Abstract
We compare methods to resum logarithms in event shape distributions as they have been used in perturbative QCD directly and in effective field theory. We demonstrate that they are equivalent. In showing this equivalence, we are able to put standard soft-collinear effective theory (SCET) formulae for cross sections in momentum space into a novel form more directly comparable with standard QCD formulae, and endow the QCD formulae with dependence on separated hard, jet, and soft scales, providing potential ways to improve estimates of theoretical uncertainty. We show how to compute cross sections in momentum space to keep them as accurate as the corresponding expressions in Laplace space. In particular, we point out that that care is required in truncating differential distributions at NkLL accuracy to ensure they match the accuracy of the corresponding cumulant or Laplace transform. We explain how to avoid such mismatches at NkLL accuracy, and observe why they can also be avoided by working to NkLL′ accuracy.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.J. Gross and F. Wilczek, Ultraviolet behavior of nonabelian gauge theories, Phys. Rev. Lett. 30 (1973) 1343 [INSPIRE].
H.D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett. 30 (1973) 1346 [INSPIRE].
G.F. Sterman and S. Weinberg, Jets from quantum chromodynamics, Phys. Rev. Lett. 39 (1977) 1436 [INSPIRE].
M. Dasgupta and G.P. Salam, Event shapes in e + e − annihilation and deep inelastic scattering, J. Phys. G 30 (2004) R143 [hep-ph/0312283] [INSPIRE].
A. Hornig, C. Lee, I.W. Stewart, J.R. Walsh and S. Zuberi, Non-global structure of the \( O\left( {\alpha_s^2} \right) \) dijet soft function, JHEP 08 (2011) 054 [arXiv:1105.4628] [INSPIRE].
M. Dasgupta and G. Salam, Resummation of nonglobal QCD observables, Phys. Lett. B 512 (2001) 323 [hep-ph/0104277] [INSPIRE].
E. Farhi, A QCD test for jets, Phys. Rev. Lett. 39 (1977) 1587 [INSPIRE].
V. Mateu, I.W. Stewart and J. Thaler, Power corrections to event shapes with mass-dependent operators, Phys. Rev. D 87 (2013) 014025 [arXiv:1209.3781] [INSPIRE].
G. Salam and D. Wicke, Hadron masses and power corrections to event shapes, JHEP 05 (2001) 061 [hep-ph/0102343] [INSPIRE].
G.F. Sterman, Mass divergences in annihilation processes. 2. Cancellation of divergences in cut vacuum polarization diagrams, Phys. Rev. D 17 (1978) 2789 [INSPIRE].
G. Sterman, Infrared divergences in perturbative QCD, in the proceedings of the Perturbative quantum chromodynamics, March 25-28, Thallassee, U.S.A. (1981).
C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy flow correlations, Phys. Rev. D 68 (2003) 014012 [hep-ph/0303051] [INSPIRE].
C.F. Berger and G.F. Sterman, Scaling rule for nonperturbative radiation in a class of event shapes, JHEP 09 (2003) 058 [hep-ph/0307394] [INSPIRE].
C.F. Berger and L. Magnea, Scaling of power corrections for angularities from dressed gluon exponentiation, Phys. Rev. D 70 (2004) 094010 [hep-ph/0407024] [INSPIRE].
S. Catani, L. Trentadue, G. Turnock and B. Webber, Resummation of large logarithms in e + e − event shape distributions, Nucl. Phys. B 407 (1993) 3 [INSPIRE].
H. Contopanagos, E. Laenen and G.F. Sterman, Sudakov factorization and resummation, Nucl. Phys. B 484 (1997) 303 [hep-ph/9604313] [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].
C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in B → X(sγ) 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, 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] [INSPIRE].
C.W. Bauer, A.V. Manohar and M.B. Wise, Enhanced nonperturbative effects in jet distributions, Phys. Rev. Lett. 91 (2003) 122001 [hep-ph/0212255] [INSPIRE].
C.W. Bauer, S.P. Fleming, C. Lee and G.F. Sterman, Factorization of e + e − event shape distributions with hadronic final states in soft collinear effective theory, Phys. Rev. D 78 (2008) 034027 [arXiv:0801.4569] [INSPIRE].
A.V. Manohar, Effective field theories, hep-ph/9606222 [INSPIRE].
A. Hornig, C. Lee and G. Ovanesyan, Effective predictions of event shapes: factorized, resummed and gapped angularity distributions, JHEP 05 (2009) 122 [arXiv:0901.3780] [INSPIRE].
A. Hornig, C. Lee and G. Ovanesyan, Infrared safety in factorized hard scattering cross-sections, Phys. Lett. B 677 (2009) 272 [arXiv:0901.1897] [INSPIRE].
A. Banfi, G.P. Salam and G. Zanderighi, Principles of general final-state resummation and automated implementation, JHEP 03 (2005) 073 [hep-ph/0407286] [INSPIRE].
T. Becher, M. Neubert and B.D. Pecjak, Factorization and momentum-space resummation in deep-inelastic scattering, JHEP 01 (2007) 076 [hep-ph/0607228] [INSPIRE].
T. Becher and M. Neubert, Threshold resummation in momentum space from effective field theory, Phys. Rev. Lett. 97 (2006) 082001 [hep-ph/0605050] [INSPIRE].
A.V. Manohar, Deep inelastic scattering as x → 1 using soft collinear effective theory, Phys. Rev. D 68 (2003) 114019 [hep-ph/0309176] [INSPIRE].
M. Bonvini, S. Forte, M. Ghezzi and G. Ridolfi, Threshold resummation in SCET vs. perturbative QCD: an analytic comparison, Nucl. Phys. B 861 (2012) 337 [arXiv:1201.6364] [INSPIRE].
M. Bonvini, S. Forte, M. Ghezzi and G. Ridolfi, The scale of soft resummation in SCET vs perturbative QCD, Nucl. Phys. Proc. Suppl. 241-242 (2013) 121 [arXiv:1301.4502] [INSPIRE].
G. Sterman and M. Zeng, Quantifying comparisons of threshold resummations, arXiv:1312.5397 [INSPIRE].
G.P. Korchemsky and G.F. Sterman, Power corrections to event shapes and factorization, Nucl. Phys. B 555 (1999) 335 [hep-ph/9902341] [INSPIRE].
G. Korchemsky and S. Tafat, On power corrections to the event shape distributions in QCD, JHEP 10 (2000) 010 [hep-ph/0007005] [INSPIRE].
A.H. Hoang and I.W. Stewart, Designing gapped soft functions for jet production, Phys. Lett. B 660 (2008) 483 [arXiv:0709.3519] [INSPIRE].
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 ), Phys. Rev. D 83 (2011) 074021 [arXiv:1006.3080] [INSPIRE].
J.-y. Chiu, F. Golf, R. Kelley and A.V. Manohar, Electroweak corrections in high energy processes using effective field theory, Phys. Rev. D 77 (2008) 053004 [arXiv:0712.0396] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
J. Thaler and K. Van Tilburg, Identifying boosted objects with N-subjettiness, JHEP 03 (2011) 015 [arXiv:1011.2268] [INSPIRE].
C.W. Bauer, F.J. Tackmann, J.R. Walsh and S. Zuberi, Factorization and resummation for dijet invariant mass spectra, Phys. Rev. D 85 (2012) 074006 [arXiv:1106.6047] [INSPIRE].
A. Altheimer et al., Jet Substructure at the Tevatron and LHC: new results, new tools, new benchmarks, J. Phys. G 39 (2012) 063001 [arXiv:1201.0008] [INSPIRE].
A. Abdesselam et al., Boosted objects: a probe of beyond the standard model physics, Eur. Phys. J. C 71 (2011) 1661 [arXiv:1012.5412] [INSPIRE].
A. Altheimer et al., Boosted objects and jet substructure at the LHC, arXiv:1311.2708 [INSPIRE].
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] [INSPIRE].
L. Clavelli, Jet invariant mass in quantum chromodynamics, Phys. Lett. B 85 (1979) 111 [INSPIRE].
L. Clavelli and D. Wyler, Kinematical bounds on jet variables and the heavy jet mass distribution, Phys. Lett. B 103 (1981) 383 [INSPIRE].
T. Chandramohan and L. Clavelli, Consequences of second order QCD for jet structure in e + e − annihilation, Nucl. Phys. B 184 (1981) 365 [INSPIRE].
S. Catani, G. Turnock and B. Webber, Jet broadening measures in e + e − annihilation, Phys. Lett. B 295 (1992) 269 [INSPIRE].
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 [INSPIRE].
T. Becher, G. Bell and M. Neubert, Factorization and resummation for jet broadening, Phys. Lett. B 704 (2011) 276 [arXiv:1104.4108] [INSPIRE].
J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, The rapidity renormalization group, Phys. Rev. Lett. 108 (2012) 151601 [arXiv:1104.0881] [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].
T. Becher and G. Bell, NNLL resummation for jet broadening, JHEP 11 (2012) 126 [arXiv:1210.0580] [INSPIRE].
Y.L. Dokshitzer, A. Lucenti, G. Marchesini and G. Salam, On the QCD analysis of jet broadening, JHEP 01 (1998) 011 [hep-ph/9801324] [INSPIRE].
A.J. Larkoski, D. Neill and J. Thaler, Jet shapes with the broadening axis, JHEP 04 (2014) 017 [arXiv:1401.2158] [INSPIRE].
S.M. Freedman, Subleading corrections to thrust using effective field theory, arXiv:1303.1558 [INSPIRE].
C. Lee and G.F. Sterman, Universality of nonperturbative effects in event shapes, eConf C 0601121 (2006) A001 [hep-ph/0603066] [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].
A. Idilbi and T. Mehen, On the equivalence of soft and zero-bin subtractions, Phys. Rev. D 75 (2007) 114017 [hep-ph/0702022] [INSPIRE].
A. Idilbi and T. Mehen, Demonstration of the equivalence of soft and zero-bin subtractions, Phys. Rev. D 76 (2007) 094015 [arXiv:0707.1101] [INSPIRE].
S. Moch, J. Vermaseren and A. Vogt, The quark form-factor at higher orders, JHEP 08 (2005) 049 [hep-ph/0507039] [INSPIRE].
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] [INSPIRE].
C.W. Bauer and A.V. Manohar, Shape function effects in B → X s γ and B → X u ln¯u decays, Phys. Rev. D 70 (2004) 034024 [hep-ph/0312109] [INSPIRE].
T. Becher and M. Neubert, Toward a NNLO calculation of the \( \overline{B} \) → X s γ 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] [INSPIRE].
R. Kelley, M.D. Schwartz, R.M. Schabinger and H.X. Zhu, The two-loop hemisphere soft function, Phys. Rev. D 84 (2011) 045022 [arXiv:1105.3676] [INSPIRE].
P.F. Monni, T. Gehrmann and G. Luisoni, Two-loop soft corrections and resummation of the thrust distribution in the dijet region, JHEP 08 (2011) 010 [arXiv:1105.4560] [INSPIRE].
T. Becher and M.D. Schwartz, A precise determination of α s from LEP thrust data using effective field theory, JHEP 07 (2008) 034 [arXiv:0803.0342] [INSPIRE].
G. Korchemsky and A. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].
S. Moch, J. Vermaseren and A. Vogt, The three loop splitting functions in QCD: the nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
G. Korchemsky and G. Marchesini, Resummation of large infrared corrections using Wilson loops, Phys. Lett. B 313 (1993) 433 [INSPIRE].
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] [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].
H. Contopanagos and G.F. Sterman, Normalization of the Drell-Yan cross-section in QCD, Nucl. Phys. B 400 (1993) 211 [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].
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].
J.C. Collins and D.E. Soper, Back-to-back jets in QCD, Nucl. Phys. B 193 (1981) 381 [Erratum ibid. B 213 (1983) 545] [INSPIRE].
T. Becher and M. Neubert, Drell-Yan production at small q T , transverse parton distributions and the collinear anomaly, Eur. Phys. J. C 71 (2011) 1665 [arXiv:1007.4005] [INSPIRE].
S. Catani and L. Trentadue, Comment on QCD exponentiation at large x, Nucl. Phys. B 353 (1991) 183 [INSPIRE].
V. Ahrens, A. Ferroglia, M. Neubert, B.D. Pecjak and L.-L. Yang, RG-improved single-particle inclusive cross sections and forward-backward asymmetry in \( t\overline{t} \) production at hadron colliders, JHEP 09 (2011) 070 [arXiv:1103.0550] [INSPIRE].
T. Becher and M.D. Schwartz, Direct photon production with effective field theory, JHEP 02 (2010) 040 [arXiv:0911.0681] [INSPIRE].
M.D. Schwartz, Resummation and NLO matching of event shapes with effective field theory, Phys. Rev. D 77 (2008) 014026 [arXiv:0709.2709] [INSPIRE].
Y.-T. Chien and M.D. Schwartz, Resummation of heavy jet mass and comparison to LEP data, JHEP 08 (2010) 058 [arXiv:1005.1644] [INSPIRE].
P. Achard et al., Generalized event shape and energy flow studies in e+e− annihilation at \( \sqrt{s} \) = 91.2 GeV-208.0 GeV, JHEP 10 (2011) 143 [INSPIRE].
O. Tarasov, A. Vladimirov and A.Y. Zharkov, The Gell-Mann-Low function of QCD in the three loop approximation, Phys. Lett. B 93 (1980) 429 [INSPIRE].
S. Larin and J. Vermaseren, The three loop QCD β-function and anomalous dimensions, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208] [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: 1401.4460
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
Almeida, L.G., Ellis, S.D., Lee, C. et al. Comparing and counting logs in direct and effective methods of QCD resummation. J. High Energ. Phys. 2014, 174 (2014). https://doi.org/10.1007/JHEP04(2014)174
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2014)174