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BNL-NT-06/4 DFF 432/01/06 LPTHE-06-01 hep-ph/0601200 January 2006 6 0 Minimal Subtraction vs. Physical Factorisation Schemes 0 2 in Small-x QCD n a J 5 M. Ciafaloni(a), D. Colferai(a), G.P. Salam(b) and A.M. Sta´sto(c) 2 1 (a) Dipartimento di Fisica, Universita` di Firenze, 50019 Sesto Fiorentino (FI), Italy; v 0 INFN Sezione di Firenze, 50019 Sesto Fiorentino (FI), Italy 0 2 (b) LPTHE, Université Pierre et Marie Curie – Paris 6, Université Denis Diderot – Paris 7, CNRS 1 UMR 7589, 75252 Paris 75005, France 0 6 (c) Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA; 0 / and h H. Niewodniczan´ski Institute of Nuclear Physics, Krako´w, Poland p - p e h : Abstract v i X Weinvestigatetherelationshipof“physical”partondensitiesdefinedbyk-factorisation, to r those in the minimal subtraction scheme, by comparing their small-x behaviour. We first a summarize recent results on the above scheme change derived from theBFKL equation at NLxlevel, andwethen proposeasimpleextension totherenormalisation-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 0 results for P and approximate ones for P . We find that, due to the stability of the gg qg RGI approach, small-x resummation effects are not much affected by the scheme-change in the gluon channel, while they are relatively more sensitive for the quark–gluon mixing. Predictions of perturbative Quantum Chromodynamics for DGLAP [1] evolution kernels in hard processes have been substantially improved in the past few years, both by higher order calculations for any Bjorken x [2] and by resummation methods in the small-x region [3–14]. However, higher order splitting functions are factorisation-scheme dependent: while the NNLO results and standard parton densities [15,16] are obtained in the minimal subtraction (MS) scheme, the resummed ones areintheso-called Q -scheme, inwhich the gluondensity is defined 0 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. ThestartingpointinthisdirectionistheworkofCatani,Hautmannandoneofus(M.C.)[17, 18], who calculated the leading-logx (LLx) coefficient function R of the gluon density in the (dimensional) Q -scheme1 versus the minimal subtraction one, namely 0 α¯ (t) k2 N g(Q0)(t,ω) = R γ s g(MS)(t,ω) , t ≡ log , α¯ ≡ α c (1) L ω µ2 s s π (cid:16) (cid:17) (cid:0) (cid:1) where ω is the moment index conjugated to x, the MS density 1 α¯s(t)/ω da g(MS)(t,ω) = exp γ (a) (2) L ε a " Z0 # factorises a string of minimal-subtraction 1/ε poles starting from an on-shell massless gluon, γ (α¯ /ω) is the LLx BFKL [21] anomalous dimension, and the explicit form of R will be given L s shortly. The purpose of the present Letter is to show how to generalise the relation (1) to possibly resummed subleading-log levels and to quarks. We first summarize the essentials of such gen- eralisation at next-to-leading log(x) (NLx) level, following the detailed analysis of the BFKL equation in 4 + 2ε dimensions of two of us [20]. We then propose a simple extension of the method to the renormalisation group improved (RGI) approach [4,6]. On this basis, we show the effect of a Q to MS scheme change on a toy resummed gluon distribution and we calculate 0 a full MS small-x resummed P evolution kernel as well as a small-x resummed P evolution gg qg kernel in an approximation to the MS scheme. x 1 Scheme change to the MS gluon at NL level Let us first summarize the results of [20] for the gluon channel only (N = 0). The starting f point is the BFKL equation [21] with NLx corrections [22,23] continued to 4+2ε dimensions, as described in more detail in [20]. In particular, we consider running coupling evolution at the level of the one-loop β-function 11N β(α ,ε) = εα −bα2 , b = c (3) s s s 12π so that 1 b 1 b α eεt − = e−εt − , α (t) = µ , (4) α (t) ε α ε s 1+bα eεt−1 s (cid:18) µ (cid:19) µ ε 1The label Q0 referred originally [19] to the fact that the initial gluon, defined by k-factorisation, was set off-mass-shell(k2 =Q2)inordertocutofftheinfraredsingularities. Itturnsout[20],however,thattheeffective 0 anomalousdimension at scale k2 ≫Q2 is independent of the cut-off procedure,whether of dimensionaltype or 0 of off-mass-shell one. 2 where α ≡ (gµε)2e−εψ(1)/(4π)1+ε is normalised according to the MS scheme. µ Note first that, the ultraviolet (UV) fixed point of Eq. (3) at α = ε/b separates the evolu- s tion (4) into two distinct regimes, according to whether (i) α < ε/b or (ii) α > ε/b. In the µ µ regime (i) α (t) runs monotonically from α = 0 to α = ε/b for −∞ < t < +∞ — and is thus s s s infrared (IR) free and bounded —, while in the regime (ii) α (t) starting from ε/b in the UV s limit, goes through the Landau pole at t = log(1 − ε/bα )/ε < 0, and reaches α = 0 from Λ µ s below in the IR limit. The main result of [20] is a factorisation formula for the BFKL gluon density in 4 + 2ε dimensions. If the gluon is initially on-shell, and the ensuing IR singularities are regulated by ε > 0 in the (unphysical) running coupling regime (i) mentioned above, then the gluon density at scale k2 = µ2et factorises in the form t g(Q0)(t,ω) = N α (t),ω exp dτ γ¯ α (τ),ω;ε , (5) ε ε s s −∞ Z (cid:0) (cid:1) (cid:0) (cid:1) where γ¯ is the saddle point value of the anomalous dimension variable γ conjugated to t and the factor N — which is perturbative in α (t) and ε — is due to fluctuations around the saddle ε s point. The expression of γ¯ for ε > 0 is determined by the analogue of the BFKL eigenvalue function, namely at NLx level by the equation (1) χ (γ¯) α¯ (t) χ(0)(γ¯)+ω ε = ω , (6) s ε (0) " χ (γ¯ −ε)# ε where, by definition, K(0) (k2)γ−1−ε = χ(0)(γ)(k2)γ−1 , (7) ε ε K(1) (k2)γ−1−2ε = χ(1)(γ)(k2)γ−1 , (8) ε ε and the detailed form of the kernels is found in Refs. [22,23] on the basis of Refs. [24,25]. The result (5) is proven in [20] 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 ), which singles out γ¯ as in s Eq. (6). Let us now make the key observation that — due to the infinite IR evolution down to α (−∞) = 0 — the exponent in Eq. (5) develops 1/ε singularities according to the identity s t αs(t) dα dτ γ¯ α (τ),ω;ε = γ¯(α,ω;ε) , (9) s α(ε−bα) Z−∞ Z0 (cid:0) (cid:1) whichproducessingleandhigherorderε-polesintheformalbα/εexpansionofthedenominator. 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: α¯ α¯ γ¯(α ,ω;ε) = γ(0) s,ε +α γ(1) s,ε (10) s s ω ω (cid:16)α¯ (cid:17) α¯(cid:16) (cid:17) α¯ α¯ α¯ (0) s (1) s (0) s (1) s (0) s = γ +α γ +εγ +ε α γ +εγ +··· . 0 ω s 0 ω 1 ω s 1 ω 2 ω (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) h (cid:16) (cid:17) (cid:16) (cid:17)i While the ε = 0 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 bα = +1 , (11a) ε−bα ε−bα ε2 b2α2 = +bα+ε , (11b) ε−bα ε−bα 3 and similarly for the higher order terms in ε. Therefore, by replacing Eqs. (10) and (11) into Eq. (5), we are able to factor out the minimal subtraction density in the form αs(t) dα g(Q0)(t,ω) = R α (t),ω exp γ(MS)(α,ω) ε ε s α(ε−bα) Z0 = R (cid:0)α (t),ω(cid:1)g(MS)(t,ω) , (12) ε s ε (cid:0) (cid:1) where now the ε-independent MS anomalous dimension is γ(MS)(α ,ω) = γ¯(α ,ω;bα ) (13) s s s α¯ α¯ α¯ α¯ α¯ (0) s (1) s (0) s (1) s (0) s = γ +α γ +bα γ +α γ +bα γ , 0 ω s 0 ω s 1 ω s 1 ω s 2 ω (cid:16) (cid:17) (cid:16) (cid:17) h (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)i and contains some NNLx terms related to the ε-dependent ones in square brackets in Eq. (10). Correspondingly, the coefficient R in Eq. (12) has a finite ε = 0 limit, at fixed values of α ε µ and α (t) = α /(1+ bα t). Therefore, we are able to reach the physical UV-free regime (ii), s µ µ and we obtain R α (t),ω α¯s(t) dα α α α 0 s ≡ R α (t),ω = exp γ(0) + αγ(1) +bαγ(0) , (14) N α (t),ω s α 1 ω 1 ω 2 ω 0(cid:0) s (cid:1) Z0 h (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)i (cid:0) (cid:1) which(cid:0)is the r(cid:1)esult for the R factor at NLx level we were looking for. (0) (0) A few remarks are in order. Firstly, the expansion coefficients γ and γ are simply 1 2 obtainedfromtheknownformoftheε-dependence ofχ(0)(γ) ≡ χ (γ)+εχ (γ)+ε2χ (γ)+O(ε3) ε 0 1 2 (1) (1) in Eq. (7), while γ is not explicitly known, because the ε-dependence of χ (γ) in Eq. (8) 1 ε has yet to be extracted from the literature [24,25]. We quote the results α¯ s (0) χ γ = 1 (15) ω 0 0 (cid:0) (cid:1) χ (γ) (0) 1 γ = − (16) 1 χ′(γ) 0 (cid:12)γ=γ(0)(α¯s) 0 ω χ (γ)(cid:12)(cid:12) χ (γ)χ′(γ) 1χ2(γ)χ′′(γ) γ(0) = − 2 (cid:12)+ 1 1 − 1 0 . (17) 2 χ′(γ) χ′2(γ) 2 χ′3(γ) 0 0 0 (cid:12)γ=γ(0)(α¯s) 0 ω (cid:12) (cid:12) (0) (cid:12) In particular γ , together with the LLx form of 1 1 N = , γ ≡ γ(0) , (18) 0 γ −χ′(0)(γ ) L 0 L L p yields the result of Ref. [17,18] α¯ (t) 1 α¯s(t) dα α s (0) R α (t),ω = R γ = exp γ +NLx (19a) 0 s L ω γ −χ′(0)(γ ) α 1 ω (cid:16) (cid:17) L L Z0 h (cid:16) (cid:17) i (cid:0) (cid:1) (cid:0) (cid:1) Γ(1−γL)χ0(γL)p 1/2 γL ′ ψ′(1)−ψ′(1−γ′) = exp γ ψ(1)+ dγ , Γ(1+γ )[−γ χ′(γ )] L χ (γ′) (cid:26) L L 0 L (cid:27) (cid:26) Z0 0 (cid:27) (19b) (0) (1) while γ and γ provide the new NLx contribution to R of [20]. 2 1 4 Secondly, the anomalous dimension in the Q -scheme takes contributions from N only, 0 0 namely2 α¯ α¯ ∂logN (α ,ω) γ(Q0)(α ,ω) = γ(0) s +α γ(1) s −bα2 0 s , (20) s 0 ω s 0 ω s ∂α s (cid:16) (cid:17) (cid:16) (cid:17) and is therefore independent of the kernel properties for ε > 0. On the other hand, by Eqs. (13) and (14), γ(MS) is related to R by the expression α¯ α¯ ∂logR(α ,ω) γ(MS)(α ,ω) = γ(0) s +α γ(1) s +bα2 s , (21) s 0 ω s 0 ω s ∂α s (cid:16) (cid:17) (cid:16) (cid:17) whose origin is tied up to the identity (11). Indeed, we have separated terms of order ε or ε2 into minimal subtraction and coefficient contributions: therefore, their t-evolution should cancel out in the ε = 0 limit, which is the content of Eq. (21). Using Eqs. (20) and (21), the well known relations [17] for NLx anomalous dimensions can be extended to subleading levels, as generated by the ε-expansion. Thus the difference ∂logN γ(MS) −γ(Q0) = bα γ(0) +α γ(1) +bα γ(0) + 0 . (22) s 1 s 1 s 2 ∂logα (cid:20) s (cid:21) is computed up to NNLx level as outlined above, even if the dynamical ε = 0 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 0 “maximal” subtraction one) only depends on the ε = 0 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. (22) of [20] directly provides the scheme change for the gluon anomalous dimension at NNLx level. 2 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(0), γ(1), γ(0) exhibit leading 1 1 2 Pomeron singularitiesofincreasing weight, indicating thatasmall-xresummationisinprinciple 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. (19b) 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 α¯ (t)/ω) . It is then conceivable that such a singularity L s will be replaced by a much softer one if the effective anomalous dimension variable becomes (cid:0) (cid:1) γ = γ α¯ (t)/ω) at resummed level. A replacement similar to this one was used in anomalous res s dimension space in the study of the scheme-change of [10]. In our framework, we are able to (cid:0) (cid:1) ensure in general that γ is the relevant variable by assuming that the Q → MS normalisation res 0 change occurs in a k-factorised form, i.e., by taking the “ansatz” dγ 1 g(MS)(t) = eγt f(Q0)(γ) , (23) ω 2πi γR˜(γ,ω) ω Z 2The normalisation factor N0 takes NLx corrections too, which however coincide [20] with those obtained by the known fluctuation expansion at ε=0. 5 where R˜ is a properly chosen γ- and ω-dependent coefficient and f(Q0)(γ) denotes the uninte- ω grated gluon density in γ-space in the Q -scheme. The latter is directly provided by the ε = 0 0 small-x BFKL equation, possibly of resummed (RGI) type. It is then clear that, at LLx level, R˜ in Eq. (23) takes the saddle point value R˜ γ (α¯ (t)/ω),0), which therefore should coincide L s with the expression (19) in order to reproduce Eq. (1). Furthermore, the NLx expression (14) (cid:0) can be reproduced too, by a properly chosen O(ω) term in the expression of R˜; and similarly for further subleading terms in the ω-expansion of R˜. In other words, Eq. (23) can be made equivalent to Eq. (14) at any degree of accuracy in the logarithmic small-x hierarchy, but has the advantage that the effective anomalous dimension is order by order dictated by f(Q0), ω possibly in RGI resummed form, and is therefore much smoother than its LLx counterpart. By the argument above, we expect the ω-expansion of the k-factorized scheme-change (23) to be more convergent than (14) in the small-x region. This encourages us to implement it at leading level (ω = 0), in which we have +∞ g(MS)(t) = ρ(t−t′)f(Q0)(t′) dt′ , (24) ω ω −∞ Z where dγ 1 ρ(τ) = eγτ , (25) 2πi γR(γ) Z0++I is pictured in Fig. 1.a. For τ ≥ 0 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(Q0) involves a weight function distributed around t′ ≃ t which, convoluted with fω(Q0)(t′), includes automatically the RGI resummation effects of the Q -scheme. A word of caution is 0 however needed about the accuracy of (24) in the finite-x region, where subleading terms in the ω-expansion of R˜ in Eq. (23) are needed, and are left to future investigations. 1 γ 0.5 0 Ρ exactHnumericL -0.5 analyticHΤ®-¥L -1 -4 -2 0 2 4 6 Τ=logQ2(cid:144)k2 Figure 1: (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. (25) for τ ≪ −1, including the cuts and two saddle points that provide the dominant contributions to ρ for very negative τ. Even if ∆(τ) ≡ ρ(τ) − Θ(τ) is in a sense a small quantity — because the first two τ- moments of ∆(τ) must vanish — the numerical evaluation of (24) is delicate because of the large oscillations of ρ in the negative τ region. For τ ≪ −1 the fastest convergence contour is ∗ shown in Fig. 1.b, where two saddle points γ ,γ of order γ (τ) ≃ 1+iexp{|τ|+ψ(1)} are sp sp sp 6 found. A saddle point evaluation τ ′ ′ exp c(τ )+ dτ γ (τ ) ρ(τ)| = 2 Im 0 τ0 sp , (26) sp rπ  h 1 χ(0)R′ γ (τ) i  2 0 sp  q provides a good estimate for the contributions from the(cid:0)diagona(cid:1)l partsof the contour, while we integrate numerically the remaining part of the contour. The result (26) is τ -independent, but 0 the constant of integration c(τ ) is determined numerically, e.g., c(−3) ≃ −1.18+i2.15. The 0 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 0 factorisation schemes, we consider a toy gluon density obtained by inserting a valence-like inhomogeneous term f in the RGI equation of [6], as follows3 0 f (x,t) = Ax0.5(1−x)5δ(t−t ) (et0 ≡ k2 = 0.55 GeV2) , (27) 0 0 0 andsolvingthecorrespondingevolutionfortheunintegratedgluondensityf(x,t) ≡ G(Q2,k2;x) 0 where t ≡ logQ2/µ2. The normalisation A is set so that the inhomogeneous term has a momentum sum-rule equal to 1/2. 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. We then define the integrated densities xg(Q0)(x,Q2) = dt′Θ(t−t′)f(x,t′) (28) Z xg(MS)(x,Q2) = dt′ρ(t−t′)f(x,t′) , (29) Z which are shown in Fig. 2 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. One observes that the difference between the Q and MS schemes is modest compared to 0 that between the CTEQ and MRST fits and, in particular, 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) 8ζ(3) d3 xg(MS3)(x,Q2) ≡ xg(Q0)(x,Q2)− xg(Q0)(x,Q2) , (30) 3 dt3 (cid:2) (cid:3) 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 (Q2 = 20 GeV2), corresponding to an effective γ ≃ 0.25.4 s We can also use our usual techniques [26] 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 3We adopt a variant of the NLLB resummation scheme introduced in Ref. [6], where we perform the ω-shift also on the higher-twist poles of the NLx eigenvalue; we denote it NLLB′. The running coupling, as a function of the momentum transfer q2, is cutoff at q2 =1GeV2. 4At the lower Q2 value we do not show the MS3 approximation, because in general it breaks down for low Q2. 7 8 CCSS toy MSbar 16 CCSS toy MSbar 7 CCSS toy Q0 CCSS toy Q0 14 CTEQ6M1 NLO CCSS toy MSbar 6 3 MRST 2004 NNLO 12 CTEQ6M1 NLO 5 MRST 2004 NNLO 2) 2) 10 Q Q x, 4 x, 8 g( g( x 3 x 6 2 4 1 Q2 = 4 GeV2 2 Q2 = 20 GeV2 0 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 10 10 10 10 10 10 10 10 10 10 x x Figure 2: Toy gluon at scales Q2 = 4 GeV2 and 20 GeV2 in the Q and MS schemes, compared 0 to MRST2004 (NNLO) [15] and CTEQ6M1 (NLO) [16]. At the higher Q2 value we also show the MS approximation to the full MS evaluation. 3 0.5 0.4 MSbar scheme MSbar scheme Q0 scheme Q0 scheme 0.4 0.3 0.3 ) ) x x ( ( g g 0.2 g g P P x x 0.2 0.1 0.1 2 2 2 2 Q = 20 GeV Q = 100 GeV α (Q2) = 0.224 α (Q2) = 0.182 s s 0 0 -6 -5 -4 -3 -2 -1 0 -6 -5 -4 -3 -2 -1 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 x x Figure 3: Resummed P splitting function in the Q and MS schemes together with the gg 0 uncertainty band for the Q scheme that comes from varying the renormalisation scale for α 0 s by a factor x , in the range 1/2 < x < 2; shown for two Q2 values. µ µ 8 in the small-x region only (x . 10−1), because at finite x the ω-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 Q2 value the difference between the MS- scheme andtheQ0-scheme (withNLLB′ resummation) isnearly thesameastherenormalisation scale uncertainty and so might not be considered a major effect. Recall however that 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 Q2 is increased (right-hand plot), the effect of the factorisation scheme change remains non-negligible. Atsmallx, most ofthedifferencebetween theMSandQ splittingfunctionsseems tobedue 0 to the MS splitting function rising as a larger power of 1/x. One can check this interpretation by estimating the factorisation-scheme dependence of the asymptotic x−ωc behaviour, using the MS approximation. We find 3 ′′′ dω (t)8ζ(3)g t,ω (t) ∆ω (t) = ωMS−ωQ0 ≃ − c c ≃ 0.029, (31) c c c dt 3 g′ t,ω (t) (cid:0) c (cid:1) where the numerical value is given for Q2 = 20 GeV2, and(cid:0)is in ro(cid:1)ugh agreement with what is seen in the left-hand plot of Fig. 3. 3 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. In order to better specify the scheme change, we first recall [18] that the quark density in a physical Q -scheme corresponding to the measuring process p (e.g. F or F in DIS) — which 0 2 T we call p-scheme — is defined, at NLx level, by k-factorisation of some g → qq¯ impact factor (p) H , as follows: ε q(p)(t) ≡ α d2+2εk′ H(p)(k,k′)F (k′) . (32) ε µ ε ε Z By then working out this convolution in terms of the eigenvalue function H(p)(γ)/γ(γ + ε) ε (p) of H , one directly obtains a NLx resummation formula for the ε-dependent qg anomalous ε dimension in the p-scheme: g(Q0)(t) −1 d q(p)(t) ≡ γ(p) α (t),ω;ε , (33) ε dt ε qg s where, in the ε = 0 limit,(cid:2)γ(p) = α(cid:3) (t)H(p)(γ ) has b(cid:0)een obtain(cid:1)ed in closed form in various qg s 0 L cases [18,20], including sometimes the ε-dependence. The MS-scheme is then related to the p-scheme by the transformation q(p) = C(p)q(MS) +C(p)g(MS) , (34) qq qg where we set C(p) = 1 and b = 0 at NLx level, so that α (t) = α eεt. We thus obtain, by the qq s µ definition (33) and by Eq. (1), −1 d γ(p)(α ,ω;ε)R(α ,ω;ε) = g(MS) C(p)(α ,ω;ε)g(MS) +γ(MS)(α ,ω) qg s s ε dt qg s ε qg s (cid:16) α¯(cid:17) h i = γ s +εDˆ C(p)(α ,ω;ε)+γ(MS)(α ,ω) , (35) L ω qg s qg s h i (cid:0) (cid:1) 9 ˆ (MS) where D = α ∂/∂α and γ is universal and ε-independent, so that the process dependence s s qg (p) (p) of γ is carried by the perturbative, scheme-changing coefficient C . qg qg (p) The coefficient C in Eq. (35) can be formally eliminated by promoting ε to be an operator qg in α -space, as follows: s α¯ ε = −γ s Dˆ−1 , (36) L ω (cid:0) (cid:1)(p) so that the square bracket in Eq. (35) in front of C just vanishes (this procedure is rigorously qg justified in [20]). That implies that the l.h.s. of Eq. (35) and H(p)(γ) are to be evaluated at ε (MS) (p) values of ε ∼ γ = O(α /ω) which modifies γ −γ at relative LLx level. Therefore, even L s qg qg (MS) if γ is by itself a NLx quantity, the subtraction of the coefficient part in Eq. (35) affects qg the scheme change at relative leading order, contrary to the gluon case of Eqs. (13) and (22). (MS) The replacement (36) was used in [20] in order to get an “exact” expression of γ at qg NLx level (that is, resumming the series of next-to-leading ω-singularities ∼ α (t)[α (t)/ω]n). s s The main observation is that, by expanding H(p)(γ) in the γ variable around γ = −ε with ε coefficients H(p)(ε), and by converting Eq. (32) to γ-space, we obtain n d q(p)(t) = α (t) dγ eγtH(p)(γ)g˜ (γ,ω) dt ε s ε ε Z ∞ dn α¯ = H(ε)+ H(p)(ε) α (t)R s,ε g(MS)(t,ω) , (37) n dtn s ω ε ! Xn=1 (cid:16) (cid:16) (cid:17) (cid:17) where g˜ (γ,ω) is the Fourier transform of g(MS)(t,ω) and H(ε) ≡ H(p)(−ε) turns out to be the ε ε ε universal function T 2 1+ε eεψ(1)Γ2(1+ε)Γ(1−ε) H(ε) = R ≡ Hrat(ε)Htran(ε) , (38) 2π 3 (1+2ε)(1+ 2ε) Γ(1+2ε) (cid:20) 3 (cid:21)(cid:20) (cid:21) whichwefactoriseintopartswithrationalandtranscendentalcoefficients. Notethattheuniver- salityofH(ε),whichisproportionaltotheresidueofthecharacteristicfunctionH(p)(γ)/γ(γ+ε) ε at the collinear pole γ = −ε, is due to the interesting fact that one can define [18] a universal [20], off-shell g(k) → q(l) splitting function in the collinear limit k2 ∼ l2 ≪ Q2, for any ratio k2/l2 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 [20] ∞ n α¯ (t) 1 α¯ (t) 1 α¯ (t) γ(MS) α (t),ω = α (t)Hrat −γ s γ s R s , (39) qg s s L ω 1+Dˆ L ω 1+Dˆ n ω (cid:0) (cid:1) (cid:18) (cid:16) (cid:17) (cid:19)Xn=0(cid:18) (cid:16) (cid:17) (cid:19) (cid:16) (cid:17) where wehavefactorisedα (t)bythecommutator [Dˆ,α ] = α andwehavedefinedthequantity s s s ∞ α¯ α¯ Htran(ε)R s,ε = (−ε)nR s , (40) n ω ω (cid:16) (cid:17) Xn=0 (cid:16) (cid:17) which has transcendental coefficients and differs from unity by terms of order (γ )3 or higher, L in the ε ∼ γ region [18,20]. L The complicated expression (39) has been evaluated iteratively [18], but not resummed in closed form. However, it can be drastically simplified in the k-factorised framework by 10