We present recent results on finite temperature electromagnetic form factors
and the electrical conductivity in a pion gas. The standard Chiral Perturbation
Theory power counting needs to be modified for transport coefficients. We pay
special attention to unitarity and to possible applications for dilepton and
photon production.Comment: 4pp, 2 figures, talk given at "Strong and Electroweak Matter 2006",
  BNL, May 200

Fernandez-Fraile, D.

Nicola, A. Gomez

English

arXiv

© Elsevier Science Bv.
International Conference on Strong and Electroweak Matter (7. 2006. New York)We present recent results on finite temperature electromagnetic form factors and the electrical conductivity in a pion gas. The standard Chiral Perturbation Theory power counting needs to be modified for transport coefficients. We pay special attention to unitarity and to possible applications for dilepton and photon production.Depto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Gómez Nicola, Ángel

Fernández Fraile, Daniel

Docta Complutense

arXiv:hep-ph/0608273v1  24 Aug 20061Electromagnetic fields and transport coefficients in a hot pion gasA.Gómez Nicolaa ∗, D.Fernández-FraileaaDepartamentos de F́ısica Teórica I, II, Universidad Complutense, 28040 Madrid, SpainWe present recent results on finite temperature electromagnetic form factors and theelectrical conductivity in a pion gas. The standard Chiral Perturbation Theory powercounting needs to be modified for transport coefficients. We pay special attention tounitarity and to possible applications for dilepton and photon production.1. IntroductionThe most relevant properties of electromagnetic interactions in a hot and dense mediumcan be inferred from the retarded current-current correlator ΠRµν(ω, ~q). On the one hand,Linear Response Theory provides the reaction of the system to soft external fields [1].Thus, the electrical conductivity measures the response to a constant electric field, whereasthe Debye mass parametrizes the screening of a single charge placed at the origin:σ(T ) = limω→0+lim|~q|→0+Im(ΠR)ii(ω, |~q|)3ω; m2D = − lim|~q|→0+limω→0+ΠR00(ω, |~q|) (1)On the other hand, Im(ΠR)µµ(ω, |~q|) is directly related to the photon yield emanatedfrom a Relativistic Heavy Ion Collision for q2 = 0 and to the dilepton rate for q2 = M2with M the dilepton invariant mass [2].Here we are interested in a pion gas at finite temperature and zero baryon density.Below the chiral phase transition, the dynamics of such a system can be described withChiral Perturbation Theory (ChPT) [3], the most general low-energy expansion compat-ible with the spontaneous breakdown of chiral symmetry. The lagrangian is written asan expansion in pion field derivatives and masses and Weinberg’s chiral power counting[4] establishes that the perturbative contribution of a given diagram is of order (E/Λχ)D(O(ED) for short) with D = 2(NL + 1) +∑d(d − 2)Nd, NL the number of loops, Nd thenumber of vertices coming from the d-derivatives lagrangian, E a typical pion energy andΛχ ≃ 1 GeV. Diagrams with photon lines can be included by counting eΛχ = O(E) andtemperature corrections are perturbative for T well below Tc ≃ 180-200 MeV.2. Pion EM form factors and charge distributionOnly with pion degrees of freedom, the imaginary part of the current-current correlatorentering the dilepton rate is directly related, to lowest order in e, to the modulus of the∗Work supported by the Spanish research projects and grants FPA2004-02602,FPA2005-02327, PR27/05-13955-BSCH, BES-2005-6726.2(a) (b)· · ·nFigure 1. Dominant diagrams for transport coefficients (a) to one loop, (b) of ladder type.pion electromagnetic form factor squared. Physically, this gives the contribution of theannihilation of two charged pions to form a dilepton pair, which is dominant for the lowenergy part of the spectrum.The EM form factor at finite temperature has been calculated in [5] in ChPT to one loop.At zero energy, it provides the spatial Fourier transform of the pion charge distributionF (|~q|2) = QT (1 − 〈r2〉T |~q|2/6 + . . .), where we get for the net pion charge:QT = 1 −12π2f 2π∫ ∞mπdE2E2 − m2π√E2 − m2πnB(E; T ) = 1 −m2D(T )2e2f 2π(2)where the pion decay constant fπ ≃ 93 MeV, nB(E; T ) = [exp(E/T ) − 1]−1 and m2D(T )is calculated to the same order [6]. Therefore, the pion charge is screened in the thermalbath proportionally to the Debye mass. As a consequence, the pion charge radius 〈r2〉Tis notably increased from T > 100 MeV [5]. Estimating the deconfinement temperatureas (4π/3)〈r2〉3/2Td nπ(Td) = 1 with nπ the pion density, gives Td ≃ 200 MeV, about 65 MeVlower than using the same estimate with 〈r2〉T=0 [7].In order to account for the ρ-resonance contribution, we have used unitarized thermalform factors F and partial waves tIJ [5,8] satisfying unitarity in the center of mass frame:ImF (E; T ) = σT t11(E; T )F∗(E; T ) ; Imt11(E; T ) = σT |t11(E; T )|2 (3)where σT =√1 − 4m2π/E2 [1 + 2nB(E/2; T )] is the two-pion thermal space factor. Thebehaviour of |F |2(E; T ) shows a clear thermal ρ broadening while reducing only slightly itsmass. This is compatible with recent dilepton data in the CERN-SPS NA60 experiment,which rules out a Brown-Rho like dropping mass scenario with almost no broadening [9].Our results are also consistent with approaches based on Vector Meson Dominance [10].3. Transport coefficients in ChPT: the electrical conductivityTransport coefficients are intrinsically nonperturbative. Diagrammatically, ImΠR(0+,~0)shows to one loop (Figure 1a) a typical “pinching pole” behaviour GR(ω′, |~p|)GA(ω′, |~p|) ≈πδ(ω′2 − E2p)/(2EpΓp), where (ω′, ~p) is the four-momentum carried by the particle in theloop, E2p = |~p|2 + m2 and GR,A, are retarded/advanced propagators with the perturba-tively small thermal width Γp << Ep [11,12]. A dominant contribution proportional tothe inverse width is also expected from elementary kinetic theory, since transport co-efficients are proportional to mean free times. For instance, the electrical conductivityσ ∼ e2Nch/(mΓ) where Nch is the number of charge carriers of mass m and width Γ.3100 120 140 160 180T HMeVL0.0050.010.0150.020.0250.03Unitarized IAMs HTLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅmpPerturbative OHE4LFigure 2. The electrical conductivity to leading order in ChPT, with and without unita-rizing the partial waves in the thermal pion width.In ChPT, the pion width is Γp = O(E5)[13], so that the above considerations leadto a redefinition of the standard chiral power counting for the calculation of transportcoefficients [14]. For that purpose, we use the double-line notation for those internal lineswhich share the same four-momentum when the external momentum goes to zero. Thoselines carry propagators with Γp 6= 0 and give the dominant contributions mentioned above.Therefore, double lines do not count as a chiral loop suppression in this new chiral powercounting. Instead, they are assigned a nonperturbative factor Y , which we estimate bycalculating the leading order diagram in Figure 1a. Thus, we write the contribution ofthat diagram to the electrical conductivity as σ(0) = e2mπY . In the standard chiral powercounting we would have Y = O(E2). The next step is to identify the dominant diagramswith this new counting. As it happens in simple scalar theories [11], those are ladderdiagrams of the type showed in Figure 1b, which in our case are O(E2nY n+1) with n thenumber of rungs. Note that the loop rungs carry single lines.In order to understand the behaviour of the relevant diagrams in ChPT, let us considertwo different regimes: first, for T << mπ the typical spatial loop momenta p = O(√mπT )and E ≃ mπ, we find Y ≃√mπ/T , so that ladder diagrams could be increasingly im-portant. However, to the leading 1/Γ order, only the spectral functions of the rung loopscontribute [12] and every one of those is proportional to pCM/mπ ∼√T/mπ = 1/Y wherepCM is the center of mass momentum for ππ scattering of the double-pion (on-shell) linesattached to the rung loop. Therefore, σ ∝√mπ/T for very low T and ladder diagramsgive only perturbative corrections to the proportionality constant. The second regime ofinterest is T ∼ mπ, where p = O(T ) and E ∝ T . Here, unitarization effects have to betaken into account in the partial waves entering the pion width. Qualitatively, unitaritymakes the conductivity change its decreasing behaviour with T , as showed in Figure 2.Note that this change of behaviour occurs near Tc, which is reasonable since σ growswith T in the QGP phase [15]. A similar behaviour is expected for other transport co-efficients [16]. In addition, although ladder diagrams still remain formally perturbativein ChPT, the presence of derivative vertices increasing with T makes them numerically4important near Tc, where the corresponding Boltzmann-like integral equations for theeffective vertices [12] have to be solved and so we will do elsewhere.Finally, let us comment on a phenomenological application of our results regarding thephoton spectrum. From the definition of the conductivity in (1), the equilibrium photonrate at vanishing energy is proportional to Tσ(T ) [14]. This translates directly into aprediction for the photon yield at vanishing transverse momentum, where the hadron gasdominates over the QGP. Using a simple cylindrical Bjorken expansion with parameterstypical of the CERN WA98 experiment [17] gives ωdNγ/d3~q(qT → 0+) ≃ 5.6×102 GeV−2.This value is reasonably close to a linear extrapolation of the two closest experimentalpoints to the origin in [17] and is also compatible with recent theoretical analysis [18].Our result highlights the importance of considering a nonzero pion width and resonancesin ππ scattering for the photon spectrum near zero energy.4. ConclusionsWe have presented a recent analysis of electromagnetic properties of a pion gas. Whilethe pion electromagnetic form factor can be computed and unitarized in standard ChPT,the calculation of transport coefficients requires a redefinition of the standard chiral powercounting in order to account for typical contributions proportional to the inverse particlewidth. Our results show phenomenological and theoretical consistency in the context ofdilepton and photon production at low energies and open up the possibility of studyingin ChPT other physically relevant transport coefficients such as viscosities.REFERENCES1. M.Le Bellac, Thermal Field Theory, Cambridge University Press 1996.2. J.Alam et al, Ann.Phys.286, 159-248 (2001) and references therein.3. J. Gasser and H. Leutwyler, Phys.Lett. B184, 83 (1987).4. J.F.Donoghue, E.Golowich and B.R.Holstein, Dynamics of the Standard Model, Cam-bridge University Press 1994.5. A.Gómez Nicola, F.J.LLanes-Estrada and J.R.Peláez, Phys.Lett. B606, 351 (2005).6. J.I.Kapusta, Phys.Rev. D 46, 4749 (1992).7. J.I.Kapusta, Finite Temperature Field Theory, Cambridge University Press 1993.8. A.Dobado, A.Gómez Nicola, F.Llanes-Estrada and J.R.Peláez, Phys. Rev. C66,055201 (2002).9. H.van Hees and R.Rapp, hep-ph/0604269.10. R.Rapp and J.Wambach, Adv.Nucl.Phys. 25, 1 (2000) and references therein.11. S.Jeon, Phys.Rev. D52, 3591 (1995).12. M.A.Valle Basagoiti, Phys.Rev. D66, 045005 (2002).13. J.L.Goity and H.Leutwyler, Phys.Lett. B228, 517 (1989).14. D. Fernandez-Fraile and A. Gomez Nicola, Phys.Rev.D73 (2006) 045025.15. P.Arnold, G.D.Moore and L.G.Yaffe, JHEP 0011:001 (2000).16. L. P. Csernai, J. I. Kapusta and L. D. McLerran, nucl-th/0604032.17. M. M. Aggarwal et al. [WA98 Collaboration], Phys. Rev. Lett. 93, 022301 (2004).18. S.Turbide, R.Rapp and C.Gale, Phys.Rev. C69, 014903 (2004). W.Liu and R.Rapp,nucl-th/0604031.

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Electromagnetic fields and transport coefficients in a hot pion gas

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Electromagnetic fields and transport coefficients in a hot pion gas

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