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

TL;DR

This paper introduces a novel experimental approach to measure the photon content of the proton at LHC energies using dilepton production in proton-lead collisions, providing a cleaner environment for probing the photon PDF.

Contribution

It proposes a new method based on photon-induced dilepton production in p+Pb collisions to directly measure the photon PDF inside the proton at LHC energies.

Findings

Calculated cross sections using collinear photon PDFs.

Identified optimal scale choices for the process.

Estimated event rates for existing LHC data.

Abstract

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.

Figures

Tables

Table 1. Table 1: Definition of the fiducial region used in the studies.

Variable	Requirement
lepton transverse momentum, $p_{\text{T}}^{\ell }$	$>4$ GeV
lepton pseudorapidity, $\left|{\eta }^{\ell }\right|$
dilepton invariant mass, $m_{{\ell }^{+}{\ell }^{-}}$	$>10$ GeV

Table 2. Table 2: Integrated fiducial cross sections for p + Pb → Pb + ℓ + ​ ℓ − + X → 𝑝 Pb Pb superscript ℓ superscript ℓ 𝑋 p+\textrm{Pb}\rightarrow\textrm{Pb}+\ell^{+}\ell^{-}+X production at s N ​ N = 8.16 subscript 𝑠 𝑁 𝑁 8.16 \sqrt{s_{NN}}=8.16   TeV  for different collinear photon PDF sets. The effect of applying only p T ℓ superscript subscript 𝑝 𝑇 ℓ p_{T}^{\ell} requirement is shown in second column. The uncertainties denote the PDF uncertainties (if available) calculated at 68% CL. For comparison, the cross section for purely elastic contribution is also shown.

Contribution	$p_T^{\ell }>4$ GeV	$p_T^{\ell }>4$ GeV, ,
		$m_{{\ell }^{+}{\ell }^{-}}>10$ GeV
${\gamma }_{\mathrm{el}}^p$	44.9 nb	17.5 nb
${\gamma }_{\mathrm{el}}^p+{\gamma }_{\mathrm{inel}}^p$ [CT14qed_inc]	$98\pm 4$ (PDF) nb	$40\pm 2$ (PDF) nb
${\gamma }_{\mathrm{el}}^p+{\gamma }_{\mathrm{inel}}^p$ [LUXqed17]	$105.8\pm 0.2$ (PDF) nb	$44.1\pm 0.1$ (PDF) nb
${\gamma }_{\mathrm{el}}^p+{\gamma }_{\mathrm{inel}}^p$ [NNPDF3.1luxQED]	$115.6\pm 0.6$ (PDF) nb	$45.9\pm 0.3$ (PDF) nb
${\gamma }_{\mathrm{el}}^p+{\gamma }_{\mathrm{inel}}^p$ [HKR16qed]	121.6 nb	49.4 nb

Table 3. Table 3: Integrated fiducial cross sections for inelastic p + Pb → Pb + ℓ + ​ ℓ − + X → 𝑝 Pb Pb superscript ℓ superscript ℓ 𝑋 p+\textrm{Pb}\rightarrow\textrm{Pb}+\ell^{+}\ell^{-}+X production at s N ​ N = 8.16 subscript 𝑠 𝑁 𝑁 8.16 \sqrt{s_{NN}}=8.16   TeV  for different proton structure functions. The effect of applying only p T ℓ superscript subscript 𝑝 𝑇 ℓ p_{T}^{\ell} requirement is shown in second column.

Contribution	$p_T^{\ell }>4\text{GeV}$	$p_T^{\ell }>4\text{GeV}$, , $m_{{\ell }^{+}{\ell }^{-}}>10\text{GeV}$
${\gamma }_{\mathrm{el}}^p$	47.9 nb	18.3 nb
${\gamma }_{\mathrm{inel}}^p$ [LUX-like $F_2$]	43.6 nb	17.4 nb
${\gamma }_{\mathrm{inel}}^p$ [LUX-like $F_2+F_L$]	42.6 nb	17.1 nb
${\gamma }_{\mathrm{inel}}^p$ [ALLM97 $F_2$]	41.7 nb	16.4 nb
${\gamma }_{\mathrm{inel}}^p$ [SU $F_2$]	41.7 nb	16.7 nb
${\gamma }_{\mathrm{inel}}^p$ [SY $F_2$]	40.4 nb	16.0 nb

Table 4. Table 4: Expected number of events for p + Pb → Pb + ℓ + ​ ℓ − + X → 𝑝 Pb Pb superscript ℓ superscript ℓ 𝑋 p+\textrm{Pb}\rightarrow\textrm{Pb}+\ell^{+}\ell^{-}+X production at s N ​ N = 8.16 subscript 𝑠 𝑁 𝑁 8.16 \sqrt{s_{NN}}=8.16   TeV  assuming ∫ L ​ 𝑑 t = 200 ​ nb − 1 𝐿 differential-d 𝑡 200 superscript nb 1 \int Ldt=200\leavevmode\nobreak\ \textrm{nb}^{-1} . Shown are several contributions: purely elastic, inelastic with collinear LUXqed17 PDF and inelastic with k T subscript 𝑘 𝑇 k_{T} -factorization and LUX-like F 2 + F L subscript 𝐹 2 subscript 𝐹 𝐿 F_{2}+F_{L} proton structure function parameterization. An effect of possible experimental efficiencies is shown in last column.

Contribution	Expected events ($C=1$)	Expected events ($C=0.7$)
${\gamma }_{\mathrm{el}}^p$	3600	2500
${\gamma }_{\mathrm{inel}}^p$ [LUXqed17 collinear]	5600	3900
${\gamma }_{\mathrm{inel}}^p$ [LUX-like $F_2+F_L$]	3400	2400

Equations

$$\displaystyle\gamma^{p}_{el}(x,Q^{2})=\frac{\alpha_{\rm{em}}}{\pi}\Big{[}\Big{(}1-{x\over 2}\Big{)}^{2}\,{4m_{p}^{2}G_{E}^{2}(Q^{2})+Q^{2}G_{M}^{2}(Q^{2})\over 4m_{p}^{2}+Q^{2}}+{x^{2}\over 4}G_{M}^{2}(Q^{2})\Big{]}\leavevmode\nobreak\ ,$$

$$\frac{4m_p^2{G}_E^2(Q^2)+Q^2{G}_M^2(Q^2)}{4m_p^2+Q^2}⟶Z^2{F}_{\mathrm{em}}^2(Q^2)\text{leavevmode},$$

$$\displaystyle F_{\rm em}(Q^{2})={3\over(QR_{A})^{3}}\Big{[}\sin(QR_{A})-QR_{A}\cos(QR_{A})\Big{]}{1\over 1+a^{2}Q^{2}}\leavevmode\nobreak\ ,$$

$${\gamma }_{el}^{(p,Pb)}(x)=\int d{Q}^2{\gamma }_{el}^{(p,Pb)}(x,Q^2).$$

$$\frac{d{\gamma }_{inel}^p(x,{\mu }^2)}{d\log {\mu }^2}=$$

$$\sigma=S^{2}\int dx_{p}dx_{\rm Pb}\Big{[}\gamma^{p}_{el}(x_{p})+\gamma^{p}_{inel}(x_{p},\mu^{2})\Big{]}\gamma^{\rm Pb}_{el}(x_{\rm Pb})\sigma_{\gamma\gamma\rightarrow\ell^{+}\ell^{-}}(x_{p},x_{\rm Pb})\,,$$

$${\gamma }_{inel}^p(x,Q^2)=\frac{1}{x}\frac{1}{\pi Q^2}{\int }_{M_{\mathrm{thr}}^2}d{M}_X^2{\mathcal{F}}_{{\gamma }^{\ast }\leftarrow p}^{\mathrm{in}}(x,{\vec{q}}_T^2,M_X^2),$$

$${\mathcal{F}}_{{\gamma }^{\ast }\leftarrow p}^{\mathrm{in}}(x,{\vec{q}}_T^2,M_X)$$

$$Q^2=\frac{{\vec{q}}_T^2+x(M_X^2-m_p^2)+x^2{m}_p^2}{(1-x)}.$$

$$x_{\mathrm{Bj}}=\frac{Q^2}{Q^2+M_X^2-m_p^2}.$$

$$\displaystyle F_{L}(x_{\rm Bj},Q^{2})=\Big{(}1+{4x_{\rm Bj}^{2}m_{p}^{2}\over Q^{2}}\Big{)}F_{2}(x_{\rm Bj},Q^{2})-2x_{\rm Bj}F_{1}(x_{\rm Bj},Q^{2})$$

$$\sigma=S^{2}\int dx_{p}dx_{\rm Pb}d\vec{q}_{T}\Big{[}\gamma^{p}_{el}(x_{p},Q^{2})+\gamma^{p}_{inel}(x_{p},Q^{2})\Big{]}\gamma^{\rm Pb}_{el}(x_{\rm Pb})\sigma_{\gamma^{*}\gamma\rightarrow\ell^{+}\ell^{-}}(x_{p},x_{\rm Pb},\vec{q_{T}})\,,$$

$${\gamma }_{el}^p(x)=\frac{{\alpha }_{\mathrm{em}}}{\pi }\left(\frac{1-x+0.5x^2}{x}\right)\left(\frac{F+3}{F-1}\log F-\frac{17}{6}-\frac{4}{3F}+\frac{1}{6F^2}\right)\text{leavevmode},$$

$$G_E(Q^2)$$

$$G_M(Q^2)$$

Full text

††preprint: DESY 19-010

I Introduction

Precise calculations of various electroweak reactions in $pp$ collisions at the LHC need to account for, on top of the higher-order corrections, the effects of photon-induced processes. The relevant examples are the production of lepton pairs Aad:2014qja ; Aad:2016zzw ; Accomando:2016tah ; Luszczak:2015aoa ; Harland-Lang:2016apc or pairs of electroweak bosons Luszczak:2014mta ; Denner:2015fca ; Dyndal:2015hrp ; Ababekri:2016kkj ; Biedermann:2016guo ; Biedermann:2016yvs ; Yong:2016njr ; Luszczak:2018ntp .

Recently, a precise photon distribution inside the proton has been evaluated in Ref. Manohar:2016nzj . 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) Schmidt:2015zda or from inclusive $pp\rightarrow\ell^{+}\ell^{-}+X$ Ball:2013hta ; Aad:2016zzw ; Giuli:2017oii is limited due to large QCD background. On the contrary, the elastic part of the photon PDF is verified via exclusive $\gamma\gamma\rightarrow\ell^{+}\ell^{-}$ process, measured in $pp$ collisions by ATLAS Aad:2015bwa ; Aaboud:2017oiq , CMS Chatrchyan:2011ci ; Chatrchyan:2012tv and recently by CMS+TOTEM Cms:2018het 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 $p+\textrm{Pb}$ 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.

II Formalism

II.1 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

[TABLE]

where $x$ is the momentum fraction of the proton taken by the photon, $Q^{2}$ is the photon virtuality, $\alpha_{\rm{em}}$ is the electromagnetic structure constant and $m_{p}$ is the proton mass.

To express the elastic photon flux for the nucleus ($\gamma^{\rm Pb}_{el}$), we follow Ref. Budnev:1974de and replace

[TABLE]

where $F_{\rm em}(Q^{2})$ 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 Klein:2016yzr :

[TABLE]

where $R_{A}=1.1A^{1/3}$ fm, $a=0.7$ fm and $Q=\sqrt{Q^{2}}$.

The elastic photon PDFs of the proton and lead nucleus can be integrated over $Q^{2}$ as

[TABLE]

This is useful for the collinear-factorization approach since the $Q^{2}$ dependence factorizes in this case.

II.2 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 $\mu^{2}$, the photon parton distribution $\gamma^{p}_{inel}(x,\mu^{2})$ of photons carrying a fraction $x$ of the proton’s momentum, obeys the DGLAP equation:

[TABLE]

where $q(x,\mu^{2})$ is the quark PDF, $P_{\gamma\leftarrow q}$ is the $q\rightarrow\gamma$ splitting function, and $P_{\gamma\leftarrow\gamma}$ corresponds to the virtual self-energy correction to the photon propagator. This is the basis for colinear photon-PDFs in the initial Gluck:2002fi ; Martin:2004dh and more recent Ball:2013hta ; Martin:2014nqa ; Schmidt:2014aba ; Harland-Lang:2016kog ; Giuli:2017oii ; Manohar:2016nzj ; Bertone:2017bme analyses.

The computation of the photon-induced dilepton production cross section requires definition of the scale ($\mu^{2}$) at which the photon PDFs are convoluted. The usual choice for $\mu$ 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 $t$- and $u$-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 $\gamma\gamma\rightarrow\ell^{+}\ell^{-}$ process. This is achieved by taking the virtuality of the massive $t$- or $u$-channel propagator (Fig. 1b or c). Hence, $\mu^{2}=-(p^{\gamma^{Pb}}-p^{\ell^{-}})^{2}$ for the $t$-channel diagram and $\mu^{2}=-(p^{\gamma^{Pb}}-p^{\ell^{+}})^{2}$ for the $u$-channel exchange, where $p^{\gamma^{Pb}}$ is the four momentum of the photon emitted by lead and $p^{\ell^{\pm}}$ 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 $p+\textrm{Pb}\rightarrow\textrm{Pb}+\ell^{+}\ell^{-}+X$ production cross section can be written as

[TABLE]

where $\sigma_{\gamma\gamma\rightarrow\ell^{+}\ell^{-}}$ is the elementary cross section for the $\gamma\gamma\rightarrow\ell^{+}\ell^{-}$ 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.

II.3 $k_{T}$-factorization approach

At lowest order, the calculations with collinear photons produce leptons that are back-to-back in transverse kinematics. The transverse momentum appears at higher orders, however to describe full transverse momentum spectrum one needs to match the calculations to resummation or dedicated parton shower algorithms. This approach is not considered in this paper.

In the $k_{T}$ factorization approach (also named as high-energy factorization), one can parametrize the $\gamma^{*}p\rightarrow X$ 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 daSilveira:2014jla ; Luszczak:2015aoa :

[TABLE]

and we use the functions ${\cal{F}}^{\mathrm{in}}_{\gamma^{*}\leftarrow p}$ from Budnev:1974de ; Luszczak:2018ntp :

[TABLE]

The virtuality $Q^{2}$ of the photon depends on the photon transverse momentum ($\vec{q}_{T}^{2}$) and the proton remnant mass ($M_{X}$):

[TABLE]

Moreover, the proton structure functions $F_{1}(x_{\rm Bj},Q^{2})$ and $F_{2}(x_{\rm Bj},Q^{2})$ require the argument

[TABLE]

Note that in Eq. 8 instead of using $F_{2}(x_{\rm Bj},Q^{2}),F_{1}(x_{\rm Bj},Q^{2})$, we in practice use the pair $F_{2}(x_{\rm Bj},Q^{2}),F_{L}(x_{\rm Bj},Q^{2})$, where

[TABLE]

is the longitudinal structure function of the proton.

These unintegrated photon fluxes enter the $p+\textrm{Pb}\rightarrow\textrm{Pb}+\ell^{+}\ell^{-}+X$ production cross section as

[TABLE]

where $\sigma_{\gamma^{*}\gamma\rightarrow\ell^{+}\ell^{-}}$ is the off-shell elementary cross-section (for details see Refs. daSilveira:2014jla ; Catani:1990eg ) and for $x_{p}\ll 1$ we have $Q^{2}\approx\vec{q}_{T}^{2}$ (see Eq. 9). One should note that while the fluxes do not depend on the direction of $\vec{q}_{T}$, averaging over directions of $\vec{q}_{T}$ in the off-shell cross section replaces the average over photon polarizations in the collinear case.

III Example experimental configuration and possible background sources

We assume collision setup from recent $p+\textrm{Pb}$ run at the LHC, carried out at the centre-of-mass energy per nucleon pair $\sqrt{s_{NN}}=8.16$ 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 Aad:2008zzm and CMS Chatrchyan:2008aa detectors in the following. We consider only the dimuon channel, however the integrated results for $ee$ and $\mu\mu$ 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 ($\eta^{\ell}$) that satisfies $|\eta^{\ell}|<2.4$. Our calculations are carried out for a minimum dilepton invariant mass of $m_{\ell^{+}\ell^{-}}=10$ GeV. Such a choice is due to removal of possible contamination from $\Upsilon(\rightarrow\ell^{+}\ell^{-})$ photoproduction. A summary of all selection requirements is presented in Table 1.

Possible background for this process can arise from inclusive lepton-pair production, e.g. from Drell–Yan process Drell:1970wh ; Aad:2015gta ; Khachatryan:2015pzs ; Alice:2016wka . This processes would lead to disintegration of the incoming ion, and zero-degree calorimeters (ZDC) Dellacasa:1999ke ; ATLAS:2007aa 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. $\gamma p\rightarrow Z/\gamma^{*}+X$ 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, $p+\textrm{Pb}\rightarrow p+\textrm{Pb}+\ell^{+}\ell^{-}$ for which the following parameterization is used Budnev:1974de :

[TABLE]

where $F=1+\frac{Q_{0}^{2}(1-x)}{x^{2}m_{p}^{2}}$ and $Q_{0}^{2}=0.71$ GeV2. 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-sections is found: $\sigma_{fid}^{\textrm{el}}=17.5$ nb, whereas $\sigma_{fid}^{\textrm{STARlight}}=17.0$ nb. Both calculations are also corrected by a factor $S^{2}=0.96$ which is calculated using STARlight, where the hard-sphere proton–nucleus requirement Klein:2016yzr is used.

Next, for the inelastic case ($\gamma p\rightarrow\ell^{+}\ell^{-}+X$), several recent parameterizations of the photon parton distributions are studied: CT14qed Schmidt:2015zda , HKR16qed Harland-Lang:2016kog , LUXqed17 Manohar:2017eqh and NNPDF3.1luxQED Bertone:2017bme . 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 $p+\textrm{Pb}\rightarrow\textrm{Pb}+\ell^{+}\ell^{-}+X$ production at $\sqrt{s_{NN}}=8.16$ TeV for different collinear photon PDF sets are summarized in Tab. 2. 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 single-lepton 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 20% with each other. The differences are mainly due to overall PDF normalization, as no variation in the shape of various kinematic distributions is observed.

To check the sensitivity to the nuclear form factor modelling (Eq. 3), different values of $R_{A}$ ($R_{A}=7.1$ fm) and $a$ ($a=0.55$ fm) parameters are used, in a similar way as in Ref. Azevedo:2019fyz . These variations change the fiducial cross-sections by 4% and 3% respectively.

V Results using $k_{T}$-factorization approach

Several different parametrizations of proton strucure functions are used. Those are labeled as:

• •

ALLM Abramowicz:1991xz ; Abramowicz:1997ms : This parametrization gives a good fit to $F_{2}$ in most of the measured regions.

• •

SY Suri:1971yx : This parameterization of Suri and Yennie from the early 1970’s does not include DGLAP evolution. It is still used as one of the defaults in the LPAIR event generator Vermaseren:1982cz .

• •

SU Szczurek:1999rd : A parametrization which concentrates to give a good description at small and intermediate $Q^{2}$ for $x>0.01$. At large $Q^{2}$, it is complemented by the NNLO calculation of $F_{2}$ and $F_{L}$ from NNLO MSTW 2008 PDF analysis Martin:2009iq .

• •

LUX-like: a recently constructed parametrization, described in details in Ref. Luszczak:2018ntp . This setup closely follows the LUXqed work from Ref.Manohar:2017eqh .

To model $\gamma^{p}_{el}(x,Q^{2})$ we use Eq. 1 with so-called dipole parametrization of the proton form factors:

[TABLE]

where $\mu_{p}$ is the proton magnetic moment.

Table 3 shows the comparison of integrated fiducial cross sections for inelastic $p+\textrm{Pb}\rightarrow\textrm{Pb}+\ell^{+}\ell^{-}+X$ production at $\sqrt{s_{NN}}=8.16$ 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 2). 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. Luszczak:2018ntp .

Figure 3 presents differential cross sections for several lepton kinematic distributions: invariant mass of lepton pair, leading lepton transverse momentum, lepton pseudorapidity difference and leading lepton pseudorapidity. The shapes of the distributions obtained with various proton structure functions are very similar. For completeness, differential cross sections as a function of lepton pair transverse momentum and azimuthal angle difference between the pair are shown in Fig. 4. Quite large (small) transverse momenta (angle differences) are possible, in contrast to leading-order calculations with collinear photons where the corresponding distributions are just Dirac delta functions. The $k_{T}$-factorization approach should be considered more appropriate here. It is also visible that the SY parametrization gives lower predictions at larger pair-$p_{T}$, comparing to the other parametrizations used. This is because SY parametrization does not include explicit DGLAP evolution terms, which are relevant for large photon virtualities.

Based on Fig. 4, it is also possible to separate experimentally the elastic part ($p+\textrm{Pb}\rightarrow p+\textrm{Pb}+\ell^{+}\ell^{-}$), 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.

VI Discussion

Figure 6 compares several differential distributions computed using the two approaches. For the collinear approach pure inelastic contribution is estimated by subtracting elastic part computed following Eq. 13. For the invariant mass distribution and lepton pseudorapidity the shapes are similar and the main difference between the two predictions is observed in the normalization. For the distribution of the lepton pair rapidity the two predictions agree at larger rapidities while disagreement concentrates in the central region. The biggest difference is observed for the transverse momentum distribution of the lepton where at low $p_{T}$ collinear approximation exceeds the estimate from $k_{T}$-factorization approach while at high $p_{T}$ the ordering is reversed. This suggests that at low $p_{T}$ (close to the boundary of the fiducial region) the difference is due to the smearing of dilepton transverse momentum introduced by the $k_{T}$-factorization approach.

We also take the opportunity to calculate expected number of events for realistic assumption on total integrated luminosity. Based on the previous $p+\textrm{Pb}$ runs at the LHC, we assume $\int Ldt=200\leavevmode\nobreak\ \textrm{nb}^{-1}$. 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 4 shows the expected number of events for $p+\textrm{Pb}\rightarrow\textrm{Pb}+\ell^{+}\ell^{-}+X$ production at $\sqrt{s_{NN}}=8.16$ 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.

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 $p+\textrm{Pb}\rightarrow\textrm{Pb}+\ell^{+}\ell^{-}+X$, 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 one can expect O(3000) inelastic events with the existing datasets recorded by ATLAS/CMS at $\sqrt{s_{NN}}=8.16$ TeV for each lepton flavour.

Acknowledgements

We would like to thank James Ferrando for useful suggestions. The work of M.L. was partially supported by the Center for Innovation and Transfer of Natural Sciences and Engineering Knowledge in Rzeszów. The work of R.S. was partially supported by the BMBF-JINR cooperation. M.L. and R.S. acknowledge the hospitality of DESY where a portion of this work was performed.