Forward Elastic Amplitude Research Articles

Proton-proton forward scattering at the LHC

Recently the TOTEM experiment at the LHC has released measurements at s=13 TeV of the proton-proton total cross section, σtot, and the ratio of the real to imaginary parts of the forward elastic amplitude, ρ. Since then an intense debate on the C-parity asymptotic nature of the scattering amplitude was initiated. We examine the proton-proton and the antiproton-proton forward data above 10 GeV in the context of an eikonal QCD-based model, where nonperturbative effects are readily included via a QCD effective charge. We show that, despite an overall satisfactory description of the forward data is obtained by a model in which the scattering amplitude is dominated by only crossing-even elastic terms, there is evidence that the introduction of a crossing-odd term may improve the agreement with the measurements of ρ at s=13 TeV. In the Regge language the dominant even(odd)-under-crossing object is the so called Pomeron (Odderon).

Conformal couplings and “azimuthal matching” of QCD pomerons

Using the asymptotic conformal invariance of perturbative QCD we derive the expression of the coupling of external states to all conformal spin p components of the forward elastic amplitude. Using the wave function formalism for structure functions at small x, we derive the perturbative coupling of the virtual photon for \(p= 1\), which is maximal for linear transverse polarization. The non-perturbative coupling to the proton is discussed in terms of “azimuthal matching” between the proton color dipoles and the \(q\bar q\) configurations of the photon. As an application, the recent conjecture of a second QCD pomeron related to the conformal spin-1 component is shown to rely upon a strong azimuthal matching of the \(p= 1\) component in \(\gamma^*\)–proton scattering.

The photon-photon interaction at low x in the theory of reggeized gluons with a running coupling and Nc → ∞

The forward elastic amplitude for scattering of real and weakly virtual photons is studied in the framework of the theory of reggeized gluons, with large N c and a running coupling constant introduced in the manner which preserves the bootstrap condition. Transition from a single to multiple pomeron exchanges is observed as x gets smaller. For very low x the amplitude acquires an eikkonal form. The photon structure function reveals a strong violation of scaling: it grows a Q 2. As a function of x it behaves as ( ln( 1 x ) ln ln( 1 x) ) 2 . Correspondingly the cross-section for physical photons grows with energy as (ln s ln ln s) 2. Hadronic structure functions and cross-sections are also briefly discussed.

Comparative analysis of phenomenological models for pomeron att = 0

Four models for the pomeron contribution to the forward elastic amplitudes for MN and NN interactions are compared. A very good fit of experimental data is obtained within the framework of Regge approach. It is found that available experimental data att = 0 do not seem to indicate a growth of the total cross-sections faster than the first power in logarithm of the energy.

The phenomenology of the fractal pomeron

The effect of intermittent multihadronic states in p p collisions is studied within a two-component model. The origin of these states lies in the process of a phase transition in the Feynman-Wilson fluid and their production threshold is argued to appear at CERN collider energies. The shadow of these multiparticle states generates a new, non-conventional diffractive component which may give, in the vicinity of this new threshold, a significant, local contribution to the forward elastic amplitude (UA4 experiment). The size of this component is restricted by the measurement of the factorial moments in the same energy region (UA1 experiment).

Rigorous bounds on the reduced coupling constant of the K±p forward elastic amplitude

The positivity of the imaginary part of the forward K-p elastic amplitude on the unphysical region allows the construction of a Stieltjes function H1(z) closely related to a discrepancy function Delta -( omega ), evaluated from experimental data on the real and imaginary parts of the amplitude. A set of 159 data points, compatible with the positivity hypothesis, has been selected and all calculations and results are based on practically the whole set. The Stieltjes character of H1(z) imposes constraints on the coefficients of the formal expansion of H1(z), determined using the Gronwall conformal transformations, which limit the universe of approximant functions and act as stabilisers of the extrapolation. Then the type-I Pade approximants to H1(z) have been built with the coefficients of the formal expansion, providing rigorous bounds on the reduced coupling constant and the real parts of the amplitude, in particular at omega =0. The addition of the hypothesis of unimodality of the imaginary part of the amplitude on the unphysical region provides tighter rigorous bounds. The consistency of the calculated real parts has been successfully checked by taking different absorption points with the latter values of the real parts. The stability of the method of extrapolation has been confirmed using as a model function the discrepancy function evaluated with the calculated real parts, perturbed randomly according to the experimental errors.

Real part of the forward elastic nuclear amplitude forpp,p¯p,π+p,π−p,K+p, andK−pscattering between 70 and 200 GeV/c

We have measured the elastic cross section for $\mathrm{pp}$, $\overline{p}p$, ${\ensuremath{\pi}}^{+}p$, ${\ensuremath{\pi}}^{\ensuremath{-}}p$, ${K}^{+}p$, and ${K}^{\ensuremath{-}}p$ scattering at incident momenta of 70, 100, 125, 150, 175, and 200 GeV/c. The range of the four-momentum transfer squared $t$ varied with the beam momentum from $0.0016\ensuremath{\le}\ensuremath{-}t\ensuremath{\le}0.36$ (${\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$ at 200 GeV/c to $0.0018\ensuremath{\le}\ensuremath{-}t\ensuremath{\le}0.0625$ (${\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$ at 70 GeV/c. The conventional parametrization of the $t$ dependence of the nuclear amplitude by a simple exponential in $t$ was found to be inadequate. An excellent fit to the data was obtained by a parametrization motivated by the additive quark model. Using this parametrization we determined the ratio of the real to the imaginary part of the nuclear amplitude by the Coulomb-interference method.

On the ratio of the real-to-imaginary parts of the two-component pomeron

The two-component structure of the so-called pomeron has been investigated in previous papers (1,2). The hypothesis that any seatterings occur by way of hadronhadron to reggeon-reggeon processes is assumed. The contribution of the various inelastic collisions to the forward elastic amplitude through the unitarity relation has been studied with reference to the quark line diagrams. At wo-component structure similar to Lipkin's (3) has been found for pomeron contribution in the total cross-section. The components of the pomeron have been estimated from ~J~, K~ and J~J~ totalcross-section data at an incident momentum ranging from 35 to 200 GeV/e (~). The first component a~ increases slowly with energy, while the second one a s decreases slowly. The least-square fits have been performed with the following functions: (IA) a(logpT. ~- b) , (IA') a(1ogpL + b) 2 -b e , (IB) a -- bp'~ c for al,

Exchange amplitudes and forward dispersion relations for electron-atomic hydrogen scattering

The analytic properties of the forward elastic exchange amplitude g(E) are studied as a function of the incident electron energy E for electron scattering by atomic hydrogen. The authors first prove that the bound-state of gB2(E), the second Born contribution to g(E), has an infinite number of singularities due to the analytic structure of the off-shell first-order exchange matrix element. These singularities are (i) poles of order (n-1) at E=-1/2n2 (n>or=2) associated with excited intermediate states of the target and (ii) cuts at E=-1/2n2 (n>or=1) coming from all intermediate states. In addition, the bound-state part of gB2(E) also exhibits poles and cuts at E=-2(1+1/2n)2, n>or=1, arising from the integration over the intermediate momentum of the projectile. They then show that the continuum contributions to gB2(E) lead to singularities having the same location as those coming from the bound states, so that the interplay between bound and continuum singularities is very important.

The pion-deuteron forward elastic amplitude in the non-overlapping potentials model

The pion deuteron forward elastic amplitude has been calculated in the nonoverlapping potentials model which enables a description of off-shell propagation effects in terms of on-shell amplitudes. Calculations include spin, isospin and deuteron D-state probability effects. Two energy regions are considered. First the pion-nucleon P33 resonance region, where, using a formalism developed by Agassi and Gal, the full multiple-scattering series is summed in an approximation of P33 wave dominance of the higher-order scatterings. Second, for the subsequent higher-energy region, the double-scattering term only is calculated. Fermi smearing effects are included in both cases. Predictions for the total cross section, its dependence on the deuteron alignment and the real part of the forward elastic amplitude are compared with those of Glauber theory, and data where available. Convergence of the multiple-scattering series is also discussed.

A note on the exchange amplitude for elastic forward electron-hydrogen-atom scattering

The forward elastic exchange amplitude for electron-hydrogen scattering is singular at k2=-1. The dominant contribution to the difference between this amplitude and its first Born approximation is given by the second Born term in the neighbourhood of this singularity. It is shown that this second Born term has a third-order pole at k2=-1.

Exchange amplitudes and dispersion relations for electron-atom scattering

The analytic structure of the second Born approximation to the forward elastic exchange amplitude is analyzed for e −+H scattering, and the importance of the interplay between bound and continuum intermediate state contributions is emphasized.

Operator formulation of the Coulomb-interference problem at SPS and ISR energies

An operator formulation of the Coulomb-interference effect at high energies is presented. This provides a gauge-and translation-invariant solution to the indeterminacy problem of the Bethe phase. Given the shape of electromagnetic form factors, this technique permits extraction of electromagnetic effects with a theoretical error less than 1.5% in the hadronic phase of the forward elastic amplitude.

Phenomenological analysis of hadron total cross-sections

Total cross-sections of hadron collisions are phenomenologically analysed from several to 300 GeV/c. In the momentum range of 10 to 50 GeV/c, they are found to be represented by a fine straight line\(\sqrt 8 \sigma _T = \sqrt 8 P + R\), when\(\sqrt 8 \sigma _T \) is plotted against\(\sqrt 8 \) (i.e. Lipkin plot), and above 100 GeV/c they clearly deviate from the straight line by [log (s/s0)]∼1.5 universally for all processes of pp,\(\bar pp\), π±p, K±p, pn,\(\bar pn\) and K±n scatterings. The vector- and tensor-reggeon exchange contributions are well separated from the pomeron exchange one by the use of the difference of theirs-dependences, and the various relations about σT's due to the vector and the tensor exchange are examined. These reg geon exchange contributions are also checked by experimental quantities, which are connected with a real part of amplitudes, such as the real-to-imaginary ratio of forward elastic amplitudes and the forward differential cross-sections of two-body processes.

Determination of theJψ(3100)-nucleon total cross section using modulus dispersion relations

Using the energy dependence of photoproduction data from Cornell, Fermilab, and SLAC, we calculate the phase of the forward elastic $\frac{J}{\ensuremath{\psi}}(3100)\ensuremath{-}N$ scattering amplitude by applying modulus dispersion relations. The total cross section thus obtained suggests that the $\frac{J}{\ensuremath{\psi}}(3100)$ is a hadron. A possible threshold due to production of a new quantum number is also discussed. Similar analysis is carried out for the $\ensuremath{\varphi}p$ forward elastic amplitude.

On the precision with which the phase of the forward elastic amplitude can be determined in coulomb interference measurements

The error in the phaseϱ has been calculated by fitting the Coulomb interference expression to pre-calculated data on (dσ/dt)elastic. The error is obtained from the error matrix of this fit. The variation of the error with the following parameters has been studied: the total number of events, the error in the absolute normalization in dσ/dt, the error in the hadronic amplitude parameters and the upper and lower limits of the range over which dσ/dt is measured. One important conclusion is that the precision in the absolute normalization is of critical importance for the precision inϱ.

Real parts of forward elastic hadronic amplitudes

We compute the real parts of forward elastic hadronic scattering amplitudes using derivative analyticity relations and recent measurements of total cross sections at the Fermi National Accelerator Laboratory. The results on the ratios of real to imaginary parts, $\ensuremath{\rho}(s)$, for $pp$, $\overline{p}p$, ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$, and ${K}^{\ifmmode\pm\else\textpm\fi{}}p$ are compared with the available experimental data. The agreement with the data is reasonably good. The real parts are predicted to change signs at energies $s\ensuremath{\simeq}180,430,120,250, <20 \mathrm{and} 145$, ${\mathrm{GeV}}^{2}$ for $\overline{p}p$, $pp$, ${\ensuremath{\pi}}^{\ensuremath{-}}p$, ${\ensuremath{\pi}}^{+}p$, ${K}^{\ensuremath{-}}p$, and ${K}^{+}p$, respectively.

Rising cross sections and scaling

A parametrization for hadronic total cross sections, which incorporates a logarithmically rising term, is proposed. The new parameters present in it are restricted, using a scaling assumption for ${q}^{2}$ in the timelike region. The resulting expression contains only one parameter in addition to the usual Regge parameters. In the energy range between $10 \mathrm{and} 400 \frac{\mathrm{GeV}}{c}$ this model gives very good fits for ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$, ${K}^{\ifmmode\pm\else\textpm\fi{}}p$, $pp$, and $\overline{p}p$ total cross sections. Above $400 \frac{\mathrm{GeV}}{c}$, the agreement with recent data on the $pp$ total cross section is only qualitative. The model also predicts the vanishing of the ratio of real to imaginary parts of the $pp$ forward elastic amplitude at $360 \frac{\mathrm{GeV}}{c}$ in agreement with experimental data, the increase of this ratio to a maximum value, and a vanishing asymptotic limit.

Complex Regge poles and high energy scattering data

For the high energy data on σ tot pp the proton-proton total cross section, and on ϱ, the ratio of real to imaginary part of the forward elastic amplitude, including the recent ISR results, an excellent fit is obtained with P, P', ω Regge poles plus a pair of complex conjugate Regge poles. The complex poles describe the possible oscillatory behavior of the quantities σ tot pp and ϱ. The physical meaning of these complex poles and extrapolation to future measurements of the above quantities are discussed.

Multiperipheral dynamics for the grand partition function in high energy collisions

The generating function connecting exclusive and inclusive integrated quantities (the grand partition function in the statistical analogy), is studied in the framework of multiperipheral dynamics: the ABFST model is used as a guide. An integral equation for this function is derived in a form in which the input is given directly by the absorptive part of the forward elastic amplitude. On the basis of simple hypotheses on the structure of this input the behaviour of the integrated correlation functions is discussed in asymptotic as well as moderate energy regions. The possibility of long-range correlations, within the model, is also considered.