Abstract
In hard collisions at a hadron collider the most appropriate description of the initial state depends on what is measured in the final state. Parton distribution functions (PDFs) evolved to the hard collision scale Q are appropriate for inclusive observables, but not for measurements with a specific number of hard jets, leptons, and photons. Here the incoming protons are probed and lose their identity to an incoming jet at a scale μ B ≪ Q, and the initial state is described by universal beam functions. We discuss the field-theoretic treatment of beam functions, and show that the beam function has the same RG evolution as the jet function to all orders in perturbation theory. In contrast to PDF evolution, the beam function evolution does not mix quarks and gluons and changes the virtuality of the colliding parton at fixed momentum fraction. At μ B , the incoming jet can be described perturbatively, and we give a detailed derivation of the one-loop matching of the quark beam function onto quark and gluon PDFs. We compute the associated NLO Wilson coefficients and explicitly verify the cancellation of IR singularities. As an application, we give an expression for the next-to-next-to-leading logarithmic order (NNLL) resummed Drell-Yan beam thrust cross section.
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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] [SPIRES].
V.N. Gribov and L.N. Lipatov, Deep inelastic ep scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438 [SPIRES].
H. Georgi and H.D. Politzer, Electroproduction scaling in an asymptotically free theory of strong interactions, Phys. Rev. D 9 (1974) 416 [SPIRES].
D.J. Gross and F. Wilczek, Asymptotically free gauge theories. 2, Phys. Rev. D 9 (1974) 980 [SPIRES].
G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298 [SPIRES].
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 [SPIRES].
S.D. Ellis, A. Hornig, C. Lee, C.K. Vermilion and J.R. Walsh, Consistent Factorization of Jet Observables in Exclusive Multijet Cross-Sections, Phys. Lett. B 689 (2010) 82 [arXiv:0912.0262] [SPIRES].
T.T. Jouttenus, Jet Function with a Jet Algorithm in SCET, Phys. Rev. D 81 (2010) 094017 [arXiv:0912.5509] [SPIRES].
S.D. Ellis, C.K. Vermilion, J.R. Walsh, A. Hornig and C. Lee, Jet Shapes and Jet Algorithms in SCET, arXiv:1001.0014 [SPIRES].
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] [SPIRES].
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] [SPIRES].
C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [SPIRES].
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] [SPIRES].
J.C. Collins, D.E. Soper and G.F. Sterman, Soft Gluons and Factorization, Nucl. Phys. B 308 (1988) 833 [SPIRES].
S.M. Aybat and G.F. Sterman, Soft-Gluon Cancellation, Phases and Factorization with Initial-State Partons, Phys. Lett. B 671 (2009) 46 [arXiv:0811.0246] [SPIRES].
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] [SPIRES].
J.C. Collins and F. Hautmann, Infrared divergences and non-lightlike eikonal lines in Sudakov processes, Phys. Lett. B 472 (2000) 129 [hep-ph/9908467] [SPIRES].
C. Lee and G.F. Sterman, Momentum flow correlations from event shapes: Factorized soft gluons and soft-collinear effective theory, Phys. Rev. D 75 (2007) 014022 [hep-ph/0611061] [SPIRES].
A. Idilbi and T. Mehen, On the equivalence of soft and zero-bin subtractions, Phys. Rev. D 75 (2007) 114017 [hep-ph/0702022] [SPIRES].
J. Chay and C. Kim, Collinear effective theory at subleading order and its application to heavy-light currents, Phys. Rev. D 65 (2002) 114016 [hep-ph/0201197] [SPIRES].
A.V. Manohar, T. Mehen, D. Pirjol and I.W. Stewart, Reparameterization invariance for collinear operators, Phys. Lett. B 539 (2002) 59 [hep-ph/0204229] [SPIRES].
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] [SPIRES].
J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B 194 (1982) 445 [SPIRES].
C. Balzereit, T. Mannel and W. Kilian, Evolution of the light-cone distribution function for a heavy quark, Phys. Rev. D 58 (1998) 114029 [hep-ph/9805297] [SPIRES].
M. Neubert, Renormalization-group improved calculation of the B → X s γ branching ratio, Eur. Phys. J. C 40 (2005) 165 [hep-ph/0408179] [SPIRES].
S. Fleming, A.H. Hoang, S. Mantry and I.W. Stewart, Top Jets in the Peak Region: Factorization Analysis with NLL Resummation, Phys. Rev. D 77 (2008) 114003 [arXiv:0711.2079] [SPIRES].
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] [SPIRES].
S. Fleming, A.K. Leibovich and T. Mehen, Resumming the color octet contribution to e + e − → J/ψ + X, Phys. Rev. D 68 (2003) 094011 [hep-ph/0306139] [SPIRES].
S. Moch, J.A.M. Vermaseren and A. Vogt, The three-loop splitting functions in QCD: The non-singlet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [SPIRES].
S. Moch, J.A.M. Vermaseren, and A. Vogt, The quark form factor at higher orders, JHEP 08 (2005) 049 [hep-ph/0507039] [SPIRES].
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] [SPIRES].
A.D. Martin, W.J. Stirling, R.S. Thorne and G. Watt, Uncertainties on α s in global PDF analyses and implications for predicted hadronic cross sections, Eur. Phys. J. C 64 (2009) 653 [arXiv:0905.3531] [SPIRES].
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 [SPIRES].
S. Catani, M.L. Mangano and P. Nason, Sudakov resummation for prompt photon production in hadron collisions, JHEP 07 (1998) 024 [hep-ph/9806484] [SPIRES].
T. Becher, M. Neubert and G. Xu, Dynamical Threshold Enhancement and Resummation in Drell-Yan Production, JHEP 07 (2008) 030 [arXiv:0710.0680] [SPIRES].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, The Beam Thrust Cross section for Drell-Yan at NNLL Order, arXiv:1005.4060 [SPIRES].
C.F. Berger, C. Marcantonini, I.W. Stewart, F.J.Tackmann, and W.J. Waalewijn, Higgs Production with an Inclusive Jet Veto at NNLL+NNLO, MIT-CTP 4122, to appear.
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] [SPIRES].
R.A. Brandt, F. Neri and M.-a. Sato, Renormalization of Loop Functions for All Loops, Phys. Rev. D 24 (1981) 879 [SPIRES].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B 283 (1987) 342 [SPIRES].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [SPIRES].
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] [SPIRES].
M.D. Schwartz, Resummation and NLO Matching of Event Shapes with Effective Field Theory, Phys. Rev. D 77 (2008) 014026 [arXiv:0709.2709] [SPIRES].
A.V. Manohar, Deep inelastic scattering as x → 1 using soft-collinear effective theory, Phys. Rev. D 68 (2003) 114019 [hep-ph/0309176] [SPIRES].
C.W. Bauer, C. Lee, A.V. Manohar and M.B. Wise, Enhanced nonperturbative effects in Z decays to hadrons, Phys. Rev. D 70 (2004) 034014 [hep-ph/0309278] [SPIRES].
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 [SPIRES].
S.A. Larin and J.A.M. Vermaseren, The Three loop QCD β-function and anomalous dimensions, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208] [SPIRES].
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] [SPIRES].
T. Becher, M. Neubert and B.D. Pecjak, Factorization and momentum-space resummation in deep- inelastic scattering, JHEP 01 (2007) 076 [hep-ph/0607228] [SPIRES].
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Stewart, I.W., Tackmann, F.J. & Waalewijn, W.J. The quark beam function at NNLL. J. High Energ. Phys. 2010, 5 (2010). https://doi.org/10.1007/JHEP09(2010)005
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DOI: https://doi.org/10.1007/JHEP09(2010)005