The QCD transition in external magnetic fields

Falk Bruckmann(Univ. Regensburg)

Trento, Nov. 2012

G. Bali, G. Endrődi, Z. Fodor, S. Katz, S. Krieg, A. Schäfer, K.Szabó

JHEP 1202 (2012) 044, 1111.4956PoS LATTICE2011 (2011) 192, 1111.5155

PRD 86 (2012) 071502, 1206.4205

Falk Bruckmann The QCD transition in external magnetic fields 0 / 17

Motivation(?!)

◦ magn. fields in QCD

◦ magn. field on the lattice: not too hard

here QCD lattice simulations:

up + down + strange, physical masses & continuum limit

Eucl. space & constant external magn. field caveats cf. talk by B. Müller√

eB ' 0.1 . . . 1 GeV (quantized and bounded)

condensate at T = 0: magn. catalysis

transition and observables at finite T

Falk Bruckmann The QCD transition in external magnetic fields 1 / 17

Main result: Phase diagrampseudo-critical temperature as a function of magn. field:

renorm. condensate of light quarks (red), strange number susc. (blue)similar for chiral susceptibility⇒ Tc decreases by O(10) MeV

Falk Bruckmann The QCD transition in external magnetic fields 2 / 17

Nature of the transition

vanishing B: crossover Aoki et al. 06nonzero B:

volume dependence of u-condensate and susceptibility⇒ no effect

Falk Bruckmann The QCD transition in external magnetic fields 3 / 17

Nature of the transition

the relative change of the light susceptibility with temperature aroundthe critical one: not becoming divergent for stronger magn. fields

⇒ transition remains a crossover up to√

eB ' 1 GeV

Falk Bruckmann The QCD transition in external magnetic fields 4 / 17

Condensate at T =0

grows with B = ‘magn. catalysis’: Gusynin, Miransky, Shovkovy 96

add. and mult. renorm.: ∆Σu,d ≡ 2mu=dM2πF 2π[ψ̄ψu,d (B)− ψ̄ψu,d (0)

]⇒magn. catalysis confirmed, linear dependence for large B

Falk Bruckmann The QCD transition in external magnetic fields 5 / 17

Condensate at T =0: comparisonto χPT Cohen, McGady, Werbos 07, Andersen 12& NJL model Gatto, Ruggieri 10

⇒ well approximated

unless eB >0.10.3

GeV2 (should we trust these approaches there?!)

Falk Bruckmann The QCD transition in external magnetic fields 6 / 17

Inverse magnetic catalysisagain condensate, for finite T :

⇒ non-monotonic behavior:magn. catalysis turns into inverse magn. catalysis around Tc

captured by models? understanding? cf. talk by T. Kovács!Falk Bruckmann The QCD transition in external magnetic fields 7 / 17

Consequence of inverse magnetic catalysis

again condensate, now as function of T at fixed B:

magn. catalysis and inverse magnetic catalysis (vertically) X⇒ crit. temperature decreases

Falk Bruckmann The QCD transition in external magnetic fields 8 / 17

Condensate at finite T : comparisonto χPT Andersen 12& PNJL model Gatto, Ruggieri 10

⇒ χPT ok unless T > 100 MeV (should we trust this approach there?!)PNJL? this one uses Polyakov loop potential from Nf = 2 lattice data

Falk Bruckmann The QCD transition in external magnetic fields 9 / 17

Isospin condensate

difference ūu − d̄d : only through el. charge (vanishes at B = 0)

add. and mult. renorm. again: Σu − Σd ≡ 2mu=dM2πF 2π[ūu(B)− d̄d(B)

]⇒ order parameter

Falk Bruckmann The QCD transition in external magnetic fields 10 / 17

Polyakov loop

near Tc for different B’s:

⇒monotonically growing with B ( condensate)⇒ crit. T decreases again

Falk Bruckmann The QCD transition in external magnetic fields 11 / 17

Mass sensitivitywhat if we put (mlight,mlight,mstrange)→ (mstrange,mstrange,mstrange)?

T -dep. of u-susceptibility (top) and change of u-condensate (bottom)⇒ effects of decreasing Tc & inverse magn. catalysis disappear

light quark masses are important

Falk Bruckmann The QCD transition in external magnetic fields 12 / 17

Comparison to other lattice simulations1 + 1 + 1 (staggered, smeared) usat phys. masses

⇒ crit. temperature decreases

1 + 1 (staggered) D’Elia et al. 10at higher-than-phys. masses

5.27 5.28 5.29

β

0.05

0.1

0.15

0.2

0.25 Chiral cond. b= 0Chiral cond. b = 8Chiral cond. b = 16Chiral cond. b = 24Pol. loop b = 0

Pol. loop b = 8

Pol. loop b = 16

Pol. loop b = 24

similar in SU(2) Ilgenfritz et al. 12

! Tc(B) different! monotonicity different

rationale: masses as before,cont. limit (diff. lattice actions)

Falk Bruckmann The QCD transition in external magnetic fields 13 / 17

Remarks on renormalization

empirically: charged pion mass and Sommer scale as function of B

as expected

⇒ B does not change the lattice spacing

Falk Bruckmann The QCD transition in external magnetic fields 14 / 17

Remarks on renormalization

B always enters as (eB) and (eB)bare = (eB)r

local U(1) gauge invariance still present:

ψr =√

Z2 ψb

↓ gauge trafo ↓

eiαrψr =

√Z2 ψbeiα

b ⇒ αr = αb

er Arµ = Ze√

Z3ebAbµ (1)

↓ gauge trafo ↓

∂µαr +er Arµ = Ze

√Z3(

ebAbµ + ∂µαb)⇒ Ze

√Z3 = 1

(1)⇒ er Br = 1 · ebBb

Falk Bruckmann The QCD transition in external magnetic fields 15 / 17

Remarks on renormalizationdivergence of partition function Z /free energy density f : Leutwyler, Smilga 92

log ZV4

=− f r +O( 1a4

) +O(m2

a2) +O(m4 log a)

no new divergences by temperature or chem. pot.

ψ̄ψ = ∂mlog Z

V4=− ∂mf r + 0 + O(

ma2

) + O(m3 log a)

add. divergence removed by e.g. ψ̄ψ − ψ̄ψ(T = 0)mult. divergence removed by mult. back by mass

new divergences with B-field? free energy of free quarks (one flavor):

f − f (B = 0) = Nc(qB)2

24π2(

finite− log(m2a2))

thisistoshifttheequationtotheleft , yes

related to charge renorm. (background field method)

ψ̄ψ − ψ̄ψ(B = 0) = Nc(qB)2

24π2(∂m finite− finite)

⇒ divergence only in free energy cf. talk by G. EndrődiFalk Bruckmann The QCD transition in external magnetic fields 16 / 17

Summary

magn. catalysis turns into inverse magn. catalysis around Tc

Tc(B)↘

transition sensitive to light quark mass← understanding?

◦ χPT and (P)NJL models fine in some range of validity

◦ isospin condensate

◦ renormalization almost unchanged by magn. field

more observables from B cf. talk by G. Endrődi

open: more realistic B-fields on the lattice (still Euclidean)

Falk Bruckmann The QCD transition in external magnetic fields 17 / 17