First determination of the \({\rho }\) parameter at \({\sqrt{s} = 13}\) TeV: probing the existence of a colourless Codd threegluon compound state
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Abstract
The TOTEM experiment at the LHC has performed the first measurement at \(\sqrt{s} = 13\,\mathrm{TeV}\) of the \(\rho \) parameter, the real to imaginary ratio of the nuclear elastic scattering amplitude at \(t=0\), obtaining the following results: \(\rho = 0.09 \pm 0.01\) and \(\rho = 0.10 \pm 0.01\), depending on different physics assumptions and mathematical modelling. The unprecedented precision of the \(\rho \) measurement, combined with the TOTEM total crosssection measurements in an energy range larger than \(10\,\mathrm{TeV}\) (from 2.76 to \(13\,\mathrm{TeV}\)), has implied the exclusion of all the models classified and published by COMPETE. The \(\rho \) results obtained by TOTEM are compatible with the predictions, from other theoretical models both in the Reggelike framework and in the QCD framework, of a crossingodd colourless 3gluon compound state exchange in the tchannel of the proton–proton elastic scattering. On the contrary, if shown that the crossingodd 3gluon compound state tchannel exchange is not of importance for the description of elastic scattering, the \(\rho \) value determined by TOTEM would represent a first evidence of a slowing down of the total crosssection growth at higher energies. The very lowt reach allowed also to determine the absolute normalisation using the Coulomb amplitude for the first time at the LHC and obtain a new total proton–proton crosssection measurement \(\sigma _{\mathrm{tot}} = (110.3 \pm 3.5)\,\mathrm{mb}\), completely independent from the previous TOTEM determination. Combining the two TOTEM results yields \(\sigma _{\mathrm{tot}} = (110.5 \pm 2.4)\,\mathrm{mb}\).
1 Introduction
The TOTEM experiment at the LHC has measured the differential elastic proton–proton scattering crosssection as a function of the fourmomentum transfer squared, t, down to \(t = 8\times 10^{4}\,\mathrm{GeV^2}\) at the centreofmass energy \(\sqrt{s} = 13\,\mathrm{TeV}\) using a special \(\beta ^* = 2.5\,\mathrm{km}\) optics. This allowed to access the Coulombnuclear interference (CNI) and to determine the \(\rho \) parameter, the realtoimaginary ratio of the forward hadronic amplitude, with an unprecedented precision.
Measurements of the total proton–proton crosssection and \(\rho \) have been published in the literature from the low energy range of \(\sqrt{s} \sim 10\,\mathrm{GeV}\) up to the LHC energy of \(8\,\mathrm{TeV}\) [1]. Such experimental measurements have been parametrised by a large variety of phenomenological models in the last decades, and were analysed and classified by the COMPETE collaboration [2].
It is shown in the present paper that none of the abovementioned models can describe simultaneously the TOTEM \(\rho \) measurement at \(13\,\mathrm{TeV}\) and the ensemble of the total crosssection measurements by TOTEM ranging from \(\sqrt{s} = 2.76\) to \(13\,\mathrm{TeV}\) [3, 4, 5, 6]. The exclusion of the COMPETE published models is quantitatively demonstrated on the basis of the pvalues reported in this work. Such conventional modelling of the lowt nuclear elastic scattering is based on various forms of Pomeron exchanges and related crossingeven scattering amplitudes (not changing sign under crossing, cf. Section 4.5 in [7]).
Other theoretical models exist both in terms of Reggelike or axiomatic field theories [8] and of QCD [9, 10, 11] – they are capable of predicting or taking into account several effects confirmed or observed at LHC energies: the existence of a sharp diffractive dip in the proton–proton elastic tdistribution also at LHC energies [12], the deviation of the elastic differential crosssection from a pure exponential [4], the deviation of the elastic diffractive slope, B, from a linear \(\log (s)\) dependence as a function of the centreofmass energy [6], the variation of the nuclear phase as a function of t, the larget powerlaw behaviour of the elastic tdistribution with no oscillatory behaviour and the growth rate of the total crosssection as a function of \(\sqrt{s}\) at LHC energies [6]. These theoretical frameworks foresee the possibility of more complex tchannel exchanges in the proton–proton elastic interaction, including crossingodd scattering amplitude contributions (changing sign under crossing).
The crossingodd contributions relevant for high energies (where secondary Reggeons are expected to be negligible [13]) were associated with the concept of the Odderon (the crossingodd counterpart of the Pomeron [14]) invented in the 70’s [15, 16] and later confirmed as an essential QCD prediction [9, 10, 11, 17, 18]. They are quantified in QCD (see e.g. Refs. [19, 20]) where they are represented (in the most basic form) by the exchange of a colourless 3gluon compound state in the tchannel in the nonperturbative regime (t ranging from 0 up to roughly the diffractive dip and bump). Such a state would naturally have \(J^{PC}=1^{}\) quantum numbers and is predicted by lattice QCD with a mass of about 3 to \(4\,\mathrm{GeV}\) (also referred to as vector glueball) [21, 22] as required by the st channel duality [23]. For completeness, an exchange of a 3gluon state may also be crossing even, e.g. in case the state evolves (collapses) into 2 gluons [24, 25, 26]. However hereafter, unless specified differently, we will refer only to crossingodd 3gluon exchanges – the crossingeven 3gluon exchanges will be included in the Pomeron amplitude as a subleading contribution (suppressed by \(\alpha _{\mathrm{s}}\) with respect to the 2gluon exchanges).
Experimental searches for a 3gluon compound state have used various channels. In central production the 3gluon state emitted by one proton may fuse with a Pomeron (photon) emitted from the other proton (electron/positron) and create a detectable meson system [26, 27]. However, such processes are dominated by pomeron–photon (photon–photon) fusion, making the observation of a 3gluon state difficult. In elastic scattering at low energy [28], the observation of 3gluon compound state is complicated by the presence of secondary Regge trajectories influencing the potential observation of differences between the proton–proton and proton–antiproton scattering. At high energy (gluonicdominated interactions) [29], one could investigate for both proton–proton and proton–antiproton scattering the diffractive dip, where the imaginary part of the Pomeron amplitude vanishes; however there are no measurements nor facilities allowing a comparison at the same fixed \(\sqrt{s}\) energy.
The Coulombnuclear interference at the LHC is an ideal laboratory to probe the exchange of a virtual oddgluons compound state, because it selects the required quantum numbers in the trange where the interference terms cannot be neglected with respect to the QED and nuclear amplitudes squared. The highest sensitivity is reached in the trange where the QED and nuclear amplitudes are of similar magnitude, thus this has been the driving factor in designing the acceptance requirements then achieved via the \(2.5\,\mathrm{km}\) optics of the LHC. The \(\rho \) parameter being an analytical function of the nuclear phase at \(t=0\), it represents a sensitive probe of the interference terms into the evolution of the real and imaginary parts of the nuclear amplitude.
Consequently theoretical models have made sensitive predictions via the evolution of \(\rho \) as a function of \(\sqrt{s}\) to quantify the effect of the possible 3gluon compound state exchange in the elastic scattering tchannel [20, 30]. Those, currently nonexcluded, theoretical models systematically require significantly lower \(\rho \) values at \(13\,\mathrm{TeV}\) than the predicted Pomerononly evolution of \(\rho \) at \(13\,\mathrm{TeV}\), consistently with the \(\rho \) measurement reported in the present work.
The confirmation of this result in additional channels would bring, besides the evidence for the existence of the QCDpredicted 3gluon compound state, theoretical consequences such as the generalization of the Pomeranchuk theorem (i.e. the total crosssection of proton–proton and proton–antiproton asymptotically having their ratio converging to 1 rather than their difference converging to 0).
On the contrary, if the role of the 3gluon compound state exchange is shown insignificant, the present TOTEM results at \(13\,\mathrm{TeV}\) would imply by the dispersion relations the first experimental evidence for total crosssection saturation effects at higher energies, eventually deviating from the asymptotic behaviour proposed by many contemporary models (e.g. the functional saturation of the Froissart compound [31]).
The two effects, crossingodd contribution and crosssection saturation, could both be present without being mutually exclusive.
Besides the extraction of the \(\rho \) parameter, the very low t elastic scattering can be used to determine the normalisation of the differential crosssection – a crucial ingredient for measurement of the total crosssection, \(\sigma _{\mathrm{tot}}\). In its ideal form, the normalisation can be determined as the proportionality constant between the Coulomb crosssection known from QED and the data measured at such low t that other than Coulomb crosssection contributions can be neglected. This “Coulomb normalisation” technique opens the way to another total crosssection measurement at \(\sqrt{s} = 13\,\mathrm{TeV}\), completely independent of previous results. This publication presents the first successful application of this method to LHC data.
2 Experimental apparatus
The TOTEM experiment, located at the LHC Interaction Point (IP) 5 together with the CMS experiment, is dedicated to the measurement of the total crosssection, elastic scattering and diffractive processes. The experimental apparatus, symmetric with respect to the IP, detects particles at different scattering angles in the forward region: a forward proton spectrometer composed of detectors in Roman Pots (RPs) and the magnetic elements of the LHC and, to measure at larger angles, the forward tracking telescopes T1 and T2. A complete description of the TOTEM detector instrumentation and its performance is given in [32, 33]. The data analysed here come from the RPs only.
A RP is a movable beampipe insertion which houses the tracking detectors that are thus capable of approaching the LHC beam to a distance of less than a millimetre, and to detect protons with scattering angles of only a few microradians. The proton spectrometer is organised in two arms: one on the left side of the IP (LHC sector 45) and one on the right (LHC sector 56), see Fig. 1. In each arm, there are two RP stations, denoted “210” (about \(210\,\mathrm{m}\) from the IP) and “220” (about \(220\,\mathrm{m}\) from the IP). Each station is composed of two RP units, denoted “nr” (near to the IP) and “fr” (far from the IP). The presented measurement is performed with units “210fr” (approximately \(213\,\mathrm{m}\) from the IP) and “220fr” (about \(220\,\mathrm{m}\) from the IP). The 210fr unit is tilted by \(8^\circ \) in the transverse plane with respect to the 220fr unit. Each unit consists of 3 RPs, one approaching the outgoing beam from the top, one from the bottom, and one horizontally. Each RP houses a stack of 5 “U” and 5 “V” silicon strip detectors, where “U” and “V” refer to two mutually perpendicular strip orientations. The special design of the sensors is such that the insensitive area at the edge facing the beam is only a few tens of micrometres [34]. Due to the \(7\,\mathrm{m}\) long lever arm between the two RP units in one arm, the local track angles can be reconstructed with an accuracy of about \(2.5\,\mathrm{\mu rad}\).
Since elastic scattering events consist of two collinear protons emitted in opposite directions, the detected events can have two topologies, called “diagonals”: 45 bottom–56 top and 45 top–56 bottom, where the numbers refer to the LHC sector.
This article uses a reference frame where x denotes the horizontal axis (pointing out of the LHC ring), y the vertical axis (pointing against gravity) and z the beam axis (in the clockwise direction).
3 Beam optics
Optical functions for elastic proton transport for the \(\beta ^{*} = 2500\,\)m optics. The values refer to the left arm, for the right one they are very similar
RP unit  \(L_x\)  \(v_x\)  \(L_y\)  \(v_y\) 

210fr  \(73.05\,\mathrm{m}\)  \(0.634\)  \(244.68\,\mathrm{m}\)  \(+0.009\) 
220fr  \(51.10\,\mathrm{m}\)  \(0.540\)  \(282.96\,\mathrm{m}\)  \(0.018\) 
4 Data taking
The results reported here are based on data taken in September 2016 during a sequence of dedicated LHC proton fills (5313, 5314, 5317 and 5321) with the special beam properties described in the previous section.
The vertical RPs approached the beam centre to only about 3 times the vertical beam width, \(\sigma _{y}\), thus roughly to \(0.4\,\mathrm{mm}\). The exceptionally close distance was required in order to reach very low t values and was possible due to the low beam intensity in this special beam operation: each beam contained only four or five colliding bunches and one noncolliding bunch, each with about \(5\times 10^{10}\) protons.
The horizontal RPs were only needed for the trackbased alignment and therefore placed at a safe distance of \(8\,\sigma _{x} \approx 5\) mm, close enough to have an overlap with the vertical RPs.
The collimation strategy applied in the previous measurement [5] with carbon primary collimators was first tried, however, this resulted in too high beam halo background. To keep the background under control, a new collimation scheme was developed, with more absorbing tungsten collimators closest to the beam in the vertical plane, in order to minimise the outscattering of halo particles. As a first step, vertical collimators TCLA scraped the beam down to \(2\,\sigma _{y}\), then the collimators were retracted to \(2.5\,\sigma _{y}\), thus creating a \(0.5\,\sigma _{y}\) gap between the beam edge and the collimator jaws. A similar procedure was performed in the horizontal plane: collimators TCP.C scraped the beam to \(3\,\mathrm{\sigma _{x}}\) and then were retracted to \(5.5\,\mathrm{\sigma _{x}}\), creating a \(2.5\,\mathrm{\sigma _x}\) gap. With the halo strongly suppressed and no collimator producing showers by touching the beam, the RPs at \(3\,\sigma _{y}\) were operated in a backgrounddepleted environment for about one hour until the beamtocollimator gap was refilled by diffusion, as diagnosed by the increasing shower rate (red graph in Fig. 2). When the background conditions had deteriorated to an unacceptable level, the beam cleaning procedure was repeated, again followed by a quiet datataking period.
The events collected were triggered by a doublearm proton trigger (coincidence of any RP left of IP5 and any RP right of IP5) or a zerobias trigger (random bunch crossings) for calibration purposes.
In total, a data sample with an integrated luminosity of about \(0.4\,{\mathrm{nb}}^{1}\) was accumulated in which more than 7 million of elastic event candidates were tagged.
5 Differential crosssection
The analysis method is very similar to the previously published one [5]. The only important difference stems from using different RPs for the measurement: unit 210fr instead of 220nr as in [5] since the latter was not equipped with sensors anymore. Due to the optics and beam parameters the unit 210fr has worse lowt acceptance, further deteriorated by the tilt of the unit (effectively increasing the RP distance from the beam). Consequently, in order to maintain the lowt reach essential for this study, the main analysis (denoted “2RP”) only uses the 220fr units (thus 2 RPs per diagonal). Since not using the 210nr units may, in principle, result in worse resolution and background suppression, for control reasons, the traditional analysis with 4 units per diagonal (denoted “4RP”) was pursued, too. In Sect. 5.5 the “2RP” and “4RP” will be compared showing a very good agreement. In what follows, the “2RP” analysis will be described unless stated otherwise.
Section 5.1 covers all aspects related to the reconstruction of a single event. Section 5.2 describes the steps of transforming a raw tdistribution into the differential crosssection. The tdistributions are analysed separately for each LHC fill and each diagonal, and are only merged at the end as detailed in Sect. 5.3. Section 5.4 describes the evaluation of systematic uncertainties and Sect. 5.5 presents several comparison plots used as systematic cross checks.
5.1 Event analysis
The event kinematics are determined from the coordinates of track hits in the RPs after proper alignment (see Sect. 5.1.2) using the LHC optics (see Sect. 5.1.3).
5.1.1 Kinematics reconstruction
In the “4RP” analysis, the same reconstruction as in [5] is used which allows for stronger elasticselection cuts, see Sect. 5.2.1.
5.1.2 Alignment
TOTEM’s usual threestage procedure (Section 3.4 in [33]) for correcting the detector positions and rotation angles has been applied: a beambased alignment prior to the run followed by two offline methods. The first method uses straight tracks to determine the relative position among the RPs by minimising trackhit residuals. The second method exploits the symmetries of elastic scattering to determine the positions of RPs with respect to the beam. This determination is repeated in 20minute time intervals to check for possible beam movements.
5.1.3 Optics
It is crucial to know with high precision the LHC beam optics between IP5 and the RPs, i.e. the behaviour of the spectrometer composed of the various magnetic elements. The optics calibration has been applied as described in [35]. This method uses RP observables to determine fine corrections to the optical functions presented in Eq. (1).
5.1.4 Resolution
Two kinds of resolution can be distinguished: the resolution of the singlearm angular reconstruction, Eq. (2), used for selection cuts and nearedge acceptance correction, and the resolution of the doublearm reconstruction, Eq. (3), used for the unsmearing correction of the final tdistribution. Since the singlearm reconstruction is biased by the vertex term in the horizontal plane, the corresponding resolution is significantly worse than the doublearm reconstruction.
The singlearm resolution can be studied by comparing the angles reconstructed from the left and right arm, see an example in Fig. 3. The width of the distributions was found to grow slightly during the fills, compatible with the effect of beam emittance growth. The typical range was from 10.0 to \(14.5\,\mathrm{\mu rad}\) for the horizontal projection and from 0.36 to \(0.38\,\mathrm{\mu rad}\) for the vertical. The associated uncertainties were 0.3 and 0.007, respectively. As illustrated in Fig. 3, the shape of the distributions is very close to Gaussian, especially at the beginning of each fill.
5.2 Differential crosssection reconstruction
The elastic selection cuts. The superscripts R and L refer to the right and left arm. The rightmost column gives a typical standard deviation of the cut distribution
Number  Cut  Std. dev. (\(\equiv 1\sigma \)) 

1  \(\theta _x^{*\mathrm{R}}  \theta _x^{*\mathrm{L}}\)  \(14\,\mathrm{\mu rad}\) 
2  \(\theta _y^{*\mathrm{R}}  \theta _y^{*\mathrm{L}}\)  \(0.38\,\mathrm{\mu rad}\) 
5.2.1 Event tagging
Within the “2RP” analysis one may apply the cuts requiring the reconstructedtrack collinearity between the left and the right arm, see Table 2. The correlation plots corresponding to these cuts are shown in Fig. 4.
In order to limit the selection inefficiency, the thresholds for the cuts are set to \(4\,\mathrm{\sigma }\). Applying the cuts at the \(5\,\mathrm{\sigma }\)level would yield about \(0.1\,\mathrm{\%}\) more events almost uniformly in every tbin. This kind of inefficiency only contributes to a global scale factor, which is irrelevant for this analysis because the normalisation is taken from a different data set (cf. Sect. 5.2.6).
5.2.2 Background
As the RPs were very close to the beam, one may expect an enhanced background from coincidence of beam halo protons hitting detectors in the two arms. Other background sources (pertinent to any elastic analysis) are central diffraction and pileup of two single diffraction events.
5.2.3 Acceptance correction
The full acceptance correction, \({\mathcal {A}}\), has a value of 12 in the lowestt bin and decreases smoothly towards about 2.1 at \(t = 0.2\,\mathrm{GeV^2}\). Since a single diagonal cannot cover more than half of the phase space, the minimum value of the correction is 2.
The uncertainties related to \({\mathcal {A}}_{\mathrm{fluct}}\) follow from the uncertainties of the resolution parameters: standard deviation and distribution shape, see Sect. 5.1.4. Since \({\mathcal {A}}_{\mathrm{geom}}\) is calculated from a trivial trigonometric formula, there is no uncertainty directly associated with it. However biases can arise indirectly from effects that break the assumed azimuthal symmetry like misalignments or optics perturbations already covered above.
5.2.4 Inefficiency corrections
Since the overall normalisation will be determined from another dataset (see Sect. 5.2.6), any inefficiency correction that does not alter the tdistribution shape does not need to be considered in this analysis (trigger, data acquisition and pileup inefficiency discussed in [36, 37]). The remaining inefficiencies are related to the inability of a RP to resolve the elastic proton track.
Proton interactions in a RP affecting simultaneously another RP downstream represent another source of inefficiency. The contribution from these correlated inefficiencies, \({\mathcal {I}}_2\), is determined by evaluating the rate of events with high track multiplicity (\(\gtrsim \) 5) in both 210fr and 220fr RP units. Events with high track multiplicity simultaneously in the top and bottom RP of the 210fr units are discarded as such a shower is likely to have started upstream from the RP station and thus be unrelated to the elastic proton interacting with detectors. The value, \({\mathcal {I}}_2 \approx (1.5 \pm 0.7)\,\mathrm{\%}\), is compatible between left/right arms and top/bottom RP pairs and compares well to MonteCarlo simulations (e.g. section 7.5 in [38]).
5.2.5 Unfolding of resolution effects
 1.
The differential crosssection data are fitted by a smooth curve.
 2.
The fit is used in a numericalintegration calculation of the smeared tdistribution (using the resolution parameters determined in Sect. 5.1.4). The ratio between the smeared and the nonsmeared tdistributions gives a set of perbin correction factors.
 3.
The corrections are applied to the observed (yet uncorrected) differential crosssection yielding a better estimate of the true tdistribution.
 4.
The corrected differential crosssection is fed back to step 1.
The final correction \({\mathcal {U}}\) is significantly different from 1 only at very low t (where a rapid crosssection growth occurs, see Fig. 9). The relative effect is never greater than \(0.4\,\mathrm{\%}\).
Several fit parametrisations were tested, however yielding negligible difference in the final correction \({\mathcal {U}}\) for \(t \lesssim 0.3\,\mathrm{GeV^2}\). Figure 9 shows the case for two of those.
The elastic differential crosssection as determined in this analysis (medium binning). The three leftmost columns describe the bins in t. The representative point gives the t value suitable for fitting [40]. The other columns are related to the differential crosssection. The five rightmost columns give the leading systematic biases in \(\mathrm{d}\sigma /\mathrm{d}t\) for \(1\sigma \)shifts in the respective quantities, \(\delta s_q\), see Eqs. (12) and (13). The contribution due to optics corresponds to the third vector in Eq. (7). In order to avoid undesired interplay between statistical and systematic uncertainties, the latter are calculated from the relative uncertainties (Sect. 5.4) by multiplying by a smooth fit (Fig. 14) evaluated at the representative point
t bin \((GeV^2)\)  \(\mathrm{d}\sigma /\mathrm{d}t \quad (mb/GeV^2)\)  

Left  Right  Represent.  Value  Statist.  System.  Normal.  Alignment  Optics  Vert. beam  Beam 
edge  edge  point  uncert.  uncert.  vert. shift  mode 3  divergence  mom.  
0.000800  0.000966  0.000879  868.726  12.518  48.472  \(+\,46.865\)  \(+\,9.265\)  \(\,0.175\)  \(\,5.360\)  \(+\,0.548\) 
0.000966  0.001144  0.001051  784.894  7.252  42.786  \(+\,42.318\)  \(+\,5.098\)  \(\,0.252\)  \(\,1.279\)  \(+\,0.750\) 
0.001144  0.001335  0.001236  716.217  5.943  39.656  \(+\,39.476\)  \(+\,2.900\)  \(\,0.299\)  \(\,0.660\)  \(+\,0.876\) 
0.001335  0.001540  0.001434  696.283  5.279  37.685  \(+\,37.603\)  \(+\,1.722\)  \(\,0.330\)  \(\,0.435\)  \(+\,0.963\) 
0.001540  0.001759  0.001646  655.272  4.710  36.358  \(+\,36.313\)  \(+\,1.059\)  \(\,0.350\)  \(\,0.327\)  \(+\,1.012\) 
0.001759  0.001995  0.001874  643.657  4.346  35.415  \(+\,35.385\)  \(+\,0.670\)  \(\,0.363\)  \(\,0.259\)  \(+\,1.047\) 
0.001995  0.002248  0.002118  634.502  4.047  34.713  \(+\,34.689\)  \(+\,0.435\)  \(\,0.370\)  \(\,0.212\)  \(+\,1.068\) 
0.002248  0.002519  0.002380  617.090  3.764  34.166  \(+\,34.144\)  \(+\,0.287\)  \(\,0.375\)  \(\,0.180\)  \(+\,1.080\) 
0.002519  0.002809  0.002661  611.317  3.552  33.720  \(+\,33.699\)  \(+\,0.193\)  \(\,0.377\)  \(\,0.156\)  \(+\,1.085\) 
0.002809  0.003117  0.002960  606.121  3.374  33.341  \(+\,33.320\)  \(+\,0.132\)  \(\,0.377\)  \(\,0.137\)  \(+\,1.085\) 
0.003117  0.003444  0.003279  601.057  3.212  33.005  \(+\,32.984\)  \(+\,0.092\)  \(\,0.375\)  \(\,0.122\)  \(+\,1.080\) 
0.003444  0.003791  0.003616  594.143  3.064  32.695  \(+\,32.675\)  \(+\,0.065\)  \(\,0.373\)  \(\,0.109\)  \(+\,1.073\) 
0.003791  0.004155  0.003972  589.140  2.945  32.402  \(+\,32.382\)  \(+\,0.047\)  \(\,0.369\)  \(\,0.099\)  \(+\,1.062\) 
0.004155  0.004538  0.004346  581.891  2.827  32.117  \(+\,32.097\)  \(+\,0.033\)  \(\,0.365\)  \(\,0.090\)  \(+\,1.050\) 
0.004538  0.004940  0.004738  577.737  2.726  31.836  \(+\,31.816\)  \(+\,0.024\)  \(\,0.360\)  \(\,0.082\)  \(+\,1.035\) 
0.004940  0.005361  0.005150  575.008  2.636  31.553  \(+\,31.534\)  \(+\,0.019\)  \(\,0.354\)  \(\,0.075\)  \(+\,1.019\) 
0.005361  0.005801  0.005581  560.883  2.526  31.266  \(+\,31.248\)  \(+\,0.016\)  \(\,0.349\)  \(\,0.066\)  \(+\,1.002\) 
0.005801  0.006260  0.006030  563.968  2.468  30.974  \(+\,30.956\)  \(+\,0.014\)  \(\,0.342\)  \(\,0.059\)  \(+\,0.984\) 
0.006260  0.006737  0.006498  554.645  2.387  30.676  \(+\,30.659\)  \(+\,0.012\)  \(\,0.335\)  \(\,0.053\)  \(+\,0.965\) 
0.006737  0.007232  0.006984  551.682  2.323  30.372  \(+\,30.355\)  \(+\,0.011\)  \(\,0.329\)  \(\,0.048\)  \(+\,0.945\) 
0.007232  0.007746  0.007488  547.232  2.260  30.060  \(+\,30.043\)  \(+\,0.010\)  \(\,0.321\)  \(\,0.042\)  \(+\,0.924\) 
0.007746  0.008279  0.008012  543.798  2.202  29.739  \(+\,29.723\)  \(+\,0.009\)  \(\,0.314\)  \(\,0.036\)  \(+\,0.903\) 
0.008279  0.008833  0.008556  534.391  2.133  29.410  \(+\,29.395\)  \(+\,0.008\)  \(\,0.306\)  \(\,0.032\)  \(+\,0.881\) 
0.008833  0.009407  0.009120  527.706  2.076  29.073  \(+\,29.059\)  \(+\,0.008\)  \(\,0.299\)  \(\,0.029\)  \(+\,0.859\) 
0.009407  0.009999  0.009703  523.040  2.027  28.729  \(+\,28.715\)  \(+\,0.007\)  \(\,0.291\)  \(\,0.026\)  \(+\,0.836\) 
0.009999  0.010608  0.010303  514.667  1.976  28.377  \(+\,28.364\)  \(+\,0.006\)  \(\,0.283\)  \(\,0.023\)  \(+\,0.813\) 
0.010608  0.011237  0.010922  507.673  1.925  28.019  \(+\,28.006\)  \(+\,0.005\)  \(\,0.274\)  \(\,0.020\)  \(+\,0.789\) 
0.011237  0.011887  0.011562  501.645  1.877  27.653  \(+\,27.641\)  \(+\,0.005\)  \(\,0.266\)  \(\,0.018\)  \(+\,0.765\) 
0.011887  0.012556  0.012221  498.095  1.840  27.280  \(+\,27.269\)  \(+\,0.004\)  \(\,0.258\)  \(\,0.015\)  \(+\,0.741\) 
0.012556  0.013242  0.012898  487.164  1.791  26.902  \(+\,26.891\)  \(+\,0.003\)  \(\,0.249\)  \(\,0.013\)  \(+\,0.717\) 
0.013242  0.013948  0.013594  482.155  1.753  26.519  \(+\,26.509\)  \(+\,0.003\)  \(\,0.241\)  \(\,0.011\)  \(+\,0.692\) 
0.013948  0.014674  0.014311  475.608  1.712  26.130  \(+\,26.120\)  \(+\,0.003\)  \(\,0.233\)  \(\,0.009\)  \(+\,0.668\) 
0.014674  0.015421  0.015047  465.619  1.668  25.735  \(+\,25.726\)  \(+\,0.002\)  \(\,0.224\)  \(\,0.007\)  \(+\,0.643\) 
0.015421  0.016186  0.015803  460.386  1.635  25.335  \(+\,25.327\)  \(+\,0.002\)  \(\,0.215\)  \(\,0.005\)  \(+\,0.618\) 
0.016186  0.016969  0.016577  455.279  1.605  24.933  \(+\,24.925\)  \(+\,0.001\)  \(\,0.207\)  \(\,0.004\)  \(+\,0.594\) 
0.016969  0.017771  0.017370  447.960  1.570  24.527  \(+\,24.519\)  \(+\,0.001\)  \(\,0.198\)  \(\,0.003\)  \(+\,0.569\) 
0.017771  0.018597  0.018183  437.466  1.526  24.116  \(+\,24.109\)  \(+\,0.001\)  \(\,0.190\)  \(\,0.002\)  \(+\,0.545\) 
0.018597  0.019443  0.019020  430.342  1.493  23.702  \(+\,23.695\)  \(+\,0.000\)  \(\,0.181\)  \(\,0.002\)  \(+\,0.521\) 
0.019443  0.020308  0.019874  423.167  1.463  23.285  \(+\,23.279\)  \(+\,0.000\)  \(\,0.173\)  \(\,0.001\)  \(+\,0.496\) 
0.020308  0.021189  0.020748  414.878  1.432  22.868  \(+\,22.862\)  \(\,0.000\)  \(\,0.164\)  \(\,0.001\)  \(+\,0.472\) 
0.021189  0.022087  0.021638  406.158  1.402  22.450  \(+\,22.444\)  \(\,0.001\)  \(\,0.156\)  \(\,0.000\)  \(+\,0.448\) 
0.022087  0.023007  0.022547  400.652  1.374  22.030  \(+\,22.025\)  \(\,0.001\)  \(\,0.148\)  \(+\,0.000\)  \(+\,0.425\) 
0.023007  0.023942  0.023475  393.124  1.348  21.610  \(+\,21.605\)  \(\,0.001\)  \(\,0.140\)  \(+\,0.001\)  \(+\,0.402\) 
0.023942  0.024899  0.024421  384.736  1.317  21.189  \(+\,21.186\)  \(\,0.001\)  \(\,0.132\)  \(+\,0.001\)  \(+\,0.379\) 
0.024899  0.025878  0.025389  376.243  1.286  20.768  \(+\,20.764\)  \(\,0.001\)  \(\,0.124\)  \(+\,0.001\)  \(+\,0.356\) 
0.025878  0.026875  0.026377  372.189  1.266  20.346  \(+\,20.343\)  \(\,0.002\)  \(\,0.116\)  \(+\,0.002\)  \(+\,0.334\) 
0.026875  0.027895  0.027385  362.930  1.235  19.925  \(+\,19.922\)  \(\,0.002\)  \(\,0.108\)  \(+\,0.002\)  \(+\,0.312\) 
0.027895  0.028932  0.028413  357.126  1.214  19.505  \(+\,19.502\)  \(\,0.002\)  \(\,0.101\)  \(+\,0.002\)  \(+\,0.290\) 
0.028932  0.029988  0.029460  348.345  1.186  19.086  \(+\,19.084\)  \(\,0.002\)  \(\,0.094\)  \(+\,0.003\)  \(+\,0.269\) 
0.029988  0.031067  0.030528  339.830  1.158  18.668  \(+\,18.666\)  \(\,0.002\)  \(\,0.086\)  \(+\,0.003\)  \(+\,0.248\) 
0.031067  0.032162  0.031615  333.025  1.137  18.252  \(+\,18.250\)  \(\,0.002\)  \(\,0.079\)  \(+\,0.003\)  \(+\,0.228\) 
0.032162  0.033279  0.032720  323.442  1.109  17.838  \(+\,17.837\)  \(\,0.002\)  \(\,0.072\)  \(+\,0.003\)  \(+\,0.208\) 
0.033279  0.034415  0.033846  316.769  1.087  17.427  \(+\,17.426\)  \(\,0.002\)  \(\,0.066\)  \(+\,0.003\)  \(+\,0.189\) 
0.034415  0.035568  0.034989  309.514  1.066  17.019  \(+\,17.018\)  \(\,0.002\)  \(\,0.059\)  \(+\,0.003\)  \(+\,0.170\) 
0.035568  0.036742  0.036154  300.609  1.040  16.614  \(+\,16.613\)  \(\,0.002\)  \(\,0.053\)  \(+\,0.003\)  \(+\,0.151\) 
0.036742  0.037930  0.037335  295.114  1.024  16.213  \(+\,16.213\)  \(\,0.002\)  \(\,0.047\)  \(+\,0.003\)  \(+\,0.133\) 
0.037930  0.039138  0.038533  288.375  1.003  15.817  \(+\,15.816\)  \(\,0.003\)  \(\,0.041\)  \(+\,0.004\)  \(+\,0.116\) 
0.039138  0.040369  0.039752  280.807  0.979  15.423  \(+\,15.423\)  \(\,0.003\)  \(\,0.035\)  \(+\,0.004\)  \(+\,0.099\) 
0.040369  0.041618  0.040990  271.508  0.955  15.033  \(+\,15.033\)  \(\,0.003\)  \(\,0.029\)  \(+\,0.004\)  \(+\,0.083\) 
0.041618  0.042887  0.042251  266.133  0.938  14.647  \(+\,14.647\)  \(\,0.003\)  \(\,0.024\)  \(+\,0.004\)  \(+\,0.067\) 
0.042887  0.044177  0.043531  258.862  0.917  14.266  \(+\,14.266\)  \(\,0.003\)  \(\,0.018\)  \(+\,0.004\)  \(+\,0.052\) 
0.044177  0.045487  0.044830  253.719  0.900  13.889  \(+\,13.888\)  \(\,0.003\)  \(\,0.013\)  \(+\,0.004\)  \(+\,0.038\) 
0.045487  0.046815  0.046149  245.394  0.879  13.516  \(+\,13.516\)  \(\,0.003\)  \(\,0.008\)  \(+\,0.004\)  \(+\,0.024\) 
0.046815  0.048165  0.047489  238.906  0.860  13.148  \(+\,13.148\)  \(\,0.003\)  \(\,0.004\)  \(+\,0.004\)  \(+\,0.010\) 
0.048165  0.049528  0.048844  232.195  0.843  12.786  \(+\,12.786\)  \(\,0.003\)  \(+\,0.001\)  \(+\,0.004\)  \(\,0.003\) 
0.049528  0.050917  0.050221  226.191  0.824  12.430  \(+\,12.430\)  \(\,0.003\)  \(+\,0.005\)  \(+\,0.004\)  \(\,0.015\) 
0.050917  0.052322  0.051619  220.655  0.809  12.078  \(+\,12.078\)  \(\,0.002\)  \(+\,0.009\)  \(+\,0.004\)  \(\,0.027\) 
0.052322  0.053748  0.053031  212.493  0.787  11.732  \(+\,11.732\)  \(\,0.002\)  \(+\,0.013\)  \(+\,0.003\)  \(\,0.039\) 
0.053748  0.055193  0.054470  207.171  0.772  11.391  \(+\,11.391\)  \(\,0.002\)  \(+\,0.017\)  \(+\,0.003\)  \(\,0.049\) 
0.055193  0.056660  0.055923  200.154  0.752  11.056  \(+\,11.056\)  \(\,0.002\)  \(+\,0.021\)  \(+\,0.003\)  \(\,0.060\) 
0.056660  0.058145  0.057401  194.826  0.737  10.726  \(+\,10.726\)  \(\,0.002\)  \(+\,0.024\)  \(+\,0.003\)  \(\,0.069\) 
0.058145  0.059649  0.058894  189.250  0.722  10.403  \(+\,10.402\)  \(\,0.002\)  \(+\,0.027\)  \(+\,0.003\)  \(\,0.079\) 
0.059649  0.061175  0.060411  184.095  0.706  10.085  \(+\,10.084\)  \(\,0.002\)  \(+\,0.030\)  \(+\,0.003\)  \(\,0.087\) 
0.061175  0.062717  0.061942  177.115  0.689  9.773  \(+\,9.773\)  \(\,0.002\)  \(+\,0.033\)  \(+\,0.003\)  \(\,0.096\) 
0.062717  0.064277  0.063496  171.504  0.674  9.468  \(+\,9.467\)  \(\,0.002\)  \(+\,0.036\)  \(+\,0.003\)  \(\,0.103\) 
0.064277  0.065859  0.065065  165.886  0.658  9.169  \(+\,9.168\)  \(\,0.002\)  \(+\,0.038\)  \(+\,0.003\)  \(\,0.110\) 
0.065859  0.067461  0.066659  160.981  0.644  8.875  \(+\,8.874\)  \(\,0.002\)  \(+\,0.041\)  \(+\,0.003\)  \(\,0.117\) 
0.067461  0.069082  0.068270  155.821  0.629  8.588  \(+\,8.587\)  \(\,0.002\)  \(+\,0.043\)  \(+\,0.003\)  \(\,0.123\) 
0.069082  0.070723  0.069900  150.892  0.615  8.307  \(+\,8.306\)  \(\,0.002\)  \(+\,0.045\)  \(+\,0.003\)  \(\,0.129\) 
0.070723  0.072392  0.071556  145.575  0.599  8.031  \(+\,8.030\)  \(\,0.002\)  \(+\,0.047\)  \(+\,0.003\)  \(\,0.134\) 
0.072392  0.074077  0.073232  141.394  0.587  7.762  \(+\,7.760\)  \(\,0.002\)  \(+\,0.048\)  \(+\,0.003\)  \(\,0.139\) 
0.074077  0.075777  0.074923  136.424  0.574  7.499  \(+\,7.497\)  \(\,0.002\)  \(+\,0.050\)  \(+\,0.003\)  \(\,0.144\) 
0.075777  0.077497  0.076635  131.196  0.560  7.242  \(+\,7.241\)  \(\,0.002\)  \(+\,0.051\)  \(+\,0.003\)  \(\,0.148\) 
0.077497  0.079239  0.078366  126.732  0.546  6.992  \(+\,6.990\)  \(\,0.002\)  \(+\,0.053\)  \(+\,0.003\)  \(\,0.151\) 
0.079239  0.080997  0.080115  123.202  0.536  6.747  \(+\,6.745\)  \(\,0.001\)  \(+\,0.054\)  \(+\,0.003\)  \(\,0.155\) 
0.080997  0.082783  0.081887  118.162  0.521  6.509  \(+\,6.507\)  \(\,0.001\)  \(+\,0.055\)  \(+\,0.003\)  \(\,0.157\) 
0.082783  0.084581  0.083680  114.017  0.510  6.276  \(+\,6.274\)  \(\,0.001\)  \(+\,0.056\)  \(+\,0.003\)  \(\,0.160\) 
0.084581  0.086404  0.085491  110.242  0.497  6.050  \(+\,6.047\)  \(\,0.001\)  \(+\,0.056\)  \(+\,0.002\)  \(\,0.162\) 
0.086404  0.088239  0.087319  105.794  0.486  5.830  \(+\,5.827\)  \(\,0.001\)  \(+\,0.057\)  \(+\,0.002\)  \(\,0.164\) 
0.088239  0.090099  0.089166  101.754  0.473  5.615  \(+\,5.613\)  \(\,0.001\)  \(+\,0.057\)  \(+\,0.002\)  \(\,0.165\) 
0.090099  0.091976  0.091033  98.155  0.462  5.407  \(+\,5.404\)  \(\,0.001\)  \(+\,0.058\)  \(+\,0.002\)  \(\,0.166\) 
0.091976  0.093868  0.092918  95.128  0.453  5.204  \(+\,5.201\)  \(\,0.001\)  \(+\,0.058\)  \(+\,0.002\)  \(\,0.167\) 
0.093868  0.095784  0.094823  91.683  0.442  5.008  \(+\,5.005\)  \(\,0.001\)  \(+\,0.058\)  \(+\,0.002\)  \(\,0.168\) 
0.095784  0.097721  0.096750  87.967  0.430  4.816  \(+\,4.813\)  \(\,0.001\)  \(+\,0.058\)  \(+\,0.002\)  \(\,0.168\) 
0.097721  0.099679  0.098697  84.557  0.420  4.630  \(+\,4.627\)  \(\,0.001\)  \(+\,0.058\)  \(+\,0.002\)  \(\,0.168\) 
0.099679  0.101659  0.100666  80.860  0.408  4.450  \(+\,4.446\)  \(\,0.001\)  \(+\,0.058\)  \(+\,0.002\)  \(\,0.168\) 
0.101659  0.103658  0.102656  77.695  0.398  4.275  \(+\,4.271\)  \(\,0.001\)  \(+\,0.058\)  \(+\,0.002\)  \(\,0.168\) 
0.103658  0.105679  0.104666  75.105  0.389  4.105  \(+\,4.101\)  \(\,0.001\)  \(+\,0.058\)  \(+\,0.002\)  \(\,0.167\) 
0.105679  0.107705  0.106689  71.021  0.378  3.940  \(+\,3.936\)  \(\,0.001\)  \(+\,0.058\)  \(+\,0.002\)  \(\,0.166\) 
0.107705  0.109766  0.108732  68.777  0.368  3.781  \(+\,3.777\)  \(\,0.001\)  \(+\,0.057\)  \(+\,0.002\)  \(\,0.165\) 
0.109766  0.111845  0.110802  66.428  0.360  3.627  \(+\,3.622\)  \(\,0.001\)  \(+\,0.057\)  \(+\,0.002\)  \(\,0.164\) 
0.111845  0.113945  0.112891  62.555  0.348  3.477  \(+\,3.473\)  \(\,0.001\)  \(+\,0.056\)  \(+\,0.002\)  \(\,0.162\) 
0.113945  0.116056  0.114995  60.954  0.342  3.332  \(+\,3.328\)  \(\,0.000\)  \(+\,0.056\)  \(+\,0.002\)  \(\,0.161\) 
0.116056  0.118185  0.117116  57.750  0.332  3.193  \(+\,3.189\)  \(\,0.000\)  \(+\,0.055\)  \(+\,0.002\)  \(\,0.159\) 
0.118185  0.120342  0.119260  55.934  0.324  3.058  \(+\,3.054\)  \(\,0.000\)  \(+\,0.055\)  \(+\,0.001\)  \(\,0.157\) 
0.120342  0.122517  0.121427  52.788  0.314  2.928  \(+\,2.923\)  \(\,0.000\)  \(+\,0.054\)  \(+\,0.001\)  \(\,0.155\) 
0.122517  0.124719  0.123615  51.158  0.307  2.801  \(+\,2.797\)  \(\,0.000\)  \(+\,0.053\)  \(+\,0.001\)  \(\,0.153\) 
0.124719  0.126932  0.125821  48.734  0.299  2.680  \(+\,2.675\)  \(\,0.000\)  \(+\,0.052\)  \(+\,0.001\)  \(\,0.150\) 
0.126932  0.129175  0.128048  46.061  0.288  2.562  \(+\,2.557\)  \(\,0.000\)  \(+\,0.051\)  \(+\,0.001\)  \(\,0.148\) 
0.129175  0.131421  0.130294  44.625  0.284  2.449  \(+\,2.444\)  \(\,0.000\)  \(+\,0.051\)  \(+\,0.001\)  \(\,0.146\) 
0.131421  0.133681  0.132548  42.645  0.276  2.340  \(+\,2.335\)  \(\,0.000\)  \(+\,0.050\)  \(+\,0.001\)  \(\,0.143\) 
0.133681  0.135974  0.134823  40.137  0.266  2.236  \(+\,2.230\)  \(\,0.000\)  \(+\,0.049\)  \(+\,0.001\)  \(\,0.140\) 
0.135974  0.138285  0.137125  38.674  0.260  2.134  \(+\,2.129\)  \(\,0.000\)  \(+\,0.048\)  \(+\,0.001\)  \(\,0.138\) 
0.138285  0.140614  0.139446  36.488  0.252  2.037  \(+\,2.032\)  \(\,0.000\)  \(+\,0.047\)  \(+\,0.001\)  \(\,0.135\) 
0.140614  0.142962  0.141784  35.400  0.247  1.943  \(+\,1.938\)  \(\,0.000\)  \(+\,0.046\)  \(+\,0.001\)  \(\,0.132\) 
0.142962  0.145328  0.144140  33.650  0.240  1.853  \(+\,1.848\)  \(\,0.000\)  \(+\,0.045\)  \(+\,0.001\)  \(\,0.129\) 
0.145328  0.147710  0.146515  32.302  0.234  1.766  \(+\,1.761\)  \(\,0.000\)  \(+\,0.044\)  \(+\,0.001\)  \(\,0.126\) 
0.147710  0.150118  0.148909  30.473  0.226  1.683  \(+\,1.678\)  \(+\,0.000\)  \(+\,0.043\)  \(+\,0.001\)  \(\,0.123\) 
0.150118  0.152551  0.151330  28.634  0.218  1.603  \(+\,1.597\)  \(+\,0.000\)  \(+\,0.042\)  \(+\,0.001\)  \(\,0.121\) 
0.152551  0.155000  0.153771  27.551  0.213  1.525  \(+\,1.520\)  \(+\,0.000\)  \(+\,0.041\)  \(+\,0.001\)  \(\,0.118\) 
0.155000  0.157452  0.156221  26.250  0.208  1.451  \(+\,1.446\)  \(+\,0.000\)  \(+\,0.040\)  \(+\,0.001\)  \(\,0.115\) 
0.157452  0.159942  0.158693  25.092  0.202  1.380  \(+\,1.375\)  \(+\,0.000\)  \(+\,0.039\)  \(+\,0.001\)  \(\,0.112\) 
0.159942  0.162445  0.161189  23.721  0.195  1.312  \(+\,1.307\)  \(+\,0.000\)  \(+\,0.038\)  \(+\,0.001\)  \(\,0.109\) 
0.162445  0.164974  0.163705  22.677  0.190  1.246  \(+\,1.241\)  \(+\,0.000\)  \(+\,0.037\)  \(+\,0.001\)  \(\,0.106\) 
0.164974  0.167515  0.166239  21.752  0.186  1.183  \(+\,1.178\)  \(+\,0.000\)  \(+\,0.036\)  \(+\,0.001\)  \(\,0.103\) 
0.167515  0.170078  0.168791  20.011  0.177  1.123  \(+\,1.118\)  \(+\,0.000\)  \(+\,0.035\)  \(+\,0.001\)  \(\,0.100\) 
0.170078  0.172669  0.171369  19.180  0.173  1.065  \(+\,1.060\)  \(+\,0.000\)  \(+\,0.034\)  \(+\,0.001\)  \(\,0.097\) 
0.172669  0.175277  0.173966  18.237  0.168  1.010  \(+\,1.005\)  \(+\,0.000\)  \(+\,0.033\)  \(+\,0.001\)  \(\,0.094\) 
0.175277  0.177899  0.176583  17.122  0.162  0.957  \(+\,0.952\)  \(+\,0.000\)  \(+\,0.032\)  \(+\,0.001\)  \(\,0.091\) 
0.177899  0.180548  0.179217  16.427  0.158  0.906  \(+\,0.901\)  \(+\,0.000\)  \(+\,0.031\)  \(+\,0.001\)  \(\,0.088\) 
0.180548  0.183212  0.181874  15.434  0.153  0.858  \(+\,0.853\)  \(+\,0.000\)  \(+\,0.030\)  \(+\,0.000\)  \(\,0.085\) 
0.183212  0.185903  0.184552  14.965  0.149  0.812  \(+\,0.807\)  \(+\,0.000\)  \(+\,0.029\)  \(+\,0.000\)  \(\,0.082\) 
0.185903  0.188606  0.187249  13.905  0.144  0.768  \(+\,0.763\)  \(+\,0.000\)  \(+\,0.028\)  \(+\,0.000\)  \(\,0.079\) 
0.188606  0.191329  0.189960  12.957  0.138  0.726  \(+\,0.721\)  \(+\,0.000\)  \(+\,0.027\)  \(+\,0.000\)  \(\,0.077\) 
0.191329  0.194088  0.192702  12.445  0.135  0.685  \(+\,0.681\)  \(+\,0.000\)  \(+\,0.026\)  \(+\,0.000\)  \(\,0.074\) 
0.194088  0.196855  0.195466  11.711  0.130  0.647  \(+\,0.643\)  \(+\,0.000\)  \(+\,0.025\)  \(+\,0.000\)  \(\,0.071\) 
0.196855  0.199646  0.198244  10.987  0.126  0.611  \(+\,0.606\)  \(+\,0.000\)  \(+\,0.024\)  \(+\,0.000\)  \(\,0.069\) 
0.199646  0.202452  0.201041  10.371  0.122  0.576  \(+\,0.572\)  \(+\,0.000\)  \(+\,0.023\)  \(+\,0.000\)  \(\,0.066\) 
5.2.6 Normalisation
The normalisation factor \({\mathcal {N}}\) is determined by requiring the integrated nuclear elastic crosssection to be \(\sigma _{\mathrm{el}} = 31.0\,\mathrm{mb}\) as obtained by TOTEM from a \(\beta ^* = 90\,\mathrm{m}\) dataset at the same energy [6]. The elastic crosssection is extracted from the data in two parts. The first part sums the \(\mathrm{d}\sigma /\mathrm{d}t\) histogram bins for \(0.01< t < 0.5\,\mathrm{GeV^2}\). The second part corresponds to the integral over \(0< t < 0.01\,\mathrm{GeV^2}\) of an exponential fitted to the data on the interval \(0.01< t < 0.05\,\mathrm{GeV^2}\).
The uncertainty of \({\mathcal {N}}\) is dominated by the \(5.5\,\mathrm{\%}\) uncertainty of \(\sigma _{\mathrm{el}}\) from Ref. [6].
5.2.7 Binning
The bin sizes are set according to the t resolution. Three different binnings are considered in this analysis: “dense” where the bin size is as large as the standard deviation of t, “medium” with bins twice as large and “coarse” with bins three times larger than the standard deviation of t.
5.3 Data merging
After analysing the data in each diagonal and LHC fill separately, the individual differential crosssection distributions are merged. This is accomplished by a perbin weighted average, with the weight given by inverse squared statistical uncertainty. The final crosssection values are listed in Table 3 and are visualised in Fig. 10. The figure clearly shows a rapid crosssection rise below \(t \lesssim 0.002\,\mathrm{GeV^2}\) which, as interpreted later, is an effect due to the electromagnetic interaction.
5.4 Systematic uncertainties

Alignment: shifts in \(\theta ^*_{x,y}\) (see Sect. 5.1.2). Both leftright symmetric and antisymmetric modes have been considered. In the vertical plane, both contributions correlated and uncorrelated between the diagonals have been considered.

Alignment xy tilts and optics: mixing between \(\theta ^*_{x}\) and \(\theta ^*_{y}\) (see Sect. 5.1.2). Both leftright symmetric and antisymmetric modes have been considered.

Optics uncertainties: scaling of \(\theta ^*_{x,y}\) (see Sect. 5.1.3). The three relevant modes in Eq. (6) have been considered.

Background subtraction (see Sect. 5.2.2): the tdependent uncertainty of the correction factor \({\mathcal {B}}\).

Acceptance correction (see Sect. 5.2.3): the uncertainty of resolution parameters, nongaussianity of the resolution distributions, leftright asymmetry of the beam divergence.

Inefficiency corrections (see Sect. 5.2.4): for the uncorrelated inefficiency \({\mathcal {I}}_1\) both uncertainties of the fitted slope and intercept have been considered. For the correlated inefficiency \({\mathcal {I}}_2\) the uncertainty of its value has been considered.

The beammomentum uncertainty: considered when the scattering angles are translated to t, see Eq. (4). The uncertainty was estimated by LHC experts as \(0.1\,\mathrm{\%}\) [39] in agreement with a previous assessment by TOTEM (Section 5.2.8. in [4]).

Unsmearing (see Sect. 5.2.5): uncertainty of resolution parameters and model dependence of the fit.

Normalisation (see Sect. 5.2.6): overall multiplicative factor.
The systematic uncertainty corresponding to the final differential crosssection merged from all the analysed LHC fills and both diagonals is propagated according to the same method as applied to the data, see Sect. 5.3. To be conservative, the systematic errors are assumed fully correlated among the four analysed LHC fills. The correlations between the two diagonals are respected for each systematic effect. This is particularly important for the vertical (mis)alignment, as already noted in Ref. [5]. The relative position between the top and bottom RPs is known precisely from trackbased alignment (see Sect. 5.1.2) and the leading component of residual misalignment is thus between the beam and a RP. Furthermore, whenever the beam was closer to a top RP, it would be further away from the corresponding bottom RP and vice versa. Consequently, the effect of the misalignment is predominantly anticorrelated between the diagonals. While the misalignment uncertainty in the lowest t bin reaches about \(7\,\mathrm{\%}\) for a single diagonal, once the diagonals are merged the impact drops to about \(1.2\,\mathrm{\%}\).
5.5 Systematic crosschecks
Compatible results have been obtained by analysing data subsets of events from different bunches, different diagonals (Fig. 12, top left), different fills and different time periods – in particular those right after and right before the beam cleanings (Fig. 12, top right). Figure 12, bottom left, shows that both analysis approaches, “2RP” and “4RP”, yield compatible results. The relatively large difference between the diagonals at very low t (Fig. 12, top left) is fully within the uncertainty due to the vertical misalignment, see Sect. 5.4.
6 Determination of \(\rho \) and total crosssection
The value of the \(\rho \) parameter can be extracted from the differential crosssection thanks to the effects of Coulombnuclear interference (CNI). Explicit treatment of these effects allows also for a conceptually more accurate determination of the total crosssection.
Our modelling of the CNI effects is summarised in Sects. 6.1, 6.2 and 6.3 describe data fits and results. In Sect. 6.2 the differential crosssection normalisation is fixed by the \(\beta ^* = 90\,\mathrm{m}\) data [6] (see Sect. 5.2.6). In Sect. 6.3 the normalisation is adjusted or entirely determined from the \(\beta ^* = 2500\,\mathrm{m}\) data presented in this publication. This allows for different or even completely independent total crosssection determination with respect to Ref. [6].
6.1 Coulombnuclear interference
A detailed overview of different CNI descriptions was given in Ref. [5], Section 6. Here we briefly summarise the choices used for the presented analysis.
6.2 Data fits with fixed normalisation
Summary of results for various fit configurations (medium binning)
\(N_b\)  \(t_{\mathrm{max}} = 0.07\,\mathrm{GeV^2}\)  \(t_{\mathrm{max}} = 0.15\,\mathrm{GeV^2}\)  

\(\chi ^2/\hbox {ndf}\)  \(\rho \)  \(\sigma _{\mathrm{tot}}\quad (mb)\)  \(\chi ^2/\hbox {ndf}\)  \(\rho \)  \(\sigma _{\mathrm{tot}}\quad (mb)\)  
1  0.9  \(0.09\pm 0.01\)  \( 111.8 \pm 3.1\)  2.1  –  – 
2  0.9  \(0.10\pm 0.01\)  \( 111.9 \pm 3.1\)  1.0  \(0.09\pm 0.01\)  \( 111.9 \pm 3.1\) 
3  0.9  \(0.09\pm 0.01\)  \( 111.9 \pm 3.0\)  0.9  \(0.10\pm 0.01\)  \( 112.1 \pm 3.1\) 
The complete fit procedure has been validated with a MonteCarlo study confirming that it has negligible bias. It also indicates the composition of the fit parameter uncertainties. For example, for a fit with \(N_b = 1\) using data in the “coarse binning” up to \(t = 0.07\,\mathrm{GeV^2}\), the \(\rho \) uncertainty due to the statistical uncertainties is about 0.004, due to the systematic uncertainties is about 0.003 and due to the normalisation uncertainty is about 0.009.
The fits have been found to have negligible dependence on the binning used (see Sect. 5.2.7), the choice of electromagnetic form factor (see text below Eq. (14)), the hight nuclear amplitude (see text below Eq. (15)), the choice of the nuclear amplitude phase (see text above Eq. (16)), the number of fit iterations and the choice of start parameter values for the \(\chi ^2\) minimisation.
The fit configuration \(N_b=1\) with \(t_{\mathrm{max}} = 0.07\,\mathrm{GeV^2}\) has another important meaning. Considering the shrinkage of the “forwardcone”, this t range is similar to the one used in the UA4/2 analysis [46]. This fact may suggest why UA4/2 could not observe deviations of the differential crosssection from pure exponential: the t range was too narrow, as it would be for the present data, had the acceptance stopped at \(t = 0.07\,\mathrm{GeV^2}\), see Fig. 15. Beyond the t range, this fit combination shares more similarities with the UA4/2 fit (and in general with many other past experiments): purely exponential fit and assumption of constant hadronic phase. Moreover, as shown in Ref. [5], the “KL” interference formula [43] used in this report gives for this fit configuration very similar \(\rho \) results as the “SWY” interference formula [47] used in many past data analyses. From this point of view this fit combination corresponds to the most fair comparison to previous \(\rho \) determinations and their extrapolations, as e.g. in Fig. 16. It is worth noting that this fit configuration yields a \(\rho \) value incompatible at the level of about \(4.7\,\mathrm{\sigma }\) with the preferred COMPETE model (blue curve in the figure).
Figure 17 illustrates a small correction due to a conceptual improvement in combining the data from this publication and from Ref. [6]. The latter assumes certain values of \(\rho \) in order to evaluate crosssection estimates which are in turn used in this analysis (see Sect. 5.2.6) to estimate \(\rho \). This circular dependence can be resolved by considering simultaneously the \(\rho \) dependence of \(\sigma _{\mathrm{el}}\) in Ref. [6] (blue line) and the \(\sigma _{\mathrm{el}}\) dependence of \(\rho \) determined in this analysis (red line). The latter is done as linear interpolation of \(\rho \) values extracted assuming \(\sigma _{\mathrm{el}} = 30.9\) and \(31.1\,\mathrm{mb}\). The linear dependence is confirmed with MonteCarlo studies. The solution consistent with both datasets (the crossing of the red and blue curves) brings negligible correction to \(\rho \) and \(0.03\,\mathrm{\%}\) correction to the value of \(\sigma _{\mathrm{el}}\) published in Ref. [6] for \(\rho =0.10\).
6.3 Data fits with variable normalisation

approach 1: normalisation from \(90\,\mathrm{m}\) data, results presented in the previous section (in particular Table 4),

approach 2: normalisation estimated with \(2500\,\mathrm{m}\) data under the constraint (mean and standard deviation) from the \(90\,\mathrm{m}\) data,

approach 3: normalisation estimated only from \(2500\,\mathrm{m}\) data.
Summary of \(\rho \) and total crosssection results
Data  Method  \(\rho \)  \(\sigma _{\mathrm{tot}}\quad (mb)\) 

\(\beta ^* = 90\,\mathrm{m}\)  Ref. [6]  –  \(110.6 \pm 3.4\) 
\(\beta ^* = 2500\,\mathrm{m}\)  Approach 1  \(0.09 \pm 0.01\)  \(111.8 \pm 3.1\) 
Approach 2  \(0.09 \pm 0.01\)  \(111.3 \pm 3.2\)  
Approach 3  \(0.08(5) \pm 0.01\)  \(110.3 \pm 3.5\)  
Approach 3 (single fit)  \(0.10 \pm 0.01\)  \(109.3 \pm 3.5\)  
\(\beta ^* = 90\) and \(2500\,\mathrm{m}\)  Ref. [6] \(\oplus \) approach 3  \(110.5 \pm 2.4\) 
Since the Coulomb normalisation is performed at very low t, the presentation in this section will focus on fits with \(N_b = 1\). Fits with \(N_b = 3\) were tested, too, without significant changes in the results. For the sake of simplicity, only the medium binning will be used in this section. The previous section has shown that results do not depend on the choice of binning.
Since the nuclearamplitude component cannot be neglected even at the lowest t points of the available dataset, Table 3, the normalisation determination must be performed with care. It has been found preferable to make the fits in sequence of three steps, using dedicated and physicsmotivated fit configurations for each parameter. The parameters of the nuclear amplitude are determined from a “golden nuclear t range” where t is large enough for CNI effects to be small while t is small enough for the \(N_b = 1\) parametrisation to be suitable. For example, analysing Eq. (17) one can find that CNI effects modify the nuclear crosssection by less than \(1\,\mathrm{\%}\) for \(t \gtrsim 0.007\,\mathrm{GeV^2}\). This range agrees with what is empirically found when trying to go as low as possible in t with the nuclear range without finding significant deviations from the exponential with \(N_b = 1\) either due to the destructive interference with the Coulomb interaction or due to the nonexponentiality of the nuclear amplitude [4]. In the nuclear range, the CNI effects can be ignored (charging the residual effects on systematics), making the fit independent of the interference modelling. The normalisation \(\eta \), in contrary, is determined from the lowest t points which are the only ones having sensitivity to the Coulombamplitude component. The \(\rho \) parameter is derived from a t range where CNI effects are significant, thus including at least the complement of the nuclear range, \(t \lesssim 0.007\,\mathrm{GeV^2}\). Note that overlapping t ranges are used for determination of \(\eta \) and \(\rho \).

Step a (determination of \(b_1\)): fit over range \(0.005< t < 0.07\,\mathrm{GeV^2}\), the CNI effects are ignored. The fit gives a pvalue of 0.75.

Step b (determination of \(\eta \)): fit over range \(t < 0.0015\,\mathrm{GeV^2}\), with \(b_1\) fixed from step a. The overall \(\chi ^2\) receives an additional term \((\eta  1)^2/ \sigma _\eta ^2\), \(\sigma _\eta = 0.055\), which reflects the constraint from the \(\beta ^* = 90\,\mathrm{m}\) data. The fit gives negligible average pull and yields a pvalue of 0.11.

Step c (determination of \(\rho \) and a): fit over range \(t < 0.07\,\mathrm{GeV^2}\), with \(b_1\) fixed from step a and \(\eta \) fixed from step b. The fit gives a pvalue of 0.73.

Step a (determination of \(\eta a^2\) and \(b_1\)): fit over range \(0.0071< t < 0.026\,\mathrm{GeV^2}\). The CNI effects are ignored, therefore the fit is only sensitive to the product \(\eta a^2\), cf. Eqs. (20) and (15). The fit yields a pvalue of 0.91.

Step b (determination of \(\eta \)): fit over range \(t < 0.0023\,\mathrm{GeV^2}\), with \(b_1\) and product \(\eta a^2\) fixed from step a. Since \(\eta \) is determined and the product \(\eta a^2\) is fixed, a is also determined in this step. The fit gives negligible average pull and yields a pvalue of 0.14.

Step c (determination of \(\rho \)): fit over range \(t < 0.0071\,\mathrm{GeV^2}\), with \(b_1\) fixed from step a and \(\eta \) and a fixed from step b. The fit yields a pvalue of 0.23.
As a test we tried approach 3 implementation with a single fit over \(t < 0.05\,\mathrm{GeV^2}\), where all parameters (\(\eta \), a, \(b_1\) and \(\rho \)) are free and initialised to the values obtained in the previous paragraph. As anticipated above, such fit might have encountered problems due to nonoptimal parameter sensitivities on the available t range, however, the results listed in Table 5 are reasonable. \(\eta \) was found to be 1.05 thus deviating by less than a sigma (\(\sigma _\eta \)) from the \(\beta ^* = 90\,\mathrm{m}\) normalisation. The fit quality is good: pvalue of 0.70, see also the illustration in Fig. 18. The single fit is also able to show the correlations between the fitted parameters. As expected, \(\eta \) and a are essentially fully anticorrelated. Both \(\eta \) and a are strongly correlated with \(\rho \) with correlation coefficients of about 0.85, whereas the correlation of these parameters to \(b_1\) is weak, the correlation coefficient is about 0.4. Finally the correlation coefficient between \(\rho \) and \(b_1\) is in between with a correlation coefficient about 0.6. These correlations confirm the necessity of the stepwise determination of the parameters using the ranges with most sensitivity for the parameter concerned to minimize the influence of the value of the other parameters to the determination.
The uncertainties for the fits presented above were determined with the following procedure. The experimentally determined \(\mathrm{d}\sigma /\mathrm{d}t\) histogram was modified by adding randomly generated fluctuations reflecting the statistical, systematic and normalisation uncertainties (see Sect. 5.4). This was done 100 times with different random seeds. Each of the modified histograms was fitted by the above sequences, yielding fit parameter samples to determine the parameter fluctuations, i.e. uncertainties. Histogram modifications resulting in excessive parameter deviations from the unmodified fit (\(\varDelta \rho > 0.05\) or \(\varDelta \sigma _\mathrm{tot} > 10\,\mathrm{mb}\)) were disregarded since such cases would not be accepted in the analysis. This estimation method gives consistent results with Sect. 6.2 (for approach 1) and \(\chi ^2\)based estimate (from approach 3, single fit). The \(\rho \) and \(\sigma _{\mathrm{tot}}\) uncertainties were crosschecked and adjusted by varying one of the variables with its uncertainty at a time for the steps where several variables were determined.
7 Discussion of physics implications

The blue band is compatible (pvalue 0.990 to 0.995) with the \(\sigma _{\mathrm{tot}}\) data, but incompatible (pvalue \(3\times 10^{6}\)) with the \(\rho \) point.

The magenta band is incompatible (pvalue \(1\times 10^{5}\) to \(5\times 10^{4}\)) with the \(\sigma _{\mathrm{tot}}\) data and incompatible (pvalue \(9\times 10^{3}\)) with the \(\rho \) point.

The green band is incompatible (pvalue \(3\times 10^{18}\) to \(5\times 10^{12}\)) with the \(\sigma _{\mathrm{tot}}\) data, but compatible (pvalue 0.4) with the \(\rho \) point.
Another, even less modeldependent, relation between \(\sigma _{\mathrm{tot}}\) and \(\rho \) can be obtained from dispersion relations [7, 49]. If only the crossingeven component of the amplitude is considered, it can be shown that \(\rho \) is proportional to the rate of growth of \(\sigma _{\mathrm{tot}}\) with energy. Therefore, the low value of \(\rho \) determined in Sect. 6 indicates that either the total crosssection growth should slow down at higher energies or that there is a need for an oddsignature object being exchanged by the protons. While at lower energies such contributions may naturally come from secondary Reggeons, their contribution is generally considered negligible at LHC energies due to their Regge trajectory intercept lower than unity.
A variety of oddsignature exchanges relevant at high energies have been discussed in literature, within different frameworks and under different names, see e.g. the reviews [17, 26]. The “Odderon” was introduced within the axiomatic theory [8, 15, 30] as an amplitude contribution responsible for \(\mathrm{p}{{\bar{\mathrm{p}}}}\) vs. \({\mathrm{pp}}\) differences in the total crosssection as well as in the differential crosssection, particularly in the dip region. Crossingodd trajectories (with \(J=1\) at \(t=0\)) were also studied within the framework of Regge theory as a counterpart of the crossingeven Pomeron. It has also been shown that such an object should exist in QCD, as a colourless compound state of three reggeised gluons with quantum numbers \(J^{PC} = 1^{}\) (see e.g. [24]). The binding strength among the 3 gluons is greater than the strength of their interaction with other particles. There is also evidence for such a state in QCD lattice calculations, known under the name “vector glueball” (see e.g. [21]). Such a state, on one hand, can be exchanged in the tchannel and contribute, e.g., to the elasticscattering amplitude. On the other hand it can be created in the schannel and thus be observed in spectroscopic studies. QCDlike studies based on the AdS/CFT correspondence show that the Odderon emerges on equally firm footing as the Pomeron [50].
There are multiple ways how an oddsignature exchange component may manifest itself in observable data. Focussing on elastic scattering at the LHC (unpolarised beams), there are 3 regions often argued to be sensitive. In general, the effects of an oddsignature exchange (3gluon compound) are expected to be much smaller than those of evensignature exchanges (2gluon compound). Consequently, the sensitive regions are those where the contributions from 2gluon exchanges cancel or are small. At very low t the 2gluon amplitude is expected to be almost purely imaginary, while a 3gluon exchange would make contributions to the real part and therefore \(\rho \) is a very sensitive parameter. The effects on \(\rho \) in \({\mathrm{pp}}\) and \(\mathrm{p}{{\bar{\mathrm{p}}}}\) are opposite so that for \({\mathrm{pp}}\) the oddsignature exchange component is expected to decrease the \(\rho \) value and for \(\mathrm{p}{{\bar{\mathrm{p}}}}\) to increase its value, see e.g. [30]. Another such example is the dip, often described as the imaginary part of the amplitude crossing zero, thus ceding the dominance to the real part to which a 3gluon exchange may contribute. In agreement with such predictions, the observed dips in \(\mathrm{p}{{\bar{\mathrm{p}}}}\) scattering are shallower than those in \({\mathrm{pp}}\). At \(\sqrt{s} = 53\,\mathrm{GeV}\), there are data showing a very significant difference between the \({\mathrm{pp}}\) and \(\mathrm{p}{{\bar{\mathrm{p}}}}\) dip [28]. The interpretation of this difference is, however, complicated due to nonnegligible contribution from secondary Reggeons. These are not expected to give sizeable effects at the Tevatron energies (see e.g. [13]), which thus gives weight to the D0 observation of a very shallow dip in \(\mathrm{p}{{\bar{\mathrm{p}}}}\) elastic scattering [52] compared to the very pronounced dip measured by TOTEM at \(7\,\mathrm{TeV}\) [12]. The \({\mathrm{pp}}\) vs. \(\mathrm{p}{{\bar{\mathrm{p}}}}\) dip difference is also predicted to be energydependent which presents another experimental observable (see e.g. [53]). Sometimes the hight region is also argued to be sensitive to 3gluon exchanges. Actually the original “Odderon” concept was general to include any crossingodd contribution. Beside the solution discussed earlier, a solution to the Odderon equation exists in QCD for a leading order 3freegluons approximation. In fact in the larget range (perturbative QCD) models (e.g. [54]) predict coherent exchange of 3 individual gluons as opposed to the 3gluon compound state exchanged at low t (nonperturbative QCD).
Figure 21 compares the TOTEM data with two compatible models: by Nicolescu et al. [51] and the extended Durham model [20] (original model [55] plus crossingodd contribution from [19]). The 2007 version of the Nicolescu model (dashed blue) is based only on preLHC data and predicts \(\sigma _{\mathrm{tot}}\) overestimating the TOTEM measurements – as argued in Ref. [51] it might be due to the ambiguities in prolonging the amplitudes in the nonforward region. The 2017 version (solid blue) includes also LHC measurements up to \(13\,\mathrm{TeV}\) and describes the \(\sigma _{\mathrm{tot}}\) data well. Both versions yield similar results for \(\rho \), with a pronounced energy dependence. This comes from the fact that the crossingodd component is almost negligible at \(\sqrt{s} \approx 500\,\mathrm{GeV}\) but very significant at \(13\,\mathrm{TeV}\). Conversely, in the Durham model the effect is sizeable at \(\sqrt{s} \approx 500\,\mathrm{GeV}\) and gently diminishes with energy. The Durham model also predicts a mild energy dependence of the \(\rho \) parameter. Therefore, precise \(\rho \) measurements at \(\sqrt{s} \approx 900\,\mathrm{GeV}\) and \(14\,\mathrm{TeV}\) would be valuable for discrimination between these models. For both models, the inclusion of a crossingodd exchange component was essential to reach the agreement between the data and model. In particular, the Durham model without such a contribution (black line) is not so well compatible (pvalue 0.02) with the (rhs.) \(\rho \) point obtained with \(N_b=1\) and \(t_{\mathrm{max}} = 0.07\,\mathrm{GeV^2}\).
8 Summary
The measurement of elastic differential crosssection disfavours the purelyexponential lowt behaviour at \(\sqrt{s} = 13\,\mathrm{TeV}\), similarly to the previous observation at \(8\,\mathrm{TeV}\). Thanks to the very lowt reach, the first extraction of the \(\rho \) parameter at \(\sqrt{s} = 13\,\mathrm{TeV}\) was made by exploiting the Coulombnuclear interference. The fit with conditions similar to past experiments yields \(\rho = 0.09 \pm 0.01\), one of the most precise \(\rho \) determinations in history. The fit over the maximum of data points and with maximum reasonable flexibility of the fit function gives \(\rho = 0.10 \pm 0.01\).
Also thanks to the very low t reach, it was possible to apply the “Coulomb normalisation” technique for the first time at the LHC and obtain another total crosssection measurement \(\sigma _{\mathrm{tot}} = (110.3 \pm 3.5)\,\mathrm{mb}\) completely independent from the previous TOTEM measurement at \(\sqrt{s} = 13\,\mathrm{TeV}\) [6] but well compatible with it. Since these two measurements are independent, it is possible to calculate the weighted average yielding \(\sigma _{\mathrm{tot}} = (110.5 \pm 2.4)\,\mathrm{mb}\).
The updated collection of TOTEM’s \(\sigma _{\mathrm{tot}}\) and \(\rho \) data presents a stringent test of model descriptions. For an indicative example, none of the models considered by the COMPETE collaboration is compatible with both \(\sigma _{\mathrm{tot}}\) and \(\rho \).
For both models found to be consistent with TOTEM’s data, the inclusion of a crossingodd 3gluonstate exchange in the tchannel was essential for reaching the good agreement with the data.
If it is demonstrated in future that the crossingodd exchange component is unimportant for elastic scattering, the low \(\rho \) value determined in this publication represents the first experimental evidence for slowing down of the total crosssection growth at higher energies, leading to a deviation from most current model expectations.
We observe significant incompatibilities between \({\mathrm{pp}}\) and \(\mathrm{p} {{\bar{\mathrm{p}}}}\) differential crosssection (in the nonperturbative trange): this implies experimental evidence of crossingodd exchange in the tchannel, hence of a colourless Codd 3gluon compound state exchange [24, 25].
Notes
Acknowledgements
This work was supported by the institutions listed on the front page and also by the Magnus Ehrnrooth foundation (Finland), the Waldemar von Frenckell foundation (Finland), the Academy of Finland, the Finnish Academy of Science and Letters (The Vilho, Yrjö and Kalle Väisälä Fund), the OTKA NK 101438 and the EFOP3.6.116201600001 Grants (Hungary) and the grant MNiSW DIR/WK/2018/13 by the Polish Ministry of Science and Higher Education. Individuals have received support from Nylands nation vid Helsingfors universitet (Finland), from the MŠMT ČR (Czech Republic) and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and the NKP174 New National Excellence Program of the Hungarian Ministry of Human Capacities.
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