Recursion relations for scattering amplitudes with massive particles

Abstract

We use the recently developed massive spinor-helicity formalism [1] of Arkani-Hamed et al. to study a new class of recursion relations for tree-level amplitudes in gauge theories. These relations are based on a combined complex deformation of massless as well as massive external momenta. We use these relations to study tree-level amplitudes in scalar QCD as well as amplitudes involving massive vector bosons in the Higgsed phase of Yang-Mills theory. We prove the validity of our proposal by showing that in the limit of infinite momenta of two of the external particles, the amplitude once again is controlled by an enhanced Spin-Lorentz symmetry paralleling the proof of BCFW shift for massless gauge theories. Simple examples illustrate that the proposed shift may lead to an efficient computation of tree-level amplitudes.

All-multiplicity amplitudes with four massive quarks and identical-helicity gluons

“…For lower spins s ≤ 1, the validity of eq. (2.4) is guaranteed by the previously established results for scalars [29] and quarks [19,30], as well as by the boundary-behavior arguments of [44,51] for vector particles in the context of a spontaneously broken gauge theory.…”

“…At first the corresponding massive spinors used to be defined using the massless spinor-helicity formalism with the help of an additional massless reference momentum [14][15][16]. The new massive spinor-helicity formalism [13] allows for a more elegant formulation of such shifts, as explored in [46,[48][49][50][51]. In the rest of this paper, we will be picking one massive particle j and one massless particle k and shift their on-shell spinors as follows [43,49]:…”

“…The general-spin formula (2.4) first appeared in an explicit form in the lectures[31] and implicitly already in[32] in the context of heavy-particle effective theory. Apart from matching the s = 0, 1/2 results of[19,29,30], it has also been recently derived for s = 1 in[33].…”

Scattering of gravitons and spinning massive states from compact numerators

“…As an alternative to the diagrammatic rules and to gain more insight into the structure of amplitudes, we will now discuss an extension of the recurrence formula obtained in [106][107][108][109] (for formulas derived from on-shell recursion, see [126][127][128][129][130]). We found the recurrence formula by studying the double-cover approach and the properties of the reduced Pfaffian for gluons.…”

Color-kinematics duality, double copy and the unitarity method for higher-derivative QCD and quadratic gravity

“…As argued, this is a strong constraint in that it imposes a kind of special kinematics for the amplitudes. By resorting to recursion relations involving massive particles [115,116,143] and applying BCJ relations with massive particles [119], the tree-level proof of the squaring relations given in Ref. [3] can be generalized in a straightforward way -one just have to remember that the gauge numerators now are gauge invariant, so the product of left and right numerators of subamplitudes in the BCFW approach must be equal to the numerator of the original representation of the amplitude, but evaluated at shifted momenta, that is (in the notation of Ref.…”