Appendix A: Cross-checking the \(\chi ^2\) definition: symmetric treatment
This Appendix summarizes our final, conservative and robust estimate of the significance of the Odderon observation in the compared D0 [8] and TOTEM [4, 28, 74] datasets, at \(\sqrt{s} = 1.96\) TeV for \(p\bar{p}\) and at \(\sqrt{s} = 2.76\) and 7 TeV for pp elastic scattering. Here we compare the considered data sets in a symmetric manner, and also mention some of the several robustness and quality tests that we have performed.
As a cross-check and a robustness test, we have validated the method with the help of a Levy-fit of Ref. [9], confirming that both methods (the fit with the full covariance matrix and the method described below) gave within one standard deviation the same minima with MINOS errors, error matrix accurate, fit in converged status and a statistically acceptable confidence level of CL \( \ge \) 0.1%. As a robustness test, the same analysis was repeated by two different co-authors of this manuscript using two different programming codes written in two different programming languages, providing the same results. In order to test the robustness of the results, we have tried different possible definitions of \(\chi ^2\) and the values reported in this Appendix correspond to the lowest possible significances, that we obtained when we used all the measured data in the signal region. Further reduction seems to be possible only by removing data from the Odderon signal region.
Let us also stress that our investigations were model-independent, as they were based on the direct comparison of the H(x) scaling functions of various experimentally measured data sets. However, we could not determine the domain of validity of the H(x) scaling, as a function of s. We have shown in the beginning of Sect. 4, that at low values of x, in the diffractive cone, for an analytic scattering amplitude \(H(x) \approx \exp (-x)\), and in these cases, the lower limit of validity of the H(x) scaling corresponds to \(x_{min}(s) \equiv 0\). However, \(x_{max}\), the upper limit of the domain of validity of the H(x) scaling at a given value of s is an s-dependent function, \(x_{max} \equiv x_{max}(s)\), that can be determined only in a model dependent manner. We have evaluated \(x_{max}(s)\) at \(\sqrt{s} = 1.96\) and 0.546 TeV, where \(p\bar{p}\) data are available, based on the evaluation of the pp differential cross-sections from the model of Ref. [42] and the H(x) scaling limit of the same model (Fig. 13).
As the measurements were performed at different values of the horizontal axis \(x = - Bt\), some interpolations were however inevitable. The technical aspects of these interpolations were detailed in Sect. 6. Let us illustrate these interpolations by the two panels of Fig. 14. We hope, that these pictures illustrate the model independent nature of our analysis more clearly than the technical description of these interpolations.
Our final quantification of the Odderon significance is based on a method developed by the PHENIX collaboration in Ref. [75] using a specific \(\chi ^2\) definition that effectively diagonalizes the covariance matrix. We utilized the measured differential cross-section of elastic pp scattering and its published covariance matrix at \(\sqrt{s} = 13\) TeV, as measured by TOTEM in Ref. [3], for a validation of this method. We have adopted the PHENIX method of the diagonalization of the covariance matrix using type A, B and C errors [75].
In its original form, the experimental data that have statistical and systematic errors are compared to a theoretical calculation that is assumed to be a function of the fit parameters. In our final analysis, presented in this Appendix, we adapted the PHENIX method for comparison of a dataset that contains data with errors directly to another dataset, that also contains central data values with errors. So our method is defined without referring to any theoretical model or parameter dependent function. Due to this reason, the most conservative definition described in this Appendix is defined to be symmetric for the exchange of the two datasets.
We classify the experimental errors of a given data set into three different types: (i) type A, point-to-point fluctuating (uncorrelated) systematic and statistical errors, (ii) type B errors that are point-to-point dependent, but 100% correlated systematic errors, and (iii) type C errors, that are point-to-point independent, but fully correlated systematic errors [75] to evaluate the significance of correlated data, when the covariance matrix is not publicly available.
Suitably generalizing the method of Ref. [75], that was developed originally for a comparison of a dataset with a theoretical, parametric fit curve, for a comparison of two data sets in this case, where the datasets have type A, B and cancelling type C errors we obtain the significance of a projection of the data set \(D_{2}\) to data set \(D_{1}\) determined by the following \(\chi ^2\) definition:
$$\begin{aligned} \tilde{\chi }^2_{21}= & {} \sum _{j=1}^{n_{21}} \frac{ (d_{1}(j) +\epsilon _{b,1} e_{B,1}(j) - d_{21}(j) - \epsilon _{b,2 1} e_{B,21}(j) )^2 }{\tilde{e}_{A,1}^2(j) + \tilde{e}_{A,21}^2(j)} \nonumber \\&+ \epsilon _{b,1}^2 + \epsilon _{b,21}^2 . \end{aligned}$$
(A.1)
In this equation, \(\tilde{e}_{A,1}(j)\) is the type A uncertainty of the data point j of the data set \(D_{1}\) in the united acceptance, while \(\tilde{e}_{A,21}(j)\) is the same for the \(D_{21}\) data set obtained from the \(D_2\) dataset by interpolation to point j of dataset \(D_1\) (Fig. 15). Both uncertainties are scaled by a multiplicative factor such that the fractional uncertainty is unchanged under multiplication by a point-to-point varying factor:
$$\begin{aligned} \tilde{e}_{A,1}(j)= & {} e_{A,1}(j) \left( \frac{d_{1}(j) + \epsilon _{b,1} e_{B,1}(j)}{d_{1}(j)}\right) , \, \end{aligned}$$
(A.2)
$$\begin{aligned} \tilde{e}_{A,21}(j)= & {} e_{A,21}(j) \left( \frac{d_{21}(j) +\epsilon _{b,21} e_{B,21}(j)}{d_{21}(j)}\right) . \end{aligned}$$
(A.3)
In these equations, \(\epsilon _{b,1}\) and \(\epsilon _{b,21}\) stand for the overall correlation coefficient of the j-dependent, point-to-point correlated type B \(D_{2}\) to the measured values in data set \(D_{1}\), \(e_{B,21}(j)\) of the projected data set \(D_{21}\). Note that \(\epsilon _{b,1}\) and \(\epsilon _{b,21}\) are independent of the point j, while the B-type errors have a point-to-point changing values \(e_{B,1}(j)\) and \(e_{B,21}(j)\) in both \(D_1\) and in the projected dataset in \(D_{21}\). For our comparison of H(x) scaling functions, where the absolute normalization and type C errors cancel, we have \(\epsilon _{c,1} = \epsilon _{c,2} = 0\), so have not indicated these terms for the sake of simplicity. For the sake of clarity and to demonstrate the importance of scaling out these type C errors, we have also included Fig. 16. That plot indicates that if the overall correlated, type C errors are added (incorrectly) in quadrature with the point-to-point fluctuating type A errors, the significance of the Odderon signal is decreased from 6.26 \(\sigma \) to 3.64 \(\sigma \).
We have utilized the scaled variance method of ROOT to include the horizontal errors, adding in quadrature to the type A errors also the type A error coming form the type A uncertainty of x, denoted as \(\delta _A x\). Similarly, we have added in quadrature to the type B error the type B uncertainty of x, denoted by \(\delta _B x\). Using a notation where M may stand for any of A or B, the errors are given as
$$\begin{aligned} e_{M,1}^2(j)= & {} \sigma _{M,1}^2(j) + [d^{\prime }_1(j) \delta _{M,1} x(j) ]^2 , \end{aligned}$$
(A.4)
$$\begin{aligned} e_{M,21}^2(j)= & {} \sigma _{M,21}^2(j) + [d^{\prime }_{21}(j) \delta _{M,21} x(j) ]^2 , \end{aligned}$$
(A.5)
where \(\sigma _{M}(j)\) indicates the type \(M \in \left\{ A, B\right\} \) error of the value of the vertical error on data point j, and it is added in quadrature to \(d^{\prime }(j) \delta _M x(j)\), the corresponding vertical error that is associated with the same uncertainty of type M originating from the measurement error on the horizontal axis x in Eq. (A.4). The notations \( d^{\prime }_1(j)\) and \( d^{\prime }_{21}(j)\) stand for the numerical derivative of the data points at point j in the datasets \(D_1\) and \(D_{21}\), respectively.
The errors on the projected data set (\(D_{21}\)) are also obtained by a linear-exponential interpolation between the projections of data set 2 (\(D_2\)) to data set 1 (\(D_1\)). Their type A and type B errors, indicated by \(e_{A,21}(j)\) and \( e_{B,21}(j)\) are also added in quadrature with the other A or B type of errors. These errors on the interpolated and on the measured values of (x, H(x)) through Eqs. (A.4) and (A.5) provided our most stringent significance estimate for the Odderon effects. We have cross-checked that several variations on the \(\chi ^2\) definition, for example the frequently adopted negligence of the horizontal errors and their contribution to the vertical errors through the scaled variance method, or perturbing the central values or the errors of the elastic cross-sections within the allowed limits, may only increase the significance reported in this Appendix.
We have evaluated Eq. (63) as a function of \(\epsilon _{b,1}\) and \(\epsilon _{b,21}\). However, for the critical test of the projection of the \(\sqrt{s} = 7.0\) TeV TOTEM data on H(x) to that of D0 at \(\sqrt{s} = 1.96\) TeV, we found that D0 did not publish any error on \(-t\) and cross-checked that the D0 value on B contains only type A, uncorrelated statistical and systematic errors only. We also noticed that there are no published type-B errors on the published differential cross-section data of D0 [8]. Hence for the D0 dataset, all the type B errors are zero and as a consequence, we have fixed the correlation coefficient of type B errors to zero for the \(\sqrt{s} = 1.96\) TeV D0 dataset.
Table 6 Summary table of the elastic cross-sections \(\sigma _\mathrm{el}\), the nuclear slope parameters B, with references. We have indexed with superscript A the type A, point-to-point fluctuating systematic and statistical errors, that can be added in quadrature, while type B errors (point-to-point changing, fully correlated systematic errors) are indicated with superscript B and type C errors (overall correlated, but \(-t\) independent errors) are indicated with superscript C. Note that the value and the type A error of the elastic cross-section \(\sigma _\mathrm{el}\) at \(\sqrt{s} = 1.96\) TeV [*] is obtained from a low \(-t\) exponential fit to data of Ref. [8], while the type C error is directly taken from the publication [8] Table 7 Summary table of the significant Odderon signal in the one-way comparison of the H(x) scaling functions of proton–proton collisions at \(\sqrt{s} = 7\) TeV, as measured by the TOTEM experiment at CERN LHC [28, 74], and proton–antiproton elastic collisions at \(\sqrt{s} = 1.96\) TeV as measured by the D0 experiment at Tevatron [8]. This table indicates that the Odderon signal is observed in this comparison with at least a 6.26 \(\sigma \) significance, corresponding to an Odderon discovery Let us now denote by subscript 21 the projection of the H(x) scaling function at \(\sqrt{s} = 7\) TeV measured by TOTEM for pp reaction to the D0 dataset on \(p\bar{p}\) elastic scattering at \(\sqrt{s} = 1.96\) TeV. We found a minimum for \(\epsilon _{b,21}\equiv \epsilon _{b, \, \mathrm{\tiny 7\, TeV}}\) within the \(-1 \le \epsilon _{b,21} \le 1\) domain, with the best value of \(\epsilon _{b, 7~TeV}\) and the lowest value of \(\chi ^2 \equiv \tilde{\chi }^2_{21}\) of Eq. (A.1) indicated in Fig. 15. Table 6 summarizes the input values and the appropriate references to the utilized elastic cross-section \(\sigma _\mathrm{el}\) and the nuclear slope B(s). The final and most stringent result of this cross check, corresponding to the lowest values of significance for the Odderon observation is summarized in Table 7. We found that the significance of the Odderon observation in the 7 TeV \(\rightarrow \) 1.96 TeV projection is at least \(6.26\sigma \), corresponding to a \(\chi ^2/\mathrm{NDF} = 80.1 / 17\) and CL of not larger than \(3.7 \times 10^{-8}\)%. We notice that all variations of the procedure may only increase this significance. We conclude, that the probability of the Odderon observation in this t-channel mode is statistically significant, at least \(P = 1-\mathrm{CL} = 0.99999999963\), if the H(x) scaling is valid at \(\sqrt{s} = 1.96\) TeV in the range of \(0 < x \le 20.2\) . As Fig. 13 indicates that model dependently this range might be smaller, only \(0 < x \le 15.1 = x_{max}(s_1)\) for \(\sqrt{s_1} = 1.96\) TeV, we have also performed several x-range stability studies in Appendix E.
From the experimental side, a very strong argument to support for the domain of validity of the H(x) scaling can be obtained from the observation that the bump/dip ratio is s-independent, if the H(x) scaling holds up to the bump position, \(x_{bump}\). If the H(x) scaling is valid for \(x_{max}(s) \ge x_{bump} = 13\), we find that
$$\begin{aligned} \frac{\frac{d\sigma (s)}{dt}|_{bump} }{\frac{d\sigma (s)}{dt}|_{dip}} =\frac{ H(x_{bump})}{H(x_{dip})} = \mathrm{const}(s) . \end{aligned}$$
(A.6)
Recent TOTEM data indicate that this ratio is, within the energy range available for TOTEM, and within experimental errors, is indeed independent of s [80] and a diffractive minimum-maximum with an approximately s-dependent ratio of \(R = 1.7 \pm 0.2\), \(1.7 \pm 0.1\) at \(\sqrt{s} = 2.76\) and 7.0 TeV, respectively. Apparently, an s-independent R(s) is within errors a permanent feature of elastic pp scattering at these energies [80]. This experimental result supports, that the H(x) scaling holds up to at least \(x = x_{bump} = 13\) at energies close to the \(2.76 \le \sqrt{s} \le 7\) TeV range. Indeed, at 13 TeV, \(R = 1.77 \pm 0.01\) [80], so we expect a similar value of R(s) at \(\sqrt{s} = 1.96\) TeV, too, as this scale is logarithmically close to 2.76 TeV. Due to the observation of a diffractive cone in pp collisions both at 2.76 and 7.0 TeV [80], we expect a similar diffractive cone in pp elastic scattering at \(\sqrt{s} = 1.96\) TeV, too, which corresponds to \(H(x) \simeq \exp (-x)\). Thus the available experimental data also suggests that the expected domain of validity of H(x) scaling at \(\sqrt{s} = 1.96\) TeV includes the \(7.0 < x \le 13.5\) domain. These experimental observations combined with our model independent x-range stability studies in Appendix E indicate, that in this domain the significance of the Odderon observation is at least 5.0 \(\sigma \). This \(7.0 < x \le 13.5\) interval physically begins with the “swing”, where the differential cross-section of pp elastic scattering starts to bend below the exponential shape and ends just after the diffractive maximum or “bump”, located at \(x_{bump} \approx 13.0\) . For one of the theoretically expected domains of validity, the limited \(0 < x \le 15.1\) range, we find that our method provides an Odderon significance of at least 5.3 \(\sigma \), as detailed in Appendix E. Another theoretical argument is shown in Fig. 27, where the H(x) scaling limit of the ReBB model of Ref. [42] is shown to describe the experimental data in the whole D0 acceptance, corresponding to the domain of validity of \(0 < x \le 20.2\) range and a 6.25 \(\sigma \) Odderon effect. This is one of the indications of the robustness of our results.
For the sake of completeness, we also evaluate the asymmetry parameter from the H(x) scaling functions to demonstrate the level of agreement between the experimental data and our full model calculations and also to have a better insight on the magnitude of the scaling violations. In order to measure the size of the Odderon effect, let us define the following asymmetry measure or asymmetry parameter:
$$\begin{aligned} A(x|p\bar{p},s_1| pp,s_2)= & {} \frac{H(x| p\bar{p},s_1) - H(x| pp,s_2)}{H(x| p\bar{p},s_1) + H(x| pp,s_2)}, \end{aligned}$$
(A.7)
$$\begin{aligned} A(x|pp,s_1| pp,s_2)= & {} \frac{H(x| pp,s_1) - H(x| pp,s_2)}{H(x| pp,s_1) + H(x| pp,s_2)}. \end{aligned}$$
(A.8)
If the crossing-odd component of elastic scattering amplitude vanishes at high energies, it implies that \(H(x| p\bar{p},s_1) \) \(=\) \( H(x| pp,s_1)\). In this case, we find that \(A(x|p\bar{p},s_1| pp,s_1) \) \( \equiv \) \( 0\) for \(\sqrt{s_1} \ge 1\) TeV. Additionally, if H(x) scaling holds for elastic pp scattering at high energies, then \(H(x| pp,s_1) \) \( = \) \( H(x| pp,s_2)\), hence \(A(x|pp,s_1| pp,s_2)\) \( = \) 0. Indeed, as Fig. 17 indicates, this asymmetry parameter vanishes within experimental errors. There are small systematic deviations that are well described by theoretical model calculations based on the Real Extended Bialas–Bzdak model of Ref. [42]. The agreement between the small asymmetries and the theoretical calculations indicates that the scaling violations are under precise theoretical control.
On the other hand, if the H(x) scaling holds for elastic pp scattering and the crossing-odd component of elastic scattering amplitude is not vanishing at high energies, then \(H(x| p\bar{p},s_1) \ne H(x| pp,s_1) \), hence \(A(x|p\bar{p},s_1| pp,s_2) \ne 0\) for \(\sqrt{s_1}, \sqrt{s_2} \ge 1\) TeV. Indeed, as shown in Fig. 18, this asymmetry parameter is significantly different from zero. Similarly to Fig. 17, the solid line in Fig. 18 is the result of a model calculation based on the full version of the Real Extended Bialas–Bzdak model of Ref. [42]. The solid line describes the experimental data well, within errors, which indicates that the scaling violations are well under theoretical control in this calculation.
Appendix B: Pomeron and Odderon at the TeV energy scale: model independent properties
In this Appendix we discuss some model independent properties of the Pomeron and Odderon that utilize their correspondence with the crossing-even and crossing-odd components of the elastic scattering amplitude at the TeV energies. In the TeV energy range, we identify the crossing-even and crossing-odd components with the Pomeron and the Odderon amplitude, given that the Reggeon contributions in this energy range are generally expected to be negligibly small, as confirmed also by explicit calculations for example in Ref. [29]. These results are obtained by a straightforward generalization, from a model dependent to a model independent class, of the Pomeron and Odderon properties obtained in Ref. [42] for the Real Extended Bialas–Bzdak model of Ref. [30].
The proton–proton (pp) as well as the proton–antiproton (\(p\bar{p}\)) elastic scattering amplitudes can always be written as the difference as well as the sum of the C-even and C-odd amplitudes, as detailed in Eqs. (14) and (15). These amplitudes depend on Mandelstam variables \(s = (p_1 + p_2)^2\) and \(t = (p_1 - p_3)^2\), but we suppress these dependencies in our notation throughout this Appendix B.
With the help of the pp and the \(p\bar{p}\) scattering amplitudes, the crossing even and the crossing odd components of the elastic scattering amplitude can be expressed as Eqs. (18) and (19). The pp and the \(p\bar{p}\) scattering amplitudes can be evaluated based for example on R. J. Glauber’s theory of multiple diffractive scattering [82,83,84], and various models of the structures inside the protons. However, in this Appendix we focus on the model-independent properties so we do not specify a model yet. Model dependent calculations are subject of Appendix C and Appendix D.
The differential elastic cross section is defined by Eq. (2). The elastic, the total and the inelastic cross sections are given by Eqs. (1), (5) and (6), respectively. The real to imaginary ratio is given by Eq. (8), and the nuclear slope parameter is given by the model independent relation of Eq. (4).
These measurable quantities are utilized to characterize experimentally the (s, t)-dependent elastic scattering amplitudes, \(T_\mathrm{el}(s,t)\) discussed above. These quantities can be evaluated for a specific process like elastic pp or elastic \(p\bar{p}\) collisions. If we can evaluate the elastic scattering amplitude for both pp and \(p\bar{p}\) scattering in the TeV energy range, that straightforwardly yields also the (s, t)-dependent elastic scattering amplitude also for the Pomeron and the Odderon exchange.
Let us focus on the model independent properties of the crossing-even Pomeron (\(\mathbb P\)) and for the crossing-odd Odderon (\(\mathbb O\)) exchange.
The scattering amplitude \(T_\mathrm{el}(s,t)\), for spin independent processes, is related to a Fourier–Bessel transform of the impact parameter dependent elastic scattering amplitudes \(t_\mathrm{el}(s,b)\):
$$\begin{aligned} T(s,t) = 2\pi \int \limits _0^{\infty }{J_{0}\left( \varDelta \cdot b\right) t_\mathrm{el}(s,b)b\, \mathrm{d}b}. \end{aligned}$$
(B.1)
Here, \(b=|{\mathbf {b}}|\) is the modulus of the impact parameter vector \(\mathbf {b}\), \(\varDelta \) stands for the modulus of the transverse momentum vector and \(J_{0}\) is the is the zeroth order Bessel-function of the first kind. In the considered high energy limit, \(\sqrt{s} \gg m_p\) and in this case the modulus of the transverse momentum transfer is calculated as \(\varDelta (t) \simeq \sqrt{-t}\).
This impact parameter dependent amplitude is constrained by the unitarity of the scattering matrix S,
$$\begin{aligned} S S^{\dagger } = I , \end{aligned}$$
(B.2)
where I is the identity matrix. Its decomposition is \(S = I + iT\), where the matrix T is the transition matrix. In terms of T, unitarity leads to the relation
$$\begin{aligned} T - T^{\dagger } = i T T^{\dagger } , \end{aligned}$$
(B.3)
which can be rewritten in terms of the impact parameter or b dependent amplitude \(t_\mathrm{el}(s,b)\) as
$$\begin{aligned} 2 \, {\mathcal Im} \, t_\mathrm{el}(s, b) = |t_\mathrm{el}(s, b)|^2 + \tilde{\sigma }_\mathrm{inel}(s,b) , \end{aligned}$$
(B.4)
where \(\tilde{\sigma }_\mathrm{inel}(s,b)\) is a non-negative contribution. If the process is completely elastic, this quantity is zero, and the elastic amplitude lies on the Argand circle, while in case there are also inelastic collisions present, the elastic amplitude lies within the Argand circle [44]. It can be equivalently expressed from the above unitarity relation as
$$\begin{aligned} \tilde{\sigma }_\mathrm{inel}(s,b) = 1 - ({\mathcal Re} \, t_\mathrm{el}(s, b))^2 - ({\mathcal Im} \, t_\mathrm{el}(s, b) - 1 )^2 . \end{aligned}$$
(B.5)
It follows that
$$\begin{aligned} 0\le \tilde{\sigma }_\mathrm{inel}(s,b)\le & {} 1 \, \qquad 0\le | 1 + i t_{el}(s,b)|^2 \, \le \, 1 \end{aligned}$$
(B.6)
as a consequence of unitarity. Thus probabilistic interpretation can be given only to the inelastic scattering and to the sum of the elastic scattering amplitude and the amplitude for propagation without interaction. This is why elastic scattering is a genuine quantum or wave-like process, and this is also the reason why elastic scattering, in contrast to inelastic scattering, has no classical interpretation. Thus \(\tilde{\sigma }_\mathrm{inel}(s,b) \) is interpreted as the impact parameter and s-dependent probability of inelastic scattering.
The elastic scattering amplitude has the unitary form of Eq. (12) as the function of the center-of-mass energy squared s and impact parameter b. This form is expressed with the help of the opacity (or, eikonal) function, denoted by \(\Omega (s,b)\), which in the general case is a complex valued function.
The imaginary part of \(\Omega (s,b)\) determines the real part of the scattering amplitude, while the real part of \(\Omega (s,b)\) determines the dominant, imaginary part of the scattering amplitude. Let us introduce \(\tilde{\sigma }_\mathrm{inel}(s,b)\) with the help of the real part of the complex opacity function as follows:
$$\begin{aligned} \sqrt{1 - \tilde{\sigma }_\mathrm{inel}(s,b)}= & {} \exp \left( - {\mathcal Re} \, \Omega (s,b)\right) . \end{aligned}$$
(B.7)
This leads to Eq. (41).
The shadow profile function is defined with the help of the opacity function, which yields
$$\begin{aligned} P(s,b) = 1 - |\exp (-\Omega )|^2 = \tilde{\sigma }_\mathrm{inel}(s,b) . \end{aligned}$$
(B.8)
This relation indicates that \(\tilde{\sigma }_\mathrm{inel}(s,b)\) is interpreted as the probability of inelastic collisions at a given value of the squared center of mass energy s and impact parameter b. The inelastic profile function can in general be evaluated with the help of Glauber’s multiple diffraction theory [83], using various model assumptions, for example the assumptions of Ref. [30].
The impact parameter dependent elastic scattering amplitudes for elastic pp and \(p\bar{p}\) scatterings are given in terms of the complex opacity or eikonal functions \(\Omega (s,b)\). The defining relations are
$$\begin{aligned} t_\mathrm{el}^{pp}(s,b)= & {} i \, (1 - \exp (-\Omega ^{pp}(s,b) ) , \ \end{aligned}$$
(B.9)
$$\begin{aligned} t_\mathrm{el}^{p\bar{p}}(s,b)= & {} i \, (1 - \exp (-\Omega ^{p\bar{p}}(s,b) ) . \end{aligned}$$
(B.10)
The explicit expressions for the C-even and the C-odd components of the impact parameter dependent elastic scattering amplitude are detailed below. These relations are explicit consequences of unitarity and do not depend on model details. However, it is important to note that at the TeV energy range in \(\sqrt{s}\), the C-even exchange corresponds to the Pomeron exchange while the C-odd amplitude corresponds to the Odderon exchange, while the corrections due to the exchange of Reggeons or hadronic resonances is smaller than the experimental errors as detailed recently in Ref. [29]. Thus, in the TeV energy range, the S-matrix unitarity and the dominance of the Pomeron and Odderon amplitudes constrains their form as follows:
$$\begin{aligned} t_\mathrm{el}^{\mathbb P}(s,b)= & {} i \, \left( 1 - \frac{1}{2} (\exp (-\Omega ^{pp}(s,b) ) + \exp (-\Omega ^{p\bar{p}}(s,b) ))\right) , \\ t_\mathrm{el}^{\mathbb O}(s,b)= & {} i \, \frac{1}{2} (\exp (-\Omega ^{pp}(s,b) ) - \exp (-\Omega ^{p\bar{p}}(s,b) )) . \end{aligned}$$
It is obvious to note that the Pomeron amplitude given above is crossing-even, while the Odderon amplitude is crossing-odd: \(C t_\mathrm{el}^{\mathbb P}(s,b) = t_\mathrm{el}^{\mathbb P}(s,b)\) and \(C t_\mathrm{el}^{\mathbb O}(s,b) = - t_\mathrm{el}^{\mathbb O}(s,b)\) .
Under two assumptions, these relations can be further simplified for a broad class of models as follows.
- (i):
-
If the imaginary part of the complex opacity function in elastic pp and \(p\bar{p}\) collisions has the same b-dependent factor, denoted here in general by \(\Sigma (s,b)\), and
- (ii):
-
if these imaginary parts of the complex opacity function also include an s-dependent but b independent prefactor that is a different function in elastic pp and in \(p\bar{p}\) collisions,
then these assumptions correspond mathematically to the following relations:
$$\begin{aligned} {\mathcal Im} \Omega ^{pp}(s,b)= & {} - \alpha ^{pp}(s) \Sigma (s,b) , \end{aligned}$$
(B.11)
$$\begin{aligned} {\mathcal Im} \Omega ^{p\bar{p}}(s,b)= & {} - \alpha ^{p\bar{p}}(s) \Sigma (s,b), \end{aligned}$$
(B.12)
yielding the following simple expressions for the impact parameter dependent elastic pp and \(p\bar{p}\) scattering amplitudes
$$\begin{aligned} t_\mathrm{el}^{pp}(s,b) = i\left( 1-e^{i\, \alpha ^{pp}(s) \, \Sigma (s,b)} \, \sqrt{1-\tilde{\sigma }_\mathrm{in}(s,b)}\right) , \end{aligned}$$
(B.13)
$$\begin{aligned} t_\mathrm{el}^{p\bar{p}}(s,b) = i\left( 1-e^{i\, \alpha ^{p\bar{p}}(s) \, \Sigma (s,b)} \, \sqrt{1-\tilde{\sigma }_\mathrm{in}(s,b)}\right) . \end{aligned}$$
(B.14)
These relations can be equivalently rewritten for the Pomeron amplitude, using \(\tilde{\sigma }_\mathrm{in}\equiv \tilde{\sigma }_\mathrm{inel}(s,b)\) and \(\Sigma \equiv \Sigma (s,b)\) as shorthand notations while also suppressing the s-dependence of \(\alpha ^{pp}(s)\) and \(\alpha ^{p\bar{p}}(s)\):
$$\begin{aligned} {\mathcal Im }\, t_\mathrm{el}^{\mathbb P}(s,b)= & {} 1 - \sqrt{1-\tilde{\sigma }_\mathrm{in} }\\&\times \cos \left( \frac{\alpha ^{pp} + \alpha ^{p\bar{p}}}{2} \Sigma \right) \cos \left( \frac{\alpha ^{p\bar{p}} - \alpha ^{pp}}{2} \Sigma \right) , \\ {\mathcal Re}\, t_\mathrm{el}^{\mathbb P}(s,b)= & {} \sqrt{1-\tilde{\sigma }_\mathrm{in} } \\&\times \sin \left( \frac{\alpha ^{pp} + \alpha ^{p\bar{p}}}{2}\Sigma \right) \cos \left( \frac{\alpha ^{p\bar{p}} - \alpha ^{pp}}{2}\Sigma \right) . \end{aligned}$$
This form of the Pomeron amplitude is explicitly C-even, and satisfies unitarity. Thus, if the difference between the opacity parameters \(\alpha \) for pp and \(p\bar{p}\) elastic collisions is small, the Pomeron is predominantly imaginary, with a small real part that is proportional to \(\sin \left( \frac{\alpha ^{pp} + \alpha ^{p\bar{p}}}{2}\tilde{\Sigma }\right) \). Similarly, for the Odderon we have, under the conditions i) and ii), real and imaginary parts of the amplitude
$$\begin{aligned} {\mathcal Re }\, t_\mathrm{el}^{\mathbb O}(s,b)= & {} \sqrt{1-\tilde{\sigma }_\mathrm{in} } \nonumber \\&\times \sin \left( \frac{\alpha ^{p\bar{p}} - \alpha ^{pp}}{2}\Sigma \right) \cos \left( \frac{\alpha ^{p\bar{p}} + \alpha ^{pp}}{2}\Sigma \right) , \nonumber \\ \end{aligned}$$
(B.15)
$$\begin{aligned} {\mathcal Im}\, t_\mathrm{el}^{\mathbb O}(s,b)= & {} \sqrt{1-\tilde{\sigma }_\mathrm{in} } \nonumber \\&\times \sin \left( \frac{\alpha ^{p\bar{p}} - \alpha ^{pp}}{2}\Sigma \right) \sin \left( \frac{\alpha ^{pp} + \alpha ^{p\bar{p}}}{2}\Sigma \right) .\nonumber \\ \end{aligned}$$
(B.16)
This form of the Odderon amplitude is explicitly C-odd and satisfies unitarity, if the above two assumptions (i) and (ii) are satisfied, without any further reference to the details of the model. In this class of models, if the difference between the opacity parameters \(\alpha \) for pp and \(p\bar{p}\) elastic collisions becomes vanishingly small, both the real and the imaginary parts of the Odderon amplitude vanish, as they are both proportional to \(\sin \left( \frac{\alpha ^{p\bar{p}} - \alpha ^{pp}}{2}\Sigma \right) \). If this term is non-vanishing, but \((\alpha ^{p\bar{p}} + \alpha ^{pp}) \Sigma \) remains small, the above Odderon amplitude remains predominantly real, with a small, linear in \((\alpha ^{p\bar{p}}+\alpha ^{pp}) \Sigma \) at the leading order, imaginary part.
Appendix C: Emergence of the H(x) scaling from the real extended Bialas–Bzdak model
With the help of the ReBB model of Ref. [30], we have recently described the pp and \(p\bar{p}\) differential cross-sections in a limited kinematic range of \(0.546 \le \sqrt{s} \le 8\) TeV and \( 0.372 \le -t \le 1.2\) GeV\(^2\), in a statistically acceptable manner with CL \(\ge \) 0.1%, as detailed in Ref. [42].
It is important to realize that within the ReBB model, the pp elastic scattering dependence on s comes only through four energy-dependent quantities, as specified recently in Ref. [42]. Let us recapitulate the general formulation, for the sake of clarity, denoting the s-dependent quantities as \(R_q^{pp}(s)\), \(R_d^{pp}(s)\), \(R_{qd}^{pp}(s)\) and \(\alpha ^{pp}(s)\):
$$\begin{aligned} T_\mathrm{el}^{pp}(s,t) = F(R_q^{pp}(s), R_d^{pp}(s), R_{qd}^{pp}(s),\alpha ^{pp}(s); t) . \end{aligned}$$
(C.1)
Similarly, the scattering amplitude of the elastic \(p\bar{p}\) scattering is found in terms of four energy-dependent quantities, that we denote here for the sake of clarity as \(R_q^{p\bar{p}}(s)\), \(R_d^{p\bar{p}}(s)\), \(R_{qd}^{p\bar{p}}(s)\) and \(\alpha ^{p\bar{p}}(s)\):
$$\begin{aligned} T_\mathrm{el}^{p\bar{p}}(s,t) = F(R_q^{p\bar{p}}(s), R_d^{p\bar{p}}(s), R_{qd}^{p\bar{p}}(s), \alpha ^{p\bar{p}}(s); t) . \end{aligned}$$
(C.2)
Here, F stands for a symbolic short-hand notation for a function, that indicates how the left hand side of the pp and \(p\bar{p}\) scattering amplitude depends on s through their s-dependent quantities. Among those, \(R_q\), \(R_d\), and \(R_{qd}\) correspond to the Gaussian sizes of the constituent quarks, diquarks and their separation in the scattering (anti)protons. These scales are physically expected to be the same in pp and in \(p\bar{p}\) elastic collisions.
Indeed, the trends of \(R_q(s)\), \(R_d(s)\) and \(R_{qd}(s)\) follow, within errors, the same excitation functions in both pp and \(p\bar{p}\) collisions, as indicated on panels a, b and c of Fig. 6 of Ref. [42]. Due to this reason, let us denoted these – in principle, different – scale parameters with the same symbols in the body of the manuscript, as they are found to be independent of the type of the collision, i.e.
$$\begin{aligned} R_q(s)\equiv & {} R_q^{pp}(s) \, = \, R_q^{p\bar{p}}(s) , \end{aligned}$$
(C.3)
$$\begin{aligned} R_d(s)\equiv & {} R_d^{pp}(s) \, = \, R_d^{p\bar{p}}(s) , \end{aligned}$$
(C.4)
$$\begin{aligned} R_{qd}(s)\equiv & {} R_{qd}^{pp}(s) \, = \, R_{qd}^{p\bar{p}}(s) . \end{aligned}$$
(C.5)
On the other hand, the opacity or dip parameter \(\alpha (s)\) is different for elastic pp and \(p\bar{p}\) reactions: if they were the same too, then the scattering amplitude for pp and \(p\bar{p}\) reactions were the same, and correspondingly the differential cross-sections were the same in these reactions, while the experimental results indicate that they are qualitatively different [42]. Hence,
$$\begin{aligned} \alpha ^{pp}(s)\ne & {} \alpha ^{p\bar{p}}(s) . \end{aligned}$$
(C.6)
The ReBB model [30] provides a statistically acceptable description of the elastic scattering amplitude, both for pp and \(p\bar{p}\) elastic scattering, in the kinematic range that extends to at least \( 0.372 \le -t \le 1.2\) GeV\(^2\) and \(0.546 \le \sqrt{s} \le 8\) TeV. For the sake of clarity, let us also note that the s-dependence of the Pomeron and Odderon components of the scattering amplitude thus happens through the s-dependence of five parameters only. Based on Ref. [42], we write
$$\begin{aligned} T_\mathrm{el}^{\mathbb P}(s,t)= & {} G(R_q(s), R_d(s), R_{qd}(s),\alpha ^{pp}(s), \alpha ^{p\bar{p}}(s);t) , \nonumber \\&\end{aligned}$$
(C.7)
$$\begin{aligned} T_\mathrm{el}^{\mathbb O}(s,t)= & {} H(R_q(s), R_d(s), R_{qd}(s),\alpha ^{pp}(s), \alpha ^{p\bar{p}}(s);t) . \nonumber \\&\end{aligned}$$
(C.8)
Here, G and H are just symbolic short-hand notations that summarize how the left hand sides of the above equations depend on s through their s-dependent parameters.
As detailed in Ref. [42], within the ReBB model there is a deep connection between the \(t=0\) and the dip region. This supports the findings that the recently observed decrease in \(\rho _0(s)\) around \(\sqrt{s}=\)13 TeV, the dip-bump structure in pp scattering and its absence in \(p\bar{p}\) scattering are both the consequences of the Odderon contribution. In the ReBB model, this Odderon contribution is encoded in the difference between \(\alpha ^{pp}(s)\) and \(\alpha ^{p\bar{p}}(s)\). This conclusion is supported also by the detailed calculations of the ratio of the modulus-squared Odderon to Pomeron scattering amplitudes. Thus, if \(\rho _0^{pp}(s)\ne \rho _0^{p\bar{p}}(s)\), within the ReBB model it follows that \(\alpha ^{pp}(s)\ne \alpha ^{p\bar{p}}(s)\) or, equivalently, \(t_\mathrm{el}^{\mathbb O}(s,b)\ne 0\) in the TeV region.
Within the framework of the ReBB model, we have proved in Ref. [42] an interesting Odderon theorem. The weaker, original form of this theorem was formulated above in Sect. 3 as follows:
Theorem 1
If the pp differential cross sections differ from that of \(p\bar{p}\) scattering at the same value of s in a TeV energy domain, then the Odderon contribution to the scattering amplitude cannot be equal to zero, i.e.
$$\begin{aligned} \frac{d\sigma ^{pp}}{dt} \ne \frac{d\sigma ^{p\bar{p}}}{dt},\quad \mathrm{for}\ \sqrt{s}\ge 1 \,\mathrm{~TeV} \implies T_\mathrm{el}^O(s,t) \ne 0 . \end{aligned}$$
(C.9)
This theorem is model-independently true as it depends only on the general structure of the theory of elastic scattering. Within the ReBB model, this theorem has been sharpened in Ref. [42] as follows:
Theorem 2
In the framework of the unitary ReBB model, the elastic pp differential cross sections differ from that of elastic \(p\bar{p}\) scattering at the same value of s in a TeV energy domain, if and only if the Odderon contribution to the scattering amplitude is not equal to zero. This happens if and only if \(\alpha ^{pp}(s) \ne \alpha ^{p\bar{p}}(s)\) and as a consequence, if and only if \(\rho _0^{pp} \ne \rho _0^{p\bar{p}}{:}\)
$$\begin{aligned} \frac{d\sigma ^{pp}}{dt}\ne & {} \frac{d\sigma ^{p\bar{p}}}{dt} \iff T_\mathrm{el}^O(s,t) \ne 0 \, \\\iff & {} \rho _0^{pp}(s) \ne \rho _0^{p\bar{p}}(s) \\\iff & {} \alpha ^{pp}(s) \ne \alpha ^{p\bar{p}}(s) \\&\,\,\, { \mathrm for}\,\, \sqrt{s}\ge 1 \,\, \mathrm{~TeV} . \end{aligned}$$
In this work, we extend these theorems to the emergence of H(x) scaling within the ReBB model, as follows:
Theorem 3
In the framework of the unitary ReBB model, the elastic pp differential cross sections obey a H(x) scaling in a certain kinematic region, if and only if in that region the opacity parameter is approximately energy independent, \(\alpha ^{pp}(s) \approx \mathrm{const}\) and the geometrical scale parameters evolve with the same s-dependent, but radius parameter independent factor b(s). Thus, the conditions of validity of H(x) scaling in elastic pp collisions, within the framework of the ReBB model are the simultaneous validity of the following four equations :
$$\begin{aligned} R_q(s)= & {} b(s) R_q(s_0) , \end{aligned}$$
(C.10)
$$\begin{aligned} R_d(s)= & {} b(s) R_d(s_0) , \end{aligned}$$
(C.11)
$$\begin{aligned} R_{qd}(s)= & {} b(s) R_{qd}(s_0) , \end{aligned}$$
(C.12)
$$\begin{aligned} \alpha ^{pp}(s)= & {} \alpha ^{pp}(s_0) , \end{aligned}$$
(C.13)
where b(s) is the same function of s for each of \(R_q,\) \(R_d\) and \(R_{qd}\).
The key point of the proof of Theorem 3 is that B(s) is related, for an analytic amplitude, to the variance of the scattering amplitude in the impact parameter space. If this variance depends on the evolution of the scale parameters \(R_q(s)\), \(R_d(s)\) and \(R_{qd}(s)\) only, as \(\alpha (s)\) is within the experimental errors a constant, and if these scale parameters all evolve with the same s-dependent factor, then the nuclear slope parameter must also scale as
$$\begin{aligned} B(s) = b^2(s) B(s_0) . \end{aligned}$$
(C.14)
At the same time, the elastic and the differential cross-sections must also scale as
$$\begin{aligned} \sigma _\mathrm{el}(s)= & {} b^2(s) \sigma _\mathrm{el}(s_0) , \end{aligned}$$
(C.15)
$$\begin{aligned} \frac{d\sigma }{dt}(s,t)= & {} b^2(s) \frac{d\sigma }{dt}\left( s_0, t_0 = \frac{t}{b(s)^2}\right) . \end{aligned}$$
(C.16)
Hence, in such an s and \(x = - t B\), range the H(x) scaling, defined by Eq. (54) must hold: \(H(x,s) = H(x, s_0)\) and vice versa.
We have cross-checked the scaling properties of the ReBB model at both ISR and LHC energies. At the ISR energies of 23.5 \(\le \sqrt{s} \le 62.5\) GeV, \(\rho _0 \equiv \rho _0(s)\) is not a constant as reviewed recently in [2], so our Theorem 3 suggests, that the \(H(x,s) \equiv H(x,s_0) \) scaling cannot be interpreted in terms of the ReBB model, and in particular we expect that R(s), the bump-to-dip ratio decreases with increasing values of s as ref. [2] suggests that \(\rho _0(s)\) is an increasing function of s at ISR energies. Thus, the approximate H(x) scaling, indicated in Fig. 1 actually is expected to be violated in the dip regions and also perhaps also in the tail region at ISR energies. A more detailed investigation of the scaling violations at ISR energies goes well beyond the scope of this manuscript.
At LHC energies, let us summarize the main results of the ReBB model studies, as given in Ref. [42] in greater details. If we do not utilize the validity of the H(x) scaling at the LHC energies, we obtain Figs. 16, 17 and 18 of Ref. [42]. In these figures a yellow band indicates the uncertainty of the model prediction, without the assumption of the validity of the H(x) scaling. Fig. 16 of Ref. [42] indicates that the extrapolation without the H(x) scaling is rather uncertain in the region of the diffractive shoulder as compared to \(\sqrt{s} = 1.96\) TeV \(p\bar{p}\) elastic scattering data and correspondingly, no significant difference is observed in this model comparison at this energy. However, the model allows for the investigation of H(x) scaling violations and the extrapolation of \(p\bar{p}\) data to the LHC energies. As the \(p\bar{p}\) data do not obey a H(x) scaling, their extrapolation to the LHC energies without such a model is not possible: the H(x) scaling works only for pp but not for \(p \,\bar{p} \) collisions. Comparing the ReBB model extrapolations of \(p\bar{p}\) differential cross-sections with TOTEM data on pp differential cross-sections at \(\sqrt{s} = 2.76\) TeV, we obtained in Ref. [42] an Odderon effect with a significance of 7.12\(\sigma \), as indicated on Fig. 17 of Ref. [42]. Combining this value with the model dependent results at \(\sqrt{s} = 1.96\) TeV, the combined significance is hardly reduced, changes only to 7.08 \(\sigma \). On the other hand, if we extrapolate \(p\bar{p}\) data also up to \(\sqrt{s} = 7\) TeV, the significance increases further, to values greater than 10\(\sigma \). In practical terms, extrapolating \(p\bar{p}\) data theoretically up to 7 TeV, we obtain a certainty for the Odderon contribution. The quoted 6.26 \(\sigma \) model independent significance is thus a safe, model independent lower limit for the observation of a crossing-odd component of elastic pp and \(p\bar{p}\) scattering in the TeV energy range.
Appendix D: Model-dependent estimation of the range of validity and violations of H(x) scaling
In this Appendix, we summarize our model-dependent results on the estimated range of validity of H(x) scaling in elastic pp collisions. We find that this scaling might be extended in \(\sqrt{s}\) from 7 and 8 TeV at LHC down to \(\sqrt{s} = 200\) GeV at RHIC, as detailed below.
The \(R_q(s)\), \(R_d(s)\) and \(R_{qd}(s)\) parameters of the ReBB model were determined in Ref. [42] using both an affine linear and a quadratic dependence on \(\ln (s)\). This allowed us to test if the scale parameters of the ReBB model obey, within one standard deviation, the same energy dependence or not. For the reference point, we have chosen \(\sqrt{s_0} = 7\) TeV. Figure 19 indicates that such an affine linear scaled energy dependence of the ReBB model parameters \(R_q(s)\), \(R_d(s)\) and \(R_{qd}(s)\) by the values of the same parameters at 7 TeV suggests that the one \(\sigma \) systematic error-bands overlap down to \(\sqrt{s}=2436\) GeV. However, this result does not yet take into account the possible quadratic dependence of these parameters on \(\ln (s)\) and it also neglects the correlations between the model parameters. However, the validation of this linear in \(\ln (s)\) energy dependence of the b(s) factor of the H(x) scaling by an explicit calculation failed, indicating that a quadratic extrapolation in \(\ln (s)\) is apparently necessary.
Taking into account the possible quadratic in \(\ln (s)\) dependence of the excitation function of the ReBB model parameters \(R_q(s)\), \(R_d(s)\) and \(R_{qd}(s)\) pushes this limit further down to \(\sqrt{s}=500\) GeV, as demonstrated in Fig. 20. This plot utilizes the parameters of quadratic dependence in \(\ln (s)\) as indicated in Fig. 23 of Ref. [42], and collected in Table 3 of that manuscript, but without taking into account the correlations between these model parameters. When we tried to validate such a quadratic in \(\ln (s)\) but uncorrelated dependence of the b(s) parameter of the H(x) scaling, the validation plots did not result in acceptable confidence levels with CL \(\ge 0.1\)% for \(\sqrt{s} = 2.76\), 1.96 and 0.546 TeV. Hence we had to take into account the correlations between \(R_q\), \(R_d\) and \(R_{qd}\) together with their quadratic in \(\ln (s)\) behaviour as detailed below.
Our final estimate for the range of the validity of the H(x) scaling, in particular, the possible lowest value for the validity of this scaling is based on the quadratic in \(\ln (s)\) dependence of the model parameters \(R_q(s)\), \(R_d(s)\) and \(R_{qd}(s)\), taking also into account their correlations. This we have studied so that we determined these parameters at 5 different energies, at \(\sqrt{s} = 23\) GeV, as well as at 0.546, 1.96, 2.76 and 7 TeV, and fitted the resulting 5 points with a 3-parameter quadratic formula of \(R_i(s) = p_0 + p_1 \ln (s/s_0) + p_2 \ln ^2(s/s_0)\). This line is our best estimate for the quadratic energy dependence for these parameters. However, the parameters are correlated so we have repeated these fits by shifting up (or down) by one standard deviation each of these model parameters at each energy and fixed their values, while re-fitting all the other parameters of the ReBB model at the same energy to find their best estimate. This way we perturbed in two different directions four model parameters at five different energies and re-fitted each set with the quadratic in \(\ln (s/s_0)\) evolution, resulting in \(2\times 4\times 5 = 40\) curves around the central line. The area between these curves is our best estimate for the systematic error band of the energy evolution, that takes into account not only the errors of the ReBB model parameters but also the correlations between the ReBB model parameters at each energy.
The one \(\sigma \) systematic error band on the \(R_q(s)\) parameter, that takes into account both the quadratic in \(\ln (s)\) evolution and the correlations between the model parameters is presented as an orange band in Fig. 21. Similarly, the one \(\sigma \) systematic error band on the \(R_d(s)\) parameter, that takes into account both the quadratic in \(\ln (s)\) evolution and the correlations between the model parameters is presented as a darker orange band in Fig. 22. The same error band is shown for the \(R_{qd}(s)\) parameter in yellow band in Fig. 23. These error bands are actually overlapping and the region of their overlap determines the domain of validity of the H(x) scaling.
The overlaid one \(\sigma \) systematic error-bands of the scale parameters of the ReBB model are shown in Fig. 24. This figure indicates that from \(\sqrt{s} = 7\) TeV down to \(\sqrt{s} = 200\) GeV these one standard deviation error-bands overlap within one standard deviation. This implies, that one of the necessary conditions for the validity of the H(x) scaling in elastic pp collisions is satisfied in the kinematic range of \(0.2 \le \sqrt{s} \le 7.0\) TeV. The domain of validity of the ReBB model was limited in \(-t\) as well, to \(0.375 \le -t \le 1.2\) GeV\(^2\) as detailed in Ref. [42]. Thus, this model-dependent study cannot be applied at very low or very high values of \(-t\) and the additional condition for the domain of validity of the H(x) scaling, the constancy of the parameter \(\alpha ^{pp}(s)\) can be cross-checked if experimental data on elastic pp collisions are becoming available in the lower end of this energy range.
Let us first cross-check if indeed the H(x) scaling can be valid in elastic pp or \(p\bar{p} \) collisions in such a broad energy range, or not. We have demonstrated that this H(x) scaling is violated if we go with energy up to \(\sqrt{s} = 13\) TeV. This is easily understood within the framework of the ReBB model. Condition ii) indicates that one of the necessary condition for the H(x) scaling is the constancy, the approximately energy independence of the parameter \(\alpha ^{pp}(s)\). In Ref. [42] we have shown that within the ReBB model this corresponds to the energy independence of the real to imaginary ratio at \(t=0\), the parameter \(\rho _0(s)\). The TOTEM Collaboration recently demonstrated that at the top LHC energy of \(\sqrt{s} = 13\) TeV, the \(\rho _0\) parameter starts to decrease significantly [2]. This decrease increases the dip at these energies corresponding to Theorem 2 of Ref. [42] and thus the decrease of \(\rho _0(s)\) leads also to a violation of the H(x) scaling at the top LHC energies.
At the lower energies, the H(x) scaling of elastic pp collisions imposes a condition also on the differential cross-sections of elastic \(p\bar{p}\) collisions. Although the value of \(\alpha ^{p\bar{p}}(s)\) is not constrained, the scale parameters in pp and in \(p\bar{p}\) elastic collisions were found to follow the same trends. Hence, if H(x) scaling is valid down to \(\sqrt{s} = 200\) GeV, then the scale parameters of elastic \(p\bar{p}\) collisions at \(\sqrt{s} = 0.546\) and 1.96 TeV have also to follow the common energy dependencies as specified by Eqs. (C.10)–(C.12). So the validity of the H(x) scaling in elastic pp collisions constrains the possible shape of elastic \(p\bar{p}\) collisions as well within the framework of the ReBB model and these constraints can be tested both theoretically and experimentally.
Let us present the tests of the upper limit of the domain of validity in x of the H(x) scaling on experimental data first.
The test of the validity of the H(x) scaling, using elastic pp data at \(\sqrt{s} = 2.76\) TeV LHC energy is shown in Fig. 25. The agreement at 2.76 TeV is excellent and needs no comments or explanations. The upper limit of the validity of the H(x) scaling, \(x_{max}\) at this \(\sqrt{s} = 2.76\) TeV can not be determined from this plot, as apparently \(x_{max}(2.76) \gg 12.7\) that is this upper limit is clearly larger, than the upper end of the acceptance in x of the TOTEM experiment at this energy.
At \(\sqrt{s} = 0.546\) TeV, \(x_{max}(s)\), the upper limit of the domain of the validity of the H(x) scaling is investigated in Fig. 26. For \(p\bar{p}\) collisions, \(\alpha ^{p\bar{p}}\) remains the only free fit parameter, except the overall normalization parameters. In this case, the \(p\bar{p}\) differential cross-sections are constrained, because of the requirement \(R_q(s)/R_q(s_0) = R_d(s)/R_d(s_0) = R_{qd}(s)/R_{qd}(s_0) = b(s)\) is a prescribed function of s and the parameters at \(\sqrt{s_0} = 7\) TeV are already determined at \(\sqrt{s} = \) 7 TeV. As indicated in Fig. 26, these constraints are satisfied with a CL = 0.2% > 0.1% for the measured \(p\bar{p}\) data, but only in a rather narrow kinematic region of \(0.375 \le -t \le 0.56\) GeV\(^2\).
An important test of the validity of the H(x) scaling \(p\bar{p}\) collisions at the Tevatron energy of \(\sqrt{s} = 1.96\) TeV is indicated in Fig. 27. In this plot, the model parameters \(R_q(s)\), \(R_d(s)\) and \(R_{qd}(s)\) are constrained by the H(x) scaling. Solid line indicates that one parameter, \(\alpha ^{p\bar{p}}(s)\) can be fitted to describe the D0 differential cross-section in the whole acceptance of the D0 experiment in \(-t\). According to this plot, at the D0 energy of \(\sqrt{s} = 1.96\), the domain of the H(x) scaling extends to the \(-t \le 1.2\) GeV\(^2\) domain, which corresponds to \(x_{max}(s) = 20.2 \) at \(\sqrt{s} = 1.96\) TeV. In pp collisions, the other condition of validity of the H(x) scaling is that \(\alpha ^{pp}\) is independent of the energy of the collision in the \(\sqrt{s} = 1.96 - 7.0\) TeV range, however, for \(p\bar{p}\) collisions, \(\alpha ^{p\bar{p}}\) is a free fit parameter. For \(p\bar{p}\) collisions, the H(x) scaling limit of the ReBB model of Ref. [42] describes the D0 data in a statistically acceptable manner, with a CL = 0.7%, corresponding to an agreement with a \(\chi ^2/NDF = 25.7/11\), and no significant deviation, an agreement at the 2.69 \(\sigma \) level.
As the diffractive minimum in pp is deeper than in \(p\bar{p}\) if the H(x) scaling is valid, the plot also indicates that a signal of Odderon exchange is also present in the ReBB model if extrapolated with the H(x) scaling for pp collisions to \(\sqrt{s} = 1.96 \) TeV. However, some significances are lost, due to two reasons: (i) the ReBB model-dependent extrapolations are limited to the \(-t \ge 0.375\) GeV\(^2\) region, while the model-independent comparisons can be utilized in the whole \(-t\) region; and (ii) the comparison is done on the level of the differential cross-sections so the overall correlated, type C errors do not cancel. In this case, for the Odderon signal we find a \(\chi ^2/NDF = 40.57/12\), corresponding to a statistical significance of 4.02 \(\sigma \).
In addition to these experimental tests, that are obtained from fitting measured \(p\bar{p}\) data with the help of one free parameter, \(\alpha ^{p\bar{p}}(s)\) and using the results for the s-dependence of the other 3 physical parameters of the ReBB model, \(R_q(s)\), \(R_d(s)\) and \(R_{qd}(s)\), we can have a theoretical test as well. This corresponds to the evaluation of the differential cross-section of elastic pp collisions from both the H(x) scaling limit of the ReBB model of Refs. [30, 42] and from the fully fledged version of the same model, that includes also terms that result in scaling violations, and makes the full H(x, s) functions weakly s-dependent. Evaluating the error bands from the uncertainty of the model parameters, we can evaluate \(x_{max}(s)\) at a given s by determining the domain in x, where the two calculations agree within 1 standard deviation of the model parameters. Such a calculation is performed in Fig. 28. This calculation does not allow for a compensation of the modification of the shape by a possible overall normalization factor and so it results in a more stringent upper limit for the domain of the validity of the H(x) scaling, \(x_{max}(s) = 15.1\) at \(\sqrt{s} = 1.96\) TeV.
Recently, the STAR collaboration measured the differential cross-section of elastic pp collisions at the center-of-mass energy of 200 GeV [81]. This measurement has resulted in a straight exponential differential cross-section in the range of \( 0.045 \le -t \le 0.135\) GeV\(^2\). This range is the range where \(H(x) = \exp (-x)\) and the conditions of the validity of the H(x) scaling are indeed satisfied by this dataset, that is, however, limited to a rather low \(-t\) range. It is thus a very interesting and most important experimental cross-check for the validity of the H(x) scaling to push forward the experimental data analysis of elastic pp collisions at the top RHIC energy of \(\sqrt{s} = 510\) GeV including if possible a larger \(-t\) range extending to the non-exponential domain of \(\frac{d\sigma }{dt}\) as well.
Appendix E: Study of the stability of the Odderon signal for the variation of x-range
In this Appendix, we summarize our \(x = -tB\) range stability results. The central topic of this Appendix is the clarification of the role of the correlation between the correlation coefficient \(\epsilon _{b,21}\) and the domain of x, over which the \(\chi ^2\) or the statistical significance \(\sigma \) is optimized. This correlation coefficient shifts all the projected datapoints together, up or down, and its best value for all the datapoints is \(\epsilon _{b,21} = -0.56\) if the H(x) scaling function of the densest pp dataset of \(\sqrt{s} = 7\) TeV is projected to the \(\sqrt{s} = 1.96\) TeV \(p\bar{p}\) data. If we fixed this value, and started to limit the x-range of comparison by removing 1, 2, 3, 4, 5 and 6 D0 datapoints with the largest values of x, we obtain the results summarized in Table 8.
In this case of a constant, x-range independent \(\epsilon _{b,21} = -0.56\) , the \(\chi ^2\) for all the points and the 6.26 \(\sigma \) overall significance are both constants, but the partial contributions to the \(\chi ^2\) and NDF are x-range dependent. Note, that even for the \(x \le 10.8\) region (left from the dip) we obtain a significant contribution, with a statistical significance of 5.46 \(\sigma \). This is seen visually in Fig. 8 as well as on our final H(x) comparison plots, Fig. 15 in Appendix A.
However, if we applied a more conservative approach, and re-optimized the correlation coefficient to obtain an x-range dependent \(\epsilon _{b,21} \), that minimizes the partial contributions to the \(\chi ^2\) in the investigated x-range, we obtain the results summarized in Table 9.
As shown in Table 9, the correlation coefficient \(\varepsilon _{B21}\) turns out to be strongly x-range dependent. The importance of the contribution from large values of x can be formulated as follows: if we optimize the correlation coefficient \(\varepsilon _{B21}\) for the range of investigation, the Odderon signal is stable for the removal of the top 1, 2, 3 and 4 D0 datapoints with \(x \ge 14.8\), as even with the x-range dependent minimization of the \(\chi ^2\), the statistical significance of the Odderon observation remains at least 5.33 \(\sigma \) . However, if we remove 5 or more of the D0 datapoints at large values of \(x = - tB\), then the remaining D0 \(p\bar{p}\) data and the H(x) scaling function of pp at \(\sqrt{s} = 7\) TeV can be renormalized to the top of each other. TOTEM data at \(\sqrt{s} = 2.76 \) TeV extend only to the \(x \le 12.1\) region, where even a 2.5 \(\sigma \) level, statistically not significant apparent agreement can be reached by totally distorting the value of the correlation coefficient \(\varepsilon _{B21}\), even if using only the more dense and more precise TOTEM 7 TeV pp data. Orange colored fields indicate that if we apply a local optimalization to the correlation coefficient \(\varepsilon _{B21}\), this increases the overall significance of the Odderon, if all the datapoints are used. As indicated by the last column of Table 9, these values of the correlation coefficient \(\varepsilon _{B21}\) are gradually becoming rather unreasonable, when all the datapoints are considered. As more and more D0 data points are removed at high x, the global statistical significance increases drastically, even above the model-dependent limit of 7.08 \(\sigma \), as indicated by the red-colored fields in the rightmost column of the above table. This 7.08 \(\sigma \) limit corresponds to the combined significance of pp vs \(p\bar{p}\) comparisons at both \(\sqrt{s} = 1.96\) and 2.76 TeV, as summarized in Table 4 below, and detailed in Ref. [42].
Table 8 Stability of the Odderon signal on the variation \(x_{max}\), the upper limit of the domain of validity of the H(x) scaling using \(\sqrt{s} = 7\) TeV TOTEM pp data projected to \(\sqrt{s} = 1.96\) TeV D0 \(p\bar{p} \) data in \(x = -tB\) . If the correlation coefficient kept its value of \(\epsilon _{b,21} = -0.56\), corresponding to the “global” minimum of \(\chi ^2\), the Odderon signal would remain at least 5.4 \(\sigma \), even if the last \(1,2,\ldots 5\) and 6 D0 points were discarded (by hand) in the x-range stability analysis Table 9 Stability of the Odderon signal on the variation \(x_{max}\), the upper limit of the domain of validity of the H(x) scaling using \(\sqrt{s} = 7\) TeV TOTEM pp data projected to \(\sqrt{s} = 1.96\) TeV D0 \(p\bar{p} \) data in \(x = -tB\), for a fit range dependent, minimized correlation coefficient \(\epsilon _{b,21}\), corresponding to the minimum of \(\chi ^2\) in the considered x-range. The Odderon signal remains at least 5.3 \(\sigma \), if the last 1,2, 3 and 4 D0 points were discarded (by hand) in this x-range stability analysis. In this sense, the Odderon signal remains significant, if the H(x) scaling is valid at last up to x = 14.8. The last column indicates that these locally optimized \(\epsilon _{b,21}\) coefficients globally increase the Odderon significance If we fully utilize the results of the model dependent analysis of Ref. [42], we find a larger, combined significance of 7.08 \(\sigma \) as shown in Table 10. This result is obtained by comparing simultaneously the pp and \(p\bar{p}\) differential cross-sections at \(\sqrt{s} = 1.96\) and 2.76 TeV with the help of the ReBB model. The increased significance is due to the fact that with the help of the same model, the \(p\bar{p}\) data can also be extrapolated to 2.76 TeV and result in an overwhelming Odderon signal. Let us stress, that this model dependent significance is obtained even without using the 7 TeV TOTEM dataset, to evaluate the \(\chi ^2\) and the Odderon significance and to bridge the energy gap between 2.76 and 1.96 TeV. If the \(p\bar{p}\) differential cross-section is evaluated and extrapolated also up to 7 TeV with the help of the same ReBB model, the probability of Odderon observation becomes very much larger than a 7.08 \(\sigma \) effect [42], practically it becomes a certainty.
Table 10 The trade-off effect of using model dependent results: If we utilize the ReBB model instead of H(x) scaling, it decreases the significance of the pp prediction vs \(p\bar{p}\) data at \(\sqrt{s} = 1.96\) TeV, from 6.26 down to 2.19 \(\sigma \). However, as a trade-off, the same model allows for an extrapolation of the \(p\bar{p}\) data to \(\sqrt{s} = 2.76\) TeV, which was not possible with the help of the H(x) scaling. This \(p\bar{p}\) differential cross-section vs pp data at \(\sqrt{s} = 2.76\) TeV results in a 7.12 \(\sigma \) effect. The combined significance of the ReBB model on both the 1.96 TeV D0 and 2.76 TeV TOTEM data is found to be 7.08 \(\sigma \) . Thus the lower limit of significance from the ReBB model to data comparison, 7.08 is larger than the model independent estimate of 6.26 \(\sigma \), that was obtained using the full x-range of D0 and the assumption of the validity of the H(x) scaling in this range We have made another cross-check and divided the final result of our calculations to four different regions: We have 2 D0 datapoint in Region 0, the diffractive cone with \(0 < x = -tB \le 5.1\) . The remaining 15 D0 datapoints can be divided into 3 regions with 5-5 D0 data points as follows. Region I corresponds to the “swing” region, just to the left of the dip, corresponding to \(5.1 < x \le 8.4\); Region II, including the dip and the bump, to \(8.4 < x \le 13.5\); and Region III, the tail corresponds to \(13.5 < x \le 20.2\). We evaluated their partial contributions to our final Odderon significance of 6.26 \(\sigma \). For a cross-check we have also evaluated their combined significance and also the contribution of the first two D0 datapoint from the diffractive cone.
Table 11 Stability of the Odderon signal in various regions of \(x_{min}\) and \(x_{max}\), for a constant value of the correlation coefficient \(\epsilon _{b,21}\), minimized on all the 17 available D0 data points. In this case, the greatest partial contribution to the Odderon significance comes from the swing region, \(5.1 < x \le 8.4\) the second most important contribution comes from the diffractive interference (dip and bump) region with \(8.4 < x \le 13.5\), and for this value of the correlation coefficient, the tail with \(13.5 < x \le 20.2\) has a relatively small contribution Table 12 Stability of the Odderon signal in various regions of \(x_{min}\) and \(x_{max}\), in the case, when the value of the correlation coefficient \(\epsilon _{b,21}\) locally minimized for the data in the \(x_{min} < x \le x_{max}\) range. In this case, the greatest partial contribution to the Odderon significance comes from the diffractive interference (dip and bump) region with \(8.4 \le x \le 13.5\), the second most important contribution comes from the swing region, \(5.1 < x \le 8.4\), while the relatively least important contribution comes from the tail with \(13.5 < x \le 20.2\). It is important to recognise, that the 10 D0 datapoints in the swing and diffractive interference region already provide a statistically significant, more than 5 \(\sigma \) Odderon effect. The interference and the tail, taken together, also indicate an Odderon signal, as a 3.91 \(\sigma \), indicative but statistically not yet sufficient effect. When all these three regions are combined together, they dominate the final Odderon significance, providing 6.23 out of the 6.26 \(\sigma \) Odderon effect for all x The results for a fixed \(\varepsilon _{B21} = -0.56\) – optimized on all the 17 D0 datapoints – are shown in Table 11. In this case, the greatest partial contribution to the Odderon significance comes from the swing region, \(5.1 < x \le 8.4\) the second most important contribution comes from the diffractive interference (dip and bump) region with \(8.4 < x \le 13.5\), and for this value of the correlation coefficient, the tail with \(13.5 < x \le 20.2\) has a relatively small contribution.
In contrast, similar results for a regionally optimized correlation coefficient, an x-range dependent \(\varepsilon _{B21} \) is shown on Table 12. In this case, the greatest partial contribution to the Odderon significance comes from the diffractive interference (dip and bump) region with \(8.4 < x \le 13.5\), the second most important contribution comes from the swing region, \(5.1 < x \le 8.4\), while the relatively least important contribution comes from the tail with \(13.5 < x \le 20.2\). It is important to recognise, that the 10 D0 datapoints in the swing and diffractive interference region already provide a statistically significant, more than 5 \(\sigma \) Odderon effect. The interference and the tail together also indicate an Odderon effect, with a significance that is between a 3 and a 5 \(\sigma \) effect. When all these three regions are combined together, they dominate the final Odderon significance, providing 6.23 out of the 6.26 \(\sigma \) Odderon effect for all x.
Table 13 In the \(9.0 < x = - tB \le 15.1 \) kinematic domain we obtain a lower, model dependent limit of significance of 3.82 \(\sigma \). Note, that the ReBB model is not validated in the \(-t \le 0.372\) kinematic range [42], corresponding to low values of x. At \(\sqrt{s} = 1.96\) TeV, this lower limit of validity of the ReBB model is \(x_{min} = 4.4\). In the corresponding \(4.4 < x \le 15.1\) domain, the lowest limit of Odderon significance is an 5.37 \(\sigma \) effect. Within this domain, in the range of \(7.0 < x \le 13.5\), the Odderon signal is still above the 5 \(\sigma \), discovery level, even when 9 out of the 17 D0 datapoints are removed from the analysis It is an intriguing problem if one can determine model independently the region, from where the dominant contribution to the Odderon signal is coming. To reach that goal, we have developed a so called sliding window technique, and determined the minimum size of this sliding window that still provides an Odderon signal on the discovery level of at least 5.0 \(\sigma \). Namely, D0 published 17 datapoints. We have taken the first n of these datapoints, with n varied from 2 to 17, and then locally optimized the correlation coefficients \(\epsilon _B\) for both projections of 7 \(\rightarrow \) 1.96 TeV and 1.96 \(\rightarrow \) 7 TeV, and determined where the minimum sized sliding window is wherein the observation of the Odderon is an at least 5 \(\sigma \) effect. As one picture is worth ten thousand words according to a Chinese word of wisdom, we have summarized our results in Fig. 29. For the sake of clarity, and only on this plot, we have shifted the TOTEM 7 TeV datapoints by their type B errors (properly multiplied by the correlation coefficient \(\epsilon _{B,7~TeV}\) that minimalized \(\chi ^2\) for that given sliding acceptance window) and show only the type A vertical and horizontal errors. Figure 29 indicates that there are 8 D0 datapoints in the minimal sized sliding acceptance window, where an at least 5 \(\sigma \), statistically significant Odderon signal is observed. Thus 9 out of the 17 datapoints can be removed (5 from the tail and 4 points from the diffractive cone region), without destroying the greater than 5 \(\sigma \) level of the Odderon significance.
If we take into account the evolution of H(x, s) as a function of s, this evolution becomes model dependent, but allows for the estimation of the domain of validity of the \(H(x,s) = H(x,s_0)\) scaling law, as detailed in Appendix D.
We obtained our model-dependent results within the framework a Glauber-type calculation using the ReBB model of refs. [30, 42]. This model is validated at \(\sqrt{s} = 1.96 \) TeV in the \(x_{min} = 4.4 < x\) region [42]. According to the calculations of Appendix D, the H(x) scaling at this 1.96 TeV energy is expected to be valid at least in the \(9.0 < x = - tB \le 15.1\) kinematic domain. As we cannot estimate reliably the validity of the lower limit of the H(x) scaling at low values of x with this model and in addition, in the diffractive cone, \(H(x) \approx \exp (-x)\) and the scaling is expected to hold, the \(9.0 < x = - tB \le 15.1\) kinematic domain seems to give the worst possible, model dependent limit for the domain of validity of the H(x) scaling at 1.96 TeV. As shown on Table 13, this interval corresponds to a significance of at least 3.82 \(\sigma \). In this very limited x range, the corresponding best correlation coefficient, - 0.62, is rather close to the best correlation coefficient, -0.56, that minimizes significance for the case of the complete, \(0 < x = - tB \le 20.2\) kinematic domain of all the D0 data, thus for this correlation coefficient, the significance for the \( 0 < x = - tB \le 20.2\) kinematic domain is nearly unchanged from 6.27 to 6.28 \(\sigma \). If we assume that the lower limit \(x_{min}\) corresponds to the lower limit of the validation of the ReBB model [42], then we find that in the \(4.4 < x \le 15.1 \) kinematic domain the significance of the Odderon signal is at least 5.3 \(\sigma \), as detailed on Table 13. However, we know that in the low \(x = -t B\) region, there is a diffractive cone, where \(H(x) \approx \exp (-x)\) for scattering amplitudes that are analytic at \(t=0\), and for experimental data that are not indicating a non-exponential behaviour at low values of |t|. This is the case for the \(\sqrt{s} = 1.96\) TeV D0 \(p\bar{p}\) and for the \(\sqrt{s} = 2.76\) TeV TOTEM pp data, so for closing the energy gap between 1.96 and 2.76 TeV, the lower limit of the applicability of the H(x) scaling is actually \(x_{min} = 0\).
If we fully utilize the results of this model dependent analysis we find that the model dependent, combined Odderon significance on pp prediction versus \(p\bar{p}\) data at \(\sqrt{s} = 1.96\) TeV data and \(p\bar{p}\) prediction versus pp data at \(\sqrt{s} = 2.76\) TeV is 7.08 \(\sigma \), as shown on Table 10. This increased significance is due to the fact that with the help of the same ReBB model [42], the \(p\bar{p}\) data can also be extrapolated to the lowest TOTEM energy of \(\sqrt{s} = 2.76\) TeV and they result in a dominant Odderon signal, as detailed also in Table 10.
In this Appendix, we thus find a hierarchy of the Odderon significances. When we take theoretical modelling into account, we find that the significance of an Odderon observation on elastic pp collisions at \(\sqrt{s} = 2.76\) TeV versus elastic \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96\) TeV, in the corresponding TOTEM and D0 acceptances, is at least 7.08 \(\sigma \). If we do not utilize fully the model dependent information, but use only the theoretical limits for the validity of the H(x) scaling at \(\sqrt{s} = 1.96\) TeV, we find that the Odderon signal is greater than 5 \(\sigma \) if the H(x) scaling is valid in the range of \(5.1 < x \le 13.1\) . We cannot reliably estimate the lower limit of the H(x) scaling, but demonstrated on available data that in the diffraction cone, \(H(x) \approx \exp (-x)\) is satisfied. We have validated the model of Ref. [42] in the range of \(4.4 < x\) at 1.96 TeV, and find that this model dependent domain of validity of the H(x) scaling extends up to \(x_{max} = 15.1\) .
The theoretically and model dependently validated range of H(x) scaling at \(\sqrt{s} = 1.96\) TeV, \(4.4 < x \le 15.1\) includes the interval \(7.0 < x \le 13.5\), where we find that the Odderon signal is greater than a 5 \(\sigma \) effect.
We conclude this x-range stability investigations as follows:
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Model independently, we find that the significance of the Odderon is greater than 5 \(\sigma \) in the \(7 < x \le 13.5\) domain at \(\sqrt{s} = 1.96\) TeV.
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In our model independent analysis, we relied only on already published D0 and TOTEM datapoints, without relying on preliminary data. We have evaluated the \(\chi ^2\) of the Odderon signal using arithmetic operations only (addition, substraction, multiplication, division) but we did not impose any model dependent fits.
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In our model independent analysis, we utilized a newly introduced H(x) scaling function, that is not sensitive to the dominant, and overall correlated normalization errors of the differential cross-sections. As a trade-off, the domain of validity of this H(x) scaling became an energy dependent \((0, x_{max}(s)) \) interval.
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This H(x) scaling function scales out the trivial energy dependencies that appear due to the energy dependence of the elastic cross-section \(\sigma _{el}(s)\) and the nuclear slope parameter B(s).
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Using a model, validated in the \(4.4 < x\) domain, we find that the validity in x of the H(x) scaling at \(\sqrt{s} = 1.96\) TeV is extending up to \(x \le 15.1\). Using published D0 data, and the same model, we validated in Fig. 27 that the H(x) scaling at \(\sqrt{s} = 1.96\) is extending even up to the end of D0 acceptance, to \(x \le 20.2\) .
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In the very limited interval \(7.0 < x = -B t \le 13.5\), we find that the Odderon signal is greater than a 5 \(\sigma \) effect at \(\sqrt{s} = 1.96\) TeV.
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Thus the model independent and at least 5 \(\sigma \), discovery level Odderon signal is remarkably stable for the variations of the domain of validity of the H(x) scaling at \(\sqrt{s} = 1.96\) TeV.