Presentation on theme: "Strong and Electroweak Matter 2004. Helsinki, 16-19 June. Angel Gómez Nicola Universidad Complutense Madrid."— Presentation transcript:
Strong and Electroweak Matter 2004. Helsinki, 16-19 June. Angel Gómez Nicola Universidad Complutense Madrid
Motivation T>0 ChPT pion electromagnetic form factors Thermal and poles Motivation
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
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
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.
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
Motivation T>0 ChPT pion electromagnetic form factors Thermal and poles T>0 ChPT pion electromagnetic form factors
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)
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
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)
Model independent ! Confirms Dominguez et al ’94 (QCD sum rules) The pion electromagnetic charge radius at T>0 (rough) deconfinement estimate: Charge screening Kapusta
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
Motivation T>0 ChPT pion electromagnetic form factors Thermal and poles
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).
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.
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)
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,...
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)
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
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 )
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