Three dimensional conformal sigma models

Collaborated with Takeshi Higashi and Kiyoshi Higashijima

(Osaka U.)

Etsuko Itou (Kyoto U. YITP)

hep-th/0702188

To appear in Prog. Theor. Phys.

2007/08/10 QFT, Kinki University

Plan to talk

1.Introduction 2.Two dimensional cases

3.Renormalizability of three dimensional sigma model

4.Conformal sigma models5.Summary

We consider three dimensional nonlinear sigma models using the Wilsonian renormalization group method.

In particularly, we investigate the renormalizability and the fixed point of the models.

1.Introduction

Bosonic Non-linear sigma model

The target space ・・・ O(N) model

Toy model of 4-dim. Gauge theory

(Asymptotically free, instanton, mass gap etc.)

Polyakov action of string theory

2-dim. Non-linear sigma model

3-dim. Non-linear sigma model

(perturbatively renormalizable)

Non-Linear Sigma Model

We divide all fields into two groups,

high frequency modes and low frequency modes.

The high frequency mode is integrated out.

Infinitesimal change of cutoff

The partition function does not depend on .

Wilsonian Renormalization Group Equation

There are some Wilsonian renormalization group equations.

• Wegner-Houghton equation (sharp cutoff)

• Polchinski equation (smooth cutoff)

• Exact evolution equation ( for 1PI effective action)

K-I. Aoki, H. Terao, …

T.Morris, K. Itoh, Y. Igarashi, H. Sonoda, M. Bonini,…

C. Wetterich, M. Reuter, N. Tetradis, J. Pawlowski,…

quantum gravity, Yang-Mills theory,

higher-dimensional gauge theory…

YM theory, QED, SUSY…

local potential, Nambu-Jona-Lasinio, NLσM

Canonical scaling ： Normalize kinetic terms

Quantum correction

Wegner-Houghton eq

In this equation, all internal lines are the shell modes which have nonzero values in small regions.

More than two loop diagrams vanish in the 　　　 　 limit.

This is exact equation. We can consider (perturbatively) nonrenormalizable theories.

Approximation method:

Symmetry and Derivative expansionConsider a single real scalar field theory that is invariant under

We expand the most generic action as

In this work, we expand the action up to second order in derivative and assume it =2 supersymmetry.

K.Aoki Int.J.Mod.Phys. B14 (2000) 1249

T.R.Morris Int.J.Mod.Phys. A9 (1994) 2411Local potential approximation :

Local Potential term

Non-linear sigma model

“Renormalizability” and Continuum limit

●critical line ・・・ EAB

●renormalized trajectory

●fixed point ・・・ A, B,

line

The renormalization group flow which can be extrapolated back to critical surface defines a renormalized theory.

Two dimensional theory space

Renormalizable UV fixed point

2. Two dimensional casesNon-linear sigma models with N=2 SUSY in 3D (2D) is defined by Kaehler potential.

The scalar field has zero canonical dimension.

In perturbative analysis, the 1-loop function is proportional to the Ricci tensor of target spaces. The perturbative results

Alvarez-Gaume, Freedman and Mukhi Ann. of Phys. 134 (1982) 392

Perturbatively renormalizable

Beta function from WRGK.Higashijima and E.I. Prog. Theor. Phys. 110 (2003) 107

Here we introduce a parameter which corresponds to the anomalous dimension of the scalar fields as follows:

When N=1, 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

Ricci Flat solution

Fixed Point Theories

3.Three dimensional cases(renormalizability)

The scalar field has nonzero canonical dimension.

We need some nonperturbative renormalization methods.

WRG approach

Large-N expansion

Inami, Saito and Yamamoto Prog. Theor. Phys. 103 （ 2000 ） 1283

Our works

The CPN-1 model :SU(N)/[SU(N-1) ×U(1)]

From this Kaehler potential, we derive the metric and Ricci tensor as follow:

Beta fn. from WRG (Ricci soliton equation)

Renormalization condition

When the target space is an Einstein-Kaehler manifold, the βfunction of the coupling constant is obtained.

Einstein-Kaehler condition:

The constant h is negative (example Disc with Poincare metric)

We have only IR fixed point at

i, j=1IR

If the constant h is positive, there are two fixed points:

At UV fixed point

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.

Renormalizable

IR

IR

4.Conformal Non-linear sigma models

Fixed point theory obtained by solving an equation

At

Fixed point theories have Kaehler-Einstein mfd. with the special value of the radius.

Ｃ is a constant which depends on models.

Hermitian symmetric space (HSS)・・・・ A special class of Kaehler- Einstein manifold with higher symmetry

New fixed points (γ≠-1/2)Two dimensional fixed point target space for

The line element of target space

RG equation for fixed point e(r)

It is convenient to rewrite the 2nd order diff.eq. to a set of 1st order diff.eq.

　　　 　　　 : Sphere　 S2

（ CP1）　 　　　 : Deformed sphere

　　　 : Flat R2e(r)

Deformed sphere

At the point, the target mfd. is not locally flat.

It has deficit angle. Euler number is equal to S2

Summary• We study a perturbatively nonrenormalizable

theory (3-dim. NLSM) using the WRG method.

• Some three dimensional nonlinear sigma models are renormalizable within a nonperturbative sense.

• We construct a class of 3-dim. conformal sigma models.