Quantum Field Theory in Hot and Dense Media

1. Quantum Field Theory in Hot and Dense Media MahnazQaderHaseeb SumayyaYaqeen Department of Physics COMSATS Institute of Information Technology Islamabad ISS-2017, NCP, Islamabad, March 13-17, 2017

2. Motivation • Extremely high temperatures in the early universe. • Nuclear processes in the very hot and dense cores of stars. • Very high densities and high temperatures in heavy-ion collisions probed through particle accelerators. • Production of QED plasma with Laserbeams. • Studies on Quark Gluon Plasma (QGP).

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5. Finite temperature and density present in • Supernovae explosions • T ~ 3 x1011 K (30 MeV) are possible: • in the vicinity of neutron stars, and • in accretion disks around black holes. • Creation ultra-relativistic e+e– plasmas with high-intensity lasers (≈1018 W/cm2) • two opposite laser pulses hitting a thin gold foil can heat up the foil up to several MeV leading to e+e–pair creation • The latest accelerators at LHC and RHIC detect QGP at • temperatures >150 MeV and • density 5-10 times nuclear densities.

6. Supernova 1987A • Tychoand Kepler observed that tremendous amount of energy is released in a supernova explosion. • The only supernova in modern time, visible to the naked eye detected on Feb. 23, 1987 (known as SN1987A). • SN1987A emitted more than 1011 times visible light as the Sun for over one month! Temperatures ~ 21011K. Sanduleak

7. Time QGP • A hot and dense fireball (“little bang”) ~ atomic nucleus, is produced which could exist for <10-22 sec in lab and is a form of matter at transition temperatures ~100-200 MeV. • The hot and dense environment in QGP and the studies of its reproduction in nucleus-nucleus collisions require TFT methods for more detailed understanding. • With the increased feasibility of creation of QGP in latest particle accelerators: • RHIC and • LHC the methods developed in thermal field theory have their specific significance in QCD at finite temperature - Extensive research on various aspects since last few decades.

8. Special Relativity Quantum Mechanics Q. Statistical Mechanics Finite Temperature and Density Quantum Field Theory Quantum Field Theory

9. Quantum Statistical Mechanics Ensembles • define statistical thermodynamic properties of a system. • Three categories • Micro canonical Ensemble • Canonical Ensemble • Grand canonical Ensemble • Crucial mathematical elements in QSM are • Statistical density element • Partition function • Any observable can be defined by

10. Quantum Statistical Mechanics Suitable choice for Finite Temperature Field Theory (FTFT) is canonical ensemble: • A system which is in contact with the heat reservoir at Temperature T • Fixed particle number and volume • Energy exchange between system and reservoir is allowed • Mean energy of a system • The partition function is

11. Finite Temperature Field Theory • Relativistic generalization of finite temperature non-relativistic quantum statistical mechanics General Idea • The general formulation of FTFT was developed by Weinberg, Dolan and Jackiw and Duncan. • The transition amplitude of QM is replaced by path integral in field theory • The partition function of quantum statistical mechanics could be represented as the functional integral. • Non-interacting systems of particles (QSM) set the foundation for the functional integral representation of partition function.

12. At finite temperature, the path integral description of a quantum mechanical system proves itself to be intrinsically unique. • It provides multiple techniques with various pros attached to each evaluation and yet in the end results are equivalent. • These techniques are inter-convertible as well. • FT Formalisms • Imaginary-time Formalism (ITF) • Real-time Formalism (RTF)

13. Finite Temperature Formalisms • Imaginary-time Formalism • Dynamical time is traded with temperature. • In QSM, the observables are ensemble averages, in terms of partition function • Thus defining imaginary-time variable Kubo-Martin Schwinger Relation (KMS)

14. Imaginary-time Formalism • The two point correlation function for a generalized field operator has KMS relation • In terms of imaginary time variable • It is observable here that Periodicity or anti-periodicity

15. FTFT Imaginary-time Formalism Matsubara Frequencies In frequency domain the field is In order to justify KMS, field can attain only discrete frequencies where

16. FTFT Canonical partition function for a QM system, in coordinate basis is Comparison of transition amplitude via path integral technique yields Thus the path integral representation of partition function becomes where Euclidean imaginary-time action is defined over an interval as To satisfy a trace (partition function) initial and boundary condition must be same:

17. FTFT Path Integral and Partition Function The boson and fermion field analogous for partition function in path integral form are respectively with respective boundary conditions

18. FTFT Real-time Formalism • Mostly utilized for non-equilibrium conditions • Provides both time and temperature • Time-like four velocity of the heat bath is introduced: • The exponentials in the propagators take the form • The energies are continuous in RTF. • RTF provides explicit zero temperature terms and finite temperature terms.

19. FTFT Wick’s Rotation • It is a method which acquires solution in Minkowski space for some mathematical problem, from a solution to a related problem in Euclidean space. • It is done by transformation of real time variable to an imaginary time variable. Euclidean Metric Minkowski Metric

20. FTFT Propagators Scalar Propagator • Two-point correlation function is defined as • In complete set of eigen states, considering only time-argument • Similarly • In defined range, KMS relation is

21. FTFT Propagators Scalar Propagator • In Fourier frequency space • Spectral density At and differentiating the following equation w.r.t. time yields Bose-Einstein Distribution factor Odd in B

22. FTFT Propagators Scalar Propagator • Under Euclidean space, in terms of imaginary time variable, • With Fourier transform • Similarly, time-ordered Feynman correlation function is Matsubara Propagator

23. FTFT Propagators Scalar Propagator • Using spectral density definition Feynman correlator becomes • At zero temperature • Explicit calculation of spectral density shows that • The free scalar Feynman Propagator is

24. FTFT Propagators Fermion Propagator • Two point correlation functions are • Spectral density is • Euclidean propagator • Matsubara propagator Fermi-Dirac Distribution factor

25. Gauge theories at FTFT QED • The Lagrangian describing the system of electrons and photons is • The partition function for photons is • Gauge transformation should not change anything physically thus the invariance demands • Thus

26. Gauge theories at FTFT QED • The full partition function of QED is • Evaluation of this yields • In imaginary-time variable • unitary photon transformation is exploited

27. QED at FTFT Photon Propagator • By definition • The free photon propagator is • In the presence of medium i.e. temperature/ heat bath the propagator adapts the form • Photon self energy is Lorentz covariant and obeys Ward identity Photon self energy

28. QED at FTFT Photon Propagator • Medium introduces a 4-vector such that • Combination provides various tensors such as with and follows All components are independent, thus Transverse Projector Longitudinal Projector

29. QED at FTFT Photon Self -energy • Feynman rules lead to • Since fermion propagator is • Further mathematical evaluation leads to

30. QED at FTFT Full Photon Propagator • Full photon propagator is • Photon self energy can be defined as Thus Full propagator is F and G are determinable scalar decomposition functions • Full propagator requires self-energy • Self-energy subsequently demands F and G evaluation

31. Photon Self-energy Computation of F • Projectors • Since Hard Thermal Loop Limit • High temperature limit • Temperature is much higher than any mass scale at zero temperature • Mass terms can be dropped • External momenta can be neglected in comparison to loop momenta. 00-components are

32. Photon Self-energy Computation of F • Dropping external momenta and mass terms • In HTL approximation • With the only non-zero terms are

33. Photon Self-energy Computation of F • In HTL limit • After the evaluation of integrals the only thermal contributors yield • Thus where Photon Thermal mass squared and

34. Photon Self-energy Computation of G • Along almost the similar line the decomposition function G can also be evaluated • Here two different terms are encountered which contribute to self-energy • xx components of projectors are

35. Photon Self-energy Computation of G • Final result for the computation of G • From the evaluation of F and G it is clear that photon self-energy thus is both temperature and momentum dependent.

36. Electron Self-energy • Electron self energy • In HTL corrections, Electron thermal mass squared

37. Breakdown of Perturbative Theory • In QFT, the theory is expanded perturbatively in terms of dimensionless coupling constant in progressive order. • TFT due to contributions from HTL, higher order Feynman diagrams may sometimes share same magnitude with the lower order ones (in coupling constant). • This problem arises because of the fact that massless scalar particles acquire a thermal mass in thermal background. For example the full propagator of scalar theory is • Free propagator is of same magnitude as thermal mass. • This gives us the hint that perturbative theory breaks down for such soft momenta thus it must be resummed.

38. Breakdown of Perturbative Theory Solution • Since the only HTL contribution to is from two-point correlation functions thus we may use effective propagator to obtain an improved perturbative expansion • Gauge theories however are more complicated due to the dependence of self energies on temperature, momentum and energy. • So for gauge theories one may start from the effective Lagrangian. In QED, the effective Lagrangian for photon and fermion HTL are

39. QED • All n-point functions can be generated from effective Lagrangians. • For example photon-electron vertex, by evaluating • Lorentz covariance requires the strict condition of obeying Ward identities. This could be used as test to check the correct behavior of effective perturbative theory. • Resummation procedures enable to calculate thermodynamicals to higher orders

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41. System Fermions or Bosons Hot and Dense System • Consider a system completely out of equilibrium with lots of kinetic energy. • Use the Grand Canonical Ensemble to calculate the abundances of all the final measured particles. Depends on Temperature T and Chemical Potential μ.