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Euclidean coordinate-space perturbation theory with a single mass scale

Christoph L. Schröder, Harvey B. Meyer

TL;DR

This work develops a robust coordinate-space perturbation theory framework for massive quantum field theories in $d=2\lambda+2$ dimensions by leveraging Gegenbauer polynomials to handle angular integrations. It provides analytic expressions for three-point functions involving one massive and one massless field and systematically studies antiderivatives of products of modified Bessel functions via Lommel and Schafheitlin-type recursions, culminating in algorithms for both integer and certain non-integer powers. The authors apply the formalism to key building blocks, including the leading-order three-point function $W(x,y)$ and the one-loop two-point function $\delta_2 G_m^{(\lambda)}(x)$, and extend the framework to finite temperature and mixed momentum/coordinate-space amplitudes. These results yield re-usable, semi-analytic components that can simplify high-dimensional or high-vertex perturbative calculations, including finite-volume and lattice-QCD contexts. The methods offer a pathway to systematic, dimensionally regularized coordinate-space calculations with clear connections to their momentum-space counterparts and potential for practical lattice implementations.

Abstract

We develop elements of coordinate-space perturbation theory for massive quantum field theories in general $d$-dimensional Euclidean space. Using the expansion in Gegenbauer polynomials, we provide analytic expressions for several three-point correlation functions in theories with one massive and one massless field. To this end, a class of antiderivatives of products of two Bessel functions multiplied by a power of their common argument are studied systematically. We expect these results to be useful in perturbative calculations involving vertices of high degree, at finite temperature and/or in finite volume, as well as in auxiliary perturbative computations for a lattice QCD treatment of hadronic effects in precision observables. As an illustration, we compute the one-loop coordinate-space propagator of a massive particle coupled to a massless one, both in the vacuum and at finite temperature.

Euclidean coordinate-space perturbation theory with a single mass scale

TL;DR

This work develops a robust coordinate-space perturbation theory framework for massive quantum field theories in dimensions by leveraging Gegenbauer polynomials to handle angular integrations. It provides analytic expressions for three-point functions involving one massive and one massless field and systematically studies antiderivatives of products of modified Bessel functions via Lommel and Schafheitlin-type recursions, culminating in algorithms for both integer and certain non-integer powers. The authors apply the formalism to key building blocks, including the leading-order three-point function and the one-loop two-point function , and extend the framework to finite temperature and mixed momentum/coordinate-space amplitudes. These results yield re-usable, semi-analytic components that can simplify high-dimensional or high-vertex perturbative calculations, including finite-volume and lattice-QCD contexts. The methods offer a pathway to systematic, dimensionally regularized coordinate-space calculations with clear connections to their momentum-space counterparts and potential for practical lattice implementations.

Abstract

We develop elements of coordinate-space perturbation theory for massive quantum field theories in general -dimensional Euclidean space. Using the expansion in Gegenbauer polynomials, we provide analytic expressions for several three-point correlation functions in theories with one massive and one massless field. To this end, a class of antiderivatives of products of two Bessel functions multiplied by a power of their common argument are studied systematically. We expect these results to be useful in perturbative calculations involving vertices of high degree, at finite temperature and/or in finite volume, as well as in auxiliary perturbative computations for a lattice QCD treatment of hadronic effects in precision observables. As an illustration, we compute the one-loop coordinate-space propagator of a massive particle coupled to a massless one, both in the vacuum and at finite temperature.

Paper Structure

This paper contains 41 sections, 199 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Elements of coordinate-space perturbation theory
  3. Propagators
  4. Convolution of two propagators
  5. Gegenbauer polynomials
  6. Antiderivatives of products of two modified Bessel functions
  7. Known results due to Lommel, et al.
  8. Extension of Schafheitlin's recursion relation and further special results
  9. The case $\mu^2=\nu^2$
  10. The milestone relation in the case $\rho=-2$
  11. The general case, excluding $\rho= -1$
  12. The special case $\rho=\sigma-1$, $\nu=\mu + \sigma$
  13. An algorithm for finding antiderivatives for non-negative integer powers
  14. Example 1
  15. Example 2
  16. ...and 26 more sections

Figures (11)

  • Figure 1: The coordinate-space three-point function. Full (dashed) lines represent massive (massless) propagators.
  • Figure 2: Feynman diagram corresponding to the convolution of two coordinate-space propagators.
  • Figure 3: Representation of the $\ell$-$n$-plane including the two shorelines where the result is known, see in particular Eq. \ref{['eq:schaf:spcl_II']} and following, and Lommel 1. For the point ($n = 0,\ell = -1)$, indicated by the star, consider Eqs. \ref{['eq:lom:ell-1_equal_int_order_II']} and following. Two regions are defined, I above the $\ell = n+1$-magic-line, and II below this line. Applying the recursion relation Eq. (\ref{['eq:schafheitlin_full']}) corresponds to taking a two-unit vertical step. The triangles signal that the $\ell=-1$ lines cannot be reached by taking a downward vertical step, and the $\ell=n+1$ line cannot be reached by taking an upward vertical step.
  • Figure 4: Allowed operations to be taken in the $\ell$-$n$-plane represented by arrows. As before, the shorelines are drawn in red. Starting from the top right and going clockwise: our extended integral recursion relation originally due to Schafheitlin, see Eq. (\ref{['eq:schafheitlin_full']}) (see the main text for caveats), the usual recursion relation for the modified Bessel function, which always creates two strands which either both move to the left or both to the right, and finally the special recursion relation only valid when $n=1$; see Eq. \ref{['eq:algpos:rec0_II']}.
  • Figure 5: Investigation of our algorithm for the indefinite integral $\int z^4 \mathcal{D}_\mu(z) \bar{\mathcal{D}}_{\mu+1}(z) \,d z$. The starting point is marked as a black dot and lies in region I. One application of the recursion relation (creating two strands represented by the green arrows) is necessary so that all paths reach a shoreline.
  • ...and 6 more figures