Abstract
The coupling of a pseudo-scalar Higgs boson to gluons is mediated through a heavy quark loop. In the limit of large quark mass, it is described by an effective Lagrangian that only admits light degrees of freedom. In this effective theory, we compute the three-loop massless QCD corrections to the form factor that describes the coupling of a pseudo-scalar Higgs boson to gluons. Due to the axial anomaly, the pseudo-scalar operator for the gluonic field strength mixes with the divergence of the axial vector current. Working in dimensional regularization and using the ’t Hooft-Veltman prescription for the axial vector current, we compute the three-loop pseudo-scalar form factors for massless quarks and gluons. Using the universal infrared factorization properties, we independently derive the three-loop operator mixing and finite operator renormalisation from the renormalisation group equation for the form factors, thereby confirming recent results in the operator product expansion. The finite part of the three-loop form factor is an important ingredient to the precise prediction of the pseudo-scalar Higgs boson production cross section at hadron colliders. We discuss potential applications and derive the hard matching coefficient in soft-collinear effective theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Altarelli, R.K. Ellis and G. Martinelli, Large perturbative corrections to the Drell-Yan process in QCD, Nucl. Phys. B 157 (1979) 461 [INSPIRE].
R. Hamberg, W.L. van Neerven and T. Matsuura, A complete calculation of the order α 2 s correction to the Drell-Yan K factor, Nucl. Phys. B 359 (1991) 343 [Erratum ibid. B 644 (2002) 403] [INSPIRE].
S. Dawson, Radiative corrections to Higgs boson production, Nucl. Phys. B 359 (1991) 283 [INSPIRE].
A. Djouadi, M. Spira and P.M. Zerwas, Production of Higgs bosons in proton colliders: QCD corrections, Phys. Lett. B 264 (1991) 440.
D. Graudenz, M. Spira and P.M. Zerwas, QCD corrections to Higgs boson production at proton proton colliders, Phys. Rev. Lett. 70 (1993) 1372 [INSPIRE].
M. Spira, A. Djouadi, D. Graudenz and P.M. Zerwas, Higgs boson production at the LHC, Nucl. Phys. B 453 (1995) 17 [hep-ph/9504378] [INSPIRE].
A. Djouadi, M. Spira and P.M. Zerwas, QCD corrections to hadronic Higgs decays, Z. Phys. C 70 (1996) 427 [hep-ph/9511344] [INSPIRE].
M. Spira, QCD effects in Higgs physics, Fortsch. Phys. 46 (1998) 203 [hep-ph/9705337] [INSPIRE].
S. Catani, D. de Florian and M. Grazzini, Higgs production in hadron collisions: soft and virtual QCD corrections at NNLO, JHEP 05 (2001) 025 [hep-ph/0102227] [INSPIRE].
R.V. Harlander and W.B. Kilgore, Next-to-next-to-leading order Higgs production at hadron colliders, Phys. Rev. Lett. 88 (2002) 201801 [hep-ph/0201206] [INSPIRE].
C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
V. Ravindran, J. Smith and W.L. van Neerven, NNLO corrections to the total cross-section for Higgs boson production in hadron hadron collisions, Nucl. Phys. B 665 (2003) 325 [hep-ph/0302135] [INSPIRE].
V. Ravindran, J. Smith and W.L. van Neerven, Two-loop corrections to Higgs boson production, Nucl. Phys. B 704 (2005) 332 [hep-ph/0408315] [INSPIRE].
R. Harlander and P. Kant, Higgs production and decay: analytic results at next-to-leading order QCD, JHEP 12 (2005) 015 [hep-ph/0509189] [INSPIRE].
C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Higgs boson gluon-fusion production in QCD at three loops, Phys. Rev. Lett. 114 (2015) 212001 [arXiv:1503.06056] [INSPIRE].
V.V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics, Sov. Phys. JETP 3 (1956) 65 [INSPIRE].
J.C. Collins, Algorithm to compute corrections to the Sudakov form-factor, Phys. Rev. D 22 (1980) 1478 [INSPIRE].
A. Sen, Asymptotic behavior of the Sudakov form-factor in QCD, Phys. Rev. D 24 (1981) 3281 [INSPIRE].
L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [INSPIRE].
S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form-factors, Phys. Lett. B 625 (2005) 245 [hep-ph/0508055] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [arXiv:0901.0722] [INSPIRE].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].
O. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multi-leg scattering, arXiv:1507.00047 [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, Higher-order corrections in threshold resummation, Nucl. Phys. B 726 (2005) 317 [hep-ph/0506288] [INSPIRE].
E. Laenen and L. Magnea, Threshold resummation for electroweak annihilation from DIS data, Phys. Lett. B 632 (2006) 270 [hep-ph/0508284] [INSPIRE].
A. Vogt, S. Moch and J.A.M. Vermaseren, The three-loop splitting functions in QCD: the singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The three loop splitting functions in QCD: the nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519] [INSPIRE].
T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [arXiv:1004.3653] [INSPIRE].
T. Gehrmann, J.M. Henn and T. Huber, The three-loop form factor in N = 4 super Yang-Mills, JHEP 03 (2012) 101 [arXiv:1112.4524] [INSPIRE].
T. Gehrmann and D. Kara, The \( Hb\overline{b} \) form factor to three loops in QCD, JHEP 09 (2014) 174 [arXiv:1407.8114] [INSPIRE].
T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, The quark and gluon form factors to three loops in QCD through to O(ϵ 2), JHEP 11 (2010) 102 [arXiv:1010.4478] [INSPIRE].
F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP 07 (2004) 046 [hep-ph/0404258] [INSPIRE].
A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
C. Studerus, Reduze-Feynman integral reduction in C++, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546] [INSPIRE].
A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
T. Gehrmann, T. Huber and D. Maître, Two-loop quark and gluon form-factors in dimensional regularisation, Phys. Lett. B 622 (2005) 295 [hep-ph/0507061] [INSPIRE].
T. Gehrmann, G. Heinrich, T. Huber and C. Studerus, Master integrals for massless three-loop form-factors: one-loop and two-loop insertions, Phys. Lett. B 640 (2006) 252 [hep-ph/0607185] [INSPIRE].
G. Heinrich, T. Huber and D. Maître, Master integrals for fermionic contributions to massless three-loop form-factors, Phys. Lett. B 662 (2008) 344 [arXiv:0711.3590] [INSPIRE].
G. Heinrich, T. Huber, D.A. Kosower and V.A. Smirnov, Nine-propagator master integrals for massless three-loop form factors, Phys. Lett. B 678 (2009) 359 [arXiv:0902.3512] [INSPIRE].
R.N. Lee, A.V. Smirnov and V.A. Smirnov, Analytic results for massless three-loop form factors, JHEP 04 (2010) 020 [arXiv:1001.2887] [INSPIRE].
T. Ahmed, G. Das, P. Mathews, N. Rana and V. Ravindran, Spin-2 form factors at three loop in QCD, arXiv:1508.05043 [INSPIRE].
P. Fayet, Supergauge invariant extension of the Higgs mechanism and a model for the electron and its neutrino, Nucl. Phys. B 90 (1975) 104 [INSPIRE].
P. Fayet, Supersymmetry and weak, electromagnetic and strong interactions, Phys. Lett. B 64 (1976) 159 [INSPIRE].
P. Fayet, Spontaneously broken supersymmetric theories of weak, electromagnetic and strong interactions, Phys. Lett. B 69 (1977) 489 [INSPIRE].
S. Dimopoulos and H. Georgi, Softly broken supersymmetry and SU(5), Nucl. Phys. B 193 (1981) 150 [INSPIRE].
N. Sakai, Naturalness in supersymmetric guts, Z. Phys. C 11 (1981) 153 [INSPIRE].
K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Aspects of grand unified models with softly broken supersymmetry, Prog. Theor. Phys. 68 (1982) 927 [Erratum ibid. 70 (1983) 330] [INSPIRE].
K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Renormalization of supersymmetry breaking parameters revisited, Prog. Theor. Phys. 71 (1984) 413 [INSPIRE].
K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Low-energy parameters and particle masses in a supersymmetric grand unified model, Prog. Theor. Phys. 67 (1982) 1889 [INSPIRE].
ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
J.R. Ellis, M.K. Gaillard and D.V. Nanopoulos, A phenomenological profile of the Higgs boson, Nucl. Phys. B 106 (1976) 292 [INSPIRE].
M.A. Shifman, A.I. Vainshtein, M.B. Voloshin and V.I. Zakharov, Low-energy theorems for Higgs boson couplings to photons, Sov. J. Nucl. Phys. 30 (1979) 711 [INSPIRE].
B.A. Kniehl and M. Spira, Low-energy theorems in Higgs physics, Z. Phys. C 69 (1995) 77 [hep-ph/9505225] [INSPIRE].
R.P. Kauffman and W. Schaffer, QCD corrections to production of Higgs pseudoscalars, Phys. Rev. D 49 (1994) 551 [hep-ph/9305279] [INSPIRE].
A. Djouadi, M. Spira and P.M. Zerwas, Two photon decay widths of Higgs particles, Phys. Lett. B 311 (1993) 255 [hep-ph/9305335] [INSPIRE].
R.V. Harlander and W.B. Kilgore, Production of a pseudoscalar Higgs boson at hadron colliders at next-to-next-to leading order, JHEP 10 (2002) 017 [hep-ph/0208096] [INSPIRE].
C. Anastasiou and K. Melnikov, Pseudoscalar Higgs boson production at hadron colliders in NNLO QCD, Phys. Rev. D 67 (2003) 037501 [hep-ph/0208115] [INSPIRE].
K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser and W.A. Bardeen, Effective QCD interactions of CP odd Higgs bosons at three loops, Nucl. Phys. B 535 (1998) 3 [hep-ph/9807241] [INSPIRE].
M. Spira, A. Djouadi, D. Graudenz and P.M. Zerwas, SUSY Higgs production at proton colliders, Phys. Lett. B 318 (1993) 347 [INSPIRE].
S. Catani, D. de Florian, M. Grazzini and P. Nason, Soft gluon resummation for Higgs boson production at hadron colliders, JHEP 07 (2003) 028 [hep-ph/0306211] [INSPIRE].
S. Moch and A. Vogt, Higher-order soft corrections to lepton pair and Higgs boson production, Phys. Lett. B 631 (2005) 48 [hep-ph/0508265] [INSPIRE].
V. Ravindran, On Sudakov and soft resummations in QCD, Nucl. Phys. B 746 (2006) 58 [hep-ph/0512249] [INSPIRE].
V. Ravindran, Higher-order threshold effects to inclusive processes in QCD, Nucl. Phys. B 752 (2006) 173 [hep-ph/0603041] [INSPIRE].
A. Idilbi, X.-d. Ji, J.-P. Ma and F. Yuan, Threshold resummation for Higgs production in effective field theory, Phys. Rev. D 73 (2006) 077501 [hep-ph/0509294] [INSPIRE].
V. Ahrens, T. Becher, M. Neubert and L.L. Yang, Renormalization-group improved prediction for Higgs production at hadron colliders, Eur. Phys. J. C 62 (2009) 333 [arXiv:0809.4283] [INSPIRE].
D. de Florian and M. Grazzini, Higgs production through gluon fusion: updated cross sections at the Tevatron and the LHC, Phys. Lett. B 674 (2009) 291 [arXiv:0901.2427] [INSPIRE].
M. Bonvini and S. Marzani, Resummed Higgs cross section at N 3 LL, JHEP 09 (2014) 007 [arXiv:1405.3654] [INSPIRE].
S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, Threshold resummation at N 3 LL accuracy and soft-virtual cross sections at N 3 LO, Nucl. Phys. B 888 (2014) 75 [arXiv:1405.4827] [INSPIRE].
D. de Florian and J. Zurita, Soft-gluon resummation for pseudoscalar Higgs boson production at hadron colliders, Phys. Lett. B 659 (2008) 813 [arXiv:0711.1916] [INSPIRE].
T. Schmidt and M. Spira, Higgs boson production via gluon fusion: soft-gluon resummation including mass effects, arXiv:1509.00195 [INSPIRE].
T. Ahmed, M.C. Kumar, P. Mathews, N. Rana and V. Ravindran, Pseudo-scalar Higgs boson production at threshold N 3 LO and N 3 LL QCD, arXiv:1510.02235 [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
S.A. Larin, The renormalization of the axial anomaly in dimensional regularization, Phys. Lett. B 303 (1993) 113 [hep-ph/9302240] [INSPIRE].
S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].
O.V. Tarasov, A.A. Vladimirov and A.Yu. Zharkov, The Gell-Mann-Low function of QCD in the three loop approximation, Phys. Lett. B 93 (1980) 429 [INSPIRE].
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279.
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
R.N. Lee, Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals, JHEP 07 (2008) 031 [arXiv:0804.3008] [INSPIRE].
D.A. Akyeampong and R. Delbourgo, Dimensional regularization, abnormal amplitudes and anomalies, Nuovo Cim. A 17 (1973) 578 [INSPIRE].
J. Kodaira, QCD higher order effects in polarized electroproduction: flavor singlet coefficient functions, Nucl. Phys. B 165 (1980) 129 [INSPIRE].
S.L. Adler and W.A. Bardeen, Absence of higher order corrections in the anomalous axial vector divergence equation, Phys. Rev. 182 (1969) 1517 [INSPIRE].
M.F. Zoller, OPE of the pseudoscalar gluonium correlator in massless QCD to three-loop order, JHEP 07 (2013) 040 [arXiv:1304.2232] [INSPIRE].
G.F. Sterman and S. Weinberg, Jets from quantum chromodynamics, Phys. Rev. Lett. 39 (1977) 1436 [INSPIRE].
A.H. Mueller, On the asymptotic behavior of the Sudakov form-factor, Phys. Rev. D 20 (1979) 2037 [INSPIRE].
S. Catani and L. Trentadue, Comment on QCD exponentiation at large x, Nucl. Phys. B 353 (1991) 183 [INSPIRE].
A. Vogt, Next-to-next-to-leading logarithmic threshold resummation for deep inelastic scattering and the Drell-Yan process, Phys. Lett. B 497 (2001) 228 [hep-ph/0010146] [INSPIRE].
T. Ahmed, N. Rana and V. Ravindran, Higgs boson production through \( b\overline{b} \) annihilation at threshold in N 3 LO QCD, JHEP 10 (2014) 139 [arXiv:1408.0787] [INSPIRE].
A.V. Kotikov and L.N. Lipatov, On the highest transcendentality in N = 4 SUSY, Nucl. Phys. B 769 (2007) 217 [hep-th/0611204] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [Erratum ibid. B 632 (2006) 754] [hep-th/0404092] [INSPIRE].
A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL evolution equations in the N = 4 supersymmetric gauge theory, hep-ph/0112346 [INSPIRE].
R.N. Lee and V.A. Smirnov, Analytic ϵ-expansions of master integrals corresponding to massless three-loop form factors and three-loop g − 2 up to four-loop transcendentality weight, JHEP 02 (2011) 102 [arXiv:1010.1334] [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].
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].
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].
G.P. Korchemsky and A.V. Radyushkin, Loop space formalism and renormalization group for the infrared asymptotics of QCD, Phys. Lett. B 171 (1986) 459 [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].
G.P. Korchemsky, Double logarithmic asymptotics in QCD, Phys. Lett. B 217 (1989) 330 [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. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog et al., Higgs boson gluon-fusion production at threshold in N 3 LO QCD, Phys. Lett. B 737 (2014) 325 [arXiv:1403.4616] [INSPIRE].
C. Anastasiou et al., Higgs boson gluon-fusion production beyond threshold in N 3 LO QCD, JHEP 03 (2015) 091 [arXiv:1411.3584] [INSPIRE].
T. Ahmed, M. Mahakhud, N. Rana and V. Ravindran, Drell-Yan production at threshold to third order in QCD, Phys. Rev. Lett. 113 (2014) 112002 [arXiv:1404.0366] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1510.01715
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ahmed, T., Gehrmann, T., Mathews, P. et al. Pseudo-scalar form factors at three loops in QCD. J. High Energ. Phys. 2015, 169 (2015). https://doi.org/10.1007/JHEP11(2015)169
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2015)169