The Froissart bound on the total cross section is subjected to test against very high energy data. We have found no clear evidence for its violation. The scaling property of differential cross section in the diffraction region is investigated. It exhibits scaling in the ISR, SPS, Tevatron and LHC energy domain which had hitherto remained unexplored. The slope of the diffraction peak is fitted and the data are tested against the rigorous bounds.
This Commentary on the paper "Total Hadronic Cross Section Data and the Froissart-Martin Bound", by Fagundes, Menon and Silva, to be published in Braz. J. of Phys., Vol. 42 (2012) (arXiv 1112.4704), was invited by the Editors of the Brazilian Journal of Physics to appear directly after the above authors' printed version, in the same journal issue. We here challenge that paper's conclusions that the Froissart bound was violated. We will show that this conclusion follows from a statistical methodology that we question, and will present compelling supplementary evidence that the latest ultra-high…
Experimentally observed slow growth of hadron cross-sections at high energies is a very intriguing but poorly understood property of QCD. It is tempting to explain the slow growth by saturation of Froissart bound or another similar universal mechanism. We reconsider derivation of Froissart bound in QCD in chiral limit and argue it can not justify experimentally observed behavior. Although the conventional Froissart-Martin bound should impose non-trivial constraint on the growth of hadron cross-sections, because of the small value of pion masses it will become restrictive only at currently unac…
A reply to the above mentioned commentary by M.M. Block and F. Halzen on our quoted work is presented. We answer to each point raised by these authors and argument that our data reductions, strategies and methodology are adequate to the nonlinear-fit-problem in focus. In order to exemplify some arguments, additional information from our subsequent analysis is referred to. A brief commentary on the recent results by Block and Halzen is also presented. We understand that this reply gives support to the results and conclusions presented in our quoted work.
In this note my personal point of view on the question brought up for a discussion at the Conference "Diffraction 2008" by Andre Martin has been presented.
The energy dependence of the total hadronic cross section at high energies is investigated with focus on the recent experimental result by the TOTEM Collaboration at 7 TeV and the Froissart-Martin bound. On the basis of a class of analytical parametrization with the exponent $γ$ in the leading logarithm contribution as a free parameter, different variants of fits to $pp$ and $\bar{p}p$ total cross section data above 5 GeV are developed. Two ensembles are considered, the first comprising data up to 1.8 TeV, the second also including the data collected at 7 TeV. We shown that in all fit variants…
The Froissart bounds for amplitudes and cross sections are explained and reconsidered to clarify the role of different assumptions. It is the physical conditions of unitarity and of no massless exchanges, together with mathematical properties of the Legendre functions, that imply much softer high-energy asymptotics for elastic amplitudes at physical angles than for the same amplitudes at nonphysical angles. The canonical log-squared boundary for σ_{tot} appears only under the additional hypothesis that the amplitude at any nonphysical angle cannot grow faster than some power of energy. The Fro…
We re-examine V. Gribov's theorem of 1960 according to which the total cross-section cannot approach a finite non-zero limit with, at the same time, a diffraction peak having a finite slope. We are very close to proving by an explicit counter-example that elastic unitarity in the elastic region is an essential ingredient of the proof. By analogy, we raise the question of the saturation of the Froissart-Martin bound, for which no examples incorporating elastic unitarity exist at the present time.
Proof of the Froissart theorem is reconsidered in a different way to extract its necessary conditions. Two physical inputs, unitarity and absence of massless intermediate hadrons, are indisputable. Also important are mathematical properties of the Legendre functions. Assumptions on dispersion relations, single or double, appear to be excessive. Instead, one should make assumptions on possible high-energy asymptotics of the amplitude in nonphysical configurations, which have today no firm basis. Asymptotics for the physical amplitude always appear essentially softer than for the nonphysical one…
A previously successful model for purely hadronic total cross-sections, based on QCD minijets and soft-gluon resummation, is here applied to the total photoproduction cross section. We find that our model in the gamma p case predicts a rise with energy stronger than in the pp -pbarp case.
The diffraction production of many hadron showers separated by large rapidity gaps, when calculated within the standard pomeron approach, lead to cross sections rising much faster than Froissart-Martin bound. This is the point of Finkelstein-Kajantie problem. We consider the unitarization procedure based on Dyson-Schwinger equations with input froissaron propagators and 3-froissaron vertex (3f-vertex) depending on angular momenta of froissarons in it. The developed diffraction production model allows to resolve Finkelstein-Kajantie problem.
Fits to high energy data alone cannot cleanly discriminate between asymptotic $\ln s$ and $\ln^2s$ behavior of total hadronic cross sections. We demonstrate that this is no longer true when we require that these amplitudes also describe, on average, low energy data dominated by resonances.
We study the effect of gluon number fluctuations (Pomeron loops) on the impact parameter behavior of the scattering amplitude in the fixed coupling case. We demonstrate that the dipole-hadron cross-section computed from gluon number fluctuations saturates the Froissart bound and the growth of the radius of the black disk with rapidity is enhanced by an additional term as compared to the single event case. We find that the physical amplitude has a Gaussian impact parameter dependence once the gluon number fluctuations are included. This indicates that the fluctuations may be the microscopic ori…
At very high energies we expect that the hadronic cross sections satisfy the Froissart bound, which is a well-established property of the strong interactions. In this energy regime we also expect the formation of the Color Glass Condensate, characterized by gluon saturation and a typical momentum scale: the saturation scale $Q_s$. In this paper we show that if a saturation window exists between the nonperturbative and perturbative regimes of Quantum Chromodynamics (QCD), the total cross sections satisfy the Froissart bound. Furthermore, we show that our approach allows us to describe the high …
We demonstrate that the dipole-hadron cross-section computed from the non-linear evolution equation for the Colour Glass Condensate saturates the Froissart bound in the case of a fixed coupling and for a small dipole (Q^2 >> Lambda_{QCD}^2). That is, the cross-section increases as the logarithm squared of the energy, with a proportionality coefficient involving the pion mass and the BFKL intercept (alpha_s N_c/pi)4 ln 2. The pion mass enters via the non-perturbative initial conditions at low energy. The BFKL equation emerges as a limit of the non-linear evolution equation valid in the tail of …
In this paper we consider the issue of the Froissart bound on the high energy behaviour of total cross sections. This bound, originally derived using principles of analyticity of scattering amplitudes, is seen to be satisfied by all the available experimental data on total hadronic cross sections. At strong coupling, gauge/gravity duality has been used to provide some insights into this behaviour. In this work, we find the subleading terms to the so-derived Froissart bound from AdS/CFT. We find that a $(\ln \frac{s}{s_0})$ term is obtained, with a negative coefficient. We see that the fits to …
The multiperipheral ladder structure of the Pomeron leads to the quite natural conclusion that the elastic slope Bel is not simple linear function of the colliding particles energy logarithm. The existing experimental data on the diffraction cone shrinkage points to such "complicated" dependence indeed. The shrinkage diffraction cone speed up with the beam energy is directly connected with an extreme rise of total cross-section (Froissart limit).
In the previous work hep-ph/0112140, hep-ph/0204277, we showed that while the nonlinear QCD evolution equation of Balitsky and Kovchegov (BK) leads to saturation of the scattering amplitude locally in impact parameter space, it does not unitarize the total cross section. This result was recently challenged by Ferreiro, Iancu, Itakura and McLerran (FIIM) who claim that the dipole-hadron cross section computed from the BK equation saturates the Froissart bound. In this comment, we point to a fundamental error in the argument of FIIM which invalidates their conclusion. We show that if the total c…
We show how universal, Froissart-like hadronic total cross sections can be obtained in QCD in the functional-integral approach to soft high-energy scattering, and we discuss indications of this behaviour obtained from lattice simulations.
We study soft gluon kt-resummation and the relevance of zero momentum gluons for the energy dependence of total hadronic cross-sections. We discuss a model in which consistency of the energy dependence of the cross-section with the limitation of the Froissart bound, is directly related to the behaviour of the strong coupling constant in the infrared region. Our predictions for the asymptotic behaviour are shown to be related to the ansatz that the infrared behaviour of the QCD strong coupling constant follows an inverse power law.
In a previous paper, we have analyzed high energy QCD from AdS-CFT and proved the saturation of the Froissart bound (a purely QCD proof of which is still lacking). In this paper we describe the calculation in more physical terms and map it to QCD language. We find a remarkable agreement with the 1952 Heisenberg description of the saturation (pre-QCD!) in terms of shockwave collisions of pion field distributions. It provides a direct map between gauge theory physics and the gravitational physics on the IR brane of the Randall-Sundrum model. Saturation occurs through black hole production on the…
We argue that the deep inelastic structure function $F_2^{γp}(x, Q^2)$, regarded as a cross section for virtual $γ^*p$ scattering, is hadronic in nature. This implies that its growth is limited by the Froissart bound at high hadronic energies, giving a $\ln^2 (1/x)$ bound on $F_2^{γp}$ as Bjorken $x\rightarrow 0$. The same bound holds for the individual quark distributions. In earlier work, we obtained a very accurate global fit to the combined HERA data on $F_2^{γp}$ using a fit function which respects the Froissart bound at small $x$, and is equivalent in its $x$ dependence to the function u…
We compute total and differential elastic cross sections of high-energy hadronic collisions in the loop-loop correlation model that provides a unified description of hadron-hadron, photon-hadron, and photon-photon reactions. The impact parameter profiles of pp and gamma*p collisions are calculated. For ultra-high energies the hadron opacity saturates at the black disc limit which tames the growth of the hadronic cross sections in agreement with the Froissart bound. We compute the impact parameter dependent gluon distribution of the proton xG(x,Q^2,b) and find gluon saturation at small Bjorken …
We discuss how hadronic total cross sections at high energy depend on the details of QCD, namely on the number of colours $N_c$ and the quark masses. We find that while a "Froissart"-type behaviour $σ_{\rm tot}\sim B\log^2s$ is rather general, relying only on the presence of higher-spin stable particles in the spectrum, the value of $B$ depends quite strongly on the quark masses. Moreover, we argue that $B$ is of order ${\cal O}(N_c^0)$ at large $N_c$, and we discuss a bound for $B$ which does not become singular in the $N_f=2$ chiral limit, unlike the Froissart-Łukaszuk-Martin bound.
The gauge/string-gravity duality correspondence opened renewed hope and possibility to address some of the fundamental and non-perturbative QCD problems in particle physics, such as hadron spectrum and Regge behavior of the scattering amplitude at high energies. One of the most fundamental and long-standing problem is the high energy behavior of total cross-sections. According to a series of exhaustive tests by the COMPETE group, (1). total cross-sections have a universal Heisenberg behavior in energy corresponding to the maximal energy behavior allowed by the Froissart bound, i.e., $A + B ln^…
We present evidence for the saturation of the Froissart bound at high energy for {\em all hadronic} total cross sections at high energies, and use this to unify $pp$ (and $\bar p p$) total cross sections over the energy range from cyclotrons to colliders to ultra-high energy cosmic rays, an energy span from $\sqrt s = 4$ GeV to 80 TeV.
In Part I (in this journal) we argued that the structure function $F_2^{γp}(x,Q^2)$ in deep inelastic $ep$ scattering, regarded as a cross section for virtual $γ^*p$ scattering, has a saturated Froissart-bounded form behaving as $\ln^2 (1/x)$ at small $x$. This form provides an excellent fit to the low $x$ HERA data, including the very low $Q^2$ regions, and can be extrapolated reliably to small $x$ using the natural variable $\ln(1/x)$. We used our fit to derive quark distributions for values of $x$ down to $x=10^{-14}$. We use those distributions here to evaluate ultra-high energy (UHE) cros…
We will show how a universal and Froissart-like (i.e., of the kind "B log^2 s") hadron-hadron total cross section can emerge in QCD asymptotically at high energy, finding indications for this behavior from the lattice. The functional integral approach provides the natural setting for achieving this result, since it encodes the energy dependence of hadronic scattering amplitudes in a single elementary object, i.e., a proper correlation function of two Wilson loops.
Assuming that axiomatic local field theory results hold for hadron scattering, André Martin and S. M. Roy recently obtained absolute bounds on the D-wave below threshold for pion-pion scattering and thereby determined the scale of the logarithm in the Froissart bound on total cross sections in terms of pion mass only. Previously, Martin proved a rigorous upper bound on the inelastic cross-section $σ_{inel}$ which is one-fourth of the corresponding upper bound on $σ_{tot}$, and Wu, Martin,Roy and Singh improved the bound by adding the constraint of a given $σ_{tot}$. Here we use unitarity and a…
Hadron production in single and central diffraction dissociation is studied in a model which includes soft hadron interaction as controlled by a supercritical pomeron parametrization and hard diffraction. Within this model, particle production in collisions with pomerons exhibit properties like multiple soft interactions and multiple minijets, quite similar to hadron production in non-diffractive hadronic collisions at high energies. However, important differences occur in transverse momentum jet and hadron distributions. It is shown that the model is able to describe data from the CERN-SPS co…
It is argued that the breakdown of factorization observed recently in the diffractive dijet production in deep inelastic lepton induced and hadron induced processes is naturally explained in the Good-Walker picture of diffraction dissociation. The explicit formula for the hadronic cross-section is given and successfully compared with the existing data.
We show how to obtain rising hadronic total cross sections in QCD, in the framework of the nonperturbative approach to soft high-energy scattering based on Wilson-loop correlators. Total cross sections turn out to be of "Froissart"-type, i.e., the leading energy dependence is of the form $σ_{\rm tot} \sim B \log^2 s$, in agreement with experiments. The observed universality of the prefactor $B$ is obtained rather naturally in this framework. In this case, $B$ is entirely determined by the stable spectrum of QCD, and predicted to be $B_{th} = 0.22\, {\rm mb}$, in fair agreement with experiments…
Current stage of development of the high-energy elastic diffractive scattering phenomenology is reviewed. Verification of various theoretical models via comparison of their predictions with the recent D0 and TOTEM data on the nucleon-nucleon total and differential cross-sections is presented.
We review a broad range of phenomena in diffraction in the context of hadron-hadron, hadron-nucleus collisions and deep inelastic lepton-proton/nucleus scattering focusing on the interplay between the perturbative QCD and non-perturbative models. We discuss inclusive diffraction in DIS, phenomenology of dipole models, resummation and parton saturation at low $x$, hard diffractive production of vector mesons, inelastic diffraction in hadron-hadron scattering, formalism of color fluctuations, inclusive coherent and incoherent diffraction as well as soft and hard diffraction phenomena in hadron-h…
A QCD-based treatment of projectile size fluctuations is used to compute inelastic diffractive cross sections $σ_{diff}$ for coherent hadron-nuclear processes. We find that fluctuations near the average size give the major contribution to the cross section with $ \le few \%$ contribution from small size configurations. The computed values of $σ_{diff}$ are consistent with the limited available data. The importance of coherent diffraction studies for a wide range of projectiles for high energy Fermilab fixed target experiments is emphasized. The implications of these significant color fluctuati…
Hadron structure is considered in the frame of Strongly Correlated Quark Model (SCQM). It is shown that quark correlations result in fluctuations of hadronic matter distributions and single diffractive dissociation processes in hadronic collisions, hard or soft, are manifestation of these fluctuations inside colliding hadrons.
We discuss a collective effect, which can be possible in hadron--hadron collisions at superhigh energies, that is, a quasi-diffraction production of several white clusters of quarks and gluons. Being transformed into hadrons, such clusters are sources of the fastest particles, which move forward nearly parallel to each other.
Hard single diffractive processes are studied within the framework of the triple--Pomeron approximation. Using a Pomeron structure function motivated by Regge--theory we obtain parton distribution functions which do not obey momentum sum rule. Based on Regge-- factorization cross sections for hard diffraction are calculated. Furthermore, the model is applied to hard diffractive particle production in photoproduction and in $p\bar{p}$ interactions.
I review the physical and mathematical foundations for the theoretical description of the hadron wavefunction at small $x$ as a Colour Glass Condensate. In this context, I discuss the phenomenon of gluon saturation and some of its remarkable consequences: a new ``geometric scaling'' for $F_2$, which has been recently identified at HERA, and the unitarization of the hadronic cross-sections at high energy. I show that by combining saturation and confinement one obtains cross-sections which saturate the Froissart bound.
SCI model gives a good and natural description of DDIS cross-sections. Howevere, this model is pure phenomenological, and does not explain the nature of soft color rearrangment. In this paper we argue, that the most capabilities of SCI model can be derived from low constituent model and overview applicability of low constituent model to DDIS processes.
Hadron diffraction dissociation is considered in the dipole Pomeron model where the Pomeron is represented by a double pole in the J-plane. We find that unitarity is satisfied without decoupling of the triple Pomeron vertex. The reaction bar{p}+p --> bar{p}+X is analysed.
This is a cursory review of diffractive studies ( mostly elastic scattering)at the LHC with physical interpretation of the experimental data and comments to related problems of theory.
We use an almost model-independent analytical parameterization for $pp$ and $\bar{p}p$ elastic scattering data to analyze the eikonal, profile, and inelastic overlap functions in the impact parameter space. Error propagation in the fit parameters allows estimations of uncertainty regions, improving the geometrical description of the hadron-hadron interaction. Several predictions are shown and, in particular, the prediction for $pp$ inelastic overlap function at $\sqrt{s}=14$ TeV shows the saturation of the Froissart-Martin bound at LHC energies.
The rates of multiparton collisions in high energy hadronic interactions provide information on the typical transverse distances between partons in the hadron structure. The different configurations of the hadron in transverse space are, on the other hand, at the origin of hadron diffraction. The relation between the two phenomena is exploted in an eikonal model of hadronic interactions.
The rôle of inelastic diffraction in elastic scattering of nuclei is studied in the formalism of \emph{diffractive limit}. The results obtained for scattering of the $α$--particles on light nuclei show that the nucleonic diffraction is especially important at large momentum transfers where the Glauber model of geometric diffraction fails.
We note that it is premature to make a conclusion on the multiplicity of the Pomerons on the basis of the available experimental data since the interactions have a preasymptotic nature
We consider a multi-channel correction accounting for intermediate transitions inside an equivalence class of diffractive states. Our formalism of 'diffractive limit' provides an excellent description of elastic and inclusive inelastic scattering of high energy (53 - 546 GeV) protons and antiprotons.
In this talk we review the models describing the hard diffractive production of jets or more generally high-mass states in presence of rapidity gaps in hadron-hadron and lepton-hadron collisions. By rapidity gaps we mean regions on the lego plot in (pseudo)-rapidity and azimuthal angle where no hadrons are produced, between the jet(s) and an elastically scattered hadron (single hard diffraction) or between two jets (double hard diffraction).
The relationships between the fundamental dynamics and diffraction phenomena in scattering from two-body composite system are discussed. A new simple formula for the shadow corrections to the total cross-section in scattering from deuteron has been derived and new scaling characteristics with a clear physical interpretation have been established. The effect of weakening the inelastic screening at super-high energies is theoretically discovered. A comparison with the experimental data on proton(antiproton)-deuteron total cross-sections has been performed. It is shown that there is quite a remar…
We show how to obtain the leading energy dependence of hadronic total cross sections, in the framework of the nonperturbative approach to soft high-energy scattering based on Wilson-loop correlation functions, if certain nontrivial analyticity assumptions are satisfied. The total cross sections turn out to be of "Froissart" type, $σ_{\rm tot}^{(hh)}(s) \mathop\sim B\log^2 s$ for ${s \to \infty}$. We also discuss under which conditions the coefficient $B$ is universal, i.e., independent of the hadrons involved in the scattering process. In the most natural scenarios for universality, $B$ can be…
It is proposed, that ``the colorless objects'' which manifest themselves in large-rapidity-gap events are color-singlet gluon-clusters due to self-organized criticality (SOC), and that optical-geometrical concepts and methods are useful in examing the space-time properties of such objects. A simple analytical expression for the $t$-dependence of the inelastic single diffractive cross section $dσ/dt$ ($t$ is the four-momentum transfer squared) is derived. Comparison with the existing data and predictions for future experiments are presented. The main differences and similarities between the SOC…
Experimental results on hadronic soft and hard diffractive processes are reviewed with emphasis on aspects of the data that point to the underlying QCD mechanism for diffraction. Diffractive differential cross sections are shown to be factorized into two terms, one representing the total cross section at the reduced energy, corresponding to the rapidity region(s) in which there is particle production, and another interpreted as the probability of formation of the rapidity gap(s) characterizing diffraction. By (re)normalizing the term of gap formation probability to unity, cross sections for si…
The electromagnetic (e.m.) current conservation and renormalizability of QCD are used to calculate the amplitude of energetic hadron(photon) diffraction into several jets with large relative transverse momenta off the nucleon(nucleus) Coulomb field. Numerical estimates of the ratio of e.m. and strong amplitudes show that within the kinematic range where the leading twist approximation for the strong amplitude is applicable, the e.m. contribution can be neglected. In pA scattering at LHC and in the fragmentation of a photon into two jets in ultraperipheral AA collisions in the black limit (whic…
We argue that the increase of the ratio baryon/meson due to the presence of strong colour fields and percolation in ultra-high energy hadronic collisions, helps to explain some of the global features of ultra-high energy cosmic ray cascades at E>10^18 eV and, in particular the observed excess in the number of muons with respect to current models of hadronic interactions. A reasonable agreement with the small value and slope of the average depth of shower maximum Xmax vs shower energy -- as seen in data collected at the Pierre Auger Observatory -- can be obtained with a fast increase of the p-A…
The status of hadron-hadron interactions is reviewed, with emphasis on the forward and near-forward scattering regions. From analyticity, Finite Energy Sum Rules are introduced from which new analyticity constraints are derived that exploit the many very accurate low energy experimental cross sections, i.e., they constrain the values of the asymptotic cross sections and their derivatives at low energies just above the resonance regions, allowing us new insights into duality. A new robust fitting technique is introduced in order to `clean up' large data samples that are contaminated by outliers…
A low mass single diffraction dissociation mode based on the quark-diquark representation of hadron-nucleon interaction is proposed. The mass spectra of nucleon-pion system are compared with experimental data and predictions of other models. The hadron-nucleus single-diffraction cross-sections are calculated in the framework of the Glauber-Gribov model for integral cross-sections. The model predictions are compared with experimental data for the different distributions of secondary particles produced in the hadron-nucleus interactions in the momentum range 31-320 GeV/c.
In standard Regge theory with a pomeron intercept a(0)=1+ε, the contribution of the tripe-pomeron amplitude to the t=0 differential cross section for single diffraction dissociation has the form dσ/dM^2(t=0) \sim s^{2ε}/(M^2)^{1+ε}. For ε>0, this form, which is based on factorization, does not scale with energy. From an analysis of p-p and p-pbar data from fixed target to collider energies, we find that such scaling actually holds, signaling a breakdown of factorization. Phenomenologically, this result can be obtained from a scaling law in diffraction, which is embedded in the hypothesis of po…
In the framework of our model (GLM) for soft interaction with $α'_{\pom}(0)=0$, we propose a procedure based on Gribov's partonic interpretation of the Pomeron, which enables one to calculate the diffractive mass distributions in hadron-hadron scattering. Using the analogy with deep-inelastic scattering, we associate the Pomeron-quark interaction with the Good-Walker sector of the hadron-hadron scattering, and the Pomeron-gluon sector with the t-channel Pomeron interactions. We present predicted mass distributions for the LHC energies
A phenomenological analysis of diffractive dissociation of nuclei in proton-nucleus and meson-nucleus collisions is presented. The theoretical approach employed here is able to take into account at once data of the HELIOS and EHS/NA22 collaborations that exhibit quite different atomic mass dependences. Possible extensions of this approach to hard diffraction in nuclear processes are also discussed.
