Effective Quark Models in the Presence of a Magnetic Field

Pablo Guillermo Allen PhD Student University of Buenos Aires, Argentina Director: Dr. N.N. Scoccola Comisión Nacional de Energía Atómica

MOTIVATION Background

Effect of magnetic fields on hadronic matter

- Generated in Heavy ion collisions

High Temperature, Low density regime

B ∼ 1. 1019

- Neutron stars (Magnetars)

Low Temperature, High density regime

B ∼ 1. 1015 - 1. 1018

Nambu Jona Lasinio (NJL) Lagrangian

INTRODUCTION The Model - Basics

CHIRAL SYMMETRY

Effective Interaction among quark fields

Parameters: G: Coupling constant, Л Momentum Cut-off m0: Bare mass

Quark fields 2 flavours

Spontaneous symmetry breaking driven by G

MEAN FIELD APPROXIMATION

INTRODUCTION: The Model - Treatment

CHIRAL CONDENSATE ORDER PARAMETER

Gap Equation

Self-Consistent for M

INTRODUCTION: The Model - Extension

Chemical Potential

Introduced as Lagrange Multiplier

Dispersion Relation

Temperature

Magnetic Field

Matsubara frequency

* Flavour * Colour * 4-Momentum

Includes sum over:

Sum over Landau Levels (Constant and homogeneous)

Time component integration

Gap Equation solved for M, for different

values of T,mu and B

Phase Diagram

INTRODUCTION: Model Behaviour

B=1.1019

Broken Symmetry

Symmetry Restored

Cross over and first order transitions

RESULTS:

Enhancement of Symmetry Breaking for higher B

Behaviour for T=0 & mu=0

Mass dependance on Magnetic Field

More complicated behaviour for finite temperature and chemical potential

B=0 & B=1.1018 – 1.1019 B=1.1019 – 1.1020

Phase Diagram – Distinct Ranges RESULTS:

Criticial Temp vs Magnetic field at µ=0

Criticial µ vs Magnetic field at T=0

CRITICAL END POINT – Dependance on Magnetic Field RESULTS:

Introduction of Polyakov Loop

Non Local Interaction

WHAT’S NEXT?

Background Colour Field

Effective coupling term with quarks

Study deconfinement transition

Interaction through Non Local Currents

Sharp cut-off Removed

Introduction of regulators

More realistic interaction

Thanks!

Masa vs Parámetro adimensional G*L^2

Transición de Fase – Cross over

Parámetros de ajuste

RESULTADOS:

NJL ELEMENTAL

RESULTADOS:

TEMPERATURA Y POTENCIAL QUIMICO FINITO Mass vs Temperatura - Mu Parámetro

Diagrama de fases-

M vs T para diferentes valores de B

mu=0 MeV

Cambio en el diagrama de fases a diferentes valores de B

Algo vs algo a algo fijo con algo como parámetro

m0=0 Transicion 2do orden

m0= finite Cross-over

INTRODUCCION DE CAMPO MAGNETICO:

Acoplamiento a B externo, uniforme y constante

Derivada Covariante

Resolución de Ecuación de Dirac: Relación de Dispersión modificada

Ecuacion del gap….

Integral x-y en el propagador Suma de niveles de Landau

INTRODUCCION:

Poner T,mu,B, resolver la ecuación del gap. En general se hace fijando dos parámetros, B,mu ponele… y corriendo sobre T. Graficar potenciales… poner algunas curvas tipo que representan la transicion de fase que nos interesa y que con los potenciales se pueden clasificar las transiciones posibles y el punto que separa las dos transiciones