Open problems in forward-backward multiplicity correlations in hadron-hadron collisions in the TeV region

Continuing previous work on forward-backward multiplicity correlation properties in proton-proton collisions in the framework of the weighted superposition model of two components (each one described by a negative binomial multiplicity distribution) with the addition of the leakage parameter which controls clan spreading from one hemisphere to the opposite one, we examine E735 data on the c.m. energy dependence of the total correlation strength and of the forward variance at fixed total multiplicity. A comparison with the Chou-Yang approach to the problem is presented and extrapolations of the mentioned variables at LHC c.m. energy in possible scenarios in the new energy domain are discussed.

In the case of e + e − annihilation, the correct value of b total at LEP energy was reproduced under the experimental conditions b 2-jet ≈ b 3-jet ≈ 0. This fact was considered as a successful test of the superposition mechanism itself.
The situation was found to be quite different in pp collisions. Here it was shown that the correct energy dependence of b total can be determined by assuming, in addition to the above mentioned parameters, the explicit form of the MD in the soft and semi-hard components (i.e., two negative binomial (NB) MD's with different parameters) and related clan structure, and by introducing the corresponding particle leakage parameters p soft and p semi-hard , which control clan spreading over both hemispheres. The leakage parameter is indeed defined as the fraction of particles within one clan which remain in the same hemisphere where the clan was produced.
Within this framework, the energy dependence of b total was then extrapolated in the TeV region by examining three different scenarios (see [2] for details) characterised by the same soft component structure satisfying KNO scaling, and by different semi-hard component behaviours: 1) obeying again KNO scaling, or 2) with strong KNO scaling violation or 3) with a QCD inspired behaviour. A clear bending of b total was visible in all examined scenarios. It was remarked that an early saturation of b total toward 1 in the semi-hard component would require a fast increase with energy of particle leakage, i.e, a decrease of the corresponding leakage parameter p semi-hard , and would favour strong KNO scaling violation.
The aim of this paper is to discuss in the mentioned framework the E735 Collaboration results [3] on c.m. energy dependence of the FBMC strength, b total , and of the forward variance at fixed total multiplicity n, d 2 n F (n), obtained at Tevatron. It should be pointed out that where indicates an average over all events, and D n is the dispersion of the MD. Furthermore, let us introduce the variable where n indicates the average over all events at fixed n. It is clear that variable (2) works at a deeper level of investigation, being variable (1) related to the average of (2) over all multiplicities. Variable (1) is particularly interesting for global properties of the collisions related to average n.
Variable (1), in the weighted two-component superposition model summarised above, can be expressed as follows: where the single component FBMC strength is with i = soft, semi-hard; k is the parameter of the NBMD which is related to the dispersion by k −1 = (D 2 n −n)n −2 .
Variable (2) in turn can be written as where is the total MD, with P 1 (n) and P 2 (n) the two component MD's, respectively.
It is found that the points at 1000 GeV and 1800 GeV from E735 Collaboration have the same energy dependence (they lie on the same straight line) as the other data in the GeV region [3]. In order to include the point at 1800 GeV the three extrapolated scenarios for the semi-hard component discussed in Ref. [1] are reexamined. Results are shown in Fig. 1.
In general, one can conclude that the leakage parameter for the semi-hard component, p semi-hard , must decrease, and accordingly particle leakage increase, in all scenarios. To be quantitative, we have found that satisfactory results are obtained by taking the leakage parameter for the soft component energy independent and equal to 0.8, as argued in [1], and taking tentatively p semi-hard = 0.84−0.07 log( √ s/200) for √ s > 200 GeV; keeping this energy dependence for p semi-hard at all energies, the curves in Fig. 1 have been extrapolated to 14 TeV. It should be noticed that in scenario 2, characterised by a semi-hard component with strong KNO scaling violation, the FBMC strength becomes less steep with the increase of the c.m. energy and its saturation toward 1 (as that of b total ) quicker than in the other two scenarios.
In conclusion, a linear behaviour of b total with c.m. energy is incompatible with our approach above 2.5 TeV in scenario 1, above 3.5 TeV in scenario 2 and above 5 TeV in scenario 3, i.e., the leakage parameter energy dependence cannot be adjusted to such situation in the various scenarios without spoiling the model itself. On the contrary, if such a linear behaviour were found experimentally at LHC, it could be indicative of the onset of a third component (class of events). z 2 n vs n dependence Figure 2 shows experimental data from UA5 Collaboration [4] at 900 GeV in 1 < |η| < 4 together with the result of calculations in the present approach (solid line); the dashed line is a linear fit according to the cluster model of Chou and Yang [5], already discussed in [1]. Below n ≈ 40, where data are available, it is quite hard to distinguish the two model predictions; at n ≈ 40 our approach shows a "hump". In view of the lack of sufficiently precise data in this domain, no conclusions can be drawn.
It should be pointed out that at higher c.m. energy (1.8 TeV), the E735 data also show at n ≈ 40 a qualitative picture like a hump. However, a quantitative comparison is problematic for two reasons: a) our calculations are based on extrapolations in full phasespace while data refer to the interval |η| < 3.25, for which no MD has been published (on the contrary, available MD data from UA5 allowed us to compare our model's predictions at SppS); b) at lower c.m. energy (546 GeV, see Fig. 3) we noticed a discrepancy between UA5 results [6] and E735 results; for completeness, in Fig. 3 our results and a linear fit are also shown.
The three scenarios we have discussed previously do not show any remarkable difference in the GeV region as far as the z 2 n vs n dependence is concerned (see Fig 4a, where z 2 n vs n is plotted in the three scenarios at 900 GeV in full phase-space). In Fig. 4b the same plot is shown at 1800 GeV. Differences in the three scenario predictions become more evident for n larger than 40 and the hump is more visible. In Fig. 4c at 14 TeV the hump becomes even more visible and in addition its maximum varies with the scenario.
In conclusion, the behaviour of z 2 n , i.e., the two different sides of the hump appearing in the plot of z 2 n vs n, which is remarkable at 1800 GeV, confirms in our view the presence of two components (samples of two classes of events) and the importance of the role of the semi-hard component at this energy. The question remains whether this is also the indication of the occurrence of a phase transition [7].