Methods from soft-collinear effective theory are used to perform the
threshold resummation of Sudakov logarithms for the deep-inelastic structure
function F_2(x,Q^2) in the endpoint region x->1 directly in momentum space. An
explicit all-order formula is derived, which expresses the short-distance
coefficient function C in the convolution F_2=C*phi_q in terms of Wilson
coefficients and anomalous dimensions defined in the effective theory.
Contributions associated with the physical scales Q^2 and Q^2(1-x) are
separated from non-perturbative hadronic physics in a transparent way. A
crucial ingredient to the momentum-space resummation is the exact solution to
the integro-differential evolution equation of the jet function, which is
derived. The methods developed in this Letter can be applied to many other hard
QCD processes.Comment: 4 pages, 1 figure. Version to appear in PR

Becher, Thomas

Fermi National Accelerator Laboratory

Neubert, Matthias

English

arXiv

Thomas Becher

Matthias Neubert

Crossref

Threshold Resummation in Momentum Space from Effective Field Theory

Derives a crucial ingredient to the momentum-space resummation

UNT Digital Library

arXiv:hep-ph/0605050v2  23 Aug 2006Threshold Resummation in Momentum Space from Effective Field TheoryThomas Bechera and Matthias Neubertba Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, U.S.A.b Institute for High-Energy Phenomenology, Laboratory for Elementary-Particle Physics,Cornell University, Ithaca, NY 14853, U.S.A.Methods from soft-collinear effective theory are used to perform the threshold resummation ofSudakov logarithms for the deep-inelastic structure function F2(x,Q2) in the endpoint region x→ 1directly in momentum space. An explicit all-order formula is derived, which expresses the short-distance coefficient function C in the convolution F2 = C ⊗ φq in terms of Wilson coefficients andanomalous dimensions defined in the effective theory. Contributions associated with the physicalscales Q2 and Q2(1 − x) are separated from non-perturbative hadronic physics in a transparentway. A crucial ingredient to the momentum-space resummation is the exact solution to the integro-differential evolution equation for the jet function, which is derived. The methods developed in thisLetter can be applied to many other hard QCD processes.Preprint: CLNS 06/1963, FERMILAB-PUB-06-101-TI. INTRODUCTIONA generic problem in applications of perturbative QCDto collider physics or heavy-quark physics is to disen-tangle contributions associated with different momentumscales, and to resum large logarithms of ratios of suchscales to all orders in the perturbative expansion. Inprocesses containing hadronic jets, a scale hierarchy iscreated by the fact that the invariant mass of a colli-mated jet is typically much smaller than the hard scaleof the process (e.g., the center-of-mass energy). The in-tricate interplay of soft and collinear emissions then leadsto large Sudakov double logarithms. The resummationof these logarithms is conventionally performed in mo-ment space, and predictions for differential cross sectionsin momentum space are obtained by an inverse Mellintransformation. This procedure is cumbersome and of-ten leads to unphysical singularities, because the resum-mation formulae involve integrals over the Landau poleof the running coupling. These singularities are dealtwith by means of ad hoc prescriptions, or by introducingartificial infrared cutoffs.In this Letter we develop an approach based on ef-fective field theory, which allows us to resum Sudakovlogarithms for a large class of processes directly in mo-mentum space. The starting point is a factorization the-orem for the cross section, in which contributions fromdifferent momentum scales are separated in a transparentway. Evolution equations for the various components inthe factorization formula are solved exactly in momen-tum space, in such a way that one never encounters inte-grals over the Landau pole. We illustrate the procedurewith the example of deep-inelastic scattering. However,the same methods can be applied to many other hardQCD processes, such as Drell-Yan lepton-pair produc-tion, prompt photon production in hadron-hadron col-lisions, Higgs-boson production in gluon-gluon fusion,heavy-quark fragmentation, event shapes, and others.Technical details of our derivations are presented in [1].II. FACTORIZATION FORMULAWe focus on the flavor non-singlet component of thestructure function F2(x,Q2) in deep-inelastic scattering(DIS) of electrons off a nuclear target, e− + N(p) →e−+X(P ), denoting by q = P − p the momentum of thevirtual photon. We are interested in the region where theBjorken scaling variable x = Q2/(2p ·q) is near 1, so thatthere is a hierarchy of scales Q2 ≫ Q2(1 − x) ≫ Λ2QCD.The intermediate scale Q2(1 − x) ≈ M2X is set by theinvariant mass MX of the final-state jet. In this regionthe structure function can be written in the factorizedform [2–4] (with µf the factorization scale and eq thequark electric charge)F ns2 (x,Q2) =∑qe2q |CV (Q2, µf )|2× Q2∫ 1xdξ J(Q2(ξ − x), µf)φnsq (ξ, µf ) .This formula is valid to all orders in perturbation theoryand at leading power in (1−x) and Λ2QCD/M2X . Here CVis a hard matching coefficient, J is a jet function, andφnsq is the non-singlet component of the quark distribu-tion function in the nucleon. As shown in [1], a simplederivation of the factorization formula can be given usingthe technology of soft-collinear effective theory (SCET)[5] (see [6–8] for earlier investigations).In SCET the hard function CV is identified with theWilson coefficient in the matching relation of the QCDvector current onto the unique leading-power current op-erator in the effective theory. To calculate the Wilson co-efficient one must compare perturbative expressions forthe photon vertex function in the two theories. The cal-culation can be simplified by performing the matchingon-shell, in which case all loop graphs in the effective the-ory are scaleless and vanish in dimensional regularization.The bare on-shell vertex function in QCD (called the on-shell quark form factor) has been studied at two-loop1order and beyond [9–11]. The form factor is infrared di-vergent and must be regularized. When the SCET graphsare subtracted from the QCD result, the infrared polesin 1/ǫ get replaced by ultraviolet poles. To obtain thematching coefficient we introduce a renormalization fac-tor ZV , which absorbs these poles. At one-loop orderthis gives [6]CV (Q2, µ) = 1 +CFαs4π(−L2 + 3L− 8 + π26),where L = ln(Q2/µ2) and αs = αs(µ). The two-loopexpression for CV can be found in [1]. The scale depen-dence of the Wilson coefficient is governed by the evolu-tion equationdCV (Q2, µ)d lnµ=[Γcusp(αs) lnQ2µ2+ γV (αs)]CV (Q2, µ) ,(1)where Γcusp is the universal cusp anomalous dimensionof Wilson loops with light-like segments [12], which isassociated with the appearance of Sudakov double loga-rithms. The quantity γV accounts for single-logarithmicevolution effects. The anomalous dimension can be ob-tained from the coefficient of the 1/ǫ pole term in therenormalization factor ZV . Using the results of [11] itcan be calculated at three-loop order [1].The jet function J is defined in terms of the disconti-nuity of a vacuum correlator of two quark fields, madegauge invariant by the introduction of Wilson lines. Itobeys the integro-differential evolution equation [13]dJ(p2, µ)d lnµ= −[2Γcusp(αs) lnp2µ2+ 2γJ(αs)]J(p2, µ)− 2Γcusp(αs)∫ p20dp′2J(p′2, µ)− J(p2, µ)p2 − p′2 .We encounter again the cusp anomalous dimension, andin addition a new function γJ , which has been calculatedin [13] at two-loop order, and whose three-loop coefficientis determined in [1].III. SOLUTIONS OF THE RENORMALIZATIONGROUP EQUATIONSThe exact solution to the evolution equation (1) isCV (Q2, µ) = exp[2S(µh, µ)− aγV (µh, µ)]×(Q2µ2h)−aΓ(µh,µ)CV (Q2, µh) , (2)where µh ∼ Q is a hard matching scale, at which thevalue of the coefficient CV is calculated using fixed-orderperturbation theory. The Sudakov exponent S and theexponents aγ are given byS(ν, µ) = −αs(µ)∫αs(ν)dαΓcusp(α)β(α)α∫αs(ν)dα′β(α′),aΓ(ν, µ) = −αs(µ)∫αs(ν)dαΓcusp(α)β(α), (3)and similarly for aγV , where β(αs) = dαs/d lnµ isthe β-function. The explicit perturbative expansions ofthese expressions valid at next-to-next-to-leading order(NNLO) in renormalization-group (RG) improved per-turbation theory are given in [1].An important object in the derivation of the solutionto the evolution equation for J is the associated jet func-tion j˜, which has originally been defined in terms of anintegral over the jet function followed by a certain re-placement rule [14]. More elegantly, it can be obtainedby the Laplace transformationj˜(lnQ2µ2, µ)=∫∞0dp2 e−sp2J(p2, µ) ,where s = 1/(eγEQ2). The inverse transformation isJ(p2, µ) =12πi∫ c+i∞c−i∞ds esp2j˜(ln1eγEs µ2, µ). (4)Using the evolution equation for the jet function we findthat the associated jet function obeysdd lnµj˜(lnQ2µ2, µ)= −[2Γcusp(αs) lnQ2µ2+ 2γJ(αs)]j˜(lnQ2µ2, µ),which is analogous to the evolution equation (1) for thehard function. Inserting the solution to this equation intothe inverse transformation (4) we obtainJ(p2, µ) = exp[−4S(µi, µ) + 2aγJ (µi, µ)]× j˜(∂η, µi) e−γEηΓ(η)1p2(p2µ2i)η, (5)where η = 2aΓ(µi, µ), and ∂η denotes a derivative withrespect to this quantity. The above form of the result isvalid as long as η > 0 (i.e., µ < µi). For negative η thesingularity at p2 = 0 must be regularized using a stardistribution [1]. Relation (5) is one of the main results ofthis Letter. It relates J to the associated jet function j˜evaluated at a scale µi, where it can be computed usingfixed-order perturbation theory. At one-loop orderj˜(L, µ) = 1 +CFαs4π(2L2 − 3L+ 7− 2π23),where in (5) the argument L is replaced by the deriva-tive operator ∂η. The two-loop expression for j˜ can beextracted from [13].2IV. MOMENTUM-SPACE RESUMMATIONWe are now ready to write down a resummed expres-sion for the structure function F ns2 (x,Q2) valid to allorders in perturbation theory and at leading power in(1− x) and Λ2QCD/M2X . When combining the results (2)and (5) the Sudakov exponents can be simplified. Intro-ducing the short-hand notation aγφ = aγJ −aγV , we findafter a straightforward calculationF ns2 (x,Q2) =∑qe2q |CV (Q2, µh)|2 U(Q,µh, µi, µf )×j˜(lnQ2µ2i+ ∂η, µi) e−γEηΓ(η)∫ 1xdξφnsq (ξ, µf )(ξ − x)1−η , (6)whereU(Q,µh, µi, µf ) = exp [4S(µh, µi)− 2aγV (µh, µi)]×(Q2µ2h)−2aΓ(µh,µi)exp[2aγφ(µi, µf )],and as before η = 2aΓ(µi, µf ). The remaining integralcan be performed noting that, on general grounds, thebehavior of the parton distribution function near the end-point can be parameterized asφnsq (ξ, µf )∣∣ξ→1= N(µf ) (1− ξ)b(µf )[1 +O(1 − ξ)],where b(µf ) > 0. This leads to the final expressionF ns2 (x,Q2)∑q e2q xφnsq (x, µf )= |CV (Q2, µh)|2 U(Q,µh, µi, µf )×(1− x)η j˜(lnQ2(1− x)µ2i+ ∂η, µi)×e−γEη Γ(1 + b(µf))Γ(1 + b(µf ) + η). (7)The exact all-order results (6) and (7) are independentof the scales µh and µi, at which the matching coefficientCV and the associated jet function j˜ are calculated. Theanswers simplify further if we choose the “natural” val-ues µh = Q and µi = Q√1− x (for fixed x). In practi-cal calculations the residual dependence on these scalesintroduced by the truncation of the perturbative expan-sions of the various objects can be used as an estimatorof yet unknown higher-order corrections.Above we have accomplished the resummation ofthreshold logarithms for F2 directly in momentum space.The resulting formulae are simpler than correspondingexpressions in the literature (see e.g. [15]) in that theydo not require a Mellin inversion and in that the de-pendence on x and Q is explicit. The right-hand sidesof (6) and (7) can be evaluated at any desired order inresummed perturbation theory. Using currently avail-able results, it is possible to include terms at NNLO[1], which is equivalent to the so-called next-to-next-to-next-to-leading double-logarithmic (N3LL) approxima-tion. The resummation is under perturbative control aslong as (1−x)≫ Λ2QCD/Q2, since only then the interme-diate scale µi ∼ Q√1− x is a short-distance scale. Whilethe theoretical description thus breaks down very closeto the endpoint, we note that weighted integrals of thestructure function over an interval x0 ≤ x ≤ 1 can becalculated as long as Q√1− x0 is in the short-distancedomain.It is instructive to compare our result (7) with theconventional approach to threshold resummation in DIS,which proceeds via moment space [2,3]. One definesF ns2,N (Q2) =∫ 10dxxN−1F ns2 (x,Q2)= CN (Q2, µf )∑qe2q φnsq,N+1(µf ) ,where the moments of φnsq (ξ, µ) are defined in analogywith those of F ns2 (x,Q2). For large values of N the inte-gral is dominated by the endpoint region (1− x) ∼ 1/N .The short-distance coefficient CN is decomposed asCN (Q2, µf ) = g0(Q2, µf ) exp[GN (Q2, µf )],where the prefactor g0 collects all N -independent terms,and the exponent is written in the form (see [15] for themost up-to-date discussion)GN (Q2, µf ) =∫ 10dzzN−1 − 11− z (8)×[∫ (1−z)Q2µ2fdk2k2Aq(αs(k)) +Bq(αs(Q√1− z))] .The resummation for the momentum-space structurefunction F2(x,Q2) itself is obtained from that for the mo-ments F2,N (Q2) by an inverse Mellin transformation. Itis possible to show (see [1] for details) that the outcome ofthis procedure is equivalent, at any finite order in the per-turbative expansion, to the result (7) derived from effec-tive field theory, provided we identify Aq(αs) = Γcusp(αs)and(1 +π212∇2 + . . .)Bq(αs) = γJ(αs) +∇ ln j˜(0, µ)−(π212∇− ζ33∇2 + . . .)Γcusp(αs) ,where ∇ = d/d lnµ2. It follows from this relation thatthe quantities Bq and γJ agree at first order in αs (asobserved in [6]), but they differ starting from two-looporder.There are a few unpleasant features of the conventionalapproach which are worth pointing out. First, note thatthe integrals over the functions Aq and Bq in (8) runover the Landau pole of the running coupling αs(µ),311/2 1/√2 2µh/Q√2F ns2 /(∑q e2qxφnsq )0.60.70.80.911/2 1/√2 2µi/(Q√1− x)√2F ns2 /(∑q e2qxφnsq )0.60.70.80.9FIG. 1. Dependence of the resummed result for F ns2 (x,Q2)on the hard (left) and intermediate (right) scales, at differ-ent orders in RG-improved perturbation theory: LO (dot-ted), NLO (dashed), and NNLO (solid). We use Q = 30GeV,x = 0.9, µf = 5GeV, and b(µf ) = 4.introducing an infrared renormalon ambiguity of orderΛ2QCD/M2X . No such problem arises for the integrals (3)in our approach. The particular integral representationof the solution (8) results from the fact that in the con-ventional approach one solves a set of partial differen-tial equations derived by diagramatic methods instead ofthe RG evolution equations in SCET [2,3]. The use ofRG methods for the resummation avoids the Landau-polesingularity in the exponent [16,17]. Secondly, when per-forming the inverse Mellin transform one needs to inte-grate the function GN over N along a contour parallel tothe imaginary axis. This integration involves arbitrarilysmall physical scales |k2| ∼ Q2/|N |, leading to a secondencounter with the Landau pole. Different prescriptionsto deal with this problem have been proposed in the lit-erature. In our approach integrals over the Landau polenever arise, because factorization and resummation areperformed directly in momentum space. The singulari-ties appearing in the conventional approach are an arti-fact of the way the resummation of large logarithms isimplemented and they cannot be used to assess whethera corresponding renormalon pole is present [17]. In ourapproach infrared renormalons are expected to affect thelarge-order perturbative behavior of the matching coeffi-cients CV and j˜. The corresponding infrared ambiguitieswill be commensurate with power corrections from sub-leading operators in the effective theory. The evolution,on the other hand, is driven by anomalous dimensions,which are expected to be free of renormalons.Figure 1 shows the scale dependence of our result forF ns2 obtained with Q = 30GeV, x = 0.9, µf = 5GeV,and nf = 5 light flavors. Varying the scales µh and µiabout their default values, we observe that the residualscale dependence is strongly reduced when going to suc-cessively higher orders in perturbation theory. In theliterature the matching scales are often held fixed, andthe perturbative uncertainties can only be estimated bycomparing results at different orders in the expansion.V. CONCLUSIONSUsing methods from effective field theory we have in-troduced a new approach to the resummation of large Su-dakov logarithms in hard QCD processes. Factorizationand resummation are performed directly in momentumspace, such that the resulting formulae are free of un-physical infrared sensitivities and provide a transparentseparation of the different scales in the problem. As anapplication we have derived an exact all-order expressionfor the resummed deep-inelastic structure function F2,which is much simpler than corresponding results foundin the literature.Acknowledgments: We are grateful to Ben Pecjak foruseful discussions. The research of T.B. was supportedby the Department of Energy under Grant DE-AC02-76CH03000, and that of M.N. by the National ScienceFoundation under Grant PHY-0355005. Fermilab is op-erated by Universities Research Association Inc., undercontract with the U.S. Department of Energy.[1] T. Becher, M. Neubert and B. D. Pecjak, hep-ph/0607228.[2] G. Sterman, Nucl. Phys. B 281, 310 (1987).[3] S. Catani and L. Trentadue, Nucl. Phys. B 327, 323(1989).[4] G. P. Korchemsky and G. Marchesini, Nucl. Phys. B 406,225 (1993).[5] C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart,Phys. Rev. D 63, 114020 (2001); C. W. Bauer, D. Pirjoland I. W. Stewart, Phys. Rev. D 65, 054022 (2002).[6] A. V. Manohar, Phys. Rev. D 68, 114019 (2003).[7] B. D. Pecjak, JHEP 0510, 040 (2005).[8] J. Chay and C. Kim, hep-ph/0511066.[9] G. Kramer and B. Lampe, Z. Phys. C 34, 497 (1987)[Erratum-ibid. C 42, 504 (1989)].[10] T. Matsuura and W. L. van Neerven, Z. Phys. C 38, 623(1988); T. Matsuura, S. C. van der Marck and W. L. vanNeerven, Nucl. Phys. B 319, 570 (1989).[11] S. Moch, J. A. M. Vermaseren and A. Vogt, JHEP 0508,049 (2005).[12] G. P. Korchemsky and A. V. Radyushkin, Nucl. Phys.B 283, 342 (1987); I. A. Korchemskaya and G. P. Ko-rchemsky, Phys. Lett. B 287, 169 (1992).[13] T. Becher and M. Neubert, Phys. Lett. B 637, 251(2006).[14] M. Neubert, Phys. Rev. D 72, 074025 (2005).[15] S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys.B 726, 317 (2005).[16] G. P. Korchemsky and G. Marchesini, Phys. Lett. B 313,433 (1993).[17] M. Beneke and V. M. Braun, Nucl. Phys. B 454, 253(1995).4

Threshold Resummation in Momentum Space from Effective Field Theory

Abstract

Similar works

Full text

Available Versions

Crossref

UNT Digital Library