Predictions of pp, ¯ pp total cross section and ρ ratio at LHC and cosmic-ray energies

We propose to use rich informations on the pp, ¯ pp total cross sections σ tot below N ( ∼ 10)GeV in order to predict the total cross section and ρ ratio at very high energies. Using the FESR as a constraint for high energy parameters, we search for the simultaneous best ﬁt to the data points of σ tot and ρ ratio up to some energy (e.g., ISR, Tevatron) to determine the high-energy parameters. We then predict σ tot and ρ in the LHC and high-energy cosmic-ray regions. Using the data up to √ s = 1 . 8TeV(Tevatron), we predict σ pp tot and ρ pp at the LHC energy( √ s = 14TeV) as 106 . 3 ± 5 . 1 syst ± 2 . 4 stat mb and 0 . 126 ± 0 . 007 syst ± 0 . 004 stat , respectively. The predicted values of σ tot in terms of the same parameters are in good agreement with the cosmic-ray experimental data sample up to P lab ∼ 10 8 ∼ 9 GeV by Block, Halzen, and Stanov.

Recently [1], we have proposed to use rich informations on πp total cross sections below N(∼10 GeV) in addition to high-energy data to discriminate whether these cross sections increase like log ν or log 2 ν at high energies [2]. The FESR which was derived in the spirit of the P ′ sum rule [3] as well as the n = 1 moment FESR( [4], [5]) have been required to constrain the high-energy parameters. We then searched for the best fit of σ (+) tot above 70GeV in terms of high-energy parameters constrained by the two FESR. We then arrived at the conclusion that our analysis prefers the log 2 ν behaviours consistent with the Froissart-Martin unitarity bound [6].
As for thepp and pp total cross sections, there are a lot of data including cosmic-ray data up to √ s ∼ several times of 10 4 GeV compared with data up to √ s ∼30GeV for πN scattering. Therefore, it is very valuable if one could investigate the high-energy behaviours at LHC and cosmic-ray regions [8] using the similar approach as ref. [1].
(The purpose of this Letter): The purpose of this Letter is to predict σ (+) tot , thepp, pp total cross sections and ρ (+) , the ratio of the real to imaginary part of the forward scattering amplitude at the LHC and the higher-energy cosmic-ray regions, using the experimental data for σ (+) tot and ρ (+) for 70GeV< P lab < P large as inputs. We first choose P large = 2100GeV corresponding to ISR region( √ s ≃ 60GeV). Secondly we choose P large = 2 × 10 6 GeV corresponding to the Tevatron collider ( √ s ≃ 2TeV). In a recent paper, Block and Halzen [7] emphasized the importance of ρ for the evidence for saturation of the Froissart-Martin bound [6]. We also use the ρ ratio as input data in addition to FESR as a constraint. We searched for the simultaneous best fit of σ (+) tot and ρ (+) in terms of high-energy parameters c 0 , c 1 , c 2 and β P ′ constrained by the FESR. It turns out that the prediction of σ (+) tot agrees with pp experimental data at these cosmic-ray energy regions [8,22] within errors in the first case ( ISR ). It has to be noted that the energy range of predicted σ (+) tot , ρ (+) is several orders of magnitude larger than the energy region of σ (+) tot , ρ (+) input (see Fig. 1). If we use data up to Tevatron (the second case), the situation is much improved, although there are some systematic uncertainties coming from the data at √ s = 1.8TeV (see Fig. 2). FESR(1): Firstly we derive the FESR in the spirit of the P ′ sum rule [3]. Let us consider the crossing-even forward scattering amplitude defined by and subsequently obtain substituting α P ′ = 1 2 in Eq. (4). Let us definẽ Using the similar technique to ref. [1], we obtain Let us call Eq. (8) as the FESR(1).
(The ρ (+) ratio): The ρ (+) ratio, the ratio of the real to imaginary part of F (+) (ν) is obtained from Eqs. (2), (5) and (6) as (General approach): The FESR(1)(Eq. (8)) has some problem. i.e., there are the so-called unphysical regions coming from boson poles below thepp threshold. So, the contributions from unphysical regions of the first term of the right-hand side of Eq. (8) have to be calculated. Reliable estimates, however, are difficult. Therefore, we will not adopt the FESR(1).
On the other hand, contributions from the unphysical regions to the first term of the left-hand side of FESR(2)(Eq. (9)) can be estimated to be an order of 0.1% compared with the second term. 1 Thus, it can easily be neglected.
Therefore, the FESR(2)(Eq. (9)), the formula of σ (+) tot (Eqs. (1) and (2)) and the ρ (+) ratio (Eq. (10)) are our starting points. Armed with the FESR(2), we express high-energy parameters c 0 , c 1 , c 2 , β P ′ in terms of the integral of total cross sections up to N. Using this FESR(2) as a constraint for β P ′ = β P ′ (c 0 , c 1 , c 2 ), the number of independent parameters is three. We then search for the simultaneous best fit to the data points of σ (+) tot (k) and ρ (+) (k) for 70GeV≤ k ≤ P large to determine the values of c 0 , c 1 , c 2 giving the least χ 2 . We thus predict the σ tot and ρ (+) in LHC energy and high-energy cosmic-ray regions.
(Data): We use rich data [9] of σp p and σ pp to evaluate the relevant integrals of cross sections appearing in FESR (2). We connect the each data point 2 of 1 The average of the imaginary part from boson resonances below thepp threshold is the smooth extrapolation of the t-channel qqqq exchange contributions from high energy to ν ≤ M due to FESR duality [4,5]. Since Im F 3GeV. So, resonance contributions to the first term of Eq. (9) is less than 0.1% of the second term. Besides boson resonances, there may be additional contributions from multi-pion contributions belowpp threshold. In thepp annihilation,pp → ππ could give comparable contributions with ρ-meson, but multi-pion contributions are suppressed due to the phase volume effects. Therefore, the first term of Eq. (9) will still be negligible even if the above contributions are included. 2 We take the error ∆y for each data point y as ∆y = (∆y) 2 stat + (∆y) 2 syst . When several data points, denoted y i with error ∆y i (i = 1, · · · , n), are listed at the same value of k, these points are replaced byȳ with ∆ȳ, given byȳ = k 2 σp p tot and k 2 σ pp tot with the next point by a straight line in order, from k = 0 to k = N , and regard the area of this polygonal line graph as the relevant integral in the region 0 ≤ k ≤ N . The integral of k 2 σ (+) tot (k) is given by averaging those of k 2 σp p tot (k) and k 2 σ pp tot (k). We have obtained for N = 10GeV (which corresponds to √ s = E cm = 4.54GeV). 3 The error of the integral, which is from the error of each data point, is very small (less than 1%), and thus, we regard the central value as an exact one in the following analysis.
It is necessary to pay special attention to treat the data with the maximum k = 1.7266 × 10 6 GeV( √ s = 1.8TeV) in this energy range, which comes from the three experiments E710 [13]/E811 [14] and CDF [15]. The former two experiments are mutually consistent and their averagedpp cross section is σp p tot = 72.0 ± 1.7mb, which deviates from the result of CDF experiment σp p tot = 80.03 ± 2.24mb.
In the actual analyses, we use Re F (+) instead of ρ (+) (= Re F (+) /Im F (+) ). The data points of Re F (+) (k) are made by multiplying ρ (+) (k) by Im F (+) (k) = k 8π (σp p tot (k) + σ pp tot (k)). The values of σp p tot and σ pp tot at the relevant values of k are obtained as follows: For k < 1500GeV, they are determined by the formula given in ref. [9](see the footnote 4). Two experimental values [12,13] of σp p in the Tevatron region are used.
We have done for the following three cases: fit 1): The fit to the data up to ISR energy region, 70GeV ≤ k ≤ 2100GeV, which includes 12 points of σ (Results of the fit): The results are shown in Fig. 1(Fig. 2) for the fit 1(fit 4 Here the values of Im fp p (k) and Im f pp (k) at the relevant values of k are determined through the formula given in ref. [9], σp p/pp = Z + Blog 2 (s/s 0 ) +  Table 1 The values of χ 2 for the fit 1 (fit up to ISR energy) and the fit 2 and fit 3 (fits up to Tevatron-collider energy). N F and N σ (N ρ ) are the degree of freedom and the number of σ (+) tot (ρ (+) ) data points in the fitted energy region.  Table 2 The best-fit values of parameters in the fit 1, fit 2 and fit 3. tot and ρ (+) are less than or equal to unity. The fits are successful in all cases. There are some systematic differences between fit 2 and fit 3, which come from the experimental uncertainty of the data at √ s = 1.8TeV mentioned above.
The best-fit values of the parameters are given in Table 2. Here the errors of one standard deviation are also given. 6 (Predictions for σ (+) and ρ (+) at LHC and cosmic-ray energy region): By using the values of parameters in Table 2, we can predict the σ (+) tot and ρ (+) in higher energy region, as are shown, respectively in (c) and (d) of Fig. 1 and  2. The thin dot-dashed lines represent the one standard deviation.
As is seen in (c) and (d) of Fig. 1, the fit 1 leads to the prediction of σ (+) tot and ρ (+) with somewhat large errors in the Tevatron-collider energy region, although the best-fit curves are consistent with the present experimental data in this region. Furthermore, the predicted values of σ (+) tot agree with pp experimental data at the cosmic-ray energy regions [8,22] within errors (see (a),(c) of Fig. 1). The best-fit curve gives χ 2 /(number of data) to be 13.0/16, and 6 The c 2 log 2 (ν/M )-term in Eq. (2) is most relevant for predicting σ (+) tot in high energy region, and thus, the error estimation is done as follows: The c 2 is fixed with a value deviated a little from the best-fit value, and then the χ 2 -fit is done by two parameters c 0 and c 1 . When the resulting χ 2 is larger than the least χ 2 of the three-parameter fit by one, the corresponding values of parameters give one standard deviation. the prediction is successful. As was mentioned in the purpose of this Letter, it has to be noted that the energy range of predicted σ (+) tot is several orders of magnitude larger than the energy region of the σ (+) tot , ρ (+) input. If we use data up to Tevatron-collider energy region as in the fit 2 and fit 3, the situation is  The best-fit curve gives χ 2 /(number of data) from cosmic-ray data, 1.3/7(1.0/7) for fit 2(fit 3). Table 3 The predictions of σ (+) tot and ρ (+) at LHC energy √ s = E cm = 14TeV(P lab =1.04×10 8 GeV), and at a very high energy P lab = 5 · 10 20 eV ( √ s=E cm =967TeV.) in cosmic-ray region.
The prediction by the fit 1 in which data up to the ISR energy are used as input has somewhat large(fairly large) errors at LHC energy(at high energy of cosmic ray). By including the data up to the Tevatron collider, the prediction of fit 2(using E710/E811 datum) is smaller than that of fit 3(using CDF datum). We regard the difference between the results of fit 2 and fit 3 as the systematic uncertainties of our predictions. As a result, we predict σ pp tot = 106.3 ± 5.1 syst ± 2.4 stat mb, ρ pp = 0.126 ± 0.007 syst ± 0.004 stat (13) at LHC energy( √ s = E cm = 14TeV). We obtain fairly large systematic errors coming from the experimental unceratinty at √ s = 1.8 TeV.
The predicted central value of σ pp tot is in good agreement with Block and Halzen [18] σ pp tot = 107.4±1.2 mb, ρ pp = 0.132±0.001. In contrary to our results( see Fig. 2(a), (c)), however, their values are not affected so much about CDF, E710/E811 discrepancy. Our prediction has also to be compared with Cudell et al. [23]  Finally we emphasize that precise measurements of both σ pp tot and ρ pp in the coming LHC experiments will resolve the FNAL discrepancy of σ pp tot ( Fig. 2(a), (c)). The LHC measurements would also clarify which is the best solution among the three high-energy cosmic-ray samples [20,21,22]. Note added in proof: After completion of the hep-ph/0505058, we were informed that M. M. Block and F. Halzen [18] have also done the similar work based on the same spirit of duality using different method independently. We were also informed by M. J. Menon [19] about other cosmic-ray analyses by Gaisser et al. [20] and N. N. Nikolaev [21] besides M. M. Block et al. [22] which are used as cosmic-ray data in this Letter.