Temperature-dependent cross sections for meson-meson nonresonant reactions in hadronic matter
aa r X i v : . [ nu c l - t h ] D ec Temperature-dependent cross sectionsfor meson-meson nonresonant reactionsin hadronic matter
Yi-Ping Zhang Xiao-Ming Xu Hui-Jun Ge
Department of Physics, Shanghai University, Baoshan, Shanghai 200444, China
Abstract
We present a potential of which the short-distance part is given by one gluonexchange plus perturbative one- and two-loop corrections and of which the large-distance part exhibits a temperature-dependent constant value. The Schr¨odingerequation with this temperature-dependent potential yields a temperature depen-dence of the mesonic quark-antiquark relative-motion wave function and of mesonmasses. The temperature dependence of the potential, the wave function and themeson masses brings about temperature dependence of cross sections for the non-resonant reactions ππ → ρρ for I = 2, KK → K ∗ K ∗ for I = 1, KK ∗ → K ∗ K ∗ for I = 1, πK → ρK ∗ for I = 3 / πK ∗ → ρK ∗ for I = 3 / ρK → ρK ∗ for I = 3 / πK ∗ → ρK for I = 3 /
2. As the temperature increases, the rise orfall of peak cross sections is determined by the increased radii of initial mesons, theloosened bound states of final mesons, and the total-mass difference of the initialand final mesons. The temperature-dependent cross sections and meson masses areparametrized.
PACS: 25.75.-q; 13.75.Lb; 12.38.MhKeywords: Meson-meson nonresonant reaction; Cross section; Quark-interchange mecha-nism. 1 . Introduction
In spite of scarce experimental data of cross sections for inelastic meson-meson scat-tering, these cross sections can be calculated from quark potential models, parton dis-tributions or effective meson Lagrangians that respect various symmetries. This hasbeen seen from the dissociation cross sections of
J/ψ in collisions with mesons, whichhave received much attention. For example, cross sections for reactions like π + J/ψ → D ¯ D ∗ + D ∗ ¯ D + D ∗ ¯ D ∗ and ρ + J/ψ → D ¯ D ∗ + D ∗ ¯ D + D ∗ ¯ D ∗ were obtained with thequark-interchange mechanism [1, 2] in different quark potential models [3–6]. The quark-interchange models give the characteristic that cross sections for endothermic reactionsfirst increase from threshold energies and then decrease as the center-of-mass energy of J/ψ and hadron increases. A very small low-energy nucleon-
J/ψ break-up cross sectionwas obtained by using the operator product expansion for the elastic scattering amplitudeof
J/ψ [7–9]. The dissociation cross section for π − J/ψ , evaluated in QCD sum rules [10]in the soft-pion limit, increases gradually in contrast to rapid growth near the thresholdenergy obtained in meson exchange [11–15] and quark interchange models [3–6].In comparison to meson-
J/ψ reactions, however, cross sections for inelastic scatteringof a light meson by another light meson are rarely studied. Inelastic scattering betweenlight mesons does occur in hadronic matter and the cross sections involved influence thetime dependence of meson momentum distributions and flavor dependence of the measuredmomentum distributions at kinetic freeze-out. In order to understand this influence, thecross sections for inelastic scattering and their relevant characteristics must be studied.Since hadronic matter produced in Au-Au collisions at the Relativistic Heavy Ion Collidermainly consists of pions, rhos and kaons [16–23], in this work we pay attention only to thenonresonant reactions of the four mesons π , ρ , K and K ∗ , which are taken to be governedby quark-interchange processes. In Ref. [24] we have calculated in-vacuum cross sectionsfor the nonresonant reactions ππ → ρρ for I = 2, KK → K ∗ K ∗ for I = 1, KK ∗ → K ∗ K ∗ for I = 1, πK → ρK ∗ for I = 3 / πK ∗ → ρK ∗ for I = 3 / ρK → ρK ∗ for I = 3 / πK ∗ → ρK for I = 3 /
2. The cross sections for the seven endothermic reactions depend2n the center-of-mass energy √ s of the two initial mesons and the energy where themaximum of cross section occurs is mainly determined by the maximum of | ~P ′ | /s | ~P | ,where ~P and ~P ′ are the momenta of the initial and final mesons in the center-of-massframe, respectively. The endothermic reactions have maximum cross sections rangingfrom 0.47 mb to 1.41 mb. These cross sections are obtained with a quark-quark potentialthat includes the linear confinement and one-gluon-exchange potential plus perturbativeone- and two-loop corrections [6]. At nonzero temperature the linear confinement ismodified to become weaker. At a temperature below the critical temperature T c of theQCD phase transition, medium effects show up in the region where the quark-antiquarkdistance is larger than 0.3 fm [25]. When the quark-antiquark distance is large enough, thequark-antiquark potential at a given temperature becomes constant. This temperature-dependent potential must change the wave function of quark-antiquark relative motion in ameson and cross sections for inelastic meson-meson scattering are expected to change withtemperature as well. Therefore, in this work we study the dependence of cross sectionson the temperature for the nonresonant reactions ππ → ρρ for I = 2, KK → K ∗ K ∗ for I = 1, KK ∗ → K ∗ K ∗ for I = 1, πK → ρK ∗ for I = 3 / πK ∗ → ρK ∗ for I = 3 / ρK → ρK ∗ for I = 3 / πK ∗ → ρK for I = 3 /
2. Until now the temperaturedependence of the cross sections has not been studied in experiments or theory.In the next section, we present a potential of which the short-distance part is given byperturbative QCD and the large-distance part is displayed in the lattice gauge results ofRef. [25]. The Schr¨odinger equation with the potential is solved to get the quark-antiquarkrelative-motion wave function of a meson. A convenient framework for application of thetemperature-dependent potential to inelastic meson-meson scattering is the Born approx-imation for the quark-interchange processes. In Section 3 we review formulas of crosssections for meson-meson nonresonant reactions that are based on the Born approxima-tion. In Section 4 we show numerical results of cross sections and give relevant discussions.Parametrizations of the cross sections are given. Conclusions are in the last section.
2. Potential and wave functions of quark-antiquark relative motion
3n Ref. [25] the heavy quark potential was assumed to be equal to the free energy ofheavy quark-antiquark pair and the lattice calculations provided the temperature depen-dence of the potential. When the distance r between quark and antiquark is large enough,the potential at a given temperature exhibits a constant value and becomes a plateau.This value depends on the temperature T and decreases with increasing temperature.The constant value at large distances can be parametrized as V ab ( ~r ) = − ~λ a · ~λ b D " . − (cid:18) TT c (cid:19) , (1)with D = 0 . T c = 0 .
175 GeV. Here ~λ are the Gell-Mann matrices for thecolor generators. D is a fit parameter, but happens to equal the height of the plateau of T = 0 . T c .The relative color orientation of two constituents (quarks and antiquarks here) canbe indicated by wave functions of the two constituents. The expectation value of ~λ a · ~λ b depends on the relative color orientation, so does the heavy quark potential given in Eq.(1). The dependence of the potential on the relative color orientation is supported byhadronic physics and lattice calculations. We discuss evidence in the three cases: (a) T = 0, (b) T > T c and (c) 0 < T < T c .(a) Since the establishment of QCD the dependence of confinement with ~λ a · ~λ b on therelative color orientation has been widely used in hadronic physics, for instance, meson-baryon scattering [26] and meson-meson scattering [5]. The expectation value of ~λ a · ~λ b is negative in the color singlet state and positive in any color octet state. This complieswith the fact that the color singlet state is observed and the color octet states are notobserved.(b) When temperature is above T c , it is expected in Ref. [27] that the interaction inan antitriplet diquark system is only half as strong as in a color-singlet quark-antiquarksystem, i.e. obeys Casimir scaling, and the interaction depends on the relative colororientation.(c) When temperature is between 0 and T c , the situation is complicated. From Figs.15, 17 and 20 in Ref. [28], we can derive that the color-singlet free energy does not equal4he color-octet free energy at large distances in the region 0 . T c ≤ T < T c . It can alsobe expected from Fig. 3 of Ref. [27] that the interaction in an antitriplet diquark systemis nearly half as strong as in a color-singlet quark-antiquark system, i.e. the interactiondepends on the relative color orientation in the region 0 . T c ≤ T < T c .Much attention has been paid to construction of the color octet potential from per-turbative QCD and lattice data in the study of hybrid mesons. The color octet potentialrelies on the spatial extension of gluon field confined in hybrid mesons. In the stringdescription the symmetry of the gluon field is labeled by the spins Σ, Π and ∆ about theaxis connecting the quark and antiquark, a P C value and an additional reflection sym-metry for Σ states. It was obtained in vacuum that the color octet potential is differentwith respect to different symmetries of the gluon field [29, 30]. Temperature dependenceof the gluon field affects temperature dependence of the color octet potential. How doesthe gluon field confined in a meson depend on temperature at 0 < T < T c ? This is adifficult problem.In summary, we still lack of the lattice study of the color octet potential at 0 < T < T c and the heavy quark potential depends on the relative color orientation.The large-distance plateau of the heavy quark potential indicates the onset of stringbreaking at a certain distance [27]. The string breaking occurs when the energy stored inthe string exceeds the mass of a light quark-antiquark pair [31]. The combination of thelight quark and the heavy antiquark and the combination of the light antiquark and theheavy quark form open heavy flavors. When the temperature is higher, light mesons inmedium more effectively take a kind of flip-flop recoupling of quark constituents [32–34]and string breaking occurs at a shorter distance. Hence, the plateau appears at a shorterdistance and confinement gets weaker. Consequently, the wave function of the heavyquark-antiquark pair becomes wider in space.However, uncertainty of the heavy quark potential exists because the potential is de-rived from Polyakov loop correlation functions. The Polyakov loop at the space coordinate ~r is indicated by L ( ~r ). The Polyakov loop correlation functions are related to the free5nergy of heavy quark-antiquark pair F ( T, r ) − T ln < L ( ~r ) L + ( ~r ) > = F ( T, r ) + C (2)where r = | ~r − ~r | and C is a normalization constant. The free energy leads to theinternal energy by U ( T, r ) = F ( T, r ) +
T S ( T, r ) (3)The entropy S ( T, r ) = − ∂F ( T, r ) /∂T is independent of r at large distances and dependson r at the other distances. If T S is small, U ( T, r ) ≈ F ( T, r ); otherwise, the internalenergy deviates from the free energy. Taking the free energy as the heavy quark potentiallike Ref. [25] is an approximation. Then the internal energy was suggested in Refs. [35,36]as the heavy quark potential. Very recently the heavy quark potential is argued to be theinternal energy which is the expectation value of the difference of the Hamiltonians withand without the heavy quark-antiquark pair at rest [37,38], and this results in dissociationtemperatures of heavy quarkonia that agree quite well with the values from lattice studies.How to find the correct heavy quark potential is an important problem but has notbeen solved so far. It has been proposed that the heavy quark potential takes the form ξU ( T, r ) + (1 − ξ ) F ( T, r ) where the quantity ξ is between 0 and 1 and was determinedin models [39–45]. Since the present lattice calculations provide the internal energy U that includes the internal energy of the heavy quark-antiquark pair and the gluon in-ternal energy difference U g ( T, r ) − U g0 ( T ) where U g ( T, r ) and U g0 ( T ) correspond to thegluon internal energies in the presence and absence of the heavy quark-antiquark pair,respectively, it was proposed by Wong [41] that the heavy quark potential should be U ( T, r ) − [ U g ( T, r ) − U g0 ( T )]. In the local energy-density approximation that adopts anequation of state for the quark-gluon plasma, the heavy quark potential can be repre-sented as a linear combination of F and U . With F and U obtained in lattice calculationsin quenched QCD, the potential gives spontaneous dissociation temperatures of heavyquarkonia that agree with those obtained from spectral analyses in quenched QCD [43].This is an improvement in comparison to the use of F as a potential. The work of Wongimplies that the heavy quark potential properly defined gives reliable results. Even though6he method of Wong has not been applied to hadronic matter, we still expect that theheavy quark potential for hadronic matter, if obtained, can produce more reliable crosssections for the meson-meson nonresonant reactions in the nonzero temperature region ofhadronic matter than those from F .At very small distances r < .
01 fm, the potential obtained from one-gluon exchangeplus one- and two-loop corrections in perturbative QCD is [46] V ab ( ~r ) = ~λ a · ~λ b π rw (cid:20) (cid:18) γ E + 5375 (cid:19) w − ww (cid:21) , (4)where γ E is the Euler’s constant and w = ln(1 / Λ r ) with the QCD scale parameterΛ MS determined by Eq. (2.13) in Ref. [46].An interpolation between the constant confinement at large distances given in Eq.(1) and the spin-independent perturbative potential given in Eq. (4) produces a centralspin-independent potential V ab ( ~r ) = − ~λ a · ~λ b D " . − (cid:18) TT c (cid:19) tanh( Ar ) + ~λ a · ~λ b π v ( λr ) r exp( − Er ) , (5)where A = 1 . .
75 + 0 . TT c ) ] GeV , (6)and E = 0 . , (7)are fit parameters, and λ = p b / π α ′ , (8) α ′ = 1 .
04 GeV − is the Regge slope and b = 11 − N f with the quark flavor number N f = 4 [46]. This potential is different from the parametrizations given by Digal et al. [39]and Wong [47]. The potential contains the dimensionless function [46] v ( x ) = 4 b π Z ∞ dQQ ( ρ ( ~Q ) − K~Q ) sin( Qλ x ) , (9)with K = 3 / π α ′ , where Q is the absolute value of gluon momentum ~Q and ρ ( ~Q ) isgiven by Buchm¨uller and Tye [46]. The quantity ρ − K~Q arises from one-gluon exchange and7erturbative one- and two-loop corrections. exp( − Er ) is a medium modification factorto the potential of one-gluon exchange plus perturbative one- and two-loop corrections.The temperature correction to the one-gluon-exchange potential with the limit shownin Eq. (4) is the difference between the second term in Eq. (5) and the perturbativepotential ~λ a · ~λ b π v ( λr ) r [46]. The temperature correction is completely negligible at veryshort distances and obvious at intermediate and large distances.The parametrization in Eq. (5) versus lattice gauge results is plotted in Fig. 1. It isclearly seen that a lower plateau at large distances corresponds to a higher temperature.Plateaus at T /T c = 0 .
97, 0.94, 0.9, 0.84 approximately begin at r = 1 .
15 fm, 1.18 fm, 1.26fm, 1.38 fm, respectively. Hence, a higher plateau begins at a larger distance. Confinementcan be assumed to be flavor-independent in hadronic physics. For example, it was shown inRefs. [48–53] that quark-quark potentials with flavor-independent confinement, a Coulombterm and hyperfine interactions can consistently describe a large body of data like massesfrom light to heavy hadrons. The light hadrons may consist of only up and down quarksand the heavy hadrons may contain charm and bottom quarks. The flavor dependence ofthe quark-quark potentials is relevant to quark masses in the hyperfine interactions. Thesuccess of the potentials renders that the flavor independence of confinement is universaland the hyperfine interactions must be flavor-dependent. Therefore, the potential in Eq.(5) is reasonably flavor-independent. The potential obtained by Karsch et al. for heavyquarks is applied to light quarks. It is shown by the lattice calculations that screening setsin at distances r ≈ . r < .
01 fm originates from one-gluon exchange plus loop corrections in perturbativeQCD. The degree of freedom is removed at r < .
01 fm.The medium screening obtained in the lattice gauge calculations at present affectsonly the central spin-independent potential. This allows us to keep using the spin-spininteraction relevant to perturbative QCD, as done by Wong [47]. In our work, the spin-8pin interaction arises not only from one-gluon exchange but also from perturbative one-and two-loop corrections [6] V ss = − ~λ a · ~λ b π δ ( ~r ) ~s a · ~s b m a m b + ~λ a · ~λ b π
25 1 r d v ( λr ) dr ~s a · ~s b m a m b , (10)where ~s a and m a are the spin and mass of constituent quarks or antiquarks labeled as a ,respectively. This expression of the spin-spin interaction comes from the application of Eq.(7e) in the transformed Hamiltonian obtained by Chraplyvy [54] from the two-constituentHamiltonian that includes the relativistic potential originating from one-gluon exchangeplus perturbative one- and two-loop corrections [6]. The transformed Hamiltonian wasobtained from an application of the Foldy-Wouthuysen canonical transformation to arelativistic two-particle Hamiltonian. The spin-spin interaction is related to the terms ofthe direct product of two Dirac α matrices [55] in the relativistic potential.Given the masses m u = m d = 0 .
32 GeV for the up and down quarks, the Schr¨odingerequation with the central spin-independent potential in Eq. (5) is solved at T = 0 toobtain a radial wave function R q ¯ q ( T = 0 , r ) for the quark-antiquark relative motion of π and ρ mesons. Assuming all the mesons in the ground-state pseudoscalar octet andthe ground-state vector nonet taking the same spatial wave function of quark-antiquarkrelative motion as the π and ρ mesons, the spin-spin interaction leads to the mass splittingbetween a pseudoscalar meson and a vector meson with the same isospin quantum number < V ss > = 16 π m a m b [ R q ¯ q ( T = 0 , r = 0) − Z ∞ drr d v ( λr ) dr R q ¯ q ( T = 0 , r )] , (11)where R q ¯ q satisfies the normalization condition R ∞ drr R q ¯ q ( T, r ) = 1. At m s = 0 . m ρ − m π =0 . m K ∗ − m K = 0 . m ω + m φ − m η = 0 . , where m i ( i = π, ρ, K, K ∗ , η, ω, φ ) represent the masses of π, ρ, K, K ∗ , η, ω and φ , respectively. Thesemass splittings can be compared to the experimental values 0.6304 GeV, 0.3963 GeV and0.3930 GeV, respectively.The spin-averaged mass of a spin-0 meson and a spin-1 meson with the same isospinis one-fourth of the spin-0 meson mass plus three-fourths of the spin-1 meson mass [3]. In9able 1 we list vacuum masses of π , ρ , K and K ∗ , the spin-averaged mass m πρ of π and ρ and the one m KK ∗ of K and K ∗ . The theoretical values of m πρ , m KK ∗ , m ρ , m K and m K ∗ approach the corresponding experimental data. The pion mass from our calculationsalmost doubles the experimental value. These are understandable. The potential givenin Eq. (5) has the behavior of tanh( Ar ) at large distances and cannot mimic the linearconfinement. In vacuum the pion with the lightest mass among mesons has the largestradius and is sensitive to the potential behavior at large distances. Therefore, the exper-imental datum of pion mass must differ from the value derived from the potential. For ρ , K and K ∗ with masses larger than π , less sensitivity to the potential behavior at largedistances does not lead to a large separation of the experimental and theoretical masses.If the theoretical pion mass is used, cross sections for pions in collisions with mesonsare not reliable. Hence, the experimental values of meson masses are used in calculatingmeson-meson cross sections at T = 0 GeV in this work and, as will see, the cross sectionsare similar to those obtained from the potential with the linear confinement in Ref. [24].If the quark masses are set equal to zero, the QCD Hamiltonian is symmetric under thechiral group SU (3) × SU (3). Although the spontaneous breakdown of the chiral symmetryis widely expected, the problem of calculating the spontaneous chiral symmetry breakingin QCD has not yet been satisfactorily solved [56,57]. However, the spontaneous symmetrybreaking can be related to the quark condensate which vanishes in perturbative QCDand does not equal zero in the non-perturbative region. With the explicit breakdownof the chiral symmetry due to the current quark masses, the square of pion mass isproportional to the product of the quark condensate and the sum of the up and downquark masses at the lowest order in chiral perturbation theory. Therefore, the small pionmass comes from the small quark masses and the non-vanishing quark condensate in thenon-perturbative region where confinement sets in. While only the spontaneous symmetrybreaking occurs, Goldstone bosons and quarks are massless and the soft-pion theoremsobtained in the vanishing pion four-momentum are exact [58]. While the explicit chiralsymmetry breaking also occurs, bosons get masses and the soft-pion theorems becomeapproximate. The deviation of the theorems from the experimental data has been studied10n chiral perturbation theory and the soft-pion theorems have been corrected [59–62]. Inour work all the mesons in the ground-state pseudoscalar octet and the ground-statevector nonet are assumed to take the same spatial wave function of quark-antiquarkrelative motion as the pion and rho mesons, but the quark masses are the constituentmasses which are much larger than the current quark masses used in chiral perturbationtheory. The constituent quark masses show the explicit chiral symmetry breaking so thatmesons get masses and the soft-pion theorems are violated. The cross sections that wewill obtain from the constituent quark masses and the nonzero meson masses must deviatefrom the cross sections obtained from vanishing masses of quarks and mesons. Therefore,the deviation of our results from the soft-pion theorems should be expected.
3. Formulas for cross sections
Let m i and P i = ( E i , ~P i ) be the mass and the four-momentum of meson i ( i = q ¯ q , q ¯ q , q ¯ q , q ¯ q ), respectively. The Mandelstam variables for the reaction A ( q q ) + B ( q q ) → C ( q q ) + D ( q q ) are s = ( E q ¯ q + E q ¯ q ) − ( ~P q ¯ q + ~P q ¯ q ) and t = ( E q ¯ q − E q ¯ q ) − ( ~P q ¯ q − ~P q ¯ q ) . In the center-of-mass frame the meson A ( q q ) has the momentum ~P = ~P q ¯ q = − ~P q ¯ q and the meson C ( q q ) has the momentum ~P ′ = ~P q ¯ q = − ~P q ¯ q . ~P and ~P ′ are expressed in terms of s by | ~P ( √ s ) | = 14 s n(cid:2) s − (cid:0) m q ¯ q + m q ¯ q (cid:1)(cid:3) − m q ¯ q m q ¯ q o , (12) | ~P ′ ( √ s ) | = 14 s n(cid:2) s − (cid:0) m q ¯ q + m q ¯ q (cid:1)(cid:3) − m q ¯ q m q ¯ q o . (13)Denote the angle between ~P and ~P ′ by θ . The cross section for A ( q q ) + B ( q q ) → C ( q q ) + D ( q q ) is σ = 132 πs | ~P ′ ( √ s ) || ~P ( √ s ) | Z π dθ |M fi ( s, t ) | sin θ. (14)This formula provides the dependence on the total energy √ s of the two initial mesonsin the center-of-mass frame and is valid for the interchange of the two quarks ( q and q ) or of the two antiquarks (¯ q and ¯ q ). The interchange of quarks brings about twoforms of scattering that may lead to different values of the transition amplitude M fi . The11orms are known as the prior form and the post form [63–65]. Scattering in the prior formmeans that gluon exchange takes place prior to the quark or antiquark interchange. Thetransition amplitude in the prior form is [24] M priorfi = p E q ¯ q E q ¯ q E q ¯ q E q ¯ q Z d p q ¯ q (2 π ) d p q ¯ q (2 π ) ψ + q ¯ q ( ~p q ¯ q ) ψ + q ¯ q ( ~p q ¯ q )( V q ¯ q + V ¯ q q + V q q + V ¯ q ¯ q ) ψ q ¯ q ( ~p q ¯ q ) ψ q ¯ q ( ~p q ¯ q ) , (15)where ψ ab ( ~p ab ) is the product of color, spin, flavor and momentum-space wave functions ofthe relative motion of constituents a and b and satisfies R d p ab (2 π ) ψ + ab ( ~p ab ) ψ ab ( ~p ab ) = 1. Therelative momentum of a and b is ~p ab . Scattering in the post form means that the quarkor antiquark interchange is followed by gluon exchange. The transition amplitude in thepost form is [24] M postfi = p E q ¯ q E q ¯ q E q ¯ q E q ¯ q (cid:18)Z d p q ¯ q (2 π ) d p q ¯ q (2 π ) ψ + q ¯ q ( ~p q ¯ q ) ψ + q ¯ q ( ~p q ¯ q ) V q ¯ q ψ q ¯ q ( ~p q ¯ q ) ψ q ¯ q ( ~p q ¯ q )+ Z d p q ¯ q (2 π ) d p q ¯ q (2 π ) ψ + q ¯ q ( ~p q ¯ q ) ψ + q ¯ q ( ~p q ¯ q ) V ¯ q q ψ q ¯ q ( ~p q ¯ q ) ψ q ¯ q ( ~p q ¯ q )+ Z d p q ¯ q (2 π ) d p q ¯ q (2 π ) ψ + q ¯ q ( ~p q ¯ q ) ψ + q ¯ q ( ~p q ¯ q ) V q q ψ q ¯ q ( ~p q ¯ q ) ψ q ¯ q ( ~p q ¯ q )+ Z d p q ¯ q (2 π ) d p q ¯ q (2 π ) ψ + q ¯ q ( ~p q ¯ q ) ψ + q ¯ q ( ~p q ¯ q ) V ¯ q ¯ q ψ q ¯ q ( ~p q ¯ q ) ψ q ¯ q ( ~p q ¯ q ) (cid:19) . (16)The transition amplitudes in the prior form and in the post form are equal to oneanother when the potential and the wave function of quark-antiquark relative motion arethose used in the Schr¨odinger equation [63–65]. Otherwise, M priorfi = M postfi for inelasticscattering. The inequality yields different cross sections corresponding to the two forms,which is the so-called post-prior discrepancy [63–65].In the Schr¨odinger equation we have only used the central spin-independent potentialin Eq. (5) since the spin-spin interaction in Eq. (10) contains the delta function thatcan not be correctly dealt with in the equation. But in the transition amplitudes wherethe Fourier transform of the spin-spin interaction can be correctly dealt with, we use12he Fourier transform of both the central spin-independent potential and the spin-spininteraction: V ab (cid:16) ~Q (cid:17) = − ~λ a · ~λ b D " . − (cid:18) TT c (cid:19) (2 π ) δ ( ~Q ) − πQ Z ∞ dr r sin( Qr )exp(2 Ar ) + 1 (cid:21) + ~λ a · ~λ b πE Z ∞ dq ρ ( q ) − Kq ( E + Q + q ) − Q q − ~λ a · ~λ b π ~s a · ~s b m a m b + ~λ a · ~λ b π λ Q Z ∞ dx d v ( x ) dx sin (cid:18) Qλ x (cid:19) ~s a · ~s b m a m b . (17)Therefore, the post-prior discrepancy occurs in our calculations. We take the averageof the cross section in the prior form and the one in the post form. Each of the twocross sections related to the prior form and the post form is the unpolarized cross sectionobtained from the cross section in Eq. (14) σ unpol ( √ s ) = 1(2 S A + 1)(2 S B + 1) X S (2 S + 1) σ ( S, m S , √ s ) , (18)where S A and S B are the spins of A and B , respectively, and S is the total spin of thetwo mesons allowed by the reaction A + B → C + D . The cross section σ unpol ( √ s ) isindependent of the magnetic projection quantum number m S of S .
4. Numerical results and discussions
The quark masses m u = m d = 0 .
32 GeV and m s = 0 . S -wave solution of the Schr¨odingerequation is a temperature-dependent radial wave function of the quark-antiquark rela-tive motion of mesons in the ground-state pseudoscalar octet and the ground-state vectornonet. When the temperature increases, the peak of r times the S -wave radial wave func-tion, rR q ¯ q ( T, r ) , moves to larger quark-antiquark distances and the meson’s root-mean-square radius increases. This reflects the phenomenon that with increasing temperatureany bound state becomes looser and looser while confinement gets weaker, i.e. the poten-13ial plateau at large distances decreases with increasing temperature and a higher plateaubegins at a larger distance.Another consequence of the temperature-dependent potential in Eq. (5) is that massesof π , ρ , K and K ∗ decrease with increasing temperature in the region 0 . ≤ T /T c ≤ . < V ss > = 16 π m a m b [ R q ¯ q ( T, r = 0) − Z ∞ drr d v ( λr ) dr R q ¯ q ( T, r )] . (19)The spin-averaged mass of a spin-0 meson and a spin-1 meson with the same isospinis one-fourth of the spin-0 meson mass plus three-fourths of the spin-1 meson mass [3].The spin-averaged mass of π and ρ equals the sum of quark mass, antiquark mass andthe energy of the relative motion obtained from the Schr¨odinger equation. Since for thequark-antiquark relative motion we take the same spatial wave functions of K and K ∗ asthat of π and ρ mesons, the spin-averaged mass of K and K ∗ equals the sum of quarkmass, antiquark mass and the nonrelativistic-Hamiltonian expectation value of the wavefunction. After mass splittings and spin-averaged masses are obtained, we find mesonmasses of which the mass of the spin-0 meson is the spin-averaged mass minus three-fourths of the mass splitting of the spin-0 and spin-1 mesons, and the mass of the spin-1meson is the spin-averaged mass plus one-fourth of the mass splitting. The temperature-dependent meson masses are plotted in Fig. 2. The reason for the falloff of masses withincreasing temperature is that the first term of the potential related to the large-distanceplateau gives a smaller contribution at higher temperature. When T → T c , the masses ofthe π and ρ mesons approach 0 and 0.006 GeV, respectively. This indicates that the π and ρ mesons are almost massless at a temperature very close to T c . In addition to thefalloff of masses, the mass splitting of π and ρ and the one of K and K ∗ are both close tozero at T → T c . When T → T c , the masses of K and K ∗ approach finite values 0.179 GeVand 0.183 GeV, respectively. The kaons and vector kaons become degenerate in mass near T c . Similar to Ref. [3], a constant of 0.88857 GeV is subtracted from the energy of theSchr¨odinger equation for π and ρ and from the nonrelativistic-Hamiltonian expectation14alue of the wave function for K and K ∗ . This subtraction makes the theoretical valuesof the spin-averaged masses at T = 0 approach the experimental data and the masses of π and ρ go to zero at T → T c . The subtraction does not influence the sizes of mesonbound states and the mass splittings.The meson masses in units of GeV are parametrized as m π = 0 . " − (cid:18) T . T c (cid:19) . . , (20) m ρ = 0 . " − (cid:18) TT c (cid:19) . . , (21) m K = 0 . " − (cid:18) T . T c (cid:19) . . , (22) m K ∗ = 0 . " − (cid:18) T . T c (cid:19) . , (23)which are valid in the region 0 . ≤ T /T c ≤ .
99. Since the temperature of hadronicmatter is generally larger than 0.11 GeV and smaller than T c , the meson masses and crosssections shown in the next paragraph in the temperature region are sufficient for studiesconcerned with hadronic matter.The wave function of quark-antiquark relative motion of the π and ρ mesons is takento be the same as those of the other ground-state mesons. With m u = m d = 0 .
32 GeVobtained in Section 2, the experimental data of S -wave I = 2 elastic phase shifts for ππ scattering in vacuum [66–69] are reproduced with the potential in Eq. (17). Keepingquark masses at the values determined in the fit to the experimental mass splittings,the dependence of the transition amplitude M fi on temperature comes from the wavefunction, the potential in Eq. (17) and the meson masses. The transition amplitudecontains also color, spin and flavor matrices that are not affected by the temperature.Since the meson masses depend on temperature, threshold energies for inelastic meson-meson scattering depend on temperature. The temperature dependence of the potential,the quark-antiquark wave function and the meson masses leads to temperature-dependentcross sections for inelastic meson-meson scattering. Unpolarized cross sections for the15even reactions ππ → ρρ for I = 2, KK → K ∗ K ∗ for I = 1, KK ∗ → K ∗ K ∗ for I = 1, πK → ρK ∗ for I = 3 / πK ∗ → ρK ∗ for I = 3 / ρK → ρK ∗ for I = 3 /
2, and πK ∗ → ρK for I = 3 / σ unpol = a (cid:18) √ s − √ s b (cid:19) c exp (cid:20) c (cid:18) − √ s − √ s b (cid:19)(cid:21) + a (cid:18) √ s − √ s b (cid:19) c exp (cid:20) c (cid:18) − √ s − √ s b (cid:19)(cid:21) . (24)Time-consuming computations determine values of the parameters a , b , c , a , b and c , which are shown in Tables 2 and 3.At the threshold energy of an endothermic reaction at a given temperature the mo-menta of final mesons in the center-of-mass frame equal zero. It is shown by Eqs. (12)and (13) that the absolute values of the momenta of initial and final mesons ( | ~P | and | ~P ′ | ) increase as √ s increases. The rise of | ~P ′ | causes a rapid increase of the crosssection close to the threshold energy. The relative momentum ~p ab is a linear combinationof ~P and ~P ′ . Thus, | ~p ab | increases with increasing √ s , while ψ ab ( ~p ab ) is reduced by theincrease of | ~p ab | . The wave function ψ ab ( ~p ab ) and the absolute value of the transitionamplitude | M fi | thus decrease with increasing √ s . This decreases the cross section withincreasing √ s . The rising | ~P ′ | and falling ψ ab ( ~p ab ) produce a peak in the cross sectionnear the threshold energy. At a higher temperature the constituents a and b have largersupport of relative motion in coordinate space and so ψ ab ( ~p ab ) gets narrower in momentumspace. This results in a cross section that decreases faster from the peak and forms a morenarrow peak. The exception is that the width of the peak of a reaction at T /T c = 0 .
95 isequal to or slightly larger than the one of the same reaction at
T /T c = 0 . T /T c = 0 to 0.65 but decrease from T /T c = 0 . T /T c = 0 to 0.65 as the radii of initial mesons increase.While temperature goes to a higher value, bound states of mesons become looser due to16eaker confinement. At a temperature near T c the bound states are very loose. Eventhough mesons are easily broken in the reaction A ( q q ) + B ( q q ) → C ( q q ) + D ( q q )as T /T c →
1, it is more difficult to combine final quarks and antiquarks into mesonsthrough quark rearrangement. Hence, peak cross sections decrease from
T /T c = 0 . ππ → ρρ for I = 2 , KK → K ∗ K ∗ for I = 1 and πK → ρK ∗ for I = 3 / T /T c = 0 . πK ∗ → ρK for I = 3 / T /T c = 0 .
85. The third class is comprised of thethree reactions KK ∗ → K ∗ K ∗ for I = 1 , πK ∗ → ρK ∗ for I = 3 / ρK → ρK ∗ for I = 3 / T /T c = 0 .
65 or 0.75. In the firstor third class the meson-flavor dependence of masses distinguishes the peak values of thethree reactions at a given temperature. In the first class the difference of the total massof the initial mesons and the total mass of the final mesons is larger than the ones in theother classes. When
T /T c changes from 0.6 to 0.99, the mass difference of the first classis reduced faster than the other two classes. The quicker reduction causes a more rapidincrease of the peak value so that another maximum of the peak cross sections appearsat T /T c = 0 . T /T c = 0 . r < .
01 fm and the lattice gauge results at large distances. Theinterpolation is in fact this procedure of adjusting the parameters A and E . We nowbegin to examine the sensitivity of mass splittings at T = 0 GeV, temperature-dependent17eson masses and cross sections to the interpolation procedure by adopting two new setsof A and E . The first set (named Set I) is obtained by reducing A in Eq. (6) by 5%and increasing E in Eq. (7) by 10% while the second set (named Set II) is obtained byincreasing A in Eq. (6) by 5% and reducing E in Eq. (7) by 10%. The two new sets givegood fits to the lattice gauge results. Mass splittings at T = 0 GeV and meson massesat five temperatures resulted from the two sets are listed in Tables 4 and 5, respectively.The change of the mass splittings or of the meson masses from Eqs. (6) and (7) to Set Ior Set II is very small. The largest change in meson mass is 2.88% at T = 0 . T c . Butsuch a change only leads to negligible changes in cross section. For example, the peakcross section obtained from Set I for KK → K ∗ K ∗ for I = 1 at T /T c = 0 .
65 is 1.2735mb in comparison to the value 1.2686 mb obtained from Eqs. (6) and (7), and the peakcross section obtained from Set II for πK → ρK ∗ for I = 3 / T /T c = 0 .
65 is 0.4787mb very close to the value 0.4930 mb resulted from Eqs. (6) and (7). Eventually, weunderstand that the mass splittings at T = 0 GeV, the temperature-dependent massesand the cross sections are not sensitive to the interpolation procedure for the constructionof the potential given in Eq. (5).In Figs. 3-9 we plot the average of the cross section in the prior form and the one inthe post form. To see the uncertainty in this prescription, as an example, we draw in Fig.10 cross sections obtained in the prior form and in the post form for the reaction ππ → ρρ for I = 2. The solid curves in Fig. 3 are between the dashed curves corresponding to theprior form and the dotted curves corresponding to the post form. At a given temperaturethe dashed curve and the dotted curve almost overlap at the center-of-mass energies veryclose to or far away from the threshold energy. The post-prior discrepancy can be clearlymarked by the difference of peak cross sections obtained in the two forms. We thus list thepeak cross sections in Tables 6-8. The three tables are enough to display the discrepancy.Denote the peak cross section obtained in the prior (post) form by σ priormax ( σ postmax ). Toindicate the discrepancy, we define χ = σ priormax − σ postmax σ priormax + σ postmax (25)18ince σ priormax > σ postmax >
0, the values of χ are between -1 and 1. The smaller theabsolute value of χ , the smaller the discrepancy. If χ is positive, σ priormax > σ postmax ; otherwise, σ priormax ≤ σ postmax . The quantity χ is also presented in Tables 6-8. From Fig. 10 we notethat the peak cross sections obtained in the two forms at a given temperature may notcorrespond to the same center-of-mass energy.
5. Summary
We have given the temperature-dependent central-spin-independent potential that in-terpolates between the perturbative-QCD potential with loop corrections at short dis-tances and the potential data offered by lattice gauge calculations at large distances. Fromthe potential we obtain: (1) experimental mass splittings of the ground-state mesons withthe same isospin when the masses of up, down and strange quarks are determined; (2)meson masses that decrease from
T /T c = 0 . ππ → ρρ for I = 2, KK → K ∗ K ∗ for I = 1, KK ∗ → K ∗ K ∗ for I = 1, πK → ρK ∗ for I = 3 / πK ∗ → ρK ∗ for I = 3 / ρK → ρK ∗ for I = 3 / πK ∗ → ρK for I = 3 /
2. The temperature dependence of the cross sections is determined by the tem-perature dependence of the potential, the quark-antiquark wave function and the mesonmasses. Peak cross sections are affected by three factors: larger sizes of initial mesons ata higher temperature give larger peak cross sections, looser bound states of final mesonsat a higher temperature lead to smaller peak cross sections, and a smaller total-massdifference of the initial mesons and the final mesons yields larger peak cross sections. Thenumerical cross sections are parametrized for future studies.
Acknowledgements
This work was supported in part by the National Natural Science Foundation of Chinaunder Grant No. 10675079 and in part by Shanghai Leading Academic Discipline Project19project number S30105). We thank H. J. Weber for a careful reading of the manuscript.20 eferences [1] T. Barnes, E.S. Swanson, Phys. Rev. D 46 (1992) 131.[2] E.S. Swanson, Ann. Phys. (N.Y.) 220 (1992) 73.[3] K. Martins, D. Blaschke, E. Quack, Phys. Rev. C 51 (1995) 2723.[4] C.-Y. Wong, E.S. Swanson, T. Barnes, Phys. Rev. C 65 (2001) 014903.[5] T. Barnes, E.S. Swanson, C.-Y. Wong, X.-M. Xu, Phys. Rev. C 68 (2003) 014903.[6] X.-M. Xu, Nucl. Phys. A 697 (2002) 825.[7] D. Kharzeev, H. Satz, Phys. Lett. B 334 (1994) 155.[8] M.E. Peskin, Nucl. Phys. B 156 (1979) 365.[9] G. Bhanot, M.E. Peskin, Nucl. Phys. B 156 (1979) 391.[10] F.D. Duraes, et al., Phys. Rev. C 68 (2003) 035208.[11] S.G. Matinyan, B. M¨uller, Phys. Rev. C 58 (1998) 2994.[12] Z. Lin, C.M. Ko, J. Phys. G 27 (2001) 617.[13] K. Haglin, Phys. Rev. C 61 (2000) 031902.[14] K. Haglin, C. Gale, Phys. Rev. C 63 (2001) 065201.[15] Y. Oh, T. Song, S.H. Lee, Phys. Rev. C 63 (2001) 034901.[16] S.S. Adler, et al., PHENIX Collaboration, Phys. Rev. C 69 (2004) 034909.[17] I. Arsene, et al., BRAHMS Collaboration, Nucl. Phys. A 757 (2005) 1.[18] B.B. Back, et al., PHOBOS Collaboration, Nucl. Phys. A 757 (2005) 28.[19] J. Adams, et al., STAR Collaboration, Nucl. Phys. A 757 (2005) 102.[20] K. Adcox, et al., PHENIX Collaboration, Nucl. Phys. A 757 (2005) 184.[21] J. Adams, et al., STAR Collaboration, Phys. Rev. Lett. 92 (2004) 092301.[22] I.G. Bearden, et al., BRAHMS Collaboration, Phys. Rev. Lett. 90 (2003) 102301.[23] I.G. Bearden, et al., BRAHMS Collaboration, Phys. Rev. Lett. 94 (2005) 162301.[24] Y.-Q. Li, X.-M. Xu, Nucl. Phys. A 794 (2007) 210.[25] F. Karsch, E. Laermann, A. Peikert, Nucl. Phys. B 605 (2001) 579.2126] I. Bender, H.G. Dosch, H.J. Pirner, H.G. Kruse, Nucl. Phys. A 414 (1984) 359.[27] M. D¨oring, K. H¨ubner, O. Kaczmarek, F. Karsch, Phys. Rev. D 75 (2007) 054504.[28] S. Gupta, K. H¨ubner, O. Kaczmarek, Phys. Rev. D 77 (2008) 034503.[29] C. Michael, arXiv:hep-ph/9809211.[30] G.S. Bali, A. Pineda, Phys. Rev. D 69 (2004) 094001.[31] T. Sj¨ostrand, Comput. Phys. Commun. 39 (1986) 347.[32] H. Miyazawa, Phys. Rev. D 20 (1979) 2953.[33] V. Goloviznin, H. Satz, Yad. Fiz. 60N3 (1997) 523.[34] H. Satz, hep-ph/0602245.[35] O. Kaczmarek, F. Karsch, P. Petreczky, F. Zantow, Phys. Lett. B 543 (2002) 41.[36] F. Zantow, O. Kaczmarek, F. Karsch, P. Petreczky, hep-lat/0301015.[37] H. Satz, arXiv:0812.3829.[38] L. Kluberg, H. Satz, arXiv:0901.3831.[39] S. Digal, P. Petreczky, H. Satz, Phys. Lett. B 514 (2001) 57.[40] E. Shuryak, I. Zahed, Phys. Rev. D 70 (2004) 054507.[41] C.-Y. Wong, Phys. Rev. C 72 (2005) 034906.[42] C.-Y. Wong, hep-ph/0509088.[43] C.-Y. Wong, Phys. Rev. C 76 (2007) 014902.[44] W.M. Alberico, A. Beraudo, A. De Pace, A. Molinari, Phys. Rev. D 72 (2005) 114011.[45] H. Satz, J. Phys. G 32 (2006) R25.[46] W. Buchm¨uller, S.-H.H. Tye, Phys. Rev. D 24 (1981) 132.[47] C.-Y. Wong, Phys. Rev. C 65 (2002) 034902.[48] A. De R´ujula, H. Georgi, S.L. Glashow, Phys. Rev. D 12 (1975) 147.[49] N. Isgur, G. Karl, Phys. Rev. D 18 (1978) 4187.[50] N. Isgur, G. Karl, Phys. Rev. D 19 (1979) 2653.[51] N. Isgur, G. Karl, Phys. Rev. D 20 (1979) 1191.2252] S. Godfrey, N. Isgur, Phys. Rev. D 32 (1985) 189.[53] S. Capstick, N. Isgur, Phys. Rev. D 34 (1986) 2809.[54] Z.V. Chraplyvy, Phys. Rev. 91 (1953) 388.[55] J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York,1964.[56] S. Weinberg, The Quantum Theory of Fields, Vol. II, Cambridge University Press,Cambridge, 1996.[57] S. Pokorski, Gauge Field Theories, Cambridge University Press, Cambridge, 2000.[58] J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model, Cam-bridge University Press, Cambridge, 1992.[59] J. Gasser, H. Leutwyler, Ann. Phys. 158 (1984) 142.[60] J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465.[61] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M.E. Sainio, Nucl. Phys. B 508 (1997)263.[62] G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603 (2001) 125.[63] N.F. Mott, H.S.W. Massey, The Theory of Atomic Collisions, Clarendon Press, Ox-ford, 1965.[64] T. Barnes, N. Black, E.S. Swanson, Phys. Rev. C 63 (2001) 025204.[65] C.-Y. Wong, H.W. Crater, Phys. Rev. C 63 (2001) 044907.[66] E. Colton, et al., Phys. Rev. D 3 (1971) 2028.[67] N.B. Durusoy, et al., Phys. Lett. B 45 (1973) 517.[68] W. Hoogland, et al., Nucl. Phys. B 126 (1977) 109.[69] M.J. Losty, et al., Nucl. Phys. B 69 (1974) 185.23able 1: Vacuum masses of π , ρ , K and K ∗ and their spin-averaged masses. m πρ (GeV) m π (GeV) m ρ (GeV) m KK ∗ (GeV) m K (GeV) m K ∗ (GeV)model 0.7112 0.2620 0.8609 0.8339 0.5465 0.9298experiment 0.6124 0.1396 0.7700 0.7929 0.4957 0.892024able 2: Values of parameters in the parametrization given in Eq. (24).reaction T /T c a (mb) b (GeV) c a (mb) b (GeV) c I = 2 ππ → ρρ I = 2 ππ → ρρ I = 2 ππ → ρρ I = 2 ππ → ρρ I = 2 ππ → ρρ I = 1 KK → K ∗ K ∗ I = 1 KK → K ∗ K ∗ I = 1 KK → K ∗ K ∗ I = 1 KK → K ∗ K ∗ I = 1 KK → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ T /T c a (mb) b (GeV) c a (mb) b (GeV) c I = πK → ρK ∗ I = πK → ρK ∗ I = πK → ρK ∗ I = πK → ρK ∗ I = πK → ρK ∗ I = πK ∗ → ρK ∗ I = πK ∗ → ρK ∗ I = πK ∗ → ρK ∗ I = πK ∗ → ρK ∗ I = πK ∗ → ρK ∗ I = ρK → ρK ∗ I = ρK → ρK ∗ I = ρK → ρK ∗ I = ρK → ρK ∗ I = ρK → ρK ∗ I = πK ∗ → ρK I = πK ∗ → ρK I = πK ∗ → ρK I = πK ∗ → ρK I = πK ∗ → ρK m ρ − m π m K ∗ − m K m ω + m φ − m η Eqs. (6) and (7) 0.5989 0.3833 0.3622Set I 0.5890 0.3770 0.3530Set II 0.6090 0.3898 0.366627able 5: Masses of π , ρ , K and K ∗ in units of GeV at various temperatures. T /T c m π m ρ m K m K ∗ Eqs. (6) and (7) 0.65 0.4042 0.5776 0.6161 0.7271Eqs. (6) and (7) 0.75 0.3839 0.4710 0.5789 0.6347Eqs. (6) and (7) 0.85 0.2958 0.3183 0.4776 0.4921Eqs. (6) and (7) 0.9 0.2105 0.2221 0.3902 0.3976Eqs. (6) and (7) 0.95 0.1021 0.1095 0.2809 0.2857Set I 0.65 0.3926 0.5761 0.6063 0.7237Set I 0.75 0.3775 0.4708 0.5736 0.6334Set I 0.85 0.2956 0.3182 0.4774 0.4919Set I 0.9 0.2110 0.2221 0.3905 0.3976Set I 0.95 0.1026 0.1095 0.2813 0.2858Set II 0.65 0.4118 0.5787 0.6162 0.7230Set II 0.75 0.3873 0.4713 0.5791 0.6328Set II 0.85 0.2952 0.3184 0.4772 0.4920Set II 0.9 0.2098 0.2221 0.3899 0.3978Set II 0.95 0.1014 0.1095 0.2807 0.285928able 6: Post-prior discrepancyreaction
T /T c σ priormax (mb) σ postmax (mb) χI = 2 ππ → ρρ I = 2 ππ → ρρ I = 2 ππ → ρρ I = 2 ππ → ρρ I = 2 ππ → ρρ I = 2 ππ → ρρ I = 1 KK → K ∗ K ∗ I = 1 KK → K ∗ K ∗ I = 1 KK → K ∗ K ∗ I = 1 KK → K ∗ K ∗ I = 1 KK → K ∗ K ∗ I = 1 KK → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ I = 1 KK ∗ → K ∗ K ∗ T /T c σ priormax (mb) σ postmax (mb) χI = πK → ρK ∗ I = πK → ρK ∗ I = πK → ρK ∗ I = πK → ρK ∗ I = πK → ρK ∗ I = πK → ρK ∗ I = πK ∗ → ρK ∗ I = πK ∗ → ρK ∗ I = πK ∗ → ρK ∗ I = πK ∗ → ρK ∗ I = πK ∗ → ρK ∗ I = πK ∗ → ρK ∗ I = ρK → ρK ∗ I = ρK → ρK ∗ I = ρK → ρK ∗ I = ρK → ρK ∗ I = ρK → ρK ∗ I = ρK → ρK ∗ T /T c σ priormax (mb) σ postmax (mb) χI = πK ∗ → ρK I = πK ∗ → ρK I = πK ∗ → ρK I = πK ∗ → ρK I = πK ∗ → ρK I = πK ∗ → ρK κ V (r) / κ / Figure 1: Temperature-dependent potential. From top to bottom the temperatures corre-sponding to the different data sets [25] are
T /T c = 0 . , . , . , . , . , . , . √ κ = 2 .
154 fm − = 0 .
425 GeV. 32 .6 0.7 0.8 0.9 1T/T c m a ss ( G e V ) m π m ρ m K m K* Figure 2: Meson masses as functions of