We present recent results on the calculation of transport coefficients for a
pion gas at zero chemical potential in Chiral Perturbation Theory using Linear
Response Theory. More precisely, we show the behavior of DC conductivity and
shear viscosity at low temperatures. To compute transport coefficients, the
standard power counting of ChPT has to be modified. The effects derived from
imposing unitarity are also analyzed. As physical applications in Relativistic
Heavy Ion Collisions, we show the relation of the DC conductivity to
soft-photon production and phenomenological effects related to a nonzero shear
viscosity. In addition, our values for the shear viscosity to entropy ratio
satisfy the KSS bound.Comment: 3 pages, 6 figures, EPJA style. Talk given at the QNP06 conference in
  Madrid, and accepted for publication in EPJ

A. Gómez Nicola

Aggarwal)

Arnold

Basagoiti

Csernai

D. Fernández-Fraile

Davesne

Dobado

Fernández-Fraile

Gasser

Gerber

Goity

Jeon

Kovtun

Nakamura

Nicola

Prakash

Teaney

Turbide

English

arXiv

Crossref

Transport coefficients in Chiral Perturbation Theory

We present recent results on the calculation of transport coefficients for a pion gas at zero chemical potential in Chiral Perturbation Theory (ChPT) using the Linear Response Theory (LRT). More precisely, we show the behavior of DC conductivity and shear viscosity at low temperatures. To compute transport coefficients, the standard power counting of ChPT has to be modified. The effects derived from imposing unitarity are also analyzed. As physical applications in relativistic heavy-ion collisions, we show the relation of the DC conductivity to soft-photon production and phenomenological effects related to a non-zero shear viscosity. In addition, our values for the shear viscosity to entropy ratio satisfy the KSS bound

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© Società Italiana di Fisica.
Conferencia: Hot Quarks Conference 2008 (2008. Aspen, Colorado). We are grateful to A. Dobado and F. Llanes-Estrada for their useful comments. We also acknowledge financial support from the Spanish research projects FPA2004-02602, PR27/05-13955-BSCH, FPA2005-02327 and FPI fellowship BES-2005-6726.We present recent results on the calculation of transport coefficients for a pion gas at zero chemical potential in Chiral Perturbation Theory (ChPT) using the Linear Response Theory (LRT). More precisely, we show the behavior of DC conductivity and shear viscosity at low temperatures. To compute transport coefficients, the standard power counting of ChPT has to be modified. The effects derived from imposing unitarity are also analyzed. As physical applications in relativistic heavy-ion collisions, we show the relation of the DC conductivity to soft-photon production and phenomenological effects related to a non-zero shear viscosity. In addition, our values for the shear viscosity to entropy ratio satisfy the KSS bound.Spanish research projectsBSCHFPI programmeDepto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Fernández Fraile, Daniel

Gómez Nicola, Ángel

Docta Complutense

arXiv:hep-ph/0610197v1  16 Oct 2006EPJ manuscript No.(will be inserted by the editor)Transport coefficients in Chiral Perturbation TheoryD. Fernández-Frailea and A. Gómez NicolabDepartamentos de F́ısica Teórica I y II, Universidad Complutense, 28040 Madrid, Spain.Oct. 15, 2006Abstract. We present recent results on the calculation of transport coefficients for a pion gas at zerochemical potential in Chiral Perturbation Theory using Linear Response Theory. More precisely, we showthe behavior of DC conductivity and shear viscosity at low temperatures. To compute transport coefficients,the standard power counting of ChPT has to be modified. The effects derived from imposing unitarity arealso analyzed. As physical applications in Relativistic Heavy Ion Collisions, we show the relation of the DCconductivity to soft-photon production and phenomenological effects related to a nonzero shear viscosity.In addition, our values for the shear viscosity to entropy ratio satisfy the KSS bound.PACS. 11.10.Wx Finite-temperature field theory – 12.39.Fe Chiral lagrangians – 25.75.-q Relativisticheavy-ion collisions1 IntroductionWe are interested in studying transport properties in apion gas with zero baryon density. Pions are the lightesthadronic degrees of freedom and their dynamics is welldescribed for low-enough energies and temperatures byChiral Perturbation Theory (ChPT) [1]. ChPT is a low-energy expansion performed in terms of p2 (p represents amomentum, a mass or a temperature) against some scaleΛ2χ ∼ (1 GeV)2. However, as we shall see, the compu-tation of transport coefficients in ChPT is intrinsicallynon-perturbative, leading to a modification of the stan-dard ChPT power counting scheme to take into accountcollisions in the plasma (i.e. a non-zero pion width).Considering small external perturbations which putthe system slightly out of equilibrium, we can employ Lin-ear Response Theory (LRT) [2,3,4], in which a transportcoefficient t is given by:t = C limq0→0+lim|q|→0+∂ρ(q0, q)∂q0, (1)with C a constant and ρ the spectral density of a current-current correlator.In the calculation of transport coefficients, there is adominant contribution coming from products of the kindGAGR ∝ 1/Γ (GA and GR are the advanced and re-tarded propagators respectively), called pinching poles [3,4], where Γ is the particle width (which is perturbativelysmall: Γ = O(p5) in ChPT [5]). From Kinetic Theory(KT), a behavior ∼ 1/Γ for transport coefficients is alsoexpected, since 1/Γ represents the mean time betweena Electronic address: danfer@fis.ucm.esb Electronic address: gomez@fis.ucm.estwo collisions of the particles in the plasma. Each of thesepinching poles comes from a pair of lines in a Feynman di-agram contributing to the spectral density that share thesame four-momentum when the limit in (1) is taken. Ac-cordingly, there are some classes of diagrams potentiallynon-perturbative, being the most harmful the type calledladder diagrams (see Fig. 1). Another type, the so calledbubble diagrams (see Fig. 2), which in principle would bethe most harmful, for n ≥ 2 turn out to give a negligiblecontribution to the electrical conductivity and the shearviscosity. Thus, the standard power counting scheme fromChPT has to be modified to compute transport coeffi-cients [6]. In the case of a ladder diagram with n rungs,its contribution is in principle of order O(p2nY n+1), whereY is the contribution from a single-bubble diagram. ForT ≪ Mπ ≃ 140 MeV, a detailed analysis of the spec-tral function [6] shows that it contributes O(p2nY ). ForT & Mπ, Y becomes of order 1 or larger (due to uni-tarity effects) so that diagrams with one or more rungscould be non-negligible in this regime. Also note that, forhigh enough temperatures where typical momenta p ∼ T ,diagrams with derivative vertices become potentially dom-inant against those with constant vertices.nFig. 1. Generic ladder diagram with n rungs (n ≥ 1). Doublelines represent particles with Γ 6= 0.2 D. Fernández-Fraile, A. Gómez Nicola: Transport coefficients in Chiral Perturbation TheorynFig. 2. Generic bubble diagram with n bubbles (n ≥ 1). Dou-ble lines represent particles with Γ 6= 0.2 Electrical conductivityWe are interested in the DC electrical conductivity, whichrepresents the response of the gas under the action of aconstant electric field. The expression for this transportcoefficient in LRT is:σ = − limq0→0+lim|q|→0+ρσ(q0, q)6q0,withρσ(q0, q) = 2 Im i∫d4x eiq·xθ(t)〈[Ji(x), J i(0)]〉 ,and J i is the electric current density in the gas. For thistransport coefficient, at T ≪ Mπ, Y ≃√Mπ/T . So thecontribution from a single-bubble diagram to the conduc-tivity at these temperature is σ(0) ≃ e2Mπ√Mπ/T , wheree is the charge of the electron. The pion width is calculatedin ChPT [5] from the pion-pion scattering amplitudes.To consider the unitarity effects on the transport coeffi-cients, we impose unitarity on these amplitudes by meansof the Inverse Amplitude Method (IAM) [7]. This unita-rized scattering amplitudes have also been calculated atfinite temperature [8]. We have implemented unitarizationin the dilute gas approximation, where we neglect O(n2B)terms, with nB the Bose-Einstein distribution evaluatedat a pion energy. This is approximately valid for T ≤ Mπ.In Fig. 3 we represent the behavior with temperature ofthe σ(0) contribution to conductivity. We see that uni-tarity makes conductivity grow near T = 170 MeV, asis expected in the QGP phase [9]. The contribution fromladder diagrams with derivative vertices has to be takeninto account for temperatures T & Mπ.The DC conductivity is related to the zero-energy (soft)photon spectrum emitted by the pion gas. Considering apion gas produced after a Relativistic Heavy Ion Colli-sion which expands cylindrically (Bjorken’s hydrodynam-ical model), we obtain [6] the estimate EdNγ/d3p(pT →0+) ≃ 5.6 × 102 GeV−2 for the photon spectrum at zerotransverse momentum. In Fig. 4 we see that this estimateis compatible with the experimental results obtained bythe experiment WA98 [10] and other theoretical analy-sis [11]. This is reasonable since it is expected that thehadronic contribution dominates at low pT .3 Shear viscosityThe expression for shear viscosity in LRT isη = limq0→0+lim|q|→0+ρη(q0, q)20q0,0 50 100 150 200 25000.0050.010.0150.020.0250.030.0350.040.0450.05σ(0)MπT (MeV)unitarized O(p4)T -unitarized O(p4)O(p4)Fig. 3. Contribution to the electrical conductivity from thesingle-bubble diagram as a function of temperature. Threecurves for the pion width are shown: non-unitarized O(p4)(dashed), T = 0 unitarized and finite T unitarized.10-710-610-510-410-310-210-11101020 1 2 3 4Correlation methodSubtraction methodUpper limit,subtraction methodPredictionsHadr. GasQGPpQCDSumpT  (GeV/c)E dN/d3 p (GeV-2)Fig. 4. Photon spectrum obtained by the experiment WA98[10]. We show the linear extrapolation from the two lowestenergy points, which reaches a point at the origin compatiblewith our estimate. Note that theoretical predictions tend tounderestimate the lowest energy points [11].whereρη(q0, q) = 2 Im i∫d4x eiq·xθ(t)〈[πij(x), πij(0)]〉 ,and πij is the traceless part of the spatial energy-momentumtensor, i.e.πij ≡ Tij −13gijTll .The shear viscosity has been previously calculated fora meson gas in the KT framework [12,13,14]. For instance,in the work [12] (referred as DLE), a treatment based on arelativistic version of the Boltzmann equation is followed,using unitarized amplitudes from ChPT. In Fig. 5 we showour leading order result for η, where we see that unitarityalso makes the shear viscosity change its behavior with T .Following the same steps as in [6], we readily get the be-havior for T ≪ Mπ as η ≃ 37F 4π√T/M3/2π (Fπ ≃ 93 MeVis the pion decay constant). This behavior is consistentwith nonrelativistic KT [12].D. Fernández-Fraile, A. Gómez Nicola: Transport coefficients in Chiral Perturbation Theory 3In Fig. 5 we compare our leading order result with [12,13]. The analysis in [12] is performed for fixed fugacityz ≡ eβ(µπ−Mπ). We work at µπ = 0, so that for the twovalues of z shown in Fig. 5, we must compare at temper-atures T ≃ 30 MeV (for z = 0.01) and T ≃ 200 MeV (forz = 0.5). We get a reasonable agreement with [13] and alower η than [12] for T > Mπ. The results in [14], wherekaons are also included in the gas, also agree with oursfor low and moderate temperatures. Nevertheless, our re-sults should be taken with care, since for the shear viscos-ity, ladder diagrams with derivative vertices could becomeimportant already at T ≃ 50 MeV.0 20 40 60 80 100 120 140 160 180 20000.0020.0040.0060.0080.010.0120.0140.016η(0)(GeV3)T (MeV)O(p4)unitarized O(p4)[Davesne] 1st order[DLE] z = 0.01[DLE] z = 0.5Fig. 5. Behavior of the unitarized shear viscosity with tem-perature and comparison with the results obtained in [12,13].An interesting quantity is the ratio η/s, where s is theentropy density. It has been conjectured [15] that the value1/(4π) is a universal lower bound for this quantity in anyphysical system. On the other hand, the sound attenuationlength Γs ≡ 4η/(3sT ) (neglecting bulk viscosity) enters di-rectly in phenomenological effects observed in RelativisticHeavy Ion Collisions, such as elliptic flow or HBT radii[16]. In Fig. 6 we plot this ratio for the pion gas. We seethat in our analysis η/s decreases monotonously, but itrespects the KSS bound for temperatures up to the chiralphase transition Tc ≃ 180−200 MeV. On general grounds,one expects η/s to decrease below Tc and increase loga-rithmically for very high temperatures [17]. The numericalvalues we get for Γs(≃ 1.1 fm at T = 180 MeV) are in re-markable agreement with those used in [16] and at highT our curve is not far from recent lattice and model esti-mates giving η/s ∼ 0.4 − 0.5 above Tc [18].In conclusion, we have presented a systematic methodfor evaluating transport coefficients in ChPT. The stan-dard ChPT has to be modified, and as we go to highertemperatures, more diagrams have to be taken into ac-count and eventually resummed. Even though the limi-tations of our approach are important, we get a reason-able agreement for physical quantities such as the very lowpT photon spectrum or the viscosity to entropy ratio. Weare currently analyzing the high-temperature diagrams,as well as the inclusion of kaons and other transport co-0 20 40 60 80 100 120 140 160 180 20000.511.522.53η(0)sT (MeV)unitarized O(p4)1/(4π)Fig. 6. Dependence of the η/s ratio with temperature inChPT. The entropy density s is calculated to O(p5) [19].efficients such as bulk viscosity and thermal conductivity.Detailed results will be reported elsewhere.We are grateful to A. Dobado and F. Llanes-Estrada for theiruseful comments. We also acknowledge financial support fromthe Spanish research projects FPA2004-02602, PR27/05-13955-BSCH, FPA2005-02327 and FPI fellowship BES-2005-6726.References1. J. Gasser and H. Leutwyler, Ann. Phys. 158, (1984) 142.2. M. Le Bellac, Thermal field theory (Cambridge Univ. Press,2000).3. S. Jeon, Phys. Rev. D 52, (1995) 3591-3642.4. M. A. Valle Basagoiti, Phys. Rev. D 66, (2002) 045005.5. J. L. Goity and H. Leutwyler, Phys. Lett. B 228, (2002)517.6. D. Fernández-Fraile and A. Gómez Nicola, Phys. Rev. D73, (2006) 045025.7. A. Gómez Nicola and J. R. Pelaez, Phys. Rev. D 65, (2002)054009.8. A. Dobado, A. Gómez Nicola, F. J. Llanes-Estrada and J.R. Pelaez, Phys. Rev. C 66, (2002) 055201.9. P. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0011, (2000)001.10. M. M. Aggarwal et al. (WA98 Collaboration), Phys. Rev.Lett. 93, (2004) 022301.11. S. Turbide, R. Rapp and C. Gale, Phys. Rev. C 69, (2004)014903; W. Liu and R. Rapp, nucl-th/0604031.12. A. Dobado and F. J. Llanes-Estrada, Phys. Rev. D 69,(2004) 116004.13. D. Davesne, Phys. Rev. C 53, (1996) 3069-3084.14. M. Prakash, M. Prakash, R. Venugopalan and G. M.Welke, Phys. Rev. Lett. 70, (1993) 1228.15. P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett.94, (2005) 1 11601.16. D. Teaney, Phys. Rev. D 68, (2003) 034913.17. L. P. Csernai, J. I. Kapusta and L. D. McLerran, Phys.Rev. Lett. 97, (2006) 152303.18. A. Nakamura and S. Sakai, Phys. Rev. Lett. 94, (2005)72305; B. A. Gelman, E. V. Shuryak and I. Zahed,nucl-th/0601029.19. P. Gerber and H. Leutwyler, Nucl. Phys. B 321, (1989)387.

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Transport coefficients in chiral perturbation theory

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