Abstract
We compute explicitly the two-dimensional version of Basso-Dixon type integrals for the planar 4-point correlation functions given by conformal “fishnet” Feynman graphs. These diagrams are represented by a fragment of a regular square lattice of power-like propagators, arising in the recently proposed integrable bi-scalar fishnet CFT. The formula is derived from first principles, using the formalism of separated variables in integrable SL(2, ℂ) spin chain. It is generalized to anisotropic fishnet, with different powers for propagators in two directions of the lattice.
Highlights
We proved the integrability of our open spin chain since both
We derived an explicit formula for the two-dimensional analogue of BassoDixon integral given by conformal fishnet Feynman graph represented by regular square lattice of rectangular L × N shape, presented on figure 1 and figure 2(left)
Our result represents a slightly more general quantity Basso-Dixon graph: it concerns the anisotropic fishnet, i.e. with different powers for vertical and horizontal propagators, corresponding to arbitrary spins s, sof principal series representation of SL(2, C) group, or for the analytic continuation to s = sbelonging to the real interval
Summary
The functions Ψ(x|z) form a complete orthonormal basis in the Hilbert space HN. Any function Φ ∈ HN can be expanded w.r.t. this basis as follows. Depending on the value of spin in the quantum space, ns = s−s, the sum over nk goes over all integers (integer ns) or half-integers (half-integer ns). The coefficient function C(x) is given by the scalar product. The completeness condition for the functions Ψ(x|z) has the following form (2π)−N π−N2 N!. A similar formula was proven in the case of SL(2, R) Toda spin chain by [25], in the case of modular XXZ magnet in [26] and for b-Whittaker functions in [27]. It is commonly believed to work for our SL(2, C) spin chain as well, though the proof is still missing
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