Abstract
On-shell helicity methods provide powerful tools for determining scattering amplitudes, which have a one-to-one correspondence with leading power helicity operators in the Soft-Collinear Effective Theory (SCET) away from singular regions of phase space. We show that helicity based operators are also useful for enumerating power suppressed SCET operators, which encode subleading amplitude information about singular limits. In particular, we present a complete set of scalar helicity building blocks that are valid for constructing operators at any order in the SCET power expansion. We also describe an interesting angular momentum selection rule that restricts how these building blocks can be assembled.
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L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
H. Elvang and Y.-T. Huang, Scattering amplitudes, arXiv:1308.1697 [INSPIRE].
L.J. Dixon, A brief introduction to modern amplitude methods, arXiv:1310.5353 [INSPIRE].
J.M. Henn and J.C. Plefka, Scattering amplitudes in gauge theories, Lect. Notes Phys. 883 (2014) 1 [INSPIRE].
C. Cheung and C.-H. Shen, Nonrenormalization theorems without supersymmetry, Phys. Rev. Lett. 115 (2015) 071601 [arXiv:1505.01844] [INSPIRE].
R. Alonso, E.E. Jenkins and A.V. Manohar, Holomorphy without supersymmetry in the standard model effective field theory, Phys. Lett. B 739 (2014) 95 [arXiv:1409.0868] [INSPIRE].
I. Moult, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Employing helicity amplitudes for resummation, Phys. Rev. D 93 (2016) 094003 [arXiv:1508.02397] [INSPIRE].
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].
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].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Factorization for short distance hadron-hadron scattering, Nucl. Phys. B 261 (1985) 104 [INSPIRE].
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].
J.C. Collins, D.E. Soper and G.F. Sterman, Soft gluons and factorization, Nucl. Phys. B 308 (1988) 833 [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Transverse momentum distribution in Drell-Yan pair and W and Z boson production, Nucl. Phys. B 250 (1985) 199 [INSPIRE].
S. Catani, G. Turnock, B.R. Webber and L. Trentadue, Thrust distribution in e + e − annihilation, Phys. Lett. B 263 (1991) 491 [INSPIRE].
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 [INSPIRE].
G.F. Sterman, Summation of large corrections to short distance hadronic cross-sections, Nucl. Phys. B 281 (1987) 310 [INSPIRE].
S. Catani and L. Trentadue, Resummation of the QCD perturbative series for hard processes, Nucl. Phys. B 327 (1989) 323 [INSPIRE].
F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].
T.H. Burnett and N.M. Kroll, Extension of the low soft photon theorem, Phys. Rev. Lett. 20 (1968) 86 [INSPIRE].
V. Del Duca, High-energy bremsstrahlung theorems for soft photons, Nucl. Phys. B 345 (1990) 369 [INSPIRE].
E. Casali, Soft sub-leading divergences in Yang-Mills amplitudes, JHEP 08 (2014) 077 [arXiv:1404.5551] [INSPIRE].
B.U.W. Schwab and A. Volovich, Subleading soft theorem in arbitrary dimensions from scattering equations, Phys. Rev. Lett. 113 (2014) 101601 [arXiv:1404.7749] [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a new soft graviton theorem, arXiv:1404.4091 [INSPIRE].
A.J. Larkoski, Conformal invariance of the subleading soft theorem in gauge theory, Phys. Rev. D 90 (2014) 087701 [arXiv:1405.2346] [INSPIRE].
S. He, Y.-T. Huang and C. Wen, Loop corrections to soft theorems in gauge theories and gravity, JHEP 12 (2014) 115 [arXiv:1405.1410] [INSPIRE].
Z. Bern, S. Davies and J. Nohle, On loop corrections to subleading soft behavior of gluons and gravitons, Phys. Rev. D 90 (2014) 085015 [arXiv:1405.1015] [INSPIRE].
J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, Constraining subleading soft gluon and graviton theorems, Phys. Rev. D 90 (2014) 065024 [arXiv:1406.6574] [INSPIRE].
M. Zlotnikov, Sub-sub-leading soft-graviton theorem in arbitrary dimension, JHEP 10 (2014) 148 [arXiv:1407.5936] [INSPIRE].
V. Lysov, S. Pasterski and A. Strominger, Low’s subleading soft theorem as a symmetry of QED, Phys. Rev. Lett. 113 (2014) 111601 [arXiv:1407.3814] [INSPIRE].
C.D. White, Diagrammatic insights into next-to-soft corrections, Phys. Lett. B 737 (2014) 216 [arXiv:1406.7184] [INSPIRE].
A.J. Larkoski, D. Neill and I.W. Stewart, Soft theorems from effective field theory, JHEP 06 (2015) 077 [arXiv:1412.3108] [INSPIRE].
D. Kapec, M. Pate and A. Strominger, New symmetries of QED, arXiv:1506.02906 [INSPIRE].
A. Strominger, Magnetic corrections to the soft photon theorem, Phys. Rev. Lett. 116 (2016) 031602 [arXiv:1509.00543] [INSPIRE].
K.S.M. Lee and I.W. Stewart, Factorization for power corrections to B → X s γ and \( B\to {X}_u\ell \overline{\nu} \), Nucl. Phys. B 721 (2005) 325 [hep-ph/0409045] [INSPIRE].
T. Mannel and F.J. Tackmann, Shape function effects in \( B\to {X}_c\ell {\overline{\nu}}_{\ell } \), Phys. Rev. D 71 (2005) 034017 [hep-ph/0408273] [INSPIRE].
S.W. Bosch, M. Neubert and G. Paz, Subleading shape functions in inclusive B decays, JHEP 11 (2004) 073 [hep-ph/0409115] [INSPIRE].
M. Beneke, F. Campanario, T. Mannel and B.D. Pecjak, Power corrections to \( \overline{B}\to {X}_u\ell \overline{nu}\left({X}_s\gamma \right) \) decay spectra in the ‘shape-function’ region, JHEP 06 (2005) 071 [hep-ph/0411395] [INSPIRE].
E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [arXiv:0811.2067] [INSPIRE].
E. Laenen, L. Magnea and G. Stavenga, On next-to-eikonal corrections to threshold resummation for the Drell-Yan and DIS cross sections, Phys. Lett. B 669 (2008) 173 [arXiv:0807.4412] [INSPIRE].
G. Grunberg and V. Ravindran, On threshold resummation beyond leading 1 − x order, JHEP 10 (2009) 055 [arXiv:0902.2702] [INSPIRE].
E. Laenen, L. Magnea, G. Stavenga and C.D. White, Next-to-eikonal corrections to soft gluon radiation: a diagrammatic approach, JHEP 01 (2011) 141 [arXiv:1010.1860] [INSPIRE].
A.A. Almasy, G. Soar and A. Vogt, Generalized double-logarithmic large-x resummation in inclusive deep-inelastic scattering, JHEP 03 (2011) 030 [arXiv:1012.3352] [INSPIRE].
S.M. Freedman, Subleading corrections to thrust using effective field theory, arXiv:1303.1558 [INSPIRE].
S.M. Freedman and R. Goerke, Renormalization of subleading dijet operators in soft-collinear effective theory, Phys. Rev. D 90 (2014) 114010 [arXiv:1408.6240] [INSPIRE].
D. Bonocore, E. Laenen, L. Magnea, L. Vernazza and C.D. White, The method of regions and next-to-soft corrections in Drell-Yan production, Phys. Lett. B 742 (2015) 375 [arXiv:1410.6406] [INSPIRE].
D. de Florian, J. Mazzitelli, S. Moch and A. Vogt, Approximate N 3 LO Higgs-boson production cross section using physical-kernel constraints, JHEP 10 (2014) 176 [arXiv:1408.6277] [INSPIRE].
D. Bonocore, E. Laenen, L. Magnea, S. Melville, L. Vernazza and C.D. White, A factorization approach to next-to-leading-power threshold logarithms, JHEP 06 (2015) 008 [arXiv:1503.05156] [INSPIRE].
M. Beneke and T. Feldmann, Multipole expanded soft collinear effective theory with non-Abelian gauge symmetry, Phys. Lett. B 553 (2003) 267 [hep-ph/0211358] [INSPIRE].
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].
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].
D. Pirjol and I.W. Stewart, A complete basis for power suppressed collinear ultrasoft operators, Phys. Rev. D 67 (2003) 094005 [Erratum ibid. D 69 (2004) 019903] [hep-ph/0211251] [INSPIRE].
M. Beneke, A.P. 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].
C.W. Bauer, D. Pirjol and I.W. Stewart, On power suppressed operators and gauge invariance in SCET, Phys. Rev. D 68 (2003) 034021 [hep-ph/0303156] [INSPIRE].
C.M. Arnesen, Z. Ligeti, I.Z. Rothstein and I.W. Stewart, Power corrections in charmless nonleptonic B-decays: annihilation is factorizable and real, Phys. Rev. D 77 (2008) 054006 [hep-ph/0607001] [INSPIRE].
C.M. Arnesen, I.Z. Rothstein and I.W. Stewart, Three-parton contributions to B → M 1 M 2 annihilation at leading order, Phys. Lett. B 647 (2007) 405 [Erratum ibid. B 653 (2007) 450] [hep-ph/0611356] [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].
M. Benzke, S.J. Lee, M. Neubert and G. Paz, Factorization at subleading power and irreducible uncertainties in \( \overline{B}\to {X}_s\gamma \) decay, JHEP 08 (2010) 099 [arXiv:1003.5012] [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].
I. Feige, D.W. Kolodrubetz, I. Moult and I.W. Stewart, Helicity operators for subleading factorization, forthcoming publication.
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. Marcantonini and I.W. Stewart, Reparameterization invariant collinear operators, Phys. Rev. D 79 (2009) 065028 [arXiv:0809.1093] [INSPIRE].
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].
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].
A.V. Manohar and M.B. Wise, Inclusive semileptonic B and polarized Λ b decays from QCD, Phys. Rev. D 49 (1994) 1310 [hep-ph/9308246] [INSPIRE].
M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, QCD factorization for exclusive, nonleptonic B meson decays: general arguments and the case of heavy light final states, Nucl. Phys. B 591 (2000) 313 [hep-ph/0006124] [INSPIRE].
M. Beneke and T. Feldmann, Symmetry breaking corrections to heavy to light B meson form-factors at large recoil, Nucl. Phys. B 592 (2001) 3 [hep-ph/0008255] [INSPIRE].
M. Beneke, T. Feldmann and D. Seidel, Systematic approach to exclusive B → V ℓ + ℓ − , V γ decays, Nucl. Phys. B 612 (2001) 25 [hep-ph/0106067] [INSPIRE].
C.W. Bauer, D. Pirjol and I.W. Stewart, A proof of factorization for B → Dπ, Phys. Rev. Lett. 87 (2001) 201806 [hep-ph/0107002] [INSPIRE].
S.W. Bosch and G. Buchalla, The radiative decays B → V γ at next-to-leading order in QCD, Nucl. Phys. B 621 (2002) 459 [hep-ph/0106081] [INSPIRE].
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].
M. Beneke and T. Feldmann, Factorization of heavy to light form-factors in soft collinear effective theory, Nucl. Phys. B 685 (2004) 249 [hep-ph/0311335] [INSPIRE].
M. Beneke, T. Feldmann and D. Seidel, Exclusive radiative and electroweak b → d and b → s penguin decays at NLO, Eur. Phys. J. C 41 (2005) 173 [hep-ph/0412400] [INSPIRE].
M.C. Arnesen, B. Grinstein, I.Z. Rothstein and I.W. Stewart, A precision model independent determination of |V ub | from B → πeν, Phys. Rev. Lett. 95 (2005) 071802 [hep-ph/0504209] [INSPIRE].
K.S.M. Lee, Z. Ligeti, I.W. Stewart and F.J. Tackmann, Universality and m X cut effects in B → X s ℓ + ℓ −, Phys. Rev. D 74 (2006) 011501 [hep-ph/0512191] [INSPIRE].
K.S.M. Lee, Z. Ligeti, I.W. Stewart and F.J. Tackmann, Extracting short distance information from b → sℓ + ℓ − effectively, Phys. Rev. D 75 (2007) 034016 [hep-ph/0612156] [INSPIRE].
A.J. Buras and P.H. Weisz, QCD nonleading corrections to weak decays in dimensional regularization and ’t Hooft-Veltman schemes, Nucl. Phys. B 333 (1990) 66 [INSPIRE].
M.J. Dugan and B. Grinstein, On the vanishing of evanescent operators, Phys. Lett. B 256 (1991) 239 [INSPIRE].
S. Herrlich and U. Nierste, Evanescent operators, scheme dependences and double insertions, Nucl. Phys. B 455 (1995) 39 [hep-ph/9412375] [INSPIRE].
G. Bozzi, S. Catani, D. de Florian and M. Grazzini, The q T spectrum of the Higgs boson at the LHC in QCD perturbation theory, Phys. Lett. B 564 (2003) 65 [hep-ph/0302104] [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. Banfi, G.P. Salam and G. Zanderighi, Resummed event shapes at hadron-hadron colliders, JHEP 08 (2004) 062 [hep-ph/0407287] [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].
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].
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. 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] [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].
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].
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].
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].
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].
I. Feige, M.D. Schwartz, I.W. Stewart and J. Thaler, Precision jet substructure from boosted event shapes, Phys. Rev. Lett. 109 (2012) 092001 [arXiv:1204.3898] [INSPIRE].
T. Becher and M. Neubert, Factorization and NNLL resummation for Higgs production with a jet veto, JHEP 07 (2012) 108 [arXiv:1205.3806] [INSPIRE].
A. Banfi, P.F. Monni, G.P. Salam and G. Zanderighi, Higgs and Z-boson production with a jet veto, Phys. Rev. Lett. 109 (2012) 202001 [arXiv:1206.4998] [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].
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].
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].
I.W. Stewart, F.J. Tackmann, J.R. Walsh and S. Zuberi, Jet p T resummation in Higgs production at NNLL’+NNLO, Phys. Rev. D 89 (2014) 054001 [arXiv:1307.1808] [INSPIRE].
A.H. Hoang, D.W. Kolodrubetz, V. Mateu and I.W. Stewart, C-parameter distribution at N 3 LL’ including power corrections, Phys. Rev. D 91 (2015) 094017 [arXiv:1411.6633] [INSPIRE].
A.J. Larkoski, I. Moult and D. Neill, Analytic boosted boson discrimination, arXiv:1507.03018 [INSPIRE].
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Kolodrubetz, D.W., Moult, I. & Stewart, I.W. Building blocks for subleading helicity operators. J. High Energ. Phys. 2016, 139 (2016). https://doi.org/10.1007/JHEP05(2016)139
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DOI: https://doi.org/10.1007/JHEP05(2016)139