Neda Sadooghi Department of Physics Sharif University of Technology Tehran-Iran

1. 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

2. 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. 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

4. Electroweak (EW) baryogenesis • In EWSM there are processes that violate C and CP • EW phase transition  Out ofequilibrium 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

5. 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 isenough to generate sizable baryon number • Other methods …  Electroweak baryogenesis in a constant magnetic field

6. The Relation between Baryogenesis and Magnetogenesis • The sphaleron decay changes the baryon number and produces helical magnetic field • The helicity of the magneticfield 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

7. 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

8. EW Baryogenesis in Strong Hypermagnetic Field 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: • The phase transition is of 1st order for magnetic field • The baryogenesis condition is not satisfied !!!

9. Improved ring potential of QED at finite temperature and in the presence of weak/strong magnetic fieldThe Critical T of Dynamical Symmetry Breaking in the LLL 0805.0078 [hep-ph] N. S. & K. Sohrabi

10. 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

11. Part 1: QED at B = 0 and finite T Ring Diagrams

12. Ring (Plasmon) Potential • Partition Function at finite Temperature • Bosonic partition function

13. Partition function of interacting fields: Perturbative Series: In the theory the free propagator is given by Bosonic Matsubara frequencies

14. In higher orders of perturbation  Full photon propagator is the self energy • QED free photon propagator • Photon self energy

15. 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

16. QED Ring Diagrams at zero B and non-zero T • Using the free propagator and the photon self energy 

17. QED Ring potential

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

19. Effects of Ring Potential • In the MSM  EW phase transition • Changing the type of phase transition • Decreasing the critical T

20. 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?

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

22. 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

23. 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 effectivequantum field theory (QFT) replaces the full QFT

24. Properties of effective QED in the LLL (I) A) Dimensional reduction • Fermion propagator  Dimensional Reduction • Photon acquires a finite mass

25. 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

26. Part 3: QED at Finite B and T QED Effective Potential at nonzero T and B

27. 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

28. QED One-Loop Effective Potential at Finite T and B • T independent part • T dependent part

29. 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

30. 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)

31. The integrals

32. IR vs. Static Limit • Ring potential in the IR limit • In the static limitk 0 

33. QED Ring Potential in Weak Magnetic Field Limit

34. 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

35. 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)

36. QED Ring Potential in Strong Magnetic Field Limit

37. 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:

38. QED Ring Potential in Strong B Field limit at nonzero T • Conditions: • Evaluating in limit with

39. 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)

40. Dynamical Chiral Symmetry Breaking in the LLL

41. 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

42. 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

43. One-loop Contribution • Dynamical mass • Critical temperature Tc is determined by

44. Ring Contribution • Using and • Dynamical mass • Critical temperature of Dynamical Symmetry Breaking (DSB)

45. 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

46. Critical Temperature of DSB in the Static Limit • Using • The critical temperature Tc in the static limit

47. 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

48. Define the efficiency factor where and the Lambert W(z) function, staisfying • It is known that

49. Numerical Results • Choosing , and • Astrophysics of neutron stars • RHIC experiment (heavy ion collisions)