Evidence of Odderon-exchange from scaling properties of elastic scattering at TeV energies

One of the most important and critical tests of quantum chromodynamics (QCD) in the infrared regime is provided by the ongoing studies of elastic differential hadron–hadron scattering cross section at various energies and momentum transfers. The characteristics of the elastic amplitude, its both real and imaginary parts, carry a plenty of information about the inner proton structure, the proton profile in the impact parameter space and its energy dependence, as well as about the properties of QCD exchange interaction at low momentum transfers.

The first and most precise measurement of the total, elastic and differential cross sections of elastic pp collisions, together with the \(\rho \)-parameter, has recently been performed by the TOTEM Collaboration at the Large Hadron Collider (LHC) at CERN at the highest energy frontier of \(\sqrt{s} = 13\) TeV (for the corresponding recent TOTEM publications, see Refs. [1,2,3,4]). A correct theoretical interpretation of the LHC data, together with the lower-energy Tevatron and ISR data, is a subject of intense debates and ongoing research development in the literature, see e.g. Refs. [5, 6]. Among the important recent advances, data by the TOTEM Collaboration [4] for the first time have indicated the presence of an odd-under-crossing (or C-odd) contribution to the elastic scattering amplitude known as the Odderon [7]. In particular, a comparison of the differential cross-section of elastic proton–proton (pp) scattering obtained by the TOTEM Collaboration at \(\sqrt{s} = 2.76\) TeV with D0 results on elastic proton–antiproton (\(p\bar{p}\)) scattering at 1.96 TeV [8] indicates important qualitative differences that can be attributed to the Odderon effect [4, 9]. In the more rigorous language of QCD, an Odderon exchange is usually associated with a quarkless odd-gluon (e.g. three-gluon, to the lowest order) bound state such as a vector glueball, and a vast literature is devoted to theoretical understanding of its implications. An increase of the total cross section, \(\sigma _\mathrm{tot}(s)\), associated with a decrease of the real-to-imaginary ratio, \(\rho (s)\), with energy, first identified at \(\sqrt{s} = 13\) TeV [1, 2], also indicated a possible Odderon effect.

The TOTEM measurements have recently triggered intense theoretical studies in the literature. In particular, the Phillips–Barger parameterisation of the elastic amplitude has been found to describe the recent pp data in Refs. [10, 11]. Several other Regge parameterisations have also been found to describe the LHC data reasonably well (see e.g. Refs. [6, 12, 13]), while the Pomeron dominance has been explored in a generic Regge theory set-up in Refs. [14, 15]. In Ref. [5], a new feature of the second diffractive cone in the differential cross-section of elastic scattering at large t and s has been identified arguing about the existence of two stationary points in \(d\sigma /dt\) at the LHC energies and relating those to the two-scale structure of protons at these energies. Remarkably, this rules out the dominance of perturbative exchanges of a few non-interacting gluons pointing towards a core-like proton substructure found also in the framework of the so-called Lévy imaging technique in Refs. [9, 16]. For a thorough discussion of general properties of the s-dependence of \(\rho (s)\) in the light of the TOTEM data and its connections to the growing energy dependence of the elastic-to-total cross-sections ratio, see Ref. [17]. A number of studies based upon a QCD-based analysis of the Odderon signatures considering the non-linear QCD evolution have also been triggered recently (see e.g. Refs. [18,19,20,21]).

Important statements about the maximal nature of the Odderon effect were made in Refs. [6, 22,23,24] but apparently these studies still lack a rigorous statistical significance analysis. Although the s-dependence of both \(\sigma _\mathrm{tot}(s)\) and \( \rho (s)\) is consistent with an Odderon effect, this indication is not a unique Odderon signal as the same effect can also be attributed to the secondary Reggeon effects [18], reinforcing the elusiveness of the Odderon. As it was argued in Ref. [25] any conclusions about the magnitude of the Odderon effects based upon the \(\rho (s)\) measurement alone have to be made with special care due to a zero in the real part of the elastic amplitude at very small t, as the latter can affect the Coulomb-Nuclear Interference (CNI) region at high energies.

In earlier studies of Refs. [26, 27], the Odderon signatures have been identified and qualitatively described in a model-independent way using the power of the Lévy imaging technique [9]. One of such signatures concern the presence of a dip-and-bump structure in the differential cross section of elastic pp collisions and the lack of such a structure in elastic \(p\bar{p}\) collisions. The latter effectively emerges in the t-dependence of the elastic slope B(t), that crosses zero for elastic pp collisions and remains non-negative for all values of t in elastic \(p\bar{p}\) collisions. Besides, Ref. [9] noted that the position of the node of the nuclear phase \(\phi (t)\), as reconstructed with the help of the Lévy expansion method, is characteristically and qualitatively different for elastic pp from \(p\bar{p}\) collisions, thus, indicating the Odderon exchange. In addition, the presence of a smaller substructure of the proton has been revealed in the data that is imprinted in the behaviour of the t-dependent elastic slope B(t), apparent at large values of t. In particular, in Refs. [9, 16, 26, 27] two substructures of two distinct sizes have been identified in the low (a few tens of GeV) and high (a few TeV) energy domains, respectively. Besides, a new statistically significant feature in the b-dependent shadow (or inelasticity) profile has been found at the maximal available energy \(\sqrt{s} = 13\) TeV and represents a long-debated hollowness, or “black-ring” effect that emerges instead of the conventionally anticipated “black-disk” regime [16, 26].

In this paper, in order to further unveil the important characteristics of elastic hadron–hadron scattering we study the scaling properties of the existing data sets available from the ISR and Tevatron colliders as well as those provided by the TOTEM Collaboration in a TeV energy range [1,2,3,4, 28]. We investigate a generic scaling behavior of elastic differential proton-(anti)proton scattering cross section, with the goal of transforming out the trivial colliding energy dependent variation of the key observables like that of the total and elastic cross-sections \(\sigma _\mathrm{tot}(s)\) and \(\sigma _\mathrm{el}(s)\), the elastic slope B(s) and the real-to-imaginary ratio \(\rho (s)\). We search successfully for a universal scaling function and the associated data-collapsing behaviour that is valid not only in the low-|t| domain, but also in the dip-and-bump region. We discuss the physics implications of such a scaling behaviour and explore its consequences for understanding of the Odderon effect as well as the high-energy behaviour of the proton structure.

The paper is organised as follows. In Sect. 2, we recapitulate the formalism that is utilized for evaluation of the observables of elastic proton-(anti)proton scattering in the TeV energy range. In Sect. 3, we connect this formalism to a more general strategy of the experimental Odderon search, namely, to the search for a crossing-odd component in the differential cross-section of elastic proton-(anti)proton scattering. In Sect. 4, we study some of the scaling functions of elastic scattering already existing in the literature as well as propose a new scaling function denoted as H(x) that is readily measurable in pp and \(p \bar{p}\) collisions. In particular, in Sect. 4.3 we introduce a new scaling function for the diffractive cone or low values of the square of the four-momentum (\(-t\)) region. We generalize this scaling function for larger values of \(-t\) in Sect. 4.4 and present a first test of the H(x) scaling in the ISR energy range of 23.5–62.5 GeV in the same subsection. Subsequently, in Sect. 5 we extend these studies to the TeV (Tevatron and LHC) energy range, where the possible residual effects of Reggeon exchange are expected to be below the scale of the experimental errors [29]. In Sect. 6, we present a method of how to quantify the significance of our findings, giving the formulas that are used to evaluate \(\chi ^2\), confidence level (CL), and significance in terms of the standard deviation, \(\sigma \). In Sect. 7, we discuss how to employ the newly found scaling behavior of the differential cross-section to extrapolate the differential cross-sections of elastic pp scattering within the domain of the validity of the new H(x) scaling. Let us note, that this method of comparing differential cross-sections is a possible strategy for the Odderon search. However, as we detail later, the overall normalization uncertainties are large and reduce the statistical significance of these kind of results: practically it is a better strategy to compare scaling functions, evaluated from the differential cross-sections in such a way, that the overall normalization constants (including their large errors) cancel. In Sect. 8, we present further, more detailed results of our studies with the help of H(x) and compare such a scaling function for pp differential cross-sections at the LHC energies with the \(p\bar{p}\) scaling function at the Tevatron energy. In Sect. 9 we evaluate the significance of the Odderon-effect, and find that it is at least a 6.26\(\sigma \)-significant effect, taking into account also the improvements detailed in Appendix A. Subsequently, we present several cross-checks in Sect. 10 and discuss the main results and its implications in Sect. 11. Finally, we summarize and conclude our work in Sect. 12.

This manuscript is completed with several Appendices that highlight various aspects of this analysis. Appendix A details the robustness and symmetry properties of the \(\chi ^2\) definition and provides the final Odderon significance of at least \(6.26\sigma \) from a model-independent comparison of the H(x) scaling functions of already published data. In Appendix B we discuss the model-independent properties of the Pomeron and Odderon exchanges at the TeV energy scale, under the condition that this energy is sufficiently large: as the effects from the exchange of known hadronic resonances decreases as an inverse power of s, at large enough energies Pomeron and Odderon exchanges can be identified with the crossing-even and the crossing-odd contributions to the elastic scattering, respectively. We demonstrate here that S-matrix unitarity constrains the possible form of the impact-parameter dependence of the Pomeron and Odderon amplitudes. In Appendix C, we discuss model-dependent properties of the Pomeron and Odderon exchanges at the TeV energy scale and derive, how the H(x) scaling emerges within a specific model, defined in Ref. [30]. This model is one of the possible models in the class considered in Appendix B. We evaluate the experimentally observable consequences of the H(x) scaling in Appendix D, where we estimate the domain of validity of the H(x) scaling also in a model-dependent manner, based on Ref. [30]. Finally, in Appendix E we cross-check the stability and robustness of the Odderon signal for the variation of the x-range, the domain or support in x where the signal is determined. We also identify here a minimal set of only 8 out of 17 D0 datapoints, close to the diffractive interference region, that alone carry an at least 5 \(\sigma \) Odderon signal, when compared to the TOTEM datapoints in the same region.

For the sake of completeness and clarity, let us start first with recapitulating the connection between the scattering amplitude and the key observables of elastic scattering, following the conventions of Refs. [30,31,32,33].

The Mandelstam variables s and t are defined as usual \(s = (p_1 + p_2)^2\), \(t = (p_1 - p_3)^2\) for an elastic scattering of particles a and b with incoming four-momenta \(p_1\) and \(p_2\), and outgoing four-momenta \(p_3\) and \(p_4\), respectively.

The elastic cross-section is given as an integral of the differential cross-section of elastic scattering:

The elastic differential cross section is

The t-dependent slope parameter B(s, t) is defined as

and in the experimentally accessible low-t region this function is frequently assumed or found within errors to be a constant. In this case, a t-independent slope parameter B(s) is introduced as

where the \(t\rightarrow 0\) limit is taken within the experimentally probed region. Actually, experimentally the optical \(t=0\) point can only be approached by extrapolations from the measurements in various \(-t > 0\) kinematically accessible regions that depend on the optics and various settings of the particle accelerators and colliding beams.

According to the optical theorem, the total cross section is also found by a similar extrapolation. Its value is given by

while the inelastic cross-section is defined by

The ratio of the real to imaginary parts of the elastic amplitude is found as

and its measured value at \(t=0\) reads

Here, the \(t\rightarrow 0\) limit is taken typically as an extrapolation in dedicated differential cross section measurements at very low \(-t\), where the parameter \(\rho _0\) can be measured using various CNI methods. The differential cross section at the optical \((t = 0)\) point is thus represented as

In the impact-parameter b-space, we have the following relations:

This Fourier-transformed elastic amplitude \(t_{el}(s,b)\) can be represented in the eikonal form

where \(\Omega (s,b)\) is the so-called opacity function (known also as the eikonal function), which is complex in general. The shadow profile function is then defined as

For clarity, let us note that other conventions are also used in the literature and for example the shadow profile P(b, s) is also referred to as the inelasticity profile function since it corresponds to the probability distribution of inelastic proton–proton collisions in the impact parameter b with \(0\le P(b,s) \le 1\). When the real part of the scattering amplitude is neglected, P(b, s) is frequently denoted as \(G_\mathrm{inel}(s,b)\), see for example Refs. [34,35,36,37,38].

As noted in Refs. [10, 39], the only direct way to see the Odderon is by comparing the particle and antiparticle scattering at sufficiently high energies provided that the high-energy pp or \(p\bar{p}\) elastic scattering amplitude is a sum or a difference of even and odd C-parity contributions, respectively,

Here, the even-under-crossing part consists of the Pomeron P and the Reggeon f trajectories, while the odd-under-crossing part contains the Odderon O and a contribution from the Reggeon \(\omega \).

At sufficiently high collision energies \(\sqrt{s}\), the relative contributions from secondary Regge trajectories are suppressed since they decay as negative powers of \(\sqrt{s}\). In Ref. [10], the authors argued that the LHC energy scale is already sufficiently large to suppress the Reggeon contributions, and they presented the (s, t)-dependent contributions of an Odderon exchange to the differential and total cross-sections at typical LHC energies. More recently, this observation was confirmed in Ref. [29], suggesting that indeed the relative contribution of the Reggeon trajectories is well below the experimental precision in elastic pp scattering in the TeV energy range. The analysis of Ref. [10] relies on a model-dependent, phenomenological picture formulated in the framework of the Phillips–Barger model [40] and is focused primarily on fitting the dip region of elastic pp scattering, but without a detailed analysis of the tail and cone regions. In Ref. [29], a phenomenological Reggeon + Pomeron + Odderon exchange model is employed to study, in particular, the possible hollowness effect in the high-energy elastic pp collisions. A similar study of the Philips-Barger model was performed in Ref. [11] using the most recent TOTEM data on elastic pp scattering. Similarly, Ref. [41] has also argued that the currently highest LHC energy of \(\sqrt{s} = \) 13 TeV is sufficiently high to observe the Odderon effect.

In this paper, we follow Refs. [10, 29, 41] and assume that the Reggeon contributions to the elastic scattering amplitudes for \(\sqrt{s} \ge \) 1.96 TeV and at higher energies are negligibly small. We search for an odd-under-crossing contribution to the scattering amplitude, in a model independent way, and find that such a non-vanishing contribution is present at a TeV scale that is recognised as an Odderon effect. The vanishing nature of the Reggeon contributions offers a direct way of extracting the Odderon as well as the Pomeron contributions, \(T_\mathrm{el}^{O}(s,t)\) and \(T_\mathrm{el}^{P}(s,t)\), respectively, from the elastic pp and \(p\bar{p}\) scattering data at sufficiently high colliding energies as follows

These kind of studies rely on the extrapolation of the fitted model parameters of pp and \(p\bar{p}\) reactions to an exactly the same energy, given that the elastic pp and \(p\bar{p}\) scattering data have not been measured at the same (or close enough) energies in the TeV region so far. Another problem is a lack of precision data at the low- and high-|t|, primarily, in \(p\bar{p}\) collisions. Recently, the TOTEM Collaboration noted in Ref. [4] that “Under the condition that the effects due to the energy difference between TOTEM and D0 can be neglected, the result” (namely the differential cross-section measured by TOTEM at \(\sqrt{s} = 2.76 \) TeV) “provides evidence for a colourless 3-gluon bound state exchange in the t-channel of the proton–proton elastic scattering”. In other words, if the effects due to the energy difference between TOTEM and D0 measurements can be neglected, the direct comparison of the differential cross section of elastic pp scattering at \(\sqrt{s} = 2.76\) with that of \(p\bar{p}\) scattering at \(\sqrt{s}= 1.96\) TeV provides a conditional evidence for a colourless three-gluon state exchange in the t-channel.

In this paper, we show that the conditional evidence stated by TOTEM can be turned to an unconditional evidence, i.e. a discovery of the Odderon, by closing the energy gap as much as possible at present, without a direct measurement, based on a re-analysis of already published TOTEM and D0 data.

Our main result, an at least 6.26\(\sigma \) Odderon effect, is obtained by taking the data at a face value as given in published sources, without any attempt to extrapolate them with a help of a model, or using phenomenological, s-dependent parameters and extrapolating them towards their unmeasured values (in unexplored energy domains). Nevertheless, we have tested what happens if one employs this kind of model as detailed in a different manuscript, see Ref. [42]. These model-dependent results lead to a higher than 7.08\(\sigma \) combined significance for the Odderon effect, based on the results of Appendix C. The experimentally observable signs of the newly found H(x) scaling are detailed in Appendix D, where we also determine the model-dependent domain of validity of this new scaling and find that this domain of validity is model-dependently, but sufficiently large so that the Odderon signal remains well above the discovery threshold of a 5\(\sigma \) effect, as detailed in Appendix E. As the 7.08\(\sigma \) combined significance is based only on model-dependent results, evaluated and combined at two energies, \(\sqrt{s}=1.96\) and 2.76 TeV (detailed in both Appendix C and Appendix E), we find that the model-independent approach, summarized in the body of this manuscript and detailed in Appendix A and Appendix B, provides a more conservative, 6.26 \(\sigma \) estimate for the Odderon significance.

Our main result is based on the validity of a new kind of scaling relation, called as the H(x)-scaling. We test this scaling on the experimental data and show their data-collapsing behaviour in a limited energy range. We demonstrate that such a data-collapsing behaviour can be used to close the small energy gap between the highest-energy elastic \(p\bar{p}\) collisions, \(\sqrt{s}= 1.96\) TeV and the lowest-energy elastic pp collisions at the LHC where the public data are available, \(\sqrt{s} = 2.76\) TeV. We investigate the stability of this result on the x-range or the domain of validity of the H(x) scaling in Appendix E. We find that the result is extremely stable for the removal of data points at the beginning or at the end of the acceptance of the D0 experiment. Namely, 9 out of the 17 D0 data points can be removed without decreasing the significance of the Odderon signal below the 5\(\sigma \) discovery threshold.

We look for the even-under-crossing and odd-under-crossing contributions by comparing the scaling functions of pp and \(p\bar{p}\) collisions in the TeV energy range. In other words, we look for and find a robust Odderon signature in the difference of the scaling functions of the elastic differential cross-section between pp and \(p\bar{p}\) collisions. We thus discuss the Odderon features that can be extracted in a model-independent manner by directly comparing the corresponding data sets to one another.

Let us start with three general remarks as direct consequences of Eqs. (18) and (19):

• If the Odderon exchange effect is negligibly small (within errors, equal to zero) or if it does not interfere with that of the Pomeron at a given energy, then the differential cross sections of the elastic pp and \(p\bar{p}\) scattering have to be equal:

  $$\begin{aligned} T_\mathrm{el}^O(s,t) = 0 \implies \frac{d\sigma ^{pp}}{dt} = \frac{d\sigma ^{p\bar{p}}}{dt}\quad \mathrm{for}\ \sqrt{s}\ge 1\, \mathrm{TeV}. \end{aligned}$$

  (20)

• If the differential cross sections of elastic pp and \(p\bar{p}\) collisions are equal within the experimental errors, this does not imply that the Odderon contribution has to be equal to zero. Indeed, the equality of cross sections does not require the equality of complex amplitudes:

  $$\begin{aligned} \frac{d\sigma ^{pp}}{dt} = \frac{d\sigma ^{p\bar{p}}}{dt}\quad \mathrm{for}\ \sqrt{s}\ge 1 \,\mathrm{TeV} \nRightarrow T_\mathrm{el}^O(s,t) = 0 . \end{aligned}$$

  (21)

• 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.

  $$\begin{aligned} \frac{d\sigma ^{pp}}{dt} \ne \frac{d\sigma ^{p\bar{p}}}{dt}\quad \mathrm{for}\ \sqrt{s}\ge 1 \,\mathrm{TeV} \implies T_\mathrm{el}^O(s,t) \ne 0 . \end{aligned}$$

  (22)

Such a difference is thus a clear-cut signal for the Odderon-exchange, if the differential cross sections were measured at exactly the same energies. However, currently such data are lacking in the TeV energy range. Our research strategy in this paper is to scale out the known s-dependencies of the differential cross section by scaling out its dependencies on \(\sigma _\mathrm{tot}(s)\), \(\sigma _\mathrm{el}(s)\), B(s) and \(\rho (s)\) functions. The residual scaling functions will be compared for the pp and \(p\bar{p}\) elastic scattering to see if any difference remains.

In what follows, we introduce and discuss the newly found scaling function H(x) in Sect. 4 and subsequently evaluate the significance of these observations as detailed in Sects. 6 and 9.

In this section, let us first investigate the scaling properties of the experimental data based on a simple Gaussian model elaborating on the discussion presented in Ref. [43]. The motivation for this investigation is that we would like to work out a scaling law that works at least in the simplest, exponential diffractive cone approximation, and scales out the trivial s-dependencies of \(\sigma _\mathrm{tot}(s)\), \(\sigma _\mathrm{el}(s)\), \(\rho (s)\), and B(s). Based on the results of such a frequently used exponential approximation, we gain some intuition and experience on how to generalize such scaling laws for realistic non-exponential differential cross sections.

Experimentally, the low-|t| part of the measured distribution is usually approximated with an exponential,

where it is explicitly indicated that both the normalization parameter \(A \equiv A(s) \) and the slope parameter \(B \equiv B(s)\) are the functions of the center-of-mass energy squared s. If the data deviate from such an exponential shape, that can be described if one allows for a t-dependence of the slope parameter \(B \equiv B(s,t)\) as defined in Eq. (3). For simplicity, we would like to scale out the energy dependence of the elastic slope \(B(s) \equiv B(s,t=0)\) from the differential cross section of elastic scattering, together with the energy dependence of the elastic and total cross sections, \(\sigma _\mathrm{el}(s)\) and \(\sigma _\mathrm{tot}(s)\), as detailed below. For this purpose, let us follow the lines of a similar derivation in Refs. [29, 43].

It is clear that Eq. (23) corresponds to an exponential “diffractive cone” approximation, that may be valid in the low-t domain only. This equation corresponds to the so called “Grey Gaussian” approximation that suggests a relationship between the nuclear slope parameter B(s), the real-to-imaginary ratio \(\rho _0(s)\), the total cross section \(\sigma _\mathrm{tot}(s)\), and the elastic cross section \(\sigma _\mathrm{el}(s)\) as follows [29, 44, 45]:

Such relations for A and B parameters in terms of the elastic and total cross sections are particularly useful when studying the shadow profile function as detailed below. The above relationships, in a slightly modified form, have been utilized by TOTEM to measure the total cross section at \(\sqrt{s} = \) 2.76, 7, 8 and 13 TeV in Refs. [1, 46,47,48], using the luminosity independent method. In what follows, we do not suppress the s-dependence of the observables, i.e. \(\sigma _\mathrm{tot} \equiv \sigma _\mathrm{tot}(s)\), \(\sigma _\mathrm{el} \equiv \sigma _\mathrm{el}(s)\).

4.1 Scaling properties of the shadow profiles

In the exponential approximation given by Eqs. (23)–(25), the shadow profile function introduced in Eq. (13) has a remarkable and very interesting scaling behaviour, as anticipated in Ref. [29]:

Thus, the shadow profile at the center, \(P_0(s) \equiv P(b=0,s)\) reads as

which cannot become maximally absorptive (or black), i.e. \(P_0(s) = 1\) is not reached at those colliding energies, where \(\rho _0\) is not negligibly small. The maximal absorption corresponds to \(P_0(s) \, = \, \frac{1}{1+\rho _0^2(s)}\), which is rather independent of the detailed b-dependent shape of the inelastic collisions [29]. It is achieved when r(s) of Eq. (27) approaches the value \(r(s) = 1/(1+\rho _0^2(s))\). Thus, at such a threshold, we have the following critical value of the ratio

As \(\rho _0 \le 0.15\) for the existing measurements and \(\rho _0(s)\) seems to decrease with increasing energies at least in the 8 \(\le \sqrt{s} \le 13\) TeV region, the critical value of the elastic-to-total cross section ratio (29) corresponds to, roughly, \(\sigma _\mathrm{el}/\sigma _\mathrm{tot} \approx 24.5{-}25.0\)%. Evaluating the second derivative of P(b, s) at \(b=0\), one may also observe that it changes sign from a negative to a positive one exactly at the same threshold given by Eq. (29). Such a change of sign can be interpreted as an onset of the hollowness effect [29]. The investigation of such a hollowness at \(b=0\) is a hotly debated topic in the literature. For early papers on this fundamental feature of pp scattering at the LHC and asymptotic energies, see Refs. [35, 36, 45, 49,50,51,52], as well as Refs. [29, 34, 37, 38, 53,54,55,56,57,58,59,60] for more recent theoretical discussions.

As pointed out in Ref. [43], the threshold (29), within errors, is reached approximately already at \(\sqrt{s} = 2.76 \) TeV. The threshold behavior saturates somewhere between 2.76 and 7 TeV and a transition may happen around the threshold energy of \(\sqrt{s_\mathrm{th}} \approx 2.76{-}4\) TeV. The elastic-to-total cross section ratio becomes significantly larger than the threshold value at \(\sqrt{s} = 13 \) TeV. As a result, the shadow profile function of the proton undergoes a qualitative change in the region of \(2.76< \sqrt{s} < 7 \) TeV energies. At high energies, with \(\sigma _\mathrm{el} \ge \sigma _\mathrm{tot}/4\), the hollowness effect may become a generic property of the impact parameter distribution of inelastic scatterings. However, the expansion at low impact parameters corresponds to the large-|t| region of elastic scattering, where the diffractive cone approximation of Eqs. (23)–(25) technically breaks down, and more refined studies are necessary (see below). For the most recent, significant and model-independent analysis of the hollowness effect at the LHC and its extraction directly from the TOTEM data, see Ref. [16].

4.2 Scaling functions for testing the black-disc limit

When discussing the scaling properties of the differential cross section of elastic scattering, let us mention that various scaling laws have been proposed to describe certain features and data-collapsing behaviour of elastic proton–proton scattering already in the 1970s. One of the early proposals was the so called geometric scaling property of the inelastic overlap function [61, 62]. The concept of geometric scaling was based on a negligibly small ratio of the real-to-imaginary parts of the scattering amplitude at \(t=0\), \(\rho _0 \le 0.01\) and resulted in an s-independent ratio of the elastic-to-total cross-sections, \(\sigma _\mathrm{el}/\sigma _\mathrm{tot} \approx \mathrm{const}(s)\), while at the LHC energies, \(\rho _0\) is not negligibly small and the elastic-to-total cross section ratio is a strongly rising function of s. Here, we just note about the geometric scaling as one of the earliest proposals to have a data-collapsing behavior in elastic scattering, but we look in detail for other kind of scaling laws that are more in harmony and consistency with the recent LHC measurements [43].

Let us first detail the following two dimensionless scaling functions proposed in Ref. [33] and denoted as F(y) and G(z) in what follows. These scaling functions were introduced in order to cross-check if elastic pp collisions at the LHC energies approach the so-called black-disc limit, expected at ultra-high energies, or not. In a strong sense, the black disc limit corresponds to the shadow profile \(P(b) = \theta (R_b - b)\) that results in \(\sigma _\mathrm{el} /\sigma _\mathrm{tot} = 1/2\), independently of the black disc radius \(R_b\). This limit is clearly not yet approached at LHC energies, but in a weak sense, a black-disc limit is considered to be reached also if the shadow profile function at \(b=0\) reaches unity, i.e. \(P(b=0) = 1\), corresponding to black disc scattering at zero impact parameter. This kind of black disc scattering might have been approached at \(\sqrt{s} = 7\) TeV LHC energy [30].

The first scaling function of the differential cross-section is defined as follows:

In the diffractive cone approximation, the s-dependence in F(y) does not cancel, but it can be approximately written as

This result clearly indicates that in the diffractive cone, generally the F(y) scaling is violated by energy-dependent factors, while in the black-disc limit of elastic scattering, corresponding to \(\frac{\sigma _{tot}(s)}{\sigma _{el}(s)} \rightarrow 2\) and \(\rho _0(s) \rightarrow 0\), the F(y) scaling becomes valid as detailed and discussed in Ref. [33]. Indeed, the aim to introduce the scaling function F(y) was to clarify that even at the highest LHC energies we do not reach the black-disk limit (in the strong sense). As discussed in the previous section, the deviations from the black-disc limit might be due to the effects of the real part and the hollowness, i.e. reaching a black-ring limit instead of a black-disc one at the top LHC energies.

Since in the F(y) scaling function the position of the diffractive minimum (dip) remains s-dependent, yet another scaling function denoted as G(z) was proposed to transform out such s-dependence of the dip. This function was introduced also in Ref. [33] as follows:

In principle, all black-disc scatterings, regardless of the value of the total cross section, should show a data-collapsing behaviour to the same G(z) scaling function. As observed in Ref. [33], such an asymptotic form of the G(z) scaling function is somewhat better approached at the LHC energies as compared to the lower ISR energies but still not reproduced it exactly. This is one of the key indications the black-disc limit in the elastic pp scattering is not achieved at the LHC, up to \(\sqrt{s} = 13\) TeV. This may have several other important implications. For example, this result indicates that in simulations of relativistic heavy-ion collisions at the LHC energies, more realistic profile functions have to be used to describe the impact parameter dependence of the inelastic pp collisions: a simple gray or black-disc approximation for the inelastic interactions neglects the key features of elastic pp collisions at the TeV energy scales.

One advantage of the scaling variables y and z mentioned above is that they are dimensionless. Numerically, G(z) corresponds to the F(y) function if the scaling variable y is rescaled to z. As indicated in Fig. 23 of Ref. [33], indeed the main difference between F(y) and G(z) is that the diffractive minimum is rescaled in G(z) to the \(z=1\) position, so G(z) has less evolution with s as compared to F(y). However, as it is clear from the above discussion, the function

is well-defined only for pp elastic scattering, where a unique dip structure is observed experimentally.

Even the dip region is not always measurable in pp reactions if the experimental acceptance is limited to the cone region, which is a sufficient condition for the total cross section measurements. If the acceptance was not large enough in |t| to observe the diffractive minimum, or, in the case when the diffractive minimum did not clearly exist, then neither the F(y) nor the G(z) scaling functions would be usable. So, the major disadvantage of these scaling functions for extracting the Odderon signatures from the data is that in \(p\bar{p}\) collisions no significant diffractive minimum is found by the D0 collaboration at 1.96 TeV [8]. Besides, even if z variable were defined, the above expressions indicate, in agreement with Fig. 23 of Ref. [33], that the G(z) scaling function has a non-trivial energy-dependent evolution in the cone (\(z \ll 1\)) region. Due to these reasons, variables z and y are not appropriate scaling variables for a scale-invariant analysis of the crossing-symmetry violations at high energies.

Having recapitulated the considerations in Ref. [43], with an emphasis on the s-dependence of the parameters, let us now consider, how these s-dependencies can be scaled out at low values of |t|, where the diffraction cone approximation is valid, by evaluating the scaling properties of the experimental data on the differential elastic pp and \(p\bar{p}\) cross sections. For this purpose, let us look into the scaling properties of the differential cross sections and their implications related to the Odderon discovery in a new way.

4.3 A new scaling function for the elastic cone

In the elastic cone region, all the pp and \(p\bar{p}\) differential cross sections can be rescaled to a straight line in a linear-logarithmic plot, when the horizontal axis is scaled by the slope parameter to \(-t B(s)\) while the vertical axis is simultaneously rescaled by \(B(s) \sigma _\mathrm{el}(s)\), namely,

This representation, in the diffractive cone, scales out the s-dependencies of the total and elastic cross section, \(\sigma _\mathrm{tot}(s)\) and \(\sigma _\mathrm{el}(s)\), and also that of the slope parameter, B(s). As a function of the scaling variable \(x = - tB\), it will correspond to the plot of \(\exp (-x)\) i.e. a straight line with slope \(-1\) on a linear-logarithmic plot. It is well-known that the elastic scattering is only approximately exponential in the diffractive cone, but by scaling out this exponential feature one may more clearly see the scaling violations on this simple scaling plot. We will argue that such a scaling out of the trivial energy-dependent terms can be used as a powerful method in the search for the elusive Odderon effects in the comparison of elastic pp and \(p\bar{p}\) data in the TeV energy range.

In what follows, we investigate the scaling properties of the new scaling function,

This simple function has four further advantages summarized as follows:

1. 1.

   First of all, it satisfies a sum-rule or normalization condition rather trivially, \(\int dx H(x) = 1\), as follows from the definition of the elastic cross section.

2. 2.

   Secondly, if almost all of the elastically scattered particles belong to the diffractive cone, the differential cross-section at the optical point is also given by \( \left. \frac{d\sigma }{dt}\right| _{t=0} \, = \, A(s)\, = \, B(s) \sigma _\mathrm{el}(s)\), and in these experimentally realized cases we have another (approximate) normalization condition, namely, \(H(0) = 1.\)

3. 3.

   Third, in the diffractive cone, all the energy dependence is scaled out from this function, i.e., \(H(x) = \exp (-x)\) that shows up as a straight line on a linear-logarithmic plot with a trivial slope \(-1\).

4. 4.

   Last, but not least, the slope parameter B(s) is readily measurable not only for pp but also for \(p\bar{p}\) collisions, hence the pp and the \(p\bar{p}\) data can be scaled to the same curve without any experimental difficulties.

Let us first test these ideas by using the ISR data in the energy range of \(\sqrt{s} = 23.5{-}62.5\) GeV. The results are shown in Fig. 1 which indicates that the ISR data indeed show a data-collapsing behaviour.

At low values of x, the scaling function is indeed, approximately, \(H(x) \simeq \exp (-x)\), that remains a valid approximation over, at least, five orders of magnitude in the decrease of the differential cross section. However, at the ISR energies, the scaling seems to be valid, within the experimental uncertainties, not only at low values of \(x = - B t\), but extended to the whole four-momentum transfer region, including the dip and bump region \((15 \le x \le 30)\) as well. Even at large-|t| after the bump region, corresponding to \(x \ge 30\), the data can approximately be scaled to the same, non-exponential scaling function: \(H(x) \ne \exp (-x)\) in the tails of the distribution. Thus, Fig. 1 indeed indicates a non-trivial data-collapsing behaviour to the same, non-trivial scaling function at the ISR energy range of \(\sqrt{s} = 23.5{-}62.5\) GeV.

This observation motivated us to generalize the derivation presented above in this section, to arbitrary positively definite non-exponential scaling functions H(x). Such a generalisation is performed in the next subsection, in order to give a possible explanation of the data-collapsing behaviour in Fig. 1.

Scaling behaviour of the differential cross section \(d\sigma /dt\) of elastic pp collisions in the ISR energy range of \(\sqrt{s} = 23.5{-}62.5\) GeV. The measured differential cross section data are taken from Ref. [63] and references therein. These data are rescaled to \(H(x) = \frac{1}{B\sigma _\mathrm{el}} \frac{d\sigma }{dt}\) as a function of \(x = - t B\). This figure indicates a clear, better than expected data-collapsing behaviour.

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4.4 Generalized scaling functions for non-exponential differential cross-sections

In this section, we search for a novel type of scaling functions of pp elastic data that may be valid not only in the diffractive cone, but also in the crucial dip and bump region, as well. In Fig. 1, we have noticed that the data-collapsing behaviour may extend well above the small \(x = - tB\) region significantly beyond the diffractive maximum, indicating a clear deviation of the scaling function H(x) from the exponential shape.

In addition, a recent detailed study of the low-|t| behaviour of the differential elastic pp cross section at \(\sqrt{s} = 8\) TeV observed a more than 7\(\sigma \)-significant deviation from the exponential shape [64, 65], which also corresponds to a non-exponentiality in the scaling function H(x) even in the low-|t|, or small x, range.

In this section, we thus further generalize the derivation of the \(H(x) = \exp (-x)\) scaling function, in order to allow for arbitrary positively definite functions with \(H(x=0) = 1\) normalisation, and to develop a physical interpretation of the experimental observations.

Let us start the derivation from the relation of the elastic scattering amplitude in the impact parameter space \(t_\mathrm{el}(s,b)\) and the complex opacity function \(\Omega (s,b)\) based on Eq. (12), using the same notation as in Ref. [30]:

The shadow profile function P(s, b) is equal to the inelastic scattering profile \(\tilde{\sigma }_{in}(s,b)\) as follows from Eq. (13), \(P(s,b) = \tilde{\sigma }_\mathrm{in}(s,b)\). The imaginary part of the opacity function \(\Omega \) is generally not known or less constrained by the data, but it is experimentally known that \(\rho _0(s)\) is relatively small at high energies: at all the measured LHC energies and below, \(\rho _0 \le 0.15\), hence, \(\rho ^2 \le 2.3 \)%.

Here, we thus follow the choice of Ref. [30], that has demonstrated that the ansatz

gives a satisfactory description of the experimental data in the \(-t \le 2.5\) GeV\(^2\) region, with a small coefficient of proportionality that was denoted in Ref. [30] by \(\alpha \propto \rho _0\) parameter. This ansatz assumes that the inelastic collisions at low four-momentum transfers correspond to the cases when the parts of proton suffer elastic scattering but these parts are scattered to different directions, not parallel to one another. This physical interpretation is actually due to \(\rho _0 \ll 1\) and \(\hbox { Im} \, \Omega (s,b) \ll 1\). We will use this approximation below to demonstrate that the H(x) scaling function can have more complex shapes, that differ from \( H(x) = \exp (-x)\).

Based on the results of the previous section obtained in the diffractive cone in the \(\rho _0 \ll 1\) and \(\tilde{\sigma }(s,b) \ll 1\) limit, we have the following scaling property of the opacity function:

where r(s) is four times the ratio of the elastic to the total cross section, as given in Eq. (27), and \(E(\tilde{x})\) describes the distribution of the inelastic collisions as a function of the dimensionless impact parameter b normalised to \(\sqrt{B(s)}\), the characteristic length-scale of the pp collisions at a given value of the center-of-mass energy \(\sqrt{s}\).

This ansatz allows for a general shape of the impact parameter b-dependent scattering amplitude, that leads to a H(x) scaling. Under the assumption that the b-dependence may occur only through the two-dimensional scaling variable \(\tilde{x}\), as described by the scaling function \(E(\tilde{x})\),

a general form of the H(x) scaling can be obtained. Here we assume that \(E(\tilde{x})\) is a real function that depends on the modulus of the dimensionless impact parameter \(\tilde{x} = b/R(s)\). For normalization, we choose that the Fourier-transform \(\tilde{E}({0}) = 1\), which also corresponds to the condition

keeping in mind that we have two-dimensional Fourier-transform which at zero is equal to the integral over the two different directions in the impact-parameter space.

Let us investigate first the consequences of the scaling ansatz of Eq. (47) for the shadow profile function P(s, b). The algebra is really very similar to that of the exponential cone approximation that was implemented above. We obtain the following result:

Evaluating the above relation at \(b=0\) and using the normalization condition \(E({0}) = 1\), we obtain again that the shadow profile at zero impact parameter value has a maximum that is slightly less than unity: \(P(s,0) \le 1/(1+\rho _0^2)\). It is interesting to note that the maximum in the profile function is reached at the same threshold (29) as in the case of the exponential cone approximation, corresponding to

Thus a threshold-crossing behaviour seems to happen if the elastic-to-total cross-section ratio exceeds 0.25. Remarkably, in the domain of validity of our derivation, this threshold crossing point is independent of the detailed shape of the H(x) scaling function for a broad class of models. However, it is also clear from Eq. (49) that the shape of \(E(\tilde{x})\) function plays an important role in determining the hollowness effect, so a detailed precision shape analysis is necessary to obtain the significance of this effect.

Starting from the definition, Eq. (2), the scattering amplitude in the b-space (47) yields the following form of the differential cross section in the momentum space:

Utilizing Eq. (46), we find that this form of the differential cross section is dependent on the four-momentum transfer squared, t, indeed only through the variable \(x \equiv - B(s) t = R^2(s) \varDelta ^2\), so it is a promising candidate to be a scaling variable.

Now, if we consider the function (52) at the optical point, \(t = 0\), we find

If the impact parameter dependent elastic amplitude has an s-dependent internal scale and s-dependent strength, we thus obtain the following generalized scaling relation for arbitrary elastic scattering amplitudes that satisfy Eq. (47):

This scaling is derived for \(\rho _0 \ll 1\) and \(\tilde{\sigma }(s,b) \ll 1\), and it indicates that the H(x) with a non-exponential scaling function is a very interesting theoretical possibility. Further generalizations of this derivation are possible and interesting but go clearly well beyond the scope of this manuscript, that aims to look for Odderon effects using the experimentally available information on this H(x) scaling and its possible violations.

In addition to providing an insight to the meaning of the non-exponential behaviour in the interference (dip and bump) region, the above derivation also clarifies meaning of the normalization of H(x). In particular, the normalization of H(x) scaling function on the left hand side of Eq. (54) should be made by the value of the differential cross section at the optical (\(t = 0\)) point as given by Eq. (53). This value for differential cross sections with nearly exponential diffractive cone is indeed approximately equal to \(A(s) = B(s) \sigma _\mathrm{el}(s)\). In this case, the normalization condition \(H(0) = 1\) is maintained, while the integral of H(x) becomes unity only for differential cross sections dominated by the exponential cone (i.e. when the integral contribution from the non-exponential tails is several orders of magnitude smaller as compared to the integral of the cone region).

For the total cross section, we find from Eq. (5)

Note that here we have indicated the normalization just for clarity, but one should keep in mind that in our normalization, \(\tilde{E}(0) = 1\), and correspondingly, \(H(x=0)=1\) by definition.

As clarified by Eq. (54), the scaling function H(x) coincides with the modulus squared of the normalized Fourier-transform of the scaling function \(E(\tilde{x})\), if the elastic amplitude depends on the impact parameter b only through its scale invariant combination \(x = \frac{b}{R(s)}\) and if \(\rho (s,t) \equiv \rho _0(s)\). In this case, the H(x) scaling is directly connected to the impact parameter dependence of the elastic amplitude and transforms out the trivial s-dependencies coming from \(\sigma _\mathrm{tot}(s)\), \(\sigma _\mathrm{el}(s)\), B(s), and \(\rho _0(s)\) functions. This approximation has enabled us to establish possible physical reasons of this new scaling, and to derive non-exponential shapes for the H(x) scaling function and to connect violations of the H(x) scaling to the hollowness effect in the shadow profile function of the proton at ultra-high energies. At the time of closing this manuscript, the generalization of the above derivation to a t-dependent \(\rho (s,t)\) function is still incomplete, and will be the subject of a separate study. Nevertheless, in our numerical analysis of the H(x) scaling, detailed in the subsequent sections, in the comparisons of the scaled differential cross-sections and the deduced Odderon significance we have not imposed any \(\rho (s,t) \equiv \rho (s)\) condition. Our analysis is generic and has been done using the published experimental data sets only, without imposing any theoretical assumptions such as a t-independent \(\rho (s,t)\) etc.

The above derivation also indicates that it is a promising possibility to evaluate the H(x) scaling function directly from the experimental data. It has a clear normalization condition, \(H(0) = 1\). Furthermore, in the diffractive cone, for nearly exponential cone distributions, \(H(x) \approx \exp (-x)\). We have shown in this section, that even if one neglects the possible t dependence of \(\rho (s,t)\), arbitrary positively definite H(x) scaling functions can be introduced if the elastic amplitude is a product of s-dependent functions, and its impact parameter dependence originates only through an s-dependent scaling variable which can be conveniently defined as \(\tilde{x}^2 = \frac{b^2}{B(s)}\). Thus, the violations of the H(x) scaling may happen if not only the slope parameter B(s), the real-to-imaginary ratio \(\rho _0(s)\) and the integrated elastic and total cross sections \(\sigma _\mathrm{el}(s)\) and \(\sigma _\mathrm{tot}(s)\) depend on s, but also the b-dependence of the elastic scattering amplitude starts to change noticeably. Namely, the H(x) scaling breaks if the scaling relation \(t_\mathrm{el}(b,s) = C(s) E(b/R(s))\) gets violated in the above mentioned case.

Let us also note that the leading-order exponential shape of \(H(x) \approx \exp (-x)\) can be derived as a consequence of the analyticity of \(T_\mathrm{el}(s,\varDelta )\) at \(\varDelta = 0\) corresponding to the \(t =0\) optical point, as follows. By leading order we mean the result of a first-order Taylor series expansion at \(x = 0\), so that \(H(x) \approx \exp (-x) \approx 1 - x\), although beyond this approximation the functional behaviour of the H(x) function cannot be determined from analyticity. If \(T_{\mathrm el}(s,\varDelta )\) is an analytic function at \(\varDelta = 0\), then its leading-order behaviour is \(T_{\mathrm el}(s,0) + c(s) \varDelta \), where c(s) is a complex coefficient that is in general dependent on s. Hence, in this approximation the differential cross-section behaves as \(d\sigma /dt \simeq A(s) \exp \left( B(s) t\right) \approx A(s) (1 + B(s) t + \cdots )\) corresponding to the scaling function \(H(x) \approx \exp (-x)\) in the diffractive cone. Similar considerations, related to (non)-analyticity of modulus squared amplitudes and Lévy stable source distributions were introduced to Bose–Einstein correlations in high energy physics in Ref. [66].

On the other hand, our recent analysis of the differential elastic cross sections in the LHC energy range [9, 26] suggests that the approximation \(H(x) \approx \exp (-x)\) breaks down since the TOTEM experiment observed a significant non-exponential behaviour already in the diffractive cone. In this case, at low values of |t|, nearly Lévy stable source distributions can be introduced, that lead to an approximate \(H(x)\propto \exp (-x^{\alpha })\) behaviour, where \(\alpha = \alpha _\mathrm{Levy}/2 \le 1.\) In this case, the leading order behaviour is non-analytic, \(H(x) \approx 1 -x^{\alpha }\). We have shown in Refs. [9, 26], at low |t|, such a stretched exponential form with \(\alpha \simeq 0.9\) describes the elastic scattering data from ISR to LHC energies reasonably well in a very broad energy range from 23.5 GeV to 13 TeV.

The main limitation of the above derivation is that although it leads to a H(x) scaling, the real-to-imaginary ratio \(\rho (s,t) \rightarrow \rho _0(s)\) is independent of t in this approximation. So let us consider a generalization, where the real to imaginary ratio is not only s but also t dependent. We will discuss, model independently, such a scenario in terms of the impact parameter dependent elastic scattering amplitude in Appendix B. Such a t dependence of \(\rho (s,t) \) can actually be realized in a number of physical models. In greater details, we consider one particular model, that has a H(x, s) type of scaling limit and the s-dependent scaling violations are related to the s-dependence of the opacity parameter in this model. We discuss the emergence of the H(x) scaling within a physical model, the so-called Real Extended Bialas–Bzdak model of Refs. [30, 31, 33, 67,68,69,70] in Appendix C. We evaluate the domain of validity of this ReBB model in \((s,x = -tB)\) in Appendix D, in order to determine if this domain is including (or not) a kinematic region, where the H(x) scaling indicates the Odderon signal.

We established that the H(x) scaling holds within experimental errors at the ISR center-of-mass energies varying from 23.5 to 62.5 GeV, i.e. less than by a factor of three. Let us also investigate the same scaling function at the LHC energies, where the TOTEM measurements span, on a logarithmic scale, a similar energy range, from 2.76 to 13 TeV, i.e. slightly more than by a factor of four. The TOTEM data at 13, 7 and 2.76 TeV are collected from Refs. [1, 28], and Ref. [4], respectively, and plotted in Fig. 2. Note that the possible scaling violating terms are small in the \(\sqrt{s} = 2.76 - 7\) TeV region: they are within the statistical errors, when increasing \(\sqrt{s}\) from 2.76 to 7 TeV, i.e. by about a factor of 2.5. Let us also stress that we do not claim the validity of the H(x) scaling up to the top LHC energy of \(\sqrt{s} = 13\) TeV, as scaling violating terms start to be significant at that energy, in particular, close to the diffractive dip region.

Let us look into the scaling behaviour in the energy range of \(\sqrt{s} = 2.76{-}7\) TeV in more detail.

Scaling behaviour of the differential cross section \(d\sigma /dt\) of elastic pp collisions at LHC energies. Elastic scattering data are measured by the TOTEM Collaboration at \(\sqrt{s} = 13 \) TeV [1], at \(\sqrt{s}= 7\) TeV [28], and at \(\sqrt{s} = 2.76 \) TeV [4]. Left panel shows the 2.76 and 7 TeV data points with statistical errors only, while the right panel shows the 7.0 and 13.0 TeV data with statistical and t-dependent systematic errors added in quadrature. The left panel indicates, that the H(x) scaling is within statistical errors valid between \(\sqrt{s} = 2.76\) and 7.0 TeV, so the H(x) scaling works from 7 TeV downwards. The right panel indicates that the H(x) scaling is violated, when the colliding energy is increased from \(\sqrt{s} = 7.0\) to 13 TeV: the right panel indicates scaling violations that go well beyond the combined statistical and systematic errors

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The left panel of Fig. 2 indicates that the H(x) scaling valid within statistical errors in the \(\sqrt{s} = 2.76{-}7\) TeV energy range. The confidence level of this comparison corresponds to a CL = 99% (statistical errors only). The right panel of the same Fig. 2 indicates that this scaling is violated, beyond systematic errors, if the \(\sqrt{s} = 13\) TeV data are also included into this comparison: the violation of the H(x) scaling by the 13 TeV data is focused to the region of the diffractive dip. However, in the \(x < 10\) region, the H(x) scaling is approximately valid at each of these LHC energies of \(\sqrt{s} = 2.76\), 7 and 13 TeV. Instead of being approximately valid in the whole measurable x region, at the LHC this scaling remains valid at all these three LHC energies only through about 3-4 orders of magnitude drop in the differential cross-section at lower values of x. The so called “swing” effect becomes clear at \(\sqrt{s} = 13\) TeV: the scaling function starts to decrease faster than exponential before the diffractive minimum, and also the diffractive minimum moves to lower values in x as compared to its position at lower LHC energies. This swing effect, apparent in Fig. 2, can be interpreted in terms of changes in the shadow profile of protons at the LHC energies as the energy range increases from 2.76 through 7 to 13 TeV. Indeed, such small s-dependent scaling violations in the H(x) scaling function show the same qualitative picture as what has been observed by the direct reconstruction of the P(s, b) shadow profiles in the TeV energy range in several earlier papers, see for example Refs. [37, 38, 71] or our Refs. [9, 26, 30].

Inspecting the left panel of Fig. 2, we find, that the H(x) scaling functions agree within statistical errors, if the colliding energy is increased from \(\sqrt{s} = 2.76\) to 7 TeV. The right panel of the same figure shows that these data change significantly if the colliding energy increases further to \(\sqrt{s } = 13\) TeV. This implies that the possible scaling violating terms are small as they are within the statistical errors, when increasing \(\sqrt{s}\) from 2.76 to 7 TeV, by about a factor of 2.5. We have checked that TOTEM preliminary data at \(\sqrt{s}\) \(=\) 8 TeV also satisfy this H(x) scaling [72, 73].

However, this H(x) scaling is violated by s-dependent terms when increasing \(\sqrt{s}\) from 8 to 13 TeV, and such a scaling violation is significantly larger than the quadratically (maximally) added statistical and t-dependent systematic errors, as indicated on the right panel of Fig. 2.

This behaviour may happen due to approaching a new domain, where the shadow profile function of pp scattering changes from a nearly Gaussian form to a saturated shape, that in turn may develop hollowness at 13 TeV and higher energies. The experimental indications of such a threshold-crossing behaviour were summarized recently in Ref. [43], and are also described above: a new domain may be indicated by a sudden change of B(s) in between 2.76 and 7 TeV and, similarly, the crossing of the critical \(\sigma _\mathrm{el}(s)/\sigma _\mathrm{tot}(s) = 1/4\) line in multi-TeV range of energies, somewhere between 2.76 and 7 TeV. From the theoretical side, we have previously noted such as drastic change in the size of the proton substructure between the ISR and LHC energy domains from a dressed quark-like to a dressed di-quark type of a substructure [9, 26] which may be, in principle, connected to such a dramatic change in the scaling behaviour of the elastic cross section. However, in this work we focus on the scaling properties of the experimental data, and do not intend to draw model-dependent conclusions. Nevertheless, we use the model-dependent results as well in order to cross-check our model-independent conclusions. Some details of the model-independent calculations are summarized in Appendix A and Appendix B, while our model-dependent estimates are described in Appendix C, Appendix D and Appendix E.

In Fig. 3 we directly compare the H(x) scaling functions of the differential cross sections, using the same ISR and LHC data, as in Figs. 1 and 2, respectively. This range of data now spans nearly a factor of about 500, about a three orders of magnitude increase in the range of available colliding energies, from 23.5 GeV to 13 TeV. As can be seen in the corresponding Fig. 3, the scaling works approximately in the diffractive cone, however, the H(x) scaling function cannot be considered as an approximately constant if such a huge change in the colliding energies is considered.

Scaling behaviour of the differential cross section \(d\sigma /dt\) of elastic pp collisions from ISR to LHC energies. Data points are the same as shown in Figs. 1 and 2. (Left panel): Data points are shown with statistical errors only. (Right panel): Same data set, but now showing both statistical and t-dependent systematic errors added in quadrature

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Comparing Figs. 1, 2 and 3, we find that the s-dependence of the H(x) scaling functions is rather weak if s changes within a factor of two, however, there are very significant changes if the range of energies is changing by a factor of a few hundred, from the ISR energy range of \(\sqrt{s} = 23.5{-}62.5\) GeV to the LHC energy range of 2.76–7.0–13.0 TeV.

In the left panel of Fig. 4, the H(x) function of the \(\sqrt{s} = 2.76 \) TeV TOTEM data set of Ref. [4] is compared with that of the \(p\bar{p}\) collisions measured by the D0 collaboration at \(\sqrt{s} = 1.96 \) TeV Tevatron energy [8]. The right panel of Fig. 4 compares the H(x) scaling functions of elastic pp collision at \(\sqrt{s} = 7\) TeV LHC energy [28, 74] to that of the elastic \(p\overline{p}\) collisions at the Tevatron energy, \(\sqrt{s} = 1.96\) TeV. On both panels, the statistical errors and t-dependent systematic errors are added in quadrature. Lines are shown to guide the eye corresponding to fits with the model-independent Lévy series studied in Refs. [9, 26]. These plots suggest that the comparison of the H(x) scaling functions or elastic pp to \(p\bar{p}\) collisions in the TeV energy range is a promising method for the Odderon search, and a precise quantification of the difference between the H(x) scaling functions for pp to \(p\bar{p}\) collisions data sets is important. But how big is the difference between the H(x) scaling functions of elastic pp collisions at similar energies?

Left panel: Scaling function \(H(x) = \frac{1}{B \sigma _\mathrm{el} }\frac{d\sigma }{dt}\) of the differential cross section of elastic pp collisions at \(\sqrt{s} = 2.76\) TeV LHC (red), as compared to that of the elastic \(p\overline{p}\) collisions at the Tevatron energy of \(\sqrt{s} = 1.96\) TeV (blue), shown as a function of \(x = -tB\). Right panel: Same as the left panel, but now using elastic pp data at \(\sqrt{s} = 7\) TeV (red), as compared to elastic \(p\overline{p}\) collisions at \(\sqrt{s} = 1.96\) TeV (blue). On both panels, statistical errors and t-dependent systematic errors are added in quadrature. Lines are shown to guide the eye, corresponding to fits with the model-independent Lévy series from Refs. [9, 26]

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Same as Fig. 4, but now the H(x) scaling of the differential cross section \(d\sigma /dt\) of elastic pp collisions is compared at the nearby \(\sqrt{s} = 2.76\) and 7 TeV LHC energies. Left panel shows the data with statistical errors only, while on the right panel, statistical errors and t-dependent systematic errors are added in quadrature. The two H(x) scaling functions are, within statistical errors, apparently the same

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The H(x) scaling of the differential cross section \(d\sigma /dt\) of elastic pp collisions is compared at the nearby \(\sqrt{s} = 2.76\) and 7 TeV LHC energies in Fig. 5. These plots are similar to the panels of Fig. 4. The H(x) scaling functions are remarkably similar, in fact, they are the same within the statistical errors of these measurements. Due to their great similarity, it is important to quantify precisely how statistically significant their difference is.

We stress in particular that the possible scaling violations are small, apparently within the statistical errors, when pp results are compared at LHC energies and \(\sqrt{s}\) is increased from 2.76 to 7 TeV, by about a factor of 2.5. This makes it very interesting to compare the differential cross-sections of pp and \(p\bar{p}\) elastic scattering at the nearest measured energies in the TeV range, where crossing-odd components are associated with Odderon effects. Actually, the largest \(\sqrt{s}\) of \(p\bar{p}\) elastic scattering data is 1.96 TeV [8] while at the LHC the public data set on the elastic pp scattering is available at \(\sqrt{s} = 2.76\) TeV [4], corresponding to a change in \(\sqrt{s}\) by a factor of \(2.76/1.96 \approx 1.4\). This is a rather small multiplicative factor on the logarithmic scale, relevant to describe changes both in high energy pp and \(p\bar{p}\) collisions. Given that the H(x) scaling function is nearly constant between 2.76 and 7 TeV within the statistical errors of these data sets, we will search for a significant difference between the H(x) scaling function of elastic pp collisions at \(\sqrt{s} = 2.76 \) and 7 TeV as well as that of the elastic \(p\bar{p}\) scattering at \(\sqrt{s} = 1.96 \) TeV. If such a difference is observed, then there must be a crossing-odd (Odderon) component in the scattering amplitude of elastic pp and \(p\bar{p}\) scatterings.

Approximate \(H(x) = \frac{1}{B \sigma _\mathrm{el}} \frac{d \sigma }{dt}\) scaling of the differential cross section \(d\sigma /dt\) of elastic \(p\overline{p}\) collisions at \(\sqrt{s} = 0.546\) to 1.96 TeV. The scaling behaviour is valid in the exponential cone region, with the scaling function \(H(x) = \exp (-x)\). The scaling domain starts at \(x = 0\) and extends up to \(x = -tB \simeq 10\). Scaling violations are evident in the \(-t B \ge 10\) region, when the colliding energy increases from 546 GeV to 1.96 TeV, nearly by a factor of four

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Let us now consider Fig. 6. This plot compares the H(x) scaling functions for \(p\bar{p}\) collisions at various energies from \(\sqrt{s} = 546\) GeV to 1.96 TeV. Within experimental errors, an exponential cone is seen that extends to \(x = - t B \approx 10\) at each measured energies, while for larger values of x the scaling law breaks down in an energy dependent manner. At lower energies, the exponential region extends to larger values of \(x \approx 13\), and the tail regions are apparently changing with varying colliding energies. Due to this reason, in this paper we do not scale the differential cross section of elastic \(p\bar{p}\) collisions to different values of \(\sqrt{s}\) as this cannot be done model-independently. This property of elastic \(p\bar{p}\) collisions is in contrast to that of the elastic pp collisions, where we have demonstrated in Figs. 1, 2 that in a limited energy range between \(\sqrt{s} = 23.5\) and 62.5 GeV, as well as at the LHC in the energy range between \(\sqrt{s} = 2.76\) and 7 TeV, the H(x) scaling works well. Due to these experimental facts and the apparent violations of the H(x) scaling for \(p\bar{p}\) collisions in the \(x = -t B \ge 10\) region, in this paper we do not attempt to evaluate the energy dependence of the differential cross sections for \(p\bar{p}\) collisions. However, based on the observed H(x) scaling in pp collisions, we do find a model-independent possibility to rescale the differential cross sections of elastic pp collisions in limited energy ranges.

Rescaling of the differential cross section of elastic pp collisions at the ISR and LHC energies, using Eq. (67). This demonstrates that our method can also be used to get the differential cross sections at other energies by such a rescaling procedure, provided that the nuclear slope and the elastic cross sections are known at the new energy as well as at the energy from where we start to rescale the differential cross section. In all panels, we have evaluated the level of agreement between the rescaled and measured data with the help of Eq. (60). Left panel: Rescaling of the differential cross sections from the lowest ISR energy of \(\sqrt{s} = 23.5 \) to the highest ISR energy of 62.5 GeV. The level of agreement between the rescaled 23.5 GeV pp data and the measured 62.5 GeV pp data corresponds to \(\chi ^2/\mathrm{NDF} = 111.0/110\) with a CL = 21.3% , that indicates an agreement within 1.3\(\sigma \). Middle panel: Rescaling of the differential cross section of elastic pp collisions from the energy of \(\sqrt{s} = 7\) TeV [28, 74] down to 2.76 TeV [4]. The level of agreement between the rescaled 7.0 TeV pp data and the measured 2.76 TeV pp data corresponds to \(\chi ^2/\mathrm{NDF} = 39.3/63\) with a CL = 99.2%, that indicates an agreement, within 0.01\(\sigma \), corresponding to a nearly vanishing deviation. Right panel: Rescaling of the differential cross section of elastic pp collisions from the energy of \(\sqrt{s} = 2.76\) TeV, measured by TOTEM [4], down to 1.96 TeV, where it is compared to the D0 dataset of Ref. [8]. The level of agreement between the rescaled 2.76 TeV pp data and the measured 1.96 TeV \(p\overline{p}\) data is quantified by a \(\chi ^2/\mathrm{NDF} = 18.1/11\) and a CL = 7.9%, that indicates an agreement within 1.76\(\sigma \).

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After the above qualitative discussion of H(x) scaling for both pp and \(p\bar{p}\) elastic collisions, let us work out the details of the possibility of rescaling the measured differential cross sections to other energies in the domain where H(x) indicates a scaling behaviour within experimental errors.

The left panel of Fig. 7 indicates the result of rescaling of the differential cross sections of elastic pp scattering from the lowest \(\sqrt{s} = 23.5\) GeV to the highest 62.5 GeV ISR energy, using Eq. (67). We have evaluated the level of agreement of the rescaled 23.5 GeV pp data with the measured 62.5 GeV pp data with the help of Eq. (60). The result indicates that the data measured at \(\sqrt{s} = 23.5\) GeV and duly rescaled to 62.5 GeV are, within the errors of the measurements, consistent with the differential cross section of elastic pp collisions as measured at \(\sqrt{s} = 62.5\) GeV. This demonstrates that our method can also be used to extrapolate the differential cross sections at other energies by rescaling, provided that the H(x) scaling is not violated in that energy range and that the nuclear slope and the elastic cross sections are known at a new energy as well as at the energy from where such a rescaling starts.

A similar method is applied at the LHC energies in the middle panel of Fig. 7. This plot also indicates a clear agreement between the 2.76 TeV data and the rescaled 7 TeV data, which corresponds to a \(\chi ^2/\mathrm{NDF} = 39.3/63\) and a CL of 99.2% and a deviation on the 0.01 \(\sigma \) level only. This suggests that indeed the rescaling of the differential cross section of elastic scattering can be utilized not only in the few tens of GeV range but also in the few TeV energy range. Most importantly, this plot indicates that there is a scaling regime in elastic pp collisions, that includes the energies of \(\sqrt{s} = \) 2.76 and 7 TeV at LHC, where the H(x) scaling is within errors, not violated. This is in a qualitative contrast to the elastic \(p\bar{p}\) collisions at TeV energies, where the validity of the H(x) scaling is limited only to the diffractive cone region with \(x \le 10\), while at larger values of x, the H(x) scaling is violated.

The right panel of Fig. 7 indicates a surprising agreement: after rescaling of the differential cross section of elastic pp collisions from 2.76 to 1.96 TeV, we find no significant difference between the rescaled 2.76 TeV pp data with the \(p\bar{p}\) data at the same energy, \(\sqrt{s} = 1.96 \) TeV. The agreement between the extrapolated pp and the measured \(p\bar{p}\) differential cross sections correspond to an agreement at a CL of 7.9%, i.e. a surprising agreement at the \(1.76\sigma \) level. It can be seen on the right panel of Fig. 7 that in the swing region, before the dip, the rescaled pp differential cross section seems to differ qualitatively with the \(p\bar{p}\) collisions data. However, according to our \(\chi ^2\) analysis that also takes into account the horizontal errors of the TOTEM data, we find that this apparent qualitative difference between these two data sets is quantitatively not significant: it is characterized as an agreement within less than 2\(\sigma \).

These plots suggest that the H(x) scaling functions of elastic pp and \(p\bar{p}\) collisions differ at similar energies, while the same scaling functions for elastic pp collisions are similar at similar energies, thus the comparison of the H(x) scaling functions of elastic pp and \(p\bar{p}\) collisions is a promising candidate for an Odderon search. Due to this reason, it is important to quantify how significant is this difference, given that the H(x) scaling functions scale out the dominant s-dependent terms, that arise from the energy-dependent \(\sigma _\mathrm{el}(s)\) and B(s) functions. Such a quantification is the subject of the next section.

Before going into more details, we can already comment on a new Odderon effect qualitatively. When comparing the H(x) scaling function of the differential cross section of elastic pp collisions at 2.76 and 7.0 TeV colliding energies, we see no qualitative difference. By extrapolation, we expect that the H(x) scaling function may be approximately energy independent in a bit broader interval, that extends down to 1.96 TeV. Such a lack of energy evolution of the H(x) scaling function of the pp collisions is in a qualitative contrast with the evolution of the H(x) scaling functions of \(p\bar{p}\) collisions at energies of \(\sqrt{s} = 0.546{-} 1.96\) TeV, where a qualitative and significant energy evolution is seen in the \(x = -t B > 10 \) kinematic range. Thus, our aim is to quantify the Odderon effect in particular in this kinematic range of \(x = -t B > 10 \) in order to evaluate the significance of this qualitative difference between elastic pp and \(p\bar{p}\) collisions.

In this section, we investigate the question of how to compare the two different scaling functions \(H(x) = \frac{1}{B\sigma _{el}}\frac{d\sigma }{dt}\) with \(x = - t B\) introduced above measured at two distinct energies. We would like to determine if two different measurements correspond to significantly different scaling functions H(x), or not. In what follows, we introduce and describe a model-independent, simple and robust method, that enables us to quantify the difference of datasets or H(x) measurements. The proposed method takes into account the fact that the two distinct measurements may have partially overlapping acceptance in x and their binning might be different, so the datasets may correspond to two different sets of x values.

Let us first consider two different datasets denoted as \(D_i\), with \(i = 1, 2\). In the considered case, \(D_i = \big \{x_i(j), H_i(j), e_i(j)\big \}\), \(j = 1,\ldots n_i\) consists of a set of data points located on the horizontal axis at \(n_i\) different values of \(x_i\), ordered as \(x_i(1)< x_i(2)< \cdots < x_i(n_i)\), \(H_i(j) \equiv H_i(x_i(j))\) are the measured values of H(x) at \(x=x_i(j)\) points, and \(e_i(j)\equiv e_i(x_i(j))\) is the corresponding error found at \(x_i(j)\) point.

In general, two different measurements have data points at different values of x. Let us denote as \(X_1 = \big \{x_1(1),\ldots x_1(n_1)\big \}\) the domain of \(D_1\), and similarly \(X_2 = \big \{x_2(1), \ldots , x_2(n_2)\big \}\) stands for the domain of \(D_2\). Let us choose the dataset \(D_1\) which corresponds to \(x_1(1) < x_2(1)\). In other words, \(D_1\) is the dataset that starts at a smaller value of the scaling variable x as compared to the second dataset \(D_2\). If the first dataset ends before the second one starts, i.e. when \(x_1(n_1) < x_2(1)\), their acceptances would not overlap. In this limiting case, the two datasets cannot be compared with our method. Fortunately, however, the relevant cases e.g. the D0 data on elastic \(p\overline{p}\) collisions at \(\sqrt{s} = 1.96 \) TeV have an overlapping acceptance in x with the elastic pp collisions of TOTEM at \(\sqrt{s} = 2.76\), 7 and 13 TeV. So from now on we consider the case with \(x_1(n_1) > x_2(1)\).

If the last datapoint in \(D_2\) satisfies \(x_2(n_2) < x_1(n_1)\), then \(D_2\) is within the acceptance of \(D_1\). In this case, let us introduce \(f_2 = n_2\) as the final point with the largest value of \(x_f\) from \(D_2\). If \(D_2\) has \(x_2(n_2) > x_1(n_1)\), then the overlapping acceptance ends at the largest (final) value of index \(f_2\) such that \(x_2(f_2)< x_1(n_1) < x_2(f_2+1)\). This means that the point \(f_2\) of \(D_2\) is below the largest value of x in \(D_1\), but the next point in \(D_2\) is already above the final, largest value of \(x(n_1)\) in \(D_1\).

The beginning of the overlapping acceptance can be found in a similar manner. Due to our choice of \(D_1\) as being a dataset that starts at a lower value, \(x_1(1) < x_2(1)\), let us determine the initial point \(i_1\) in \(D_1\) that already belongs to the acceptance domain of \(D_2\). This is imposed by the criterion that \(x_1(i_1-1)< x_2(1) < x_1(i_1)\).

We compare the \(D_1\) and \(D_2\) datasets in the region of their overlapping acceptance, defined above, either in a one-way or in a two-way projection method. The projection \(1 \rightarrow 2\) has the number of degrees of freedom NDF\((1 \rightarrow 2)\) equal to the number of points of \(D_2\) in the overlapping acceptance. For any of such a point \(x_i(2)\), we used linear interpolation of the nearest points from \(D_1\) such that \(x_j(1) < x_i(2) \le x_{j+1}(1)\) in order to evaluate the data and the errors of \(D_1\) at this particular value of \(x = x_i(2)\). This is done employing a default (linear, exponential) scale in the (x, H(x)) plane, that is expected to work well in the diffraction cone, where the exponential cone is a straight line. However, for safety and due to the unknown exact structure at the dip and bump region, we have also tested the linear interpolation utilizing the (linear, linear) scales in the (x, H(x)) plane.

Similarly, the projection \(2 \rightarrow 1\) has the number of degrees of freedom NDF\((2\rightarrow 1)\) as the number of points of dataset \(D_1\) that fell into the overlapping common acceptance. A linear extrapolation was used for each \(x_i(1)\) points in this overlapping acceptance, so that \(x_j(2) < x_i(1) \le x_{j+1}(2)\), using both (linear, exponential) and (linear, linear) scales in the (x, H(x))-planes. For the two-way projections, for example \(1 \longleftrightarrow 2\), the number of degrees of freedom is the sum of the points of \(D_1\) and \(D_2\) in the overlapping acceptance, defined as NDF\((1\longleftrightarrow 2)\) = NDF\((1\rightarrow 2)\) + NDF\((2\rightarrow 1)\).

Let us describe the two-way projections in more detail as the one-way projections can be considered as special cases of this method. A common domain \(X_{12} = \big \{ x_{12}(1), \ldots , x_{12}(n_{12})\big \}\) in the region of the overlap of the \(X_1\) and \(X_2\) domains can be introduced as follows. Take the data points in the interval \([i_1\dots n_1]\) from the \(D_1\) set and the data points in the interval \([1\dots f_2]\) from the \(D_2\) set. This selection procedure provides a total of \(n_{12} = n_1+f_2-i_1 + 1\) points. Let us order this new set of points and denote such a united domain as \(X_{12}\). This domain corresponds to a common acceptance region which has \(n_{12}\) data points on the horizontal axis denoted as \(\big \{ x_{12}(1), \ldots , x_{12}(n_{12})\big \}\).

In order to compare the datasets \(D_1\) and \(D_2\), one needs to build two analog datasets that are both extrapolated to the same common domain \(X_{12}\) starting from \(D_1\) and \(D_2\) as if the data in both analog datasets were measured at the same values of x. So far, either \(D_1\) or \(D_2\) has some data value on any element of the domain \(X_{12}\), but only one of them is determined.

Let us take first those points from \(X_{12}\) that belong to \(D_1\), and label them with j index. There are \(n_1 - i_1 +1 \) such points. For such points, the data and error-bars of the extrapolated data set \(D_{12}\) will be taken from \(D_1\): \(d_{12}(x_{12}(j)) = d_1(x_1(j))\), \(e_{12}(x_{12}(j) = e_1(x_1(j))\). However, for the same points, \(D_2\) has no measured value. But we need to compare the data of \(D_1\) and \(D_2\) at common values of x. So \(D_2\) data and errors can be interpolated using linear or more sophisticated interpolation methods. If the binning is fine enough, linear interpolation between the neighbouring datapoints can be used.

At this point, let us consider that in the diffractive cone, when an exponential approximation to the differential cross section can be validated, the shape of the scaling function is known to be \(H(x) \approx \exp (-x)\). This function is linear on a (linear, logarithmic) plot of (x, H(x)). In what follows, we will test both a (linear, exponential) interpolation in the (x, H(x)) plots (that is expected to give the best results in the diffractive cone) and a (linear, linear) interpolation that has the least assumptions and that may work better than the (linear, exponential) interpolation technique around the diffractive minimum. These two different interpolation methods also allow us to estimate the systematic error that comes from the interpolation procedure itself. If the data points are measured densely enough in the (x, H(x)) plot, both methods are expected to yield similar results. We present our final results using both techniques and note that indeed we find similar results with both methods.

Suppose that for the j-th point of data set \(D_{12}\) and for some i value of \(D_2\), \(x_2(i)< x_{12}(j) < x_2(i+1)\). Then a linear interpolation between the i-th and \(i+1\)-th point of \(D_2\) yields the following formula:

Similarly, the errors can also be determined by linear interpolation as

This way, one extends \(D_2\) to the domain \(X_{12}\), corresponding to the overlapping acceptance of two measurements. If there is a measured value in \(D_2\), we use that value and its error bar. If there is no measurement in \(D_2\) precisely at that given value of x that is part of the overlapping acceptance (corresponding to a value x from \(D_1\)) then we use the two neighbouring points from \(D_2\) and use a (linear) interpolation to estimate the value at this intermediate point. This method works if the binning of both data sets is sufficiently fine so that non-linear structures are well resolved.

This way, for those \(j= 1, \ldots , n_1 - i_1 +1 \) points from \(X_{12}\) that belonged to \(D_1\), we have defined the data values from \(D_1\) by identity and defined the data points from \(D_2\) by linear interpolation from the neighbouring bins, so for these points both data sets are defined.

A similar procedure works for the remaining points in \(D_{12}\) that originate from \(D_2\). The number of such points is \(f_2\). Let us index them with \(k = 1, \ldots , f_2\). For these points, data and error-bars of the extrapolated data set \(D_{12}\) will be taken from \(D_2\): \(d_{21}(x_{12}(k)) = d_2(x_2(k))\), while the errors are given as \(e_{12}(x_{12}(k)) = e_2(x_2(k))\). However, for the same points, \(D_1\) has no measured value. As we need to compare the data of \(D_1\) and \(D_2\) at common values of x, for these points, \(D_1\) data and errors can be extrapolated using the linear or more sophisticated interpolation methods based on the nearest measured points. If the binning is fine enough, linear interpolation between the neighbouring data-points can be appropriately used. For broader bins, more sophisticated interpolation techniques may also be used that take into account non-linear interpolations based on more than two nearby bins, for example interpolations using Levy series expansion techniques of Ref. [9]. However, in the present manuscript such refinements are not necessary as the (linear, linear) and the (linear, exponential) interpolations in (x, H(x)) give similar results.

Consider now that for the k-th point of data set \(D_{12}\) and for some l-th value of \(D_2\), \(x_1(l)< x_{12}(k) < x_1(l+1)\). Then linear interpolation between the l-th and \(l+1\)-th point of \(D_2\) yields the following formula:

Similarly, the errors can also be determined by linear interpolation as

This way, using the linear interpolation techniques between the neighbouring data points, we can now compare the extended \(D_1\) and \(D_2\) to their common kinematic range: \(D_1\) was embedded and extrapolated to data points and errors denoted as \(d_{12}(x_{12})\) and \(e_{12}(x_{12})\) while \(D_2\) was embedded and extrapolated to data points and errors denoted as \(d_{21}(x_{12})\) and \(e_{21}(x_{12})\), respectively. Note that the domain of both of these extended data sets is the same \(X_{12}\) domain. The index “12” indicates that \(D_1\) was extended to \(X_{12}\), while index “21” indicates that \(D_2\) was extended to domain \(X_{12}\).

Now, we are done with the preparations to compare the two data sets, using the following \(\chi ^2\) definition:

In this comparison, there are no free parameters, so the number of degrees of freedom is NDF \(= n_{12} = n_1+f_2-i_1+1\), the number of data points in the unified data sample.

Based on the above Eq. (60) we get the value of \(\chi ^2\) and NDF, which can be used to evaluate the p-value, or the confidence level (CL), of the hypothesis that the two data sets represent the same H(x) scaling function. If CL satisfies the criteria that CL \( > 0.1\%\), the two data sets do not differ significantly. In the opposite case, if CL \( < 0.1\%\) the hypothesis that the two different measurements correspond to the same a priori H(x) scaling function, can be rejected.

The advantage of the above \(\chi ^2\) definition by Eq. (60) is that it is straightforward to implement it, however, it has a drawback that it does not specify how to deal with the correlated t or \(x = -t B\) dependent errors, and horizontal or x errors. The t measurements at \(\sqrt{s}=7\) TeV are published with their horizontal errors according to Table 5 of Ref. [28]. These errors should be combined with the published errors on the nuclear slope parameter B to get a horizontal error on x indicated as \(\delta x\). Such a horizontal error has to be taken into account in the final calculations of the significance of the Odderon observation.

Regarding the correlations among the measured values, and the measured errors, the best method would be to use the full covariance matrix of the measured differential cross section data. However, this covariance matrix is typically unknown or unpublished, with an exception of the \(\sqrt{s} = 13\) TeV elastic pp measurement by TOTEM [3]. Given that this TOTEM measurement of \(d\sigma /dt\) at 13 TeV indicates already the presence of small scaling violating terms in H(x) according to Fig. 2, this 13 TeV dataset cannot be used directly in our Odderon analysis, that is based on the s-independence of the scaling function of the differential elastic pp cross section \(H(x) \ne H(x,s)\) in a limited range that includes \(\sqrt{s} = \) 2.76 and 7 TeV, but does not extend up to 13 TeV. However, we can utilize this TOTEM measurement of \(d\sigma /dt\) at 13 TeV, to test the method of diagonalization of the covariance matrix that we apply in our final analysis of the Odderon significance.

Our analysis of the covariance matrix relies on a method developed by the PHENIX Collaboration and described in detail in Appendix A of Ref. [75]. This method is based on the following separation of the various types of experimental uncertainties:

Type A errors are point-to-point uncorrelated systematic uncertainties.

Type B errors are point-to-point varying but correlated systematic uncertainties, for which the point-to-point correlation is 100%, as the uncorrelated part is separated and added to type A errors in quadrature.

Type C systematic errors are point-independent, overall systematic uncertainties, that scale all the data points up and down by exactly the same, point-to-point independent factor.

Type D errors are point-to-point varying statistical errors. These type D errors are uncorrelated statistical errors, hence they can be added to the also uncorrelated, type A systematic errors in quadrature.

In this paper, where we apply this method to compare two different H(x) scaling functions, we also consider a fifth kind of error, type E that corresponds to the theoretical uncertainty, which we identify with the error of the interpolation of one of the (projected) data sets to the x values that are compared at some (measured) values of x to a certain measured data point at a measured x value. This type E error is identified with the value calculated from the linear interpolation, described above, as given for each A, B, C and D type of errors similarly by Eq. (59). Type D errors are added in quadrature to type A errors, and in what follows we index these errors with the index of the data point as well as with subscripts a, b and c, respectively.

Using this notation, Eq. (A16) of Ref. [75] yields the following \(\chi ^2\) definition, suitable for the projection of dataset \(D_2\) to \(D_1\), or \(2 \rightarrow 1\):

where \(\tilde{e}_{a,12}(j)\) is the type A uncertainty of the data point j of the united data set \(D_{12}\) scaled by a multiplicative factor such that the fractional uncertainty is unchanged under multiplication by a point-to-point varying factor:

In these sums, there are NDF\(_1 = f_1 - i_1 - 1\) number of data points in the overlapping acceptance from dataset \(D_1\). A similar sum describes the one-way projection \(1 \rightarrow 2\), but there are NDF\(_2 = f_2\) points in the common acceptance. For the two-way projections, not only the number of degrees of freedom add up, \(\mathrm{NDF}_{12} = \mathrm{NDF}_1+\mathrm{NDF}_2\), but also the \(\chi ^2\) values are added as \(\chi ^2 (1 \leftrightarrow 2) = \chi ^2(1 \rightarrow 2) + \chi ^2( 2 \rightarrow 1)\).

Let us note at this point, that H(x) is a scaling function that is proportional to the differential cross section normalized by the integrated cross section. In this ratio, the overall, type C point-independent normalization errors multiply both the numerator and the denominator, hence these type C errors cancel out in H(x). Given that these type C errors are typically rather large, for example, 14.4% for the D0 measurement of Ref. [8], it is an important advantage in the significance computation that we use a normalized scaling function H(x). So in what follows, we set \(\epsilon _{c,1} = 0\) and rewrite the equation for the \(\chi ^2\) definition accordingly. This effect increases the significance of a H(x)-scaling test.

The price we have to pay for this advantage is that we have to take into account the horizontal errors on x in order to not overestimate the significance of our \(\chi ^2\) test. In this step, we follow the propagation of the horizontal error to the \(\chi ^2\) as utilized by the so-called effective variance method of the CERN data analysis programme ROOT. This yields the following \(\chi ^2\) definition that we have utilized in our significance analysis for the case of symmetric errors in x:

where \(\delta x_{12}(j)\) is the (symmetric) error of x in the j-th data point of the data set \(D_{1}\), and \(d^{\prime }_{1}(j))^2\) is the numerically evaluated derivative of the extrapolated value of the projected data point obtained with the help of a linear interpolation using Eq. (58). Such definition is valid when the type B errors are known and are symmetric for the data set \(D_1\) and the errors on x are also symmetric. When the data set \(D_1\) corresponds to the D0 measurement of elastic \(p\bar{p}\) collisions, Ref. [8], we have to take into account that D0 did not publish the separated statistical and |t|-dependent systematic errors, but decided to publish their values added in quadrature. So we use these errors as type A errors and with this method, we underestimate the significance of the results as we neglect the correlations among the errors of the data points in the D0 dataset. The TOTEM published the |t|-dependent statistical type D errors and the |t|-dependent systematic errors both for the 2.76 and 7 TeV measurements of the differential cross sections [4, 28, 74], with the note that the |t|-dependent systematic errors are almost fully correlated. In these works, TOTEM did not separate the point-to-point varying uncorrelated part of the |t|-dependent systematic errors. We thus estimate the type A errors by the statistical errors of these TOTEM measurements, we then slightly underestimate them, hence overestimate the \(\chi ^2\) and the difference between the compared data sets. Given that they are almost fully correlated, we estimate the type B errors by the point-to-point varying almost fully correlated systematic errors published by the TOTEM. We have tested this scheme by evaluating the \(\chi ^2\) from a full covariance matrix fit and from the PHENIX method of diagonalizing the covariance matrix at \(\sqrt{s} = 13\) TeV, using the Lévy expansion method of Ref. [9]. We find that the fit with the full covariance matrix results in the same minimum within one standard deviation of the fit parameters, hence the same significance as the fit with the PHENIX method of Appendix A of Ref. [75].

We have thus validated the PHENIX method of Ref. [75] for the application of the analysis of differential cross section at \(\sqrt{s} = 13 \) TeV, together with the effective variance method of the ROOT package. This validation is important as the full covariance matrix of the \(\sqrt{s} = 2.76 \) TeV and 7 TeV measurements by TOTEM is not published, but the PHENIX method appended with the ROOT method of effective variances can be used to effectively diagonalize the covariance matrix and to get similar results within the errors of the analysis. In Sect. 9, we employ the preliminary \(\chi ^2\) definition of Eq. (63) to estimate the significance of the Odderon signal in comparison of the H(x) scaling functions for elastic pp and \(p\bar{p}\) collisions. Our final \(\chi ^2\) definition and the corresponding final results are described in Appendix A.

In this section, we discuss how to extrapolate the data points to energies where measurements are missing. We emphasize that this method is not our best method to evaluate the significance of the Odderon signal, but we include this section for the sake of completeness, as other groups follow this method. The obvious reason for this is that a large, 14.4% overall correlated, type C error of the D0 measurement does not cancel from the differential cross-sections, and their significances, while it simply cancels from the H(x) scaling functions, that are normalized to the integral of the differential cross-section. A quantitative estimate of the importance of this effect is shown in Appendix A, and we detail the results from the comparison of the H(x) scaling functions starting from the next section. We recommend this section to those readers, who are motivated to understand how to extrapolate the differential cross-sections to a new, not measured energy in a domain of (s, t) where the H(x) scaling is known to be valid from already performed measurements.

We have found, for example, that in the ISR energy range of \(\sqrt{s} = 23.5\)–62.5 GeV the H(x) scaling function is approximately independent of \(\sqrt{s}\) within errors, and with a possible exception at a small region around the diffractive minimum. We show how to extrapolate data points to unmeasured energies, under the condition that in a given energy range, H(x) is independent of the collision energy, \(H(x) \ne H(x,s) \). In general, such a feature has to be established or cross-checked experimentally. This case is important, given that we have shown before, for example in Fig. 5, that H(x) for pp collisions stays energy-independent within errors between the LHC energies of 2.76 TeV \(\le \sqrt{s}\le \) 7 TeV. Furthermore, we have already shown that for \(p\overline{p}\) collisions, \(H(x) = H(x,s)\) in the energy range of 0.546 \(\le \sqrt{s} \le 1.96\) TeV, as indicated in Fig. 6.

Let us denote two different center-of-mass energies between which \(H(x)=\mathrm{const}(\sqrt{s})\) within the experimental errors as \(\sqrt{s_1}\) and \(\sqrt{s_2}\). Analogically, we denote various observables as \(B_i \equiv B(s_i)\), \(\sigma _i \equiv \sigma _{el,i}\equiv \sigma _{el}(s_i)\), \(x_i \equiv B_i t\).

The energy independence of the H(x) scaling function formally can be written as

This simple statement has tremendous experimental implications. The equality \(x_1 = x_2\) means that the scaling function is the same, if at center-of-mass energy \(\sqrt{s_1}\) it is measured at \(t_1\) and at energy \(\sqrt{s_2}\) it is measured at \(t_2\), so that

The equality \(H_1(x_1) = H_2(x_2) = H(x)\) is expressed as

(66)

Putting these equations together, this implies that the experimental data can be scaled to other energies in an energy range where H(x) is found to be independent of \(\sqrt{s}\) as follows:

(67)

With the help of this equation, the data points on differential cross sections can be scaled to various different colliding energies, if in a certain energy region the H(x) scaling holds within the experimental errors. In other words, the differential cross section can be rescaled from \(\sqrt{s_1}\) to \(\sqrt{s_2}\) by rescaling the |t|-variable using the ratio of \(B_1/B_2=B(s_1)/B(s_2)\), and by multiplying the cross section with the ratio \(\frac{B_1 \sigma _1}{B_2 \sigma _2}\).

In this section, we present our results and close the energy gap, as much as possible without a direct measurement, between the TOTEM data on elastic pp collisions at \(\sqrt{s} = 2.76\) and 7.0 TeV and D0 data on elastic \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96 \) TeV. This section is based on the application of Eq. (67) in this energy range. After the rescaling procedure, the resulting data set at the new energy is compared with the measured data quantitatively with the help of Eq. (60).

We have used the rescaling equation, Eq. (67) first to test and to cross-check, if the rescaling of the \(\sqrt{s} = 23.5 \) GeV ISR data to other ISR energies works, or not. The left panel of Fig. 7 indicates that such a rescaling of the differential cross sections from the lowest ISR energy of \(\sqrt{s} = 23.5\) to the highest ISR energy of 62.5 GeV actually works well. The level of agreement of the rescaled 23.5 GeV pp data with the measured 62.5 GeV pp data has been evaluated with the help of Eq. (60). We found an agreement with a \(\chi ^2/\mathrm{NDF} = 111/100\), corresponding to a CL = 21.3% and a difference is at the level of 1.25\(\sigma \) only. This result demonstrates that our rescaling method can also be used to get the differential cross sections at other energies, provided that the nuclear slope and the elastic cross sections are known at the new energy as well as at the energy from where we start the rescaling procedure.

Rescaling of the differential cross section of elastic pp collisions from the energy of \(\sqrt{s} = 7\) to 1.96 TeV using Eq. (67). We have evaluated the confidence level of the comparison between the rescaled 7 TeV pp data set and the 1.96 TeV \(p\bar{p}\) data set with the help of Eq. (60), that does not take into account the horizontal errors of x coming from the slopes B and the type C point-to-point correlated errors on the vertical scale. Without these important effects, the difference between the datasets provides a \(\chi ^2/\mathrm{NDF} = 73.6/17 \), equivalent to a confidence level of \(\hbox {CL}=5.13 \times 10^{-7}\)% and a statistically significant, \(5.84\sigma \) effect

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Subsequently, one can also rescale the TOTEM data at \(\sqrt{s} = 2.76\) or 7 to 1.96 TeV, given that H(x) is (within errors) energy independent in the range of \(2.76{-}7 \) TeV, corresponding to nearly a factor of 2.5 change in \(\sqrt{s}\), while the change in \(\sqrt{s}\) from 1.96 to 2.76 TeV is only a factor of 1.4. The right panel of Fig. 7 indicates that rescaling of the differential elastic pp cross section from \(\sqrt{s} = 2.76\) to 1.96 TeV also gives valuable results. We have evaluated the confidence level of the comparison of the rescaled 2.76 TeV pp data with the 1.96 TeV \(p\bar{p}\) data with the help of Eq. (60). As was already mentioned above, we have found a surprising agreement with a \(\chi ^2/\mathrm{NDF} = 18.1/11\), corresponding to a CL = 7.93%, and a difference at the level of 1.75 \(\sigma \) only.

Another important result is illustrated in Fig. 8. This comparison indicates a difference between the rescaled \(\sqrt{s} = \) 7 TeV elastic pp differential cross-section [28, 74] to the \(\sqrt{s} = \) 1.96 TeV energy and to the corresponding \(p\bar{p}\) data measured at \(\sqrt{s}= 1.96 \) TeV [8]. To obtain a first estimate, this difference is quantified with the help of Eq. (60) yielding a CL of \(5.13\cdot 10^{-7}\)%, which corresponds to a difference at the 5.84 \(\sigma \) level. As this method adds the statistical and the point-to-point varying systematic errors in quadrature, it underestimates the actual significance of the difference between the two data sets. Although this estimate already provides a significant, greater than 5\(\sigma \) effect for the Odderon observation, corresponding to a significant, 5.84\(\sigma \) difference between the pp dataset and the 1.96 TeV \(p\overline{p}\) dataset, however, the evaluation of this significance does not yet take into account the rather large overall normalization error of 14.4% that has been published by the D0 collaboration.

This Fig. 8 indicates that not only the diffractive interference, the dip and the bump may carry an Odderon signal, but also the so called swing region, where the pp differential cross-section bends below the straight exponential diffractive cone of the \(p\bar{p}\) result. See also Fig. 16 of Appendix A for more details on how the type C errors reduce the significance of the Odderon signal to a 3.64\(\sigma \), if the comparison is done directly at the level of the differential cross-sections and if these type C, overall correlated errors are added in quadrature to the point-to-point correlated, type A errors. This is only a lower bound of the significance as type C errors are not point-to-point fluctuating, but shift the whole dataset up or down in a correlated way, see the end of Appendix A for more details on this lower bound. The point is that it is advantageous to use the H(x) scaling function instead of the differential cross-sections, as the rather large type C errors cancel from H(x) while they may lead to an important reduction of the significance of the signal when they are considered on the differential cross-sections.

It can be seen in Fig. 8 that in the swing region, before the dip, the rescaled pp differential cross section differ significantly from that of \(p\bar{p}\) collisions. Looking by eye, the swing and the diffractive interference (dip and bump) regions both seem to provide an important contribution. We have dedicated Appendix E to evaluate the significances of various regions, to determine precisely how much do they contribute to the significance of this Odderon signal.

The estimates of statistical significances given in the present section are based on a \(\chi ^2\) test that includes the |t|-dependent statistical errors and the |t|-dependent systematic errors added in quadrature. Thus the values of \(\chi ^2/\)NDF and significances given in this section can only be considered as estimates. Indeed, although the |t|-dependent systematic errors on these \(\sqrt{s} = 7\) TeV data are known to be almost fully correlated, the covariance matrix is not publicly available at the time of closing this manuscript from the TOTEM measurement at \(\sqrt{s} = 7\) TeV. It is clear that the \(\chi ^2\) is expected to increase if the covariance matrix is taken into account, and this effect would increase the disagreement between the measured \(p\bar{p}\) and the extrapolated pp differential cross sections at \(\sqrt{s} = 1.96\) TeV.

So this indicates that we have to consider the proposed rescaling method as conservatively as possible, that allows us to take into account the statistical and |t|-dependent correlated systematic errors, as well as the |t|-independent correlated systematic errors. Such an analysis is presented in the next section, where we quantify the differences between the scaling functions H(x) of elastic pp and \(p\bar{p}\) collisions using the fact that H(x) is free of |t|-independent normalisation errors, and our final results are summarized in Appendix A.

In this section, we estimate a preliminary, 6.55\(\sigma \) significance for the Odderon signal, while Appendix A determines and summarizes our final Odderon signal of an at least 6.26\(\sigma \) effect. Both results are obtained by comparing the H(x) scaling functions of pp and \(p\bar{p}\) collisions.

We have found a significant Odderon signal by comparing the H(x) scaling functions of the differential cross section of elastic pp collisions with \(\sqrt{s} = 7\) TeV to that of \(p\bar{p}\) collisions with \(\sqrt{s} = 1.96\) TeV, as indicated in Fig. 10. The comparison is made in both possible ways, by comparing the pp data to the \(p\bar{p}\) data, and vice versa. The difference between these two datasets corresponds to at least a \(\chi ^2/\mathrm{NDF} = 84.6/17\), giving rise to a CL of \(5.8 \times 10^{-9}\)% and to a preliminary, 6.55\(\sigma \) significance, obtained with the help of Eq. (63). The overall, |t|-independent normalization error of 14.4% on the D0 data set cancels from this H(x), and does not propagate to our conclusions.

These results are obtained for the \(\sigma _\mathrm{el} = 17.6 \pm 1.1\) mb value of the elastic \(p\bar{p}\) cross section at \(\sqrt{s} = 1.96 \) TeV, and for the linear-exponential interpolation in (x, H(x)). Using this method of interpolation, the nearest points were connected with a linear-exponential line, that corresponds to a straight line on a linear-logarithmic plot in (x, H(x)). We have used the published values of the differential cross sections \(\frac{d\sigma }{dt}\), that of the nuclear slope parameter B and the measured value of the elastic cross section \(\sigma _\mathrm{el}\) for 7 TeV pp elastic collisions. For the elastic cross section of \(p\bar{p}\) collisions at \(\sqrt{s}\) \( = 1.96\) TeV, we have numerically integrated the differential cross section with an exponential approximation at very low-|t| that provided us with \(\sigma _\mathrm{el} = 20.2 \pm 1.4\) mb.

We have systematically checked the effect of variations in our interpolation method by switching from the (linear-exponential) in (x, H(x)) interpolation to a linear-linear one and by changing the value of the elastic \(p\bar{p}\) collisions from the numerically integrated differential cross-section value of \(\sigma _\mathrm{el} = 20.2 \pm 1.4\) mb, which is an unusually large value, but equals within the quoted 14.4% systematic error to the \(\sigma _\mathrm{el} = 17.6 \pm 1.1\) mb value, that corresponds to the trend published by the Particle Data Group, see the Fig. 51.6, bottom panel, yellow line of Ref. [76]. The input values of the nuclear slope parameter B and the elastic cross-section \(\sigma _\mathrm{el}\) are summarized in Table 1, the corresponding results are shown in Tables 2, 3,  4 and  5.

As part of our systematic studies, we have also changed the direction of the projection. The results are summarized in Table 2. They indicate that the improved version of Fig. 8, shown as the top left panel of Fig. 10 and evaluated with the help of our improved \(\chi ^2\) definition of Eq. (63) corresponds to a conservative case of Odderon observation based on the \(\sqrt{s} = 7\) TeV TOTEM and the \(\sqrt{s} = 1.96\) TeV D0 data sets. This panel indicates that the Odderon signal is observed in this comparison with a preliminary, at least a 6.55\(\sigma \) significance, indicating the power of our method of Odderon observation. In addition to this, our final result includes a symmetry requirement and a robustness test described in Appendix A. These effects decreased the significance of our Odderon observation, from a preliminary, \(\ge \) 6.55 \(\sigma \) effect to a final and statistically significant, \(\ge \) 6.26 \(\sigma \) effect.

We have checked the robustness of this result for several possible variations of the \(\chi ^2\) definition. The consideration that was most successful in decreasing this significance was related to the fact that unlike the original PHENIX method of Ref. [75], that was worked out for a theory to data comparison, in this manuscript we compare data to data. So we have adapted the PHENIX method of Ref. [75], from a situation where there was a theoretical function without errors compared to data with errors to a situation where we compare two datasets and both of these datasets have the same type of errors. This slightly decreased the significance of the Odderon signal, from the value of a preliminary, at least 6.55 \(\sigma \) to the final value of 6.26 \(\sigma \), as detailed in Appendix A of this manuscript. Given that both significances of the preliminary 6.55 \(\sigma \), detailed in this section, and 6.26 \(\sigma \), detailed in Appendix A are clearly and safely above the 5 \(\sigma \) discovery threshold, this robustness test did not change our conclusions.

The detailed figures, that show the \(\chi ^2(\epsilon _b)\) functions for each of these cases are summarized in the left and right panels of Fig. 9 for the comparison of the 7 TeV TOTEM data set with the 1.96 TeV D0 data set. Each plot indicates a clear, nearly quadratic minimum. The values of \(\chi ^2\) at the minima are summarized in Table 2, together with other characteristics of significance, like the confidence level and the significance in terms of standard variations. Similarly, the \(\chi ^2(\epsilon _b)\) functions for the comparison of the 2.76 TeV TOTEM data set with the 1.96 TeV D0 data set are summarized in Fig. 11. The values of \(\chi ^2\) at the minima are given in Table 3, together with other relevant characteristics.

Dependence of \(\chi ^2\) on the coefficient of the correlated but point-to-point varying systematic errors, \(\epsilon _b\), for the comparison of the H(x) scaling functions of elastic \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96\) TeV with that of pp collisions at \(\sqrt{s} = 7.0\) TeV. Each of the four cases are shown together corresponding to the direction of the projection. Upper panel indicates the results of the 1.96 \(\rightarrow \) 7.0 TeV projection. Lower panel indicates the results of the 7.0 \(\rightarrow \) 1.96 TeV projection. Both cases indicate four \(\chi ^2(\epsilon _b)\) curves corresponding to the choice of linear-linear or linear-exponential interpolations in (x, H(x)), as well as to the choice of the elastic \(p\bar{p}\) cross section at \(\sqrt{s} = 1.96\) TeV (20.2 ± 1.4 mb vs 17.6 ± 1.1 mb). A parabolic structure is seen in each case with a clear minimum, and the fit quality corresponding to these minima in \(\epsilon _b\) is summarized in Table 2

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As summarized in Fig. 10, a significant Odderon signal is found in the comparison of the H(x) scaling functions of the differential elastic pp (at \(\sqrt{s} = 7.0\) TeV) vs \(p\bar{p}\) (\(\sqrt{s} = 1.96\) TeV) cross sections. The horizontal error bars are indicated by a properly scaled horizontal line or “−” at the data point. The statistical (type A, point-to-point fluctuating) errors are indicated by the size of the vertical error bars (|), while shaded boxes indicate the size of the (asymmetric) type B (point-to-point varying, correlated) systematic errors. The overall normalization errors (|t|-independent, type C errors) cancel from the H(x) scaling functions since they multiply both the numerator and the denominator of H(x) in the same way. The correlation coefficient of the |t|-dependent systematic errors, \(\epsilon _b\), is optimized to minimize the \(\chi ^2\) based on Eq. (63), and the values indicated in Fig. 10 correspond to the minimum of the \(\chi ^2(\epsilon _b)\). The location of these minima and the best values of \(\epsilon _b\) depend on the domain in x or x-range from where the contributions to the \(\chi ^2(\epsilon _b)\) are added up. The stability of our final results with respect to the variation of the x-range, together with the correlations between the best value of the \(\epsilon _b\) and the x-range are detailed in Appendix E. These \(\chi ^2\) values, as well as the numbers of degrees of freedom (NDFs) and the corresponding confidence levels (CLs) are indicated on both panels of Fig. 10, for both projections. The \(\chi ^2(\epsilon _b)\) functions are summarized in Fig. 9. The 7 TeV \(\rightarrow \) 1.96 TeV projection has a preliminary statistical significance of 6.55\(\sigma \) of an Odderon signal, corresponding to a \(\chi ^2/\mathrm{NDF} = 84.6 / 17\) and CL = \(5.78 \times 10^{-9}\)%. Appendix A presents the robustness test of this result, and summarizes the result of our tests of various possible modifications of our \(\chi ^2 \) definition. It turns out that the symmetry requirement discussed in Appendix A slightly reduces this significance from a 6.55 \(\sigma \) level to a 6.26 \(\sigma \) level, safely above the 5.0 \(\sigma \) discovery threshold, corresponding to a \(\chi ^2/\mathrm{NDF}\) \(=\) 80.1/17 and CL = \(3.7 \times 10^{-8}\)%. Thus the probability of Odderon observation in this analysis is at least \(P = 1-CL = 0.99999999963\).

Figure 10 illustrates some of the results of our systematic studies in four different panels described as follows. The top-left panel of this figure uses a linear-exponential interpolation in the (x, H(x)) plane and uses the value of 17.6 ± 1.1 mb for the elastic \(p\bar{p}\) cross section at \(\sqrt{s} = 1.96\) TeV. This case gives the lowest (6.55\(\sigma \)) significance for the Odderon observation from among the possible cases that we have considered in Fig. 10. The top-right panel is similar but for a linear-linear interpolation in the (x, H(x)). The bottom-left panel is similar to the top-left panel, but now using 20.2 ± 1.4 mb for the elastic \(p\bar{p}\) cross section at \(\sqrt{s} = 1.96 \) TeV and also using a linear-exponential interpolation in (x, H(x)). The bottom-right panel is similar to the bottom-left panel, but using a linear-linear interpolation method.

Odderon signal in the comparison of the H(x) scaling functions of pp collisions at \(\sqrt{s} = 7\) TeV, measured by the TOTEM experiment at the LHC [28, 74], and \(p\bar{p}\) elastic collisions at \(\sqrt{s} = 1.96\) TeV measured by the D0 experiment at Tevatron [8]. The results of this preliminary Odderon observation are indicated on the plots, where the CL is evaluated without the rounding of the \(\chi ^2\) values to the printed level of precision. The rounded values of \(\chi ^2 \) and the corresponding CL values are summarized in Table 2. The final Odderon significance results are given in Appendix A. Top-left panel: This comparison uses \(17.6 \pm 1.1\) mb for the elastic \(p\bar{p}\) cross section at \(\sqrt{s} = 1.96 \) TeV, and a linear-exponential interpolation technique in (x, H(x)). This corresponds to the smallest difference between the two data sets. Top-right panel: Same as the top-left panel but for linear-linear interpolations in the horizontal and vertical directions. For these interpolations, the nearest data points are connected with lines that correspond to a straight line on a linear-linear plot. Bottom-left panel: Same as the top-left panel but now using \(20.2 \pm 1.4\) mb for the elastic \(p\bar{p}\) cross section at \(\sqrt{s} = 1.96 \) TeV. Bottom-right panel: Same as the bottom-left panel but using a linear-linear interpolation method

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Dependence of \(\chi ^2\) on the coefficient of the correlated but point-to-point varying systematic errors, \(\epsilon _b\), for the comparison of the H(x) scaling functions of elastic \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96 \) TeV, measured by the D0 experiment at Tevatron [8], with that of elastic pp collisions at \(\sqrt{s} = 2.76 \) TeV, measured by the TOTEM experiment at the LHC [4]. All the eight cases are shown together corresponding to the choice of linear-linear or linear-exponential interpolations in H(x), to a different choice of the elastic cross section of \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96\) TeV (\(20.2 \pm 1.4\) mb vs \(17.6 \pm 1.1\) mb), and to the direction of the projection (\(1.96 \rightarrow 2.76\) TeV, or 2.76 TeV \(\rightarrow \) 1.96 TeV). A clear parabolic structure is seen in each case and the fit quality of the results that belong to these minima in \(\epsilon _b\) is summarized in Table 3

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The results of the scaling studies for a comparison of elastic pp collisions at \(\sqrt{s} = 2.76 \) TeV, measured by the TOTEM experiment at the LHC [4] to that of \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96 \) TeV, measured by D0 at the Tevatron [8] are summarized in Figs. 11 and 12. The top-left panel of Fig 12 uses \(\sigma _\mathrm{el} = 17.6 \pm 1.1\) mb and a linear-exponential interpolation method in (x, H(x)). The top-right panel is the same as the top-left panel, but for a linear-linear interpolation in (x, H(x)). The bottom-left panel is nearly the same as the top-right panel, but for \(\sigma _\mathrm{el} = 20.2 \pm 1.4\) mb. The bottom-right panel is the same as the bottom-left panel, but for a linear-linear interpolation in (x, H(x)). Neither of these comparisons shows a significant difference between the H(x) scaling function of elastic pp collisions at \(\sqrt{s} = 2.76 \) TeV as compared to that of \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96 \) TeV. It seems that the main reason for such a lack of significance is the acceptance limitation of the TOTEM dataset at \(\sqrt{s} = 2.76\) TeV, which extends up to \(x = - t B \approx 13\), in contrast to the acceptance of the 7 TeV TOTEM measurement that extends up to \(x = - tB \approx 20\). We have cross-checked this by limiting the 7 TeV data set also to the same acceptance region of \(4.4< -Bt < 12.7\) as that of the 2.76 TeV data set. This artificial acceptance limitation has resulted in a profound loss of significance, down a to \(\chi ^2/\mathrm{NDF} = 25.7 / 11\), that corresponds to a CL = 0.71% and to a deviation at the 2.69 \(\sigma \) level only. This result indicates that if we limit the acceptance of the 7 TeV TOTEM measurement to the acceptance of the 2.76 TeV TOTEM measurement, the significance of the Odderon observation decreases well below the 5\(\sigma \) discovery threshold. This result can be understood if we consider, that the diffractive maximum (“bump”) is located, if the H(x) scaling is valid, at \(x \approx 13\), which is very close but slightly above the value of the \(x_{max} = 12.7\) upper limit of the acceptance in x of the TOTEM data published in Ref. [4]. Fig. 8 of Ref. [4] indicates that indeed the precise location of the diffractive maximum can not be determined from these TOTEM data, it may be just close to the upper limit of the TOTEM acceptance at \(\sqrt{s} = 2.76\) TeV.

We have thus dedicated Appendix E to the scrutiny of the x-range dependence of the Odderon signal. In particular, we have investigated how important is the contribution from the large values of x. We developed and tested our most conservative \(\chi ^2\) definition in Appendix A. We have investigated the domain of validity of the H(x) scaling with the help of a model and detailed in Appendix D, that at \(\sqrt{s} = 2.76\) TeV, the H(x) scaling is expected to hold up to \(x = 15.1\), well above the TOTEM acceptance of \(x < 12.7 \). In Appendix E, we find that it is sufficient to include a small shift to the investigated x range: already for only 10 datapoints from the D0 acceptance the significance of the Odderon signal is greater than 5 \(\sigma \) in the \(5.1 < x \le 13.1\) domain at \(\sqrt{s} = 1.96\) TeV. In Appendix E we also show that the minimum size of subsequent D0 datapoints for a greater than 5 \(\sigma \) Odderon signal is actually 8 out of 17, corresponding to the \(7.0 \le x \le 13.5\) range.

We have performed several cross-checks: this topic is detailed in the next section.

Lack of a significant Odderon signal in the comparison of the H(x) scaling functions of the differential cross section of elastic pp collisions with \(\sqrt{s} = 2.76\) TeV, measured by the TOTEM [4], to that of \(p\bar{p}\) collisions with \(\sqrt{s} = 1.96\) TeV, measured by D0 [8]. The correlation coefficient of the |t|-dependent systematic errors, \(\epsilon _b\), is optimized to minimize the \(\chi ^2\) based on Eq. (63), and the value indicated on the plot corresponds to the minimum of \(\chi ^2(\epsilon _b)\). The results of our Odderon search are summarized in Table 3. See also Table 5 for a summary of the results of the two-way comparisons of these H(x) scaling functions. Top-left panel: Using \(\sigma _\mathrm{el} = 17.6 \pm 1.7\) mb and a linear-exponential interpolation method. Top-right panel: Same as the top-left panel but for a linear-linear interpolation in (x, H(x)). Bottom-left panel: Same as the top-left panel but for \(\sigma _\mathrm{el} = 20.2 \pm 1.4\) mb. Bottom-right panel: Same as the bottom-left panel but for a linear-linear interpolation in (x, H(x))

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In this section, we summarize some of the most important cross-checks that we performed using our methods and results.

We have cross-checked what happens if one rescales the differential cross section of elastic pp scattering form the lowest ISR energy of \(\sqrt{s} = 23.5\) GeV to the top ISR energy of \(\sqrt{s} = 62.5\) GeV. As can be expected based on the approximate equality of all the H(x) scaling functions at the ISR energies, as indicated on the left panel of Fig. 7, the rescaled 23.5 GeV pp data coincide with the measured 62.5 GeV pp data. The resulting \(\chi ^2/\mathrm{NDF} = 111/100\) corresponds to a CL = 21.3%, or a lack of significant difference – a 1.3\(\sigma \) effect. Within errors, our quantitative analysis thus indicates that the two data sets at the ISR energies of 23.5 and 62.5 GeV correspond to the same H(x) scaling function, but with possible small deviations in a small x-region around the dip position. This indicates that the method that we applied to extrapolate the 2.76 and 7 TeV data sets to lower energies satisfied the cross-checks at the ISR energies, i.e. our method works well. As one of the critical cross-checks of these calculations, two different co-authors coded the same formulae with two different codes using two different programming languages, and these codes were cross-checked against one another until both provided the same values of significances.

We have validated the PHENIX method of Ref. [75] implemented in the form of the \(\chi ^2\) definition of Eq. (63) for the diagonalization of the covariance matrix on fits to the \(\sqrt{s} = 13 \) TeV TOTEM data of Ref. [3]. This PHENIX method resulted, within one standard deviation, the same minimum, hence the same significances, as the use of the full covariance matrix at \(\sqrt{s} = 13 \) TeV elastic pp collisions. At the lower LHC energies of \(\sqrt{s} = 2.76\) and 7.0 TeV, due to the lack of publicly available information on the covariance matrix, only the PHENIX method of Ref. [75] was available for our final significance analysis.

We have also explored the main reason of the observation of a significant Odderon signal in the comparison of the H(x) scaling functions of elastic pp collisions at \(\sqrt{s} = 7 \) TeV with that of the elastic \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96 \) TeV. The question was rather intriguing as we have found no significant difference between the H(x) scaling functions of elastic pp collisions at \(\sqrt{s} = 2.76\) and 7 TeV. At the same time, we also see that the comparison of the 2.76 TeV pp dataset to the 1.96 TeV \(p\bar{p}\) dataset does not indicate a significant Odderon effect. We have found that the Odderon signal vanishes from the comparison of the 7 TeV pp and the 1.96 TeV \(p\bar{p}\) datasets too, if we limit the acceptance of the 7 TeV dataset to the acceptance in \(x = -tB\) as that of the 2.76 TeV pp dataset: the significance of the Odderon observation decreased from an at least 6.26 \(\sigma \) discovery effect, detailed in Appendix A, to a 2.69\(\sigma \) level agreement. We may note that a similar observation was made already in Ref. [10] that pointed out a strong |t| dependence of the Odderon contribution.

Table 4 summarises the search for an Odderon signal in the two-way comparison, for the significance of an Odderon signal in the comparison of the H(x) scaling functions of pp collisions at \(\sqrt{s} = 7\) TeV and \(p\bar{p}\) collisions at \(\sqrt{s} = 1.96\) TeV. Applying this method the Odderon signal is observed with at least a 13\(\sigma \) significance, when both projections are combined from Table 2, by adding the \(\chi ^2\) and the NDF values of both directions of the comparisons.

We have explored the scaling properties of the elastic differential cross sections at various energies, from the ISR up to the highest LHC energy. We have recalled that the earlier proposals for the F(y) and G(z) scaling functions were useful to explore if elastic scattering of protons in the LHC energy range is already close to the black-disc limit or not. After investigating several possible new dimensionless scaling variables and scaling function candidates, we have realized that in order to look for scaling violations in the low |t| kinematic range, corresponding to the diffractive cone it is advisable to scale all the diffractive cones to the same dimensionless scaling function, \(H(x) \approx \exp (-x)\). This function can be obtained as the differential cross section normalized to its value at the \(x = -tB = 0\) optical point, which also for nearly exponential distributions equals to the elastic cross section \(\sigma _\mathrm{el}\) multiplied by the slope parameter B. Both are readily measurable in elastic pp and \(p\bar{p}\) collisions, while other scaling variables that we have investigated may depend on \(t_\mathrm{dip}\) values – the location of the diffractive minimum. The latter however is not readily accessible neither in elastic \(p\bar{p}\) collisions (where there is no significant dip) nor in the acceptance limited elastic pp differential cross section (where the diffractive minimum or maximum may be located outside the acceptance of the experiment for that particular data set).

The scaling function H(x) of elastic proton-(anti)proton scattering transforms out the energy dependence of the elastic slope B(s) and the elastic cross section \(\sigma _\mathrm{el}(s)\), and due to the relation \([1 + \rho _0^2(s)] \sigma ^2_\mathrm{tot}(s) = 16 \pi \sigma _\mathrm{el} (s)\) they also scale out a combination of the total cross sections and the real-to-imaginary ratio. As was discussed above, for analytic scattering amplitudes and for differential cross-sections starting with a diffractive cone at low values of \(x = - tB\) the scaling function will have a universal, \(H(x) \approx \exp (-x)\) shape. The price for the removal of these trivial s-dependencies from the scaling function is paid by an s-depended domain of validity, \(x_\mathrm{max}(s)\) which is found to be typically above the position of the diffractive interference region. Without direct experimental observations, or without theoretical, model-dependent calculations, it is not possible to determine model-independently this \(x_\mathrm{max}(s)\) function, the s-dependent upper limit of the domain of validity of this H(x) scaling.

Figures 8 and 10 clearly indicate a crossing-odd component of the elastic scattering amplitude. At the \(\sim \) 2 TeV energy scale, where the Reggeon contributions to the scattering amplitude are suppressed by their power-law decays, this is apparently a clear Odderon effect, a characteristic difference in the shape of the scaling function of elastic scattering between pp and \(p\bar{p}\) collisions at the logarithmically similar energies of 7 and 1.96 TeV, respectively.

The effects due to the energy-induced difference between TOTEM and D0 data sets can be estimated by the lack of change of the H(x) scaling function for pp scattering between 2.76 TeV and 7 TeV, within the statistical errors of these TOTEM data sets. However, the H(x) scaling function of elastic pp scattering at \(\sqrt{s} = 7.0\) TeV is significantly different from the corresponding result of elastic \(p\bar{p}\) scattering at \(\sqrt{s} = 1.96\) TeV. These qualitative and quantitative differences, first, show up well below the diffractive minimum of the pp elastic scattering, namely, the H(x) function for pp collisions indicates a strong “swing” or faster than exponential decrease effect, before developing a characteristic interference pattern consisting of a diffractive minimum and subsequent maximum. In contrast, the D0 data on \(p\bar{p}\) elastic scattering features a structureless exponential decrease that in turn changes to a plateaux or a shoulder-like structure at higher values of the scaling variable x. No clear indication of a diffractive maximum is seen in the \(p\bar{p}\) elastic scattering data [8], while the TOTEM data sets at each LHC energies of 2.76, 7 and 13 TeV clearly indicate a diffractive minimum followed by an increasing part of the differential cross section before the edge of the TOTEM acceptance is reached, respectively [3, 4, 74].

These qualitative and quantitative differences between the H(x) scaling functions of elastic pp and \(p\bar{p}\) scatterings provide a clear-cut and statistically significant evidence for a crossing-odd component in the scattering amplitude in the TeV energy range. This corresponds to the observation of the Odderon exchange in the t-channel of the elastic scattering. The Odderon in this context is a crossing-odd component of the amplitude of elastic pp and \(p\bar{p}\) scattering, that remains significant even in the large s limit. In Regge phenomenology, the Odderon is a trajectory that at \(J=1\) contains a \(J^\mathrm{PC} = 1^{--}\) vector glueball as well as other glueball states with higher angular momentum. Hence, one of the implications of our result is that not only one but several glueball states should exist in Nature [77].

Due to the presence of the faster-than exponentially decreasing (swing) region in elastic pp scatterings, high-statistic pp elastic scattering data at \(\sqrt{s} = 1.96 \) TeV may be taken as an additional measurement clearly closing the energy gap. However, the aperture limitation of the LHC accelerator is already resulting in a loss of significance of the comparison of the H(x) scaling function at 2.76 TeV with that of the D0 data at 1.96 TeV. Due to this reason, we propose an additional measurement of the dip and bump region of elastic pp collisions in the domain where the H(x) scaling was shown to work, in between 2.76 and 7 TeV, if that can be harmonized with the LHC running schedule and scenarios.

The current TOTEM acceptance (the upper end of the last bin of the published TOTEM data) ends at \(-t B \approx 13\) at \(\sqrt{s} = 2.76\) TeV. This value almost coincides with the bump position of the H(x) scaling function. It seems that including at least one D0 point to the comparison of the H(x) scaling functions of pp and \(p\bar{p}\) data above the \(x =13\) bump position is sufficient for reaching an at least 5 \(\sigma \) significance for the Odderon observation, as detailed in Appendix E.

New elastic pp scattering data around \(\sqrt{s} \approx \) 4 – 5 TeV could be particularly useful to determine more precisely any possible residual dependence of these Odderon effects as a function of \(\sqrt{s}\).

The current significance of the Odderon observation may be further increased from the 6.26 \(\sigma \) effect, but only by a tedious experimental re-analysis of some of the already published data, for example, by separating the point-to-point uncorrelated statistical and systematic errors (type A errors) from the point-to-point correlated systematic errors in elastic \(p\bar{p}\) collisions by D0, or, by the publication of the covariance matrix of the elastic cross section measurement of pp collisions at 2.76 and 7 TeV colliding energies by TOTEM. So taking more TOTEM data in special runs at new energies between \(\sqrt{s} = 2.76\) and 7.0 TeV seems to be a more enlightening and inspiring scenario, if it can be harmonized with LHC schedule and other ongoing experimental efforts.

11.1 Discussion of some of our model-dependent results

As noted above, the upper limit of the domain of validity of the H(x) scaling, the \(x_{max}(s)\) as a function of s cannot be determined model independently, it has to be taken either from extrapolations between different measured points, or from theoretical, model-dependent and validated calculations. Such calculations are presented in Appendix A, Appendix C and Appendix D. One of the most interesting characteristic features of elastic pp scattering at TeV energies is the presence of a single diffractive minimum and maximum in the experimental data on the differential cross-section of elastic pp scattering at TeV energies. In terms of the theory of multiple diffraction a single diffractive minimum is obtained if the scattering structures have a two-component internal structure [78]. The model that we have utilized for the evaluation of \(x_{max}(s)\) is based on Refs. [31, 67, 68], where the proton is assumed to have a quark–diquark structure, \(p = (q,d)\) and in one variant of this picture, the diquark is further resolved as a correlated \(d = (q,q) \) structure, corresponding to the \(p = (q, (q,q))\) case. As detailed in Ref. [30], this scenario indeed gives too many diffractive minima in the experimental acceptance, so it can be excluded. Thus our model-dependent results actually also reveal the effective sizes of constituent (dressed) quarks and diquarks inside the protons.

Concerning the quark and diquark sizes, let us note that our values are in qualitative agreement with those obtained first by Bialas and Bzdak in Refs. [31, 67, 68] at the ISR energies. Already in those papers, the binding energy of the diquark was found to be negligibly small, corresponding to the mass ratio of quarks to diquarks as 1:2. In the model, this mass ratio is reflected in a fixed value for the \(\lambda = \frac{1}{2}\) parameter, that determines the location of their center of mass to the center of the proton. The correlated motion of the quark and the diquark gives an important contribution to the description of the differential cross-section of elastic pp scattering, as the \(p = (q,q,q)\) model of three uncorrelated quarks inside the proton is in a disagreement with the experimental data.

However, the size and existence of the diquarks is a well-known controversy in the literature, related to the interpretation of diquarks. Many scientists theorize that diquarks should not be considered as particles. Even though they may contain two correlated quarks, they are not colour neutral, and therefore cannot exist as isolated bound states. So instead they tend to float freely inside protons as composite entities; while free-floating they have a size of the order of 1 fm. This also happens to be the same size as the proton itself. Other theorists analyzing elastic pp scattering in the energy range of \(\sqrt{s} = 23.5{-}62.5\) GeV suggest [79], that the size of the diquark is much smaller as compared to the size of the protons.

From our Levy studies, published recently in [16], it follows, that inside the protons the substructure increases in size, when going from the ISR energies of \(\sqrt{s} = 23.5{-}62.5\) GeV to the LHC energy of 7.0 TeV, see Fig. 3 and Tables 1, 2 of that paper. So part of the difference of the diquark size in our current manuscript and the sizes obtained in Ref.  [79] might be the difference of the investigated energy range, \(\sqrt{s} = 23.5{-}62.5\) GeV versus our results on the TeV energy scale. Another part of these quantitative differences might be due to the more precise, quantitative, statistically significant level of data description as presented in our paper. The comparison of the diquark sizes is a quantitative question, and it is difficult to make a quantitative comparison with models that were used to describe certain qualitative features of the experimental data, without aiming at a data description on statistically acceptable, significant level.

We have introduced a new, straightforwardly measurable scaling function H(x) of elastic proton-(anti)proton scattering. This scaling function transforms out the trivial energy-dependent factors, in particular, the effects due to the s-dependencies stemming from the elastic slope B(s), from the real-to-imaginary ratio \(\rho _0(s)\), as well as from the total and elastic cross sections, \(\sigma _\mathrm{tot}(s)\) and \(\sigma _\mathrm{el}(s)\), respectively. In our numerical re-analysis of already published TOTEM data, the H(x) scaling is observed from a comparison of the pp elastic scattering data at \(\sqrt{s} = 2.76\) and 7 TeV, without theoretical assumptions. TOTEM preliminary data at \(\sqrt{s} = 8\) TeV are also in the scaling limit, however, published TOTEM data at \(\sqrt{s} = 13\) TeV indicate significant violations of this new scaling. The theoretical background of this scaling law is simple and straightforward in the diffraction cone, where \(H(x) \approx \exp (-x)\), as detailed in Sect. 4.3. However, the range of the validity of this scaling extends well beyond the diffraction cone already at ISR energies, as shown in Fig. 1. A straightforward theoretical derivation for non-exponential H(x) scaling functions was presented in Sect. 4.4.

When comparing the H(x) scaling function of the differential cross section of elastic pp collisions at \(\sqrt{s} = 2.76\) and 7.0 TeV colliding energies, we find no qualitative differences. At ISR energies, in a limited energy region of \(23.5 \le \sqrt{s} \le 62.5\) GeV, the H(x) scaling curves are approximately s-independent, with a possible small scaling violation in the region of the diffractive minimum.

Such a lack of energy evolution of the H(x) scaling function of the pp collisions, even outside the diffractive cone, is in a qualitative contrast with the evolution of the H(x) scaling functions of \(p\bar{p}\) collisions at energies of \(\sqrt{s} = 0.546{-}1.96\) TeV, where a qualitative and significant energy evolution is seen in the \(x = -t B > 8 \) kinematic range for all the investigated energies. This way, we have found a qualitative difference between elastic pp and \(p\bar{p}\) collisions in terms of their H(x, s) scaling functions: these functions are not s-independent outside the diffractive cone for \(p\bar{p}\) collisions, while they are approximately s-independent in elastic pp collisions even outside the diffractive cone.

Such a lack of energy evolution of the H(x) scaling function of the pp collisions, the H(x) scaling as a property of the data in the few TeV energy range provides a strong constraint on model-building. Several simple models, like the simple eikonal amplitude of one-Pomeron-exchange lead to the violation of such a H(x) scaling. It follows that in the few TeV energy range, where the H(x) scaling is found to be valid, one-Pomeron exchange cannot be the only contribution to the scattering amplitude.

The main part of our manuscript deals with the quantification of this qualitative Odderon signal, to determine if it is statistically significant, or not.

Figures 8 and 10 clearly illustrate a qualitative and a quantitative difference between the scaling properties of the elastic pp and \(p\bar{p}\) collisions, corresponding to a crossing-odd component of the elastic scattering amplitude at the TeV energy scale. As in this kinematic region the Reggeon contributions to the scattering amplitude are suppressed by their power-law decays, a significant characteristic difference between the H(x) scaling functions of elastic pp and \(p\bar{p}\) collisions at the logarithmically similar energies of 7, 2.76 and 1.96 TeV is a clear-cut Odderon effect, because the trivial energy dependences of \(\sigma _\mathrm{el}(s)\) and B(s) as well as that of \((1+\rho ^2_0(s))\sigma ^2_\mathrm{tot}(s)\) are scaled out from H(x) by definition.

A comparison in Fig. 10 indicates a significant difference between the rescaled 7 TeV pp data set down to 1.96 TeV with the corresponding \(p\bar{p}\) data measured at \(\sqrt{s} = 1.96 \) TeV. Thus the re-analyzed D0 and TOTEM data, taken together with the verified energy independence of the H(x) scaling function in the \(\sqrt{s} = 2.76{-}7.0\) TeV energy range amount to the closing of the energy gap between 2.76 and 1.96 TeV in model-independent way, as much as reasonably possible without a direct measurement, provided that the H(x) scaling is valid for pp scattering in the kinematic range, where D0 measured the differential cross-section of elastic \(p\bar{p}\) scattering at \(\sqrt{s} = 1.96 \) TeV.

We have dedicated Appendix B to relate the crossing-odd and crossing-even contributions to the elastic pp and \(p\bar{p}\) scattering to a model independent and unitary framework, formulated in the impact parameter space. We have specialized the results of Appendix B using the model of Ref. [42] in Appendix C and demonstrated how the H(x) scaling and H(x, s), the collision energy dependent violations of this scaling can be evaluated with the help of this model [42]. Note that these calculations are based on R. J. Glauber’s multiple diffraction theory of elastic scattering, assuming that elastic scattering reveals a quark–diquark structure inside the scattered protons. In Appendix D, we have determined, model dependently, the domain of validity of the H(x) scaling in \(x = -tB\) at \(\sqrt{s} = 1.96\) TeV. According to Fig. 27, the upper limit for the domain of validity of the H(x) scaling at \(\sqrt{s} = 1.96\) TeV may include the whole D0 acceptance, with \(x_{max}(s) \ge 20.2\). This plot is directly obtained by fitting the H(x) scaling limit of the ReBB model to D0 data. In this fit, three out of the four model parameters are constrained by the H(x) scaling, so only one physical parameter had to be fitted to achieve a beautiful agreement in a statistically acceptable manner. Another estimate for the upper limit for the domain of validity of the H(x) scaling was obtained by comparing the predicted 1 standard deviation error bands of the H(x) scaling limit of the ReBB model with the same error band, obtained from the full model calculations that included scaling violations too. This result gave a more conservative upper limit, \(x_{max}(s) = 15.1\) at \(\sqrt{s} = 1.96\) TeV. Due to these theoretical uncertainties of the upper limit in x of the domain of validity of the H(x) scaling at \(\sqrt{s} = 1.96\) TeV, the highest colliding energy, where \(p\bar{p}\) elastic scattering data are available, we have investigated the stability of the Odderon signal within the theoretically determined domain of validity of the H(x) scaling in Appendix E.

Our final significance analysis is presented in Appendix A, resulting in an at least 6.26\(\sigma \), discovery level Odderon effect, if the H(x) scaling is valid in the full kinematic range of the D0 measurement, \(0 < x = -tB \le 20.2\), corresponding to a \(\chi ^2/\mathrm{NDF}\) \(=\) 80.1/17 and CL = \(3.7 \times 10^{-8}\)%. The probability of this Odderon signal is at least \(P = 1-\mathrm{CL} = 0.99999999963\). According to our model dependent calculations, as presented in Fig. 27, the assumption that the domain of validity of the H(x) scaling at \(\sqrt{s} = 1.96\) TeV includes the whole D0 acceptance with \(x_{max}(s) \ge 20.2\), is consistent with the D0 data at the 2.69 \(\sigma \) level.

We have also performed an x-range stability analysis, also model independently, in Appendix E. We established, that the significance of the Odderon is greater than 5 \(\sigma \), if the H(x) scaling is valid in the \(7 < x = - tB \le 13.5\) kinematic domain at \(\sqrt{s} = 1.96\) TeV. However, we could not determine model independently, what is the domain of validity of the H(x) scaling at this energy, as there are no measured pp data at \(\sqrt{ s} = 1.96\) TeV. So we have included a model dependent estimate of this x-range. Using the model of Refs. [30, 42, 67], that was shown to describe all the experimental data in elastic pp and \(p\bar{p}\) scattering in the \(0.546 \le \sqrt{s} \le 8\) TeV energy interval and in the \(4.4 < x = -tB \) domain, we found that the validity in x of the H(x) scaling at \(\sqrt{s} = 1.96\) TeV may extend up to \(x \le 15.1\), as detailed in Appendix D. This interval or domain of validity of the H(x) scaling includes the smaller \(7 < x \le 13.5\) domain, where the signal is larger than 5 \(\sigma \). 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: 9 out of the 17 D0 datapoints can be discarded, 5 at large x and 4 points at low x, and the signal still remains significant enough for a discovery. When we have fitted the H(x) scaling limit of the same model to \(p\bar{p}\) experimental data, as presented in Fig. 27, we found that \(x_{max}(s) \ge 20.2\) is also consistent with the description of the D0 data at the 2.69 \(\sigma \) level. In general, the H(x) scaling is not valid for \(p\bar{p}\), but in the Real Extended Bialas–Bzdak model, the H(x) scaling of elastic pp collisions induces a one parameter fit to elastic \(p\bar{p}\) collisions, as three out of four physical model parameters are the same in this model for pp and \(p\bar{p}\) collisions.

Our x-range stability analysis, detailed in Appendix E, indicates that part of the statistically significant contribution to this Odderon signal is coming from the kinematic range of \(x < 10 \): excluding this region decreases the significance of the crossing-odd signal below the discovery level. It is thus important to measure elastic scattering cross-sections at LHC at large \(-t\), well beyond the diffractive cone. Elastic pp scattering data in a vicinity of \(\sqrt{s}\approx \) 2 TeV as well as in between 2.76 and 7 TeV would be most useful for further detailing the Odderon properties. Similarly, part of the signal is coming from large x region, as excluding the kinematic range \(x > 12.1\) also results in a loss of significance.

In order to determine where the important contributions to this signal are originating from, we have divided the \(0< x \le 20.2\) kinematic range of acceptance to four regions, the diffractive cone, the swing, the diffractive interference and the tail regions, corresponding to \(0 < x \le 5.1\), \(5.1 < x \le 8.4\), \(8.4 < x \le 13.5\) and \(13.5 < x \le 20.2\), respectively, with 2 D0 points associated with the diffractive cone, and 5-5 D0 points in each of the remaining three regions. We have shown that the type B, point dependent but overall correlated errors and their correlation coefficient plays an important role in this analysis and that the best value of the correlation coefficient is x-range dependent. This means that locally shifting the datapoints up or down in a specific interval the agreement between the pp and \(p\bar{p}\) measurements can be improved in that particular interval. We have performed the interval dependent optimizations and found that the locally optimized contributions from the swing and from the tail are not as important as the contributions from the diffractive interference region, that includes the diffractive minimum and the maximum. The second most important contribution comes from the swing region, where the H(x) scaling function for elastic pp collisions bends below the exponential. The combination of the swing and the diffractive interference region already provided a more than 5 \(\sigma \) effect, as detailed in Appendix E. If we consider that the interpolations needed to compare the H(x) scaling functions in a model-independent manner at different energies behave as theoretical curves (i.e. have only one kind of error) then we obtain the significance of at least 6.55\(\sigma \) as discussed in the body of this manuscript. This significance further decreases to 6.26\(\sigma \) if we consider that these interpolation lines do not have theoretical type of errors but both type A and type B, point-to-point fluctuating and point-to-point correlated systematic experimental errors as well. The only way to further decrease the significance of the Odderon signal is to limit the kinematic range of the comparison to narrower and narrower ranges in \(x =-tB\).

Our final significance for an Odderon signal of \(6.26\sigma \) is presented from the model-independent analysis in Appendix A, which relies on the validity of the H(x) scaling in the \(0 < x \le 20.2\) kinematic range at \(\sqrt{s} = 1.96 \) TeV. The self-consistency of this assumption is shown in Fig. 27.

Let us emphasize, for the sake of completeness, that we find an interesting hierarchy of significances.

Based on model dependent considerations, the x-range of the \(p\bar{p}\) data of the D0 experiment might be narrowed and correspondingly, the significances may decrease, as more and more datapoints are removed. Model dependently, we estimate, that the H(x) scaling may be valid at the D0 energy of \(\sqrt{s} = 1.96 \) TeV up to \(x_{max} = 15.1\). In the \(0 < x \le 14.8\), theoretically limited x-range the Odderon signal remains greater than 5.3 \(\sigma \), according to Appendix E. We have investigated how far this domain can be narrowed down under the condition that the Odderon signal remains greater than a 5 \(\sigma \) effect? We found that in the very much narrowed from below and from above domain, corresponding to the \(7.0 < x \le 13.5\) interval that includes only 8 out of the 17 D0 points, the Odderon signal that we analyzed has a significance that is greater than a 5 \(\sigma \), discovery level effect. This range is well below the theoretically estimated \(x_{max} = 15.1\) limit.

In Appendix B, we discuss the model-independent properties of the Pomeron and Odderon exchange at the TeV energy scale, under the condition that this energy is large enough that the Pomeron and Odderon exchange can be identified with the crossing-even and the crossing-odd contributions to the elastic scattering, respectively. We demonstrated here that S-matrix unitarity strongly constrains the possible form of the impact parameter dependence of the Pomeron and Odderon amplitudes.

In Appendix C, we demonstrated how the H(x) scaling emerges within a specific model, defined in Ref. [30]. This model is one of the possible models in the class discussed in Appendix B. We have demonstrated that four conditions must be satisfied simultaneously. One of the conditions for the validity of the H(x) scaling is found to be the s-independence of the experimentally measured \(\rho (s)\) parameter. The decrease of \(\rho (s)\) at the currently maximal LHC energy of \(\sqrt{s} = 13 \) TeV as measured by the TOTEM Collaboration in Ref. [2] thus provides a natural explanation for the violation of the H(x) scaling at these energies.

In Appendix D we estimate the domain of validity of the H(x) scaling also in a model-dependent manner, using the same model of Ref. [30]. Surprisingly, we found that after carefully taking into account the possible quadratic in \(\ln (s)\) energy dependencies of the scale parameters of the model of Ref. [30] and after taking into account the correlations between these model parameters, the kinematic range for the domain of validity of this new H(x) scaling may extend to a very broad range of 200 GeV \(\le \sqrt{s}\le \) 8 TeV, however, with a range that gradually narrows in \(-t = x/B(s)\) with decreasing energies.

Another key point to recognize is that if we allow for a model-dependent analysis, the significance goes further up to 7.08\(\sigma \) as detailed in Ref. [42] and summarized in Appendix E. There is a trade-off effect in the background of this. Model dependent calculations lead to a reduction of significance at 1.96 TeV, as the extrapolation of pp differential cross-section becomes more uncertain, as compared to the extrapolation with the help of the H(x) scaling. However, this loss in significance is overcompensated by the gain in the possibility to extrapolate the \(p\bar{p}\) differential cross-sections up in energy. If we utilize only the H(x) scaling, it allows only to compare pp data with \(p\bar{p}\) data at decreasing energies, but \(p\bar{p}\) data do not obey a H(x) scaling law, so it cannot be used to compare them with pp data at 2.76 TeV. But this extrapolation becomes possible with the help of a model calculation and it results in a huge increase in the Odderon significance, as detailed in Appendix E and in Ref. [42].

TOTEM data on an approximately energy independent ratio of the differential cross-section at the diffractive maximum and minimum  [80] indicate, that the expected upper limit for the domain of validity of the H(x) scaling is at least 13.5 at \(\sqrt{s} = 1.96\) TeV, as detailed at the end of Appendix A. This experimental insight also suggests, that the domain of validity of H(x) scaling at \(\sqrt{s} = 1.96\) TeV includes the \(7.0 < x \le 13.5\) domain. These observations combined with our model independent x-range stability studies in Appendix E indicate, that in this \(7.0 < x \le 13.5\) 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 the theoretically expected domain 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.

Recently, the STAR collaboration measured the differential cross-section of elastic pp collisions at \(\sqrt{s} = 200\) GeV [81]. This measurement 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.

In conclusion, we find from a model-independent re-analysis of the scaling properties of the differential cross sections of already published D0 and TOTEM data sets a statistically significant, more than a 6.26 \(\sigma \) Odderon effect, based on the assumption of the validity of the H(x) scaling at \(\sqrt{s} = 1.96\) TeV in the \(0 < x \le 20.2\) kinematic range. Based on theoretical considerations we estimated that the domain of validity of the H(x) scaling at this particular energy might be actually smaller, \(0 < x \le 15.1\). So we have also determined what is the minimum size of the domain of validity of the H(x) scaling that corresponds to the 5\(\sigma \) level Odderon significance. As detailed in Appendix E, any interval that fully includes the \(7.0 < x \le 13.5\) range results in a greater than 5 \(\sigma \), discovery level Odderon signal. The experimentally observed [4, 43] energy independence of the diffractive maximum-to-minimum ratio R(s) also supports that the domain of the H(x) scaling at \(\sqrt{s} = 1.96\) TeV extends above the diffractive maximum, which is at \(x_{bump} = 13\) in this scaling limit.

We thus find a statistically significant, greater than 5\(\sigma \) signal of t-channel Odderon exchange, that is robust for variation of the lower or upper limit of the domain of validity of the H(x) scaling at \(\sqrt{s} = 1.96 \) TeV. We have highlighted a hierarchy of significances, including experimental and theoretical, model dependent results too. If theoretical modelling is also taken into account, the combined significance of Odderon observation increases to at least 7.08\(\sigma \), as shown in Appendix E.

Whatever we tried the significance of the Odderon observation remained safely above the 5\(\sigma \) discovery threshold, with the most conservative significance estimate detailed in Appendix A. An x-range stability analysis, summarized in Appendix E indicates, that the only way to decrease this signal is to decrease the \(-t = x/B(s)\) range of the comparison i.e. deleting data from the signal region.

We have validated the surprisingly large domain of H(x) scaling with already published data both in elastic proton–proton and in proton–antiproton collisions and are eagerly waiting for upcoming results from the STAR collaboration to test our new scaling in elastic proton–proton collisions at the top RHIC energy of \(\sqrt{s} = 510\) GeV.