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Loop-Level Structure

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Scattering Amplitudes in Gauge Theories

Part of the book series: Lecture Notes in Physics ((LNP,volume 883))

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Abstract

In chapter three the loop-level structure of amplitudes is reviewed. Here the reduction of one-loop Feynman integrals to a basis of scalar integrals is discussed. The idea of (generalized) unitarity in constructing one-loop amplitudes from tree-level data is reviewed and a number of concrete examples are computed in detail. In the second part of chapter three we give an introduction to the evaluation of Feynman integrals at one and higher loop order. After reviewing their definition and useful parameter representations, such as the Feynman and Mellin representations, we give an introduction to the integration by parts techniques in conjunction with differential equations.

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Notes

  1. 1.

    Note that i0 is always to be understood as a positive infinitesimal quantity. Strictly speaking the i0 in Eq. (3.1) differs from the i0 here by a positive factor of \(2\sqrt{ \mathbf {k}^{2}+m^{2}}\) and terms of order (i0)2 are suppressed. The same will be true when we introduce Feynman parametrization below.

  2. 2.

    For this counting one simply adds the powers of the loop-momentum l in the denominator and numerator of the integral including the measure. A positive or vanishing result indicates a UV-divergent integral: ∫d 4 ll a/l b is UV-divergent for 4+a≥b. Equality in this relation entails a logarithmic divergence, the other cases yield power-like divergences. In fact dimensional regularization is only sensitive to logarithmic divergencies, power-like divergences are set to zero.

  3. 3.

    The definition will be clarified below in the Feynman representation.

  4. 4.

    In fact \(q_{N}^{\mu}=0\) so in the last equation the sum effectively only runs up to i=N−1, but this does not change the argument.

  5. 5.

    With the upper indices [i] we denote the spins of the respective particles running in the loop.

  6. 6.

    Recall that ε i ⋅ε i =0 and that in our gauge choice p i ⋅ε j =0.

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Authors and Affiliations

  1. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA

    Dr. Johannes M. Henn

  2. Institut für Physik, Math.-Naturw. Fakultät, Humboldt Universität zu Berlin, Berlin, Germany

    Prof. Dr. Jan C. Plefka

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  1. Dr. Johannes M. Henn
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  2. Prof. Dr. Jan C. Plefka
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Henn, J.M., Plefka, J.C. (2014). Loop-Level Structure. In: Scattering Amplitudes in Gauge Theories. Lecture Notes in Physics, vol 883. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54022-6_3

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