Extrema bounds for the soft Pomeron intercept

By using an extended Regge parametrization and taking into account the discrepancies in the high-energy pp and p(bar)p total cross section data, in both accelerator and cosmic-ray regions, we estimate extrema bounds for the soft Pomeron intercept. First we consider two ensembles of data with either the CDF or the E710 and E811 results for the p(bar)p total cross section at 1.8 TeV, from which we obtain the bounds 1.102 and 1.081, respectively. These ensembles are then combined with the highest and lowest estimations for the pp total cross section from cosmic-ray experiments (6-40 TeV), leading to the upper and lower bounds 1.109 and 1.082, respectively. The effects of simultaneous fits to total cross sections and the rho parameter, individual fits to total cross sections, and the influence of the subtraction constant in the dispersion relations are also presented. Our global results favor the E710 and E811 data.


I. INTRODUCTION
Analytic models for hadron-hadron scattering are characterized by simple parametrizations for the forward amplitude F and the use of dispersion relation techniques to study the total cross section σ tot and the ρ parameter (the ratio of the real to the imaginary part of the amplitude), σ tot (s) = Im F (s, t = 0) s , ρ = Re F (s, t = 0) Im F (s, t = 0) , where t is the four-momentum transfer squared and s the center-of-mass energy squared. In a recent work several aspects concerning the application of the analytic models to pp andpp elastic scattering have been studied [1]. In particular, in the case of the Donnachie-Landshoff parametrization, investigation of discrepant estimations for the total cross sections from cosmic-ray experiments allowed to infer an upper bound for the soft Pomeron intercept, namely, 1 + ǫ = 1.094. In addition, the effects of global vs individual fits to σ tot and ρ, and the effects of the subtraction constant in the dispersion relations have also been analyzed and discussed.
In this report we extend the previous analysis in several ways, with focus on new upper/lower bounds for the soft Pomeron intercept: (1) we investigate the effect of discrepant values for σ tot from accelerator experiments at 1.8 TeV, by selecting different ensembles of data that include either the highest (CDF) or the lowest (E710/E811) results; (2) these ensembles are then combined with discrepant estimations for σ tot from cosmic-ray experiments, now using as the lowest estimations the results by Block, Halzen, and Stanev; (3) we use here, as a framework, an extended parametrization with non-degenerate C = +1 and C = −1 meson trajectories. As in the previous analysis we also present the effects of individual and global fits to σ tot and ρ, and the effect of the subtraction constant. Since the soft Pomeron exchange dominates the high energy behavior of the total cross sections and the pp andpp scattering correspond to the highest energy interval with available data (including information from cosmic-ray experiments), we shall limit our analysis to these processes.
The paper is organized as follows. In Sec. II the essential formula of the analytic approach with the extended Regge parametrization are presented. In Sec. III the discrepancies at both accelerator and cosmic-ray domains are reviewed, and it the fit results through four different ensembles of experimental information are presented. The conclusions and some final remarks are the contents of Sec. IV.
The forward effective Regge amplitude introduced by Donnachie and Landshoff has two contributions, one from a single Pomeron and the other from secondary Reggeons exchanges [2]. The model assumes degeneracies between the secondary reggeons, imposing a common intercept for the C = +1 (a 2 , f 2 ) and the C = −1 (ω, ρ) trajectories. This was the parametrization adopted in the previous paper [1].
Although the original fits by Donnachie and Landshoff have been performed only to the σ tot data, more recent analysis, treating global fits to σ tot and ρ, have indicated that the best results are obtained with non-degenerate meson trajectories [3,4]. In this case the forward scattering amplitude is decomposed into three reggeon exchanges, F (s) = F IP (s) + F a2/f2 (s) + τ F ω/ρ (s), where the first term represents the exchange of a single Pomeron, the other two the secondary Reggeons and τ = +1 (−1) for pp (pp) amplitudes. Using the notation α IP (0) = 1 + ǫ, α + (0) = 1 − η + and α − (0) = 1 − η − for the intercepts of the Pomeron and the C = +1 and C = −1 trajectories, respectively, the total cross sections, Eq. (1), for pp andpp interactions are written as The connection with the ρ parameter is obtained by means of dispersion relations and for the above parametrization convergence is ensured by using analyticity relations with one subtraction. Defining 2F ± ≡ F pp ± Fp p these relations read [1] Re where K is the subtraction constant. Within this formalism, Eqs. (1), (2) and (3) lead to the following connection between ρ(s) and σ tot (s):

III. DISCREPANCIES, STRATEGIES AND FITTING RESULTS
The experimental information on pp andpp total cross sections at the highest energies are characterized by discrepant results. As is well known, in the accelerator region, the conflit concerns the results for σp p tot at √ s = 1.8 TeV reported by the CDF Collaboration [5] and those reported by the E710 [6] and the E811 [7,8] Collaborations (Fig. 1). In the cosmic-ray region, 6 TeV < √ s ≤ 40 TeV, the discrepancies are due to both experimental and theoretical uncertainties in the determination of σ pp tot from p-air cross sections. The situation has been recently reviewed in detail in our previous paper [1], where a complete list of references, numerical results and discussions are presented. As showed there, the highest predictions for σ pp tot concern the result by Gaisser, Sukhatme, and Yodh [9] together with those by Nikolaev [10]. In the other extreme, the lowest values come from the results by Block, Halzen, and Stanev [11]. These extrema estimations are displayed in Fig. 1 (numerical values may be found in ref. [1]).
Although, in principle, all available data in the accelerator region could be used, it should be stressed that the difference between the CDF and the E710/E811 results involves two standard deviations [7]. This strong disagreement certainly indicates the possibility of distinct scenarios for the rise of the total cross section and consequently for the value of the Pomeron intercept. Moreover, despite the large error bars in the extracted values of σ pp tot from cosmic-ray experiments, the discrepancies also presented can corroborate the distinction between the different scenarios. It is expected that answers to these questions will be provided by the new values for σ tot and ρ coming from the BNL RHIC, the Fermilab Tevatron-run II and the CERN LHC.
Based on these facts, we consider important at the moment to investigate these experimental discrepancies and examine its consequences in terms of extrema bounds for the Pomeron intercept.
Since recent analysis showed that the parameters of Regge fits are stable for a cutoff √ s ∼ 9 GeV [13], in what follows we consider experimental data on σ tot and ρ above √ s = 10 GeV. We use the data sets compiled and analyzed by the Particle Data Group [14], to which we add the new E811 data on σ tot and ρ at 1.8 TeV [8]. The statistic and systematic errors have been added in quadrature.
In order to investigate the effects of the discrepancies in a quantitative way, we select different ensembles of σ tot data that include all the ρ results above 10 GeV. First we only consider accelerator data in two ensembles with the pp pp √ √  II: σ pp tot and σp p tot data (10 ≤ √ s ≤ 900 GeV) + E710/E811 data ( √ s = 1.8 TeV). Ensemble I represents the faster increase scenario for the rise of σ tot from accelerator data and ensemble II the slowest one. These ensembles are then combined with the highest and lowest estimations for σ pp tot from cosmic-ray experiments, namely, the Nikolaev and Gaisser, Sukhatme, and Yodh (NGSY) results and the Block, Halzen, and Stanev (BHS) results, respectively. These new ensembles are denoted by Ensemble I + NGSY and Ensemble II + BHS.
As in the previous paper, we consider both individual fits to σ tot , and simultaneous fits to σ tot and ρ, either in the case where the subtraction constant is considered as a free fit parameter or assuming K = 0 in Eq. (3). The fits have been performed with the program CERN-MINUIT and the errors in the fit parameters correspond to an increase of the χ 2 by one unit.
In the case of accelerator data only the fit results for σ tot and ρ with ensembles I and II are displayed in Table I and Figs. 2, 3 and 4. The results concerning the combination of these ensembles with the estimations from cosmic-ray experiments, namely, ensembles I + NGSY and II + BHS, are shown in Table II and Figs. 5, 6 and 7. In the last two cases we present the curves and experimental information in the region from 500 GeV to 50 TeV, since the results are the same at lower energies, as can be seen in the corresponding results for ρ(s).

IV. CONCLUSIONS AND FINAL REMARKS
In this analysis we have used the experimental information presently available in the accelerator domain, including the recent E811 results on σp p tot and ρ at 1.8 TeV, and also the highest and lowest estimations for σ pp tot from cosmic-ray experiments.
From Table I ( Adding the cosmic-ray information, Table II, we infer α upper IP (0) = 1.104 ± 0.005 (individual fit to σ tot from ensemble I + NGSY) and α lower IP (0) = 1.085±0.003 (global fits to ensemble II + BHS, and K as a free fit parameter or individual fit to σ tot from this ensemble), with bounds 1.109 and 1.082, respectively.
Our approach, and the above values and bounds, may be compared with some representative results obtained by other authors, which are displayed in Table III   1.8 TeV. The CDF Collaboration (CDF), based on their result for the σp p tot , obtained a higher value for the intercept [5]. Further analyses, through the extended Regge parametrization, included also the E811 result. In the work by Cudell, Kang, and Kim (CKK) [3] only pp andpp data above 10 GeV have been fitted. The analysis by Covolan, Montanha and Goulianos (CMG) [4] (using both a Born level and Eikonal parametrizations) involved global fits to pp,pp, π ± p and k ± p at √ s ≥ 6 GeV. The COMPETE Collaboration (COMPETE) [13] treated simultaneous fits to σ tot and ρ in global fits to pp,pp, meson-p, γp and γγ above 9 GeV. All these results concerned only accelerator data. In Ref. [1],Ávila, Luna and Menon (ALM) included also some cosmic-ray estimations for the σ pp tot and made use of the original DL parametrization. It is also shown in Table III a recent theoretical result by R. A. Janik (Janik) [15] through a nonperturbative approach and using the AdS/CFT correspondence.
In this work we have presented all the possible fits to pp andpp data, above 10 GeV, through the extended Regge parametrization, exploring the contrasting data and the faster and the slower increase scenarios for the rise of the total cross section, allowed by the experimental information presently available.
From Table III, our results exclude the values for the Pomeron intercept obtained by CMG (in the case of the eikonal parametrization), the lower bounds by ALM and Janik and the mean value by the CDF Collaboration. The  [2] 0.0808 -CDF [5] 0.112 ± 0.013 -CKK [3] 0.096 +0.012 −0.009 -CMG [4] 0.104 ± 0.002 (Born) -0.122 ± 0.002 (Eikonal) -COMPETE [13] 0.093 ± 0.002 -ALM [1] -0.0790 − 0.0940 Janik [15] -0.0729 − 0.083 This work 0.085 ± 0.004 (lower) 0.081 0.104 ± 0.005 (upper) 0.109 DL result is barely compatible with our lower limit. It should be noted that, if the same ensemble is fitted, the introduction of nondegenerate trajectories results in a slightly increase of the Pomeron intercept. For example, fit to all the accelerator data above 10 GeV (including the CDF and the E710/E811 values), leads to ǫ = 0.086 ± 0.003 and ǫ = 0.089 ± 0.004, in the cases of degenerate (DL) and nondegenerate parametrizations, respectively. From figures (2) to (7), we see that in all the cases investigated the ρ parameter is better described with ensembles I and I + BHS, a result that is also roughly supported by the χ 2 /DOF (Tables I and II). We understand that this picture favors the E710/E811 results. This conclusion is contrary to that obtained by CMG [4] and more recently by the COMPETE Collaboration [16].
As a next step it may be important to investigate the consequences of the above extrema bounds in fittings to p-mesons, pγ, and γγ scattering, with focus in the ratio of strengths of the Pomeron exchange (quark counting and factorization).