Wilson action for the O(N) model

Abstract

In this paper the fixed-point Wilson action for the critical O(N ) model in D = 4 − ǫ dimensions is written down in the ǫ expansion to order ǫ 2 . It is obtained by solving the fixed-point Polchinski Exact Renormalization Group equation (with anomalous dimension) in powers of ǫ. This is an example of a theory that has scale and conformal invariance despite having a finite UV cutoff. The energy-momentum tensor for this theory is also constructed (at zero momentum) to order ǫ 2 . This is done by solving the Ward-Takahashi identity for the fixed point action. It is verified that the trace of the energy-momentum tensor is proportional to the violation of scale invariance as given by the exact RG, i.e., the β function.The vanishing of the trace at the fixed point ensures conformal invariance. Some examples of calculations of correlation functions are also given.

Fixed Point Structure of Gradient Flow Exact Renormalization Group for Scalar Field Theories

“…By setting D = 4 − and solving the fixed point condition Eq. (2.24) of the WP equation with the expansion [56], we get the action S * WF at the WF fixed point up to O( ) as…”

“…As was seen in Section 2.2, the asymptotic behavior of λ(τ ) at the large flow time is controlled by the signature of the quantity D − 2 + η. The anomalous dimension η can be explicitly calculated with the expansion at this fixed point [56] and is given to…”

“…Let us see the relationship between the signature of D − 2 + η and the cluster decomposition principle concretely from the two-point function of the WF fixed point action. According to [56], the connected two-point function of φ a is given to O 2 by…”

Finite Cutoff CFT's and Composite Operators

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Recently a conformally invariant action describing the Wilson-Fischer fixed point in D = 4−ǫ dimensions in the presence of a finite UV cutoff was constructed [41]. In the present paper we construct two composite operator perturbations of this action with definite scaling dimension also in the presence of a finite cutoff.Thus the operator (as well as the fixed point action) is well defined at all momenta 0 ≤ p ≤ ∞ and at low energies they reduce tox φ 2 and x φ 4 respectively.

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“…Recently a conformally invariant action describing the Wilson-Fischer fixed point in D = 4−ǫ dimensions in the presence of a finite UV cutoff was constructed [41]. In the present paper we construct two composite operator perturbations of this action with definite scaling dimension also in the presence of a finite cutoff.…”