Rescattering corrections in elastic scattering

A detailed study of the rescattering series is performed within a model, using a generalized procedure of eikonalization and fitted to the $pp$ and $\bar pp$ elastic scattering data. We estimate and compare the various rescattering corrections to be added to the Born contribution in the amplitude. We find that their number is finite, whereas it increases with the energy and the transfer, like does their importance. In the domain where data exist, we find also that a correct computation must include, at least, all two- and three-Reggeon exchanges and some four- and five-Reggeon exchanges. Any approximation aiming to reduce this (large) number of exchange would be hazardous, especially when extrapolating. We extend our estimates in the domain of future experiments.Comment: LaTeX2e, 15 pages, 14 eps figure

Eikonalization and Unitarity Constraints

An extensive generalization of the ordinary and quasi-eikonal methods is presented for the $pp$ and $\bar pp$ elastic scattering amplitudes, which takes into account in a phenomenological way all intermediate multiparticle states involving the crossing even and crossing odd combinations of Reggeons. The formalism in this version involves a maximum of three parameters corresponding to the intermediate states which are possible in this configuration. The unitarity restriction is investigated and particular cases are discussed. An interesting result that emerges concerns the Odderon trajectory intercept: we find that unitarity dictates that this quantity {\it must} be below or equal unity unless a very peculiar equality exists between the coupling of the particles to the Pomeron and the Odderon.Comment: Latex2e, 22 pages, 4 eps figures included, version accepted for publication in Eur. Phys. J. C (some comments in Sect.2 and in Conclusion and one reference are added

Of Dips, Structures and Eikonalization

We have investigated several models of Pomeron and Odderon contributions to high energy elastic $pp$ and $\bar p p$ scattering. The questions we address concern their role in this field, the behavior of the scattering amplitude (or of the total cross-section) at high energy, and how to fit all high energy elastic data. The data are extremely well reproduced by our approach at all momenta and for sufficiently high energies. The relative virtues of Born amplitudes and of different kinds of eikonalizations are considered. An important point in this respect is that secondary structures are predicted in the differential cross-sections at increasing energies and these phenomena appear quite directly related to the procedure of eikonalizing the various Born amplitudes. We conclude that these secondary structures arise naturally within the eikonalized procedure (although their precise localization turns out to be model dependent). The fitting procedure naturally predicts the appearance of a zero at small $|t|$ in the real part of the even amplitude as anticipated by general theorems. We would like to stress, once again, how important it would be to have at LHC both $pp$ and $p \bar p$ options for many questions connected to the general properties of high energy hadronic physics and for a check of our predictions.Comment: 28 pages, LaTeX, 7 figure

The Pomeron in Elastic and Deep Inelastic Scattering

We discuss some properties of the Pomeron in high energy elastic hadron-hadron and deep inelastic lepton-hadron scattering. A number of issues concerning the nature and the origin of the Pomeron are briefly recalled here. The novelty in this paper resides essentially in its presentation; we strive at discussing all these various issues in the following unifying perspective : it is our contention that the Pomeron is one and the same in all reactions. Various examples will be provided illustrating why we do not believe that one should invoke additional tools to describe the data. For pedagogical convenience, we list below the topics to be covered in the following. -- 1. Introduction. How many Pomerons? -- 2. The Pomeron in the $S$-matrix theory -- 3. The Pomeron in QCD -- 4. The Pomeron in deep inelastic scattering -- 5. The Pomeron structure -- 6. (Temporary?) ConclusionsComment: 32 pages in TeX; 27 figures (available on request from [email protected]

Of dips and structures

A detailed analysis of the existence of high energy secondary diffractive dips and structures in the extrapolations of the fits to the data is given. The existence of these dips and a fortiori their position is found to be rather model-dependent: present in all eikonalized models including Pomeron, Odderon and secondary Reggeons they disappear when an additional large- t term is added (as sometimes advocated).Comment: 7 pages (LaTeX), 1 figure (eps), submitted to Physics Letter

Elastic pp and pˉp\bar pp scattering in the Modified Additive Quark Model

The modified additive quark model, proposed recently, allows to improve agreement of the standard additive quark model with the data on the $pp, \bar pp, \pi^{\pm} p, \gamma p$ and $\gamma \gamma$ total cross-sections, as well as on the ratios of real to imaginary part of $pp$ and $\bar pp$ amplitudes at $t=0$. Here, we extend this model to non forward elastic scattering of protons and antiprotons. A high quality reproduction of angular distributions at 19.4 GeV $\leq \sqrt{s}\leq$ 1800 Gev is obtained. A zero at small $|t|$ in the real part of even amplitude in accordance with a recently proved high energy general theorem is found.Comment: 14 pages, LaTeX2e, 4 eps figures. Revised version accepted for publication in European Physical Journal

QuicK-means: Acceleration of K-means by learning a fast transform

K-means -- and the celebrated Lloyd algorithm -- is more than the clustering method it was originally designed to be. It has indeed proven pivotal to help increase the speed of many machine learning and data analysis techniques such as indexing, nearest-neighbor search and prediction, data compression, Radial Basis Function networks; its beneficial use has been shown to carry over to the acceleration of kernel machines (when using the Nyström method). Here, we propose a fast extension of K-means, dubbed QuicK-means, that rests on the idea of expressing the matrix of the $K$ centroids as a product of sparse matrices, a feat made possible by recent results devoted to find approximations of matrices as a product of sparse factors. Using such a decomposition squashes the complexity of the matrix-vector product between the factorized $K \times D$ centroid matrix $\mathbf{U}$ and any vector from $\mathcal{O}(K D)$ to $\mathcal{O}(A \log A+B)$, with $A=\min (K, D)$ and $B=\max (K, D)$, where $D$ is the dimension of the training data. This drastic computational saving has a direct impact in the assignment process of a point to a cluster, meaning that it is not only tangible at prediction time, but also at training time, provided the factorization procedure is performed during Lloyd's algorithm. We precisely show that resorting to a factorization step at each iteration does not impair the convergence of the optimization scheme and that, depending on the context, it may entail a reduction of the training time. Finally, we provide discussions and numerical simulations that show the versatility of our computationally-efficient QuicK-means algorithm