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Wilsonian renormalization group approach to the lower dimensional nonlinear sigma models

Wilsonian renormalization group approach to the lower dimensional nonlinear sigma models. Prog. Theor. Phys. 110 (2003) 563 With K.Higashijima hep-th/0505056 With K.Higashijima and M.Tsuzuki. Etsuko Itou (KANAZAWA University). Main Topics:

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Wilsonian renormalization group approach to the lower dimensional nonlinear sigma models

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  1. Wilsonian renormalization group approach to the lower dimensional nonlinear sigma models Prog. Theor. Phys. 110 (2003) 563 With K.Higashijima hep-th/0505056 With K.Higashijima and M.Tsuzuki Etsuko Itou (KANAZAWA University) Main Topics: We study the 3-dimensional non-linear sigma models using Wilsonian renormalization group approach.

  2. 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

  3. In the point of view of 3-dim NLσM NLσM are quantum field theories on the manifolds. The 3-dim. NLσM are nonrenormalizable within the perturbative method. We need some nonperturbative renormalization methods. target spaces Ex) SN, CP(N) WRG approach Large-N expansion Our works Inami, Saito and Yamamoto Prog. Theor. Phys. 103 （2000）1283

  4. 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] .

  5. 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.

  6. 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.

  7. D=3 N =2 supersymmetric non linear sigma model i=1～N：N is the dimensions of target spaces Where K is Kaehler potential and F is chiral superfield. We expand the action around the scalar fields. where : the metric of target spaces (function of scalar fields and coupling const.) Considering only Kaehler potential term corresponds to second derivative interactions for scalar field. There is not local potential term of scalar field only.

  8. 3.Renormalizability In 3-dimension, the b function for Kaehler metric is written: One-loop corrections Rescaling terms The CPNmodel :SU(N+１)/[SU(N) ×U(1)] From this Kaehler potential, we derive the metric and Ricci tensor as follow:

  9. The b function and anomalous dimension of scalar field are given by Renormalizable! Large-N result: Prog. Theor. Phys. 103 （2000）1283; Inami, Saito and Yamamoto hep-th/0505056; Kiyoshi Higashijima, E.I. and Makoto Tsuzuki Massive phase IR There are two fixed points: Massless phase IR At UV fixed point, it is possible to take the continuum limit by choosing the cutoff dependence of the bare coupling constant as

  10. When the target spaces are Einstein-Kaehler manifolds, the βfunction of the coupling constant is obtained. Einstein-Kaehler condition: β（λ） ｈ＜０ ｈ＞０ IR IR IR λ Ex) Target space is CP(N) mfd. Ex) Target space is Poincare disc

  11. 4.Fixed point theories 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.

  12. The following function satisfies b=0 for any values of parameter The coupling constant correspond to the anomalous dimension of the scalar fields. If we fix the value of , we obtain a conformal field theory.

  13. 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.

  14. In this work, we investigate the renormalizability of 3-dim. NLσMand derive a class of fixed point theories using the WRG. Because of SUSY and taking only Kaehler potential, we can drop the local potential terms. We find that some 3-dim. NLσMare renormalizable within nonperturbative method. The fixed point theories have one free parameter which corresponds to the anomalous dimension of scalar field. The parameter describes a marginal deformation from the IR to UV fixed points of the CPNmodel in the theory spaces. 5. Summary and Discussions Future Problems: ●The effects of the higher derivative interactions? ●The property of 3-dimensional conformal field theory?

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