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

2. QCD School in Les Houches, April/20082 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 ~ 10 14 -10 15 G at the surface, much higher in the core “Magnetars”: B ~ 10 14 -10 15 G at the surface, much higher in the core [Duncan & Thompson (1992/1993)] [Duncan & Thompson (1992/1993)] Stable stacks of  0 domain walls or axial scalars ( ,  ’) domain walls in nuclear matter: B ~ 10 17 -10 19 G [Son & Stephanov (2008)] Stable stacks of  0 domain walls or axial scalars ( ,  ’) domain walls in nuclear matter: B ~ 10 17 -10 19 G [Son & Stephanov (2008)]

3. QCD School in Les Houches, April/20083  High temperature and low density Early universe (relevant for nucleosynthesis): B ~ 10 24 G for the EWPT epoch [Grasso & Rubinstein (2001)] Early universe (relevant for nucleosynthesis): B ~ 10 24 G for the EWPT epoch [Grasso & Rubinstein (2001)] Non-central RHIC collisions: eB ~ 10 4 -10 5 MeV 2 ~ 10 19 G Non-central RHIC collisions: eB ~ 10 4 -10 5 MeV 2 ~ 10 19 G [Kharzeev, McLerran & Warringa (2007)] [Au-Au, 200 GeV] [Au-Au, 62 GeV]

4. QCD School in Les Houches, April/20084

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6. 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” ?) 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 ? Are there modifications in the nature of phase transitions ? Does it affect significantly time scales for phase conversion ? Does it affect significantly time scales for phase conversion ? Are there any new phenomena ? 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.

7. QCD School in Les Houches, April/20087 Other approaches (usually concerned about vacuum effects): NJL: Klevansky & Lemmer (1989) Klevansky & Lemmer (1989) Gusynin, Miransky & Shovkovy (1994/1995) Gusynin, Miransky & Shovkovy (1994/1995) Klimenko et al. (1998-2008) Klimenko et al. (1998-2008) Hiller, Osipov, … (2007-2008) Hiller, Osipov, … (2007-2008) … …  PT: Shushpanov & Smilga (1997) Shushpanov & Smilga (1997) Agasian & Shushpanov (2000) Agasian & Shushpanov (2000) Cohen, McGady & Werbos (2007) Cohen, McGady & Werbos (2007) … … Large-N QCD: Miransky & Shovkovy (2002) Miransky & Shovkovy (2002) Quark model: Kabat, Lee & Weinberg (2002) Kabat, Lee & Weinberg (2002)

8. QCD School in Les Houches, April/20088 Outline  Effective theory for the chiral transition: the linear  model the linear  model  Incorporating a strong magnetic field background  The modified effective potential  Some phenomenological consequences  Final remarks

9. QCD School in Les Houches, April/20089 Effective theory for the chiral transition (L  M) Symmetry: for massless QCD, the action is invariant under SU(N f ) L x SU(N f ) R Symmetry: for massless QCD, the action is invariant under SU(N f ) L x SU(N f ) R “Fast” degrees of freedom: quarks “Fast” degrees of freedom: quarks “Slow” degrees of freedom: mesons “Slow” degrees of freedom: mesons Typical energy scale: hundred of MeV Typical energy scale: hundred of MeV We choose SU(N f =2), for simplicity: we have pions and the sigma We choose SU(N f =2), for simplicity: we have pions and the sigma Framework: coarse-grained Landau-Ginzburg effective potential Framework: coarse-grained Landau-Ginzburg effective potential SU(2)  SU(2) spontaneously broken in the vacuum SU(2)  SU(2) spontaneously broken in the vacuum Also accommodates explicit breaking by massive quarks Also accommodates explicit breaking by massive quarks [Gell-Mann & Levy (1960); Scavenius, Mócsy, Mishustin & Rischke (2001); …] [Gell-Mann & Levy (1960); Scavenius, Mócsy, Mishustin & Rischke (2001); …]

10. QCD School in Les Houches, April/200810 Building the effective lagrangian Kinetic terms: Fermion-meson interaction: Chiral self-interaction: Explicit chiral symmetry breaking term: [with scalars allowed by  symmetry]

11. QCD School in Les Houches, April/200811 Parameters should be fixed such that: SU(2) L  SU(2) R is spontaneously broken in the vacuum, with = f , = 0 SU(2) L  SU(2) R is spontaneously broken in the vacuum, with = f , = 0 h should be related to the nonzero pion mass (plays a role analogous to an external magnetic field for a spin system) h should be related to the nonzero pion mass (plays a role analogous to an external magnetic field for a spin system) f  = 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 f  = 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): The small but nonzero pion mass breaks “softly” the axial current (PCAC):

12. QCD School in Les Houches, April/200812 Including a term ~ h  brings the following consequences: Including a term ~ h  brings the following consequences: - The true vacuum (in the  direction) is shifted [redefine f  such that it coincides with the experimental value] [redefine f  such that it coincides with the experimental value] - The  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.

13. QCD School in Les Houches, April/200813 - The connection with the quark mass is given by the Gell-Mann--Oakes--Renner (GOR) relation: “by construction”, since one wants this term to mimic the QCD explicit breaking of chiral symmetry Connection not only between m  and m q, but also between the  field condensate and the chiral condensate - In a medium, one can use (T) in the effective theory to describe the melting of the chiral condensate at high T.

14. QCD School in Les Houches, April/200814 Putting  and  i together in an O(4) field  =( , i ), we have Lagrangian: Partition function: Integrating over the fermions (heat bath for the long wavelength chiral fields), we obtain an effective thermodynamic potential for =(,)

15. QCD School in Les Houches, April/200815 Example of an effective potential in the  direction (modulo inhomogeneity corrections which tend to reduce the barrier) For a 1st order chiral transition [Aguiar, ESF & Kodama (2006)] [Aguiar, ESF & Kodama (2006)]

16. QCD School in Les Houches, April/200816 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) : charged mesons (new dispersion relation) : Landau levels:

17. QCD School in Les Houches, April/200817 quarks (new dispersion relation) : quarks (new dispersion relation) : integration measure: integration measure: T = 0: T > 0: l: Matsubara index n: Landau level index

18. QCD School in Les Houches, April/200818 The modified effective potential 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:  symm. broken; reproduce usual L  M &  PT results) Quark d.o.f. are absent (excited only for T > 0) Quark d.o.f. are absent (excited only for T > 0) The  is heavy (M  ~ 600 MeV) and treated classically The  is heavy (M  ~ 600 MeV) and treated classically Pions are light: fluctuations in  + and  - couple to B; Pions are light: fluctuations in  + and  - couple to B; fluctuations in  0 give a B-independent contribution (ignored here) fluctuations in  0 give a B-independent contribution (ignored here) [ESF & Mizher, in prep.]

19. QCD School in Les Houches, April/200819 At T > 0 (plasma:  symm. approximately restored) Quarks are relevant (fast) degrees of freedom: incorporate their thermal fluctuations in the effective potential for  (integrate over quarks) Quarks are relevant (fast) degrees of freedom: incorporate their thermal fluctuations in the effective potential for  (integrate over quarks) Pions become rapidly heavy only after T c, so we incorporate their thermal contribution Pions become rapidly heavy only after T c, so we incorporate their thermal contribution Later: ZPT, CJT resummations, etc - here, the simplest phenomenological approach

20. QCD School in Les Houches, April/200820 Vacuum effective potential (  direction): Classical:  + and  - fluctuations: Computing the contribution from pions in the MSbar scheme, we obtain (ignoring  -independent terms):  now means ) using the assumption of large magnetic field, |q|B >> m   , in the expansion of generalized Zeta functions of the form

21. QCD School in Les Houches, April/200821 Condensate grows with increasing magnetic field Condensate grows with increasing magnetic field Minimum deepens with increasing magnetic field Minimum deepens with increasing magnetic field Relevant effects for equilibrium thermodynamics and nonequilibrium process of phase conversion ? Relevant effects for equilibrium thermodynamics and nonequilibrium process of phase conversion ? Results in line with calculations in  PT 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:

22. QCD School in Les Houches, April/200822 Thermal corrections:  + and  - fluctuations: quarkfluctuations: Computing in the MSbar scheme (ignoring  -independent terms and assuming a large magnetic field - also compared to T - in the expansion of zeta functions), we obtain:

23. QCD School in Les Houches, April/200823 Ignoring exponentially suppressed contributions (besides ZPT, etc), the effective potential is given by Remarks: Exponential suppresions come as Exponential suppresions come as In what follows, we take N c =3, g=3.3 (to reproduce the nucleon mass), and eB given in units of m  2 : In what follows, we take N c =3, g=3.3 (to reproduce the nucleon mass), and eB given in units of m  2 : Other parameters are fixed to fit vacuum conditions, as customary. 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. For very large B, the n = 0 Landau level dominates. Corrections can be incorporated systematically.

24. QCD School in Les Houches, April/200824 B = 0: For g=3.3, one has a crossover at  =0 For g=3.3, one has a crossover at  =0 [g=5.5, e.g., yields a 1st order transition] [g=5.5, e.g., yields a 1st order transition] Critical temperature: T c ~ 140-150 MeV Critical temperature: T c ~ 140-150 MeV [Scavenius et al. (2001)]

25. QCD School in Les Houches, April/200825 eB = 5 m  2 : Tiny barrier: very weakly 1st order chiral transition! Tiny barrier: very weakly 1st order chiral transition! Higher critical temperature: Higher critical temperature: T c > 200 MeV T c > 200 MeV

26. QCD School in Les Houches, April/200826 eB = 10 m  2 : Critical temperature goes down again due to the larger hot fermionic contribution (T c < 140 MeV) Critical temperature goes down again due to the larger hot fermionic contribution (T c < 140 MeV) Larger barrier: clear 1st order chiral transition! Larger barrier: clear 1st order chiral transition! Non-trivial balance between T and B… one needs to explore the phase diagram Non-trivial balance between T and B… one needs to explore the phase diagram

27. QCD School in Les Houches, April/200827 eB = 20 m  2 : Even lower critical temperature Even lower critical temperature Large barrier persists: 1st order chiral transition Large barrier persists: 1st order chiral transition

28. QCD School in Les Houches, April/200828 Phenomenological consequences At RHIC, estimates by Kharzeev, McLerran and Warringa (2007) give: At RHIC, estimates by Kharzeev, McLerran and Warringa (2007) give: For LHC, we have a factor (Z Pb /Z Au = 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 For LHC, we have a factor (Z Pb /Z Au = 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 [ESF & Mizher, in prep.]

29. QCD School in Les Houches, April/200829 B = 0: eB = 6 m  2 : Rapid crossover (no barrier) Rapid crossover (no barrier) T c ~ 140-150 MeV T c ~ 140-150 MeV System smoothly drained to the true vacuum: no bubbles or spinodal instability System smoothly drained to the true vacuum: no bubbles or spinodal instability Weak 1st order (tiny barrier) Weak 1st order (tiny barrier) T c > 200 MeV T c > 200 MeV Part of the system kept in the false vacuum: some bubbles and spinodal instability, depending on the intensity of supercooling Part of the system kept in the false vacuum: some bubbles and spinodal instability, depending on the intensity of supercooling

30. QCD School in Les Houches, April/200830 Comparing barriers: eB = 6 m  2 : [Taketani & ESF (2006)] g = 5.5 - clear 1st order phase transition for  =0 and B=0 g = 5.5 - clear 1st order phase transition for  =0 and B=0 barrier ~ 0.25 close to T c barrier ~ 0.25 close to T c System mostly apprisionated in the false vacuum until the spinodal System mostly apprisionated in the false vacuum until the spinodal explosive phase conversion explosive phase conversion g = 3.3 - crossover for B=0; very weak 1st order phase transition for B > 0 g = 3.3 - crossover for B=0; very weak 1st order phase transition for B > 0 barrier ~ 0.025 close to T c barrier ~ 0.025 close to T c But even such small barriers can hold the system in the false vacuum until the spinodal for a fast enough supercooling ! But even such small barriers can hold the system in the false vacuum until the spinodal for a fast enough supercooling ! explosive phase conversion ? explosive phase conversion ?

31. QCD School in Les Houches, April/200831 Final remarks Lattice QCD indicates a crossover instead of a 1st order chiral transition at finite temperature and  =0. However, a strong magnetic background might invert this situation. Lattice QCD indicates a crossover instead of a 1st order chiral transition at finite temperature and  =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. 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. However: B falls off rapidly in the case of RHIC - early-time dynamics might be affected.

32. QCD School in Les Houches, April/200832 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. 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. 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)]. 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. Nevertheless, clean results for the case of constant field are encouraging.

33. QCD School in Les Houches, April/200833 To do list: More realistic treatment of the effective model (ZPT, resummation, etc) 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. 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. 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? Possible signatures of these features in heavy ion collisions? Situation at high density and applications to compact stars: phase structure inside magnetars. Situation at high density and applications to compact stars: phase structure inside magnetars.