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Can we trust small x resummation? 9 0 Stefano Forte,a Guido Altarelli,b Richard D. Ball c 0 2 aDipartimento di Fisica, Università di Milano and INFN, Sezione di Milano, n Via Celoria 16, I-20133 Milan, Italy a J bDipartimento di Fisica “E.Amaldi”, Università Roma Tre and INFN, Sezione di Roma Tre 9 Via della Vasca Navale 84, I–00146Roma, Italy, CERN, Department of Physics, Theory Division ] h CH-1211 Genève 23, Switzerland p - cSchool of Physics, University of Edinburgh p Edinburgh EH9 3JZ, Scotland e h [ Wereviewthecurrentstatusofsmallxresummation ofevolutionofpartondistributionsandofdeep–inelastic coefficient functions. We show that the resummed perturbative expansion is stable, robust upon different treat- 1 ments of subleading terms, and that it matches smoothly to the unresummed perturbative expansions, with v corrections which are of the same order as the typical NNLO ones in the HERA kinematic region. We discuss 4 different approaches to small x resummation: we show that the ambiguities in the resummation procedure are 9 2 small, provided all parametrically enhanced terms are included in theresummation and properly matched. 1 . 1 0 9 1. The need for small x resummation that typical x values probed at the LHC in the 0 central rapidity region are almost two orders of : The so-called small x regime of QCD is the v magnitudesmallerthanxvaluesprobedatHERA kinematical region in which hard scattering pro- i at the same scale. Hence, small x corrections X cesses happen at a center-of-mass energy which start being relevant even for a final state with r is much larger than the characteristic hard scale a a characteristic electroweak scale M ∼ 100 GeV of the process. An understanding of strong in- (see Fig. 1). teractions in this region is therefore necessary to As is by now well known, perturbative correc- dophysicsathigh–energycolliders. Inthissense, tions become large at small x. Due to the ac- HERAwasthefirstsmallxmachine,andLHCis cidental vanishing of some coefficients, the lead- going to be even more of a small x accelerator. ing large corrections cannot be seen in NLO and In deep–inelastic lepton–hadron interactions, NNLO splitting functions; however,the firstsub- the scale is set by the virtuality of the photon leading correction can already be seen in the Q2, and x = 2Qp.2q = Qs2(1 + O(x)), where s is NNLO splitting functions which have been com- the center–of–massenergy ofthe virtualphoton– puted recently, as well as in NNLO coefficient hadron collision, and the distribution of partons functions: they are large enough to make recent which carry a fraction x of the incoming nucleon NNLOpartonfitsunstableatsmallx(seeFig.2). energyisprobed. Inhadron–hadroninteractions, This suggests dramatic effects from yet higher the scale is set by the invariant mass M2 of the orders,sothesuccessofNLOperturbationtheory finalstate,andx=x1x2,withxi = √Mse±y,sthe at HERA, as demonstrated by the scaling laws center–of–mass energy and y the rapidity of the it predicts [3,4], has been for a long time very hadron–hadron collision. Here, the distribution hard to explain. In the last severalyears this sit- of partons which carry fractions xi of the two in- uation has been clarified, mostly thanks to the coming nucleon energies are probed. This means effortoftwogroups(ABF[5]-[10]andCCSS[11]- 1 2 Stefano Forte, Guido Altarelli, Richard D. Ball Gluon LO , NLO and NNLO LHC parton kinematics 109 10 15 Q2=2 GeV2 Q2=5 GeV2 x = (M/14 TeV) exp(–y) 108 Q1 ,2= M M = 10 TeV NNLNOL Ofi tfit 5 LO fit 10 107 2Q) 106 M = 1 TeV xg(x, 0 5 2V) 105 2Q (Ge 104 M = 100 GeV 10-5 10-4 10-3 10-2 10-1 1 010-5 10-4 10-3 10-2 10-1 30 50 103 Q2=20 GeV2 Q2=100 GeV2 y = 6 4 2 0 2 4 6 40 102 M = 10 GeV 20 101 HERA ftaixrgeedt 2xg(x,Q) 2300 100 10 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 x 10 Figure 1. Comparison of the HERA and LHC 010-5 10-4 10-3x10-2 10-1 1 010-5 10-4 10-3x10-2 10-1 kinematical regions (from Ref. [1]). Figure2. ComparisonoftheLO,NLOandNNLO gluon distributions in the MSTW08 parton fit (from Ref. [2]). [16]),whichhavepresentedafullresummationof evolution equations in the gluon sector, thereby showing that, once all relevant large terms are sions. A comparisonof resummedpredictions for included, the effect of the resummation of terms deep-inelastic structurefunctions obtainedin the which are enhanced at small x is perceptible but TW and ABF schemes for resummation is pre- moderate —comparableinsize to typicalNNLO sented in Ref. [21]. fixed order GLAP corrections in the HERA re- Here,weshallreviewthemainingredientsthat gion. A detailed comparison of the ABF and gointosmallxresummation,withthe aimofun- CCSS approach in the pure gluon (n = 0) case derstandingtheimpactofvariouscontributionon f was presented in Ref. [1], where excellent agree- the final result and its stability upon higher or- ment was found. der perturbative corrections and upon the inclu- This approach has now been generalized by sionofvarioussubleadingcontributions: we shall ABF[17]toafullresummationincludingquarks, taketheABFapproachasabaseline,anddiscuss and including the resummation of deep-inelastic the impact of alternative options. We shall show coefficient functions, so that resummed expres- that small x resummation is very constrained by sions for deep-inelastic structure functions can various requirements, which include momentum be obtained. Progress towards the inclusion conservation, the inclusion of collinear contribu- of quarks has also been made by the CCSS tion and matching to GLAP evolution, and con- group [18], though full results are not yet avail- sistency with the renormalization group. Once able. Meanwhile,analternativesomewhatsimpli- all these requirements are met, further sublead- fied approach has also been developed [19], and ing ambiguities are quite small; however, if some used to perform a fit to deep-inelastic scatter- ofthecorrespondingtermsaremissedout,effects ing data [20]. In this approach, the factoriza- are not negligible as we shall see. The resummed tion scheme is not defined in a fully consistent corrections thus obtained are perturbatively sta- way at the resummed level, and also some con- ble, as demonstrated by the fact that renormal- tributions to the resummation are either not in- ization and factorization scale dependence are cluded, or treated by means of truncated expan- moderate, and decrease with increasing pertur- Can we trust small x resummation? 3 bative order as they ought to. The typical effect t=t by solving the GLAP equation 0 of the resummation in the HERA and LHC re- d gionsiscomparabletothatofNNLOcorrections, G(N,t)=γ(N,α ) G(N,t) (2) s dt but with the opposite sign. In the next section, we shall review the ingre- which at the NkLO sums all terms of order dients which are necessary in order to perform αntn k, to all orders in α . The first step of re- s − s the resummation in the gluon sector. In the sub- summation consists of including, to the NkL log sequent section, we shall summarize the generic level, all contributions to the anomalous dimen- features of the resummed results. In the last sion γ(N,α ) of order αnN (n k), to all orders s s − − section, we shall discuss how quarks and deep- in α , since they correspond to contributions of s inelastic coefficient functions may be included in order αns lnn−k x1 to G(ξ,t). the resummation, and discuss resummed results This inclusion is straightforward at the fixed for deep-inelastic physical observables. coupling level, thanks to the fact that the gluon distribution G(ξ,M) can be expressed in terms 2. The three ingredients of stable resum- of the gluon distribution at ξ =ξ0 by solving the mation BFKL equation d In this section, we discuss the resummation of G(ξ,M)=χ(M,α ) G(ξ,M), (3) s dξ evolution equations when n = 0. In this case, f there is a single parton distribution, the gluon whose kernel χ(M,α ) is simply the inverse distribution G(ξ,t), with ξ ≡ lnx1, t ≡ lnQΛ22. It function of the GLAsP anomalous dimension is convenient to define the Mellin transforms γ(N,α ) [23,5]: s G(N,t) ≡ ∞dξe Nξ G(ξ,t) χ(γ(N,αs),αs)=N. (4) − Z 0 The duality equation (4) maps the perturba- G(ξ,M) ≡ Z ∞dte−Mt G(ξ,t) (1) tive expansion of γ in powers of αs at fixed N in the expansion of χ in powers of α at fixed M −∞ s 2.1. Double–leading expansion and conversely. One can thus construct a double leadingexpansion[22](seeFig.3)whichincludes in χ all terms up to a given order in the expan- sioninpowersofα bothatfixedM andatfixed s αs. Its dual γ can be shown to include terms up M to the same order in the expansion in powers of α both at fixed α and at fixed αs. s N Usingeitherthedouble–leadingχorthedouble leadingγ intheBFKLorGLAPequationrespec- tively leadstoasolutionwhichincludesallterms which are logarithmically enhanced either in 1 x or in Q2 to the given order [6]. The result (see Fig. 4) is close to the GLAP one when M → 0, andclosetotheBFKLonewhenN →0. Because Figure3. Double leadingexpansionoftheGLAP theperturbativeexpansionoftheBFKLkernelis anomalousdimensionγ (left) andthe BFKL ker- very poorly behaved, this resummed result has nel χ (right). poor perturbative stability as N →0. 2.2. Exchange symmetry The gluon distribution G(N,t) can be ex- The perturbative instability of the kernel as pressed in terms of the gluon distribution at N →0canbecuredbyobservingthattheBFKL 4 Stefano Forte, Guido Altarelli, Richard D. Ball Figure 5. The LO and NLO resummed symmetrized double–leadingkernels comparedto the LOandNLOkernelsintheBFKLexpansionand the NLO GLAP kernel. CCSS denotes the corre- Figure 4. The BFKL kernel and its dual GLAP sponding result of Ref. [15] (from Ref. [1]) anomalous dimension computed at LO and NLO intheBFKLexpansion,theGLAPexpansionand follows that χ always has a minimum, because the double–leading expansion. it must go through the value χ = 1 at a pair of values of M which are symmetric about the minimum, located at M = 1. The minimum kernel must be symmetric upon the interchange 2 is preserved even when the symmetry is broken, M → 1−M, due to the symmetry of the three– in which case the two “momentum conservation” gluonvertexupontheinterchangeoftheradiated points at which χ = 1 get shifted to M = 0 and and radiating gluon [13]. This symmetry, which M = 2. One can further show that when the is manifest in the LO BFKL kernel, is however symmetry breaking is removed, the kernel is an broken beyond the LO of the BFKL expansion entirefunctionintheM plane;afterrestoringthe by running coupling correctionand by the choice symmetrybreakingitremainsfreeofsingularities ofDISkinematics[11,13]. Nevertheless,the sym- for ReM >−1 [10]. metry breaking terms can be computed exactly. The perturbative instability of the BFKL ker- Hence, one can symmetrize the double–leading nel is thus completely removed: see Fig. 5. In expansion by undoing the symmetry breaking this Figure we also compare results with those terms (by changing kinematics and argument of obtained by CCSS through a procedure which the running coupling),then symmetrizing the re- is rather different, but shares the following fea- sults,andfinallyrestoringtheoriginalsymmetry– tures: (a) all logarithmically enhanced terms in breaking kinematics and choice of argument of Q2 and 1 are included up to NLO (b) the under- α [10]. x s lying symmetry is implemented up to sublead- The result turns out to be surprisingly stable ing terms. The extreme similarity of the results because of two features: (a) the anomalous di- demonstrates the stability of the procedure. mensionmustsatisfythemomentumconservation constraint γ(1,α ) =0, which by duality implies 2.3. Running coupling s χ(0,α ) = 1; the anomalous dimension γ(N,α ) The double–leading expansion upon which the s s decreases monotonically as N increases as a con- resummation has been based so far sums up all sequence of the fact that a gluon looses momen- terms which are large when α is small, but s tum when radiating. Combining these two, it α ξ = α ln 1 ∼ 1: namely, a contribution of the s s x Can we trust small x resummation? 5 form F(αsξ) to the splitting function is consid- pling correction γβ(k0) is also asymptotically large ered to be of order α0s. Thus, if the computation as ξ → ∞: in fact f(k) Eq. (5) grows as αnξn in is performed at NkL order, relative corrections β0 s comparisontothesplittingfunctioncomputedat are of order αk+1 when α →0 while either x or s s LO in the double–leading expansion. Hence, in α ξ are kept fixed. However, this does not guar- s order to determine the correct ξ → ∞ limit we anteethatcorrectionsremainsmallifξ →∞(i.e. must resum all these terms. This resummation x → 0) with α fixed. It is easy to see [5] that s can be performed exactly for the series of terms any correction which changes the leading (right- which grow fastest as ξ → ∞ [8]: the result can most in the N plane) small N singularity of the be expressedinterms ofAiry functions for a ker- anomalous dimension leads to a contribution to nel which is linear in α , and Bateman functions s the splitting function which blows up as ξ → ∞ for a kernelwith a generic dependence ofα [10]. s in comparison to the lower order, because such a The (asymptotic, divergent) expansion of these contribution changes the asymptotic x → 0 be- functions in powers of α at fixed αs gives back, haviour of the splitting function, and thus of the s N to each order in α , the contribution which upon s parton distribution. inverse Mellin transform grows fastest as ξ →∞ The leading singularity of the anomalous di- to the terms γ(k) Eq. (5). mensionisasimplepoleatN =0intheLO(and β0 NLO) GLAP case. After double leading resummation it becomes a square-root branch cut: the inverse dual Eq. (4) of the quadratic behaviour of the kernel near its minimum. As discussed in Sect. 2.2, the presence of a minimum of the kernel is a generic feature which follows from its symmetry properties and momentum conservation. Theinterceptofthekernelhoweverchanges order by order, and it is only its all–order posi- tionwhichdeterminestheasymptoticsmallxbe- haviour. Hence, higher order corrections to the positionoftheminimumareasymptoticallylarge as ξ → ∞: this suggests that the all-order lo- cation of the minimum of the kernel should be treated as a non-perturbative parameter, to be fitted to the data [6,7]. Figure 6. Quadratic approximation to the NLO However, running coupling corrections change resummed kernel of Fig. 5 and its Bateman this state of affairs. Because the running of the running–coupling resummation. coupling is subleading in ln 1, running coupling x corrections only affect the duality equation (4) It can then be seen that the resummation through a finite number of terms: at the NkL of running coupling corrections changes the na- logarithmiclevel,thedualityrelationiscorrected tureoftheleadingsingularity: the fixed-coupling by a term of the form square-rootbranchcut isturned backinto asim- α ple pole. This is shown in Fig. 6, where we dis- γβ(k0) =(αsβ0)kfβ(k0)(cid:16)Ns(cid:17), (5) play the quadratic approximationto the double– leadingNLOkernelofFig.5,anditsBatemanre- wherethe functionf(k) αs canbe calculatedat summation, i.e. the anomalous dimension which β0 N any given order using su(cid:0)ita(cid:1)ble operator methods isobtainedfromitwhenrunningcouplingcorrec- in terms of the NkLO BFKL kernel [24]. tionstoEq.(5)itareincludedtoallorders(com- Now,it turns out that whenever the fixed cou- puted using the dependence on α of the NLO s pling kernel has a minimum, the running cou- resummed result of Fig. 5). 6 Stefano Forte, Guido Altarelli, Richard D. Ball Hence, after running coupling resummation, in the anomalous dimension on top of its stan- theminimumofthekernelnolongerprovidesthe dard GLAP fixed–order expansion in powers of leadingsmallxsingularity,whichisinsteadgiven α at fixed N can be expanded out perturba- s bythepoleintheBatemananomalousdimension. tively, so that it is possible to obtain a fully re- ThelocationandresidueoftheBatemanpoleare summed expressionby simply combining all con- fully determined by the intercept and curvature tributions, and subtracting the double counting of the minimum of the original kernel, and their terms. The procedure canbe performedorderby dependence on α . orderinperturbationtheory,bystartingwiththe s double–leading expansion of Fig. 3 to any given 3. General features of resummed results order, and then improving it as discussed in the previous section. The resummed anomalous dimension has then schematically the form γrc (α (t),N)=γrc,pert(α (t),N) ΣNLO s ΣNLO s +γB(α (t),N)−γB(α (t),N) s s s −γB(α (t),N)−γB (α (t),N) (6) ss s ss,0 s +γ (α (t),N)+γ (α (t),N), match s mom s where • γrc,pert(α (t),N) is the fixed–coupling re- ΣNLO s summed anomalous dimension displayed in Fig. 5, obtained by duality Eq. (4) from the kernel which in turn is found by symmetrization (Sect. 2.2) of the NLO double– leading kernel (Figs. 3,4); • γB(α (t),N)isthe Batemananomalousdi- s mension Fig. 6; • γB(α (t),N), γB(α (t),N) γB (α (t),N) s s ss s ss,0 s are double counting subtractions between the previous two, namely the contributions to the LO and NLO terms γ(k) Eq. (5) β0 which grow fastest as ξ →∞; • γ subtracts subleading terms which mom wouldotherwisespoilmomentumconserva- tion; • γ subtractsanycontributionwhichde- match viates from NLO GLAP and at large N Figure 7. The resummed splitting function with (whichcorrespondstolargex)doesn’tdrop n = 0 and α = 0.12 with fixed coupling (top) f s at least as 1. and running coupling (bottom). N Note that the last two contributions are formally The crucial feature of the resummation proce- subleading: they are included in order to im- dure summarized in the previous section is that prove the matching to the GLAP anomalous di- ateachstepthe contributionswhichareincluded mension, inthatthey removefromthe resumma- Can we trust small x resummation? 7 tion subleading contributions which may be non- sult. Because this effect is only present at the negligible in the large x region where the resum- running coupling level, it is likely to be due to mation is not supposed to have any effect. the fact that running coupling terms leadto con- In Figure 7 we display the splitting functions tributions which survive in the large x region, obtained from Mellin inversion of the resummed where these termsareformallysubleading (recall anomalous dimension Eq. (7) at the fixed cou- that the running coupling contributions were se- pling level and at the running coupling level — lected on the basis of their behaviour as ξ → ∞ i.e. respectively without and with the inclusion i.e. x → 0). In the ABF approach (but not oftheBatemancontributionγB(α (t),N)andits in the CCSS approach) these contributions are s associate double counting subtractions. The re- subtracted through the term γ (α (t),N) in match s sultisalsocomparedtotheCCSSresultRef.[15] Eq. (7). We have found this subtraction to be (from Ref. [1]). The resummed expansion is seen necessary in order for the resummed results not > to be stable (the LO and NLO results are close), to differ from the fixed–order ones in the x∼0.1 allthemoresoattherunningcouplinglevel. The region. resummedresultmatchessmoothlytothe GLAP > result in the large x∼ 0.1 region, but at small x it is free of the instability which the GLAP expansion shows already at NNLO. The comparisonbetween results obtained with the ABF method discussed here and those by CCSS[15]isilluminatinginvariousrespects. The CCSS approach also includes the three ingredients discussed in the previous section — double– leading resummation, symmetrization of the kernel, and running coupling corrections — but the implementation is rather different. In particular, the resummation of running coupling corrections isobtainedbynumericalsolutionoftherunning– coupling BFKL equation (3) (see Ref. [1]). The closeness of results obtained by CCSS and ABF 0.2 at the fixed coupling level reflects the closeness 0.1 GLAP’NLO of the kernels Fig. 5. The fact that CCSS and 0 ABF results are even closer at the running coupling levels follows from the softening of resum- -0.1 P’rc,res NLO P’Bateman mation effects due to asymptotic freedom. Also, -0.2 the fact that exact numerical resolution of the -0.3 running–coupling BFKL equation (in the CCSS approach) followed by a numerical extraction of (xP)’ 1 2 3 4 5 6 log(1/x) the associate anomalous dimension, and the resummation of the leading running coupling cor- Figure 8. The leading small x singularity vs. α s rections Eq. (5) in terms of a Bateman function (top) and the slope of the splitting function vs. (in the ABF approach) lead to a result which ξ =ln 1 (bottom, from Ref. [25]). < x manifestly coincides for allx∼0.03supports the accuracy of both procedures. More detailed insight into the generic features It is interesting to observe however that the of the resummation can be obtained by study- CCSS resummed result shows a significant devi- ing its small x behaviour. As already mentioned, ation from the unresummed GLAP result even this is largely determined by the position of the > for x ∼ 0.1, which is not seen in the ABF re- leading (rightmost) singularity of the anomalous 8 Stefano Forte, Guido Altarelli, Richard D. Ball dimension in the N plane. For the LO and NLO upon perturbative corrections as x→0. GLAP anomalous dimension this is located at N = 0, while at the resummed level it is plot- 4. Quarks and phenomenology ted as a function of α in Fig. 8: the large and s Going from a resummation of the evolution perturbatively unstable corrections at the BFKL equations at N = 0 to fully resummed physical level(compareFig.4)areturnedintomoremod- f observables requires two ingredients: the inclu- erate and stable correction (compare Fig. 5) by sion of the quark sector in the resummation, and thesymmetrization,andfurtherreducedandsta- resummed partonic cross sections. bilized by the running of the coupling. The relative importance of various contribu- 4.1. The resummed splitting function ma- tions to the resummation is elucidated by com- trix paring in Fig. 8 the slope of the resummed An extension of small x BFKL-like evolution running–coupling splitting function (Fig. 7) with equationstothequarksectorhasbeenattempted respect to ξ to that of the NLO GLAP result, in Ref. [18], but results in a fully consistent fac- and that of the splitting function obtained by in- torization scheme are still not available. How- verseMellintransformoftheBatemananomalous ever, it turns out that this construction is not dimension γB(α (t),N) in Eq. (7). It is clear s necessary for the determination of the matrix of > that for x ∼ 0.1 the slope (and thus, up to a resummed splitting functions. This is due to the constant,thesplitting function)isdeterminedby fact that the resummationonly affects one of the the GLAP result, and for x<∼10 2 by the Bate- − two eigenvectorsof the singlet anomalousdimen- man result[25]. Note that this is despite the fact sionmatrix. Therefore,inordertoobtaincoupled that it is only for tiny values of x <∼ 10−15 that resummed evolution equations for singlet quarks the Batemanasymptoticslopereduces(towithin and gluons it is sufficient to fix the factorization about 10%)to its asymptotic value N (the loca- 0 scheme at the resummed level [17]. tion of the leading singularity) [25]. Indeed,callingγ+ andγ the twoeigenvectors − Hencethe qualitativefeaturesoftheresumma- of the anomalous dimension matrix, if only con- tionareessentiallydeterminedbymatchingtothe tributions to γ which are singular as N → 0 are GLAPresulttheBatemananomalousdimension, included, γ =0 and thus − whichinturnisfullydeterminedbyresummation of the running coupling terms Eq. (5), and thus γgg +γqq =γ+; by the intercept and shape of the minimum of γggγqq =γqgγgq. (7) the symmetrized kernel displayed in Fig. 5 [10]. The insensitivity of these to the details of the re- uptononsingularterms. Furthermore,intheMS summationprocedureexplainsthestabilityofthe andrelatedschemes, γgq = CFγgg (upto nonsin- CA results. Thedominantfeatureoftheresultinthe gular terms) [27]. Hence, combining the determi- HERA region 10−2 >∼ x >∼ 10−4 is a dip in the nationofγ+ withtheknowledgeofγqg intheMS splitting function which results from this match- scheme, which was computed at the resummed ing (see Ref. [16] for an alternative discussion of level in Ref. [27], the anomalous dimension ma- this feature of resummed results). trix is fully determined. A resummed gluon splitting function was also The actual computation of the full splitting presented in Refs. [19,20]. The agreement is rea- function matrix with n 6=0 entails some techni- f sonableatthequalitativelevel,buttheresummed cal complications, which have only recently been splitting function appears to display a stronger solved in Ref. [17]. Firstly, when n 6= 0, the f rise atsmallx anda somewhatmorepronounced eigenvectors of the anomalous dimension matrix dipatintermediatex. Thismaybeduetothefact develop an unphysical singularity at the value of that inRef.[19,20]no symmetrizationofthe ker- N where the two eigenvectors cross. This singu- nel is performed: this, as discussed above, leads larity cancels in the solution to evolution equa- to resummed results which tend to be unstable tions, and the cancellation must be enforced ex- Can we trust small x resummation? 9 an amount which is undetermined beyond fixed NLO, a qualitative comparison shows reasonable agreement. Quarksectorsplitting functions havealsobeen giveninRef.[20](seealsoRef.[21]). Theiragree- ment with those of Fig. 9 is not so good: the P qg splitting function shows a much stronger small x rise, and a sizable deviation from the NLO re- > sult at large x ∼ 0.1. The latter feature can be understood as a consequence of the fact that in Refs.[19,20]nomatchingtolargexisperformed: contributions from the small x resummation in the large x region are not subtracted. The for- mer feature is likely to be due to the fact that results in Ref. [20] are not determined in a fully Figure 9. The splitting function matrix with consistent factorization scheme. In fact, the re- n = 4, α = 0.2. The curves correspond to f s summationis performedinthe Q scheme,but it 0 LO (dotted), NLO (dashed), NNLO (dot-dash), is then combined with MS (or DIS) scheme coef- resummed (solid). The two different resummed ficient functions: in the TW approach, the issue curves (in the gluon sector) correspond to the of the scheme transformation from Q to MS is 0 MS (steeper at small x) and Q MS factorization 0 still unresolved. Because of the aforementioned schemes. interplay between the scheme choice and the resummation of running coupling singularity, this actly throughout the double–leading resumma- inconsistency is likely to affect stronglythe small tion and symmetrization: if it were spoiled by x behaviour. subleading terms, this would lead to a spurious smallxriseofpartondistributions. Furthermore, 4.2. Coefficient function resummation the quark–sector anomalous dimension γ was The leading small x contributions to partonic qg determined in Ref. [27] in the MS scheme, while cross sections are known to all orders for a small the double–leading resummation is most natu- butincreasingnumberofphysicalprocesses: they rally performed in the Q scheme [29,28]: in the were first computed for heavy quark photo–and 0 MSschemetherunningcouplingtermsEq.(5)are electroproduction in Ref. [31] (later extended to factoredorderbyorderinthecoefficientfunction. hadroproductioninRef.[32]),theyhavebeende- Because their resummation determines the lead- termined for deep-inelastic scattering in Ref. [27] ingsmallxbehaviour,asdiscussedinSect.2.3,it and more recently for Higgs production [33] and is more convenient to perform the resummation the Drell-Yan processes [34]. in a scheme where they are included in the split- ExpressionsforcoefficientfunctionsintheNLO ting function. Whereas the scheme change from of the double–leading expansion were already the Q scheme to MS in the pure–gluoncase was constructed in Ref. [7], where, however, the run- 0 workedout previously [29,28], its construction in ning coupling terms discussed in Sect. 2.3 were the presence of quarks is nontrivial [22,17]. still left unresummed, thereby simplifying issues The resummed matrix of splitting functions of scheme dependence, but at the cost of not re- in the MS and Q MS factorization schemes is producing the correct small x behaviour. How- 0 compared to the unresummed result in Fig. 9. ever, running coupling corrections to the resum- Whereas a detailed comparison with the CCSS mation of coefficient functions also grow as ξ → splitting function matrix [18] is not possible, ∞, analogously to the running coupling correc- because CCSS results are given in a scheme tions to splitting functions Eq. 5: they must be which differs from the standard MS scheme by resummed lest physical observables develop un- 10 Stefano Forte, Guido Altarelli, Richard D. Ball Figure 10. The matrix of coefficient functions Figure 11. The ratio of the resummed (solid) or with nf = 4, αs = 0.2. The curves correspond NNLO (dot-dashed) to NLO singlet quark and to NLO (dashed), NNLO (dot-dash), resummed gluondistributions as afunction ofx atthe scale (solid). The two different resummed curves (in Q =5GeV(top)andasafunctionofQatfixed 0 the gluon sector) correspond to the MS (steeper x = 10 2, 10 4, x = 10 6 (bottom; the small- − − − at small x) and Q0MS factorization schemes. est x is the lowest curve in the resummed case and the highest at NNLO). The ratios are deter- physical singularities which leads to a spurious mined assuming that the structure functions F 2 small x growth [30]. and F are kept fixed at the scale Q = 5 GeV. L 0 The resummation can be performed ex- Thetwodifferentresummedcurvescorrespondto actly [30] when the double–leading expression of the MS (smalleratsmallx) andQ MS factoriza- 0 the coefficient function is known in closed form. tion schemes. This is however not the case for the F deep- 2 inelastic coefficient function in the MS scheme, expressions for other physical processes, for the for which only a series expansion generated by a determination of parton distributions at the re- recursion relation is available [27]. In such case, summed level. the dominantrunningcouplingcorrectionstothe An estimate of the impact of resummation on coefficient function can be resummed through a partondistributionscanbeobtainedbyfirstcom- divergent asymptotic expansion, which may be puting the structure functions F and F with summed by the Borel method [17]. The ensu- 2 L some typical “toy” set of NLO parton distribu- ing resummed coefficient functions are displayed in Fig. 10. Resummed coefficient functions were tions (PDFs), and then assuming that the struc- turefunctionsarekeptfixedatsomescale: thisis alsopresentedinRef.[20](alsoincludingrunning then enough to determine the resummed singlet couplingresummation)andarequalitativelysim- quark and gluon distribution at that scale. The ilar,thoughadetailedcomparisonishamperedby effect on PDFs is close to that which would be the fact that the factorization scheme used there obtained if PDFs were determined from a fit to is different (DIS instead of MS). DISdatamostlyclusteredaroundthatscale. Re- 4.3. Parton distributions and structure sultsforthetypicalHERA(compareFig.1)scale functions choice of Q2 = 25 GeV2 are shown in Fig. 11, Combiningtheingredientsdiscussedsofaritis wherewedisplaythesingletquarkandgluondis- possible to determine resummed predictions for tributionsasafunctionofxatthisstartingscale, deep–inelastic structure functions. Eventually, andthenasafunctionofQ2 forvariousxvalues. these should be used, together with resummed Results are shown as a ratio of the resummed or