Nonperturbative and analytical approach to Yang-Mills thermodynamics

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?


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


Typical situation in thermal perturbation theory

taken from Kajantie et al. 2002


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


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?

• ...


Outline




• Comparison of thermodynamical potentials with

lattice results

• Application: A strongly interacting gauge theory

underlying QED?


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


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 ,


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


taken from van Baal & Kraan 1998

SU(2)


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


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


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


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


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


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)


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


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)


.

Does fluctuate?

quantum mechanically:

)2

( 13 332

2

EEE

l

E Tl

Vl

No !

thermodynamically:

compositeness scale

112 222

2

lT

VEl

No !


.

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


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

...


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


.

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


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


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


Evolution of effective gauge-coupling

PE ,logarithmic singularity

cE ,

plateau value

(independent of ) PE ,

(independent of ) PE ,


• 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


.

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


.

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


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 ,,


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


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


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


Relaxation to the minima


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


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.


Outline





lattice results

• Application: a strongly interacting theory underlying

QED?


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):


.

Pressure

(0.88)

J. Engels et al. (1982)

(0.97)

G. Boyd et al. (1996)


Energy density

(0.85)

J. Engels et al. (1982)

(0.93)

G. Boyd et al. (1996)


Entropy density


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


Outline





lattice results

• Application: a strongly interacting theory

underlying QED?


)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


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


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


Coarse-grained contribution to cos

visible Universe

local ‘fireballs’ from high-energy particle collisions

CMBSU )2(

eSU )2(e.g.

ee collision


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


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


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


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 !

Documents

Nonperturbative and analytical approach to Yang-Mills thermodynamics