Note on Identities Inspired by New Soft Theorems

The new soft theorems, for both gravity and gauge amplitudes, have inspired a number of works, including the discovery of new identities related to amplitudes. In this note, we present the proof and discussion for two sets of identities. The first set includes an identity involving the half-soft function which had been used in the soft theorem for one-loop rational gravity amplitudes, and another simpler identity as its byproduct. The second set includes two identities involving the KLT momentum kernel, as the consistency conditions of the KLT relation plus soft theorems for both gravity and gauge amplitudes. We use the CHY formulation to prove the first identity, and transform the second one into a convenient form for future discussion.Comment: 17 page

Derivation of Feynman Rules for Higher Order Poles Using Cross-ratio Identities in CHY Construction

In order to generalize the integration rules to general CHY integrands which include higher order poles, algorithms are proposed in two directions. One is to conjecture new rules, and the other is to use the cross-ratio identity method. In this paper,we use the cross-ratio identity approach to re-derive the conjectured integration rules involving higher order poles for several special cases: the single double pole, single triple pole and duplex-double pole. The equivalence between the present formulas and the previously conjectured ones is discussed for the first two situations.Comment: 29 pages, 11 figure

On Multi-step BCFW Recursion Relations

In this paper, we extensively investigate the new algorithm known as the multi-step 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.Comment: 38 pages, 13 figures, 3 appendice

Explaining the DAMPE data with scalar dark matter and gauged U(1)Le−LμU(1)_{L_e-L_\mu} interaction

Inspired by the peak structure observed by recent DAMPE experiment in $e^+e^-$ cosmic-ray spectrum, we consider a scalar dark matter (DM) model with gauged $U(1)_{L_e-L_\mu}$ symmetry, which is the most economical anomaly-free theory to potentially explain the peak by DM annihilation in nearby subhalo. We utilize the process $\chi \chi \to Z^\prime Z^\prime \to l \bar{l} l^\prime \bar{l}^\prime$, where $\chi$, $Z^\prime$, $l^{(\prime)}$ denote the scalar DM, the new gauge boson and $l^{(\prime)} =e, \mu$, respectively, to generate the $e^+e^-$ spectrum. By fitting the predicted spectrum to the experimental data, we obtain the favored DM mass range $m_\chi \simeq 3060^{+80}_{-100} \, {\rm GeV}$ and $\Delta m \equiv m_\chi - m_{Z^\prime} \lesssim 14 \, {\rm GeV}$ at $68\%$ Confidence Level (C.L.). Furthermore, we determine the parameter space of the model which can explain the peak and meanwhile satisfy the constraints from DM relic abundance, DM direct detection and the collider bounds. We conclude that the model we consider can account for the peak, although there exists a tension with the constraints from the LEP-II bound on $m_{Z^\prime}$ arising from the cross section measurement of $e^+e^- \to Z^{\prime\ast} \to e^+ e^-$.Comment: 15 pages, 4 figure

Determination of Boundary Contributions in Recursion Relation

In this paper, we propose a new algorithm to systematically determine the missing boundary contributions, when one uses the BCFW on-shell recursion relation to calculate tree amplitudes for general quantum field theories. After an instruction of the algorithm, we will use several examples to demonstrate its application, including amplitudes of color-ordered phi-4 theory, Yang-Mills theory, Einstein-Maxwell theory and color-ordered Yukawa theory with phi-4 interaction.Comment: 20 pages, 1 appendi