Higher-Order Corrections in Threshold Resummation

Transcription

1 DESY 5-5, SFB/CPP-5-5 DCPT/5/6, IPPP/5/3 NIKHEF 5- June 5 hep-ph/5688 Higher-Order Corrections in Threshold Resummation S. Moch a, J.A.M. Vermaseren b and A. Vogt c a Deutsches Elektronensynchrotron DESY Platanenallee 6, D 5735 Zeuthen, Germany b NIKHEF Theory Group Kruislaan 49, 98 SJ Amsterdam, The Netherlands c IPPP, Department of Physics, University of Durham South Road, Durham DH 3LE, United Kingdom Abstract We extend the threshold resummation exponents G N in Mellin-N space to the fourth logarithmic N 3 LL order collecting the terms α s α s lnn n to all orders in the strong coupling constant α s. Comparing the results to our previous three-loop calculations for deep-inelastic scattering DIS, we derive the universal coefficients B q and B g governing the final-state jet functions to order α 3 s, extending the previous quark and gluon results by one and two orders. A curious relation is found at second order between these quantities, the splitting functions and the large-angle soft emissions in Drell-Yan type processes. We study the numerical effect of the N 3 LL corrections using both the fully exponentiated form and the expansion of the coefficient function in towers of logarithms.

2 Introduction Coefficient functions, or partonic cross sections, form the backbone of perturbative QCD. These quantities are defined in terms of power expansions in the strong coupling constant α s. In general, only a few terms in this expansion can be calculated. It is however possible, and necessary, to resum the dominant contributions to all orders in α s close to exceptional kinematic points. Close to threshold, for example, where real emissions are kinematically suppressed, the resummation takes the form of an exponentiation in Mellin-N space 4, with the moments N defined with respect to the appropriate scaling variable, like Bjorken-x in deep-inelastic scattering DIS and x T = p T / S in direct photon and inclusive hadron production. The resummation exponents are given by integrals over functions in turn defined by a power series in α s. Besides by dedicated calculations, the corresponding expansion coefficients can be obtained by expanding the exponentials and comparing to the results of fixed-order calculations. Hence progress in the latter sector also facilitates improved resummation predictions. At present the next-to-leading order NLO is the standard approximation for many important observables, facilitating a resummation with next-to-leading logarithmic NLL accuracy. For recent introductory overviews see, for instance, Refs. 5, 6. The next-to-next-to-leading order NNLO corrections have been completed so far only for the coefficient functions for inclusive lepton-proton DIS 7, the Drell-Yan process 4 and the related Higgs boson production 3, 5 7 in proton-proton collisions. Consequently, the threshold resummation has been carried out at the next-to-next-to-leading logarithmic NNLL accuracy only for these processes 8, 9. Recently we have computed the complete three-loop coefficient functions for inclusive photonexchange DIS,. Moreover, in the course of the calculation of the third-order splitting functions governing the NNLO evolution of the parton distributions,, we have also computed DIS by exchange of a scalar directly coupling only to gluons. Together these results enable us to extend two more universal functions entering the resummation exponents, the quark and gluon jet functions B q and B g collecting final-state collinear emissions, to the third order in α s. In fact, already the second-order coefficient for B g represents a new result, relevant for future NNLL resummations of processes with final-state gluons at the Born level. Making use also of the results of Refs. 3, 4 we can furthermore effectively, i.e., up to the small contribution of the four-loop cusp anomalous dimension, extend the threshold resummation for inclusive DIS to an unprecedented next-to-next-to-next-to-leading logarithmic N 3 LL accuracy. The remainder of this article is organized as follows: after recalling the general structure of the resummation exponents in Section, we extend the required integrations in Section 3 to the fourth logarithmic N 3 LL order. In Section 4 we determine the relevant expansion coefficients by comparison to our three-loop results for DIS and illustrate the numerical effect of the N 3 LL contributions to the resummation exponent. In Section 5 we present the resulting predictions for the leading seven large-x terms of the four-loop coefficient function and discuss higher-order effects in terms of the expansion in towers of threshold logarithms. Our results are briefly summarized in Section 6. Some basic relations for the integrations of Section can be found in the Appendix.

3 The general structure For processes with only one colour structure at the Born level, the resummed Mellin-space coefficient functions C N defined in the MS scheme are given by a single exponential, C N Q /C N LO Q = g Q expg N Q +ON ln n N.. Here CLO N denotes the lowest-order coefficient function for the process under consideration, e.g., CLO N = for DIS. The prefactor g collects, order by order in the strong coupling constant α s, all N-independent contributions. The exponent G N contains terms of the form ln k N to all orders in α s. Besides the physical hard scale Q = q in DIS, with q the four-momentum of the exchanged gauge boson, both functions also depend on the renormalization scale µ r and the massfactorization scale µ f. The reference to these scales will be often suppressed for brevity. The exponential in Eq.. is build up from universal radiative factors for each initial- and final-state parton p, p and J p, together with a process-dependent contribution int. For example, the resummation exponents for inclusive deep-inelastic scattering, Drell-Yan DY lepton-pair production and direct photon production via q q gγ and qg qγ 5 take the form G N DIS G N DY G N ab cγ = ln q + lnj q + ln int DIS, = ln q + ln int DY, = ln a + ln b + lnj c + ln int ab cγ.. The radiation factors are given by integrals over functions of the running coupling. Specifically, the effects of collinear soft-gluon radiation off an initial-state parton p = q,g are collected by ln p Q, µ f = Z dz zn z Z z Q µ f dq q A pα s q..3 Collinear emissions from an unobserved final-state parton lead to the so-called jet function, Z Z lnj p Q = dz zn zq dq z z Q q A pα s q +B p α s zq..4 Finally the process-dependent contributions from large-angle soft gluons are resummed by ln int Q = Z dz zn z Dα s z Q..5 The functions g in Eq.. and A p, B p and D in Eqs..3.5 are given by the expansions Fα s = l=l F l αl s 4π l=l F l a l s,.6 where l = with g = for F = g, and l = else. Here we have also taken the opportunity to specify the reduced coupling a s employed for the rest of this article. The extent to which these functions are known defines the accuracy to which the threshold logarithms can be resummed.

4 The situation for inclusive DIS is actually simpler than indicated in Eq.., as the function.5 is found to vanish to all orders in α s 3, 4, D DIS k =, int DIS =..7 On the other hand, the resummation of processes with four or more partons at the Born level, like inclusive hadron production in pp collisions 6, is more complicated than Eq.. due to colour interferences and correlations in the large-angle soft gluon emissions 3, 4. However, the process-independent functions A p and B p retain their relevance also for such cases. 3 The resummation exponent to fourth logarithmic order After performing the integrations in Eqs..3.5, the function G N in Eq.. takes the form G N Q = lnn g λ + g λ + a s g 3 λ + a s g 4 λ +..., 3. where λ = β a s lnn. For the actual computation of the functions g i it is convenient to employ the following representation for the scale dependence of a s up to N 3 LO: a s q = a s X a s β X lnx + a3 s β β X 3 β ln X lnx +X+ β X β β 3 XlnX + 5 ln X ln 3 X + X X + a4 s X 4 β 3 + β 3 X + β β β β X lnx 3lnX X X +Oas 5 3. with a s a s µ r and the abbreviation X = +a s β lnq /µ r. The terms up to the n-th order in a s in Eq. 3. contribute to g n in Eq. 3.. Thus the calculation of g 4 requires the highest known coefficient of the beta function of QCD, β 3 7, 8. Generalizing the approach of Ref. 8, the functions g i λ can be obtained using well-known methods for Mellin transforms based on properties of harmonic sums and harmonic polylogarithms 9, 3 in addition to algorithms for the evaluation of nested sums 3. The basic relations for this approach, suitable for the evaluation of Eq. 3. to any accuracy, are presented in the Appendix. As a check we have also carried out the integrations along the lines of Ref. 9. For the convenience of the reader, we first recall the known results for g, g and g 3,, 8, 9. For brevity suppressing factors of β see below and using the short-hand notations L qr = lnq /µ r and L fr = lnµ f /µ r, these functions can be written as g DIS λ = A ln λ+λ ln λ, 3.3 g DIS λ = A β A λ+ln λ+ A β ln λ A γ e B ln λ+l qr A ln λ+l fr A λ, 3.4 3

5 g3 DIS λ = A β A β + A β A 3 +λ λ ln λ + A β λ + ln λ + A β A β ln λ λ +A β γ e + A β B β ln λ λ λ A β + A γe + ζ +A γ e B γ e B λ + L qr A γ e A β + A B ln λ + A β λ λ + L fr A λ Lqr A L fr λ A λ. 3.5 The dependence on β is recovered by A k A k /β k, B k B k /β k, β k β k /β k+ and multiplication of g 3 by β. In the same notation the new function g 4 to be multiplied by β is given by g DIS 4 λ = 6 A β 3 ln 3 λ λ + A β γ e + A β B β λ ln λ + A β 3 A β β A β γ e + ζ +A β A β γ e A 3 β + B β γ e + B β A β 3 A ln λ β β λ + A β 3 A β β + A β 3 ln λ λ ln λ+a β 3 A β β A β γ e + A β γ e A β + A β + B β B β λ + + λ 3 A β 3 6 A β β 6 A β 3 3 A 3γ e ζ + γe 3 + ζ 3 + A β γ e A γe + ζ 5 6 A β + 3 A β A 3β A 3 γ e 3 A 4 B β + B γe + ζ +B γ e + B 3 + λ A β 3 3 A β β + A β 3 + A β A β + A 3β A 4 λ+l qr A β A β λ + λ + A γe + ζ A β + A γ e + A 3 B γ e B ln λ +A β γ e + A β B β + A B + L fr A 3 λ L fr λ A β λ + A ln λ β λ + Lqr 3 A + A β λ+lfr 3 λ ln λ λ Lqr A γ e 6 A λ 3 A λ. 3.6 The results gi DY for the Drell-Yan process and, with slightly different coefficients A i and D i, Higgs production via gluon-gluon fusion, are related to Eqs as follows: the function corresponding to Eq. 3.3 reads g DY λ = g DIS λ, while the functions g DY,g3 DY and g4 DY are 4

6 obtained from Eqs by replacing λ λ everywhere and substituting B i D i / in all terms. Finally, the constants have to be changed according to γ e γ e and ζ n n ζ n. The generalization of Eqs to other processes involving Eqs..3.5 is obvious. The functions g λ collecting the leading logarithms La s L k depend on A only and are finite for all λ. The N n LL contributions g n> λ to Eq. 3. including A n, B n and D n, on the other hand, exhibit Landau poles at the moments N = exp/β a s for DIS and N = exp/β a s for the DY case and at both values in general. As obvious from Eq the strength of these singularities increases with the logarithmic order, reaching λ at the N 3 LL level. 4 Resummation coefficients and numerical stability Since observables are independent, order by order in α s, of the factorization scale, the functions A p in Eqs..3 and.4 are given by the large-n coefficients of the diagonal splitting functions for µ r = µ f, P pp α s = A p α s lnn +O, 4. which in turn are identical to the anomalous dimension of a Wilson line with a cusp 3. The first and second order coefficients have been known for a long time, the third order has been recently completed by us. The expansion coefficients.6 for the quark case read, 33 A q, = 4C F 67 A q, = 8C F 8 ζ A q,3 = 6C F C A C A 5 9 n f ζ + 6 ζ C A n f ζ 7 3 ζ 3 5 ζ + nf + C F n f ζ Here n f denotes the number of effectively massless quark flavours, and C F and C A are the usual colour factors, with C F = 4/3 and C A = 3 in QCD. The gluonic quantities are given by A g,i = C A /C F A q,i. 4.3 The perturbative expansion of A p is very benign. For n f = 4, for example, Eqs. 4. lead to A q α s =.444 α s α s +.5 α s Consequently, already the effect of A 3 on the resummed coefficient functions is very small 8,9, and a simple estimate suffices for the presently unknown fourth-order coefficients A 4 entering g 4. With the / results differing by less than %, we will employ the / Padé approximants A q,4 7849, 433, 553 for n f = 3, 4, 5, 4.5 corresponding to an estimate of α 3 s for the next term in Eq

7 For the determination of the coefficients B i we also need the constant-n piece, g in Eq.., for inclusive DIS. The corresponding expansion coefficients g k can be obtained by Mellin inverting the +-distribution and δ x parts of the k-th order coefficient function c k,q. Again using the expansion parameter a s = α s /4π, the presently known terms 7,, are given by g DIS = C F 9 ζ + γe + 3γ e, g DIS = CF 8 5 γ e 7 γ e + 6γe 3 + γe 4 + ζ 8γ e ζ 4γe ζ 66ζ 3 + 4γ e ζ ζ +C A C F γ e γ e + 9 γ3 e 39 8 ζ 3 γ eζ 4γ e ζ ζ 3 4γ e ζ C F n f γ e 9 9 γ e 4 9 γ3 e ζ γ eζ ζ 3 5 ζ 4.7 and g DIS 3 = CF γ e γ e 93 γ3 e 9γe 4 + 6γe γ6 e 697 ζ γ eζ + 66γe ζ 36γe 3 ζ 4γe 4 ζ 4ζ 3 346γ e ζ 3 6γe ζ γe 3 ζ ζ + 84γ eζ γ e ζ + 556ζ ζ 3 8γ e ζ ζ ζ C A CF γ e γ e 4γ e ζ ζ ζ γ3 e γ4 e γ5 e ζ γ eζ 59 3 γ e ζ 8γ 4 e ζ 84 9 γ3 e ζ ζ γ e ζ γ e ζ 3 8γe 3 ζ ζ γ eζ γ e ζ 88ζ ζ γ e ζ ζ ζ 5 + γ e ζ ζ ζ 3 +C A C F γ e γ e γ3 e + 7 γ4 e ζ γ eζ γ e ζ 88 9 γ3 e ζ + 5 ζ γ4 e ζ γ eζ γ e ζ 88 9 γ3 e ζ ζ ζ γ eζ ζ ζ 5 + 3γ e ζ ζ ζ 3 +CF n f γ e γ e γ3 e 7 9 γ4 e 8 9 γ5 e ζ γ eζ + 3 γ e ζ γ3 e ζ ζ 3 9 γ eζ γ e ζ γ eζ 6

8 8 35 ζ 4 3 ζ ζ ζ 5 +C A C F n f γ e γ e 55 8 γ3 e 44 7 γ4 e ζ γ eζ γ e ζ γ3 e ζ 48 8 ζ γ eζ 3 + 8γe ζ ζ 8 5 γ eζ 64 9 ζ ζ ζ 5 +C F nf γ e γ e γ3 e γ4 e 8 ζ 8 9 γ e ζ 6 7 γ eζ ζ γ eζ ζ + dabc d abc f l 64+6ζ n c ζ ζ 8 3 ζ Note the new flavour structure f l,34 in g 3. This contribution, introducing the colour factor d abc d abc /n c, for the first time leads to a difference between the flavour-singlet and non-singlet coefficient functions for the photon-exchange structure function F in the soft-gluon limit, with f l ns = 3 e and f l s = e / e, where e k represents the average of the charge e k for the active quark flavours, e k = n f n f i= ek i. Correspondingly, the large-n coefficient functions for Z- and W -exchange DIS will differ from each other and from F e.m. at order α 3 s. Based on the size of the f l term in Eq. 4.8, however, we expect these differences to be numerically insignificant. Now the coefficients B q,k entering the jet function.4 can be derived successively from the ln N terms of the k-loop DIS coefficient functions c k,q. Expansion of Eqs. 3. and in powers of a s yields c,q = A γ e B lnn c,q = lnn A β γe + ζ +A γ e B β γ e B + g A γ e B c 3 = lnn 3 A β γ3 e + 3γ e ζ + ζ 3 + A β γe + ζ +A β γe + ζ,q + A 3 γ e B β γ e + β ζ + β γ e B β γ e B 3 + g A γ e B +g A β γ e + ζ +A γ e B β γ e B. 4.9 The coefficients of ln l N, l k in c k,q, on the other hand, are completely fixed by lower-order resummation coefficients, thus providing an explicit k-loop check of the exponentiation formula. Comparison of the relations 4.9 with the corresponding results from the fixed-order calculations of Refs. 7,,, using Eqs. 4., 4.6 and 4.7, leads to B q, = 3C F, 4. B q, = CF 3 + ζ 4 ζ 3 +C F C A ζ + 4 ζ C F n f ζ, 4. 7

9 B q,3 = C 3 F +C A C F +C A C F 9 8 ζ 68 ζ ζ + 3 ζ ζ ζ ζ 7 3 ζ ζ 6 ζ ζ 3 ζ 5 5 ζ 76 3 ζ ζ 3 3 ζ 5 +C F nf ζ 3 7 ζ ζ ζ C Fn f ζ ζ 3 +C A C F n f ζ ζ ζ Eq. 4. is, of course, a well-known result,. Eq. 4. has been derived by us before 35, establishing D DIS = from the n f ln N term at three loops. For our new result 4., on the other hand, we have to rely on the subsequent all-order proofs of Eq..7 in Refs. 3, 4. The QCD expansion of B q analogous to Eq. 4.4 appears far less stable than that for A q, B q α s,n f =4 =.383 α s.7 α s 3.45 α s The ingredients for the resummation of inclusive DIS are now complete, and in the left part of Fig. we show the corresponding LL, NLL, N LL and N 3 LL approximations to the exponent 3. resulting, for α s =. and three flavours, from Eqs , 4., 4.5 and For these parameters the expansion 3. is stable in the N-range shown in the figure. For example, the relative N 3 LL corrections amount to % at N = λ =.33 and 4% at N = 4 λ =.53, whereas the corresponding N LL figures read 9% and %. The large third-order contribution to B q actually stabilizes g 4 λ: for B q,3 = the N 3 LL term at N = 4 would instead reach %, i.e., the size of the previous order. The effect of both A q,4 and β 3, on the other hand, is very small, as their respective nullification would change the result even at N = 4 by only.6% and.%. In the right part of Fig. and in Fig. the exponentiated results are convoluted with the typical input shape x f = x.5 x 3 for a couple of values for α s and n f. The Mellin inversion is in principle ambiguous due to the Landau poles briefly addressed at the end of Section 3. We employ the standard minimal prescription thus adopting the usual fixed-order contour of Ref. 36, to which the reader is referred for a detailed discussion. For total soft-gluon enhancements up to almost an order of magnitude, as shown in the figures, the resulting N 3 LL corrections remain far smaller than their N LL counterparts and amount to less than % even for α s =.3. Note that the dependence on n f is larger than the effect of g 4. Thus, at this level of accuracy, a reliable understanding of heavy-quark mass effects is called for also in the limit x. The gluonic coefficients corresponding to Eqs can be obtained in the same manner from DIS by exchange of a scalar φ with a pointlike coupling to gluons, like the Higgs boson in limit of a heavy top quark. We have derived the corresponding coefficient functions c k φ,p up to k = 3 already during the calculations for Ref., as a process of this type is required to access the lower row of the flavour-singlet splitting function matrix. Comparing those results to 8

10 8.5 G N DIS exp G DIS f / f N 3 LL N LL NLL 6 4 N 3 LL N LL NLL LL.5 LL α S =., N f = N xf = x.5 x Figure : Left: the LL, NLL, N LL and N 3 LL approximations for the resummation exponent for standard DIS. Right: the convolutions of the exponentiated results with a typical input shape. x 8 8 exp G DIS f / f exp G DIS f / f 6 N 3 LL 6 xf = x.5 x 3 N LL 4 NLL LL 4 α S =., N f = x α S =.3, N f = Figure : As the right part of the previous figure, but for a different value of n f left and α s right. x 9

11 Eqs. 4.9 for c k φ,g yields B g, = 3 C A + 3 n f = β, 4.4 B g, = CA ζ + 6 ζ 3 + C A n f ζ + C F n f 7 n f, 4.5 B g,3 = CA ζ ζ ζ 8 3 ζ ζ 3 ζ CA n f ζ 96 9 ζ ζ CF n f C A C F n f 54 6 ζ 83 9 ζ ζ + C F nf ζ 3 + C A nf ζ 76 7 ζ 3 + nf ζ, 4.6 where Eqs. 4.5 and 4.6 are new results. For n f = 4 the numerical expansion of B g reads B g α s =.663 α s.765 α s.696 α s +..., 4.7 exhibiting an enhanced third order correction similar to that of B q in Eq The gluonic threshold resummation resulting from Eqs. 4.3 and is illustrated in Fig. 3 using the practically irrelevant scalar-exchange process, with same parameters as in Fig. for direct comparison. The soft and collinear radiation effects are much larger here due to the larger colour charge of the gluons, but the qualitative pattern is rather similar to normal inclusive DIS. As mentioned above, the closely related Drell-Yan process and Higgs boson production via gluon-gluon fusion presently represent the only other processes for which the NNLL threshold resummation is known, with D = and 8, 9 D {DY,H} = {C F,C A } C A ζ + 56ζ n f ζ. 4.8 Extending a result given in Ref. 8, we notice the following conspicuous relation between these coefficients and B p, : DDY B q, P q,δ = 7β C F DH B g, P g,δ = 3 β 4C A + 5β, 4.9 where P p,δ denotes the coefficients of δ x in the diagonal two-loop splitting functions, and the colour structures on the right-hand sides are those of A p, and B p, = P p,δ, multiplied by β. Note especially the non-trivial cancellation of all ζ-function terms between the three contributions on the left-hand sides of Eqs. 4.9 and the vanishing of the right-hand sides for β.

12 4 G N φdis 8 exp G φdis f / f 3 N 3 LL N LL 6 N 3 LL N LL NLL 4 NLL LL LL α S =., N f = N xf = x.5 x Figure 3: As Fig., but for inclusive DIS by exchange of a scalar φ directly coupling to gluons. x 5 Fourth-order predictions and tower expansion Another manner to organize the all-order information encoded in Eqs...5 is to re-expand the exponential, C N Q /C N LO Q = + k= a k s k l= c kl ln k l+ N, 5. and retain only those terms in the second sum which are completely fixed by the available information on the expansion coefficients in Eq..6. Using the notation g i λ = g ik λ k 5. k= for the expansion of Eqs together with Eq..6 for g, the quantities c kl in Eq. 5. receive contributions from the following coefficients: c k : g c k : + g, g c k3 : + g 3, g, g c k4 : + g 4, g 3, g 3

13 c k5 : + g 5, g 4, g 3, g c k6 : + g 6, g 5, g 33, g 4 c k7 : + g 7, g 6, g 34, g 4, g 3 c k8 : + g 8, g 7, g 35, g 43, g The complete relations for the first four terms c k... c k4 can be found in Ref. 37 in a slightly different notation, g 3 g 3 i.e., the second index denoting the total power of a s in Eq. 3.. Note that the quantities c kl vanish factorially for k and fixed l. Taking into account Eq..7 and considering the coefficient A i as either known or irrelevant, the function g i for inclusive DIS is completely specified by its leading term, obtained by matching to the i-th order calculation of the coefficient functions as in Eqs The same holds for other processes, at least for cases like Eqs.., once the required coefficients B i are known from DIS. The leading three towers of logarithms, c kl for any k and l =,, 3, are fixed by a one-loop calculation providing g i for i =,, together with the NLL resummation adding g k and g k for k. This is the status for many important observables 5, 6. Correspondingly, a two-loop computation of the process under consideration specifies g 3 and g and hence fixes, together with the NNLL resummation, also the next two towers. This is the accuracy reached for the Drell-Yan process and Higgs production 8,9. Finally a three-loop computation combined with the N 3 LL resummation fixes the first seven towers, c kl for l =,..., 7. With the results of Ref. and Sections 3 and 4, we have now reached this point for the structure function F in DIS. The resulting four-loop predictions, in x-space expressed in terms of the coefficients of the +-distributionsd k = x ln x +, for the six highest terms read c 4,q c 4,q D7 = 6 3 C F 4, 5.4 D6 = 8CF C ACF C Fn 3 f, = CF ζ + C A CF ζ C A C F c 4,q D5 c 4,q D C Fn 3 f 74 7 C ACFn f C Fn f, 5.6 = CF 4 +6 ζ ζ 3 + C A CF ζ + 4 ζ 3 + CA C F ζ C A 3 C F + CF 3 n f ζ 8 + C A CF n f ζ C A C Fn f 64 7 C F n f 44 9 C AC F nf C Fn 3 f, 5.7

14 c 4,q D3 c 4,q D 3 = CF ζ 7 ζ C A CF CA C F ζ ζ 88 9 ζ ζ CA 3 C F ζ 59 7 ζ 34 ζ ζ + CFn 3 f ζ ζ 3 + C A C Fn f ζ C A CFn f ζ ζ 3 + C A C F nf ζ CFn f ζ 3 8 C Fnf 3, = CF 4 88 ζ + 39 ζ ζ 64 ζ ζ ζ C A CF ζ ζ 3 96 ζ ζ ζ 3 7 ζ CA C F 8684 ζ 4544 ζ ζ + 83 ζ ζ ζ 5 + CA 3 C F ζ + 45 ζ ζ C A CFn f ζ 6 ζ ζ CA C Fn f C 3 F n f C A C F n f ζ 35 ζ ζ 936 ζ ζ 6 ζ 3 5 ζ 5 ζ C F n f 94 + C F nf ζ ζ 34 9 ζ The seventh term withd is not exactly known, since the fourth-order contribution to A q has not been computed so far. Inserting the numerical values for the ζ-functions and the QCD colour factors, including d abc d abc /n c = 5n f /8, the resummation prediction is given by c 4,q = n f 9.4 nf +.66 n3 f 63.4 f l n f + A q,4. 5. D As mentioned above, numerically insignificant are both the uncertainty due to A q,4 estimated in Eq. 4.5 and the singlet/non-singlet difference introduced by the f l contribution of Eq It is also interesting to note that the fourth coefficient β 3 of the beta function 7,8 with its quartic group invariants d abcd F and d abcd A only enters the eighth tower, starting at the fifth order in α s. 3

15 k c k c k c k3 c k4 c k5 c k6 c k Table : Numerical values of the four-flavour coefficients c kl in Eq. 5. for the quark coefficient function in DIS. The first six columns are exact up to the numerical truncation, and the same for F, F and F 3. The seventh column refers to F and F for the photon-exchange flavour-singlet case, f l = / 7, and uses the estimate 4.5 for the four-loop cusp anomalous dimension. The numerical values of the N-space coefficients c kl in Eq. 5. are presented in Table for l 7 and k. Recall that also these coefficients refer to an expansion in a s = α s /4π. Whatever the normalization of the expansion parameter, however, the coefficients in each column tower finally vanish for k, as mentioned below Eq Thus the series 5. converges at all N for any finite number of towers l max, i.e., with the upper limit in the second sum replaced by l max. The Mellin inversion of the product with the parton distributions f N is therefore well-defined, in contrast to the fully exponentiated result discussed above. Before we turn to the higher-order predictions, it is instructive to compare the approximations by the leading large-x and large-n terms to the completely known two- and three-loop coefficient functions 7,,. This is done in Fig. 4 for the successive approximations in terms of the +-distributionsd k defined above Eq The corresponding results for the expansion in powers of ln N are presented in Fig. 5. Obviously both expansions reproduce the exact large-x behaviour up to terms not increasing as x for the ratios shown in the figures at order α n s once all enhanced terms, D k with k =,..., n or ln l N with l =,..., n, have been taken into account. The x-space expansion, however, would lead to a gross overestimate if only the terms up to k n+ were known. The convergence in the N-space approach, on the other hand, is much smoother, with a good approximation already reached after n terms. A similar pattern is found for the fourth-order coefficient function c 4,q illustrated in the same manner in the left part of Fig. 6: the expansion in decreasing powers of ln N stabilizes after the fourth term. Based on these results and the higher-order coefficients shown in Table, we expect that the first l logarithms should provide a good estimate up to about the l-th order in α s, but severely underestimate the effect of the coefficient functions of much higher orders. Consequently, 4

16 .5 a c f / f S,q.5 a 3 c 3 f / f S,q xf = -x 3 xf = x.5 -x D k terms full.5 / 3 / 4 5 / 6 D k terms full α S =., N f = x Figure 4: Convolution of the two-loop left and three-loop right contributions to the DIS coefficient functions C,q with a typical input shape. Shown are the full results 7,, and the large-x expansion by successively including the +-distributionsd k, respectively starting withd 3 andd 5. x.5 a c f / f S,q.5 a 3 c 3 f / f S,q xf = -x 3 xf = x.5 -x ln N terms full.5 / 3 / 4 5 / 6 ln N terms full α S =., N f = x Figure 5: As the previous figure, but for large-n expansion in terms of decreasing powers of ln N. x 5

17 the tower expansion should underestimate the corrections towards x, where more and more orders become relevant. This is exactly the pattern shown in the right part of Fig. 6, where the predictions of all effects beyond order α 3 s are compared between the tower expansion and the full exponentiation for the latter again using a minimal-prescription contour 36. Both approaches agree very well, for the chosen input parameters, at x <.93, but start to diverge at x >.95 where the exponentiation is also intrinsically more stable..5 a 4 c 4 f / f S,q xf = x.5 -x 3 3 k=4 k a S c k f / f,q xf = x.5 -x ln N terms 5 7 towers NNLL N 3 LL α S =., N f = x Figure 6: Left: the successive approximations of the four-loop coefficient function c 4,q by the large-n terms specified in Table, illustrated by the convolution with a typical quark distribution. Right: corresponding results for the effect of all orders beyond αs 3 as obtained from the tower expansion with up to seven towers and from the exponentiation up to N 3 LL accuracy. x 6 Summary We have extended the threshold resummation exponents,, 8, 9 for few-parton processes to the fourth logarithmic N 3 LL order collecting the terms α s α s lnn k to all orders in α s. For our reference process, inclusive deep-inelastic scattering DIS, the N 3 LL contributions are specified by two universal expansion parameters: the four-loop cusp anomalous dimension A q,4 and the third-order quantity B q,3 which defines the jet function resumming collinear radiation off an unobserved final-state quark. The former coefficient has not been computed so far, but can be safely expected to have a very small effect of less than %. In fact, the perturbative expansion up to 6

18 A 3, does not exhibit enhanced higher-order corrections, and A 4 can be estimated by Padé approximations. We have calculated the more important second coefficient B q,3 by comparison of the expanded resummation result to our recent third-order calculation of electromagnetic DIS. The perturbative expansion of B q seems to indicate, as far as this can be judged from the first three terms, the onset of a factorial enhancement of the higher-order coefficients. However, the rather large size of the coefficient B q,3 actually stabilizes the logarithmic expansion of the coefficient functions. In fact, the N 3 LL corrections are very small at large scales Q, and even facilitate a reliable prediction of the soft-gluon effects at scales as low as Q 4 GeV corresponding to α s.3 down to very small invariant masses W of the hadronic final state in ep ex, W m p.5 GeV. Thus we expect our results to be useful also for low-scale data analyses using parton-hadron duality concepts. The threshold resummation can also be employed to predict, order by order in α s, the leading ln N contributions to the higher-order coefficient functions. At the level of accuracy reached in the present article for inclusive DIS, the exponentiation fixes the seven highest terms, ln n N with l 6 n k, at all orders k 4 of α s. Already the highest k powers of ln N provide a good estimate of the soft-gluon enhancement of the k-loop coefficient functions at least for k 7, in contrast to the expected, see Ref. 36 worse behaviour of the corresponding expansion in x-space +-distributions. Except very close to threshold, where too many orders in α s become important, the summation of the above seven N-space logarithms to all orders yields a good agreement with the exponentiated coefficient function. This agreement further confirms the minimal prescription 36 used for defining the in principle ambiguous Mellin inversion of the resummation exponential. Besides the standard gauge-boson exchange process, we have also considered DIS by exchange of a scalar directly coupling to gluons. By comparison of the resummation to our unpublished three-loop coefficient function for this process we have derived, for the first time, the second and third order contributions to the coefficient B g governing the jet function of a final-state gluon. The quantity B g, will be employed to extend the NNLL resummation to more processes, once the corresponding NNLO results required to fix the process-dependent large-angle soft contribution become available. Finally we would like to draw attention to the curious relation 4.9 which connects, for both the quark and gluon channels, the second-order splitting functions, jet functions and the two-parton Drell-Yan large-angle soft emissions in a non-trivial manner. Acknowlegments We thank E. Laenen and W. Vogelsang for stimulating discussions. The work of S.M. has been supported in part by the Helmholtz Gemeinschaft under contract VH-NG-5 and by the Deutsche Forschungsgemeinschaft in Sonderforschungsbereich/Transregio 9. The work of J.V. has been part of the research program of the Dutch Foundation for Fundamental Research of Matter FOM. 7

19 Appendix Here we show some key elements of the calculation of the resummation exponents g i presented in Section 3. Useful auxiliary relations for x < are x n ε = ln k x x n ε = i= ε Γn ε+i Γn ε k. A. x n ε x i i!, A. The Mellin transforms of the +-distributions follow from the results for harmonic polylogarithms 3 and are given by Z dz zn z lnk z = k+ k!s,..., N, }{{} k+ A.3 where S m,...,m k N denotes the harmonic sums 9. Eqs. A. A.3 lead to the master formula for the derivation of the functions g i, Z dz zn z i= +aln z n ε = Γn ε+i Γn ε a s β i S,..., N }{{} i+ i+ j j j= j a s β ln Q A.4 with a = a s β /+a s β lnq /µ r. The double sum in Eq. A.4 can be solved to the desired logarithmic accuracy with the algorithms for the summation of nested sums 3 coded in FORM 38. The expansion of the Gamma function in powers of ε for positive integers n reads Γn++ε n!γ+ε = +εs n+ε S, n S n+ε 3 S,, n S, n S, n+s 3 n+ε 4 S,,, n S,, n S,, n + S,3 n S,, n+s, n+s 3, n S 4 n+oε 5. A.5 Finally, with θ i j = for i j and θ i j = else, the sums S,..., N are factorized according to i! S,..., N = S N i + }{{} ii S NS N i + 3 ii i S 3NS N i 3 i + 4 ii i i 3 S 4 N+ S N S N i A.6 µ r θ i ln i Ñ + θ i3ii ζ ln i Ñ + 3 θ i4ii i ζ 3 ln i 3 Ñ + 4 θ i5ii i i 3 ζ 4 + ζ ln i 4 Ñ +..., A.7 where Ñ = Ne γ e and the algebraic properties of harmonic sums have been used 9. 8

20 References G. Sterman, Nucl. Phys. B S. Catani and L. Trentadue, Nucl. Phys. B N. Kidonakis and G. Sterman, Nucl. Phys. B , hep-ph/ R. Bonciani, S. Catani, M.L. Mangano and P. Nason, Nucl. Phys. B , hep-ph/ E. Laenen, Pramana E. Laenen, AIP Conf. Proc W.L. van Neerven and E.B. Zijlstra, Phys. Lett. B E.B. Zijlstra and W.L. van Neerven, Phys. Lett. B E.B. Zijlstra and W.L. van Neerven, Phys. Lett. B S. Moch and J.A.M. Vermaseren, Nucl. Phys. B , hep-ph/99355 S. Moch, J.A.M. Vermaseren and A. Vogt, Phys. Lett. B66 5 3, hep-ph/4 R. Hamberg, W.L. van Neerven and T. Matsuura, Nucl. Phys. B Erratum ibid. B R.V. Harlander and W.B. Kilgore, Phys. Rev. Lett. 88 8, hep-ph/6 4 C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, Phys. Rev. D , hep-ph/366 5 C. Anastasiou and K. Melnikov, Nucl. Phys. B646, hep-ph/74 6 V. Ravindran, J. Smith and W.L. van Neerven, Nucl. Phys. B , hep-ph/335 7 C. Anastasiou, K. Melnikov and F. Petriello, hep-ph/53 8 A. Vogt, Phys. Lett. B497 8, hep-ph/46 9 S. Catani, D. de Florian, M. Grazzini and P. Nason, JHEP , hep-ph/36 J.A.M. Vermaseren, A. Vogt and S. Moch, hep-ph/544, to appear in Nucl. Phys. B S. Moch, J.A.M. Vermaseren and A. Vogt, Nucl. Phys. B688 4, hep-ph/439 A. Vogt, S. Moch and J.A.M. Vermaseren, Nucl. Phys. B69 4 9, hep-ph/44 3 S. Forte and G. Ridolfi, Nucl. Phys. B65 3 9, hep-ph/954 4 E. Gardi and R. G. Roberts, Nucl. Phys. B , hep-ph/49 5 S. Catani, M.L. Mangano and P. Nason, JHEP , hep-ph/ D. de Florian and W. Vogelsang, Phys. Rev. D7 5 44, hep-ph/558 7 T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, Phys. Lett. B , hep-ph/ M. Czakon, Nucl. Phys. B , hep-ph/46 9 J.A.M. Vermaseren, Int. J. Mod. Phys. A , hep-ph/ E. Remiddi and J.A.M. Vermaseren, Int. J. Mod. Phys. A5 75, hep-ph/ S. Moch, P. Uwer and S. Weinzierl, J. Math. Phys , hep-ph/83 S. Moch and P. Uwer, to appear 3 G. P. Korchemsky, Mod. Phys. Lett. A J. Kodaira and L. Trentadue, Phys. Lett. B S.A. Larin, P. Nogueira, T. van Ritbergen and J.A.M. Vermaseren, Nucl. Phys. B , hep-ph/ S. Moch, J.A.M. Vermaseren and A. Vogt, Nucl. Phys. B646 8, hep-ph/9 36 S. Catani, M.L. Mangano, P. Nason and L. Trentadue, Nucl. Phys. B , hep-ph/ A. Vogt, Phys. Lett. B , hep-ph/ J.A.M. Vermaseren, math-ph/5 9