Derivative Dispersion Relations Research Articles

Optical theorem, crossing property, and derivative dispersion relations: implications on the asymptotic behavior of and

In this paper, we present some results on the behavior of the total cross section and ρ-parameter at asymptotic energies in proton–proton ( ) and antiproton–proton ( ) collisions. Hence, we consider three of the main theoretical results in high energy physics: the crossing property, derivative dispersion relation, and optical theorem. The use of such machinery facilitates the derivation of analytic formulas for a wide set of the measured global scattering parameters and some important relations between them. The suggested parameterizations approximate the energy dependence for the total cross section and ρ-parameter for and with a statistically acceptable quality in the multi-TeV region. Additionally, the qualitative description is obtained for important interrelations, namely difference, sum, and ratio of the antiparticle–particle and particle–particle total cross sections. Despite the reduced number of experimental data for the total cross section and ρ-parameter at the TeV-scale, which complicates any prediction for the beginning of the asymptotic domain, the fitting procedures indicates that asymptotia occur in the energy range 25.5–130 TeV. Moreover, in the asymptotic regime, we obtain . A detailed quantitative study of the energy behavior of the measured scattering parameters and their combinations in the ultra–high energy domain indicates that the scenario with the generalized formulation of the Pomeranchuk theorem is more favorable with respect to the original formulation of this theorem.

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

Forward amplitude analyses constitute an important approach in the investigation of the energy dependence of the total hadronic cross-section [Formula: see text] and the [Formula: see text] parameter. The standard picture indicates for [Formula: see text] 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 [Formula: see text], with [Formula: see text] as a real free fit parameter. In this case, analytic connections with [Formula: see text] 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 [Formula: see text] and L2 leading terms. We also develop new Regge–Gribov fits with updated dataset on [Formula: see text] and [Formula: see text] from [Formula: see text] and [Formula: see text] 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 at 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 L[Formula: see text] indicate [Formula: see text] (AU approach) and [Formula: see text] (DDR approach); (3) by including the ATLAS data the fits provide [Formula: see text] (AU) and [Formula: see text] (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 [Formula: see text] and [Formula: see text] from the Regge–Gribov, DDR and AU approaches is presented. Formal and practical aspects related to forward amplitude analyses are also critically discussed.

Bounds on the rise of total cross section from LHC7 and LHC8 data

Recent measurements of the proton–proton total cross section σtot at 7 and 8 TeV by the TOTEM and ATLAS Collaborations are characterized by some discrepant values: the TOTEM data suggest a rise of the cross section with the energy faster than the ATLAS data. Attempting to quantify these different behaviors, we develop new analytical fits to σtot and ρ data from pp and p¯p scattering in the energy region 5 GeV–8 TeV. The dataset comprises all the accelerator data below 7 TeV and we consider three ensembles by adding: either only the TOTEM data (T), or only the ATLAS data (A), or both sets (T + A). For the purposes, we use our previous RRPLγ parametrization for σtot(s), consisting of two Reggeons (RR), one critical Pomeron (P) and a leading log-raised-to-gamma (Lγ) contribution (with γ as a free fit parameter), analytically connected to ρ(s) through singly-subtracted derivative dispersion relations and energy scale fixed at the physical threshold. The data reductions with ensembles T and A present good agreement with the experimental data analyzed and cannot be distinguished on statistical grounds. The quality of the fit is not as good with ensemble T + A. The fit results provide γ∼2.3±0.1 (T), 2.0±0.1 (A), 2.2±0.2 (T+A), with χ2/DOF∼1.07 (T), 1.09 (A), 1.14 (T + A), suggesting extrema bounds for γ given by 1.9 and 2.4. Fits with γ=2 (fixed) are also developed and discussed.

AN UPDATED ANALYSIS ON THE RISE OF THE HADRONIC TOTAL CROSS-SECTION AT THE LHC ENERGY REGION

A forward amplitude analysis on pp and [Formula: see text] elastic scattering above 5 GeV is presented. The dataset includes the recent high-precision TOTEM measurements of the pp total and elastic (integrated) cross-sections at 7 TeV and 8 TeV. Following previous works, the leading high-energy contribution for the total cross-section (σ tot ) is parametrized as ln γ(s/sh), where γ and sh are free real fit parameters. Singly-subtracted derivative dispersion relations are used to connect σ tot and the rho parameter (ρ) in an analytical way. Different fit procedures are considered, including individual fits to σ tot data, global fits to σ tot and ρ data, constrained and unconstrained data reductions. The results favor a rise of the σ tot faster than the log-squared bound by Froissart and Martin at the LHC energy region. The parametrization for σ tot is extended to fit the elastic cross-section (σ el ) data with satisfactory results. The analysis indicates an asymptotic ratio σ el /σ tot consistent with 1/3 (as already obtained in a previous work). A critical discussion on the correlation, practical role and physical implications of the parameters γ and sh is presented. The discussion confronts the 2002 prediction of σ tot by the COMPETE Collaboration and the recent result by the Particle Data Group (2012 edition of the Review of Particle Physics). Some conjectures on possible implications of a fast rise of the proton–proton total cross-section at the highest energies are also presented.

On the rise of proton–proton cross-sections at high energies

The rise of total, elastic and inelastic hadronic cross sections at high energies is investigated by means of an analytical parametrization, with the exponent of the leading logarithm contribution as a free fit parameter. Using derivative dispersion relations with one subtraction, two different fits to proton–proton and antiproton–proton total cross section and ρ parameter data are developed, reproducing well the experimental information in the energy region 5 GeV–7 TeV. The parametrization for the total cross sections is then extended to fit the elastic (integrated) cross section data in the same energy region, with satisfactory results. From these empirical results we extract the energy dependence of several physical quantities: inelastic cross section, ratios elastic/total, inelastic/total cross sections, ratio total-cross-section/elastic-slope, elastic slope and optical point. All data, fitted and predicted, are quite well described. We find a statistically consistent solution indicating: (1) an increase of the hadronic cross sections with the energy faster than the log-squared bound by Froissart and Martin; (2) asymptotic limits 1/3 and 2/3 for the ratios elastic/total and inelastic/total cross sections, respectively; a result in agreement with unitarity. These indications corroborate recent theoretical arguments by Azimov on the rise of the total cross section.

Two-point derivative dispersion relations

A new derivation is given for the representation, under certain conditions, of the integral dispersion relations of scattering theory through local forms. The resulting expressions have been obtained through an independent procedure to construct the real part and consist of new mathematical structures of double infinite summations of derivatives. In this new form the derivatives are calculated at the generic value of the energy E and separately at the reference point E = m that is the lower limit of the integration. This new form may be more interesting in certain circumstances and directly shows the origin of the difficulties in convergence that were present in the old truncated forms called standard-derivative dispersion relations (DDR). For all cases in which the reductions of the double to single sums were obtained in our previous work, leading to explicit demonstration of convergence, these new expressions are seen to be identical to the previous ones. We present, as a glossary, the most simplified explicit results for the DDR’s in the cases of imaginary amplitudes of forms (E/m)λ[ln (E/m)]n that cover the cases of practical interest in particle physics phenomenology at high energies. We explicitly study the expressions for the cases with λ negative odd integers, that require identification of cancelation of singularities, and provide the corresponding final results.

Dispersion Relations for Meson–Proton and Proton–Proton Forward Elastic Scattering

An analysis of the data on forward \({pp, \bar pp, \pi^{\pm}p}\) and \({K^{\pm}p}\) scattering is performed making use of the single- and double-subtraction integral and comparing with derivative dispersion relations for amplitudes. Various pomeron and odderon models for the total cross sections are considered and compared. The real part of the amplitude is calculated via dispersion relations. It is shown that the integral dispersion relations lead to a better description of the data for \({\sqrt{s} > 5\,{\rm GeV}}\). Predictions of the considered models for the TOTEM experiment at LHC energies are given.

ANALYSIS OF pp AND $\bar pp$ ELASTIC SCATTERING BASED ON HIGH ENERGY BOUNDS

Considering the Froissart–Martin bound, Jin–Martin–Cornille bound and the optical theorem, we propose a novel parametrization for the total cross-section of proton–proton and antiproton–proton elastic scattering data. Using derivative dispersion relations we obtain the real part of the elastic scattering amplitude and thus the ρ parameter. Simultaneous fits to σtot and ρ are performed allowing very good statistical descriptions of the available data. Furthermore, predictions to σtot and ρ at energies not used in the fit procedures are presented. For σtot we obtain predictions at RHIC, LHC and future hadron collider energies.

Pp and [formula omitted] scattering at high energies

A discussion is presented of pp and p p ¯ scattering at high energies which satisfies the requirements of dispersion relations and of a theorem by A. Martin about the zero of the real amplitude. In particular the use of Derivative Dispersion Relations is extended to the control of the slopes of the real and imaginary amplitudes.

Representation of integral dispersion relations by local forms

The representation of the usual integral dispersion relations (IDRs) of scattering theory through series of derivatives of the amplitudes is discussed, extended, simplified, and confirmed as mathematical identities. Forms of derivative dispersion relations (DDRs) valid for the whole energy interval, recently obtained and presented as double infinite series, are simplified through the use of new sum rules of the incomplete Γ functions, being reduced to single summations, where the usual convergence criteria are easily applied. For the forms of the imaginary amplitude used in phenomenology of hadronic scattering at high energies, we show that expressions for the DDR can represent, with absolute accuracy, the IDR of scattering theory, as true mathematical identities. Besides the fact that the algebraic manipulation can be easily understood, numerical examples show the accuracy of these representations up to the maximum available machine precision. As consequence of our work, it is concluded that the standard forms, sDDR, originally intended for high energy limits are an inconvenient and incomplete separation of terms of the full expression, leading to wrong evaluations. Since the correspondence between IDR and the DDR expansions is linear, our results have wide applicability, covering more general functions, built as combinations of well studied basic forms.

DERIVATIVE DISPERSION RELATIONS FOR THE AMPLITUDE SLOPES IN pp AND ${\rm \bar{p}p}$ SCATTERING

We extend the use of derivative dispersion relations to the study of slopes of the real and imaginary amplitudes in pp and [Formula: see text] elastic scattering. The new relations are tested against the solutions for the amplitudes obtained in the analysis of the high energy data. Extensions beyond the forward direction are investigated.

From integral to derivative dispersion relations

We demonstrate that integral dispersion relations for hadron-hadron scattering amplitudes can be replaced by differential relations, without the usual high-energy approximation. We obtain analytical expressions for the corrections associated with the low energy region and exemplify the applicability of the novel relations in the context of an analytical parametrization for proton-proton and antiproton-proton total cross sections.

Derivative dispersion relations above the physical threshold

We discuss some formal and practical aspects related to the replacement of Integral Dispersion Relations (IDR) by derivative forms, without high-energy approximations. We first demonstrate that, for a class of functions with physical interest as forward scattering amplitudes, this replacement can be analytically performed, leading to novel Extended Derivative Dispersion Relations (EDDR), which, in principle, are valid for any energy above the physical threshold. We then verify the equivalence between the IDR and EDDR by means of a popular parametrization for total cross sections from proton-proton and antiproton-proton scattering and compare the results with those obtained through other representations for the derivative relations. Critical aspects on the limitations of the whole analysis, from both formal and practical points of view, are also discussed in some detail.

Critical analysis of derivative dispersion relations at high energies

We discuss some formal and fundamental aspects related with the replacement of integral dispersion relations by derivative forms, and their practical uses in high energy elastic hadron scattering, in particular pp and p ¯ p scattering. Starting with integral relations with one subtraction and considering parametrizations for the total cross sections belonging to the class of entire functions in the logarithm of the energy, a series of results is deduced and our main conclusions are the following: (1) except for the subtraction constant, the derivative forms do not depend on any additional free parameter; (2) the only approximation in going from integral to derivative relations (at high energies) concerns to assume as zero the lower limit in the integral form; (3) the previous approximation and the subtraction constant affect the fit results at both low and high energies and therefore, the subtraction constant cannot be disregarded; (4) from a practical point of view, for single-pole Pomeron and secondary Reggeons parametrizations and center-of-mass energies above 5 GeV, the derivative relations with the subtraction constant as a free fit parameter are completely equivalent to the integral forms with finite (non-zero) lower limit. A detailed review on the conditions of validity and assumptions related with the replacement of integral by derivative relations is also presented and discussed.

Integral and derivative dispersion relations, analysis of the forward scattering data

Integral and derivative dispersion relations (DR) are considered for the forward scattering $pp$ and $\bar pp$ amplitudes. A new representation for the derivative DR, valid not only at high energy, is obtained. The data on the total cross sections for $pp (\bar pp)$ interaction as well as the data on the parameter $\rho $ are analyzed within the various forms of the DR and high-energy Regge models. It is shown that three models for the Pomeron, Simple pole Pomeron, Tripole Pomeron and Dipole Pomeron (the both with the intercept equal unit) lead to practically equivalent description of the data at $\sqrt{s}>$5 GeV. It is also shown that the correctly calculated low-energy part of the dispersion integral (from the two-proton threshold up to $\sqrt{s}=$5 GeV) allows to reproduce well the $\rho $ data at low energies without additional free parameters.

Differential dispersion relations and elementary amplitudes in a multiple diffraction model

We discuss the evaluation of the real part of the elementary amplitudes in the context of a multiple diffraction model for $\mathrm{pp}$ elastic scattering earlier developed. The framework is based on the concepts of analyticity and polynomial boundedness, and the techniques of dispersion relations. Novel results concern the use of derivative dispersion relations at the elementary level (constituent-constituent interactions) and an optimization of these relations in terms of one free parameter. In addition to a theoretical improvement, we achieved a satisfactory description of the physical quantities.

Differential dispersion relations with an arbitrary number of subtractions: a recursive approach

Making use of a recursive approach, derivative dispersion relations are generalized for an arbitrary number of subtractions. The results for both cross even and odd amplitudes are theoretically consistent at sufficiently high energies and in the region of small momentum transfer.

Elastic scattering with and without the odderon

We present the results of fitting all data for pp and p p scattering at √ s ≥ 10 GeV and up to the collider energy with various analytic parametrizations of the forward scattering amplitudes involving the Pomeron, Odderon, and Reggeon components. These parametrizations can be thought of as solutions to the simplest derivative dispersion relations that satisfy the maximal asymptotic behaviors or as the unitarized color-singlet multigluon exchange contributions. It is found that the simplets model containing the Pomeron and an exchange-degenerate Reggeon term can describe the data equally well as any other model, thus making the role of the odderon rather insignificant.

Extensions of the derivative dispersion relations for amplitude analyses

It is shown that analyticity properties of scattering amplitudes as expressed by the derivative analyticity relations can be combined with measured cross-section and polarization data to extract the full scattering amplitudes in a simple and practical way. The method should be particularly useful in problems involving a coupled amplitude analysis. The measurement of a complete set of spin parameters, or a phase-shift analysis at only one energy, is sufficient to eliminate ambiguities in the extraction at all other energies. Our procedure is illustrated by considering both simple mathematical examples and real data. (AIP)

Low-energy pion-pion parameters from forward dispersion relations

Values of the ππ scattering lengths and low-energy phase shifts are tabulated corresponding to various assumptions on the mass and width of the sigma meson. The method uses forward and first derivative dispersion relations. The I = 0 S-wave scattering length a o comes out in the range −0.3 to 0.6 m π −1. The P-and D-wave scattering lengths are relatively independent of the sigma parameters. Solutions are compared with results on Ke 4 decay and with the predictions of current algebra. Super broad sigmas seem to be preferred. Such solutions are also more self consistent in a bootstrap sense.