High-energy resummation in direct photon production
aa r X i v : . [ h e p - ph ] J un IFUM-943-FT
High-energy resummation in direct photonproduction
Giovanni Diana
Dipartimento di Fisica, Universit`a di Milano andINFN, sezione di Milano, via Celoria 16, I-20133 Milan, ItalyDecember 7, 2018
Abstract
We present the computation of the direct photon production cross-section inperturbative QCD to all orders in the limit of high partonic center-of-mass energy.We show how the high-energy resummation can be performed consistently in thepresence of a collinear singularity in the final state, we compare our results to thefixed order NLO cross-section in MS scheme, and we provide predictions at NNLOand beyond.
Perturbative QCD provides accurate theoretical predictions for hard processes at high-energy colliders. Logarithmic corrections to the lowest-order cross-sections can be system-atically computed in the region of large hard scale Q , Λ ≪ Q ∼ S , by a renormalizationgroup approach which leads to the factorization theorem of mass singularities [1]. How-ever, the TeV energy range opens up the two scale region Λ ≪ Q ≪ S , where the usualperturbative expansion receives large contributions characterized by logarithms of the ra-tio x = Q /S . In order to recover the accuracy of the perturbative results, logarithmicallyenhanced small- x contributions to the hard cross-sections, associated to multiple gluonemission, must be resummed to all orders.Prompt photon production [2] is a relevant processes for the study of hard interactionsin high-energy collisions. For example, it is the most important reducible background forhe H → γγ signal in the light Higgs scenario [3]. A thorough understanding of thisprocess in the small- x limit is thus relevant to make predictions for the LHC.Currently the direct photon cross-section is known up to O ( αα s ) [4] and Sudakovresummation effects have been computed up to NLL accuracy [5]. Prompt photon pro-duction is especially useful to probe the gluon parton density over a wide range of x [6],since the initial state gluon appears already at leading order, and in particular the small- x ( x & − ) region. For this reason we expect that, in prompt photon production,high-energy resummation affects significantly the fixed-order results.The general procedure for the small- x leading-log (LL x ) resummation of hard coef-ficient functions is well established in perturbative QCD within the framework of the k t -factorization theorem [7, 8], and involves the computation of the leading amplitude ofthe process with off-shell incoming gluons. This technique has been used to obtain re-summed cross-sections for heavy quarks photo- and hadro-production [7, 9, 10, 11], deepinelastic scattering [8, 12], Higgs production [13, 14, 15] and recently for the Drell-Yanprocess [16].Prompt photon cross-section contains two different contributions: the direct compo-nent, where the photon participates at leading order to the hard process, and a frag-mentation component, which is needed to take account of the hadronic component of thephoton. From a phenomenological point of view, at high-energy both terms are impor-tant [17]; in this work we will consider the direct contribution, leaving the fragmentationcomponent to a future work.All the processes for which small- x resummation has been performed so far are free ofcollinear singularities in the final state since the corresponding cross-sections are totallyinclusive; on the contrary such a divergence does appear in direct photon productionbecause the process is exclusive with respect to the final state photon, which from thispoint of view must be viewed as another hadronic state [18]. In this work we performthe high-energy resummation of the direct photon coefficient function consistently withthe MS scheme of subtraction of the final state singularity to all orders in perturbationtheory. The prompt photon process is characterized by a hard event involving the production ofa single photon. Let us consider the hadronic process H ( P ) + H ( P ) → γ ( q ) + X. (1)2ccording to perturbative QCD, the direct and the fragmentation component of the in-clusive cross-section at fixed transverse momentum q of the photon can be written as [5] q dσ γ ( x ⊥ , q ) d q = X a,b Z x ⊥ dx f a/H ( x , µ F ) Z x ⊥ /x dx f b/H ( x , µ F ) ×× Z dx (cid:26) δ (cid:18) x − x ⊥ x x (cid:19) C γab ( x, α s ( µ ); q , µ F , µ f )++ X c Z dz z d c/γ ( z, µ f ) δ (cid:18) x − x ⊥ zx x (cid:19) C cab ( x, α s ( µ ); q , µ F , µ f ) ) , (2)where we have introduced the customary scaling variable: x ⊥ = 4 q S , < x ⊥ < . (3)in terms of the hadronic center-of-mass energy S = ( P + P ) . In the factorization formulaeq. (2) we have used the short-distance cross-sections C γ ( c ) ab ≡ q d ˆ σ ab → γ ( c ) ( x, α s ( µ ); q , µ F , µ f ) d q , (4)where a , b and c are parton indices ( q , ¯ q , g ) while f i/H j ( x i , µ F ) is the parton density at thefactorization scale µ F . The fragmentation component is given in terms of a convolutionwith the fragmentation function d c/γ ( z, µ f ).If we define the Mellin moments σ γ ( N ) = Z dx ⊥ x N − ⊥ (cid:18) q dσ γ ( x ⊥ , q ) d q (cid:19) ,F i/H ( N, µ F ) = Z dx x N − xf i/H ( x, µ F ) ,D c/γ ( N, µ f ) = Z dx x N − x d c/γ ( x, µ f ) , ˜ C γ ( c ) ab ( N ) = Z dx x N − C γ ( c ) ab ( x )the collinear factorization theorem in N -space becomes σ γ ( N ) = X a,b F a/H ( N ) F b/H ( N ) ˜ C γab ( N ) + X c D c/γ ( N ) ˜ C cab ( N ) ! . (5)3 .2 Leading-order coefficient functions and beyond At leading order the processes that contribute to the direct component of the promptphoton cross-section are q ¯ q → γg, q (¯ q ) g → γq (¯ q ) . (6)The corresponding leading-order hard coefficient functions are given by [2, 18] C γ,LOq ¯ q ( x ) = q d ˆ σ q ¯ q → γg d q = αα s Q q π C F C A x √ − x (2 − x ) , (7) C γ,LOq (¯ q ) g ( x ) = q d ˆ σ q (¯ q ) g → γq (¯ q ) d q = αα s Q q π C A x √ − x (1 + x x = 4 q / ˆ s , where ˆ s is the partonic center-of-mass energy.QCD corrections to the Born coefficient functions eqs. (7) and (8) have been computedin ref. [4] up NLO accuracy.In the high-energy limit, the leading coefficient function is logarithmically enhancedby contributions of the form C γqg ( x ) = C γ,LOqg ( x ) + αα s ∞ X k =0 c ( k ) qg ( α s log x ) k + O (cid:0) αα s ( α s log x ) n (cid:1) (9)which, in N -Mellin space, becomes a sum of poles at N = 0 of increasing order:˜ C γqg ( N ) = ˜ C γ,LOqg ( N ) + αα s ∞ X k =1 ˜ c ( k ) qg (cid:16) α s N (cid:17) k + O (cid:16) αα s (cid:16) α s N (cid:17) n (cid:17) , n ≥ . (10)As in the case of heavy quark production (HQ) and of the Drell-Yan processes (DY), theBorn coefficients ˜ C γ,LOab are regular as N →
0: indeed, eqs. (7,8) vanish when x approacheszero. The first singular term in eq. (10) is a simple pole in N = 0 given by the NLOcontribution to the perturbative series; in x -space, this pole corresponds to a constantvalue, while the small- x logarithms arise from the poles of increasingly higher order.The NLO contribution to the expansion given in eq. (9) αα s c (0) qg has been computed forvarious processes in ref. [19], in particular for direct photon production, by considering theFeynman diagrams in fig. (1) where an extra gluon is radiated from the initial state. Thisis a general feature of the high-energy limit to all orders in perturbation thory: dominantcontributions at high-energy (LL x ) are given by the exchange of spin-1 particles in the t -channel therefore all the relevant Feynman diagrams in the small- x limit are given bythe BFKL ladders in fig. (2).From the NLO onwards, the direct photon cross-section acquires a final state diver-gence when the photon becomes collinear to the outgoing parton; this collinear divergencecannot be removed by adding the virtual corrections, rather it must be absorbed in the4igure 1: In this picture are shown the dominant Feynman graphs in the high-energylimit. Both of the two diagram determine the constant behaviour of the hard coefficientfunction C γq ¯ g at small- x .fragmentation component of eq. (2) as it happens for the initial state collinear divergenceswhich are properly absorbed in the definition of the parton densities [18]. In the nextsections we will show how to remove this divergence in the MS scheme consistently withthe resummation procedure of the gluon ladder and we will give the analytic expressionfor the resummed hard coefficient function C γq ¯ q and C γq ¯ g . In this section we compute the leading logarithmic corrections to the direct photon cross-section at high-energy. The high-energy resummation of gluon ladders arises from thegeneral formalism of the k t -factorization theorem [7, 8] and can be performed by followingthe computational procedure outlined in ref. [14] which allows us to compute the coefficientof the dominant log x to all orders in perturbation theory. This procedure requires thecalculation of the Born cross-section with off-shell (transverse) incoming gluons and off-shellness fixed in terms of their transverse momenta k i (impact factor).For a single off-shell gluon, we can parametrize the dependence on the virtuality k through the dimensionless variable ξ = k Q , (11)where Q is the hard scale of the process which determines the argument of the run-ning coupling α ( Q ). In direct photon production Q is the magnitude of the transversemomentum of the photon q . The LL x resummation is performed by taking a Mellintransform of the off-shell cross-section h ( N, M ) ≡ Z ∞ dξ ξ M − Z dx x N − ˆ σ ( x, ξ ) , (12)5igure 2: Multiple gluon emission (BFKL ladder) from the initial state in direct photonproduction. kp qp ′ Figure 3: Diagrams for direct photon production.and by identifying M as the sum of leading singularities of the largest eigenvalue of thesinglet anomalous dimension matrix [20] (BFKL anomalous dimension) M = γ s (cid:16) α s N (cid:17) + O (cid:18) α s N (cid:19) (13) γ s (cid:16) α s N (cid:17) = ∞ X n =1 c n (cid:18) C A α s πN (cid:19) n , c n = 1 , , , ζ (3) . (14)This corresponds to the sum of the high energy contributions coming from all diagrams offig. (2). Finally, the N k LO coefficient of the maximum power of log x is given by expandingthe impact factor in powers of α s . 6 .1 The off-shell cross-section Let us consider the process g ⋆ ( k ) + q ( p ) → γ ( q ) + q ( p ′ ) , (15)with an off-shell incoming gluon. We use a Sudakov parametrization for both incomingand outgoing momenta of the diagrams in fig. 3, thus we have p = z p , (16) k = z p + k ⊥ , (17) q = x z p + (1 − x ) z p + q ⊥ , (18) p ′ = (1 − x ) z p + x z p + k ⊥ − q ⊥ , (19)where k ⊥ = −| ~k ⊥ | = − k ,q ⊥ = −| ~q ⊥ | = − q , with light-like vectors p and p such that p · p = S/ S is the energy in thecenter-of-mass frame. The relevant scalar products are p · k = s/ ,p · q = x s/ ,k · q = (1 − x ) s/ − ~k ⊥ · ~q ⊥ = q x − ~k ⊥ · ~q ⊥ , where we introduced the longitudinal energy s = z z S . The d -dimensional phase space d Φ ( d ) = 1(2 π ) d − d d qd d p ′ δ ( p ′ ) δ ( q ) δ ( d ) ( k + p − q − p ′ ) == 1(2 π ) d − d d q δ (( p + k − q ) ) δ ( q ) (20)can be rewritten in terms of the Sudakov parameters since d d q = s d d − q dx dx . Weobtain d Φ ( d ) = 1(2 π ) s d d − q dx dx δ ( − k + s − x s − (1 − x ) s + 2 k · q ) ·· δ ( − q + x (1 − x ) s ) == (4 π ) − ǫ √ π (2 π ) (sin θ ) ǫ Γ(1 / ǫ ) dx x q ǫ d q δ ( − k + s (1 − x ) − q x + 2 k · q ) (21)where θ is the angle between q and k and we used the last δ -function to perform theintegration in x , now fixed to x = 1 − q x s , which implies x < x .7ince we are interested in the differential cross-section ˆ σ ( x, ξ ) = q dσd q , our phase spacein four dimension reduces to q d Φ (4) d q = dφ (4) = 1(2 π ) dx x q dθ δ (cid:18) − k + s (1 − x ) − q x + 2 k · q (cid:19) Θ( s − k ) , (22)which in terms of the dimensionless partonic variables x = 4 q /s and ξ = k / q reads dφ (4) = 1(2 π ) xsdx x dθ δ (cid:18) x (1 − x ) − ξ − x + 2 p ξ cos θ (cid:19) Θ( 4 x − ξ )Θ(4 x − x ) == 1(2 π ) sxdx x (1 − x ) dθ δ x − ξ + x − √ ξ cos θ − x ) ! Θ( 4 x − ξ )Θ(4 x − x ) . In the resummation procedure of refs. [7, 8, 14] the computation of Feynman diagramsis performed by using the eikonal rule for the gluon polarization sum X λ ǫ λµ ( k ) ǫ λν ( k ) = k µ k ν k , k µ ≡ (0 , k , , (23)understood as the projector P over the high-energy singularities, analogously to the ap-proach of refs. [21], which factorizes the gluon ladder from the Born coefficient. Thechannels s and t lead to the simple result for the amplitude in d = 4 + 2 ǫ dimensions A ( d ) ( x, x , ξ ) = X M = 4 e g s · C A (cid:20) ( q − sx ) + s x + ǫ q sx ( s − k ) (cid:21) = 16 e g s · C A (cid:16) − x x (cid:17) + x x ǫxx (cid:0) x − ξ (cid:1) (24)averaged over color and helicity (of the quark) and summed over the final states. Asshown in eq. (10), the high-energy enhancement appears as a series of poles in N = 0,therefore we are interested in the most singular term in the small- N limit.Since the off-shell cross-section is well behaved at N = 0, all the singular terms comefrom the substitution shown in eq. (13), hence at this level we can reduce the computationto the N = 0 moment of the impact factor in the ( N, M ) space h (0 , M ) = 1(2 π ) s Z π d θ Z s d x x (1 − x ) Z x d x ·· Z /x d ξ ξ M − A (4) ( x, x , ξ ) δ x − ξ + x − √ ξ cos θ − x ) ! == 1(2 π ) Z π d θ Z d x x (1 − x ) Z ∞ d ξ ξ M − ·· Z + ∞ max( x , ξ ) d ρρ A (4) (1 /ρ, x , ξ ) δ ρ − ξ + x − √ ξ cos θ − x ) ! , (25)8here we introduced the variable ρ = 1 /x and we exchanged the order of integration of ξ and ρ . By using the delta function we obtain: h (0 , M ) = 1(2 π ) Z π d θ Z d x x (1 − x ) Z ∞ d ξ ξ M − ·· ρ A (4) (1 / ¯ ρ, x , ξ )Θ (cid:18) ¯ ρ − max( 14 x , ξ (cid:19) , (26)where we have defined ¯ ρ = ξ + x − √ ξ cos θ − x ) . (27)The argument of the Heaviside Θ-function in eq. (26) is always positive since¯ ρ − x = ξx + 1 − x √ ξ cos θ − x x (1 − x ) = ξ − √ ξ cos θ + 14 x (1 − x ) > ( √ ξ − x (1 − x ) > /x > ξ , and¯ ρ − ξ ξ + x − √ ξ cos θ − ξ (1 − x )4(1 − x ) = x − √ ξ cos θ + ξx − x ) > ( √ x − √ ξx ) − x ) > h (0 , M ) = 1(2 π ) Z ∞ d ξ ξ M − Z π d θ Z d x x (1 − x ) 1¯ ρ A (4) (1 / ¯ ρ, x , ξ ) . (28)The integration over the region 0 < θ < π , 0 < x < ξ > i.e. | k | > | q | ; indeed the latter condition defines the kinematical region wherethe photon can be radiated collinearly to the quark in the final state. In the collinearlimit, the amplitude in eq. (24) is singular and the divergence is given by the fermionicpropagator in the s -channel. In the collinear limit we have:4 ¯ ρ − ξ = ξx − √ ξx cos θ + 1 x − x = 0 , (29)which happens when (cid:26) θ = 0 x = √ ξ . (30)9 .2 Subtraction of the collinear singularity in MS scheme In order to cancel the collinear divergence in the MS scheme, first, we regularize theintegrations of eq. (28) by subtracting the collinear limit of the four dimensional amplitudebefore doing any integration, second we recover the pole in ǫ = 0 and the remaining finiteparts by adding back the same quantity computed in d dimensions.We can do this by writing the impact factor in 4 + 2 ǫ dimensions as h ( d ) ( x, ξ ) = Z dφ ( d ) A ( d ) . (31)We then remove the singularity of the integrand by introducing a function D ( d ) which hasthe same singular behaviour of the squared amplitute A ( d ) in the collinear limit eq. (30).In four dimension we have D (4) = e g s C A P qγ (1 / p ξ ) √ ξ − − √ ξx ) + θ Θ( ξ − , (32)where P ( z ) = 1 + (1 − z ) z . (33)By adding and subtracting the phase space integral of the function D ( d ) to the d -dimensionalimpact factor we obtain h ( d ) = lim ǫ → (cid:18)Z dφ ( d ) A ( d ) − Z dφ ( d ) D ( d ) (cid:19) + Z dφ ( d ) D ( d ) + O ( ǫ ) == Z dφ (4) (cid:0) A (4) − D (4) (cid:1) + f A + Z dφ ( d ) D ( d ) + O ( ǫ ) (34)where the first integral is finite in four dimensions and the finite part f A comes from thelinear term in ǫ in the d -dimensional amplitude eq. (24).By using the d -dimensional phase space, the last term in eq. (34) is dφ ( d ) D ( d ) = αα s C A √ π (4 π ) − ǫ Γ(1 / ǫ ) (cid:18) µ q (cid:19) − ǫ ξ P qγ (1 / p ξ ) Θ( ξ − δ (cid:0) x − x (cid:1)(cid:2) (1 − √ ξx ) + θ (cid:3) θ ǫ dx dθ, (35)where the dimensional scale µ (introduced by dimensional regularization) from now onwill be identified with q . By using this result in eq. (34) and performing the Mellinintegrations with N = 0 we have h ( d ) (0 , ξ ) = 1(2 π ) Z d x (cid:18)Z π − π d θ ρ A (4) ( ¯ ρ, x , ξ )32 x (1 − x ) − Z ∞−∞ d θ ξ D (4) (cid:19) + f A ++ αα s C A √ π (4 π ) − ǫ Γ(1 / ǫ ) 1 ξ Z dx Z + ∞−∞ dθ θ ǫ P qγ (1 / √ ξ )Θ( ξ − (cid:2) (1 − √ ξx ) + θ (cid:3) (36)10here we have extended the limits of the angular integration of the function D in orderto simplify the results of the integration while the finite part f A is f A = lim ǫ → αα s ǫC A Z d ξ ξ M − Z d x p ξ Z + ∞−∞ d θ ξ − θ ǫ (cid:2) (1 − √ ξx ) + θ (cid:3) ! == παα s C A − M . (37)All the integrations in eq. (36) can be performed in closed form, thus by subtractingthe pole in ǫ = 0 with the usual combination 1 /ǫ + γ E − log 4 π we obtain h (0 , ξ ) = παα s C A (cid:26) Θ( ξ − (cid:18) − ξ + 8 √ ξ (1 − log 2) (cid:19) ++ sign( ξ − − (cid:18) ξ (cid:19) log(1 − ξ ) − √ ξ log (cid:18) √ ξ + 11 − √ ξ (cid:19) !) , (38)while the M-Mellin moments are h (0 , M ) = αα s πC A (cid:26) (7 − M + 2 M )( M − M − M − (cid:18) π cot( M π ) + 2 H M − + 2 M − (cid:19) + 12 − M (cid:27) , (39)where H M − is the harmonic number of argument M − By expanding the impact factor obtained in the previous section around M = 0, we obtain M h (0 , M ; α s ) = παα s C A (cid:26)
76 + 6736 M + 385 M
216 + (cid:18) ζ (3)3 (cid:19) M ++ (cid:18) ζ (3)18 (cid:19) M + (cid:18) ζ (3)108 + 7 ζ (5)3 (cid:19) M + O (cid:0) M (cid:1)(cid:27) . (40) In ref. [19] the impact factor h q ( a ) was only computed in the region 0 < ξ < h q , given in eq. (3.7) of ref. [19], is seen to agree with our resultrecalling that in the region 0 < ξ < h q is related to h eq. (38) by ξ dh q ( ξ ) dξ = h ( ξ ) , (0 < ξ < . Note however that the expression for the cross-section σ qg ( q > p T ) given in eq. (3.11) of ref. [19] is toolarge by a factor 2 [22]. M = 0. The resummed coefficient function in the MS factorization schemeis given by the relation ˜ C γqg ( N, α s ) = M h ( N, M ; α s ) R ( M ) (cid:12)(cid:12) M = γ s (41)in terms of the impact factor eq. (39) and the function R ( M ) = 1 + 83 ζ M − ζ M + O ( M ) (42)which takes into account finite parts coming from the MS subtraction of initial statecollinear singularities and where M is identified as the BFKL anomalous dimension γ s (cid:16) ¯ α s N (cid:17) = ¯ α s N + 2 ζ (cid:16) ¯ α s N (cid:17) + 2 ζ (cid:16) ¯ α s N (cid:17) + . . . , ¯ α s = α s C A π (43)We have˜ C γqg ( N, α s ) = παα s C A (cid:26)
76 + 6736 (cid:16) ¯ α s N (cid:17) + 385216 (cid:16) ¯ α s N (cid:17) + (cid:18) ζ (3)9 (cid:19) (cid:16) ¯ α s N (cid:17) ++ (cid:18) − π
720 + 308 ζ (3)27 (cid:19) (cid:16) ¯ α s N (cid:17) + O (cid:18)(cid:16) ¯ α s N (cid:17) (cid:19)(cid:27) . (44)The NLO term in eq. (44) gives, in the x -space, the constant value 67 / αα s which is inagreement with the fixed order calculation of refs. [4, 19]. By using the high-energy colorcharge relation between the hard coefficient functions˜ C γq ¯ q ( q ) ( N, α s ) = C F C A (cid:16) ˜ C γqg ( N, α s ) − ˜ C γ,LOqg (0 , α s ) (cid:17) (45)we can also obtain the LL x contributions coming from the process q ¯ q → γg ˜ C γq ¯ q ( q ) ( N, α s ) = α α s N C F C A (cid:26) (cid:16) ¯ α s N (cid:17) + (cid:18) ζ (3)9 (cid:19) (cid:16) ¯ α s N (cid:17) ++ (cid:18) − π
720 + 308 ζ (3)27 (cid:19) (cid:16) ¯ α s N (cid:17) + O (cid:18)(cid:16) ¯ α s N (cid:17) (cid:19)(cid:27) . (46) In fig. 4 we compare in x -space the coefficient function C qg ( x, α s ) at LO, NLO and N LO inthe high-energy limit. The large contributions at small- x spoil the perturbative expansionand must be resummed in order to recover accurate results. The resummation of these12 -5 -4 -3 -2 -1 x00.10.20.30.40.50.60.70.8 C q g ( x ) LONLON LO Figure 4: The picture shows a comparison between the leading order direct photon coeffi-cient function (black solid) and the relative small- x corrections up to NLO (green dashed)and N LO (red dashed).logarithmic terms in the hadronic cross-section can be performed to all orders [12, 23]including running coupling. A full phenomenological study could be performed by com-bining the resummed hard cross section computed here with the resummation of GLAPevolution equations, following the formalism of refs [12, 23] (see also ref. [24] for an al-ternative approach). However, we can get a feeling for the size of resummation effects bycomparing the result in eq. (44) with the DIS coefficient function C g ( N ). As shown infig. 5 the ratio between C g and C γqg (both of them normalized to the respective LO values)is of order 1, therefore we expect that, at low- x , resummation effects may be as importantas those obtained in the DIS case. In this work we performed the computation of the hard coefficient function for directphoton production in the high center-of-mass energy limit (the small- x limit) to all or-ders in perturbative QCD. We have shown how to resum the leading logarithms of theBjorken variable x which stem from the BFKL gluon ladders, and how the high-energyresummation technique can be joined with the MS scheme of subtraction of a final statecollinear singularities.High-energy resummation for both PDFs evolution and coefficient functions are wellestablished tools which are needed to quantify the small- x effects on perturbative results.13 -5 -4 -3 -2 -1 x051015202530 C D I S / C γ DISDPhP
Figure 5: Ratio between F ( ¯ C DIS and direct photon (DPhP) ( ¯ C γ ) coefficient functions,both of them normalized to their LO valuesA clear understanding of this high-energy effects will be crucial expecially for the LHC. We thank Stefano Forte, who suggested this work, for useful discussions and critical read-ing of the manuscript, G. Bozzi, A. Vicini and S. Marzani for several comments, F. Caolafor discussions and encouragement and A. Sportiello for mathematical discussions. Spe-cial thanks are to R. K. Ellis and D. A. Ross for correspondence. This work was partlysupported by the European network HEPTOOLS under contract MRTN-CT-2006-035505and by a PRIN2006 grant (Italy). 14 eferences [1] G. Altarelli and G. Parisi,
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