Abstract
We discuss recursion relations for scattering amplitudes with massive particles of any spin. They are derived via a two-parameter shift of momenta, combining a BCFW-type spinor shift with the soft limit of a massless particle involved in the process. The technical innovation is that spinors corresponding to massive momenta are also shifted. Our recursions lead to a reformulation of the soft theorems. The well-known Weinberg’s soft factors are recovered and, in addition, the subleading factors appear reshaped such that they are directly applicable to massive amplitudes in the modern on-shell language. Moreover, we obtain new results in the context of non-minimal interactions of massive matter with photons and gravitons. These soft theorems are employed for practical calculations of Compton and higher-point scattering. As a by-product, we introduce a convenient representation of the Compton scattering amplitude for any mass and spin.
Highlights
The amplitude to a lower-point one with the soft particle removed
We discuss recursion relations for scattering amplitudes with massive particles of any spin. They are derived via a two-parameter shift of momenta, combining a BCFWtype spinor shift with the soft limit of a massless particle involved in the process
We introduced a class of complex momentum shifts combining a BCFW-like spinor shift with a soft limit of a massless particle present in the process
Summary
We begin with a brief overview of our conventions (readers well-versed in the massless and massive spinor formalism are invited to skip this section). Massive momentum satisfying the on-shell condition p2 = m2 can be represented by four spinors χα, χα2, χβ 1, χβ 2 They can be collected into χJ and χJ , where J = 1, 2 is identified with the SU(2) little group index which, in complete analogy to spinor indices, can be raised and lowered by epsilon tensors. Where summation over the little group indices is implicit This is a natural and convenient generalization of the spinor helicity formalism to massive particles [22] Massive outgoing particles can be represented by spinors χi Jk or χi Jk with lowered little group indices, in this paper we treat all external particles as incoming. Where we adhere to the bold notation introduced in ref. [22]
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