Odderon effects in the differential cross-sections at Tevatron and LHC energies

In the present paper, we extend the Froissaron-Maximal Odderon (FMO) approach at $t$ different from 0. Our extended FMO approach gives an excellent description of the 2148 experimental points considered in a wide range of energies and momentum transferred. We show that the very interesting TOTEM results for proton-proton differential cross-section in the range 2.76-13 TeV, together with the Tevatron data for antiproton-proton at 1.8 and 1.96 TeV give further experimental evidence for the existence of the Odderon. One spectacular theoretical result is the fact that the difference in the dip-bump region between $\bar pp$ and $pp$ differential cross-sections is diminishing with increasing energies and for very high energies (say 100 TeV), the difference between $\bar pp$ and $pp$ in the dip-bump region is changing its sign: $pp$ becomes bigger than $\bar pp$ at $|t|$ about 1 GeV$^2$. This is a typical Odderon effect. Another important - phenomenological - result of our approach is that the slope in $pp$ scattering has a different behavior in $t$ than the slope in $\bar pp$ scattering. This is also a clear Odderon effect.


Introduction
The Odderon is certainly one the most important problems in strong interaction physics. It was introduced [1] in 1973 on the basis of asymptotic theorems [2], [3] and was rediscovered later in QCD [4], [5], [6]. In spite of the fact that its theoretical status is very solid, its experimental evidence from half a century is still scarce. This situation is not astonishing, The clear evidence for Odderon has to come by comparing the data at the same energy in hadron-hadron and antihadron-hadron scatterings. But we have not such accelerators! We therefore have to limit our search for evidence for the Odderon only in an indirect way. The search for the Odderon is crucial in order to confirm the validity of QCD. It is very fortunate that the TOTEM datum ρ pp = 0.1±0.01 at 13 TeV [7] is the first experimental discovery of the Odderon, namely in its maximal form [8]. Moreover, we checked recently that the Froissaron-Maximal Odderon (FMO) approach is the only model in agreement with the LHC data. We generalized the FMO approach by relaxing the ln 2 s constraints both in the even-and oddunder-crossing amplitude and we show that, in spite of a considerable freedom of a large class of amplitudes, the best fits bring us back to the maximality of strong interaction [9].
In the present paper, we extend the FMO approach at t different from 0. We show that the very interesting TOTEM results for proton-proton differential crosssection in the range 2.76-13 TeV, together with the D0 data for antiproton-proton at 1.96 TeV give further experimental evidence for the existence of the Odderon.

Extension of the FMO approach at t different from zero -General definitions
In general amplitude of pp forward scattering is F pp (s, t) = F + (s, t) + F − (s, t) (1) and the amplitude of antiproton-proton scattering is Fp p (s, t) = F + (s, t) − F − (s, t). (2) In this model we used the following normalization of the physical amplitudes.
In the FMO model CE and CO terms of amplitudes are defined as sums of the asymptotic contributions where F H (z t , t) denotes the Froissaron contribution and F M O (z t , t) denotes the Maximal Odderon contribution. Their specified form will be defined below.

Regge poles and their double rescatterings
In the FMO model in the terms F R± (s, t) we consider not only single Regge pole contributions but also their double rescatterings or double cuts. Their contributions, F R pp (z t , t), F R pp (z t , t), are the following where . For a convenience in further work with parameterizations in FMO model at t = 0 and t = 0 contrary to standard definition of z t we put opposite sign for it.
Here F P (z t , t), F O (z t , t) are simple j-pole Pomeron and Odderon contributions and F R+ (z t , t), F R− (z t , t) are effective f and ω simple j-pole contributions, where j is an angular momenta of these reggeons. F P P (z t , t), F OO (z t , t), F P O (z t , t), are double P P, OO, P O cuts, correspondingly. We consider the model at t = 0 and at energy √ s > 19 GeV, so we neglect the rescatterings of secondary reggeons with P and O. In the considered kinematical region they are small. Besides, because f and ω are effective, they can take into account small effects from the cuts. The standard Regge pole contributions have the form where The factor 2m 2 = z t /z t (t = 0) (at s m 2 ) is inserted in amplitudes F R± (z t , t) in order to have the normalization for amplitudes and dimension of coupling constants (in mb) coinciding with those in the [8]. The same is made for all other amplitudes, including Froissaron and Maximal Odderon (see below). For the coupling function C R± (t) we have considered two possibilities. The first one is a simple exponential form The second case is a linear combination of exponents for Standard Pomeron and Odderon terms which allow to take into account some possible effects of nonexponential behavior of coupling function. Secondary reggeons still are parameterized in the simplest exponential form, because we did not consider low energies where terms R ± (s, t) are more important.
The double cuts are written in a simplified form as compared with the exact form of a cut. They can be considered also as effective P P, OO, P O cuts. Namely, 4 Froissaron and Maximal Odderon at t = 0

Partial amplitudes for Froissaron and Odderon
Let us start from the Froissaron amplitude in (s, t)representation at high s. The amplitude can be expanded in the series of partial amplitudes φ(ω, t). In accordance with the standard definition of partial amplitude With such definition partial amplitude satisfies the unitarity equation in the form We use of the Sommerfeld-Watson transform amplitude (here and in what follows ω = j − 1 and j is complex angular momentum) which can be written as follows where ξ is the signature of the term, contour C is a straight line parallel to imaginary axis and at the right of all singularities of φ ξ (ω, t), ζ = ln(z t ) − iπ/2 ≡ ln(−iz t ) and Thus for crossing even amplitude (ξ=+1) we have and for crossing odd amplitude (ξ=-1) Inverse transformation is One can show that in order to have maximal growth of total cross section σ tot (s) ∝ ξ 2 at s → ∞, to have a growing elastic cross section bounded by and to provide the correct analytical properties of amplitude at t ≈ 0 necessary to write the partial amplitude φ(ω, t) in the following form (more details are given in the Appendics, Section A) where r ± are some constants, q 2 ⊥ = −t and β(ω, t) has not singularity at ω 2 + R 2 q 2 ⊥ = 0. In fact a choice of the sign in φ − (ω, t) does nor matter because the crossing odd terms contribute to pp andpp amplitude with the opposite signs. In order to have agreement with parametrization and parameters which we used in the papers devoted to analysis of the data at t = 0, we should replace -1 for for +1 in front of φ − (ω, t) At ω = 0, function ϕ − (ω, t) has singularity in t if β − (0, t) = 0, namely, φ − (0, t) ∝ (−t) 3/2 . One of arguments against the Maximal Odderon is that this singularity in partial amplitude means the existing of massless particle in the model. However as we seen above ϕ − (ω, t) is not the real physical partial amplitude which is and it equals to 0 at ω = 0 because of sin(πω/2) coming from signature factor. Now let us suppose that in accordance with the structure of the singularity of ϕ ± (ω, t) at ω 2 + ω 2 0± = 0 (ω 2 0± = R 2 ± q 2 ⊥ ) the functions β ± (ω, t), depending on ω through the variable κ ± = (ω 2 + ω 2 0± ) 1/2 , can be expanded in powers of κ ± ± ϕ ± (ω, t) = i 1 Then making use the table integrals (see the Section A we obtain the expressions for F ± (z t , t) which are written in the next Section.
The Froissaron and the Maximal Odderon defined at t = 0 by above Eqs. (24,25) allow various extensions to analytical t-dependences. Probably it is impossible a priory to choose the best of them. In the present work we consider an extension of Eqs. (24, 25). Comparing the various cases we have found that the best description of the data is achieved if third terms do not contain Bessel functions of r ± τ ξ, while they have a more complicated than simple exponents functions of t.
In above equations ζ = ln(−iz t ). Due to the factor z (instead of z t ) the amplitudes F H (z t , t) and F M O (z t , t) have the required normalization with additional factor 2m 2 .
In the Froissaron and Maximal Odderon contributions the couplings H 3 and O 3 are redefined. The standard Pomeron and Odderon were included to constant terms H 3 and O 3 , correspondingly, in the fit at t = 0. At t = 0 contribution of P and O are important and they have to be added to amplitudes. In order to save parameter H 3 (O 3 ) as sum of Froissaron and standard Pomeron (Maximal Odderon and standard Odderon) at t = 0 but distinguish them at t = 0 we made the above mentioned replacement. Thus the real coulpings of Froissaron and Maximal Odderon are H 3 − C P and O 3 + C O correspondingly.

Comparison of the FMO model with the data
We give here the results of the fit to the data in the following region of s and |t|.
The t-region is chosen in such a way that we can ignore the contribution of the Coulumb part of amplitudes which in given region are one order of magnitude or less than 1% of the nuclear amplitude. For 13 TeV TOTEM data we used the data at t = 0 for σ tot [10] and ρ [7] and data for dσ/dt at t = 0 presented at the 4th Elba Workshop on Forward Physics @ LHC Energy by F.Nemes [11] and at the 134th open LHCC meeting by F. Ravera [12].

A new method of minimization for global fits
By the term "global fit" we mean a simultaneous fit at t = 0 (experimental observables σ pp tot , σp p tot and ρ pp .ρp p ) and at t = 0 (dσ pp /dt, dσp p /d). In the standard method in which the sum of all χ 2 for each experimental point is minimized, we face with the following problem.
The number of experimental points at t = 0 in the chosen kinematic region is about 250, whereas number of points at t = 0 is about 2000. It is clear that in such a case an influence of 2000 points for differential cross sections is much stronger than those for cross sections and ratios of real to imaginary part of amplitude at t − 0. As result we have all parameters mainly fixed by differential cross sections. Thus we see some destruction of the good fit at t = 0 which can be performed separately, the best parameters controlling behavior of the forward observables are changed and a worse value of χ 2 for data at t = 0.
There are two ways to fix the problem. In the first method we fix all parameters which are critical for t = 0 and then perform the fit at t = 0. It works for simple models where free parameters can be split for two relatively independent groups: parameters responsible for t − 0 and parameters responsible for t different from 0. However this method does not work, for example, in the models which take into account multiple rescatterings. In this case parameters such as slopes of Regge trajectories, slopes B-s of residual functions (in the simplest case they are proportional to exp(Bt) and so on) are important for value of amplitude at t = 0. At the same time, the expressions for the total cross sections and ratios ρ also contain these parameters. For such models we can't do separate fits at t = 0 and at t = 0. Thus we come back to the above mentioned problem in a global fit.
To solve the problem we propose a new method of minimization. In this method we minimize not the total sum of the individual χ 2 -s but another combination of these χ 2 -s. Namely the most adapted quantity for minimization is the weighted χ 2 w . Let we have M kinds of observables and for each them i = 1, 2, ..., M we have N i experimental points. We define the weighted χ 2 as In fact we sum for the given kind of observable, their number being N i . In the ideal case after minimization all In the next Sections we present and shortly comment the results of the standard minimization and of weighted one performed in accordance with Eq. (28).
We would like to notice that it is very easy to calculate the usual total χ 2 and χ 2 /dof or to present the results of the fit in the set of χ 2 (i)/N i along with χ 2 w .

Total pp andpp cross sections and parameters ρ pp and ρp p
In this Section we give the results of the standard minimization at t = 0. The obtained in FMO model values of χ 2 in the fit at t = 0 with double rescattering of the standard Pomeron and Odderon (Eqs. (11)(12)) are given in the The values of parameters obtained from the fit at t = 0 are given in the Table 2. Experimental data and theoretical curves are presented in the Fig. 1.

pp andpp differential cross sections dσ/dt
Here we present results for both methods of minimization, the standard method and the weighted one.
Number of experimental points in pp andpp differential cross sections used in the standard fit (all parameters listed in the Table 1 are fixed in this case) and quality of fit are shown in the Table 3.
The values of χ 2 for the both methods of fit at t = 0 are presented in the Table 4. One can see that the both methods of minimization lead to comparable description of the data at t = 0 but the weighted minimization gives lower (by about 3%)) total χ 2 .  Table 3 Number of experimental points for differential cross sections and the quality of their description in FMO model by the standard method of minimization (parameters determining the amplitudes at t = 0 are fixed in this fi)  The values of parameters and their errors obtained in two fits, standard and weighted, within the FMO model are given in the Table 5 (parameters of the Froissaron and Maximal Odderon terms) and in the Table 6 (parameters of the standard Pomeron and Odderon, of their double rescatterings and of secondary reggeons) In Figs. 2 and 3 we show the differential cross-sections at energies bigger than 19 GeV. In Fig. 4 we show the differential cross-sections at the LHC energies 7, 8 and 13 TeV and in Fig. 5 we show our predictions at 2.76 TeV. In Fig. 6 we show in a magnified way the differential cross-sections at 53 GeV. In Fig. 7 we show thē pp differential cross section at 1.18-1.96 TeV and pp differential cross section at 7 TeV.
As one can see from these figures our description of the data in a wide range of energies is very good. In Fig. 8 we show the evolution of the dip-bump structure in pp andpp differential cross sections with increasing energy. In Fig. 9 we show in a magnified way the dipbump region at different energies and in Fig. 10 we show the evolution of the ratio R σ = (dσ(pp)/dt)/(dσ(pp)/dt) with increasing energy. A remarkable prediction can be seen from these last three figures: the difference in the dip-bump region betweenpp and pp differential cross sections is diminishing with increasing energies and, for very high energies (say 100 TeV, see Fig. 9), the ratio in the dip-bump region goes to 1. At ISR energies until ∼ 60 GeV the ratio R σ > 1 and then it becomes less than 1 but increases to maximum at some t m . After maximum the value of R σ decreasing and equals to 1 at t 0 which is going to lower t with increasing energy. At higher t however R σ is oscillating around of 1 when t increase. This is a spectacular Odderon effect.     pap-19(X10 16 ) pap-53GeV(X10 14 ) pap-62GeV(X10 12 ) pap-546GeV(X10 10 ) pap-630GeV(X10 8 ) pap-1.96TeV(X10 6 ) pap-2.76TeV(X10 6 ) pap-7.0TeV(X10 4 ) pap-13.0TeV(X10 2 ) pap-100TeV pp-19(X10 16 ) pp-53GeV(X10 14 ) pp-62GeV(X10 12 ) pp-546GeV(X10 10 ) pp-630GeV(X10 8 ) pp-1.96TeV(X10 6 ) pp-2.76TeV(X10 6 ) pp-7.0TeV(X10 4 ) pp-13.0TeV(X10 2 ) pp-100TeV The slope B(s, t) is a very interesting quantity in the search for Odderon effects. It is defined by If we consider the dependence of slope on energy and compare this dependence with available experimental data we have to take into account that slopes in any realistic model depend on t. Dependence of slope on t at various energies in the FMO model is illustrated in Fig. 11 (left panel). Therefore we must to calculate the slope < B(s) > averaged in some interval of t. We did that in the interval |t| ∈ (0.05, 0.2)GeV 2 which approximately is in agreement to the intervals from which the experimental data on B is determined.
where ∆ t = t max − t min . We show in Table 7 our predictions for the averaged slopes in the TeV region of energy as compared with experiments at Tevatron and LHC.
In Fig. 11 (right panel) we show the increasing of the averaged slopes at t=0 with increasing energy. One can see that the slopes are approaching the ln 2 s increase at high energies.  Table 7 Experimetal values of slopes of pp andpp differential cross sections at TeV energies and the averaged slopes calculated in FMO model In Fig. 12 we plot the slopes as function of t in pp andpp scatterings. We discover from the t-dependence of the slopes an extremely interesting phenomenon. The slope in pp scattering has a different behaviour in t than the slope inpp scattering. In the left panel of Fig. 12 we see that in pp scattering the slopes are first nearly constant and after that they fall sharply, they cut a first time the B(t) = 0 line, reach a deep minimum negative value, after that they increase and cut a second time the B(t) = 0 line and finally they reach an approximately constant value for higher t. The two crossing points of the B(t) = 0 line move towards smaller t when energy increases. In the right panel of Fig. 12 we see a very different behaviour inpp scattering. In this case, at energies higher than ISR ones, B(t) marginally crosses zero, but no so deeply and sharply as in pp scattering. For completeness, we show in Fig. 13 the slope parameter for pp scattering at 7 and 13 TeV as compared with the slope parameter inpp scattering at 1.96 TeV, where we can see the same phenomenon.
This phenomenon is a clear Odderon effect. The odd-under crossing amplitude makes the difference between pp andpp scatterings and this amplitude is dominated at high energy by the Maximal Odderon.

Comparison with other approaches
To our knowledge, the present model is the only model which fits forward and non forward data in a wide range of energies (including TeV region), without explicit theoretical defects (like the violation of the unitarity).
However, it is important to note that our results concerning the slopes are in complete agreement with those obtained recently by Csörgö et al. [13], who performed a very useful mirroring between the discontinuous experimental data (points) and continuous analytic functions (scattering amplitudes) by using an expansion in terms of Lévy polynomials. In such a way they get a very clear Odderon effect concerning the slopes. Their analysis have no dynamical content: it is a parametrization of experimental data in terms of big number of parameters.
This agreement is very important from two points of view. On one side, the Odderon existence is reinforced by two quite different analysis, one model-independent and the other one having a dynamical content.
On another side, the fact that the Maximal Odderon is in agreement with a model-independent analysis reinforce the status of the Maximal Odderon.

Conclusion
In our paper we present an extension of the Froissaron-Maximal Odderon (FMO) approach for t different from zero, which satisfies rigorous theoretical constraints. Our extended FMO approach gives an excellent description of the 2148 1 experimental points considered in a wide range of energies and momentum transferred. One spectacular theoretical result is the fact that the difference in the dip-bump region betweenpp and pp differential cross sections is diminishing with increasing energies and for very high energies (say 100 TeV), the difference in the dip-bump region betweenpp and pp is changing its sign: pp becomes bigger thanpp at |t| about 1 GeV 2 . This is a typical Odderon effect.
Another important -phenomenological -result of our approach is that the slope in pp scattering has a different behaviour in t than the slope inpp scattering. This is a clear Odderon effect.
New ways of detecting Odderon effects, e. g. in an Electron-Ion Collider, were recently explored on the basis of a general QCD light front formalism [15].
Acknowledgment. The authors thank Prof. Simone Giani for a careful reading of the manuscript. One of us (E.M.) thanks the Department of Nuclear Physics and Power Engineering of the National Academy of Sciences of Ukraine for support.

A.1 General constraints
Let us reiterate here that the model with σ t (s) ∝ ln 2 s is not compatible with a linear pomeron trajectory having the intercept 1. Indeed, let us assume that and the partial wave amplitude has the form For Pomeron (simple or double pole) and Froissaron signature is positive, ξ = +1.
Then, we have pomeron contribution at large s as If as usuallyβ(t) =β exp(bt) then we obtain σ t (s) ∝ ln n−1 s, According to the obvious inequality, we have Thus we come to the conclusion that the a model with σ t (s) ∝ ln 2 s (n=3) is incompatible with a linear pomeron trajectory. In other words the partial amplitude Eq. (32) with n = 3 is incorrect.
If n = 1 we have a simple j-pole leading to constant total cross section and vanishing at s → ∞ elastic cross section. However such a behaviour of the cross sections is not supported by experimental data.
If n = 2 we have the model of dipole pomeron (σ t (s) ∝ ln(s)) and would like to emphasize that double j-pole is the maximal singularity of partial amplitude settled by unitarity bound (38) if its trajectory is linear at t ≈ 0.
We would like to notice here that TOTEM data for the pp total cross section exclude the dipole pomeron model which is unable to describe with a reasonable χ 2 the high values of σ pp tot (s) at LHC energies.
Thus, constructing the model leading to cross section which increases faster than ln(s), we need to consider a more complicated case (we consider at the moment a region of small t and j ≈ 1): iβ(1, t) j − 1 + r(−t) 1/µ n .
However in this case amplitude a(s, t) has a branch point at t = 0 which is forbidden by analyticity of amplitude a(s, t).
A proper form of amplitude leading to t ef f 2 decreasing faster than ln −1 s (it is necessary for σ t rising faster than ln s) is the following Now we have m branch points colliding at t = 0 in j-plane and creating the pole of order mn at j = 1 (but there is no branch point in t at t = 0). At the same time t ef f ∝ 1/ ln m s and from σ el ∝ ln 2mn−2−m s ≤ σ t ∝ ln mn−1 s ≤ ln 2 s one obtains If σ el ∝ σ t then n = 1 + 1/m. Furthermore, if σ t ∝ ln s then m = 1 and n = 2 which corresponds just to the dipole pomeron model. In the Froissaron (or tripole pomeron) model m = 2 and n = 3/2. It means that σ t ∝ ln 2 s.
Now let us suppose that in agreement with the structure of the singularity of φ ± (ω, t) at ω 2 + ω 2 0± = 0 the functionsβ ± (ω, t) depend on ω through the variable κ ± = (ω 2 + ω 2 0± ) 1/2 and it can be expanded in powers of κ ± φ ± (ω, t) = There are a different ways to add to partial amplitude ϕ(j, t) terms which at s → ∞ are small corrections (they can be named as subasymptotic terms).