On the Generalized Harmonic Polylogarithms of One Complex Variable

1 Introduction

• Accurate collider predictions have driven development of methods for multi-loop Feynman-diagram calculations.The introduction notes that analytic calculations once considered infeasible became possible.Source: §1 · p. 2
• The introduction places master-integral reduction and analytic evaluation in the broader development of multi-loop calculation methods.It describes Laporta reduction using IBP identities and the use of differential equations with Mellin-Barnes techniques [4].Source: §1 · p. 2
• Identifying a function basis for master integrals is another ingredient in analytic higher-order calculations.The basis is related to the threshold structure of the Feynman diagrams.Source: §1 · p. 2
• GHPLs extend HPLs for Feynman diagrams with richer threshold structures, while HPLs are suited to simpler thresholds [6].The extensions include one- and two-dimensional GHPLs [13].Source: §1 · p. 2
• The paper presents (G)HPLs as a basis for analytic results, connected to differential-equation solutions and control of master-integral analytical properties.It also notes their linear independence and the availability of numerical routines in FORTRAN, Mathematica, and C++.Source: §1 · p. 2
• The paper gives a general framework for evaluating single-variable GHPLs with square-root weights using existing numerical routines.It shows that all GHPLs introduced in can be evaluated with the VW routines.Source: §1 · p. 2
• The paper applies its numerical strategy to NLO light-fermion electroweak corrections for Higgs production in gluon fusion with complex W and Z masses.The paper also covers the corrections’ analytic continuation, complex-variable behavior, and an extended weight set.Source: §1 · p. 3

2.1 Harmonic Polylogarithms

• HPLs are defined by repeated integration of three weight functions, with an all-zero-weight exception defined using a logarithm.Their weight-1 functions have logarithmic singularities and branch cuts, so continuation prescriptions determine complex values outside their real domains.Source: §2.1 · pp. 3–4
• The HPL shuffle algebra expresses products as sums over interleavings that preserve each factor’s weight order, with resulting weight n1 + n2.For continuation across cuts, the algebra can expose rightmost zeros or ones and isolate the associated logarithmic imaginary parts.Source: §2.1 · p. 4, §2.1.1 · pp. 5–6

2.1.1 Analytic Continuation

• HPL analytic continuation uses logarithmic branch-cut prescriptions, with imaginary parts arising from the relevant logarithms.For x < 0 the continuation includes x → −x′ − i0+; for x′ > 1, log(1 − x′ − i0+) = log(x′ − 1) − iπ.Source: §2.1.1 · pp. 5–6

2.1.2 Transformations of Variables and Numerical Evaluation

• HPL numerical evaluation extracts logarithmic behavior near x = 0, expands the remaining analytic functions, and maps other real-axis domains to a central interval.The cited implementation uses a Taylor expansion with Bernoulli numbers and Chebyshev polynomials in its central region.Source: §2.1.2 · pp. 6–7

2.2 Generalized Harmonic Polylogarithms

• GHPLs extend HPLs with additional weight functions when the HPL weights do not suffice to describe a result’s analytic structure.The discussed application uses weights {−r, −4, −1, 0}, with branch behavior arising from logarithms and square roots.Source: §2.2 · p. 7

2.2.1 Analytic Continuation

• For negative real x, GHPL continuation depends on the interval because logarithms and square roots acquire different imaginary parts.The discussion distinguishes −1 ≤ x < 0, −4 ≤ x < −1, and x < −4.Source: §2.2.1 · p. 8

2.2.2 Numerical evaluation

• Direct numerical evaluation of square-root-weight GHPLs is cumbersome, so the paper converts them into linear-weight GHPLs for evaluation with public routines [19].The GHPLs inherit the shuffle algebra, and the transformed functions can use the routines for generalized polylogarithms with generic linear weights.Source: §2.2 · p. 8, §2.2.2 · p. 9

3 Linearization

• The weights {−r, −4, 0} transform to {−1, 0, 1}, reducing these GHPLs to the usual HPLs.This covers the simpler case before the section treats simultaneous square-root and linear weights.Source: §3 · p. 10
• The weights {−r, −4, −1, 0} can be transformed together into linear weights, although no all-real transformation linearizes them simultaneously.The transformation permits complex linear weights for this combination.Source: §3 · pp. 10–11
• The procedure extracts trailing zeroes before changing variables, linearizes weight-1 functions, then proceeds weight by weight for higher weights.Extracted logarithms are rewritten and transformed in the new variable to avoid endpoint singularities.Source: §3 · p. 11

3.1 Analytic Continuation of the Linearized GHPLs and their Numerical Evaluation

• For 0 ≤ x, conjugate-weight terms occur in pairs whose imaginary parts cancel; for −4 ≤ x < 0, rescaling converts the pure-phase argument to real argument 1.The resulting complex weights can be evaluated with the VW routines.Source: §3.1 · p. 13

• For an unstable particle with mass m, its width Γ plays an active role in determining the corresponding physical observable, so the parameter x becomes complex in this situation.Here, x is a dimensionless ratio between squared momentum and squared mass.Source: §4 · pp. 13–14
• In the non-physical region s < 0, x has positive real and imaginary parts, and the GHPL value is path-independent there because the functions are analytic.The integration is along a curve in the complex plane.Source: §4 · p. 14
• Scale invariance rescales the GHPL weights and argument together, letting the complex-variable problem be evaluated at real argument 1 with complex weights.The rescaled weights are ratios of the original weights and x.Source: §4 · pp. 14–16

5 Generalizations and Additional Weights

• The expanded alphabet is {−1 −r, −r, −4, −1, 0, 1, 4, r, 1 + r, c, c̄}, covering GHPLs needed for the master integrals in.The guidelines can also be used for more complicated sets.Source: §5 · pp. 16–17

5.1 The set { −1, 0, 1, c, c̄}

• The GHPLs in the referenced set can be evaluated straightforwardly using the routines in [19], without further variable transformation; when x is complex, scale invariance of the GHPLs is used.Source: §5.1 · p. 17

5.2 The set { −1 −r, −r, −4, −1, 0}

• The additional weight −1 −r linearizes into {−1, 0, 1, c, c̄}; a complex difference of conjugate GHPLs is made real by the prefactor i.Source: §5.2 · p. 17

5.3 The set {0, 1, 4, r, 1 + r}

• The positive weights {0, 1, 4, r, 1 + r} require a different variable change, and trailing zeroes must be extracted first to avoid a possible divergence at η = −1.The extracted logarithms can be transformed directly using the relation given for log(x).Source: §5.3 · pp. 17, 19

5.4 Mixed Weights

• Wider sets can be linearized when square roots with different signs are not mixed, including {−1 −r, −r, −4, −1, 0, 1, 4}.The section gives a separate transformation for {−4, −1, 0, 1, 4, r, 1 + r}.Source: §5.4 · p. 19

6 Two-loop Light-Fermion contributions to the Higgs Production in Gluon Fusion

• Finite W and Z widths smear the correction peaks at mH ∼ 2mW and mH ∼ 2mZ, resizing their relative importance near those thresholds.The section recomputes the corrections with complex W and Z masses, unlike the earlier evaluation with real masses and ad hoc FORTRAN routines.Source: §6 · p. 19
• The section evaluates δlf with the VW routines and reports numerical values as a function of Higgs-boson mass.The table caption identifies mH in GeV; the supplied passage does not include its numerical entries.Source: §6 · p. 21, Table 1 · p. 21
• The calculation uses the listed top- and bottom-quark, W- and Z-boson, and electroweak input parameters.The supplied parameter list includes mt = 173.1 GeV, mW = 80.398 GeV, and mZ = 91.1876 GeV.Source: §6 · p. 21

7 Conclusions

• The analyzed one-variable GHPL set includes the weights {−1 −r, −r, −4, −1, 0, 1, 4, r, 1 + r, c, c̄}.The paper connects this set to analytic expressions for master integrals entering the electroweak form factor [20].Source: §7 · pp. 21–22
• Complex weights allow GHPLs with square roots in their weights to be expressed as combinations of GHPLs with linear weights.The specified set {−1 −r, −r, −4, −1, 0, 1, 4, r, 1 + r, c, c̄} can be linearized using two variable transformations.Source: §7 · p. 22
• The linearized GHPLs can be evaluated numerically with the existing C++/GiNaC routines by Vollinga and Weinzierl [19].The paper describes these routines as a fast and precise numerical-evaluation tool.Source: §7 · p. 22
• The strategy is applied to NLO light-fermion electroweak corrections to Higgs production in gluon fusion, using complex mW and mZ.The paper notes that the resulting expressions are more realistic than the earlier calculation neglecting W and Z widths, while GHPLs with square-root weights keep the analytic expressions compact.Source: §7 · p. 22

A Some Examples at Weight 2

• The appendix applies the paper’s procedure to weight-2 GHPL examples.The examples are G(−r, −r; x) and G(−r, −1; x).Source: A Some Examples at Weight 2 · p. 22
• One example is G(−r, −r; x), a weight-2 GHPL treated by the appendix’s procedure.Source: A Some Examples at Weight 2 · p. 22
• The second example is G(−r, −1; x), also at weight 2.Source: A Some Examples at Weight 2 · p. 22

B Some Examples at Weight 3

• The appendix provides expressions for five weight-3 GHPLs containing square-root weights that enter the corrections of.Source: B Some Examples at Weight 3 · p. 23
• The displayed expressions include terms in η and ξ with weights c, c̄, −1, 0, and 1.The cited formulas contain products and nested GHPL terms.Source: B Some Examples at Weight 3 · p. 23
• K1 = 0.278425076639727748441973590814 and K2 = −0.152226248227607546589100778278.The appendix says these constants have cumbersome expressions in known transcendental constants and gives their numerical values.Source: B Some Examples at Weight 3 · p. 24