chiral transition in a strong magnetic background

Chiral Transition in a Chiral Transition in a Strong Magnetic BackgroundStrong Magnetic Background

Eduardo S. FragaEduardo S. Fraga

Instituto de Física Instituto de Física Universidade Federal do Rio de JaneiroUniversidade Federal do Rio de Janeiro

QCD School in Les Houches, April/2008QCD School in Les Houches, April/2008 22

Introduction and MotivationIntroduction and Motivation

Strong interactions under intense magnetic fields can be found,Strong interactions under intense magnetic fields can be found,in principle, in a variety of systems:in principle, in a variety of systems:

High density and low temperatureHigh density and low temperature

• ““Magnetars”: B Magnetars”: B ~ ~ 10101414-10-101515 G at the surface, much higher in the G at the surface, much higher in the corecore

[Duncan & Thompson (1992/1993)][Duncan & Thompson (1992/1993)]

• Stable stacks of Stable stacks of 00 domain walls or axial scalars ( domain walls or axial scalars (,,’) domain ’) domain walls in nuclear matter: B walls in nuclear matter: B ~ ~ 10101717-10-101919 G G [Son & Stephanov (2008)][Son & Stephanov (2008)]


High temperature and low densityHigh temperature and low density

• Early universe (relevant for nucleosynthesis): BEarly universe (relevant for nucleosynthesis): B~~ 10102424 G for the EWPT epoch G for the EWPT epoch [Grasso & Rubinstein (2001)][Grasso & Rubinstein (2001)]

• Non-central RHIC collisions: eBNon-central RHIC collisions: eB~ ~ 101044-10-1055 MeV MeV2 2 ~ ~ 10101919 GG

[Kharzeev, McLerran & [Kharzeev, McLerran & Warringa (2007)]Warringa (2007)]

[Au-Au, 200 GeV][Au-Au, 200 GeV][Au-Au, 62 GeV][Au-Au, 62 GeV]


Besides, there are several theoretical/phenomenological Besides, there are several theoretical/phenomenological interesting questions:interesting questions:

• How does the QCD phase diagram looks like including a How does the QCD phase diagram looks like including a nonzero uniform B ? (another interesting “control nonzero uniform B ? (another interesting “control parameter” ?) parameter” ?)

• Are there modifications in the nature of phase Are there modifications in the nature of phase transitions ?transitions ?

• Does it affect significantly time scales for phase Does it affect significantly time scales for phase conversion ?conversion ?

• Are there any new phenomena ?Are there any new phenomena ?

Some of these questions have already been addressed in Some of these questions have already been addressed in different ways. different ways. Here, we consider effects on the chiral Here, we consider effects on the chiral transition at finite temperature and zero density in the transition at finite temperature and zero density in the Linear Sigma Model with Quarks.Linear Sigma Model with Quarks.


Other approachesOther approaches (usually concerned about (usually concerned about vacuum effectsvacuum effects):):

NJL: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:PT:• Shushpanov & Smilga (1997)Shushpanov & Smilga (1997)• Agasian & Shushpanov (2000)Agasian & Shushpanov (2000)• Cohen, McGady & Werbos (2007)Cohen, McGady & Werbos (2007)• … …

Large-N QCD:Large-N QCD:• Miransky & Shovkovy (2002)Miransky & Shovkovy (2002)

Quark model:Quark model:• Kabat, Lee & Weinberg (2002)Kabat, Lee & Weinberg (2002)


OutlineOutline

Effective theory for the chiral transition:Effective theory for the chiral transition: the linear the linear model model

Incorporating a strong magnetic field backgroundIncorporating a strong magnetic field background

The modified effective potentialThe modified effective potential

Some phenomenological consequencesSome phenomenological consequences

Final remarksFinal remarks


Effective theory for the chiral transition (LEffective theory for the chiral transition (LM)M)

• Symmetry: for massless QCD, the action is invariant under Symmetry: for massless QCD, the action is invariant under SU(NSU(Nff))L L x x SU(NSU(Nff))RR

• “ “Fast” degrees of freedom: quarksFast” degrees of freedom: quarks “ “Slow” degrees of freedom: mesonsSlow” degrees of freedom: mesons

• Typical energy scale: hundred of MeVTypical energy scale: hundred of MeV

• We choose SU(NWe choose SU(Nff=2), for simplicity: we have pions and the sigma=2), for simplicity: we have pions and the sigma

• Framework: coarse-grained Landau-Ginzburg effective potentialFramework: coarse-grained Landau-Ginzburg effective potential

• SU(2) SU(2) SU(2) spontaneously broken in the vacuum SU(2) spontaneously broken in the vacuum

• Also accommodates explicit breaking by massive quarksAlso accommodates explicit breaking by massive quarks

[Gell-Mann & Levy (1960); Scavenius, Mócsy, Mishustin & Rischke (2001); …][Gell-Mann & Levy (1960); Scavenius, Mócsy, Mishustin & Rischke (2001); …]


Building the effective lagrangianBuilding the effective lagrangian

Kinetic terms:

Fermion-meson interaction:

Chiral self-interaction:

Explicit chiral symmetry breaking term:

[with scalars allowed by symmetry]


Parameters should be fixed such that:Parameters should be fixed such that:

• SU(2)SU(2)LL SU(2) SU(2)RR is spontaneously broken in the vacuum, is spontaneously broken in the vacuum, with <with <> = f> = f , <, <> = 0> = 0

• hh should be related to the nonzero pion mass (plays a role should be related to the nonzero pion mass (plays a role analogous to an external magnetic field for a spin system)analogous to an external magnetic field for a spin system)

• ff = 93 MeV is the pion decay constant, determined = 93 MeV is the pion decay constant, determined experimentally. It comes about when one computes the experimentally. It comes about when one computes the weak decay of the pion, which is proportional to the weak decay of the pion, which is proportional to the amplitudeamplitude

a,b: isospina,b: isospin

• The small but nonzero pion mass breaks “softly” The small but nonzero pion mass breaks “softly” the axial current (PCAC):the axial current (PCAC):


• Including a term Including a term ~~ hh brings the following brings the following consequences:consequences:

- The true vacuum (in the - The true vacuum (in the direction) is shifted direction) is shifted [redefine f[redefine f such that it coincides with the experimental value] such that it coincides with the experimental value]

- The - The mass is modified mass is modified

- Pions acquire a nonzero mass- Pions acquire a nonzero mass

which fixes which fixes hh to be: to be:

All parameters can be chosen to reproduce the vacuum features of All parameters can be chosen to reproduce the vacuum features of mesons.mesons.


- The connection with the quark mass is given by - The connection with the quark mass is given by the Gell-Mann--Oakes--Renner (GOR) relation:the Gell-Mann--Oakes--Renner (GOR) relation:

““by construction”, since by construction”, since one wants this term to one wants this term to mimic the QCD explicit mimic the QCD explicit breaking of chiral breaking of chiral symmetrysymmetry

Connection not only between mConnection not only between m and mand mqq, but also between the , but also between the field condensate and the chiral field condensate and the chiral condensatecondensate

- In a medium, one can use <- In a medium, one can use <>(T) in the effective >(T) in the effective theory to describe the melting of the chiral condensate at theory to describe the melting of the chiral condensate at high T.high T.


Putting Putting and and ii together in an O(4) field together in an O(4) field =(=(,,ii), we have), we have

Lagrangian:Lagrangian:

Partition function:Partition function:

Integrating over the fermions (heat bath for the long wavelength Integrating over the fermions (heat bath for the long wavelength chiral fields), we obtain an effective thermodynamic potential forchiral fields), we obtain an effective thermodynamic potential for =(=(,,))


Example of an effective potential in the Example of an effective potential in the direction direction (modulo inhomogeneity corrections which tend to (modulo inhomogeneity corrections which tend to reduce the barrier) reduce the barrier) For a 1st order chiral transitionFor a 1st order chiral transition

-200 -100 0 100 200

(MeV)

-200

-100

0

100

200

Ueff(MeV)

μB=0T=75,100,125,150MeV

[Aguiar, ESF & Kodama (2006)][Aguiar, ESF & Kodama (2006)]


Incorporating a strong magnetic field backgroundIncorporating a strong magnetic field background

Let us assume the system is in the presence of a strong Let us assume the system is in the presence of a strong magnetic field background that is constant and magnetic field background that is constant and homogeneous:homogeneous:

choice of gaugechoice of gauge

• charged mesons charged mesons (new dispersion relation)(new dispersion relation)::

Landau levels:Landau levels:


• quarks quarks (new dispersion relation)(new dispersion relation)::

• integration measure:integration measure:

T = 0:T = 0:

T > 0:T > 0:

l: Matsubara indexl: Matsubara indexn: Landau level indexn: Landau level index


The modified effective potentialThe modified effective potential

Simple mean-field treatment with the following customary Simple mean-field treatment with the following customary simplifying assumptions simplifying assumptions [Scavenius, Mócsy, Mishustin & Rischke (2001); Dumitru & Paech [Scavenius, Mócsy, Mishustin & Rischke (2001); Dumitru & Paech

(2005); …] (2005); …] ::

Quarks constitute a Quarks constitute a thermalized gasthermalized gas that provides a background in that provides a background in which the long wavelength modes of the chiral condensate evolve. which the long wavelength modes of the chiral condensate evolve. Hence:Hence:

At T = 0 (vacuum: At T = 0 (vacuum: symm. broken; reproduce usual L symm. broken; reproduce usual LM & M & PT PT results)results)

• Quark d.o.f. are absent (excited only for T > 0)Quark d.o.f. are absent (excited only for T > 0)

• The The is heavy (M is heavy (M~~600 MeV) and treated classically600 MeV) and treated classically

• Pions are light: fluctuations in Pions are light: fluctuations in ++ and and -- couple to B; couple to B; fluctuations in fluctuations in 00 give a B-independent contribution (ignored give a B-independent contribution (ignored here)here)

[ESF & Mizher, in prep.][ESF & Mizher, in prep.]


At T > 0 (plasma: At T > 0 (plasma: symm. approximately restored) symm. approximately restored)

• Quarks are relevant (fast) degrees of freedom: Quarks are relevant (fast) degrees of freedom: incorporate their thermal fluctuations in the effective incorporate their thermal fluctuations in the effective potential for potential for (integrate over quarks) (integrate over quarks)

• Pions become rapidly heavy only after TPions become rapidly heavy only after Tcc, so we , so we incorporate their thermal contributionincorporate their thermal contribution

Later: Later: ZPT, CJT resummations, etc - ZPT, CJT resummations, etc - herehere, the simplest , the simplest phenomenological approachphenomenological approach


Vacuum effective potential (Vacuum effective potential ( direction): direction):

Classical:Classical:

++ and and --

fluctuations:fluctuations:

Computing the contribution from pions in the MSbar scheme, we Computing the contribution from pions in the MSbar scheme, we obtain (ignoring obtain (ignoring -independent terms):-independent terms):

now means <now means <>) >)

using the assumption of large magnetic field,using the assumption of large magnetic field, |q|B >> m |q|B >> m22, in the , in the

expansion of generalized Zeta functions of the formexpansion of generalized Zeta functions of the form


• Condensate grows with increasing magnetic Condensate grows with increasing magnetic fieldfield

• Minimum deepens with increasing magnetic Minimum deepens with increasing magnetic fieldfield

• Relevant effects for equilibrium Relevant effects for equilibrium thermodynamics and nonequilibrium process of thermodynamics and nonequilibrium process of phase conversion ?phase conversion ?

Results in line with calculations Results in line with calculations in in PT and NJL, as in e.g.PT and NJL, as in e.g.

- Shushpanov & Smilga (1997)- Shushpanov & Smilga (1997)- Cohen, McGady & Werbos - Cohen, McGady & Werbos (2007)(2007)- Hiller, Osipov et al. Hiller, Osipov et al. (2007/2008)(2007/2008)- … …

However:However: for very large B, for very large B, effects from the quarks could effects from the quarks could become important - non-trivial become important - non-trivial transition? [Kabat, Lee & transition? [Kabat, Lee & Weinberg (2002)]Weinberg (2002)](later…)(later…)

““Large” (“critical”) B for QCD:Large” (“critical”) B for QCD:


Thermal corrections:Thermal corrections:

++ and and --

fluctuations:fluctuations:

quark quark fluctuations:fluctuations:

Computing in the MSbar scheme (ignoring Computing in the MSbar scheme (ignoring -independent terms -independent terms and assuming a large magnetic field - also compared to T - in the and assuming a large magnetic field - also compared to T - in the expansion of zeta functions), we obtain:expansion of zeta functions), we obtain:


Ignoring exponentially suppressed contributions (besides ZPT, etc), Ignoring exponentially suppressed contributions (besides ZPT, etc), the effective potential is given bythe effective potential is given by

Remarks:Remarks:

• Exponential suppresions come as Exponential suppresions come as

• In what follows, we take NIn what follows, we take Ncc=3, g=3.3 (to reproduce the nucleon =3, g=3.3 (to reproduce the nucleon mass), and eB given in units of mmass), and eB given in units of m

2 2 ::

• Other parameters are fixed to fit vacuum conditions, as Other parameters are fixed to fit vacuum conditions, as customary.customary.

• For very large B, the n = 0 Landau level dominates. Corrections For very large B, the n = 0 Landau level dominates. Corrections can be incorporated systematically.can be incorporated systematically.


B = 0:B = 0:

• For g=3.3, one has a crossover at For g=3.3, one has a crossover at μμ=0=0 [g=5.5, e.g., yields a 1st order [g=5.5, e.g., yields a 1st order transition]transition]

• Critical temperature: TCritical temperature: Tcc~~ 140-150 140-150 MeVMeV [Scavenius et al. (2001)][Scavenius et al. (2001)]


eB = 5 meB = 5 m22::

• Tiny barrier: very Tiny barrier: very weakly 1st order chiral weakly 1st order chiral transition!transition!

• Higher critical temperature: Higher critical temperature: TTc c >> 200 MeV 200 MeV


eB = 10 meB = 10 m22::

• Critical temperature goes down again due to the larger Critical temperature goes down again due to the larger hot fermionic contribution (Thot fermionic contribution (Tcc < 140 MeV) < 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 Non-trivial balance between T and B… one needs to explore the phase diagramexplore the phase diagram


eB = 20 meB = 20 m22::

• Even lower critical temperatureEven lower critical temperature

• Large barrier persists: 1st order chiral Large barrier persists: 1st order chiral transitiontransition


Phenomenological consequencesPhenomenological consequences

• At RHIC, estimates by Kharzeev, McLerran and Warringa (2007) At RHIC, estimates by Kharzeev, McLerran and Warringa (2007) give:give:

• For LHC, we have a factor (ZFor LHC, we have a factor (ZPbPb/Z/ZAuAu = 82/79) and some small = 82/79) and some small increase in the maximum value of eB due to the higher CM energy increase in the maximum value of eB due to the higher CM energy (as observed for RHIC). So, it is reasonable to consider(as observed for RHIC). So, it is reasonable to consider

[ESF & Mizher, in prep.][ESF & Mizher, in prep.]


B = 0:B = 0: eB = 6 meB = 6 m22::

• Rapid crossover (no barrier)Rapid crossover (no barrier)

• TTcc ~~ 140-150 MeV 140-150 MeV

• System smoothly drained to System smoothly drained to the true vacuum: no bubbles the true vacuum: no bubbles or spinodal instabilityor spinodal instability

• Weak 1st order (tiny barrier)Weak 1st order (tiny barrier)

• TTcc > 200> 200 MeV MeV

• Part of the system kept in the Part of the system kept in the false vacuum: some bubbles false vacuum: some bubbles and spinodal instability, and spinodal instability, depending on the intensity of depending on the intensity of supercoolingsupercooling


Comparing barriers:Comparing barriers: eB = 6 meB = 6 m22::

[Taketani & ESF (2006)][Taketani & ESF (2006)]

• g = 5.5 - clear 1st order phase g = 5.5 - clear 1st order phase transition for transition for μμ=0 and B=0=0 and B=0

• barrier barrier ~~ 0.25 close to T 0.25 close to Tcc

• System mostly apprisionated in System mostly apprisionated in the false vacuum until the the false vacuum until the spinodalspinodal

• explosive phase conversionexplosive phase conversion

• g = 3.3 - crossover for B=0; very g = 3.3 - crossover for B=0; very weak 1st order phase transition for B weak 1st order phase transition for B > 0> 0

• barrier barrier ~~ 0.025 close to T 0.025 close to Tcc

• But even such small barriers can But even such small barriers can hold the system in the false vacuum hold the system in the false vacuum until the spinodal for a fast enough until the spinodal for a fast enough supercooling !supercooling !

• explosive phase conversion ?explosive phase conversion ?


Final remarksFinal remarks

• Lattice QCD indicates a crossover instead of a 1st order Lattice QCD indicates a crossover instead of a 1st order chiral transition at finite temperature and chiral transition at finite temperature and μμ=0. =0. However, However, a strong magnetic background might invert this situation.a strong magnetic background might invert this situation.

• For RHIC and LHC heavy ion collisions, the barrier in the For RHIC and LHC heavy ion collisions, the barrier in the effective potential seems to be quite small. Nevertheless, effective potential seems to be quite small. Nevertheless, it can probably hold most of the system in a metastable it can probably hold most of the system in a metastable state down to the spinodal explosion. -> state down to the spinodal explosion. -> Different Different dynamics of phase conversion.dynamics of phase conversion.

• However: B falls off rapidly in the case of RHIC - However: B falls off rapidly in the case of RHIC - early-early-timetime dynamics might be affected. dynamics might be affected.


• The phenomenlogy resulting from varying T and B The phenomenlogy resulting from varying T and B seems to be rich: seems to be rich: competition between strengthening competition between strengthening the chiral symmetry breaking via vacuum effects and the chiral symmetry breaking via vacuum effects and its restoration by the thermal (magnetic) bath.its restoration by the thermal (magnetic) bath.

• Non-central heavy ion collisions might show features Non-central heavy ion collisions might show features of a 1st order transition when contrasted to central of a 1st order transition when contrasted to central collisions.collisions.

• Caveat:Caveat: treatment still admittedly very simple - in treatment still admittedly very simple - in heavy ion collisions, B varies in space and time. It can, heavy ion collisions, B varies in space and time. It can, e.g., induce an electric field that might play a role e.g., induce an electric field that might play a role [Cohen [Cohen et al. (2007)].et al. (2007)].

• Nevertheless, clean results for the case of constant Nevertheless, clean results for the case of constant field are encouraging.field are encouraging.


To do list:To do list:

• More realistic treatment of the effective model (ZPT, More realistic treatment of the effective model (ZPT, resummation, etc)resummation, etc)

• Investigation of the low magnetic field regime at finite T, Investigation of the low magnetic field regime at finite T, for B < T and B for B < T and B ~~ T. T.

• Simulation of time evolution of the phase conversion Simulation of time evolution of the phase conversion process to compare relevant time scales to those in the process to compare relevant time scales to those in the crossover picture.crossover picture.

• Possible signatures of these features in heavy ion Possible signatures of these features in heavy ion collisions?collisions?

• Situation at high density and applications to compact Situation at high density and applications to compact stars: phase structure inside magnetars.stars: phase structure inside magnetars.

chiral transition in a strong magnetic background

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