PROTON STRUCTURE FUNCTIONS IN THE DIPOLE PICTURE OF BFKL DYNAMICS a Ch. ROYON DAPNIA-SPP, Centre d'etudes de Saclay, F-9 9 Gif-sur-Yvette Cedex, France The F, F G, R = F L =F T proton structure functions are derived in the QCD dipole picture. Assuming k T factorisation, we get a three parameter t describing the 994 H proton structure function F data in the low x, moderate Q range. Without any additional parameters, the gluon density and the longitudinal structure functions are predicted. The purpose of this contribution is to show that the QCD dipole picture which contains the BFKL dynamics provides a pertinent model for describing the proton structure function at HERA in the low x and moderate Q range. The recently published H data might provide an opportunity to distinguish between the dierent QCD evolution equations (DGLAP and BFKL equations ) for small x physics. This is why itisvery important to test the BFKL dynamics based on the dipole model and k T factorisation 3.We can then get predictions for F, F G, and R = F L =F T proton structure functions. BFKL dynamics in the QCD dipole framework To obtain the structure function F,we use the k T factorisation theorem which is valid at high energy (small x). In a rst step, we calculate the deep inelastic cross section onium of a photon of virtuality Q on an onium state (heavy qq state). This onium state can be described by dipole congurations. The photon-onium cross-section reads onium = R d rdz () (r;z)(x; Q ; r) where () (b; z) is the probability distribution of dipole congurations of transverse coordinate r. In the k T -factorization scheme 3, one writes Q (x; Q ; r) = Q d ~ k dz z ^(x=z; ~ k =Q ) F (z;(kr) ) () where ^=Q is the ( g(k)!qq) Born cross section for an o-shell gluon of transverse momentum ~ k. F (z;(kr) ) is the unintegrated gluon distribution of an onium state of size r and contains the physics of the BFKL pomeron. a Invited talk given at the Workshop on Deep Inelastic Scattering and QCD (DIS96), 5- April 996, Rome, Italy
After doing two Mellin transforms in x, and in the k -space, and taking the unintegrated gluon distribution for an onium of radius r derived in the QCD dipole picture, one obtains: d where Q (x; Q ; r) = N C v() = i v() ^() (r Q ) N C e () ln( x B j ) () v(lr)(lr) d(lr) = ( ) ( + ) : (3) The detailed calculations can be found in reference 5. The expression of v() was derived using once more the k T factorisation in extracting a gluon from a dipole of transverse radius r. In order to average over the wave function of the onium state, one denes: <r > = d r(r ) dz () (r;z)=(m ) (4) where M is a perturbative scale characterizing the average onium size. Thus F onium (x; Q =M ) = N c d Q h() v() i M e Nc () ln( x ) d Q F (x; ) (5) i M In order to deal with deep inelastic scattering on a proton target, we substitute in formula 5 F () by F()!(; M; Q ) assuming the k T factorisation properties to be valid for high energy scattering o a proton target, where! can be interpreted as the Mellin transformed probability of nding an onium of transverse mass M in the proton. Q isatypically non perturbative proton scale. Assuming the renormalisation group properties,!(; M ; Q )=!()(M =Q ), one gets: F proton (x; Q ; Q Nc ) e ()ln( x ) : (6) = Nc R d v() i h() w() Q Q We can use this generic result to get some predictions for F T, F L, and F G (respectively the transverse, longitudinal, and gluon structure functions) corresponding to h T, h L, and h G : @ h T h L A = 3 h G ( ( ) ( + )) 3 ( ) ( + ) ( ) 3 @ ( + )( ) A (7)
The integral in is performed by the steepest descent method. The saddle point isat C = ( a ln Q Q ) where a = Nc x 7(3) ln Formula 6 shows that the considered approximation is valid when ln Q=Q = ln(=x) <<, that is small x, moderate Q=Q kinematical domain. We nally obtain: F F T + F L = Ca = e (P ) ln x Q Q e a ln QQ ; (8) 4NC ln where P =. C, P and Q will be taken as free parameters for the t of the H data, it will be then possible to compare with the values of P predicted by theory. We get nally R, and F G =F, which are independant of the overall normalisation C: F G F = h T + h L 3 c =c R = h L h T ( c )= 3 c + 3 c F t and prediction for F G and R 3 c ( c ) ( + c ) ( ( c ) ( + c )) 3 (9) c ( c ) ( + c )( c ) : () In order to test the accuracy of the F parametrisation obtained in formula 8, a t using the recently published data from the H experiment 4 has been performed 5. Wehave only used the points with Q 5GeV to remain in the domain of validity of the QCD dipole model. The is for 3 points, and the values of the parameters are P =:8, Q =:63GeV, C =:77. The result of the t is shown in gure a. The obtained value for the hard pomeron intercept P is in agreement with other determinations applying BFKL dynamics to F data at HERA. The corresponding eective coupling constant is=:, close to (M ) used in the H QCD t. The value of Q corresponds to a tranverse size of.3 fm which is a non perturbative scale as expected. Deviations from the t at high x and high Q are observed. Indeed, the valence contribution is not contained in our model, and it should contribute at high x. Relation (9) provides a paramater-free prediction for the gluon density (not shown in the gure) which is in good agreement with the results obtained by the H QCD ts based on a NLO DGLAP evolution equation 5. We also give a prediction for the value of R, which is given in gure b. The only parameters which enters this prediction is Q, determined by the F t. The corresponding curve (full line) is compared with the one loop approximation 3
.5.8.6.4.5..5.8.6.4.5..5.8.6.4.5..5.8.6.4.5. -4 - x -4 - x -4 - x -4 - x -4 - x -4 - x -4 - x -4 - x Figure : a: Results of the 3-parameter t of the H proton structure function for Q 5GeV - b: Predictions on R (continuous line : resummed prediction) of the h functions (dashed curve) of formula (7). The comparison of the two curves exhibits the ln =x terms resummation eects on the coecient functions h T and h L. The measurement ofrmight be an opportunity to distinguish between the BFKL and DGLAP mechanisms, R being expected to be much higher with the DGLAP mechanism. We thus think that a measurement ofr in this region would be useful. Acknowledgments The results described in the present contribution comes from a fruitful collaboration with A.Bialas, H.Navelet, R.Peschanski and S.Wallon. References. A.H.Mueller and B.Patel, Nucl. Phys. B45 (994) 47., A.H.Mueller, Nucl. Phys. B437 (995) 7., N.N.Nikolaev and B.G.akharov, eit. fur. Phys. C49 (99) 67., A.H.Mueller, Nucl. Phys. B45 (994) 373.. G. Altarelli and G. Parisi, Nucl. Phys. B6 (977) 98; V.N. Gribov and L.N. Lipatov, Sov. Journ. Nucl. Phys. 5 (97) 438 and 675., V.S.Fadin, E.A.Kuraev and L.N.Lipatov Phys. Lett. B6 (975) 5; I.I.Balitsky and L.N.Lipatov, Sov.J.Nucl.Phys. 8 (978) 8. 3. S.Catani, M.Ciafaloni and Hautmann, Phys. Lett. B4 (99) 97; Nucl. Phys. B366 (99) 35; J.C.Collins and R.K.Ellis, Nucl. P hys. 4
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