Generalized Option Pricing Formulas

1

Center-Symmetric 1/N Expansion
Phys.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

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The center-symmetric vacuum
Center-symmetric expansion
center-symmetric ground state Lattice
center-symmetry
Contractible Wilson loops (including plaquettes)
are invariant (they contain pairs of oppositely
oriented links of the slice)
contraction on lattice generated by
for
instance

But (non-contractible) Polyakov loops of
non-vanishing N-ality are sensitive to the
center symmetry
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Center symmetric orbit Under gauge trafo
Use gauge freedom to diagonalize ALL Us and
order the eigenphases
independent of x!
Center-symmetric orbits in this gauge have
constant Abelian temporal links, i.e.
correspond to a constant temporal Abelian
connection
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.
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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
direction For 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 links1 except for those on
slice. Vanishing Wilson
action spatial links do not depend on
?. On the slice we
then have
or
Since none of the phases of are degenerate,
is Abelian as well.
Consider therefore the perturbative expansion of
SU(N) gauge theory about the center-symmetric
background connection .
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t Hooft diagrams
j i
k l
j
i
etc.
Color loops are FACES, gluon propagators are
EDGES, vertices are VERTICES and fundamental
loops are HOLES of oriented complexes
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determined by
topology of vacuum diagram
In 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,
Factorization
non-trivial multiplets contribute
NON-CONFINING PHASE
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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
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Color- and momentum- loops coincide for planar
diagrams (with and without perimeter )
Perimeter contributes a factor of N



By dimensional analysis, BEFORE evaluating the
(divergent) loops
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Regularization and Renormalization
Since the pressure superficially is quartically
divergent, summations and integrals must be
regularized gauge invariantly. Because,
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

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
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Non-planar Contributions to the Pressure
Question are non-planar contributions to the
pressure also suppressed? Answer No, because
Implies 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
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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 gt3 loops.
Pisarski et al Consider hard thermal loops
only.
2
r
1
There are NO such IR-divergences in a
center-symmetric background
Infrared-singular perturbative modes are Abelian
(ab). 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.) .
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1-loop Free Energy

Eigenvalues of
2 N2 bosons 4 NF fermions but at temperature
T/N !
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A more careful calculation indeed gives
fermions
bosons
Effectively rather heavy at large N !
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 Venezianos vector ghost??
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Perturbative (In)Stability and the Hessian
The 1-loop Hessian at the center-symmetric
configuration (CS) is
All EVs are negative the CS-background
maximizes the 1-loop free energy The N-1
tachyonic modes are Abelian and have masses
But the CS-planar model is TACHYON-FREE and
perturbatively stable at any temperature
the phase transition in this case is first
order.
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Confinement at Weak Coupling?
A non-trivial large-N model that confines at weak
coupling is obtained by modifying the Wilson
action to

Since adjoint Polyakov loops
, the
CS-phase of this (renormalizable) model is
perturbatively stable for
Z(N) unbroken
are conserved, CS-symmetric charges

Broken Z(N)
0
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Note For SU(2) SU(3) only k1 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).

• The CS-symmetric phase also is perturbatively
  stable for
• adjoint fermions and even
  N and !
• is N1 SYM (See
  Hosotani)

Large N Mass gap?
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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.

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Coupling of Tachyonic Modes
To leading order N0 , the non-planar polarization
of order gives 1PR contributions of the form
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.
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Outlook