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Separation of soft and collinear singularities from one-loop N-point integrals

Stefan Dittmaier

TL;DR

The work addresses infrared mass singularities in general one-loop $N$-point integrals by analyzing collinear and soft regions to derive a compact formula that expresses the singular part in terms of 3-point integrals. A key result is the definition of coefficients $A_{nk}$, drawn from collinear and soft categories with overlap subtraction, yielding a unified expression for the singular part of any tensor $N$-point integral in terms of 3-point functions. This formulation is regularization-scheme independent and can translate singular results between schemes or serve as a momentum-space subtraction to render one-loop integrals IR finite before evaluation. The approach also facilitates the Sudakov limit analysis and provides explicit 5-point (pentagon) integral examples, demonstrating broad applicability to NLO computations and higher-point reductions in $D$ dimensions. Overall, the paper offers a practical, analytically transparent method to predict and manage soft and collinear singularities across a wide class of one-loop integrals.

Abstract

The soft and collinear singularities of general scalar and tensor one-loop N-point integrals are worked out explicitly. As a result a simple explicit formula is given that expresses the singular part in terms of 3-point integrals. Apart from predicting the singularities, this result can be used to transfer singular one-loop integrals from one regularization scheme to another or to subtract soft and collinear singularities from one-loop Feynman diagrams directly in momentum space.

Separation of soft and collinear singularities from one-loop N-point integrals

TL;DR

The work addresses infrared mass singularities in general one-loop -point integrals by analyzing collinear and soft regions to derive a compact formula that expresses the singular part in terms of 3-point integrals. A key result is the definition of coefficients , drawn from collinear and soft categories with overlap subtraction, yielding a unified expression for the singular part of any tensor -point integral in terms of 3-point functions. This formulation is regularization-scheme independent and can translate singular results between schemes or serve as a momentum-space subtraction to render one-loop integrals IR finite before evaluation. The approach also facilitates the Sudakov limit analysis and provides explicit 5-point (pentagon) integral examples, demonstrating broad applicability to NLO computations and higher-point reductions in dimensions. Overall, the paper offers a practical, analytically transparent method to predict and manage soft and collinear singularities across a wide class of one-loop integrals.

Abstract

The soft and collinear singularities of general scalar and tensor one-loop N-point integrals are worked out explicitly. As a result a simple explicit formula is given that expresses the singular part in terms of 3-point integrals. Apart from predicting the singularities, this result can be used to transfer singular one-loop integrals from one regularization scheme to another or to subtract soft and collinear singularities from one-loop Feynman diagrams directly in momentum space.

Paper Structure

This paper contains 17 sections, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. One-loop $N$-point integrals and mass singularities
  3. Separation of mass singularities
  4. Asymptotic behaviour in collinear regions
  5. Asymptotic behaviour in soft regions
  6. Final result
  7. Discussion and applications
  8. Possible applications of the final result
  9. Sudakov limit of the one-loop box integral
  10. Singular structure of some 5-point integrals
  11. A 5-point integral for the process $gg\to{\rm t}$ t$\bar{{\rm t}$ t$}{\rm H}$ H$$
  12. A 5-point integral for the process $gg\to{\rm t}$ t$\bar{{\rm t}$ t$} g$
  13. A 5-point integral for the process ${\rm e^+}$ e^+${\rm e^-}$ e^-$\to\nu_{\rm e}$ e$\bar{\nu}_{\rm e}$ e${\rm H}$ H$$
  14. A 5-point integral for the process ${\rm e^+}$ e^+${\rm e^-}$ e^-$\to{\rm t}$ t$\bar{{\rm t}$ t$}{\rm H}$ H$$
  15. Summary
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Momentum and mass assignment for a general one-loop $N$-point integral (\ref{['eq:Npointint']})
  • Figure 2: Examples for pentagon diagrams contributing to the one-loop corrections to the processes $gg\to{\rm t}$ t$\bar{{\rm t}$ t$}{\rm H}$ H$$ and $gg\to{\rm t}$ t$\bar{{\rm t}$ t$} g$
  • Figure 3: Examples for pentagon diagrams contributing to the one-loop corrections to the processes ${\rm e^+}$ e^+${\rm e^-}$ e^-$\to\nu_{\rm e}$ e$\bar{\nu}_{\rm e}$ e${\rm H}$ H$$ and ${\rm e^+}$ e^+${\rm e^-}$ e^-$\to{\rm t}$ t$\bar{{\rm t}$ t$}{\rm H}$ H$$ (The number $n$ of propagator $N_n$ is indicated in parentheses.)