Abstract
It is well known that under a BCFW-deformation, there is a boundary contribution when the amplitude scales as $$ \mathcal{O}\left({z}^0\right) $$ or worse. We show that boundary contributions have a similar recursion relation as scattering amplitude. Just like the BCFW recursion relation, where scattering amplitudes are expressed as the products of two on-shell subamplitudes (plus possible boundary contributions), our new recursion relation expresses boundary contributions as products of sub-amplitudes and boundary contributions with less legs, plus yet another possible boundary contribution. In other words, the complete scattering amplitude, including boundary contributions, can be obtained by multiple steps of recursions, unless the boundary contributions are still non-zero when all possible deformations are exploited. We demonstrate this algorithm by several examples. Especially, we show that for standard model like renormalizable theory in 4D, i.e., the theory including only gauge boson, fermions and scalars, the complete amplitude can always be computed by at most four recursive steps using our algorithm.
Highlights
Boundary contributions, which can be evaluated directly or recursively afterwards
We show that boundary contributions have a similar recursion relation as scattering amplitude
Just like the BCFW recursion relation, where scattering amplitudes are expressed as the products of two on-shell subamplitudes, our new recursion relation expresses boundary contributions as products of sub-amplitudes and boundary contributions with less legs, plus yet another possible boundary contribution
Summary
The key idea of BCFW recursion is determining scattering amplitudes by their poles. To read out B 1|n], a good way is to do the large z expansion in the first line of (2.3). B 1|n] can be read out by selecting same power of z in numerator f (z) and denominators. It is worth to notice that, in principle, B 1|n] can have terms which are pure polynomials in momentum (i.e. they do not have any pole), and our method is not applicable. This can happen in many effective theories with higher dimension operators
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