Direct extraction of one loop rational terms

By Simon Badger. 2008

187 citations recorded by the source corpus.

We present a method for the direct extraction of rational contributions to one-loop scattering amplitudes, missed by standard four-dimensional unitarity techniques. We use generalised unitarity in $D=4-2\e$ dimensions to write the loop amplitudes in terms of products of massive tree amplitudes. We find that the rational terms in $4-2\e$ dimensions can be determined from quadruple, triple and double cuts without the need for independent pentagon contributions using a massive integral basis. The additional mass-dependent integral coefficients may then be extracted from the large mass limit which can be performed analytically or numerically. We check the method by computing the rational parts of all gluon helicity amplitudes with up to six external legs. We also present a simple application to amplitudes with external massless fermions.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/a1c8b7580a7068c7521baeda8ef6909cc1c29f58

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In Nuclear and High Energy Physics

Citations in the corpus

Integrand reduction of one-loop scattering amplitudes through Laurent series expansion (2012, methodology): “[16] S. Badger, H. Frellesvig, and Y. Zhang, Hepta-Cuts of Two-Loop Scattering Amplitudes, arXiv:1202.2019.”
Spinning black hole binary dynamics, scattering amplitudes, and effective field theory (2020, methodology): “[31, 74], which extracts the coefficients of box and triangle scalar integrals from quadruple and triple cuts, respectively, shown in Fig.”
Uniqueness of two-loop master contours (2012, unknown): “I+− ≡ I1,1,0,1,1,0 [ [1|/̀1|2〉〈4|/̀2|3] ] × [24]〈13〉 (5.”
Bounding the Higgs width at the LHC using full analytic results for gg → e−e+μ−μ+ (2013, background): “It is obtained by exploiting the relation between mass dependent terms in the box and triangle coefficients, and the rational term [36].”
Colour-Kinematics duality for one-loop rational amplitudes (2013, methodology): “Using the method of D-dimensional cuts of [48], an all-plus amplitude is given by a sum over cut boxes.”
Scattering amplitudes for all masses and spins (2017, background): “These terms can be obtained by considering the states in the internal loops to be massive [51, 52], where the mass m2 is identified with the extra −2 dimension piece of `2, denoted as μ2.”
One-loop calculations in quantum field theory: From Feynman diagrams to unitarity cuts (2011, methodology): “The rational part The rational part can be obtained with a number of methods [30, 79, 122] For this elementary example, the simplest method is to retain the dimensiondependent terms in a Passarino-Veltman decomposition.”
The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations (2014, background): “The latter problem can be bypassed by means of a mass shift [167], which however might imply additional complications in the case of axial couplings in massive theories.”
Electroweak radiative corrections for collider physics (2019, background): “The calculations in additional dimensions can be avoided by using appropriate mass shifts and exchanging the D-dimensional cuts by 4-dimensional massive cuts [251].”
Scattering amplitudes from unitarity-based reduction algorithm at the integrand-level (2010, methodology): “nd rational terms at once. This technique has been neatly systematized for numerical purposes by Ellis, Giele, Kunszt and Melnikov [87,88], and later proposed also for the on-shell approach by Badger [60]. In this paper we present samurai, a tool based on a hybrid algorithm for the numerical computa”