Probing the photonic content of the proton using photon-induced dilepton production in $p+\textrm{Pb}$ collisions at the LHC

We propose a new experimental method to probe the photon parton distribution function inside the proton (photon PDF) at LHC energies. The method is based on the measurement of dilepton production from the $\gamma p\rightarrow\ell^+\ell^-+X$ reaction in proton--lead collisions. These experimental conditions guarantee a clean environment, both in terms of reconstruction of the final state and in terms of possible background. We firstly calculate the cross sections for this process with collinear photon PDFs, where we identify optimal choice of the scale, in analogy to deep inelastic scattering kinematics. We then perform calculations including the transverse-momentum dependence of the probed photon. Finally we estimate rates of the process for the existing LHC data samples.

Recently, a precise photon distribution inside the proton has been evaluated in Ref. [14].
This approach provides a model-independent determination of the photon PDF (embedded in the so-called LUXqed distribution) and it is based on proton structure function and elastic form factor fits in electron-proton scattering.
To date, there are no experimentally clean processes identified that would allow verification or strong constraint of the calculations. For example, the extraction of the photon PDF from isolated photon production in deep inelastic scattering (DIS) [15] or from inclusive pp → + − + X [2,16,17] is limited due to large QCD background. On the contrary, the elastic part of the photon PDF is verified via exclusive γγ → + − process, measured in pp collisions by ATLAS [18,19], CMS [20,21] and recently by CMS+TOTEM [22] collaborations.
We therefore propose a new experimental method to constrain the photonic content of the proton. Due to the large fluxes of quasi-real photons from the lead ion (Pb) at the LHC, the photon-induced dilepton production in p + Pb collision configuration (where Pb serves as a source of elastic photons) is a very clean way to probe the photon PDF inside a proton. This process is shown schematically in Fig. 1, where by analogy to DIS, two leading-order diagrams can be identified. Since the photon flux from the ion scales with Z 2 (Z is the charge of the ion) and QCD-induced cross-sections scale approximately with the atomic number A, the amount of QCD background is greatly reduced comparing to the pp case.
Moreover, as this process does not involve the exchange of color with the photon-emitting nucleus, no significant particle production is expected in the rapidity region between the dilepton system and the nucleus. The photon-emitting nucleus is also expected to produce no neutrons because the photons couple to the entire nucleus. Thus a combination of requirements on rapidity gap and zero neutrons in the same direction provide straightforward criteria to identify these events experimentally.

A. Elastic photon fluxes
To get the distribution of the elastic photons from the proton, one can express the equivalent photon flux through the electric and magnetic form factors G E (Q 2 ) and G M (Q 2 ) of the proton. This contribution is obtained as where x is the momentum fraction of the proton taken by the photon, Q 2 is the photon virtuality, α em is the electromagnetic structure constant and m p is the proton mass.
To express the elastic photon flux for the nucleus (γ Pb el ), we follow Ref. [23] and replace where F em (Q 2 ) is the electromagnetic form factor of the nucleus and Z is its charge. We also neglect the magnetic form factor of the ion in the following.
For the Pb nucleus, we use the form factor parameterization from the STARlight MC generator [24]: where R A = 1.1A 1/3 fm, a = 0.7 fm and Q = Q 2 .

B. Collinear-factorization approach and choice of the scale
The inelastic processes, with breakup of a proton, can be also considered. At LO and at a given scale µ 2 , the photon parton distribution γ p inel (x, µ 2 ) of photons carrying a fraction x of the proton's momentum, obeys the DGLAP equation: where q(x, µ 2 ) is the quark PDF, e q is the quark charge, P γ←q is the q → γ splitting function, and P γ←γ corresponds to the virtual self-energy correction to the photon propagator. This is the basis for colinear photon-PDFs in the initial [25,26] and more recent [14,16,17,[27][28][29][30] analyses.
The computation of the photon-induced dilepton production cross section requires definition of the scale (µ 2 ) at which the photon PDFs are convoluted. The usual choice for µ is the mass of the system (motivated by the s-channel quark-antiquark annihilation process) or the transverse momentum of the leading object. These choices are however not optimal for the s-and t-channel initiated photon-induced process. By analogy to DIS (Fig. 1), where the scale is associated with the virtuality of the exchanged photon, it is possible to define the scale in case of the γγ → + − process. This is achieved by taking the virtuality of the massive t-or u-channel propagator (Fig. 1b or c). Hence, µ 2 = −(p γ P b − p − ) 2 for the t-channel diagram and µ 2 = −(p γ P b −p + ) 2 for the u-channel exchange, where p γ P b is the four momentum of the photon emitted by lead and p ± is the four momentum of the lepton of the corresponding charge. Note that the u-and t-channel diagrams have vanishing interference in the zero lepton mass limit. Therefore, they can be separated while convoluting PDFs with the partonic cross section.
In the collinear approach, the p + Pb → Pb + + − + X production cross section can be written as where σ γγ→ + − is the elementary cross section for the γγ → + − subprocess and S 2 is the so-called survival factor which takes into account the requirement that there be no hadronic interactions between the proton and the ion.

C. k T -factorization approach
At lowest order, the calculations with collinear photons produce leptons that are backto-back in transverse kinematics. Therefore, to take the effect of transverse momentum smearing into account, dedicated parton shower algorithms are usually used.
In the k T factorization approach, one can parametrize the γ * p → X vertices in terms of the proton structure functions. The photons from inelastic production have transverse momenta and non-zero virtualities Q 2 and the unintegrated photon distributions are used, in contrast to collinear distributions. In the DIS limit, the unintegrated inelastic photon flux can be obtained using the following equation [4,31]: and we use the functions F in γ * ←p from [13,23]: The virtuality Q 2 of the photon depends on the photon transverse momentum ( q 2 T ) and the proton remnant mass (M X ): Moreover, the proton structure functions F 1 (x Bj , Q 2 ) and F 2 (x Bj , Q 2 ) require the argument Note that in Eq. 7 instead of using F 2 (x Bj , Q 2 ), F 1 (x Bj , Q 2 ), we in practice use the pair is the longitudinal structure function of the proton.
These unintegrated photon fluxes enter the p + Pb → Pb + + − + X production cross section as where σ γ * γ→ + − is the off-shell elementary cross-section [31] and for x p 1 we have Q 2 ≈ q 2 T (see Eq. 8). One should note that while the fluxes do not depend on the direction of q T , averaging over directions of q T in the off-shell cross section replaces the average over photon polarizations in the collinear case.

GROUND SOURCES
We assume collision setup from recent p + Pb run at the LHC, carried out at the centreof-mass energy per nucleon pair √ s N N = 8.16 TeV. Since the energy per nucleon in the proton beam is larger than in the lead beam, the nucleon-nucleon centre-of-mass system has a rapidity in the laboratory frame of y = 0.465.
As an example of method's applicability, we will use the geometry of ATLAS [32] and CMS [33] detectors in the following. We consider only the dimuon channel, however the integrated results for ee and µµ channels can be obtained by simply multiplying the dimuon cross-sections by a factor of two.
We start by applying a minimum transverse momentum requirement of 4 GeV to both muons. This requirement is imposed to ensure high lepton reconstruction and triggering efficiency. Moreover, due to limited acceptance of the detectors, each muon is required to have a pseudorapidity (η ) that satisfies |η | < 2.4. Our calculations are carried out for a minimum dilepton invariant mass of m + − = 10 GeV. Such a choice is due to removal of possible contamination from Υ(→ + − ) photoproduction. A summary of all selection requirements is presented in Table I.
Possible background for this process can arise from inclusive lepton-pair production, e.g.
from Drell-Yan process [34][35][36][37]. This processes would lead to disintegration of the incoming ion, and zero-degree calorimeters (ZDC) [38,39] can be used to veto very-forward-going neutral fragments which would allow this background to be reduced fully. Another background can arise from diffractive interactions, hence possibly mimicking the signal topology.
However, since the Pb nucleus is a fragile object (with the nucleon binding energy of just  8 MeV) even the softest diffractive interaction will likely result in the emission of a few nucleons from the ion, detectable in the ZDC.
Another background category is the photon-induced process with a resolved photon, i.e.
γp → Z/γ * + X reaction. Here, the rapidity gap is expected to be smaller than in the signal process due to the additional particle production associated with the "photon remnant".
Any other residual contamination of this process can be controlled using a dedicated sample, with a dilepton invariant mass around the Z-boson mass.

IV. RESULTS WITH COLLINEAR PHOTON-PDFS
We start with the calculation of the elastic contribution, p + Pb → p + Pb + + − . In this case the photon flux becomes: and the following parameterization is used [23]: where F = 1 + This parameterization is a good analytical approximation of Eq. 1 integrated over Q 2 . The results for the elastic case are cross-checked with the calculation from STARlight MC and a good agreement between the fiducial cross- proton-nucleus requirement [24] is used.
Next, for the inelastic case (γp → + − + X), several recent parameterizations of the photon parton distributions are studied: CT14qed [15], HKR16qed [29], LUXqed17 [40] and NNPDF3.1luxQED [30]. All predictions are scaled by S 2 = 0.95, again derived from STARlight. This value of S 2 is lower than for the purely elastic case, due to slightly smaller average impact parameter between the proton and the ion in the inelastic reaction. One should note that all of these PDF sets include both elastic and inelastic parts of the photon spectrum. We keep the elastic part now (as provided by each group), but we subtract it later in Sec. VI for the comparison with k T -factorization results.
The integrated fiducial cross-sections for p + Pb → Pb + + − + X production at √ s N N = 8.16 TeV for different collinear photon PDF sets are summarized in Tab. II. Comparison of several lepton kinematic distributions between different photon-PDFs is shown in Fig. 2, including invariant mass and rapidity of lepton pair, and single-lepton transverse momentum/pseudorapidity distributions. The asymmetry visible in pair rapidity and singlelepton pseudorapidity distributions is due to expermental setup, which assumes a difference in the energy per nucleon between the proton beam and the lead beam (see Sec. III). All photon PDF parameterizations agree within 10% with each other. The small differences are mainly due to overall PDF normalization, as no variation in the shape of various kinematic distributions is observed.

V. RESULTS USING k T -FACTORIZATION APPROACH
Several different parametrizations of proton strucure functions are used. Those are labeled as: • ALLM [41,42]: This parametrization gives a good fit to F 2 in most of the measured regions.
This setup closely follows the LUXqed work from Ref. [40].
To model γ p el (x, Q 2 ) we use Eq. 1 with so-called dipole parametrization of the proton form factors: where µ p is the proton magnetic moment. Table III shows the comparison of integrated fiducial cross sections for inelastic p + Pb → Pb + + − + X production at √ s N N = 8.16 TeV for different proton structure functions.
All structure functions provide similar fiducial cross-section, at the level of 16-18 nb. These inelastic cross-sections are also similar in size to the elastic contribution (18 nb) and are slightly lower than the numbers from collinear analysis, subtracted for elastic part (see Table II). A comparison is also made with LUX-like parametrization when the longitudinal structure function (F L ) is explicitly considered. This leads to the decrease of the cross section by 2%, similarly to Ref. [13].    Based on Fig. 4, it is also possible to separate experimentally the elastic part (p + Pb → p + Pb + + − ), with striking back-to-back topology, from the inelastic contribution. With k T -factorization, one can also calculate the mass of the proton remnants (M X ). This is shown in Fig. 5; in contrast to the elastic case (M X = m p ) quite large masses of the remnant system can be achieved.       We also take the opportunity to calculate expected number of events for realistic assumption on total integrated luminosity. Based on the previous p + Pb runs at the LHC, we assume Ldt = 200 nb −1 . We also assume possible experimental efficiencies, mainly due to trigger and reconstruction of leptons, which we embed in a single correction factor C = 0.7. Table IV shows the expected number of events for p + Pb → Pb + + − + X production at √ s N N = 8.16 TeV and configuration described above. Approximately 2500 elastic dilepton events are expected. Depending on the calculations, 3400 (collinear with LUXqed17 PDF) or 2400 (k T -factorization with LUX-like F 2 + F L ) reconstructed inelastic events are predicted.
The data should be therefore sensitive to discriminate between the predictions based on collinear and k T -factorization approaches, using existing datasets collected by ATLAS and CMS. 10

VII. SUMMARY
In summary, we propose a method that would provide an unambiguous test of the photon parton distribution at LHC energies, and allow constraints to be placed on it. This method is based on the measurement of the cross-section for the reaction p + Pb → Pb + + − + X, where the expected background is small compared to the analogous process in pp collisions.
Results are shown for different choices of collinear photon PDFs, and a comparison is made with unintegrated photon distributions that include non-zero photon transverse momentum.
Due to the smearing of dilepton transverse momentum introduced by the k T -factorization approach, these two approaches lead to the cross sections that differ by about 30%. Moreover, for collinear approach and by analogy to DIS, an optimal choice of the scale is identified.
Using simple (realistic) experimental requirements on lepton kinematics, it is shown that