Fragmentation functions beyond fixed order accuracy

Anderle, Daniele P.; Kaufmann, Tom; Stratmann, Marco; Ringer, Felix

doi:10.1103/PhysRevD.95.054003

Title: Fragmentation functions beyond fixed order accuracy

Abstract

Here, we give a detailed account of the phenomenology of all-order resummations of logarithmically enhanced contributions at small momentum fraction of the observed hadron in semi-inclusive electron-positron annihilation and the timelike scale evolution of parton-to-hadron fragmentation functions. The formalism to perform resummations in Mellin moment space is briefly reviewed, and all relevant expressions up to next-to-next-to-leading logarithmic order are derived, including their explicit dependence on the factorization and renormalization scales. We discuss the details pertinent to a proper numerical implementation of the resummed results comprising an iterative solution to the timelike evolution equations, the matching to known fixed-order expressions, and the choice of the contour in the Mellin inverse transformation. First extractions of parton-to-pion fragmentation functions from semi-inclusive annihilation data are performed at different logarithmic orders of the resummations in order to estimate their phenomenological relevance. To this end, we compare our results to corresponding fits up to fixed, next-to-next-to-leading order accuracy and study the residual dependence on the factorization scale in each case.

Authors:

Anderle, Daniele P. ^[1]; Kaufmann, Tom ^[2]; Stratmann, Marco ^[2]; Ringer, Felix ^[3]

The Univ. of Manchester, Manchester (United Kingdom)
Univ. of Tubingen, Tubingen (Germany)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: Mon Mar 06 00:00:00 EST 2017

Research Org.:: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1460626

Alternate Identifier(s):: OSTI ID: 1345854

Report Number(s):: LA-UR-16-27763
Journal ID: ISSN 2470-0010; PRVDAQ; TRN: US1901881

Grant/Contract Number:: AC52-06NA25396; 2012LANL7033

Resource Type:: Accepted Manuscript

Journal Name:: Physical Review D

Additional Journal Information:: Journal Volume: 95; Journal Issue: 5; Journal ID: ISSN 2470-0010

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 74 ATOMIC AND MOLECULAR PHYSICS; Atomic and Nuclear Physics

Citation Formats


                    Anderle, Daniele P., Kaufmann, Tom, Stratmann, Marco, and Ringer, Felix. Fragmentation functions beyond fixed order accuracy.  United States: N. p., 2017. 
Web.  doi:10.1103/PhysRevD.95.054003.

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                    Anderle, Daniele P., Kaufmann, Tom, Stratmann, Marco, & Ringer, Felix. Fragmentation functions beyond fixed order accuracy.  United States.  https://doi.org/10.1103/PhysRevD.95.054003

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                    Anderle, Daniele P., Kaufmann, Tom, Stratmann, Marco, and Ringer, Felix. Mon .  
"Fragmentation functions beyond fixed order accuracy".  United States.  https://doi.org/10.1103/PhysRevD.95.054003.  https://www.osti.gov/servlets/purl/1460626.

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@article{osti_1460626,

  title        = {Fragmentation functions beyond fixed order accuracy},

  author       = {Anderle, Daniele P. and Kaufmann, Tom and Stratmann, Marco and Ringer, Felix},

  abstractNote = {Here, we give a detailed account of the phenomenology of all-order resummations of logarithmically enhanced contributions at small momentum fraction of the observed hadron in semi-inclusive electron-positron annihilation and the timelike scale evolution of parton-to-hadron fragmentation functions. The formalism to perform resummations in Mellin moment space is briefly reviewed, and all relevant expressions up to next-to-next-to-leading logarithmic order are derived, including their explicit dependence on the factorization and renormalization scales. We discuss the details pertinent to a proper numerical implementation of the resummed results comprising an iterative solution to the timelike evolution equations, the matching to known fixed-order expressions, and the choice of the contour in the Mellin inverse transformation. First extractions of parton-to-pion fragmentation functions from semi-inclusive annihilation data are performed at different logarithmic orders of the resummations in order to estimate their phenomenological relevance. To this end, we compare our results to corresponding fits up to fixed, next-to-next-to-leading order accuracy and study the residual dependence on the factorization scale in each case.},

  doi          = {10.1103/PhysRevD.95.054003},

  journal      = {Physical Review D},

  number       = 5,

  volume       = 95,

  place        = {United States},

  year         = {Mon Mar 06 00:00:00 EST 2017},

  month        = {Mon Mar 06 00:00:00 EST 2017}

}

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Physical Review D, Vol. 57, Issue 9
DOI: 10.1103/PhysRevD.57.5811

Fragmentation functions from flavor-inclusive and flavor-tagged $e^{+} e^{-}$ annihilations
journal, July 2000

Kretzer, S.
Physical Review D, Vol. 62, Issue 5
DOI: 10.1103/PhysRevD.62.054001

Generalized threshold resummation in inclusive DIS and semi-inclusive electron-positron annihilation
text, January 2016

Almasy, A. A.; Lo Presti, N. A.; Vogt, A.
Springer
DOI: 10.5167/uzh-121809

The evolution of unpolarized singlet structure functions at small x
text, January 1998

Blümlein, Johannes; Vogt, Andreas
Deutsches Elektronen-Synchrotron, DESY, Hamburg
DOI: 10.3204/pubdb-2017-08393

AKK Update: Improvements from New Theoretical Input and Experimental Data
text, January 2008

Albino, S.; Kniehl, B. A.; Kramer, G.
arXiv
DOI: 10.48550/arxiv.0803.2768

Quark Fragmentation within an Identified Jet
text, January 2009

Procura, Massimiliano; Stewart, Iain W.
arXiv
DOI: 10.48550/arxiv.0911.4980

Leading logarithmic large-x resummation of off-diagonal splitting functions and coefficient functions
text, January 2010

Vogt, A.
arXiv
DOI: 10.48550/arxiv.1005.1606

Fully double-logarithm-resummed cross sections
text, January 2011

Albino, S.; Bolzoni, P.; Kniehl, B. A.
arXiv
DOI: 10.48550/arxiv.1104.3018

QCD resummation for semi-inclusive hadron production processes
text, January 2012

Anderle, Daniele P.; Ringer, Felix; Vogelsang, Werner
arXiv
DOI: 10.48550/arxiv.1212.2099

Average gluon and quark jet multiplicities at higher orders
text, January 2013

Bolzoni, Paolo; Kniehl, Bernd A.; Kotikov, Anatoly V.
arXiv
DOI: 10.48550/arxiv.1305.6017

Probing fragmentation functions from same-side hadron-jet momentum correlations in p-p collisions
text, January 2013

Arleo, Francois; Fontannaz, Michel; Guillet, Jean-Philippe
arXiv
DOI: 10.48550/arxiv.1311.7356

Fragmentation in Jets at NNLO
text, January 2014

Ritzmann, Mathias; Waalewijn, Wouter J.
arXiv
DOI: 10.48550/arxiv.1407.3272

Measurement of hadron composition in charged jets from pp collisions with the ALICE experiment
text, January 2014

Lu, Xianguo
arXiv
DOI: 10.48550/arxiv.1407.8385

Parton-to-Pion Fragmentation Reloaded
text, January 2014

de Florian, D.; Sassot, R.; Epele, M.
arXiv
DOI: 10.48550/arxiv.1410.6027

Interplay of Threshold Resummation and Hadron Mass Corrections in Deep Inelastic Processes
text, January 2014

Accardi, Alberto; Anderle, Daniele P.; Ringer, Felix
arXiv
DOI: 10.48550/arxiv.1411.3649

Hadron Fragmentation Inside Jets in Hadronic Collisions
text, January 2015

Kaufmann, Tom; Mukherjee, Asmita; Vogelsang, Werner
arXiv
DOI: 10.48550/arxiv.1506.01415

Fragmentation Functions at Next-to-Next-to-Leading Order Accuracy
text, January 2015

Anderle, Daniele P.; Ringer, Felix; Stratmann, Marco
arXiv
DOI: 10.48550/arxiv.1510.05845

The role of heavy quarks in light hadron fragmentation
text, January 2016

Epele, M.; Canal, C. A. Garcia; Sassot, R.
arXiv
DOI: 10.48550/arxiv.1604.08427

Jet substructure using semi-inclusive jet functions within SCET
text, January 2016

Kang, Zhong-Bo; Ringer, Felix; Vitev, Ivan
arXiv
DOI: 10.48550/arxiv.1606.07063

Fragmentation Functions from Flavour-inclusive and Flavour-tagged e^+e^- Annihilations
text, January 2000

Kretzer, S.
arXiv
DOI: 10.48550/arxiv.hep-ph/0003177

Next-to-leading order QCD corrections to one hadron-production in polarized pp collisions at RHIC
text, January 2002

de Florian, Daniel
arXiv
DOI: 10.48550/arxiv.hep-ph/0210442

Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS
text, January 2004

Vogt, A.
arXiv
DOI: 10.48550/arxiv.hep-ph/0408244

Crossing Heavy-Flavour Thresholds in Fragmentation Functions
text, January 2005

Cacciari, Matteo; Nason, Paolo; Oleari, Carlo
arXiv
DOI: 10.48550/arxiv.hep-ph/0504192

The Quark Form Factor at Higher Orders
text, January 2005

Moch, S.; Vermaseren, J. A. M.; Vogt, A.
arXiv
DOI: 10.48550/arxiv.hep-ph/0507039

Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions
text, January 2006

Mitov, A.; Moch, S.; Vogt, A.
arXiv
DOI: 10.48550/arxiv.hep-ph/0604053

Determination of fragmentation functions and their uncertainties
text, January 2007

Hirai, M.; Kumano, S.; Nagai, T. -H.
arXiv
DOI: 10.48550/arxiv.hep-ph/0702250

Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron-positron annihilation
text, January 1996

Rijken, P. J.; van Neerven, W. L.
arXiv
DOI: 10.48550/arxiv.hep-ph/9609377

Works referencing / citing this record:

First determination of $D^{*}$ -meson fragmentation functions and their uncertainties at next-to-next-to-leading order
journal, April 2018

Soleymaninia, Maryam; Khanpour, Hamzeh; Nejad, S. Mohammad Moosavi
Physical Review D, Vol. 97, Issue 7
DOI: 10.1103/physrevd.97.074014

First determination of $D^{*+}$-meson fragmentation functions and their uncertainties at next-to-next-to-leading order
text, January 2017

Soleymaninia, Maryam; Khanpour, Hamzeh; Nejad, S. Mohammad Moosavi
arXiv
DOI: 10.48550/arxiv.1711.11344

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Title: Fragmentation functions beyond fixed order accuracy

Abstract

Citation Formats

Multiplicity and hadron distributions in QCD jets journal, March 1983

On the Drell–Levy–Yan relation to O(α2) journal, October 2000

Scaling violation in e+e− fragmentation functions: QCD evolution, hadronization and heavy quark mass effects [Nucl. Phys. B 421 (1994) 473] journal, December 1996

Soft-gluon resummation for the fragmentation of light and heavy quarks at large x journal, December 2001

Measurement of the jet fragmentation function and transverse profile in proton–proton collisions at a center-of-mass energy of 7 TeV with the ATLAS detector journal, November 2011

Evolution of parton densities beyond leading order journal, November 1980

Probing fragmentation functions from same-side hadron-jet momentum correlations in p-p collisions journal, April 2014

Fragmentation functions at next-to-next-to-leading order accuracy journal, December 2015

Quark fragmentation within an identified jet journal, April 2010

Measurement of hadron composition in charged jets from pp collisions with the ALICE experiment journal, November 2014

Production of charged pions, kaons, and protons in e + e − annihilations into hadrons at s = 10.54 GeV journal, August 2013

Parton fragmentation within an identified jet at NNLL journal, May 2011

Perspectives on Higgs Physics II book, November 1997

AKK update: Improvements from new theoretical input and experimental data journal, November 2008

Fragmentation in jets at NNLO journal, September 2014

QCD corrections to semi-inclusive hadron production in electron–positron annihilation at two loops journal, September 2006

Precision Measurement of Charged Pion and Kaon Differential Cross Sections in e + e − Annihilation at s = 10.52 GeV journal, August 2013

Leading logarithmic large-x resummation of off-diagonal splitting functions and coefficient functions journal, July 2010

Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron-positron annihilation journal, March 1997

Scaling violation in e+e− fragmentation functions: QCD evolution, hadronization and heavy quark mass effects journal, June 1994

Fragmentation in jets: Cone and threshold effects journal, June 2012

Parton-to-pion fragmentation reloaded journal, January 2015

Pion and Kaon multiplicities in heavy quark jets from e+e− annihilation at 29 GeV journal, January 1987

Charged-Hadron Inclusive Cross Sections and Fractions in e + e − Annihilation at s = 29 GeV journal, September 1988

Inclusive ?�, K� and $$(p,\bar p)$$ differential cross-sections at the Z resonance journal, September 1995

Higher-order corrections to in quantum chromodynamics journal, August 1979

Gluon- and Quark-Jet Multiplicities with NNNLO and NNLL Accuracy journal, December 2012

corrections to particle multiplicity ratios in gluon and quark jets journal, July 1984

Determination of fragmentation functions and their uncertainties journal, May 2007

Next-to-leading order evolution of polarized and unpolarized fragmentation functions journal, July 1997

Next-to-next-to-leading order evolution of non-singlet fragmentation functions journal, June 2006

Efficient evolution of unpolarized and polarized parton distributions with QCD-Pegasus journal, July 2005

Jet substructure using semi-inclusive jet functions in SCET journal, November 2016

Next-to-leading order QCD corrections to high- p T pion production in longitudinally polarized pp collisions journal, March 2003

On the next-to-next-to-leading order evolution of flavour-singlet fragmentation functions journal, January 2012

On third-order timelike splitting functions and top-mediated Higgs decay into hadrons journal, January 2008

Production of π + , K + , K 0 , K * 0 , φ , p and Λ 0 in hadronic Z 0 decays journal, January 1999

Global analysis of fragmentation functions for protons and charged hadrons journal, October 2007

Next-to-leading order QCD corrections to one-hadron production in polarized pp collisions at BNL RHIC journal, March 2003

Parton distributions in the LHC era: MMHT 2014 PDFs journal, May 2015

On the multiplicity of hadrons in QCD jets journal, August 1981

Global analysis of fragmentation functions for pions and kaons and their uncertainties journal, June 2007

Fragmentation functions for light charged hadrons with complete quark flavour separation journal, October 2005

Crossing heavy-flavour thresholds in fragmentation functions journal, October 2005

Jet-fragmentation properties in p ¯ p collisions at √ s =1.8 TeV journal, August 1990

Role of heavy quarks in light hadron fragmentation journal, August 2016

One-particle inclusive processes in deeply inelastic lepton-nucleon scattering journal, December 1994

Parton model interpretation of the cut vertex formalism journal, April 1980

Higher-Order Quantum Chromodynamic Corrections in e + e − Annihilation journal, September 1979

Erratum: Quark fragmentation within an identified jet [Phys. Rev. D 81 , 074009 (2010)] journal, February 2011

QCD corrections to parton-parton scattering processes journal, November 1989

O(αS2) contributions to the longitudinal fragmentation function in e+e− annihilation journal, October 1996

Processes involving fragmentation functions beyond the leading order in QCD journal, December 1979

Back-to-back jets in QCD journal, December 1981

NNLO evolution of deep-inelastic structure functions: the singlet case journal, November 2000

Energy evolution of the moments of the hadron distribution in QCD jets including NNLL resummation and NLO running-coupling corrections journal, August 2014

QCD resummation for semi-inclusive hadron production processes journal, February 2013

Problem of kinematic mass corrections for unpolarized semi-inclusive deep inelastic scattering journal, November 2016

Lepton-hadron processes beyond leading order in quantum chromodynamics journal, December 1982

Measurement of jet fragmentation into charged particles in pp and PbPb collisions at $ \sqrt{{{s_{\mathrm{NN}}}}} = 2.76 $ TeV journal, October 2012

Towards Three-loop QCD Corrections to the Time-like Splitting Functions journal, January 2015

Measurement of the production rates of charged hadrons ine + e ?-annihilation at theZ 0 journal, June 1994

Fully double-logarithm-resummed cross sections journal, October 2011

The three-loop QCD β-function and anomalous dimensions journal, April 1993

Asymptotic freedom in parton language journal, August 1977

Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation journal, October 2011

Jet multiplicity and soft gluon factorization journal, November 1982

Average gluon and quark jet multiplicities at higher orders journal, October 2013

timelike Wilson coefficients for parton–fragmentation functions in Mellin space journal, August 2006

Hadron fragmentation inside jets in hadronic collisions journal, September 2015

First Monte Carlo analysis of fragmentation functions from single-inclusive e + e − annihilation journal, December 2016

Generalized threshold resummation in inclusive DIS and semi-inclusive electron-positron annihilation journal, January 2016

Uncertainties in pion and kaon fragmentation functions journal, October 2012

Multiplicity and hadron distributions in QCD jets journal, November 1983

Resummed small-x and first-moment evolution of fragmentation functions in perturbative QCD journal, October 2012

Analytic Calculation of Higher-Order Quantum-Chromodynamic Corrections in e + e − Annihilation journal, March 1980

The quark form factor at higher orders journal, August 2005

Multiplicity and hadron distributions in QCD jets
journal, March 1983

On the Drell–Levy–Yan relation to O(α2)
journal, October 2000

Scaling violation in e+e− fragmentation functions: QCD evolution, hadronization and heavy quark mass effects [Nucl. Phys. B 421 (1994) 473]
journal, December 1996

Soft-gluon resummation for the fragmentation of light and heavy quarks at large x
journal, December 2001

Measurement of the jet fragmentation function and transverse profile in proton–proton collisions at a center-of-mass energy of 7 TeV with the ATLAS detector
journal, November 2011

Evolution of parton densities beyond leading order
journal, November 1980

Probing fragmentation functions from same-side hadron-jet momentum correlations in p-p collisions
journal, April 2014

Fragmentation functions at next-to-next-to-leading order accuracy
journal, December 2015

Quark fragmentation within an identified jet
journal, April 2010

Measurement of hadron composition in charged jets from pp collisions with the ALICE experiment
journal, November 2014

Production of charged pions, kaons, and protons in $e^{+} e^{-}$ annihilations into hadrons at $\sqrt{s} = 10.54 GeV$
journal, August 2013

Parton fragmentation within an identified jet at NNLL
journal, May 2011

Perspectives on Higgs Physics II
book, November 1997

AKK update: Improvements from new theoretical input and experimental data
journal, November 2008

Fragmentation in jets at NNLO
journal, September 2014

QCD corrections to semi-inclusive hadron production in electron–positron annihilation at two loops
journal, September 2006

Precision Measurement of Charged Pion and Kaon Differential Cross Sections in $e^{+} e^{-}$ Annihilation at $\sqrt{s} = 10.52 GeV$
journal, August 2013

Leading logarithmic large-x resummation of off-diagonal splitting functions and coefficient functions
journal, July 2010

Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron-positron annihilation
journal, March 1997

Scaling violation in e+e− fragmentation functions: QCD evolution, hadronization and heavy quark mass effects
journal, June 1994

Fragmentation in jets: Cone and threshold effects
journal, June 2012

Parton-to-pion fragmentation reloaded
journal, January 2015

Pion and Kaon multiplicities in heavy quark jets from e+e− annihilation at 29 GeV
journal, January 1987

Charged-Hadron Inclusive Cross Sections and Fractions in $e^{+} e^{-}$ Annihilation at $\sqrt{s} = 29$ GeV
journal, September 1988

Inclusive ?�, K� and $$(p,\bar p)$$ differential cross-sections at the Z resonance
journal, September 1995

Higher-order corrections to in quantum chromodynamics
journal, August 1979

Gluon- and Quark-Jet Multiplicities with NNNLO and NNLL Accuracy
journal, December 2012

corrections to particle multiplicity ratios in gluon and quark jets
journal, July 1984

Determination of fragmentation functions and their uncertainties
journal, May 2007

Next-to-leading order evolution of polarized and unpolarized fragmentation functions
journal, July 1997

Next-to-next-to-leading order evolution of non-singlet fragmentation functions
journal, June 2006

Efficient evolution of unpolarized and polarized parton distributions with QCD-Pegasus
journal, July 2005

Jet substructure using semi-inclusive jet functions in SCET
journal, November 2016

Next-to-leading order QCD corrections to high- $p_{T}$ pion production in longitudinally polarized $pp$ collisions
journal, March 2003

On the next-to-next-to-leading order evolution of flavour-singlet fragmentation functions
journal, January 2012

On third-order timelike splitting functions and top-mediated Higgs decay into hadrons
journal, January 2008

Production of $π^{+},$ $K^{+},$ $K^{0},$ $K^{* 0},$ $φ,$ $p$ and $Λ^{0}$ in hadronic $Z^{0}$ decays
journal, January 1999

Global analysis of fragmentation functions for protons and charged hadrons
journal, October 2007

Next-to-leading order QCD corrections to one-hadron production in polarized $pp$ collisions at BNL RHIC
journal, March 2003

Parton distributions in the LHC era: MMHT 2014 PDFs
journal, May 2015

On the multiplicity of hadrons in QCD jets
journal, August 1981

Global analysis of fragmentation functions for pions and kaons and their uncertainties
journal, June 2007

Fragmentation functions for light charged hadrons with complete quark flavour separation
journal, October 2005

Crossing heavy-flavour thresholds in fragmentation functions
journal, October 2005

Jet-fragmentation properties in p ¯ p collisions at √ s =1.8 TeV
journal, August 1990

Role of heavy quarks in light hadron fragmentation
journal, August 2016

One-particle inclusive processes in deeply inelastic lepton-nucleon scattering
journal, December 1994

Parton model interpretation of the cut vertex formalism
journal, April 1980

Higher-Order Quantum Chromodynamic Corrections in $e^{+} e^{-}$ Annihilation
journal, September 1979

Erratum: Quark fragmentation within an identified jet [Phys. Rev. D 81 , 074009 (2010)]
journal, February 2011

QCD corrections to parton-parton scattering processes
journal, November 1989

O(αS2) contributions to the longitudinal fragmentation function in e+e− annihilation
journal, October 1996

Processes involving fragmentation functions beyond the leading order in QCD
journal, December 1979

Back-to-back jets in QCD
journal, December 1981

NNLO evolution of deep-inelastic structure functions: the singlet case
journal, November 2000

Energy evolution of the moments of the hadron distribution in QCD jets including NNLL resummation and NLO running-coupling corrections
journal, August 2014

QCD resummation for semi-inclusive hadron production processes
journal, February 2013

Problem of kinematic mass corrections for unpolarized semi-inclusive deep inelastic scattering
journal, November 2016

Lepton-hadron processes beyond leading order in quantum chromodynamics
journal, December 1982

Measurement of jet fragmentation into charged particles in pp and PbPb collisions at $ \sqrt{{{s_{\mathrm{NN}}}}} = 2.76 $ TeV
journal, October 2012

Towards Three-loop QCD Corrections to the Time-like Splitting Functions
journal, January 2015

Measurement of the production rates of charged hadrons ine + e ?-annihilation at theZ 0
journal, June 1994

Fully double-logarithm-resummed cross sections
journal, October 2011

The three-loop QCD β-function and anomalous dimensions
journal, April 1993

Asymptotic freedom in parton language
journal, August 1977

Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation
journal, October 2011

Jet multiplicity and soft gluon factorization
journal, November 1982

Average gluon and quark jet multiplicities at higher orders
journal, October 2013

timelike Wilson coefficients for parton–fragmentation functions in Mellin space
journal, August 2006

Hadron fragmentation inside jets in hadronic collisions
journal, September 2015

First Monte Carlo analysis of fragmentation functions from single-inclusive $e^{+} e^{-}$ annihilation
journal, December 2016

Generalized threshold resummation in inclusive DIS and semi-inclusive electron-positron annihilation
journal, January 2016

Uncertainties in pion and kaon fragmentation functions
journal, October 2012

Multiplicity and hadron distributions in QCD jets
journal, November 1983

Resummed small-x and first-moment evolution of fragmentation functions in perturbative QCD
journal, October 2012

Analytic Calculation of Higher-Order Quantum-Chromodynamic Corrections in $e^{+} e^{-}$ Annihilation
journal, March 1980

The quark form factor at higher orders
journal, August 2005

The gell-mann-low function of QCD in the three-loop approximation
journal, June 1980

Interplay of threshold resummation and hadron mass corrections in deep inelastic processes
journal, February 2015

Parton distribution and decay functions
journal, January 1982