I present a systematic approach to calculating soft-gluon corrections through
next-to-next-to-next-to-leading order for arbitrary hard-scattering cross
sections. Using a unified approach, master formulas are derived for processes
with both simple and complex color flows. Applications and numerical results
are given for some processes of interest, including charged Higgs production
and top quark production in the Standard Model and beyond.Comment: 4 pages, presented at the International Europhysics Conference on
  High Energy Physics, HEP-EPS 2005, Lisbon, Portugal, July 21-27, 200

Kidonakis, Nikolaos

English

arXiv

I present a systematic approach to calculating soft-gluon corrections through next-to-next-to-nextto-leading order for arbitrary hard-scattering cross sections. Using a unified approach, master formulas are derived for processes with both simple and complex color flows. Applications and numerical results are given for some processes of interest, including charged Higgs production and top quark production in the Standard Model and beyond

DigitalCommons@Kennesaw State University

Kennesaw State UniversityDigitalCommons@Kennesaw State UniversityFaculty Publications12-1-2005Soft-gluon Corrections to Hard-scattering CrossSections Through NNNLONikolaos KidonakisKennesaw State University, nkidonak@kennesaw.eduFollow this and additional works at: https://digitalcommons.kennesaw.edu/facpubsPart of the Physics CommonsThis Article is brought to you for free and open access by DigitalCommons@Kennesaw State University. It has been accepted for inclusion in FacultyPublications by an authorized administrator of DigitalCommons@Kennesaw State University. For more information, please contactdigitalcommons@kennesaw.edu.Recommended CitationKidonakis, Nikolaos, "Soft-gluon Corrections to Hard-scattering Cross Sections Through NNNLO" (2005). Faculty Publications. 3877.https://digitalcommons.kennesaw.edu/facpubs/3877arXiv:hep-ph/0512017v1  1 Dec 2005Soft-gluon corrections to hard-scattering crosssections through NNNLONikolaos Kidonakis∗Kennesaw State University, Physics #1202, Kennesaw, GA 30144-5591, USAE-mail: nkidonak@kennesaw.eduI present a systematic approach to calculating soft-gluon corre tions through next-to-next-to-next-to-leading order for arbitrary hard-scattering cross sections. Using a unified approach, masterformulas are derived for processes with both simple and complex color flows. Applications andnumerical results are given for some processes of interest,including charged Higgs productionand top quark production in the Standard Model and beyond.International Europhysics Conference on High Energy PhysicsJuly 21st - 27th 2005Lisboa, Portugal∗Speaker.PoS(HEP2005)055c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licen. http://pos.sissa.it/Soft-gluon corrections through NNNLO Nikolaos Kidonakis1. Soft-gluon correctionsPhysical cross sections can be calculated in perturbative QCD [1] as convolutions of pertur-bative hard-scattering cross sectionsσ̂ with non-perturbative parton distribution functionsφ . Nearthreshold for the production of a specified system there is restricted phase space for real gluonemission. This results in an incomplete cancellation of infrared divergences between real and vir-tual graphs, which gives rise to large logarithms. If we defines4 = s+t1+t2−∑m2, where theti arestandard kinematical invariants and we sum over the particle masses, thens4 → 0 at threshold andthe soft and collinear corrections take the form of plus distribu ionsDl(s4) ≡ [(lnl(s4/M2))/s4]+with M a relevant hard scale andl ≤ 2n − 1 for the n-th order corrections. The terms withl = 2n− 1 are the leading logarithms (LL), those withl = 2n− 2 are the next-to-leading loga-rithms (NLL), and so on [2, 3]. These corrections exponentiate n moment space. If we definemoments of the cross section byσ̂(N) =∫ ∞0 ds4 e−Ns4/M2 σ̂(s4) then the soft corrections becomeDl(s4) →(−1)l+1l+1 lnl+1 N + · · · We can formally resum these logarithms to all orders inαs by fac-torizing the soft gluons from the hard scattering. To inverta resummed cross section back tomomentum space one needs to employ a prescription to deal with the Landau pole, and this createsunavoidable ambiguities. However if one expands the resummed cross section at any fixed order,no matter how high, then the inversion can be done without resorting to prescriptions [4].At next-to-leading order (NLO) inαs we haveD1(s4) (LL) andD0(s4) (NLL) terms. At next-to-next-to-leading order (NNLO) we haveD3(s4) (LL), D2(s4) (NLL), D1(s4) (NNLL), andD0(s4)(NNNLL) terms. At next-to-next-to-next-to-leading order(NNNLO) we haveD5(s4) (LL), D4(s4)(NLL), D3(s4) (NNLL), D2(s4) (NNNLL), D1(s4) (NNNNLL), andD0(s4) (NNNNNLL) terms.Threshold resummation has by now been applied to a large number of processes, most recentlyto top quark pair hadroproduction [5], charged Higgs production [6], large-QT W [7] and Higgs [8]production, and FCNC top production [9]. The numerical results typically show that the higher-order corrections are sizable and they dramatically decrease the scale dependence.2. NNNLO correctionsA unified expression for the resummed cross section for an arbitr ry process is given by [4, 10]σ̂ res(N) = exp[∑iE fi(Ni)+ Efiscale(µF ,µR)]exp[∑jE ′ f j(N j)]×Tr{H fi f j exp[∫dµ ′µ ′Γ† fi f jS]S fi f j exp[∫dµ ′µ ′Γ fi f jS]}, (2.1)whereµF is the factorization scale andµR is the renormalization scale. The exponentsE fi andE ′ f j resum contributions from incoming and outgoing partons, repectively,H are hard-scatteringmatrices in color space, andS are soft matrices that describe noncollinear soft-gluon emission andwhose evolution is given in terms of the soft anomalous dimension matricesΓS. See Refs. [4, 10]for details.PoS(HEP2005)055055 / 2Soft-gluon corrections through NNNLO Nikolaos KidonakisThe expansion of the resummed cross section to NLO provides us with a master formula forthe NLO soft-gluon correctionsσ̂ (1) = σ Bαs(µ2R)π{c3D1(s4)+ c2D0(s4)+ c1δ (s4)}+αdαs+1s (µ2R)π[Ac D0(s4)+ Tc1 δ (s4)] (2.2)with c3 = ∑i 2Ci−∑ j C j, where for quarksCq =CF = (N2c −1)/(2Nc) and for gluonsCg =CA = Nc.Also c2 = cµ2 +T2, with cµ2 =−∑iCi ln(µ2F/M2) andT2 =−∑i[Ci+2Ci ln(−ti/M2)+Ci ln(M2/s)]−∑ j[B(1)j +C j +C j ln(M2/s)], Ac = tr(H(0)Γ′(1)†S S(0) + H(0)S(0)Γ′(1)S), andc1 = cµ1 + T1, with cµ1 =∑i[Ci ln(−ti/M2)−γ(1)i ] ln(µ2F/M2)+dαs(β0/4) ln(µ2R/M2), whereB(1)q = γ(1)q = 3CF/4 andB(1)g =γ(1)g = β0/4.The master formula for the NNLO soft-gluon corrections isσ̂ (2) = σ Bα2s (µ2R)π212c23 D3(s4)+ σ Bα2s (µ2R)π2{32c3 c2−β04c3 +∑jC jβ08}D2(s4)+αdαs+2s (µ2R)π232c3 AcD2(s4)+ σ Bα2s (µ2R)π2C(2)D1 D1(s4)+αdαs+2s (µ2R)π2{(2c2−β02)Ac + c3Tc1 + Fc}D1(s4)+ σ Bα2s (µ2R)π2C(2)D0 D0(s4)+αdαs+2s (µ2R)π2{[c1−ζ2c3 +β04ln(µ2RM2)+β04ln(M2s)]Ac+(c2−β02)T c1 + Fc ln(M2s)+ Gc}D0(s4) . (2.3)HereC(2)D1 = c3c1 + c22− ζ2c23 − (β0/2)T2 + (β0/4)c3 ln(µ2R/M2)+ c3K/2−∑ j(β0/4)B(1)j , Fc =tr[H(0)(Γ(1)†S )2S(0) + H(0)S(0)(Γ(1)S )2 + 2H(0)Γ(1)†S S(0)Γ(1)S ], andC(2)D0 , Gc can be found in [10] andinvolve two-loop [11] corrections.The master formula for the NNNLO soft-gluon corrections isσ̂ (3) = σ Bα3s (µ2R)π318c33 D5(s4)+ σ Bα3s (µ2R)π3{58c23 c2−52c3 X3}D4(s4)+αdαs+3s (µ2R)π358c23 AcD4(s4)+ σ Bα3s (µ2R)π3{c3 c22 +12c23 c1−ζ2c33 +(β0−4c2)X3 +2c3X2−∑jC jβ 2048}D3(s4)+αdαs+3s (µ2R)π3{12c23 Tc1 +[2c3 c2−β02c3−4X3]Ac + c3Fc}D3(s4)+ σ Bα3s (µ2R)π3{32c3 c2 c1 +12c32−3ζ2c23 c2 +52ζ3c33 +(−3c1 +272ζ2c3)X3+(3c2−β0)X2−32c3X1−∑iCiβ18+∑jC jβ016[β0 ln(µ2RM2)+2K]+∑jβ 2016B′(1)j +∑j332C j β1}D2(s4)+αdαs+3s (µ2R)π3{(32c3 c2−3X3)T c1 +32[c2 + c3 ln(M2s)]Fc +32c3 Gc +12Kc3PoS(HEP2005)055055 / 3Soft-gluon corrections through NNNLO Nikolaos Kidonakis+[32c22 +32c3 c1−3ζ2c23 +3X2+β 204−34β0(c2−c32ln(µ2RM2))−3β08c3 ln(M2s)]Ac}D2(s4)+ · · · (2.4)where for brevity we have omitted theD1(s4) and D0(s4) terms (full expressions are providedin [10]). HereX3 = (β0/12)c3 −∑ j C jβ0/24, X2 = −(β0/4)T2 + (β0/8)c3 ln(µ2R/M2)+ c3K/4−∑ j(β0/8)B(1)j , andX1,Kc3 can be found in [10].The NNNLO soft-gluon corrections for top quark production at the Tevatron in theqq̄ channelwere calculated in [10]. The corrections are small but they dcrease further the scale dependenceof the cross section, which is already small when the NNLO soft corrections (for which full resultsfor both partonic channels and in two different kinematics are available [5]) are taken into account.When the NNNLO soft corrections are included, the cross section varies by only a few percent aswe vary the ratioµ/mt by two orders of magnitude. The theoretical prediction is ingood agreementwith data from the Tevatron [12, 13].For charged Higgs production viabg → tH− the NNNLO soft corrections were found to bequite large [6, 10]; the soft corrections through NNNLO provide an increase of over 80% over theleading order cross section at the LHC for a charged Higgs mass of 1000 GeV.References[1] The QCD/SM Working Group: Summary Report, hep-ph/0403100.[2] N. Kidonakis and G. Sterman, Phys. Lett. B387, 867 (1996); Nucl. Phys. B505, 321 (1997)[hep-ph/9705234]; N. Kidonakis, G. Oderda, and G. Sterman,Nucl. Phys. B531, 365 (1998)[hep-ph/9803241]; N. Kidonakis, Int. J. Mod. Phys. A15 1245 (2000)[hep-ph/9902484].[3] E. Laenen, G. Oderda, and G. Sterman, Phys. Lett. B438, 173 (1998) [hep-ph/9806467]; N.Kidonakis and J.F. Owens, Phys. Rev.D 63, 054019 (2001) [hep-ph/0007268].[4] N. Kidonakis, Int. J. Mod. Phys. A19, 1793 (2004) [hep-ph/0303186]; Mod. Phys. Lett. A19, 405(2004) [hep-ph/0401147]; inDIS 2003, hep-ph/0306125.[5] N. Kidonakis, Phys. Rev. D64, 014009 (2001) [hep-ph/0010002]; Int. J. Mod. Phys. A16, s1A, 363(2001) [hep-ph/0009013]; N. Kidonakis and R. Vogt, Phys. Rev. D 68, 114014 (2003)[hep-ph/0308222]; Eur. Phys. J. C33, s466 (2004) [hep-ph/0309045]; Int. J. Mod. Phys. A20, 3171(2005) [hep-ph/0410367].[6] N. Kidonakis, JHEP05, 011 (2005) [hep-ph/0412422]; in these proceedings, hep-ph/0511235.[7] N. Kidonakis and A. Sabio Vera, JHEP02, 027 (2004) [hep-ph/0311266]; R.J. Gonsalves, N.Kidonakis, and A. Sabio Vera, Phys. Rev. Lett.95, 222001 (2005) [hep-ph/0507317].[8] D. de Florian, A. Kulesza, and W. Vogelsang, hep-ph/0511205.[9] A. Belyaev and N. Kidonakis, Phys. Rev. D65, 037501 (2002) [hep-ph/0102072]; N. Kidonakis andA. Belyaev, JHEP12, 004 (2003) [hep-ph/0310299].[10] N. Kidonakis, hep-ph/0509079; inDPF 2004, hep-ph/0410116; inDIS 2005, hep-ph/0506299.[11] N. Kidonakis, hep-ph/0208056; inDIS 2003, hep-ph/0307145.[12] CDF Coll., hep-ex/0504053; Phys. Rev. D72, 032002 (2005) [hep-ex/0506001].[13] D0 Coll., hep-ex/0504043; Phys. Lett. B626, 35 (2005) [hep-ex/0504058]; hep-ex/0505082.PoS(HEP2005)055055 / 4

Soft-gluon corrections to hard-scattering cross sections through NNNLO

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