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Scalar One-Loop Integrals using the Negative-Dimension Approach

C. Anastasiou, E. W. N. Glover, C. Oleari

TL;DR

The paper develops a unified negative-dimension (NDIM) framework to evaluate massive one-loop integrals with arbitrary propagator powers in D, recasting loop integrals as finite sums of generalized hypergeometric functions. A template solution plus a constraint system yields representations valid across all kinematic regions, demonstrated explicitly for massive bubbles and triangles and extended to new two-variable hypergeometric results for a vertex with one off-shell leg and two masses. The approach reproduces known results (e.g., via Mellin-Barnes methods) and provides analytic continuations between regions, offering a coherent pathway toward more complex graphs and potential two-loop extensions. Overall, NDIM delivers a principled, systematic method for expressing one-loop integrals in terms of well-studied hypergeometric functions, with practical steps for obtaining region-specific results and analytic continuations.

Abstract

We study massive one-loop integrals by analytically continuing the Feynman integral to negative dimensions as advocated by Halliday and Ricotta and developed by Suzuki and Schmidt. We consider n-point one-loop integrals with arbitrary powers of propagators in general dimension D. For integrals with m mass scales and q external momentum scales, we construct a template solution valid for all n which allows us to obtain a representation of the graph in terms of a finite sum of generalised hypergeometric functions with m+q-1 variables. All solutions for all possible kinematic regions are given simultaneously, allowing the investigation of different ranges of variation of mass and momentum scales. As a first step, we develop the general framework and apply it to massive bubble and vertex integrals. Of course many of these integrals are well known and we show that the known results are recovered. To give a concrete new result, we present expressions for the general vertex integral with one off-shell leg and two internal masses in terms of hypergeometric functions of two variables that converge in the appropriate kinematic regions. The kinematic singularity structure of this graph is sufficiently complex to give insight into how the negative-dimension method operates and gives some hope that more complicated graphs can also be evaluated.

Scalar One-Loop Integrals using the Negative-Dimension Approach

TL;DR

The paper develops a unified negative-dimension (NDIM) framework to evaluate massive one-loop integrals with arbitrary propagator powers in D, recasting loop integrals as finite sums of generalized hypergeometric functions. A template solution plus a constraint system yields representations valid across all kinematic regions, demonstrated explicitly for massive bubbles and triangles and extended to new two-variable hypergeometric results for a vertex with one off-shell leg and two masses. The approach reproduces known results (e.g., via Mellin-Barnes methods) and provides analytic continuations between regions, offering a coherent pathway toward more complex graphs and potential two-loop extensions. Overall, NDIM delivers a principled, systematic method for expressing one-loop integrals in terms of well-studied hypergeometric functions, with practical steps for obtaining region-specific results and analytic continuations.

Abstract

We study massive one-loop integrals by analytically continuing the Feynman integral to negative dimensions as advocated by Halliday and Ricotta and developed by Suzuki and Schmidt. We consider n-point one-loop integrals with arbitrary powers of propagators in general dimension D. For integrals with m mass scales and q external momentum scales, we construct a template solution valid for all n which allows us to obtain a representation of the graph in terms of a finite sum of generalised hypergeometric functions with m+q-1 variables. All solutions for all possible kinematic regions are given simultaneously, allowing the investigation of different ranges of variation of mass and momentum scales. As a first step, we develop the general framework and apply it to massive bubble and vertex integrals. Of course many of these integrals are well known and we show that the known results are recovered. To give a concrete new result, we present expressions for the general vertex integral with one off-shell leg and two internal masses in terms of hypergeometric functions of two variables that converge in the appropriate kinematic regions. The kinematic singularity structure of this graph is sufficiently complex to give insight into how the negative-dimension method operates and gives some hope that more complicated graphs can also be evaluated.

Paper Structure

This paper contains 31 sections, 108 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. The negative-dimension approach
  4. The general case: different masses
  5. The general form of the solutions
  6. An example: the massless bubble
  7. Massive bubble integrals
  8. Identification of the groups of solutions using the convergence regions
  9. Identification of the groups of solutions using the system of constraints
  10. The limiting case: $\boldsymbol{M_1 \neq 0,\, M_2 = 0}$
  11. The limiting case: $\boldsymbol{Q_1^2 \,\hbox{$\rightarrow$}\, 0}$
  12. The special case: all masses equal
  13. Massive Vertex integrals
  14. Massless propagators: $\boldsymbol{M_1 =M_2 = M_3= 0}$
  15. Two massive propagators and one off-shell leg
  16. ...and 16 more sections

Figures (4)

  • Figure 1: The generic one-loop graph
  • Figure 2: The one-loop vacuum graph
  • Figure 3: The one-loop vertex diagram
  • Figure 4: The kinematic regions for the one-loop triangle with $Q_2^2 = Q_3^2 = M_3^2=0$. The solid line shows the threshold in the Landau surface at $M_2^2 = Q_1^2 + M_1^2$, together with the reflections $M_1^2 + M_2^2 = Q_1^2$ and $M_1^2 = M_2^2 + Q_1^2$. The reflections are relevant for the convergence properties of the hypergeometric functions which only involve the absolute values of ratios of the scales. The dashed lines show the boundaries $M_1^2 = M_2^2$ and $M_2^2 = Q_1^2$.