Analytical and numerical methods for massive two-loop self-energy diagrams

F. A. Berends; M. Boehm; M. Buza; S. Bauberger

REVIEW

Analytical and numerical methods for massive two-loop self-energy diagrams

S. Bauberger, F. A. Berends, M. Boehm, M. Buza

High Energy Physics - Phenomenology · Submitted 22 Sep 1994 · 17 references

Pith's verdict: A solid, checkable methods paper: three usable representations for the massive two-loop self-energies, with one real presentational gap in the convergence conditions.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 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.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 6 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-ph/9409388 (version 2)

Pith ID5PE6YI44, Pith's permanent record of this paper and its review

CitationF. A. Berends, M. Boehm, M. Buza, S. Bauberger. "Analytical and numerical methods for massive two-loop self-energy diagrams." arXiv:hep-ph/9409388 (1994). https://pith.science/paper/5PE6YI44

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Bottom line: this is a competent methods paper that does what its abstract says. Three independent representations for T123 and T1234 — multivariable hypergeometric series at small and large p2 in arbitrary D (Eqs. 17, 19, 24–27), complete elliptic integrals for the imaginary parts (Eqs. 53, 60), and one-dimensional integrals with elementary integrands valid for all p2 (Eqs. 70, 73, 74, 75). The last of these is the practically useful one: it replaces two-dimensional numerical integration with a single quadrature, and the integrand is elementary. That is a real gain for anyone doing two-loop electroweak self-energies, and it is the piece I would actually use. The master diagram T12345 is explicitly left open, and the authors say so rather than burying it — that is the honest framing of the paper's scope, and it should be read as such, not as an omission discovered by a referee. The derivations are written out step by step and are checkable: the dispersion integrals (5, 15), the Mellin-Barnes route (22–23), the partial-fraction reduction (65). Table 1 puts four routes against each other on a stated benchmark and they agree to about five digits, with the stochastic column visibly noisier and shown as such. That is the right kind of evidence for a paper of this sort. The self-citation is heavy — [3], [6], [7] all include present authors — but those results came from different techniques (x-space, Mellin-Barnes, two-dimensional numerics), so it is cross-validation, not circularity. On the elliptic continuation to c > 1 (Eqs. 60–63): the reader flagged this as the weakest assumption, and I think that is over-weighted. The continuation is cited to a published monograph [14], the imaginary jump is written out explicitly in Eq. 63, and the numerical checks against [3, 6] would catch a mishandled i-epsilon. That is supported. The one thing I would actually press on is smaller but real: the convergence condition stated in Eq. 18 is not satisfied by the Table 1 masses (m1=1, m2=3, m3=5, m4=7 makes m2 + sqrt(p2) < m1 impossible), yet column B from the appendix series agrees with columns C and D. So Eq. 18 is evidently not the operative condition for the subtracted combination T1234N, and the paper never says what is. The appendix result (86) also carries no stated convergence domain. That is a presentation gap, not a correctness problem, but it is the kind of gap that costs a reader an afternoon. Worth a paragraph in revision. Who should read it: anyone computing massive two-loop self-energies, and anyone who wants a worked example of the dispersion-plus-Mellin-Barnes toolkit applied to a nontrivial diagram. Send it out.

Referee report

Significance

If the results hold, this is a genuinely useful methods paper. The D-dimensional hypergeometric series unify a large number of previously case-by-case massless and equal-mass results into single expressions, and the elliptic-integral form of the imaginary parts gives a fast, well-conditioned route to a quantity that otherwise requires two-dimensional numerical integration. The one-dimensional representations with elementary integrands are the most immediately practical contribution, since they are stated to be valid for all p2 and are directly implementable. The paper is derivation-first: masses and p2 are free inputs, no quantity is fitted, and the numerical cross-checks in Table 1 against independent methods (columns C and D) provide real, if partial, validation. The explicit statement in §5 that the master diagram T12345 is not covered is appropriately candid and correctly scopes the claim.

Major objections (2)

[§2, Eq. (18) and Table 1] The stated convergence condition for the small-p2 series, Eq. (18), requires m2 + sqrt|p2| < m1 and m1 + m3 < m4. Table 1 uses m1 = 1, m2 = 3, m3 = 5, m4 = 7, so the first inequality is impossible for any p2 >= 0, yet column B — computed from Eq. (86), which descends from Eq. (17) — agrees with columns C and D to five digits across the whole table. Either Eq. (18) is not the operative convergence condition for the subtracted combination T1234N defined in Eq. (20), or the series is being evaluated outside its stated domain. Please state the convergence condition that actually applies to the subtracted result and to the appendix formula (86), and reconcile it with the masses used in Table 1.
[§5, Table 1] The text states that the comparison in Table 1 'gives confidence in the various calculational methods', but column A (VEGAS two-dimensional integration) visibly fails at small p2: at p2 = 0.006090 it gives -7.793835 against -7.99308 from column B, and at p2 = 0.013702 it gives -7.900682 against -7.99427. Columns B, C, and D agree with each other to five digits throughout. The table therefore demonstrates that the new methods succeed where the two-dimensional integration degrades, which is a stronger and more useful statement than the one made. Please say so explicitly rather than describing the table as uniform agreement.

Minor objections (6)

[§3.2, Eqs. (60)–(63)] The validity of Eq. (60) in all parameter regions rests on the analytic continuation of the third-kind integral Pi(c, kappa) to c > 1, Eq. (62). Only the imaginary jump of that continuation is written out, Eq. (63), with the real part deferred to the Tricomi–Krafft monograph [14]. Since the cancellation making Delta T1234 purely imaginary depends on this step, it would help to state the continuation formula explicitly, or at least to record the numerical check of the cancellation itself rather than only the agreement with [3, 6].
[§2, Eq. (46)] The representation for the m3 = m4 = 0 case contains a '±' in (1 ± sqrt(y)) and a corresponding '±' in z2 = ±4 sqrt(y)/(1 ± sqrt(y))^2, but the text does not say which sign is to be taken or whether the two signs are correlated. Please specify.
[§3.1, Eqs. (53) and (55)] Please verify the mass dimensions of the individual terms inside the braces of Eqs. (53) and (55). As printed, the K term, the E term, and the two Pi terms appear to carry different overall powers of mass, which would prevent them from being combined. This may be a transcription issue, but it should be checked against the derivation from Eq. (49).
[§2 and Appendix, Eqs. (17), (82)–(83)] The variables z_i are defined differently in Eq. (17) (z_i = m_i^2/m_4^2, i = 1,2,3, z_4 = p^2/m_4^2) and in the appendix (x, y, u, v in Eqs. 82–83). Please add a short dictionary connecting the two conventions so that Eq. (86) can be checked directly against Eq. (17).
[§5 and Appendix] The convergence domain of the appendix series C1–C7 in Eq. (86) is not stated anywhere. Given that these series are the practical output of the small-p2 analysis, a statement of their region of convergence would be valuable.
[Whole manuscript] A number of typographical and typesetting issues should be corrected: 'greatfully' for 'gratefully' in the Acknowledgement, 'considerd' for 'considered' after Eq. (13), 'the electroweak theor y' in the abstract, and the inconsistent spacing in 'two-loopFeynman' in the abstract. The reference list is otherwise complete and appropriately points to the published sources for quoted results.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The paper gives three explicit ways to compute massive two-loop self-energy Feynman diagrams — hypergeometric series in the external momentum, complete elliptic integrals for the imaginary parts, and a one-dimensional integral with an integ

The reading

Two-loop Feynman diagrams with several different internal masses are the bottleneck for the next generation of electroweak precision predictions, and for arbitrary masses nobody expects a closed answer in terms of ordinary polylogarithms. This paper attacks the two simplest such diagrams — the three-propagator 'London transport' diagram $T_{123}$ and the four-propagator diagram $T_{1234}$ — and offers three separate handles on them. First, exact results in $D$ dimensions as multivariable hypergeometric series, with explicit series that converge for small and for large external momentum $p^2$. Second, the imaginary parts in four dimensions written as complete elliptic integrals of the first, second and third kind, valid in all parameter regions. Third, one-dimensional integral representations whose integrands are only elementary functions, obtained by inserting a one-loop self-energy into a one-loop diagram and using a dispersion relation. The payoff a sympathetic reader should care about is practical: a class of integrals that previously needed two-dimensional numerical integration, or a fresh expansion per mass case, now has a fast and checkable route, and the paper's numerical table shows the three routes agreeing.

What carries the argument

Three pieces of machinery carry the argument. The dispersion relation converts a two-loop diagram into a one-dimensional integral over the discontinuity of a one-loop self-energy, $B_0$, contracted with a second one-loop function; partial fractioning in the subloop mass then reduces the two-loop object to a difference of simpler diagrams (Eq. 65). The Mellin–Barnes representation of a massive propagator, $1/(k^2-m^2)^α = (1/Γ(α))(1/2πi)∫ ds (-m^2)^s (k^2)^{-α-s} Γ(-s)Γ(α+s)$, relates the massive diagram to the known massless one with raised propagator powers, which is how the large-$p^2$ series and the partial differential equations (Eqs. 29–32) in the variables $x_i = m_i^2/p^2$ are generated. Finally, reduction of the cut integrals to Legendre normal form by partial fractions and partial integration turns the discontinuities into complete elliptic integrals, with the third-kind integral $Π(c,κ)$ continued to characteristic $c>1$ so that the two-particle and three-particle cuts cancel into a purely imaginary discontinuity.

Extended reading notes

Core claim

The central claim is that the massive two-loop self-energy diagrams $T_{123}$ and $T_{1234}$ are computable in closed or semi-closed form without any restriction to a special mass hierarchy. Concretely: (i) $T_{1234}$ is given in arbitrary $D$ as a small-$p^2$ multiple series (Eq. 17) plus a product of one-loop self-energies, and as a large-$p^2$ representation (Eq. 24) built from a vacuum-diagram factor times four-variable series $B_1$–$B_6$; (ii) the imaginary parts of $T_{123}$ and $T_{1234}$ in four dimensions reduce to complete elliptic integrals $K$, $E$ and $Π$ with explicit arguments in terms of the Källén variables, Eqs. 53 and 60, valid in all parameter regions; (iii) every two-loop diagram containing a self-energy subloop admits a one-dimensional representation with an elementary integrand, Eq. 68, specialised to Eqs. 70 and 73 for the two diagrams at hand. The authors also state plainly that the five-propagator master diagram is not covered: it is not of self-energy-insertion type, the massless case with arbitrary propagator powers is unavailable, and the Mellin–Barnes route is therefore blocked for it.

Load-bearing premise

The claim that the elliptic-integral formula for the three-particle cut holds in every parameter region depends on continuing the third-kind elliptic integral to characteristic $c>1$; the paper writes down only the imaginary jump of that continuation and defers the real part to a cited monograph, so the cancellation that makes the total discontinuity purely imaginary is asserted rather than demonstrated here.

Editorial extensions

If this is right

Any electroweak two-loop self-energy with a self-energy insertion can be evaluated numerically from a single integral over elementary functions, replacing two-dimensional Monte Carlo integration with a fast deterministic quadrature.
The imaginary parts of these diagrams can be computed from standard elliptic-integral algorithms in all parameter regions, so thresholds and particle-production cut contributions become cheap to evaluate.
The arbitrary-$D$ hypergeometric results unify the many previously published massless and equal-mass special cases into one expression, since setting masses to zero or equal reduces the series to known ${}_3F_2$ and ${}_2F_1$ forms.
The large-$p^2$ series is organised by subgraphs — the diagram itself, one-line removals, two-line removals, and the vacuum diagram — giving a direct correspondence between each series and a region of the loop-momentum integration.
The numerical table comparing two-dimensional integration, the hypergeometric series, a Taylor expansion and the one-dimensional representation provides cross-validation that any follow-up calculation can reuse.

Reading between the lines

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

The one-dimensional representation is the piece most likely to outlive the paper: because the integrand is elementary, it can be pushed to higher precision and to complex $p^2$ (timelike and spacelike) without re-deriving series, which is what a full electroweak two-loop calculation would actually need.
The elliptic-integral results suggest a structural statement the authors do not make explicitly: for these topologies the imaginary part lives in a two-dimensional space of elliptic periods, so any threshold expansion should be expressible as a sum over those periods rather than as a generic hypergeometric series.
Since the master diagram is excluded precisely because it lacks a self-energy subloop, a natural extension is to look for a topology where a two-loop insertion plays the role that $B_0$ plays here — the same dispersion-plus-partial-fraction trick would then apply one level up.
The agreement in Table 1 between four independent methods is a template: any new method for massive two-loop diagrams can be benchmarked against these numbers, which makes the paper useful as a test set even to readers uninterested in the analytic forms.

Original abstract

Motivated by the precision results in the electroweak theory studies of two-loopFeynman diagrams are performed. Specifically this paper gives a contribution to the knowledge of massive two-loop self-energy diagrams in arbitrary and especially four dimensions.This is done in three respects firstly results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta. Secondly the imaginary parts of these integrals are expressed as complete elliptic integrals.Finally one-dimensional integral representations with elementary functions are derived.They are very well suited for the numerical evaluations.

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, 3,030 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Motivated by the precision results in the electroweak theory studies of two-loopFeynman diagrams are performed. Specifically this paper gives a contribution to the knowledge of massive two-loop self-energy diagrams in arbitrary and especially four dimensions.This is done in three respects firstly results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta. Secondly the imaginary parts of these integrals are expressed as complete elliptic integrals.Finally one-dimensional integral representations with elementary functions are derived.They are very well suited for the numerical evaluations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Motivated by the precision results in the electroweak theory studies of two-loopFeynman diagrams are performed.	background 0.80	0.65	0.05	asserted 0.79
s1	Specifically this paper gives a contribution to the knowledge of massive two-loop self-energy diagrams in arbitrary and especially four dimensions.This is done in three respects firstly results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta.	contribution 0.99	0.92	0.29	asserted 1.00
s2	Secondly the imaginary parts of these integrals are expressed as complete elliptic integrals.Finally one-dimensional integral representations with elementary functions are derived.They are very well suited for the numerical evaluations.	result 0.48	0.95	0.19	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

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

Figures

**Figure 1.** Topologies In order to classify our conventions we give the explicit form of the so-called master diagram: T12345(p 2 ; m2 1 , m2 2 , m2 3 , m2 4 , m2 5 ) =<< 1 k 2 1 − m2 1 . . . 1 k 2 5 − m2 5 >> (1) where each bracket denotes a D dimensional integration over one loop momentum q: < . . . >= Z d Dq iπ2(2πµ)D−4 (. . .), (2) where µ is an arbitrary mass. The momenta of the propagators k1, . . . , k5 are determined by momentum conservation in terms of the external momentum p and the integration momenta q1 and q2. It can be immediately seen that the diagram T11234(p 2 ; m2 1 , m′2 1 ,…

Figures reproduced from arXiv:hep-ph/9409388 by the authors.

References

Works cited (17)

[3]

F. A. Berends, M. Buza, M. B¨ ohm and R. Scharf, Z. Phys. C 63 (1994) 227

View on Pith 1994
[6]

F. A. Berends and J. B. Tausk, Nucl. Phys. B 421 (1994) 456

View on Pith 1994
[7]

R. Scharf and J. B. Tausk, Nucl. Phys. B 412 (1994) 423

View on Pith 1994
[14]

F. Tricomi, M. Kraﬀt, Elliptische Funktionen , Akademische Verlagsgesellschaft, Leipzig (1948)

View on Pith 1948
[1]

D. Kreimer, Phys. Lett. B 273 (1991) 277

View on Pith 1991
[2]

A. Czarnecki, U. Kilian and D. Kreimer, Mainz preprint MZ-TH/94-13

View on Pith
[4]

A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series , Gordon and Breach Science Publishers, New York (1986)

View on Pith 1986
[5]

H. Exton, Multiple Hypergeometric Functions and Applications , Ellis Horwood Limited, Chicester (1976)

View on Pith 1976
[8]

A. I. Davydychev, V. A. Smirnov and J. B. Tausk, Nucl. Phys. B 410 (1993) 325

View on Pith 1993
[9]

A. I. Davydychev and J. B. Tausk, Nucl. Phys. B 397 (1993) 123

View on Pith 1993
[10]

P. Appell, J. Kamp´ e de F´ eriet,Fonctions Hyperg´ eom´ etriques, Gauthier-Villars, Paris (1926)

View on Pith 1926
[11]

H. M. Srivastava, P. W. Karlson, Multiple Gaussian Hypergeometric Series , Halstead Press, John Wiley, New York (1985)

View on Pith 1985
[12]

A. Erd´ elyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions , vol. II, McGraw-Hill, New York (1953)

View on Pith 1953
[13]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions , Dover Publications, New York (1972)

View on Pith 1972
[15]

J. B. Tausk, Top quark eﬀects on multijet production and electroweak par ameters, Leiden (1993)

View on Pith 1993
[16]

D. J. Broadhurst, Z.Phys., C 47 (1990) 115

View on Pith 1990
[17]

G. K¨ all´ en, A. Sabry,Dan. Met. Fys. Medd. , 29 (1955) no. 17 24

View on Pith 1955

Forward citations

Cited by 37 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Higgs Boson Production at Hadron Colliders at N3LO in QCD
hep-ph 2018-02 accept novelty 8.0 of 10

The paper computes the exact next-to-next-to-next-to-leading-order QCD correction to Higgs boson production via gluon fusion, the first exact hadron collider cross section at this order and the first to require ellipt...
Fermionic Electroweak Two-Loop Corrections to Drell-Yan and Related Processes
hep-ph 2025-12 conditional novelty 7.0 of 10

The full NNLO electroweak corrections with closed fermion loops to e+e- → ffbar (and Drell-Yan) were calculated semi-numerically and found to modify NLO cross-sections by about 1%.
Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics
hep-ph 2019-08 conditional novelty 7.0 of 10

The thesis computes three-loop and two-loop QCD corrections to Higgs processes with full quark mass dependence, including a new analytic series-expansion treatment of elliptic two-loop master integrals for Higgs-plus-...
An analytic solution for the equal-mass banana graph
hep-th 2019-07 conditional novelty 7.0 of 10

All three master integrals of the equal-mass three-loop banana graph in two dimensions are expressed as linear combinations of iterated integrals of modular forms for the congruence subgroup Gamma1(6), with a numerica...
The unequal mass sunrise integral expressed through iterated integrals on $\overline{\mathcal M}_{1,3}$
hep-th 2019-07 conditional novelty 7.0 of 10

The unequal-mass two-loop sunrise integral is solved to all orders in the dimensional regularization parameter as iterated integrals on the moduli space of a genus-one curve with three marked points.

Show 32 more citing papers

Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
hep-ph 2018-06 accept novelty 7.0 of 10

The planar double box integral for top-pair production with a closed top loop is solved analytically to all orders in epsilon, revealing three distinct elliptic curves and yielding explicit iterated-integral results t...
The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
hep-ph 2018-04 accept novelty 7.0 of 10

The planar double box integral for top pair production with a closed top loop is computed analytically to all orders in the dimensional regularisation parameter, via a differential-equation system made linear in epsil...
Complete electroweak two-loop corrections to Z boson production and decay
hep-ph 2018-04 accept novelty 7.0 of 10

The authors compute the bosonic two-loop electroweak corrections to all Z-boson partial widths, branching ratios, and the hadronic peak cross-section, completing the Standard Model two-loop description of Z-pole observables.
Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
hep-th 2018-03 conditional novelty 7.0 of 10

A symbol and coaction calculus for elliptic polylogarithms is constructed and used to show that the equal-mass two-loop sunrise integral in two dimensions equals a short sum of iterated integrals of weight-three Eisen...
The kite integral to all orders in terms of elliptic polylogarithms
hep-ph 2016-07 accept novelty 7.0 of 10

The two-loop kite Feynman integral's Laurent expansion is expressed at every order in epsilon in terms of elliptic generalisations of polylogarithms, via an iterative differential-equation method.
High precision black hole scattering
hep-th 2026-10 conditional novelty 6.0 of 10

A dissertation presenting the first analytic two-black-hole scattering observables at fifth Post-Minkowskian order, whose integral solutions contain Calabi-Yau threefold periods.
The ice cone family and iterated integrals for Calabi-Yau varieties
hep-th 2022-12 conditional novelty 6.0 of 10

The ice cone Feynman integrals in two dimensions are solved as iterated integrals of Calabi-Yau periods, and the periods are rewritten as iterated integrals of geometric invariants whose quadratic relations become shu...
Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
hep-th 2021-08 conditional novelty 6.0 of 10

Griffiths transversality yields quadratic relations among maximal cuts that reduce l-loop banana integrals in two dimensions to an integral over a Calabi-Yau period, with an extension to dimensional regularization.
Algorithms and tools for iterated Eisenstein integrals
hep-th 2019-11 conditional novelty 6.0 of 10

Modular transformations of iterated Eisenstein integrals are turned into a working algorithm for analytic continuation and fast numerical evaluation, demonstrated on an elliptic hypergeometric family and on the two-lo...
Elliptic Feynman integrals and pure functions
hep-th 2018-09 conditional novelty 6.0 of 10

A new basis of elliptic polylogarithms makes several two-loop elliptic Feynman integrals evaluate to pure, uniform-weight combinations, extending a key structural principle beyond ordinary polylogarithms.
The $\varepsilon$-form of the differential equations for Feynman integrals in the elliptic case
hep-ph 2018-02 accept novelty 6.0 of 10

The kite family's differential equations admit an ε-form via a transcendental basis change dividing by an elliptic period, with all kernels becoming modular forms of Γ1(6).
Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
hep-ph 2017-12 conditional novelty 6.0 of 10

The authors express all master integrals of the two-loop sunrise Feynman diagram, including the previously uncomputed unequal-mass second master, as iterated integrals on an elliptic curve.
Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
hep-th 2017-12 accept novelty 6.0 of 10

A root-valued kernel basis with at most simple poles is shown to span the same space as multiple elliptic polylogarithms, giving physicists a practical handle on genus-one Feynman integrals.
An Elliptic Generalization of Multiple Polylogarithms
hep-ph 2017-09 conditional novelty 6.0 of 10

A new class of elliptic polylogarithms is defined, given a weight that a differential operator lowers, and used to write the imaginary part of the two-loop massive sunrise graph compactly through order epsilon squared.
Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
hep-ph 2017-05 conditional novelty 6.0 of 10

A monodromy-corrected definition of the two elliptic periods makes the ELi-function q-series for kite-family Feynman integrals valid for all real t, with |q|<1 away from the singular points.
Feynman integrals and iterated integrals of modular forms
hep-ph 2017-04 conditional novelty 6.0 of 10

The equal-mass sunrise and kite Feynman integrals are expressed to all orders in epsilon as iterated integrals of modular forms, with an explicit alphabet and a closed all-orders formula for the sunrise.
A non-planar two-loop three-point function beyond multiple polylogarithms
hep-ph 2017-01 conditional novelty 6.0 of 10

A two-loop non-planar three-point Feynman integral with a massive top quark is evaluated analytically in terms of elliptic integrals and one-fold integral representations, extending the class of calculable diagrams be...
The two-loop electroweak bosonic corrections to $\sin^2\theta_{\rm eff}^{\rm b}$
hep-ph 2016-07 accept novelty 6.0 of 10

The bosonic two-loop electroweak correction to the Z b bbar vertex is computed for the first time, completing the two-loop prediction of sin²θ_eff^b with a relative correction of -0.9855 × 10⁻⁴.
The iterated structure of the all-order result for the two-loop sunrise integral
hep-ph 2015-12 conditional novelty 6.0 of 10

The two-loop sunrise integral with equal masses can be expanded to arbitrary order in epsilon using a new class of elliptic multiple polylogarithms, with explicit results given through order epsilon squared.
The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case
hep-ph 2015-04 conditional novelty 6.0 of 10

The O(ε) term of the two-loop sunrise integral near two dimensions is expressed in elliptic generalisations of Clausen and Glaisher functions, including a new depth-two object, and used to obtain the four-dimensional ...
The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
hep-ph 2014-05 conditional novelty 6.0 of 10

The two-loop sunrise integral with arbitrary masses in two dimensions equals a period times a sum of three elliptic dilogarithms whose arguments are images of the integration region's corner points.
The two-loop sunrise graph with arbitrary masses
hep-ph 2013-02 conditional novelty 6.0 of 10

The two-loop sunrise integral with arbitrary masses in two dimensions is solved analytically as an iterated integral whose homogeneous solutions are periods of an elliptic curve.
TSIL: a program for the calculation of two-loop self-energy integrals
hep-ph 2005-01 accept novelty 6.0 of 10

The paper documents TSIL, a C library that computes all two-loop self-energy basis integrals for arbitrary masses and momentum, by analytic polylogarithms where known and by Runge-Kutta integration of their differenti...
Simple one-dimensional integral representations for two-loop self-energies: the master diagram
hep-ph 1995-01 accept novelty 6.0 of 10

A one-dimensional integral representation using only elementary functions is derived for the two-loop scalar self-energy master diagram with arbitrary masses, enabling fast numerical evaluation.
On a procedure to derive $\epsilon$-factorised differential equations beyond polylogarithms
hep-th 2023-05 conditional novelty 5.0 of 10

A five-step procedure produces epsilon-factorised differential equations for Feynman integrals on elliptic and Calabi-Yau geometries, demonstrated on the sunrise, several triangles, a double box, the ice cone, and the...
TVID 2: Evaluation of planar-type three-loop self-energy integrals with arbitrary masses
hep-ph 2019-08 conditional novelty 5.0 of 10

TVID 2 computes all master integrals for planar three-loop self-energy diagrams with two closed fermion loops using at most two-dimensional numerical integrations.
Elliptic polylogarithms and Feynman parameter integrals
hep-ph 2019-02 conditional novelty 5.0 of 10

Three families of two-loop Feynman integrals are evaluated in closed form as elliptic polylogarithms, and each is shown to admit a basis of pure master integrals.
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
hep-ph 2017-04 accept novelty 5.0 of 10

All independent homogeneous solutions of the third-order differential equation for the three-loop massive banana graph are obtained by evaluating its maximal cut on three independent contours, and expressed as product...
Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral
hep-ph 2016-02 conditional novelty 5.0 of 10

Combining dispersion relations with the differential-equations method gives explicit one-fold-integral representations for the two-loop massive sunrise and the QED kite graph up to first order in (d-4).
Feynman integrals, toric geometry and mirror symmetry
hep-th 2018-07 conditional novelty 3.0 of 10

The two-loop sunset Feynman integral is the Legendre transform of the local Gromov-Witten prepotential of the degree-6 del Pezzo surface, and its elliptic-dilogarithm and trilogarithm forms are related by a local mirror map.
Functions Beyond Multiple Polylogarithms for Precision Collider Physics
hep-ph 2022-03 accept novelty 2.0 of 10

A community white paper maps which Feynman integrals escape multiple polylogarithms, which elliptic and Calabi-Yau geometries they encode, and what tooling is still missing to evaluate them for collider predictions.
The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
hep-th 2022-03 accept

The chapter organizes modern methods for multiloop Feynman integrals, from integration-by-parts reduction and canonical differential equations to iterated integrals and twisted cohomology.

How the literature uses it

51 arXiv papers cite this in 57 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 · 25 sentences without a confident label

Uses its method or tool 7 papers

The two-loop self-energy integrals needed for the renormalization procedure and the vertex integrals with self-energy sub-loops have been computed using the dispersion relation technique described in Refs. [ref] .

Complete electroweak two-loop corrections to Z boson production and decay · Calculation of two-loop vertex corrections · 1.00
We evaluate these two-loop integrals using the numerical dispersion integral technique [ref] (see also Refs. [ref] ).

Parity-Violating M{\o}ller Scattering at NNLO: Closed Fermion Loops · Method · 1.00
Two-loop self-energy integrals and vertex integrals with self-energy subloops have been computed using the dispersion relation techniques described in Refs. [ref] .

The two-loop electroweak bosonic corrections to $\sin^2\theta_{\rm eff}^{\rm b}$ · Strictly numerical two-loop integration techniques · 0.99

Uses its data or values 1 paper

Also all $$-divergent parts and a few finite parts of the two-loop integrals appearing in the calculation are known from Refs. [ref] .

Evaluation of multi-loop multi-scale integrals and phenomenological two-loop applications · Analytically known integrals · 0.85

Cites it as background 23 papers

These integrals have long been known to involve elliptic curves [ref] .

Cuts and Isogenies · Introduction · 0.94
It is now time to introduce the two-loop sunset integral [ref] .

A walk on sunset boulevard · The two-loop sunset integral · 0.98
Many procedures for a precise evaluation of the Master Integrals (MI's) can be found in the literature [ref] .

BOKASUN: a fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams · Introduction · 0.98

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [3]

F. A. Berends, M. Buza, M. B¨ ohm and R. Scharf, Z. Phys. C 63 (1994) 227

View on Pith 1994

[2] [6]

F. A. Berends and J. B. Tausk, Nucl. Phys. B 421 (1994) 456

View on Pith 1994

[3] [7]

R. Scharf and J. B. Tausk, Nucl. Phys. B 412 (1994) 423

View on Pith 1994

[4] [14]

F. Tricomi, M. Kraﬀt, Elliptische Funktionen , Akademische Verlagsgesellschaft, Leipzig (1948)

View on Pith 1948

[5] [1]

D. Kreimer, Phys. Lett. B 273 (1991) 277

View on Pith 1991

[6] [2]

A. Czarnecki, U. Kilian and D. Kreimer, Mainz preprint MZ-TH/94-13

View on Pith

[7] [4]

A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series , Gordon and Breach Science Publishers, New York (1986)

View on Pith 1986

[8] [5]

H. Exton, Multiple Hypergeometric Functions and Applications , Ellis Horwood Limited, Chicester (1976)

View on Pith 1976

[9] [8]

A. I. Davydychev, V. A. Smirnov and J. B. Tausk, Nucl. Phys. B 410 (1993) 325

View on Pith 1993

[10] [9]

A. I. Davydychev and J. B. Tausk, Nucl. Phys. B 397 (1993) 123

View on Pith 1993

[11] [10]

P. Appell, J. Kamp´ e de F´ eriet,Fonctions Hyperg´ eom´ etriques, Gauthier-Villars, Paris (1926)

View on Pith 1926

[12] [11]

H. M. Srivastava, P. W. Karlson, Multiple Gaussian Hypergeometric Series , Halstead Press, John Wiley, New York (1985)

View on Pith 1985

[13] [12]

A. Erd´ elyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions , vol. II, McGraw-Hill, New York (1953)

View on Pith 1953

[14] [13]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions , Dover Publications, New York (1972)

View on Pith 1972

[15] [15]

J. B. Tausk, Top quark eﬀects on multijet production and electroweak par ameters, Leiden (1993)

View on Pith 1993

[16] [16]

D. J. Broadhurst, Z.Phys., C 47 (1990) 115

View on Pith 1990

[17] [17]

G. K¨ all´ en, A. Sabry,Dan. Met. Fys. Medd. , 29 (1955) no. 17 24

View on Pith 1955