QED at Finite Temperature and Constant Magnetic Field: The Standard Model of Electroweak Interaction at Finite Temperature and Strong Magnetic Field Neda Sadooghi Department of Physics Sharif University of Technology Tehran-Iran Prepared for CEP seminar, Tehran, May 2008

Summary of the 1 st 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 2 nd order phase transition at Tc=225 GeV  One loop approx 1 st order phase transition at Tc= 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

Strong Magnetic Field ; Experiment Magnetic fields in the compact stars: Experiment:  In the Little Bang (heavy ion collisions at RHIC) [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

EW Baryogenesis in Strong Hypermagnetic Field Series of papers by: Skalozub & Bordag ( ), Ayala et al. ( )  Electroweak phase transition in a strong magnetic field  Effective potential in one-loop + ring contributions  Higgs massResult: The phase transition is of 1 st 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 [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 [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:  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

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)

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 Evaluatingin 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)

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 2 nd 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)