pith. sign in

arxiv: 2507.07175 · v3 · submitted 2025-07-09 · ✦ hep-ph

Polarized Deep Inelastic Scattering as x to 1 using Soft Collinear Effective Theory

Jaipratap Singh Grewal , Aneesh V. Manohar , Jyotirmoy Roy This is my paper

Pith reviewed 2026-05-19 05:29 UTC · model grok-4.3

classification ✦ hep-ph
keywords polarized deep inelastic scatteringsoft collinear effective theorystructure functions g1 g2Sudakov resummationlarge-x limittwist-three operatorsparton distribution functions
0
0 comments X p. Extension

The pith

Soft Collinear Effective Theory factorizes the polarized DIS structure functions g1 and g2 near x=1 and resums Sudakov double logarithms.

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

The paper shows how Soft Collinear Effective Theory organizes the polarized deep inelastic scattering structure functions g1(x) and g2(x) when the momentum fraction x approaches one. Factorization separates the hard scattering from the soft and collinear parts, allowing all-order resummation of large logarithms of (1-x). The treatment uses both the x-space distributions and their moments, with g2 requiring subleading operators that include gluon fields. One-loop matching coefficients between QCD and these SCET operators are computed, along with the anomalous dimension of the resulting bilocal PDF operator. As x approaches one this anomalous dimension reduces to a single-variable evolution equation.

Core claim

SCET factorization theorems apply to the polarized structure functions g1 and g2, with the parton distribution function represented by a bilocal operator rather than a trilocal one. One-loop matching from QCD onto the subleading gluon-containing operators is performed, and the one-loop anomalous dimension of the PDF operator is shown to factor into single-variable evolution as x approaches 1. A side result establishes the 1/N dependence of the QCD coefficient functions for F1, FL, and g1 in the N to infinity limit.

What carries the argument

Soft Collinear Effective Theory factorization using subleading operators that contain gluons to handle the g2 structure function.

If this is right

  • The structure functions g1 and g2 near x=1 can be written in terms of SCET matrix elements whose logarithms are resummed to all orders.
  • The bilocal SCET operator for the polarized PDF obeys a simpler evolution equation than the corresponding trilocal QCD operator when x is close to one.
  • The large-N moments of the coefficient functions for unpolarized and polarized structure functions share a common 1/N suppression pattern that holds to all orders in the strong coupling.
  • Comments on the equation-of-motion relations among twist-three operators in QCD follow directly from the SCET matching.

Where Pith is reading between the lines

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

  • The same SCET operators could be used to derive higher-order resummation for other polarized observables that involve twist-three contributions.
  • The bilocal operator formulation may simplify lattice calculations of polarized distributions in the large-x regime compared with the trilocal QCD version.
  • The single-variable evolution suggests that threshold resummation techniques developed for unpolarized DIS can be adapted with only minor changes for the polarized case.

Load-bearing premise

The SCET factorization theorems and power counting continue to hold for polarized scattering once subleading gluon operators are included at large x.

What would settle it

An explicit two-loop calculation of the anomalous dimension of the bilocal PDF operator that fails to reduce to single-variable evolution as x approaches 1 would contradict the claimed factorization.

Figures

Figures reproduced from arXiv: 2507.07175 by Aneesh V. Manohar, Jaipratap Singh Grewal, Jyotirmoy Roy.

Figure 1
Figure 1. Figure 1: Loop graphs that generate the PDF running or a Wilson line. Graph (l) is one-particle reducible, but contributes for on-shell matching. [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
read the original abstract

We use Soft Collinear Effective Theory (SCET) to factorize the polarized Deep Inelastic Scattering (DIS) structure functions $g_1(x)$ and $g_2(x)$, and to sum Sudakov double logarithms of $1-x$. The analysis is done both in terms of lightcone parton distributions and their moments. Computing $g_2$ requires subleading SCET operators which contain gluons. We calculate the one-loop matching coefficients from QCD onto these subleading SCET operators, and the one-loop matching from SCET onto the parton distribution function (PDF). The PDF in SCET is given by a bilocal operator, rather than the trilocal operator used in the QCD analysis of $g_2$ for generic $x$. We compute the one-loop anomalous dimension of the PDF operator for any $x$, and show that as $x \to 1$, it factors into a single-variable evolution. We comment on the QCD anomalous dimensions of twist-three operators, their equation-of-motion relation, and connection to the SCET analysis. We briefly discuss the definition of axial operators in the BMHV scheme. As a side result, we derive the $1/N$ dependence of the QCD coefficient functions for $F_1$, $F_L$ and $g_1$ in the $N \to \infty$ limit, where $N$ is the moment, which is expected to hold to all orders in $\alpha_s$.

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.

Referee Report

1 major / 2 minor

Summary. The paper uses Soft Collinear Effective Theory (SCET) to factorize the polarized DIS structure functions g1(x) and g2(x) as x → 1, summing Sudakov double logarithms. It computes one-loop matching coefficients from QCD to subleading SCET operators (including gluon-containing operators for g2), one-loop matching from SCET to the PDF, and the one-loop anomalous dimension of the bilocal SCET PDF operator for arbitrary x, demonstrating that it factors into a single-variable form as x → 1. The work also discusses the relation to QCD twist-3 operators via equations of motion, the BMHV scheme for axial operators, and derives the 1/N dependence of QCD coefficient functions for F1, FL, and g1 in the N → ∞ limit.

Significance. If the SCET factorization and operator mappings hold, the paper offers a systematic approach to resumming large logarithms in polarized DIS near x=1, with explicit one-loop calculations supporting the central claims. The demonstration that the anomalous dimension factors at x→1 and the side result on coefficient functions are useful contributions. The explicit treatment of subleading operators for g2 and the connection to QCD twist-3 via EOM provide a bridge between SCET and standard QCD analyses.

major comments (1)
  1. [Factorization and matching sections] The factorization theorem for g2 relies on the equivalence of the bilocal SCET PDF operator to the trilocal QCD twist-3 operator via an equation-of-motion relation at x→1 (invoked in the factorization and matching sections). While one-loop matching coefficients and the anomalous dimension are computed explicitly, it is unclear whether this mapping holds without additional mixing or power corrections when subleading gluon operators are included under polarized axial currents in the BMHV scheme. A concrete check of the power counting and any residual terms at large x would be needed to confirm the central claim.
minor comments (2)
  1. The abstract states that the analysis is performed both in terms of lightcone parton distributions and their moments; ensure the moment-space results are cross-referenced clearly with the x-space factorization to aid readability.
  2. The side result on the 1/N dependence of coefficient functions for F1, FL, and g1 is mentioned briefly; a short appendix or explicit derivation sketch would help readers verify the all-orders expectation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment on the factorization theorem for g2. We address the point below and have incorporated additional clarification in the revised version.

read point-by-point responses

  1. Referee: [Factorization and matching sections] The factorization theorem for g2 relies on the equivalence of the bilocal SCET PDF operator to the trilocal QCD twist-3 operator via an equation-of-motion relation at x→1 (invoked in the factorization and matching sections). While one-loop matching coefficients and the anomalous dimension are computed explicitly, it is unclear whether this mapping holds without additional mixing or power corrections when subleading gluon operators are included under polarized axial currents in the BMHV scheme. A concrete check of the power counting and any residual terms at large x would be needed to confirm the central claim.

    Authors: We thank the referee for highlighting this aspect. The factorization theorem for g2 is derived from the equation-of-motion relation that identifies the bilocal SCET PDF operator with the trilocal QCD twist-3 operator at leading power as x → 1. This relation is applied consistently in the factorization and matching sections. The one-loop matching coefficients for the subleading gluon-containing operators are computed explicitly within the BMHV scheme for the polarized axial currents, and these calculations show no additional mixing at this order. Power corrections are suppressed by (1-x) and do not enter the leading-power result. To address the request for a concrete check, we have added a new paragraph in Section 3.2 that explicitly discusses the power counting for the subleading operators and verifies the absence of residual leading-power terms at large x. This revision confirms that the mapping remains valid under the stated conditions. revision: partial

Circularity Check

0 steps flagged

SCET factorization, matching, and anomalous dimension computed directly from Lagrangian and operators

full rationale

The derivation begins from the SCET Lagrangian and power counting, proceeds to explicit one-loop matching coefficients from QCD onto subleading SCET operators containing gluons, then to SCET-to-PDF matching, and finally to the one-loop anomalous dimension of the bilocal PDF operator for arbitrary x. The x to 1 factorization of this anomalous dimension is shown to follow from the computed operator structure rather than any external input or fit. The paper comments on the QCD twist-3 EOM relation and its connection to the SCET bilocal operator but does not use that relation as a load-bearing step to derive the central results; those results are obtained independently within SCET. No parameters are fitted to data, no self-citations justify uniqueness or ansatze for the main claims, and the side result on 1/N coefficient functions is derived separately. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard SCET power counting and factorization theorems applied to polarized operators; no new free parameters are introduced beyond the perturbative order, and no invented entities are postulated.

axioms (2)
  • domain assumption SCET factorization and power counting hold for polarized DIS structure functions including subleading operators as x approaches 1
    Invoked to justify the operator matching and resummation of Sudakov logs in the x to 1 limit.
  • standard math One-loop matching coefficients can be computed by equating QCD and SCET matrix elements in the appropriate kinematic regime
    Used for the explicit calculations of coefficients from QCD onto SCET operators and SCET onto the PDF.

pith-pipeline@v0.9.0 · 5816 in / 1391 out tokens · 34519 ms · 2026-05-19T05:29:17.293930+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We use Soft Collinear Effective Theory (SCET) to factorize the polarized Deep Inelastic Scattering (DIS) structure functions g1(x) and g2(x), and to sum Sudakov double logarithms of 1−x. ... We calculate the one-loop matching coefficients from QCD onto these subleading SCET operators, and the one-loop matching from SCET onto the parton distribution function (PDF).

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The PDF in SCET is given by a bilocal operator, rather than the trilocal operator used in the QCD analysis of g2 for generic x. We compute the one-loop anomalous dimension of the PDF operator for any x, and show that as x→1, it factors into a single-variable evolution.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · 29 internal anchors

  1. [1]

    Combination of Measurements of Inclusive Deep Inelastic $e^{\pm}p$ Scattering Cross Sections and QCD Analysis of HERA Data

    H1, ZEUS collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep inelastic e±p scattering cross sections and QCD analysis of HERA data , Eur. Phys. J. C 75 (2015) 580, [ 1506.06042]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  2. [2]

    Final COMPASS results on the deuteron spin-dependent structure function $g_1^{\rm d}$ and the Bjorken sum rule

    COMPASS collaboration, C. Adolph et al., Final COMPASS results on the deuteron spin-dependent structure function gd 1 and the Bjorken sum rule , Phys. Lett. B 769 (2017) 34–41, [1612.00620]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  3. [3]

    Ruth et al., Proton spin structure and generalized polarizabilities in the strong quantum chromodynamics regime , Nature Phys

    Jefferson Lab Hall A g2p collaboration, D. Ruth et al., Proton spin structure and generalized polarizabilities in the strong quantum chromodynamics regime , Nature Phys. 18 (2022) 1441–1446, [ 2204.10224]

    work page arXiv 2022

  4. [4]

    Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report

    R. Abdul Khalek et al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report , Nucl. Phys. A 1026 (2022) 122447, [ 2103.05419]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  5. [5]

    C. W. Bauer, S. Fleming and M. E. Luke, Summing Sudakov logarithms in B → Xsγ in effective field theory, Phys. Rev. D63 (2000) 014006, [ hep-ph/0005275]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  6. [6]

    C. W. Bauer, D. Pirjol and I. W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D65 (2002) 054022, [ hep-ph/0109045]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  7. [7]

    C. W. Bauer and I. W. Stewart, Invariant operators in collinear effective theory , Phys. Lett. B516 (2001) 134–142, [ hep-ph/0107001]. – 41 –

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  8. [8]

    C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein and I. W. Stewart, Hard scattering factorization from effective field theory , Phys. Rev. D 66 (2002) 014017, [ hep-ph/0202088]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  9. [9]

    A. V. Manohar, Deep inelastic scattering as x → 1 using soft-collinear effective theory , Phys. Rev. D68 (2003) 114019, [ hep-ph/0309176]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  10. [10]

    Inglis-Whalen, M

    M. Inglis-Whalen, M. Luke, J. Roy and A. Spourdalakis, Factorization of power corrections in the Drell-Yan process in EFT , Phys. Rev. D 104 (2021) 076018, [ 2105.09277]

    work page arXiv 2021

  11. [11]

    M. Luke, J. Roy and A. Spourdalakis, Factorization at subleading power in deep inelastic scattering in the x → 1 limit, Phys. Rev. D 107 (2023) 074023, [ 2210.02529]

    work page arXiv 2023

  12. [12]

    Renormalization of Dijet Operators at Order $1/Q^2$ in Soft-Collinear Effective Theory

    R. Goerke and M. Inglis-Whalen, Renormalization of dijet operators at order 1/Q2 in soft-collinear effective theory, JHEP 05 (2018) 023, [ 1711.09147]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  13. [13]

    A. Ali, V. M. Braun and G. Hiller, Asymptotic solutions of the evolution equation for the polarized nucleon structure function g2(x, Q2), Phys.Lett. B266 (1991) 117–125

    work page 1991

  14. [14]

    The evolution of the nonsinglet twist-3 parton distribution function

    B. Geyer, D. Mueller and D. Robaschik, The evolution of the nonsinglet twist - three parton distribution function, in 3rd Meeting on the Prospects of Nucleon-Nucleon Spin Physics at HERA, 11, 1996, hep-ph/9611452

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  15. [15]

    S. A. Larin and J. A. M. Vermaseren, The α3 s corrections to the Bjorken sum rule for polarized electroproduction and to the Gross-Llewellyn Smith sum rule , Phys. Lett. B259 (1991) 345–352

    work page 1991

  16. [16]

    S. A. Larin, The Renormalization of the axial anomaly in dimensional regularization , Phys. Lett. B303 (1993) 113–118, [ hep-ph/9302240]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  17. [17]

    R. L. Jaffe and A. Manohar, Deep Inelastic Scattering from Arbitrary Spin Targets , Nucl. Phys. B 321 (1989) 343

    work page 1989

  18. [18]

    Hoodbhoy, R

    P. Hoodbhoy, R. L. Jaffe and A. Manohar, Novel Effects in Deep Inelastic Scattering from Spin 1 Hadrons , Nucl. Phys. B 312 (1989) 571–588

    work page 1989

  19. [19]

    Power Counting and Modes in SCET

    R. Goerke and M. Luke, Power Counting and Modes in SCET , JHEP 02 (2018) 147, [1711.09136]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  20. [20]

    A. V. Manohar, An introduction to spin dependent deep inelastic scattering , hep-ph/9204208

    work page internal anchor Pith review Pith/arXiv arXiv

  21. [21]

    A. V. Manohar, Polarized parton distribution functions , Phys. Rev. Lett. 66 (1991) 289–292

    work page 1991

  22. [22]

    A. V. Manohar, Parton distributions from an operator viewpoint , Phys. Rev. Lett. 65 (1990) 2511–2514

    work page 1990

  23. [23]

    Burkhardt and W

    H. Burkhardt and W. N. Cottingham, Sum rules for forward virtual Compton scattering , Annals Phys. 56 (1970) 453–463

    work page 1970

  24. [24]

    E. V. Shuryak and A. I. Vainshtein, Theory of Power Corrections to Deep Inelastic Scattering in Quantum Chromodynamics. 2. Q−4 Effects: Polarized Target, Nucl. Phys. B 201 (1982) 141

    work page 1982

  25. [25]

    Kodaira, S

    J. Kodaira, S. Matsuda, T. Muta, K. Sasaki and T. Uematsu, QCD Effects in Polarized Electroproduction, Phys. Rev. D20 (1979) 627

    work page 1979

  26. [26]

    Kodaira, S

    J. Kodaira, S. Matsuda, K. Sasaki and T. Uematsu, QCD Higher Order Effects in Spin Dependent Deep Inelastic Electroproduction, Nucl. Phys. B159 (1979) 99–124. – 42 –

    work page 1979

  27. [27]

    Kodaira, QCD Higher Order Effects in Polarized Electroproduction: Flavor Singlet Coefficient Functions, Nucl

    J. Kodaira, QCD Higher Order Effects in Polarized Electroproduction: Flavor Singlet Coefficient Functions, Nucl. Phys. B165 (1980) 129–140

    work page 1980

  28. [28]

    Spin Structure Function $g_2$ and Twist-3 Operators in QCD

    J. Kodaira, Y. Yasui and T. Uematsu, Spin structure function g2(x, Q2) and twist - three operators in QCD, Phys. Lett. B 344 (1995) 348–354, [ hep-ph/9408354]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  29. [29]

    A. P. Bukhvostov, E. A. Kuraev and L. N. Lipatov, Deep inelastic electron scattering by a polarized target in quantum chromodynamics , JETP Lett. 37 (1983) 482–486

    work page 1983

  30. [30]

    P. G. Ratcliffe, Transverse Spin and Higher Twist in QCD , Nucl. Phys. B 264 (1986) 493–512

    work page 1986

  31. [31]

    Ji and C.-h

    X.-D. Ji and C.-h. Chou, QCD radiative corrections to the transverse spin structure function g2(x, Q2): 1. Nonsinglet operators , Phys. Rev. D 42 (1990) 3637–3644

    work page 1990

  32. [32]

    I. I. Balitsky and V. M. Braun, Evolution Equations for QCD String Operators , Nucl. Phys. B 311 (1989) 541–584

    work page 1989

  33. [33]

    Jaffe, Parton Distribution Functions for Twist Four , Nucl.Phys

    R. Jaffe, Parton Distribution Functions for Twist Four , Nucl.Phys. B229 (1983) 205

    work page 1983

  34. [34]

    A. V. Manohar and I. W. Stewart, The Zero-Bin and Mode Factorization in Quantum Field Theory, Phys. Rev. D76 (2007) 074002, [ hep-ph/0605001]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  35. [35]

    S. M. Freedman and M. Luke, SCET, QCD and Wilson Lines , Phys. Rev. D 85 (2012) 014003, [1107.5823]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  36. [36]

    A. V. Manohar, T. Mehen, D. Pirjol and I. W. Stewart, Reparameterization invariance for collinear operators, Phys. Lett. B 539 (2002) 59–66, [ hep-ph/0204229]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  37. [37]

    X.-D. Ji, W. Lu, J. Osborne and X.-T. Song, One loop factorization of the nucleon g2 structure function in the nonsinglet case , Phys. Rev. D 62 (2000) 094016, [hep-ph/0006121]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  38. [38]

    S. M. Freedman and R. Goerke, Renormalization of Subleading Dijet Operators in Soft-Collinear Effective Theory, Phys. Rev. D 90 (2014) 114010, [ 1408.6240]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  39. [39]

    Goerke, A New Formalism for Soft Collinear Effective Theory with Applications , Ph.D

    R. Goerke, A New Formalism for Soft Collinear Effective Theory with Applications , Ph.D. thesis, Toronto U., 2018

    work page 2018

  40. [40]

    Ji, Gluon correlations in the transversely polarized nucleon , Phys

    X.-D. Ji, Gluon correlations in the transversely polarized nucleon , Phys. Lett. B 289 (1992) 137–142

    work page 1992

  41. [41]

    De Rujula, H

    A. De Rujula, H. Georgi and H. D. Politzer, Demythification of Electroproduction, Local Duality and Precocious Scaling, Annals Phys. 103 (1977) 315

    work page 1977

  42. [42]

    W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Deep Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories , Phys. Rev. D18 (1978) 3998

    work page 1978

  43. [43]

    Ahmed and G

    M. Ahmed and G. G. Ross, Polarized Lepton - Hadron Scattering in Asymptotically Free Gauge Theories, Nucl.Phys. B111 (1976) 441

    work page 1976

  44. [44]

    Bl¨ umlein,The theory of deeply inelastic scattering , Progress in Particle and Nuclear Physics 69 (Mar., 2013) 28–84

    J. Bl¨ umlein,The theory of deeply inelastic scattering , Progress in Particle and Nuclear Physics 69 (Mar., 2013) 28–84

    work page 2013

  45. [45]

    The Calculation of the Two-Loop Spin Splitting Functions $P_{ij}^{(1)}(x)$

    R. Mertig and W. L. van Neerven, The Calculation of the two loop spin splitting functions P(ij)(1)(x), Z. Phys. C70 (1996) 637–654, [ hep-ph/9506451]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  46. [46]

    A Rederivation of the Spin-dependent Next-to-leading Order Splitting Functions

    W. Vogelsang, A Rederivation of the spin dependent next-to-leading order splitting functions , Phys. Rev. D 54 (1996) 2023–2029, [ hep-ph/9512218]. – 43 –

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  47. [47]

    S. A. Larin, P. Nogueira, T. van Ritbergen and J. A. M. Vermaseren, The Three loop QCD calculation of the moments of deep inelastic structure functions , Nucl. Phys. B 492 (1997) 338–378, [hep-ph/9605317]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  48. [48]

    S. Moch, J. A. M. Vermaseren and A. Vogt, The three-loop splitting functions in QCD: The non-singlet case, Nucl. Phys. B688 (2004) 101–134, [ hep-ph/0403192]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  49. [49]

    A. Vogt, S. Moch and J. A. M. Vermaseren, The three-loop splitting functions in QCD: The singlet case, Nucl. Phys. B691 (2004) 129–181, [ hep-ph/0404111]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  50. [50]

    S. Moch, J. A. M. Vermaseren and A. Vogt, The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case , Nucl. Phys. B 889 (2014) 351–400, [ 1409.5131]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  51. [51]

    Bl¨ umlein, P

    J. Bl¨ umlein, P. Marquard, C. Schneider and K. Sch¨ onwald,The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements , JHEP 01 (2022) 193, [2111.12401]

    work page arXiv 2022

  52. [52]

    Bl¨ umlein, P

    J. Bl¨ umlein, P. Marquard, C. Schneider and K. Sch¨ onwald,The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements , Nucl. Phys. B 971 (2021) 115542, [ 2107.06267]

    work page arXiv 2021

  53. [53]

    Bl¨ umlein, P

    J. Bl¨ umlein, P. Marquard, C. Schneider and K. Sch¨ onwald,The massless three-loop Wilson coefficients for the deep-inelastic structure functions F 2, F L, xF 3 and g 1, JHEP 11 (2022) 156, [2208.14325]

    work page arXiv 2022

  54. [54]

    The Three-Loop Splitting Functions $P_{qg}^{(2)}$ and $P_{gg}^{(2, N_F)}$

    J. Ablinger, A. Behring, J. Bl¨ umlein, A. De Freitas, A. von Manteuffel and C. Schneider, The three-loop splitting functions P (2) qg and P (2,NF ) gg , Nucl. Phys. B 922 (2017) 1–40, [1705.01508]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  55. [55]

    Evolution Kernels of Twist-3 Light-Ray Operators in Polarized Deep Inelastic Scattering

    B. Geyer, D. Mueller and D. Robaschik, Evolution kernels of twist - three light ray operators in polarized deep inelastic scattering , Nucl. Phys. B Proc. Suppl. 51 (1996) 106–110, [hep-ph/9606320]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  56. [56]

    ’t Hooft and M

    G. ’t Hooft and M. J. G. Veltman, Regularization and Renormalization of Gauge Fields , Nucl. Phys. B 44 (1972) 189–213

    work page 1972

  57. [57]

    Breitenlohner and D

    P. Breitenlohner and D. Maison, Dimensional Renormalization and the Action Principle , Commun. Math. Phys. 52 (1977) 11–38

    work page 1977

  58. [58]

    Trueman, Chiral symmetry in perturbative QCD , Physics Letters B 88 (1979) 331–334

    T. Trueman, Chiral symmetry in perturbative QCD , Physics Letters B 88 (1979) 331–334

    work page 1979

  59. [59]

    Two-loop operator matrix elements calculated up to finite terms for polarized deep inelastic lepton-hadron scattering

    Y. Matiounine, J. Smith and W. L. van Neerven, Two loop operator matrix elements calculated up to finite terms for polarized deep inelastic lepton - hadron scattering , Phys. Rev. D 58 (1998) 076002, [ hep-ph/9803439]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  60. [60]

    J. B. Kogut and D. E. Soper, Quantum Electrodynamics in the Infinite Momentum Frame , Phys. Rev. D 1 (1970) 2901–2913

    work page 1970

  61. [61]

    Jaffe and X.-D

    R. Jaffe and X.-D. Ji, Chiral odd parton distributions and Drell-Yan processes , Nucl.Phys. B375 (1992) 527–560

    work page 1992

  62. [62]

    An Analysis of the Next-to-Leading Order Corrections to the g_T(=g_1+g_2) Scaling Function

    X.-D. Ji and J. Osborne, An Analysis of the next-to-leading order corrections to the gT (= g1 + g2) scaling function, Nucl. Phys. B 608 (2001) 235–278, [ hep-ph/0102026]. – 44 –

    work page internal anchor Pith review Pith/arXiv arXiv 2001