Polyakov chiral quark model N. N. Scoccola Tandar Lab -CNEA Buenos - PowerPoint PPT Presentation

Strong magnetic fields in a non-local Polyakov chiral quark model N. N. Scoccola Tandar Lab -CNEA – Buenos Aires PLAN OF THE TALK • Introduction • Magnetic field in non-local Polyakov chiral quark models • Results • Outlook & Conclusions Refs: Pagura, Gomez Dumm, Noguera & NNS, Phys.Rev. D95 (2017) 034013 Gomez Dumm, Izzo Villafañe, Noguera, Pagura & NNS, in preparation

Introduction Recently, there has been quite a lot of interest in investigating how the QCD phase diagram is affected by the presence of strong magnetic fields. Motivation: their possible existence in physically relevant situations: High magnetic fields in non-central relativistic heavy ion collisions Magnetic field at t=0 L or B Voloshin, QM2009 (A. Bzdak, V. Skokov (12)) Compact Stellar Objects: magnetars are estimated to have B ~10 14 -10 15 G at the surface. It could be much higher in the interior (Duncan and Thompson (92/93))

Features of strongly interacting matter under intense magnetic fields has been investigated in a variety of approaches. For example [certainly incomplete list !] • NJL and relatives (Klevansky, Lemmer (89); Klimenko et al. (92,..); Gusynin, Miransky, Shokovy (94/95); Ferrer, Incera et al (03..), Hiller, Osipov (07/08); Menezes et al (09);Fukushima, Ruggieri, Gatto (10) [PNJL ]; … ) •  PT (Shushpanov, Smilga (97); Agasian, Shushpanov (00); Cohen, McGady, Werbos (07); … . ) • Linear Sigma Model and MIT bag model: (Fraga, Mizher (08), Fraga, Palhares (12) … ) • Lattice QCD (D’Elia (10/11), Bali et al (11/12), … ) Recent reviews: Kharzeev, Landsteiner, Schmitt,Yee, Lect. Notes Phys. 871, 1 (2013). Miransky, Shovkovy, Phys. Rept. 576, 1 (2015). Andersen, W. R. Naylor, A. Tranberg, Rev. Mod. Phys. 88, 025001 (2016).

Magnetic catalysis (  =T=0) 1,0 [ M(B) - M 0 ] / M 0 Set A Set B 0,5 0,0 0,0 0,2 0,4 0,6 0,8 1,0 2 ] eB [GeV Lattice Bali et al (12) Typical NJL model result  PT Cohen et al(07) At T =0 there is an enhancement of the condensate with B: Magnetic catalysis (Gusynin, Miransky, Shokovy (94/95))

Critical temperatures for deconfinement and chiral transitions LSM Mizhner, Fraga (10) E-PNJL Gatto-Ruggieri (12) Most models and early LQCD results foresee an enhancement of critical temperatures for chiral transition with B LQCD D’Elia et al (10)

Lattice results with smaller lattice spacings ( Bali et al (12) ) Condensates as functions of B Critical temperatures as for various T functions of B At that time most models failed to predict these lattice results for the behavior of the condensates as functions of B for T close and above T c

Many scenarios have been considered in the last few years to account for the Inverse Magnetic Catalysis (IMC). E.g. [certainly incomplete list !] • T and B dependence on the NJL coupling constant (Ayala et al (14); Farias et al (14), Ferrer et al (15)) • B dependence of PL parameters in EPNJL models (M. Ferreira et al (14)) • Holography: (Rougemont, R. Critelli and J. Noronha (16)) • Effects beyond MFA (K. Fukushima and Y. Hidaka (13), S. Mao (16) … ) • Schwinger-Dyson methods (N. Mueller and J. M. Pawlowski (15), Braun, W. A. Mian and S. Rechenberger (16)) Yet, the physics behind IMC at finite T is not fully understood.

Non-local quark models Compared to NJL, non-local quark models represent a step towards a more realistic modeling of the QCD interactions: Nonlocal quark couplings present in the many approaches to low-energy q dynamics: i.e. instanton liquid model, Schwinger-Dyson resummation techniques, etc. Also in LQCD. Some advantages over the local NJL model: • No need to introduce sharp momentum cut -offs • Small next -to-leading order corrections • Successful description of meson properties at T =  = B= 0   G            Euclidean action 4   ( ) ( ) ( ) ( ) S d x x i m x j x j x E  c a a  2 for two flavors z z        4 Where ( ) ( ) ( ) ( ) j x d z z x x a a 2 2 nonlocal, well behaved ( ) z   i   (1, ) covariant form factors 5 a

Since we are interested in studying the influence of a magnetic field, we introduce in the effective action a coupling to an external electromagnetic gauge field  For a local theory this can be done by performing the replacement ˆ     ( ) i Q x    ˆ where = diag( q u , q d ), with q u / 2 = - q d = e / 3. Q In the case of the nonlocal model the situation is more complicated since the inclusion of gauge interactions implies a change not only in the kinetic terms of the Lagrangian but also in the nonlocal currents. One has         ( / 2) , / 2 ( / 2) x z W x x z x z where   ˆ  t   ( , ) P exp ( ) W s t iQ dr r       s r runs over an arbitrary path connecting s with t . We take a straight line path.

We bosonize the fermionic theory introducing scalar and pseudoscalar field and integrating out the fermion fields. The gauged bosonized action is 1            4 lndet ( ) ( ) ( )· ( ) S d x x x x x bos 2 G where     z z z        ,   ,  x x W x x 0 0     2 2 2     z     ( ) [ ]            (4) ( ) ( ) · ,   z i m z x i x W x x c   2 For constant and homogenous magnetic field along the 3-axis in the Landau   gauge we have . We work in MFA assuming that σ(x) has a B x   1 2    0 nontrivial translational invariant MF value , while . Then, i     i ˆ ˆ                  (4) MFA ( , ) ( ) ( ) exp ( )( ) x x x x i QBx m x x QB x x x x     1 2 2 2 1 1 c   2

To deal with this operator we introduced its Ritus transform      4 4 MFA MFA ( ) ( , ) ( ) d x d x x x x x  p  , p p p ( ) p x where are the usual Ritus matrices, with p = ( k , p 2 , p 3 , p 4 ). After some calculation we find that is diagonal not only in flavor space but also in p-space. Thus, the corresponding “ ln Det ” can be readily determined. In this way we obtain   2   MFA 2 | | q B d p   S  2     , || s f f 2 bos ln N p M f       || 0, c p (4) 2 2 2 (2 ) V G ||  , f u d          2 2          2 , , 2 , , f f f f ln 2 | | k q B p M M p M M Here     || , , || , , f k p k p k p k p || || || ||  1 k  2   4 d p            , k 2 2 2 2 f ( 1) exp / | | ( ) (2 / | |)  M p q B m g p p L p q B where       k p , f c || k f 2  | | (2 ) q B f s f        Here, 1 ( , ) , ( , ), sign( ), p p p p p p s q B k k  1 2 || 3 4 f f 2 2 L n ( x ): Laguerre polynomials

In the extension to finite temperature with consider the coupling of the quarks to the Polyakov loop (order parameter for deconfinement) 1   • Polyakov loop 1/   T     ( ) ( , ) x Tr i d A x    pure gauge Z(3) symmetry 4 Polyakov, PLB (78) N 0 c Effective potential deconfinement : Z(3) symmetry confinement : spontaneously Z(3) symmetry broken not broken Fukushima (03), Megias,Ruiz Arriola, Salcedo (06), Ratti, Thaler, Weise (06),… For the Polyakov Loop effective potential we take (Ratti, Thaler, Weise (06))  ( , ) ( ) T b T b b        2 3 4 2 3 4 , 4 2 3 4 T            where 2 3 ( ) ; / ; 1 2cos( / ) / 2 b T a a t a t a t t T T T 2 0 1 2 3 0 3 and a i , b i chosen to fit Quenched LQCD results. Due finite quark mass effects we consider T 0 = 210 MeV (Schaefer, Pawlowski, Wambach (07))

To obtain the thermodynamical potential  MFA at finite temperature we use the Matsubara formalism in the quark sector        3 ,(2 1) p p n T 2  d p dp     || nc c  || 2 2 3 ( ) ( ) N F p T F p   || || c nc 2 (2 ) 2         3 ; 0   , , c r g b n r g b and include the contribution of the Polyakov loop potencial  ( , ) T Given  MFA we obtain the gap equations as To be solved        / 0 ; / 0 numerically MFA MFA t      f MFA / q q m and the quark condensates , f f B T c To compare with LQCD calculations of Bali et al we also define 2 m           S   f 1/2 c 1 (135 86) MeV q q q q   , , 0,0 B T f f B T f f 4 S and          f f f u d an d ( ) / 2. , , 0, , , , B T B T T B T B T B T