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HPL, a Mathematica implementation of the harmonic polylogarithms

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D. Maitre

High Energy Physics - Phenomenology · Submitted 13 Jul 2005 · 47 references

Pith's verdict: A solid, honest tool paper: the Mathematica HPL package is real and useful, but its numerical validation is a single unquantified sentence and the option defaults contradict each other in the documentation.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
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  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

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arXiv IDhep-ph/0507152 (version 2)
Pith IDMBZOWK3V, Pith's permanent record of this paper and its review
CitationD. Maitre. "HPL, a Mathematica implementation of the harmonic polylogarithms." arXiv:hep-ph/0507152 (2005). https://pith.science/paper/MBZOWK3V
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The paper

The paper claims

A Mathematica package implements the harmonic polylogarithms end to end.

The reading

Harmonic polylogarithms are a family of special functions built by iterated integration of the three kernels $1/x$, $1/(1-x)$ and $1/(1+x)$; they generalise ordinary logarithms and polylogarithms and appear throughout multi-loop calculations in high-energy physics, from splitting functions to two-loop vertex form factors. The paper's aim is to give Mathematica users a working implementation of these functions rather than a new theorem about them: the package supplies the product algebra, the derivative rules, the extraction of logarithmic divergences at $x=0$ and $x=1$, series expansions around 0 and 1, reduction to a minimal basis, rewriting in terms of more familiar functions, and numerical evaluation. A deliberate design point is analytic continuation: rather than letting the computer algebra system's own branch conventions decide, the package lets the user specify the sign of the infinitesimal imaginary part, so that results on the regions $(-1,0)$, $(1,∞)$ and $(-∞,-1)$ carry the prescription the calculation needs. The author also translates the existing FORM tables of multiple zeta values up to weight 8, so that harmonic polylogarithms evaluated at argument 1 reduce to known constants. The payoff, if the implementation is faithful, is that a Mathematica session can carry a full HPL calculation from algebra to a number without leaving the notebook.

What carries the argument

Two structures carry the whole package. The first is the shuffle, or product, algebra: $H(p;x)H(q;x)$ equals the sum of $H(r;x)$ over all arrangements $r$ of the combined index vector that preserve the internal order of $p$ and of $q$. This single identity generates the extraction of divergences (solve it for the term with a leading 1 or a trailing 0), the reduction to a minimal set, and the conversion of products into single functions of higher weight. The second is the recursion for the Taylor coefficients $Z_j$ of the series expansion, which follows directly from the integral definition: for a leftmost index $n>0$ one has $Z_j(n,m_1,…) = j^{-n}∑_{i=2}^{j} Z_{i-1}(m_1,…)$, with the sign pattern $(-1)^{j}$ for negative indices, and the recursion is closed by $H(0;x)=log x$, $H(1;x)=-log(1-x)$ and $H(n;x)=Li_n(x)$. Around these two sit a set of argument transformations ($x→-x$, $x→x^2$, $x→1-x$, $x→x/(x-1)$, $x→(1-x)/(1+x)$) and the continuation formulae that map each real interval back to $(0,1)$, plus the 'm'-notation, which drops internal zeros and increments the next non-zero index, so that the same function can be written in whichever form makes a given identity short.

Extended reading notes

Core claim

The claim is that the package reproduces, in Mathematica, the mathematical properties that the harmonic polylogarithms are known to obey, and that it does so accurately enough to be used in practice. Concretely: products of two harmonic polylogarithms are expanded into the shuffle sum over all order-preserving interleavings of the two index vectors, giving a linear combination of functions of the summed weight; derivatives reduce the weight by one through the kernel attached to the leftmost index; leftmost 1s and rightmost 0s are traded for explicit logarithms, which makes the singular behaviour at $x=1$ and $x=0$ visible; and the remaining 'irreducible' functions are projected onto a minimal basis whose dimension the paper tabulates up to weight 8 (3, 3, 8, 18, 48, 116, 312, 810). The author reports numerical agreement with the existing FORTRAN implementation at double precision. Two limitations are stated in the paper rather than left for a reader to find: the numerical evaluation relies on series expansions and, in the author's words, 'the convergence is not very good for values near 1, where not all digits displayed are accurate', with users pointed to the FORTRAN package for precision there; and enabling the automatic conversion to known functions overrides the user's chosen sign of the imaginary part, because Mathematica's branch conventions then take over, an effect the paper illustrates with a worked example.

Load-bearing premise

The numerical evaluation rests on systematic use of the series expansion, and the paper itself states in section 11.4 that this converges poorly near $x=1$, where not all displayed digits are accurate.

Editorial extensions

If this is right

  • A user can carry out a multi-loop calculation inside Mathematica, taking harmonic polylogarithms through multiplication, differentiation, expansion and numerical evaluation without exporting expressions to FORM or FORTRAN.
  • Because the sign of the imaginary part is a user-set option, results in the physical regions above $x=1$ and below $x=-1$ can be produced with a chosen $i\epsilon$ prescription and checked against a second choice.
  • Harmonic polylogarithms at argument 1 reduce to multiple zeta values, and the translated tables up to weight 8 make those reductions automatic for the weights most two- and three-loop calculations reach.
  • The reduction to a minimal basis gives a canonical form for expressions, which is what makes results from different diagrams, or different groups, comparable term by term.
  • The package can serve as a backend for other Mathematica tools that need these functions, as the paper notes it already does for an expansion package for hypergeometric functions.

Reading between the lines

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

  • The honest boundary of the numerical evaluation is the endpoint $x=1$: a reader who needs many digits there is better served by the FORTRAN code the paper cites, and a useful test would be to chart the number of correct digits against $x$ for a fixed weight.
  • The interaction the paper documents between automatic conversion to known functions and analytic continuation suggests a design rule that generalises beyond this package: any layer that hands a function to the computer algebra system's built-in branch cuts silently discards a user-supplied $i\epsilon$ prescription, so such rewrites should be opt-in and applied last.
  • Since the minimal-set choice is a convention rather than a canonical object, two users reducing the same expression with different orderings could obtain different-looking but equal answers; a cross-check between two independent implementations of the reduction would settle whether the convention is being applied consistently.
  • The tables of multiple zeta values stop at weight 8, which matches current multi-loop needs; extending the same machinery to higher weight is a bookkeeping problem rather than a conceptual one, but the number of independent constants grows quickly.
Original abstract

In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren for Mathematica. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. The analytic continuation has been treated carefully, allowing the user to keep the control over the definition of the sign of the imaginary parts. Many options enables the user to adapt the behavior of the package to his specific problem.

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, 3,498 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.06 · mentions code or data availability 0.41.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren for Mathematica. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. The analytic continuation has been treated carefully, allowing the user to keep the control over the definition of the sign of the imaginary parts. Many options enables the user to adapt the behavior of the package to his specific problem.

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 In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren for Mathematica. contribution 1.00 0.90 0.36 asserted 1.00
s1 It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. contribution 0.88 0.87 0.09 asserted 1.00
s2 The analytic continuation has been treated carefully, allowing the user to keep the control over the definition of the sign of the imaginary parts. contribution 0.79 0.86 0.09 asserted 1.00
s3 Many options enables the user to adapt the behavior of the package to his specific problem. contribution 0.88 0.74 0.07 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.85.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
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

References
How later papers used it

173 later papers cite it

Evidence: substantial. 172 of the citing papers share no author with it, and 134 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 214 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 10 of 134 papers with this label
How the literature uses it

164 arXiv papers cite this in 204 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 · 51 sentences without a confident label

Uses its method or tool 125 papers
Compares with it 2 papers
  • By contracting Eq. [ref] with appropriate polarisation vectors, we have verified with the help of the software package S@M [ref] that the above expressions agree with the results of Ref. [ref] with the exception of a sign difference in the term proportional to the colour structure[We have replaced this colour structure using Eqs. [ref] and [ref], while it has been kept as is in Ref. [ref] .] $if^a_1a_2c T^c_i$ for the $++$ helicity configuration.

    Revisiting the double-soft asymptotics of one-loop amplitudes in massless QCD · Expansion accurate at O(E0) · 0.85

  • As far as the numerical implementation is concerned, we perfectly reproduced Fig. (2) of [ref] , and we verified that our implementation matches the numerical results obtained using and the package HPL [ref] .

    Fiducial Higgs and Drell-Yan distributions at N$^3$LL$^\prime$+NNLO with RadISH · Extraction of hard and collinear coefficient functions at $O(^3)$ · 0.96

Cites it as background 6 papers

Discussion

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