A New Class of Conformal Field Theories with Anomalous Dimensions

1. A New Class of Conformal Field Theories with Anomalous Dimensions Prog. Theor. Phys. 109 (2003) 751 (2-dim.) Prog. Theor. Phys. 110 (2003) 563 (3-dim.) Etsuko Itou (Osaka University) With K.Higashijima Main Topics: We find a class of fixed point theory for 2- and 3-dimensional non-linear sigma models using Wilsonian renormalization group approach.

2. 1. Introduction The point view of Wilsonian renormalization group ●We consider the Wilsonian effective action which has derivative interactions. It corresponds to next-to-leading order approximation in derivative expansion. Local potential term Non-linear sigma model

3. Wilsonian Renormalization Group 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] .

4. The Wilsonian RG equation is written as follow: Field rescaling effects to normalize kinetic terms. To obtain the WRG eq. , we integrate shell mode. only

5. Approximation method: Symmetry and Derivative expansion Consider 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) 2411 Local potential approximation :

6. The point of view of non-linear sigma model: ●In perturbative analysis, the 1-loop b function for 2-dimensional non-linear sigma model is proportional to Ricci tensor of target spaces. ⇒Ricci Flat Is there the other fixed point? ●The 3-dimensional non-linear sigma models are nonrenormalizable within the perturbative method. We need some nonperturbative renormalization methods.

7. D=2 (3) N =2 supersymmetric non linear sigma model i=1～N：N is the dimensions of target spaces : the metric of target spaces Considering only Kaehler potential term corresponds to second order to derivative for scalar field. There is not local potential term.

8. 2. Fixed points with U(N) symmetry 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

9. The function f(x) have infinite number of coupling constants. The Kaehler potential gives the Kaehler metric and Ricci tensor as follows:

10. The solution of the β＝０ equation satisfies the following equation: Here we introduce a parameter which corresponds to the anomalous dimension of the scalar fields as follows: When N=1, the function f(x) is givenin closed form The target manifold takes the form of a semi-infinite cigar with radius . It is embedded in 3-dimensional flat Euclidean spaces.

11. This solution has been discussed in other context. They consider the non-linear sigma model coupled with dilaton. Witten Phys.Rev.D44 (1991) 314 Kiritsis, Kounnas and Lust Int.J.Mod.Phys.A9 (1994) 1361 Hori and Kapustin :JHEP 08 (2001) 045 In k>>1 region, we can use the perturbative renormalization method and obtain 1-loop b function: If one prefers to stay on a flat world-sheet, one may say that a non-trivial dilaton gradient in space-time is equivalent to assigning a non-trivial Weyl transformation law to target space coordinates. Our parameter a (anomalous dim.) corresponds to k as follow.

12. ３.3-dimensional case Similarly to 2-dimenion, we obtain the nonperturbative b function for 3-dimensional non-linear sigma models.

13. The CPNmodel :SU(N+1)/[SU(N) ×U(1)] The b function and anomalous dimension of scalar field are given by Large-N result: Inami, Saito and Yamamoto Prog. Theor. Phys. 103 （2000）1283 There are two fixed points:

14. We derive the action of the conformal field theory corresponding to the fixed point of the 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.

15. The following function satisfies b=0 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.

16. We take the specific values of the parameter, the function takes simple form. ● This theory is equal to IR fixed point of CPNmodel ● This theory is equal to UV fixed point of CPNmodel. Then the parameter describes a marginal deformation from the IR to UV fixed points of the CPNmodel in the theory spaces.

17. ４. Summary and Discussions In this work, we derive a class of fixed point theories of 2- and 3-dimensional N =2 supersymmetric nonlinear sigma models. These theory have one free parameter corresponding to the anomalous dimension of scalar fields. In the 2-dimensional case, these theory coincide with perturbative 1-loop βfunction solution for NLσM coupled with dilaton. In the 3-dimensional case, the free parameter describes a marginal deformation from the IR to UV fixed points of the CPNmodel in the theory spaces. Future Problems: ●The effects of the higher derivative interactions? ●The property of 3-dimensional conformal field theory?