Gluing operation and form factors of local operators in N = 4 Super Yang-Mills theory

The gluing operation is an effective way to get form factors of both local and non-local operators starting from different representations of on-shell scattering amplitudes. In this paper it is shown how it works on the example of form factors of operators from stress-tensor operator supermultiplet in Grassmannian and spinor helicity representations.


Introduction
N = 4 Super Yang-Mills theory (SYM) turns out to be a very popular object for theoretical research. Mainly this is due to the development of new techniques, such as recursive methods for tree-level amplitudes, on-shell diagrams and so on (see [3] for review), allowing to obtain analytical results for the matrix elements of the S-matrix. The Grassmannian integral representation for the amplitude is of particular interest, since it makes symmetry properties of the amplitudes manifest and relates scattering amplitudes to on-shell diagrams [4]. After a while, many of the techniques, developed for scattering amplitudes, were generalized to more complicated class of quantities -form factors, which are matrix elements of the form 0|O(x)|1, ..., n , on which our attention is focused on. In particular, it was especially interesting to obtain the Grassmannian integral representation for form factors, which was done in [1] and [2] in slightly different ways. The common thing for both these two approaches is gluing of the minimal formfactor to the corresponding amplitude, whose integral representation is known [4]. The purpose of this article is to check formulae, obtained in [1] and [2] for different representations, i.e. for the Grassmannian integral and spinor helicity representations.

Form factors of the Stress-Tensor Supermultiplet operator
For the purposes, claimed in the previous section, only form factors of the chiral part of the stresstensor supermultiplet operator would be considered. With the help of the so-called harmonic superspace [5] the operator itself can be written as A are projections of coordinates on the superspace to the harmonic superspace with projectors u +a A and u −a A . The lowest component of T (x, θ + ) is scalar operator tr(φ ++ φ ++ ) with φ ++ = 1 2 ab u +a A u +b B φ AB . The indices a, a and ± correspond to S U(2) × S U(2) × U(1) ⊂ S U (4). Let q and γ −α a be the momentum and supermomentum, carried by the operator O(x) respectively. Then the super form factor is defined by For k = 2, which corresponds to the Maximal Helicity Violating (MHV) case, the form factor is given by the following equation: where

Gluing operation & Grassmannian integral representation
As it was pointed out in the beginning, there were some difficulties with obtaining the grassmannian integral representation for the form factor, which was explained by the fact that form factors are partially off-shell quantities. A conjecture was made, that such a representation could be obtained using known integral representation for the scattering amplitude. Graphically it can be illustrated by the figure 3. The idea of the gluing operation is very simple, i.e. to take known representation of the amplitude with two additional external legs and glue to it the minimal form factor, which contains delta functions, imposing proper conditions on kinematics, by integrating out these two extra degrees of freedom. Since the minimal form factor for the operator, considered here, has two external legs, one needs to glue it to A k,n+2 . On mathematical language gluing reduces to the on-shell phase-space integration: A k,n+2 + other gluing positions, (5) where F 2,2 = δ 2 (λ 4 )δ 4 (η 4 )δ 2 (λ 5 )δ 4 (η 5 ) and underscored variables impose twisted kinematics conditions (see eq. 2.21 of [1]): After performing the integration, one arrives at the formula, valid for arbitrary k, n [1], [2]: (6) where ξ A and ξ B are arbitrary reference spinors and and M i are consecutive minors of the C matrix.

Examples of evaluation
In this section several examples of the application of the eq. (5), (6) will be provided. The first example is 3-point MHV form factor in spinor helicity representation. Since F 2,3 has 3 external legs, there are 3 gluing positions, but due to the fact, that all top-cell diagrams are equivalent in this case, all gluing positions should give the same results, so, to obtain answer for F 2,3 one has to take into account only one top-cell diagram (i.e. one gluing position) since they are all equal.

Substituting the expression for
12 23 34 45 51 to the integral (5) and considering gluing between legs 3 and 1, one has to integrate over λ 4,5 ,λ 4,5 and η 4,5 . Integrations overλ 4,5 and η 4,5 could be done with the help of delta functions of the minimal form factor F 2,2 which in turn mean that corresponding variables are just to be relabelled (see definitions of the underlined variables in the previous section). Thus, at this stage one can write: To do the last interation one has to eliminate GL(1) 2 invariance, reparametrizing spinors as follows, introducing arbitrary reference spinors: This reparametrization is just change of variables, thus one can rewrite the result for F 2,3 as integral over parameters β 1,2 : The integral I is evaluated via the Cauchy residue theorem. For the case, when the minimal form factor is glued between legs i and j, one obtains for I: Thus the full answer reads [6]: The same answer could be derived via grassmannian integral (6). As can be easily shown, there are no integrations over grassmannian, since the number of integration variables is equal to zero in this case: which means that all coefficients of the C matrix are determined by the delta functions. Sincê on the support of these delta functions one arrives at eq. (7) again, where double underscore mens twisted kinematics conditions (see eq. 3.9 of [1]). The next example is 3-point next-to-MHV form factor. As in the previous case, all top-cell diagrams are equivalent here, so [23] 12 4 · I, .
Taking residues in simple poles of the integrand one gets Let's extract a specific component from this super form factor, namely the component at the maximal η degree ( [6]), which is 12 in this case: .
This quantity would be necessary to check that results, derived using different approaches, agree.
In the full analogy with the first example it is possible to compute the same quantity via integral over the Grassmannian. Again, for F 3,3 there are no integrations, so on the support of thê one can solve the constraints to obtain coefficients of the C matrix: Substituting these values to the integrand yields: one can extract the component with the maximal η degree, which agrees with eq. (9). The simplest example, containing non-trivial integrations over the Grassmannian manifold is Nextto-MHV form factor with 4 legs, or NMHV 4 . In the case of 4 external legs there are 4 possible gluing positions, shown on the picture: It's worth mentioning here that a priori there is no prescription for choosing contour of integration, which gives the correct answer for arbitrary n, k. The integral over Grassmannian, corresponding to gluing between legs 1 and 4 has the form: (3)] Ω 3,4δ (6) (C ·λ)δ (12) (C · η)δ (6) (C ⊥ · λ).
Using the formula #(integrations) = (k − 2)(n − k), one notices that in this case the integral involves one non-trivial integration. The strategy of integration is described in [3], [7]: since the integral is 1-dimensional, there must exist one-parameter family of solutionsĉ i j = c i j (τ) that solve constraints, imposed by the delta functions of the integrand on the coefficients of the C-matrix. Then bosonic delta functions could be replaced by the momentum conserving delta function by localizing the integral on the solution of the constraints. Thus, the whole thing could be rewritten as follows: i=2,4,6δ In the latter formula the C-matrix is gauge-fixed to be Hats here indicate τ-dependence. Let's denote (i jk) -minor of the C-matrix, composed of i-th, j-th and k-th columns of C. The integrand has poles where (ii + 1i + 2) = 0. With the help of this condition one can express c i j in terms of helicity spinors, assigned to external particles (see [3] for details). As it has already been mentioned, there is no prescription in which poles one has to take residues, so it has to be figured out somehow. It turns out that in this particular case only 2 of 4 gluing positions contribute to the answer, and only in 2 of 4 poles one must take residues in each gluing position, namely F 3,4 = Res 41 (123) + Res 41 (561) + Res 23 (123) + Res 23 (561).
It's worth metioning that eq. (16) agrees with eq. 3.76 of [1]. Contributing diagrams are shown on the figure 3. Here the subscript on Res i j (klm) denotes legs, between which the minimal form factor is  .

Conclusion
In recent papers [1], [2] it was shown that gluing operation could be used to derive different represen- in the present article. The obtained results agree with those derived earlier. Also it's worth mentioning that presented results can be further generalized to the case of gluing non-local operators, such as Wilson line operators, to on-shell amplitudes to obtain Grassmannian integral representation for amplitudes with one leg off-shell, which will be the next step.