Abstract
A classic result, originally due to Kluberg-Stern and Zuber, states that operators that vanish by the classical equation of motion (eom) do not mix into “physical” operators. Here we show that and explain why this result does not hold in soft-collinear effective theory (SCET) for the renormalization of power-suppressed operators. We calculate the non-vanishing mixing of eom operators for the simplest case of N -jet operators with a single collinear field in every direction. The result implies that — for the computation of the anomalous dimension but not for on-shell matrix elements — there exists a preferred set of fields that must be used to reproduce the infrared singularities of QCD scattering amplitudes. We identify these fields and explain their relation to the gauge-invariant SCET Lagrangian. Further checks reveal another generic property of SCET beyond leading power, which will be relevant to resummation at the next-to-leading logarithmic level, the divergence of convolution integrals with the hard matching coefficients. We propose an operator solution that allows to consistently renormalize such divergences.
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References
W.S. Deans and J.A. Dixon, Theory of gauge invariant operators: their renormalization and S matrix elements, Phys. Rev.D 18 (1978) 1113 [INSPIRE].
H.D. Politzer, Power corrections at short distances, Nucl. Phys.B 172 (1980) 349 [INSPIRE].
H. Kluberg-Stern and J.B. Zuber, Renormalization of nonabelian gauge theories in a background field gauge. 2. Gauge invariant operators, Phys. Rev.D 12 (1975) 3159 [INSPIRE].
S.D. Joglekar and B.W. Lee, General theory of renormalization of gauge invariant operators, Annals Phys.97 (1976) 160 [INSPIRE].
D. Espriu, Renormalization of gauge invariant operators and the axial anomaly, Phys. Rev.D 28 (1983) 349 [INSPIRE].
J.C. Collins and R.J. Scalise, The renormalization of composite operators in Yang-Mills theories using general covariant gauge, Phys. Rev.D 50 (1994) 4117 [hep-ph/9403231] [INSPIRE].
A.V. Manohar, Introduction to effective field theories, talk given at the Les Houches summer school: EFT in Particle Physics and Cosmology, July 3–28, Les Houches, Chamonix Valley, France (2018), arXiv:1804.05863 [INSPIRE].
J.C. Criado and M. Pérez-Victoria, Field redefinitions in effective theories at higher orders, JHEP03 (2019) 038 [arXiv:1811.09413] [INSPIRE].
P. Gambino, M. Gorbahn and U. Haisch, Anomalous dimension matrix for radiative and rare semileptonic B decays up to three loops, Nucl. Phys.B 673 (2003) 238 [hep-ph/0306079] [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.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].
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].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett.102 (2009) 162001 [Erratum ibid.111 (2013) 199905] [arXiv:0901.0722] [INSPIRE].
M. Beneke, F. Campanario, T. Mannel and B.D. Pecjak, Power corrections to \( \overline{B} \) → X ul \( \overline{\nu} \) (X sγ) decay spectra in the ‘shape-function’ region, JHEP06 (2005) 071 [hep-ph/0411395] [INSPIRE].
A.J. Larkoski, D. Neill and I.W. Stewart, Soft theorems from effective field theory, JHEP06 (2015) 077 [arXiv:1412.3108] [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.W. Kolodrubetz, I. Moult and I.W. Stewart, Building blocks for subleading helicity operators, JHEP05 (2016) 139 [arXiv:1601.02607] [INSPIRE].
I. Feige, D.W. Kolodrubetz, I. Moult and I.W. Stewart, A complete basis of helicity operators for subleading factorization, JHEP11 (2017) 142 [arXiv:1703.03411] [INSPIRE].
I. Moult, I.W. Stewart and G. Vita, A subleading operator basis and matching for gg → H, JHEP07 (2017) 067 [arXiv:1703.03408] [INSPIRE].
M. Beneke, M. Garny, R. Szafron and J. Wang, Subleading-power N -jet operators and the LBK amplitude in SCET, PoS(RADCOR2017) 048 [arXiv:1712.07462] [INSPIRE].
M. Beneke, M. Garny, R. Szafron and J. Wang, Anomalous dimension of subleading-power N-jet operators, JHEP03 (2018) 001 [arXiv:1712.04416] [INSPIRE].
M. Beneke, M. Garny, R. Szafron and J. Wang, Anomalous dimension of subleading-power N -jet operators. Part II, JHEP11 (2018) 112 [arXiv:1808.04742] [INSPIRE].
I. Moult, I.W. Stewart, G. Vita and H.X. Zhu, First subleading power resummation for event shapes, JHEP08 (2018) 013 [arXiv:1804.04665] [INSPIRE].
M. Beneke et al., Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power, JHEP03 (2019) 043 [arXiv:1809.10631] [INSPIRE].
M.A. Ebert et al., Subleading power rapidity divergences and power corrections for q T, JHEP04 (2019) 123 [arXiv:1812.08189] [INSPIRE].
M. Beneke, Y. Kiyo and D.s. Yang, Loop corrections to subleading heavy quark currents in SCET, Nucl. Phys.B 692 (2004) 232 [hep-ph/0402241] [INSPIRE].
R.J. Hill, T. Becher, S.J. Lee and M. Neubert, Sudakov resummation for subleading SCET currents and heavy-to-light form-factors, JHEP07 (2004) 081 [hep-ph/0404217] [INSPIRE].
M. Beneke and D. Yang, Heavy-to-light B meson form-factors at large recoil energy: Spectator-scattering corrections, Nucl. Phys.B 736 (2006) 34 [hep-ph/0508250] [INSPIRE].
M. Beneke, C. Bobeth and R. Szafron, Enhanced electromagnetic correction to the rare B-meson decay B s,d→ μ +μ −, Phys. Rev. Lett.120 (2018) 011801 [arXiv:1708.09152] [INSPIRE].
S. Alte, M. König and M. Neubert, Effective field theory after a new-physics discovery, JHEP08 (2018) 095 [arXiv:1806.01278] [INSPIRE].
M. Beneke, M. Garny, R. Szafron and J. Wang, in preparation.
S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett.B 427 (1998) 161 [hep-ph/9802439] [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].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys.B 522 (1998) 321 [hep-ph/9711391] [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].
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].
M. Beneke, Perturbative heavy quark-anti-quark systems, hep-ph/9911490 [INSPIRE].
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ArXiv ePrint: 1907.05463
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Beneke, M., Garny, M., Szafron, R. et al. Violation of the Kluberg-Stern-Zuber theorem in SCET. J. High Energ. Phys. 2019, 101 (2019). https://doi.org/10.1007/JHEP09(2019)101
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DOI: https://doi.org/10.1007/JHEP09(2019)101