Abstract
Jet shapes are weighted sums over the four-momenta of the constituents of a jet and reveal details of its internal structure, potentially allowing discrimination of its partonic origin. In this work we make predictions for quark and gluon jet shape distributions in N-jet final states in e + e − collisions, defined with a cone or recombination algorithm, where we measure some jet shape observable on a subset of these jets. Using the framework of Soft-Collinear Effective Theory, we prove a factorization theorem for jet shape distributions and demonstrate the consistent renormalization-group running of the functions in the factorization theorem for any number of measured and unmeasured jets, any number of quark and gluon jets, and any angular size R of the jets, as long as R is much smaller than the angular separation between jets. We calculate the jet and soft functions for angularity jet shapes τ α to one-loop order \( \left( {\mathcal{O}\left( {{\alpha_s}} \right)} \right) \) and resum a subset of the large logarithms of τ α needed for next-to-leading logarithmic (NLL) accuracy for both cone and kT-type jets. We compare our predictions for the resummed τ α distribution of a quark or a gluon jet produced in a 3-jet final state in e + e − annihilation to the output of a Monte Carlo event generator and find that the dependence on a and R is very similar.
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M.H. Seymour, Searches for new particles using cone and cluster jet algorithms: A Comparative study, Z. Phys. C 62 (1994) 127 [SPIRES].
J.M. Butterworth, B.E. Cox and J.R. Forshaw, WW scattering at the CERN LHC, Phys. Rev. D 65 (2002) 096014 [hep-ph/0201098] [SPIRES].
J.M. Butterworth, J.R. Ellis and A.R. Raklev, Reconstructing sparticle mass spectra using hadronic decays, JHEP 05 (2007) 033 [hep-ph/0702150] [SPIRES].
G. Brooijmans, High p T hadronic top quark identification. Part I: Jet mass and Ysplitter, ATLAS report ATL-PHYS-CONF-2008-008 [SPIRES].
J.M. Butterworth, A.R. Davison, M. Rubin and G.P. Salam, Jet substructure as a new Higgs search channel at the LHC, Phys. Rev. Lett. 100 (2008) 242001 [arXiv:0802.2470] [SPIRES].
J.M. Butterworth, J.R. Ellis, A.R. Raklev and G.P. Salam, Discovering baryon-number violating neutralino decays at the LHC, Phys. Rev. Lett. 103 (2009) 241803 [arXiv:0906.0728] [SPIRES].
T. Plehn, G.P. Salam and M. Spannowsky, Fat Jets for a Light Higgs, Phys. Rev. Lett. 104 (2010) 111801 [arXiv:0910.5472] [SPIRES].
G.D. Kribs, A. Martin, T.S. Roy and M. Spannowsky, Discovering the Higgs Boson in New Physics Events using Jet Substructure, Phys. Rev. D 81 (2010) 111501 [arXiv:0912.4731] [SPIRES].
S.D. Ellis, C.K. Vermilion and J.R. Walsh, Recombination Algorithms and Jet Substructure: Pruning as a Tool for Heavy Particle Searches, Phys. Rev. D 81 (2010) 094023 [arXiv:0912.0033] [SPIRES].
S.D. Ellis, C.K. Vermilion and J.R. Walsh, Techniques for improved heavy particle searches with jet substructure, Phys. Rev. D 80 (2009) 051501 [arXiv:0903.5081] [SPIRES].
D. Krohn, J. Thaler and L.-T. Wang, Jet Trimming, JHEP 02 (2010) 084 [arXiv:0912.1342] [SPIRES].
L.G. Almeida et al., Substructure of high-p T Jets at the LHC, Phys. Rev. D 79 (2009) 074017 [arXiv:0807.0234] [SPIRES].
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] [SPIRES].
S.D. Ellis, Z. Kunszt and D.E. Soper, Jets in hadron colliders at order α s 3, Conf. Proc. C910725V1 (1991) 417 [SPIRES].
S.D. Ellis, Z. Kunszt and D.E. Soper, Jets at hadron colliders at order $α s 3 : A Look inside, Phys. Rev. Lett. 69 (1992) 3615 [hep-ph/9208249] [SPIRES].
CDF collaboration, F. Abe et al., A Measurement of jet shapes in \( p\bar{p} \) collisions at \( \sqrt {s} = 1.8\;TeV \), Phys. Rev. Lett. 70 (1993) 713 [SPIRES].
C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy flow correlations, Phys. Rev. D 68 (2003) 014012 [hep-ph/0303051] [SPIRES].
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].
G.F. Sterman, Partons, factorization and resummation, hep-ph/9606312 [SPIRES].
H. Contopanagos, E. Laenen and G.F. Sterman, Sudakov factorization and resummation, Nucl. Phys. B 484 (1997) 303 [hep-ph/9604313] [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].
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].
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] [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].
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].
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] [SPIRES].
C.W. Bauer, A. Hornig and F.J. Tackmann, Factorization for generic jet production, Phys. Rev. D 79 (2009) 114013 [arXiv:0808.2191] [SPIRES].
M. Trott, Jets in effective theory: Summing phase space logs, Phys. Rev. D 75 (2007) 054011 [hep-ph/0608300] [SPIRES].
W. M.-Y. Cheung, M. Luke and S. Zuberi, Phase Space and Jet Definitions in SCET, Phys. Rev. D 80 (2009) 114021 [arXiv:0910.2479] [SPIRES].
C.W. Bauer and M.D. Schwartz, Event generation from effective field theory, Phys. Rev. D 76 (2007) 074004 [hep-ph/0607296] [SPIRES].
C.W. Bauer and M.D. Schwartz, Improving jet distributions with effective field theory, Phys. Rev. Lett. 97 (2006) 142001 [hep-ph/0604065] [SPIRES].
T. Becher and M.D. Schwartz, Direct photon production with effective field theory, JHEP 02 (2010) 040 [arXiv:0911.0681] [SPIRES].
M. Dasgupta, L. Magnea and G.P. Salam, Non-perturbative QCD effects in jets at hadron colliders, JHEP 02 (2008) 055 [arXiv:0712.3014] [SPIRES].
G.P. Korchemsky and S. Tafat, On power corrections to the event shape distributions in QCD, JHEP 10 (2000) 010 [hep-ph/0007005] [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].
Y.L. Dokshitzer and B.R. Webber, Calculation of power corrections to hadronic event shapes, Phys. Lett. B 352 (1995) 451 [hep-ph/9504219] [SPIRES].
Y.L. Dokshitzer and B.R. Webber, Power corrections to event shape distributions, Phys. Lett. B 404 (1997) 321 [hep-ph/9704298] [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].
C. Lee and G.F. Sterman, Universality of nonperturbative effects in event shapes, hep-ph/0603066 [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].
S. Catani, G. Turnock, B.R. Webber and L. Trentadue, Thrust distribution in e + e − annihilation, Phys. Lett. B 263 (1991) 491 [SPIRES].
S. Catani, G. Turnock and B.R. Webber, Heavy jet mass distribution in e + e − annihilation, Phys. Lett. B 272 (1991) 368 [SPIRES].
M. Dasgupta and G.P. Salam, Resummation of non-global QCD observables, Phys. Lett. B 512 (2001) 323 [hep-ph/0104277] [SPIRES].
S.J. Burby and E.W.N. Glover, Resumming the Light Hemisphere Mass and Narrow Jet Broadening distributions in e + e − annihilation, JHEP 04 (2001) 029 [hep-ph/0101226] [SPIRES].
S. Catani, G. Turnock and B.R. Webber, Jet broadening measures in e + e − annihilation, Phys. Lett. B 295 (1992) 269 [SPIRES].
Y.L. Dokshitzer, A. Lucenti, G. Marchesini and G.P. Salam, On the QCD analysis of jet broadening, JHEP 01 (1998) 011 [hep-ph/9801324] [SPIRES].
S. Catani and B.R. Webber, Resummed C-parameter distribution in e + e − annihilation, Phys. Lett. B 427 (1998) 377 [hep-ph/9801350] [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].
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].
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] [SPIRES].
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] [SPIRES].
N. Brown and W.J. Stirling, Jet cross-sections at leading double logarithm in e + e − annihilation, Phys. Lett. B 252 (1990) 657 [SPIRES].
A. Banfi, G. Marchesini and G. Smye, Away-from-jet energy flow, JHEP 08 (2002) 006 [hep-ph/0206076] [SPIRES].
R.B. Appleby and G.P. Salam, Theory and phenomenology of non-global logarithms, hep-ph/0305232 [SPIRES].
Y.L. Dokshitzer and G. Marchesini, On large angle multiple gluon radiation, JHEP 03 (2003) 040 [hep-ph/0303101] [SPIRES].
A. Banfi, M. Dasgupta, K. Khelifa-Kerfa and S. Marzani, Non-global logarithms and jet algorithms in high-pT jet shapes, JHEP 08 (2010) 064 [arXiv:1004.3483] [SPIRES].
L.G. Almeida, S.J. Lee, G. Perez, I. Sung and J. Virzi, Top Jets at the LHC, Phys. Rev. D 79 (2009) 074012 [arXiv:0810.0934] [SPIRES].
N. Kidonakis, G. Oderda and G.F. Sterman, NLL resummation for dijet production, hep-ph/9805279 [SPIRES].
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].
G.P. Salam and G. Soyez, A practical Seedless Infrared-Safe Cone jet algorithm, JHEP 05 (2007) 086 [arXiv:0704.0292] [SPIRES].
G.P. Salam, Towards Jetography, Eur. Phys. J. C 67 (2010) 637 [arXiv:0906.1833] [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].
C. Marcantonini and I.W. Stewart, Reparameterization Invariant Collinear Operators, Phys. Rev. D 79 (2009) 065028 [arXiv:0809.1093] [SPIRES].
C.M. Arnesen, J. Kundu and I.W. Stewart, Constraint equations for heavy-to-light currents in SCET, Phys. Rev. D 72 (2005) 114002 [hep-ph/0508214] [SPIRES].
G.P. Korchemsky, G. Oderda and G.F. Sterman, Power corrections and nonlocal operators, hep-ph/9708346 [SPIRES].
A.V. Belitsky, G.P. Korchemsky and G.F. Sterman, Energy flow in QCD and event shape functions, Phys. Lett. B 515 (2001) 297 [hep-ph/0106308] [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].
A. Hornig, C. Lee and G. Ovanesyan, Infrared Safety in Factorized Hard Scattering Cross-Sections, Phys. Lett. B 677 (2009) 272 [arXiv:0901.1897] [SPIRES].
J.-y. Chiu, A. Fuhrer, A.H. Hoang, R. Kelley and A.V. Manohar, Soft-Collinear Factorization and Zero-Bin Subtractions, Phys. Rev. D 79 (2009) 053007 [arXiv:0901.1332] [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].
A. Idilbi and T. Mehen, Demonstration of the Equivalence of Soft and Zero-Bin Subtractions, Phys. Rev. D 76 (2007) 094015 [arXiv:0707.1101] [SPIRES].
C.W. Bauer and A.V. Manohar, Shape function effects in B → X s γ and \( B \to {X_u}l\bar{\nu } \) decays, Phys. Rev. D 70 (2004) 034024 [hep-ph/0312109] [SPIRES].
S.W. Bosch, B.O. Lange, M. Neubert and G. Paz, Factorization and shape-function effects in inclusive B-meson decays, Nucl. Phys. B 699 (2004) 335 [hep-ph/0402094] [SPIRES].
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].
C.W. Bauer, C.-W. Chiang, S. Fleming, A.K. Leibovich and I. Low, Resumming the color-octet contribution to radiative Upsilon decay, Phys. Rev. D 64 (2001) 114014 [hep-ph/0106316] [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. 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].
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].
G.P. Korchemsky and G. Marchesini, Resummation of large infrared corrections using Wilson loops, Phys. Lett. B 313 (1993) 433 [SPIRES].
A.V. Belitsky, Two-loop renormalization of Wilson loop for Drell-Yan production, Phys. Lett. B 442 (1998) 307 [hep-ph/9808389] [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].
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].
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, Advanced predictions for moments of the \( \bar{B} \to {X_s}\gamma \) photon spectrum, Phys. Rev. D 72 (2005) 074025 [hep-ph/0506245] [SPIRES].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B 283 (1987) 342 [SPIRES].
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] [SPIRES].
N. Kidonakis, G. Oderda and G.F. Sterman, Threshold resummation for dijet cross sections, Nucl. Phys. B 525 (1998) 299 [hep-ph/9801268] [SPIRES].
N. Kidonakis, G. Oderda and G.F. Sterman, Evolution of color exchange in QCD hard scattering, Nucl. Phys. B 531 (1998) 365 [hep-ph/9803241] [SPIRES].
Y.L. Dokshitzer and G. Marchesini, Hadron collisions and the fifth form factor, Phys. Lett. B 631 (2005) 118 [hep-ph/0508130] [SPIRES].
M. Sjodahl, Color evolution of 2 → 3 processes, JHEP 12 (2008) 083 [arXiv:0807.0555] [SPIRES].
M. Sjodahl, Color structure for soft gluon resummation - a general recipe, JHEP 09 (2009) 087 [arXiv:0906.1121] [SPIRES].
J. Alwall et al., MadGraph/MadEvent v4: The New Web Generation, JHEP 09 (2007) 028 [arXiv:0706.2334] [SPIRES].
T. Sjöstrand, S. Mrenna and P.Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026 [hep-ph/0603175] [SPIRES].
M. Cacciari and G.P. Salam, Dispelling the N 3 myth for the k t jet-finder, Phys. Lett. B 641 (2006) 57 [hep-ph/0512210] [SPIRES].
T.T. Jouttenus, Jet Function with a Jet Algorithm in SCET, Phys. Rev. D 81 (2010) 094017 [arXiv:0912.5509] [SPIRES].
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