On form factors in N = 4 $$ \mathcal{N}=4 $$ SYM theory and polytopes

Abstract

In this paper we discuss different recursion relations (BCFW and all-line shift) for the form factors of the operators from the N = 4 SYM stress-tensor current supermultiplet T AB in momentum twistor space. We show that cancelations of spurious poles and the equivalence between different types of recursion relations can be naturally understood using geometrical interpretation of the form factors as a special limit of the volumes of polytopes in CP 4 in close analogy with the amplitude case. We also show how different relations for the IR pole coefficients can be easily derived using the momentum twistor representation. This raises an intriguing question -which of powerful on-shell methods and ideas can survive off-shell?

On soft theorems and form factors in N = 4 $$ \mathcal{N}=4 $$ SYM theory

“…It is still unclear for us whether it is possible to write some consistent deformation of Grassmannian integral (4.1), which will give us all tree-level form factors of stress tensor operator supermultiplet. Relations between different terms of BCFW and BCFW/CSW recursion observed in [53] suggest that such deformation exists. It is likely, that it would be easier to find such deformation in the case when the momentum q carried by operator is lightlike q 2 = 0.…”

“…13 Here we use notations indentical to [66], which are slightly different from [53]. One can convert one notation into another using mnemonic rule X +a,here = X − a,there , X −a ,here = X + a ,there , X +a,here = X +a,there ,X −a ,here = X −a ,there .…”

“…all kinematical invariants one can encounter in form factor computation it is necessary to consider different closed contours [36,53,66,67] Γ k where momentum q is inserted at different positions among p i momenta (see figure 14). It is convenient to parametrize these contours with different sets of coordinates y k i :…”

“…So, one can think of y k i as of points on large periodical contour x i [36,53,66,67]. Sum over k runs through all inequivalent contours Γ k .…”

“…Let us consider the BCFW recursion for NMHV form factor of stress tensor operator supermultiplet in momentum supertwistor notation [53]. Performing the shift of momentum supertwistor as [5,68] The momentum supertwistor is defined as…”