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IPPP/09/78 DCPT/09/156 23rd November 2009 NLO prescription for unintegrated parton distributions 0 1 A.D. Martina, M.G. Ryskina,b and G. Watta 0 2 a Institute for Particle Physics Phenomenology, University of Durham, DH1 3LE, UK n a b Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg, 188300, Russia J 6 ] h p - Abstract p e h Weshowhowpartondistributionsunintegratedoverthepartontransversemomentum, [ kt,maybegenerated,atNLOaccuracy,fromtheknownintegrated(DGLAP-evolved)par- 2 v ton densities determined from global data analyses. A few numerical examples are given, 9 which demonstrate that sufficient accuracy is obtained by keeping only the LO splitting 2 functions together with the NLO integrated parton densities. However, it is important to 5 5 keep the precise kinematics of the process, by taking the scale to be the virtuality rather . 9 than the transverse momentum, in order to be consistent with the calculation of the NLO 0 splitting functions. 9 0 : v i 1 Introduction X r a Conventionally, hardprocessesinvolving incomingprotons, suchasdeep-inelasticlepton–proton scattering, are described in terms of scale-dependent parton distribution functions (PDFs), a(x,µ2) = xg(x,µ2) or xq(x,µ2). These distributions correspond to the density of partons in the proton with longitudinal momentum fraction x, integrated over the parton transverse momentum up to k = µ. They satisfy DGLAP evolution in the factorisation scale µ, and are t determined from global analyses of deep-inelastic and related hard-scattering data. However, for semi-inclusive processes, parton distributions unintegrated over k are more appropriate. t For example, unintegrated parton distributions play an important rôle in the description of the transverse momentum dependence of different inclusive hard processes, such as inclusive jet production in deep-inelastic scattering (DIS) [1], electroweak boson production [2], prompt photon production [3], azimuthal correlations in high-p dijet production [4], etc. Moreover, T the exclusive cross sections for vector meson photoproduction [5] or central exclusive diffractive (cid:3) (cid:3) (cid:3) (cid:3) (cid:13) (a) (cid:13) (b) (cid:13) (cid:13) ' + + 2 2 fq(x;z;kt;(cid:22) ) Figure 1: A schematic diagram of inclusive jet production in DIS at LO which shows the approximate equality between, on the left-hand side (a), the formalism based on the doubly-unintegrated quark distribution, f (x,z,k2,µ2), where the off-shell quark has virtuality q t k2/(1 z), and on the right-hand side (b), the conventional QCD approach using integrated − t − parton densities, a(x,µ2), where the incoming partons are on-shell. Higgs bosonproduction[6] arealso calculated in termsof the unintegrated partondistributions. Infact, so-called‘k -factorisation’wasoriginallyestablished[7]forheavy-quarkpairproduction, t so that the cross section for pp QQ¯X is of the form: → dx dx dk2 dk2 σ(pp QQ¯X) = 1 2 1,t 2,t f (x ,k2 )f (x ,k2 ) σˆ(sˆ,M2,k2 ,k2 ), (1) → x x k2 k2 g 1 1,t g 2 2,t 1,t 2,t Z 1 Z 2 Z 1,t Z 2,t where the f are the gluon densities of the incoming protons, unintegrated over k2 , such that g i,t f (x,k2)(dx/x)(dk2/k2) is the number of gluons in the longitudinal and transverse momentum g t t t intervals from x to x + dx and from k2 to k2 + dk2, respectively, and σˆ is the gg QQ¯ t t t → subprocess cross section. In general, the unintegrated distributions, f (x,k2,µ2), depend on two hard scales, k and a t t µ, and so the evolution is much more complicated. The additional scale µ plays a dual rôle. On the one hand it acts as the factorisation scale, while on the other hand it controls the angular ordering of the partons emitted in the evolution. In Refs. [1, 8] a prescription was given which allows the unintegrated distributions to be determined from the well-known integrated distributions. The prescription was based on the fact that due to strong k ordering, inherent in t DGLAP evolution, the transverse momentum of the final parton is obtained, to leading-order (LO) accuracy, just at the final step of theevolution. Thus thek -dependent distribution canbe t calculated directly from the DGLAP equation keeping only the contribution which corresponds to a single real emission, while all the virtual contributions from a scale equal to k up to the t final scale µ of the hard subprocess are resummed into a Sudakov-like T-factor. The factor T describes the probability that during the evolution there are no parton emissions. Theidea isthat, considering inclusive jet productioninDIS,forexample, theLOdiagramat (α )computedusing k -factorisationwill alreadyinclude, toagoodapproximation, themain em t O effects (which are of kinematical origin) of the conventional LO QCD diagrams at (α α ) em S O computed using collinear factorisation. This approximate equality is shown schematically in 2 Fig. 1. The cross section for any hard process is then determined by convoluting the unintegrated parton distributions with the off-shell subprocess cross sections where the incoming partons have virtuality k2. To be precise, it is necessary to also take account of the frac- − t tion z of the light-cone momentum of the parent parton carried by the ‘unintegrated’ parton, that is, to use ‘doubly-unintegrated’ parton distributions [1], f (x,z,k2,µ2), where the off- a t shell parton now has virtuality k2/(1 z). The doubly (or ‘fully’) unintegrated distributions − t − preserve the exact kinematics of the partonic subprocess (see also Ref. [9]) and we speak of (z,k )-factorisation. Here, we wish to extend the ‘last step’ LO prescription for determining t unintegrated parton distributions to next-to-leading order (NLO).1 First, in Section 2 we recall the LO prescription, then we extend it to NLO in Section 3. We show numerical results in Section 4 and conclude in Section 5. More details of the NLO derivation are given in an Appendix. 2 LO prescription for unintegrated parton distributions It is useful to review how LO unintegrated parton distributions, f (x,k2,µ2), may be calcu- a t lated from the conventional (integrated) parton densities, a(x,µ2), in the case of pure DGLAP evolution. As usual, we adopt a physical (axial) gauge, which sums over only the transverse gluon polarisations, so that the ladder-type diagrams dominate the evolution. Recall that the number of partons in the proton with longitudinal (or, to be precise, light-cone plus2) momentum fraction between x and x + dx and transverse momentum k between zero and the t factorisation scale µ is a(x,µ2)(dx/x), whereas the number of partons with longitudinal momentum fraction between x and x + dx and transverse momentum squared between k2 and t k2 + dk2 is f (x,k2,µ2)(dx/x)(dk2/k2). Thus the unintegrated distributions must satisfy the t t a t t t normalisation relation,3 µ2dk2 a(x,µ2) = t f (x,k2,µ2), (2) k2 a t Z0 t where a(x,µ2) = xq(x,µ2) or xg(x,µ2). We start from the LO DGLAP equations evaluated at a scale4 k2: t ∂xq(x,k2) α (k2) 1 x 1 t = S t dzP (z)b ,k2 xq(x,k2) dζ P (ζ) , (3) ∂logk2 2π qb z t − t qq t "bX=q,gZx (cid:16) (cid:17) Z0 # 1We do not consider fully unintegrated distributions at NLO. This would require the recalculation of the NLO DGLAP splitting kernels in fully unintegrated form, which is a necessary ingredient for a NLO parton shower Monte Carlo (see Ref. [10] for work in this direction). 2The plus and minus light-cone components of a parton with 4-momentum k are k± k0 k3. 3Note that the exact value of the upper limit in Eq. (2), and the possible non-logar≡ithmi±c tail for k > µ, t are beyond NLO accuracy. 4Usually DGLAP evolution is written in terms of the virtuality k2, but at LO level this is the same. The difference is a NLO effect. We examine the difference in detail in Section 4. 3 ∂xg(x,k2) α (k2) 1 x 1 t = S t dzP (z)b ,k2 xg(x,k2) dζ (ζP (ζ)+n P (ζ)) , ∂logk2 2π gb z t − t gg F qg t "bX=q,gZx (cid:16) (cid:17) Z0 # (4) where b(x,k2) = xq(x,k2) or xg(x,k2) and P (z) are the unregulated LO DGLAP splitting t t t ab kernels. The two terms on the right-hand sides of Eqs. (3) and (4) correspond to real emission and virtual contributions respectively. The virtual (loop) contributions may be resummed to all orders by the Sudakov form factor, µ2dκ2 α (κ2) 1 T (k2,µ2) exp t S t dζ P (ζ) , (5) q t ≡ −Zkt2 κ2t 2π Z0 qq ! µ2dκ2 α (κ2) 1 T (k2,µ2) exp t S t dζ (ζP (ζ)+n P (ζ)) , (6) g t ≡ −Zkt2 κ2t 2π Z0 gg F qg ! which, recall, give the probability of evolving from a scale k to a scale µ without parton t emission. Thus, from Eqs. (2)–(4), the unintegrated distributions have the form: α (k2) 1 x f (x,k2,µ2) = T (k2,µ2) S t dzP (z)b ,k2 . (7) a t a t · 2π ab z t bX=q,g Zx (cid:16) (cid:17) This equation can be rewritten as ∂ f (x,k2,µ2) = a(x,k2)T (k2,µ2) , (8) a t ∂logk2 t a t t (cid:2) (cid:3) since the derivative ∂T /∂logk2 cancels the last terms in Eqs. (3) and (4). a t This definition is meaningful for k > µ , where µ 1 GeV is the minimum scale for which t 0 0 ∼ DGLAP evolution of the conventional parton distributions, a(x,µ2), is valid. Integrating over transverse momentum up to the factorisation scale we find that µ2dk2 t f (x,k2,µ2) = a(x,k2)T (k2,µ2) kt=µ Zµ20 kt2 a t t a t kt=µ0 = a(cid:2)(x,µ2) a(x,µ2)T(cid:3)(µ2,µ2). (9) − 0 a 0 Thus, the normalisation condition Eq. (2) will be exactly satisfied if we assume5 1 1 f (x,k2,µ2) = a(x,µ2)T (µ2,µ2), (10) k2 a t µ2 0 a 0 t (cid:12)kt<µ0 0 (cid:12) (cid:12) so that the density of partons in the pro(cid:12)ton is constant for kt < µ0 at fixed x and µ. So far, we have ignored the singular behaviour of the unregularised splitting kernels, P (z) qq and P (z), at z = 1, corresponding to soft gluon emission. These soft singularities cancel gg 5A more complicated extrapolation of the contribution from kt < µ0, which ensures continuity of fa at kt =µ0, was given in Ref. [2]. 4 between the real and virtual parts of the original DGLAP equations, Eqs. (3) and (4). So it was enough to replace an explicit cutoff by the so-called ‘+’ prescription. Now, when real emission (which changes the k of the parton) and the virtual loop contribution (which does not t change the kinematics of the process) are treated separately, we must introduce the infrared cutoff explicitly. The singularities indicate a physical effect that we have not yet accounted for. It is the angular ordering caused by colour coherence [11], which implies an infrared cutoff on the splitting fraction z for those splitting kernels where a real gluon is emitted in the s-channel. Indeed, the polar angle θ ordering ... < θi−1 < θi < θi+1 < ... (11) implies the ordering of the ratios ξ = p−/p+, that is, of the rapidities η = (1/2)logξ = i i i i − i logtan(θi/2). The light-cone components of the massless gluon-i momentum (pi = ki−1 ki) − − satisfy p− = p2 /p+, while the ratio of the momenta in the proton direction is given by the i i,t i ratio of the momentum fractions carried by the t-channel gluons (see Fig. 2 of Ref. [1]), z = i ki+/ki+−1 = xi/xi−1. If we denote p¯i = pi,t/(1−zi) then we obtain ξi/ξi−1 = (p¯i/zi−1p¯i−1)2 > 1 that is zi−1p¯i−1 < p¯i. (12) In the last step the angle is limited by the value of factorisation scale µ. If we choose µ = Q for DIS in the Breit frame, then the last inequality reads [1] z p¯ < µ, (13) n n which leads to z < µ/(µ+k ) and provides the inequality θ < θ . t i µ Thus the factorisation scale, µ, is entirely determined from the kinematics of the subprocess at the ‘top’ of the evolution ladder [1]. So we define the infrared cutoff to be k t ∆ , (14) ≡ µ+k t then the precise expressions for the unintegrated quark and gluon distributions are α (k2) 1 x x x x f (x,k2,µ2) = T (k2,µ2) S t dz P (z) q ,k2 Θ(1 ∆ z)+P (z) g ,k2 q t q t 2π qq z z t − − qg z z t Zx h (cid:16) (cid:17) (cid:16) (1(cid:17)5)i and α (k2) 1 x x x x f (x,k2,µ2) = T (k2,µ2) S t dz P (z) q ,k2 +P (z) g ,k2 Θ(1 ∆ z) . g t g t 2π gq z z t gg z z t − − Zx "Xq (cid:16) (cid:17) (cid:16) (cid:17) # (16) By unitarity the same form of the cutoff, ∆(κ ), must be chosen in the virtual part. Thus t we insert Θ(1 ∆ ζ) into the Sudakov factor for those splitting functions where a gluon is − − 5 emitted in the s-channel and Θ(ζ ∆) where a gluon is emitted in the t-channel.6 Then − µ2dκ2 α (κ2) 1 T (k2,µ2) = exp t S t dζP (ζ)Θ(1 ∆ ζ) . (17) q t −Zkt2 κ2t 2π Z0 qq − − ! Recall that the exponent of the gluon Sudakov factor was already simplified by exploiting the symmetry P (1 ζ) = P (ζ), so that the gluon Sudakov factor is qg qg − µ2dκ2 α (κ2) 1 T (k2,µ2) = exp t S t dζ [ ζP (ζ)Θ(1 ∆ ζ)Θ(ζ ∆)+n P (ζ) ] , g t −Zkt2 κ2t 2π Z0 gg − − − F qg ! (18) where n is the active number of quark–antiquark flavours into which the gluon may split. F Note that in the expression for the unintegrated distribution, Eq. (8), the derivative of a with respect to logk2 gives the right-hand side of the usual DGLAP equation, which contains t both the real emission (which we are looking for) and the virtual loop contribution. However the virtual contribution is cancelled by the derivative of T with respect to logk2 . a t In the next section we introduce an analogous prescription for unintegrated parton distributions which may be justified at NLO accuracy. 3 Unintegrated parton distributions at NLO First, we discuss the structure of the NLO contribution. The original definition was that the amplitude was decomposed into the perturbative sum ∞ ∞ A = A αk (α logQ2)n, (19) n,k S S k n=1 XX where k = 0,1,2,... denote the LO, NLO, NNLO, ... contributions. However, nowadays the analogousdecompositionisusedfortheanomalousdimensions, γ , whichdescribetheevolution N in terms of the OPE operators, e.g. Mellin moments, M , N M (Q) = M (Q )(Q2/Q2)γN, (20) N N 0 0 rather than for the amplitude. That is, M is written in the form N n 1 M (Q) = M (Q ) exp log(Q2/Q2) cNαk = M (Q ) log(Q2/Q2) cNαk . N N 0 0 k S N 0 0 n! k S ! ! k n k X X X (21) Thus the Feynman diagrams which describe DGLAP evolution have the structure shown in 6The lower cutoff is beyond the LO accuracy and was simply introduced to make the formulation more symmetric. It is not included in the NLO prescription of Section 3. 6 Figure 2: The ladder diagram describing DGLAP evolution at NLO. C(1) is the appropriate NLOcoefficient function, andP(0) andP(1) aretheappropriateLOandNLOsplittingfunctions. Fig. 2. We have strong ordering of the transverse momenta, ki,t ki−1,t, between the splitting ≫ functions, but no ordering in the transverse momenta of the t-channel partons which enter the NLO (NNLO, ...) splitting functions. ′ Here, we discuss NLO. If we denote the transverse momentum in the NLO loop as k , then i,t we have the ordering ′ k k k k . (22) i,t ≪ i,t ∼ i+1,t ≪ i+2,t The ‘parton’ of the k -factorisation approach at NLO is the t-channel quark or gluon placed t between the coefficient and splitting functions. That is, the parton labelled as k in Fig. 2. n There should be strong k ordering between the loops which correspond to the NLO coefficient t function and to the NLO splitting function. Otherwise the uppermost two loops should be assigned to the NNLO coefficient function. We seek the prescription which gives the unintegrated k distribution of the parton to n,t NLO accuracy. To do this we have to extend the formalism of the last section in two steps. First, we have to replace the LO splitting functions P (z) by the corresponding ‘LO+NLO’ ab splitting functions [12, 13] and the LO ‘global’ parton densities a by the NLO parton densities. Secondly, we must take care of the precise value of the scale at which the parton densities a are measured, and of the limits in the integrations over the terms which include the LO splitting functions. In the DGLAP evolution equation the current scale k2 is not exactly equal to the 7 parton transverse momentum k , instead we have7 t k2 k2 = t . (23) 1 z − At LO level, or at very small z, this scale difference is negligible. However, to reach NLO accuracy we have to account for this effect, at least in the LO part of the splitting functions. Thus, recalling Eq. (7), at NLO we have 1 α (k2) x k2 fNLO(x,k2,µ2) = dz T (k2,µ2) S P˜(0+1)(z) bNLO ,k2 = t Θ(1 z k2/µ2), a t a 2π ab z 1 z − − t Zx b=q,g (cid:18) − (cid:19) X (24) where P˜(0+1) = P˜(0) +(α /2π)P˜(1), and S P˜(i)(z) = P(i)(z) Θ(z (1 ∆)) F(i) δ p (z), (25) ab ab − − − ab ab ab (i) with strength of the z 1 singularity in P given by [13] → ab F(0) = C , F(1) = C T N 10 +C (π2 67) , (26) qq F qq − F R F 9 A 6 − 18 F(0) = 2C , F(1) = 2C (cid:16) T N 10 +C (π2 67)(cid:17) . (27) gg A gg − A R F 9 A 6 − 18 (cid:16) (cid:17) Here i = 0,1 denote the LO and NLO contributions, respectively, and p (z) = (1+z2)/(1 z) qq − and p (z) = z/(1 z)+(1 z)/z +z(1 z). More details are given in the Appendix. gg − − − The last term in Eq. (25) accounts for the coherence and eliminates the 1/(1 z) singularity − inthesplittingfunctioncausedbytheemissionofonesoftgluon,whichviolatesangularordering Eqs. (13,14). Recall that at NLO, the only strong singularity in P comes from the vertex and ab self-energy corrections to single gluon emission, see Table 1 of Ref. [12]. The contribution from the emission of two ‘real’ gluons has (after the usual subtraction of that generated by two LO splitting functions) no 1/(1 z) singularity, but only the ‘soft’ integrable term proportional to − log(1 z). Note that, in contrast to Eq. (7), at NLO T and α occur inside the z integration a S − as k2 = k2/(1 z). t − The final point is to consider the Sudakov factors8 T which resum the virtual DGLAP a contributions during the evolution from k2 to µ2. At the NLO we get µ2dκ2 α (κ2) 1 T (k2,µ2) = exp S dζ ζ[P˜(0+1)(ζ)+P˜(0+1)(ζ)] , (28) q − κ2 2π qq gq Zk2 Z0 ! 7Eq. (23) assumes that the mass of the partonic system produced by the splitting is zero. This is true at LO, but not at NLO, where a pair of massless partons may be created. However, to account for non-zero mass of the pair, in the NLO splitting function, is a NNLO correction, which is beyond our NLO accuracy. 8Note that, at NLO, the splitting function P contains terms which change the flavour of the quark, P . qq qiqj So now T of Eqs. (17,28) must include a sum over the flavour of the new quark i, and similarly for the sum q over the initial quark b=q in Eq. (24). j 8 µ2dκ2 α (κ2) 1 T (k2,µ2) = exp S dζ ζ[P˜(0+1)(ζ)+2n P˜(0+1)(ζ)] , (29) g − κ2 2π gg F qg Zk2 Z0 ! where, at NLO, we must take P˜(0+1) = P˜(0) +(α /2π)P˜(1) and S κ t ∆ = with κ = κ2(1 ζ). (30) t µ+κ − t p The theta functions in Eq. (25) again provide the correct angular ordering for soft gluon emission. If κ µ, the effect of ∆ in the non-singular part of P˜ is negligible. The region of κ close t t ≪ to µ should be specified by the convention of how to separate the NLO coefficient function (that is, the hard matrix element) and the unintegrated parton distributions of the k -factorisation t approach. In particular, the diagram corresponding to the parton self-energy, or to large angle gluon emission, could be assigned either way; that is, to the parton distribution or to the coefficient function. Our Θ functions correspond to strong angular ordering. Our prescription means that the contribution coming from angles θ > θ should be included in the coefficient µ function, while the lower θ domain should be assigned to the unintegrated parton distribution. The angle θ is defined below Eq. (13). µ 4 Numerical results at NLO For simplicity, we concentrate on the singlet evolution with n = 3 and we use the corre- F sponding MSTW 2008 PDFs [14] and α evolved with three fixed flavours. First, in Fig. 3 S we show the DGLAP splitting functions at LO and NLO given by Eq. (25). It is seen that the NLO corrections to the splitting functions are relatively small in comparison with the LO contributions. The largest NLO correction is perhaps in P at small z, where at LO there is qg no 1/z term as z 0. In Fig. 4 we present the unintegrated PDFs as a function of k2 at → t µ2 = 104 GeV2 for x = 0.1, 0.01, 0.001 and 0.0001. The corresponding plot at µ2 = 100 GeV2 ¯ is shown in Fig. 5. Here, q is the quark singlet distribution, q = u+d+s+u¯ +d+s¯. Since the LO and NLO splitting functions are similar, the unintegrated PDFs calculated using the full NLO framework are close to the results keeping only the LO part of the splitting functions: compare the dotted and dashed curves in Figs. 4 and 5. If we were to use the LO (rather than NLO) integrated PDFs, then we obtain the unintegrated distributions shown by the continuous curves. We see that there is a sizeable enhancement of the unintegrated gluon at very small x, that is x . 10−3. This simply reflects the well-known difference between the integrated LO and NLO gluon distributions at small x.9 The kink in the k distributions at relatively large t k is due to the presence of two cutoffs in the expressions for the unintegrated distributions. t One cutoff, Θ(1 ∆ z), accounts for the coherence effect, which leads to angular ordering. − − The other cutoff is due to the bound on the virtuality, k2 < µ2, in the DGLAP evolution. For k µ, the angular ordering of the soft gluon emissions is the stronger bound, while at large k t t ≪ the bound onthe virtuality takes over, with the transition point at k = (1/2)(√5 1)µ 0.6µ. t − ≃ 9The LOintegratedgluondistributionsobtainedinthe globalanalysesarelargerinordertocompensate for the absence of the 1/z pole in P at LO and the absence of a photon–gluon coefficient function at LO. qg 9 70 4 LO qq LO gq 60 NLO qq NLO gq LO gg 3 LO qg 50 n NLO gg n NLO qg o o i i t t c 40 c n n u u f f 2 g g 30 n n i i t t t t i i l l p 20 p S S 1 10 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 z z Figure 3: DGLAP splitting functions, zP˜ (z), given by Eq. (25), at LO and NLO, after the ab subtraction for z > µ/(µ + k ) due to angular ordering, where we take µ2 = 104 GeV2 and t k2 = 100 GeV2. t 10