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. After introducing the natural dimensionless variable 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] 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: 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 ). t = 1 2 πb ln [ 1+ g $^{2}$( µ $^{2}$) 4 π b ln ( Q 2 µ 2 )] = 1 2 πb ln [ ln( Q 2 Λ 2 QCD ) ln( µ 2 Λ 2 QCD ) ] , dD j $_{i}$( x, t ) dt = ∑ j ′ 1 ∫ x dx ′ x ′ D j ′ i ( x $^{′}$, t ) P$_{j}$ ′ → j ( x x ′ ) . (3) R ( x, t ) = D gg p (QCD , corr . $_{)}$( x$_{1}$, x$_{2}$, t ) D g $_{p}$( x$_{1}$, t ) D g $_{p}$( x$_{2}$, t $^{)(1}$− x$_{1}$ − x$_{2}$ ) 2 ∣ x$_{1}$ = x$_{2}$ = x (5) D j$_{1}$j$_{2}$ i ( x$_{1}$, x$_{2}$, t ) = D j$_{1}$ i ( x$_{1}$, t ) D j$_{2}$ i ( x$_{2}$, t ) . (4) 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: 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). 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). 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 III. POSSIBLE MANIFESTATION OF CORRELATIONS INDUCED BY QCD EVOLUTION