Two-Loop Quark and Gluon Form Factors in Dimensional Regularisation

D. Maitre; T. Gehrmann; T. Huber

REVIEW

Two-Loop Quark and Gluon Form Factors in Dimensional Regularisation

T. Gehrmann, T. Huber, D. Maitre

High Energy Physics - Phenomenology · Submitted 5 Jul 2005 · 22 references

Pith's verdict: The exact crossed-triangle integral A6 is the real deliverable; the ε² gluon form factor is new but only numerically checked, and the check is reported in one acknowledgement sentence.

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arXiv IDhep-ph/0507061 (version 3)

Pith IDZCPYU36F, Pith's permanent record of this paper and its review

CitationD. Maitre, T. Gehrmann, T. Huber. "Two-Loop Quark and Gluon Form Factors in Dimensional Regularisation." arXiv:hep-ph/0507061 (2005). https://pith.science/paper/ZCPYU36F

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Editor's note

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The thing worth your time here is Eq. 5: an exact closed form for the two-loop crossed-triangle master integral A6 in terms of 3F2 and 4F3 functions of unit argument. Before this, A6 was known only as a Laurent series down to the finite term [14]. That is a genuine, reusable object, and it is what makes the rest of the paper possible. The ε² expansions of the quark and gluon form factors (Eqs. 17, 18) follow from it via HypExp, and they are what the three-loop infrared factorization program actually needs. The reduction chain is transparent and exact in ε, and the note added handles the simultaneous MVV paper [7] openly rather than burying it. Credit where due: this is clean, standard multi-loop work done properly, and the quark form factor to ε² is explicitly confirmed against MVV's (B.1). That cross-check is real and it is the strongest evidence in the paper for the A6 expansion being right at higher orders, since the quark coefficients carry A6 too, just with different weights from the gluon ones. So the stress-test's framing is a bit overstated: the A6 expansion is not unanchored, it is anchored through the quark channel. What is genuinely unanchored externally is the gluon-specific reduction in Eq. 11 combined with A6 at ε¹ and ε². The ε⁰ term of Eq. 18 is at least implicitly pinned by Harlander [4], so the gap is narrower than 'no anchor at all'. Still, the only stated validation of the higher orders is one acknowledgement sentence about G. Heinrich's sector-decomposition checks, with no numbers, no error bars, no table. In a letter that is forgivable on space grounds, but a small table would have cost half a page and closed the question. Minor point: HypExp is the authors' own package, but it is an expansion algorithm, not a fitted input, so I do not weight that heavily. Verdict: solid enabling paper, worth citing if you work on three-loop amplitudes or IR structure, and worth a serious referee. I would ask for the numerical comparison numbers, not for a rederivation.

Referee report

Significance

If correct, this is a valuable and durable result. The exact expression for A6 (Eq. 5) closes the last gap in the set of massless two-loop vertex master integrals and turns the two-loop form factors into objects known to arbitrary order in ε, limited only by the order to which one chooses to expand. The explicit ε² coefficients of Eqs. (17)–(18) are exactly the input needed for the infrared factorisation of the three-loop form factors, and the paper makes a genuine, checkable prediction there. Several features give independent confidence: the ε⁰ term of Eq. (6) reproduces the finite part known from [14]; the ε⁰ limits of Eqs. (17)–(18) should reproduce the classic results of [3,4]; and the coefficients of Eq. (6) are uniformly of maximal transcendental weight at each order, which is a nontrivial structural consistency check on the HypExp output. The quark form factor is explicitly confirmed against the independent published result (B.1) of [7], and the overlap with that paper is disclosed transparently in the note added. The paper also demonstrates a reusable computational tool (HypExp) on a real multi-loop problem, which is a strength in its own right.

Major objections (1)

[Eq. (18) and Eq. (6)] The gluon form factor F_g^(2) through ε² in Eq. (18) is the one headline result that carries no external cross-check in the manuscript. The quark case is anchored by the explicit statement in the note added that Eq. (10) expanded to ε² confirms Eq. (B.1) of [7], but no analogous anchor is offered for the gluon case, whose ε¹ and ε² coefficients depend on the ε-expansion of A6 (Eq. 6) together with the gluon-specific reduction coefficients of Eq. (11). The only stated validation of the A6 expansion is the acknowledgement that G. Heinrich performed independent numerical checks by sector decomposition, with no numbers, no error estimates, and no comparison table. Since the ε¹–ε⁴ terms of Eq. (6) are new and are the sole new mathematical input on which Eq. (18) rests, the reader has no quantitative way to judge the accuracy of the highest-order coefficients. I would ask the authors to either (a) report the numerical check quantitatively for at least the ε¹ and ε² coefficients of A6 and of F_g^(2), or (b) state explicitly that the ε⁰ limit of Eq. (18) reproduces the published result of [4] and, if possible, that the ε⁰ limit of Eq. (6) reproduces [14], so that the chain has at least two documented anchor points. This is a documentation gap rather than an internal inconsistency: the reduction is exact in ε and the derivation chain is transparent, so the request is proportionate and local.

Minor objections (8)

[§2, Eq. (6)] The text says 'we obtain the eighth-order expansion' but the displayed series runs from ε⁻⁴ to ε⁴, which is nine terms. Please state the counting convention explicitly (for example, 'through order ε⁴') to avoid ambiguity.
[§2, Eq. (6)] The ε⁴ coefficient contains the multiple zeta value ζ_{5,3}. Please give its definition (or a reference to the convention used), since conventions for multiple zeta values differ in the literature and the coefficient cannot otherwise be checked.
[§2, Eq. (5)] The individual terms of Eq. (5) carry explicit 1/ε⁴ and 1/ε² prefactors whose poles must cancel in the sum. It would be helpful to state that this cancellation has been verified, since it is a strong and cheap consistency check on the relative normalisations of the 3F2 and 4F3 terms.
[§2, master integrals] The master integrals are labelled A2,LO, A3, A4 and A6, with no A5 appearing anywhere. Please clarify whether A5 is absent because it is not needed for these form factors, because it is a subtopology of one of the listed integrals, or because of the labelling inherited from [14].
[§3, Eqs. (10)–(11)] The reduction coefficients are given in d dimensions but no statement is made about how they were checked. It would strengthen the paper to note that in the ε→0 limit Eqs. (10)–(11) reproduce the known two-loop results of [3,4], which would independently validate the reduction coefficients separately from the master integrals.
[§4, Eq. (13)] The definition Δ(q²)=(−sgn(q²)−i0)^(−ε) deserves one clarifying sentence on the sign convention for space-like versus time-like q², since the subsequent reality properties of the form factors depend on it and the notation −sgn(q²) is easy to misread.
[§4, Eq. (12)] The renormalisation constant is displayed only through O(α_s), which is sufficient here but should be stated explicitly so that the reader does not look for the two-loop coefficient.
[§1] There is a typographical error in the phrase 'the corresponding three-loop from factors' in the final sentence of the introduction; it should read 'form factors'.

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The paper

The paper claims

The two-loop quark and gluon form factors are obtained in closed form to all orders in the dimensional-regularisation parameter, after the last missing two-loop master integral is expressed through generalised hypergeometric functions of单位.

The reading

In dimensional regularisation the dimension of space-time is continued away from four, $d=4-2epsilon$, and every loop integral becomes a Laurent series in $epsilon$. Physical predictions need only the first few terms of that series, but higher-order terms are exactly what infrared factorisation formulas consume: the pole operators at three loops act on the $epsilon^2$ pieces of the two-loop form factors. The obstacle was that one of the three two-loop vertex master integrals, the crossed triangle $A_6$, had no exact expression — only a series known up to the finite term. The authors give $A_6$ in closed form as $Gamma$-functions plus generalised hypergeometric functions of unit argument, expand it with a purpose-built package, and then write out the one-loop form factors to $epsilon^4$ and the two-loop quark and gluon form factors to $epsilon^2$. If the expansions are right, the two-loop form factors are no longer limited by integral evaluation at all: any order in $epsilon$ can be produced on demand.

What carries the argument

The load-bearing object is the exact expression for $A_6$ in Eq. (5): a sum of four terms built from $Gamma$-functions of $epsilon$-dependent arguments times ${}_3F_2(1,-4epsilon,-2epsilon;1-3epsilon,1-2epsilon;1)$, ${}_3F_2(1,1,1+2epsilon;2+epsilon,2+2epsilon;1)$ and ${}_4F_3(1,1-epsilon,-4epsilon,-2epsilon;1-3epsilon,1-2epsilon,1-2epsilon;1)$. Generalised hypergeometric functions of unit argument have parameters that here carry the regulator $epsilon$, so expanding in $epsilon$ means expanding in the parameters of the function, not just in its argument; that is what the HypExp package does, and it is the mechanism that converts a closed-form integral into a Laurent series to any desired order. The second piece of machinery is the reduction of the full two-loop form factors onto the master integrals $A_{2,LO}$, $A_3$, $A_4$ and $A_6$ using integration-by-parts and Lorentz-invariance identities, which is carried out exactly in $d$ so that no accuracy is lost before the final expansion.

Extended reading notes

Core claim

The paper's central claim is that the two-loop crossed-triangle integral $A_6 = int frac{d^dk}{(2pi)^d}frac{d^dl}{(2pi)^d} frac{1}{k^2 l^2 (k-p_1-p_2)^2 (k-l)^2 (k-l-p_2)^2 (l-p_1)^2}$ admits an exact expression in $d$ dimensions as a combination of $Gamma$-functions and a ${}_3F_2$ and a ${}_4F_3$ of unit argument (Eq. 5). With this, every master integral entering the massless quark form factor $gamma^* to qbar q$ and the gluon form factor $H to gg$ is known to all orders in $epsilon$, so the form factors themselves are known to all orders once the reduction coefficients are inserted. The authors give the eighth-order Laurent expansion of $A_6$ (Eq. 6), the renormalised one-loop form factors through $epsilon^4$ (Eqs. 15 and 16), and the renormalised two-loop form factors through $epsilon^2$ (Eqs. 17 and 18), the latter being the order required for the ultraviolet renormalisation and infrared factorisation of the corresponding three-loop quantities. A note added at the end states that expanding the unrenormalised two-loop quark form factor to $epsilon^2$ reproduces Eq. (B.1) of the concurrent work by Moch, Vermaseren and Vogt.

Load-bearing premise

Everything rests on the correctness of the HypExp-generated Laurent expansions of the ${}_3F_2$ and ${}_4F_3$ functions in Eq. (5), and the only validation stated for those expansions is an acknowledgement of independent numerical checks by sector decomposition, with no numbers, error estimates or comparison tables shown.

Editorial extensions

If this is right

The $epsilon^2$ coefficients of the two-loop quark and gluon form factors supply exactly the input that the three-loop infrared singularity operators project onto, so the previously unknown hard radiation constants in the three-loop infrared factorisation formula for these form factors become fixable.
Because the two-loop vertex master integrals reappear as subtopologies in three-loop reductions whenever a loop is pinched off, the $epsilon^2$ terms given here can be inserted directly into three-loop form-factor calculations.
Any other integral that reduces to $A_6$ can now be expanded to arbitrary order in $epsilon$ rather than only to the finite term, since the closed form plus HypExp removes the order-by-order bottleneck.
The paper argues the same technique should apply to three- and four-loop two-point functions and to multi-particle phase-space integrals, which are cuts of such two-point functions, for fully inclusive observables.

Reading between the lines

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

The methodological bet worth watching is that exact-in-$d$ master integrals plus a parameter-expansion package beat order-by-order numerical sector decomposition once high $epsilon$ orders are needed; if that holds generally, the bottleneck in multi-loop work shifts from evaluating integrals to reducing diagrams onto them.
The quark half of the result is cross-checked against an independent published expansion, but the gluon half is not: the paper offers no analogous external comparison for Eq. (18), so the gluon $epsilon^2$ coefficients are the more exposed part of the claim.
The appearance of the multiple zeta value $zeta_{5,3}$ already at order $epsilon^4$ in $A_6$ hints that pushing to higher orders will require a working calculus of multiple zeta values, which is a real constraint on any 'expand to any order' promise.
Since the form factors are now known as exact functions of $d$, one could ask whether the $epsilon to 0$ limit of the product of two-loop form factors and the infrared operators exponentiates in a way that is visible before expansion, which would turn the order-by-order matching into a structural statement rather than a coefficient hunt.

Original abstract

We compute the two-loop corrections to the massless quark form factor $\gamma^* \to q\bar q$ and gluon form factor $H\to gg$ to all orders in the dimensional regularisation parameter $\epsilon=(4-d)/2$. The two-loop contributions to the form factors are reduced to linear combinations of master integrals, which are computed in a closed form, expressed as $\Gamma$-functions and generalised hypergeometric functions of unit argument. Using the newly developed HypExp-package, these can be expanded to any desired order, yielding Laurent expansions in $\epsilon$. We provide expansions of the form factors to order $\epsilon^2$, as required for ultraviolet renormalisation and infrared factorisation of the three-loop form factors.

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, 3,723 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.36 · compares with prior results 0.05 · mentions code or data availability 0.23.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We compute the two-loop corrections to the massless quark form factor $\gamma^* \to q\bar q$ and gluon form factor $H\to gg$ to all orders in the dimensional regularisation parameter $\epsilon=(4-d)/2$. The two-loop contributions to the form factors are reduced to linear combinations of master integrals, which are computed in a closed form, expressed as $\Gamma$-functions and generalised hypergeometric functions of unit argument. Using the newly developed HypExp-package, these can be expanded to any desired order, yielding Laurent expansions in $\epsilon$. We provide expansions of the form factors to order $\epsilon^2$, as required for ultraviolet renormalisation and infrared factorisation of the three-loop form factors.

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	We compute the two-loop corrections to the massless quark form factor $\gamma^* \to q\bar q$ and gluon form factor $H\to gg$ to all orders in the dimensional regularisation parameter $\epsilon=(4-d)/2$.	contribution 1.00	0.96	0.41	asserted 1.00
s1	The two-loop contributions to the form factors are reduced to linear combinations of master integrals, which are computed in a closed form, expressed as $\Gamma$-functions and generalised hypergeometric functions of unit argument.	method 0.76	0.94	0.22	asserted 1.00
s2	Using the newly developed HypExp-package, these can be expanded to any desired order, yielding Laurent expansions in $\epsilon$.	method 0.99	0.73	0.07	asserted 0.96
s3	We provide expansions of the form factors to order $\epsilon^2$, as required for ultraviolet renormalisation and infrared factorisation of the three-loop form factors.	result 0.73	0.95	0.34	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.88.

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sources read

arXiv listing abstract v3

not examined

full text
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References

Works cited (22)

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The C-parameter distribution in electron-positron annihilation is computed with N3LL-prime resummation plus a renormalon-free power correction, reaching about 2.5 percent perturbative uncertainty at Q = mZ.
Cutting through form factors and cross sections of non-protected operators in N=4 SYM
hep-th 2014-10 accept novelty 6.0 of 10

A correction rule lets on-shell unitarity compute Konishi form factors in D=4-2epsilon, yielding two-loop two-point and one-loop three-point results and an infrared-finite decay rate whose logarithm gives the known tw...
Analytic two-loop form factors in N=4 SYM
hep-th 2012-01 accept novelty 6.0 of 10

The two-loop three-point MHV form factor in N=4 SYM is computed and reduced to a compact classical-polylogarithm remainder that matches the maximally transcendental part of a two-loop QCD H to ggg amplitude.
Antenna subtraction at NNLO with hadronic initial states: initial-final configurations
hep-ph 2009-12 conditional novelty 6.0 of 10

The authors derive and integrate all initial-final antenna functions needed for NNLO antenna subtraction with one hadronic initial state, and cross-check the results against known deep inelastic coefficient functions.
Virtual three-loop corrections to Higgs boson production in gluon fusion for finite top quark mass
hep-ph 2009-07 conditional novelty 6.0 of 10

Five terms of the expansion of the three-loop virtual corrections to gluon-fusion Higgs production in M_H^2/M_t^2 are computed, and the truncated series appears to converge up to near the top-antitop threshold.
Dynamical Threshold Enhancement and Resummation in Drell-Yan Production
hep-ph 2007-10 conditional novelty 6.0 of 10

The steep fall-off of parton densities at high x dynamically enhances the partonic threshold region in Drell-Yan production, producing an effective soft scale about 13 times smaller than M(1-τ) and modest but nonzero ...
HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters
hep-ph 2007-08 conditional novelty 6.0 of 10

A new reduction algorithm plus a public Mathematica package, HypExp 2, extends automated hypergeometric-function expansion from integer to half-integer parameters, with worked examples up to four loops.
Heavy-quark production in gluon fusion at two loops in QCD
hep-ph 2007-07 accept novelty 6.0 of 10

The authors derive the two-loop interference of the Born and two-loop amplitudes for gluon fusion into a heavy-quark pair, exact in the high-energy limit up to power corrections in the quark mass.
Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering
hep-ph 2006-07 accept novelty 6.0 of 10

Threshold logarithms in deep-inelastic scattering are resummed analytically in momentum space using soft-collinear effective theory, avoiding the Landau-pole ambiguities of the standard Mellin-moment approach while ag...
Three-Loop Results for Quark and Gluon Form Factors
hep-ph 2005-08 accept novelty 6.0 of 10

The authors complete the three-loop infrared pole structure of the gluon form factor and the finite fermion-loop terms of the quark form factor, then use them to bound higher-order corrections to Higgs production.
HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters
hep-ph 2005-07 accept novelty 6.0 of 10

The authors release HypExp, a Mathematica package that computes epsilon-expansions of hypergeometric functions around integer parameters to arbitrary order, plus new routines for the integrals that appear in the integ...
On the Computation of Form Factors in Massless QCD with Finite Master Integrals
hep-ph 2015-10 accept novelty 5.0 of 10

In a basis of finite master integrals, the cusp anomalous dimensions of massless QCD receive contributions only from integrals with fewer than the maximal number of propagators.
Higgs boson pair production: top quark mass effects at NLO and NNLO
hep-ph 2015-08 conditional novelty 5.0 of 10

Finite top quark mass effects change the NNLO gluon-fusion Higgs pair production cross section by at most about 5%, so the infinite-top-mass approximation is adequate at that level.
On form factors in N=4 sym
hep-th 2010-11 conditional novelty 5.0 of 10

Two-loop form factors for half-BPS operators in N=4 super Yang-Mills exponentiate their infrared divergences with a universal cusp anomalous dimension and a process-dependent collinear anomalous dimension.
NNLO vertex corrections to non-leptonic B decays: Tree amplitudes
hep-ph 2009-11 conditional novelty 5.0 of 10

Two-loop vertex corrections to the colour-allowed and colour-suppressed tree amplitudes in B to pi pi, pi rho and rho rho decays are computed analytically, and the updated predictions agree with experiment except for ...
On the Resolution of Singularities of Multiple Mellin-Barnes Integrals
hep-ph 2009-01 conditional novelty 5.0 of 10

A modified, optimization-driven version of Strategy A for resolving epsilon singularities in multifold Mellin-Barnes integrals, implemented as the public Mathematica code MBresolve.m and demonstrated on a tenfold four...
The two-loop helicity amplitudes for $q \bar q' \to V_1 V_2 \to 4~\mathrm{leptons}$
hep-ph 2015-03 accept novelty 4.0 of 10

The authors independently recompute the two-loop helicity amplitudes for off-shell vector-boson pair production, confirm a prior calculation after correcting one of its form factors, and release a fast public numerical code.
The Path forward to N$^3$LO
hep-ph 2022-03 accept novelty 3.0 of 10

N3LO QCD corrections shift key LHC cross sections by a few percent and cut scale uncertainty to the percent level, but widespread use still needs four-loop splitting functions, N3LO PDFs, and better subtraction schemes.

How later papers used it

94 later papers cite it

Evidence: substantial. 86 of the citing papers share no author with it, and 32 of those carry a confident label from Pith's citation classifier.

Pith read 11 of the 121 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

1 later paper by other authors is listed below: 1 used its data or values.

Uses its data or values 1 of 22 papers with this label

Took the 1-loop and 2-loop gluon form factors relevant for the calculation of NLO and NNLO corrections to the signal amplitude from this paper.

Signal-background interference effects in Higgs-mediated diphoton production beyond NLO (2022) · 0.80 · paraphrased: the licence does not allow quoting

How the literature uses it

93 arXiv papers cite this in 120 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 · 72 sentences without a confident label

Uses its method or tool 6 papers

The back-to-back asymptotics of the EEC can be described using Soft and Collinear Effective Theory (SCET) [ref] via the following factorization theorem [ref] [equation] In EEC_fact_thm_q, $J_0$ is the Bessel function arising from the Fourier transform due to the azymuthal symmetry of the EEC measurement, $H_qq̅$ is the quark color singlet SCET hard function, which is related to the IR finite part of the quark form factors [ref] and can be extracted up to 4 loops from the recent result of Lee:2022nhh, and $_q$ is the quark EEC jet function which is known up to N$^3$LO [ref] .

The Four-Loop Rapidity Anomalous Dimension and Event Shapes to Fourth Logarithmic Order · EEC in the back-to-back limit · 0.87
Amongst the resulting terms are integrals in $(D+2n)$ dimensions (where $n ∈ℤ$) and with raised powers of propagators. * Higher-dimensional scalar integrals are shifted back to $D$ dimensions using dimensional recurrence relations [ref] . * Using the Mathematica package LiteRed [ref] , the resulting $D$-dimensional scalar integrals are reduced to a basis of masters using integration-by-parts relations (IBPs) [ref] . * Known expressions for the four-point two-loop master integrals [ref] are inserted.

Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes · Integration & transcendental weight · 1.00
In this way the three-loop vertex integrals are reduced to two-, and three-loop vacuum integrals which are treated with MATAD [ref] , and to massless one- and two-loop vertex contributions which are reduced to master integrals (see, e.g., Ref. [ref] ) with the help of FIRE [ref] and an independent program based of the Laprota-method.

Three-loop vertex corrections · Virtual NNLO corrections to Higgs production in gluon fusion · 1.00

Uses its data or values 21 papers

Written in terms of the two-loop MIs specified in Appendix A of Ref. [ref] , the unrenormalized two-loop form factor is given by [equation] As in the one-loop case, the all-order result is obtained by replacing $B_4,2$, $B_3,1$, $C_4,1$ and $C_6,2$ with $A_2,LO^2$, $A_3$, $A_4$ and $A_6$ from Ref. [ref] , respectively.

Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics · Results at Two Loops · 0.82
The 1-loop and 2-loop gluon form factors which are relevant for the calculation of NLO and NNLO corrections to the signal amplitude have been taken from Ref. [ref] .

Signal-background interference effects in Higgs-mediated diphoton production beyond NLO · Details on the signal amplitude · 0.80
The first three non-cusp anomalous dimensions of the hard function (quark form factor) are [ref] : [equation] The first two constants of the hard function are [ref] : [equation]

Two- and Three-Loop Data for Groomed Jet Mass · Hard Function · 0.97

Compares with it 4 papers

We have evaluated them in terms of a pure combination of eMPLs of uniform weight four [ref] , in agreement with the case of massless propagators, which is known to give rise to non-elliptic functions of uniform weight four [ref] .

Elliptic Feynman integrals and pure functions · Elliptic Feynman integrals and elliptic purity · 0.84
Using the Taylor expansion of this function in Eq. [ref] we fully reproduce the leading flavor structure of full QCD up to $𝒪(α_s^3)$ [ref] , see first column of Table [ref].

Towards massless and massive event shapes in the large-$\beta_0$ limit · Hard function · 0.82
After collecting the coefficients of master integrals, we find the parameters cancel and the coefficients we obtain are consistent with the known result [ref] .

Color-Kinematics Duality for Sudakov Form Factor in Non-Supersymmetric Pure Yang-Mills Theory · Integral reduction · 0.99

Cites it as background 12 papers

The three-loop form factor master integrals were first computed numerically to $𝒪(ϵ^2)$ in reference [ref] using dimensional recurrence relations [ref] but, with the help of the celebrated PSLQ algorithm [ref] , the authors were able to recover the analytical solutions.[At the time, it was conjectured that zeta and multiple zeta values of, at most, weight eight would appear in the higher-order results required to expand the three-loop form factors through to $𝒪(ϵ^2)$.] The explicit higher-order results for the three-loop masters are, regrettably, scattered over three different articles [ref] .

On the Computation of Form Factors in Massless QCD with Finite Master Integrals · Notation and Conventions · 0.81
The two- and three-loop QCD corrections to $F_q$ and $F_g$ are available from Refs. [ref] , At four loops, only partial results have been obtained so far.

Quark and gluon form factors in four-loop QCD · Introduction · 0.84
Within QCD such form factors are known up to three loops for some years now [ref] , where a vector current with no association to the gauge group has been used for the fermions.

Infrared Anomalous Dimensions at Three-Loop in the SM from Conserved Currents · Introduction · 0.89

Discussion

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