content_image/440412.jpg content_image/440413.jpg content_image/440415.jpg content_image/440414.jpg
After introducing the natural dimensionless variable 

<img src='content_image/440412.jpg'>

where b $^{= (33}$− 2 n$_{f}$ ) / 12 π in QCD , g ( µ $^{2}$) is the running coupling constant at the reference scale µ $^{2}$, n$_{f}$ is the number of active flavors, Λ$_{QCD}$ is the dimensional QCD parameter, the GLAPD equations read [15, 18] 

<img src='content_image/440413.jpg'>

They describe the scaling violation of the parton distribu- tions D j $_{i}$( x, t ) inside a dressed quark or gluon ( i, j = q/g ). 

We will not write the kernels P explicitly and will not derive the generalized equations for two-parton distribu- tions D j$_{1}$j$_{2}$ i ( x$_{1}$, x$_{2}$, t ), representing the probability that in a dressed constituent i one finds two bare partons of types j$_{1}$ and j$_{2}$ with the given longitudinal momentum fractions x$_{1}$ and x$_{2}$ (referring to [14–18] for details). We note only that their solutions can be represented as the convolution of single distributions [16, 17]. This convolu- tion coincides with the jet calculus rules [19] as mentioned above and is the generalization of the well-known Gribov- Lipatov relation installed for single functions [15] (the distribution of bare partons inside a dressed constituent is identical to the distribution of dressed constituents in the fragmentation of a bare parton in the leading loga- rithm approximation). The obtained solution shows also that the double distribution of partons is correlated in the leading logarithm approximation: 

<img src='content_image/440415.jpg'>

Of course, it is interesting to find out the phenomeno- logical issue of this parton level consideration. This can be done within the well-known factorization of soft and hard stages (physics of short and long distances). As a result, the equations (3) describe the evolution of parton distributions in a hadron ( h ) with t ( Q $^{2}$), if one replaces the index i by index h only. However, the initial condi- tions for new equations at t = 0 ( Q 2 = µ $^{2}$) are unknown a priori and must be introduced phenomenologically or must be extracted from experiments or some models deal- ing with physics of long distances [at the parton level: D j $_{i}$( x, t = 0) = δ$_{ij}$δ ( x − 1); D j$_{1}$j$_{2}$ i ( x$_{1}$, x$_{2}$, t = 0) = 0]. Nevertheless the solution of the generalized GLAPD evo- lution equations with a given initial condition may be written as before via the convolution of single distribu- tions [14, 17]. This result shows that if the two-parton distributions are factorized at some scale µ $^{2}$, then the evolution violates this factorization inevitably at any different scale ( Q 2 ̸ = µ $^{2}$), apart from the violation due to the kinematic correlations induced by the momentum conservation, which is the analogue of the momentum conserving phase space factor in Eq. (2). 

For a practical employment it is interesting to know the degree of this violation. Partialy this problem was investigated theoretically in Refs. [17, 20] and for the two-particle correlations of fragmentation functions in Ref. [21]. That technique is based on the Mellin trans- formation of distribution functions and the asymptotic behavior can be estimated. Namely, with the growth of t ( Q $^{2}$) the correlation term becomes dominant for finite x$_{1}$ and x$_{2}$ [20] and thus the two-parton distribution func- tions “forget” the initial conditions unknown a priori and the perturbatively calculated correlations appear. 

The asymptotic prediction “teaches” us a tendency only and tells nothing about the values of x$_{1}$, x$_{2}$, t ( Q $^{2}$) beginning from which the correlations are significant. Naturally numerical estimations can give an answer to this specific question using the CTEQ fit [22] for single distributions as an input. The nonperturbative initial conditions D j $_{h}$( x, 0) are specified in a parametrized form at a fixed low-energy scale Q$_{0}$ = µ = 1 . 3 GeV. The par- ticular function forms and the value of Q$_{0}$ are not cru- cial for the CTEQ global analysis at a flexible enough parametrization. The results of numerical calculations were obtained in Ref. [14] for the ratio: 

<img src='content_image/440414.jpg'>

At the hard process scale of the CDF measurement [4] ( Q ∼ 6 GeV) the ratio (5) is nearly 10% and increases right up to 30% at much higher scale ( Q ∼ 100 GeV) for the longitudinal momentum fractions x ≤ 0 . 1 acces- sible to these measurements. For the finite longitudinal momentum fractions x ∼ 0 . 2 ÷ 0 . 4 the correlations may increase right up to 90%. They become important for almost all x with growing t in accordance with the pre- dicted QCD asymptotic behavior [17, 20]. The more com- prehensive analysis of double parton distributions based on the direct numerical integration of generalized equa- tions can be found in the quite recent paper [23] (see also Ref. [11] for estimation therein of joint interaction probability due to evolution in comparison with the fac- torization contribution). 

## III. POSSIBLE MANIFESTATION OF CORRELATIONS INDUCED BY QCD EVOLUTION 

As an effect of evolution, the double distribution func- tions become strongly correlated in longitudinal momen- tum fractions at large Q 2 and finite x as mentioned above. On the other hand, the indications from the ex- perimental observation of double scatterings at CDF [4] are not in favor of strong correlations effects in longi- tudinal momentum fractions. The most likely reason is that the studied kinematical domain, with relatively small x values and low resolution (energy) scale, is far from the asymptotic QCD prediction. The possibility of testing double collisions at much higher resolution scales 

