The Odderon remains an elusive object, 33 years after its invention. The Odderon is now a fundamental object in QCD and CGC and it has to be found experimentally if QCD and CGC are right. In the present paper, we show how to find it at RHIC and LHC. The most spectacular signature of the Odderon is the predicted difference between the differential cross-sections for proton-proton and antiproton-proton at high s and moderate t. This experiment can be done by using the STAR detector at RHIC and by combining these future data with the already present UA4/2 data. The Odderon could also be found by ATLAS experiment at LHC by performing a high-precision measurement of the real part of the hadron elastic scattering amplitude at small t. Paris 6 et Paris 7, Associée au CNRS The Odderon is defined as a singularity in the complex J-plane, located at J = 1 when t = 0 and which contributes to the odd-under-crossing amplitude F − . The concept of Odderon first emerged in 1973 in the context of asymptotic theorems [1]. 7 years later, it was possibly connected with 3-gluon exchanges in perturbative QCD [2-4], but it took 27 years to firmly rediscover it in the context of pQCD [5]. The Odderon was also rediscovered recently in the Color Glass Condensate (CGC) approach [6,7] and in the dipole picture [8].One can therefore assert that the Odderon is a crucial test of QCD.On experimental level, there is a strong evidence for the non-perturbative Odderon: the discovery, in 1985, of a difference between (dσ/dt)p p and (dσ/dt) pp in the dip-shoulder region 1.1 < |t| < 1.5 GeV 2 at √ s = 52.8 GeV [9,10]. Unfortunately, these data were obtained in one week, just before ISR was closed and therefore the evidence, even if it is strong (99,9 % confidence level), is not totally convincing. Moderate evidence for the existence of the non-perturbative Odderon also comes from the dramatic change of shape in the polarization in π − p → π 0 n, in going form p L = 5 GeV/c [11,12] to p L = 40 GeV/c [13], but this Odderon corresponds to a different type of Odderon as compared with the one identified in pQCD. Finally, weak evidence for the non-perturbative Odderon comes from a strange structure seen in the UA4/2 dN/dt data forpp scattering at √ s = 541 GeV, namely a bump centered at |t| = 2 · 10 −3 GeV 2 [14]. This structure could correspond to oscillations of a very small period due to the presence of the Odderon [15]. All the above mentioned experimental results point towards the maximal Odderon [1, 16], a special case corresponding to the maximal asymptotic (s → ∞) behavior allowed by the general principles of strong interactions: σ T (s) ∝ ln 2 s, as s → ∞ (1) and ∆σ(s) ≡ σp p T (s) − σ pp T (s) ∝ ln s, as s → ∞ .Interestingly enough, an important stream of theoretical papers concern precisely the maximal behavior [1], which was first discovered by Heisenberg in 1952 [17] and later proved, in a more rigorous way by Froissart and Martin [18,19]. Half a century after the discovery of Heisenberg, this maximal behavior (1) was also proved in the contex...

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 $\bar{p}p$ 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 can not 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.Comment: Revised version, 30 pages, 16 eps-figures, elsart.cls (included), to appear in Nucl Phys.

By means of improved empirical fits to the differential cross section data on pp elastic scattering at 19.4 ≤ √ s ≤ 62.5 GeV and making use of a semi-analytical method, we determine the eikonal in the momentum transfer space (the inverse scattering problem). This method allows for the propagation of the uncertainties from the fit parameters up to the extracted eikonal, providing statistical evidence that the imaginary part of the eikonal (real part of the opacity function) presents a zero (change of signal) in the momentum space, at q 2 ≈ 7 ± 1 GeV 2 . We discuss the implication of this change of signal in the phenomenological context, showing that eikonal models with one zero provide good descriptions of the differential cross sections in the full momentum transfer range, but that is not the case for models without zero. Empirical connections between the extracted eikonal and results from a recent global analysis on the proton electric form factor are also discussed, in particular the Wu-Yang conjecture. In addition, we present a critical review on the pp differential cross section data presently available at high energies.PACS. 13.85.Dz Elastic scattering -13.85.-t Hadron-induced high-energy interactions