1. Strong and Electroweak Matter 2004. Helsinki, 16-19 June. Angel Gómez Nicola Universidad Complutense Madrid

2. Motivation T>0 ChPT pion electromagnetic form factors Thermal  and  poles Motivation

3. After QGP hadronization and  SB, the description of the meson gas must rely on Chiral Perturbation Theory (model independent, chiral power counting p,T << 1 GeV ) Only NGB mesons (and photons) involved. L 2 loops are O(p 2 ), divergences absorbed in L 4 and so on. L2L2 L4L4 L22L22 L6L6 Derivative and mass expansion nonlinear  -model S.Weinberg, ‘79 J.Gasser&H.Leutwyler ’84,’85

4. point towards Chiral Symmetry Restoration: J.Gasser&H.Leutwyler ‘87 P.Gerber&H.Leutwyler ‘89 A.Bochkarev&J.Kapusta ‘96 A.Dobado, J.R.Peláez ’99 ’01. T T=0 T

5. J.L.Goity&H.Leutwyler, ‘89 A.Schenk, ‘93 R.Pisarski&M.Tytgat, ‘96 D.Toublan, ‘97 J.M.Martinez Resco&M.A.Valle, ‘98 Pion dispersion law: Nonequilibrium ChPT: f  (t),  amplification via parametric resonance. AGN ‘01

6. However, ChPT alone cannot reproduce the light resonances ( , ,...) Needed to explain observed phenomena in RHIC. K.Kajantie et al ’96 C.Gale, J.Kapusta ’87 ‘91 G.Q.Li,C.M.Ko,G.E.Brown ‘95 H.J.Schulze, D.Blaschke ‘96,’03 V.L.Eletsky et al ‘01 Enhancement consistent with a dropping M  and a significant broadening    in the hadron gas at freeze-out.

7. CHIRAL SYMMETRY BREAKING UNITARITY + Inverse Amplitude Method “Thermal” poles Dynamically generated (no explicit resonance fields) OUR APPROACH AGN, F.J.Llanes-Estrada, J.R.Peláez PLB550, 55 (2002), hep-ph/0405273 A.Dobado, AGN, F.J.Llanes-Estrada, J.R.Peláez, PRC66, 055201 (2002)  scattering amplitude and  form factors in T > 0 SU(2) ChPT

8. Motivation T>0 ChPT pion electromagnetic form factors Thermal  and  poles T>0 ChPT pion electromagnetic form factors

9. Pion form factors enter directly in the dilepton rate: In the central region the dominant channel is pion annihilation:    e+e+ e-e-    e+e+ e-e-  ~+... (thermal equilibrium)

10. At T>0 a more general structure is allowed: k = p 1 - p 2 S = p 1 + p 2 ChPT to O(p 4 ) (At T = 0, F t (S 2 )= F s (S 2 ), G s = 0) Related by gauge invariance to  dispersion law in hot matter

11. T  0 limit (J.Gasser&H.Leutwyler 1984). Gauge invariance condition.Thermal perturbative unitarity in the c.o.m. frame (see later) T>0 ChPT calculation to O(p 4 ): L 2 one loop L 4 tree level (including renormalization)

12. Model independent ! Confirms Dominguez et al ’94 (QCD sum rules) The pion electromagnetic charge radius at T>0 (rough) deconfinement estimate: Charge screening Kapusta

13. H.A.Weldon ’92 Enhancement Absorption Thermal perturbative unitarity: Likewise, for the thermal amplitude: Consider c.om. frame (, back to back dileptons) 2  thermal phase space: (1+n B ) 2  n B 2 I=J=1  scattering partial wave 1 to lowest order

14. Motivation T>0 ChPT pion electromagnetic form factors Thermal  and  poles

15. Excellent T=0 data description up to 1 GeV energies and resonance generation as s poles in the complex amplitude. T.N.Truong, ‘88 A.Dobado, M.J.Herrero,T.N.Truong, ‘90 A.Dobado&J.R.Peláez, ’93,’97 J.A.Oller, E.Oset, J.R.Peláez, ’99 A.Dobado, M.J.Herrero, E.Ruiz Morales ‘00 AGN&J.R.Peláez ‘02 Unitarization: The Inverse Amplitude Method Exact unitarity at T>0 + ChPT matching at low energies Valid to O(n B ) (only 2  thermal states, dilute gas).

16. SU(2) L 4 constants from T=0 fit of phase shifts:  (770)  Thermal  and  poles I=J=0I=J=1 2n B (M  /2)  0.3 Consistent with Chiral Symmetry Restoration: : M   M   m  (m  (T) much softer)    first by phase space but decreases as M   m  suppresses    2  decay. (similar results to T.Hatsuda, T.Kunihiro et al, ’98,’00 ) * Small M  change at low T (VMD*). Further decrease consistent with phenomenological estimates and observed behaviour (STAR  ) * M.Dey, V.L.Eletsky&B.L.Ioffe, 1990 Significant  broadening as required by dilepton data.

17. The unitarized form factor Peak reduction and spreading around M  compatible with dilepton spectrum (n B contributions alone overestimate data) and other calculations including explicitly resonances under VMD assumption (C.Song and V.Koch, ’96) m  = 139.6 MeV f  = 92.4 MeV (T=0 form factor fit)

18. Chiral Perturbation Theory provides model-independent predictions for meson gas properties. In one-loop ChPT, we have calculated scattering amplitudes and the two independent form factors, checking gauge invariance and thermal unitarity. The electromagnetic pion radius grows for T>100 MeV, favouring a deconfinement temperature T c ~200 MeV. Imposing unitarity in SU(2) allows to describe the thermal  and  poles in the amplitudes and form factors. Our results show a clear increase of   (T) and a slow M  (T) reduction consistently with theoretical and experimental analysis, including dilepton data.   (T) and M  (T) behave according to Chiral Symmetry Restoration. Angular dependence, plasma expansion,  +  -    e + e , baryon density, hadronic photon spectrum,...

19.  VMD coupling For s  M T 2,  T <<  T and a  at rest: (Breit-Wigner) IAM + Thermal Unitarity (expected very low-T behaviour:  T  only by phase space increase) M T  M 0 R T  R 0 With our calculated a (4) (s;  ) Higher T  T,M T, R T corrections  and  thermal poles for s   (appropriate analytic structure)

20. T-dependence of the phase shifts: Temperature enhances the interaction strength in all channels  11 gives the  tail  Compatible with             at low T The enhancement is dominated by phase space  T (  SR,T.Hatsuda et al) (not a strong | a 00 | enhancement near threshold) H.A.Weldon, 1983 M.Dey, V.L.Eletsky&B.L.Ioffe, 1990 C.Gale&J.Kapusta, 1991 R.D.Pisarski, 1995 VMD prediction

21. From the IAM poles  (770)  Not a BW ! From the IAM I=J=1 phase shift shape SU(2) L 4 constants from T=0 fit: M 0 = 770 MeV  0 = 159 MeV Consistent with Chiral Symmetry Restoration: : M   M   m  (m  (T) much softer, A.Schenk, 1993 )    at first by  T but decreases as M   m  disallows    2  decay. (similar results to T.Hatsuda, T.Kunihiro et al )

22. 2n B (M  /2)  0.3 M T decrease consistent with phenomenological analysis Little M T change at low T, as predicted by VMD Effective VMD  vertex (g 0  6.2). Expected low T behaviour (C.Song&V.Koch 1996). Significant deviations from pure thermal phase space broadening as T increases

24. m  = 139.6 MeV f  = 92.4 MeV (T=0 fit)