Factorization breaking in diffractive dijet production

We study diffractive hard dijet production, with one or two rapidity gaps, at high energies. We emphasize that both hard and Regge factorization are broken in these processes. We show that a multi-Pomeron-exchange model for screening effects gives a specific pattern for the breakdown of factorization, which is in good agreement with diffractive dijet data collected at the Tevatron.

The investigation of diffractive processes at high energies gives important information on the structure of hadrons and their interaction mechanisms. Hard diffractive processes, such as the diffractive production of dijets, allow the study of the interplay of small-and largedistance dynamics within QCD. The existence of a hard scale provides the normalization of the Born term diagram, which is shown for single-diffractive dissociation in Fig. 1(a) and for the double-diffractive process 1 in Fig. 1(b). These processes are respectively characterized by the existence of one and two large rapidity gaps, each of which is represented by Pomeron exchange. At high energies, there are important contributions from unitarization effects. In the t channel Reggeon framework, these effects are described by multi-Pomeron exchange diagrams, As has been known for a long time (see, for example, [1]), factorization does not necessarily hold for diffractive production processes; for recent studies see Refs. [2,3,4] and references therein. The suppression of the single-Pomeron Born cross section due to the multi-Pomeron contributions 2 depends, in general, on the particular hard process. At the Tevatron energy, √ s = 1.8 TeV, the suppression is in the range 0.05-0.2 [6,7,8,2]. In fact, a computation of this effect was found to give a quantitative understanding [2] of the experimentally-observed 2 Another approach to the phenomenon of factorization breaking has been proposed by Goulianos within the so-called gap probability renormalization model, see [5] and references therein.
suppression of the single diffractive dijet cross sections at the Tevatron [9] as compared to the predictions based on HERA results [10]. The comparison relies on the partonic distributions in the Pomeron determined from HERA data. These parton densities have some uncertainty (especially for the gluonic content of the Pomeron). Interestingly, when the new fit to the H1 diffractive data [11] is used in the approach of Ref. [2], even better agreement with the CDF Tevatron data [9] is achieved [12].
Nowadays diffractive processes are attracting more attention as a way of extending the physics programme at proton colliders, including novel ways of searching for New Physics; see, for example, [13,14,15]. Clearly, the correct treatment of the screening effects is crucial for the reliability of the theoretical predictions of the cross sections for these diffractive processes. As mentioned above, some tests of the mechanism of diffractive dijet production have been made [16,2,12], but further, model-independent, checks are highly desirable.
Double-diffractive dijet production provides the attractive possibility to test factorization, in a parameter-free way, using only data from hadronic collisions. This allows an important consistency test of the whole approach. Indeed, a test of factorization has been performed recently by the CDF collaboration at √ s = 1.8 TeV [17].
since the hard subprocess is dominated by gluon t channel exchange. Here g(x) and q(x) denote the gluon and the sum of quark, antiquark densities, and 4/9 is the appropriate colour factor. The subscript a = p (p) or P indicates whether the 'gluon' belongs to the proton (antiproton) or the Pomeron, as seen, for example, in the upper and lower parts of Fig. 1(a) respectively.
Following [17], we consider first the ratio of SD dijet production to non-diffractive (ND) dijet production. Then, in the ratio of the cross sections (for the same kinematical characteristics of jets, E T i > 7 GeV), the quantity f p (x i )σ jj (η j1 , η j2 , E T 1 , E T 2 ) cancels out, where σ jj is the partonic cross section to produce dijets with pseudorapidities η j1 , η j2 and transverse energies E T 1 , E T 2 . Moreover the two processes are measured by the same detector so, in the ratio, experimental uncertainties are reduced. Allowing for the above cancellation, the ratio can be written in the form where F P (ξp) is the Pomeron 'flux factor', ξp is the fraction of the initial momentum carried by the Pomeron (experimentally ξp < 0.1), f P (β) is the effective distribution of partons in the Pomeron and xp = βξp. The suppression factor S 1 , which, following Bjorken [19], is often called the survival probability, accounts for the screening effects caused by diagrams of the type shown in Fig. 1(c). It is normalized so that S 1 ≡ 1 for the Born diagram of Fig. 1(a).
In order to provide a quantitative check of the 'factorization-violating' suppression factors, it is possible to remove, to a large extent, the uncertainties associated with the Pomeron flux factor and the parton distributions and detector effects, by studying a second ratio of measured cross sections. Considering the ratio of cross sections for dijet production by the double-Pomeron-exchange (DP) and the single diffractive (SD) processes, we have where S 2 is the suppression factor for the DP process (in general S 2 = S 1 ). Then the ratio of the two ratios, (2) and (3), becomes In the case when ξp = ξ p and β = β 1 (xp = x p ), the double ratio becomes If there were no multi-Pomeron effects (S 1 = S 2 = 1) then D = 1. Thus a deviation of D from unity would signal a failure of factorization. We emphasize that, although we use Regge factorization (with the cross sections written as products of flux factors and the corresponding partonic distributions), the result is practically insensitive to this assumption. The breakdown of factorization for hard diffractive processes is naturally expected in QCD; see, for example, [20,2,4].
To make a quantitative evaluation of the ratio D we use the model predictions of Ref. [7] for the multi-Pomeron screening effects, where the suppression factors S i were calculated for a range of hard diffractive processes at the various pp (and pp) collider energies. For our processes at √ s = 1.8 TeV they were found to be S 1 = 0.10, S 2 = 0.05.
The difference between S 1 and S 2 is due to the difference in the impact parameter profiles for the processes of diagrams 1(a) and 1(b). Thus the prediction of the model is D = 0.2. It was emphasized in Ref. [2] that the suppression factor 4 S 1 can depend also on xp and β. When more precise data become available, these dependences can be taken into account. However, in the kinematical range of the CDF measurement [17], the average value of S 1 turns out to be very close to 0.10.
The experimental value for the double ratio obtained by the CDF Collaboration [17] is D = 0.19 ± 0.07, in good agreement with the theoretical prediction. It clearly demonstrates the presence of factorization breaking and the importance of unitarization effects due to multi-Pomeron exchanges.
It is worth commenting briefly on one aspect of the experimental determination of the ratio D [17]. The range of ξp covered in the CDF measurements was 0.035 < ξp < 0.095, while ξ p was in the interval 0.01 < ξ p < 0.03. It was therefore necessary to extrapolate the second ratio R DP SD in ξp to the value ξp = 0.02 in order to have the same values of ξ i in Eq. (4). This was done on the assumption that this ratio is independent of ξp (for a given xp), as was in fact seen in the data in the observed range of ξp. This assumption is confirmed by the model of Ref. [2], where a detailed description of the CDF data on SD dijet production was performed. It was shown that the ξp-dependence, arising from the Pomeron and secondary exchange contributions in the region 0.01 ≤ ξp ≤ 0.1, can be approximated by 1/ξ n 1 p with n 1 ≃ 1. Moreover, the βdependence in the region β < 0.2 is also close to 1/β n 2 with n 2 ≃ 1; a behaviour which well summarizes several contributing β behaviours [2]. Thus, the total dependence on β and ξp is 1/βξp = 1/xp. Therefore, for fixed xp there is no dependence on ξp. The approximate equality of the values of n 1 and n 2 may not hold at very small ξp or at much higher energies [2].
In conclusion, we have demonstrated that a theoretical framework, which takes into account unitarization effects due to multi-Pomeron exchanges, predicts a substantial breakdown of factorization in hard diffractive processes. The multi-Pomeron effects suppress the Born cross section by different factors for different diffractive processes, due to the different impact parameter profiles of the various cross sections. We have proposed a check of the factorizationviolating suppression factors which depends only on measured cross sections at the Tevatron, namely the cross sections for dijet production in single diffractive and double-Pomeron-exchange processes. The prediction and the data are in good agreement. A detailed comparison, along the lines proposed, of forthcoming precision dijet data from the Tevatron or LHC will be very informative.