T-dependence of pseudoscalar and scalar correlations P. Maris,† C.D. Roberts,∗ S.M. Schmidt∗ and P.C. Tandy† †Center for Nuclear Research, Department of Physics, Kent State University, Kent OH 44242-0001 ∗Physics Division, Argonne National Laboratory, Argonne IL 60439-4843 InhomogeneouspseudoscalarandscalarBethe-Salpeterequationssolvedusingarenormalisation- group-improvedrainbow-laddertruncationexhibitboundstatepolesbelowandaboveT ,thecritical c temperature for chiral symmetry restoration. Above T the bound state amplitudes are identical, c as are the positions and residues of the pseudoscalar and scalar poles in the vertices. In the chiral limit theπ0 →γγ coupling vanishes at T , as do f , m , g . Forlight current-quarkmasses the c π σ σππ 2π decay channel of the isoscalar-scalar meson remains open until very near T , and the widths of c thedominant pion decay modes remain significant in the vicinity of the crossover. Pacs Numbers: 11.10.Wx, 11.30.Rd, 12.38.Mh, 24.85.+p 0 0 0 I. INTRODUCTION is observedin numericalsimulations of lattice-QCD[12], 2 couldbeattributedtoapersistenceintotheQGPofnon- n Numerical simulations of 2-light-flavour lattice-QCD, perturbativeeffectsinthequark’svectorselfenergy[13].1 a J andthestudyofmodelsthataccuratelydescribedynami- Also, using a more sophisticated model, the present in- 1 calchiralsymmetrybreakingandtheπ-ρmass-difference compatibilitybetweenlatticeestimatesofthecriticalex- 3 atT =0bothindicatethataquark-gluonplasma(QGP) ponents[1]wasidentifiedaslikelyanartefactofworking is reachedviaasecondordertransitionata criticaltem- too far from the chiral limit [17]. 1 perature T ≈ 0.15GeV [1–3]. This QGP existed as a The calculation of the T-dependence of hadron prop- c v stageintheearlyevolutionoftheuniverseanditsterres- ertieshashithertousedonlysimplemembersofthisclass 4 trial recreation is a primary goal of current-generation [10,18,19]. Herein we employ a version that provides for 6 ultrarelativistic heavy-ion experiments. The plasma is renormalisability and the correct one-loop renormalisa- 0 1 characterisedby the free propagationof quarks and glu- tiongroupevolutionofscale-dependentmatrixelements, 0 ons over distances ∼ 10-times larger than the proton. andfocus onscalarandpseudoscalarproperties. Mesons 0 However, its formation can only be observed indirectly appear as simple poles in 3-point vertices. Importantly, 0 by searching for a modification of particle yields and/or however, these vertices also provide information about h/ hadron properties in the debris of the collisions. thepersistenceofcorrelationsawayfromtheboundstate t The plasma phase is also characterised by chiral sym- pole; e.g., Refs. [20,21], which can be useful in studying - l metryrestoration. Hence,sincethepropertiesofthepion the T-evolutionof a system with deconfinement. Conse- c (mass, decay constant, other vertex residues, etc.) are quently our starting point is the inhomogeneous Bethe- u n tied to the dynamical breaking of chiral symmetry, an Salpeter equation (BSE). We then employ the homoge- : elucidation of the T-dependence of these properties is neous equation when appropriate and useful. v important; particularly since a prodigious number of pi- In Sec. II we describe the T 6= 0 quark DSE and in- i X onsisproducedinheavyioncollisions. Alsoimportantis homogeneous BSEs, and introduce our model. This also r understanding theT-dependence ofthe propertiesofthe servestomakeclearournotation. SectionIIIreportsour a scalar analogues and chiral partners of the pion in the results for the T-dependence of scalar and pseudoscalar strong interaction spectrum. For example, should the correlations while Sec. IV is a brief recapitulation and mass of a putative light isoscalar-scalar meson [4–7] fall epilogue. An appendix contains selected formulae. below 2m , the strong decay into a two pion final state π can no longer provide its dominant decay mode. In this II. DYSON-SCHWINGER AND case electroweak processes will be the only open decay BETHE-SALPETER EQUATIONS channels below T and the state will appear as a narrow c resonance. Analogous statements are true of isovector- scalar mesons. The renormalised quark DSE is The Dyson-Schwinger equations (DSEs) [8] provide S−1(p ):=i~γ·~pA(p )+iγ ω C(p )+B(p ) (1) a nonperturbative, continuum framework for analysing ωk ωk 4 k ωk ωk quantum field theories and the class of rainbow-ladder truncation DSE models yields a qualitative understand- ing of the thermodynamic properties of the QGP phase transition at T 6= 0 [9] (and also at µ 6= 0 [9–11]). For 1T = 0 DSE studies characteristically yield momentum- example, using a simple element of this class the pres- dependent vector and scalar quark self energies that remain sure’s slow approach to its ultrarelativistic limit, which largeuntilk2 =1GeV2;e.g., Fig.8ofRef.[14]andRef.[15]. These features are also observed in lattice simulations [16]. 1 =ZAi~γ·~p+Z (iγ ω +m ) +Σ′(p ), (2) µ,ν = 1,...,4, and Ω := 2πkT. A Debye mass for the 2 2 4 k bm ωk k gluon appears as a T-dependent contribution to ∆ . F where pωk := (p~,ωk) with ωk = (2k + 1)πT the The ultraviolet behaviour of the kernel in Eq. (4) is fermionMatsubarafrequency,andmbmistheLagrangian fixed by perturbative QCD because the DSEs yield per- current-quark bare mass. The regularisedself energy is turbation theory in the weak coupling limit. Our model is defined by specifying a form for the kernel’s infrared Σ′(p )=i~γ·p~Σ′ (p )+iγ ω Σ′ (p )+Σ′ (p ), ωk A ωk 4 k C ωk B ωk behaviour: (3) Λ¯ ∆F(pΩk) = D(pΩk;mg), ∆G(pΩk)=D(pΩk;0), (12) Σ′F(pωk)=Zl,q 34g2Dµν(p~−~q,ωk−ωl) D(pΩk;mg):=2π2D 2Tπδ0kδ3(p~)+DM(pΩk;mg), (13) ×14tr[PFγµS(qωl)Γν(qωl;pωk)] , (4) where D =(0.881GeV)2 is a mass-scale parameter and where: F =A,B,C; A,B,C are functions of (|p~|2,ωk2); DM(pΩk;mg)= 4ωπ62DsΩke−sΩk/ω2 PA :=−(Z1A/|p~|2)i~γ·p~,PB :=Z1,PC :=−(Z1/ωk)iγ4; 8π2γm 1−e−sΩk/(4m2t) + , (14) (5) 2 s ln τ + 1+s /Λ2 Ωk Ωk QCD and Λ¯ := T ∞ Λ¯d3q/(2π)3, with Λ¯ represent- (cid:20) (cid:16) (cid:17) (cid:21) itneggraaRll,tqarnadnsΛl¯attiPohnela=rl−leyg∞uilnRavraisraiatniotnremgauslsa-rsicsaalteiRo.nTohfethreenoinr-- wmitth=sΩ0k.5G=eVp,2Ωγkm+=m2g1,2/τ25,=m2ge2=−(11,6/ω5)π=2T12.,2amntd, malised self energies are ΛNf=4 = 0.234GeV. This model, which is motivated by QCD Refs. [23,24], incorporates the one-loop logarithmic sup- F(p ;ζ) = ξ +Σ′ (p ;Λ¯)−Σ′ (ζ−;Λ¯), (6) ωk F F ωk F ω0 pression identified in perturbative calculations. Its pa- ζ is the renormalisation point, (ζ−)2 := ζ2 −ω2, ξ = rameters were fixed at T =0 by fitting a range of π and ω0 0 A K meson properties: a renormalisation point invariant 1=ξ , and ξ =m (ζ). C B R light current-quark mass mˆ = 5.7MeV, correspond- u,d ing to m (1GeV) = 4.8MeV, gives m = 0.14GeV, R π A. The Model fπ =0.092MeV. The model exhibits a second order chi- ral symmetry restoring transition at [17] Γν(qωl;pωk) in Eq. (4) is the renormalised dressed- Tc =0.15GeV. (15) quark-gluonvertex. Itisaconnected,irreducible3-point function that should not exhibit light-cone singularities in covariant gauges [22]. A number of Ansa¨tze with this property have been proposed and it has become clear B. Pseudoscalar Channel that the judicious use of the rainbow truncation Ward-Takahashi identities relate the 3-point vector Γν(qωl;pωk)=γν (7) and axial-vector vertices to the dressed-quark propaga- tor. Hence, once a truncation of the kernel in the quark inLandaugaugeprovidesphenomenologicallyreliablere- DSE has been selected, requiring the preservation of sults so we employ it herein. A mutually consistent con- these identities constrains the kernel in the BSE. This straint is isexploredinRef.[25],whereasystematicprocedurefor Z =Z and ZA =ZA. (8) constructing the kernels is introduced that ensures the 1 2 1 2 order-by-orderpreservation of these identities. The rainbow truncation is the leading term in a 1/N Using this procedure the inhomogeneous BSE for the c expansion of Γ (q ;p ). zerothMatsubaramode ofthe isovector0−+ vertex con- ν ωl ωk D (p )istherenormaliseddressed-gluonpropagator sistent with the rainbow truncation of Eq. (2) is µν Ωk g2Dµν(pΩk)=PµLν(pΩk)∆F(pΩk)+PµTν(pΩk)∆G(pΩk), Γips(pωk;P0;ζ)=Z4 12τiγ5 Λ¯ (9) − 4g2D (p −q ) 3 µν ωk ωl Zl,q ×γ S(q+)Γi (q ;P ;ζ)S(q−)γ , (16) 0, µand/orν =4, µ ωl ps ωl 0 ωl ν PµTν(pΩk):=( δij − ppip2j, µ,ν =i,j =1,2,3, (10) where {τi, i = 1,2,3} are the Pauli matrices, qω±l = p p q ±P /2, P = (P~,0), and Z = Z (ζ,Λ¯) is the mass PL(p ) + PT (p )=δ − µ ν , (11) ωl 0 0 4 4 µν Ωk µν Ωk µν p2 renormalisationconstant: 2 m (ζ)Γi (p ;P ;ζ) (17) It is the gauge-invariant pseudovector projection of R ps ωk Ωn Bethe-Salpeter wave function at the origin, which com- is renormalisation point independent. pletely determines the strong interactioncontributionto Equation (16) is a dressed-ladder BSE. At T = 0 its the leptonic decay of the pion: solution exhibits poles, and their positions and residues provide a good description of light vector and flavour- Γ = 1 f2G2 |V |2m m2 1−m2/m2 2 , (23) π→ℓνℓ 4π π F ud π ℓ ℓ π nonsingletpseudoscalarmesonswhenS isobtainedfrom the rainbow quark DSE [14,26]. This truncation is reli- GF = 1.166× 10−5GeV−2, |Vud|(cid:0)= 0.975, an(cid:1)d me = able because of cancellations between vertex corrections 0.511MeV, mµ =0.106GeV. andcrossed-boxcontributionsateachhigherorderinthe It is a model independent consequence of the axial- quark-antiquark scattering kernel. vector Ward-Takahashi identity that [27] The solution of Eq. (16) has the form (hereafter the f m2 =2m (ζ)r (ζ). (24) ζ-dependence is often implicit) π π R π In the chiral limit Γi (p ;P~)= ps ωk 1 lim r (ζ)=− hq¯qi0, (25) 12τiγ5 iEps(pωk;P~)+~γ·P~ Fps(pωk;P~) mˆ→0 π fπ0 ζ +~γ·p~h~p·P~Gkps(pωk;P~)+γ4ωk ~p·P~G⊥ps(pωk;P~) , (18) mwhoedreel fπ0 is the chiral limit decay constant, and in this i where we have neglected terms involving σµν-like con- f0 =0.088GeV, −hq¯qi0 =(0.235GeV)3 tributions, which play a negligible role at T = 0 [14]. π 1GeV2 The scalar functions in Eq. (18) exhibit a simple pole at ⇒rπ0(1GeV2)=(0.384GeV)2. (26) P~2+m2 =0 so that π r (ζ) Γi (p ;P~)= π Γi(p ;P~)+regular, (19) C. Scalar Channel ps ωk P~2+m2 π ωk π where “regular” means terms regular at this pole and The analogue of Eq. (16) for the 0++ vertex is pre- Γi(p ;P~) is the canonically normalised, bound state sentedinEq.(A1). However,thecombinationofrainbow π ωk and ladder truncations is not certain to provide a reli- pion Bethe-Salpeter amplitude: able approximationin the scalar sector because here the Λ¯ ∂S(q+) cancellations described above do not occur [28]. This is 2δijP~ =tr Γi(q ;−P~) ωl Γj(q ;P~)S(q−) Zl,q (cid:26) π ωl ∂P~ π ωl ωl etenrteadngilneduwnditehrstthaendpihnegnotmheencoolmogpiocasiltdioinfficoufltsiceaslaenrcroeusno-- ∂S(q−) + Γi(q ;−P~)S(q+)Γj(q ;P~) ωl , (20) nances below 1.4GeV [4–6]. For the isoscalar-scalarver- π ωl ωl π ωl ∂P~ (cid:27)(cid:12)P~2=−m2π textheproblemisexacerbatedbythepresenceoftimelike (cid:12) gluonexchangecontributionstothekernel,whicharethe (cid:12) withthetraceovercolour,Diracandisosp(cid:12)inindices,and analogue of those diagrams expected to generate the η- the residue is η′ mass splitting in BSE studies [29]. Nevertheless, in Λ¯ the absence of an improved, phenomenologically effica- δijir =Z tr 1τiγ χj(q ;P~), (21) cious kernel we employ Eq. (A1) in the expectation that π 4 2 5 π ωl Zl,q it will provide some qualitatively reliable insight. (This is justified a posteriori.) where χ (q ;P~) := S(q+)Γ (q ;P~)S(q−) is the un- π ωl ωl π ωl ωl The scalar functions in Eq. (A1) exhibit a simple pole amputated Bethe-Salpeter wave function. Substituting at P~2+m2 =0, Eqs. (A3,A4), with residue Eq. (19) into Eq. (16) and equating pole residues yields σ the homogeneouspion BSE, which provides the simplest Λ¯ way to obtain the bound state amplitude. δαβr =Z tr 1ταχβ(q ;P~), (27) σ 4 2 σ ωl At T = 0, rπ(ζ) is the gauge-invariant pseudoscalar Zl,q projection of the pion Bethe-Salpeter wave function at where χα is an obvious analogue of χi. Since a V −A the originin configurationspace; i.e, it is a field theoret- σ π current cannot connect a 0++ state to the vacuum the icalanalogueof the “wavefunction at the origin,”which scalar meson does not appear as a pole in the vector describesthe decayofbound statesinquantummechan- vertex; i.e, ics. The residue of the pion pole in the axial-vectorvertex Λ¯ is the pion decay constant: δαβP~fσ =Z2Atr 21τα~γχβσ(qωl;P~)≡0. (28) Zl,q Λ¯ δijP~f =ZAtr 1τiγ ~γχj(q ;P~). (22) The homogeneous equation for the scalar bound state π 2 2 5 π ωl amplitude is obtained from Eqs. (A1,A3). Zl,q 3 D. Two-body Decays l = −4,...,4. At the critical temperature this corre- spondstoω >4.0GeV,whichisgreaterthantheother l=4 The σ and π bound state amplitudes along with the mass-scales in the problem, and yields results that are dressed-quark propagators are necessary elements in the numerically accurate to within ∼ 5%. Reproducing the definition of the impulse approximationto hadronic ma- T = 0 limit requires many more Matsubara modes [36], trix elements. For example, the isoscalar-scalar-ππ cou- whichprecludesthatasacheckofournumericalmethod. pling is described by the matrix element in Eq. (A5). Insteadwe estimate the errorby reducing the number of This yields the width modes and comparing the results. Our discretisation of the three-momentum grids in the quark DSE and BSE Γσ→(ππ) = 32gσ2ππ 11−64πmm2π/m2σ , (29) iGsaleussssisaenriqouuasldyralitmuriteedp,oianntds.we typically employ ∼>1000 p σ which vanishes if m < 2m . A contemporary analysis σ π of ππ data identifies a u¯u+d¯d scalar with [5] A. Chiral limit mσ ≈0.46GeV, Γσ ∼0.22−0.47GeV, (30) For mˆ =0 the T =0 analogues of the inhomogeneous BSEs, Eqs. (16,A1), exhibit poles at which corresponds to g ∼2.1–3.0GeV=4.5–6.6m . σππ σ Wealsoconsidertheelectromagneticdecayoftheneu- m =0 and m =0.56GeV, (33) π σ tral pion, for which the dressed-quark-photon vertex is alsorequiredincalculatingtheimpulseapproximationto with m =0.59GeVatmˆ =5.7MeV.A low-massscalar σ the coupling. Quantitativelyreliable numericalsolutions is typical of the rainbow-ladder truncation. However, oftheT =0vectorvertexequationarenowavailable[20]. there is some model sensitivity; e.g., cf. this result with However,thisanomalouscouplingisinsensitivetodetails m = 0.59GeV in Ref. [24], m = 0.67GeV using the σ σ andanaccurateresultrequiresonlythatthedressedver- model of Ref. [26] and m = 0.72GeV in the separable σ tex satisfy the vector Ward-Takahashi identity. This is model of Ref. [37]. These four independent calculations similar to the pion form factor: the q2 =0 value is fixed give an average-m = 0.64GeV with a standard devia- σ by current conservation and only the charge radius re- tion of 10%. The rainbow-ladder truncation yields de- sponds to changes in the vertex [20], and the γ∗π0 → γ generate isoscalar and isovector bound states, and ideal transition form factor whose value is fixed at the real flavour mixing in the 3-flavour case. Hence the distri- photon point [30] but whose q2-evolution is sensitive to bution of mass estimates between that of the isoscalar details of the vertex [31]. σ and isovector a (980) might be anticipated. In con- 0 AnefficaciousT =0vertexAnsatzisgiveninEq.(A6) trast the three comparison studies, Refs. [24,26,37], give [32,33] and using this in calculating the mˆ =0 π0 →γγ m =m =0.75GeVwithastandarddeviationof<2%, ω ρ coupling (Eq. (A9) for T →0) one obtains [30,34] illustratingthedependabilityofthetruncationinthevec- tor channel. gπ00γγ = 21 (31) As already remarked, improvements to the kernel are required in the scalar channel. In the isoscalar-scalar independent of the model parameters. Using this chiral channel, because Γ /m is large, it may even be nec- σ σ limit coupling on-shell essary to include couplings to the dominant ππ mode, which can be handled perturbatively in the ω-ρ sector Γπ0→γγ ≈ 1m63ππ απ2e2m gπ0f0γγ 2 = 6m43ππ παefm 2 ; (32) [e6lu,3c8id].aItnedthheearebisnenacreeosftsruiccthlycoornrelycttiohnoss,ethofeσanpridopeaelritsieeds π ! (cid:18) π(cid:19) chiral partner of the π. Hitherto no model bound-state i.e., 7.7keV cf. the experimental value [35]: 7.7±0.6. description escapes this caveat. The evolutionwithT ofthe pole positions inthe solu- tion of the inhomogeneous BSEs is illustrated in Fig. 1, III. MESON PROPERTIES fromwhichitisclearthat: 1)atthecriticaltemperature, T ,wehavedegenerate,masslesspseudoscalarandscalar c bound states; and 2) the bound states persist above T , Solving the (in)homogeneous BSE at T = 0 is a de- c becoming increasingly massive with increasing T. These mandingnumericaltaskbecauseanaccuratesolutionre- features are also observed in numerical simulations of quiresa largeamountofcomputermemoryand/ortime. lattice-QCD [1]. These problems are exacerbated at T 6= 0 because of Weobtaintheboundstateamplitudesfromthehomo- the loss of O(4) invariance, which is manifest in the sep- geneous Bethe-Salpeter equations, which Fig. 1 demon- aration of the four-momentum into a three-momentum strates are certain to have a solution. Their T-evolution and a Matsubara frequency. Hence in the sum over is depicted in Fig. 2, which indicates that: 1) in both fermion Matsubara modes we usually limit ourselves to cases allbut the leading Dirac amplitude vanishes above 4 T ;and2)thesurvivingpseudoscalaramplitudeispoint- chiral limit. B vanishes above T eliminating the in- c 0 c wise identical to the surviving scalar one. These re- homogeneity and allowing a trivial, identically zero so- sultsindicatethatthechiralpartnersarelocallyidentical lution for each of these amplitudes. Additionally, with above T , they do not just have the same mass. B ≡ 0 the kernels in the equations for the dominant c 0 pseudoscalar and scalar amplitudes are identical, and 100 hence so are the solutions. T = 0.10 GeV It follows from these results that the Goldberger- T = 0.15 GeV Treiman-like relation [27] T = 0.16 GeV 80 T = 0.20 GeV f0E (p ;0)=B (p ), (34) π π ωk 0 ωk ω)0 60 0, issatisfiedforallT onlybecausebothf0andB (p )are E(sca equivalentorderparametersforchiralsπymmetr0yrωesktora- ω), 0 40 tion. This possibility was overlooked in Ref. [36]. Fur- 0, E(ps ther, above Tc, the other constraints on the chiral-limit pion Bethe-Salpeter amplitude derived in Ref. [27] from 20 the axial-vector Ward-Takahashi identity are trivially satisfied. 0 Using the bound state amplitudes and dressed-quark −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 P2 [GeV2] propagators we calculate the matrix elements discussed in Sec. II. Their chiral limit T-dependence is depicted FIG. 1. Leading Dirac amplitude for the pseudoscalar in Fig. 3. As indicated by Fig. 1, the pseudoscalar and (shaded symbols) and scalar (open symbols) vertices, E in scalarboundstatesaremassiveanddegenerateaboveT . c Eqs.(18,A2),evaluatedat(p~=0,ω0)andplottedasfunction Below T the scalar meson residue in the scalar ver- of P~2 in the chiral limit; i.e., E(p~ = 0,ω0;P~2), for a range tex, r in Ecq. (27), is a little larger than the residue of σ of temperature values. The bound state poles are evident in the pseudoscalarmesoninthe pseudoscalarvertex,r in each case. (Weuse ζ =19GeV throughout.) π Eq. (21). However, they are nonzero and equal above T , which is an algebraic consequence of B ≡0 and the c 0 vanishing of the subleading Dirac amplitudes. As a bona 25 fide order parameter for chiral symmetry restoration f ∝(1−T/T )β, T/T <1, (35) π c c ∼ 20 where β is the zero-external-field critical-exponent for E π chiral symmetry restoration (β = 1/2 in rainbow-ladder 15 Fπ Gperp models)[17]. fπ =0andrπ 6=0forT >Tc demonstrates π thatthepiondisappearsasapoleintheaxial-vectorver- E 10 σ tex [36]butpersists asapole inthe pseudoscalarvertex. F σ Our analysis yields Gperp σ 5 Esym mσ ∝(1−T/Tc)β, T/Tc ∼<1, (36) 0 within numerical errors, which is evident in Fig. 3: m2 0.1 0.12 0.14 0.16 0.18 0.2 σ T [GeV] follows a linear trajectory in the vicinity of Tc. Such behaviour in the isoscalar-scalarchannel might be antic- FIG. 2. Dirac amplitudes characterising the mˆ = 0 pseu- ipated because this channel has vacuum quantum num- doscalar and scalar bound states, evaluated at (p~ = 0,ω0). bers and hence the bound state is a strong interaction G⊥, Gk and F behave similarly: as T → T− their value at analogue of the electroweak Higgs boson [6]. π π σ c (p~ = 0,ω0) increases rapidly while the domain on which the m2sym in Fig. 3 is the mass obtained when the chirally functions are nonzero shrinks quickly. Esym is the amplitude symmetricsolutionofthequarkDSEisusedintheBSE. calculated inthechirally symmetricB0 ≡0phase: aboveTc, (B0 ≡ 0 is always a solution in the chiral limit.) For Eπ =Eσ =Esym while all other amplitudes vanish. T >Tc, m2sym(T) is the unique meson mass-squaredtra- jectory. However,forT <T ,m2 <0;i.e.,thesolution c sym It is easy to understand this algebraically. The BSE of the BSE in the Wigner-Weyl phase exhibits a tachy- is a set of coupled homogeneous equations for the Dirac onic solution (cf. the Nambu-Goldstone phase masses: amplitudes. Below Tc each of the equations for the sub- m2 >m2 =0). Byanalogywiththeσ-modelthistachy- σ π leading Dirac amplitudes has an“inhomogeneity”whose onic mass indicates the instability of the Wigner-Weyl magnitude is determined by B0, the scalar piece of the phasebelowTc. Ittranslatesintothe statementthatthe quark self energy which is dynamically generated in the pressure is not maximal in this phase. 5 0.3 associated homogeneous equations. Their T-dependence is characterised in Fig. 5, which shows that for mˆ 6= 0 they are locally identical for T > 4T . ∼ 3 c 0.2 2.5 2eV] mπ nd r [G 0.1 2 mm2πσs yTm a 2m mπ2 0.0 rmπ 2 1.5 rmrσσsym2 m [GeV] −0.1 sym 1 0.1 0.12 0.14 0.16 0.18 0.2 T [GeV] 0.5 0.5 0 0 0.1 0.2 0.3 0.4 0.4 T [GeV] FIG.4. T-dependenceofforlargeT withmˆ =0,seeFig.3. GeV] 0.3 mπ = mσ for T > Tc and m/(2πT) → 1−. This behaviour nd f [π pquerirseisstssigwniitfihcamˆntl6=y m0,orheowcoemvepru,tperrotviimdien.g(Tanhaitlluissteraastiieorninrea- a 0.2 m simpler model [10,19].) m π 0.1 fπ m σ m sym E 25 π 0.0 0.1 0.12 0.14 0.16 0.18 0.2 Fπ T [GeV] Gperp π 20 FIG. 3. mˆ = 0 results. Upper panel: T-dependence of Eσ the meson masses-squared and pole-residue matrix elements, Fσ Eqs. (21,22,27), along with the mass-squared: m2sym, and 15 Gσperp residue: rsym, calculated in the chirally symmetric B0 ≡ 0 phase. For T ≤T , m2 <0, which is a signal of the insta- c sym 10 bility of the chirally symmetric phase at low T. For T >T , c m2 =m2 =m2 . Lower panel: T-dependenceofthemasses π σ sym and pion decay constant. m =0 within numerical error. 5 π Figure 4 depicts the evolutionofthe (common) meson 0 mass at large T. As expected in a gas of weakly inter- 0.1 0.12 0.14 0.16 0.18 0.2 T [GeV] acting quarks and gluons FIG. 5. Dirac amplitudes characterising the mˆ 6= 0 pseu- m meson →1−, (37) doscalarandscalarboundstates,evaluatedat(p~=0,ω0)and 2ω0 plotted as a function of temperature. where ω =πT is a quark’s zeroth Matsubara frequency 0 Figure6isthemˆ 6=0analogueofFig.3. Thatthetran- and “screening mass.” sitionhasbecome acrossoverisevidentinthe behaviour of f . The meson masses become indistinguishable at π T ∼ 1.2T , a little before the local equivalence is mani- B. Nonzero light current-quark masses c fest in Fig. 5, which is unsurprising given that the mass is an integrated quantity. The small difference between It is straightforward to repeat the calculations of r and r below T is again evident and they assume a σ π c Sec. IIIA for nonzero current-quark masses where chiral common value at the same temperature as the masses. symmetry restorationwithincreasingT is exhibitedas a Figure 7 illustrates the preservation of the axial- crossover rather than a phase transition. The solutions vector Ward-Takahashi identity via the mass formula of oftheinhomogeneousBSEsagainexhibitapoleforallT Eq.(24). ThemagnitudeandT-dependenceofbothsides and we determine the bound state amplitudes from the 6 are equal within numerical errors above and below the this can be traced to B → 0. For mˆ 6= 0, the coupling 0 crossover. m (ζ)r0(ζ)istherenormalisationpointinde- reflectsthecrossover. However,thatismootbecausethe R π pendentquantitythatappearsinthepion’scurrentalge- width vanishes just below T where the isoscalar-scalar c bramassformula: m (ζ)r0(ζ)andm (ζ)r (ζ)differby meson mass falls below 2m and the phase space factor R π R π π < 5% until T > 0.95T . This comparison illustrates the vanishes (see the lower panel of Fig. 6). c T-domain on which the current algebra formula is valid and the analysis of Ref. [39]. It was noted in Ref. [17] 0.0016 that rπ0(ζ)/fπ0 ∝(1−T/Tc)−β, T/Tc ∼<1, (38) 3GeV] s [ ue 0.0014 which is qualitatively apparent from Figs. 3 and 7. d si e nt r 0.3 de n e p nde 0.0012 2 mR rπ oint i fπ mπ2 n.p 2 mR rσ 2V] 0.2 re 2 mR <qq>0/fπ0 e G nd r [ 0.00100.1 0.12 0.14T [GeV]0.16 0.18 0.2 a 2m 0.1 FIG. 7. T-dependence of the residue of the pion pole in mr π2 the axial-vector vertex, fπm2π, and in the pseudoscalar ver- mπ 2 tex, 2mR(ζ)rπ(ζ). They are equal within numerical errors σ rσ as required by theaxial-vector Ward-Takahashi identity. For comparison wealso plot 2m (ζ)r0(ζ) andtheresidueofthe 0.0 R π 0.1 0.12 0.14 0.16 0.18 0.2 scalar meson pole in thescalar vertex, 2m (ζ)r (ζ). T [GeV] R σ 0.5 3 0.3 g mπ Γσππ 0.4 fπ chσπirπal g mσ chiral Γσππ V] mσ − 2 mπ σππ e G 0.3 2 0.2 m and f [π0.2 g [GeV] Γ [GeV] 1 0.1 0.1 0.0 0.1 0.12 0.14 0.16 0.18 0.2 T [GeV] 0 0.1 0.12 0.14 0.16 0.18 0.2 T [GeV] FIG. 6. T-dependence of the meson masses and pole-residue matrix elements, Eqs. (21,22,27), for mˆ 6=0. FIG. 8. T-dependence of the isoscalar-scalar-ππ cou- pling and width, both in the chiral limit and for real- istic light current-quark masses. The phase space factor C. Triangle Diagrams (1−(2m /m )2)1/2 is θ(T −T) in the chiral limit but non- π σ c trivial for mˆ 6= 0, vanishing at T ≈ 0.98T ; i.e., this decay c channelcloses at a temperature just 2% less-than T . We also studied the matrix elements describing the c two-body decays discussed in Sec. IID. With every ele- The T-dependence of the π0γγ coupling, which sat- ment in the calculation only known numerically this too urates the Abelian anomaly at T = 0, is calculated is a challenging numerical exercise, which we simplified from Eqs. (A9,A12) and depicted along with the width by a judicious choice of the external momenta. in Fig. 9. In the chiral limit the width is identically Thechirallimitisoscalar-scalar-ππcouplingandwidth zero because m = 0 and the interesting quantity is: obtained from Eq. (A5) are depicted in Fig. 8, which π T(0) = g0 /f0. Clear in the figure is that T(0) van- indicatesthatbothvanishatTc inthechirallimit. Again π0γγ π 7 ishes at T . It vanishes with a mean field critical expo- simulations of lattice-QCD, and find m → 2πT as c σ,π nent,asismosteasilyinferredfromFig.10. (Anaccurate T →∞. calculationispossiblebecauseEq.(34)obviatestheneed for a solution of the BSE.) Thus, in the chiral limit, the coupling to the dominant decay channel closes for both 0.5 llmax== 48 DDSSEE BB0 max 0 charged and neutral pions. These features were antici- lmax=16 DSE B0 l =32 DSE B pated in Ref. [40]. Further, as is evident in Fig. 10, our 0.4 lmax= 4 BSE fE0 max π π calculatedT(0)ismonotonicallydecreasingwithT,sup- porting the perturbative O(T2/f2) analysis in Ref. [41]. π 0.3 gπγγ 10 10 0.1 0.2 8 8 0.1 0 0.146 0.150 0.154 τ−1(0) [GeV] 46 τ(0)πγγ 64 Γ [GeV] FIG.1000..1T-dep0.e11ndenc0e.12of gTπ0 0 γ[.Gγ1e3=V] fπ00T.14(0). W0.1i5thin 0n.u16mer- Γchπγiγral τ(0) ical error, gπ0γγ ∝ (1−T/Tc). Near Tc, fπ ∝ (1−T/Tc)1/2 2 2 becauseourmodelhasmeanfieldcriticalexponentsandhence T(0)∝(1−T/T )1/2. c 0 0.1 0.12 0.14 0.16 0.18 0.2 The BSE solutions indicate a local equivalence be- T [GeV] tweenthe isovector-scalarand -pseudoscalarcorrelations FIG. 9. T-dependence of the coupling T(0) in Eq. (A12) above the chiral restoration transition/crossover;2 i.e., and the π0→γγ width. the scalar functions characterising the bound state am- plitudes are identical, and from this follows equality of Formˆ 6=0boththecoupling: gπ0γγ/fπ,andthewidth the masses and many of the matrix elements.3 Further, exhibit the crossover with a slight enhancement in the the axial-vector Ward-Takahashi identity and the pseu- width as T → T due to the increase in m . There are doscalarmass formulathat is its corollaryarevalid both c π similaritiesbetweentheseresultsandthoseofRef.[42]al- above and below the crossover. thoughtheT-dependencehereinismuchweakerbecause For realistic light current-quark masses the isoscalar- our pion mass approaches twice the T 6= 0 free-quark scalarmesonmassdoesnotfallbelow2mπ untilverynear screening-mass from below, never reaching it, Eq. (37); the transition temperature. Hence this dominant decay i.e., the continuum threshold is not crossed. channel remains open almost until the phase boundary is crossed. Once it is crossed, however, only electroweak decay channels are open. In the chiral limit the anoma- IV. SUMMARY AND CONCLUSION lous two-photoncoupling constant vanishes at Tc just as doesf ,thecouplingthatdeterminesthestrengthofthe π leptonic charged pion decay. For realistic masses, how- We employed a renormalisation group improved ever, the widths for the leptonic charged pion mode and rainbow-laddertruncationofthequarkDyson-Schwinger two-photonπ0 mode remainsignificant in the vicinity of equation, and pseudoscalar and scalar Bethe-Salpeter the crossover. equations to estimate the T-dependence of a range of properties that characterise correlations in these chan- nels. The rainbow-ladder truncation is quantitatively reliable in the pseudoscalar channel. However, that is not certain in the scalar channel where the cancellations 2We refrain from asserting this of the isoscalar-scalar cor- that assistin the pseudoscalarchannelarenotapparent. relation because we cannot anticipate the effect of timelike Nevertheless,we anticipate that many ofthe features we gluon exchange contributions present in the BSE kernel in exposed in the scalar channel are qualitatively correct. thischannel. However,theyarekindredtothoseencountered The solutions of the inhomogeneous Bethe-Salpeter inanalysingtherealisationofUA(1)symmetryabovethechi- equation (BSE) exhibit poles at all T, both above and ral transition [1,2,29]. below the critical temperature, and in the chiral limit 3The evolution to equality of the masses has been observed in numerical simulations of lattice-QCD and in the explo- and for realistic light current-quark masses. We use the ration of other models that accurately describe dynamical associated homogeneous BSEs to determine the masses, chiral symmetry breaking. which correspond to the screening masses determined in 8 The construction of a DSE-BSE truncation that al- Γi(p ;P~)= 1τα1 E (p ;P~)+i~γ·p~Gk(p ;P~) s ωk 2 s ωk s ωk lows for an improved description of the scalar channel h at T = 0 would provide the foundation for a significant + iγ ω G⊥(p ;P~)+i~γ·P~ p~·P~F (p ;P~) . (A2) 4 k s ωk s ωk improvement of our analysis. Employing an Ansatz for i thekernelofthequarkDSEwhoseinfraredformexhibits (NB: Here the requirement that the neutral mesons be some T-dependence, perhaps constrained by lattice sim- charge conjugation eigenstates shifts the p~·P~ term cf. ulationsofthestringtension[1],mayalsobeinteresting. the 0−+ amplitude.) However,giventhe results ofRefs. [17,43],we do not ex- The scalar functions in Eq.(A1) exhibit a simple pole pect such a modificationto havea significantqualitative at P~2+m2 =0: impact. In the absence of such improvements we never- σ thelessexpectthelocalequivalencewehaveelucidatedto r (ζ) be exhibitedbyallisovectorchiralpartnersinthe strong Γis(pωk;P~)= P~2σ+m2 Γiσ(pωk;P~)+regular, (A3) interaction spectrum. However, the explicit demonstra- σ tionofthisisdifficult;e.g.,intheρ-a complexthebound 1 where Γi(p ;P~) is the canonically normalised, 0++ stateamplitudeshaveeightindependentamplitudeseven σ ωk bound state Bethe-Salpeter amplitude: atT =0comparedwiththefourinthepseudoscalarand scalar amplitudes at T 6=0. Λ¯ ∂S(q+) 2δαβP~ =tr Γα(q ;−P~) ωl Γβ(q ;P~)S(q−) Zl,q (cid:26) σ ωl ∂P~ σ ωl ωl ACKNOWLEDGMENTS ∂S(q−) + Γα(q ;−P~)S(q+)Γβ(q ;P~) ωl . (A4) We acknowledge interactions with D. Blaschke and σ ωl ωl σ ωl ∂P~ (cid:27)(cid:12)P~2=−m2s (cid:12) Yu.L. Kalinovsky. C.D.R. is grateful for the support The dressed-quark propagator and ca(cid:12)(cid:12)nonically nor- and hospitality of the Special Centre for the Subatomic malisedBethe-Salpeteramplitudesmakepossiblethedef- Structure of Matter at the University of Adelaide dur- inition of the impulse approximation to the isoscalar- ing a visit in which some of this work was conducted, scalar-ππ matrix element: p~ 2 = −m2 = p~2, (p~ = 1 π 2 and both C.D.R. and S.M.S. are grateful for the same p~ +p~)2 =−m2, 1 2 σ from the Physics Department at the University of Ros- tockduringajointvisitinwhichaspectsofthisworkwere g :=hπ(p~ )π(p~ )|σ(p~)i= σππ 1 2 completed. S.M.S. acknowledges financial support from Λ¯ theA.v.Humboldtfoundation. Thisworkwassupported 2N tr Γ (k ;~p)S (k ) c D σ ωl u ++ by the US Department of Energy, Nuclear Physics Divi- Zl,q sion,undercontractno. W-31-109-ENG-38,theNational ×iΓ (k ;−p~ )S (k )iΓ (k ;−p~)S (k ), (A5) π 0+ 1 u +− π −0 2 u −− Science Foundation under grant nos. INT-9603385 and PHY97-22429, and benefited from the resources of the kαβ =kωl +(α/2)p~1+(β/2)p~2, with only the trace over National Energy Research Scientific Computing Center. Dirac indices remaining. APPENDIX: COLLECTED FORMULAE 2. Neutral Pion Decay 1. Scalar Vertex An efficacious Ansatz for the dressed-quark-photon coupling at T =0 is [32,33]: The inhomogeneous ladder-like Bethe-Salpeter equa- iΓ (ℓ ,ℓ )=iΣ (ℓ2,ℓ2)γ (A6) tion for the zeroth Matsubara mode of the 0++ vertex µ 1 2 A 1 2 µ is +(ℓ1+ℓ2)µ 21iγ·(ℓ1+ℓ2)∆A(ℓ21,ℓ22)+∆B(ℓ21,ℓ22) ; Γαs(pωk;P0;ζ)=Z4 21τα1 ΣF(ℓ21,ℓ22)=(cid:2)21[F(ℓ21)+F(ℓ22)], ((cid:3)A7) Λ¯ ∆ (ℓ2,ℓ2)= F(ℓ21)−F(ℓ22), (A8) − 4g2D (p −q ) F 1 2 ℓ2−ℓ2 3 µν ωk ωl 1 2 Zl,q ×γ S(q+)Γi(q ;P ;ζ)S(q−)γ , (A1) where F =A,B; i.e., the scalarfunctions inthe dressed- µ ωl s ωl Ωn ωl ν quark propagator: S−1(p)=iγ·pA(p2)+B(p2), so that where α = 0,1,2,3 with τ0 = diag(1,1). (NB: In this this model is completely determined by S(p). Improve- truncation the isoscalar and isovector states are degen- mentsofthisAnsatz,suchasthosecanvassedinRef.[33], erate,which exemplifies our observationthat the ladder- donothaveaqualitativelysignificanteffectinthepresent liketruncationisaccurateforvectormesonsbutrequires context. improvement before it is quantitatively reliable in the The dressed-quark-photon vertex is a necessary ele- 0++ sector.) The solution has the form ment in the calculation of the impulse approximation to 9 the π0 → γγ amplitude. At T = 0 the anomalous con- [3] Understanding Deconfinement in QCD, Proceedings of tribution to the divergence of the axial-vector vertex is the International Workshop on Understanding Decon- saturated by the pseudoscalar piece of the pion Bethe- finement in QCD, Trento, Italy, 1999, edited by D. Salpeter amplitude [34] Blaschke, F.KarschandC.D.Roberts(WorldScientific, Singapore, 2000). Λ¯ [4] M. Boglione and M.R. Pennington, Eur. Phys. J. C 9, Tˆµν(k1,k2)=tr S(q1)γ5τ3iEπ(qˆ;−P) 11 (1999). Zl,q [5] M.R. Pennington, “Riddle of the scalars: Where is the ×S(q2)iQΓµ(q2,q12)S(q12)iQΓν(q12,q1), (A9) σ?,” hep-ph/9905241. [6] M.R. Pennington, “Low Energy Hadron Physics,” hep- whereQ=diag(2/3,−1/3)andhereink =(~k ,0),k = ph/0001183. 1 1 2 (~k ,0), P = k +k , q = q −k , q = q +k , qˆ= [7] J.C.R. Bloch, M.A. Ivanov, T. Mizutani, C.D. Roberts 1(2q +q ),q 1=q 2−k1+k .ωlUsing1Eq2.(A6ω)ltoev2aluate and S.M. Schmidt, “K →ππ and a light scalar meson,” 2 1 2 12 ωl 1 2 nucl-th/9910029. Eq. (A9) for real photons at T =0 one obtains [8] C.D.RobertsandA.G.Williams,Prog.Part.Nucl.Phys. Tˆ (k ,k )= αem ǫ k k T(0), (A10) 33, 477 (1994); C.D. Roberts, “Nonperturbative QCD µν 1 2 π µνρσ 1ρ 2σ with modern tools,” in Proceedings of the 11th Physics Summer School: Frontiers in Nuclear Physics, edited by with in the chiral limit [30] S. Kuyucak(World Scientific,Singapore, 1999) p. 212. [9] C.D. Roberts, Fiz. E´lem. Chastits At. Yadra 30, 537 fπ0T(0):=gπ0γγ =1/2. (A11) (1999) (Phys. Part. Nucl. 30, 223 (1999)). [10] D. Blaschke and P.C. Tandy, “Mesonic correlations and AtnonzeroT thetensorstructureofEq.(A10)survives quark deconfinement,” in Ref. [3], p. 218. totheextentthat,withourchoiceofk1,k2,itensuresone [11] J.C.R. Bloch, C.D. Roberts and S.M. Schmidt, Phys. of the photons is longitudinal (a plasmon) and the other Rev. C 60, 065208 (1999). transverse. In this case we determine the T-dependence [12] J.Engels,R.Joswig,F.Karsch,E.Laermann,M.Lutge- using meier and B. Petersson, Phys. Lett. B 396, 210 (1997). [13] D.Blaschke,C.D.RobertsandS.M.Schmidt,Phys.Lett. α Tˆ (k ,k )= em (k~ ×k~ ) T(0) (A12) B 425, 232 (1998). i4 1 2 1 2 i π [14] P. Maris and C.D. Roberts, Phys. Rev. C 56, 3369 (1997). and a generalisationof Eq. (A6) to nonzero T: [15] L.S.Kisslinger,M.Aw,A.HareyandO.Linsuain,Phys. Rev. C 60, 065204 (1999). i~Γ(q ,q )=Σ (q2 ,q2 )i~γ ωl1 ωl2 A ωl1 ωl2 [16] J.I.SkullerudandA.G.Williams,“Thequarkpropagator +(~q +~q )[1iG(q ,q )+∆ (q2 ,q2 )], (A13) in momentum space,” hep-lat/9909142. 1 2 2 ωl1 ωl2 B ωl1 ωl2 [17] A. H¨oll, P. Maris and C.D. Roberts, Phys. Rev. C59, iΓ (q ,q )=Σ (q2 ,q2 )iγ 4 ωl1 ωl2 C ωl1 ωl2 4 1751 (1999). +(ω +ω )[1iG(q ,q )+∆ (q2 ,q2 )], (A14) [18] P.Maris, C.D.RobertsandS.M.Schmidt,Phys.Rev.C l1 l2 2 ωl1 ωl2 B ωl1 ωl2 57, 2821 (1998). G(qωl1,qωl2)=~γ·(~q1+~q2)∆A(qω2l1,qω2l2) [19] D. Blaschke, G. Burau, Yu.L. Kalinovsky, P. Maris and P.C. Tandy, “Finite T correlations and quark deconfine- +γ (ω +ω )∆ (q2 ,q2 ), (A15) 4 l1 l2 C ωl1 ωl2 ment,” preprint nos. MPG-VT-UR 192/99, KSUCNR- 101-00. which satisfies the vector Ward-Takahashiidentity [20] P.MarisandP.C.Tandy,“Thequarkphotonvertexand the pion charge radius,” nucl-th/9910033. (q −q ) iΓ (q ,q )=S−1(q )−S−1(q ). ωl1 ωl2 µ µ ωl1 ωl2 ωl1 ωl2 [21] J.C.R.Bloch,C.D.RobertsandS.M.Schmidt,“Selected (A16) nucleon form factors and a composite scalar diquark,” nucl-th/9911068. [22] F.T. Hawes, P. Maris and C.D. Roberts, Phys. Lett. B 440, 353 (1998). [23] H.J. Munczek and A.M. Nemirovsky, Phys. Rev. D 28, 181 (1983). [24] P.JainandH.J.Munczek,Phys.Rev.D48,5403(1993). [25] A. Bender, C.D. Roberts and L. v. Smekal, Phys. Lett. [1] E. Laermann, Fiz. E´lem. Chastits At. Yadra 30, 720 B 380, 7 (1996). (1999) (Phys.Part. Nucl. 30, 304 (1999)). [26] P. Maris and P C. Tandy, Phys. Rev. C 60, 055214 [2] QCD Phase Transitions, Proceedings of the XXVth In- (1999). ternational Workshopon Gross Properties ofNuclei and [27] P. Maris, C.D. Roberts and P.C. Tandy, Phys. Lett. B NuclearExcitations, Hirschegg, Austria, 1997, edited by 420, 267 (1998). H. Feldmeier, J. Knoll, W. Norenberg and J. Wambach [28] C.D. Roberts, in Quark Confinement and the Hadron (GSI,Darmstadt, 1997). 10