We present results for two-loop diagrams with massive quarks in the eikonal
approximation. Explicit expressions are given for the UV poles in dimensional
regularization of several of the required integrals.Comment: 5 pages, 3 figures; presented at DIS 2008, London, England, April
  7-11, 2008; corrections in I_2, I_2^UV, and I_ql^U

Kidonakis, Nikolaos

Stephens, Philip

English

arXiv

We present results for two-loop diagrams with massive quarks in the eikonal approximation. Explicit expressions are given for the UV poles in dimensional regularization of several of the required integrals

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Kennesaw State UniversityDigitalCommons@Kennesaw State UniversityFaculty Publications5-9-2008Status of Eikonal Two-Loop Calculations withMassive QuarksNikolaos KidonakisKennesaw State University, nkidonak@kennesaw.eduPhilip StephensKennesaw State UniversityFollow this and additional works at: https://digitalcommons.kennesaw.edu/facpubsPart of the Atomic, Molecular and Optical Physics Commons, Elementary Particles and Fieldsand String Theory Commons, and the Nuclear CommonsThis Conference Proceeding is brought to you for free and open access by DigitalCommons@Kennesaw State University. It has been accepted forinclusion in Faculty Publications by an authorized administrator of DigitalCommons@Kennesaw State University. For more information, pleasecontact digitalcommons@kennesaw.edu.Recommended CitationKidonakis N; Stephens P. 2008. Status of eikonal two-loop calculations with massive quarks. Deep Inelastic Scattering, DIS 2008.arXiv:0805.1193v3  [hep-ph]  1 Apr 2009Status of Eikonal Two-Loop Calculationswith Massive QuarksNikolaos Kidonakis1 and Philip StephensKennesaw State University, Physics #12021000 Chastain Rd., Kennesaw, GA 30144-5591AbstractWe present results for two-loop diagrams with massive quarks in the eikonal approxi-mation. Explicit expressions are given for the UV poles in dimensional regularization ofseveral of the required integrals.1 IntroductionThe calculation of threshold corrections to hard scattering cross sections beyond leading log-arithms requires the calculation of loop diagrams in the eikonal approximation [1]. One-loopcalculations have been performed for all 2 → 2 partonic processes in heavy quark [2] and jet[3] production. The soft anomalous dimension matrix ΓS at one-loop allows the resummationof soft-gluon corrections at next-to-leading logarithm (NLL) accuracy [2]. The exponentiationfollows from the renormalization group evolution of ΓS and involves the calculation of the ul-traviolet (UV) poles in dimensional regularization of one-loop diagrams with eikonal lines. Toextend resummation to next-to-next-to-leading logarithms (NNLL) two-loop calculations arerequired. For massless quark-antiquark scattering the two-loop ΓS was completed in [4]. Forheavy quark production, however, the result is not known. In this contribution we presentseveral results for two-loop diagrams involved in the calculation of the two-loop ΓS for massivequarks. In the eikonal approximation the usual Feynman rules are simplified by letting thep+ k pk → 0pi + kpikpj − kpjFigure 1: The eikonal approximation (left) and a one-loop diagram (right).1Presented at DIS 2008, London, England, April 7-11, 2008gluon momentum approach zero (left diagram in Fig. 1):u¯(p) (−igsTcF ) γµ i(p/+ k/+m)(p+ k)2 −m2 + iǫ→ u¯(p) gsTcF γµ p/+m2p · k + iǫ= u¯(p) gsTcFvµv · k + iǫwith p ∝ v, and T cF the generators of SU(3).2 One-loop and two-loop diagramsWe perform our calculation for eikonal massive quarks in Feynman gauge using dimensionalregularization with n = 4− ǫ.We begin with the one-loop diagram in Fig. 1. The momentum integral is given byI1l = g2s∫dnk(2π)n(−i)gµνk2vµivi · k(−vνj )(−vj · k).Using Feynman parametrization, followed by integration over k, and after several manipula-tions, we findI1l =αsπ(−1)−1−ǫ/2 25ǫ/2 πǫ/2 Γ(1 +ǫ2)(1 + β2)∫ 10dx x−1+ǫ(1− x)−1−ǫ×{∫ 10dz[4zβ2(1− z) + 1− β2]−1−ǫ2∫ 10dzln [4zβ2(1− z) + 1− β2]4zβ2(1− z) + 1− β2+O(ǫ2)}where β =√1− 4m2/s. The integral over x contains both UV and infrared (IR) singularities.We isolate the UV singularities,∫ 10 dx x−1+ǫ (1 − x)−1−ǫ = 1ǫ+ IR, and find the UV pole andconstant terms at one loop:IUV1l =αsπ(1 + β2)2β{1ǫln(1− β1 + β)+12(4 ln 2 + ln π − γE − iπ) ln(1− β1 + β)+14ln2(1 + β)−14ln2(1− β)−12Li2(1 + β2)+12Li2(1− β2)}.More details on this one-loop integral are given in [5]. We now continue with the two-loopdiagrams (these are the eikonal versions of the diagrams involved in the calculation of the two-loop heavy quark form factor [6]). In Fig. 2, we show a diagram with two gluons exchangedbetween the massive quarks (left) and the crossed diagram (right). We denote by I1 the integralfor the first diagram and by I2 that for the crossed diagram. We haveI1 = g4s∫dnk1(2π)ndnk2(2π)n(−i)gµνk21(−i)gρσk22vµivi · k1vρivi · (k1 + k2)(−vνj )−vj · k1(−vσj )−vj · (k1 + k2).We note that I1 is symmetric under k1 ↔ k2 as is the integral for the crossed diagram, I2.Utilizing the properties of these two integrals and the one-loop integral, I1l, we find the relationI1 =12(I1l)2− I2 .2pi + k1 + k2pi + k1pik2pj − k1 − k2pj − k1pjk1pi + k1 + k2pi + k1pik2pj − k1 − k2pj − k2pjk1Figure 2: Two loop diagrams with two-gluon exchanges.Therefore I1 is determined once we calculate I2. For the crossed diagram, we haveI2 = g4s∫dnk1(2π)ndnk2(2π)n(−i)gµνk21(−i)gρσk22vµivi · k1vρivi · (k1 + k2)(−vνj )−vj · (k1 + k2)(−vσj )−vj · k2.We begin with the k2 integral and after some work findI2 = −iα2sπ22−4+ǫ π−2+3ǫ/2 Γ(1−ǫ2)Γ(1 + ǫ) (1 + β2)2∫ 10dz×∫ 10dy (1− y)−ǫ[2β2(1− y)2z2 − 2β2(1− y)z − (1−β2)2]1−ǫ/2∫ dnk1k21 vi · k1 [((vi − vj)z + vj) · k1]1+ǫ .Now we proceed with the k1 integral and separate the UV and IR poles. After many steps, wefind the 1/ǫ2 and 1/ǫ UV poles of I2:IUV2 = −α2sπ2(1 + β2)28β21ǫ{ln(1− β1 + β) [2 Li2(2β1 + β)+ 4Li2(1− β1 + β)+ 2Li2(−(1− β)1 + β)− ln(1 + β) ln(1− β)− ζ2]− 2 ln2(1− β1 + β)ln(1 + β2β)+13ln3(1− β)−13ln3(1 + β)− Li3((1− β)2(1 + β)2)+ ζ3}.We now proceed with the diagrams in Fig. 3 that involve internal quark and gluon loops. Forthe quark loop we findIql = (−1)nfg4s∫ dnk(2π)ndnl(2π)nvµivi · k(−vρj )(−vj · k)(−i)gµνk2(−i)gρσk2Tr[−iγνil/l2(−i)γσi(l/− k/)(l − k)2].After many steps (see also [5]) we extract the UV polesIUVql = −nfα2sπ2(1 + β2)6β{1ǫ2ln(1− β1 + β)+1ǫ[−Li2(1 + β2)+ Li2(1− β2)+12ln2(1 + β)−12ln2(1− β) +(56+ 4 ln 2 + ln π − γE − iπ)ln(1− β1 + β)]}.3pi + kpikll − kkpj − kpjpi + kpikk − llkpj − kpjFigure 3: Two-loop diagrams with quark and gluon loops.For the gluon-loop integral, we haveIgl =12g4s∫dnk(2π)ndnl(2π)nvµivi · k(−vνj )(−vj · k)(−i)gµµ′k2(−i)gρρ′l2(−i)gσσ′(k − l)2(−i)gνν′k2×[gµ′ρ(k + l)σ + gρσ(k − 2l)µ′+ gσµ′(−2k + l)ρ]×[gρ′ν′(l + k)σ′+ gν′σ′(−2k + l)ρ′+ gσ′ρ′(k − 2l)ν′].We calculate the UV poles and findIUVgl = −1996α2sπ2(1 + β2)β{1ǫ2ln(1− β1 + β)+1ǫ[−Li2(1 + β2)+ Li2(1− β2)+12ln2(1 + β)−12ln2(1− β) +(5857+ 4 ln 2 + ln π − γE − iπ)ln(1− β1 + β)]}.We also have to add a diagram to those in Fig. 3 involving a ghost loop. The correspondingintegral isIgh = (−1)g4s∫dnk(2π)ndnl(2π)nvµivi · k(−vρj )(−vj · k)il2lνi(l − k)2(l − k)σ(−i)gµνk2(−i)gρσk2and a calculation of its UV poles givesIUVgh = −α2sπ2(1 + β2)96β{1ǫ2ln(1− β1 + β)+1ǫ[−Li2(1 + β2)+ Li2(1− β2)+12ln2(1 + β)−12ln2(1− β) +(43+ 4 ln 2 + lnπ − γE − iπ)ln(1− β1 + β)]}.We also note that the integral for another diagram involving an internal gluon loop with afour-gluon vertex vanishes.There are additional diagrams not discussed here, also including self-energies and countert-erms. The color factors for all diagrams have been calculated and must be accounted for in thefinal result.4AcknowledgementsThe work of N.K. was supported by the National Science Foundation under Grant No. PHY0555372.References[1] Slides:http://indico.cern.ch/contributionDisplay.py?contribId=200&sessionId=13&confId=24657[2] N. Kidonakis and G. Sterman, Phys. Lett. B387, 867 (1996); Nucl. Phys. B505 321 (1997)[hep-ph/9705234].[3] N. Kidonakis, G. Oderda and G. Sterman, Nucl. Phys. B531, 365 (1998) [hep-ph/9803241].[4] S.M. Aybat, L.J. Dixon and G. Sterman, Phys. Rev. Lett. 97, 072001 (2006)[hep-ph/0606254]; Phys. Rev. D74, 074004 (2006) [hep-ph/0607309].[5] N. Kidonakis, A. Sabio Vera and P. Stephens, arXiv:0802.4240 [hep-ph] (2008).[6] W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia andE. Remiddi, Nucl. Phys. B706, 245 (2005) [hep-ph/0406046].5

Status of Eikonal Two-Loop Calculations with Massive Quarks

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Status of Eikonal Two-Loop Calculations with Massive Quarks

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