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An effective field theory for forward scattering and factorization violation. (English) Zbl 1390.81365
Summary: Starting with QCD, we derive an effective field theory description for forward scattering and factorization violation as part of the soft-collinear effective field theory (SCET) for high energy scattering. These phenomena are mediated by long distance Glauber gluon exchanges, which are static in time, localized in the longitudinal distance, where \(|t| \ll s\). In hard scattering, Glauber gluons can induce corrections which invalidate factorization. With SCET, Glauber exchange graphs can be calculated explicitly, and are distinct from graphs with soft, collinear, or ultrasoft gluons. We derive a complete basis of operators which describe the leading power effects of Glauber exchange. Key ingredients include regulating light-cone rapidity singularities and subtractions which prevent double counting. Our results include a novel all orders gauge invariant pure glue soft operator which appears between two collinear rapidity sectors. The 1-gluon Feynman rule for the soft operator coincides with the Lipatov vertex, but it also contributes to emissions with \(\geq 2\) soft gluons. Our Glauber operator basis is derived using tree level and one-loop matching calculations from full QCD to SCET. The rapidity RGE yields gluon Reggeization at the amplitude level, and gives the BFKL equation for the soft and collinear functions in the forward scattering cross section. We derive an explicit rule for when eikonalization is valid, and provide a direct connection to the picture of multiple Wilson lines crossing a shockwave. In hard scattering operators Glauber subtractions for soft and collinear loop diagrams ensure that we are not sensitive to the directions for soft and collinear Wilson lines. Conversely, certain Glauber interactions can be absorbed into these soft and collinear Wilson lines by taking them to be in specific directions. We also discuss criteria for factorization violation.

MSC:
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
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[1] E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk singularity in non-Abelian gauge theories, Sov. Phys. JETP45 (1977) 199 [Zh. Eksp. Teor. Fiz.72 (1977) 377] [INSPIRE].
[2] I.I. Balitsky and L.N. Lipatov, The Pomeranchuk singularity in quantum chromodynamics, Sov. J. Nucl. Phys.28 (1978) 822 [Yad. Fiz.28 (1978) 1597] [INSPIRE].
[3] A.H. Mueller, Unitarity and the BFKL Pomeron, Nucl. Phys.B 437 (1995) 107 [hep-ph/9408245] [INSPIRE].
[4] I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys.B 463 (1996) 99 [hep-ph/9509348] [INSPIRE].
[5] Y.V. Kovchegov, Small x F_{2}structure function of a nucleus including multiple Pomeron exchanges, Phys. Rev.D 60 (1999) 034008 [hep-ph/9901281] [INSPIRE].
[6] J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, The BFKL equation from the Wilson renormalization group, Nucl. Phys.B 504 (1997) 415 [hep-ph/9701284] [INSPIRE].
[7] E. Iancu, A. Leonidov and L.D. McLerran, Nonlinear gluon evolution in the color glass condensate. 1, Nucl. Phys.A 692 (2001) 583 [hep-ph/0011241] [INSPIRE]. · Zbl 0971.81173
[8] G.P. Korchemsky, On near forward high-energy scattering in QCD, Phys. Lett.B 325 (1994) 459 [hep-ph/9311294] [INSPIRE].
[9] I.A. Korchemskaya and G.P. Korchemsky, High-energy scattering in QCD and cross singularities of Wilson loops, Nucl. Phys.B 437 (1995) 127 [hep-ph/9409446] [INSPIRE].
[10] I.A. Korchemskaya and G.P. Korchemsky, Evolution equation for gluon Regge trajectory, Phys. Lett.B 387 (1996) 346 [hep-ph/9607229] [INSPIRE].
[11] L.N. Lipatov, Gauge invariant effective action for high-energy processes in QCD, Nucl. Phys.B 452 (1995) 369 [hep-ph/9502308] [INSPIRE].
[12] L.N. Lipatov, Small x physics in perturbative QCD, Phys. Rept.286 (1997) 131 [hep-ph/9610276] [INSPIRE].
[13] I. Balitsky, Factorization and high-energy effective action, Phys. Rev.D 60 (1999) 014020 [hep-ph/9812311] [INSPIRE].
[14] I. Balitsky, High-energy QCD and Wilson lines, hep-ph/0101042 [INSPIRE]. · Zbl 1025.81503
[15] Caron-Huot, S., When does the gluon reggeize?, JHEP, 05, 093, (2015)
[16] Duca, V.; Duhr, C.; Gardi, E.; Magnea, L.; White, CD, An infrared approach to reggeization, Phys. Rev., D 85, 071104, (2012)
[17] Duca, V.; Duhr, C.; Gardi, E.; Magnea, L.; White, CD, The infrared structure of gauge theory amplitudes in the high-energy limit, JHEP, 12, 021, (2011) · Zbl 1306.81333
[18] Ø. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multi-leg scattering, arXiv:1507.00047 [INSPIRE].
[19] G.T. Bodwin, Factorization of the Drell-Yan cross-section in perturbation theory, Phys. Rev.D 31 (1985) 2616 [Erratum ibid.D 34 (1986) 3932] [INSPIRE].
[20] Collins, JC; Soper, DE; Sterman, GF, Transverse momentum distribution in Drell-Yan pair and W and Z boson production, Nucl. Phys., B 250, 199, (1985)
[21] Collins, JC; Soper, DE; Sterman, GF, Soft gluons and factorization, Nucl. Phys., B 308, 833, (1988)
[22] 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]. · Zbl 0961.81526
[23] G.C. Nayak, J.-W. Qiu and G.F. Sterman, Fragmentation, NRQCD and NNLO factorization analysis in heavy quarkonium production, Phys. Rev.D 72 (2005) 114012 [hep-ph/0509021] [INSPIRE].
[24] Diehl, M.; Gaunt, JR; Ostermeier, D.; Plößl, P.; Schäfer, A., Cancellation of Glauber gluon exchange in the double Drell-Yan process, JHEP, 01, 076, (2016)
[25] Aybat, SM; Sterman, GF, Soft-gluon cancellation, phases and factorization with initial-state partons, Phys. Lett., B 671, 46, (2009)
[26] 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] [INSPIRE].
[27] 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].
[28] C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett.B 516 (2001) 134 [hep-ph/0107001] [INSPIRE]. · Zbl 0971.81569
[29] 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].
[30] 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].
[31] J. Collins, Foundations of perturbative QCD, Cambridge University Press, Cambridge U.K. (2013).
[32] M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys.B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
[33] Jantzen, B., Foundation and generalization of the expansion by regions, JHEP, 12, 076, (2011) · Zbl 1306.81420
[34] Idilbi, A.; Majumder, A., Extending soft-collinear-effective-theory to describe hard jets in dense QCD media, Phys. Rev., D 80, 054022, (2009)
[35] D’Eramo, F.; Liu, H.; Rajagopal, K., Transverse momentum broadening and the jet quenching parameter, redux, Phys. Rev., D 84, 065015, (2011)
[36] Ovanesyan, G.; Vitev, I., An effective theory for jet propagation in dense QCD matter: jet broadening and medium-induced bremsstrahlung, JHEP, 06, 080, (2011) · Zbl 1298.81403
[37] G. Ovanesyan, Medium-induced splitting kernels from SCET_{\(G\)}, Nucl. Phys.A 904-905 (2013) 981c [arXiv:1210.4945] [INSPIRE].
[38] Benzke, M.; Brambilla, N.; Escobedo, MA; Vairo, A., Gauge invariant definition of the jet quenching parameter, JHEP, 02, 129, (2013) · Zbl 1342.81258
[39] I.Z. Rothstein and I.W. Stewart, Glauber gluons in soft-collinear effective theory, talk at the SCET 2010 workshop, https://indico.mpp.mpg.de/getFile.py/access?contribId=5&resId=0&materialId=slides&confId=632, Germany April 2010.
[40] Bauer, CW; Lange, BO; Ovanesyan, G., On Glauber modes in soft-collinear effective theory, JHEP, 07, 077, (2011) · Zbl 1298.81366
[41] F. Liu and J.P. Ma, Glauber gluons in soft collinear effective theory and factorization of Drell-Yan processes, arXiv:0802.2973 [INSPIRE].
[42] Donoghue, JF; Wyler, D., On Regge kinematics in SCET, Phys. Rev., D 81, 114023, (2010)
[43] Donoghue, JF; El-Menoufi, BK; Ovanesyan, G., Regge behavior in effective field theory, Phys. Rev., D 90, 096009, (2014)
[44] Fleming, S., The role of Glauber exchange in soft collinear effective theory and the balitsky-fadin-kuraev-Lipatov equation, Phys. Lett., B 735, 266, (2014)
[45] J.R. Walsh and S. Zuberi, Factorization constraints on jet substructure, arXiv:1110.5333 [INSPIRE].
[46] Collins, J.; Qiu, J-W, K_{T} factorization is violated in production of high-transverse-momentum particles in hadron-hadron collisions, Phys. Rev., D 75, 114014, (2007)
[47] J. Collins, 2-soft-gluon exchange and factorization breaking, arXiv:0708.4410 [INSPIRE].
[48] P.J. Mulders and T.C. Rogers, Gauge links, TMD-factorization and TMD-factorization breaking, arXiv:1102.4569 [INSPIRE].
[49] S.M. Aybat and T.C. Rogers, TMD-factorization, factorization breaking and evolution, arXiv:1107.3973 [INSPIRE].
[50] Catani, S.; Florian, D.; Rodrigo, G., Space-like (versus time-like) collinear limits in QCD: is factorization violated?, JHEP, 07, 026, (2012)
[51] Forshaw, JR; Seymour, MH; Siodmok, A., On the breaking of collinear factorization in QCD, JHEP, 11, 066, (2012)
[52] J.R. Forshaw, A. Kyrieleis and M.H. Seymour, Super-leading logarithms in non-global observables in QCD, JHEP08 (2006) 059 [hep-ph/0604094] [INSPIRE].
[53] V. Del Duca and E.W.N. Glover, The high-energy limit of QCD at two loops, JHEP10 (2001) 035 [hep-ph/0109028] [INSPIRE].
[54] Duca, V.; Falcioni, G.; Magnea, L.; Vernazza, L., High-energy QCD amplitudes at two loops and beyond, Phys. Lett., B 732, 233, (2014) · Zbl 1360.81293
[55] Duca, V.; Falcioni, G.; Magnea, L.; Vernazza, L., Analyzing high-energy factorization beyond next-to-leading logarithmic accuracy, JHEP, 02, 029, (2015)
[56] I.W. Stewart, 8.EFTx: a free online graduate course on effective field theory webpage, http://web.mit.edu/eftx, (2014).
[57] Chiu, J-Y; Jain, A.; Neill, D.; Rothstein, IZ, The rapidity renormalization group, Phys. Rev. Lett., 108, 151601, (2012)
[58] Chiu, J-Y; Jain, A.; Neill, D.; Rothstein, IZ, A formalism for the systematic treatment of rapidity logarithms in quantum field theory, JHEP, 05, 084, (2012) · Zbl 1348.81437
[59] 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].
[60] S. Catani and M. Grazzini, The soft gluon current at one loop order, Nucl. Phys.B 591 (2000) 435 [hep-ph/0007142] [INSPIRE].
[61] Bodwin, GT; Brodsky, SJ; Lepage, GP, Initial state interactions and the Drell-Yan process, Phys. Rev. Lett., 47, 1799, (1981)
[62] M.E. Luke, A.V. Manohar and I.Z. Rothstein, Renormalization group scaling in nonrelativistic QCD, Phys. Rev.D 61 (2000) 074025 [hep-ph/9910209] [INSPIRE].
[63] G.P. Korchemsky and G.F. Sterman, Power corrections to event shapes and factorization, Nucl. Phys.B 555 (1999) 335 [hep-ph/9902341] [INSPIRE].
[64] G.P. Korchemsky and S. Tafat, On power corrections to the event shape distributions in QCD, JHEP10 (2000) 010 [hep-ph/0007005] [INSPIRE].
[65] Ligeti, Z.; Stewart, IW; Tackmann, FJ, Treating the b quark distribution function with reliable uncertainties, Phys. Rev., D 78, 114014, (2008)
[66] Hoang, AH; Stewart, IW, Designing gapped soft functions for jet production, Phys. Lett., B 660, 483, (2008)
[67] Abbate, R.; Fickinger, M.; Hoang, AH; Mateu, V.; Stewart, IW, thrust at N\^{}{3}LL with power corrections and a precision global fit for α_{\(s\)}(\(m\)_{\(Z\)} ), Phys. Rev., D 83, 074021, (2011)
[68] Berger, CF; Marcantonini, C.; Stewart, IW; Tackmann, FJ; Waalewijn, WJ, Higgs production with a central jet veto at NNLL+NNLO, JHEP, 04, 092, (2011)
[69] Mateu, V.; Stewart, IW; Thaler, J., Power corrections to event shapes with mass-dependent operators, Phys. Rev., D 87, 014025, (2013)
[70] Stewart, IW; Tackmann, FJ; Waalewijn, WJ, N-jettiness: an inclusive event shape to veto jets, Phys. Rev. Lett., 105, 092002, (2010)
[71] Neill, D.; Rothstein, IZ; Vaidya, V., The Higgs transverse momentum distribution at NNLL and its theoretical errors, JHEP, 12, 097, (2015)
[72] T. Sjöstrand, S. Mrenna and P.Z. Skands, PYTHIA 6\(.\)4 physics and manual, JHEP05 (2006) 026 [hep-ph/0603175] [INSPIRE].
[73] Bahr, M.; etal., HERWIG++ physics and manual, Eur. Phys. J., C 58, 639, (2008)
[74] G.F. Sterman, Mass divergences in annihilation processes. 1. Origin and nature of divergences in cut vacuum polarization diagrams, Phys. Rev.D 17 (1978) 2773 [INSPIRE].
[75] Coleman, S.; Norton, RE, Singularities in the physical region, Nuovo Cim., 38, 438, (1965)
[76] Smirnov, VA, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys., 177, 1, (2002) · Zbl 1025.81002
[77] C.W. Bauer, D. Pirjol and I.W. Stewart, Factorization and endpoint singularities in heavy to light decays, Phys. Rev.D 67 (2003) 071502 [hep-ph/0211069] [INSPIRE].
[78] A.V. Manohar and I.W. Stewart, Renormalization group analysis of the QCD quark potential to order v\^{}{2}, Phys. Rev.D 62 (2000) 014033 [hep-ph/9912226] [INSPIRE].
[79] A.V. Manohar and I.W. Stewart, The QCD heavy quark potential to order v\^{}{2}: one loop matching conditions, Phys. Rev.D 62 (2000) 074015 [hep-ph/0003032] [INSPIRE].
[80] A.V. Manohar and I.W. Stewart, Running of the heavy quark production current and 1/v potential in QCD, Phys. Rev.D 63 (2001) 054004 [hep-ph/0003107] [INSPIRE].
[81] A.V. Manohar and I.W. Stewart, Logarithms of α in QED bound states from the renormalization group, Phys. Rev. Lett.85 (2000) 2248 [hep-ph/0004018] [INSPIRE].
[82] A.H. Hoang and I.W. Stewart, Ultrasoft renormalization in nonrelativistic QCD, Phys. Rev.D 67 (2003) 114020 [hep-ph/0209340] [INSPIRE].
[83] A. Pineda and J. Soto, Effective field theory for ultrasoft momenta in NRQCD and NRQED, Nucl. Phys. Proc. Suppl.64 (1998) 428 [hep-ph/9707481] [INSPIRE].
[84] N. Brambilla, A. Pineda, J. Soto and A. Vairo, Potential NRQCD: an effective theory for heavy quarkonium, Nucl. Phys.B 566 (2000) 275 [hep-ph/9907240] [INSPIRE].
[85] N. Brambilla, A. Pineda, J. Soto and A. Vairo, Effective field theories for heavy quarkonium, Rev. Mod. Phys.77 (2005) 1423 [hep-ph/0410047] [INSPIRE].
[86] Marcantonini, C.; Stewart, IW, Reparameterization invariant collinear operators, Phys. Rev., D 79, 065028, (2009)
[87] Idilbi, A.; Scimemi, I., Singular and regular gauges in soft collinear effective theory: the introduction of the new Wilson line T, Phys. Lett., B 695, 463, (2011)
[88] Garcia-Echevarria, M.; Idilbi, A.; Scimemi, I., SCET, light-cone gauge and the T -Wilson lines, Phys. Rev., D 84, 011502, (2011)
[89] 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] [INSPIRE]. · Zbl 0996.81518
[90] J. Chay and C. Kim, Collinear effective theory at subleading order and its application to heavy-light currents, Phys. Rev.D 65 (2002) 114016 [hep-ph/0201197] [INSPIRE].
[91] M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Perseus Books Publishing, U.S.A. (1995).
[92] L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
[93] E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Multi-Reggeon processes in the Yang-Mills theory, Sov. Phys. JETP44 (1976) 443 [Zh. Eksp. Teor. Fiz.71 (1976) 840] [INSPIRE].
[94] B. Grinstein and I.Z. Rothstein, Effective field theory and matching in nonrelativistic gauge theories, Phys. Rev.D 57 (1998) 78 [hep-ph/9703298] [INSPIRE].
[95] C. Lee and G.F. Sterman, Universality of nonperturbative effects in event shapes, eConfC 0601121 (2006) A001 [hep-ph/0603066] [INSPIRE].
[96] A. Idilbi and T. Mehen, On the equivalence of soft and zero-bin subtractions, Phys. Rev.D 75 (2007) 114017 [hep-ph/0702022] [INSPIRE].
[97] C.W. Bauer, D. Pirjol and I.W. Stewart, Power counting in the soft collinear effective theory, Phys. Rev.D 66 (2002) 054005 [hep-ph/0205289] [INSPIRE].
[98] I.W. Stewart, Theoretical introduction to B decays and the soft collinear effective theory, hep-ph/0308185 [INSPIRE].
[99] G.F. Sterman, Mass divergences in annihilation processes. 2. Cancellation of divergences in cut vacuum polarization diagrams, Phys. Rev.D 17 (1978) 2789 [INSPIRE].
[100] G.F. Sterman, Partons, factorization and resummation, TASI 95, hep-ph/9606312 [INSPIRE].
[101] A. Banfi, G.P. Salam and G. Zanderighi, Resummed event shapes at hadron-hadron colliders, JHEP08 (2004) 062 [hep-ph/0407287] [INSPIRE]. · Zbl 1290.81159
[102] Stewart, IW; Tackmann, FJ; Waalewijn, WJ, Factorization at the LHC: from PDFs to initial state jets, Phys. Rev., D 81, 094035, (2010)
[103] Gaunt, JR, Glauber gluons and multiple parton interactions, JHEP, 07, 110, (2014)
[104] Zeng, M., Drell-Yan process with jet vetoes: breaking of generalized factorization, JHEP, 10, 189, (2015)
[105] Forshaw, JR; Seymour, MH, Soft gluons and superleading logarithms in QCD, Nucl. Phys. Proc. Suppl., 191, 257, (2009)
[106] J.R. Forshaw and D.A. Ross, Quantum chromodynamics and the Pomeron, Cambridge University Press, Cambridge U.K. (1997).
[107] B.L. Ioffe, V.S. Fadin and L.N. Lipatov, Quantum chromodynamics, Cambridge University Press, Cambridge U.K. (2010). · Zbl 1192.81006
[108] Y.V. Kovchegov and E. Levin, Quantum chromodynamics at high energy, Cambridge University Press, Cambridge U.K. (2012). · Zbl 1271.81005
[109] 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].
[110] Weinberg, S., Dynamics at infinite momentum, Phys. Rev., 150, 1313, (1966)
[111] H. Cheng and T.T. Wu, Expanding protons, The MIT Press, U.S.A. (1987).
[112] Jackiw, R.; Kabat, DN; Ortiz, M., Electromagnetic fields of a massless particle and the eikonal, Phys. Lett., B 277, 148, (1992)
[113] G. ’t Hooft, Graviton dominance in ultrahigh-energy scattering, Phys. Lett.B 198 (1987) 61 [INSPIRE].
[114] Levy, M.; Sucher, J., Eikonal approximation in quantum field theory, Phys. Rev., 186, 1656, (1969)
[115] Kabat, DN; Ortiz, M., Eikonal quantum gravity and Planckian scattering, Nucl. Phys., B 388, 570, (1992)
[116] Chang, S-J, radiative corrections to the e\^{}{−}\(e\)\^{}{±}scattering amplitude at infinite energy, Phys. Rev., D 1, 2977, (1970)
[117] Neill, D.; Rothstein, IZ, Classical space-times from the S matrix, Nucl. Phys., B 877, 177, (2013) · Zbl 1284.83052
[118] H.L. Verlinde and E.P. Verlinde, QCD at high-energies and two-dimensional field theory, hep-th/9302104 [INSPIRE]. · Zbl 0985.81681
[119] Feige, I.; Schwartz, MD, Hard-soft-collinear factorization to all orders, Phys. Rev., D 90, 105020, (2014)
[120] J. Chay, C. Kim, Y.G. Kim and J.-P. Lee, Soft Wilson lines in soft-collinear effective theory, Phys. Rev.D 71 (2005) 056001 [hep-ph/0412110] [INSPIRE].
[121] 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] [INSPIRE].
[122] A.V. Manohar, Deep inelastic scattering as x → 1 using soft collinear effective theory, Phys. Rev.D 68 (2003) 114019 [hep-ph/0309176] [INSPIRE].
[123] 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].
[124] Chiu, J-Y; Golf, F.; Kelley, R.; Manohar, AV, Electroweak corrections in high energy processes using effective field theory, Phys. Rev., D 77, 053004, (2008)
[125] A.H. Hoang, A.V. Manohar and I.W. Stewart, The running Coulomb potential and Lamb shift in QCD, Phys. Rev.D 64 (2001) 014033 [hep-ph/0102257] [INSPIRE].
[126] Kang, D.; Labun, OZ; Lee, C., equality of hemisphere soft functions for e\^{}{+}\(e\)\^{}{−}, DIS and pp collisions at\( \mathcal{O}\left({α}_s^2\right) \), Phys. Lett., B 748, 45, (2015) · Zbl 1345.81123
[127] Stewart, IW; Tackmann, FJ; Waalewijn, WJ, The quark beam function at NNLL, JHEP, 09, 005, (2010) · Zbl 1291.81397
[128] Stewart, IW; Tackmann, FJ; Waalewijn, WJ, The beam thrust cross section for Drell-Yan at NNLL order, Phys. Rev. Lett., 106, 032001, (2011)
[129] Banfi, A.; Salam, GP; Zanderighi, G., Phenomenology of event shapes at hadron colliders, JHEP, 06, 038, (2010) · Zbl 1290.81159
[130] Papaefstathiou, A.; Smillie, JM; Webber, BR, Resummation of transverse energy in vector boson and Higgs boson production at hadron colliders, JHEP, 04, 084, (2010) · Zbl 1272.81197
[131] Tackmann, FJ; Walsh, JR; Zuberi, S., Resummation properties of jet vetoes at the LHC, Phys. Rev., D 86, 053011, (2012)
[132] Grazzini, M.; Papaefstathiou, A.; Smillie, JM; Webber, BR, Resummation of the transverse-energy distribution in Higgs boson production at the large hadron collider, JHEP, 09, 056, (2014)
[133] Banfi, A.; Monni, PF; Salam, GP; Zanderighi, G., Higgs and Z-boson production with a jet veto, Phys. Rev. Lett., 109, 202001, (2012)
[134] Becher, T.; Neubert, M.; Rothen, L., factorization and N\^{}{3}LL_{\(p\)}+NNLO predictions for the Higgs cross section with a jet veto, JHEP, 10, 125, (2013)
[135] Stewart, IW; Tackmann, FJ; Walsh, JR; Zuberi, S., Jet p_{T} resummation in Higgs production at NNLL’+NNLO, Phys. Rev., D 89, 054001, (2014)
[136] T. Becher, X. Garcia i Tormo and J. Piclum, Next-to-next-to-leading logarithmic resummation for transverse thrust, Phys. Rev.D 93 (2016) 054038 [arXiv:1512.00022] [INSPIRE].
[137] J.C. Collins and A. Metz, Universality of soft and collinear factors in hard-scattering factorization, Phys. Rev. Lett.93 (2004) 252001 [hep-ph/0408249] [INSPIRE].
[138] S. Mantry, D. Pirjol and I.W. Stewart, Strong phases and factorization for color suppressed decays, Phys. Rev.D 68 (2003) 114009 [hep-ph/0306254] [INSPIRE].
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.