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Seminar-Talk, 20 April 2004, Universität Bielfeld. Nonperturbative and analytical approach to Yang-Mills thermodynamics. Ralf Hofmann, Universität Heidelberg. Motivation for nonperturbative approach to SU(N) Yang-Mills theory. Construction of an effective theory. - PowerPoint PPT Presentation
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Nonperturbative and analytical approach to
Yang-Mills thermodynamicsSeminar-Talk, 20 April 2004,
Universität Bielfeld
Ralf Hofmann,
Universität Heidelberg
2Ralf 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?
3Ralf Hofmann, Heidelberg
on experimental groundsRHIC 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)
Motivation
analytical grasp of SU(N) YM thermodynamics
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)
T
on theoretical grounds
1/ s
cTT 54
5g
cTT
T
T
4Ralf Hofmann, Heidelberg
Typical situation in thermal perturbation theory
taken from Kajantie et al. 2002
5Ralf Hofmann, Heidelberg
gg ln6People 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
Status in unsummed TPT
6Ralf 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?
• ...
7Ralf 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?
8Ralf 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
9Ralf Hofmann, Heidelberg
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 .
Construction of an effective thermal theory
A gauge-field fluctuation in the fundamental SU(N) YM theory can always be decomposed as
aAA top
A
minimal (BPS saturated ) topologically nontrivial part topologically trivial part
NYMT ,
10Ralf Hofmann, Heidelberg
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.
0T
0GD
T
Calorons
11Ralf Hofmann, Heidelberg
taken from van Baal & Kraan 1998
SU(2)
12Ralf Hofmann, Heidelberg
g
08
2
2
g
S cal
remark 2: since action density of a caloron is dependent
modulus of caloron condensate is dependent
Remarks
remark 1: caloron condensation shown to be self-consistent by
large fundamental gauge coupling ; charge-one caloron action
TT
13Ralf Hofmann, Heidelberg
remark 3: probably defined in a nonlocal way in terms of fundamental gauge fields, possible local definition
)2( N
cal cbabca FFd remark 4: caloron BPS
0
a
ofabsence
BPS
0 calBPS
Remarks
14Ralf Hofmann, Heidelberg
remark 6: breaks gauge symmetry at most to
remark 5: ground state described by pure gauge configuration
otherwise O(3) invariance violated
0G
(TLM) modes gauge massless 1
&(TLH) modes gauge massive )1(
)1()( 1
N
NN
UNSU N
Remarks
15Ralf Hofmann, Heidelberg
remark 6: is compositeness scale
off-shellness of quantum fluctuations
is constrained as
Remarks
a
or 222222 mpmp E
Higgs-induced mass
remark 7: thermodynamical self-consistency:
temperature evolution of effective gauge
coupling such that thermodynamical
relations satisfied
)(Te
16Ralf Hofmann, Heidelberg
At large temperatures , that is, in the electric phase (E), we propose the following effective action:
) ( YMT
) tr
tr1/2(
N
N
/1
0
3
E
T
E
VDD
GGxddS
where
abba
cbabcaaa
aa
tt
aafeaaG
tGGaieD
2/1tr
,
, , ,
N
Effective action
17Ralf Hofmann, Heidelberg
How does a potential look like which is in accord with the postulate?
Let´s work in a gauge where
)2(
)2(
)2(
SU
SU
SU
EV
0
0
.
even) (N
(winding gauge)
18Ralf Hofmann, Heidelberg
We propose
2N
6N tr tr EEEE vvV
where
2
2/
2/12
1
11
3 ,,...diag N
NEE iv
and
22
2 tr
2
1ll , . ... ,
0 1
1 01
19Ralf Hofmann, Heidelberg
Ground-state thermodynamics
BPS equation for :
Ev solutions:
2/,...,1 , 2 13
3
NlTlil
El
2exp
l is traceless and hermitian and breaks symmetry maximally
(winding gauge)
20Ralf Hofmann, Heidelberg
.
Does fluctuate?
quantum mechanically:
)2
( 13 332
2
EEE
l
E Tl
Vl
No !
thermodynamically:
compositeness scale
112 222
2
lT
VEl
No !
21Ralf Hofmann, Heidelberg
.
top. trivial gauge-field fluctuations (ground-state part of caloron interaction effectively)
solve
DieGD ,2 with0solution .. sga
0 DG
)2( 2
with )0(0
3
/1
0
3
NNTV
VxddS
EE
E
T
E
22Ralf Hofmann, Heidelberg
Gauge-field fluctuations
aaa sg ..consider:
back reaction of on gauge field
taken into account
thermodynamically (TSC)
..sga
perform gauge trafo to unitary gauge,
0 and diag .. sga
involves nonperiodic gauge functions lTl 1
but: periodicity of is left intact
no Hosotani mechanism upon
integrating out in unitary gauge,
,a
a
...
23Ralf Hofmann, Heidelberg
We have:
2 22
2,
)1
1
2
/1
0
3
),(tr2)(
where)
tr2
1(
kNk
Ek
N(N-
kk
N
T
E
tTeTm
Vam
aGaGxddS
Mass spectrum
24Ralf Hofmann, Heidelberg
.
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 dependentT
PdT
dPT
relations between pressure and energy density and other thermod. potentials violated:
P
Thermodynamical self-consistency
EV
EV
derivatives involve not only explicit but also implicit dependences
T
25Ralf Hofmann, Heidelberg
cured by imposing minimal thermodynamial self-consistency (Gorenstein 1995):
0 Pkm
evolution equation for )(Te
Evolution equation
)()2()2(
24 )1(
1
2
6
4
k
NN
kk
EEa aDc
NN
a
) 2
, ( 1
EE
T
T
ma
26Ralf Hofmann, Heidelberg
Evolution with temperature
right-hand side:
evolution has two fixed points at )(aE aa and 0
there is a highest and a lowest attainable temperature in the electric phase
27Ralf Hofmann, Heidelberg
Evolution of effective gauge-coupling
PE ,logarithmic singularity
cE ,
plateau value
(independent of ) PE ,
(independent of ) PE ,
28Ralf Hofmann, Heidelberg
• at we have (condensate forms)
• calorons in condensate grow and scatter, 3 possibilities:
(a) annihilation into a monopole-antimonopole pair
(b) elastic scattering
(c) inelastic scattering (instable monopoles)
Interpretation
PE , ge
calorons action small
calorons condense
plateau value of : existence of isolated magnetic charge
e
29Ralf Hofmann, Heidelberg
.
Do we understand this in the effective theory?
)2(
)2(
)2(
SU
SU
SU
in SU(2) algebra only at isolated points in time
stable winding around isolated points in 3D space
monopole flashes
monopole-antimonopole
pair
30Ralf Hofmann, Heidelberg
.
Transition to the magnetic phase
at we have: cE ,0 callylogarithmi e
TLH modes decouple kinematically, mass
on tree level TLM modes remain massless
monopole mass 0el
e
monopoles condense in a 2nd order – like
phase transition ( continuous), symmetry breaking:
NN ZU 1)1(
a
31Ralf Hofmann, Heidelberg
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
2/,... ,1 Nii NZ
i
Mii
N
iiM ivvvV
31
1
where
i
)(Tg
2/N
iD
i a ,,
32Ralf Hofmann, Heidelberg
Evolution with temperature
evolution has two fixed points at )(aM aa and 0
there is a highest and a lowest attainable temperature in the magnetic phase
logarithmicsingularities
cM ,Continous increase with temperature possible since monopoles condensed
33Ralf 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:
• magnetic coupling has logarithmic singularity at
2
1
gSCV
g 0 cMTT
• unstable monopoles form stable dipoles which condense
• all dual Abelian gauge modes
decouple thermodynamically
• center vortices condense
)1,...,1( , Nka kD
34Ralf Hofmann, Heidelberg
Transition to center phase
3
13
1
1
where
NC
Nk
k
CkC
N
k
kC
k
CC
v
vvV
center-vortex loops are one-dimensional objects,
nonlocal definition:
D
kA
DkCk Adzigx
,,exp)(
monopole part included
in limit a discussion of the 1st order phase transition can be based on BPS saturated solutions subject to potential:
N
extrapolate to finite N
35Ralf Hofmann, Heidelberg
Relaxation to the minima
36Ralf Hofmann, Heidelberg
Relaxation to the minima
at finite there exist tangential tachyonic modes associated with dynamical and local transformations:
NNZ
relaxation to minima by generation of magnetic flux quanta (tangential) and radial excitations
37Ralf Hofmann, Heidelberg
Matching the phases
pressure continuous across a thermal phase transition
scales are related )for ( , NCME
There is a single independent scale, say , determined by a boundary condition
E
Dimensional transmutation already seen in TPT also takes place here.
38Ralf 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?
39Ralf Hofmann, Heidelberg
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)
3,2N
cTT 5
SP and and
cTT 20
pressure (electric phase):
pressure (magnetic phase):
40Ralf Hofmann, Heidelberg
.
Pressure
(0.88)
J. Engels et al. (1982)
(0.97)
G. Boyd et al. (1996)
41Ralf Hofmann, Heidelberg
Energy density
(0.85)
J. Engels et al. (1982)
(0.93)
G. Boyd et al. (1996)
42Ralf Hofmann, Heidelberg
Entropy density
43Ralf Hofmann, Heidelberg
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?
T
T
44Ralf 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?
45Ralf Hofmann, Heidelberg
)2()2()2( SUSUSU eCMB
Application: QED and strong gauge interactions
consider gauge symmetry:
naively: to interprete as solitons of respective SU(2) factors ,e
stable states
Crossing of center vortices =1/2
magnetic monopole
one unit of U(1) charge
localized zero mode
neutral and extremely light particle
46Ralf Hofmann, Heidelberg
It turns out …
local symmetry in confining phase of SU(2) gauge theory makes stable fermion states
2Z
boundary condition for :
CMBSU )2(
• we see one massless photon in the CMB
• including radiative corrections in electric phase
photon is precisely massless at a single pointphoton mass
magnetic electric
47Ralf Hofmann, Heidelberg
Homogeneous contribution to cos
CMB boundary condition determines the scale
potential can be computed at
CMB
MV CMBT
This is the homogeneous part of . cos
we have:
44homcos eV101.3
2
1 MV
This is smaller than . 43cos eV100.1 WMAP
48Ralf Hofmann, Heidelberg
Coarse-grained contribution to cos
visible Universe
local ‘fireballs’ from high-energy particle collisions
CMBSU )2(
eSU )2(e.g.
ee collision
49Ralf Hofmann, Heidelberg
Value of the fine-structure constant
naively (only one SU(2) factor and one-loop evolution):
6.93
1
4
g and
15.17
2 2
g
taking 3-photon maximal mixing into account at (one-loop): mT
433.140
1
4
g 2r
50Ralf Hofmann, Heidelberg
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
CMBSU )2(
ground state is superconducting
51Ralf 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
52Ralf 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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