1. Understanding the QGP through Spectral Functions and Euclidean Correlators BNL April 2008 Angel Gómez Nicola Universidad Complutense Madrid IN MEDIUM LIGHT MESON RESONANCES AND CHIRAL SYMMETRY RESTORATION

2. Spectral properties of light meson resonances in hot and dense matter: Motivation f 0 (600)/  → vacuum quantum numbers, chiral symmetry restoration Observed in nuclear matter experiments (CHAOS, …) through threshold enhancement? Any chance for Heavy Ions (finite T)?  → dilepton spectrum (CERES,NA60) and nuclear matter Broadening vs Mass shift (scaling?) What can medium effects tell about the nature of these states?

3. DILEPTONS NA45/CERES ( e + e - ) Compatible with both broadening and dropping-mass scenarios NA60 (  +  - ) Broadening favored, dropping mass almost excluded Rapp-WambachBrown/Rhomeson cocktail 2000 data

4.  → e + e - IN NUCLEAR MATTER Signals free of T≠0 complications Linear decrease of vector meson masses from scaling&QCD sum rules: Brown, Rho ‘91 Hatsuda, Lee ‘92 normal nuclear matter density Experiments not fully compatible: KEK-E325 (C,Fe-Ti):  = 0.092  0.002 Jlab-CLAS (C,Cu):  = 0.02  0.02 Cabrera,Oset,Vicente-Vacas ‘02 Urban, Buballa, Rapp,Wambach ‘98 Chanfray,Schuck ‘98 Other many-body approaches give negligible mass shift

5.  production in Nuclear Matter: threshold enhancement in the  (I=J=0) channel  A →  A’  A →     A’ Crystal Ball CHAOS MAMI-B

6. Threshold enhancement as a signal of chiral symmetry restoration: M   decreases, so that when M   2m , phase space is squeezed    0 and the  pole reaches the real axis Hatsuda, Kunihiro ‘85 O(N) models at finite T show that the  remains broad when M   2m  Further in-medium strength causes 2nd-sheet pole to move into 1st sheet   bound state. Hidaka et al ‘04 Patkos et al ‘02 Finite density analysis compatible with threshold enhancement of  cross section Davesne, Zhang, Chanfray ‘00 Roca et al ‘02 Narrow resonance argument !

7. CHIRAL SYMMETRY UNITARITY + Inverse Amplitude Method “Thermal” poles Dynamically generated (no explicit resonance fields) OUR APPROACH: UNITARIZED CHIRAL PERTURBATION THEORY AGN, F.J.Llanes-Estrada, J.R.Peláez PLB550, 55 (2002), PLB606:351-360,2005 A.Dobado, AGN, F.J.Llanes-Estrada, J.R.Peláez, PRC66, 055201 (2002) D.Fernández-Fraile,AGN, E.Tomás-Herruzo, PRD76:085020,2007  scattering amplitude and  form factors in T > 0 SU(2) one-loop ChPT

8. Chiral Perturbation Theory: Relevant for low and moderate temperatures below Chiral SSB Weinberg’s chiral power counting: NLSM Most general derivative and mass expansion of NGB mesons compatible with the SSB pattern of QCD model-independent low-energy predictions.

9. Perturbative Unitarity In the two-pion c.om. frame: (static resonaces): ChPT does not reproduce resonances due to the lack of exact unitarity (resonances saturate unitarity bounds). Unitarization: The Inverse Amplitude Method Two-pion thermal phase space enhancement Enhancement  Absorption

10. Exact unitarity + ChPT matching at low energies * At T>0, valid for dilute gas (only two-pion states). Thermal  and  poles (2nd Riemann sheet) * Very sucessful at T=0 for scattering data up to 1 GeV and low-lying resonance multiplets, also for SU(3) Dobado, Peláez, Oset, Oller, AGN.

11. = 20 MeV Thermal phase space enhancement + Increase of effective  vertex, small mass reduction up to T c. THE THERMAL  POLE (for a narrow Breit-Wigner resonance) (2nd Riemann sheet)

12. The unitarized EM pion form factor shows also broadening compatible with dilepton data and VMD analysis:

13. THE THERMAL f 0 (600)/  POLE Strong pole mass reduction (chiral restoration) means phase space squeezing, which overcomes low- T thermal enhancement (2nd Riemann sheet) T=100 MeV = 20 MeV However, the pole remains wide even for M ~2 m  (spectral function not peaked around the mass for broad resonances)

14. Narrow vs Broad Resonances

15. NARROW:  ( s ) strongly peaked around Phase space squeezing Threshold enhancement (R “particle” at rest) 2-particle differential phase space differential decay rate Narrow vs Broad Resonances BROAD:  ( s ) broadly distributed  pole away from the real axis  ChPT approach valid at threshold  no enhancement Generalized decay rate: H.A.Weldon, Ann.Phys.228 (1993) 43 NO phase-squeezing for wide enough  s  !

16. Narrow vs Broad Resonances:

17. REAL AXIS POLES AND ADLER ZEROS Require extra terms in the IAM to account properly for Adler zeros  t(s A )=0. Otherwise, spurious real poles below threshold in the 1st,2nd Riemann sheets. Preserving chiral symmetry+unitarity: No difference away from s A No additional poles for T  0 with the redefined amplitudes. Alternatively derived with dispersion relations. AGN,J.R.Peláez,G.Ríos PRD77, 056006 (2008) No problem for I=J=1 

18. THE NATURE OF THERMAL RESONANCES: f 0 (600)/  Does not behave as a (thermal) state, not even near the chiral limit Consistent with not- scalar nonet (tetraquark,glueball,meson-meson…) “molecule” picture J.R.Peláez ‘04 M.Alford,R.L.Jaffe ‘00

19. THE NATURE OF THERMAL RESONANCES:  No BR-like scaling with condensate. Mass dropping only very near “critical” (too high) T 0, as in BR-HLS models Harada&Sasaki ‘06 Brown&Rho ‘05 Nature of our thermal  dominated by non-restoring effects (broadening)

20. NUCLEAR CHIRAL RESTORING EFFECTS Chiral restoring effects at T=0 and finite nuclear density approx. encoded in f  Meissner,Oller,Wirzba ‘02 Thorsson,Wirzba ‘95 Justified by approximate validity of GOR (  0,T=0) Non chiral-restoring many-body effects not included (p-h, p-wave  self-energy, …) Cabrera,Oset,Vicente-Vacas ‘05 Chiral restoring expected to be important in the  -channel as density  approaches  the transition. No broadening to compete with now !

21.  bound state (“molecule” behaviour)  0 9.1 

22. Mass linear fits: Compatible with some theoretical estimates and KEK experiment. However, additional medium effects (important in this channel!) might lead to negligible mass shift  No threshold enhancement for reasonably high densities. Compatible with BR-like scaling Brown,Rho ‘04 “non-molecular” ( )

23. In-medium light meson resonances studied through scattering poles in Unitarized ChPT provide chiral symmetry predictions for their spectral properties and nature. The f 0 (600)/  shows chiral symmetry restoration features but remains as a T  0 wide not- state  no threshold enhancement at finite T. The  finite-T behaviour is dominated by thermal broadening in qualitative agreement with dilepton data. Mass dropping does not scale with the condensate. Nuclear density chiral-restoring effects encoded in f   drive the poles to the real axis giving threshold enhancement in the  -channel and BR-like scaling in the  -channel.  bound states of different nature formed near the transition. Full finite-density analysis, SU(3) extension ( ,K*,a 0,…)