A. Achilli, A. Grau, G. Pancheri, Y. N. Srivastava
We discuss the effect of infrared soft gluons on the asymptotic behaviour of the total cross-section. We use a singular but integrable expression for the strong coupling constant in the infrared limit and relate its behaviour to the satisfaction of the Froissart bound, giving a specific phenomenological example.
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Here I explain why critics of my work [arXiv:1104.5314] in the above paper [arXiv:1306.5210] is inadequate.
In this letter we derive the Froissart boundary in QCD for the deep inelastic structure function in low $x$ kinematic region. We show that the comparison of the Froissart boundary with the new HERA experimental data gives rise to a challenge for QCD to explain the matching between the deep inelastic scattering and real photoproduction process.
The Froissart bound limmits the asymptotic s->infinity behavior of crossections by (π/t_0) ln ^2 (s/{(s_0)} where t_0 is the lightest exchanged particle, or more generally the nearest ssingularity, in the t channel. We suggest that in comparing this bound with data at energies less than those of LHC, gluebaall masses raather than the small pion mass should be used for (t_0)^{1/2}.
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.
We determine the scale of the logarithm in the Froissart bound on total cross-sections using absolute bounds on the D-wave below threshold for $ππ$ scattering. E.g. for $π^0 π^0$ scattering we show that for c.m. energy $\sqrt{s}\rightarrow \infty $, $\barσ_{tot}(s,\infty)\equiv s\int_{s} ^{\infty} ds'σ_{tot}(s')/s'^2 \leq π(m_π)^{-2} [\ln (s/s_0)+(1/2)\ln \ln (s/s_0) +1]^2$ where $m_π^2/s_0= 17π\sqrt{π/2} $ .
The Froissart--Martin bound for total $ππ$ scattering cross sections is reconsidered in the light of QCD properties such as spontaneous chiral symmetry breaking and the counting rules for a large number of colours $\Nc$.
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 reanalyze high energy QCD scattering regimes from scattering in cut-off AdS via gravity-gauge dualities (a la Polchinski-Strassler). We look at 't Hooft scattering, Regge behaviour and black hole creation in AdS. Black hole creation in the gravity dual is analyzed via gravitational shockwave collisions. We prove the saturation of the QCD Froissart unitarity bound, corresponding to the creation of black holes of AdS size, as suggested by Giddings.
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…
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…
The Froissart asymptotic behavior of cross-sections is usually considered in a parton picture as corresponding to the collision of two almost black disks filled with partons. In this article we mainly concentrate on the examination of the local transparency of such F-disks. We discuss how is it possible to guarantee the boost-invariance of the reciprocal transparency of two such F-disks in a process of their collision, despite the fact that the mean area of the overlapping of these F-disks at the same impact parameter is varying with the Lorentz frame. We argue that on will always have such pr…
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…
We have investigated the detailed structure of $l$-plane singularities of scattering amplitude saturating the Froissart bound. A self-consistent analysis of these singularities provides us secondary terms in the Froissart bound. These secondary terms lead to ghosts in the $l$-plane, which can only be removed by introducing an odderon singularity. Phenomenological implications of this analysis are also discussed.
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…
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…
The Froissart bound implies that the total cross section (or, equivalently, the structure function) cannot rise faster than the logarithmic growth of $ \ln^{2} \left(\frac{1}{x} \right)$. In this work, we show that such a slow growth is not compatible with the notion of self-similarity. As a result, it calls for the modification of the defining transverse-momentum-dependent parton density function (TMD PDF) of a self-similarity based proton structure function $F_{2} \left(x,Q^{2} \right)$ at small \textit{x}. Using plausible assumptions, we obtain the Froissart saturation condition on this TMD…
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.
We explain why we hope that the Froissart bound can be improved, either qualitatively or, more likely, quantitatively, by making a better use of unitarity, in particular elastic unitarity. In other instances (Gribov's theorem) elastic unitarity played a crucial role. A preliminary requirement for this is to work with an appropriate average of the cross-section, to make the problem well defined. This is possible, without destroying the Lukaszuk--Martin bound.
Froissart Bound implies that the total proton-proton cross-section (or equivalently structure function) cannot rise faster than the logarithmic growth $\log^2 s \sim \log^2 (1/x)$, where \textit{s} is the square of the center of mass energy and \textit{x} is the Bjorken variable. In the present report, compatibility of such behavior will be shown in case of Transverse Momentum Dependent Parton Distributions Functions (TMDPDF) or TMDs for the model of proton based on self-similarity.
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.
We prove that while the total cross{}-section is bounded by $(π/m_π^2) \ln^2 s$, where $s$ is the square of the c.m. energy and $m_π$ the mass of the pion, the total inelastic cross{}-section is bounded by $(1/4)(π/m_π^2) \ln^2 s$, which is 4 times smaller. We discuss the implications of this result on the total cross{}-section itself.
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…
It is well known that fits to high energy data cannot discriminate between asymptotic ln(s) and ln^2(s) behavior of total cross section. We show that this is no longer the case when we impose the condition that the amplitudes also describe, on average, low energy data dominated by resonances. We demonstrate this by fitting real analytic amplitudes to high energy measurements of the gamma p total cross section, for sqrt(s) > 4 GeV. We subsequently require that the asymptotic fit smoothly join the sqrt(s) = 2.01 GeV cross section described by Dameshek and Gilman as a sum of Breit-Wigner resonanc…
We describe the taming effect induced by soft gluon $k_t$-resummation on the rapid rise of QCD mini-jet contributions to the total cross-sections.This results from an eikonal model in which the rise of the total cross-section is due to mini-jet contribution. We perform the calculation with current Parton Density Functions (PDFs). The impact parameter distribution we use is obtained as the Fourier transform of the resummed $k_t$-distribution of soft gluons emitted from the initial state during the collision.The emission, which is energy dependent, destroys the initial collinearity of partons.In…
The authors of a recent paper, "Ultrahigh-Energy Neutrino-Nucleon Deep-Inelastic Scattering and the Froissart Bound", A. Illarianov, B. Kniehl and A. Kotikov, Phys. Rev. Lett. 106, 231802 (2011), derive an approximate formula for the UHE limit of $σ_{νN}(s)$ in a class of models that includes our own and assert that they are led "to the important observation that $σ_{BBT}^{νN} \propto ln^3s$, which manifestly violates the Froissart bound [2] in contrast to what is stated in Refs. [6-8]", the latter reference being to our work and the $σ_{BBT}^{νN}$ to the cross sections we reported there. We h…
High-energy scattering in non-conformal gauge theories is investigated using the AdS/CFT dual string/gravity theory. It is argued that strong-gravity processes, such as black hole formation, play an important role in the dual dynamics. Further information about this dynamics is found by performing a linearized analysis of gravity for a mass near an infrared brane; this gives the far field approximation to black hole or other strong-gravity effects, and in particular allows us to estimate their shape. From this shape, one can infer a total scattering cross-section that grows with center of mass…
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…
Systematic description of hadron's response to non-local QCD probes: Froissart-Gribov projections in analysis of deeply virtual Compton scattering
本研究重新探讨Froissart-Gribov投影在深度虚康普顿散射分析中的应用，首次建立自旋1/2靶的该投影与广义部分子分布的关联，结合实验数据给出首个数值估计，为强子对非定域QCD探针响应的系统描述提供方法支撑。
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 …
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., $A + B ln^…
We fit the reduced cross section for deep-inelastic electron scattering data to a three parameter ln^2 s fit, A + beta ln^2 (s/s_0), where s= [Q^2/x] (1-x) + m^2, and Q^2 is the virtuality of the exchanged photon. Over a wide range in Q^2 (0.11 < Q^2 < 1200 GeV^2) all of the fits satisfy the logarithmic energy dependence of the Froissart bound. We can use these results to extrapolate to very large energies and hence to very small values of Bjorken x -- well beyond the range accessible experimentally. As Q^2 --> infinity, the structure function F_2^p(x, Q^2) exhibits Bjorken scaling, within exp…
Available high energy data for both pp and pp total cross sections ($ 5 \ GeV \ < \ \sqrt s \ < \ 1.8 \ TeV$) are described by means of two well-known distinct parametrizations, characteristic of theoretical (``Regge-like" expression) and experimental (``Froissart-Martin-like" expression) practices, respectively. Both are compared from the statistical point of view. For the whole set of present data, statistical analysis ($χ^2/d.o.f.$) seems to favour a ``Froissart-like" ((ln $s$)$^{γ\approx 2}$ ) rise of the total cross section rather than a ``Regge-like" ($s^ε$) one.
Froissart Bound implies that the total proton-proton cross-section (or equivalently structure function) cannot rise faster than the logarithmic growth $\log^2 s \sim \log^2 1/x$, where \textit{s} is the square of the center of mass energy and \textit{x} is the Bjorken variable. Compatibility of such behavior with the notion of self-similarity in a model of structure function suggested by us sometime back is now generalized to more recent improved self-similarity based models and compare with recent data as well as with the model of Block, Durand, Ha and McKay. Our analysis suggests that Froiss…
The Froissart (\F) asymptotic behavior of high energy cross-sections, if considered in a parton picture, is usually represented as a kind of behavior that occurs in the process of a collision of two almost black disks, filled with partons, when radiuses of these \F - disks grow proportional to log's of there energies. In this article we briefly summarize the main asymptotic properties of \F - disks that one can expect in QCD. ~~Then we consider if it is possible to guarantee the boost-invariance of transparency $T(s,b) = 1 - σ_{in}(s,b)$, where $σ_{in}(s,b)$ is the total inelastic cross-sectio…
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 …
High-energy behavior of total cross sections is discussed in experiment and theory. Origin and meaning of the Froissart bounds are described and explained. Violation of the familiar log-squared bound appears to not violate unitarity (contrary to the common opinion), but correspond to rapid high-energy increase of the amplitude in nonphysical regions.
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.
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…
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.
Cross-Section Bootstrap: Unveiling the Froissart Amplitude
本文利用S矩阵自举推导有限能量总散射截面的普适界限，发现饱和振幅具有上升总截面、收缩衍射峰和丰富雷格共振谱，并揭示散射局域于缓慢扩展的环状区域，挑战传统圆盘衍射图像。
The problem of restoring Froissart bound to the BFKL-Pomeron is studied in an extended leading-log approximation of QCD. We consider parton-parton scattering amplitude and show that the sum of all Feynman-diagram contributions can be written in an eikonal form. In this form dynamics is determined by the phase shift, and subleading-logs of all orders needed to restore the Froissart bound are automatically provided. The main technical difficulty is to find a way to extract these subleading contributions without having to compute each Feynman diagram beyond the leading order. We solve that proble…
We present a simple formula for the total cross section sigma^{nu N} of neutral- and charged-current deep-inelastic scattering of ultrahigh-energy neutrinos on isoscalar nuclear targets, which is proportional to the structure function F_2^{nu N}(M_V^2/s, M_V^2), where M_V is the intermediate-boson mass and s is the square of the center-of-mass energy. The coefficient in the front of F_2^{nu N}(x, Q^2) depends on the asymptotic low-x behavior of F_2^{nu N}. It contains an additional ln(s) term if F_2^{nu N} scales with a power of ln(1/x). Hence, an asymptotic low-x behavior F_2^{nu N} propto ln…
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…
Energy dependence of cross sections in proton-proton and antiproton-proton collisions
研究质子-质子和反质子-质子碰撞中截面的能量依赖性，利用AQFT方法描述全球散射参数，并发现其行为符合Pomeranchuk定理并接近Froissart-Martin限制。
We present the predictions of a model for proton-proton total cross-section at LHC. It takes into account both hard partonic processes and soft gluon emission effects to describe the proper high energy behavior and to respect the Froissart bound.
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.
On ultrahigh-energy neutrino-nucleon deep-inelastic scattering and the Froissart bound
本文回顾了超高能中微子深度非弹性散射总截面σ^{νN}的近似结果，指出文献中常假设的小x时F_2^{νN}∝ln^2(1/x)的渐近行为会导致σ^{νN}违反Froissart界。
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…
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.
We show that it is possible to fit all of the HERA DIS (deep inelastic scattering) data on $F_2^{γp}$ at small values of Bjorken $x$, including the data at {\em very low} $Q^2$, using a new model for $F_2^{γp}$ which both includes an asymptotic (high energy) part that satisfies a saturated Froissart bound behavior, with a vector-dominance like mass factor in the parameterization, and extends smoothly to $Q^2=0$. We require that the corresponding part of the virtual $γ^* p$ cross section match the known asymptotic part of the real $γp$ cross section at $Q^2=0$, a cross section which is determin…
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.
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…
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 show that the geometric scaling of the dipole cross section can be explained using standard DGLAP perturbative evolution. The DGLAP improved saturation model due to the Laplace transforms method is considered at LO and NNLO approximations from the experimental data by relying on a Froissart-bounded parametrization of $F_{2}(x,Q^{2})$. These results are comparable with Golec-Biernat-W$\ddot{\mathrm{u}}$sthoff (GBW) model in a wide kinematic region $rQ_{s}$ which takes into account charm mass. The successful description of $σ_{\mathrm{dip}}(x,r)/σ_{0}$ and $σ_{\mathrm{dip}}(rQ_{s})/σ_{0}$ are…
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 propose to use rich informations on pi p total cross sections below N=10 GeV in order to investigate whether these cross sections increase like log nu or log2 nu at high energies. A finite-energy sum rule (FESR) which is derived in the spirit of the P' sum rule as well as the n=1 moment FESR have been required to constrain the high-energy parameters. We then searched for the best fit of pi p total cross sections above 70 GeV in terms of high-energy parameters constrained by these two FESR. We can conclude that our analysis strongly favours the log2 nu behaviors satisfying the Froissart unit…
In this paper we perform a numerical study of the tranverse expansion of hadronic scattering amplitudes in the dipole picture of high energy QCD. We go beyond the mean field approximation by including fluctuations and also wave function saturation effects, and the evolution with both a fixed and a running coupling is investigated. We also study the nonperturbative aspects, and as has been predicted earlier, our results indicate that the Froissart-Martin bound is saturated once confinement effects are included in the evolution. Thus the total cross section increases proportional to the square o…
An attempt is made in QCD to explain the growth of total cross-sections with energy, without violating the Froissart bound. This is achieved by computing the phase shifts of elastic scatterings of partons rather than their amplitudes. To render that possible a general formalism of phase-shifts in QCD is developed. Computed to two-loop order, agreements with hadronic and virtual-photon total cross-sections are quite satisfactory. Predications for the slower rate of growth at higher energies are also presented.
We argue that if the elastic proton-proton cross section increases with energy, the Froissart-like high energy behaviour of the elastic amplitude (which corresponds to a `black disk' of radius $R(s)=c\ln s-β\ln(\ln s)$) is the only possibility to satisfy the unitarity equation at each value of the impact parameter, $b$. Otherwise the cross section of events with Large Rapidity Gaps grows faster than the total cross section at the same $b$. That is, these `gap' events require {\it maximal} growth of the high-energy (asymptotic) cross section and of the interaction radius $R(s)$ in order to be c…
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.
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