Abstract
We obtain the next-to-next-to-leading order corrections to transverse-momentum spectra of W, Z and Higgs bosons near the partonic threshold. In the threshold limit, the electroweak boson recoils against a low-mass jet and all radiation is either soft, or collinear to the jet or the beam directions. We extract the virtual corrections from known results for the relevant two-loop four-point amplitudes and combine them with the soft and collinear two-loop functions as defined in Soft-Collinear Effective Theory. We have implemented these results in a public code PeTeR and present numerical results for the threshold resummed cross section of W and Z bosons at next-to-next-to-next-to-leading logarithmic accuracy, matched to next-to-leading fixed-order perturbation theory. The two-loop corrections lead to a moderate increase in the cross section and reduce the scale uncertainty by about a factor of two. The corrections are significantly larger for Higgs production.
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References
L. Garland, T. Gehrmann, E.N. Glover, A. Koukoutsakis and E. Remiddi, The two loop QCD matrix element for e + e − → 3 jets, Nucl. Phys. B 627 (2002) 107 [hep-ph/0112081] [INSPIRE].
L. Garland, T. Gehrmann, E.N. Glover, A. Koukoutsakis and E. Remiddi, Two loop QCD helicity amplitudes for e + e − → three jets, Nucl. Phys. B 642 (2002) 227 [hep-ph/0206067] [INSPIRE].
C. Anastasiou, K. Melnikov and F. Petriello, A new method for real radiation at NNLO, Phys. Rev. D 69 (2004) 076010 [hep-ph/0311311] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann and E.N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [INSPIRE].
M. Czakon, A novel subtraction scheme for double-real radiation at NNLO, Phys. Lett. B 693 (2010) 259 [arXiv:1005.0274] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann, E. Glover and J. Pires, Second order QCD corrections to jet production at hadron colliders: the all-gluon contribution, Phys. Rev. Lett. 110 (2013) 162003 [arXiv:1301.7310] [INSPIRE].
R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze, Higgs boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, JHEP 06 (2013) 072 [arXiv:1302.6216] [INSPIRE].
T. Becher and M. Neubert, Toward a NNLO calculation of the 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].
T. Becher and G. Bell, The gluon jet function at two-loop order, Phys. Lett. B 695 (2011) 252 [arXiv:1008.1936] [INSPIRE].
T. Becher, G. Bell and S. Marti, NNLO soft function for electroweak boson production at large transverse momentum, JHEP 04 (2012) 034 [arXiv:1201.5572] [INSPIRE].
T. Gehrmann and L. Tancredi, Two-loop QCD helicity amplitudes for \( q\overline{q} \) → W ± γ and \( q\overline{q} \) → Z 0 γ, JHEP 02 (2012) 004 [arXiv:1112.1531] [INSPIRE].
T. Gehrmann, L. Tancredi and E. Weihs, Two-loop QCD helicity amplitudes for g g → Z g and g g → Z γ, JHEP 04 (2013) 101 [arXiv:1302.2630] [INSPIRE].
T. Gehrmann, M. Jaquier, E. Glover and A. Koukoutsakis, Two-loop QCD corrections to the helicity amplitudes for H → 3 partons, JHEP 02 (2012) 056 [arXiv:1112.3554] [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, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
M. Beneke, A. 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] [INSPIRE].
T. Becher and M.D. Schwartz, Direct photon production with effective field theory, JHEP 02 (2010) 040 [arXiv:0911.0681] [INSPIRE].
T. Becher, C. Lorentzen and M.D. Schwartz, Resummation for W and Z production at large p T , Phys. Rev. Lett. 108 (2012) 012001 [arXiv:1106.4310] [INSPIRE].
T. Becher, C. Lorentzen and M.D. Schwartz, Precision Direct Photon and W-Boson Spectra at High p T and Comparison to LHC Data, Phys. Rev. D 86 (2012) 054026 [arXiv:1206.6115] [INSPIRE].
N. Kidonakis and V. Del Duca, Electroweak boson hadroproduction at large transverse momentum: Factorization, resummation and NNLO corrections, Phys. Lett. B 480 (2000) 87 [hep-ph/9911460] [INSPIRE].
N. Kidonakis and A. Sabio Vera, W hadroproduction at large transverse momentum beyond next-to-leading order, JHEP 02 (2004) 027 [hep-ph/0311266] [INSPIRE].
R.J. Gonsalves, N. Kidonakis and A. Sabio Vera, W production at large transverse momentum at the large hadron collider, Phys. Rev. Lett. 95 (2005) 222001 [hep-ph/0507317] [INSPIRE].
N. Kidonakis and R.J. Gonsalves, Higher-order QCD corrections for the W-boson transverse momentum distribution, Phys. Rev. D 87 (2013) 014001 [arXiv:1201.5265] [INSPIRE].
N. Kidonakis and R.J. Gonsalves, Two-Loop Corrections to W and Z Boson Production at High p T , arXiv:1109.2817 [INSPIRE].
Ch. Lorentzen, http://peter.hepforge.org.
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [arXiv:0901.0722] [INSPIRE].
T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 1311 (2013) 024] [arXiv:0903.1126] [INSPIRE].
I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Combining fixed-order helicity amplitudes with resummation using SCET, PoS(LL2012)058 [arXiv:1211.2305] [INSPIRE].
S. Catani and M. 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. 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].
S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
T. Gehrmann and E. Remiddi, Analytic continuation of massless two loop four point functions, Nucl. Phys. B 640 (2002) 379 [hep-ph/0207020] [INSPIRE].
T. Gehrmann and E. Remiddi, Two loop master integrals for γ * → 3 jets: the planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].
T. Gehrmann and E. Remiddi, Two loop master integrals for γ * → 3 jets: The Nonplanar topologies, Nucl. Phys. B 601 (2001) 287 [hep-ph/0101124] [INSPIRE].
T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].
T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms, Comput. Phys. Commun. 144 (2002) 200 [hep-ph/0111255] [INSPIRE].
H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel, Leipzig Germany (1928).
B.L. van der Waerden, Spinoranalyse, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1929), pg. 100.
F.A. Berends and W. Giele, The Six Gluon Process as an Example of Weyl-Van Der Waerden Spinor Calculus, Nucl. Phys. B 294 (1987) 700 [INSPIRE].
M.E. Peskin, Simplifying Multi-Jet QCD Computation, arXiv:1101.2414 [INSPIRE].
L.J. Dixon, Scattering amplitudes: the most perfect microscopic structures in the universe, J. Phys. A 44 (2011) 454001 [arXiv:1105.0771] [INSPIRE].
R.K. Ellis, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts, Phys. Rept. 518 (2012) 141 [arXiv:1105.4319] [INSPIRE].
J. van der Bij and E.N. Glover, Z Boson Production and Decay via Gluons, Nucl. Phys. B 313 (1989) 237 [INSPIRE].
R. Hopker and J. van der Bij, Z 0 decay into three gluons, Phys. Rev. D 49 (1994) 3779 [INSPIRE].
T.T. Jouttenus, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Jet Mass Spectra in Higgs + One Jet at NNLL, Phys. Rev. D 88 (2013) 054031 [arXiv:1302.0846] [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].
C.W. Bauer and A.V. Manohar, Shape function effects in B → X s γ and B → \( {X_u}l\overline{\nu} \) decays, Phys. Rev. D 70 (2004) 034024 [hep-ph/0312109] [INSPIRE].
S. Bosch, B. 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] [INSPIRE].
R.J. Gonsalves, J. Pawlowski and C.-F. Wai, QCD Radiative Corrections to Electroweak Boson Production at Large Transverse Momentum in Hadron Collisions, Phys. Rev. D 40 (1989) 2245 [INSPIRE].
T. Becher and X. Garcia i Tormo, Electroweak Sudakov effects in W, Z and γ production at large transverse momentum, Phys. Rev. D 88 (2013) 013009 [arXiv:1305.4202] [INSPIRE].
A. Martin, W. Stirling, R. Thorne and G. Watt, Parton distributions for the LHC, Eur. Phys. J. C 63 (2009) 189 [arXiv:0901.0002] [INSPIRE].
T. Becher, M. Neubert and G. Xu, Dynamical threshold enhancement and resummation in Drell-Yan production, JHEP 07 (2008) 030 [arXiv:0710.0680] [INSPIRE].
S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].
M. Krämer, E. Laenen and M. Spira, Soft gluon radiation in Higgs boson production at the LHC, Nucl. Phys. B 511 (1998) 523 [hep-ph/9611272] [INSPIRE].
K. Chetyrkin, B.A. Kniehl and M. Steinhauser, Hadronic Higgs decay to order \( \alpha_S^4 \) Phys. Rev. Lett. 79 (1997) 353 [hep-ph/9705240] [INSPIRE].
T. Inami, T. Kubota and Y. Okada, Effective Gauge Theory and the Effect of Heavy Quarks in Higgs Boson Decays, Z. Phys. C 18 (1983) 69 [INSPIRE].
B. Grinstein and L. Randall, The Renormalization of g 2, Phys. Lett. B 217 (1989) 335 [INSPIRE].
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Becher, T., Bell, G., Lorentzen, C. et al. Transverse-momentum spectra of electroweak bosons near threshold at NNLO. J. High Energ. Phys. 2014, 4 (2014). https://doi.org/10.1007/JHEP02(2014)004
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DOI: https://doi.org/10.1007/JHEP02(2014)004