I present calculations of two-loop vertex corrections with massive and
massless partons in the eikonal approximation. I show that the n-loop
result for the UV poles can be given in terms of the one-loop calculation

Kidonakis, Nikolaos

English

arXiv

Apollo (Cambridge)

arXiv:hep-ph/0307145v1  10 Jul 2003TWO-LOOP AND N-LOOP EIKONAL VERTEXCORRECTIONS ∗NIKOLAOS KIDONAKISCavendish Laboratory, University of CambridgeCambridge CB3 0HE, EnglandE-mail: kidonaki@hep.phy.cam.ac.ukI present calculations of two-loop vertex corrections with massive andmassless partons in the eikonal approximation. I show that the n-loopresult for the UV poles can be given in terms of the one-loop calculation.1 IntroductionThe eikonal approximation is valid for emission of soft gluons. The approxima-tion simplifies the usual Feynman rules for the quark propagator and quark-gluon vertex as follows:u¯(p)(−iγµ)i(p/+ k/+m)(p+ k)2 −m2 + iǫ→ u¯(p)γµp/+m2p · k + iǫ= u¯(p)vµv · k + iǫ(1)with k → 0 the gluon momentum, p the quark momentum after emission ofthe gluon, v a dimensionless vector v ∝ p, and I have omitted overall factorsof gs TcF with g2s = 4παs and TcF the generators of SU(3) in the fundamentalrepresentation.The eikonal approximation has numerous phenomenological applications inQCD, including threshold resummations for a variety of QCD processes [1, 2, 3,4, 5, 6]. In these applications we are mainly interested in the ultraviolet (UV)pole structure (in dimensional regularization) of one-loop, two-loop, and higher-loop eikonal vertex corrections. In this talk, I discuss explicit calculations ofone-loop and two-loop eikonal vertex corrections for diagrams with massive andmassless partons and show that the n-loop UV poles are given simply in termsof the one-loop result [7].∗The author’s research has been supported by a Marie Curie Fellowship of the Euro-pean Community programme “Improving Human Research Potential” under contract numberHPMF-CT-2001-01221.122 One-loop calculationsLet us denote by ω(n)ij the kinematics, color-independent, part of the n-loopcorrection to the eikonal vertex with lines i and j. At one loop, the expressionpi + kpikpj − kpjFigure 1: One-loop eikonal vertex correction diagramfor ω(1)ij (see fig. 1) in axial gauge isω(1)ij (vi, vj) ≡ g2s∫dDk(2π)D(−i)k2 + iǫNµν(k)∆i vµiδivi · k + iǫ∆j vνjδjvj · k + iǫ(2)with δ = +1(−1) when k flows in the same (opposite) direction as v, andNµν(k) = gµν −nµkν + kµnνn · k+ n2kµkν(n · k)2, (3)where n is the axial gauge vector. ∆ = +1(−1) for a quark (antiquark) eikonalline, while for a gluon eikonal line ∆ = +i(−i) for a gluon located above (below)the eikonal line.Now let I(1)l denote the contribution to ω(1)ij from the l-th term in the gluonpropagator Nµν(k) (i.e. I(1)1 is the contribution from the gµν term, I(1)2 fromthe nµkν terms, and I(1)3 from the kµkν term). In dimensional regularizationwith ε = 4−D, the UV poles in ω(1)ij for the case of massive quarks, with massm, are given by [1]I(1)UV1 =αsπ1εLβ , I(1)UV2 =αsπ1ε(Li + Lj) , I(1)UV3 = −αsπ1ε, (4)3and thusω(1) UVij = S(1)ij[I(1)UV1 + I(1)UV2 + I(1)UV3]= S(1)ijαsπε[Lβ + Li + Lj − 1]with S(1)ij = ∆i∆jδiδj an overall sign andLβ =1− 2m2/sβ[ln(1− β1 + β)+ πi](5)with β =√1− 4m2/s. The functions Li and Lj depend on the axial gaugevector n and are cancelled when we include the heavy-quark self energies.When vi refers to a massive quark and vj to a massless quark we have [1]I(1)UV1 =αs2π{2ε2−1ε[γE + ln(v2ijs2m2)− ln(4π)]},I(1)UV2 =αs2π{−2ε2+1ε[2Li + γE + ln νj − ln(4π)]},I(1)UV3 = −αsπ1ε, (6)where νa = (va ·n)2/|n|2, vij = vi · vj , and γE is the Euler constant. Note thatthe double poles cancel in the sum over the I(1)’s and we getω(1) UVij (vi, vj) = S(1)ijαsπε[−12ln(v2ijs2m2)+ Li +12ln νj − 1]. (7)Finally, when both vi and vj refer to massless quarks we have [8, 1]I(1) UV1 =αsπ{2ε2−1ε[γE + ln(δiδjvij2)− ln(4π)]},I(1) UV2 =αsπ{−2ε2+1ε[γE +12ln(νiνj)− ln(4π)]},I(1) UV3 = −αsπ1ε. (8)Again, we note that the double poles cancel in the sum over the I(1)’s and wegetω(1) UVij (vi, vj) = S(1)ijαsπε[− ln(δi δjvij2)+12ln(νiνj)− 1]. (9)43 Two-loop and n-loop calculationsFor the two-loop diagram in fig. 2 we havepi + k1 + k2pi + k1pik2pj − k1 − k2pj − k1pjk1Figure 2: Two-loop eikonal vertex correction diagramω(2)ij (vi, vj) = g4s∫dDk1(2π)D(−i)k21 + iǫNµν(k1)∆1i vµiδ1ivi · k1 + iǫ∆1j vνjδ1jvj · k1 + iǫ×∫dDk2(2π)D(−i)k22 + iǫNρσ(k2)∆2i vρiδ2ivi · (k1 + k2) + iǫ∆2j vσjδ2jvj · (k1 + k2) + iǫ. (10)Let I(2)kl denote the contribution to ω(2)ij from the product of the k-th termin the axial-gauge gluon propagator Nµν(k1) with the l-th term in Nρσ(k2).Then we can rewriteω(2)ij (vi, vj) = S(2)ij∑k,l=1,2,3I(2)kl (vi, vj) . (11)When both partons are massive, an explicit calculation of the two-loop diagramgives the following results for the leading UV (1/ε2) poles at two loops:I(2),UV11 =α2sπ21ε2L2β , I(2),UV12 =α2sπ21ε2Lβ (Li + Lj) = I(2),UV21I(2),UV22 =α2sπ21ε2(Li + Lj)2 , I(2),UV13 = −α2sπ21ε2Lβ = I(2),UV31I(2),UV23 = −α2sπ21ε2(Li + Lj) = I(2),UV32 , I(2),UV33 =α2sπ21ε2. (12)5Thenω(2) UVij (vi, vj) = S(2)ijα2sπ21ε2[Lβ + Li + Lj − 1]2+O(1ε), (13)where S(2)ij = ∆1i∆1j∆2i∆2jδ1iδ1jδ2iδ2j . The calculation of the 1/ε terms isgiven in [7].We now note that the leading two-loop UV poles are simply the square ofthe one-loop result since I(2),UVmn = I(1),UVm I(1),UVn . Similar results hold for thecases when one or both partons are massless. For example, the leading polesin I(2),UV23 when one of the partons is massless and the other is massive are(α2s/π2)(1/ε3).We noted that the coefficient of the leading UV pole in ω(2)ij is simply thesquare of the coefficient of the leading UV pole in ω(1)ij . We can show by induc-tion that this generalizes to n loops [7]. Thus the leading UV pole in the n-loopcorrections for massive partons, ω(n)ij , is (αs/π)n(1/εn) [Lβ + Li + Lj − 1]n.Furthermore a similar structure holds for non-leading UV poles [7].References[1] N. Kidonakis and G. Sterman, Phys. Lett. B 387, 867 (1996); Nucl. Phys.B505, 321 (1997).[2] N. Kidonakis, G. Oderda, and G. Sterman, Nucl. Phys. B525, 299 (1998);Nucl. Phys. B531, 365 (1998).[3] E. Laenen, G. Oderda, and G. Sterman, Phys. Lett. B 438, 173 (1998).[4] E. Laenen, G. Sterman, and W. Vogelsang, Phys. Rev. D 63, 114018 (2001).[5] N. Kidonakis, Phys. Rev. D 64, 014009 (2001); Int. J. Mod. Phys. A 15,1245 (2000).[6] N. Kidonakis, Cavendish-HEP-03/02, hep-ph/0303186; in DIS03, hep-ph/0306125.[7] N. Kidonakis, hep-ph/0208056; in DPF2002 hep-ph/0207142.[8] J. Botts and G. Sterman, Nucl. Phys. B325, 62 (1989).

Two loop and N loop eikonal vertex corrections.

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Two loop and N loop eikonal vertex corrections.

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