The resummation of soft gluon exchange for QCD hard scattering requires a
matrix of anomalous dimensions. We compute this matrix directly for arbitrary 2
to n massless processes for the first time at two loops. Using color generator
notation, we show that it is proportional to the one-loop matrix. This result
reproduces all pole terms in dimensional regularization of the explicit
calculations of massless 2 to 2 amplitudes in the literature, and it predicts
all poles at next-to-next-to-leading order in any 2 to n process that has been
computed at next-to-leading order. The proportionality of the one- and two-loop
matrices makes possible the resummation in closed form of the
next-to-next-to-leading logarithms and poles in dimensional regularization for
the 2 to n processes.Comment: 5 pages, 1 figure, revte

B. Jantzen

George Sterman

J. C. Collins

Lance J. Dixon

N. Kidonakis

S. Catani

S. Mert Aybat

S. Moch

English

arXiv

The resummation of soft gluon exchange for QCD hard scattering requires a matrix of anomalous dimensions. We compute this matrix directly for arbitrary 2 {yields} n massless processes for the first time at two loops. Using color generator notation, we show that it is proportional to the one-loop matrix. This result reproduces all pole terms in dimensional regularization of the explicit calculations of massless 2 {yields} 2 amplitudes in the literature, and it predicts all poles at next-to-leading order in any 2 {yields} n process that has been computed at next-to-leading order. The proportionality of the one- and two-loop matrices makes possible the resummation in closed form of the next-to-next-to-leading logarithms and poles in dimensional regularization for the 2 {yields} n processes

Aybat, S.Mert

Dixon, Lance J.

Sterman, George

UNT Digital Library

Work supported  in part by Department of Energy contract DE-AC02-76SF00515YITP-SB-06-25SLAC–PUB–11907The Two-loop Anomalous Dimension Matrix for Soft Gluon ExchangeS. Mert Aybat1, Lance J. Dixon2, George Sterman11C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York 11794–3840, USA, and2 Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA(Dated: June 23, 2006)The resummation of soft gluon exchange for QCD hard scattering requires a matrix of anomalousdimensions. We compute this matrix directly for arbitrary 2 → n massless processes for the firsttime at two loops. Using color generator notation, we show that it is proportional to the one-loop matrix. This result reproduces all pole terms in dimensional regularization of the explicitcalculations of massless 2 → 2 amplitudes in the literature, and it predicts all poles at next-to-next-to-leading order in any 2 → n process that has been computed at next-to-leading order. Theproportionality of the one- and two-loop matrices makes possible the resummation in closed formof the next-to-next-to-leading logarithms and poles in dimensional regularization for the 2 → nprocesses.The calculation of high-energy cross sections in per-turbative quantum chromodynamics (QCD) for hadroniccollisions involves the factorization of long- and short-distance eﬀects. Sensitivity to long-distance dynamicsis enhanced by powers of logarithms whenever there isan incomplete cancellation of parton emission and vir-tual corrections. In such situations, it is useful to or-ganize, or resum, these corrections to all orders in per-turbation theory. Correspondingly, in partonic scatter-ing or production amplitudes, it is necessary to organizepoles in ε that arise in dimensional regularization (withD = 4−2ε). The resummation of these poles and relatedlogarithmic enhancements is well-understood for inclu-sive reactions mediated by electroweak interactions, suchas the Sudakov form factor [1, 2] and in Drell-Yan pro-cesses [3]. With recent advances in the computation ofsplitting functions [4], many such corrections can be re-summed explicitly to next-to-next-to-leading level. Theirstructure at arbitrary level is known to be determined bya handful of anomalous dimensions.The situation for QCD hard scattering processes con-taining four or more partons — critical to understandingmany types of backgrounds to new physics at hadron col-liders [5] — is more complex. Resummation beyond lead-ing logarithms or poles requires a matrix of additionalanomalous dimensions [6–9]. These matrices are foundin turn from the renormalization of the vacuum matrixelements of products of Wilson lines, one for each exter-nal parton in the underlying process [7]. In this paper, weinvestigate the structure of the two-loop anomalous di-mension matrix. We will ﬁnd that, remarkably, for everyhard-scattering process involving only massless partons,this matrix is proportional to the one-loop matrix. Wewill concentrate below on the role that the matrix playsin partonic amplitudes. The full calculation of the two-loop matrix will be given elsewhere [10]. In this paper,we provide the simple calculation that is at the heart ofthe main result. We will show that certain color correla-tions due to two-loop diagrams that couple three Wilsonlines vanish identically. We will also provide an explicitexpression in terms of color generators [11, 12] for allsingle-pole terms in massless 2 → n amplitudes.We consider a general process involving the scatteringof massless partons, which we denote by “f”:f : f1(p1, r1) + f2(p2, r2)→ f3(p3, r3) + · · ·+ fn+2(pn+2, rn+2) . (1)The fi are the ﬂavors of the participating partons, whichcarry momenta {pi} and color {ri}. Adopting the color-state notation of Ref. [12], we represent the amplitudefor this process as |Mf〉.It is convenient to express these amplitudes as vectorswith C elements in the space of color tensors, for somechoice of basis tensors {(cI){ri}} [7, 12, 13],∣∣∣∣Mf(βj ,Q2μ2, αs(μ), ε)〉≡C∑L=1Mf,L(βj ,Q2μ2, αs(μ), ε)(cL){ri}. (2)We will analyze these amplitudes at ﬁxed momenta pifor the participating partons, which we represent aspi = Qβi , β2i = 0 , where the βi are four-velocities,and where Q is an overall momentum scale.In dimensional regularization, on-shell amplitudes maybe factorized into jet, soft and hard functions that de-scribe the dynamics of partons collinear with the exter-nal lines, soft exchanges between those partons, and theshort-distance scattering process, respectively. This fac-torization follows from the general space-time structureof long-distance contributions to elastic processes [6, 14].The general form of the factorized amplitude, for equalfactorization and renormalization scales μ, is [13]∣∣∣∣Mf(βi,Q2μ2, αs(μ), ε)〉=n+2∏i=1J [i] (αs(μ), ε)×Sf(βi,Q2μ2, αs(μ), ε) ∣∣∣∣Hf(βi,Q2μ2, αs(μ))〉, (3)where J [i] is the jet function for external parton i, Sfis the soft function, and Hf is the hard (short-distance)function.Submitted to Physical Review LettersJune 20062The jet function for parton i can be expressed to allorders in terms of three anomalous dimensions, K[i], G[i]and γ[i]K , of which the ﬁrst is determined order-by-orderfrom the third. The general form of the jet function, andits expansion to second order is given by (expanding anyfunction as f(αs) =∑n(αs/π)n f (n)) [2],lnJ [i] (αs(μ), ε) =12∫ μ0dξξ[K[i](αs(μ), ε) + G[i] (−1, α¯s (ξ, ε) , ε) +∫ μξdμ˜μ˜γ[i]K (α¯s (μ˜, ε))](4)= −(αsπ)( 18ε2γ[i](1)K +14εG[i](1)(ε))+(αsπ)2 [β0321ε2(34εγ[i](1)K + G[i](1)(ε))− 18(γ[i](2)K4ε2+G[i](2)(ε)ε)]+ . . . .In the expansion we use the D-dimensional running-coupling, evaluated at one-loop order,α¯s (μ˜, ε) = αs(μ)(μ2μ˜2)ε ∞∑n=0[β04πε((μ2μ˜2)ε− 1)αs(μ)]n, (5)with the one-loop coeﬃcient β0 = 11CA/3− 4TFnF /3. The corresponding expression for the soft matrix isSf(βi · βju0, αs(μ), ε)= P exp[−∫ μ0dμ˜μ˜ΓSf(βi · βju0, α¯s (μ˜, ε))]= 1 +12ε(αsπ)Γ(1)Sf +18ε2(αsπ)2 (Γ(1)Sf)2− β016ε2(αsπ)2Γ(1)Sf +14ε(αsπ)2Γ(2)Sf + . . . , (6)where u0 = μ2/Q2, so that βi · βj/u0 = sij/μ2.Expanding G[i] = G[i]0 + εG[i]′ + . . . , one ﬁnds fromEq. (4) the single pole in ε in the logarithm of the jetfunction at two loops. For the quark case this term is−G[q](2)08+β0 G[q](1)′32= − 38C2F[116− 12ζ(2) + ζ(3)]− 116CACF[961216+114ζ(2)− 132ζ(3)]+116CFTFnF[6554+ ζ(2)], (7)using values of G[q](ε) from ref. [15]. Notice the contri-bution from the running of the ﬁnite term at one loop,which appears as an O(ε) contribution in G[i](1).The one-loop soft anomalous dimension in color-generator form isΓ(1)Sf =12∑i∈f∑j =iTi ·Tj ln(μ2−sij), (8)where sij = (pi + pj)2, with all momenta deﬁned to ﬂowinto (or out of) the amplitude. The Ti are given explic-itly by color generators in the representation of partoni, multiplied by ±1: plus one for an outgoing quark orgluon, or incoming antiquark; minus one for an incomingquark or gluon, or outgoing antiquark. The color gen-erator form for the anomalous dimension matrix is moreﬂexible, but less explicit, than the corresponding matrixexpressions in a chosen basis of color tensors for the am-plitude. An example of the latter for qq¯ → qq¯ scattering,in an s-channel t-channel singlet basis, isΓ(1)Sf =⎛⎝ 1Nc (U − T ) + 2CF S (S − U)(T − U) 1Nc (U − S) + 2CF T⎞⎠ ,(9)where T ≡ ln(−tμ2), and so on for the other Mandelstaminvariants, deﬁned by s = s12, t = s13, u = s14. Re-summed cross sections are determined by the eigenvaluesand eigenvectors of these matrices [8, 16].We are now ready to provide our result for the fulltwo-loop soft anomalous dimension matrix,Γ(2)Sf =K2Γ(1)Sf . (10)Here K = CA(67/18−ζ(2))−10TF nF /9 is the same con-stant appearing in the relation between the one- and two-loop Sudakov, or “cusp” anomalous dimensions [17, 18]:γ[i]K = 2Ci(αs/π)[1 + (αs/π)K/2]. Remarkably, relation-ship (10) holds for an arbitrary 2 → n process, eventhough the two-loop diagrams shown in Fig. 1 apparentlycouple together the color factors of three eikonal (Wilson)lines coherently. We derive Eq. (10) using the color gen-erator formalism; however, the result is completely gen-eral, and applies to explicit matrix representations suchas Eq. (9).Following the method described in detail at one loopin Ref. [7], and extended to two loops in Ref. [10], the3FIG. 1: Two-loop diagrams involving three eikonal lines.two-loop anomalous dimension is found from the residueof single-pole terms in suitable combinations of Wilsonlines computed at two loops. The simplicity of the re-sult (10) follows from the special properties of the di-agrams of Fig. 1, which connect three diﬀerent eikonallines. First consider Fig. 1a, where a three-gluon cou-pling ties together three eikonals labelled vA, vB and vC ,in an otherwise arbitrary eikonal process. We shall provethat this integral is zero, as long as we take the eikonalsvA and vB to be lightlike.Since vA and vB are lightlike, we choose a frame inwhich vμA = δμ− and vμB = δμ+. Components transverseto vA and vB carry the subscript T . The eikonal integralin momentum space is thenF1a(vA, vB , vC) =∫dDk1dDk21k21 + i1k22 + i1(k1 + k2)2 + i1k−1 + i1k+2 + i×[v−C(k+1 − k+2)+ v+C(k−1 − k−2)− vC,T · (k1,T − k2,T ) + v+C (k−1 + 2k−2 )+ v−C (−2k+1 − k+2 ) ]v−C(k+1 + k+2)+ v+C(k−1 + k−2)− vC,T · (k1,T + k2,T ) + i , (11)where the term in square brackets is the three-gluon ver-tex momentum factor. We now introduce a change ofvariables (with unit Jacobean) from momenta kμi to k¯μi ,(k+1 , k−1 , k1,T)=(ζ k¯−2 , ζ−1 k¯+2 , k¯2,T),(k+2 , k−2 , k2,T)=(ζ k¯−1 , ζ−1 k¯+1 , k¯1,T), (12)where ζ = v+C/v−C . It provides an expression identical toF1a(vA, vB , vC), Eq. (11), but of the opposite sign. Theintegral corresponding to Fig. 1a therefore vanishes.Regarding Fig. 1b, the same change of variables yields1/[(vC · (k1 + k2))(vC · k1)] = 1/[(vC · (k¯1 + k¯2))(vC · k¯2)],from which it is easy to show that this diagram reducesto the product of one-loop diagrams, and so does notcontribute to the two-loop anomalous dimension. In-deed, the only nontrivial contributions to Γ(2)Sf at twoloops involve only two eikonal lines. Using results fromrefs. [17, 18], the color structure of these contributionsreduces to that of a single gluon exchange. The sum ofthe diagrams then modiﬁes the one-loop result by thesame multiplicative factor as for the cusp anomalous di-mension, which gives Eq. (10).The explicit expression for single poles in 2 → n am-plitudes is easily found from Eqs. (4) and (6) using theexplicit form of the two-loop matrix (10),∣∣∣M(2)f 〉(single pole) = (13)1ε[∑i∈f(−G[i](2)08+β0 G[i](1)′32)+K8Γ(1)Sf] ∣∣∣M(0)f 〉− 14ε∑i∈fG[i](1)0∣∣∣H(1)f (0)〉−∑i∈f18εγ[i](1)K∣∣∣H(1)f ′(0)〉 .Here we normalize the one-loop hard scattering by ab-sorbing into it all ﬁnite terms from the jet functions, or-der by order, and |H(1)f (0)〉, |H(1)f ′(0)〉 are this functionand its derivative with respect to ε, respectively, evalu-ated at ε = 0. (This absorption is possible to any looporder because the jets are diagonal in color.) Explicitcomparison [10] shows that this simple result agrees withall single-pole terms found at 2 → 2 in the literature, assummarized for example in Refs. [19, 20]. It also predictsall such poles in a 2 → n process, once the one-loop hardpart∣∣∣H(1)f 〉 is known.We remark that an analogous anomalous dimensionmatrix for Wilson lines has been computed in Ref. [18],in the forward limit t → 0. This limit is a singular one,with respect to our arguments regarding Fig. 1; thus ourresults and theirs are not directly comparable.It is also worth remarking on the relationship betweenour results and the inﬂuential alternative formalism ofRef. [12], in which both pole and ﬁnite terms are putinto an exponential form to two loops. We may thinkof these as alternative schemes for organizing infraredpoles. When explicit calculations are organized accordingto the scheme of Ref. [12], more complex color productsare found, namely ifabcTai TbjTck = −[Ti · Tj , Tj · Tk],appearing in the matrix Hˆ(2) at order 1/ε [19, 20]. Suchproducts are not encountered in the resummation schemedescribed above. These diﬀerences in color structure,however, are by no means disagreements. They arisefrom a particular commutator, between the one-loop ﬁ-nite terms that are exponentiated in the formalism ofRef. [12], and the one-loop soft anomalous dimensionmatrix Γ(1)Sf . The result of performing the commuta-4tor [10] agrees with the form of Hˆ(2) in Ref. [19] for2 → 2 processes, and with that proposed in Ref. [21]for 2 gluon → n gluon processes, based on consistencyof collinear limits.Given an explicit two-loop amplitude, the strategy de-scribed here may be reversed, and Γ(2)Sf extracted di-rectly from the amplitude. This approach was adopted inRef. [22] for the case of quark-quark elastic scattering, inthe context of electroweak Sudakov corrections. The orig-inal version of Ref. [22] diﬀers from ours due to omissionof the commutator contribution described above. Theauthors have informed us that a revision is in prepara-tion.Similar remarks apply to the color-diagonal singlepoles given in Eq. (13). These coeﬃcients do not equalthe corresponding coeﬃcients H(2)i in the formalism ofRef. [12], but they are connected [23]. The diﬀerence canbe related precisely to the diﬀerent treatment of ﬁniteterms in the two approaches [10].In addition to clarifying the structure of singular termsin calculations of 2 → 2 processes at two loops, the re-sults outlined here have potentially useful consequencesand suggest further directions of research. Eq. (13) pre-dicts the two-loop pole structure for any 2 → n process,in color-generator form, for any process whose one-loophard function is known to O(ε).Another practical consequence is that, because theone- and two-loop anomalous dimensions are propor-tional, all terms in the expansion of the soft functioncommute to next-to-next-to-leading level (NNLL), andin this approximation, the ordering operator P can bedropped in Eq. (6). Thus, once the color eigenstates ofthe one-loop matrix are known, the same states will diag-onalize the two-loop matrix. A semi-numerical approach,bypassing diagonalization, is to simply exponentiate therelevant matrices in any convenient basis [9]. Given therelation (10), this is now possible at NNLL as well asNLL.The study of these matrices for processes beyond 2 →2, already begun in Ref. [24], is clearly an important chal-lenge. Another intriguing question is whether the propor-tionality (10) might extend beyond two loops, whether inQCD or any of its allied gauge theories. If so, it couldhave consequences for the interpretation of infrared di-verences in the relevant theory. The extension, and/ormodiﬁcation of the results above for the production ofmassive colored particles is another important directionfor research.AcknowledgmentsThis work was supported in part by the NationalScience Foundation, grants PHY-0098527 and PHY-0354776, and by the Department of Energy under con-tract DE–AC02–76SF00515. We thank the authors ofRef. [22] for a very helpful exchange. We also wish tothank Babis Anastasiou, Carola Berger, Zvi Bern, YuriDokshitzer, Nigel Glover, David Kosower, Gavin Salam,Jack Smith and Werner Vogelsang for very helpful con-versations. LD thanks the Kavli Institute for Theoreti-cal Physics and the Aspen Center for Physics for supportduring a portion of this work, and GS thanks SLAC forhospitality.[1] J. C. Collins and D. E. Soper, Nucl. Phys. B 193, 381(1981) [Err.-ibid. B 213, 545 (1983)]; A. Sen, Phys. Rev.D24 (1981) 3281.[2] L. Magnea and G. Sterman, Phys. Rev. D42, 4222(1990).[3] N. Kidonakis, Int. J. Mod. Phys. A 15, 1245 (2000) [hep-ph/9902484].[4] S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys.B 688, 101 (2004) [hep-ph/0403192]; Nucl. Phys. B 691,129 (2004) [hep-ph/0404111].[5] C. Buttar et al., hep-ph/0604120.[6] A. Sen, Phys. Rev. D 28, 860 (1983); J. Botts andG. Sterman, Nucl. Phys. B325, 62 (1989).[7] N. Kidonakis, G. Oderda and G. Sterman, Nucl. Phys.B531 (1998) 365 [hep-ph/9803241].[8] N. Kidonakis, G. Oderda and G. Sterman, Nucl. Phys.B525 (1998) 299 [hep-ph/9801268].[9] A. Banfi, G. P. Salam and G. Zanderighi, JHEP 0408(2004) 62 [hep-ph/0407287].[10] S. M. Aybat, L. Dixon and G. Sterman, in preparation.[11] S. Catani and M. H. Seymour, Phys. Lett. B 378, 287(1996) [hep-ph/9602277]; Nucl. Phys. B 485, 291 (1997)[Err.-ibid. B 510, 503 (1997)] [hep-ph/9605323].[12] S. Catani, Phys. Lett. B427, (1998) 161 [hep-ph/9802439].[13] G. Sterman and M. E. Tejeda-Yeomans, Phys. Lett.B552 (2003) 48 [hep-ph/0210130].[14] R. Akhoury, Phys. Rev. D19 (1979) 1250.[15] S. Moch, J. A. M. Vermaseren and A. Vogt, JHEP 0508,049 (2005) [hep-ph/0507039].[16] Y. L. Dokshitzer and G. Marchesini, Phys. Lett. B 631,118 (2005) [hep-ph/0508130]; JHEP 0601, 007 (2006)[hep-ph/0509078]; M. H. Seymour, JHEP 0510, 029(2005) [hep-ph/0508305].[17] G. P. Korchemsky and A. V. Radyushkin, Nucl. Phys.B283 (1987) 342.[18] I. A. Korchemskaya and G. P. Korchemsky, Nucl. Phys.B437 (1995) 127 [hep-ph/9409446].[19] Z. Bern, A. De Freitas and L. J. Dixon, JHEP 0306, 028(2003) [hep-ph/0304168].[20] E. W. N. Glover, JHEP 0404, 021 (2004) [hep-ph/0401119].[21] Z. Bern, L. J. Dixon and D. A. Kosower, JHEP 0408(2004) 12 [hep-ph/0404293].[22] B. Jantzen, J. H. Ku¨hn, A. A. Penin and V. A. Smirnov,Nucl. Phys. B 731, 188 (2005) [hep-ph/0509157], andprivate communication.[23] V. Ravindran, J. Smith and W. L. van Neerven, Nucl.5Phys. B704 (2005) 332 [hep-ph/0408315].[24] A. Kyrieleis and M. H. Seymour, JHEP 0601, 085 (2006)[hep-ph/0510089].

The Two-loop Anomalous Dimension Matrix for Soft Gluon Exchange

Crossref

Two-Loop Anomalous-Dimension Matrix for Soft-Gluon Exchange

The Two-loop Anomalous Dimension Matrix for Soft Gluon Exchange

Abstract

Similar works

Full text

Available Versions

UNT Digital Library

Crossref