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Chiral Transition in a Strong Magnetic Background

Chiral Transition in a Strong Magnetic Background. Eduardo S. Fraga. Instituto de Física Universidade Federal do Rio de Janeiro. Introduction and Motivation. Strong interactions under intense magnetic fields can be found, in principle, in a variety of systems:

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Chiral Transition in a Strong Magnetic Background

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  1. Chiral Transition in a Strong Magnetic Background Eduardo S. Fraga Instituto de Física Universidade Federal do Rio de Janeiro

  2. Introduction and Motivation • Strong interactions under intense magnetic fields can be found, • in principle, in a variety of systems: • High density and low temperature • “Magnetars”: B ~ 1014-1015 G at the surface, much higher in the core [Duncan & Thompson (1992/1993)] • Stable stacks of p0 domain walls or axial scalars (h,h’) domain walls in nuclear matter: B ~ 1017-1019 G[Son & Stephanov (2008)] QCD School in Les Houches, April/2008

  3. High temperature and low density • Early universe (relevant for nucleosynthesis): B~ 1024 G for the EWPT epoch [Grasso & Rubinstein (2001)] • Non-central RHIC collisions: eB~ 104-105 MeV2 ~ 1019 G [Kharzeev, McLerran & Warringa (2007)] [Au-Au, 62 GeV] [Au-Au, 200 GeV] QCD School in Les Houches, April/2008

  4. QCD School in Les Houches, April/2008

  5. QCD School in Les Houches, April/2008

  6. Besides, there are several theoretical/phenomenological interesting questions: • How does the QCD phase diagram looks like including a nonzero uniform B ? (another interesting “control parameter” ?) • Are there modifications in the nature of phase transitions ? • Does it affect significantly time scales for phase conversion ? • Are there any new phenomena ? Some of these questions have already been addressed in different ways. Here, we consider effects on the chiral transition at finite temperature and zero density in the Linear Sigma Model with Quarks. QCD School in Les Houches, April/2008

  7. Other approaches (usually concerned about vacuum effects): • NJL: • Klevansky & Lemmer (1989) • Gusynin, Miransky & Shovkovy (1994/1995) • Klimenko et al. (1998-2008) • Hiller, Osipov, … (2007-2008) • … • cPT: • Shushpanov & Smilga (1997) • Agasian & Shushpanov (2000) • Cohen, McGady & Werbos (2007) • … • Large-N QCD: • Miransky & Shovkovy (2002) • Quark model: • Kabat, Lee & Weinberg (2002) QCD School in Les Houches, April/2008

  8. Outline • Effective theory for the chiral transition: the linear s model • Incorporating a strong magnetic field background • The modified effective potential • Some phenomenological consequences • Final remarks QCD School in Les Houches, April/2008

  9. Effective theory for the chiral transition (LsM) [Gell-Mann & Levy (1960); Scavenius, Mócsy, Mishustin & Rischke (2001); …] • Symmetry: for massless QCD, the action is invariant under SU(Nf)L x SU(Nf)R • “Fast” degrees of freedom: quarks “Slow” degrees of freedom: mesons • Typical energy scale: hundred of MeV • We choose SU(Nf=2), for simplicity: we have pions and the sigma • Framework: coarse-grained Landau-Ginzburg effective potential • SU(2)  SU(2) spontaneously broken in the vacuum • Also accommodates explicit breaking by massive quarks QCD School in Les Houches, April/2008

  10. Building the effective lagrangian • Kinetic terms: • Fermion-meson interaction: • Chiral self-interaction: • Explicit chiral symmetry breaking term: [with scalars allowed by c symmetry] QCD School in Les Houches, April/2008

  11. Parameters should be fixed such that: • SU(2)L SU(2)R is spontaneously broken in the vacuum, with <s> = fp, <p> = 0 • h should be related to the nonzero pion mass (plays a role analogous to an external magnetic field for a spin system) • fp = 93 MeV is the pion decay constant, determined experimentally. It comes about when one computes the weak decay of the pion, which is proportional to the amplitude a,b: isospin • The small but nonzero pion mass breaks “softly” the axial current (PCAC): QCD School in Les Houches, April/2008

  12. Including a term ~hs brings the following consequences: - The true vacuum (in the s direction) is shifted [redefine fp such that it coincides with the experimental value] - The s mass is modified - Pions acquire a nonzero mass which fixes h to be: All parameters can be chosen to reproduce the vacuum features of mesons. QCD School in Les Houches, April/2008

  13. - The connection with the quark mass is given by the Gell-Mann--Oakes--Renner (GOR) relation: Connection not only between mp and mq, but also between the s field condensate and the chiral condensate “by construction”, since one wants this term to mimic the QCD explicit breaking of chiral symmetry - In a medium, one can use <s>(T) in the effective theory to describe the melting of the chiral condensate at high T. QCD School in Les Houches, April/2008

  14. Putting s and pi together in an O(4) field f=(s,pi), we have Lagrangian: Partition function: Integrating over the fermions (heat bath for the long wavelength chiral fields), we obtain an effective thermodynamic potential forf=(s,p) QCD School in Les Houches, April/2008

  15. Example of an effective potential in the s direction (modulo inhomogeneity corrections which tend to reduce the barrier) For a 1st order chiral transition [Aguiar, ESF & Kodama (2006)] QCD School in Les Houches, April/2008

  16. Incorporating a strong magnetic field background Let us assume the system is in the presence of a strong magnetic field background that is constant and homogeneous: choice of gauge • charged mesons (new dispersion relation): Landau levels: QCD School in Les Houches, April/2008

  17. quarks (new dispersion relation): • integration measure: T = 0: T > 0: l: Matsubara index n: Landau level index QCD School in Les Houches, April/2008

  18. The modified effective potential [ESF & Mizher, in prep.] • Simple mean-field treatment with the following customary simplifying assumptions [Scavenius, Mócsy, Mishustin & Rischke (2001); Dumitru & Paech (2005); …] : • Quarks constitute a thermalized gas that provides a background in which the long wavelength modes of the chiral condensate evolve. Hence: • At T = 0 (vacuum: c symm. broken; reproduce usual LsM & cPT results) • Quark d.o.f. are absent (excited only for T > 0) • The s is heavy (Ms~600 MeV) and treated classically • Pions are light: fluctuations in p+ and p- couple to B; fluctuations in p0 give a B-independent contribution (ignored here) QCD School in Les Houches, April/2008

  19. At T > 0 (plasma: c symm. approximately restored) • Quarks are relevant (fast) degrees of freedom: incorporate their thermal fluctuations in the effective potential for s (integrate over quarks) • Pions become rapidly heavy only after Tc, so we incorporate their thermal contribution • Later: ZPT, CJT resummations, etc - here, the simplest phenomenological approach QCD School in Les Houches, April/2008

  20. Vacuum effective potential (s direction): (s now means <s>) Classical: p+ and p- fluctuations: Computing the contribution from pions in the MSbar scheme, we obtain (ignoring s-independent terms): using the assumption of large magnetic field, |q|B >> mp 2, in the expansion of generalized Zeta functions of the form QCD School in Les Houches, April/2008

  21. Results in line with calculations in cPT and NJL, as in e.g. • - Shushpanov & Smilga (1997) • - Cohen, McGady & Werbos (2007) • Hiller, Osipov et al. (2007/2008) • … However:for very large B, effects from the quarks could become important - non-trivial transition? [Kabat, Lee & Weinberg (2002)] (later…) “Large” (“critical”) B for QCD: • Condensate grows with increasing magnetic field • Minimum deepens with increasing magnetic field • Relevant effects for equilibrium thermodynamics and nonequilibrium process of phase conversion ? QCD School in Les Houches, April/2008

  22. Thermal corrections: p+ and p- fluctuations: quark fluctuations: Computing in the MSbar scheme (ignoring s-independent terms and assuming a large magnetic field - also compared to T - in the expansion of zeta functions), we obtain: QCD School in Les Houches, April/2008

  23. Ignoring exponentially suppressed contributions (besides ZPT, etc), the effective potential is given by • Remarks: • Exponential suppresions come as • In what follows, we take Nc=3, g=3.3 (to reproduce the nucleon mass), and eB given in units of mp2 : • Other parameters are fixed to fit vacuum conditions, as customary. • For very large B, the n = 0 Landau level dominates. Corrections can be incorporated systematically. QCD School in Les Houches, April/2008

  24. B = 0: • For g=3.3, one has a crossover at m=0 [g=5.5, e.g., yields a 1st order transition] • Critical temperature: Tc~ 140-150 MeV QCD School in Les Houches, April/2008 [Scavenius et al. (2001)]

  25. eB = 5 mp2: • Higher critical temperature: Tc > 200 MeV • Tiny barrier: very weakly 1st order chiral transition! QCD School in Les Houches, April/2008

  26. eB = 10 mp2: • Critical temperature goes down again due to the larger hot fermionic contribution (Tc < 140 MeV) • Larger barrier: clear 1st order chiral transition! • Non-trivial balance between T and B… one needs to explore the phase diagram QCD School in Les Houches, April/2008

  27. eB = 20 mp2: • Even lower critical temperature • Large barrier persists: 1st order chiral transition QCD School in Les Houches, April/2008

  28. Phenomenological consequences [ESF & Mizher, in prep.] • At RHIC, estimates by Kharzeev, McLerran and Warringa (2007) give: • For LHC, we have a factor (ZPb/ZAu = 82/79) and some small increase in the maximum value of eB due to the higher CM energy (as observed for RHIC). So, it is reasonable to consider QCD School in Les Houches, April/2008

  29. B = 0: eB = 6 mp2: • Rapid crossover (no barrier) • Tc~ 140-150 MeV • System smoothly drained to the true vacuum: no bubbles or spinodal instability • Weak 1st order (tiny barrier) • Tc> 200 MeV • Part of the system kept in the false vacuum: some bubbles and spinodal instability, depending on the intensity of supercooling QCD School in Les Houches, April/2008

  30. Comparing barriers: eB = 6 mp2: [Taketani & ESF (2006)] • g = 3.3 - crossover for B=0; very weak 1st order phase transition for B > 0 • barrier ~ 0.025 close to Tc • But even such small barriers can hold the system in the false vacuum until the spinodal for a fast enough supercooling ! • explosive phase conversion ? • g = 5.5 - clear 1st order phase transition for m=0 and B=0 • barrier ~ 0.25 close to Tc • System mostly apprisionated in the false vacuum until the spinodal • explosive phase conversion QCD School in Les Houches, April/2008

  31. Final remarks • Lattice QCD indicates a crossover instead of a 1st order chiral transition at finite temperature and m=0. However, a strong magnetic background might invert this situation. • For RHIC and LHC heavy ion collisions, the barrier in the effective potential seems to be quite small. Nevertheless, it can probably hold most of the system in a metastable state down to the spinodal explosion. -> Different dynamics of phase conversion. • However: B falls off rapidly in the case of RHIC - early-time dynamics might be affected. QCD School in Les Houches, April/2008

  32. The phenomenlogy resulting from varying T and B seems to be rich: competition between strengthening the chiral symmetry breaking via vacuum effects and its restoration by the thermal (magnetic) bath. • Non-central heavy ion collisions might show features of a 1st order transition when contrasted to central collisions. • Caveat: treatment still admittedly very simple - in heavy ion collisions, B varies in space and time. It can, e.g., induce an electric field that might play a role [Cohen et al. (2007)]. • Nevertheless, clean results for the case of constant field are encouraging. QCD School in Les Houches, April/2008

  33. To do list: • More realistic treatment of the effective model (ZPT, resummation, etc) • Investigation of the low magnetic field regime at finite T, for B < T and B ~ T. • Simulation of time evolution of the phase conversion process to compare relevant time scales to those in the crossover picture. • Possible signatures of these features in heavy ion collisions? • Situation at high density and applications to compact stars: phase structure inside magnetars. QCD School in Les Houches, April/2008

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