Nonperturbative and analytical approach to Yang-Mills thermodynamics

1. Seminar-Talk, 20 April 2004, Universität Bielfeld Nonperturbative and analytical approach to Yang-Mills thermodynamics Ralf Hofmann, Universität Heidelberg

2. Motivation for nonperturbative approach to • SU(N) Yang-Mills theory • Construction of an effective theory • Comparison of thermodynamical • potentials with lattice results • Application: • A strongly interacting theory • underlying QED? Outline

3. Motivation analytical grasp of SU(N) YM thermodynamics • on experimental grounds • RHIC results: • success of hydrodynamical approach to elliptic flow, QGP most perfect fluid known in Nature: • only at large collision energy transverse expansion dominated by perturbative QGP • Why is pressure so different from SB on the lattice at ? • Cosmological expansion: • What do Hubble expansion and expansion of fire ball in early stage of HIC have in common? • (Shuryak 2003) on theoretical grounds • Thermal perturbation theory (TPT): • naive TPT only applicable up to • (weakly screened magnetic gluons, Linde 1980) • poor convergence of thermodynamical potentials • resummations: • HTL: nonlocal theory for semi-hard, soft modes, • fails to reproduce the pressure at , • Local expansion -> dependent UV div. • SPT: loss of gauge invariance • in local approximation of HTL vertices • Lattice: • strong nonperturbative effects at very large • (Hart & Philipsen 1999, private communication)

4. Typical situation in thermal perturbation theory taken from Kajantie et al. 2002

5. Status in unsummed TPT People compute pressure up to and fit an additive constant to lattice data. BUT WHAT HAVE WE LEARNED ? Try an inductiveanalytical approach to Yang-Mills thermodynamics

6. Broader Motivations • Why accelerated cosmological expansion at present • (dark energy)? • Origin of dark matter • How can pointlike fermions have spin and finite classical • self-energy? What is the reason for their apparent pointlike- • ness? • Are neutrinos Majorana and if yes why? • If theoretically favored existence of intergalactic magnetic • fields confirmed, how are they generated? • ...

7. Motivation for nonperturbative approach to • SU(N) Yang-Mills theory • Construction of an effective theory • Comparison of thermodynamical potentials with • lattice results • Application: A strongly interacting gauge theory • underlying QED? Outline

8. Conceptual similarity macroscopic theory for superconductivity (Landau-Ginzburg-Abrikosov): • introduce complex scalar field to describe condensate of Cooper • pairs macroscopically, stabilize this field by a potential • effectively introduces separation between gauge-field • configurations associated with the existence of Cooper pairs and • those that are fluctuating around them • mass for fluctuating gauge fields by Abelian Higgs mechanism

9. Construction of an effective thermal theory A gauge-field fluctuation in the fundamental SU(N) YM theory can always be decomposed as minimal (BPS saturated ) topologically nontrivial part topologically trivial part Postulate: At a high temperature, , SU(N) Yang-Mills thermodynamics in 4D condenses SU(2) calorons with varying topological charge and embedding in SU(N). The caloron condensate is described by a quantum mechanically and thermodynamically stabilized adjoint Higgs field .

10. Calorons • SU(N) calorons are(Nahm 1984, vanBaal & Kraan 1998): • (i) Bogomoln´yi-Prasad-Sommerfield (BPS) saturated solutions • to the Euclidean Yang-Mills equation • at • (ii) SU(2) caloron composed of BPS magnetic monopole • and antimonopole with increasing spatial separation as • decreases.

11. SU(2) taken from van Baal & Kraan 1998

12. Remarks remark 1: caloron condensation shown to be self-consistent by large fundamental gauge coupling ; charge-one caloron action remark 2: since action density of a caloron is dependent modulus of caloron condensate is dependent

13. Remarks remark 3:probably defined in a nonlocal way in terms of fundamental gauge fields, possible local definition BPS remark 4: caloron BPS BPS

14. Remarks remark 5: ground state described by pure gauge configuration otherwise O(3) invariance violated remark 6:breaks gauge symmetry at most to

15. Remarks remark 6: is compositeness scale off-shellness of quantum fluctuations is constrained as Higgs-induced mass thermodynamical self-consistency: temperature evolution of effective gauge coupling such that thermodynamical relations satisfied remark 7:

16. Effective action At large temperatures , that is, in the electric phase (E), we propose the followingeffective action: where

17. How does a potential look like which is in accord with the postulate? Let´s work in a gauge where 0 0 . (winding gauge)

18. We propose where and ,

19. Ground-state thermodynamics BPS equation for : (winding gauge) solutions: is traceless and hermitianandbreaks symmetrymaximally

20. Does fluctuate? quantum mechanically: No ! compositeness scale thermodynamically: No ! .

21. top. trivial gauge-field fluctuations (ground-state part of caloron interaction effectively) solve .

22. Gauge-field fluctuations consider: back reaction of on gauge field taken into account thermodynamically (TSC) perform gauge trafo to unitary gauge, involves nonperiodic gauge functions but: periodicity of is left intact no Hosotani mechanism upon integrating out in unitary gauge,

23. Mass spectrum We have:

24. Thermodynamical self-consistency pressure (one-loop): ideal gas of massless and massive particles plus ground-state contribution ( ) (correction to from quantum part of gauge-boson loop is negligibly small) however: masses and ground-state pressure are both dependent derivatives involve not only explicit but also implicit dependences relations between pressure and energy density and other thermod. potentials violated: .

25. Evolution equation cured by imposing minimal thermodynamial self-consistency (Gorenstein 1995): evolution equation for

26. Evolution with temperature right-hand side: evolution has two fixed points at there is a highest and a lowest attainable temperature in the electric phase

27. Evolution of effective gauge-coupling plateau value (independent of ) logarithmic singularity (independent of )

28. Interpretation • at we have (condensate forms) • calorons in condensate grow and scatter, calorons action small calorons condense plateau value of : existence of isolated magnetic charge 3 possibilities: • annihilation into a monopole-antimonopole pair • elastic scattering • inelastic scattering (instable monopoles)

29. Do we understand this in the effective theory? stable winding around isolated points in 3D space in SU(2) algebra only at isolated points in time monopole flashes monopole-antimonopole pair .

30. Transition to the magnetic phase at we have: TLH modes decouple kinematically, mass on tree level TLM modes remain massless monopole mass monopoles condense in a 2nd order – like phase transition ( continuous), symmetry breaking: .

31. Magnetic phase • condensates of stable monopoles described by • complex fields , • symmetry represented by local permutations of • potential • again, winding solutions to BPS equation • again, no field fluctuations • again, zero-curvature solution to Maxwell equation • now, some (dual) gauge fields massive by Abelian Higgs mech. • again, evolution equation for magnetic coupling • from TSC

32. Evolution with temperature logarithmic singularities Continous increase with temperature possible since monopoles condensed evolution has two fixed points at there is a highest and a lowest attainable temperature in the magnetic phase

33. Center vortices • form in the magnetic phase as quasiclassical, closed loops • composed of monopoles and antimonopoles (Olejnik et al. 1997) • a single vortex loop has a typical action: • magnetic coupling has logarithmic singularity at • unstable monopoles form stable dipoles which condense • all dual Abelian gauge modes • decouple thermodynamically • center vortices condense

34. Transition to center phase center-vortex loops are one-dimensional objects, nonlocal definition: monopole part included in limit a discussion of the 1st order phase transition can be based on BPS saturated solutions subject to potential: extrapolate to finite

35. Relaxation to the minima

36. Relaxation to the minima at finite there exist tangential tachyonic modes associated with dynamical and local transformations: relaxation to minima by generation of magnetic flux quanta (tangential) and radial excitations

37. Matching the phases pressure continuous across a thermal phase transition scales are related Dimensional transmutation already seen in TPT also takes place here. There is a single independent scale, say , determined by a boundary condition

38. Motivation for nonperturbative approach to • SU(N) Yang-Mills theory • Construction of an effective theory • Comparison of thermodynamical potentials with • lattice results • Application: a strongly interacting theory underlying • QED? Outline

39. Computation and comparison with the lattice • negative pressure in low-T electric and in magnetic phase • lattice data for , • (up to 40% deviation for , Stefan-Boltzmann limit • reached at but with larger number of polarizations) pressure (electric phase): pressure (magnetic phase):

40. Pressure (0.97) (0.88) . J. Engels et al. (1982) G. Boyd et al. (1996)

41. Energy density (0.93) (0.85) J. Engels et al. (1982) G. Boyd et al. (1996)

42. Entropy density

43. Possible reasons for deviations • at low : • - no radiative corrections in magnetic phase, 1-loop result exact • - integration of plaquette expectation, biased integration • constant (Y. Deng 1988)? • - finite-volume artefacts, how reliable beta-function used? • at high : • - to maintain three polarization up to arbitrarily • small masses may be unphysical • (in fits always two polarizations assumed) • - radiative corrections in electric phase? • - finite lattice cutoff?

44. Motivation for nonperturbative approach to • SU(N) Yang-Mills theory • Construction of an effective theory • Comparison of thermodynamical potentials with • lattice results • Application: a strongly interacting theory • underlying QED? Outline

45. Application: QED and strong gauge interactions consider gauge symmetry: naively: to interprete as solitons of respective SU(2) factors localized zero mode Crossing of center vortices =1/2 magnetic monopole stable states neutral and extremely light particle one unit of U(1) charge

46. It turns out… local symmetry in confining phase of SU(2) gauge theory makes stable fermion states boundary condition for: • we see one massless photon in the CMB • including radiative corrections in electric phase • photon is precisely massless at a single point photon mass magnetic electric

47. Homogeneous contribution to • CMB boundary condition determines the scale • potential can be computed at This is the homogeneous part of . we have: This is smaller than .

48. Coarse-grained contribution to local ‘fireballs’ from high-energy particle collisions visible Universe e.g. ee collision

49. Value of the fine-structure constant naively (only one SU(2) factor and one-loop evolution): taking 3-photon maximal mixing into account at (one-loop):

50. More consequences • spin-polarizations as two possible center-flux-directions in • presence of external magnetic field • intergalactic magnetic fields: in magnetic phase • neutrino single center-vortex loop • cannot be distinguished from antiparticle • neutrino is Majorana • Tokamaks ground state is superconducting