On multi-step BCFW recursion relations

Abstract

In this paper, we extensively investigate the new algorithm known as the multistep BCFW recursion relations. Many interesting mathematical properties are found and understanding these aspects, one can find a systematic way to complete the calculation of amplitude after finite, definite steps and get the correct answer, without recourse to any specific knowledge from field theories, besides mass dimension and helicities. This process consists of the pole concentration and inconsistency elimination. Terms that survive inconsistency elimination cannot be determined by the new algorithm. They include polynomials and their generalizations, which turn out to be useful objects to be explored. Afterwards, we apply it to the Standard Model plus gravity to illustrate its power and limitation. Ensuring its workability, we also tentatively discuss how to improve its efficiency by reducing the steps.

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).…”

“…Having understood poles of above three categories, we can now consider the following contour integration 28) where in the second line we have explicitly written down the above mentioned subtle factors in the denominator. Now we consider its various pole contributions,…”

Spin Physics with a fixed-target experiment at the LHC

“…A fixed-target experiment at the LHC would greatly extend its physics capabilities, offering many opportunities to study the nucleon/nuclear structure at high x, the properties of nuclear matter under extreme conditions in heavy-ion collisions, and the nucleon 3D/spin decomposition in terms of partonic degrees of freedom, which has been the topic of this talk. Extensive theoretical works have contributed to the development of a full physics program for a fixed-target experiment at the LHC, using both the multi-TeV proton and ion beams [31,19,32,13,33,20,34,35,36,23,37,38,39,40,41,31,42,43]. Several projection studies, based on the performances of ALICE and LHCb detectors in fixed-target mode [9,7,8,6,44,5,45], clearly show that unprecedented precise measurements are at reach, both on quark and gluon sensitive probes such as Drell-Yan, open heavy-flavour and quarkonium production.…”

“…They would for the first time offer Figure 3: Projected statistical uncertainty on asymmetries in DY production with LHCb (the rapidity is integrated over, as well as the mass in bins of dM = 1 GeV/c 2 ). A major strength of the LHC fixed-target mode is the large production rates for open heavyflavour mesons and quarkonia (roughly 10 6 ϒ and 10 9 J/ψ in pH collisions for a single year of data taking [6,19]). These processes are very useful probes to precisely access and constrain the gluon Sivers effect, still essentially unknown [20].…”

Form factor and boundary contribution of amplitude

“…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. Especially in paper [54], it is pointed out that the boundary contribution possesses similar BCFW recursion relation as amplitudes, and it can be computed recursively from the lower-point boundary contribution.…”