Correlation functions in scalar field theory at large charge

Abstract

We compute general higher-point functions in the sector of large charge operators φ n ,φ n at large charge in O(2) (φφ) 2 theory. We find that there is a special class of "extremal" correlators having only one insertion ofφ n that have a remarkably simple form in the doublescaling limit n → ∞ at fixed g n 2 ≡ λ, where g ∼ ǫ is the coupling at the O(2) Wilson-Fisher fixed point in 4 − ǫ dimensions. In this limit, also non-extremal correlators can be computed. As an example, we give the complete formula for φ(x 1 ) n φ(x 2 ) nφ (x 3 ) nφ (x 4 ) n , which reveals an interesting structure.

On the UV completion of the O(N) model in 6 − ϵ dimensions: a stable large-charge sector

“…These equations reproduce the ones obtained for the leading order in the perturbation series in section 2.1, which, with the current scaling, become exact. This is the precise analog of the limit considered in [40,46], with the difference that now there is an additional field η, which mediates the interaction. The saddle-point calculation in the double-scaling limit gives rise to the exponentiation of the second Feynman diagram of figure 1.…”

“…The existence of this double scaling limit was first hinted for scalar theories long ago in [41][42][43], and very recently reconsidered in [40,[44][45][46]. Even though these studies concern scalar theories (mostly the O(2) model), a similar double-scaling limit was found in N = 2 SQCD in [47] and further studied in [48][49][50][51] (see also [52,53]).…”

“…The exponentiation corresponds to the full resummation of the Feynman diagrams of figure 3, which are the only surviving Feynman diagrams in this limit, all other diagrams being suppressed by powers of 1/n. In the case of [40,46], the relevant Feynman diagram corresponds to collapsing the η propagator to a point, i.e. setting x = y in figure 2 (this gives rise to the "Kermit the frog" diagram of [40,46]).…”

$$ \mathcal{N} $$ = 2 conformal gauge theories at large R-charge: the SU(N) case

“…In this model higher order corrections in κ are associated with suppressed diagrams in the double scaling limit. 2 Although the anomalous dimension γ n is inherently associated with a two-point function, similar results have been recently extended to more general higher point functions with one anti-holomorphic insertion of ϕ n [14].…”

“…Then, for the five models we obtain (of course, only 3 expressions are independent thanks to (6.16)) 14 Notice that the successive derivation of (6.19) in [35] was done independently and with a different method strongly suggesting that there are no non-perturbative ambiguities in the reconstruction from the weak-coupling expansion, at least in the half-plane Re(κ) > 0. 15 We checked agreement with many more terms, a task that is possible due to (6.13).…”