Measurement of elastic pp scattering at \(\sqrt{\hbox {s}} = \hbox {8}\) TeV in the Coulomb–nuclear interference region: determination of the \(\mathbf {\rho }\)parameter and the total crosssection
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Abstract
The TOTEM experiment at the CERN LHC has measured elastic proton–proton scattering at the centreofmass energy \(\sqrt{s}=8\,\)TeV and fourmomentum transfers squared, t, from \(6\times 10^{4}\) to 0.2 GeV\(^{2}\). Near the lower end of the tinterval the differential crosssection is sensitive to the interference between the hadronic and the electromagnetic scattering amplitudes. This article presents the elastic crosssection measurement and the constraints it imposes on the functional forms of the modulus and phase of the hadronic elastic amplitude. The data exclude the traditional Simplified West and Yennie interference formula that requires a constant phase and a purely exponential modulus of the hadronic amplitude. For parametrisations of the hadronic modulus with second or thirdorder polynomials in the exponent, the data are compatible with hadronic phase functions giving either central or peripheral behaviour in the impact parameter picture of elastic scattering. In both cases, the \(\rho \)parameter is found to be \(0.12 \pm 0.03\). The results for the total hadronic crosssection are \(\sigma _\mathrm{tot} = (102.9 \pm 2.3)\) mb and \((103.0 \pm 2.3)\) mb for central and peripheral phase formulations, respectively. Both are consistent with previous TOTEM measurements.
Keywords
Elastic Scattering Elastic Event Beam Momentum Acceptance Correction Elastic Proton1 Introduction
The present article discusses the first measurement of elastic scattering in the CNI region at the CERN LHC by the TOTEM experiment. The data have been collected at \(\sqrt{s} = 8\,\)TeV with a special beam optics (\(\beta ^{*}=1000\,\)m) and cover a tinterval from \(6\times 10^{4}\) to 0.2 GeV\(^{2}\), extending well into the interference region. In order to strengthen the statistical power and thus enable a cleaner identification of the interference effects, the analysis also exploits another, complementary data set with higher statistics [14], taken at the same energy, but with different beam optics (\(\beta ^{*}=90\,\)m), and thus covering a different trange: \(0.027< t < 0.2\,\mathrm{GeV^2}\). The isolated analysis of the latter data set has excluded a purely exponential behaviour of the observed elastic crosssection with more than \(7\,\mathrm{\sigma }\) confidence. The new data in the CNI region allow to study the source of the nonexponentiality: nuclear component, CNI effects or both. In order to explore the full spectrum of possibilities, an interference formula without the limitations of SWY is needed. In the present study the more general and complex interference formulae of Cahn [15] and Kundrát–Lokajíček (KL) [16] are used, offering much more freedom for the choice of the theoretically unknown functional forms of the hadronic modulus and phase. Since the data cannot unambiguously determine all functional forms and their parameters, the results of this study, still representatively expressed in terms of \(\rho \), become conditional to the choice of the model describing the hadronic amplitude. This choice has implications on the behaviour of the interaction in impact parameter space. In particular, the functional form of the hadronic phase at small t determines whether elastic collisions occur predominantly at small or large impact parameters (centrality vs. peripherality). It will be shown that both options are compatible with the data, thus the central picture still prevalent in theoretical models is not a necessity.
Section 2 of this article outlines the experimental setup used for the measurement. The properties of the special beam optics are described in Sect. 3. Section 4 gives details of the datataking conditions. The data analysis and reconstruction of the differential crosssection are described in Sect. 5. Section 6 presents the study of the Coulomb–nuclear interference together with the functional form of the hadronic amplitude. The values of \(\rho \) and \(\sigma _\mathrm{tot}\) are determined.
2 Experimental apparatus
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 \(\beta ^{*} = 1000\,\)m optics. The values refer to the right arm, for the left one they are very similar
RP unit  \(L_x\) (m)  \(v_x\)  \(L_y\) (m)  \(v_y\) 

Near  59.37  \(0.867\)  255.87  0.003 
Far  45.89  \(0.761\)  284.62  \(0.017\) 
4 Data taking
The results reported here are based on data taken in October 2012 during a dedicated LHC proton fill (3216) with the special beam properties described in the previous section.
The vertical RPs approached the beam centre to only 3 times the beam width, \(\sigma _{y}\), resulting in an acceptance for tvalues down to \(6 \times 10^{4}\,\mathrm{GeV}^{2}\). The exceptionally close distance was possible due to the low beam intensity in this special beam operation: each beam contained only two colliding bunches and one noncolliding bunch for background monitoring, each with \(10^{11}\) protons. A novel collimation strategy was applied to keep the beam halo background under control. As a first step, the primary collimators (TCP) in the LHC betatron cleaning insertion (point 7) 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. 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 1 h until the beamtocollimator gap was refilled by diffusion, as diagnosed by the increasing RP trigger rate (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 beam cleaning at \(1.5\,\mathrm{h}\) from the beginning of the run employed only vertical collimators and led to a quickly increasing background rate, see Fig. 2. Therefore, the following beam cleaning operations were also performed in the horizontal plane. Altogether there were 6 beam cleaning interventions until the luminosity had decreased from initially \(1.8\times 10^{27}\) to \(0.4\times 10^{27}\,\mathrm{cm}^{2}\,\mathrm{s}^{1}\) at which point the data yield was considered as too low. During the 9 h long fill, an integrated luminosity of \(20\,\upmu \mathrm{b}^{1}\) was accumulated in 6 data sets corresponding to the calm periods between the cleaning operations.
Due to an anticollision protection system, the top and the bottom pots of a vertical RP unit could not approach each other close enough to be both at a distance of \(3\,\sigma _{y} = 780\,\upmu \)m from the beam centre. Therefore a configuration with one RP diagonal (45 top–56 bottom) at \(3\,\sigma _{y}\) (“close diagonal”) and the other (45 bottom–56 top) at \(10\,\sigma _{y}\) (“distant diagonal”) was chosen. The distant diagonal provides a systematic comparison at larger tvalues. The horizontal RPs were only needed for the databased alignment and therefore placed at a safe distance of \(10\,\sigma _{x} \approx 7.5\) mm, close enough to have an overlap with the vertical RPs (Fig. 1, right).
The events collected were triggered by a logical OR of: inelastic trigger (at least one charged particle in either arm of T2), doublearm proton trigger (coincidence of any RP left of IP5 and any RP right of IP5) and zerobias trigger (random bunch crossings) for calibration purposes.
In the close and distant diagonals a total of 190 and 162 k elastic event candidates have been tagged, respectively.
5 Differential crosssection
The analysis method is very similar to the previously published one [14]. 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 for the two diagonals are analysed separately. After comparison (Sect. 5.3) they are finally merged (Sect. 5.4).
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
5.1.2 Alignment
TOTEM’s usual threestage procedure [18] for correcting the detector positions and rotation angles has been applied: a beambased alignment prior to the run followed by two offline methods. First, trackbased alignment for relative positions among RPs, and second, alignment with elastic events for absolute position with respect to the beam – repeated in 15 min 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 [23]. This method uses RP observables to determine fine corrections to the optical functions presented in Eq. (2).
5.1.4 Resolution
5.2 Differential crosssection reconstruction
5.2.1 Event tagging
The elastic selection cuts. The superscripts R and L refer to the right and left arm. The \(\alpha \theta _x^*\) term in cut 3 absorbs the effects of residual optics imperfections, \(\alpha \) is of the order of \(0.1\,\mathrm{\upmu m/\upmu rad}\). The rightmost column gives a typical RMS of the cut distribution
Number  Cut  RMS (\(\equiv 1\sigma \)) 

1  \(\theta _x^{*\mathrm R}  \theta _x^{*\mathrm L}\)  \(3.9\,\mathrm{\upmu rad}\) 
2  \(\theta _y^{*\mathrm R}  \theta _y^{*\mathrm L}\)  \(1.0\,\mathrm{\upmu rad}\) 
3  \(x^{*\mathrm R}  x^{*\mathrm L}  \alpha \theta _x^*\)  \(250\,\mathrm{\upmu m}\) 
Since a MonteCarlo study shows that applying the three cuts at the \(3\,\mathrm{\sigma }\) level would lead to a loss of about \(0.5\,\mathrm{\%}\) of the elastic events, the cut threshold is set to \(4\,\mathrm{\sigma }\).
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.
The background rate (i.e. impurity of the elastic tagging) is estimated in two steps, both based on distributions of discriminators from Table 2 plotted in various situations, see an example in Fig. 7. In the first step, diagonal data are studied under several cut combinations. While the central part (signal) remains essentially constant, the tails (background) are strongly suppressed when the number of cuts is increased. In the second step, the background distribution is interpolated from the tails into the signal region. The form of the interpolation is inferred from nondiagonal RP track configurations (45 bottom–56 bottom or 45 top–56 top), artificially treated like diagonal signatures by inverting the coordinate signs in the arm 45; see the dashed distributions in the figure. These nondiagonal configurations cannot contain any elastic signal and hence consist purely of background which is expected to be similar in the diagonal and nondiagonal configurations. This expectation is supported by the agreement of the tails of the blue solid and dashed curves in the figure. Since the nondiagonal distributions are flat, the comparison of the signalpeak size to the amount of interpolated background yields the estimate \(1  \mathcal{B} < 10^{4}\).
5.2.3 Acceptance correction
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 [20, 22]). The remaining inefficiencies are related to the inability of a RP to resolve the elastic proton track.
Another source of inefficiency are proton interactions in a near RP affecting simultaneously the far RP downstream. The contribution from these nearfar correlated inefficiencies, \(\mathcal{I}_{2/4}\), is determined by evaluating the rate of events with high track multiplicity (\(\gtrsim \)5) in both near and far RPs. Events with high track multiplicity simultaneously in a near top and near bottom RP 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 outcome, \(\mathcal{I}_{2/4} \approx 1.5\,\mathrm{\%}\), is compatible between left/right arms and top/bottom RP pairs and compares well to MonteCarlo simulations (e.g. Section 7.5 in [25]).
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.
For the uncertainty estimate, the uncertainties of the \(\theta _x^*\) and \(\theta _y^*\) resolutions (accommodating the full time variation) as well as fitmodel dependence have been considered, each contribution giving a few permille for the lowestt bin.
5.2.6 Normalisation
The normalisation \(\mathcal{N}\) is determined by requiring the same crosssection integral between \(t = 0.014\) and \(0.203\,\mathrm{GeV^2}\) as for dataset 1 published in [22]. This publication describes a measurement of elastic and inelastic rates at the same collision energy of \(\sqrt{s} = 8\,\mathrm{TeV}\). These rates can be combined using the optical theorem in order to resolve the value of the luminosity which consequently allows for normalisation of the differential crosssection. The leading uncertainty of \(\mathcal{N}\), \(4.2\,\mathrm{\%}\), comes from the rate uncertainties in [22].
5.2.7 Binning
At very low t, where the crosssection varies the fastest (\(\approx \) \(0.001\,\mathrm{GeV^2}\)), a fine binning is used. In the middle of the t range (\(\approx \) \(0.03\,\mathrm{GeV^2}\)), the bin width is chosen to give about \(1\,\mathrm{\%}\) statistical uncertainty. This rule is abandoned at higher t (above \(0.07\,\mathrm{GeV^2}\)) in favour of bins with a constant width of \(0.01\,\mathrm{GeV^2}\) to avoid excessively large bins.
5.2.8 Systematic uncertainties
Besides the systematic uncertainties mentioned at the above analysis steps, the beam momentum uncertainty needs to be considered when the scattering angles are translated to t, see Eq. (5). The uncertainty was estimated to \(0.1\,\mathrm{\%}\) in Section 5.2.8 in [14] which is further supported by a recent review [26].
The MonteCarlo simulations show that the combined effect of several systematic errors is well approximated by linear combination of the individual contributions from Eq. (14).
5.3 Systematic crosschecks
Compatible results have been obtained by analysing data subsets of events from different bunches, different diagonals and different time periods – in particular those right after and right before the beam cleanings.
5.4 Final data merging
Finally, the differential crosssection histograms from both diagonals are merged. This is accomplished by a perbin weighted average, with the weight given by inverse squared statistical uncertainty. The statistical and systematic uncertainties are propagated accordingly. For the systematic ones, the correlation between the diagonals is taken into account. For example the vertical (mis)alignment of the RPs of one unit is almost fully correlated; thus the effect on the differential crosssection is opposite for the two diagonals and consequently its impact is strongly reduced once the diagonals are merged.
The elastic differential crosssection as determined in this analysis. The three leftmost columns describe the bins in t. The representative point gives the t value suitable for fitting [27]. The other columns are related to the differential crosssection. The six 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. (14) and (15). The two contributions due to optics correspond to the two vectors in Eq. (8)
t bin \((\mathrm{GeV^2})\)  \(\mathrm{d}\sigma /\mathrm{d}t \,(\mathrm{mb/GeV^2})\)  

Left edge  Right edge  Represent. point  Value  Statist. uncert.  System. uncert.  Normal. \(\mathcal{N}\)  Optics mode 1  Optics mode 2  Beam momentum  Alignment hor. shift  Alignment vert. shift 
0.000600  0.000916  0.000741  912.13  44.0  54.7  +36.3  +4.18  0.151  +0.029  0.791  40.7 
0.000916  0.001346  0.001110  665.09  21.0  30.4  +27.3  +1.66  +0.318  +0.579  +0.042  13.5 
0.001346  0.001930  0.001612  564.20  14.6  24.3  +23.8  +0.905  +0.500  +0.806  +0.065  4.87 
0.001930  0.002725  0.002298  529.76  11.3  22.4  +22.2  +0.663  +0.569  +0.895  +0.027  1.99 
0.002725  0.003806  0.003240  516.92  9.19  21.4  +21.4  +0.579  +0.585  +0.914  +0.004  1.05 
0.003806  0.005276  0.004525  502.29  6.24  20.8  +20.7  +0.587  +0.570  +0.891  0.008  0.216 
0.005276  0.007276  0.006266  477.43  4.83  20.0  +20.0  +0.563  +0.536  +0.840  0.010  0.126 
0.007276  0.009995  0.008628  454.13  3.86  19.1  +19.1  +0.534  +0.486  +0.763  0.010  0.089 
0.009995  0.01369  0.01183  424.90  3.09  17.9  +17.9  +0.496  +0.421  +0.663  0.004  0.061 
0.01369  0.01786  0.01576  398.49  2.75  16.5  +16.5  +0.447  +0.349  +0.552  0.005  0.048 
0.01786  0.02255  0.02019  363.33  2.44  15.1  +15.1  +0.394  +0.279  +0.443  0.002  0.035 
0.02255  0.02783  0.02517  327.03  2.15  13.7  +13.7  +0.338  +0.210  +0.337  0.006  0.031 
0.02783  0.03378  0.03077  293.88  1.90  12.2  +12.2  +0.283  +0.147  +0.238  0.005  0.025 
0.03378  0.04047  0.03709  257.86  1.67  10.8  +10.8  +0.229  +0.089  +0.148  0.005  0.020 
0.04047  0.04801  0.04419  225.35  1.49  9.34  +9.34  +0.229  +0.036  +0.068  +0.088  0.007 
0.04801  0.05650  0.05220  193.69  1.35  7.98  +7.97  +0.261  0.011  0.000  +0.232  0.006 
0.05650  0.06606  0.06121  158.48  1.18  6.69  +6.68  +0.258  0.047  0.054  +0.306  0.004 
0.06606  0.07606  0.07098  130.78  1.06  5.54  +5.52  +0.239  0.072  0.094  +0.337  0.003 
0.07606  0.08606  0.08098  107.80  0.98  4.57  +4.55  +0.214  0.087  0.118  +0.340  0.002 
0.08606  0.09606  0.09098  89.71  0.90  3.77  +3.75  +0.188  0.095  0.131  +0.328  0.001 
0.09606  0.1061  0.1010  73.41  0.83  3.12  +3.10  +0.163  0.097  0.136  +0.306  0.000 
0.1061  0.1161  0.1110  61.78  0.79  2.58  +2.56  +0.214  0.099  0.136  +0.234  +0.001 
0.1161  0.1261  0.1210  52.55  0.76  2.14  +2.11  +0.241  0.097  0.131  +0.179  +0.001 
0.1261  0.1361  0.1310  41.52  0.70  1.78  +1.75  +0.246  0.093  0.125  +0.141  +0.001 
0.1361  0.1461  0.1410  34.58  0.66  1.48  +1.44  +0.239  0.087  0.116  +0.113  +0.001 
0.1461  0.1561  0.1510  28.69  0.61  1.23  +1.19  +0.227  0.080  0.107  +0.091  +0.000 
0.1561  0.1661  0.1610  24.37  0.65  1.01  +0.99  +0.169  0.072  0.098  +0.095  +0.000 
0.1661  0.1761  0.1710  18.95  0.68  0.84  +0.81  +0.117  0.064  0.088  +0.104  +0.000 
0.1761  0.1861  0.1810  15.86  0.73  0.69  +0.67  +0.082  0.056  0.079  +0.111  +0.000 
0.1861  0.1961  0.1910  12.59  0.77  0.58  +0.55  +0.054  0.049  0.071  +0.123  +0.000 
Let us emphasize that the systematic effects with linear t dependence (see Fig. 13) cannot alter the nonpurelyexponential character of the data. This is the case for the effects of normalisation, beam momentum and to a large degree also of opticsmode 2. For the beam momentum, this can also be understood analytically: changing the value of p would yield a scaling of t, see Eq. (5), and consequently also scaling of the \(b_n\) parameters in Eq. (17). However, the nonzero \(b_2, b_3\) etc. parameters (reflecting the nonexponentiality) cannot be brought to 0 (as in purelyexponential case).
6 Coulomb–nuclear interference
The Coulomb–nuclear interference (CNI) can be used to probe the nuclear component of the scattering amplitude. Since the CNI effects are sensitive to the phase of the nuclear amplitude, both modulus and phase can be tested.
For the modulus, a relevant question is whether the earlier reported nonexponentiality of the differential crosssection [14] can be attributed solely to the nuclear component or whether Coulomb scattering gives a sizeable contribution. Concerning the phase, several parametrisations with different physics interpretations will be tested; for each of them the \(\rho \) parameter (representative for the phase value at \(t = 0\) according to Eq. (1)) will be determined.
Section 6.1 outlines the theoretical concepts needed to describe the CNI effects. Section 6.2 provides details on fitting procedures used to analyse the data. Sections 6.3 and 6.4 discuss the fit results for two relevant alternatives in the description of the nuclear modulus: either exponential functions with exponents linear in t (called “purely exponential”) or exponential functions with higherdegree polynomials of t in the exponent (called “nonexponential”).
6.1 Theoretical framework

Containing QED elements only. This amplitude can be obtained by perturbative calculations, see Sect. 6.1.1.

Containing QCD elements only. This amplitude is not directly calculable from the QCD lagrangian, Sects. 6.1.2 and 6.1.3 will propose several phenomenologically motived parametrisations.

Containing both QED and QCD elements. This contribution can neither be directly calculated from the Lagrangians, nor can ad hoc parametrisations be used – this amplitude is correlated with the previous two. Section 6.1.4 will introduce several interference formulae attempting to calculate the corresponding effects.
6.1.1 Coulomb amplitude
6.1.2 Nuclear amplitude—modulus
Since the calculation of CNI may, in principle, involve integrations (e.g. Eq. (25)), it is necessary to extend the nuclear amplitude meaningfully to \(t > 0.2\,\mathrm{GeV^2}\). Therefore the parametrisation Eq. (17) is only used for \(t < 0.2\,\mathrm{GeV^2}\) while at \(t > 0.5\,\mathrm{GeV^2}\) the amplitude is fixed to follow a preliminary crosssection derived from the same data set as in [14] which features a dipbump structure similar to the one observed at \(\sqrt{s} = 7\,\mathrm{TeV}\) [32]. In order to avoid numerical problems, the intermediate region \(0.2< t < 0.5\,\mathrm{GeV^2}\) is modelled with a continuous and smooth interpolation between the low and hight parts. It will be shown that altering the extended part of the nuclear amplitude (\(t > 0.2\,\mathrm{GeV^2}\)) within reasonable limits has negligible impact on the results presented later on.
6.1.3 Nuclear amplitude—phase
 (a)A constant phase is obviously the simplest choice:It leads to a strict proportionality between the real and the imaginary part of the amplitude at all t.$$\begin{aligned} \arg \mathcal{A}^\mathrm{N}(t) = {\pi \over 2}  \arctan \rho = \hbox {const}. \end{aligned}$$(18)
 (b)The standard phase parametrisation,describes the main features of many theoretical models – almost imaginary amplitude in the forward direction (\(\rho \) small) while almost purely real at the diffraction dip. The parameter values \(t_0 =  0.50\,\mathrm{GeV^2}\) and \(\tau = 0.1\,\mathrm{GeV^2}\) have been chosen such that the shape is similar to a number of model predictions, see Fig. 14.$$\begin{aligned} \arg \mathcal{A}^\mathrm{N}(t) =&{\pi \over 2}  \arctan \rho + \arctan \left( \frac{tt_{0}}{\tau }\right) \\& \arctan \left( \frac{t_{0}}{\tau }\right) \, , \end{aligned}$$(19)
 (c)The parametrisation by Bailly et al. [33]:where \(t_\mathrm{d} \approx 0.53\,\mathrm{GeV^2}\) gives the position of the diffractive minimum at \(8\,\mathrm{TeV}\) (preliminary result derived from the \(\beta ^* = 90\,\mathrm{m}\) data [14]). This phase has a behaviour qualitatively similar to the model of Jenkovszky et al., see Fig. 14.$$\begin{aligned} \arg \mathcal{A}^\mathrm{N}(t) = {\pi \over 2}  \arctan {\rho \over 1  {t\over t_\mathrm{d}}} \end{aligned}$$(20)
 (d)Another parametrisation was proposed in [16]:As shown in Fig. 14, it features a peak at \(t = \kappa / \nu \) and for asymptotically increasing t it returns to its value at \(t=0\). Due to a potentially rapid variation at low t, this functional form can yield an impactparameterspace behaviour that is qualitatively different from the one obtained with the above parametrisations. In order to ensure fit stability, the parameters$$\begin{aligned} \arg \mathcal{A}^\mathrm{N}(t) = {\pi \over 2}  \arctan \rho  \zeta _1 \left(  {t\over 1\,\mathrm{GeV^2}} \right) ^\kappa \mathrm{e}^{\nu t}.\nonumber \\ \end{aligned}$$(21)have been fixed to example values maintaining the desired impactparameter behaviour at \(\sqrt{s} = 8\,\mathrm{TeV}\), using a method detailed in [34]. This parametrisation with one free parameter will be denoted as peripheral phase in what follows.$$\begin{aligned} \zeta _1 = 800 ,\quad \kappa = 2.311 ,\quad \nu = 8.161\,\mathrm{GeV^{2}} \end{aligned}$$(22)
It should be noted that the nuclear phase has a strong influence on the amplitude behaviour in the space of impact parameter b (for a detailed discussion see e.g. Section 3 in [35]). A particularly decisive feature is the rate of phase variation at low t. Looking at Fig. 14 one can see that the constant, standard and Bailly phases are essentially flat at low t, thus leading to qualitatively similar pictures in the impact parameter space: elastic collisions being more central (preferring lower values of b) than the inelastic ones. Conversely, the peripheral phase parametrisation can yield a description with the opposite hierarchy, which is argued to be more natural by some authors (e.g. Section 4 in [36]). An impactparameter study of the presented data will be given at end of Sect. 6.4.
6.1.4 Coulomb–nuclear interference formulae
Since the quantities G in Eqs. (24) and (25) are complex, the interference effects in these treatments are generally more featurerich than with the SWY formula, Eq. (23), where the interference is reduced to a single additional phase \(\Phi \).
6.2 Analysis procedure

Step 1: fit of \(\beta ^* = 1000\,\mathrm{m}\) data with \(\rho \) free,

Step 2: fit of \(\beta ^* = 1000\) and \(90\,\mathrm{m}\) data with \(\rho \) fixed from the preceding step.

Choice of the form factor in Eq. (16). The options considered in [31] have been tested, none of them giving any significant difference with respect to the default choice [38].

Extension of the modulus of the nuclear amplitude to the unobserved t region, see the last paragraph in Sect. 6.1.2. No effect was observed when the hight part was altered (both shape and normalisation) nor when the size of the transition region was changed.

Use of the Cahn or KL formula. Only the latter will be used in what follows to represent both of them.

The two variants of the KL formula, Eqs. (25) and (26). The latter will be used below.

Fits with constant, standard and Bailly phase are practically indistinguishable. This can be expected from Fig. 15 showing that the corresponding CNI effects are very similar. Therefore, in the remainder of this article, these phases will be treated as a single family represented by the constant phase.

Section 6.3: fits with purely exponential nuclear modulus, that is \(N_b=1\) in Eq. (17). In this case, the nonexponentiality can come from the CNI effects only.

Section 6.4: fits with nuclear modulus flexible enough to describe the nonexponentiality without the CNI effects. Here, the nonexponentiality may be due to the nuclear modulus, CNI effects or both.
6.3 Fits with purely exponential nuclear modulus
Fit results with \(N_b=1\). Each column corresponds to a fit with different interference formula and/or nuclear phase
SWY, constant  Cahn/KL, constant  Cahn/KL, peripheral  

\(\hbox {Step 1:}\, \chi ^2/\hbox {ndf}\)  \( 48.0 / 27 = 1.78\)  \( 48.1 / 27 = 1.78\)  \( 27.7 / 27 = 1.03\) 
\(\hbox {Step 2:}\, \chi ^2/\hbox {ndf}\)  \( 180.8 / 58 = 3.12\)  \( 181.2 / 58 = 3.12\)  \( 64.3 / 58 = 1.11\) 
\(a\,(\mathrm{mb/GeV^2})\)  \(533 \pm 23\)  \(533 \pm 23\)  \(551 \pm 23\) 
\(b_1\,(\mathrm{GeV^{2}})\)  \(19.42 \pm 0.05\)  \(19.42 \pm 0.05\)  \(19.74 \pm 0.05\) 
\(\rho \)  \(0.05 \pm 0.02\)  \(0.05 \pm 0.02\)  \(0.10 \pm 0.02\) 
\(\zeta _1\)  800  
\(\kappa \)  2.311  
\(\nu \,(\mathrm{GeV^{2}})\)  8.161  
\(\sigma _\mathrm{tot}\,(\mathrm{mb})\)  \(102.0 \pm 2.2\)  \(102.0 \pm 2.2\)  \(103.4 \pm 2.3\) 
Table 4 shows that both fits with constant phase are essentially identical and have bad quality. The step2 fit using both \(\beta ^*=1000\) and \(90\,\mathrm{m}\) data can be excluded with \(7.6\,\mathrm{\sigma }\) significance. Consequently, since the combination of \(N_b=1\) and constant phase is the only one compatible with the SWY approach, that formula is experimentally excluded even on the basis of only the lowt data set discussed here. This result is complementary to the observation of a diffractive minimum at \(\sqrt{s} = 8\,\mathrm{TeV}\) (to be published in a forthcoming article) which also contradicts the assumptions of the SWY formula.

There are several theoretical reasons for the nuclear component not to be purely exponential, e.g. [39, 40, 41, 42]. Indeed, most elastic scattering models predict a nonexponential nuclear modulus, see e.g. [31] and references therein.

The value of \(\rho \) obtained in this fit may be regarded as an outlier with respect to a consistent pattern of other fits from this article and extrapolations from lower energies: e.g. [43, 44, 45] and most models in [31].
6.4 Fits with nonexponential nuclear modulus
The aim of this section is to discuss fits with enough flexibility in the nuclear modulus to describe the nonexponentiality in the data. Since a nonexponential hadronic modulus is used, the only applicable interference formula is KL. \(N_b=2\) to 5 were considered. The optimal degree was chosen according to two criteria: reasonable \(\chi ^2/\hbox {ndf}\) and stability of fit parameters (among which \(\rho \) is one of the most sensitive). For instance, with constant phase the fit (step 1) with \(N_b=2\) yields \(\chi ^2/\hbox {ndf} = 1.07\) and \(\rho = 0.10\) while the one with \(N_b=3\) gives \(\chi ^2/\hbox {ndf} = 1.03\) and \(\rho = 0.12\). Both fits have the normalised \(\chi ^2\) reasonably close to 1, but the value of \(\rho \) changes significantly between \(N_b=2\) and 3 which is unexpected should \(N_b=2\) be sufficient. On the other hand \(N_b=4\) gives \(\chi ^2/\hbox {ndf} = 0.861\) which is unreasonably low. Therefore \(N_b=3\) was chosen.
As shown in Table 5, both fits have reasonable fit quality and remarkably consistent values of \(\rho \) (identical within the resolution) which are compared to previous determinations at lower energies in Fig. 18. Take note that the obtained parameters for the nuclear amplitude (a and \(b_i\)) are consistent between step 1 (\(\beta ^* = 1000\,\mathrm{m}\) data only) and step 2 (both \(\beta ^* = 1000\) and \(90\,\mathrm{m}\) data) of the fitting procedure as already mentioned in Sect. 6.2.
Fit results with Cahn or KL formula and \(N_b=3\)
Cahn/KL, constant  Cahn/KL, peripheral  

\(\hbox {Step 1:}\, \chi ^2/\hbox {ndf}\)  \( 25.7/ 25 = 1.03\)  \( 25.0/ 25 = 1.00\) 
\(\hbox {Step 2:}\, \chi ^2/\hbox {ndf}\)  \( 57.5/ 56 = 1.03\)  \( 57.6/ 56 = 1.03\) 
\(a\,(\mathrm{mb/GeV^2})\)  \(549 \pm 24\)  \(549 \pm 24\) 
\(b_1\,(\mathrm{GeV^{2}})\)  \(20.47 \pm 0.14\)  \(19.56 \pm 0.13\) 
\(b_2\,(\mathrm{GeV^{4}})\)  \(8.8 \pm 1.6\)  \(3.3 \pm 1.5\) 
\(b_3\,(\mathrm{GeV^{6}})\)  \(20 \pm 6\)  \(13 \pm 5\) 
\(\rho \)  \(0.12 \pm 0.03\)  \(0.12 \pm 0.03\) 
\(\zeta _1\)  800  
\(\kappa \)  2.311  
\(\nu \,(\mathrm{GeV^{2}})\)  8.161  
\(\sigma _\mathrm{tot}\,(\mathrm{mb})\)  \(102.9 \pm 2.3\)  \(103.0 \pm 2.3\) 
The total crosssection results from the two fits in Table 5 are well consistent with each other and also with previous measurements [14, 22]. The slightly higher values with respect to previous analyses neglecting the Coulomb interaction are expected as long as \(\rho > 0\). This gives negative interference at low t and when separated leads to an increase of nuclear crosssection intercept a and thus also total crosssection via Eq. (28).
7 Summary and outlook
For the first time at LHC the differential crosssection of elastic proton–proton scattering has been measured at tvalues down to the Coulomb–nuclear interference (CNI) region. This was made possible by a special beam optics, a novel collimation procedure and by moving the RPs to an unprecedented distance of only \(3\,\sigma \) from the centre of the circulating beam.

Purely exponential nuclear modulus (\(N_b=1\)), constant phase: excluded with more than \(7\,\mathrm{\sigma }\) confidence. Since this is the only combination compatible with the SWY formula, the data exclude the usage of the formula.

Purely exponential nuclear modulus (\(N_b=1\)), peripheral phase: the data do not exclude this option which, however, is disfavoured from other perspectives.

Nonexponential nuclear modulus (\(N_b=3\)): both constant and peripheral phases are well compatible with the data, therefore the central impactparameter picture prevalent in phenomenological descriptions is not a necessity.
For even stronger results in the future the key point is a better distinction between the nuclear and CNI crosssection components, which can be achieved from both theoretical and experimental sides. New theory developments may narrow down the range of allowed parametrisations of the nuclear modulus and phase or better constrain the induced CNI effects. The experimental improvements include increasing statistics and reducing the lower t threshold. For the former, TOTEM has already upgraded the RP mechanics such that both vertical pots can be simultaneously placed very close to the beam. For the latter, TOTEM foresees an optics with extremely high \(\beta ^* \approx 2500\,\mathrm{m}\) which would allow to reach the CNI region even at Run II energies. Moreover, recent experience with the \(\beta ^* = 90\,\mathrm{m}\) optics at \(\sqrt{s} = 13\,\mathrm{TeV}\) shows that very low beam emittances can be achieved, thus possibly further reducing the RP distance from the beam.
Footnotes
Notes
Acknowledgments
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 Grant NK 101438 (Hungary). Individuals have received support from Nylands nation vid Helsingfors universitet (Finland) and from the MŠMT ČR (Czech Republic).
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