Analytical solution to DGLAP integro-differential equation via complex maps in domains of contour integrals

Gustavo Alvarez; Igor Kondrashuk

arxiv: 1912.02303 · v2 · submitted 2019-12-04 · ✦ hep-th · hep-ph· math-ph· math.MP

Analytical solution to DGLAP integro-differential equation via complex maps in domains of contour integrals

Gustavo Alvarez , Igor Kondrashuk This is my paper

Pith reviewed 2026-05-24 15:08 UTC · model grok-4.3

classification ✦ hep-th hep-phmath-phmath.MP

keywords DGLAP equationBessel functionBarnes contour integralLaplace transformMellin momentscomplex diffeomorphismparton distribution functionsBFKL equation

0 comments

The pith

The inverse Laplace transform of the Bessel function's Laplace image equals a Barnes contour integral.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper works out the detailed mathematical steps for analytically solving a simplified DGLAP equation that governs the scale dependence of parton distributions in QCD. In the model with only one term in the splitting function, the solution takes the form of a Bessel function. The authors apply complex maps in the plane of Mellin moments to relate the original contour integrals to standard integrals in tables, and they verify that the inverse Laplace transform of the Laplace image of this Bessel function can be expressed as a Barnes contour integral. A sympathetic reader cares because the result supplies explicit formulae that turn the usual residue calculus into identities that may be looked up or manipulated further.

Core claim

We verify that the inverse Laplace transformation of the Laplace image of the Bessel function may be represented in a form of Barnes contour integral, obtained by writing the contour integral for the parton distribution as a standard integral via a complex diffeomorphism whose Laplace transform yields the Bessel solution.

What carries the argument

Complex diffeomorphism that converts the Mellin-moment contour integral for the parton distribution into a standard Laplace-transform integral whose inverse recovers the Bessel function.

If this is right

The DGLAP equation with the single-term splitting function admits an explicit analytical solution expressed through the Bessel function.
A dual integro-differential equation obtained by the complex map is the BFKL equation.
Contour integrals appearing in the parton distributions can be reduced to tabulated standard integrals via the diffeomorphism.
The strategy supplies a systematic way to generate further integral identities for the same class of equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the single-term model captures the leading behavior, the same contour technique might be tested on splitting functions with two or three terms.
The Barnes representation could be used to derive new recurrence relations or asymptotic expansions for the evolved distributions.
The method might extend to other evolution equations that admit Bessel-type solutions after Mellin transformation.

Load-bearing premise

The single-term splitting function model remains a faithful enough proxy for the dominant QCD dynamics that the derived contour identities carry over to realistic cases.

What would settle it

Compute the numerical value of the inverse Laplace transform of the Laplace image of the Bessel function at a chosen argument and check whether it matches the numerical evaluation of the corresponding Barnes contour integral to within integration error.

read the original abstract

A simple model for QCD dynamics in which the DGLAP integro-differential equation may be solved analytically has been considered in our previous papers arXiv:1611.08787 [hep-ph] and arXiv:1906.07924 [hep-ph]. When such a model contains only one term in the splitting function of the dominant parton distribution, then Bessel function appears to be the solution to this simplified DGLAP equation. To our knowledge, this model with only one term in the splitting function for the first time has been proposed by Blumlein in hep-ph/9506403. In arXiv:1906.07924 [hep-ph] we have shown that a dual integro-differential equation obtained from the DGLAP equation by a complex map in the plane of the Mellin moment in this model may be considered as the BFKL equation. Then, in arXiv:1906.07924 we have applied a complex diffeomorphism to obtain a standard integral from Gradshteyn and Ryzhik tables starting from the contour integral for parton distribution functions that is usually taken by calculus of residues. This standard integral from these tables appears to be the Laplace transformation of Jacobian for this complex diffeomorphism. Here we write up all the formulae behind this trick in detail and find out certain important points for further development of this strategy. We verify that the inverse Laplace transformation of the Laplace image of the Bessel function may be represented in a form of Barnes contour integral.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper spells out the contour steps turning their prior Bessel solution into a Barnes integral inside a one-term DGLAP toy model, but adds no new physics or broader method.

read the letter

The core of the paper is a detailed expansion of the complex-map steps that convert the inverse Laplace transform of the Bessel function into a Barnes contour integral. They start from the single-term splitting function model introduced by Blumlein and already used in their earlier arXiv:1906.07924, then walk through the diffeomorphism, the Laplace image, and the final contour representation. The derivations follow standard residue and table-integral manipulations without visible internal contradictions on the page. That explicit write-up is the main addition; the abstract claim that the inverse Laplace equals the Barnes form is just the verification of a known identity applied to their existing solution. The BFKL duality remark also traces straight back to the same prior work. The limitation is built in from the start: everything stays inside the one-term toy model, so the identities do not carry over to realistic multi-term splitting functions or produce new predictions for data. No new data, no parameter-free derivation outside the model, and no independent check against full DGLAP numerics appear. Readers already tracking this specific line of analytical tricks will find the intermediate formulae useful for reproducing the contour deformation. Anyone outside that niche will see only a narrow mathematical identity. The work is coherent on its own terms and the math steps look reproducible from the description, so it is worth sending to a referee who knows the contour-integral literature in hep-th. I would not cite it in my own papers, but a specialist reading group could usefully spend an hour on the explicit steps.

Referee Report

1 major / 0 minor

Summary. The manuscript provides a detailed write-up of formulae showing that, in a simplified single-term splitting function model for the DGLAP equation (originally due to Blumlein), the inverse Laplace transform of the Laplace image of the Bessel function solution equals a Barnes contour integral. This is obtained via complex maps, diffeomorphisms applied to standard contour integrals for parton distributions, and builds on the authors' prior work establishing a DGLAP-BFKL duality through Mellin-moment mappings.

Significance. If the identity holds, the paper supplies an explicit analytical verification of a Laplace-Barnes relation inside a solvable QCD evolution model and identifies concrete points for extending the complex-map strategy. The detailed formulae constitute a strength, as they make the contour-integral manipulations reproducible and could serve as a template for similar techniques in high-energy QCD.

major comments (1)

[Derivation of the Barnes contour integral representation (following the complex diffeomorphism)] The central claim is the verification that the inverse Laplace of the Bessel image equals the Barnes integral. The manuscript must explicitly document the contour domains and deformation steps (including any residue contributions) used in this step, as these are load-bearing for confirming the identity without hidden assumptions on the integration paths.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive suggestion regarding the explicit documentation of contour domains and deformations. We agree that making these steps fully transparent will improve the clarity and reproducibility of the central identity.

read point-by-point responses

Referee: [Derivation of the Barnes contour integral representation (following the complex diffeomorphism)] The central claim is the verification that the inverse Laplace of the Bessel image equals the Barnes integral. The manuscript must explicitly document the contour domains and deformation steps (including any residue contributions) used in this step, as these are load-bearing for confirming the identity without hidden assumptions on the integration paths.

Authors: We appreciate the referee's emphasis on this point. The manuscript already derives the identity by applying the complex diffeomorphism to the standard DGLAP contour integral and relating it to the Laplace transform whose inverse yields the Barnes representation. However, we agree that the domains of the contours (vertical Bromwich lines and their images under the map), the precise conditions permitting deformation without crossing poles, and an explicit statement that no residue contributions arise (or their accounting if they do) should be spelled out more explicitly. In the revised version we will insert a short dedicated paragraph immediately after Eq. (the one presenting the final Barnes integral) that specifies: (i) the original contour as the vertical line Re(s) = c > 0 with |Im(s)| → ∞, (ii) the image contour after the diffeomorphism, (iii) the region in which the map is analytic and bijective, and (iv) verification that the deformation stays within a domain free of singularities of the integrand, thereby confirming the equality holds without additional residues. revision: yes

Circularity Check

0 steps flagged

Verification of standard Laplace-Barnes identity is self-contained; no load-bearing reduction to self-citation or fit

full rationale

The paper's core claim is verification that the inverse Laplace transform of the Laplace image of the Bessel function equals a Barnes contour integral. This is a standard mathematical identity (cross-referenced to Gradshteyn-Ryzhik tables) applied inside the single-term splitting-function model. While the model and BFKL duality originate in the authors' prior arXiv papers, the present derivation chain consists of explicit contour manipulations and diffeomorphisms that are independently verifiable from external tables and calculus of residues; no equation reduces by construction to a fitted parameter or to an unverified self-citation. The Blumlein reference is external. Hence the result is not forced by internal definition or self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of contour integrals, Laplace transforms, and the residue theorem together with the single-term model introduced in prior literature; no new free parameters or invented entities are introduced.

axioms (2)

standard math Residue theorem and contour deformation rules for Mellin and Laplace integrals hold in the relevant domains
Invoked when mapping the parton distribution contour integral to a Laplace transform via complex diffeomorphism.
domain assumption The single-term splitting function model yields a Bessel function solution to the simplified DGLAP equation
Taken from Blumlein (hep-ph/9506403) and the authors' prior papers; used as the starting point for the map.

pith-pipeline@v0.9.0 · 5812 in / 1338 out tokens · 23958 ms · 2026-05-24T15:08:39.599934+00:00 · methodology

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages · cited by 1 Pith paper · 28 internal anchors

[1]

Algorithm to find an all-order in the running coupling solution to an equation of the DGLAP type

I. Kondrashuk, “Algorithm to ﬁnd an all-order in the running coup ling solution to an equation of the DGLAP type,” arXiv:1906.07924 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv 1906
[2]

Analytical solution to DGLAP integro-diﬀerential equation in a simple toy-model with a ﬁx ed gauge coupling,

G. Alvarez, G. Cvetic, B.A. Kniehl, I. Kondrashuk and I. Parra-F errada, “Analytical solution to DGLAP integro-diﬀerential equation in a simple toy-model with a ﬁx ed gauge coupling,” arXiv:1611.08787 [hep-ph]. 11

work page arXiv
[3]

J. W. Brown and R. V. Churchill, Complex variables and applications , (McGraw-Hill Higher Education, New York), 8th edition, 2009

work page 2009
[4]

Extracting Parton Densities from Collider Data

D. A. Kosower, “Extracting parton densities from collider data,” Nucl. Phys. B 520 (1998) 263 [hep-ph/9708392]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[5]

Asymptotic Sum Rules at Inﬁnite Momentum,

J. D. Bjorken, “Asymptotic Sum Rules at Inﬁnite Momentum,” Phy s. Rev. 179 (1969) 1547

work page 1969
[6]

Deep inelastic e p scattering in per turbation theory,

V. N. Gribov and L. N. Lipatov, “Deep inelastic e p scattering in per turbation theory,” Sov. J. Nucl. Phys. 15 (1972) 438 [Yad. Fiz. 15 (1972) 781]

work page 1972
[7]

e+ e- pair annihilation and deep inela stic e p scattering in perturbation theory,

V. N. Gribov and L. N. Lipatov, “e+ e- pair annihilation and deep inela stic e p scattering in perturbation theory,” Sov. J. Nucl. Phys. 15 (1972) 675 [Yad. Fiz. 15 (1972) 1218]

work page 1972
[8]

The parton model and perturbation theory,

L. N. Lipatov, “The parton model and perturbation theory,” So v. J. Nucl. Phys. 20 (1975) 94 [Yad. Fiz. 20 (1974) 181]

work page 1975
[9]

Asymptotically Free Gauge Theories. 2.,

D. J. Gross and F. Wilczek, “Asymptotically Free Gauge Theories. 2.,” Phys. Rev. D 9 (1974) 980

work page 1974
[10]

Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamic s.,

Y. L. Dokshitzer, “Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamic s.,” Sov. Phys. JETP 46 (1977) 641 [Zh. Eksp. Teor. Fiz. 73 (1977) 1216]

work page 1977
[11]

Asymptotic Freedom in Parton Langua ge,

G. Altarelli and G. Parisi, “Asymptotic Freedom in Parton Langua ge,” Nucl. Phys. B 126 (1977) 298

work page 1977
[12]

On the $k_{\perp}$ dependent gluon density of the proton

J. Blumlein, “On the k(T) dependent gluon density of the proton ,” hep-ph/9506403

work page internal anchor Pith review Pith/arXiv arXiv
[13]

The Three-Loop Splitting Functions in QCD: The Singlet Case

A. Vogt, S. Moch and J. A. M. Vermaseren, “The Three-loop sp litting functions in QCD: The Singlet case,” Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111]

work page internal anchor Pith review Pith/arXiv arXiv 2004
[14]

The Three-Loop Splitting Functions in QCD: The Non-Singlet Case

S. Moch, J. A. M. Vermaseren and A. Vogt, “The Three loop split ting functions in QCD: The Nonsinglet case,” Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192]

work page internal anchor Pith review Pith/arXiv arXiv 2004
[15]

A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics

J. Ablinger, “A Computer Algebra Toolbox for Harmonic Sums Rela ted to Particle Physics,” arXiv:1011.1176 [math-ph]

work page internal anchor Pith review Pith/arXiv arXiv
[16]

Computer Algebra Algorithms for Special Functions in Particle Physics

J. Ablinger, “Computer Algebra Algorithms for Special Function s in Particle Physics,” arXiv:1305.0687 [math-ph]

work page internal anchor Pith review Pith/arXiv arXiv
[17]

The Evolution of Unpolarized Singlet Structure Functions at Small x

J. Blumlein and A. Vogt, “The Evolution of unpolarized singlet stru cture functions at small x,” Phys. Rev. D 58 (1998) 014020 [hep-ph/9712546]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[18]

Parton Densit ies from Deep Inelastic Scattering to Hadronic Processes at Super Collider Energies,

M. Diemoz, F. Ferroni, E. Longo and G. Martinelli, “Parton Densit ies from Deep Inelastic Scattering to Hadronic Processes at Super Collider Energies,” Z. Ph ys. C 39 (1988) 21

work page 1988
[19]

Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable

J. Ablinger, J. Blmlein, P. Marquard, N. Rana and C. Schneider, “ Automated Solution of First Order Factorizable Systems of Diﬀerential Equations in One Va riable,” Nucl. Phys. B 939 (2019) 253 [arXiv:1810.12261 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[20]

Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS

A. Vogt, “Eﬃcient evolution of unpolarized and polarized parton distributions with QCD- PEGASUS,” Comput. Phys. Commun. 170 (2005) 65 [hep-ph/0408244]. 12

work page internal anchor Pith review Pith/arXiv arXiv 2005
[21]

Singlet parton evolution at small x: a theoretical update

G. Altarelli, R. D. Ball and S. Forte, “Singlet parton evolution at s mall x: A Theoretical update,” hep-ph/0001157

work page internal anchor Pith review Pith/arXiv arXiv
[22]

All Order Running Coupling BFKL Evolution from GLAP (and vice-versa)

R. D. Ball and S. Forte, “All order running coupling BFKL evolutio n from GLAP (and vice- versa),” Nucl. Phys. B 742 (2006) 158 [hep-ph/0601049]

work page internal anchor Pith review Pith/arXiv arXiv 2006
[23]

Harmonic Polylogarithms

E. Remiddi and J. A. M. Vermaseren, “Harmonic polylogarithms,” Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[24]

Fast evolution of parton distributions

S. Weinzierl, “Fast evolution of parton distributions,” Comput. P hys. Commun. 148 (2002) 314 [hep-ph/0203112]

work page internal anchor Pith review Pith/arXiv arXiv 2002
[25]

QCDNUM: Fast QCD Evolution and Convolution

M. Botje, “QCDNUM: Fast QCD Evolution and Convolution,” Compu t. Phys. Commun. 182 (2011) 490 [arXiv:1005.1481 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[26]

Erratum for the time-like evolution in QCDNUM,

M. Botje, “Erratum for the time-like evolution in QCDNUM,” arXiv:1 602.08383 [hep-ph]

work page
[27]

I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products , Edited by D. Zwillinger and V. Moll, (Academic Press, New York), 8th edition, 2015

work page 2015
[28]

New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC

T. J. Hou et al. , “New CTEQ global analysis of quantum chromodynamics with high-pr ecision data from the LHC,” arXiv:1912.10053 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv 1912
[29]

New parton distribution functions from a global analysis of quantum chromodynamics

S. Dulat et al. , “New parton distribution functions from a global analysis of quant um chromo- dynamics,” Phys. Rev. D 93 (2016) no.3, 033006 [arXiv:1506.07443 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[30]

PDFs: from Run II to Run III

S. Forte “PDFs: from Run II to Run III”, http://nnpdf.mi.infn.it /documents/talks/

work page
[31]

Machine learning PDFs

S. Forte “Machine learning PDFs”, http://nnpdf.mi.infn.it/docum ents/talks/

work page
[32]

Parton Distributions in Nucleons and Nuclei,

J. J. Ethier and E. R. Nocera, “Parton Distributions in Nucleons and Nuclei,” Ann. Rev. Nucl. Part. Sci. (2020) no.70, 1 [arXiv:2001.07722 [hep-ph]]

work page arXiv 2020
[33]

The asymptotic behaviour of parton distributions at small and large $x$

R. D. Ball, E. R. Nocera and J. Rojo, “The asymptotic behaviour of parton distributions at small and large x,” Eur. Phys. J. C 76 (2016) no.7, 383 [arXiv:1604.00024 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[34]

Mellin Representation for the Heavy Flavor Contributions to Deep Inelastic Structure Functions

S. I. Alekhin and J. Blumlein, “Mellin representation for the heavy ﬂavor contributions to deep inelastic structure functions,” Phys. Lett. B 594 (2004) 299 [hep-ph/0404034]

work page internal anchor Pith review Pith/arXiv arXiv 2004
[35]

NNLO analysis of unpolarized DIS structure functions

J. Blumlein, H. Bottcher and A. Guﬀanti, “NNLO analysis of unpola rized DIS structure func- tions,” hep-ph/0606309

work page internal anchor Pith review Pith/arXiv arXiv
[36]

Parton distribution functions and benchmark cross sections at NNLO

S. Alekhin, J. Blumlein and S. Moch, “Parton Distribution Function s and Benchmark Cross Sections at NNLO,” Phys. Rev. D 86 (2012) 054009 [arXiv:1202.2281 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[37]

Evaluating Feynman integrals,

V. A. Smirnov, “Evaluating Feynman integrals,” Springer Tracts Mod. Phys. 211 (2004) 1

work page 2004
[38]

New four-dimensional integrals by Mellin-Barnes transform

P. Allendes, N. Guerrero, I. Kondrashuk and E. A. Notte Cuello , “New four-dimensional inte- grals by Mellin-Barnes transform,” J. Math. Phys. 51 (2010) 052304 [arXiv:0910.4805 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[39]

Solution to Bethe-Salpeter equation via Mellin-Barnes transform

P. Allendes, B. Kniehl, I. Kondrashuk, E. A. Notte Cuello and M. R ojas Medar, “Solu- tion to Bethe-Salpeter equation via Mellin-Barnes transform,” Nucl. Phys. B 870 (2013) 243 [arXiv:1205.6257 [hep-th]]. 13

work page internal anchor Pith review Pith/arXiv arXiv 2013
[40]

Belokurov-Usyukina loop reduction in non-integer dimension

I. Gonzalez and I. Kondrashuk, “Belokurov-Usyukina loop red uction in non-integer dimension,” Phys. Part. Nucl. 44 (2013) 268 [arXiv:1206.4763 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[41]

Box ladders in non-integer dimension

I. Gonzalez and I. Kondrashuk, “Box ladders in a noninteger dim ension,” Theor. Math. Phys. 177 (2013) 1515 [Teor. Mat. Fiz. 177 (2013) no.1, 276] [arXiv:1210.2243 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[42]

Two-fold Mellin-Barnes transforms of Usyukina-Davydychev functions

B. A. Kniehl, I. Kondrashuk, E. A. Notte-Cuello, I. Parra-Fer rada and M. Rojas-Medar, “Two- fold Mellin-Barnes transforms of Usyukina-Davydychev functions ,” Nucl. Phys. B 876 (2013) 322 [arXiv:1304.3004 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[43]

Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1

I. Gonzalez, B. A. Kniehl, I. Kondrashuk, E. A. Notte-Cuello, I . Parra-Ferrada and M. A. Rojas- Medar, “Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1,” Nucl. Phys. B 925 (2017) 607 [arXiv:1608.04148 [math.MP]]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[44]

Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles

I. Gonzalez, I. Kondrashuk, E. A. Notte-Cuello and I. Parra- Ferrada, “Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman d iagrams: orthogonality of triangles,” Anal. Math. Phys. 8 (2018) no.4, 589 [arXiv:1808.08337 [math-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[45]

Reggeization of the Vector Meson and the Vacu um Singularity in Nonabelian Gauge Theories,

L. N. Lipatov, “Reggeization of the Vector Meson and the Vacu um Singularity in Nonabelian Gauge Theories,” Sov. J. Nucl. Phys. 23 (1976) 338 [Yad. Fiz. 23 (1976) 642]

work page 1976
[46]

On the Pomeranch uk Singularity in Asymptot- ically Free Theories,

V. S. Fadin, E. A. Kuraev and L. N. Lipatov, “On the Pomeranch uk Singularity in Asymptot- ically Free Theories,” Phys. Lett. B 60 (1975) 50

work page 1975
[47]

Multi - Reggeon Pr ocesses in the Yang-Mills Theory,

E. A. Kuraev, L. N. Lipatov and V. S. Fadin, “Multi - Reggeon Pr ocesses in the Yang-Mills Theory,” Sov. Phys. JETP 44 (1976) 443 [Zh. Eksp. Teor. Fiz. 71 (1976) 840]

work page 1976
[48]

The Pomeranchuk Singularity in Nonabelian Gauge Theories,

E. A. Kuraev, L. N. Lipatov and V. S. Fadin, “The Pomeranchuk Singularity in Nonabelian Gauge Theories,” Sov. Phys. JETP 45 (1977) 199 [Zh. Eksp. Teor. Fiz. 72 (1977) 377]

work page 1977
[49]

The Pomeranchuk Singularity in Quantum Chromodynam- ics,

I. I. Balitsky and L. N. Lipatov, “The Pomeranchuk Singularity in Quantum Chromodynam- ics,” Sov. J. Nucl. Phys. 28 (1978) 822 [Yad. Fiz. 28 (1978) 1597]. 14

work page 1978

[1] [1]

Algorithm to find an all-order in the running coupling solution to an equation of the DGLAP type

I. Kondrashuk, “Algorithm to ﬁnd an all-order in the running coup ling solution to an equation of the DGLAP type,” arXiv:1906.07924 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv 1906

[2] [2]

Analytical solution to DGLAP integro-diﬀerential equation in a simple toy-model with a ﬁx ed gauge coupling,

G. Alvarez, G. Cvetic, B.A. Kniehl, I. Kondrashuk and I. Parra-F errada, “Analytical solution to DGLAP integro-diﬀerential equation in a simple toy-model with a ﬁx ed gauge coupling,” arXiv:1611.08787 [hep-ph]. 11

work page arXiv

[3] [3]

J. W. Brown and R. V. Churchill, Complex variables and applications , (McGraw-Hill Higher Education, New York), 8th edition, 2009

work page 2009

[4] [4]

Extracting Parton Densities from Collider Data

D. A. Kosower, “Extracting parton densities from collider data,” Nucl. Phys. B 520 (1998) 263 [hep-ph/9708392]

work page internal anchor Pith review Pith/arXiv arXiv 1998

[5] [5]

Asymptotic Sum Rules at Inﬁnite Momentum,

J. D. Bjorken, “Asymptotic Sum Rules at Inﬁnite Momentum,” Phy s. Rev. 179 (1969) 1547

work page 1969

[6] [6]

Deep inelastic e p scattering in per turbation theory,

V. N. Gribov and L. N. Lipatov, “Deep inelastic e p scattering in per turbation theory,” Sov. J. Nucl. Phys. 15 (1972) 438 [Yad. Fiz. 15 (1972) 781]

work page 1972

[7] [7]

e+ e- pair annihilation and deep inela stic e p scattering in perturbation theory,

V. N. Gribov and L. N. Lipatov, “e+ e- pair annihilation and deep inela stic e p scattering in perturbation theory,” Sov. J. Nucl. Phys. 15 (1972) 675 [Yad. Fiz. 15 (1972) 1218]

work page 1972

[8] [8]

The parton model and perturbation theory,

L. N. Lipatov, “The parton model and perturbation theory,” So v. J. Nucl. Phys. 20 (1975) 94 [Yad. Fiz. 20 (1974) 181]

work page 1975

[9] [9]

Asymptotically Free Gauge Theories. 2.,

D. J. Gross and F. Wilczek, “Asymptotically Free Gauge Theories. 2.,” Phys. Rev. D 9 (1974) 980

work page 1974

[10] [10]

Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamic s.,

Y. L. Dokshitzer, “Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamic s.,” Sov. Phys. JETP 46 (1977) 641 [Zh. Eksp. Teor. Fiz. 73 (1977) 1216]

work page 1977

[11] [11]

Asymptotic Freedom in Parton Langua ge,

G. Altarelli and G. Parisi, “Asymptotic Freedom in Parton Langua ge,” Nucl. Phys. B 126 (1977) 298

work page 1977

[12] [12]

On the $k_{\perp}$ dependent gluon density of the proton

J. Blumlein, “On the k(T) dependent gluon density of the proton ,” hep-ph/9506403

work page internal anchor Pith review Pith/arXiv arXiv

[13] [13]

The Three-Loop Splitting Functions in QCD: The Singlet Case

A. Vogt, S. Moch and J. A. M. Vermaseren, “The Three-loop sp litting functions in QCD: The Singlet case,” Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111]

work page internal anchor Pith review Pith/arXiv arXiv 2004

[14] [14]

The Three-Loop Splitting Functions in QCD: The Non-Singlet Case

S. Moch, J. A. M. Vermaseren and A. Vogt, “The Three loop split ting functions in QCD: The Nonsinglet case,” Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192]

work page internal anchor Pith review Pith/arXiv arXiv 2004

[15] [15]

A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics

J. Ablinger, “A Computer Algebra Toolbox for Harmonic Sums Rela ted to Particle Physics,” arXiv:1011.1176 [math-ph]

work page internal anchor Pith review Pith/arXiv arXiv

[16] [16]

Computer Algebra Algorithms for Special Functions in Particle Physics

J. Ablinger, “Computer Algebra Algorithms for Special Function s in Particle Physics,” arXiv:1305.0687 [math-ph]

work page internal anchor Pith review Pith/arXiv arXiv

[17] [17]

The Evolution of Unpolarized Singlet Structure Functions at Small x

J. Blumlein and A. Vogt, “The Evolution of unpolarized singlet stru cture functions at small x,” Phys. Rev. D 58 (1998) 014020 [hep-ph/9712546]

work page internal anchor Pith review Pith/arXiv arXiv 1998

[18] [18]

Parton Densit ies from Deep Inelastic Scattering to Hadronic Processes at Super Collider Energies,

M. Diemoz, F. Ferroni, E. Longo and G. Martinelli, “Parton Densit ies from Deep Inelastic Scattering to Hadronic Processes at Super Collider Energies,” Z. Ph ys. C 39 (1988) 21

work page 1988

[19] [19]

Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable

J. Ablinger, J. Blmlein, P. Marquard, N. Rana and C. Schneider, “ Automated Solution of First Order Factorizable Systems of Diﬀerential Equations in One Va riable,” Nucl. Phys. B 939 (2019) 253 [arXiv:1810.12261 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2019

[20] [20]

Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS

A. Vogt, “Eﬃcient evolution of unpolarized and polarized parton distributions with QCD- PEGASUS,” Comput. Phys. Commun. 170 (2005) 65 [hep-ph/0408244]. 12

work page internal anchor Pith review Pith/arXiv arXiv 2005

[21] [21]

Singlet parton evolution at small x: a theoretical update

G. Altarelli, R. D. Ball and S. Forte, “Singlet parton evolution at s mall x: A Theoretical update,” hep-ph/0001157

work page internal anchor Pith review Pith/arXiv arXiv

[22] [22]

All Order Running Coupling BFKL Evolution from GLAP (and vice-versa)

R. D. Ball and S. Forte, “All order running coupling BFKL evolutio n from GLAP (and vice- versa),” Nucl. Phys. B 742 (2006) 158 [hep-ph/0601049]

work page internal anchor Pith review Pith/arXiv arXiv 2006

[23] [23]

Harmonic Polylogarithms

E. Remiddi and J. A. M. Vermaseren, “Harmonic polylogarithms,” Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237]

work page internal anchor Pith review Pith/arXiv arXiv 2000

[24] [24]

Fast evolution of parton distributions

S. Weinzierl, “Fast evolution of parton distributions,” Comput. P hys. Commun. 148 (2002) 314 [hep-ph/0203112]

work page internal anchor Pith review Pith/arXiv arXiv 2002

[25] [25]

QCDNUM: Fast QCD Evolution and Convolution

M. Botje, “QCDNUM: Fast QCD Evolution and Convolution,” Compu t. Phys. Commun. 182 (2011) 490 [arXiv:1005.1481 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2011

[26] [26]

Erratum for the time-like evolution in QCDNUM,

M. Botje, “Erratum for the time-like evolution in QCDNUM,” arXiv:1 602.08383 [hep-ph]

work page

[27] [27]

I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products , Edited by D. Zwillinger and V. Moll, (Academic Press, New York), 8th edition, 2015

work page 2015

[28] [28]

New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC

T. J. Hou et al. , “New CTEQ global analysis of quantum chromodynamics with high-pr ecision data from the LHC,” arXiv:1912.10053 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv 1912

[29] [29]

New parton distribution functions from a global analysis of quantum chromodynamics

S. Dulat et al. , “New parton distribution functions from a global analysis of quant um chromo- dynamics,” Phys. Rev. D 93 (2016) no.3, 033006 [arXiv:1506.07443 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[30] [30]

PDFs: from Run II to Run III

S. Forte “PDFs: from Run II to Run III”, http://nnpdf.mi.infn.it /documents/talks/

work page

[31] [31]

Machine learning PDFs

S. Forte “Machine learning PDFs”, http://nnpdf.mi.infn.it/docum ents/talks/

work page

[32] [32]

Parton Distributions in Nucleons and Nuclei,

J. J. Ethier and E. R. Nocera, “Parton Distributions in Nucleons and Nuclei,” Ann. Rev. Nucl. Part. Sci. (2020) no.70, 1 [arXiv:2001.07722 [hep-ph]]

work page arXiv 2020

[33] [33]

The asymptotic behaviour of parton distributions at small and large $x$

R. D. Ball, E. R. Nocera and J. Rojo, “The asymptotic behaviour of parton distributions at small and large x,” Eur. Phys. J. C 76 (2016) no.7, 383 [arXiv:1604.00024 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[34] [34]

Mellin Representation for the Heavy Flavor Contributions to Deep Inelastic Structure Functions

S. I. Alekhin and J. Blumlein, “Mellin representation for the heavy ﬂavor contributions to deep inelastic structure functions,” Phys. Lett. B 594 (2004) 299 [hep-ph/0404034]

work page internal anchor Pith review Pith/arXiv arXiv 2004

[35] [35]

NNLO analysis of unpolarized DIS structure functions

J. Blumlein, H. Bottcher and A. Guﬀanti, “NNLO analysis of unpola rized DIS structure func- tions,” hep-ph/0606309

work page internal anchor Pith review Pith/arXiv arXiv

[36] [36]

Parton distribution functions and benchmark cross sections at NNLO

S. Alekhin, J. Blumlein and S. Moch, “Parton Distribution Function s and Benchmark Cross Sections at NNLO,” Phys. Rev. D 86 (2012) 054009 [arXiv:1202.2281 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[37] [37]

Evaluating Feynman integrals,

V. A. Smirnov, “Evaluating Feynman integrals,” Springer Tracts Mod. Phys. 211 (2004) 1

work page 2004

[38] [38]

New four-dimensional integrals by Mellin-Barnes transform

P. Allendes, N. Guerrero, I. Kondrashuk and E. A. Notte Cuello , “New four-dimensional inte- grals by Mellin-Barnes transform,” J. Math. Phys. 51 (2010) 052304 [arXiv:0910.4805 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2010

[39] [39]

Solution to Bethe-Salpeter equation via Mellin-Barnes transform

P. Allendes, B. Kniehl, I. Kondrashuk, E. A. Notte Cuello and M. R ojas Medar, “Solu- tion to Bethe-Salpeter equation via Mellin-Barnes transform,” Nucl. Phys. B 870 (2013) 243 [arXiv:1205.6257 [hep-th]]. 13

work page internal anchor Pith review Pith/arXiv arXiv 2013

[40] [40]

Belokurov-Usyukina loop reduction in non-integer dimension

I. Gonzalez and I. Kondrashuk, “Belokurov-Usyukina loop red uction in non-integer dimension,” Phys. Part. Nucl. 44 (2013) 268 [arXiv:1206.4763 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2013

[41] [41]

Box ladders in non-integer dimension

I. Gonzalez and I. Kondrashuk, “Box ladders in a noninteger dim ension,” Theor. Math. Phys. 177 (2013) 1515 [Teor. Mat. Fiz. 177 (2013) no.1, 276] [arXiv:1210.2243 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2013

[42] [42]

Two-fold Mellin-Barnes transforms of Usyukina-Davydychev functions

B. A. Kniehl, I. Kondrashuk, E. A. Notte-Cuello, I. Parra-Fer rada and M. Rojas-Medar, “Two- fold Mellin-Barnes transforms of Usyukina-Davydychev functions ,” Nucl. Phys. B 876 (2013) 322 [arXiv:1304.3004 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2013

[43] [43]

Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1

I. Gonzalez, B. A. Kniehl, I. Kondrashuk, E. A. Notte-Cuello, I . Parra-Ferrada and M. A. Rojas- Medar, “Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1,” Nucl. Phys. B 925 (2017) 607 [arXiv:1608.04148 [math.MP]]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[44] [44]

Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles

I. Gonzalez, I. Kondrashuk, E. A. Notte-Cuello and I. Parra- Ferrada, “Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman d iagrams: orthogonality of triangles,” Anal. Math. Phys. 8 (2018) no.4, 589 [arXiv:1808.08337 [math-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[45] [45]

Reggeization of the Vector Meson and the Vacu um Singularity in Nonabelian Gauge Theories,

L. N. Lipatov, “Reggeization of the Vector Meson and the Vacu um Singularity in Nonabelian Gauge Theories,” Sov. J. Nucl. Phys. 23 (1976) 338 [Yad. Fiz. 23 (1976) 642]

work page 1976

[46] [46]

On the Pomeranch uk Singularity in Asymptot- ically Free Theories,

V. S. Fadin, E. A. Kuraev and L. N. Lipatov, “On the Pomeranch uk Singularity in Asymptot- ically Free Theories,” Phys. Lett. B 60 (1975) 50

work page 1975

[47] [47]

Multi - Reggeon Pr ocesses in the Yang-Mills Theory,

E. A. Kuraev, L. N. Lipatov and V. S. Fadin, “Multi - Reggeon Pr ocesses in the Yang-Mills Theory,” Sov. Phys. JETP 44 (1976) 443 [Zh. Eksp. Teor. Fiz. 71 (1976) 840]

work page 1976

[48] [48]

The Pomeranchuk Singularity in Nonabelian Gauge Theories,

E. A. Kuraev, L. N. Lipatov and V. S. Fadin, “The Pomeranchuk Singularity in Nonabelian Gauge Theories,” Sov. Phys. JETP 45 (1977) 199 [Zh. Eksp. Teor. Fiz. 72 (1977) 377]

work page 1977

[49] [49]

The Pomeranchuk Singularity in Quantum Chromodynam- ics,

I. I. Balitsky and L. N. Lipatov, “The Pomeranchuk Singularity in Quantum Chromodynam- ics,” Sov. J. Nucl. Phys. 28 (1978) 822 [Yad. Fiz. 28 (1978) 1597]. 14

work page 1978