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Nonperturbative and analytical approach to Yang-Mills thermodynamics Seminar-Talk, 20 April 2004, Universität Bielfeld Ralf Hofmann, Universität Heidelberg Outline • 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? 2 Ralf Hofmann, Heidelberg Motivation analytical grasp of SU(N) YM thermodynamics on experimental grounds on theoretical grounds RHIC results: Thermal perturbation theory (TPT): • success of hydrodynamical approach to elliptic flow, QGP most perfect fluid known in Nature: • naive TPT only applicable up to g5 (weakly screened magnetic gluons, Linde 1980) / s 1 • poor convergence of thermodynamical potentials • resummations: • only at large collision energy transverse expansion dominated by perturbative QGP HTL: nonlocal theory for semi-hard, soft modes, fails to reproduce the pressure at T Tc, • Why is pressure so different from SB on the lattice at T 4 5 Tc ? Local expansion ->T dependent UV div. SPT: loss of gauge invariance Cosmological expansion: in local approximation of HTL vertices • What do Hubble expansion and expansion of fire ball in early stage of HIC have in common? Lattice: • strong nonperturbative effects at very large T T (Shuryak 2003) (Hart & Philipsen 1999, private communication) 3 Ralf Hofmann, Heidelberg Typical situation in thermal perturbation theory taken from Kajantie et al. 2002 4 Ralf Hofmann, Heidelberg Status in unsummed TPT 6 People compute pressure up to g ln g and fit an additive constant to lattice data. BUT WHAT HAVE WE LEARNED ? Try an inductive analytical approach to Yang-Mills thermodynamics 5 Ralf Hofmann, Heidelberg 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? • ... 6 Ralf Hofmann, Heidelberg Outline • 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? 7 Ralf Hofmann, Heidelberg 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 8 Ralf Hofmann, Heidelberg Construction of an effective thermal theory A gauge-field fluctuation A in the fundamental SU(N) YM theory can always be decomposed as A A top a minimal (BPS saturated ) topologically nontrivial part topologically trivial part Postulate: At a high temperature, T YM , N, 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 . 9 Ralf Hofmann, Heidelberg Calorons SU(N) calorons are (Nahm 1984, vanBaal & Kraan 1998): (i) Bogomoln´yi-Prasad-Sommerfield (BPS) saturated solutions to the Euclidean Yang-Mills equation D G 0 at T 0 (ii) SU(2) caloron composed of BPS magnetic monopole and antimonopole with increasing spatial separation as T decreases. 10 Ralf Hofmann, Heidelberg SU(2) taken from van Baal & Kraan 1998 11 Ralf Hofmann, Heidelberg Remarks remark 1: caloron condensation shown to be self-consistent by g ; charge-one caloron action large fundamental gauge coupling S cal remark 2: 8 2 g 2 0 since action density of a caloron is T dependent modulus of caloron condensate is 12 T dependent Ralf Hofmann, Heidelberg Remarks remark 3: probably defined in a nonlocal way in terms of fundamental gauge fields, possible local definition ( N d a abc BPS remark 4: caloron BPS 2) F F cal b c cal 0 absence of a BPS 0 13 Ralf Hofmann, Heidelberg Remarks remark 5: ground state described by pure gauge configuration G 0 otherwise O(3) invariance violated remark 6: breaks gauge symmetry at most to SU ( N ) U (1) N 1 N ( N 1) massive gauge modes (TLH) & N 1 massless gauge modes (TLM) 14 Ralf Hofmann, Heidelberg Remarks remark 6: is compositeness scale off-shellness of quantum fluctuations a is constrained as p m 2 2 2 or pE m 2 2 2 Higgs-induced mass remark 7: thermodynamical self-consistency: temperature evolution of effective gauge coupling e(T ) such that thermodynamical relations satisfied 15 Ralf Hofmann, Heidelberg Effective action At large temperatures ( T YM ) , that is, in the electric phase (E), we propose the following effective action: 1/ T S E d d x (1/2 tr NG G 3 0 tr N D D VE ) where D ie , a , G G t , a a G a a e f abca a , a a trNt t 1 / 2 a b a b c ab 16 Ralf Hofmann, Heidelberg How does a potential VE look like which is in accord with the postulate? Let´s work in a gauge (N even) SU (2) SU (2) SU (2) 0 where 0 . (winding gauge) 17 Ralf Hofmann, Heidelberg We propose VE tr N vE vE E tr N 6 2 where 3 N /2 1 vE i E diag 1 2 ,..., 1 2 1 N /2 and l 2 1 2 tr2 l 2 , 18 0 1 , ... . 1 1 0 Ralf Hofmann, Heidelberg Ground-state thermodynamics BPS equation for : vE (winding gauge) solutions: E l 3 exp 2i l T1 , 2 l 3 l 1,..., N / 2 l is traceless and hermitian and breaks symmetry maximally 19 Ralf Hofmann, Heidelberg Does fluctuate? quantum mechanically: 2 l VE l 2 compositeness scale 3l E 1 3 3 2T ( E ) E No ! thermodynamically: 2 l VE T 2 12 l 1 2 2 No ! 20 . Ralf Hofmann, Heidelberg top. trivial gauge-field fluctuations (ground-state part of caloron interaction effectively) solve D G 2ie , D solution a g .s. 0 with G D 0 1/ T SE d d 3 x (0 0 VE ) with 0 VE 2 E T N ( N 2) 3 21 . Ralf Hofmann, Heidelberg Gauge-field fluctuations consider: a a g .s. a back reaction of on gauge field a g .s. taken into account thermodynamically (TSC) perform gauge trafo to unitary gauge, diag and a g .s. 0 involves nonperiodic gauge functions but: periodicity of l 1T l a , is left intact no Hosotani mechanism upon 22 ... a in unitary gauge, integrating out Ralf Hofmann, Heidelberg Mass spectrum We have: 1/ T 1 S E d d x ( trN G a G a 2 0 3 N( N-1) k 1 mk a ,k VE ) where 2 2 mk (T ) 2e trN (T ), t k 2 2 23 2 Ralf Hofmann, Heidelberg Thermodynamical self-consistency pressure (one-loop): ideal gas of massless and massive particles plus ground-state contribution ( VE ) (correction to VE from quantum part of gauge-boson loop is negligibly small) however: masses and ground-state pressure are both T T dependent derivatives involve not only explicit but also implicit dependences relations between pressure P and energy density thermod. potentials violated: dP T P dT 24 and other . Ralf Hofmann, Heidelberg Evolution equation cured by imposing minimal thermodynamial self-consistency (Gorenstein 1995): mk P 0 evolution equation for 24E a a E 6 (2 ) N ( N 2) 4 e(T ) N ( N 1) k 1 2 ck D(ak ) m1 2T ( a , E ) T E 25 Ralf Hofmann, Heidelberg Evolution with temperature right-hand side: evolution E (a) has two fixed points at a 0 and a there is a highest and a lowest attainable temperature in the electric phase 26 Ralf Hofmann, Heidelberg Evolution of effective gauge-coupling plateau value (independent of E , P ) logarithmic singularity E , P (independent of E , P ) E ,c 27 Ralf Hofmann, Heidelberg Interpretation • at E , P we have e g plateau value of (condensate forms) calorons action small calorons condense e : existence of isolated magnetic charge calorons in condensate grow and scatter, 3 possibilities: (a) annihilation into a monopole-antimonopole pair (b) elastic scattering (c) inelastic scattering (instable monopoles) 28 Ralf Hofmann, Heidelberg Do we understand this in the effective theory? SU (2) SU (2) SU (2) in SU(2) algebra at isolated only points in time stable winding around isolated points in 3D space monopole flashes monopoleantimonopole pair . 29 Ralf Hofmann, Heidelberg Transition to the magnetic phase at 0 E,c we have: e logarithmi cally TLH modes decouple kinematically, mass e on tree level TLM modes remain massless l 0 monopole mass e monopoles condense in a 2 nd order – like phase transition (a continuous), symmetry breaking: U (1) N 1 30 ZN . Ralf Hofmann, Heidelberg Magnetic phase • condensates of N / 2 stable monopoles described by complex fields ZN i i 1, ..., N / 2 , symmetry represented by local permutations of • potential N 1 VM vi vi i 1 where vi i • again, winding solutions to BPS equation • again, no i M i , a D ,i 3 i 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 g (T ) from TSC 31 Ralf Hofmann, Heidelberg Evolution with temperature logarithmic singularities Continous increase with temperature possible since monopoles condensed M ,c evolution M (a) has two fixed points at a 0 and a there is a highest and a lowest attainable temperature in the magnetic phase 32 Ralf Hofmann, Heidelberg 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: S • magnetic coupling g CV 1 g2 has logarithmic singularity at T TM 0 c • unstable monopoles form stable dipoles which condense D • all dual Abelian gauge modes a ,k (k 1,..., N 1) decouple thermodynamically • center vortices condense 33 Ralf Hofmann, Heidelberg Transition to center phase center-vortex loops are one-dimensional objects, nonlocal definition: k ( x) C exp ig dz A ,k D A ,k D monopole part included in limit N a discussion of the 1st order phase transition can be based on BPS saturated solutions subject to potential: VC vC k N 1 vC vC k k where k 1 C k 3 N 1 k N 3 C extrapolate to finite 34 N Ralf Hofmann, Heidelberg Relaxation to the minima 35 Ralf Hofmann, Heidelberg Relaxation to the minima at finite N there exist tangential tachyonic modes associated with dynamical and local Z N transformations: relaxation to minima by generation of magnetic flux quanta (tangential) and radial excitations 36 Ralf Hofmann, Heidelberg Matching the phases pressure continuous across a thermal phase transition scales E , M (C for N ) are related Dimensional transmutation already seen in TPT also takes place here. There is a single independent scale, say a boundary condition 37 E , determined by Ralf Hofmann, Heidelberg Outline • 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? 38 Ralf Hofmann, Heidelberg Computation and comparison with the lattice • negative pressure in low-T electric and in magnetic phase • lattice data for N 2,3 , P and and S (up to 40% deviation for T 5 Tc , Stefan-Boltzmann limit reached at T 20Tc but with larger number of polarizations) pressure (electric phase): pressure (magnetic phase): 39 Ralf Hofmann, Heidelberg Pressure (0.88) . J. Engels et al. (1982) (0.97) G. Boyd et al. (1996) 40 Ralf Hofmann, Heidelberg Energy density (0.93) (0.85) J. Engels et al. (1982) G. Boyd et al. (1996) 41 Ralf Hofmann, Heidelberg Entropy density 42 Ralf Hofmann, Heidelberg Possible reasons for deviations • at low T: - 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 T : - 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? 43 Ralf Hofmann, Heidelberg Outline • 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? 44 Ralf Hofmann, Heidelberg Application: QED and strong gauge interactions consider gauge symmetry: SU (2)CMB SU (2) e SU (2) naively: to interprete e, 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 45 Ralf Hofmann, Heidelberg It turns out … Z local symmetry in confining phase of SU(2) gauge 2 theory makes stable fermion states boundary condition for SU (2)CMB : • 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 46 Ralf Hofmann, Heidelberg Homogeneous contribution to cos CMB boundary condition determines the scale CMB potential This is the homogeneous part of VM can be computed at TCMB cos . we have: 1 hom 4 VM cos 3.110 eV 2 This is smaller than W MAP cos 47 3 1.0 10 eV 4 . 4 Ralf Hofmann, Heidelberg Coarse-grained contribution to cos local ‘fireballs’ from highenergy particle collisions visible Universe e.g. SU (2) e ee collision SU (2)CMB 48 Ralf Hofmann, Heidelberg Value of the fine-structure constant naively (only one SU(2) factor and one-loop evolution): 2 g2 1 g and 17.15 4 93.6 taking 3-photon maximal mixing into account at T m (one-loop): 2 gr 1 4 140.433 49 Ralf Hofmann, Heidelberg More consequences • spin-polarizations as two possible center-flux-directions in presence of external magnetic field • intergalactic magnetic fields: SU (2) CMB in magnetic phase ground state is superconducting • neutrino single center-vortex loop cannot be distinguished from antiparticle neutrino is Majorana • Tokamaks 50 Ralf Hofmann, Heidelberg Conclusions and outlook • analytical approach to SU(N) YM thermodynamics • shortly above confining transition: negative pressure • compared with lattice data • electromagnetism (electron no infinite Coulomb self-energy) • QCD; What are quarks? • QCD thermodynamics: two-component, perfect fluid • QCD EOS input for hydrodynamical simulations of HICs 51 Ralf Hofmann, Heidelberg Literature R. H. : hep-ph/0304152 [PRD 68, 065015 (2003)], hep-ph/0312046, hep-ph/0312048, hep-ph/0312051, hep-ph/0401017, hep-ph/0404??? Thank you ! 52 Ralf Hofmann, Heidelberg