Nonperturbative and analytical approach to Yang-Mills thermodynamics

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

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Typical situation in thermal perturbation theory
taken from Kajantie et al. 2002
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Status in unsummed TPT

People compute pressure up to
and fit an additive constant to lattice data.
BUT WHAT HAVE WE LEARNED ?

Try an inductive analytical approach
to Yang-Mills thermodynamics
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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
Outline
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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

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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 .
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Calorons

• SU(N) calorons are (Nahm 1984, vanBaal Kraan
  1998)
• (i) Bogomolnyi-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.

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SU(2)
taken from van Baal Kraan 1998
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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
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Remarks
remark 3 probably defined in a
nonlocal way in terms of
fundamental gauge fields, possible local
definition
BPS
remark 4 caloron BPS
BPS
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Remarks
remark 5 ground state described by pure gauge
configuration otherwise O(3) invariance
violated
remark 6 breaks
gauge symmetry at most to
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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
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Effective action
At large temperatures , that is,
in the electric phase (E), we propose the
following effective action
where
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How does a potential look like which is
in accord with the postulate?
Lets work in a gauge where
0
0
.
(winding gauge)
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We propose
where
and
,
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Ground-state thermodynamics
BPS equation for
(winding gauge)
solutions
is traceless and hermitian and breaks
symmetry maximally
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Does fluctuate?
quantum mechanically
No !
compositeness scale
thermodynamically
No !
.
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top. trivial gauge-field fluctuations
(ground-state part of caloron interaction
effectively)
solve
.
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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,
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Mass spectrum
We have
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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
.
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Evolution equation
cured by imposing minimal thermodynamial
self-consistency (Gorenstein 1995)
evolution equation for
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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
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Evolution of effective gauge-coupling
plateau value (independent of )
logarithmic singularity
(independent of )
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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

1. annihilation into a monopole-antimonopole pair
2. elastic scattering
3. inelastic scattering (instable monopoles)

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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
.
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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
.
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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

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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
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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

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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
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Relaxation to the minima
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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
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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
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Outline
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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)
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Pressure
(0.97)
(0.88)
.
J. Engels et al. (1982)
G. Boyd et al. (1996)
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Energy density
(0.93)
(0.85)
J. Engels et al. (1982)
G. Boyd et al. (1996)
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Entropy density
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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?

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Outline
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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
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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
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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
.
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Coarse-grained contribution to
local fireballs from high-energy particle
collisions
visible Universe
e.g. ee collision
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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)
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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
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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

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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 !