Abstract
The fragmentation of a light parton i to a jet containing a light energetic hadron h, where the momentum fraction of this hadron as well as the invariant mass of the jet is measured, is described by “fragmenting jet functions”. We calculate the oneloop matching coefficients \( {\mathcal{J}_{ij}} \) that relate the fragmenting jet functions \( \mathcal{G}_i^h \) to the standard, unpolarized fragmentation functions D h j for quark and gluon jets. We perform this calculation using various IR regulators and show explicitly how the IR divergences cancel in the matching. We derive the relationship between the coefficients \( {\mathcal{J}_{ij}} \) and the quark and gluon jet functions. This provides a cross-check of our results. As an application we study the process e + e - → Xπ+ on the Υ(4S) resonance where we measure the momentum fraction of the π+ and restrict to the dijet limit by imposing a cut on thrust T. In our analysis we sum the logarithms of τ = 1-T in the cross section to next-to-next-to-leading-logarithmic accuracy (NNLL). We find that including contributions up to NNLL (or NLO) can have a large impact on extracting fragmentation functions from e + e - → dijet + h.
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J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of Hard Processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1988) 1 [hep-ph/0409313] [SPIRES].
H. Georgi and H.D. Politzer, Quark Decay Functions and Heavy Hadron Production in QCD, Nucl. Phys. B 136 (1978) 445 [SPIRES].
R.K. Ellis, H. Georgi, M. Machacek, H.D. Politzer and G.G. Ross, Perturbation Theory and the Parton Model in QCD, Nucl. Phys. B 152 (1979) 285 [SPIRES].
G. Curci, W. Furmanski and R. Petronzio, Evolution of Parton Densities Beyond Leading Order: The Nonsinglet Case, Nucl. Phys. B 175 (1980) 27 [SPIRES].
J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B 194 (1982) 445 [SPIRES].
Belle collaboration, R. Seidl et al., Measurement of Azimuthal Asymmetries in Inclusive Production of Hadron Pairs in e + e - Annihilation at \( \sqrt {s} = 10.58 \) GeV, Phys. Rev. D 78 (2008) 032011 [arXiv:0805.2975] [SPIRES].
E. Farhi, A QCD Test for Jets, Phys. Rev. Lett. 39 (1977) 1587 [SPIRES].
M. Procura and I.W. Stewart, Quark Fragmentation within an Identified Jet, Phys. Rev. D 81 (2010) 074009 [arXiv:0911.4980] [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].
S. Catani, L. Trentadue, G. Turnock and B.R. Webber, Resummation of large logarithms in e + e - event shape distributions, Nucl. Phys. B 407 (1993) 3 [SPIRES].
G.P. Korchemsky and G.F. Sterman, Power corrections to event shapes and factorization, Nucl. Phys. B 555 (1999) 335 [hep-ph/9902341] [SPIRES].
S. Fleming, A.H. Hoang, S. Mantry and I.W. Stewart, Jets from massive unstable particles: Top-mass determination, Phys. Rev. D 77 (2008) 074010 [hep-ph/0703207] [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].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, N-Jettiness: An Inclusive Event Shape to Veto Jets, Phys. Rev. Lett. 105 (2010) 092002 [arXiv:1004.2489] [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].
S.D. Ellis, C.K. Vermilion, J.R. Walsh, A. Hornig and C. Lee, Jet Shapes and Jet Algorithms in SCET, JHEP 11 (2010) 101 [arXiv:1001.0014] [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. D75 (2007) 114017 [hep-ph/0702022] [SPIRES].
J.-y. Chiu, A. Fuhrer, A.H. Hoang, R. Kelley and A.V. Manohar, Using SCET to calculate electroweak corrections in gauge boson production, PoS EFT09 (2009) 009 [arXiv:0905.1141] [SPIRES].
S.D. Drell, D.J. Levy and T.-M. Yan, A Theory of Deep Inelastic Lepton-Nucleon Scattering and Lepton Pair Annihilation Processes. 1, Phys. Rev. 187 (1969) 2159 [SPIRES].
S.D. Drell, D.J. Levy and T.-M. Yan, A Theory of Deep Inelastic Lepton-Nucleon Scattering and Lepton Pair Annihilation Processes. 3. Deep Inelastic electron-Positron Annihilation, Phys. Rev. D1 (1970) 1617 [SPIRES].
V.N. Gribov and L.N. Lipatov, Deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438 [SPIRES].
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 [SPIRES].
G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298 [SPIRES].
M. Stratmann and W. Vogelsang, Next-to-leading order evolution of polarized and unpolarized fragmentation functions, Nucl. Phys. B 496 (1997) 41 [hep-ph/9612250] [SPIRES].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B 283 (1987) 342 [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].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, The Quark Beam Function at NNLL, JHEP 09 (2010) 005 [arXiv:1002.2213] [SPIRES].
M. Hirai, S. Kumano, T.H. Nagai and K. Sudoh, Determination of fragmentation functions and their uncertainties, Phys. Rev. D 75 (2007) 094009 [hep-ph/0702250] [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].
G. Parisi, Summing Large Perturbative Corrections in QCD, Phys. Lett. B 90 (1980) 295 [SPIRES].
G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections, Nucl. Phys. B 281 (1987) 310 [SPIRES].
L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [SPIRES].
T.O. Eynck, E. Laenen and L. Magnea, Exponentiation of the Drell-Yan cross section near partonic threshold in the DIS and MS-bar schemes, JHEP 06 (2003) 057 [hep-ph/0305179] [SPIRES].
R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart, Thrust at N 3 LL with Power Corrections and a Precision Global Fit for α s (m Z), arXiv:1006.3080 [SPIRES].
A.H. Hoang and I.W. Stewart, Designing Gapped Soft Functions for Jet Production, Phys.Lett. B 660 (2008) 483 [arXiv:0709.3519] [SPIRES].
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, arXiv:1012.4480 [SPIRES].
X. Liu, SCET approach to top quark decay, arXiv:1011.3872 [SPIRES].
A.H. Mueller, Cut Vertices and their Renormalization: A Generalization of the Wilson Expansion, Phys. Rev. D 18 (1978) 3705 [SPIRES].
L. Baulieu, E.G. Floratos and C. Kounnas, Parton Model Interpretation of the Cut Vertex Formalism, Nucl. Phys. B 166 (1980) 321 [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].
C.W. Bauer and A.V. Manohar, Shape function effects in B → X s γ and B → X u ℓν decays, Phys. Rev. D 70 (2004) 034024 [hep-ph/0312109] [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].
T. Becher and M.D. Schwartz, Direct photon production with effective field theory, JHEP 02 (2010) 040 [arXiv:0911.0681] [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].
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].
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].
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].
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].
S. Moch, J.A.M. Vermaseren and A. Vogt, The quark form factor at higher orders, JHEP 08 (2005) 049 [hep-ph/0507039] [SPIRES].
A. Vogt, S. Moch and J.A.M. Vermaseren, The three-loop splitting functions in QCD: The singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [SPIRES].
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ArXiv ePrint:1101.4953