauli-blocking contributions. When speaking of high densities we have inmind the occupation of phase space measured by scalar densities which can be large even at vanishing baryon12ensity (zero chemical potential). We will generalize the standard Beth-Uhlenbeck approach to the situationin a hot and dense medium when the gap between the discrete spectrum of two-particle bound states andthe scattering continuum (defining the binding energy) diminishes and finally vanishes so that the boundstate merges the continuum. This generalized Beth-Uhlenbeck (GBU) EoS is applicable in a wide range ofdensities, improving the mean-field approach by including medium-modified two-particle correlations.The dissolution of composite particles into their constituents because of the screening of interaction in adense medium is known as the Mott effect in solid state physics and has also found numerous applicationsin semiconductor physics (transition from the exciton gas to the electron-hole liquid [53]), plasma physics(pressure ionization [54]) and nuclear physics (cluster dissociation due to Pauli blocking [55]). Here we wantto apply the concept to particle physics and formulate the problem of hadron dissociation as a Mott effect.As it is known that this effect should not lead to discontinuities in the thermodynamic functions like thedensity, one has to take care of the normalization of the spectral function of the two-particle correlations.Whenever a bound state gets dissociated, it should leave a trace in the behavior of the scattering phaseshift at the threshold of the continuum. This constraint is known as Levinson’s theorem II [56, 57]. As weshall see, it can play an important role for the formulation of a thermodynamics of hadronic matter underextreme conditions where one of the key puzzles is the mechanism of hadron dissociation at the transitionto the quark-gluon plasma (QGP) or, equivalently, the problem of hadronization of the QGP in the courseof which correlations (pre-hadrons) form in the vicinity of the quark-to-hadron matter transition. Thosecorrelations shall play a decisive role, e.g., for understanding the chemical freeze-out. In the present sectionwe will formulate the thermodynamics in the Beth-Uhlenbeck form [51, 52] which allows the discussion ofthese issues in terms of two-particle correlations (2nd virial coefficient) as expressed by scattering phaseshifts.While in a low-density system, the phase shifts can be regarded as measurable quantities which thenmay be used to express deviations from an ideal gas behaviour due to two-particle correlations in a dilutemedium, the situation changes in a dense system. Under extreme conditions, in particular in the vicinity ofthe Mott transition, the modification of the two-particle system by the influence of the medium has to betaken into account. This has been done systematically within a thermodynamic Green function approach[58] but its extension for relativistic systems within a field theoretic formulation with contributions fromantiparticles, relativistic kinematics and the role of the zero-point fluctuations has been missing. Previouswork in this direction has been done by H¨ufner et al. [21] and Abuki [59], who have chosen to introducethe scattering phase shifts as arguments of the Jost representation of the complex S matrix. Most recently,Wergieluk et al. [22], Yamazaki and Matsui [23], and Dubinin et al. [24] used a different approach where thephase shifts are encoding correlations of the relativistic two-particle propagators introduced above, whichare the analogue of the T matrix of two-body scattering theory. Following this formalism, we thus develophere the field theoretic analogue of the approach by [53, 58].To be as transparent as possible in this basic and new contribution, we choose here to present derivationsin the normal phase where ∆ MF = 0, neglecting the possibility of diquark condensation. The derivation ofthe more general case works in the same manner but involves the mathematical apparatus to deal with themixing of correlation channels (see, e.g., [58]) which shall be given elsewhere. Note that the dissolution ofcomposite particles into their constituents gives no discontinuities in the thermodynamic properties as longas homogeneous systems are considered. However, phase instabilities are provoked if the stability criteriaaccording to the second law are violated due to the contribution of two-particle correlations to the EoS. Starting point for the derivation of the GBU EoS is the thermodynamic potential (69) which separatesinto the contributions from the eigenmodes of the two-particle propagation (70) as encoded in the two-particlepropagators S X (i z n , q ) with the inverse S − (i z n , q ) = G − − Π X (i z n , q ) for a generic particle/correlationX ∈ { M , D } , defined at the Matsubara frequencies i z n . The complex function S − ( z, q ) is its analyticcontinuation into the complex z -plane. Π X (i z n , q ) is the polarisation loop in the corresponding channelwith an analytic continuation to the complex z - plane, analogous to that of the propagator. The complex13ropagator functions can be given the polar representation S X = | S X | e i δ X = S R + i S I , (84)where the scattering phase shift δ X has been introduced as δ X ( ω, q ) = − Im ln (cid:2) β S − ( ω − µ X + i η, q ) (cid:3) . (85)Note that in this definition we have shifted the energy argument in the inverse propagator by − µ X in orderto exploit the symmetry properties of the propagator, cf. Eq. (82). For the same reason, in the followingdiscussion, the (inverse) propagator and the polarization function Π X should always be understood as to beevaluated at the shifted energy z = ω − µ X + iη , unless stated otherwise. Of course, this only concerns thediquarks and anti-diquarks, as for mesons we have µ X = 0 anyway.The contribution of the mode X to the thermodynamic potential as given in Eq. (67) can be transformedusing the spectral respresentation of the function ln S − (i z n , q ) and the definition of the phase shift (85) toobtain the formΩ X ( T, µ ) = d X T X n Z d q (2 π ) ln (cid:2) β S − (i z n , q ) (cid:3) , = − d X T X n Z d q (2 π ) ∞ Z −∞ d ω π z n − ω Im ln (cid:2) β S − ( ω + i η, q ) (cid:3) , = d X T X n Z d q (2 π ) ∞ Z −∞ d ω π z n − ( ω − µ X ) δ X ( ω, q ) . (86)Performing the bosonic Matsubara summation we arrive atΩ X ( T, µ ) = − d X Z d q (2 π ) ∞ Z −∞ d ω π n − X ( ω ) δ X ( ω, q ) , (87)where we have employed the Bose function with chemical potential according to the notation n ± X ( ω ) =1 / { exp[( ω ± µ X ) /T ] − } . After splitting the ω - integration into negative and nonnegative domains andusing the fact that for the Bose distribution function holds n − X ( − ω ) = − [1 + n + X ( ω )] while the phase shift(85) is an odd function under the reflection ω → − ω as a consequence of J ± X, pair ( − ω ) = − J ∓ X, pair ( ω ) and J ± X, Landau ( − ω ) = − J ∓ X, Landau ( ω ), see App. A.3 and App. A.4, we arrive atΩ X ( T, µ ) = − d X Z d q (2 π ) ∞ Z d ω π [1 + n − X ( ω ) + n + X ( ω )] δ X ( ω, q ) . (88)Here the shift in the definition of δ X , Eq. (85), was essential to have this symmetry property for diquarksand anti-diquarks as well. Performing a partial integration over the energy variable in (87) leads toΩ X ( T, µ ) = d X Z d q (2 π ) ∞ Z d ω π n ω + T ln (cid:16) − e − ( ω − µ X ) /T (cid:17) + T ln (cid:16) − e − ( ω + µ X ) /T (cid:17)o dδ X ( ω, q ) dω . (89)This expression still contains the divergent vacuum energy contribution. We remove this term in analogyto the “no sea” approximation which is customary in relativistic mean field approaches to thermodynamics,like the Walecka model, and arrive at the Beth-Uhlenbeck formula for the pressure p X ( T, µ ) = − Ω X ( T, µ ), p X ( T, µ ) = − d X T Z d q (2 π ) ∞ Z d ω π n ln (cid:16) − e − ( ω − µ X ) /T (cid:17) + ln (cid:16) − e − ( ω + µ X ) /T (cid:17)o dδ X ( ω, q ) dω . (90)14ith Eq. (90) the medium-dependent derivative of the phase shift δ ′ X ( ω, q ) has been introduced as a spectralweight factor for the contribution of a two-particle state X with degeneracy factor d X , depending on thethree-momentum q and the two-particle energy ω . Eq. (90) differs from the standard Beth-Uhlenbeckequation in nonrelativistic [51, 52] or relativistic [57] systems by the fact that the two-particle propagatorand therefore also the phase is obtained by taking into account in-medium effects as encoded in the solutionsof the gap equations which define the quark quasiparticle (meanfield) propagators entering its definition.Note that the phase angle used here should not be confused with an observable scattering phase shiftwhich should be on the energy shell. Here the function δ X ( ω, q ) is merely a convenient parametrization of thespectral properties of the logarithm of the two-particle propagator S X ( ω − µ X + iη, q ). The latter is definedby the polarization function Π X ( z, q ), being a one-loop integral involving mean-field quark propagators andthus not selfconsistently determined. We shall come back to the issue of selfconsistency in a separate work. Now we want to show that the phase (84) can be decomposed in two parts, corresponding to a structure-less scattering continuum δ c and a resonant (collective) contribution δ R which under appropriate conditionsrepresents a bound state contribution.The following derivation holds whenever the polarization loop integral can be expressed in the formΠ X ( z, q ) = Π X, + Π X, ( z, q ) , (91)where Π X, is a 4-momentum independent, real number, which can be a function of external thermodynamicvariables. We will show that for this very general form the decomposition δ X = δ X,c + δ X,R holds, whereby δ X,R corresponds to a resonant mode which goes over to the real bound state at the Mott transition wherethe bound state energy meets the threshold of the continuum of scattering states. The latter are describedby δ X,c , a structureless continuum background phase shift with a threshold to be directly identified frominspection of Im Π X as given in App. A.3 and A.4. Here we generalize a result which has been derived forthe pion and sigma channels before in [60] for the NJL model and in [22] for the Polyakov-loop extendedNJL model.The two-particle propagator S X in Eq. (85) can be given the form S X ( z − µ X + i η, q ) = 1 G − − Π X, − Π X, ( z − µ X + i η, q ) = 1Π X, ( z − µ X + i η, q ) 1 R X ( z , q ) − , (92)where the auxiliary function R X ( z , q ) = 1 − G X Π X, G X Π X, ( z − µ X + i η, q ) (93)has been introduced. Now obviously holdsln S X ( z − µ X + i η, q ) − = ln Π X, ( z − µ X + i η, q ) + ln[ R X ( z , q ) − , (94)so that with (84) δ X ( ω, q ) = δ X,c ( ω, q ) + δ X,R ( ω, q ) , (95)where δ X,c ( ω, q ) = − arctan (cid:18) Im Π X, ( ω − µ X + i η, q )Re Π X, ( ω − µ X + i η, q ) (cid:19) , (96) δ X,R ( ω, q ) = arctan (cid:18) Im R X ( ω , q )1 − Re R X ( ω , q ) (cid:19) . (97)From this decomposition of the phase shift it becomes immediately obvious that δ X,R ( ω, q ) corresponds tothe phase shift of a resonance at ω = ω X = p q + m . The position of the resonance is found from the15ondition Re R X ( ω X ) = 1, where for brevity we drop here and in the following derivation the argument q .At this energy holds that δ X,R ( ω → ω X ) → π/ δ X,R ( ω → ω X ) → ∞ .Performing a Taylor expansion at the resonance position for real ω one obtains1 − Re R X ( ω ) ≈ − Re R X ( ω X ) | {z } =0 − ( ω − ω X ) Re dR X ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) z = ω X , (98)Im R X ( ω ) ≈ Im R X ( ω X ) . (99)From this follows 1 − Re R X ( ω )Im R X ( ω ) ≈ − ( ω − ω X ) Re dR X ( z ) dz (cid:12)(cid:12) z = ω X Im R X ( ω X ) . (100)If we now define that ω X Γ X = − Im R X ( ω X )Re dR X ( z ) dz (cid:12)(cid:12) z = ω X , (101)the resonant phase shift becomes δ X,R ( ω, q ) = arctan (cid:18) ω X Γ X ω − ω X (cid:19) , (102)which corresponds to the Breit-Wigner form for the spectral density in the Beth-Uhlenbeck EoS dδ X,R ( ω ) dω = 2 ωω X Γ X ( ω − ω X ) + ω Γ . (103)This form goes over to the spectral density of a bound state when the width parameter Γ X → Γ X → dδ X,R ( ω ) dω = π [ δ ( ω − ω X ) + δ ( ω + ω X )] , (104)with the Dirac δ distribution on the r.h.s.The continuum contribution is defined along a cut on the real axis in the complex energy plane, i.e.for ω ≥ ω thr ( q ) = p q + 4 m , where Im Π = 0. The value of the corresponding phase shift at thresholdvanishes, δ X,c ( ω thr ) = 0.If the energy of the state X is below that threshold, ω X < ω thr , it is a real bound state with vanishingwidth (Γ X = 0, infinite lifetime) and the resonant phase shift behaves as a step function which jumps by π at ω = ω X and has therefore this value at the threshold, δ X,R ( ω thr ) = π for T < T X , Mott . In Fig. 1 we showthe behaviour of the threshold, the meson masses and the pion width in sections through the phase diagramin the
T, µ − plane, for q = 0. Note that the sharp onset of the pion width at the Mott temperatures is anartefact of the present approximation which neglects π − π scattering and treats quarks in the mean fieldapproximation only, where they are on-shell quasiparticles with zero width.The parameters employed are a bare quark mass m = 5 . G S Λ = 2 . G D = G S , and a moderate coupling G D = 3 / G S . The latter is motivated bythe ratio between scalar diquark and scalar meson interaction channels arising from a Fierz transformationof the (massive) vector boson exchange interaction model (see, e.g., Eq. (A.20) in the App. A of Ref. [36]).With the above parameters one finds in vacuum a constituent quark mass of 319 MeV, a pion mass of138 MeV and pion decay constant f π = 92 . σ -meson is 644 MeV, which isthus slightly unbound. The scalar diquark is bound (pion-like) for the strong diquark coupling and slightlyunbound (sigma-like) for the Fierz value of the coupling. Masses and widths of mesons and diquarks are16 m a ss e s [ G e V ] π m σ w i d t h [ G e V ] Γ π µ [MeV] = 300 250 150 0 µ [MeV] = 300 250 150 0 m a ss e s [ G e V ] σ m π µ [GeV]00.10.2 w i d t h [ G e V ] Γ π T [MeV] = 100 75 50 0T [MeV] = 100 75500
Figure 1: (Color online). Two-particle spectrum in the scalar-pseudoscalar meson channels at rest in the medium ( q = 0).Left panel: Behaviour of the thresholds for the quark-antiquark continuum 2 m (dotted lines), meson masses ( m σ - dash-dottedlines, m π - solid lines) and pion width Γ π (dashed lines) as functions of T for different values of the chemical potential µ . Rightpanel: Same as left panel but as functions of µ for different temperatures T . determined from the solution of the corresponding Bethe-Salpeter equations, i.e., from the (complex) polesof their two-particle propagators, see Sects. 2.2 and 3.The Mott transition, when the bound state merges the continuum, can be detected also from the be-haviour of the resonant phase shift which will be subject to the same threshold as the continuum and vanishesat the continuum edge, δ X,R ( ω thr ) = 0 for T > T X , Mott . This is a manifestation of Levinson’s theorem IIwhich can be formulated as ∞ Z dω π dδ X ( ω ; T ) dω = 0 = ω thr ( T ) Z dω π dδ X ( ω ; T ) dω | {z } n B,X ( T ) + 1 π ∞ Z ω thr ( T ) dω dδ X ( ω ; T ) dω | {z } π [ δ X ( ∞ ; T ) − δ X ( ω thr ; T )] , (105)at any given temperature T . Since under very general conditions [56] holds δ X ( ∞ ; T ) = 0 for any temperatureit follows that δ X ( ω thr ; T ) = πn B,X ( T ), i.e., that decrementing the number of bound states in the channel X at the corresponding Mott temperature T X, Mott has to be accompanied by a jump by π of the phase shiftat threshold [57]. This behaviour is illustrated in Fig. 2, see also Refs. [22, 24]. In this subsection, we demonstrate first for the case of the pion as the lightest meson and then forthe scalar diquark, how the Mott dissociation of the bound state leads to a reduction of the correlationcontribution to the thermodynamical potential. All thermodynamic relations can be consistently derivedfrom the thermodynamic potential which for a homogeneous system is directly given by the pressure, werefor the following numerical investigation we focus on X = π, δ A , δ ∗ A . In order to discuss the pressure for pion correlations in quark matter we start from the expression (90).We respect that µ π = 0 so that the GBU EoS for the pion pressure takes the form p π ( T ) = − d π T Z d q (2 π ) ∞ Z d ωπ ln (cid:16) − e − ω/T (cid:17) dδ π ( ω, q ) dω . (106)17 δ π , R δ σ , R -3-2-10 δ π , c δ σ , c ]0123 δ π ] δ σ T [MeV] =
Figure 2: Phase shifts for pion (left panels) and sigma meson (right panels) channels at rest ( q = 0) as functions of the squaredmass variable s = ω for different temperatures, below ( T = 0 ,
200 MeV) and above ( T = 250 ,
600 MeV) the Mott-transition.The upper panels show the resonant phase shifts which together with the continuum phase shifts shown in the middle panelsadd up to the total phase shifts of the bottom panels. δ π ( ω, q ) which is defined by Eqs. (95)-(97) via the polarizationfunction examined in detail in Appendix A. For a recent numerical evaluation of the pion and sigma mesonthermodynamics within the PNJL model, see [23].We want to give here a qualitative discussion of the physics content of the GBU EoS (106) by consideringthe phase shifts for pions at rest in the medium, as shown in Fig. 2, and suggesting their approximate boostinvariance depending on ω and q only via the Mandelstam variable s = ω − q in the form δ π ( ω, q = 0) = δ π ( √ s, q = 0) ≡ δ π ( s ; T ). Then, the GBU EoS for the pionic pressure can be given the suggestive form p π ( T ) = ∞ Z ds D π ( s ; T ) p π ( T, s ) , (107)where p π ( T, s ) = − d π T Z d q (2 π ) ln (cid:16) − e − √ q + s/T (cid:17) (108)is the pressure of a relativistic Bose gas of pions with a ficticious mass √ s . The distribution of these massesto be integrated over for obtaining the total pressure is given by the density of states D π ( s ; T ) = 1 π dδ π ( s ; T ) ds . (109)Exploiting the analytic decomposition of the total phase shift (95) into a continuum and a resonant contri-bution, δ π = δ π,c + δ π,R , we can separate the pion pressure into a negative continuum contribution and aresonance contribution, p π ( T ) = p π,c ( T )+ p π,R ( T ), where for the latter we can make use of the Breit-Wignerapproximation (103) D BWπ,R ( s ; T ) = 1 π Γ π m π ( s − m π ) + Γ π m π , (110)for which holds that lim Γ π → D BWπ,R ( s ; T ) = D π, bound = δ ( s − m π ) . (111)In the limit of vanishing width Γ π → s − integration in (107) can be analytically performed and yields the ideal pion gas form p π ( T, m π ),here with a temperature dependent mass m π = m π ( T ). Since the pion mass is rising with temperature (seeFig. 1), in particular for T > T X, Mott , the corresponding pressure shows a slight drop (dot-double-dashedline in Fig. 3).Taking into account additionally the rapidly growing pion width Γ π ( T ) with a sharp onset at T = T X, Mott (see Fig. 1), the pion pressure is reduced stronger above the Mott temperature (dashed line in Fig. 3). Inthese approximations, the role of the continuum has been neglected. The pressure of the continuum states p π,c ( T ) on the other hand is obtained when the resonance contribution (dotted line in Fig. 3) to the densityof states is neglected, i.e., D π ( s ; T ) = D π,c ( s ; T ) = 1 π dδ π,c ( s ; T ) ds (112)is used in Eq. (107). The result is shown as the dash-double-dotted line in Fig. 3; its contribution is negativeand sets in already for T < T X, Mott . The total pion pressure (solid line in Fig. 3) with contributions from Formally, when substituting the ω − integral in (106) for a s − integral, the lower limit ω min = 0 corresponds to s min = − q .Because of the interpretation of the variable s as a squared meson mass, we restrict the range of the s − integration to allnonnegative values. P / T p π p π ,R p π ,b p π ,RBW p π ,c p π BW P / T p π p π ,R p π ,c Figure 3: Right panel: Pionic contribution to the pressure, Eq. (107), as a function of the temperature T for chemicalpotential µ = 0 (solid line) and its decomposition into resonant (dotted line) and continuum (dash-double-dotted line) scatteringcontributions according to the decomposition of the phase shift in the density of states Eq. (109). Left panel: Closeup of the rightpanel in the temperature region of the pion Mott transition at T π, Mott = 208 MeV and comparison to different approximationsfor the resonance contribution: The dashed line represents the Breit-Wigner approximation (110) for the resonance contributionand shows the effect of the resonance broadening compared to the pion bound state approximation (111), with a T − dependentmass but without broadening (dot-double-dashed line). The dash-dotted line shows the total pion pressure when for theresonance contribution the Breit-Wigner approximation is used. Note that the effect of the continuum pressure sets in alreadybefore the Mott transition temperature is reached. both resonant and continuum scattering shows a typical behaviour with increasing temperature: First a risetowards the Stefan-Boltzmann limit which, however, is never reached because the lowering of the continuumedge due to the chiral phase transition induces a reduction of the meson gas pressure already before the Motttemperature is reached. Second, above the Mott temperature, the growing pion width leads to a strongerreduction of the pressure with a rather sharp onset of this effect. The resulting pattern appears like a “sharkfin”. The introduction diquark fields is a prerequisite for the description of baryons as quark-diquark boundstates in a chiral quark model. Thereby the diquark appears not necessarily as a bound state. Actually,in particular in QCD DSE approaches the diquark is unbound while the baryon is a bound state, like in aBorromean three-particle state.Having developed in the present work the theoretical basis for the description of the thermodynamics ofmesons and diquarks in hot and dense quark matter including a Mott dissociation transition, we want todiscuss now after the pion also the diquark case. To that end we focus here on the normal quark matterphase and evaluate the diquark phase shifts for vanishing chemical potential at finite temperatures whichencode the analytic properties of the diquark propagator. The results are shown in Fig. 4 for two cases of thediquark coupling strength: the moderate one (left panels) which corresponds to the Fierz value G D = 3 / G S and the strong coupling case (right panels) with G D = G S . In the latter case the diquark is a bound stateat zero temperature and its phase shift behaves similar to that of the pion in Fig. 2 which exhibits a Motttransition to an unbound resonance in the continuum at a certain temperature. For our parametrizationthis happens at T D, Mott ≈
140 MeV. For the moderate coupling case, the diquark phase shift is shown in theleft panels of Fig. 2 and behaves similar to the sigma meson: it is already an unbound resonant scatteringstate in the vacuum at T = 0 and becomes even less correlated when the temperature is increased.The diquark contribution to the thermodynamics is evaluated according to the Beth-Uhlenbeck formula(90) for the pressure, where as in the pion case the “no sea” approximation is applied, i.e., the contribution ofthe zero-point energy in (89) is removed. The result is shown in Fig. 5 for the Fierz coupling case (left panel)which exhibits an almost complete compensation between the contrinutions of the diquarks resonance andthe scattering continuum so that the resulting pressure (solid line) in the diquark channel is much smaller20 δ D , R -3-2-10 δ D , c ]00.511.5 δ D T [MeV] = δ D , R -3-2-10 δ D , c ] δ D T [MeV] =
Figure 4: Phase shifts for scalar diquark channel with G D = 3 / G S (left panels) and G D = G S (right panels) at rest ( q = 0)as functions of the squared mass variable s = ω for different temperatures T = 0 , ,
190 MeV. The upper panels showthe resonant phase shifts which together with the continuum phase shifts shown in the middle panels add up to the totalphase shifts of the bottom panels. Comparison to Fig. 2 shows that for moderate coupling (left panels) the diquark behaves“sigma-like”, since it is unbound for all temperatures. For strong coupling (right panels) the diquark behaves “pion-like”, sinceit is a bound state for low temperatures and exhibits a Mott transition at T ≈
140 MeV where the phase shift at the continuumthreshold s thr = 4 m jumps from the value π to zero in accordance with the Levinson theorem (105). P / T P tot P res P c G D /G S = 3/4 0 0.2 0.4 0.6 0.8T [GeV]-0.4-0.200.20.4 P / T G D /G S = 1G D /G S = 3/4 Figure 5: Pressure for the scalar diquark (solid line) with its resonant (dotted line) and continuum (dash-double-dotted line)components as functions of temperature. Bold lines are for the case of the moderate (Fierz) coupling G D /G S = 3 / G D /G S = 1 (right panel). A key lesson from this investigation into the GBU EoS for the two-particle correlations in the pion anddiquark channels in hot quark matter is that accounting for the broadening of hadronic states due to theMott transition alone it is not sufficient and would violate the Levinson theorem [22, 24]. One has to accountfor the scattering state continuum which influences the thermodynamics as soon as the thermal energy isof the order of the dissociation threshold, the gap between bound and continuum states of the spectrum.This appears to be an important modification of the so-called “chemical” picture where hadron speciesform a statistical ensemble of quasiparticles which goes beyond attributing to these hadronic resonancesa finite lifetime (width). The more elaborate “physical” picture requires to account for a scattering statecontinuum which in a systematic way is given by a virial expansion and as such appears in the GBU EoS.For a discussion of the cluster virial expansion in nuclear matter see, e.g., [61].For dense hadronic matter, where hadrons play the role of clusters, a similar approach is still to bedeveloped. We suggest that this could be done along the lines of the GBU approach given here. However,the problem one is facing when one wants to build such an approach on an NJL-type model is the lack ofconfinement. In the NJL model used here, just the pions are bound states in the vacuum while the sigmameson is already an unbound resonance, decaying to quarks. In the flavor SU(3) extension of the NJLmodel, it is just the pseudoscalar meson octet which is bound while the other channels important for thephenomenology (vector, scalar and axialvector mesons) are unbound.While a solution to this severe problem has still to be worked out, there are simple extensions of the NJLmodel which can be used to remove unphysical quark decay thresholds but still keep the microscopic descrip-tion of the hadron Mott effect sufficiently simple. One of such extensions mimicks confining interactions bysuppressing long wavelength modes in quark propagation by applying a low-momentum cutoff to quark loopintegrals. Moreover, since confinement and chiral symmetry breaking shall be related, this cutoff has beenfixed to the dynamically generated quark mass gap which vanishes upon chiral symmetry restoration. Fordetails, see [24] and references therein. In this work it has been shown how this infrared cutoff increases thethreshold for quark-antiquark scattering states and thus makes the σ − meson a bound state in the chiralsymmetry broken phase for low temperatures. It could also be demonstrated in [24] within this NJL modelgeneralized by a low-momentum cutoff, that the shape of the phase shifts and their temperature dependencefor the σ − meson becomes similar to that of the pion, where the transition from a bound state to a resonance22n the continuum can be recognized in accordance with the Levinson theorem. Summarizing this section, wewould like to point out that the NJL model and its modifications provide a useful tool for a field theoreticformulation of the physics of the Mott dissociation of hadrons as quark bound states in hot, dense quarkmatter or, conversely, of the hadronization of quark correlations in an expanding and cooling QGP.
5. Conclusions
In this work, we have presented a general approach to the discussion of two-particle correlations in quarkmatter within a field theoretical chiral quark model of the NJL type. By restricting the expansion of thefermion determinant of the bosonized partition function to the gaussian approximation, the path integralover the meson and diquark fluctuations can be performed and a closed expression is obtained in the form ofthe determinant of a generalized matrix propagator for two-particle states (meson and diquark fields). Thediagonalization of this matrix defines the mode spectrum of the model. A detailed discussion of limitingcases (chiral limit, zero momentum limit) is performed and analytic expressions are given.Nambu-Goldstone (NG) modes in the 2SC phase are considered and the finite T extension of the workby Ebert and collaborators [42] is provided here for the first time. It is demonstrated that in the 2SC phasethere are only 3 instead of 5 massless NG modes, in accordance with the Nielsen-Chadha theorem.We have discussed the interplay between bound and scattering states in the medium, in particular atthe Mott transition where the bound state transforms to a resonance in the continuum. This transitionis reflected in a vanishing of the binding energy as well as a jump by π of the phase shift at threshold inacordance with the Levinson theorem.The thermodynamic potential is given the form of a generalized Beth-Uhlenbeck equation where specialemphasis is on the discussion of the role of the continuum contributions as well as of the resonant two-particlecorrelations as expressed in terms of the in-medium scattering phase shifts. It is shown that accounting forthe spectral broadening of hadrons alone is insufficient to account for the Mott dissociation. Only togetherwith the continuum states can be garanteed that the result for the pressure is in accordance with thein-medium Levinson theorem.The in-medium phase shifts for mesonic and diquark correlation channels as evaluated in this work withina NJL model description of low energy QCD contain spectral information which may be exploited to studymesonic and diquark correlation functions at finite temperature, around the chiral restoration. It will bevery interesting to investigate how the Mott transition for mesons manifests itself in these functions forwhich also ab-initio calcuations using lattice QCD simulations exist. We plan to extend our work in thisdirection and to provide guidance from this low-energy QCD model for the interpretation of those latticeQCD data.An outlook is given to the further development of the approach towards a description of meson-baryonmatter on the basis of chiral quark models of the NJL type which shall be detailed in subsequent work. Acknowledgement
We acknowledge R. Anglani for his collaboration in an early stage of this work and T. Brauner for dis-cussions and his critical reading of the manuscript. This work was supported by the Deutsche Forschungs-gemeinschaft (DFG) under contract BU 2406/1-1 and by the Polish National Science Centre within the“Maestro” programme under grant no. UMO-2011/02/A/ST2/00306. The work of D.B. was supported inpart by the Polish Ministry of Science and Higher Education (MNiSW) under grant no. 1009/S/IFT/14.G.R. and D.B. are grateful to the Erasmus programme and partnership between Universities of Rostock andWroclaw which supported mutual visits.
A. Meson and diquark polarization functions
In this appendix we give explicit expressions for the elements of the polarization matrix Π defined inEq. (62). We thereby neglect the vector fields ω µ . Moreover, since the three isospin components of the pion23re degenerate and do not mix with each other or any other mode, we only list one generic component herefor simplicity. A.1. 2SC phase
In the 2SC phase, the polarization matrix has the structureΠ = Π ππ σσ Π σδ Π σδ ∗ δ ∗ σ Π δ ∗ δ Π δ ∗ δ ∗ δ σ Π δ δ Π δ δ ∗ δ ∗ δ δ δ ∗ δ ∗ δ
00 0 0 0 0 0 0 Π δ δ ∗ . (A.1)Each matrix element depends on an external three-momentum q and an external bosonic Matsubarafrequency iz n , which after analytic continuation becomes the complex variable z . Labelling the fields in thevector (61) by i = ~π T , σ, δ ∗ , δ , δ ∗ , δ , δ ∗ , δ and j = ~π, σ, δ , δ ∗ , δ , δ ∗ , δ , δ ∗ , we haveΠ ij ≡ Π ij (i z n , q ) ≡ Π ij ( z, q ) . (A.2)The explicit expressions aslisted below also contain an internal momentum p which is integrated over. It isthen convenient to define a third momentum k = p − q .The individual elements are as followsΠ ππ = 12 N f Z d p (2 π ) X s p ,s k T + − ( s p , s k ) (cid:26) n ( s p ξ s p p ) − n ( s k ξ s k k ) z − s k ξ s k k + s p ξ s p p − n ( s p ξ s p p ) − n ( s k ξ s k k ) z + s k ξ s k k − s p ξ s p p + X t p ,t k F ( s p , s k ; t p , t k ) (cid:0) t p t k E s p p E s k k + s p s k ξ s p p ξ s k k − | ∆ MF | (cid:1) (cid:27) (A.3)Π δ δ = 14 (∆ ∗ MF ) N f Z d p (2 π ) X s p ,t p s k ,t k T ++ ( s p , s k ) F ( s p , s k ; t p , t k ) (A.4)Π δ ∗ δ = 14 N f Z d p (2 π ) X s p ,t p s k ,t k T ++ ( s p , s k ) F ( s p , s k ; t p , t k )( t p E s p p + s p ξ s p p )( t k E s k k − s k ξ s k k ) (A.5)Π σδ = 14 m ∆ ∗ MF N f Z d p (2 π ) X s p ,t p s k ,t k (cid:18) s p E p + s k E k (cid:19) F ( s p , s k ; t p , t k ) (cid:2)(cid:0) t p E s p p − t k E s k k (cid:1) + (cid:0) s p ξ s p p + s k ξ s k k (cid:1)(cid:3) (A.6)Π δ ∗ δ = 12 N f Z d p (2 π ) X s p ,t p s k ,t k T ++ ( s p , s k ) (cid:26) E s p p − s p t p ξ s p p E s p p n ( s k ξ s k k ) − n ( t p E s p p ) z − t p E s p p + s k ξ s k k + E s k k − s k t k ξ s k k E s k k n ( s p ξ s p p ) − n ( t k E s k k ) z − t k E s k k + s p ξ s p p (cid:27) (A.7)24nd the remaining elements are recast by the replacementsΠ σσ = Π ππ (cid:0) T + − → T −− (cid:1) (A.8)Π δ ∗ δ ∗ = Π δ δ (cid:0) (∆ ∗ MF ) → ∆ (cid:1) (A.9)Π δ δ ∗ = Π δ ∗ δ ( z → − z, p ↔ k ) (A.10)Π σδ ∗ = Π σδ (∆ ∗ MF → ∆ MF , p ↔ k ) (A.11)Π δ ∗ σ = Π σδ (∆ ∗ MF → ∆ MF , z → − z ) (A.12)Π δ σ = Π σδ ∗ (∆ MF → ∆ ∗ MF , z → − z ) (A.13)Π δ δ ∗ = Π δ ∗ δ ( z → − z, p ↔ k ) (A.14)Π δ δ ∗ = Π δ δ ∗ (A.15)Π δ ∗ δ = Π δ ∗ δ (A.16)Here s p , s k ; t p , t k = ± E p , E ± p , and ξ ± p are the single-quark dispersion relations, givenin Sect. 2.1, and n ( x ) denotes again the Fermi distribution function. Moreover, we have defined F ( s p , s k ; t p , t k ) = t p t k E s p p E s k k n ( t p E s p p ) − n ( t k E s k k ) z − t k E s k k + t p E s p p , (A.17)and the kinematic pre-factors T ±∓ ( s, s ′ ) = 1 ∓ ss ′ p · k ± m E p E k . (A.18) A.2. Correlations at rest
The expressions for the polarization functions get strongly simplified for correlations at rest, q = 0. Wethen have p = k , which then puts restrictions on the summation over s p and s k via T ±∓ and at last wecan carry out the summations over t p , t k . This then leaves us with only one sign operator, which is thenassociated with particle ( − ) and anti-particle (+) contributions. We findΠ ππ ( z, ) = − I π ( z ) , (A.19)Π σσ ( z, ) = − I σ ( z ) − m | ∆ MF | I ( z ) , (A.20)Π δ ∗ δ ( z, ) = 2 I ∆ + 4 zI ( z ) + (4 | ∆ MF | − z ) I ( z ) , (A.21)Π δ δ ( z, ) = 4∆ I ( z ) , (A.22)Π σδ ( z, ) = − m ∆ MF ( zI ( z ) + 2 I ( z )) , (A.23)Π δ ∗ δ ( z, ) = I ∆ + 2 zI ( z ) − ( | ∆ MF | − z ) I ( z ) , (A.24)and the remaining elements can be found by applying the symmetry relations Eqs. (A.9)-(A.16). Theconstant I ∆ and the functions I i ( z ) are defined as follows: I ∆ = I +∆ + I − ∆ , I = I +0 + I − , (A.25) I = I +1 − I − , I = I +2 + I − , (A.26) I = I +3 + I − , I = I +4 + I − , (A.27) I = I +5 + I − , I = I +7 − I − , (A.28)25here the individual terms are I ± ∆ ≡ Z d p (2 π ) (cid:2) − n ( E ± p ) (cid:3) E ± p , , I ± ( z ) ≡ Z d p (2 π ) E ± p F ± p ( z ) , (A.29) I ± ( z ) ≡ Z d p (2 π ) ξ ± p E ± p F ± p ( z ) , I ± ( z ) ≡ Z d p (2 π ) E p E ± p F ± p ( z ) , (A.30) I ± ( z ) ≡ Z d p (2 π ) E p ξ ± p E ± p F ± p ( z ) , I ± ( z ) ≡ Z d p (2 π ) E p E ± p F ± p ( z ) (A.31) I ± ( z ) ≡ Z d p (2 π ) − n ( E ± p ) E ± p ( z ± zξ ± p − | ∆ MF | ) , I ± ( z ) ≡ Z d p (2 π ) − n ( ξ ± p ) z ± zξ ± p − | ∆ MF | (A.32)and the remaining integral for the mesons is given by I π ( z ) ≡ Z d p (2 π ) (cid:26)(cid:2) − n ( ξ + p ) − n ( ξ − p ) (cid:3) E p z − E p + E + p E − p − ξ + p ξ − p − | ∆ MF | E + p E − p (cid:2) n ( E − p ) − n ( E + p ) (cid:3) E + p − E − p z − (cid:0) E + p − E − p (cid:1) + E + p E − p + ξ + p ξ − p + | ∆ MF | E + p E − p (cid:2) − n ( E + p ) − n ( E − p ) (cid:3) E + p + E − p z − (cid:0) E + p + E − p (cid:1) ) . (A.33)The integral I σ ( z ) differs from I π ( z ) only by an additional overall factor p /E p in the integrand.In the normal phase, for ∆ MF = 0, the integral (A.33) reduces to the simple form I π, ( z ) = N c Z d p (2 π ) (cid:2) − n ( ξ + p ) − n ( ξ − p ) (cid:3) E p z − E p . (A.34)The function F ± p ( z ) ≡ − n ( E ± p ) z − (cid:0) E ± p (cid:1) (A.35)is the analogue to the function F ( s p , s k ; t p , t k ) defined above in (A.17) for correlations at rest ( q = 0). A.3. Mesonic polarization functions in the normal phase
In the normal phase the meson and diquark-modes decouple. We denote the mesonic matrix elementsby Π π ≡ Π ππ , Π σ ≡ Π σσ . They are explicitly given by the expressionΠ π/σ ( z, q ) = 12 N f N c X s p ,s k Z d p (2 π ) T ±− ( s p , s k ) n − ( s p E p ) − n − ( s k E k ) z + s p E p − s k E k + ( µ ∗ → − µ ∗ ) , (A.36)where T + − ( s p , s k ) holds for the pion case while T −− ( s p , s k ) for the sigma meson. After analytic continuationto the complex z plane, the analytic properties of the polarization function are captured in its spectraldensity as defined by the imaginary part of the retarded function, Im Π π/σ ( ω + i η, q ).Results for scalar and pseudo-scalar mesons have already been obtained by e.g., [25, 38] and shall besummarized here.Im Π σ,π ( ω + i η, q ) = N π,σ N f N c π | q | n Θ( s − m ) h Θ( ω ) J +M , pair + Θ( − ω ) J − M , pair i + Θ( − s ) J M , Landau o (A.37)26ith J ± M , pair = T ln (cid:20) [1 − n ± ( E − )] n ∓ ( E − )[1 − n ± ( E + )] n ∓ ( E + ) (cid:21) = T ln (cid:20) n ∓ ( −E − ) n ∓ ( E − ) n ∓ ( −E + ) n ∓ ( E + ) (cid:21) (A.38) J M , Landau = T ln (cid:20) n + ( E − ) n − ( E − ) n + ( −E + ) n − ( −E + ) (cid:21) = − ω + T ln (cid:20) n + ( −E − ) n − ( −E − ) n + ( E + ) n − ( E + ) (cid:21) (A.39)with the Fermi distribution functions n ± ( E ) = [exp { ( E ± µ ∗ ) /T } + 1] − and E ± = ω ± | q | q − m s .The kinematic prefactors are N π = s and N σ = s − m , with s = ω − | q | . We want to point out, thatthe imaginary part is an odd function of ω and consequently the real part is even in ω so that the spectralfunction is an odd function of ω .This form has the thresholds for the occurrence of imaginary parts, given in terms of Θ-functions,explicitly; a property that has been exploited in the discussion of the Mott effect for mesons in Sect. 4. A.4. Diquark polarization functions in the normal phase
In the normal phase all diquark modes are degenerate and we denote them as Π D ≡ Π δ ∗ A δ A and Π ¯D ≡ Π δ A δ ∗ A . The diquark polarization function takes the formΠ D ( z − µ ∗ , q ) = N f X s p ,s k Z d p (2 π ) T + − ( s p , s k ) n + ( s p E p ) − n − ( s k E k ) z + s p E p − s k E k . (A.40)The anti-diquark mode is then obtained by the symmetry relation Π ¯D = Π D ( µ ∗ → − µ ∗ ).The imaginary part of the retarded diquark polarization function is evaluated to beIm Π D ( ω + i η, q ) = N f π | q | X a = ± s a (cid:2) Θ( s a − m ) J a D , pair + Θ( − s a ) J a D , Landau (cid:3) (A.41)with J ± D , pair = T ln " − n ∓ ( E −± )1 − n ∓ ( E + ± ) n ∓ ( E −± ) n ∓ ( E + ± ) = T ln " n ± ( −E −± ) n ± ( −E + ± ) n ∓ ( E −± ) n ∓ ( E + ± ) , (A.42) J ± D , Landau = 2 T ln " n ∓ ( E −± ) n ± ( −E + ± ) = − ω + 2 T ln " n ± ( −E −± ) n ∓ ( E + ± ) (A.43)where we defined E ±∓ = ω ∓ ± | q | q − m s ∓ and ω ± = ω ± µ ∗ and s ± = ω ± − | q | . Again, the real partof the spectral function is to be evaluated utilizing a Kramers-Kronig relation. This result collapses to thepion imaginary part at µ ∗ = 0 up to the prefactor 2. For the case of degenerate flavors considered here, theLandau damping term vanishes for correlations at rest ( q = 0).27 eferences [1] C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. , 477 (1994).[2] P. C. Tandy, Prog. Part. Nucl. Phys. , 117 (1997).[3] R. Alkofer and L. von Smekal, Phys. Rept. , 281 (2001).[4] C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. , S1 (2000).[5] H. Kleinert, Phys. Lett. B , 429 (1976).[6] D. Ebert and V. N. Pervushin, In *Tbilisi 1976, Proceedings, Conference On High Energy Physics, Vol.I*, Dubna 1976,C125-127[7] R. T. Cahill, Austral. J. Phys. , 171 (1989).[8] H. Reinhardt, Phys. Lett. B244 , 316-326 (1990).[9] D. Ebert, H. Reinhardt and M. K. Volkov, Prog. Part. Nucl. Phys. , 1 (1994).[10] R. T. Cahill, J. Praschifka, C. Burden, Austral. J. Phys. , 161 (1989).[11] C. J. Burden, R. T. Cahill, J. Praschifka, Austral. J. Phys. , 147 (1989).[12] J. Praschifka, C. D. Roberts, R. T. Cahill, Phys. Rev. D36 , 209 (1987).[13] U. Zuckert, R. Alkofer, H. Weigel and H. Reinhardt, Phys. Rev. C , 2030 (1997).[14] D. Ebert and H. Reinhardt, Nucl. Phys. B , 188 (1986).[15] C. V. Christov, A. Blotz, H. -C. Kim, P. Pobylitsa, T. Watabe, T. Meissner, E. Ruiz Arriola and K. Goeke, Prog. Part.Nucl. Phys. , 91 (1996).[16] B. Golli, W. Broniowski and G. Ripka, Phys. Lett. B , 24 (1998).[17] R. Alkofer, H. Reinhardt and H. Weigel, Phys. Rept. , 139 (1996).[18] R. Alkofer and H. Reinhardt, Chiral quark dynamics (Springer, Berlin, 1995).[19] G. Ripka,
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