1ILJII1 FR9703188
C.E. SACLAY SEMI Df MMXf DB DSM
DAPNIA/SPP 96-18 October 19%
PROTON STRUCTURE FUNCTIONS IN THE DIPOLE PICTURE OF BFKL DYNAMICS
Ch. Royon
Workshop on Deep Inelastic Scattering and QCD (DIS96), 15-20 April 1996, Rome, Italy PROTON STRUCTURE FUNCTIONS IN THE DIPOLE PICTURE OF BFKL DYNAMICS a
Ch. ROYON DAPNIA-SPP, Centre d'Etudes de Saclay, F-91 191 Gif-sur-Yvette Cedex, France
The Fj, FQ, R = FI/FT proton structure functions are derived in the QCD dipole picture. Assuming kf factorisation, we get a three parameter fit describing the 1994 HI proton structure function Fi data in the low x, moderate Q2 range. With- out 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 Q2 range. The recently published HI data might provide an opportunity to distinguish between the different QCD evolution equations (DGLAP and BFKL equations 2) for small x physics. This is why it is very important to test the BFKL dynamics based on the dipole model and kj factorisation3. We can then get predictions for F>, FG, and R = FI/FT proton structure functions.
1 BFKL dynamics in the QCD dipole framework
To obtain the structure function Fo, we use the kr factorisation theorem which is valid at high energy (small x). In a first step, we calculate the deep inelastic cross section aonium of a photon of virtuality Q2 on an onium state (heavy qq state) '. This onium state can be described by dipole configurations '. The photon-onium cross-section reads a-onuim = J d2rdz
Q2a(x, Q2; r) = J* d2k j' ^ a(x/z, P/Q2) F(z, (kr)2) (1) where &/Q2 is the (7 g(k) -4 q q) Born cross section for an off-shell gluon of transverse momentum k. F(z, (kr)2) is the unintegrated gluon distribution of an onium state of size r and contains the physics of the BFKL pomeron.
"Invited talk given at the Workshop on Deep Inelastic Scattering and QCD (DIS96), 15-20 April 1996, Rome, Italy 1 After doing two Mellin transforms in x, and in the A*--space, and taking the unintegrated gluon distribution for an onium of radius r derived in the QCD dipole picture, one obtains: • [ gtt (2) IT J IVK 7 where .j-27-1 rvi _ v(lr) (I (3)
The detailed calculations can be found in reference 5. The expression of ^(7) was derived using once more the k? 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 defines:
< r- >^= / rf2r(r2)^.-$(°>(r, z) = (M2)"7 (4) where A/2 is a perturbative scale characterizing the average onium size. Thus
= ^ I p- ('£X fc(7)!!Me*^xWin(i) 7T J 2lJT \M- J 7
In order to deal with deep inelastic scattering on a proton target, we substitute in formula 5 ^(7) by ^(7)^(7, M; Qi) assuming the kr factori- sation properties to be valid for high energy scattering off a proton target, where u can be interpreted as the Mellin transformed probability of find- ing an onium of transverse mass M2 in the proton. Qi is a typically non perturbative proton scale. Assuming the renormalisation group properties, 2 2 2 7 w(7, M ; Ql) = w(7)(M /Q0 ) - one gets: ^(g)"^"'. (6) We can use this generic result to get some predictions for FT, FI, and Fa (respectively the transverse, longitudinal, and gluon structure functions) cor- responding to hr, hi, and ha-
r(2-27)r(2 The integral in 7 is performed by the steepest descent method. The saddle l aln point is at 7c = i( - ^) where a= (^7C(3) In £)"' Formula6 shows that the considered approximation is valid when lnQ/Qo/ln(l/x) << 1, that is small x, moderate Q/Qo kinematical domain. We finally obtain:
Qo where ap — 1 = 4SNc'" 2 _ C, otp and Qo will be taken as free parameters for the fit of the HI data, it will be then possible to compare with the values of ap predicted by theory. We get finally R, and Fc/Fo, which are independant of the overall normalisation C:
Fa 1 -*.)rP+j*) F2 hT + hL
~fc)
2 Fo fit and prediction for FQ and R In order to test the accuracy of the Fi parametrisation obtained in formula 8, a fit using the recently published data from the HI experiment4 has been performed 5. We have only used the points with Q2 < \50GeV2 to remain in the domain of validity of the QCD dipole model. The x2 is 101 for 130 points, and the values of the parameters are ap = 1.282, Qo = 0.63(7eV, C = 0.077. The result of the fit is shown in figure la. The obtained value for the hard pomeron intercept ap is in agreement with other determinations applying BFKL dynamics to Fo data at HERA. The corresponding effective coupling constant is a = 0.11, close to a(M%) used in the HI QCD fit. The value of Qo corresponds to a tranverse size of 0.3 fm which is a non perturbative scale as expected. Deviations from the fit at high x and high Q2 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 figure) which is in good agreement with the results obtained by the HI QCD fits based on a NLO DGLAP evolution equation 5. We also give a prediction for the value of R, which is given in figure lb. The only parameters which enters this prediction is Qo, determined by the FT fit. The corresponding curve (full line) is compared with the one loop approximation
3 S, ^
UIUW* I9OM*
^ ^ —s ^ 4SSM* —"S —^s ^ MOM1 MM* IM4M
TTTf^,,v
Figure 1: a: Results of the 3-parameter fit of the HI proton structure function for 150(jeV2 - 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 In 1/x terms resummation effects on the coefficient functions hr and hi. The measurement of R might 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 of R in this region would be useful. Acknowledgments
The results described in the present contribution comes from a fruitful collab- oration with A.Bialas, TI.Navelet, R.Peschanski and S.Wallon. References
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