QED at Finite Temperature and Constant Magnetic Field:

The Standard Model of Electroweak Interaction at Finite Temperature and Strong Magnetic Field

Neda SadooghiNeda SadooghiDepartment of Physics

Sharif University of TechnologyTehran-Iran

Prepared for CEP seminar, Tehran, May 2008

Summary of the 1st Lecture:

The problem of baryogenesis:

Why is the density of baryons much less than the density of photons? 9 orders of magnitude difference between observation and

theory

Why in the observable part of the universe, the density of baryons is many orders greater than the density of antibaryons? The density of baryons is 4 orders of magnitude greater than

the density of antibaryons

3 Sakharov conditions for baryogenesis:

Violation of C and CP symmetries Deviation from thermal equilibrium Non-conservation of baryonic charge

A number of models describe baryogenesis:

Electroweak baryogenesis Affleck-Dine scenario of baryogenesis in SUSY …. Electroweak baryogenesis in a constant magnetic field

Electroweak (EW) baryogenesis

In EWSM there are processes that violate C and CP

EW phase transition Out of equilibrium process 2nd order phase transition at Tc=225 GeV

One loop approx

1st order phase transition at Tc=140.42 GeV

One loop + ring contributions

Baryon number non-conservation is related to sphaleron

decay

Although the minimal EWSM has all the necessary

ingredients for successful baryogenesis neither the amount of CP violation whithin the minimal SM,

nor the strength of the EW phase transition

is enough to generate sizable baryon number

Other methods …

Electroweak baryogenesis in a constant magnetic field

The Relation between Baryogenesis and Magnetogenesis

The sphaleron decay changes the baryon number and produces helical magnetic field

The helicity of the magnetic field is related to the number of baryons

produced by the sphaleron decay (Cornwall 1997, Vachaspati 2001)

A small seed field is generated by the EW phase transition

It is then amplified by turbulent fluid motion ( )

Observation: Background large scale cosmic magnetic field

G2910~

G610~

Strong Magnetic Field; Experiment

Magnetic fields in the compact stars: Experiment:

In the Little Bang (heavy ion collisions at RHIC) 0711.0950 [hep-ph] L.D. McLerran et al.

A new effect of charge separation (P and CP violation) in the

presence of background magnetic field Chiral magnetic effect

The estimated magnetic field in the center of Au+Au collisions

GB 178 1010~

GB 1716 1010~

EW Baryogenesis in Strong Hypermagnetic Field

Series of papers by:Series of papers by: Skalozub & Bordag (1998-2006), Ayala et al. (2004-2008)

Electroweak phase transition in a strong magnetic field Effective potential in one-loop + ring contributions Higgs mass

Result:Result: The phase transition is of 1st order for magnetic field

The baryogenesis condition is not satisfied !!!

Improved ring potential of QED at finite temperature and in the presence of weak/strong magnetic field

The Critical T of Dynamical Symmetry Breaking in the LLL

0805.0078 [hep-ph]

N. S. & K. Sohrabi

Outline:Part 1: QED at B = 0 and finite T

Ring diagrams in QED at B = 0 and finite T

Part 2: QED in a strong B field at T=0 Dynamical Chiral Symmetry Breaking (DSB)

Part 3: QED at finite B and T Results from 0805.0078 [hep-ph]; N.S. and K. Sohrabi

QED effective (thermodynamic) potential in the IR limit

QED effective potential in the limit of weak/strong magnetic field

Dynamical symmetry breaking in the lowest Landau Level (LLL)

Numerical analysis of Tc

Part 1: QED at B = 0 and finite T

Ring

Diagrams

Ring (Plasmon) Potential

Partition Function at finite Temperature

Bosonic partition function

Partition function of interacting fields:

Perturbative Series:

In the theory the free propagator is given by

Bosonic Matsubara frequencies

In higher orders of perturbation Full photon propagator

is the self energy

QED free photon propagator

Photon self energy

General form of photon self energy at zero B and non-zero T

with the projection operators are determined by Ward identity

G and F include perturbative corrections and are given by a

(analytic) series in the coupling constant e

QED Ring Diagrams at zero B and non-zero T

Using the free propagator and the photon self energy

QED Ring potential

QED ring potential in the static limit New unexpected contribution from perturbation theory

Effects of Ring Potential

In the MSM EW phase transition

Changing the type of phase transition

Decreasing the critical T

EWSM in the Presence of B Field (Skalozub + Bordag)

Ring contribution in the static limit

Our idea:Our idea: Calculate ring diagram in the improved IR limit

Look for e.g. dynamical chiral SB in the LLL

Question: What is the effect of the new approximation in changing

(decreasing) the critical temperature of phase transition?

Part 2: QED in a Strong Magnetic Field at T=0

QED in a strong B field at T=0

QED Lagrangian density

with

we choose a symmetric gauge with

Using Schwinger proper time formalism Full fermion and

photon propagators

Fermion propagator in a constant magnetic field

n labels the Landau levels are some Laguerre polynomials

In the IR region, with physics is dominated by the dynamics in the Lowest Landau Level LLL (n=0)

An effective quantum field theory (QFT) replaces the full QFT

Properties of effective QED in the LLL (I)

A) Dimensional reduction Fermion propagator Dimensional Reduction

Photon acquires a finite mass

Properties of effective QED in the LLL (II)

B) Dynamical mass generation

Dynamical chiral symmetry breaking

Start with a chirally invariant theory in nonzero B The chiral symmetry is broken in the LLL and

A finite fermion mass is generated

Part 3: QED at Finite B and T

QED Effective Potential at nonzero T and B

QED Effective (Thermodynamic) Potential

at Finite T and in a Background Magnetic

Field

Approximation beyond the static limit k 0

Full QED effective potential consists of two parts The one-loop effective potential

The ring potential

QED One-Loop Effective Potential at Finite T and B

T independent part

T dependent part

QED Ring Potential at Finite T and B

QED ring potential

Using a certain basis vectors defined by the eigenvalue

equation of the VPT (Shabad et al. ‘79)

The free photon propagator in the Euclidean space

)(ib

VPT at finite T and in a constant B field ( Shabad et al. ‘79)

Orthonormality properties of eigenvectors Ring potential

Ring potential in the IR limit (n=0)

)(ib

The integrals

IR vs. Static Limit

Ring potential in the IR limit

In the static limit k 0

QED Ring Potential in Weak Magnetic Field Limit

QED Ring Potential in Weak B Field Limit and Nonzero T Conditions: and

Evaluating in eB 0 limit

In the IR limit

In the static limit k 0

QED ring potential in the IR limit and weak magnetic field

In the high temperature expansion

In the limit

Comparing to the static limit, an additional term appears Well-known terms in QCD at finite T Hard Thermal Loop Expansion

Braaten+Pisarski (’90)

2/5

QED Ring Potential in Strong Magnetic Field Limit

Remember: QED in a Strong B Field at zero T; Properties Dynamical mass generation

Dynamical chiral symmetry breaking

Bound state formation

Dimensional reduction from D D-2 Two regimes of dynamical mass

Photon is massive in the 2nd regime:

QED Ring Potential in Strong B Field limit at nonzero T Conditions:

Evaluating in limit

with

QED ring potential in the IR limit and strong magnetic field

In the high temperature limit

Comparing to the static limit

From QCD at finite T and n=0 limit (Toimela ’83)

Dynamical Chiral Symmetry Breaking in the LLL

QED in a Strong Magnetic Field at zero T; Properties

Dynamical mass generation

Dynamical chiral symmetry breaking

Bound state formation

Dimensional reduction from D D-2

QED Gap Equation in the LLL

QED in the LLL Dynamical mass generation The corresponding gap equation

Using

Gap equation where

One-loop contribution Ring contribution

One-loop Contribution

Dynamical mass

Critical temperature Tc is determined by

Ring Contribution

Using and

Dynamical mass

Critical temperature of Dynamical Symmetry Breaking (DSB)

Critical Temperature of DSB in the IR Limit Using

The critical temperature Tc in the IR limit

where is a fixed, T independent mass (IR cutoff)

and

Critical Temperature of DSB in the Static Limit

Using

The critical temperature Tc in the static limit

IR vs. Static Limit

Question: How efficient is the ring contribution in the IR or static

limits in decreasing the Tc of DSB arising from one-loop EP?

The general structure of Tc

To compare Tc in the IR and static limits, define

IR limit

Static limit

Define the efficiency factor

where

and the Lambert W(z) function, staisfying

It is known that

Numerical Results

Choosing , and

Astrophysics of neutron stars RHIC experiment (heavy ion collisions)