Three dimensional conformal sigma models Collaborated with Takeshi Higashi and Kiyoshi Higashijima...
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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.