Boundary contributions using fermion pair deformation

Abstract

Continuing the study of boundary BCFW recursion relation of tree level amplitudes initiated in [1], we consider boundary contributions coming from fermion pair deformation. We present the general strategy for these boundary contributions and demonstrate calculations using two examples, i.e, the standard QCD and deformed QCD with anomalous magnetic momentum coupling. As a by-product, we have extended BCFW recursion relation to off-shell gluon current, where because off-shell gluon current is not gauge invariant, a new feature must be cooperated.

On multi-step BCFW recursion relations

“…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].…”

“…Applying (4.1), corresponding D min 's are 3, 2 and 1, which exclude first two cases. However, 11 Here ±1 and ±2 are independent, they do not necessarily have the same sign. the third case is also excluded even if its D min is allowed.…”

“…Since particles 1 −1 , 3 −1 and 5 −1 are symmetric in the helicity configuration (there is no color order), 3|1] is also a good deformation. In general, there is a last good deformation corollary: after the n-th step, when I = (known terms) n + C n · · · C 0 is reached, we can further expand it by one more step as 10) assume the (n + 1)-th step is the last step, for which C n+1 C n · · · C 0 = 0, and if 11) then the (n + 1)-th step is a good deformation. This corollary is powerful in practical calculations, since unnecessary steps can be saved if we incidentally encounter the condition above.…”

QCD amplitudes with 2 initial spacelike legs via generalised BCFW recursion

“…We found a full generalisation of the recursibility criteria devised in [34,35], proving independently that, exactly as in the fully on-shell case, amplitudes with at least one gluon leg can be recursed, whereas, when there are only external fermions, one still has to rely on the ordinary Feynman diagram expansion. Other types of generalisations of BCFW are underway in the on-shell case [46][47][48][49], which amount JHEP02(2017)009 Figure 11.…”

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.…”