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TSIL: a program for the calculation of two-loop self-energy integrals

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Stephen P. Martin, David G. Robertson

High Energy Physics - Phenomenology · Submitted 15 Jan 2005 · 63 references

Pith's verdict: A solid, honest reference paper for a tool the subfield already depends on; the only real gap is that the accuracy claims are stated qualitatively while the numbers that would back them ship in the tarball rather than the text.

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arXiv IDhep-ph/0501132 (version 6)
Pith ID4JA6EUHM, Pith's permanent record of this paper and its review
CitationDavid G. Robertson, Stephen P. Martin. "TSIL: a program for the calculation of two-loop self-energy integrals." arXiv:hep-ph/0501132 (2005). https://pith.science/paper/4JA6EUHM
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The paper

The paper claims

TSIL turns any two-loop self-energy diagram into a numerical function call: use polylogarithms where they are known, and integrate the basis integrals' own differential equations in the external momentum everywhere else.

The reading

Two-loop self-energy diagrams are the bottleneck of precision particle physics: each is a pair of loop integrals over up to five distinct masses and an external momentum, and no single closed formula covers the general case. This paper describes TSIL, a C library that accepts the fifteen basis integrals left behind after Tarasov's recurrence relations have reduced a diagram, and returns numerical values for arbitrary masses $x,y,z,u,v$ and arbitrary external momentum invariant $s$. The method is a hybrid. Where an answer in polylogarithms exists in the literature, the code evaluates it directly; everywhere else it solves the coupled first-order differential equations that the basis integrals satisfy as functions of $s$, integrating outward from $s=0$, where analytic values and small-$s$ expansions are known. The payoff is that a two-loop diagram stops being an individual research project and becomes a subroutine a phenomenologist can call, which is what two-loop pole-mass and self-energy calculations need in order to keep pace with measurement.

What carries the argument

The engine is the differential-equation system of eq. (2.44), $dF_i/ds = C_{ij}F_j + C_i$, taken over the sixteen functions of eq. (2.40) together with $B(x,z)$, $B(y,u)$ and their product. Its coefficients are obtained from Tarasov's recurrence relations and were tabulated in an earlier paper; their poles in $s$ sit exactly at the thresholds $t_{xz}= (x^{1/2}+z^{1/2})^2$ and pseudo-thresholds $p_{xz}=(x^{1/2}-z^{1/2})^2$ and their permutations, which is why the integration path matters as much as the integrator. Two supporting mechanisms carry the accuracy: a partial-fraction rewriting that reduces every pole in the coefficient functions to a simple pole (achievable for the $B$, $S$ and $T$ functions, and for the $U$ functions whenever their second argument is non-zero, with the vanishing case patched by an algebraic identity applied after the run), and an identity, eq. (3.19), that cancels the residual double poles in $d(sM)/ds$ when $v = (x^{1/2}+y^{1/2})^2$.

Extended reading notes

Core claim

The claim is that the complete basis set of two-loop self-energy integrals — the master integral $M(x,y,z,u,v)$ together with the $S$, $T$ and $U$ functions and their one-loop subordinates listed in eq. (2.40) — can be evaluated numerically for arbitrary masses and arbitrary $s$, to a relative accuracy the authors put at better than $10^{-10}$ for generic inputs in long double arithmetic. The argument is constructive: the sixteen quantities satisfy a closed linear system $dF_i/ds = C_{ij}F_j + C_i$ whose coefficients are rational in the masses and $s$ (with analytically known $A$ and $I$ functions in some entries), so a single Runge-Kutta march from $s=0$ produces all of them at once. Three numerical devices make that march survive the thresholds and pseudo-thresholds where the coefficients blow up: a displaced contour in the upper half complex $s$ plane, logarithmic changes of variable at the start and the end of the run, and a five-stage fourth-order Runge-Kutta whose nodes satisfy $c_i<1$ so that no derivative is ever evaluated at a singular endpoint. The paper also supplies two analytic master-integral results not previously in the literature, $M(0,0,x,y,0)$ and $M(0,y,y,0,x)|_{s=x}$, plus the threshold values $U(x,0,y,y)|_{s=x}$ and $U(y,0,y,x)|_{s=x}$, and a translation table matching this notation to the conventions of roughly a dozen other groups.

Load-bearing premise

The load-bearing premise is that the differential-equation coefficients, after the partial-fraction rewriting described in section 3, have at most simple poles in $s$, so that Runge-Kutta integration along the displaced contour gives controlled accuracy for arbitrary masses; the authors acknowledge this fails for $M(0,y,z,u,y)$ when $y=(z^{1/2}+u^{1/2})^2$ or $y=(z^{1/2}-u^{1/2})^2$, where they state there is some loss of accuracy at and very near that threshold, and they argue that no symmetry of a quantum field theory forces such a coincidence.

Editorial extensions

If this is right

  • Any two-loop self-energy contribution that can be reduced by Tarasov's algorithm becomes numerically available at any parameter point, removing the need to derive a special-case analytic result for each new mass hierarchy.
  • Two-loop pole-mass calculations — the paper's worked example is the pole squared mass of a single scalar with cubic and quartic couplings — can be assembled from library calls, with the reduction to basis integrals and the counterterm bookkeeping left as the user's symbolic task.
  • The two new analytic master integrals in Appendix B extend the set of parameter points where no numerical integration is needed at all, and therefore where the accuracy is exact up to roundoff.
  • The conventions appendix lets results computed with this library be compared term by term against the published literature, which is otherwise a recurring source of sign and normalisation errors in two-loop work.
  • Subset modes for the $S,T,U$ and $S,T$ systems give a factor of four or five in speed for users who do not need the full master topology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the integration is seeded at $s=0$ and marched outward, the cost of a parameter point scales with how far $s$ sits from zero and how close it passes to a threshold; a library organised this way would naturally be slowest exactly where precision calculations are most interesting, near pair-production thresholds.
  • The same architecture — a closed ODE system in the momentum invariant plus a displaced contour — should transfer to two-loop vertex and box topologies, provided the corresponding coefficient matrices can be generated by the same recurrence machinery; the paper does not attempt this, but nothing in the method is specific to self-energies.
  • The admitted blind spot, where two masses satisfy $y=(z^{1/2}\pm u^{1/2})^2$ with no symmetry forcing it, is a statement about measure-zero parameter sets; a user could in practice detect it by re-running with perturbed masses and watching for loss of digits, which suggests a cheap self-diagnostic the code does not currently advertise.
  • The comparison table of conventions across roughly a dozen prior papers is the kind of artefact that ages well: it makes the library's output checkable against results that were computed before the library existed.
Original abstract

TSIL is a library of utilities for the numerical calculation of dimensionally regularized two-loop self-energy integrals. A convenient basis for these functions is given by the integrals obtained at the end of O.V. Tarasov's recurrence relation algorithm. The program computes the values of all of these basis functions, for arbitrary input masses and external momentum. When analytical expressions in terms of polylogarithms are available, they are used. Otherwise, the evaluation proceeds by a Runge-Kutta integration of the coupled first-order differential equations for the basis integrals, using the external momentum invariant as the independent variable. The starting point of the integration is provided by known analytic expressions at (or near) zero external momentum. The code is written in C, and may be linked from C, C++, or Fortran. A Fortran interface is provided. We describe the structure and usage of the program, and provide a simple example application. We also compute two new cases analytically, and compare all of our notations and conventions for the two-loop self-energy integrals to those used by several other groups.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,944 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.84 · mentions code or data availability 0.55.

The abstract, sentence by sentence

arXiv listing abstract v6. Results and the main claim are underlined; .

TSIL is a library of utilities for the numerical calculation of dimensionally regularized two-loop self-energy integrals. A convenient basis for these functions is given by the integrals obtained at the end of O.V. Tarasov's recurrence relation algorithm. The program computes the values of all of these basis functions, for arbitrary input masses and external momentum. When analytical expressions in terms of polylogarithms are available, they are used. Otherwise, the evaluation proceeds by a Runge-Kutta integration of the coupled first-order differential equations for the basis integrals, using the external momentum invariant as the independent variable. The starting point of the integration is provided by known analytic expressions at (or near) zero external momentum. The code is written in C, and may be linked from C, C++, or Fortran. A Fortran interface is provided. We describe the structure and usage of the program, and provide a simple example application. We also compute two new cases analytically, and compare all of our notations and conventions for the two-loop self-energy integrals to those used by several other groups.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 TSIL is a library of utilities for the numerical calculation of dimensionally regularized two-loop self-energy integrals. contribution 1.00 0.60 0.18 asserted 1.00
s1 A convenient basis for these functions is given by the integrals obtained at the end of O.V. method 0.62 0.27 0.04 asserted 1.00
s2 Tarasov's recurrence relation algorithm. background 0.55 0.08 0.04 asserted 0.58
s3 The program computes the values of all of these basis functions, for arbitrary input masses and external momentum. contribution 0.80 0.81 0.15 asserted 1.00
s4 When analytical expressions in terms of polylogarithms are available, they are used. method 1.00 0.56 0.05 asserted 0.97
s5 Otherwise, the evaluation proceeds by a Runge-Kutta integration of the coupled first-order differential equations for the basis integrals, using the external momentum invariant as the independent variable. method 1.00 0.74 0.06 asserted 1.00
s6 The starting point of the integration is provided by known analytic expressions at (or near) zero external momentum. method 1.00 0.37 0.04 asserted 1.00
s7 The code is written in C, and may be linked from C, C++, or Fortran. method 0.78 0.75 0.02 hedged possibility 0.83
s8 A Fortran interface is provided. contribution 0.65 0.83 0.03 asserted 1.00
s9 We describe the structure and usage of the program, and provide a simple example application. contribution 0.99 0.91 0.07 asserted 0.99
s10 We also compute two new cases analytically, and compare all of our notations and conventions for the two-loop self-energy integrals to those used by several other groups. contribution 0.98 0.96 0.15 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.95.

What was examined, and what was not

sources read
  • arXiv listing abstract v6
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Feynman diagram topologies for the one- and two-lo
Figure 2. When s is greater than or equal to the smallest non-zero threshold or pseudo-threshold, the Runge-Kutta integration proceeds along a contour with the shape shown here, as suggested in ref. [58]. functions). The coefficients can be evaluated by using Tarasov’s recurrence relations, and were listed in [4]. At s = 0 the values of all of the functions and their derivatives with respect to s (and/or their expansions in small s) are known analytically in terms of dilogarithms. Therefore, one can integrate the functions¶ by the Runge-Kutta method to any other value of s. In order to avoid…
Figure 3. Feynman diagram topologies for some two-loop corr

Figures reproduced from arXiv:hep-ph/0501132 by the authors.

References
How later papers used it

51 later papers cite it

Evidence: substantial. 47 of the citing papers share no author with it, and 29 of those carry a confident label from Pith's citation classifier.

Pith read 29 of the 78 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 13 of 31 papers with this label · 1 by the same authors

By the same authors

How the literature uses it

49 arXiv papers cite this in 75 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 24 sentences without a confident label

Uses its method or tool 31 papers
Compares with it 2 papers
Cites it as background 7 papers

Discussion

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