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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 2i l T1  ,
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
2T
( 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
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
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  3.110 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