Minimal subtraction vs. physical factorisation schemes in small-x QCD

Transcription

1 Physics Letters B 635 ( Minimal subtraction vs. physical factorisation schemes in small-x QCD M. Ciafaloni ab D.Colferai ab G.P.Salam c A.M.Staśto de a Dipartimento di Fisica Università di Firenze 59 Sesto Fiorentino (FI Italy b INFN Sezione di Firenze 59 Sesto Fiorentino (FI Italy c LPTHE Université Pierre et Marie Curie Paris 6 Université Denis Diderot Paris 7 CNRS UMR Paris 755 France d Physics Department Brookhaven National Laboratory Upton NY 973 USA e H. Niewodniczański Institute of Nuclear Physics Kraków Poland Received 6 January 6; received in revised form 3 March 6; accepted 7 March 6 Available online March 6 Editor: N. Glover Abstract We investigate the relationship of physical parton densities defined by k-factorisation to those in the minimal subtraction scheme by comparing their small-x behaviour. We first summarize recent results on the above scheme change derived from the BFKL equation at NLx level and we then propose a simple extension to the renormalisation-group-improved (RGI equation. In this way we are able the examine the difference between resummed gluon distributions in the Q and MS schemes and also to show MS scheme resummed results for P gg and approximate ones for P qg. We find that due to the stability of the RGI approach small-x resummation effects are not much affected by the scheme-change in the gluon channel while they are relatively more important for the quark gluon mixing. 6 Elsevier B.V. Open access under CC BY license.. Introduction Predictions of perturbative quantum chromodynamics for DGLAP ] evolution kernels in hard processes have been substantially improved in the past few years both by higher order calculations for any Bjorken x ] and by resummation methods in the small-x region 3 4]. However higher order splitting functions are factorisation-scheme-dependent: while the NNLO results and standard parton densities 56] are obtained in the minimal subtraction (MS scheme the resummed ones are in the so-called Q -scheme in which the gluon density is defined by k-factorisation of a physical process. Therefore in order to compare theoretical results or to exploit the small-x results in the analysis of data we need a precise understanding of the relationship of physical schemes based on k-factorisation and of minimal subtraction ones with sufficient accuracy. The starting point in this direction is the work of Catani Hautmann and one of us (M.C. 78] who calculated the leadinglog x (LLx coefficient function R of the gluon density in the (dimensional Q -scheme versus the minimal subtraction one namely ( g (Q (t (t = R γ L g (MS (t t log k µ ᾱ N c s α s π ( * Corresponding author. address: colferai@fi.infn.it (D. Colferai. The label Q referred originally 9] to the fact that the initial gluon defined by k-factorisation was set off-mass-shell (k = Q in order to cutoff the infrared singularities. It turns out ] however that the effective anomalous dimension at scale k Q is independent of the cut-off procedure whether of dimensional type or of off-mass-shell one Elsevier B.V. Open access under CC BY license. doi:.6/j.physletb.6.3.4

2 M. Ciafaloni et al. / Physics Letters B 635 ( where is the moment index conjugated to x thems density g (MS (t = exp ε ᾱ s (t/ ] da a γ L(a factorises a string of minimal-subtraction /ε poles starting from an on-shell massless gluon γ L (ᾱ s / is the LLx BFKL ] anomalous dimension and the explicit form of R will be given shortly. The purpose of the present Letter is to show how to generalise the relation ( to possibly resummed subleading-log levels and to quarks. We first summarize the essentials of such generalisation at next-to-leading log(x (NLx level following the detailed analysis of the BFKL equation in 4 + ε dimensions of two of us ]. We then propose a simple extension of the method to the renormalisation-group-improved (RGI approach 46]. On this basis we show the effect of a Q to MS scheme change on a toy resummed gluon distribution and we calculate a full MS small-x resummed P gg evolution kernel as well as a small-x resummed P qg evolution kernel in an approximation to the MS scheme.. Scheme change to the MS gluon at NLx level Let us first summarize the results of ] for the gluon channel only (N f =. The starting point is the BFKL equation ] with NLx corrections 3] continued to 4 + ε dimensions as described in more detail in ]. In particular we consider running coupling evolution at the level of the one-loop β-function so that β(α s ε= εα s bα s α s (t b ε = e εt b= N c π ( b α s (t = α µ ε α µ e εt + bα µ e εt ε where α µ (gµ ε e εψ( /(4π +ε is normalised according to the MS scheme. Note first that the ultraviolet (UV fixed point of Eq. (3 at α s = ε/b separates the evolution (4 into two distinct regimes according to whether (i α µ <ε/b or (ii α µ >ε/b. In the regime (i α s (t runs monotonically from α s = toα s = ε/b for <t<+ and is thus infrared (IR free and bounded while in the regime (ii α s (t starting from ε/b in the UV limit goes through the Landau pole at t Λ = log( ε/bα µ /ε < and reaches α s = from below in the IR limit. The main result of ] is a factorisation formula for the BFKL gluon density in 4 + ε dimensions. If the gluon is initially on-shell and the ensuing IR singularities are regulated by ε> in the (unphysical running coupling regime (i mentioned above then the gluon density at scale k = µ e t factorises in the form g (Q ( ε (t = N ε αs (t exp { t dτ γ ( α s (τ ; ε } where γ is the saddle point value of the anomalous dimension variable γ conjugated to t and the factor N ε which is perturbative in α s (t and ε is due to fluctuations around the saddle point. The expression of γ for ε> is determined by the analogue of the BFKL eigenvalue function namely at NLx level by the equation ᾱ s (t χ ε ( ( γ+ χ ε ( ( γ χ ε ( ( γ ε ] = where by definition K ε ( ( k γ ε = χ ( ε (γ ( k γ K ε ( ( k γ ε = χ ( ε (γ ( k γ and the detailed form of the kernels is found in Refs. 3] on the basis of Refs. 45].Theresult(5 is proven in ] from the Fourier representation of the solutions of the NLx equation by using a saddle-point method in the limit of small ε = O(bα s which singles out γ as in Eq. (6. Let us now make the key observation that due to the infinite IR evolution down to α s ( = the exponent in Eq. (5 develops /ε singularities according to the identity t dτ γ ( α s (τ ; ε = α s (t dα γ(α; ε α(ε bα ( (3 (4 (5 (6 (7 (8 (9

3 3 M. Ciafaloni et al. / Physics Letters B 635 ( which produces single and higher order ε-poles in the formal bα/ε expansion of the denominator. However such singularities are not yet in minimal subtraction form because we can expand in the ε variable the leading and NL parts of γ as follows: γ(α s ; ε = γ ( ε + α s γ ( ε ] = γ ( + α s γ ( ( + εγ ( + ε α s γ ( + εγ ( +. While the ε = part is already in minimal subtraction form the terms O(ε and higher need to be expanded in the variable ε bα in order to cancel the series of ε-poles generated by the denominator: ε (a ε bα = bα ε bα + ε (b ε bα = b α + bα + ε ε bα and similarly for the higher order terms in ε. Therefore by replacing Eqs. ( and ( into Eq. (5 we are able to factor out the minimal subtraction density in the form g (Q ( ε (t = R ε αs (t exp { α s (t dα α(ε bα γ (MS (α where now the ε-independent MS anomalous dimension is γ (MS (α s = γ(α s ; bα s = γ ( + α s γ ( } ( = R ε αs (t g ε (MS (t + bα s γ ( + α s γ ( + bα s γ ( and contains some NNLx terms related to the ε-dependent ones in square brackets in Eq. (. Correspondingly the coefficient R ε in Eq. ( has a finite ε = limit at fixed values of α µ and α s (t = α µ /( + bα µ t. Therefore we are able to reach the physical UV-free regime (ii and we obtain R (α s (t N (α s (t R( α s (t = exp { ᾱ s (t dα α γ ( ( α ( + αγ ( ( α + bαγ ( ( ] } α which is the result for the R factor at NLx level we were looking for. A few remarks are in order. Firstly the expansion coefficients γ ( and γ ( are simply obtained from the known form of the ε-dependence of χ ε ( (γ χ (γ + εχ (γ + ε χ (γ + O(ε 3 in Eq. (7 while γ ( is not explicitly known because the ε-dependence of χ ε ( (γ in Eq. (8 has yet to be extracted from the literature 45]. We quote the results ] ( (3 (4 ᾱ s χ ( ( γ = γ ( = χ (γ χ (γ ( γ =γ ( ᾱs γ ( = χ (γ χ (γ + χ (γ χ (γ χ (γ (γ χ (γ χ χ 3(γ In particular γ ( together with the LLx form of N = γ L χ ((γl γ L γ ( yields the result of Refs. 78]. ( γ =γ ( ᾱs ( R αs (t ( (t = R γ L = γ L χ ( (γ L exp { ᾱ s (t { } Γ( γ L χ (γ L / = exp{ Γ( + γ L γ L χ (γ γ L ψ( + L] dα α γ L γ ( ( α ] } + NLx } dγ ψ ( ψ ( γ χ (γ (5 (6 (7 (8 (9a (9b

4 M. Ciafaloni et al. / Physics Letters B 635 ( while γ ( and γ ( provide the new NLx contribution to R of ]. Secondly the anomalous dimension in the Q -scheme takes contributions from N only namely γ (Q (α s = γ ( + α s γ ( bα log N (α s s α s and is therefore independent of the kernel properties for ε>. On the other hand by Eqs. (3 and (4 γ (MS is related to R by the expression γ (MS (α s = γ ( + α s γ ( ( + bα log R(α s s α s whose origin is tied up to the identity (. Indeed we have separated terms of order ε or ε into minimal subtraction and coefficient contributions: therefore their t-evolution should cancel out in the ε = limit which is the content of Eq. (. Using Eqs. ( and ( the well-known relations 7] for NLx anomalous dimensions can be extended to subleading levels as generated by the ε-expansion. Thus the difference γ (MS γ (Q = bα s γ ( + α s γ ( + bα s γ ( + log N ] ( log α s is computed up to NNLx level as outlined above even if the dynamical ε = NNLx contributions to the γ s are not investigated here. We conclude that while the anomalous dimension in the Q -scheme (which is roughly a maximal subtraction one only depends on the ε = properties of the BFKL evolution the MS coefficient and anomalous dimension both depend on higher orders in the ε-expansion of the kernel eigenvalue which generate subleading contributions. The result in Eq. ( of ] directly provides the scheme change for the gluon anomalous dimension at NNLx level. 3. Resummed results for the MS gluon splitting function A problem exists concerning the magnitude of the scheme change summarized above in the small-x region. In fact the explicit form of the coefficients N γ ( γ ( γ ( exhibit leading pomeron singularities of increasing weight indicating that a small-x resummation is in principle required for the scheme-change too. As a consequence any resummed evolution model should provide in principle information on the corresponding ε-dependence for a rigorous relation to the MS factorisation scheme a task which appears to be practically impossible. In order to circumvent this difficulty we remark that R in Eq. (9b is directly expressed as a function of the variable γ and that the leading pomeron singularity occurs because of the saddle point identification γ = γ L (ᾱ s (t/. It is then conceivable that such a singularity will be replaced by a much softer one if the effective anomalous dimension variable becomes γ = γ res (ᾱ s (t at resummed level. A replacement similar to this one was used in anomalous dimension space in the study of the scheme-change of ]. In our framework we are able to ensure in general that γ res is the relevant variable by assuming that the Q MS normalisation change occurs in a k-factorised form i.e. by taking the ansatz dγ g (MS (t = (3 πi eγt γ R(γ f (Q (γ where R is a properly chosen γ - and -dependent coefficient and f (Q (γ denotes the unintegrated gluon density in γ -space in the Q -scheme. The latter is directly provided by the ε = small-x BFKL equation possibly of resummed (RGI type. It is then clear that at LLx level R in Eq. (3 takes the saddle point value R(γ L (ᾱ s (t/ which therefore should coincide with the expression (9 in order to reproduce Eq. (. Furthermore the NLx expression (4 can be reproduced too by a properly chosen O( term in the expression of R; and similarly for further subleading terms in the -expansion of R. Therefore Eq. (3 can be made equivalent to Eq. (4 at any degree of accuracy in the logarithmic small-x hierarchy but differs from it at any given order in because the effective anomalous dimension is dictated by f (Q possibly in RGI resummed form and is therefore much smoother than its LLx counterpart. In other words the -expansion of the k-factorized scheme-change (3 contains already some resummation of subleading contributions and is expected to be more convergent than (4 in the small-x region. This encourages us to implement it at leading level ( = in which we have g (MS (t = + ρ(t t f (Q (t dt ( (4 The normalisation factor N takes NLx corrections too which however coincide ] with those obtained by the known fluctuation expansion at ε =.

5 34 M. Ciafaloni et al. / Physics Letters B 635 ( Fig.. (a The function ρ(τ (solid and its asymptotic estimates for τ (dashed. (b Sketch of the fastest convergence contour in the complex γ -plane of the integral in Eq. (5 for τ including the cuts and two saddle points that provide the dominant contributions to ρ for very negative τ. where ρ(τ = + +I dγ πi eγτ γr(γ is pictured in Fig. (a. For t t τ it is close to a Θ-function and for negative τ it oscillates with a damped amplitude for larger τ and with increasing frequency. The difference of g (MS and g (Q involves a weight function (τ ρ(τ Θ(τ distributed around t t which convoluted with f (Q (t includes automatically the RGI resummation effects of the Q -scheme. A word of caution is however needed about the accuracy of (4 in the finite-x region where subleading terms in the -expansion of R in Eq. (3 are needed and are left to future investigations. Even if (τ is in a sense a small quantity because the first two τ -moments of (τ must vanish the numerical evaluation of (4 is delicate because of the large oscillations of ρ in the negative τ region. For τ the fastest convergence contour is shown in Fig. (b where two saddle points γ sp γsp of order γ sp(τ + iexp{ τ +ψ(} are found. A saddle point evaluation ρ(τ { expc(τ sp = (6 π Im + τ τ dτ γ sp (τ } ] χ ( (γ sp (τ provides a good estimate for the contributions from the diagonal parts of the contour while we integrate numerically the remaining part of the contour. The result (6 is τ -independent but the constant of integration c(τ is determined numerically e.g. c( i.5. The function ρ(τ is also computed entirely numerically for τ 7. In order to illustrate the difference between small-x gluon distributions in the Q and MS factorisation schemes we consider a (5 toy gluon density obtained by inserting a valence-like inhomogeneous term f in the RGI equation of 6] as follows 3 f (x t = Ax.5 ( x 5 ( δ(t t µ e t k =.55 GeV (7 and solving the corresponding evolution for the unintegrated gluon density f(xt G(Q k ; x where t log Q /µ.the normalisation A is set so that the inhomogeneous term has a momentum sum-rule equal to /. The solution of the RGI equation approximately maintains the sum-rule for the full resulting gluon though not exactly because of some higher-twist violations. The motivation for using a valence-like inhomogeneous term f (i.e. one that vanishes for x is that when solving the BFKL equation the small-x growth should come from the BFKL evolution rather than from the initial condition. We then define the integrated densities xg (Q ( xq = dt Θ(t t f (x t (8 xg (MS( xq = dt ρ(t t f (x t (9 which are shown in Fig. in comparison to CTEQ (NLO and MRST (NNLO fits for the MS gluon at two scales. Note that we have not attempted to fine-tune the inhomogeneous gluon term to get good agreement at large x since in any case we neglect the quark part of the evolution which is likely to contribute non-negligibly there. 3 We adopt a variant of the NLL B resummation scheme introduced in Ref. 6] where we perform the -shift also on the higher-twist poles of the NLx eigenvalue; we denote it NLL B. The running coupling as a function of the momentum transfer q is cutoff at q = GeV.

6 M. Ciafaloni et al. / Physics Letters B 635 ( Fig.. Toy gluon at scales Q = 4GeV and GeV in the Q and MS schemes compared to MRST4 (NNLO 5] and CTEQ6M (NLO 6]. Atthe higher Q value we also show the MS 3 approximation to the full MS evaluation. Fig. 3. Resummed P gg splitting function in the Q and MS schemes together with the uncertainty band for the Q scheme that comes from varying the renormalisation scale for α s by a factor x µ in the range / <x µ < ; shown for two Q values. One observes that the difference between the Q and MS schemes is modest compared to that between the CTEQ and MRST fits. In both cases the BFKL-evolved gluon is somewhat higher at small x than the CTEQ and MRST fits however not by any more than the uncertainty as estimated from the difference between the CTEQ and MRST fits (furthermore it was our aim to illustrate the impact of the MS scheme change not to fit a particular gluon distribution and we have not attempted to tune the inhomogeneous term and non-perturbative cutoff. Another point to note is that there is no tendency for the MS density to go negative. This is despite the violently oscillatory nature of ρ which might have been expected to lead to significant corrections. This can in part be understood from the approximate form of g (MS d 3 xg (MS 3 ( xq xg (Q ( xq 8ζ(3 xg (Q ( (3 3 dt 3 xq ] which is obtained by expanding the scheme-change in the anomalous dimension variable γ d/dt to first non-trivial order. We can see that this approximation is pretty good in the small-x region for reasonable values of α s (Q = GeV corresponding to an effective γ.5. 4 We can also use our usual techniques 6] for extracting the splitting function itself in the MS-scheme. The results are shown in Fig. 3 where the MS curve is supposed to be reliable in the small-x region only (x because at finite x the -dependence of the scheme change will become important. It appears that the splitting function is more sensitive to the scheme-change than the density itself. At the lower Q value the difference between the MS-scheme and the Q -scheme (with NLL B resummation is nearly the same as the renormalisation scale uncertainty and so might not be considered a major effect. Recall however that 4 At the lower Q valuewedonotshowthems 3 approximation because in general it breaks down for low Q.

7 36 M. Ciafaloni et al. / Physics Letters B 635 ( the scale uncertainty is NNLx whereas the factorisation scheme change is a NLx effect so that while the renormalisation scale uncertainty decreases quite rapidly as Q is increased (right-hand plot the effect of the factorisation scheme change remains non-negligible. At small x most of the difference between the MS and Q splitting functions seems to be due to the MS splitting function rising as a larger power of /x. One can check this interpretation by investigating the factorisation-scheme dependence of the asymptotic behaviours of the splitting function. Using the MS 3 approximation one expects approximately Pgg MS/P Q gg = x c with c (t = MS c Q c d c(t dt 8ζ(3 g (t c (t 3 g (t c (t.9 where the numerical value is given for Q = GeV. An explicit measurement at asymptotically small x gives P MS gg /P Q gg = (.98 ±.4x (.4±..Forx<.3 this accounts for the observed difference between the splitting functions to within 5%. 4. Approximate resummed splitting function for the MS quark Let us now discuss the inclusion of quarks in the resummed small-x flavour singlet evolution. Resummation effects will be included via the unintegrated gluon density as discussed before while the scheme-change to the MS-quark will be an (approximate k-factorised form of the one arising at first non-trivial LLx level which for P qg is NLx. Keep in mind however that since the quarks couple directly to electroweak probes their contribution to DIS-type processes is by no means subleading. In order to better specify the scheme change we first recall 8] that the quark density in a physical Q -scheme corresponding to the measuring process p (e.g. F or F T in DIS which we call p-scheme is defined at NLx level by k-factorisation of some g q q impact factor H (p ε q ε (p (t α µ as follows: d +ε k H (p ε (k k F ε (k. By then working out this convolution in terms of the eigenvalue function H ε (p (γ /γ (γ + ε of H ε (p one directly obtains a NLx resummation formula for the ε-dependent qg anomalous dimension in the p-scheme: g (Q ε (t ] d dt q(p ε where in the ε = limit γ (p qg (t γ (p ( αs (t ; ε qg = α s (th (p (γ L has been obtained in closed form in various cases 8] including sometimes the ε-dependence. The MS-scheme is then related to the p-scheme by the transformation (3 (3 (33 q (p = C (p qq q (MS + C (p qg g (MS where we set C (p qq = and b = atnlx level so that α s (t = α µ e εt. We thus obtain by the definition (33 and by Eq. ( qg (α s ; εr(α s ; ε = ( g ε (MS d (p C qg (α s ; εg (MS ] ε + γ (MS qg (α s dt ] = γ L + ε ˆD C qg (p (α s ; ε + γ qg (MS (α s γ (p (34 (35 where ˆD = α s / α s and γ qg (MS is universal and ε-independent so that the process dependence of γ qg (p is carried by the perturbative scheme-changing coefficient C qg (p. The coefficient C qg (p in Eq. (35 can be formally eliminated by promoting ε to be an operator in α s -space as follows: ε = γ L ˆD so that the square bracket in Eq. (35 in front of C qg (p just vanishes (this procedure is rigorously justified in ]. That implies that the l.h.s. of Eq. (35 and H ε (p (γ are to be evaluated at values of ε γ L = O(α s / which modifies γ qg (MS γ qg (p at relative LLx level. Therefore even if γ qg (MS is by itself a NLx quantity the subtraction of the coefficient part in Eq. (35 affects the scheme change at relative leading order contrary to the gluon case of Eqs. (3 and (. The replacement (36 was used in ] in order to get an exact expression of γ qg (MS at NLx level (that is resumming the series of next-to-leading -singularities α s (tα s (t/] n. The main observation is that by expanding H ε (p (γ in the γ variable around (36

8 M. Ciafaloni et al. / Physics Letters B 635 ( γ = ε with coefficients H (p n d dt q(p ε (t = α s (t (ε and by converting Eq. (3 to γ -space we obtain ( dγ e γt H ε (p (γ g ε (γ = H(ε + H n (p (ε dn dt n n= ( α s (tr ε g ε (MS (t where g ε (γ is the Fourier transform of g ε (MS (t and H(ε H ε (p ( ε turns out to be the universal function ] TR + ε e εψ( Γ ] ( + εγ ( ε H(ε = π 3 ( + ε( + 3 ε H rat (εh tran (ε Γ( + ε which we factorise into parts with rational and transcendental coefficients. Note that the universality of H(ε which is proportional to the residue of the characteristic function H ε (p (γ /γ (γ + ε at the collinear pole γ = ε is due to the interesting fact that one can define 8] a universal ] off-shell g(k q(l splitting function in the collinear limit k l Q for any ratio k /l of the corresponding virtualities. We then realise from Eq. (37 that the terms in the r.h.s. with at least one derivative d/dt are of coefficient type and vanish by the replacement (36. The remaining one proportional to H(ε is universal and by (36 yields the NLx result ] γ (MS ( qg αs (t = α s (th ( γ rat (t ( (t n (t L γ L R n (39 + ˆD + ˆD where we have factorised α s (t by the commutator ˆDα s ]=α s and we have defined the quantity H tran (εr ε = ( ε n R n n= n= which has transcendental coefficients and differs from unity by terms of order (γ L 3 or higher in the ε γ L region 8]. The complicated expression (39 has been evaluated iteratively 8] but not resummed in closed form. However it can be drastically simplified in the k-factorised framework by neglecting the transcendental corrections O(γ 3 on the ground that the resummed anomalous dimension is small. In fact by setting R = H tran = and γ L =ᾱ s /Eq.(39 reduces to the expression γ (MS qg α s (th rat ( ᾱs + ˆD which is just the Borel transform of H rat (ε = n H rat n ( εn = T R 4π = α s (t n H rat n n! + ε ε ]. n (37 (38 (4 (4 (4 Since H rat n γ qg (MS = T R 4π n + 3 ( 3 n ] we obtain from (4 what we call the rational approximation NLx rat (first derived in 8] rat α ( s(tt R e ᾱs + 4π 3 e ᾱs 3. This result can be further interpreted in k-factorised form (43 dq (MS rat = H rat (MS f dt by using the characteristic function H rat (MS (γ = α st R 4π γ (e γ + 3 e 3 γ which yields the result in Eq. (43 at the LL saddle point. Finally since the exponentials in (45 generate translations in t our rough estimate of the resummed P qg is provided by the simple formula d dt q(ms rat (t x = α s(tt R 4π = x g(t + x+ 3 g (t + 3 x ] dz z P qg (MS ( αs (t z ( g t x. z (44 (45 (46a (46b

9 38 M. Ciafaloni et al. / Physics Letters B 635 ( Fig. 4. The MS g q splitting function for two values of α s and in various approximations: at two-loop dashed] the RGI resummed rational one in Eq. (46 solid] and with the addition of the first transcendental correction dash-dotted] the rational NLx approximation dotted] and the complete NLx one dash-dot-dotted]. By replacing in Eq. (46a the resummed gluon density 6] and by performing the necessary deconvolution 6] we obtain the results in Fig. 4 compared to NLO and NLx 8] results in the MS scheme. We are able in this way to judge the magnitude of resummation effects in P qg (MS while we postpone to later work the corresponding estimate in physical schemes of DIS type. A proper comparison of the curves in Fig. 4 can be made only in the small-x region x because Eqs. (39 and (46 do not include the finite-x perturbative terms. We then notice that small-x resummation effects in q rat (MS are sizeable even around x 3 and somewhat larger than the gluonic ones. They are anyway much smaller than the corresponding ones of the NLx result thus showing that the resummed anomalous dimension variable is pretty small as already noticed in the gluon case. We can also check how good the rational resummation is by calculating the effect of the first O(γ 3 transcendental correction which is also shown in Fig. 4. It appears that the difference is indeed not large and anyway much smaller than the difference between the results of NLx rat in Eq. (43 and NLx 8] both shown in Fig. 4. To sum up we have proposed here a k-factorised form of the Q MS scheme-change (Eqs. (3 and (44 which allows a convergent leading log hierarchy because of the smoothness of the resummed anomalous dimension. Applying the leading scheme change to the gluon density and in an approximate way to the quark density we have provided predictions for the gg and qg splitting functions in the MS-scheme and for the corresponding densities. We find that the gluon density itself is rather insensitive to the scheme change while its splitting function is somewhat sensitive. Resummation effects for the MS quark are more important but anyway much smaller than those at NLx level. We are thus confident that the scheme change can be calculated in a reliable way in a fully resummed approach as well. Acknowledgements We thank John Collins for a question about positivity of the gluon in the MS scheme which motivated us in part to push forwards with these investigations. This research has been partially supported by the Polish Committee for Scientific Research KBN Grant No. P3B 8 8 by the US Department of Energy Contract No. DE-AC-98CH886 and by MIUR (Italy. References ] V.N. Gribov L.N. Lipatov Sov. J. Nucl. Phys. 5 (97 438; G. Altarelli G. Parisi Nucl. Phys. B 6 (977 98; Yu.L. Dokshitzer Sov. Phys. JETP 46 ( ] S. Moch J.A.M. Vermaseren A. Vogt Nucl. Phys. B 688 (4 ; S. Moch J.A.M. Vermaseren A. Vogt Nucl. Phys. B 69 (4 9. 3] G.P. Salam JHEP 987 ( ] M. Ciafaloni D. Colferai Phys. Lett. B 45 ( ] M. Ciafaloni D. Colferai G.P. Salam Phys. Rev. D 6 ( ] M.CiafaloniD.ColferaiG.P.SalamA.M.Staśto Phys. Lett. B 576 (3 43; M. Ciafaloni D. Colferai G.P. Salam A.M. Staśto Phys. Rev. D 68 ( ] C.R. Schmidt Phys. Rev. D 6 ( MSUHEP-946 hep-ph/

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