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Non Abelian $q\bar{q}$ contributions to small-$x$ anomalous dimensions

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    a  r   X   i  v  :   h  e  p  -  p   h   /   9   6   0   6   4   2   7  v   2   2   8   J  u  n   1   9   9   6 DFF 250/6/96hep-ph 9606427 Non-abelian  q  ¯ q   contributions to small- x anomalous dimensions ∗ G. Camici and M. Ciafaloni Dipartimento di Fisica, Universit`a di Firenze and INFN, Sezione di Firenze  Abstract By using  k -factorization, we derive resummation formulas for the non-abelian  q  ¯ q  contributions to both heavy flavour production by gluon fusion, and to the next-to-leading BFKL kernel. By combining this result with previous ones by Fadin et al.on the virtual terms, we also compute in closed form the complete  q  ¯ q   contributionto the gluon anomalous dimension in the  Q 0 -scheme. We find that  q  ¯ q   resummationeffects are important for heavy flavour production, but are instead small in theanomalous dimension eigenvalues, because of a cancellation between abelian andnon abelian contributions. PACS 12.38.Cy ∗ Work supported in part by the EC program CHRX-CT93-0357 and by MURST (Italy).1  1 Introduction Hard processes in the small- x  regime  s  ≫  Q 2 ≫  Λ 2 play an increasingly important roleat present colliders [1] and are characterized by the fact that the effective QCD couplingconstant  α s ( Q 2 )log(1 /x ) is sizeable and the corresponding perturbative contributionsneed to be resummed.Such resummations are performed, at leading log level, on the basis of the BFKLequation [2] for the anomalous dimensions, and of the  k -factorization formulae [3-6] forthe coefficient functions. Particular attention has been devoted in the literature to heavyquark production processes [3, 6] and to the (abelian) light quark contributions to the  qg and  qq   entries of the anomalous dimension matrix[5]. Furthermore, an impressive programof evaluation of next-to-leading (NL) kernels of the BFKL equation is under way [7-10].In this note, we focus our attention on the (non-abelian)  q  ¯ q   contribution to the  k -factorization program, which is relevant in two ways: firstly, as a hard subprocess inDrell-Yan heavy-flavour production and secondly, in the massless quark limit, as an im-portant NL contribution to the  gluon   anomalous dimension. We shall then derive QCDresummation formulas in both cases and in particular we shall obtain in closed form all NLterms of the form  α s N  f  ( α s C  A  log(1 /x )) n , contributing to the gluon anomalous dimensionin the so-called  Q 0 -scheme [11, 12]. The evaluation of such contributions is particularly important for structure functions.Since the gluon couples to the proton only through  q  ¯ q   states, the NL anomalous dimension γ  qg  (determined essentially by the abelian  q  ¯ q   kernel) contributes to scaling violations atthe same level as  γ  gg . It is then important to perform a complete calculation of NL  q  ¯ q  contributions (in particular, the non-abelian one), in order to check the pattern describedabove, and to reduce the factorization scheme dependence of the theoretical estimates of scaling violations at HERA [13, 14]. 2  2  Q  ¯ Q  Hadroproduction Let us start considering the cross-section for the hadroproduction of a (heavy) quark-antiquark pair, which at high energies is dominated by the gluon fusion process in Fig. 1.This contribution may be expressed in the  k -factorized form [3]: M  2 σ  =    dz  1 z  1 dz  2 z  2 d 2 k 1 d 2 k 2 ˆ σ ( k 1 , k 2 ,M  2 ,z  1 z  2 s ) F  (1) ( z  1 , k 1 ) F  (2) ( z  2 , k 2 ) ,  (2.1)where  F   denotes the unintegrated gluon density in the hadron and ˆ σ  the high-energyprojection of the (Regge) gluon fusion process  g ( k 1 ) g ( k 2 ) → Q  ¯ Q .Since the evolution of the structure functions is simpler in the space of   z   and k momentsit is useful to express the high energy factorization formula (2.1) in terms of the double-Mellin transformed structure functions F  ( i ) ω  ( γ  ) =    10 dzz  ω − 1    ∞ 0 d 2 k π k 2  k 2 Q 20  γ  F  ( i ) ( z, k ) (2.2)and hard cross-section coefficient function  1  α s π  H  ω ( γ  1 ,γ  2 ) =    ∞ 0 d 2 k 1 k 21 d 2 k 2 k 22   k 21 M  2  γ  1    k 22 M  2  γ  2     ∞ 0 dss  M  2 s  ω ˆ σ ( k 1 , k 2 ,M  2 ,s ) , (2.3)so that it takes the form: M  2 σ  M  2 s  =  α s π    12 + i ∞ 12 − i ∞ idγ  1 dγ  2 dω (2 π ) 3   sM  2  ω  M  2 Q 20  γ  1 + γ  2 F  (1) ω  ( γ  1 ) F  (2) ω  ( γ  2 ) H  ω ( γ  1 ,γ  2 ) .  (2.4)For large enough  M  2 , the  γ   integrals will be dominated by the BFKL anomalousdimension,  γ  1  ≃ γ  2  ≃ γ  L ( ¯ α s /ω ). This means that the high-energy resummation effects onthe cross section we are considering are embodied in the  γ  1 , γ  2 -dependence of the hardcoefficient function  H  ω ( γ  1 ,γ  2 ), which is precisely what we wish to compute. 1 This function is related by the change of normalization  h  =  4 α s π γ  1 γ  2 H   to the  h -functions of Refs[3, 5, 6] 3  The squared matrix element for the process under consideration was computed in Ref.[3] (see also Ref. [8]), and contains two terms, with colour factors  C  F   and  C  A , that weshall call respectively the abelian and the non abelian contributions.The abelian contribution  H  a is known [3] and provides the quark anomalous dimension[5]. Let us then concentrate on the non abelian one  H  na .First, we rewrite the squared matrix element of Ref. [3] in terms of the Sudakovparametrization for the exchanged gluons’ momenta k µ 1  ≃ z  1  p µ 1  + k µ 1 , k µ 2  ≃ z  2  p µ 2  + k µ 2 ,  (2.5)and for the momentum transfer∆ µ =  z  1 x 1  p µ 1  − z  2 x 2  p µ 2  + ∆ µ ,  (2.6)where  p 1 ,  p 2  denote the (light-like) momenta of the incoming hadrons.By using explicit expressions for the invariantsˆ s  = ( k 1  +  k 2 ) 2 ,  ˆ t  = ∆ 2 ,  ˆ u  = ( k 1 − k 2 − ∆) 2 , ν   = ˆ s + ( k 1  + k 2 ) 2 ,  (2.7)the non abelian squared matrix element of Ref. [3] can be rewritten in the form A na =  π 2 α s   − 1( M  2 − ˆ t )( M  2 − ˆ u )+ 1ˆ s   1 M  2 − ˆ u  −  1 M  2 − ˆ t  (1 − x 1 − x 2 ) + 2 ν  ˆ s ++ 2 k 21 k 22  12  −  (1 − x 1 )(1 − x 2 ) ν M  2 − ˆ t  +  ν  2(1 − x 1 − x 2 ) −−  k 21 (1 − x 2 ) + k 22 (1 − x 1 ) − k 1 · k 2  + ∆ · ( k 2 − k 2 )ˆ s  ×  12  −  x 1 x 2 ν M  2 − ˆ u  −  ν  2(1 − x 1 − x 2 )+  k 21 (1 − x 2 ) + k 22 (1 − x 1 ) − k 1 · k 2  + ∆ · ( k 2 − k 2 )ˆ s  (2.8)4  By then integrating over the phase space and performing the moments of Eq. (2.3)we obtain the non abelian coefficient function  H  naω  ( γ  1 ,γ  2 ) as follows α s π H  naω  ( γ  1 ,γ  2 ) = 18 π 4    dν ν  2 − ω d 2 k 1 π k 21 d 2 k 2 π k 22 d 2 ∆ π dx 1 dx 2 δ  ( x 2 (1 − x 1 ) ν  − ( k 1 − ∆ ) 2 − M  2 ) ×× δ  ( x 1 (1 − x 2 ) ν  − ( k 2  + ∆ ) 2 − M  2 )   k 21 M  2  γ  1    k 22 M  2  γ  2 A na ( k 1 , k 2 , ∆ ,x 1 ,x 2 ,ν,M  2 ) (2.9)This expression can be evaluated analytically (at least in the limit  ω  = 0, relevant at highenergies) with a careful choice of the order of the integrations. We find it convenient tostart eliminating the variable  ν   by integration of one mass-shell delta function, and thencomputing the  ∆ -integral with the aid of a Feynman parametrization of denominators.The remaining integrals, first over the transverse momenta  k 1 ,  k 2  and then over theFeynman parameters and longitudinal momentum fractions, can be evaluated in terms of Gamma and Beta functions, and give, after some algebra, the result H  na ( γ  1 ,γ  2 ) ==  C  A 2 α s π  Γ(1 − γ  1 − γ  2 )Γ( γ  1 )Γ( γ  2 )  B (1 − γ  1 , 1 − γ  1 ) B (1 − γ  2 , 1 − γ  2 )4 ++ B (1 − γ  1 , 2 − γ  1 ) B (1 − γ  2 , 2 − γ  2 )(3 − 2 γ  1 )(3 − 2 γ  2 ) (1 + (1 − γ  1 )(1 − γ  2 ))   ++ B ( γ  1 , 1 − γ  1 ) B ( γ  2 , 1 − γ  2 )   Γ(2 − γ  1 − γ  2 )Γ(4 − 2 γ  1 − 2 γ  2 )  −  Γ(3 − γ  1 − γ  2 )Γ(6 − 2 γ  1 − 2 γ  2 )   ++2 B ( γ  1 , 1 − γ  1 − γ  2 ) B ( γ  2 , 1 − γ  1 − γ  2 ) B (3 − γ  1 − γ  2 , 3 − γ  1 − γ  2 )(1 − γ  1 − γ  2 )  .  (2.10)The details of the calculation are reported elsewhere [15].Let us note the triple pole singularity at  γ  1  +  γ  2  = 1 of the last term of Eq. (2.10),which is related to the collinear-singular behaviour of the partonic cross section in themassless limit. In fact, if   M  2 ≪  ( k 1  +  k 2 ) 2 ≡  q 2 ≪  k 21  ≃  k 22 , the hard cross sectionapproaches the singular limitˆ σ sω =0 ( k 1 , k 2 ,M  2 )=¯ α s α s π    10 dx 1 x 1 (1 − x 1 ) q 2  log  1 + x 1 (1 − x 1 ) q 2 M  2  ≃  ¯ α s α s 6 π  log  q 2 M  2  −  53   1 q 2   ¯ α s  =  α s C  A π  ,  (2.11)5
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