High-energy expansion of two-loop massive four-point diagrams

Abstract

We apply the method of regions to the massive two-loop integrals appearing in the Higgs pair production cross section at the next-to-leading order, in the high energy limit. For the non-planar integrals, a subtle problem arises because of the indefinite sign of the second Symanzik polynomial. We solve this problem by performing an analytic continuation of the Mandelstam variables such that the second Symanzik polynomial has a definite sign. Furthermore, we formulate the procedure of applying the method of regions systematically. As a by-product of the analytic continuation of the Mandelstam variables, we obtain crossing relations between integrals in a simple and systematic way. In our formulation, a concept of "template integral" is introduced, which represents and controls the contribution of each region. All of the template integrals needed in the computation of the Higgs pair production at the next-to-leading order are given explicitly. We also develop techniques to solve Mellin-Barnes integrals, and show them in detail. Although most of the calculation is shown for the concrete example of the Higgs pair production process, the application to other similar processes is straightforward, and we anticipate that our method can be useful also for other cases.

Asymptotic expansions through the loop-tree duality

“…It is of interest to see how the expansions developed above hold up in comparison to the established method of Expansion by Regions (EbR) [10][11][12][13][14][15][16][17]. Within this successful method the integrand of the Feynman amplitude, written in terms of Minkowski momenta, is expanded by dividing the space of the loop momenta into distinct regions.…”

“…Widely 0123456789(). : V,-vol known among them is Expansion by Regions [10][11][12][13][14][15][16]. While this technique has been shown to provide correct results a general proof is still pending [17].…”

Asymptotic Expansions Through the Loop--Tree Duality

“…It is of interest to see how the expansions developed above hold up in comparison to the established method of Expansion by Regions (EbR) [5][6][7][8][9][10][11]. Within this successful method the integrand of the Feynman amplitude, written in terms of Minkowski momenta, is expanded by dividing the space of the loop momenta into distinct regions.…”

“…The interest in asymptotic expansions becomes also clear noting that there are already well-developed methods for simplifying the integrands of Feynman amplitudes. Widely known among them is Expansion by Regions [5][6][7][8][9][10]. While this technique has been shown to provide correct results a general proof is still pending [11].…”

Determining arbitrary Feynman integrals by vacuum integrals

“…for helpful discussions. This work is in part supported by the Recruitment Program of Global Youth Experts of China.Note added : Recently, several preprints appeared, e.g., [78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94], which are aimed at solving two-loop cutting-edge problems with IBP reduction method but equipped with many advanced techniques. Even with these improvements of IBP reductions, our reduction method is still very competitive.…”