Leading components in forward elastic hadron scattering: Derivative dispersion relations and asymptotic uniqueness

Abstract

Forward amplitude analyses constitute an important approach in the investigation of the energy dependence of the total hadronic cross-section σ tot and the ρ parameter. The standard picture indicates for σ tot a leading log-squared dependence at the highest c.m. energies, in accordance with the Froissart-Lukaszuk-Martin bound and as predicted by the COMPETE Collaboration in 2002. Beyond this log-squared (L2) leading dependence, other amplitude analyses have considered a log-raised-to-gamma form (Lγ), with γ as a real free fit parameter. In this case, analytic connections with ρ can be obtained either through dispersion relations (derivative forms), or asymptotic uniqueness (Phragmén-Lindelöff theorems). In this work we present a detailed discussion on the similarities and mainly the differences between the Derivative Dispersion Relation (DDR) and Asymptotic Uniqueness (AU) approaches and results, with focus on the Lγ and L2 leading terms. We also develop new Regge-Gribov fits with updated dataset on σ tot and ρ from pp andpp scattering, including all available data in the region 5 GeV -8 TeV. The recent tension between the TOTEM and ATLAS results at 7 TeV and mainly 8 TeV is discussed and considered in the data reductions. Our main conclusions are the following: (1) all fit results present agreement with the experimental data analyzed and the goodness-of-fit is slightly better in case of the DDR approach; (2) by considering only the TOTEM data at the LHC region, the fits with Lγ indicate γ ∼ 2.0 ± 0.2 (AU approach) and γ ∼ 2.3 ± 0.1 (DDR approach); (3) by including the ATLAS data the fits provide γ ∼ 1.9 ± 0.1 (AU) and γ ∼ 2.2 ± 0.2 (DDR); (4) in the formal and practical contexts, the DDR approach is more adequate for the energy interval investigated than the AU approach. A pedagogical and detailed review on the analytic results for σ tot and ρ from the Regge-Gribov, DDR and AU approaches is presented. Formal and practical aspects related to forward amplitude analyses are also critically discussed. PACS: 13.85.-t, 13.85.Lg, 11.10.Jj Keywords: Hadron-induced high-and super-high-energy interactions, total cross-sections, asymptotic problems and properties Published in International Journal of Modern Physics A 32 (2017) 1750184 In 1993, the UA4/2 Collaboration extended the analysis up to 546 GeV (pp scattering) obtaining [15] γ = 2.24 +0.35 −0.31 . This result has been confirmed by Bueno and Velasco in 1996, who develop also fits to only σ tot data, obtaining [16] γ = 2.42 +0.4 −0.6 , for cutoff at 5 GeV and γ = 2.64 +0.50 −0.32 ,for cutoff at 10 GeV. After the first measurement of the total cross section from LHC7 by the TOTEM Collaboration, Fagundes, Menon and Silva (hereafter FMS) developed in 2012 an amplitude analysis with basis on the parametrization introduced by Amaldi et al. for the total cross section [28]. As in the above works, the data reductions were limited to pp andpp scattering. This analysis was then developed and extended in [29] and further updated and discussed by Menon and Silva [30,31], leadin...

Structure of forward pp and pp¯ elastic amplitudes at low energies

“…At low energies it is essential that in both DRA and DRS calculations the exact solutions [14] be used. The dispersion relations for the amplitudes has been effectively used in investigations of the energy dependence of total cross section and ρ parameter in pp and pp scattering, being a very important tool of control in the analysis of the data [15].…”

Soft pomerons and the forward LHC data

“…Further arguments in this direction are presented in what follows. It is important to stress a central point in our analysis and on the strategy employed (see also Appendix A.2 in [5] for further discussions and complete list of reference to the experimental data to be quoted). In the recent paper by Martynov and Nicolescu, the authors did not include the ATLAS data at 7 and 8 TeV, because these points "are incompatible with the TOTEM data and their inclusion would obviously compromise the coherence of the overall data" [15].…”

“…In the Regge-Gribov formalism [2][3][4], the singularities in the complex angular momentum J-plane (t-channel) are associated with the asymptotic behavior of the elastic scattering amplitude in terms of the energy (s-channel). In the general case, associated with a pole of order N, the contribution to the imaginary part of the forward amplitude in the s-channel is s α 0 ln N −1 (s), where α 0 is the intercept of the trajectory (see Appendix B in [5] for a recent short review). Therefore, for the total cross section we have σ tot (s) ∝ s α 0 −1 ln N −1 s, and the following possibilities connecting the singularities at J = α 0 and the asymptotic behavior: simple pole (N = 1) ⇒ σ ∝ s α 0 −1 ; double pole (N = 2) at α 0 = 1 ⇒ σ ∝ ln(s); triple pole (N = 3) at α 0 = 1 ⇒ σ ∝ ln 2 (s).…”

Forward elastic scattering and Pomeron models

“…The data above 5 GeV and below 7 TeV have been collected from the PDG database [12], without any kind of data selection or sieve procedure (we have used all the published data by the experimental collaborations). The data at 7 and 8 TeV by the TOTEM and ATLAS Collaborations can be found in [7], Table 1, together with further information and complete list of references. The TOTEM data at 13 TeV are those in (3) [13,14].…”

“…In this formalism [5,6], the singularities in the complex angular momentum J-plane (t-channel) are associated with the asymptotic behavior of the elastic scattering amplitude in terms of the energy (s-channel). In the general case, associated with a pole of order N , the contribution to the imaginary part of the forward amplitude in the s-channel is s α 0 ln N −1 (s), where α 0 is the intercept of the trajectory (see Appendix B in [7] for a recent short review). Therefore, for the total cross section we have σ tot (s) ∝ s α 0 −1 ln N −1 s, and the following possibilities connecting the singularities (J-plane) and the asymptotic behavior:…”