# A subleading power operator basis for the scalar quark current

## Abstract

Factorization theorems play a crucial role in our understanding of the strong interaction. For collider processes they are typically formulated at leading power and much less is known about power corrections in the λ ≪ 1 expansion. Here we present a complete basis of power suppressed operators for a scalar quark current at \( \mathcal{O}\left({\lambda}^2\right) \) in the amplitude level power expansion in the Soft Collinear Effective Theory, demonstrating that helicity selection rules significantly simplify the construction. This basis applies for the production of any color singlet scalar in \( q\overline{q} \) annihilation (such as \( b\overline{b}\to H \)). We also classify all operators which contribute to the cross section at \( \mathcal{O}\left({\lambda}^2\right) \) and perform matching calculations to determine their tree level Wilson coefficients. These results can be exploited to study power corrections in both resummed and fixed order perturbation theory, and for analyzing the factorization properties of gauge theory amplitudes and cross sections at subleading power.

## Keywords

Effective Field Theories Perturbative QCD## Notes

### **Open Access**

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

- [1]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]. - [2]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]. - [3]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]. - [4]C.W. Bauer and I.W. Stewart,
*Invariant operators in collinear effective theory*,*Phys. Lett.***B 516**(2001) 134 [hep-ph/0107001] [INSPIRE]. - [5]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]. - [6]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]. - [7]A.J. Larkoski, D. Neill and I.W. Stewart,
*Soft theorems from effective field theory*,*JHEP***06**(2015) 077 [arXiv:1412.3108] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [8]K.S.M. Lee and I.W. Stewart,
*Factorization for power corrections to B*→*X*_{s}*γ and*\( B\to {X}_u\ell {\overline{\nu}}_{\ell } \),*Nucl. Phys.***B 721**(2005) 325 [hep-ph/0409045] [INSPIRE]. - [9]M. Beneke, F. Campanario, T. Mannel and B.D. Pecjak,
*Power corrections to*\( \overline{B}\to {X}_u\ell \overline{\nu} \)*(X*_{s}*γ) decay spectra in the ‘shape-function’ region*,*JHEP***06**(2005) 071 [hep-ph/0411395] [INSPIRE]. - [10]R.J. Hill, T. Becher, S.J. Lee and M. Neubert,
*Sudakov resummation for subleading SCET currents and heavy-to-light form-factors*,*JHEP***07**(2004) 081 [hep-ph/0404217] [INSPIRE]. - [11]S.W. Bosch, M. Neubert and G. Paz,
*Subleading shape functions in inclusive B decays*,*JHEP***11**(2004) 073 [hep-ph/0409115] [INSPIRE]. - [12]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]. - [13]G. Paz,
*Subleading jet functions in inclusive B decays*,*JHEP***06**(2009) 083 [arXiv:0903.3377] [INSPIRE].ADSCrossRefGoogle Scholar - [14]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]. - [15]S.M. Freedman,
*Subleading corrections to thrust using effective field theory*, arXiv:1303.1558 [INSPIRE]. - [16]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]. - [17]D.W. Kolodrubetz, I. Moult and I.W. Stewart,
*Building blocks for subleading helicity operators*,*JHEP***05**(2016) 139 [arXiv:1601.02607] [INSPIRE].ADSCrossRefGoogle Scholar - [18]I. Moult, L. Rothen, I.W. Stewart, F.J. Tackmann and H.X. Zhu,
*Subleading power corrections for N -jettiness subtractions*,*Phys. Rev.***D 95**(2017) 074023 [arXiv:1612.00450] [INSPIRE]. - [19]I. Feige, D.W. Kolodrubetz, I. Moult and I.W. Stewart,
*A complete basis of helicity operators for subleading factorization*,*JHEP***11**(2017) 142 [arXiv:1703.03411] [INSPIRE].ADSCrossRefGoogle Scholar - [20]R. Goerke and M. Inglis-Whalen,
*Renormalization of dijet operators at order*1*/Q*^{2}*in soft-collinear effective theory*, arXiv:1711.09147 [INSPIRE]. - [21]I. Moult, I.W. Stewart and G. Vita,
*A subleading operator basis and matching for gg*→*H*,*JHEP***07**(2017) 067 [arXiv:1703.03408] [INSPIRE].ADSCrossRefGoogle Scholar - [22]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].ADSCrossRefGoogle Scholar - [23]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]. - [24]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].ADSCrossRefzbMATHGoogle Scholar - [25]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]. - [26]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]. - [27]I.W. Stewart, F.J. Tackmann and W.J. Waalewijn,
*N -jettiness: an inclusive event shape to veto jets*,*Phys. Rev. Lett.***105**(2010) 092002 [arXiv:1004.2489] [INSPIRE]. - [28]J.R. Gaunt,
*Glauber gluons and multiple parton interactions*,*JHEP***07**(2014) 110 [arXiv:1405.2080] [INSPIRE].ADSCrossRefGoogle Scholar - [29]I.Z. Rothstein and I.W. Stewart,
*An effective field theory for forward scattering and factorization violation*,*JHEP***08**(2016) 025 [arXiv:1601.04695] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [30]M. Zeng,
*Drell-Yan process with jet vetoes: breaking of generalized factorization*,*JHEP***10**(2015) 189 [arXiv:1507.01652] [INSPIRE].ADSCrossRefGoogle Scholar - [31]S. Fleming, A.K. Leibovich and T. Mehen,
*Resummation of large endpoint corrections to color-octet J/ψ photoproduction*,*Phys. Rev.***D 74**(2006) 114004 [hep-ph/0607121] [INSPIRE]. - [32]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]. - [33]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]. - [34]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]. - [35]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]. - [36]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]. - [37]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]. - [38]S. Catani and M. Grazzini,
*An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC*,*Phys. Rev. Lett.***98**(2007) 222002 [hep-ph/0703012] [INSPIRE]. - [39]R. Boughezal, C. Focke, W. Giele, X. Liu and F. Petriello,
*Higgs boson production in association with a jet at NNLO using jettiness subtraction*,*Phys. Lett.***B 748**(2015) 5 [arXiv:1505.03893] [INSPIRE]. - [40]J. Gaunt, M. Stahlhofen, F.J. Tackmann and J.R. Walsh,
*N -jettiness subtractions for NNLO QCD calculations*,*JHEP***09**(2015) 058 [arXiv:1505.04794] [INSPIRE].CrossRefGoogle Scholar - [41]S. Catani, L. Cieri, G. Ferrera, D. de Florian and M. Grazzini,
*Vector boson production at hadron colliders: a fully exclusive QCD calculation at NNLO*,*Phys. Rev. Lett.***103**(2009) 082001 [arXiv:0903.2120] [INSPIRE]. - [42]G. Ferrera, M. Grazzini and F. Tramontano,
*Associated W H production at hadron colliders: a fully exclusive QCD calculation at NNLO*,*Phys. Rev. Lett.***107**(2011) 152003 [arXiv:1107.1164] [INSPIRE].ADSCrossRefGoogle Scholar - [43]S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini,
*Diphoton production at hadron colliders: a fully-differential QCD calculation at NNLO*,*Phys. Rev. Lett.***108**(2012) 072001 [*Erratum ibid.***117**(2016) 089901] [arXiv:1110.2375] [INSPIRE]. - [44]M. Grazzini, S. Kallweit, D. Rathlev and A. Torre,
*Zγ production at hadron colliders in NNLO QCD*,*Phys. Lett.***B 731**(2014) 204 [arXiv:1309.7000] [INSPIRE]. - [45]F. Cascioli et al.,
*ZZ production at hadron colliders in NNLO QCD*,*Phys. Lett.***B 735**(2014) 311 [arXiv:1405.2219] [INSPIRE]. - [46]G. Ferrera, M. Grazzini and F. Tramontano,
*Associated ZH production at hadron colliders: the fully differential NNLO QCD calculation*,*Phys. Lett.***B 740**(2015) 51 [arXiv:1407.4747] [INSPIRE]. - [47]T. Gehrmann et al.,
*W*^{+}*W*^{−}*production at hadron colliders in next to next to leading order QCD*,*Phys. Rev. Lett.***113**(2014) 212001 [arXiv:1408.5243] [INSPIRE]. - [48]M. Grazzini, S. Kallweit and D. Rathlev,
*W γ and Zγ production at the LHC in NNLO QCD*,*JHEP***07**(2015) 085 [arXiv:1504.01330] [INSPIRE]. - [49]M. Grazzini, S. Kallweit and D. Rathlev,
*ZZ production at the LHC: fiducial cross sections and distributions in NNLO QCD*,*Phys. Lett.***B 750**(2015) 407 [arXiv:1507.06257] [INSPIRE]. - [50]J.M. Campbell, R.K. Ellis, Y. Li and C. Williams,
*Predictions for diphoton production at the LHC through NNLO in QCD*,*JHEP***07**(2016) 148 [arXiv:1603.02663] [INSPIRE].ADSCrossRefGoogle Scholar - [51]R. Boughezal et al.,
*Color singlet production at NNLO in MCFM*,*Eur. Phys. J.***C 77**(2017) 7 [arXiv:1605.08011] [INSPIRE]. - [52]R. Boughezal, C. Focke, X. Liu and F. Petriello,
*W -boson production in association with a jet at next-to-next-to-leading order in perturbative QCD*,*Phys. Rev. Lett.***115**(2015) 062002 [arXiv:1504.02131] [INSPIRE]. - [53]R. Boughezal, X. Liu and F. Petriello,
*Phenomenology of the Z-boson plus jet process at NNLO*,*Phys. Rev.***D 94**(2016) 074015 [arXiv:1602.08140] [INSPIRE]. - [54]R. Boughezal, X. Liu and F. Petriello,
*W -boson plus jet differential distributions at NNLO in QCD*,*Phys. Rev.***D 94**(2016) 113009 [arXiv:1602.06965] [INSPIRE]. - [55]J.M. Campbell, R.K. Ellis and C. Williams,
*Direct photon production at next-to-next-to-leading order*,*Phys. Rev. Lett.***118**(2017) 222001 [arXiv:1612.04333] [INSPIRE]. - [56]M. Grazzini, S. Kallweit and M. Wiesemann,
*Fully differential NNLO computations with MATRIX*, arXiv:1711.06631 [INSPIRE]. - [57]R. Boughezal, X. Liu and F. Petriello,
*Power corrections in the N -jettiness subtraction scheme*,*JHEP***03**(2017) 160 [arXiv:1612.02911] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [58]I. Moult, L. Rothen, I.W. Stewart, F.J. Tackmann and H.X. Zhu,
*N -jettiness subtractions for gg*→*H at subleading power*,*Phys. Rev.***D 97**(2018) 014013 [arXiv:1710.03227] [INSPIRE]. - [59]I.W. Stewart and C.W. Bauer,
*Lectures on the soft-collinear effective theory*, http://ocw.mit.edu/courses/physics/8-851-effective-field-theory-spring-2013/lecture-notes/MIT8_851S13_scetnotes.pdf. - [60]T. Becher, A. Broggio and A. Ferroglia,
*Introduction to soft-collinear effective theory*,*Lect. Notes Phys.***896**(2015) pp.1 [arXiv:1410.1892] [INSPIRE]. - [61]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]. - [62]C. Marcantonini and I.W. Stewart,
*Reparameterization invariant collinear operators*,*Phys. Rev.***D 79**(2009) 065028 [arXiv:0809.1093] [INSPIRE]. - [63]L.J. Dixon,
*Calculating scattering amplitudes efficiently*, in*QCD and beyond. Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics, TASI-*95, Boulder CO U.S.A., 4-30 June 1995, pg. 539 [hep-ph/9601359] [INSPIRE]. - [64]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]. - [65]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]. - [66]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]. - [67]M.J. Dugan and B. Grinstein,
*On the vanishing of evanescent operators*,*Phys. Lett.***B 256**(1991) 239 [INSPIRE]. - [68]S. Herrlich and U. Nierste,
*Evanescent operators, scheme dependences and double insertions*,*Nucl. Phys.***B 455**(1995) 39 [hep-ph/9412375] [INSPIRE]. - [69]F.E. Low,
*Bremsstrahlung of very low-energy quanta in elementary particle collisions*,*Phys. Rev.***110**(1958) 974 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [70]T.H. Burnett and N.M. Kroll,
*Extension of the low soft photon theorem*,*Phys. Rev. Lett.***20**(1968) 86 [INSPIRE].ADSCrossRefGoogle Scholar