REVIEW 2 major objections 6 minor 119 references
Three loop master integrals for ${\mathcal{O}} (\alpha \alpha_s^2)$ corrections to quark form factor
T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read All 303 master integrals for the three-loop quark form factor with a single massive boson are computed analytically in generalized polylogarithms, despite square roots that resist a single rationalizing transformation.
desk verdict Novel 303-master-integral computation with strong numeric checks, but the analytic claim is conditional while the w-letter boundary constants rest on an unproven PSLQ basis. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the first-order differential-equation system for the 303 master integrals, organized into an upper block-triangular form and solved bottom-up by decoupling each block into higher-order equations whose operators factorize into first-order pieces. Rather than searching for a canonical epsilon-form, the paper keeps the system in this factorized form and uses variation of constants. The rationalizing variables $x_l$, $x_n$, and $x_i$ are the central objects that make the final answer simple: each removes one of the square roots $\sqrt{(4-x)x}$, $\sqrt{1-4x}$, and the arctangent structure $x^{-3/2}\tan^{-1}(\sqrt{x})$, and the final polylogarithms have the small alphabets listed above. Boundary constants that cannot be obtained by Feynman-parameter evaluation or regularity are reconstructed with PSLQ from high-precision values supplied by an auxiliary mass flow method, using known tables for sixth-root-of-unity constants up to weight six.
What would settle it
Take one of the master integrals whose boundary constants were fixed through the $w$-letter alphabet and evaluate its closed form at a kinematic point that was not used in the PSLQ reconstruction (for example, a negative-$x$ threshold point), comparing against a direct high-precision numerical evaluation of the original Feynman integral by an independent method such as sector decomposition; disagreement at the claimed precision would show the assumed constant basis is incomplete, while agreement would corroborate it.
Extended reading notes
Core claim
The central discovery is that the square-root obstructions in these three-loop integrals are individually rationalizable, just not simultaneously. The paper defines the variables $x_l$, $x_n$, and $x_i$ through $x=(1+x_l)^2/x_l=x_n/(1+x_n)^2=-x_i^2$, uses different transformations in different sectors, and then integrates the mixed-argument polylogarithmic inhomogeneities piecewise. The upshot is that all 303 master integrals are expressible in generalized polylogarithms (iterated integrals built from logarithmic kernels) with alphabets $\{-1,0,1\}$ for $x$, $\{-1,0,1,r_3,r_4\}$ for $x_l$, $\{-1,0,1,r_3,r_4,w_3,w_4\}$ for $x_n$, and $\{-1,0,1,i,-i\}$ for $x_i$, where $r_3,r_4$ are the primitive sixth roots of unity and $w_3,w_4=(-3\pm\sqrt{5})/2$. Boundary conditions are fixed by Feynman-parameter values, regularity, and PSLQ reconstruction from high-precision numerical values; the resulting expressions agree with an independent numerical method to at least 50 digits.
Load-bearing premise
The boundary constants for integrals whose polylogarithms use the $w_3,w_4$ letters are reconstructed numerically under the assumption that they lie in a specific, not-yet-proven-complete basis of multiple zeta values and related constants; if that basis misses a constant, the affected analytic integrals would be wrong even though they currently match high-precision numerics.
Editorial extensions
If this is right
- Together with the integrals presented in [94] for the other topologies, these 303 master integrals provide the complete set needed to obtain the three-loop $O(\alpha\alpha_s^2)$ quark form factor and, from it, N3LO mixed QCD-electroweak Drell-Yan predictions.
- Because the expressions are analytic generalized polylogarithms with small alphabets, they can be evaluated numerically to high precision quickly with standard libraries, making them directly usable in cross-section codes.
- The computation demonstrates a workable alternative to canonical-form reductions: when square roots cannot all be rationalized by one transformation, solving the system bottom-up with several rationalizing variables and piecewise integration still yields compact analytic results.
- The reported agreement with an independent numerical method to at least 50 digits across several kinematic points indicates the analytic results can be used reliably in the physical region.
Reading between the lines
- A natural extension left implicit in the paper is that once the constant basis for the $w_3,w_4$ alphabet is established and proven complete, the same PSLQ-based boundary reconstruction could be applied to other multi-scale integral families with multiple square roots, avoiding canonical bases altogether.
- If the full set of master integrals for the $O(\alpha\alpha_s^2)$ quark form factor is expressible in generalized polylogarithms, the complete form factor amplitude is likely also polylogarithmic; the paper computes only the single-massive-boson sector, so this remains to be checked for the contributions built on [94].
- A testable extension would be to derive explicit analytic-continuation rules for the $w_3,w_4$ polylogarithms across the thresholds $s=0$ and $s=-4m_V^2$ and verify them numerically, since the mixed-argument representation makes continuation less transparent than in a single-variable representation.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents the computation of 303 three-loop master integrals that appear in the O(alpha alpha_s^2) corrections to the quark form factor in Feynman diagrams containing a single massive vector boson. The authors perform an IBP reduction to obtain a basis of master integrals, set up first-order differential equations in x = -s/m_V^2, decouple the subsystems into factorizable higher-order equations, and solve them by the method of variation of constants. Because the square roots appearing in the problem cannot be rationalized by a single change of variables, they employ several transformations (x, x_l, x_n, x_i) and express the results in terms of generalized polylogarithms with a simple alphabet but multiple interdependent arguments. Boundary conditions are fixed partly by Feynman-parameter evaluations and regularity conditions and partly by AMFlow numerics combined with PSLQ reconstruction. The analytic expressions are provided in ancillary files, a numerical table at x = 1/11 is given, and the results are checked against AMFlow to 50-digit precision at several kinematic points.
Significance. If correct, this is a substantial technical contribution: the 303 master integrals are a necessary ingredient for the three-loop mixed QCD-electroweak corrections to the quark form factor, with direct relevance to Drell-Yan phenomenology at the LHC. The treatment of simultaneously non-rationalizable square roots via concurrent transformations, leading to GPLs with several interdependent arguments, is methodologically interesting and likely to be useful beyond this specific calculation. The main strengths are the complete reduction to 303 master integrals, the explicit GPL alphabets, and the high-precision numerical validation against AMFlow. The principal weakness is that boundary constants for integrals involving the quadratic letters w3 and w4 are reconstructed from AMFlow numerics via PSLQ under an anticipated but unproven constant basis; the paper itself states that establishing this basis is left to future work. The numerical checks validate the numerical values of the integrals but do not by themselves establish the exact analytic form of those boundary constants.
major comments (2)
- [3.3.1] The boundary conditions for master integrals whose GPL alphabet contains the quadratic letters w3 and w4 are fixed by PSLQ under an 'anticipated' set of constants, and the paper states that establishing the full basis of constant GPLs involving the alphabet {w1, ..., w4} is planned for future investigation. Because these constants enter the analytic expressions of the affected master integrals, the central claim that all 303 master integrals admit analytic GPL representations is conditional on that constant basis being complete. If the basis is incomplete, PSLQ can produce a spurious rational relation at finite precision, and the resulting boundary constants could be incorrect even if the numerical checks at a few kinematic points pass. Please either prove the constant relations needed for the reconstruction, or determine the affected boundary constants through an independent non-PSLQ method (for example, by imposing regularity or evaluating the integrals at additional points in different kinematic regions), or clearly identify which of the 303 master integrals rely on the w-letter reconstruction and state the analytic results with that caveat. The AMFlow checks validate the numerical values of the integrals; they do not by themselves prove that the PSLQ-identified constants are the correct analytic ones.
- [4.1] The numerical checks are described only as 'perfect agreement with AMFlow output across several kinematic points', without specifying the points, the number of points, or which master integrals were checked. Given that boundary constants for a subset of the integrals are obtained by PSLQ reconstruction, the validation should report the actual kinematic points, the regions of x they cover (for example x > 4, 0 < x < 4, and negative x), and the achieved precision for a representative sample of the affected integrals. This information is necessary to assess the strength of the statement that all 303 master integrals are correct and to ensure that the branch choices and analytic continuations of the GPLs have been exercised.
minor comments (6)
- [3.1] The definitions of the integral families I13 and I14 are missing their closing braces; in the current text both lines end with a comma and no closing brace.
- [3.3.1] The label B5,5 is used twice, once for the limit q^2 -> 0 and once for the limit q^2 -> -1, with two different gamma-function expressions; please disambiguate these labels.
- [3.3.1] The boundary-condition block refers to diagrams B4,1, B5,1, T6,1, and so on with the statement that the thick line represents the massive propagator, but no figure showing these diagrams is included in the text; without a diagram-to-label correspondence the boundary conditions cannot be matched to the master integrals.
- [2] The notation for GPLs with multiple arguments should be defined more precisely; the paper should state explicitly how GPLs with argument x, x_l, x_n, and x_i are combined in the final expressions and how the alphabets containing r3, r4, w3, w4, i, and -i are handled in the GPL definition.
- [3.3] The sentence describing integrand sizes says 'tens of megabits', which is a data-size unit; it should read 'tens of megabytes' or 'millions of characters'.
- [4] The table of 303 numerical entries in the main text is very large; a smaller representative subset in the text with the full table in the ancillary files would improve readability.
Circularity Check
No circularity: the master-integral derivation is fixed by boundary constants from AMFlow/Feynman-parameter integrals and independently checked against AMFlow at other kinematic points.
full rationale
The paper's claimed derivation chain is not circular. The 303 master integrals are obtained by solving first-order coupled differential equations whose homogeneous parts are rationalized through the variables x_l, x_n, and x_i. The integration constants are fixed by two independent routes: (i) closed-form Feynman-parameter evaluations for planar boundary integrals, and (ii) high-precision numerical values from the external AMFlow program, followed by PSLQ reconstruction of analytic constants. The final results are then evaluated with GiNac and compared with AMFlow at 'several kinematic points' across 'different domains of the kinematic variable,' i.e., the checks are not performed at the same values used to fix the boundary constants. No equation in the paper reduces by construction to a fitted parameter, and no self-citation supplies a load-bearing uniqueness or existence claim. The self-citations in the reference list (e.g., Refs. [24,26,30,37,39]) are background citations to the authors' prior heavy-quark form-factor work; they are not used to justify the differential-equation method, the alphabet choices, or the boundary constants of this calculation. The acknowledged limitation concerning the basis of constant GPLs involving the w-letter alphabet (Section 3.3.1) is a correctness risk, not a circularity: if that basis were incomplete, the PSLQ-reconstructed constants could be wrong, but the derivation would not thereby be identical to its input. The analysis is therefore self-contained and benchmarked externally, and no circular step can be quoted.
Assumptions & free parameters
assumptions (4)
- domain assumption The differential equations for the master integrals can be decoupled such that all higher-order differential operators factorize into first-order operators.
- domain assumption The AMFlow auxiliary mass flow method provides correct high-precision numerical values for the master integrals at the chosen kinematic points.
- domain assumption The boundary constants can be expressed in the anticipated basis of multiple zeta values, logarithms, and cyclotomic constants, including the GPL constant basis for letters w1 through w4.
- domain assumption The integration-by-parts reduction performed with Kira and LiteRed is correct and complete.
Cite this review
Pith. "Pith review of Three loop master integrals for ${\mathcal{O}} (\alpha \alpha_s^2)$ corrections to quark form factor." pith.science (2026). https://pith.science/paper/V3NF4MMB
@misc{pith2026250615363,
author = {Pith},
title = {Pith review of: Three loop master integrals for $\mathcalO (\alpha \alpha_s^2)$ corrections to quark form factor},
year = {2026},
howpublished = {\url{https://pith.science/paper/V3NF4MMB}},
note = {Machine review of arXiv:2506.15363}
}
abstract
We consider the three-loop mixed strong-electroweak (${\mathcal{O}}(\alpha \alpha_s^2)$) corrections to the quark form factor. We compute the master integrals which are appearing in the Feynman diagrams containing a single massive boson in the loop. We use the state-of-the-art method of differential equations to compute all 303 of them, expressing the results in terms of generalized polylogarithms. We encounter multiple square roots that cannot be simultaneously rationalized using a single transformation. Applying concurrent transformations allows us to express the results through generalized polylogarithms with a simple alphabet, but with multiple interdependent arguments.
Reference graph
Works this paper leans on
-
[1]
V. Ravindran, J. Smith and W. L. van Neerven,Two-loop corrections to Higgs boson production,Nucl. Phys. B704(2005) 332–348, [hep-ph/0408315]
arXiv 2005
-
[2]
D. de Florian, M. Mahakhud, P. Mathews, J. Mazzitelli and V. Ravindran,Quark and gluon spin-2 form factors to two-loops in QCD,JHEP02(2014) 035, [1312.6528]
arXiv 2014
-
[3]
S. Moch, J. A. M. Vermaseren and A. Vogt,Three-loop results for quark and gluon form-factors,Phys. Lett. B625(2005) 245–252, [hep-ph/0508055]
arXiv 2005
-
[4]
S. Moch, J. A. M. Vermaseren and A. Vogt,The Quark form-factor at higher orders,JHEP 08(2005) 049, [hep-ph/0507039]
arXiv 2005
-
[5]
P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov and M. Steinhauser,Quark and gluon form factors to three loops,Phys. Rev. Lett.102(2009) 212002, [0902.3519]
arXiv 2009
-
[6]
T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli and C. Studerus,Calculation of the quark and gluon form factors to three loops in QCD,JHEP06(2010) 094, [1004.3653]
arXiv 2010
-
[7]
T. Gehrmann and D. Kara,TheHb ¯bform factor to three loops in QCD,JHEP09(2014) 174, [1407.8114]
arXiv 2014
- [8]
Show all 119 references
-
[9]
Ahmed, T
T. Ahmed, T. Gehrmann, P. Mathews, N. Rana and V. Ravindran,Pseudo-scalar Form Factors at Three Loops in QCD,JHEP11(2015) 169, [1510.01715]
2015 arXiv
-
[10]
Ahmed, P
T. Ahmed, P. Banerjee, P. K. Dhani, N. Rana, V. Ravindran and S. Seth,Konishi form factor at three loops inN=4 supersymmetric Yang-Mills theory,Phys. Rev. D95(2017) 085019, [1610.05317]
2017 arXiv
-
[11]
Ahmed, P
T. Ahmed, P. Banerjee, P. K. Dhani, P. Mathews, N. Rana and V. Ravindran,Three loop form factors of a massive spin-2 particle with nonuniversal coupling,Phys. Rev. D95 (2017) 034035, [1612.00024]
2017 arXiv
-
[12]
Ahmed, P
T. Ahmed, P. Banerjee, A. Chakraborty, P. K. Dhani and V. Ravindran,Form factors with two operator insertions and the principle of maximal transcendentality,Phys. Rev. D102 (2020) 061701, [1911.11886]. – 23 –
2020 arXiv
-
[13]
R. N. Lee, A. von Manteuffel, R. M. Schabinger, A. V. Smirnov, V. A. Smirnov and M. Steinhauser,The four-loopN= 4 SYM Sudakov form factor,JHEP01(2022) 091, [2110.13166]
2022 arXiv
-
[14]
R. N. Lee, A. von Manteuffel, R. M. Schabinger, A. V. Smirnov, V. A. Smirnov and M. Steinhauser,Quark and Gluon Form Factors in Four-Loop QCD,Phys. Rev. Lett.128 (2022) 212002, [2202.04660]
2022 arXiv
-
[15]
Chakraborty, T
A. Chakraborty, T. Huber, R. N. Lee, A. von Manteuffel, R. M. Schabinger, A. V. Smirnov, V. A. Smirnov and M. Steinhauser,Hbb vertex at four loops and hard matching coefficients in SCET for various currents,Phys. Rev. D106(2022) 074009, [2204.02422]
2022 arXiv
-
[16]
Bernreuther, R
W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia and E. Remiddi,Two-loop QCD corrections to the heavy quark form-factors: The Vector contributions,Nucl. Phys. B706(2005) 245–324, [hep-ph/0406046]
2005 arXiv
-
[17]
Bernreuther, R
W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia and E. Remiddi,Two-loop QCD corrections to the heavy quark form-factors: Axial vector contributions,Nucl. Phys. B712(2005) 229–286, [hep-ph/0412259]
2005 arXiv
-
[18]
Bernreuther, R
W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber and E. Remiddi, Two-loop QCD corrections to the heavy quark form-factors: Anomaly contributions,Nucl. Phys. B723(2005) 91–116, [hep-ph/0504190]
2005 arXiv
-
[19]
Bernreuther, R
W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, P. Mastrolia and E. Remiddi, Decays of scalar and pseudoscalar Higgs bosons into fermions: Two-loop QCD corrections to the Higgs-quark-antiquark amplitude,Phys. Rev. D72(2005) 096002, [hep-ph/0508254]
2005 arXiv
-
[20]
Gluza, A
J. Gluza, A. Mitov, S. Moch and T. Riemann,The QCD form factor of heavy quarks at NNLO,JHEP07(2009) 001, [0905.1137]
2009 arXiv
-
[21]
Ablinger, A
J. Ablinger, A. Behring, J. Bl¨ umlein, G. Falcioni, A. De Freitas, P. Marquard, N. Rana and C. Schneider,Heavy quark form factors at two loops,Phys. Rev. D97(2018) 094022, [1712.09889]
2018 arXiv
-
[22]
J. Henn, A. V. Smirnov, V. A. Smirnov and M. Steinhauser,Massive three-loop form factor in the planar limit,JHEP01(2017) 074, [1611.07535]
2017 arXiv
-
[23]
R. N. Lee, A. V. Smirnov, V. A. Smirnov and M. Steinhauser,Three-loop massive form factors: complete light-fermion corrections for the vector current,JHEP03(2018) 136, [1801.08151]
2018 arXiv
-
[24]
Ablinger, J
J. Ablinger, J. Bl¨ umlein, P. Marquard, N. Rana and C. Schneider,Heavy quark form factors at three loops in the planar limit,Phys. Lett. B782(2018) 528–532, [1804.07313]
2018 arXiv
-
[25]
R. N. Lee, A. V. Smirnov, V. A. Smirnov and M. Steinhauser,Three-loop massive form factors: complete light-fermion and large-N c corrections for vector, axial-vector, scalar and pseudo-scalar currents,JHEP05(2018) 187, [1804.07310]
2018 arXiv
-
[26]
Bl¨ umlein, P
J. Bl¨ umlein, P. Marquard, N. Rana and C. Schneider,The Heavy Fermion Contributions to the Massive Three Loop Form Factors,Nucl. Phys. B949(2019) 114751, [1908.00357]
2019 arXiv
-
[27]
M. Fael, F. Lange, K. Sch¨ onwald and M. Steinhauser,Massive Vector Form Factors to Three Loops,Phys. Rev. Lett.128(2022) 172003, [2202.05276]
2022 arXiv
-
[28]
M. Fael, F. Lange, K. Sch¨ onwald and M. Steinhauser,Singlet and nonsinglet three-loop massive form factors,Phys. Rev. D106(2022) 034029, [2207.00027]. – 24 –
2022 arXiv
-
[29]
M. Fael, F. Lange, K. Sch¨ onwald and M. Steinhauser,Massive three-loop form factors: Anomaly contribution,Phys. Rev. D107(2023) 094017, [2302.00693]
2023 arXiv
-
[30]
Bl¨ umlein, A
J. Bl¨ umlein, A. De Freitas, P. Marquard, N. Rana and C. Schneider,Analytic results on the massive three-loop form factors: quarkonic contributions,2307.02983
-
[31]
Bonciani and A
R. Bonciani and A. Ferroglia,Two-Loop QCD Corrections to the Heavy-to-Light Quark Decay,JHEP11(2008) 065, [0809.4687]
2008 arXiv
-
[32]
Huber,On a two-loop crossed six-line master integral with two massive lines,JHEP03 (2009) 024, [0901.2133]
T. Huber,On a two-loop crossed six-line master integral with two massive lines,JHEP03 (2009) 024, [0901.2133]
2009 arXiv
-
[33]
Bell,Higher order QCD corrections in exclusive charmless B decays, Ph.D
G. Bell,Higher order QCD corrections in exclusive charmless B decays, Ph.D. thesis, Munich U., 2006.0705.3133
2006
-
[34]
G. Bell, M. Beneke, T. Huber and X.-Q. Li,Heavy-to-light currents at NNLO in SCET and semi-inclusive ¯B→X sl+l− decay,Nucl. Phys. B843(2011) 143–176, [1007.3758]
2011 arXiv
-
[35]
Chen,Two-Loop master integrals for heavy-to-light form factors of two different massive fermions,JHEP02(2018) 066, [1801.01033]
L.-B. Chen,Two-Loop master integrals for heavy-to-light form factors of two different massive fermions,JHEP02(2018) 066, [1801.01033]
2018 arXiv
-
[36]
Engel, C
T. Engel, C. Gnendiger, A. Signer and Y. Ulrich,Small-mass effects in heavy-to-light form factors,JHEP02(2019) 118, [1811.06461]
2019 arXiv
-
[37]
Datta, N
S. Datta, N. Rana, V. Ravindran and R. Sarkar,Three loop QCD corrections to the heavy-light form factors in the color-planar limit,JHEP12(2023) 001, [2308.12169]
2023 arXiv
-
[38]
M. Fael, T. Huber, F. Lange, J. M¨ uller, K. Sch¨ onwald and M. Steinhauser,Heavy-to-light form factors to three loops,2406.08182
-
[39]
Datta and N
S. Datta and N. Rana,Three loop QCD corrections to the heavy-light form factors: fermionic contributions,JHEP10(2024) 254, [2407.14550]
2024 arXiv
-
[40]
Altarelli, R
G. Altarelli, R. K. Ellis and G. Martinelli,Large Perturbative Corrections to the Drell-Yan Process in QCD,Nucl. Phys. B157(1979) 461–497
1979
-
[41]
Hamberg, W
R. Hamberg, W. L. van Neerven and T. Matsuura,A complete calculation of the order α−s 2 correction to the Drell-YanKfactor,Nucl. Phys. B359(1991) 343–405
1991
-
[42]
R. V. Harlander and W. B. Kilgore,Next-to-next-to-leading order Higgs production at hadron colliders,Phys. Rev. Lett.88(2002) 201801, [hep-ph/0201206]
2002 arXiv
-
[43]
Anastasiou, L
C. Anastasiou, L. J. Dixon, K. Melnikov and F. Petriello,Dilepton rapidity distribution in the Drell-Yan process at NNLO in QCD,Phys. Rev. Lett.91(2003) 182002, [hep-ph/0306192]
2003 arXiv
-
[44]
Anastasiou, L
C. Anastasiou, L. J. Dixon, K. Melnikov and F. Petriello,High precision QCD at hadron colliders: Electroweak gauge boson rapidity distributions at NNLO,Phys. Rev. D69(2004) 094008, [hep-ph/0312266]
2004 arXiv
-
[45]
Melnikov and F
K. Melnikov and F. Petriello,Electroweak gauge boson production at hadron colliders throughO(α 2 s),Phys. Rev. D74(2006) 114017, [hep-ph/0609070]
2006 arXiv
-
[46]
Catani, L
S. Catani, L. Cieri, G. Ferrera, D. de Florian and M. Grazzini,Vector boson production at hadron colliders: a fully exclusive QCD calculation at NNLO,Phys. Rev. Lett.103(2009) 082001, [0903.2120]
2009 arXiv
-
[47]
Catani, G
S. Catani, G. Ferrera and M. Grazzini,W Boson Production at Hadron Colliders: The Lepton Charge Asymmetry in NNLO QCD,JHEP05(2010) 006, [1002.3115]. – 25 –
2010 arXiv
-
[48]
Dittmaier and M
S. Dittmaier and M. Kr¨ amer,Electroweak radiative corrections to W boson production at hadron colliders,Phys. Rev. D65(2002) 073007, [hep-ph/0109062]
2002 arXiv
-
[49]
Baur and D
U. Baur and D. Wackeroth,Electroweak radiative corrections top¯p→W ±→ℓ ±νbeyond the pole approximation,Phys. Rev. D70(2004) 073015, [hep-ph/0405191]
2004 arXiv
-
[50]
V. A. Zykunov,Radiative corrections to the Drell-Yan process at large dilepton invariant masses,Phys. Atom. Nucl.69(2006) 1522
2006
-
[51]
Arbuzov, D
A. Arbuzov, D. Bardin, S. Bondarenko, P. Christova, L. Kalinovskaya, G. Nanava and R. Sadykov,One-loop corrections to the Drell-Yan process in SANC. I. The Charged current case,Eur. Phys. J. C46(2006) 407–412, [hep-ph/0506110]
2006 arXiv
-
[52]
C. M. Carloni Calame, G. Montagna, O. Nicrosini and A. Vicini,Precision electroweak calculation of the charged current Drell-Yan process,JHEP12(2006) 016, [hep-ph/0609170]
2006 arXiv
-
[53]
U. Baur, O. Brein, W. Hollik, C. Schappacher and D. Wackeroth,Electroweak radiative corrections to neutral current Drell-Yan processes at hadron colliders,Phys. Rev. D65 (2002) 033007, [hep-ph/0108274]
2002 arXiv
-
[54]
V. A. Zykunov,Weak radiative corrections to Drell-Yan process for large invariant mass of di-lepton pair,Phys. Rev. D75(2007) 073019, [hep-ph/0509315]
2007 arXiv
-
[55]
C. M. Carloni Calame, G. Montagna, O. Nicrosini and A. Vicini,Precision electroweak calculation of the production of a high transverse-momentum lepton pair at hadron colliders, JHEP10(2007) 109, [0710.1722]
2007 arXiv
-
[56]
Arbuzov, D
A. Arbuzov, D. Bardin, S. Bondarenko, P. Christova, L. Kalinovskaya, G. Nanava and R. Sadykov,One-loop corrections to the Drell–Yan process in SANC. (II). The Neutral current case,Eur. Phys. J. C54(2008) 451–460, [0711.0625]
2008 arXiv
-
[57]
Dittmaier and M
S. Dittmaier and M. Huber,Radiative corrections to the neutral-current Drell-Yan process in the Standard Model and its minimal supersymmetric extension,JHEP01(2010) 060, [0911.2329]
2010 arXiv
-
[58]
C. Duhr, F. Dulat and B. Mistlberger,Drell-Yan Cross Section to Third Order in the Strong Coupling Constant,Phys. Rev. Lett.125(2020) 172001, [2001.07717]
2020 arXiv
-
[59]
X. Chen, T. Gehrmann, N. Glover, A. Huss, T.-Z. Yang and H. X. Zhu,Dilepton Rapidity Distribution in Drell-Yan Production to Third Order in QCD,Phys. Rev. Lett.128(2022) 052001, [2107.09085]
2022 arXiv
-
[60]
C. Duhr, F. Dulat and B. Mistlberger,Charged current Drell-Yan production at N 3LO, JHEP11(2020) 143, [2007.13313]
2020 arXiv
-
[61]
Camarda, L
S. Camarda, L. Cieri and G. Ferrera,Drell–Yan lepton-pair production: qT resummation at N3LL accuracy and fiducial cross sections at N3LO,Phys. Rev. D104(2021) L111503, [2103.04974]
2021 arXiv
-
[62]
X. Chen, T. Gehrmann, E. W. N. Glover, A. Huss, P. F. Monni, E. Re, L. Rottoli and P. Torrielli,Third-Order Fiducial Predictions for Drell-Yan Production at the LHC,Phys. Rev. Lett.128(2022) 252001, [2203.01565]
2022 arXiv
-
[63]
Neumann and J
T. Neumann and J. Campbell,Fiducial Drell-Yan production at the LHC improved by transverse-momentum resummation at N4LLp+N3LO,Phys. Rev. D107(2023) L011506, [2207.07056]. – 26 –
2023 arXiv
-
[64]
Campbell and T
J. Campbell and T. Neumann,Third order QCD predictions for fiducial W-boson production,JHEP11(2023) 127, [2308.15382]
2023 arXiv
-
[65]
Dittmaier, A
S. Dittmaier, A. Huss and C. Schwinn,Mixed QCD-electroweakO(α sα)corrections to Drell-Yan processes in the resonance region: pole approximation and non-factorizable corrections,Nucl. Phys. B885(2014) 318–372, [1403.3216]
2014 arXiv
-
[66]
Dittmaier, A
S. Dittmaier, A. Huss and C. Schwinn,Dominant mixed QCD-electroweak O(α sα) corrections to Drell–Yan processes in the resonance region,Nucl. Phys. B904(2016) 216–252, [1511.08016]
2016 arXiv
-
[67]
Bonciani, F
R. Bonciani, F. Buccioni, R. Mondini and A. Vicini,Double-real corrections atO(αα s)to single gauge boson production,Eur. Phys. J. C77(2017) 187, [1611.00645]
2017 arXiv
-
[68]
Bonciani, F
R. Bonciani, F. Buccioni, N. Rana, I. Triscari and A. Vicini,NNLO QCD×EW corrections to Z production in theq¯qchannel,Phys. Rev. D101(2020) 031301, [1911.06200]
2020 arXiv
-
[69]
Bonciani, F
R. Bonciani, F. Buccioni, N. Rana and A. Vicini,Next-to-Next-to-Leading Order Mixed QCD-Electroweak Corrections to on-Shell Z Production,Phys. Rev. Lett.125(2020) 232004, [2007.06518]
2020 arXiv
-
[70]
Buccioni, F
F. Buccioni, F. Caola, M. Delto, M. Jaquier, K. Melnikov and R. R¨ ontsch,Mixed QCD-electroweak corrections to on-shell Z production at the LHC,Phys. Lett. B811(2020) 135969, [2005.10221]
2020 arXiv
-
[71]
Behring, F
A. Behring, F. Buccioni, F. Caola, M. Delto, M. Jaquier, K. Melnikov and R. R¨ ontsch, Mixed QCD-electroweak corrections toW-boson production in hadron collisions,Phys. Rev. D103(2021) 013008, [2009.10386]
2021 arXiv
-
[72]
Buonocore, M
L. Buonocore, M. Grazzini, S. Kallweit, C. Savoini and F. Tramontano,Mixed QCD-EW corrections topp→ℓν ℓ +Xat the LHC,Phys. Rev. D103(2021) 114012, [2102.12539]
2021 arXiv
-
[73]
Bonciani, L
R. Bonciani, L. Buonocore, M. Grazzini, S. Kallweit, N. Rana, F. Tramontano and A. Vicini,Mixed Strong-Electroweak Corrections to the Drell-Yan Process,Phys. Rev. Lett. 128(2022) 012002, [2106.11953]
2022 arXiv
-
[74]
Bonciani, F
R. Bonciani, F. Buccioni, N. Rana and A. Vicini,On-shell Z boson production at hadron colliders throughO(αα s),JHEP02(2022) 095, [2111.12694]
2022 arXiv
-
[75]
Buccioni, F
F. Buccioni, F. Caola, H. A. Chawdhry, F. Devoto, M. Heller, A. von Manteuffel, K. Melnikov, R. R¨ ontsch and C. Signorile-Signorile,Mixed QCD-electroweak corrections to dilepton production at the LHC in the high invariant mass region,JHEP06(2022) 022, [2203.11237]
2022 arXiv
-
[76]
Armadillo, R
T. Armadillo, R. Bonciani, S. Devoto, N. Rana and A. Vicini,Two-loop mixed QCD-EW corrections to neutral current Drell-Yan,JHEP05(2022) 072, [2201.01754]
2022 arXiv
-
[77]
Dittmaier, A
S. Dittmaier, A. Huss and J. Schwarz,Mixed NNLO QCD×electroweak corrections to single-Z production in pole approximation: differential distributions and forward-backward asymmetry,JHEP05(2024) 170, [2401.15682]
2024 arXiv
-
[78]
Armadillo, R
T. Armadillo, R. Bonciani, S. Devoto, N. Rana and A. Vicini,Two-loop mixed QCD-EW corrections to charged current Drell-Yan,JHEP07(2024) 265, [2405.00612]
2024 arXiv
-
[79]
F. V. Tkachov,A theorem on analytical calculability of 4-loop renormalization group functions,Phys. Lett. B100(1981) 65–68. – 27 –
1981
-
[80]
K. G. Chetyrkin and F. V. Tkachov,Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops,Nucl. Phys. B192(1981) 159–204
1981
-
[81]
Laporta,High precision calculation of multiloop Feynman integrals by difference equations,Int
S. Laporta,High precision calculation of multiloop Feynman integrals by difference equations,Int. J. Mod. Phys. A15(2000) 5087–5159, [hep-ph/0102033]
2000 arXiv
-
[82]
A. V. Kotikov,Differential equations method: New technique for massive Feynman diagrams calculation,Phys. Lett. B254(1991) 158–164
1991
-
[83]
Remiddi,Differential equations for Feynman graph amplitudes,Nuovo Cim
E. Remiddi,Differential equations for Feynman graph amplitudes,Nuovo Cim. A110 (1997) 1435–1452, [hep-th/9711188]
1997 arXiv
-
[84]
Gehrmann and E
T. Gehrmann and E. Remiddi,Differential equations for two-loop four-point functions, Nucl. Phys. B580(2000) 485–518, [hep-ph/9912329]
2000 arXiv
-
[85]
Argeri and P
M. Argeri and P. Mastrolia,Feynman Diagrams and Differential Equations,Int. J. Mod. Phys. A22(2007) 4375–4436, [0707.4037]
2007 arXiv
-
[86]
J. M. Henn,Multiloop integrals in dimensional regularization made simple,Phys. Rev. Lett. 110(2013) 251601, [1304.1806]
2013 arXiv
-
[87]
J. M. Henn,Lectures on differential equations for Feynman integrals,J. Phys. A48(2015) 153001, [1412.2296]
2015 arXiv
-
[88]
Ablinger, A
J. Ablinger, A. Behring, J. Bl¨ umlein, A. De Freitas, A. von Manteuffel and C. Schneider, Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra,Comput. Phys. Commun.202(2016) 33–112, [1509.08324]
2016 arXiv
-
[89]
Ablinger, J
J. Ablinger, J. Bl¨ umlein, P. Marquard, N. Rana and C. Schneider,Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable,Nucl. Phys. B 939(2019) 253–291, [1810.12261]
2019 arXiv
-
[90]
Remiddi and J
E. Remiddi and J. A. M. Vermaseren,Harmonic polylogarithms,Int. J. Mod. Phys. A15 (2000) 725–754, [hep-ph/9905237]
2000 arXiv
-
[91]
A. B. Goncharov,Multiple polylogarithms and mixed Tate motives,math/0103059
-
[92]
Vollinga and S
J. Vollinga and S. Weinzierl,Numerical evaluation of multiple polylogarithms,Comput. Phys. Commun.167(2005) 177, [hep-ph/0410259]
2005 arXiv
-
[93]
Chen,Iterated path integrals,Bull
K.-T. Chen,Iterated path integrals,Bull. Am. Math. Soc.83(1977) 831–879
1977
-
[94]
Bonetti, K
M. Bonetti, K. Melnikov and L. Tancredi,Three-loop mixed QCD-electroweak corrections to Higgs boson gluon fusion,Phys. Rev. D97(2018) 034004, [1711.11113]
2018 arXiv
-
[95]
von Manteuffel and C
A. von Manteuffel and C. Studerus,Reduze 2 - Distributed Feynman Integral Reduction, 1201.4330
-
[96]
Maierh¨ ofer, J
P. Maierh¨ ofer, J. Usovitsch and P. Uwer,Kira—A Feynman integral reduction program, Comput. Phys. Commun.230(2018) 99–112, [1705.05610]
2018 arXiv
-
[97]
Klappert, F
J. Klappert, F. Lange, P. Maierh¨ ofer and J. Usovitsch,Integral reduction with Kira 2.0 and finite field methods,Comput. Phys. Commun.266(2021) 108024, [2008.06494]
2021 arXiv
-
[98]
R. N. Lee,Presenting LiteRed: a tool for the Loop InTEgrals REDuction,1212.2685
-
[99]
R. N. Lee,LiteRed 1.4: a powerful tool for reduction of multiloop integrals,J. Phys. Conf. Ser.523(2014) 012059, [1310.1145]
2014 arXiv
-
[100]
R. N. Lee,Reducing differential equations for multiloop master integrals,JHEP04(2015) 108, [1411.0911]. – 28 –
2015 arXiv
-
[101]
Meyer,Transforming differential equations of multi-loop Feynman integrals into canonical form,JHEP04(2017) 006, [1611.01087]
C. Meyer,Transforming differential equations of multi-loop Feynman integrals into canonical form,JHEP04(2017) 006, [1611.01087]
2017 arXiv
-
[102]
Gituliar and V
O. Gituliar and V. Magerya,Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form,Comput. Phys. Commun.219(2017) 329–338, [1701.04269]
2017 arXiv
-
[103]
Prausa,epsilon: A tool to find a canonical basis of master integrals,Comput
M. Prausa,epsilon: A tool to find a canonical basis of master integrals,Comput. Phys. Commun.219(2017) 361–376, [1701.00725]
2017 arXiv
-
[104]
R. N. Lee,Libra: A package for transformation of differential systems for multiloop integrals,Comput. Phys. Commun.267(2021) 108058, [2012.00279]
2021 arXiv
-
[105]
Meyer,Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA,Comput
C. Meyer,Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA,Comput. Phys. Commun.222(2018) 295–312, [1705.06252]
2018 arXiv
-
[106]
Davies, G
J. Davies, G. Mishima, M. Steinhauser and D. Wellmann,Double-higgs boson production in the high-energy limit: planar master integrals,Journal of High Energy Physics2018(2018) 1–24
2018
-
[107]
Ablinger, J
J. Ablinger, J. Bl¨ umlein and C. Schneider,Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms,J. Math. Phys.54(2013) 082301, [1302.0378]
2013 arXiv
-
[108]
Ablinger,A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics, Master’s thesis, Linz U., 2009
J. Ablinger,A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics, Master’s thesis, Linz U., 2009
2009
-
[109]
Ablinger, J
J. Ablinger, J. Blumlein and C. Schneider,Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials,J. Math. Phys.52(2011) 102301, [1105.6063]
2011 arXiv
-
[110]
Ablinger,The package HarmonicSums: Computer Algebra and Analytic aspects of Nested Sums,PoSLL2014(2014) 019, [1407.6180]
J. Ablinger,The package HarmonicSums: Computer Algebra and Analytic aspects of Nested Sums,PoSLL2014(2014) 019, [1407.6180]
2014 arXiv
-
[111]
Duhr and F
C. Duhr and F. Dulat,PolyLogTools — polylogs for the masses,JHEP08(2019) 135, [1904.07279]
2019 arXiv
-
[112]
Ferguson and D
H. Ferguson and D. Bailey,A polynomial time, numerically stable integer relation algorithm,
-
[113]
Liu, Y.-Q
X. Liu, Y.-Q. Ma and C.-Y. Wang,A Systematic and Efficient Method to Compute Multi-loop Master Integrals,Phys. Lett. B779(2018) 353–357, [1711.09572]
2018 arXiv
-
[114]
Liu and Y.-Q
X. Liu and Y.-Q. Ma,Determining arbitrary Feynman integrals by vacuum integrals,Phys. Rev. D99(2019) 071501, [1801.10523]
2019 arXiv
-
[115]
Liu and Y.-Q
X. Liu and Y.-Q. Ma,Multiloop corrections for collider processes using auxiliary mass flow, Phys. Rev. D105(2022) L051503, [2107.01864]
2022 arXiv
-
[116]
Blumlein, D
J. Blumlein, D. J. Broadhurst and J. A. M. Vermaseren,The Multiple Zeta Value Data Mine,Comput. Phys. Commun.181(2010) 582–625, [0907.2557]
2010 arXiv
-
[117]
J. M. Henn, A. V. Smirnov and V. A. Smirnov,Evaluating Multiple Polylogarithm Values at Sixth Roots of Unity up to Weight Six,Nucl. Phys. B919(2017) 315–324, [1512.08389]
2017 arXiv
-
[118]
C. W. Bauer, A. Frink and R. Kreckel,Introduction to the GiNaC framework for symbolic computation within the C++ programming language,J. Symb. Comput.33(2002) 1–12, [cs/0004015]
2002 arXiv
-
[119]
A. V. Smirnov, N. D. Shapurov and L. I. Vysotsky,FIESTA5: Numerical high-performance Feynman integral evaluation,Comput. Phys. Commun.277(2022) 108386, [2110.11660]. – 29 –
2022 arXiv
Reviewed August 15, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.