Pith. sign in

REVIEW 2 major objections 5 minor 2 cited by

Threshold Resummation for Semi-Inclusive Single-Hadron Production with Effective Field Theory

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper shows that semi-inclusive $e^+e^-$ annihilation near threshold factorizes in soft-collinear effective theory into a squared time-like form factor and a jet function, and uses that to resum large-$x$ logarithms to N4LL accuracy…

desk verdict Careful, internally consistent N4LL momentum-space resummation for SIA with genuinely new large-x N3LO/N4LO coefficients; the caveat is that four-loop order is where the crossing-based factorization identity is first exercised, and the headline cross-check is not independent there. read the letter →

arxiv 2411.11595 v1 pith:Z4N7QS3C submitted 2024-11-18 hep-ph

Zhen Xu , Hua Xing Zhu This is my paper · ORCID
classification hep-ph
keywords thresholdresummationsemi-inclusivee+e-annihilationfragmentationfunctionssoft-collineareffectivetheoryN4LLaccuracytime-likeformfactorHiggsdecaylarge-xlogarithms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper is about the large double-logarithmic corrections that appear when a single hadron is produced with nearly all the beam energy in $e^+e^-$ annihilation (semi-inclusive annihilation, SIA). It develops a momentum-space resummation formalism, based on soft-collinear effective theory, and claims that near threshold the SIA cross section factorizes into a hard function given by the squared time-like form factor and a jet function identical to the one in deep inelastic scattering, with no extra soft function. Using this factorization, the authors resum the threshold logarithms to N4LL accuracy for $\gamma^*\to q\bar q$, $H\to gg$, and $H\to b\bar b$, one logarithmic order beyond previous results, and show good perturbative convergence with reduced scale uncertainty. They also derive the leading large-$x$ fixed-order terms at N3LO for the Higgs channels and at N4LO for all three channels, with the N4LO terms not previously available. If the factorization holds, these results improve the theoretical basis for extracting quark, gluon, and heavy-quark fragmentation functions from present and future collider data.

What carries the argument

The central object is the momentum-space factorization theorem of soft-collinear effective theory, Eq. (2.10): the SIA cross section is the convolution of a hard function $|C_V(Q^2,\mu)|^2$ from the time-like form factor, a jet function $J(Q^2(1-x/\xi),\mu)$ identical to the DIS jet function, and the fragmentation function. The resummation is carried by renormalization-group evolution, with the cusp anomalous dimension $\Gamma_{\rm cusp}$ controlling the Sudakov exponent and the jet anomalous dimension $\gamma_J$ controlling collinear emissions. A key identity, Eq. (3.5), relates the Laplace-space jet function to the moment-space coefficient $B_q$, allowing the authors to transfer known DIS ingredients and extract the four-loop jet anomalous dimension from the large-$x$ endpoint of the splitting function.

What would settle it

An independent four-loop fixed-order computation of the leading large-$x$ terms for $\gamma^*\to q\bar q$ would settle the central claim: if the predicted N4LO logarithms in Appendix B do not match, the assumed identity between the time-like and space-like jet functions is wrong. A direct four-loop calculation of the time-like jet function would test the same assumption more directly.

Watch

Extended reading notes

Core claim

The central claim is that the semi-inclusive coefficient function in the large-$x$ limit factorizes into the modulus-squared time-like on-shell form factor times the same jet function that appears in deep inelastic scattering, with the fragmentation function convoluted in and no additional soft function. The resummation is performed directly in momentum space via the renormalization-group evolution of the hard and jet functions, producing the formulas in Eqs. (2.20) and (2.21) at N4LL accuracy for $\gamma^*\to q\bar q$, $H\to gg$, and $H\to b\bar b$. The paper verifies the formalism by reproducing the known NNLO large-$x$ results for all channels and the N3LO results where they exist, and by matching the momentum-space results against the moment-space exponent order by order after extracting the four-loop coefficient $B_q$. The new output is the prediction of the leading large-$x$ terms at N3LO for $H\to gg$ and $H\to b\bar b$ and at N4LO for all three channels, with the N4LO coefficients presented for the first time.

Load-bearing premise

The calculation assumes that near the production threshold the cross section is built from just the squared time-like form factor and the same collinear-emission function that appears in deep inelastic scattering, with no additional soft-gluon contribution.

Editorial extensions

If this is right

  • The momentum-space resummation extends SIA threshold resummation to N4LL accuracy, one logarithmic order beyond previous moment-space results, reducing the scale uncertainty at large $x$.
  • The N4LO large-$x$ logarithms in Appendix B are new predictions that can be used to approximate or cross-check future complete fixed-order calculations.
  • For $H\to gg$ and $H\to b\bar b$, the resummed distributions provide a basis for extracting gluon and heavy-quark fragmentation functions at future $e^+e^-$ Higgs factories.
  • The order-by-order agreement with moment-space resummation confirms that the time-like/space-like equivalence holds for the extracted four-loop coefficient $B_q$.
  • The fixed-order expansions through N3LO reproduce the known NNLO and, where available, N3LO results, validating the formalism before using it for new predictions.

Reading between the lines

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

  • If the factorization holds, the universality of the jet function means that improvements in DIS jet-function calculations transfer automatically to SIA, and likely to other time-like observables such as event shapes or energy correlators.
  • The same momentum-space method should extend to single-inclusive hadron production in proton-proton collisions, where the threshold logarithms are controlled by parton distribution functions rather than fragmentation functions.
  • A natural next test is to compute the subleading-power corrections suppressed by $(1-x)$; at N4LL these may become numerically relevant for $x$ below about 0.9.
  • The predicted N4LO logarithms could be used to estimate the residual theory uncertainty in fragmentation-function global fits, although the paper does not perform such an analysis.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops momentum-space threshold resummation for the semi-inclusive single-hadron cross section in e+e- annihilation and Higgs decay, using soft-collinear effective theory. The central object is the leading-power factorization formula in Eq. (2.10), which expresses the SIA coefficient function as a product of a time-like hard function and a jet function identical to the one appearing in DIS, with no nontrivial soft function. On this basis the authors construct the resummed coefficient function in Eqs. (2.20)-(2.21) and carry out N4LL resummation for gamma* -> q qbar, H -> gg, and H -> b bbar. They compare with known fixed-order results at lower orders, check the momentum-space result against their own moment-space resummation in Table 2, and use the formalism to predict the leading large-x terms at N3LO and N4LO, with the N4LO results assembled in Appendix B.

Significance. If the factorization input is valid, this is a useful step beyond the existing N3LL threshold resummation for SIA: it supplies explicit N4LL momentum-space formulas for three important channels and the first N4LO large-x coefficients for these processes, with relevance for fragmentation-function extraction at current and future e+e- colliders. The paper is technically detailed and transparent: the perturbative ingredients are specified in Appendix A, the analytic fixed-order expansions are written out, and the ancillary file makes the N4LO results available in machine-readable form. The internal consistency checks, especially the agreement between momentum-space and moment-space implementations in Table 2 and the reproduction of known two- and three-loop results in Secs. 3.2 and 4.3, are genuine strengths. The main weakness is that the leading-power factorization theorem Eq. (2.10) is asserted rather than derived, and the four-loop input that first exercises the time-like/space-like identity is not tested by an independent external computation.

major comments (2)
  1. [Sec. 2.2, Eq. (2.10)] The factorization theorem is the load-bearing step of the paper, but it is asserted rather than derived. The two bullets after Eq. (2.10) state that the hard function is the square of the time-like form factor and that the SIA jet function equals the DIS jet function, with crossing symmetry and the reciprocity relation as justification. The manuscript does not give an operator-level derivation, does not show explicitly that the soft function integrates to unity, and does not state the order to which the time-like/space-like identity of the jet function is known. Since the N4LL exponent, the four-loop coefficient Bq,4 in Eq. (3.11), and the N4LO predictions in Appendix B all rest on this identity, I ask the authors to supply a derivation within SCET, or to cite a published proof, and to state explicitly any assumptions about the validity of the identity beyond the orders checked by fixed-order comparisons.
  2. [Sec. 3.3, Table 2] The agreement between the momentum-space and moment-space results in Table 2 is an internal consistency check, not an independent validation of the new four-loop input. Both calculations use the same Bq and gamma_J coefficients, and the independent fixed-order results cited in Secs. 3.2 and 4.3 are at orders below the point at which gamma_J^(3) first contributes. The phrase 'perfect agreement' should therefore be qualified: it demonstrates that the two resummation schemes are algebraically equivalent, but it does not test the time-like/space-like identity at four-loop order. Please state explicitly which features of the comparison are new and which are already contained in the existing literature.
minor comments (5)
  1. [Introduction] There is a typo in the Introduction: 'has not beed addressed' should read 'has not been addressed'.
  2. [Sec. 2.2, Eq. (2.20)] The statement 'we have also approximated the prefactor x at the leading power' appears after the formula; it would be clearer to state this approximation before presenting Eq. (2.20), since the prefactor in Eq. (2.10) is not obviously leading power.
  3. [Sec. 3.1, Figures 2-7] The figure captions do not specify the intermediate scale choice mu_i = Q sqrt(1-x) or the treatment of the fragmentation-function scale mu_f; these choices are described only in the body text, and adding them to the captions would improve readability.
  4. [Appendix B] The N4LO coefficients in Appendix B contain the numerically determined gamma_J^(3) through the L0 terms, but the text does not quantify the resulting uncertainty in the coefficients; a brief statement that this uncertainty is negligible, or an explicit error estimate, would be helpful.
  5. [Sec. 3.3, Eq. (3.11)] The remark that Bq,4 has 'now been determined more accurately' relies on Refs. [117-120]; it would be useful to state explicitly which parts of those references are analytic and which are numerical, particularly for the gamma_J^(3) input used in Eq. (A.31).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the resummation inputs are independent fixed-order quantities, and the claimed predictions are algebraic consequences checked against external results.

full rationale

The paper's derivation chain is not circular. The central factorization in Eq. (2.10), HSIA(Q2,µ) times the DIS jet function JSIA with no soft function, is asserted from crossing symmetry and the reciprocity relation rather than derived, but this is a physics assumption with external support [8, 10, 98, 99], not a definition of the target in terms of itself. All logarithmic resummation ingredients — the cusp anomalous dimension, the time-like form-factor hard function, the three-loop jet functions, the four-loop jet anomalous dimensions, and the beta function — are taken from independent fixed-order calculations in the literature ([104, 111, 112–116, 117, 118–120, 151, 165, 166, 185, 186, 187–197]), not fitted to the SIA coefficient functions that the paper predicts. The N3LO large-x coefficients in Eqs. (3.3), (4.8), and (4.11) are expansions of the resummed exponent with known lower-order inputs and are checked against existing calculations ([106–110], [136, 137], [135]). The N4LO predictions in Appendix B use Bq,4 and γJ(3), which are extracted from the space-like endpoint splitting function and form-factor anomalous dimensions via Eqs. (A.31)–(A.32); these are independent inputs, not the SIA result itself. The momentum-space vs. moment-space comparison in Sec. 3.3 uses the same Bq and g0 ingredients through Eq. (3.5), so it is a consistency check of the formalism rather than an independent prediction; this is acknowledged by the paper and does not reduce any claimed coefficient to a fitted parameter. Self-citations such as [99] and [118] are supporting published results that are externally checkable and not the sole justification for a load-bearing claim. Overall, the paper is self-contained against external benchmarks, with the main caveat being the unproven identity of the time-like and space-like jet functions — a correctness risk, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard SCET factorization adapted from DIS plus perturbative ingredients taken from the literature. The only significant unknown numerical input is the estimated five-loop cusp anomalous dimension. No new particles, forces, or dimensions are introduced, and the toy fragmentation functions are illustrative rather than load-bearing.

free parameters (3)
  • 5-loop cusp anomalous dimension, quark = Gamma_4 = 50000 +/- 40000 (estimate from [186])
    Required for N4LL; exact value unknown, central value used in numerical results.
  • 5-loop cusp anomalous dimension, gluon = Gamma_4 = 30000 +/- 60000 (estimate from [186])
    Required for N4LL in H->gg; the quoted uncertainty is larger than the central value.
  • End-point model fragmentation function exponent = p=2 for gamma* -> q qbar, p=3 for H -> gg and H -> b bbar
    Used only in Figures 2 through 7 to illustrate perturbative convergence; not part of the resummation claim itself.
assumptions (4)
  • domain assumption The SIA coefficient function at leading power factorizes into the modulus-squared time-like form factor times the same jet function as DIS, with no non-trivial soft function.
    Eq. (2.10) in Section 2.2; asserted from crossing symmetry and the reciprocity relation rather than fully derived in this paper.
  • domain assumption Flavour-singlet and longitudinal SIA contributions are power suppressed by (1-x) at threshold and can be neglected.
    Section 2, after Eq. (2.4); standard for the leading-power large-x limit but restricts the resummation to the non-singlet transverse channel.
  • domain assumption The N4LL momentum-space accuracy requires hard function to three loops, jet constant to three loops, jet/vector anomalous dimensions to four loops, beta function to five loops, and cusp anomalous dimension to five loops.
    Table 1; the ordering conventions for logarithmic accuracy are standard in the SCET threshold-resummation literature.
  • domain assumption The five-loop cusp anomalous dimension is approximated by the estimate of [186] with the stated uncertainties.
    Appendix A.3, Eqs. (A.46) and (A.47); the exact value is not known and the authors claim the numerical effect on N4LL results is sub-permille.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Threshold Resummation for Semi-Inclusive Single-Hadron Production with Effective Field Theory." pith.science (2026). https://pith.science/paper/Z4N7QS3C

@misc{pith2026241111595,
  author       = {Pith},
  title        = {Pith review of: Threshold Resummation for Semi-Inclusive Single-Hadron Production with Effective Field Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4N7QS3C}},
  note         = {Machine review of arXiv:2411.11595}
}
abstract

Large double-logarithmic corrections are induced by soft gluon emissions near threshold in the semi-inclusive $e^+e^-$ annihilation (SIA) distributions, and must be resummed to all-orders in perturbation theory for reliable theoretical predictions. Building on strategy developed for threshold resummation for DIS structure function in momentum space using soft-collinear effective theory (SCET), we present the explicit formalism for SIA cross section. We then perform the resummation directly in momentum space for $\gamma^* \to q \bar q$, $H \to gg$ and $H \to b\bar b$ to N$^4$LL accuracy and demonstrate good convergence. We anticipate that these results will benefit the extraction of the light-quark, the heavy-quark as well as the gluon fragmentation functions.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Dihadron Angular Correlations in the $e^+e^-$ Collision

    hep-ph 2025-06 conditional novelty 7.0 of 10

    The paper derives the first complete O(alpha_s^2) analytic QCD corrections to the dihadron angular separation distribution in e+e- annihilation, with verified cancellation of infrared poles.

  2. Flavour Non-Singlet Splitting Functions at Four Loops in QCD -- The Fermionic Contributions

    hep-ph 2025-05 accept novelty 7.0 of 10

    The n_f-dependent pieces of the four-loop non-singlet QCD splitting functions are derived analytically, yielding the n_f parts of the quark and gluon virtual anomalous dimensions and new soft-gluon threshold coefficients.

Reference graph

Works this paper leans on

197 extracted references · 4 canonical work pages · cited by 2 Pith papers

  1. [10]

    Moch and A

    S. Moch and A. Vogt, Higher-order threshold resummation for semi-inclusive e+ e- annihilation, Phys. Lett. B 680 (2009) 239 [ 0908.2746]

    arXiv 2009

  2. [1]

    Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections , Nucl

    G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections , Nucl. Phys. B 281 (1987) 310

    1987

  3. [2]

    Appell, G.F

    D. Appell, G.F. Sterman and P.B. Mackenzie, Soft Gluons and the Normalization of the Drell-Yan Cross-section, Nucl. Phys. B 309 (1988) 259

    1988

  4. [3]

    Catani and L

    S. Catani and L. Trentadue, Resummation of the QCD Perturbative Series for Hard Processes, Nucl. Phys. B 327 (1989) 323

    1989

  5. [4]

    Korchemsky and G

    G.P. Korchemsky and G. Marchesini, Structure function for large x and renormalization of Wilson loop, Nucl. Phys. B 406 (1993) 225 [ hep-ph/9210281]

    arXiv 1993

  6. [5]

    Catani, M.L

    S. Catani, M.L. Mangano, P. Nason and L. Trentadue, The Resummation of soft gluons in hadronic collisions, Nucl. Phys. B 478 (1996) 273 [ hep-ph/9604351]

    arXiv 1996

  7. [6]

    Contopanagos, E

    H. Contopanagos, E. Laenen and G.F. Sterman, Sudakov factorization and resummation , Nucl. Phys. B 484 (1997) 303 [ hep-ph/9604313]

    arXiv 1997

  8. [7]

    Vogt, On soft gluon effects in deep inelastic structure functions , Phys

    A. Vogt, On soft gluon effects in deep inelastic structure functions , Phys. Lett. B 471 (1999) 97 [ hep-ph/9910545]

    arXiv 1999

Show all 197 references
  1. [8]

    Cacciari and S

    M. Cacciari and S. Catani, Soft gluon resummation for the fragmentation of light and heavy quarks at large x , Nucl. Phys. B 617 (2001) 253 [ hep-ph/0107138]

    2001 arXiv

  2. [9]

    Moch, J.A.M

    S. Moch, J.A.M. Vermaseren and A. Vogt, Higher-order corrections in threshold resummation, Nucl. Phys. B 726 (2005) 317 [ hep-ph/0506288]

    2005 arXiv

  3. [11]

    A. A H, P. Mukherjee, V. Ravindran, A. Sankar and S. Tiwari, On next to soft threshold corrections to DIS and SIA processes , JHEP 04 (2021) 131 [ 2007.12214]

    2021 arXiv

  4. [12]

    Becher and M

    T. Becher and M. Neubert, Threshold resummation in momentum space from effective field theory, Phys. Rev. Lett. 97 (2006) 082001 [ hep-ph/0605050]

    2006 arXiv

  5. [13]

    Becher, M

    T. Becher, M. Neubert and B.D. Pecjak, Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering , JHEP 01 (2007) 076 [ hep-ph/0607228]

    2007 arXiv

  6. [14]

    Manohar, Deep inelastic scattering as x — > 1 using soft collinear effective theory , Phys

    A.V. Manohar, Deep inelastic scattering as x — > 1 using soft collinear effective theory , Phys. Rev. D 68 (2003) 114019 [ hep-ph/0309176]. – 44 –

    2003 arXiv

  7. [15]

    Pecjak, Non-factorizable contributions to deep inelastic scattering at large x , JHEP 10 (2005) 040 [ hep-ph/0506269]

    B.D. Pecjak, Non-factorizable contributions to deep inelastic scattering at large x , JHEP 10 (2005) 040 [ hep-ph/0506269]

    2005 arXiv

  8. [16]

    Chay and C

    J. Chay and C. Kim, Deep inelastic scattering near the endpoint in soft-collinear effective theory, Phys. Rev. D 75 (2007) 016003 [ hep-ph/0511066]

    2007 arXiv

  9. [17]

    Manohar, Infrared scales and factorization in QCD , Phys

    A.V. Manohar, Infrared scales and factorization in QCD , Phys. Lett. B 633 (2006) 729 [hep-ph/0512173]

    2006 arXiv

  10. [18]

    Idilbi, X.-d

    A. Idilbi, X.-d. Ji and F. Yuan, Resummation of threshold logarithms in effective field theory for DIS, Drell-Yan and Higgs production , Nucl. Phys. B 753 (2006) 42 [ hep-ph/0605068]

    2006 arXiv

  11. [19]

    P.-y. Chen, A. Idilbi and X.-d. Ji, QCD Factorization for Deep-Inelastic Scattering At Large Bjorken x(B) ˜ 1 - O (Lambda(QCD)/Q) , Nucl. Phys. B 763 (2007) 183 [hep-ph/0607003]

    2007 arXiv

  12. [20]

    Idilbi and X.-d

    A. Idilbi and X.-d. Ji, Threshold resummation for Drell-Yan process in soft-collinear effective theory, Phys. Rev. D 72 (2005) 054016 [ hep-ph/0501006]

    2005 arXiv

  13. [21]

    Becher, M

    T. Becher, M. Neubert and G. Xu, Dynamical Threshold Enhancement and Resummation in Drell-Yan Production, JHEP 07 (2008) 030 [ 0710.0680]

    2008 arXiv

  14. [22]

    Bauer, S

    C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays , Phys. Rev. D 63 (2001) 114020 [ hep-ph/0011336]

    2001 arXiv

  15. [23]

    Bauer, S

    C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in B — > X(s gamma) in effective field theory , Phys. Rev. D 63 (2000) 014006 [ hep-ph/0005275]

    2000 arXiv

  16. [24]

    Bauer, D

    C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory , Phys. Rev. D 65 (2002) 054022 [ hep-ph/0109045]

    2002 arXiv

  17. [25]

    Bauer and I.W

    C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory , Phys. Lett. B 516 (2001) 134 [ hep-ph/0107001]

    2001 arXiv

  18. [26]

    Beneke, A.P

    M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power , Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152]

    2002 arXiv

  19. [27]

    Bender et al., Study of Quark Fragmentation at 29-GeV: Global Jet Parameters and Single Particle Distributions , Phys

    D. Bender et al., Study of Quark Fragmentation at 29-GeV: Global Jet Parameters and Single Particle Distributions , Phys. Rev. D 31 (1985) 1

    1985

  20. [28]

    Derrick et al., Study of Quark Fragmentation in e+ e- Annihilation at 29-GeV: Charged Particle Multiplicity and Single Particle Rapidity Distributions , Phys

    M. Derrick et al., Study of Quark Fragmentation in e+ e- Annihilation at 29-GeV: Charged Particle Multiplicity and Single Particle Rapidity Distributions , Phys. Rev. D 34 (1986) 3304

    1986

  21. [29]

    TASSO collaboration, Global Jet Properties at 14-GeV to 44-GeV Center-of-mass Energy in e+e− Annihilation, Z. Phys. C 47 (1990) 187

    1990

  22. [30]

    L3 collaboration, Measurement of the inclusive production of neutral pions and charged particles on the Z0 resonance , Phys. Lett. B 259 (1991) 199

    1991

  23. [31]

    TOPAZ collaboration, Measurement of inclusive particle spectra and test of MLLA prediction in e+ e- annihilation at s**(1/2) = 58-GeV , Phys. Lett. B 345 (1995) 335 [hep-ex/9412015]

    1995 arXiv

  24. [32]

    ALEPH collaboration, Studies of QCD in e+ e- — > hadrons at E(cm) = 130-GeV and 136-GeV, Z. Phys. C 73 (1997) 409

    1997

  25. [33]

    DELPHI collaboration, Tuning and test of fragmentation models based on identified particles and precision event shape data , Z. Phys. C 73 (1996) 11. – 45 –

    1996

  26. [34]

    DELPHI collaboration, Measurement of event shape and inclusive distributions at S**(1/2) = 130-GeV and 136-GeV , Z. Phys. C 73 (1997) 229

    1997

  27. [35]

    OPAL collaboration, QCD studies with e+ e- annihilation data at 130-GeV and 136-GeV , Z. Phys. C 72 (1996) 191

    1996

  28. [36]

    OPAL collaboration, QCD studies with e+ e- annihilation data at 161-GeV , Z. Phys. C 75 (1997) 193

    1997

  29. [37]

    OPAL collaboration, QCD studies with e+ e- annihilation data at 172-GeV - 189-GeV , Eur. Phys. J. C 16 (2000) 185 [ hep-ex/0002012]

    2000 arXiv

  30. [38]

    L3 collaboration, Studies of hadronic event structure in e+e− annihilation from 30-GeV to 209-GeV with the L3 detector , Phys. Rept. 399 (2004) 71 [ hep-ex/0406049]

    2004 arXiv

  31. [39]

    H1 collaboration, Evolution of e p fragmentation and multiplicity distributions in the Breit frame, Nucl. Phys. B 504 (1997) 3 [ hep-ex/9707005]

    1997 arXiv

  32. [40]

    ZEUS collaboration, Measurement of multiplicity and momentum spectra in the current and target regions of the Breit frame in deep inelastic scattering at HERA , Eur. Phys. J. C 11 (1999) 251 [ hep-ex/9903056]

    1999 arXiv

  33. [41]

    Braaten and T.C

    E. Braaten and T.C. Yuan, Gluon fragmentation into heavy quarkonium , Phys. Rev. Lett. 71 (1993) 1673 [ hep-ph/9303205]

    1993 arXiv

  34. [42]

    Bourhis, M

    L. Bourhis, M. Fontannaz and J.P. Guillet, Quarks and gluon fragmentation functions into photons, Eur. Phys. J. C 2 (1998) 529 [ hep-ph/9704447]

    1998 arXiv

  35. [43]

    DELPHI collaboration, Measurement of the quark and gluon fragmentation functions in Z0 hadronic decays, Eur. Phys. J. C 6 (1999) 19

    1999

  36. [44]

    DELPHI collaboration, Measurement of the gluon fragmentation function and a comparison of the scaling violation in gluon and quark jets , Eur. Phys. J. C 13 (2000) 573

    2000

  37. [45]

    Particle Data Groupcollaboration, Review of Particle Physics , PTEP 2022 (2022) 083C01

    2022

  38. [46]

    Mele and P

    B. Mele and P. Nason, Next-to-leading QCD calculation of the heavy quark fragmentation function, Phys. Lett. B 245 (1990) 635

    1990

  39. [47]

    Mele and P

    B. Mele and P. Nason, The Fragmentation function for heavy quarks in QCD , Nucl. Phys. B 361 (1991) 626

    1991

  40. [48]

    Braaten, K.-m

    E. Braaten, K.-m. Cheung, S. Fleming and T.C. Yuan, Perturbative QCD fragmentation functions as a model for heavy quark fragmentation , Phys. Rev. D 51 (1995) 4819 [hep-ph/9409316]

    1995 arXiv

  41. [49]

    Nason and C

    P. Nason and C. Oleari, A Phenomenological study of heavy quark fragmentation functions in e+ e- annihilation , Nucl. Phys. B 565 (2000) 245 [ hep-ph/9903541]

    2000 arXiv

  42. [50]

    Melnikov and A

    K. Melnikov and A. Mitov, Perturbative heavy quark fragmentation function through O(α2 s), Phys. Rev. D 70 (2004) 034027 [ hep-ph/0404143]

    2004 arXiv

  43. [51]

    Mitov, Perturbative heavy quark fragmentation function through O(α2 s): Gluon initiated contribution, Phys

    A. Mitov, Perturbative heavy quark fragmentation function through O(α2 s): Gluon initiated contribution, Phys. Rev. D 71 (2005) 054021 [ hep-ph/0410205]

    2005 arXiv

  44. [52]

    Neubert, Factorization analysis for the fragmentation functions of hadrons containing a heavy quark, 0706.2136

    M. Neubert, Factorization analysis for the fragmentation functions of hadrons containing a heavy quark, 0706.2136. – 46 –

    arXiv

  45. [53]

    Biello and L

    C. Biello and L. Bonino, Time-Like Heavy-Flavour Thresholds for Fragmentation Functions: the Light-Quark Matching Condition at NNLO , 2407.07623

    arXiv

  46. [54]

    ALEPH collaboration, Measurement of the effective b quark fragmentation function at the Z resonance, Phys. Lett. B 357 (1995) 699

    1995

  47. [55]

    ALEPH collaboration, First measurement of the quark to photon fragmentation function , Z. Phys. C 69 (1996) 365

    1996

  48. [56]

    SLD collaboration, Precise measurement of the b quark fragmentation function in Z0 boson decays, Phys. Rev. Lett. 84 (2000) 4300 [ hep-ex/9912058]

    2000 arXiv

  49. [57]

    SLD collaboration, Measurement of the b quark fragmentation function in Z0 decays , Phys. Rev. D 65 (2002) 092006 [ hep-ex/0202031]

    2002 arXiv

  50. [58]

    OPAL collaboration, Inclusive analysis of the b quark fragmentation function in Z decays at LEP , Eur. Phys. J. C 29 (2003) 463 [ hep-ex/0210031]

    2003 arXiv

  51. [59]

    DELPHI collaboration, A study of the b-quark fragmentation function with the DELPHI detector at LEP I and an averaged distribution obtained at the Z Pole , Eur. Phys. J. C 71 (2011) 1557 [ 1102.4748]

    2011 arXiv

  52. [60]

    CEPC Study Groupcollaboration, CEPC Conceptual Design Report: Volume 1 - Accelerator, 1809.00285

    arXiv

  53. [61]

    CEPC Study Groupcollaboration, CEPC Conceptual Design Report: Volume 2 - Physics & Detector , 1811.10545

    arXiv

  54. [62]

    CEPC Study Groupcollaboration, CEPC Technical Design Report: Accelerator, Radiat. Detect. Technol. Methods 8 (2024) 1 [ 2312.14363]

    2024

  55. [63]

    FCC collaboration, FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1 , Eur. Phys. J. C 79 (2019) 474

    2019

  56. [64]

    FCC collaboration, FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2 , Eur. Phys. J. ST 228 (2019) 261

    2019

  57. [65]

    Behnke, J.E

    T. Behnke, J.E. Brau, B. Foster, J. Fuster, M. Harrison, J.M. Paterson et al., eds., The International Linear Collider Technical Design Report - Volume 1: Executive Summary , 1306.6327

    arXiv

  58. [66]

    Bambade et al., The International Linear Collider: A Global Project , 1903.01629

    P. Bambade et al., The International Linear Collider: A Global Project , 1903.01629

    1903 arXiv

  59. [67]

    Laenen, G

    E. Laenen, G. Oderda and G.F. Sterman, Resummation of threshold corrections for single particle inclusive cross-sections, Phys. Lett. B 438 (1998) 173 [ hep-ph/9806467]

    1998 arXiv

  60. [68]

    Jager, M

    B. Jager, M. Stratmann and W. Vogelsang, Single inclusive jet production in polarized pp collisions at O(alpha3 s), Phys. Rev. D 70 (2004) 034010 [ hep-ph/0404057]

    2004 arXiv

  61. [69]

    de Florian and W

    D. de Florian and W. Vogelsang, Threshold resummation for the inclusive-hadron cross-section in pp collisions , Phys. Rev. D 71 (2005) 114004 [ hep-ph/0501258]

    2005 arXiv

  62. [70]

    de Florian, W

    D. de Florian, W. Vogelsang and F. Wagner, Single-Inclusive Hadron Production in Polarized pp Scattering at Next-to-Leading Logarithmic Accuracy , Phys. Rev. D 76 (2007) 094021 [0708.3060]

    2007 arXiv

  63. [71]

    de Florian, P

    D. de Florian, P. Hinderer, A. Mukherjee, F. Ringer and W. Vogelsang, Approximate next-to-next-to-leading order corrections to hadronic jet production , Phys. Rev. Lett. 112 (2014) 082001 [ 1310.7192]. – 47 –

    2014 arXiv

  64. [72]

    Catani, M

    S. Catani, M. Grazzini and A. Torre, Soft-gluon resummation for single-particle inclusive hadroproduction at high transverse momentum , Nucl. Phys. B 874 (2013) 720 [ 1305.3870]

    2013 arXiv

  65. [73]

    Kumar and S.-O

    M.C. Kumar and S.-O. Moch, Phenomenology of threshold corrections for inclusive jet production at hadron colliders , Phys. Lett. B 730 (2014) 122 [ 1309.5311]

    2014 arXiv

  66. [74]

    Almeida, G.F

    L.G. Almeida, G.F. Sterman and W. Vogelsang, Threshold Resummation for Di-hadron Production in Hadronic Collisions , Phys. Rev. D 80 (2009) 074016 [ 0907.1234]

    2009 arXiv

  67. [75]

    Hinderer, F

    P. Hinderer, F. Ringer, G.F. Sterman and W. Vogelsang, Toward NNLL Threshold Resummation for Hadron Pair Production in Hadronic Collisions , Phys. Rev. D 91 (2015) 014016 [1411.3149]

    2015 arXiv

  68. [76]

    Hinderer, F

    P. Hinderer, F. Ringer, G. Sterman and W. Vogelsang, Threshold Resummation at NNLL for Single-particle Production in Hadronic Collisions , Phys. Rev. D 99 (2019) 054019 [1812.00915]

    2019 arXiv

  69. [77]

    Forslund and N

    M. Forslund and N. Kidonakis, Resummation for 2 → n processes in single-particle-inclusive kinematics, Phys. Rev. D 102 (2020) 034006 [ 2003.09021]

    2020 arXiv

  70. [78]

    Ahrens, A

    V. Ahrens, A. Ferroglia, M. Neubert, B.D. Pecjak and L.-L. Yang, RG-improved single-particle inclusive cross sections and forward-backward asymmetry in t¯t production at hadron colliders, JHEP 09 (2011) 070 [ 1103.0550]

    2011 arXiv

  71. [79]

    Ferroglia, S

    A. Ferroglia, S. Marzani, B.D. Pecjak and L.L. Yang, Boosted top production: factorization and resummation for single-particle inclusive distributions , JHEP 01 (2014) 028 [1310.3836]

    2014 arXiv

  72. [80]

    Liu, S.-O

    X. Liu, S.-O. Moch and F. Ringer, Threshold and jet radius joint resummation for single-inclusive jet production , Phys. Rev. Lett. 119 (2017) 212001 [ 1708.04641]

    2017 arXiv

  73. [81]

    Liu, S.-O

    X. Liu, S.-O. Moch and F. Ringer, Phenomenology of single-inclusive jet production with jet radius and threshold resummation , Phys. Rev. D 97 (2018) 056026 [ 1801.07284]

    2018 arXiv

  74. [82]

    Procura and I.W

    M. Procura and I.W. Stewart, Quark Fragmentation within an Identified Jet , Phys. Rev. D 81 (2010) 074009 [ 0911.4980]

    2010 arXiv

  75. [83]

    A. Jain, M. Procura and W.J. Waalewijn, Parton Fragmentation within an Identified Jet at NNLL, JHEP 05 (2011) 035 [ 1101.4953]

    2011 arXiv

  76. [84]

    Z.-B. Kang, F. Ringer and I. Vitev, The semi-inclusive jet function in SCET and small radius resummation for inclusive jet production , JHEP 10 (2016) 125 [ 1606.06732]

    2016 arXiv

  77. [85]

    K. Lee, I. Moult and X. Zhang, Revisiting Single Inclusive Jet Production: Timelike Factorization and Reciprocity, 2409.19045

    arXiv

  78. [86]

    K. Lee, I. Moult and X. Zhang, Revisiting Single Inclusive Jet Production: Small- R Resummation at Next-to-Leading Logarithm, 2410.01902

    arXiv

  79. [87]

    van Beekveld, M

    M. van Beekveld, M. Dasgupta, B.K. El-Menoufi, J. Helliwell and P.F. Monni, Collinear fragmentation at NNLL: generating functionals, groomed correlators and angularities , JHEP 05 (2024) 093 [ 2307.15734]

    2024 arXiv

  80. [88]

    van Beekveld, M

    M. van Beekveld, M. Dasgupta, B.K. El-Menoufi, J. Helliwell, A. Karlberg and P.F. Monni, Two-loop anomalous dimensions for small-R jet versus hadronic fragmentation functions , JHEP 07 (2024) 239 [ 2402.05170]

    2024 arXiv

  81. [89]

    van Beekveld, M

    M. van Beekveld, M. Dasgupta, B.K. El-Menoufi, J. Helliwell, P.F. Monni and G.P. Salam, A collinear shower algorithm for NSL non-singlet fragmentation , 2409.08316. – 48 –

    arXiv

  82. [90]

    J. Gao, C. Liu, X. Shen, H. Xing and Y. Zhao, Simultaneous Determination of Fragmentation Functions and Test on Momentum Sum Rule , Phys. Rev. Lett. 132 (2024) 261903 [2401.02781]

    2024 arXiv

  83. [91]

    J. Gao, C. Liu, X. Shen, H. Xing and Y. Zhao, Global analysis of fragmentation functions to charged hadrons with high-precision data from the LHC , 2407.04422

    arXiv

  84. [92]

    Zhou and J

    B. Zhou and J. Gao, Towards ultimate fragmentation functions at future lepton colliders , 2407.10059

    arXiv

  85. [93]

    C. Liu, X. Shen, B. Zhou and J. Gao, Automated calculation of jet fragmentation at NLO in QCD, JHEP 09 (2023) 108 [ 2305.14620]

    2023 arXiv

  86. [94]

    Anderle, T

    D.P. Anderle, T. Kaufmann, M. Stratmann and F. Ringer, Fragmentation Functions Beyond Fixed Order Accuracy, Phys. Rev. D 95 (2017) 054003 [ 1611.03371]

    2017 arXiv

  87. [95]

    H. Chen, I. Moult, X. Zhang and H.X. Zhu, Rethinking jets with energy correlators: Tracks, resummation, and analytic continuation , Phys. Rev. D 102 (2020) 054012 [ 2004.11381]

    2020 arXiv

  88. [96]

    CMS collaboration, Measurement of Energy Correlators inside Jets and Determination of the Strong Coupling αS(mZ), Phys. Rev. Lett. 133 (2024) 071903 [ 2402.13864]

    2024 arXiv

  89. [97]

    Nason and B.R

    P. Nason and B.R. Webber, Scaling violation in e+ e- fragmentation functions: QCD evolution, hadronization and heavy quark mass effects , Nucl. Phys. B 421 (1994) 473

    1994

  90. [98]

    Gribov and L.N

    V.N. Gribov and L.N. Lipatov, e+ e- pair annihilation and deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 675

    1972

  91. [99]

    Chen, T.-Z

    H. Chen, T.-Z. Yang, H.X. Zhu and Y.J. Zhu, Analytic Continuation and Reciprocity Relation for Collinear Splitting in QCD , Chin. Phys. C 45 (2021) 043101 [ 2006.10534]

    2021 arXiv

  92. [100]

    Gribov and L.N

    V.N. Gribov and L.N. Lipatov, Deep inelastic e p scattering in perturbation theory , Sov. J. Nucl. Phys. 15 (1972) 438

    1972

  93. [101]

    Lipatov, The parton model and perturbation theory , Yad

    L.N. Lipatov, The parton model and perturbation theory , Yad. Fiz. 20 (1974) 181

    1974

  94. [102]

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

    Y.L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics. , Sov. Phys. JETP 46 (1977) 641

    1977

  95. [103]

    Altarelli and G

    G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language , Nucl. Phys. B 126 (1977) 298

    1977

  96. [104]

    Becher and M

    T. Becher and M. Neubert, Toward a NNLO calculation of the anti-B — > X(s) gamma decay rate with a cut on photon energy. II. Two-loop result for the jet function , Phys. Lett. B 637 (2006) 251 [ hep-ph/0603140]

    2006 arXiv

  97. [105]

    Bonvini, S

    M. Bonvini, S. Forte, M. Ghezzi and G. Ridolfi, Threshold Resummation in SCET vs. Perturbative QCD: An Analytic Comparison , Nucl. Phys. B 861 (2012) 337 [ 1201.6364]

    2012 arXiv

  98. [106]

    Rijken and W.L

    P.J. Rijken and W.L. van Neerven, O (alpha-s**2) contributions to the longitudinal fragmentation function in e+ e- annihilation , Phys. Lett. B 386 (1996) 422 [hep-ph/9604436]

    1996 arXiv

  99. [107]

    Rijken and W.L

    P.J. Rijken and W.L. van Neerven, Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron - positron annihilation , Nucl. Phys. B 487 (1997) 233 [ hep-ph/9609377]. – 49 –

    1997 arXiv

  100. [108]

    Rijken and W.L

    P.J. Rijken and W.L. van Neerven, O (alpha-s**2) contributions to the asymmetric fragmentation function in e+ e- annihilation , Phys. Lett. B 392 (1997) 207 [hep-ph/9609379]

    1997 arXiv

  101. [109]

    Mitov and S.-O

    A. Mitov and S.-O. Moch, QCD Corrections to Semi-Inclusive Hadron Production in Electron-Positron Annihilation at Two Loops , Nucl. Phys. B 751 (2006) 18 [hep-ph/0604160]

    2006 arXiv

  102. [110]

    Mitov, S

    A. Mitov, S. Moch and A. Vogt, Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions, Phys. Lett. B 638 (2006) 61 [ hep-ph/0604053]

    2006 arXiv

  103. [111]

    Br¨ user, Z.L

    R. Br¨ user, Z.L. Liu and M. Stahlhofen, Three-Loop Quark Jet Function, Phys. Rev. Lett. 121 (2018) 072003 [ 1804.09722]

    2018 arXiv

  104. [112]

    Gehrmann, G

    T. Gehrmann, G. Heinrich, T. Huber and C. Studerus, Master integrals for massless three-loop form-factors: One-loop and two-loop insertions , Phys. Lett. B 640 (2006) 252 [hep-ph/0607185]

    2006 arXiv

  105. [113]

    Moch, J.A.M

    S. Moch, J.A.M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form-factors, Phys. Lett. B 625 (2005) 245 [ hep-ph/0508055]

    2005 arXiv

  106. [114]

    Baikov, K.G

    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]

    2009 arXiv

  107. [115]

    Lee, A.V

    R.N. Lee, A.V. Smirnov and V.A. Smirnov, Analytic Results for Massless Three-Loop Form Factors, JHEP 04 (2010) 020 [ 1001.2887]

    2010 arXiv

  108. [116]

    Gehrmann, E.W.N

    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 , JHEP 06 (2010) 094 [ 1004.3653]

    2010 arXiv

  109. [117]

    Das, S.-O

    G. Das, S.-O. Moch and A. Vogt, Soft corrections to inclusive deep-inelastic scattering at four loops and beyond , JHEP 03 (2020) 116 [ 1912.12920]

    2020 arXiv

  110. [118]

    Moult, H.X

    I. Moult, H.X. Zhu and Y.J. Zhu, The four loop QCD rapidity anomalous dimension , JHEP 08 (2022) 280 [ 2205.02249]

    2022 arXiv

  111. [119]

    C. Duhr, B. Mistlberger and G. Vita, Four-Loop Rapidity Anomalous Dimension and Event Shapes to Fourth Logarithmic Order , Phys. Rev. Lett. 129 (2022) 162001 [ 2205.02242]

    2022 arXiv

  112. [120]

    C. Duhr, B. Mistlberger and G. Vita, Soft integrals and soft anomalous dimensions at N3LO and beyond , JHEP 09 (2022) 155 [ 2205.04493]

    2022 arXiv

  113. [121]

    Larin, P

    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 [hep-ph/9605317]

    1997 arXiv

  114. [122]

    Vermaseren, A

    J.A.M. Vermaseren, A. Vogt and S. Moch, The Third-order QCD corrections to deep-inelastic scattering by photon exchange , Nucl. Phys. B 724 (2005) 3 [hep-ph/0504242]

    2005 arXiv

  115. [123]

    Inami, T

    T. Inami, T. Kubota and Y. Okada, Effective gauge theory and the effect of heavy quarks in higgs boson decays, Zeitschrift f¨ ur Physik C Particles and Fields 18 (1983) 69

    1983

  116. [124]

    Spira, A

    M. Spira, A. Djouadi, D. Graudenz and P.M. Zerwas, Higgs boson production at the LHC , Nucl. Phys. B 453 (1995) 17 [ hep-ph/9504378]

    1995 arXiv

  117. [125]

    Chetyrkin, B.A

    K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser, Hadronic Higgs decay to order alpha-s**4, Phys. Rev. Lett. 79 (1997) 353 [ hep-ph/9705240]

    1997 arXiv

  118. [126]

    Baikov and K.G

    P.A. Baikov and K.G. Chetyrkin, Top Quark Mediated Higgs Boson Decay into Hadrons to Order α5 s, Phys. Rev. Lett. 97 (2006) 061803 [ hep-ph/0604194]. – 50 –

    2006 arXiv

  119. [127]

    Chetyrkin, Correlator of the quark scalar currents and Gamma(tot) (H — > hadrons) at O (alpha-s**3) in pQCD , Phys

    K.G. Chetyrkin, Correlator of the quark scalar currents and Gamma(tot) (H — > hadrons) at O (alpha-s**3) in pQCD , Phys. Lett. B 390 (1997) 309 [ hep-ph/9608318]

    1997 arXiv

  120. [128]

    Mondini, M

    R. Mondini, M. Schiavi and C. Williams, N3LO predictions for the decay of the Higgs boson to bottom quarks , JHEP 06 (2019) 079 [ 1904.08960]

    2019 arXiv

  121. [129]

    ATLAS collaboration, Constraints on Higgs boson production with large transverse momentum using H→bb¯ decays in the ATLAS detector , Phys. Rev. D 105 (2022) 092003 [2111.08340]

    2022 arXiv

  122. [130]

    CMS collaboration, Measurement of the Higgs boson production via vector boson fusion and its decay into bottom quarks in proton-proton collisions at √s = 13 TeV , JHEP 01 (2024) 173 [2308.01253]

    2024 arXiv

  123. [131]

    Idilbi, X.-d

    A. Idilbi, X.-d. Ji, J.-P. Ma and F. Yuan, Threshold resummation for Higgs production in effective field theory , Phys. Rev. D 73 (2006) 077501 [ hep-ph/0509294]

    2006 arXiv

  124. [132]

    A. A H, A. Chakraborty, G. Das, P. Mukherjee and V. Ravindran, Resummed prediction for Higgs boson production through b b annihilation at N 3LL, JHEP 11 (2019) 006 [1905.03771]

    2019 arXiv

  125. [133]

    Das and A

    G. Das and A. Sankar, Next-to-soft threshold effects on Higgs boson production via bottom quark annihilation , 2409.01553

    arXiv

  126. [134]

    Corcella, Fragmentation in H — > b anti-b processes, Nucl

    G. Corcella, Fragmentation in H — > b anti-b processes, Nucl. Phys. B 705 (2005) 363 [hep-ph/0409161]

    2005 arXiv

  127. [135]

    Blumlein and V

    J. Blumlein and V. Ravindran, QCD threshold corrections to Higgs decay and to hadroproduction in l+ l- annihilation , Phys. Lett. B 640 (2006) 40 [ hep-ph/0605011]

    2006 arXiv

  128. [136]

    Moch and A

    S. Moch and A. Vogt, On third-order timelike splitting functions and top-mediated Higgs decay into hadrons, Phys. Lett. B 659 (2008) 290 [ 0709.3899]

    2008 arXiv

  129. [137]

    Almasy, S

    A.A. Almasy, S. Moch and A. Vogt, On the Next-to-Next-to-Leading Order Evolution of Flavour-Singlet Fragmentation Functions, Nucl. Phys. B 854 (2012) 133 [ 1107.2263]

    2012 arXiv

  130. [138]

    Inami, T

    T. Inami, T. Kubota and Y. Okada, Effective Gauge Theory and the Effect of Heavy Quarks in Higgs Boson Decays , Z. Phys. C 18 (1983) 69

    1983

  131. [139]

    Davies, M

    J. Davies, M. Steinhauser and D. Wellmann, Hadronic Higgs boson decay at order α4 s and α5 s, PoS DIS2017 (2018) 295 [ 1706.00624]

    2018 arXiv

  132. [140]

    Herzog, B

    F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, On Higgs decays to hadrons and the R-ratio at N 4LO, JHEP 08 (2017) 113 [ 1707.01044]

    2017 arXiv

  133. [141]

    Spiridonov, Anomalous Dimension of G2 µν and β Function, inspirehep.net/literature/210644 (1984)

    V.P. Spiridonov, Anomalous Dimension of G2 µν and β Function, inspirehep.net/literature/210644 (1984)

    1984

  134. [142]

    Kramer, E

    M. Kramer, E. Laenen and M. Spira, Soft gluon radiation in Higgs boson production at the LHC, Nucl. Phys. B 511 (1998) 523 [ hep-ph/9611272]

    1998 arXiv

  135. [143]

    Chetyrkin, B.A

    K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser, Decoupling relations to O (alpha-s**3) and their connection to low-energy theorems , Nucl. Phys. B 510 (1998) 61 [hep-ph/9708255]

    1998 arXiv

  136. [144]

    Schroder and M

    Y. Schroder and M. Steinhauser, Four-loop decoupling relations for the strong coupling , JHEP 01 (2006) 051 [ hep-ph/0512058]

    2006 arXiv

  137. [145]

    Chetyrkin, J.H

    K.G. Chetyrkin, J.H. Kuhn and C. Sturm, QCD decoupling at four loops , Nucl. Phys. B 744 (2006) 121 [ hep-ph/0512060]. – 51 –

    2006 arXiv

  138. [146]

    Liu and M

    T. Liu and M. Steinhauser, Decoupling of heavy quarks at four loops and effective Higgs-fermion coupling, Phys. Lett. B 746 (2015) 330 [ 1502.04719]

    2015 arXiv

  139. [147]

    X. Chen, P. Jakubˇ c ´ ık, M. Marcoli and G. Stagnitto,The parton-level structure of Higgs decays to hadrons at N 3LO, JHEP 06 (2023) 185 [ 2304.11180]

    2023 arXiv

  140. [148]

    Gehrmann and D

    T. Gehrmann and D. Kara, The Hb¯b form factor to three loops in QCD , JHEP 09 (2014) 174 [1407.8114]

    2014 arXiv

  141. [149]

    Chakraborty, T

    A. Chakraborty, T. Huber, R.N. Lee, A. von Manteuffel, R.M. Schabinger, A.V. Smirnov et al., Hbb vertex at four loops and hard matching coefficients in SCET for various currents , Phys. Rev. D 106 (2022) 074009 [ 2204.02422]

    2022 arXiv

  142. [150]

    Baikov, K.G

    P.A. Baikov, K.G. Chetyrkin and J.H. Kuhn, Scalar correlator at O(alpha(s)**4), Higgs decay into b-quarks and bounds on the light quark masses , Phys. Rev. Lett. 96 (2006) 012003 [hep-ph/0511063]

    2006 arXiv

  143. [151]

    Banerjee, P.K

    P. Banerjee, P.K. Dhani and V. Ravindran, Gluon jet function at three loops in QCD , Phys. Rev. D 98 (2018) 094016 [ 1805.02637]

    2018 arXiv

  144. [152]

    Kramer and B

    G. Kramer and B. Lampe, Two Jet Cross-Section in e+ e- Annihilation , Z. Phys. C 34 (1987) 497

    1987

  145. [153]

    Matsuura and W.L

    T. Matsuura and W.L. van Neerven, Second Order Logarithmic Corrections to the Drell-Yan Cross-section, Z. Phys. C 38 (1988) 623

    1988

  146. [154]

    Matsuura, S.C

    T. Matsuura, S.C. van der Marck and W.L. van Neerven, The Calculation of the Second Order Soft and Virtual Contributions to the Drell-Yan Cross-Section , Nucl. Phys. B 319 (1989) 570

    1989

  147. [155]

    Harlander, Virtual corrections to g g — > H to two loops in the heavy top limit , Phys

    R.V. Harlander, Virtual corrections to g g — > H to two loops in the heavy top limit , Phys. Lett. B 492 (2000) 74 [ hep-ph/0007289]

    2000 arXiv

  148. [156]

    Ravindran, J

    V. Ravindran, J. Smith and W.L. van Neerven, Two-loop corrections to Higgs boson production, Nucl. Phys. B 704 (2005) 332 [ hep-ph/0408315]

    2005 arXiv

  149. [157]

    Gehrmann, T

    T. Gehrmann, T. Huber and D. Maitre, Two-loop quark and gluon form-factors in dimensional regularisation, Phys. Lett. B 622 (2005) 295 [ hep-ph/0507061]

    2005 arXiv

  150. [158]

    Heinrich, T

    G. Heinrich, T. Huber and D. Maitre, Master integrals for fermionic contributions to massless three-loop form-factors, Phys. Lett. B 662 (2008) 344 [ 0711.3590]

    2008 arXiv

  151. [159]

    Moch, J.A.M

    S. Moch, J.A.M. Vermaseren and A. Vogt, The Quark form-factor at higher orders , JHEP 08 (2005) 049 [ hep-ph/0507039]

    2005 arXiv

  152. [160]

    von Manteuffel, E

    A. von Manteuffel, E. Panzer and R.M. Schabinger, On the Computation of Form Factors in Massless QCD with Finite Master Integrals , Phys. Rev. D 93 (2016) 125014 [ 1510.06758]

    2016 arXiv

  153. [161]

    R.N. Lee, A. von Manteuffel, R.M. Schabinger, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Fermionic corrections to quark and gluon form factors in four-loop QCD , Phys. Rev. D 104 (2021) 074008 [ 2105.11504]

    2021 arXiv

  154. [162]

    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

  155. [163]

    Bauer and A.V

    C.W. Bauer and A.V. Manohar, Shape function effects in B — > X(s) gamma and B — > X(u) l anti-nu decays , Phys. Rev. D 70 (2004) 034024 [ hep-ph/0312109]. – 52 –

    2004 arXiv

  156. [164]

    Bosch, B.O

    S.W. Bosch, B.O. Lange, M. Neubert and G. Paz, Factorization and shape function effects in inclusive B meson decays , Nucl. Phys. B 699 (2004) 335 [ hep-ph/0402094]

    2004 arXiv

  157. [165]

    Becher and M.D

    T. Becher and M.D. Schwartz, Direct photon production with effective field theory , JHEP 02 (2010) 040 [ 0911.0681]

    2010 arXiv

  158. [166]

    Becher and G

    T. Becher and G. Bell, The gluon jet function at two-loop order , Phys. Lett. B 695 (2011) 252 [1008.1936]

    2011 arXiv

  159. [167]

    W.-L. Ju, Y. Xu, L.L. Yang and B. Zhou, Thrust distribution in Higgs decays up to the fifth logarithmic order, Phys. Rev. D 107 (2023) 114034 [ 2301.04294]

    2023 arXiv

  160. [168]

    G. Das, S. Moch and A. Vogt, Approximate four-loop QCD corrections to the Higgs-boson production cross section, Phys. Lett. B 807 (2020) 135546 [ 2004.00563]

    2020 arXiv

  161. [169]

    Korchemsky and A.V

    G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B283 (1987) 342

    1987

  162. [170]

    Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions, Mod

    G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions, Mod. Phys. Lett. A4 (1989) 1257

    1989

  163. [171]

    Moch, J.A.M

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

    2004 arXiv

  164. [172]

    Beneke and V.M

    M. Beneke and V.M. Braun, Power corrections and renormalons in Drell-Yan production , Nucl. Phys. B 454 (1995) 253 [ hep-ph/9506452]

    1995 arXiv

  165. [173]

    Henn, A.V

    J. Henn, A.V. Smirnov, V.A. Smirnov, M. Steinhauser and R.N. Lee, Four-loop photon quark form factor and cusp anomalous dimension in the large- Nc limit of QCD , JHEP 03 (2017) 139 [ 1612.04389]

    2017 arXiv

  166. [174]

    Henn, A.V

    J.M. Henn, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, A planar four-loop form factor and cusp anomalous dimension in QCD , JHEP 05 (2016) 066 [ 1604.03126]

    2016 arXiv

  167. [175]

    von Manteuffel and R.M

    A. von Manteuffel and R.M. Schabinger, Quark and gluon form factors in four loop QCD: The N 2 f and Nqγ Nf contributions, Phys. Rev. D99 (2019) 094014 [ 1902.08208]

    2019 arXiv

  168. [176]

    Lee, A.V

    R.N. Lee, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Four-loop quark form factor with quartic fundamental colour factor , JHEP 02 (2019) 172 [ 1901.02898]

    2019 arXiv

  169. [177]

    J.M. Henn, T. Peraro, M. Stahlhofen and P. Wasser, Matter dependence of the four-loop cusp anomalous dimension , Phys. Rev. Lett. 122 (2019) 201602 [ 1901.03693]

    2019 arXiv

  170. [178]

    Boels, T

    R.H. Boels, T. Huber and G. Yang, Four-Loop Nonplanar Cusp Anomalous Dimension in N=4 Supersymmetric Yang-Mills Theory , Phys. Rev. Lett. 119 (2017) 201601 [1705.03444]

    2017 arXiv

  171. [179]

    S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond , JHEP 10 (2017) 041 [ 1707.08315]

    2017 arXiv

  172. [180]

    S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, On quartic colour factors in splitting functions and the gluon cusp anomalous dimension , Phys. Lett. B782 (2018) 627 [1805.09638]

    2018 arXiv

  173. [181]

    Davies, A

    J. Davies, A. Vogt, B. Ruijl, T. Ueda and J.A.M. Vermaseren, Large-nf contributions to the four-loop splitting functions in QCD , Nucl. Phys. B915 (2017) 335 [ 1610.07477]

    2017 arXiv

  174. [182]

    Br¨ user, A

    R. Br¨ user, A. Grozin, J.M. Henn and M. Stahlhofen, Matter dependence of the four-loop QCD cusp anomalous dimension: from small angles to all angles , JHEP 05 (2019) 186 [1902.05076]. – 53 –

    2019 arXiv

  175. [183]

    Gracey, Anomalous dimension of nonsinglet Wilson operators at O (1 / N(f )) in deep inelastic scattering, Phys

    J.A. Gracey, Anomalous dimension of nonsinglet Wilson operators at O (1 / N(f )) in deep inelastic scattering, Phys. Lett. B322 (1994) 141 [ hep-ph/9401214]

    1994 arXiv

  176. [184]

    Grozin, Four-loop cusp anomalous dimension in QED , JHEP 06 (2018) 073 [1805.05050]

    A. Grozin, Four-loop cusp anomalous dimension in QED , JHEP 06 (2018) 073 [1805.05050]

    2018 arXiv

  177. [185]

    Henn, G.P

    J.M. Henn, G.P. Korchemsky and B. Mistlberger, The full four-loop cusp anomalous dimension in N = 4 super Yang-Mills and QCD , JHEP 04 (2020) 018 [ 1911.10174]

    2020 arXiv

  178. [186]

    Herzog, S

    F. Herzog, S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, Five-loop contributions to low-N non-singlet anomalous dimensions in QCD , Phys. Lett. B 790 (2019) 436 [ 1812.11818]

    2019 arXiv

  179. [187]

    Gross and F

    D.J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories , Phys. Rev. Lett. 30 (1973) 1343

    1973

  180. [188]

    Politzer, Reliable Perturbative Results for Strong Interactions? , Phys

    H.D. Politzer, Reliable Perturbative Results for Strong Interactions? , Phys. Rev. Lett. 30 (1973) 1346

    1973

  181. [189]

    Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order , Phys

    W.E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order , Phys. Rev. Lett. 33 (1974) 244

    1974

  182. [190]

    Jones, Two Loop Diagrams in Yang-Mills Theory , Nucl

    D.R.T. Jones, Two Loop Diagrams in Yang-Mills Theory , Nucl. Phys. B 75 (1974) 531

    1974

  183. [191]

    Egorian and O.V

    E. Egorian and O.V. Tarasov, Two Loop Renormalization of the QCD in an Arbitrary Gauge, Teor. Mat. Fiz. 41 (1979) 26

    1979

  184. [192]

    Tarasov, A.A

    O.V. Tarasov, A.A. Vladimirov and A.Y. Zharkov, The Gell-Mann-Low Function of QCD in the Three Loop Approximation , Phys. Lett. B 93 (1980) 429

    1980

  185. [193]

    Larin and J.A.M

    S.A. Larin and J.A.M. Vermaseren, The Three loop QCD Beta function and anomalous dimensions, Phys. Lett. B 303 (1993) 334 [ hep-ph/9302208]

    1993 arXiv

  186. [194]

    van Ritbergen, J.A.M

    T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The Four loop beta function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [ hep-ph/9701390]

    1997 arXiv

  187. [195]

    Czakon, The Four-loop QCD beta-function and anomalous dimensions , Nucl

    M. Czakon, The Four-loop QCD beta-function and anomalous dimensions , Nucl. Phys. B 710 (2005) 485 [ hep-ph/0411261]

    2005 arXiv

  188. [196]

    Baikov, K.G

    P.A. Baikov, K.G. Chetyrkin and J.H. K¨ uhn,Five-Loop Running of the QCD coupling constant, Phys. Rev. Lett. 118 (2017) 082002 [ 1606.08659]

    2017 arXiv

  189. [197]

    Herzog, B

    F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, The five-loop beta function of Yang-Mills theory with fermions , JHEP 02 (2017) 090 [ 1701.01404]. – 54 –

    2017 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.