Slide 2. Good morning everybody! My report is devoted to a model developed by Basarab Nicolescu and myself within the framework of an approach based on the hypothesis of the maximality of strong hadronic interaction and complex angular momenta in S-matrix. I will talk about theoretical base of this approach, the history and implementation of the model for forward elastic scattering and its generalization to non-zero momentum transfers. I will briefly present the latest results of the TOTEM collaboration obtained at the Large Hadron Collider, their description and interpretation in the Froissaron and Maximal Odderon model. Slide 4. The maximality of the strong interaction at asymptotical energy is the main assumption which we use in FMO model, was originally formulated by G. Chew in the 60s of the last century (when experiments showed approximate constancy of total cross sections). This hypothesis was strengthened in the work of L. Lukazsek and B. Nicolescu in 1973. Now it sounds like this. Froissaron is a reggeon dominating at high energies in a crossing-even amplitude. Maximal Odderon is a reggeon dominating at high energies in crossing-odd amplitude. Thus, Froissaron realizes the principle of maximality for the total hadron cross sections. Maximal Odderon realizes the idea of maximal growth of the difference of particle-particle and antiparticle-particle cross sections. Slides 6. The rigorous results of the analyitic S-matrix theory are based on the S-matrix postulates. I’ll briefly note those results that are used in our model. 1. Froissart-Martin bound and Pomeranchuk theorem. The total cross sections cannot grow faster than the square of the logarithm of the energy, while the ratio of the total cross sections for particles and antiparticles tends to unity with increasing energy. 2. According to the Eden and Cornell constraint, the cross-section difference can grow. 3. Auberson-Kinoshita-Matin theorem is a very important asymptotic result for amplitude at non-zero momentum transfers. It defines in fact, the form of the main contribution to amplitudes in the case of maximally rising total cross sections. Slide 7. 4. Dispersion relations determine the interrelation between imaginary and real parts of the Froissaron and Maximal Odderon contributions. 5. Froissaron is mainly imaginary contribution while Maximal Odderon is mainly real contribution to amplitudes (at least at low t). It is confirmed by experimental data. Slide 9. On this slide, I schematically depicted the history of the development of the hypotheses of maximal strong interactions and maximal odderon, as well as the Froissaron and Maximal Odderon models. It was first expressed in 1973. The name "Odderon" has been suggested in the paper of Joynson, Leader, Nicolescu, Lopez in 1975. Further, up to 2017, a number of works were published in which the properties of models with Froissaron and Maximal Odderon and various versions of model were compared with experimental data. Slide 10. In October 2017, at a meeting of the CERN Research and Resources Board (RRB), TOTEM presented for the first time the latest data on measurements of the total cross section and the parameter \rho (ratio of the real part to imaginary part of amplitude at t=0) at energy of 13 TeV. Nicolescu and I immediately took these numbers into work and after two weeks sent the work to NEP. In this work, we stated that the results of the TOTEM experiment can be interpreted as the discovery of odderon. At the February of 2018, the work was published in Physics. Letters B. In 2019, a paper was published in the European Physic Journal C with a generalization of the FMO model to nonzero momentum transfers. In addition, we responded to the work of T.T. Wu and A. Marten, in which they argued that the parameter \rho should be positive at asymptotically high energies. G. Tersimonov and I proved that it follows from the Eden and Cornille theorem and the dispersion relations that if the Maximal Odderon contributes to the pp and \ barpp scattering amplitudes, then one of the \rho becomes negative. This fact does not contradict the well-known rigorous results of the S-matrix. In the papers of T. Csörgő et al. the odderon effects in differential pp cross sections were confirmed by the model-independent analysis of the TOTEM data at LHC energy. Finally, in December of the last year, the common paper of the TOTEM and D0 collaborations was presented in the HEP archive. The LHC pp and Tevatron \bar pp data were reanalyzed and the presence of odderon effects at TeV energies was strongly confirmed. Slide 12. This slide shows the figures from TOTEM's paper on the total cross section and ratio pho at 13 TeV in pp scattering. In the upper figures, new points are marked with red circles. The small value of the ratio rho was quite unexpected and required an explanation. COMPETE's predictions for the total cross-section were confirmed, but the experimental value of pho turned out to be significantly lower than the theoretical one. This is illustrated in the figure below on the left, which compiles predictions for a large number of models that have been analyzed by the COMPETE team. None of the models were able to describe the new data on sigma t\rho otal and at the same time. A successful description of the latest data (figure below, right) and interpretation of the result were first given in our 2017 paper. It so happened that our work appeared earlier (but after the TOTEM data were presented at the RRB meeting) than the full TOTEM paper in HEP arxive. Slide 13. COMPETE analysis and the many other descriptions of pp and \bar pp forward data before LHC era did not detect the Odderon contribution. When our paper was already in HEP ArXiv we asked ourselves ”Why odderon was neglected in the most part of the models? Why the various estimations allowed to do that? “ Searching an answer we plotted the partial contributions to imaginary and real parts of amplitudes at t = 0, which we obtained in the FMO model for forward scattering. We have found that only at LHC energies the ReF MO (s) (solid blue line in the figure) becomes to be visible! Slide 15. The standard assumption about Regge poles and it trajectories is that they are the single poles in the complex j-plane. Trajectories are approximately liner in t. That is confirmed in spectroscopy of the hadronic resonances. Froissart-Martin bound requires that intercepts of all trajectories with vacuum quantum numbers could not exceed 1. How to describe the rising total cross section with such restriction? Which kind of reggeons is suitable for that? As it is shown at the slide Pomeron singularity with linear trajectory violates unutarity. Let’s write a general enough form of reggeon contribution with two parameters m and n and taking into account that amplitude at t=0 has not any t-singularity. If total cross section and integrated elastic cross section are rising proportionally each to other we obtain m=2, n=3/2. This form is used in FMO model, imaginary and real for main terms of Froissaron and Maximal Odderon correspondingly. Slide 16. Thus, the pp and \bar pp amplitudes are written as contribution of the Froissaron and Odderom Maximal and effective crossing-even and crossing-odd reggeons with intercepts about ½ or less. The main terms in Froissaron and Maximal Odderon correspond to j(omega=j-1) form given in the previous slide. Other terms in F and OM are corrections ~ \lns and constant. The standard Pomeron and Odderon (with intercepts equal to 1) are not considered at t=0 because their contributions to A_pp and A_\bar pp (z) are constants, they are absorbed in H_3 ,O_3 which as well are constant. These terms would be distinguished at t nonequal 0. The amplitudes have 10 free parameters. Slide 17. Free parameters of the model were found by the standard minimization of \chi^2. Results of the data description is shown on the Figures. From the fit of data in the energy region from 5 GeV up to 13 TeV we obtain \chi^2/(degree of freedom) equal approximately 1.067. The best prediction of the COMPETE (without Odderon) is shown by dashed lines for a comparison. Slide 19. In order to verify a stability of the considered FMO model we performed fit procedure within a more general version of the model, We consider the original Eden and Cornille bounds for pp and \bar pp sigma, their difference with free parameters \beta_F and \beta O. The intercepts of the standard pomeron and odderon were free as well but less than 1. After the fits we has been found the following results. Slide 21. 1. β_F and β_MO come back to the saturation values β _F = β+O = 2. 2. Pomeron intercept comes back to 1 from any input value < 1. 3. More details concerning the standard odderon (the single j-pole with an intercept less or equal 1) are given in our paper 4. If we put parameter O_1 equal zero (no Maximal Odderon) description of the data is failed and χ^2 /ndf ≈ 1.07 (at O_1= 0) is increased up to 1.18 (at O_1 = 0). 5. Thus, we can conclude that the Froissaron and Maximal Odderon are neccessary contributions to amplitudes for successful description of the forward LHC data. Slide 22. Let’s me come back for the moment to the simple reasoning and assumption in order to choose the generalization of the FMO model for t= 0. We take into account that in transformation from φ_{±}(ω,t) to A_{±}(s,t) at high energy, the main contribution to the integra comes frov region around of singularity of φ_{±}(ω,t). We suppose that in agreement with the structure of this singularity at zero in the denominator of φ_{±} (ω,t) the numerator of φ_{±} (ω,t), β_{±}(ω,t), is a function of the \kappa_{±}, as well. Then the numerator can be expanded in powers of kappa at small κappa. Slide 23. At this slide I just remind our definitions for pp and \bar pp amplitudes and for their crossing-even and crossing-odd terms. Slide 24. Thus, following the above arguments, we used in the FMO model for Froissaron and Maximal Odderon the parametrizations shown in this slide. The explicit form for the vertex functions is chosen after comparison of the results, obtained in the fits with exponential (in t) vertices and those written at this slide. Slide 25. Froissaron and Maximal Odderon are important in whole kinematic region. However they start to dominate in TeV region. Standard Pomeron and Odderon are important in GeV region. The cuts tune the $d\sigma/dt$ region around dip-bump structure and in the second diffraction cones. A role of "hard" terms is not clear enough, however. Nevertheless they help to obtain a better description of the considered data despite that we consider region where $|t|/s are small Secondary reggeons at t nonequal zero are important at FNAL and ISR energies. But they are negligible starting from energy about 100 GeV. Slides 26-28. I show at these slides experimental data at curves calculated in the FMO model for differential pp and \bar pp cross sections at energies higher than 19 GeV. A few words about quality of the description. Result of the fit is \chi^2/ndf equals approximately 1.61. We know some defects in this description. We hope they are not crucial. We are working now in order to fix the problems (or to make it less) improving the model. Slide 30. Here the total, integrated elastic and inelastic pp cross sections calculated in the presented FMO model are shown. Conclusion. I’d like to note first of all the following. When we say “discovery of Odderon” we do not mean glueball resonance with quantum numbers of odderon. Generally, glueball resonances, either of Pomeron or Odderon kind, can not be detected in pp elastic scattering. They must be searched in inelastic processes. In elastic pp processes Pomeron (or Froissaron) and Odderon only manifests as a reggeon. From the beginning we consider Froissaron and Maximal Odderon are as more complicated singularities in the complex angular momentum plane then single Regge poles. 1.The main properties of the Froissaron and Maximal Odderon model are determined by the hypothesis of maximality of strong interactions and follow from the rigorous theorems of the analytic S-matrix theory and theory of the complex Regge singularities. From the beginning Froissaron and Maximal Odderon are considered as more complicated singularities in the complex angular momentum plane than single Regge poles. 2. TOTEM The values of pp total cross section and the ratio \rho measured by TOTEM at 13TeV can be described and treated as Odderon effects only if the Maximal Odderon contribution to amplitude exists, i.e. parameter O_1 in MO contribution is not zero. 3. Until now the FMO model is the only physical model (not just parametrization) which not violating the main unitarity bounds for physical observables can describe the data in a wide kinematic region and explain the latest TOTEM experimental data at highest LHC energy, Slide 31. Thank for attention.