Center-Symmetric 1/N ExpansionPhys.Rev.D71,105012 (hep-th/0410254v2)

Outline :• A center-symmetric background at finite temperature • ‘t Hooft diagrams and U(N) at large N (essentials).• Confinement and the free energy at large N. • The center-symmetric perturbative expansion at large N

-Composite loop momenta-Background gauge + perturbative renormalization-1-loop contribution to the free energy-no O(N2) nor O(N) contributions to the free energy-absence of (severe) perturbative IR-divergences-confinement at weak coupling

• Outlook

The center-symmetric vacuumCenter-symmetric expansion center-symmetric “ground” stateLattice center-symmetry:

)(),(

),,0(),0(

4

44

NSUZzNSUU

xzUxU

N

Contractible Wilson loops (including plaquettes) are invariant(they contain pairs of oppositely oriented links of the slice)0

[contraction on lattice: generated by

for instance: ]

But (non-contractible) Polyakov loops of non-vanishing N-ality are sensitive to the center symmetry:

),1(),1(),(),(

)(),(Tr )(~

),,(Tr)(

444

/2

xnUxUxUxU

xLexUzxLxUxL

T

Fn

Niknnnk

Fn

nFn

Center symmetric orbit

Under gauge trafo:

NnxxLFn mod0if,0)(

)()()()(),()( 1 xgxUxgxUNSUxg Use gauge freedom to diagonalize ALL U’s and order the eigenphases:

)()(),,,( diag )( 1)()(1 xxeexU N

xixi N

N

NjxxU j

n )12()(0)(Tr

independent of x!

Center-symmetric orbits in this gauge have constant Abelian temporal links, i.e. correspond to a constant temporal Abelian connection:

)2

1,,

2

3,

2

1(diag

2)(4

NNN

N

Txga

Since none of the eigenphases of a center-symmetric orbit are degenerate, the gauge transformation g(x) is unique up to Abelian gauge transformations. Spatial links further specify the configuration there are MANY center-symmetric orbits.

)()(),,,( diag )( 1)()(1 xxeexU N

xixi N

Perturbatively (near the critical point of LGT), one is interested in a center-symmetric orbit of minimal classical (Wilson) action, i.e. vanishing curvature: All links are Abelian AND constant in each spatial directionFor a spatially cubic lattice (i.e. the lattice analog of ), the center-symmetric orbit with vanishing Wilson action is unique(there are no non-contractible spatial loops in this case.)

Proofsketch: Make all temporal links=1 except for those on slice.

Vanishing Wilson action spatial links do not depend on . On the slice we then have

same spatial link

spatial link

Abelian temporal link on slice

0

0

0

41

44 Abelian for , UUUUU iior:

Since none of the phases of are degenerate, is Abelian as well. 4U iUConsider therefore the perturbative expansion of [S]U(N) gauge theory about the center-symmetric background connection . ga

,0iga )2

1,,

2

3,

2

1(diag

2)(4

NNN

N

Txga

13 SS

‘t Hooft diagrams

lk

ji

ji NSUNSUNsuAd

)()( ))((ji

kl j

i j i

nm

lk

ji

ij

lk

m n

etc.

Color loops are FACES, gluon propagators are EDGES, vertices are VERTICES and fundamental loops are HOLES of oriented complexes:

diagram oforder 1

loops momentum#

escolor trac #

1(complex)

p

p

c

pc

L

L

FL

LLVEF

determined by topology of vacuum diagram1 pc LLIn the usual perturbative analysis with broken center symmetry, the ground state does not depend on N. Absorbing traces over color in the definition of the ‘t Hooft coupling,

1pL

Ng 2:vacuum diagrams of leading order in N have the topology of . Since , such planar contributions to the free energy are of order N 2 !!!

2S2)( 2 S

))/(()( 42 TfTNTF non-trivial multiplets contribute NON-CONFINING PHASE

BUT: Dyson-Schwinger equations for Wilson loops + factorization free-energy does not depend on T (Gocksch+Neri 1983)

Factorization:

0/1

)(

)1(2

21

21

221

s

s

Cs

Cs

NOGGG

GGG

NOGGG

)()(2

)()(,)()(2)(

)(2

)(244

xbNmaNnN

Tie

xexxbaN

TixD

bman

n

mniT

bman

n

mniTba

ba

b

a

How are these Statements Compatible? Dyson-Schwinger equations respect symmetries consider perturbative contributions to the pressure at finite T for a center-symmetric background. Use the center-symmetric background for the background gauge field i.e. extremize - suitable for perturbative analysis; - background does not renormalize

2

aA

ALL (charged) FIELDS COUPLE COVARIANTLY TO ga

)()2/1(2

)()(,)()2

1(2)(

)2/1(2

)2/1(244

xaNnN

Tie

xexxN

aN

TixD

ann

niT

ann

niTaab

ba

NOTE: The Matsubara frequency and the color appear in the combination only. The “composite” Matsubara frequency runs over ALL integers iff color- and momentum- loops coincide – the effective temperature in this case is not T, but T/N.

aNnl a

ln

Color- and momentum- loops coincide for planar diagrams (with and withoutperimeter ):

Perimeter contributes a factor of N

)(~

),/(),,(

2

22

2

4

22

S

SS

fN

T

TgNTNTfgNTF

)(~

),/(),,(

2

22

3

4

22

D

DD

fN

T

TgNTTfgNTF

By dimensional analysis, BEFORE evaluating the (divergent) loops

Regularization and RenormalizationSince the pressure superficially is quartically divergent, summations and integrals must be regularized gauge invariantly. Because,

0)()0()0(0

TT TFTSTF

it suffices to regulate and renormalize the superficially quadratically divergent specific heat capacity at constant volume . This may be achieved by dimensional regularization and renormalization of spatial loop momenta only. For spatial dimensions, the renormalized planar contributions to the free energy density scale as

FTC TV2

3d

),/;ˆ(~

),;ˆ,/(),;ˆ,,(222 2

42 TNf

N

TNTNTfgNTF SSS

2ˆ gN is the dimensionless ‘t Hooft coupling;

is the renormalization point, a (spatial) momentum scale that should

not depend on N. Large N Large , i.e. weak coupling

)/ln(11

24~)(ˆ0

~ 2

2

RSS N

Nfd

d

Non-planar Contributions to the Pressure

0)( 2 T

Question: are non-planar contributions to the pressure also suppressed?Answer: No, because

01)( pc LLdiagramImplies that there are MORE momentum- than color- loops. The two cannot be combined to a composite momentum. Only contributions with a toroidal or cylindrical topology survive the large N-limit. The summations over color in this limit can be replaced by integration over a Brillioun momentum

the N-dependence is fully absorbed in the ‘t Hooft coupling – and the diagram nevertheless remains T-dependent, due to an extra momentum loop. Lowest perturbative order are the paperclip + rings:

2

1

p

c

L

L

Absence of IR-divergences

Linde 1980: contributions of vanishing Matsubara frequency scale as

perturbative QCD at finite temperature is plagued by uncontrollable IR-divergences for >3 loops. Pisarski et al: Consider “hard thermal loops” only.

1 2 r rrr 43)1(4~

There are NO such IR-divergences in a center-symmetric background:

2222

3

/)(4

)'()',(),(

kNbanT

kkkmVknV adbcnmd

cb

a

Infrared-singular perturbative modes are Abelian (a=b). Abelian dominance?Abelian modes do not interact directly – perturbative IR-divergences therefore are at most logarithmic and (probably) cancel in physical S-matrix elements (a la Kinoshita et al.) .

1-loop Free EnergyEigenvalues of :

= 2 N2 bosons + 4 NF fermions but at temperature T/N !

Nba ,,1,

A more careful calculation indeed gives:

bosons fermionsEffectively rather heavy at large N !

TnNm je /~

suppressed as before

Non-planar contributions to the free energy of order N 0 are far more important.

The Gluonic Center-Symmetric Planar Model is UNSTABLE at large N: The free energy of U(N) includes a decoupled “photon”: It must (at least) include the contribution from a vector ghost that compensates the contribution of this “photon” to the free energy of U(N) at large N. Could this be Veneziano’s vector ghost??

Perturbative (In)Stability and the Hessian The 1-loop Hessian at the center-symmetric configuration (CS) is:

All EV’s are negative – the CS-background maximizes the 1-loop free energyThe N-1 tachyonic modes are Abelian and have masses:

1,...,1),(2)/(sin

31

6)(0 0

2

2

22

22

NkNO

T

NkN

TkmTach

N

jin

Nn

TF ij

nCSji

)(2cos

1

2

1

122

22

With EV’s: 0, and 1,,2,1for ,)/(sin

31

6 2

2

Nk

NkN

T

+Non-planar !

Planar

i j ij

But the CS-planar model is TACHYON-FREE and perturbatively stable at any temperature the phase transition in this case is first order.

Confinement at Weak Coupling?

)(

)(

isitesspatial

Akk iLQ

A non-trivial large-N model that confines at weak coupling is obtained by modifying the Wilson action to

0)()(2

xLxL Fk

Ak

sites

N

k

Ak

kWilson iLSS

2/

1

)()(),(

22

4

22

4

2)/(sin2 k

T

NkN

Tkcrit

k

d

crit

Broken Z(N)

Z(N) unbroken

0

Since adjoint Polyakov loops ,

the CS-phase of this (renormalizable) model is perturbatively stable for

are conserved, CS-symmetric “charges” :

)(

OQF

CSkk

The CS-symmetric phase also is perturbatively stable for : • adjoint fermions and even N and !

[ is N=1 SYM ] (See Hosotani)2/1. Adj

FN2/1. Adj

FN

Note: For SU(2) & SU(3) only k=1 need be considered. The phase transition at is of second order for SU(2). But (at ) the phase transition is of first order for SU(3).

crit 0

Large N Mass gap?

)()()()()( 111

3

111yLzLxL

T

zdyLxL FAFFF

)/1()()()()( 21111

3

NOyLzLzLxLT

zd FFFF

11

11

211111

with ,4

)0()(

)()0()(..)(

critmrFF

critFF

mer

TLxL

ppTGLxLTFpG

adjmT

Highlights A diagrammatic expansion about a center symmetric orbit of U(N) gauge theory

is possible at finite temperature without additional technical difficulties.

at large N gives no contributions of O(N2) and O(N) to the free energy.

does NOT describe a conventional Higgs phase because there are no colored asymptotic states at any order in perturbation theory.

avoids the severe infrared problem of ordinary perturbation theory.

A U(N) model defined by the PLANAR diagrams of the perturbative expansion about the CS-orbit at finite temperature

has a Matsubara frequency spacing of , i.e. exhibits dimensional reduction similar to Eguchi-Kawai models.

confines, but is purely topological with contributions to the free energy of order 1/N2. [leading O(1) contributions of U(N) gauge theory arise from toroidal & cylinder diagrams.]

is perturbatively stable and tachyon-free.

in the purely gluonic case includes a (perhaps decoupled) vector ghost.

N

T2

that apparently resum to a tachyonic Abelian gluon mass of . However, this resummation is not gauge invariant ! To conclude that the Abelian gluons are tachyonic is premature.

To leading order N0 , the non-planar polarization of order gives 1PR contributions of the form

2TO

)( 05NO

Coupling of Tachyonic Modes

Outlook

Glueball contributionsMesonic contributions

SSB and anomalies: For fixed T but large N, the “free” center-symmetric Dirac-operator has a (much) denser spectrum with a lowest frequency of rather than . At large N one effectively has a 3-dimensional action for low-frequency Dirac modes…..

0/ NT T

Stability: Can one estimate the (torus+cylinder) contributions of order to the pressure? To leading order it is suffices to replace summations over color by Brillioun-zone integrals – all loops apart from one do not depend on temperature!

0N0N