Calculation of the quark and gluon form factors to three loops in QCD

We describe the calculation of the three-loop QCD corrections to quark and gluon form factors. The relevant three-loop Feynman diagrams are evaluated and the resulting three-loop Feynman integrals are reduced to a small set of known master integrals by using integration-by-parts relations. Our calculation confirms the recent results by Baikov et al. for the three-loop form factors. In addition, we derive the subleading $$ \mathcal{O}\left( \varepsilon \right) $$ terms for the fermion-loop type contributions to the three-loop form factors which are required for the extraction of the fermionic contributions to the four-loop quark and gluon collinear anomalous dimensions. The finite parts of the form factors are used to determine the hard matching coefficients for the Drell-Yan process and inclusive Higgs-production in soft-collinear effective theory.

[1]  R. Harlander,et al.  Next-to-next-to-leading order Higgs production at Hadron Colliders. , 2002, Physical review letters.

[2]  F. Wilczek Decays of Heavy Vector Mesons into Higgs Particles , 1977 .

[3]  A. Smirnov Algorithm FIRE—Feynman Integral REduction , 2008, 0807.3243.

[4]  Michal Czakon,et al.  Automatized analytic continuation of Mellin-Barnes integrals , 2005, Computer Physics Communications.

[5]  V. Ravindran On Sudakov and soft resummations in QCD , 2005, hep-ph/0512249.

[6]  J. Maldacena,et al.  Comments on operators with large spin , 2007, 0708.0672.

[7]  G. Altarelli,et al.  Large Perturbative Corrections to the Drell-Yan Process in QCD , 1979 .

[8]  C. Bauer,et al.  Soft collinear factorization in effective field theory , 2001, hep-ph/0109045.

[9]  V. Ravindran,et al.  NNLO corrections to the total cross section for Higgs boson production in hadron-hadron collisions , 2003, hep-ph/0302135.

[10]  D. Maitre,et al.  Master integrals for fermionic contributions to massless three-loop form-factors , 2007, 0711.3590.

[11]  E. Gardi,et al.  Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes , 2009, 0901.1091.

[12]  G. Kramer,et al.  Two-jet cross section ine+e− annihilation , 1987 .

[13]  T. Becher,et al.  Infrared singularities of scattering amplitudes in perturbative QCD. , 2009, Physical review letters.

[14]  W. Neerven,et al.  The calculation of the second order soft and virtual contributions to the Drell-Yan cross section , 1989 .

[15]  A. Vogt,et al.  Higher-order corrections in threshold resummation , 2005, hep-ph/0506288.

[16]  C. Studerus,et al.  Reduze - Feynman integral reduction in C++ , 2009, Comput. Phys. Commun..

[17]  S. Laporta,et al.  difference equations , 2001 .

[18]  V. A. Smirnov,et al.  Analytic results for massless three-loop form factors , 2010, 1001.2887.

[19]  J. Frenkel,et al.  Non-abelian eikonal exponentiation , 1984 .

[20]  F. Wilczek,et al.  Ultraviolet Behavior of Non-Abelian Gauge Theories , 1973 .

[21]  Robert Harlander,et al.  Higgs production and decay: analytic results at next-to-leading order QCD , 2005 .

[22]  C. Anastasiou,et al.  Higgs Boson Production at Hadron Colliders in NNLO QCD , 2002, hep-ph/0207004.

[23]  G. Korchemsky Double logarithmic asymptotics in QCD , 1989 .

[24]  S. Catani,et al.  Resummation of the QCD perturbative series for hard processes , 1989 .

[25]  M. Grazzini,et al.  Soft-gluon resummation for Higgs boson production at hadron colliders , 2003, hep-ph/0306211.

[26]  F. Tkachov,et al.  Integration by parts: The algorithm to calculate β-functions in 4 loops , 1981 .

[27]  S. Catani The singular behaviour of QCD amplitudes at two-loop order , 1998, hep-ph/9802439.

[28]  A. Vogt,et al.  Three-loop results for quark and gluon form-factors , 2005, hep-ph/0508055.

[29]  A. Vogt,et al.  The Three-loop splitting functions in QCD: The Singlet case , 2004, hep-ph/0404111.

[30]  Lorenzo Magnea,et al.  Threshold resummation for electroweak annihilation from DIS data , 2005, hep-ph/0508284.

[31]  A. Radyushkin,et al.  Renormalization of the Wilson loops beyond the leading order , 1987 .

[32]  G. Sterman Summation of Large Corrections to Short Distance Hadronic Cross-Sections , 1987 .

[33]  Li Lin Yang,et al.  Renormalization-group improved prediction for Higgs production at hadron colliders , 2008, 0809.4283.

[34]  W. E. Caswell Asymptotic Behavior of Non-Abelian Gauge Theories to Two-Loop Order , 1974 .

[35]  Stefan Bekavac Calculation of massless Feynman integrals using harmonic sums , 2006, Comput. Phys. Commun..

[36]  M. Spira,et al.  QCD corrections to hadronic Higgs decays , 1995 .

[37]  A. Vainshtein,et al.  Remarks on Higgs-Boson Interactions with Nucleons , 1978 .

[38]  M. Beneke,et al.  Soft-collinear effective theory and heavy-to-light currents beyond leading power , 2002, hep-ph/0206152.

[39]  S. Moch,et al.  Higher-order soft corrections to lepton pair and Higgs boson production , 2005 .

[40]  G. Sterman,et al.  Analytic continuation of the Sudakov form factor in QCD. , 1990, Physical review. D, Particles and fields.

[41]  P. Nogueira Automatic Feynman graph generation , 1993 .

[42]  R. Harlander Virtual corrections to /gg-->H to two loops in the heavy top limit , 2000, hep-ph/0007289.

[43]  T. Becher,et al.  Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering , 2006, hep-ph/0607228.

[44]  X. Ji,et al.  Resummation of Threshold Logarithms in Effective Field Theory for DIS, Drell-Yan and Higgs Production , 2006, hep-ph/0605068.

[45]  B. Kniehl,et al.  Decoupling Relations to O(alpha_s^3) and their Connection to Low-Energy Theorems , 1997, hep-ph/9708255.

[46]  T. Huber,et al.  Two-loop quark and gluon form factors in dimensional regularisation , 2005, hep-ph/0507061.

[47]  J. Ellis,et al.  Is the Mass of the Higgs Boson About 10-GeV? , 1979 .

[48]  L. Dixon,et al.  The Two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole , 2006, hep-ph/0607309.

[49]  C. Bauer,et al.  Hard scattering factorization from effective field theory , 2002, hep-ph/0202088.

[50]  C. Bauer,et al.  Summing Sudakov logarithms in B ---> X(s gamma) in effective field theory , 2000, hep-ph/0005275.

[51]  B. A. Kniehl,et al.  Higgs Decay to Order \alpha_s^4 , 1997 .

[52]  F. Tkachov,et al.  The FORM version of MINCER , 1991 .

[53]  F. Tkachov,et al.  Mincer: Program for multiloop calculations in quantum field theory for the Schoonschip system , 1989 .

[54]  G. Sterman,et al.  Multi-loop amplitudes and resummation☆ , 2002, hep-ph/0210130.

[55]  Matthias Steinhauser,et al.  Hadronic Higgs Boson Decay to Order as4 , 1997 .

[56]  Bernd A. Kniehl,et al.  Low-energy theorems in Higgs physics , 1995 .

[57]  C. Anastasiou,et al.  Automatic integral reduction for higher order perturbative calculations , 2004 .

[58]  D. Florian,et al.  Higgs production in hadron collisions: soft and virtual QCD corrections at NNLO , 2001, hep-ph/0102227.

[59]  Tobias Huber,et al.  HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters , 2006, Comput. Phys. Commun..

[60]  David H. Bailey,et al.  Analysis of PSLQ, an integer relation finding algorithm , 1999, Math. Comput..

[61]  D. Jones Two-loop diagrams in Yang-Mills theory , 1974 .

[62]  W. Neerven,et al.  Second order logarithmic corrections to the Drell-Yan cross-section , 1988 .

[63]  D. Graudenz,et al.  MSSM Higgs Boson Production at the LHC , 1997, hep-ph/9703355.

[64]  H. Politzer,et al.  Reliable Perturbative Results for Strong Interactions , 1973 .

[65]  C. Bauer,et al.  Invariant operators in collinear effective theory , 2001, hep-ph/0107001.

[66]  S. Moch,et al.  The third-order QCD corrections to deep-inelastic scattering by photon exchange , 2005, hep-ph/0504242.

[67]  J. Vermaseren,et al.  The three-loop QCD β-function and anomalous dimensions , 1993, hep-ph/9302208.

[68]  T. Huber,et al.  Nine-propagator master integrals for massless three-loop form factors , 2009, 0902.3512.

[69]  A. Vogt,et al.  The quark form factor at higher orders , 2005, hep-ph/0507039.

[70]  T. Becher,et al.  Erratum: On the structure of infrared singularities of gauge-theory amplitudes , 2009, Journal of High Energy Physics.

[71]  O. Tarasov,et al.  Two Loop Renormalization of the {QCD} in an Arbitrary Gauge , 1978 .

[72]  Zerwas,et al.  QCD corrections to Higgs-boson production at proton-proton colliders. , 1993, Physical review letters.

[73]  C. Bauer,et al.  An Effective field theory for collinear and soft gluons: Heavy to light decays , 2000, hep-ph/0011336.

[74]  Yasuhiro Okada,et al.  Effective gauge theory and the effect of heavy quarks in Higgs boson decays , 1983 .

[75]  S. Dawson Radiative corrections to Higgs boson production , 1991 .

[76]  P. Nason,et al.  The Resummation of soft gluons in Hadronic Collisions , 1996 .

[77]  Roelof Hamberg,et al.  A complete calculation of the order αs2 correction to the Drell-Yan K-factor , 1991 .

[78]  G. Korchemsky ASYMPTOTICS OF THE ALTARELLI-PARISI-LIPATOV EVOLUTION KERNELS OF PARTON DISTRIBUTIONS , 1989 .

[79]  A. Buras,et al.  Deep-inelastic scattering beyond the leading order in asymptotically free gauge theories , 1978 .

[80]  A. Vladimirov,et al.  The gell-mann-low function of QCD in the three-loop approximation , 1980 .

[81]  H. Contopanagos,et al.  Sudakov factorization and resummation , 1996, hep-ph/9604313.

[82]  L. Dixon,et al.  Two-loop anomalous-dimension matrix for soft-gluon exchange. , 2006, Physical review letters.

[83]  S. Catani,et al.  Comment of QCD exponentiation at large x , 1991 .

[84]  M. Spira QCD Effects in Higgs Physics , 1997, hep-ph/9705337.

[85]  M. Steinhauser,et al.  Quark and gluon form factors to three loops. , 2009, Physical review letters.

[86]  B. Kniehl,et al.  Decoupling relations to O(a s 3) and their connection to low-energy theorems , 1998 .

[87]  A. Radyushkin,et al.  Loop Space Formalism and Renormalization Group for the Infrared Asymptotics of {QCD} , 1986 .

[88]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[89]  L. Dixon,et al.  On soft singularities at three loops and beyond , 2009, 0910.3653.

[90]  Master integrals for massless three-loop form factors: One-loop and two-loop insertions , 2006, hep-ph/0607185.

[91]  Differential Equations for Two-Loop Four-Point Functions , 1999, hep-ph/9912329.