Abstract
We consider the class of jet shapes known as angularities in dijet production at hadron colliders. These angularities are modified from the original definitions in e + e − collisions to be boost invariant along the beam axis. These shapes apply to the constituents of jets defined with respect to either k T -type (anti-k T , C/A, and k T ) algorithms and cone-type algorithms. We present an SCET factorization formula and calculate the ingredients needed to achieve next-to-leading-log (NLL) accuracy in kinematic regions where non-global logarithms are not large. The factorization formula involves previously unstudied “unmeasured beam functions,” which are present for finite rapidity cuts around the beams. We derive relations between the jet functions and the shape-dependent part of the soft function that appear in the factorized cross section and those previously calculated for e + e − collisions, and present the calculation of the non-trivial, color-connected part of the soft-function to \( \mathcal{O}\left({\alpha}_s\right) \). This latter part of the soft function is universal in the sense that it applies to any experimental setup with an out-of-jet p T veto and rapidity cuts together with two identified jets and it is independent of the choice of jet (sub-)structure measurement. In addition, we implement the recently introduced soft-collinear refactorization to resum logarithms of the jet size, valid in the region of non-enhanced non-global logarithm effects. While our results are valid for all 2 → 2 channels, we compute explicitly for the qq′ → qq′ channel the color-flow matrices and plot the NLL resummed differential dijet cross section as an explicit example, which shows that the normalization and scale uncertainty is reduced when the soft function is refactorized. For this channel, we also plot the jet size R dependence, the p cut T dependence, and the dependence on the angularity parameter a.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of hard processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1989) 1 [hep-ph/0409313] [INSPIRE].
G.F. Sterman, Partons, factorization and resummation, TASI 95, hep-ph/9606312 [INSPIRE].
C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in \( \overrightarrow{B}{X}_s\gamma \) 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, 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, 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].
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] [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 and I.W. Stewart, The zero-bin and mode factorization in quantum field theory, Phys. Rev. D 76 (2007) 074002 [hep-ph/0605001] [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].
Y.-T. Chien, R. Kelley, M.D. Schwartz and H.X. Zhu, Resummation of jet mass at hadron colliders, Phys. Rev. D 87 (2013) 014010 [arXiv:1208.0010] [INSPIRE].
M. Dasgupta, A. Fregoso, S. Marzani and G.P. Salam, Towards an understanding of jet substructure, JHEP 09 (2013) 029 [arXiv:1307.0007] [INSPIRE].
T.T. Jouttenus, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Jet mass spectra in Higgs boson plus one jet at next-to-next-to-leading logarithmic order, Phys. Rev. D 88 (2013) 054031 [arXiv:1302.0846] [INSPIRE].
A.J. Larkoski, S. Marzani, G. Soyez and J. Thaler, Soft drop, JHEP 05 (2014) 146 [arXiv:1402.2657] [INSPIRE].
A.J. Larkoski, D. Neill and J. Thaler, Jet shapes with the broadening axis, JHEP 04 (2014) 017 [arXiv:1401.2158] [INSPIRE].
A.J. Larkoski and J. Thaler, Unsafe but calculable: ratios of angularities in perturbative QCD, JHEP 09 (2013) 137 [arXiv:1307.1699] [INSPIRE].
A.J. Larkoski, G.P. Salam and J. Thaler, Energy correlation functions for jet substructure, JHEP 06 (2013) 108 [arXiv:1305.0007] [INSPIRE].
M. Dasgupta, K. Khelifa-Kerfa, S. Marzani and M. Spannowsky, On jet mass distributions in Z+jet and dijet processes at the LHC, JHEP 10 (2012) 126 [arXiv:1207.1640] [INSPIRE].
M. Dasgupta, A. Fregoso, S. Marzani and A. Powling, Jet substructure with analytical methods, Eur. Phys. J. C 73 (2013) 2623 [arXiv:1307.0013] [INSPIRE].
T. Becher and X. Garcia i Tormo, Factorization and resummation for transverse thrust, JHEP 06 (2015) 071 [arXiv:1502.04136] [INSPIRE].
Y.-T. Chien and I. Vitev, Jet shape resummation using soft-collinear effective theory, JHEP 12 (2014) 061 [arXiv:1405.4293] [INSPIRE].
A.J. Larkoski, I. Moult and D. Neill, Analytic boosted boson discrimination, arXiv:1507.03018 [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].
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].
S. Alioli and J.R. Walsh, Jet veto clustering logarithms beyond leading order, JHEP 03 (2014) 119 [arXiv:1311.5234] [INSPIRE].
R. Kelley, J.R. Walsh and S. Zuberi, Disentangling clustering effects in jet algorithms, arXiv:1203.2923 [INSPIRE].
R. Kelley, J.R. Walsh and S. Zuberi, Abelian non-global logarithms from soft gluon clustering, JHEP 09 (2012) 117 [arXiv:1202.2361] [INSPIRE].
M. Dasgupta and G.P. Salam, Resummation of nonglobal QCD observables, Phys. Lett. B 512 (2001) 323 [hep-ph/0104277] [INSPIRE].
A. Banfi, G. Marchesini and G. Smye, Away from jet energy flow, JHEP 08 (2002) 006 [hep-ph/0206076] [INSPIRE].
R.B. Appleby and G.P. Salam, Theory and phenomenology of nonglobal logarithms, hep-ph/0305232 [INSPIRE].
Y.L. Dokshitzer and G. Marchesini, On large angle multiple gluon radiation, JHEP 03 (2003) 040 [hep-ph/0303101] [INSPIRE].
M. Dasgupta, F. Dreyer, G.P. Salam and G. Soyez, Small-radius jets to all orders in QCD, JHEP 04 (2015) 039 [arXiv:1411.5182] [INSPIRE].
T. Becher, M. Neubert, L. Rothen and D.Y. Shao, An effective field theory for jet processes, arXiv:1508.06645 [INSPIRE].
Y.-T. Chien, A. Hornig and C. Lee, Soft-collinear mode for jet cross sections in soft collinear effective theory, Phys. Rev. D 93 (2016) 014033 [arXiv:1509.04287] [INSPIRE].
K. Khelifa-Kerfa and Y. Delenda, Non-global logarithms at finite N c beyond leading order, JHEP 03 (2015) 094 [arXiv:1501.00475] [INSPIRE].
A. Hornig, C. Lee, J.R. Walsh and S. Zuberi, Double non-global logarithms in-n-out of jets, JHEP 01 (2012) 149 [arXiv:1110.0004] [INSPIRE].
A. Hornig, C. Lee, I.W. Stewart, J.R. Walsh and S. Zuberi, Non-global structure of the O(α 2 s ) dijet soft function, JHEP 08 (2011) 054 [arXiv:1105.4628] [INSPIRE].
R. Kelley, M.D. Schwartz, R.M. Schabinger and H.X. Zhu, Jet mass with a jet veto at two loops and the universality of non-global structure, Phys. Rev. D 86 (2012) 054017 [arXiv:1112.3343] [INSPIRE].
A.J. Larkoski, I. Moult and D. Neill, Non-global logarithms, factorization and the soft substructure of jets, JHEP 09 (2015) 143 [arXiv:1501.04596] [INSPIRE].
D. Neill, The edge of jets and subleading non-global logs, arXiv:1508.07568 [INSPIRE].
L.G. Almeida et al., Substructure of high-p T jets at the LHC, Phys. Rev. D 79 (2009) 074017 [arXiv:0807.0234] [INSPIRE].
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. Lee and G.F. Sterman, Universality of nonperturbative effects in event shapes, eConf C 0601121 (2006) A001 [hep-ph/0603066] [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].
S.D. Ellis, Z. Kunszt and D.E. Soper, Jets at hadron colliders at order α 3 s : a look inside, Phys. Rev. Lett. 69 (1992) 3615 [hep-ph/9208249] [INSPIRE].
S.D. Ellis, Z. Kunszt and D.E. Soper, Jets in hadron colliders at order α 3 s , Conf. Proc. C 910725V1 (1991) 417 [INSPIRE].
Y.-T. Chien and I. Vitev, Towards the understanding of jet shapes and cross sections in heavy ion collisions using soft-collinear effective theory, arXiv:1509.07257 [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].
Z.L. Liu, C.S. Li, J. Wang and Y. Wang, Resummation prediction on the jet mass spectrum in one-jet inclusive production at the LHC, JHEP 04 (2015) 005 [arXiv:1412.1337] [INSPIRE].
L.G. Almeida et al., Comparing and counting logs in direct and effective methods of QCD resummation, JHEP 04 (2014) 174 [arXiv:1401.4460] [INSPIRE].
C.W. Bauer, N.D. Dunn and A. Hornig, Subtractions for SCET soft functions, arXiv:1102.4899 [INSPIRE].
G.P. Salam, Towards jetography, Eur. Phys. J. C 67 (2010) 637 [arXiv:0906.1833] [INSPIRE].
M. Procura and W.J. Waalewijn, Fragmentation in jets: cone and threshold effects, Phys. Rev. D 85 (2012) 114041 [arXiv:1111.6605] [INSPIRE].
M. Ritzmann and W.J. Waalewijn, Fragmentation in jets at NNLO, Phys. Rev. D 90 (2014) 054029 [arXiv:1407.3272] [INSPIRE].
M. Procura and I.W. Stewart, Quark fragmentation within an identified jet, Phys. Rev. D 81 (2010) 074009 [Erratum ibid. D 83 (2011) 039902] [arXiv:0911.4980] [INSPIRE].
R.K. Ellis, W.J. Stirling and B.R. Webber, QCD and collider physics, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 8 (1996) 1 [INSPIRE].
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] [INSPIRE].
S. Catani and M.H. Seymour, A general algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [Erratum ibid. B 510 (1998) 503] [hep-ph/9605323] [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. Fuhrer, R. Kelley and A.V. Manohar, Factorization structure of gauge theory amplitudes and application to hard scattering processes at the LHC, Phys. Rev. D 80 (2009) 094013 [arXiv:0909.0012] [INSPIRE].
T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [INSPIRE].
R. Kelley and M.D. Schwartz, 1-loop matching and NNLL resummation for all partonic 2 to 2 processes in QCD, Phys. Rev. D 83 (2011) 045022 [arXiv:1008.2759] [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].
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].
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, A formalism for the systematic treatment of rapidity logarithms in quantum field theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, The beam thrust cross section for Drell-Yan at NNLL order, Phys. Rev. Lett. 106 (2011) 032001 [arXiv:1005.4060] [INSPIRE].
R. Bain et al., Fragmenting jet functions with angularities, in preparation.
X. Liu, SCET approach to top quark decay, Phys. Lett. B 699 (2011) 87 [arXiv:1011.3872] [INSPIRE].
A. Jain, M. Procura and W.J. Waalewijn, Parton fragmentation within an identified jet at NNLL, JHEP 05 (2011) 035 [arXiv:1101.4953] [INSPIRE].
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].
G.P. Korchemsky and G. Marchesini, Resummation of large infrared corrections using Wilson loops, Phys. Lett. B 313 (1993) 433 [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].
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].
M. Neubert, Advanced predictions for moments of the \( \overline{B}\to {X}_s\gamma \) photon spectrum, Phys. Rev. D 72 (2005) 074025 [hep-ph/0506245] [INSPIRE].
S. Fleming, A.H. Hoang, S. Mantry and I.W. Stewart, Top jets in the peak region: factorization analysis with next-to-leading-log resummation, Phys. Rev. D 77 (2008) 114003 [arXiv:0711.2079] [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [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: 1601.01319
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
Hornig, A., Makris, Y. & Mehen, T. Jet shapes in dijet events at the LHC in SCET. J. High Energ. Phys. 2016, 97 (2016). https://doi.org/10.1007/JHEP04(2016)097
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2016)097