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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
• Why is pressure so different from SB
on the lattice at T  4 5 Tc ?
T  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)
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Ralf Hofmann, Heidelberg
Typical situation in thermal perturbation theory
taken from Kajantie et al. 2002
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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
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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

.
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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.
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Ralf Hofmann, Heidelberg
SU(2)
taken from van Baal & Kraan 1998
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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
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T

dependent
Ralf Hofmann, Heidelberg
Remarks
remark 3:
 probably defined in a nonlocal way in terms of
fundamental gauge fields, possible local definition ( N  2)
 d
a
abc
BPS
remark 4: caloron BPS
 F F  cal
b
c
  cal 0
absence
of
a
 
BPS
   0
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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) massivegauge modes(T LH)&
N 1
massless gauge modes(T LM)
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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
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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 trN G G 
3
0
trN D D  VE )
where


D    ie  , a , G  G t ,
a a
G    a   a  e f
a
a
a
abc
b
c
a a ,
trNt at b  1 / 2  ab
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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) where
 SU (2)

SU (2)


SU (2)

 


0




0













.
(winding gauge)
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Ralf Hofmann, Heidelberg
We propose
VE  trN vE vE   E trN
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
,
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 0 1
 , ....
1  
1 0 
Ralf Hofmann, Heidelberg
Ground-state thermodynamics
BPS equation for  :
   vE
(winding gauge)
solutions:
E
 l
3 exp 2i l T1  ,
2 l
3
l  1,...,N / 2
 l is traceless and hermitian and  breaks symmetry
maximally
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Ralf Hofmann, Heidelberg
Does

fluctuate?
quantum mechanically:

2
l
VE
l
2
compositeness scale
 3l E  1
3

3
2T
(E 
)
E
No !
thermodynamically:

2
l
VE
T
2
 12 l  1
2 2

No !
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.
Ralf Hofmann, Heidelberg
top. trivial gauge-field fluctuations (ground-state
part of caloron interaction effectively)
solve
D G  2ie , D  
  solut iona
g .s.
 0 wit h
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
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 ...
a in unitary gauge,

integrating out
Ralf Hofmann, Heidelberg
Mass spectrum
We have:
1/ T
1
S E   d  d x ( t rN G a  G a  
2
0
3
N(N-1)

k 1
mk a ,k  VE ) where
2
2

mk (T )  2e t rN  (T ), t k
2
2
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
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
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 and other
.
Ralf Hofmann, Heidelberg
Evolution equation
cured by imposing minimal thermodynamial self-consistency
(Gorenstein 1995):
 mk P  0

evolution equation for
24E a
 a E  
6
(2 ) N ( N  2)
4
e(T )
N ( N 1)

k 1
2
ck D(ak )
m1
2T
( a
, E 
)
T
E
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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
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Ralf Hofmann, Heidelberg
Evolution of effective gauge-coupling
plateau value
(independent ofE , P )
logarithmic singularity
E, P
(independent ofE , P )
E ,c
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Ralf Hofmann, Heidelberg
Interpretation
•
at E, P we have e  g (condensate forms)


plateau value of
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)
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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
.
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Ralf Hofmann, Heidelberg
Transition to the magnetic phase
at
0  E,c
we have:
e   logarithmically
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
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 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
where vi  i
i 1
• 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
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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
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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
 2
g
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
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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
k 1
C

k
3
k
vC
k
wh ere
N 1
k

N 3
C
extrapolate to finite
34
N
Ralf Hofmann, Heidelberg
Relaxation to the minima
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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):
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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)
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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  3.110 eV
2
This is smaller than

WMAP
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 !
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Ralf Hofmann, Heidelberg