Maximal unitarity at two loops

By David A. Kosower, Kasper J. Larsen. 2011

149 citations recorded by the source corpus.

The study of scattering amplitudes beyond one loop is necessary for precision phenomenology for the Large Hadron Collider and may also provide deeper insights into the theoretical foundations of quantum field theory. In this thesis we develop a new method for computing two-loop amplitudes, based on unitarity rather than Feynman diagrams. In this approach, the two-loop amplitude is first expanded in a linearly independent basis of integrals. The process dependence thereby resides in the coefficients of the integrals. These expansion coefficients are then the object of calculation.Our main results include explicit formulas for a subset of the integral coefficients, expressing them as products of tree-level amplitudes integrated over specific contours in the complex plane. We give a general selection principle for determining these contours. This principle is then applied to obtain the coefficients of integrals with the topology of a double box. We show that, for four-particle scattering, each double-box integral in the two-loop basis is associated with a uniquely defined complex contour, referred to as its master contour. We provide a classification of the solutions to setting all propagators of the general double-box integral on-shell. Depending on the number of external momenta at the vertices of the graph, these solutions are given as a chain of pointwise intersecting Riemann spheres, or a torus. This classification is needed to define master contours for amplitudes with arbitrary multiplicities.We point out that a basis of two-loop integrals with as many infrared finite elements as possible allows substantial technical simplications, in terms of obtaining the coefficients of the integrals, as well as for the analytic evaluation of the integrals themselves. We compute two such integrals at four points, obtaining remarkably compact expressions. Finally, we provide a check on a recently developed recursion relation for the all-loop integrand of the amplitudes of N=4 supersymmetric Yang-Mills theory, examining the two-loop six-gluon MHV amplitude and finding agreement. The validity of the approach to two-loop amplitudes developed in this thesis extends to all four-dimensional gauge theories, in particular QCD. The approach is suited for obtaining compact analytical expressions as well as for numerical implementations.

Subfields: Atomic and Molecular Physics, and Optics, Electrical and Electronic Engineering, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/7b3b67522e9802891624fc99991f8efbbd188b48

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Citations in the corpus

Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence (2016, background): “F Maximal cut of the elliptic sectors We show that the maximal cut [35, 89] of IA 1,1,0,1,1,1,1,0,0 provides useful information about the class of functions needed to represent the result.”
Decomposition of Feynman integrals on the maximal cut by intersection numbers (2019, background): “Earlier considerations on possible relations between the number of MIs and geometric properties of differential manifolds can be found in [89, 90].”
Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report (2017, methodology): “The maximal unitarity method [107], which incorporates IBP identities, has been applied to a variety of two-loop examples in four dimensions [108–112].”
Two-loop integrand decomposition into master integrals and surface terms (2015, methodology): “This approach does not require integration over multi-dimensional phase spaces and thus differs from the direct extraction of integral coefficients [46] or possible extensions using the duality between master integrals and homology cycles of the phase spaces [47, 48, 52].”
The elliptic dilogarithm for the sunset graph (2013, methodology): “s dilogarithms and logarithms can as well be understood motivitically [6,7], but the status of two-loop order scattering amplitude is far less well understood for generic amplitudes (see for instance [8,9,10,11] for some recent progress). The sunset integral arises as the two-loop self-energy diagra”
Cuts of Feynman Integrals in Baikov representation (2017, background): “In the recent years, progress has been made also towards the extension of these reduction methods for two-loop amplitudes at the integral [17, 18, 19, 20, 21, 22, 23, 24, 25, 26] as well as the integrand [27, 28, 29, 30, 31, 32] level.”
Uniqueness of two-loop master contours (2012, unknown): “[34], contours satisfying this consistency condition are guaranteed to produce correct results for scattering amplitudes in any quantum field theory.”
Les Houches 2015: Physics at TeV Colliders Standard Model Working Group Report (2016, methodology): “The maximal unitarity method [63], which incorporates integration-by-parts (IBP) identities, has been applied to a variety of two-loop examples in four dimensions [64–68].”