Finite remainders of the Konishi at two loops in ${\cal N}=4$ SYM

Abstract

We present three point form factors (FF) in ${\cal N}=4$ Super Yang Mills theory for both the half-BPS and the Konishi operators at two loop level in the `t Hooft coupling using Feynman diagrammatic approach. We have chosen on shell final states consisting of $g ϕϕ$ and $ϕλλ$, where $ϕ,λ, g$ are scalar, Majorana fermion and gauge fields respectively. The computation is done both in the modified dimensional reduction as well as in the four dimensional helicity scheme. We have studied the universal structure of infrared (IR) singularities in these FFs using Catani's IR subtraction operators. Exploiting the factorisation property of the IR singularities and following BDS like ansatz for the IR sensitive terms in FFs, we determine the finite remainders of them. We find that the finite remainders of FFs of the half-BPS for both the choices of final states give not only identical results but also contain terms of uniform transcendentality of weight two and four at one and two loop levels, respectively. In the case of the Konishi operator, the finite remainders depend on the external states and do not exhibit uniform transcendentality. However, surprisingly, the leading transcendental terms for $g ϕϕ$ agree with that of the half-BPS. We have demonstrated the role of on shell external states for the FFs in the context of maximum transcendentality principle.