×

Two-loop soft corrections and resummation of the thrust distribution in the dijet region. (English) Zbl 1298.81487

Summary: The thrust distribution in electron-positron annihilation is a classical precision QCD observable. Using renormalization group (RG) evolution in Laplace space, we perform the resummation of logarithmically enhanced corrections in the dijet limit, \(T \to 1\) to next-to-next-to-leading logarithmic (NNLL) accuracy. We independently derive the two-loop soft function for the thrust distribution and extract an analytical expression for the NNLL resummation coefficient \(g_{3}\). Our findings confirm earlier NNLL resummation results for the thrust distribution in soft-collinear effective theory. To combine the resummed expressions with the fixed-order results, we derive the \(\log(R)\)-matching and \(R\)-matching of the NNLL approximation to the fixed-order NNLO distribution.

MSC:

81V22 Unified quantum theories
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T18 Feynman diagrams
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] ALEPH collaboration, D. Buskulic et al., Studies of QCD in e+e− → hadrons at E(cm) = 130 GeV and 136 GeV, Z. Phys.C 73 (1997) 409 [SPIRES].
[2] ALEPH collaboration, A. Heister et al. Studies of QCD at e+e−centre-of-mass energies between 91 GeV and 209 GeV, Eur. Phys. J.C 35 (2004) 457 [SPIRES].
[3] OPAL collaboration, P.D. Acton et al., A Determination of αs(M(Z0)) at LEP using resummed QCD calculations, Z. Phys.C 59 (1993) 1 [SPIRES].
[4] OPAL collaboration, G. Alexander et al., QCD studies with e+e−annihilation data at 130 GeV and 136 GeV, Z. Phys.C 72 (1996) 191 [SPIRES].
[5] OPAL collaboration, K. Ackerstaff et al., QCD studies with e+e−annihilation data at 161 GeV, Z. Phys.C 75 (1997) 193 [SPIRES].
[6] OPAL collaboration, G. Abbiendi et al., QCD studies with e+e−annihilation data at 172 GeV to 189 GeV, Eur. Phys. J.C 16 (2000) 185 [hep-ex/0002012] [SPIRES].
[7] OPAL collaboration, G. Abbiendi et al., Measurement of event shape distributions and moments in e+e− → hadrons at 91 GeV - 209 GeV and a determination of αs, Eur. Phys. J.C 40 (2005) 287 [hep-ex/0503051] [SPIRES].
[8] OPAL collaboration, G. Abbiendi et al., Measurement of αswith radiative hadronic events, Eur. Phys. J.C 53 (2008) 21 [SPIRES].
[9] L3 collaboration, M. Acciarri et al., Study of the structure of hadronic events and determination of αsat \(\sqrt{s} = 130\) GeV and 136 GeV, Phys. Lett.B 371 (1996) 137 [SPIRES].
[10] L3 collaboration, M. Acciarri et al., QCD studies and determination of αsin e+e−collisions at \(\sqrt{s} = 161\) GeV and 172 GeV, Phys. Lett.B 404 (1997) 390 [SPIRES].
[11] L3 collaboration, M. Acciarri et al., QCD results from studies of hadronic events produced in e+e−annihilations at \(\sqrt{s} = 183\) GeV, Phys. Lett.B 444 (1998) 569 [SPIRES].
[12] L3 collaboration, P. Achard et al., Determination of αsfrom hadronic event shapes in e+e−annihilation at 192 GeV ≤ \( \sqrt{s} \) ≤ 208 GeV, Phys. Lett.B 536 (2002) 217 [hep-ex/0206052] [SPIRES].
[13] L3 collaboration, P. Achard et al., 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] [SPIRES].
[14] DELPHI collaboration, P. Abreu et al., Energy dependence of event shapes and of αsat LEP-2, Phys. Lett.B 456 (1999) 322 [SPIRES].
[15] DELPHI collaboration, J. Abdallah et al., A study of the energy evolution of event shape distributions and their means with the DELPHI detector at LEP, Eur. Phys. J.C 29 (2003) 285 [hep-ex/0307048] [SPIRES].
[16] DELPHI collaboration, J. Abdallah et al., The measurement of αsfrom event shapes with the DELPHI detector at the highest LEP energies, Eur. Phys. J.C 37 (2004) 1 [hep-ex/0406011] [SPIRES].
[17] SLD collaboration, K. Abe et al., Measurement of αs(MZ2) from hadronic event observables at the Z0resonance, Phys. Rev.D 51 (1995) 962 [hep-ex/9501003] [SPIRES].
[18] JADE collaboration, P. Pfeifen Schneider et al., QCD analyses and determinations of αsin e+e−annihilation at energies between 35-GeV and 189-GeV, Eur. Phys. J.C 17 (2000) 19 [hep-ex/0001055] [SPIRES].
[19] S. Brandt, C. Peyrou, R. Sosnowski and A. Wroblewski, The Principal axis of jets. An Attempt to analyze high-energy collisions as two-body processes, Phys. Lett.12 (1964) 57 [SPIRES].
[20] E. Farhi, A QCD Test for Jets, Phys. Rev. Lett.39 (1977) 1587 [SPIRES].
[21] R.K. Ellis, D.A. Ross and A.E. Terrano, The Perturbative Calculation of Jet Structure in e+e−Annihilation, Nucl. Phys.B 178 (1981) 421 [SPIRES].
[22] R.K. Ellis, D.A. Ross and A.E. Terrano, Calculation of Event Shape Parameters in e+e−Annihilation, Phys. Rev. Lett.45 (1980) 1226 [SPIRES].
[23] Z. Kunszt, Comment on the O(αS2) Corrections to Jet Production in e+e−Annihilation, Phys. Lett.B 99 (1981) 429 [SPIRES].
[24] J.A.M. Vermaseren, K.J.F. Gaemers and S.J. Oldham, Perturbative QCD Calculation of Jet Cross-Sections in e+e−Annihilation, Nucl. Phys.B 187 (1981) 301 [SPIRES].
[25] K. Fabricius, I. Schmitt, G. Kramer and G. Schierholz, Higher Order Perturbative QCD Calculation of Jet Cross-Sections in e+e−Annihilation, Zeit. Phys.C 11 (1981) 315 [SPIRES].
[26] Z. Kunszt, P. Nason, G. Marchesini and B.R Webber, QCD at LEP, in Workshop on Z Physics at LEP1: General Meetings. Vol. 1: Standard Physics, CERN Yellow Report ETH-PT-89-39, p.373 [SPIRES].
[27] W.T. Giele and E.W.N. Glover, Higher order corrections to jet cross-sections in e+e−annihilation, Phys. Rev.D 46 (1992) 1980 [SPIRES].
[28] S. Catani and M.H. Seymour, The Dipole Formalism for the Calculation of QCD Jet Cross Sections at Next-to-Leading Order, Phys. Lett.B 378 (1996) 287 [hep-ph/9602277] [SPIRES].
[29] S. Catani and M.H. Seymour, A general algorithm for calculating jet cross sections in NLO QCD, Nucl. Phys.B 485 (1997) 291 [hep-ph/9605323] [SPIRES].
[30] 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].
[31] Y.L. Dokshitzer, A. Lucenti, G. Marchesini and G.P. Salam, On the QCD analysis of jet broadening, JHEP01 (1998) 011 [hep-ph/9801324] [SPIRES].
[32] A. Banfi, G.P. Salam and G. Zanderighi, Semi-numerical resummation of event shapes, JHEP01 (2002) 018 [hep-ph/0112156] [SPIRES].
[33] R.W.L. Jones, M. Ford, G.P. Salam, H. Stenzel and D. Wicke, Theoretical uncertainties on αsfrom event-shape variables in e+e−annihilations, JHEP12 (2003) 007 [hep-ph/0312016] [SPIRES].
[34] A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Antenna Subtraction at NNLO, JHEP09 (2005) 056 [hep-ph/0505111] [SPIRES].
[35] A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, Infrared structure of e+e− → 3 jets at NNLO, JHEP11 (2007) 058 [arXiv:0710.0346] [SPIRES].
[36] A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, NNLO corrections to event shapes in e+e−annihilation, JHEP12 (2007) 094 [arXiv:0711.4711] [SPIRES].
[37] A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, Second-order QCD corrections to the thrust distribution, Phys. Rev. Lett.99 (2007) 132002 [arXiv:0707.1285] [SPIRES].
[38] S. Weinzierl, The infrared structure of e+e− → 3 jets at NNLO reloaded, JHEP07 (2009) 009 [arXiv:0904.1145] [SPIRES].
[39] S. Weinzierl, Event shapes and jet rates in electron-positron annihilation at NNLO, JHEP06 (2009) 041 [arXiv:0904.1077] [SPIRES].
[40] S. Weinzierl, Jet algorithms in electron-positron annihilation: Perturbative higher order predictions, Eur. Phys. J.C 71 (2011) 1565 [arXiv:1011.6247] [SPIRES].
[41] A. Denner, S. Dittmaier, T. Gehrmann and C. Kurz, Electroweak corrections to three-jet production in electron-positron annihilation, Phys. Lett.B 679 (2009) 219 [arXiv:0906.0372] [SPIRES].
[42] A. Denner, S. Dittmaier, T. Gehrmann and C. Kurz, Electroweak corrections to hadronic event shapes and jet production in e+e−annihilation, Nucl. Phys.B 836 (2010) 37 [arXiv:1003.0986] [SPIRES]. · Zbl 1206.81132
[43] G. Dissertori et al., First determination of the strong coupling constant using NNLO predictions for hadronic event shapes in e+e−annihilations, JHEP02 (2008) 040 [arXiv:0712.0327] [SPIRES].
[44] G. Dissertori et al., Precise determination of the strong coupling constant at NNLO in QCD from the three-jet rate in electron-positron annihilation at LEP, Phys. Rev. Lett.104 (2010) 072002 [arXiv:0910.4283] [SPIRES].
[45] T. Gehrmann, M. Jaquier and G. Luisoni, Hadronization effects in event shape moments, Eur. Phys. J.C 67 (2010) 57 [arXiv:0911.2422] [SPIRES].
[46] T. Gehrmann, G. Luisoni and H. Stenzel, Matching NLLA + NNLO for event shape distributions, Phys. Lett.B 664 (2008) 265 [arXiv:0803.0695] [SPIRES].
[47] R.A. Davison and B.R. Webber, Non-Perturbative Contribution to the Thrust Distribution in e+e−Annihilation, Eur. Phys. J.C 59 (2009) 13 [arXiv:0809.3326] [SPIRES].
[48] JADE collaboration, S. Bethke, S. Kluth, C. Pahl and J. Schieck, Determination of the Strong Coupling αsfrom hadronic Event Shapes with O(αs3) and resummed QCD predictions using JADE Data, Eur. Phys. J.C 64 (2009) 351 [arXiv:0810.1389] [SPIRES].
[49] G. Dissertori et al., Determination of the strong coupling constant using matched NNLO + NLLA predictions for hadronic event shapes in e+e−annihilations, JHEP08 (2009) 036 [arXiv:0906.3436] [SPIRES].
[50] OPAL collaboration and others, Determination of αSusing OPAL hadronic event shapes at \(\sqrt{s} = 91-209\) GeV and resummed NNLO calculations, arXiv:1101.1470 [SPIRES].
[51] G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections, Nucl. Phys.B 281 (1987) 310 [SPIRES].
[52] D. de Florian and M. Grazzini, The back-to-back region in e+e−energy energy correlation, Nucl. Phys.B 704 (2005) 387 [hep-ph/0407241] [SPIRES]. · Zbl 1119.81404
[53] 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].
[54] 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].
[55] 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] [SPIRES]. · Zbl 0998.81538
[56] M.D. Schwartz, Resummation and NLO Matching of Event Shapes with Effective Field Theory, Phys. Rev.D 77 (2008) 014026 [arXiv:0709.2709] [SPIRES].
[57] 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].
[58] 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].
[59] T. Becher and M.D. Schwartz, A Precise determination of αsfrom LEP thrust data using effective field theory, JHEP07 (2008) 034 [arXiv:0803.0342] [SPIRES].
[60] R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart, Thrust at N3LL with Power Corrections and a Precision Global Fit for alphas(mZ), Phys. Rev.D 83 (2011) 074021 [arXiv:1006.3080] [SPIRES].
[61] Y.-T. Chien and M.D. Schwartz, Resummation of heavy jet mass and comparison to LEP data, JHEP08 (2010) 058 [arXiv:1005.1644] [SPIRES]. · Zbl 1291.81385
[62] J.-y.Chiu, A. Jain, D. Neill and I.Z. Rothstein, The Rapidity Renormalization Group, arXiv:1104.0881 [SPIRES]. · Zbl 1348.81437
[63] T. Becher, G. Bell and M. Neubert, Factorization and Resummation for Jet Broadening, arXiv:1104.4108 [SPIRES].
[64] S. Moch, J.A.M. Vermaseren and A. Vogt, The quark form factor at higher orders, JHEP08 (2005) 049 [hep-ph/0507039] [SPIRES].
[65] 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 [arXiv:0902.3519] [SPIRES].
[66] R.N. Lee, A.V. Smirnov and V.A. Smirnov, Analytic Results for Massless Three-Loop Form Factors, JHEP04 (2010) 020 [arXiv:1001.2887] [SPIRES]. · Zbl 1272.81196
[67] 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, JHEP06 (2010) 094 [arXiv:1004.3653] [SPIRES]. · Zbl 1288.81146
[68] T. Becher and M. Neubert, Toward a NNLO calculation of the \(\bar{B} \to{X_s}\gamma\) 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] [SPIRES].
[69] T. Becher, M. Neubert and B.D. Pecjak, Factorization and momentum-space resummation in deep-inelastic scattering, JHEP01 (2007) 076 [hep-ph/0607228] [SPIRES].
[70] A.H. Hoang and S. Kluth, Hemisphere Soft Function at O(αs2) for Dijet Production in e+e−Annihilation, arXiv:0806.3852 [SPIRES].
[71] R. Kelley, R.M. Schabinger, M.D. Schwartz and H.X. Zhu, The two-loop hemisphere soft function, arXiv:1105.3676 [SPIRES].
[72] A. Hornig, C. Lee, I.W. Stewart, J.R. Walsh and S. Zuberi, Non-global Structure of the O(αs2) Dijet Soft Function, arXiv:1105.4628 [SPIRES]. · Zbl 1298.81390
[73] Y. Li, S. Mantry and F. Petriello, An Exclusive Soft Function for Drell-Yan at Next-to-Next-to-Leading Order, arXiv:1105.5171 [SPIRES].
[74] P.A. Baikov, K.G. Chetyrkin and J.H. Kuhn, Order αs4QCD Corrections to Z and τ Decays, Phys. Rev. Lett.101 (2008) 012002 [arXiv:0801.1821] [SPIRES].
[75] J.C Collins, D.E. Soper and G. Sterman, in Perturbative Quantum Chromodynamics, A.H. Mueller eds., World Scientific, Singapore (1989).
[76] C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy flow correlations, Phys. Rev.D 68 (2003) 014012 [hep-ph/0303051] [SPIRES].
[77] C.F. Berger, T. Kucs and G.F. Sterman, Interjet energy flow/event shape correlations, Int. J. Mod. Phys.A 18 (2003) 4159 [hep-ph/0212343] [SPIRES]. · Zbl 1027.81523
[78] G.P. Korchemsky and G.F. Sterman, Infrared factorization in inclusive B meson decays, Phys. Lett.B 340 (1994) 96 [hep-ph/9407344] [SPIRES].
[79] J. Frenkel and J.C. Taylor, Non-abelian eikonal exponentiation, Nucl. Phys.B 246 (1984) 231 [SPIRES].
[80] J.G.M. Gatheral, Exponentiation of eikonal cross sections in nonabelian gauge theories, Phys. Lett.B 133 (1983) 90 [SPIRES].
[81] S. Catani and M. Grazzini, The soft-gluon current at one-loop order, Nucl. Phys.B 591 (2000) 435 [hep-ph/0007142] [SPIRES].
[82] 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] [SPIRES]. · Zbl 1196.68326
[83] T. Huber and D. Maître, HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters, Comput. Phys. Commun.178 (2008) 755 [arXiv:0708.2443] [SPIRES]. · Zbl 1196.81024
[84] J. Carter and G. Heinrich, SecDec: A general program for sector decomposition, Comput. Phys. Commun.182 (2011) 1566 [arXiv:1011.5493] [SPIRES]. · Zbl 1262.81119
[85] G. Heinrich, Sector Decomposition, Int. J. Mod. Phys.A 23 (2008) 1457 [arXiv:0803.4177] [SPIRES]. · Zbl 1153.81522
[86] T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multi-loop integrals, Nucl. Phys.B 585 (2000) 741 [hep-ph/0004013] [SPIRES]. · Zbl 1042.81565
[87] S. Kawabata, A New version of the multidimensional integration and event generation package BASES/SPRING, Comp. Phys. Commun.88 (1995) 309 [SPIRES]. · Zbl 0888.65019
[88] G.P. Lepage, A New Algorithm for Adaptive Multidimensional Integration, J. Comput. Phys.27 (1978) 192 [SPIRES]. · Zbl 0377.65010
[89] T. Hahn, CUBA: A library for multidimensional numerical integration, Comput. Phys. Commun.168 (2005) 78 [hep-ph/0404043] [SPIRES]. · Zbl 1196.65052
[90] S. Catani and M. Grazzini, Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond, Nucl. Phys.B 570 (2000) 287 [hep-ph/9908523] [SPIRES].
[91] G.P. Korchemsky and G. Marchesini, Resummation of large infrared corrections using Wilson loops, Phys. Lett.B 313 (1993) 433 [SPIRES].
[92] 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].
[93] I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett.B 287 (1992) 169 [SPIRES].
[94] G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys.B 283 (1987) 342 [SPIRES].
[95] G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions, Mod. Phys. Lett.A 4 (1989) 1257 [SPIRES].
[96] G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys.B 126 (1977) 298 [SPIRES].
[97] 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]. · Zbl 1109.81374
[98] 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]. · Zbl 1149.81371
[99] S. Catani, M.L. Mangano, P. Nason and L. Trentadue, The Resummation of Soft Gluon in Hadronic Collisions, Nucl. Phys.B 478 (1996) 273 [hep-ph/9604351] [SPIRES].
[100] T. Becher, private communication.
[101] Y.L. Dokshitzer, G. Marchesini and B.R. Webber, Dispersive Approach to Power-Behaved Contributions in QCD Hard Processes, Nucl. Phys.B 469 (1996) 93 [hep-ph/9512336] [SPIRES].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.