The analytic structure of the fixed charge expansion

Abstract

We investigate the analytic properties of the fixed charge expansion for a number of conformal field theories in different space-time dimensions. The models investigated here are O(N) and QED3. We show that in d = 3 − ϵ dimensions the contribution to the O(N) fixed charge Q conformal dimensions obtained in the double scaling limit of large charge and vanishing ϵ is non-Borel summable, doubly factorial divergent, and with order $$ \sqrt{Q} $$ Q optimal truncation order. By using resurgence techniques we show that the singularities in the Borel plane are related to worldline instantons that were discovered in the other double scaling limit of large Q and N of ref. [1]. In d = 4 − ϵ dimensions the story changes since in the same large Q and small E regime the next order corrections to the scaling dimensions lead to a convergent series. The resummed series displays a new branch cut singularity which is relevant for the stability of the O(N) large charge sector for negative ϵ. Although the QED3 model shares the same large charge behaviour of the O(N) model, we discover that at leading order in the large number of matter field expansion the large charge scaling dimensions are Borel summable, single factorial divergent, and with order Q optimal truncation order.

Identifying Large Charge operators

“…Remarkably there then exists a non-trivial overlap for the application of the two methods, which was exploited in [11] both to validate the semiclassical computation and to boost the available finite order Feynman loop computations. 1 These results have been exploited to investigate the analytic structure of the large charge expansion [16] and are consistent with convexity conditions proposed as a formulation the Weak Gravity Conjecture [17].…”

“…N equals 1 or 2. For simplicity we will focus on insertions of just one specific type of neutral operators 16 O(x) = ( φφ) k (x).…”

“…(5.6) 16 The insertion of conserved currents between spinning phonon states has been studied, in EFT, in [28]. 17 For large n, (5.3) can be computed semiclassically by expanding around the saddle point (3.21).…”

Stability analysis of a non-unitary CFT

“…Secondly, there could be much bigger non-perturbative corrections of order O(exp(− √ Q)) to the one-loop correction to the static saddle-point already. 8 As discussed in [75,76], this comes from the worldline instanton of the massive particle on top of the saddle-point. 9 Additionally, even though we found a phase transition, it does not tell us how the degrees of freedom reorganises itself to give a large imaginary part of ∆(Q) there.…”

“…We thank Domenico Orlando for discussions 9. In[76] it is argued that the one-loop determinant is convergent around D = 4. This is indeed correct, but there still exist non-perturbative corrections from the worldline instanton 10.…”