Abstract
We extend the analysis of [1] to study the Regge trajectories of the Mellin amplitudes of the 0- and 1-magnon correlators of the generalized Fishnet theory in d dimensions and one type of correlators of chiral fishnet theory in 4 dimensions. We develop a systematic procedure to perturbatively study the Regge trajectories and subsequently perform the spectral integral. Our perturbative method is very generic and in principle can be applied to correlators whose perturbative Regge trajectories obey some structural conditions which we list down. Our d dimensional results reduce to previously known results in d = 4 for 0-magnon and 1-magnon. As a non-trivial check, we show that the results for 1-magnon correlator in d = 8, when evaluated using the exact techniques in [1, 2] are in perfect agreement with our d dimensional perturbative results. We also perturbatively compute the Regge trajectories and Regge-Mellin amplitudes of the chiral fishnet correlator leftlangle mathrm{Tr}left[{phi}_1left({x}_1right){phi}_1left({x}_2right)right]mathrm{Tr}left[{phi}_1^{dagger}left({x}_3right){phi}_1^{dagger}left({x}_4right)right]rightrangle using the techniques developed in this paper. Since this correlator has two couplings κ and ω, we have obtained closed-form results in the limit κ → 0, ω → 0 with κ/ω held constant. We verify this computation with an independent method of computing the same and obtain perfect agreement.
Highlights
Refer to [6,7,8,9]).1 This serves as a playground for studying a host of physical phenomenon which would not have been possible in the parent theory
We extend the analysis of [1] to study the Regge trajectories of the Mellin amplitudes of the 0- and 1-magnon correlators of the generalized Fishnet theory in d dimensions and one type of correlators of chiral fishnet theory in 4 dimensions
As a non-trivial check, we show that the results for 1-magnon correlator in d = 8, when evaluated using the exact techniques in [1, 2] are in perfect agreement with our d dimensional perturbative results
Summary
We give a general outline of perturbative analysis for obtaining Regge trajectories in the d-dimensional conformal fishnet theory. The Regge trajectories are yielded by the poles in spin J of the spectral function bnJ (ν) which, for d-dimensional fishnet theories, has the generic structure. We are looking for singular behavior of E∆(n,)JRegge in the limit ξ → 0 This can be achieved by considering the ansatz in (2.2) to be a weak coupling perturbation about the free Regge trajectory Jf(n): Jo(n)(ν) := Jf(n)(ν) + fk(ν)ξαk. Similar arguments can be put into use, if required, for other structures of the spectral functions This has been done for our analysis of Regge trajectories in chiral fishnet theories
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