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Abstract |
We study integrability of fishnet-type Feynman graphs arising in planar four-dimensional bi-scalar chiral
theory recently proposed in arXiv:1512.06704 as a special double scaling limit of gamma-deformed N=4 SYM
theory. We show that the transfer matrix "building" the fishnet graphs emerges from the R-matrix of non-
compact conformal SU(2,2) Heisenberg spin chain with spins belonging to principal series representations of
the four-dimensional conformal group. We demonstrate explicitly a relationship between this integrable spin
chain and the Quantum Spectral Curve (QSC) of N=4 SYM. Using QSC and spin chain methods, we construct
Baxter equation for Q-functions of the conformal spin chain needed for computation of the anomalous
dimensions of operators of the type tr(phi_1^J) where phi1 is one of the two scalars of the theory. For J=3 we
derive from QSC a quantization condition that fixes the relevant solution of Baxter equation. The scaling
dimensions of the operators only receive contributions from wheel-like graphs. We develop integrability
techniques to compute the divergent part of these graphs and use it to present the weak coupling expansion
of dimensions to very high orders. Then we apply our exact equations to calculate the anomalous dimensions
with J=3 to practically unlimited precision at any coupling. These equations also describe an infinite tower of
local conformal operators all carrying the same charge J=3. The method should be applicable for any J and,
in principle, to any local operators of bi-scalar theory. We show that at strong coupling the scaling
dimensions can be derived from semiclassical quantization of finite gap solutions describing an integrable
system of noncompact SU(2,2) spins. This bears similarities with the classical strings arising in the strongly
coupled limit of N=4 SYM. |
