Abstract
We present the first calculation of a fully-unintegrated parton distribution (beam function) at next-to-next-to-leading order (NNLO). We obtain the fully-differential beam function for quark-initiated processes by matching it onto standard parton distribution functions (PDFs) at two loops. The fully-differential beam function is a universal ingredient in resummed predictions of observables probing both the virtuality as well as the transverse momentum of the incoming quark in addition to its usual longitudinal momentum fraction. For such double-differential observables our result provides the part of the NNLO singular cross section related to collinear initial-state radiation (ISR), and is important for the resummation of large logarithms through N3LL.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Jain, M. Procura and W.J. Waalewijn, Fully-unintegrated parton distribution and fragmentation functions at perturbative k T , JHEP 04 (2012) 132 [arXiv:1110.0839] [INSPIRE].
A.J. Larkoski, I. Moult and D. Neill, Toward multi-differential cross sections: measuring two angularities on a single jet, JHEP 09 (2014) 046 [arXiv:1401.4458] [INSPIRE].
C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in B → X s γ in effective field theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE].
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] [INSPIRE].
C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Factorization at the LHC: from PDFs to initial state jets, Phys. Rev. D 81 (2010) 094035 [arXiv:0910.0467] [INSPIRE].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, The quark beam function at NNLL, JHEP 09 (2010) 005 [arXiv:1002.2213] [INSPIRE].
S. Mantry and F. Petriello, Factorization and resummation of Higgs boson differential distributions in soft-collinear effective theory, Phys. Rev. D 81 (2010) 093007 [arXiv:0911.4135] [INSPIRE].
J.R. Gaunt, M. Stahlhofen and F.J. Tackmann, The quark beam function at two loops, JHEP 04 (2014) 113 [arXiv:1401.5478] [INSPIRE].
A.V. Manohar and I.W. Stewart, The zero-bin and mode factorization in quantum field theory, Phys. Rev. D 76 (2007) 074002 [hep-ph/0605001] [INSPIRE].
J.-y. Chiu, A. Fuhrer, A.H. Hoang, R. Kelley and A.V. Manohar, Soft-collinear factorization and zero-bin subtractions, Phys. Rev. D 79 (2009) 053007 [arXiv:0901.1332] [INSPIRE].
T. Becher and G. Bell, Analytic regularization in soft-collinear effective theory, Phys. Lett. B 713 (2012) 41 [arXiv:1112.3907] [INSPIRE].
J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A formalism for the systematic treatment of rapidity logarithms in quantum field theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE].
J.C. Collins, T.C. Rogers and A.M. Stasto, Fully unintegrated parton correlation functions and factorization in lowest-order hard scattering, Phys. Rev. D 77 (2008) 085009 [arXiv:0708.2833] [INSPIRE].
T.C. Rogers, Next-to-leading order hard scattering using fully unintegrated parton distribution functions, Phys. Rev. D 78 (2008) 074018 [arXiv:0807.2430] [INSPIRE].
S. Fleming, A.K. Leibovich and T. Mehen, Resummation of large endpoint corrections to color-octet J/ψ photoproduction, Phys. Rev. D 74 (2006) 114004 [hep-ph/0607121] [INSPIRE].
S. Mantry and F. Petriello, Transverse momentum distributions from effective field theory with numerical results, Phys. Rev. D 83 (2011) 053007 [arXiv:1007.3773] [INSPIRE].
C.F. Berger, C. Marcantonini, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Higgs production with a central jet veto at NNLL+NNLO, JHEP 04 (2011) 092 [arXiv:1012.4480] [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The three loop splitting functions in QCD: the nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
J. Gaunt, M. Stahlhofen and F.J. Tackmann, The gluon beam function at two loops, JHEP 08 (2014) 020 [arXiv:1405.1044] [INSPIRE].
T. Gehrmann, T. Lubbert and L.L. Yang, Transverse parton distribution functions at next-to-next-to-leading order: the quark-to-quark case, Phys. Rev. Lett. 109 (2012) 242003 [arXiv:1209.0682] [INSPIRE].
T. Gehrmann, T. Luebbert and L.L. Yang, Calculation of the transverse parton distribution functions at next-to-next-to-leading order, JHEP 06 (2014) 155 [arXiv:1403.6451] [INSPIRE].
J.C. Collins and X. Zu, Initial state parton showers beyond leading order, JHEP 03 (2005) 059 [hep-ph/0411332] [INSPIRE].
J. Collins and H. Jung, Need for fully unintegrated parton densities, hep-ph/0508280 [INSPIRE].
G. Watt, A.D. Martin and M.G. Ryskin, Unintegrated parton distributions and inclusive jet production at HERA, Eur. Phys. J. C 31 (2003) 73 [hep-ph/0306169] [INSPIRE].
G. Watt, A.D. Martin and M.G. Ryskin, Unintegrated parton distributions and electroweak boson production at hadron colliders, Phys. Rev. D 70 (2004) 014012 [Erratum ibid. D 70 (2004) 079902] [hep-ph/0309096] [INSPIRE].
S. Alioli et al., Combining higher-order resummation with multiple NLO calculations and parton showers in GENEVA, JHEP 09 (2013) 120 [arXiv:1211.7049] [INSPIRE].
S. Alioli t al., Matching fully differential NNLO calculations and parton showers, JHEP 06 (2014) 089 [arXiv:1311.0286] [INSPIRE].
F.J. Tackmann, J.R. Walsh and S. Zuberi, Resummation properties of jet vetoes at the LHC, Phys. Rev. D 86 (2012) 053011 [arXiv:1206.4312] [INSPIRE].
J.R. Gaunt, Glauber gluons and multiple parton interactions, JHEP 07 (2014) 110 [arXiv:1405.2080] [INSPIRE].
D. Kang, C. Lee and I.W. Stewart, Using 1-jettiness to measure 2 jets in DIS 3 ways, Phys. Rev. D 88 (2013) 054004 [arXiv:1303.6952] [INSPIRE].
V. Antonelli, M. Dasgupta and G.P. Salam, Resummation of thrust distributions in DIS, JHEP 02 (2000) 001 [hep-ph/9912488] [INSPIRE].
Z.-B. Kang, X. Liu, S. Mantry and J.-W. Qiu, Probing nuclear dynamics in jet production with a global event shape, Phys. Rev. D 88 (2013) 074020 [arXiv:1303.3063] [INSPIRE].
Z.-B. Kang, X. Liu and S. Mantry, The 1-jettiness DIS event shape: NNLL + NLO results, Phys. Rev. D 90 (2014) 014041 [arXiv:1312.0301] [INSPIRE].
Z. Ligeti, I.W. Stewart and F.J. Tackmann, Treating the b quark distribution function with reliable uncertainties, Phys. Rev. D 78 (2008) 114014 [arXiv:0807.1926] [INSPIRE].
J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].
T. Huber and D. Maître, HypExp: a Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun. 175 (2006) 122 [hep-ph/0507094] [INSPIRE].
T. Huber and D. Maître, HypExp 2, expanding hypergeometric functions about half-integer parameters, Comput. Phys. Commun. 178 (2008) 755 [arXiv:0708.2443] [INSPIRE].
R. Mertig, M. Böhm and A. Denner, FEYN CALC: computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].
D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun. 180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1409.8281
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Gaunt, J.R., Stahlhofen, M. The fully-differential quark beam function at NNLO. J. High Energ. Phys. 2014, 146 (2014). https://doi.org/10.1007/JHEP12(2014)146
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2014)146