The analysis procedure is almost identical to the one published in Ref. [1]. Here only a brief overview is given, for details the reader is referred to the original publication.
For a given t bin, the differential cross-section is evaluated by selecting and counting elastic events:
$$\begin{aligned} {\mathrm{d}\sigma \over \mathrm{d}t}\left( \hbox {bin}\right) = {{{\mathcal {N}}}}\, \mathcal{U}(t)\, {{{\mathcal {B}}}} \, \frac{1}{\Delta t} \sum \limits _{t\, \in \, \hbox {bin}} {{{\mathcal {A}}}}(t, t_y)\, {{{\mathcal {E}}}}(t_y) \, , \end{aligned}$$
(1)
where \(\Delta t\) is the width of the bin, \({{{\mathcal {N}}}}\) is a normalisation factor, and the other symbols stand for correction factors: \({{{\mathcal {U}}}}\) for unfolding of resolution effects, \({{{\mathcal {B}}}}\) for background subtraction, \({{{\mathcal {A}}}}\) for acceptance correction and \({{{\mathcal {E}}}}\) for detection and reconstruction efficiency. \(t_y \equiv - p^{2} \theta _{y}^{*\, 2}\) represents the component of the four-momentum transfer squared related to the vertical scattering angle, relevant for some of the corrections.
The candidate events are tagged with cuts that enforce the elastic-event kinematics: two collinear protons (one in each arm of the experiment) emerging from the same vertex. In addition, the optics-imposed correlation between the vertical track position and angle at the RPs is required. All the cuts are applied at the \(4\,\mathrm{\sigma }\) level where Monte Carlo studies indicate a tolerable loss of about \(0.1\,\mathrm{\%}\) of the events.
The background, i.e. non-elastic events passing the tagging cuts, is determined by analysing the distributions of several selection discriminators (e.g. the difference between the reconstructed scattering angles from the two arms, \(\theta _{x,y}^{*\,56} - \theta _{x,y}^{*\,45}\), see Table 2 in Ref. [1]) for two complementary data sets: (a) the events with diagonal topology, containing both elastic signal and non-elastic background, and (b) the events with anti-diagonal topology (i.e. 45 top–56 top, or 45 bottom–56 bottom), which cannot contain any elastic signal. For the diagonal events, the tails of the discriminator distributions, containing only background, are interpolated into the signal region to estimate the contamination of that region. The shape used for this interpolation is taken from the anti-diagonal events. In the tail region, the discriminator distributions of the anti-diagonal and the diagonal events have been confirmed to agree well. Hence it is expected that also in the signal regions of the discriminators the distributions of the anti-diagonal events are similar to the background part of the diagonal events. This procedure yields a background estimate of \(1 - {{{\mathcal {B}}}} < 10^{-4}\).
The acceptance correction, \({{{\mathcal {A}}}}\), receives two contributions. The “geometrical” correction reflects the fraction of events with a given value of |t| that fall within the geometrical acceptance of the sensors. The second contribution corrects for fluctuations around the sensor edges mainly due to the beam divergence.
The normalisation factor, \({{{\mathcal {N}}}}\), is determined by matching the present data to the reference data from Ref. [19] (there labelled as “dataset 1”), where the luminosity-independent calibration was applied – a technique based on the optical theorem and a measurement of the elastic and inelastic collision rates. The matching is performed by requiring the same cross-section integral between \(|t| = 0.027\) and \(0.083\,\mathrm{GeV^2}\), a range which is available in both datasets.
Since the normalisation is determined from another dataset, in the present analysis it is sufficient to consider only inefficiency effects, \({{{\mathcal {E}}}}\), that may modify the t-distribution shape. These are caused by the inability of the system to reconstruct an elastic proton track. Two cases are distinguished. In the first case, a single RP does not show one unique proton track (it may have either zero or several tracks, which cannot be resolved in a strip detector system). Such inefficiencies are evaluated by removing the RP from the tagging cuts, repeating the selection and calculating the fraction of events recovered. In the second case, multiple RPs in the same arm do not show the proton track, which typically results from showers, initiated in the upstream RP and affecting also the downstream one. The related inefficiency is studied by examining the rate of events with high track multiplicity.
The scattering-angle resolution is studied by comparing the protons in the two arms of the RP system. For elastic events the angles should be identical, but fluctuations arise due to the beam divergence and partly due to the finite RP sensor resolution. The scattering-angle resolution was found to deteriorate slightly with time, at a rate compatible with the beam emittance growth.
Because of the richer structure of the differential cross-section in the full |t| range, the unfolding of resolution effects is more complex than in Ref. [1]. Consequently, an alternative determination is used besides the original method. The original method (denoted “CF” in the present article) consists of fitting the observed t-distribution with a smooth curve, which serves as an input to a Monte Carlo simulation. This is performed once with and once without simulating the scattering-angle resolution. The ratio of the output histograms gives a set of per-bin corrections factors. Applying them to the yet uncorrected differential cross-section yields a better estimate of the true t-distribution and serves as an input to the next iteration. The iterations stop when the difference between the input and output t-distributions is negligible (below \(0.1\,\mathrm{\%}\)), typically after two iterations. The alternative method performs a regularised resolution-matrix inversion (denoted “RRMI”), adapted from Chapter 11 in Ref. [20]. The regularisation is needed since the inverted resolution matrix tends to over-amplify statistical fluctuations. It is implemented via minimisation of \(\chi ^2\) which receives two contributions: one corresponding to the exact resolution-matrix inversion and one proportional to the integral of \({\mathrm{d}^2\over \mathrm{d}|t|^2} \log {\mathrm{d}\sigma \over \mathrm{d}t}\) over the full |t| range. A result comparison is given in Fig. 1, where the blue and red curves correspond to different parametrisations of the smoothing fit. The red curve is used for correcting the differential cross-section, the others contribute to the systematic-uncertainty estimate.
The systematic uncertainties considered include:
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alignment: RP horizontal and vertical shifts, rotation about beam axis,
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optics calibration,
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acceptance correction: uncertainty of the resolution parameters including their possible left-right asymmetry and non-gaussian distribution,
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uncertainties of the efficiency estimate,
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uncertainty of the beam momentum [21],
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unfolding: method and fit dependence, uncertainty of the resolution parameters including their full time variation,
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uncertainty of the normalisation [19].
The systematic uncertainties were propagated to the differential cross-section using a Monte-Carlo simulation where the correlations between the diagonals were taken into account. The leading systematic effects are evaluated in Fig. 2.
The final differential cross-section with its uncertainties is presented in Table 1 and plotted in Fig. 3.