Charging the Walking U(N)$\times$U(N) Higgs Theory as a Complex CFT

Abstract

We apply a semi-classical method to compute the conformal field theory (CFT) data for the U(N)xU(N) non-abelian Higgs theory in four minus epsilon dimensions at its complex fixed point. The theory features more than one coupling and walking dynamics. Given our charge configuration, we identify a family of corresponding operators and compute their scaling dimensions which remarkably agree with available results from conventional perturbation theory validating the use of the state-operator correspondence for a complex CFT.

Anomalous dimensions at large charge for $U(N)\times U(N)$ theory in three and four dimensions

“…2 The U (N ) × U (N ) model Our discussion in this section follows closely that of Ref. [11], where the semiclassical calculation was first performed for the U (N ) × U (N ) model. In this model we have an n × n complex matrix H, and the Lagrangian is given by…”

“…In recent years, considerable progress [4][5][6][7][8][9][10][11][12][13] has been made for this class of theories, using a semi-classical expansion in the path integral formulation of the theory 3 . In particular, this approach gives a useful handle on n-point amplitudes for large n.…”

“…Most recently, the classically scale-invariant U (N ) × U (N ) model has been investigated in Ref. [11]. Here once again the leading and non-leading terms in a large charge expansion have been derived by a semi-classical calculation, and compared with perturbative results up to two loop order.…”

“…The paper is organised as follows: In Section 2 we describe the semiclassical calculation in the U (N ) × U (N ) case, following Ref. [11]. Then in Section 3 we compare the result of this calculation with perturbative calculations up to and including 4 loops.…”

“…We do not need to give α * h , α * y explicitly, but as pointed out in Ref. [11], the fixed point is complex for N > √ 3, which may be associated with "walking" behaviour of the couplings near the fixed point.…”