BCFW recursion relation with nonzero boundary contribution

Abstract

The appearance of BCFW on-shell recursion relation has deepen our understanding of quantum field theory, especially the one with gauge boson and graviton. To be able to write the BCFW recursion relation, the knowledge of boundary contributions is needed. So far, most applications have been constrained to the cases where the boundary contribution is zero. In this paper, we show that for some theories, although there is no proper deformation to annihilate the boundary contribution, its effects can be analyzed in simple way, thus we do able to write down the BCFW recursion relation with boundary contributions. The examples we will present in this paper include the λφ 4 theory and Yukawa coupling between fermions and scalars. *

On multi-step BCFW recursion relations

“…But this is not the end of the story. When proceeding with the calculation 9) while it is just shown that C 5|1] C 3|1] = 0, incidentally we also find C 5|1] P 3|1] = 0. This means 5|1] is a good deformation and hence one step is enough.…”

“…Many related studies have been achieved including: introducing auxiliary fields to eliminate boundary terms [7,8], analyzing Feynman diagrams to isolate boundary terms [9][10][11], expressing boundary terms as roots of amplitudes [12][13][14], collecting factorization limits to interpolate boundary terms [15] and using other deformations for better large z behavior [16].…”

Note on recursion relations for the Q $$ \mathcal{Q} $$ -cut representation

“…It is well-known for tree-level amplitudes that for Yang-Mills and gravity theories, the BCFW deformation can be chosen such that the boundary contribution vanishes. While for some other theories, the boundary contribution would appear and require more careful analysis [22][23][24][25][26][27][28][29][30]. Here we shall assume B = 0 for simplicity (but the similar consideration can be generalized to the case with non-zero boundary contributions).…”

Form factor and boundary contribution of amplitude

“…However this assumption is not always true, for example, it fails in the theories involving only scalars and fermions or under the "bad" momentum deformation. Many solutions have been proposed (by auxiliary fields [41,42], analyzing Feynman diagrams [43][44][45], studying the zeros [46][47][48], the factorization limits [49], or using other deformation [50][51][52]) to deal with the boundary contribution in various situations. Most recently, a new multi-step BCFW recursion relation algorithm [53][54][55] is proposed to detect the boundary contribution through certain poles step by step.…”

“…There are three spin-1 operators 43) and their complex conjugates. In order to construct the Lagrangian, we need to product them with spin-1 trace term.…”