The nonperturbative analyses for lower dimensional non-linear sigma models

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The nonperturbative analyses for lower
dimensional non-linear sigma models

• Etsuko Itou (Kanazawa University)

Based on Progress of Theoretical Physics 110
(2003) 563 hep-th/0304194 (with Kiyoshi
Higashijima) hep-th/0505056 (with Kiyoshi
Higashijima and Makoto Tsuzuki)
Yesterday!
We consider the three dimensional non-linear
sigma model within two nonperturbative analyses
WRG and large-N expansion.
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Plan to talk 1.Introduction 2.The WRG equation
for 3-dim. NLsM 3.Renormalizability of some
models 4.Large-N analysis 5.Fixed point theory 6.
Summary
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1. Introduction
In the point of view of WRG study

• We consider the Wilsonian effective action which
  has derivative interactions.
• It corresponds to the non-linear sigma model
  action, so we compare the results with other
  analysis.

It corresponds to next-to-leading order
approximation in derivative expansion.
Local potential term
Non-linear sigma model
Local potential approximation
K.Aoki Int.J.Mod.Phys. B14 (2000) 1249 T.R.Morris
Int.J.Mod.Phys. A9 (1994) 2411
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In the point of view of 3-dim NLsM
? 2-dim. case string Polyakov action
toy model of 4-dim QCD ? 3-dim.
case membrane theory
quantum hall effects model 3-dim. NLsMs are
nonrenormalizable within the perturbative method.
We need some nonperturbative renormalization
methods.
WRG approach Large-N expansion
Our works
Inami, Saito and Yamamoto Prog. Theor. Phys. 103
(2000)1283
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2.The WRG equation
The Euclidean path integral is
K.Aoki Int.J.Mod.Phys. B14 (2000) 1249
The Wilsonian effective action has infinite
number of interaction terms.
The WRG equation (Wegner-Houghton equation)
describes the variation of effective action when
energy scale L is changed to L(dt)L exp-dt .
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To obtain the WRG eq. , we integrate shell mode.
only
The Wilsonian RG equation is written as follow
Field rescaling effects to normalize kinetic
terms.
We use the sharp cutoff equation. It corresponds
to the sharp cutoff limit of Polchinski equation
at least local potential level.
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Approximation method Symmetry and Derivative
expansion
Consider a single real scalar field theory that
is invariant under We expand the most general
action as
In this work, we expand the action up to second
order in derivative and constraint it 2
supersymmetry.
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D3 N 2 supersymmetric non linear sigma model
i1NN is the dimensions of target spaces
Where K is Kaehler potential and F is chiral
superfield.
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We expand the action around the scalar fields.
where
the metric of target spaces
From equation of motion, the auxiliary filed F
can be vanished.
Considering only Kaehler potential term
corresponds to second order to derivative for
scalar field. There is not local potential term.
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The WRG equation for non linear sigma model
Consider the bosonic part of the action. The
second term of the right hand side vanishes in
this approximation O( ) .
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The first term of the right hand side From the
bosonic part of the action From the fermionic
kinetic term
Non derivative term is cancelled.
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Finally, we obtain the WRG eq. for bosonic part
of the action as follow
The b function for the Kaehler metric is
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The other part of the action To keep
supersymmetry, we derive the WRG eq. for the
other part of the action from bosonic part.
Recall that the scalar part of the action is
derived as follow And the one-loop correction
term for scalar part is In Kaehler manifold,
the Ricci tensor is given from the metric as
follow The metric is given from the Kaehler
potential
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Using this property, we can obtain the
supersymmetric WRG eq. for Kaehler potential
The fermionic part of the WRG eq. is
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3.Renormalizability of some models
In 3-dimension, the b function for Kaehler metric
is written
The CPN-1 model SU(N)/SU(N-1) ﾗU(1)
From this Kaehler potential, we derive the metric
and Ricci tensor as follow
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Einstein-Kaehler manifolds
The Einstein-Kaehler manifolds satisfy the
condition
If h is positive, the manifold is compact.
Using the relation of the metric and Ricci
tensor, the b function can be rewritten only the
metric
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The value of h for hermitian symmetric spaces.
G/H Dimensions(complex) h
SU(N)/SU(N-1)ﾗU(1) N-1 N
SU(N)/SU(N-M)ﾗU(M) M(N-M) N
SO(N)/SO(N-2)ﾗU(1) N-2 N-2
Sp(N)/U(N) N(N1)/2 N1
SO(2N)/U(N) N(N1)/2 N-1
E6/SO(10) ﾗU(1) 16 12
E7/E6ﾗU(1) 27 18
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When the target space is an Einstein-Kaehler
manifold, the ﾟfunction of the coupling constant
is obtained.
Einstein-Kaehler condition
?The constant h is negative (example Disc with
Poincare metric)
b(l)
IR
i, j1
l
We have only IR fixed point at l0.
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?If the constant h is positive, there are two
fixed points
Renormalizable
IR
At UV fixed point
IR
It is possible to take the continuum limit by
choosing the cutoff dependence of the bare
coupling constant as
M is a finite mass scale.
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4.Large-N analysis
At UV fixed point
Do third derivative interaction terms disturb the
UV fixed point?
To check the existence of the UV fixed point, we
consider two models using large-N expansion.
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The cases of
CPN-1 model
Inami, Saito and Yamamoto Prog. Theor. Phys. 103
(2000)1283
U(1) gauge auxiliary field
Integrated out V
Using the component fields
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After integrating out dynamical fields, we obtain
the effective potential.
The stationary condition of the effective
potential gives the following equations.
or
This model has two phases
SU(N)-symmetric, massive phase
SU(N) broken, massless phase
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The gap eq.
The ﾟ function of this model has no
next-to-leading corrections.
Symmetric and massive phase
Broken and massless phase
Inami, Saito and Yamamoto Prog. Theor. Phys. 103
(2000)1283
Is supersymmetry responsible for the
vanishing of the next-to-leading order
corrections to ﾟfunction of the model?
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WRG result
The examples of the Einstein-Kaehler cases
model
ｴt Hooft coupling
model
1/N next-to-leading?
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Q(N-2) model
model
O(N) condition
There are two multiplier superfields.
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There are three phases.
?
SO(N)-symmetric ,massive theory
?
New phase
SO(N) broken, massless theory
The differences between ? and ? phases
In effective action, the gauge fields have
Chern-Simons interaction. The Parity is broken.
?
?
The gauge field acquires mass through Higgs
mechanism.
From eq. of motions,
Symmetric and massive phase
(??)
Broken and massless phase
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We calculate the ﾟfunction to next-to-leading
order and obtain the next-to-leading order
correction.
Next-to-leading order corrections to the
propagator.
Some 3-dimesional supersymmetric
sigma models have the next-to-leading order
correction of the ﾟfunction.
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5.Two dimensional fixed point
Prog. Theor. Phys. 109 (2003) 751 K.Higashijima
and E.I
The perturbative ﾟfunction follows the Ricci-flat
target manifolds.
Ricci-flat
We derive the action of the conformal field
theory corresponding to the fixed point of the b
function.
To simplify, we assume U(N) symmetry for Kaehler
potential.
where
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The function f(x) have infinite number of
coupling constants.
The Kaehler potential gives the Kaehler metric
and Ricci tensor as follows
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The solution of the ﾟ0 equation satisfies the
following equation
Here we introduce a parameter which corresponds
to the anomalous dimension of the scalar fields
as follows
When N1, the function f(x) is given in closed
form
The target manifold takes the form of a
semi-infinite cigar with radius . It is
embedded in 3-dimensional flat Euclidean spaces.
Witten Phys.Rev.D44 (1991) 314
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Three dimensional case
We derive the action of the conformal field
theory corresponding to the fixed point of the
WRG b function.
To simplify, we assume SU(N) symmetry for Kaehler
potential.
We substitute the metric and Ricci tensor given
by this Kaehler potential for following equation.
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The following function satisfies b0 for any
values of parameter
A free parameter, , is proportional to the
anomalous dimension.
If we fix the value of , we obtain a
conformal field theory.
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We take the specific values of the parameter, the
function takes simple form.
?
This theory is equal to IR fixed point of CPN
model
?
This theory is equal to UV fixed point of CPN
model.
Then the parameter describes a marginal
deformation from the IR to UV fixed points of the
CPN model in the theory spaces.
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6. Summary
In this work, we consider the three dimensional
non-linear sigma model within two nonperturbative
analyses WRG and large-N expansion. We can
discuss a large class of 3-dim. NLsM with WRG
method. We found the NLsMs on Einstein-Kaehler
manifolds with positive scalar curvature are
renormalizable in three dimensions. We also
construct the fixed point theory which vanishes
the WRG ﾟ function. The conformal field theory
has one free parameter which corresponds to the
anomalous dimension of the scalar fields. The
free parameter describes a marginal deformation
from the IR to UV fixed points of the CPN model
in the theory spaces. The large-N analysis
reveals the phase structures and the mass gap of
CP(N-1) and Q(N-2).