1 Introduction

Lepton pairs produced in ultraperipheral collisions (UPC) of protons at the Large Hadron Collider (LHC) are accompanied by forward scattering of the protons. Previous measurements of this process were performed by the ATLAS collaboration without proton tagging [1]. We have calculated the cross section for this reaction with the help of the equivalent photon approximation (EPA) in [2], taking into account the so-called survival factor which addresses the diminishing of the cross section because of proton disintegration due to strong interactions. The result obtained agrees with the measurement within the experimental accuracy. Predictions for the cross section were also made with the help of Monte-Carlo calculations [3,4,5,6].

The CMS and TOTEM collaborations have reported statistically significant proton-tagged dilepton production [7], but the cross sections have not been measured. The ATLAS collaboration has managed to measure the cross sections [8]. In the events selected for the analysis, one of the scattered protons is detected by the ATLAS Forward Proton Spectrometer. The other proton could remain intact, in which case it could or could not be detected by the opposite forward detector, or it could disintegrate. The transversal momentum of the lepton pair \(p_T^{\ell \ell }\) was required to be less than 5 GeV. This momentum equals to the sum of transversal momenta of the photons emitted by the protons. The transversal momentum of the photon that was emitted by the proton that survived the collision cannot be much higher than \({\hat{q}} = 0.2\) GeV [9]. Therefore, the transversal momentum of the second photon has to be less than 5 GeV.

In what follows we derive analytical formulas which describe the fiducial cross sections measured in [8]. These formulas allow for simple numerical integration instead of the usual Monte Carlo approach and thus can provide intuitive insights into the process targeted by the experiment. Our numerical results are in the ballpark of experimental data, while their substantial deviation would signal New Physics. This paper is a continuation of the work presented in our previous paper [10] where we have derived formulas for the cross section without experimental cuts on the leptons phase space and with no requirement for any of the protons to hit the forward detectors.

In Sect. 2 we calculate the fiducial cross section for the case of elastic scattering of the second proton. The contribution to the fiducial cross section of the processes when the second proton disintegrates is calculated in Sect. 3. We conclude in Sect. 4.

2 Fiducial cross section for the reaction \(pp \rightarrow p + \ell ^+ \ell ^- + p\)

The spectrum of photons radiated by proton is [9]

$$\begin{aligned} n_p(\omega ) = \frac{2 \alpha }{\pi \omega } \int \limits _0^\infty \frac{D(Q^2)}{Q^4} q_\perp ^3 \, \textrm{d} q_\perp , \end{aligned}$$
(1)

where \(\alpha \) is the fine structure constant, \(Q^2 \equiv -q^2 = q_\perp ^2 + \omega ^2 / \gamma ^2\), q is the photon 4-momentum, \(q_\perp \) is the photon transverse momentum, \(\omega \) is the photon energy, \(\gamma = E_p / m_p \approx 6.93 \cdot 10^3\) is the Lorentz factor of the proton, \(E_p = 6.5\) TeV is the proton energy, \(m_p\) is the proton mass,Footnote 1

$$\begin{aligned} D(Q^2) = \frac{G_E^2(Q^2) + \frac{Q^2}{4 m_p^2} G_M^2(Q^2)}{1 + \frac{Q^2}{4 m_p^2}}, \end{aligned}$$
(2)

\(G_E(Q^2)\) and \(G_M(Q^2)\) are the Sachs electric and magnetic form factors. For the Sachs form factors we use the dipole approximation:

$$\begin{aligned}{} & {} G_E(Q^2) = \frac{1}{(1 + Q^2 / \Lambda ^2)^2}, ~ G_M(Q^2) = \frac{\mu _p}{(1 + Q^2 / \Lambda ^2)^2}, ~ \nonumber \\{} & {} \quad \Lambda ^2 = \frac{12}{r_p^2} = 0.66~\text {GeV}^2, \end{aligned}$$
(3)

where \(\mu _p = 2.79\) is the proton magnetic moment, and \(r_p = 0.84\) fm is the proton charge radius [11]. Substituting (2), (3) into (1), we obtain the following analytical expression:

(4)

where

$$\begin{aligned} u = \left( \frac{\omega }{\Lambda \gamma } \right) ^2, \ v = \left( \frac{2 m_p}{\Lambda } \right) ^2. \end{aligned}$$
(5)

The cross section for the lepton pair production in the case when both of the protons survive is

$$\begin{aligned}{} & {} \sigma (pp \rightarrow p + \ell ^+ \ell ^- + p) \nonumber \\{} & {} \quad = \int \limits _0^\infty \int \limits _0^\infty \sigma (\gamma \gamma \rightarrow \ell ^+ \ell ^-) \, n_p(\omega _1) \, n_p(\omega _2) \, \textrm{d} \omega _1 \, \textrm{d} \omega _2, \end{aligned}$$
(6)

where \(\sigma (\gamma \gamma \rightarrow \ell ^+ \ell ^-)\) is the cross section for the production of a lepton pair in a collision of two real photons with energies \(\omega _1\) and \(\omega _2\). The plus signs in the reaction notation indicate large rapidity gaps between the produced particles.

In [8], as well as in all other measurements made at the LHC, the phase space is constrained by the following requirements: \(p_{i,T} > {\hat{p}}_T\), \(|\eta _i | < \hat{\eta }\), where \(p_{i,T}\) is the transversal momentum of lepton \(\ell _i\), and \(\eta _i\) is its pseudorapidity. In the case of muons \(\hat{p}_T = 15\) GeV and \(\hat{\eta }= 2.4\), while in the case of electrons \({\hat{p}}_T = 18\) GeV and \(\hat{\eta }= 2.47\). The transversal momentum of the lepton pair is equal to the sum of the transversal momenta of the photons and is limited by \(\sqrt{Q^2} \lesssim {\hat{q}} = 0.2\) GeV, so transversal momenta of the leptons are equal with good accuracy: \(p_T \equiv p_{1,T} \approx p_{2,T}\).

Neglecting the lepton mass, the differential fiducial cross section in this case is

$$\begin{aligned}{} & {} \frac{\textrm{d} \sigma _\text {fid.}(pp \rightarrow p + \ell ^+ \ell ^- + p)}{\textrm{d} W} \nonumber \\{} & {} \quad = \int \limits _{\max \left( {\hat{p}}_T, \frac{W}{2 \cosh \hat{\eta }} \right) } ^{W / 2} \textrm{d} p_T \, \frac{\textrm{d} \sigma (\gamma \gamma \rightarrow \ell ^+ \ell ^-)}{\textrm{d} p_T} \, \frac{\textrm{d} {\hat{L}}}{\textrm{d} W}, \end{aligned}$$
(7)

where \(W = \sqrt{4 \omega _1 \omega _2}\) is the invariant mass of the lepton pair,

$$\begin{aligned} \frac{\textrm{d} \sigma (\gamma \gamma \rightarrow \ell ^+ \ell ^-)}{\textrm{d} p_T} = \frac{8 \pi \alpha ^2}{W^2 p_T} \cdot \frac{1 - 2 p_T^2 / W^2}{\sqrt{1 - 4 p_T^2 / W^2}} \end{aligned}$$
(8)

[12], \(\textrm{d} {\hat{L}} / \textrm{d} W\) is the photon–photon luminosity taking into account the limits on the phase space,

$$\begin{aligned} \frac{\textrm{d} {\hat{L}}}{\textrm{d} W} = \frac{W}{2} \int \limits _{-{\hat{y}}}^{{\hat{y}}} n_p \left( \tfrac{W}{2} \, \textrm{e}^y \right) \, n_p \left( \tfrac{W}{2} \, \textrm{e}^{-y} \right) \, \textrm{d} y, \end{aligned}$$
(9)

\(y = \tfrac{1}{2} \ln \tfrac{\omega _1}{\omega _2}\) is the rapidity of the lepton pair, and the integration over \(\omega _1\) and \(\omega _2\) in (6) is changed to the integration over W and y: \(\textrm{d} \omega _1 \textrm{d} \omega _2 = \frac{W}{2} \textrm{d} W \textrm{d} y\). The pseudorapidities of charged leptons for given values of W and \(p_T\) are determined by the value of y. In this way the cut on \(\eta \) \((-\hat{\eta }< \eta < \hat{\eta })\) is transformed to the cut on y: \(-{\hat{y}}< y < {\hat{y}}\), where [9, (B.9)]Footnote 2

$$\begin{aligned} {\hat{y}} = \hat{\eta }+ \frac{1}{2} \ln \frac{1 - \sqrt{1 - 4 p_T^2 / W^2}}{1 + \sqrt{1 - 4 p_T^2 / W^2}}. \end{aligned}$$
(10)

Here \({\hat{y}}\) has to be greater than zero; this requirement leads to the inequality \(p_T > \frac{W}{2 \cosh \hat{\eta }}\) in the lower limit of the integration with respect to \(p_T\) in (7).

Another requirement imposed in [8] is that one of the protons hits the forward detector. To do that, the proton must lose a fraction of its energy : , where \(\xi _\text {min} = 0.035\), \(\xi _\text {max} = 0.08\). This translates to limits on the energy of the photon emitted by this proton:

$$\begin{aligned} 227~\text {GeV} \equiv \omega _\text {min}< \omega _1 < \omega _\text {max} \equiv 520~\text {GeV}. \end{aligned}$$
(11)

To take that into account, the integration limits in eq. (9) have to be narrowed:

$$\begin{aligned} \frac{\textrm{d} {\hat{L}}_\text {FD}}{\textrm{d} W} = \frac{W}{2} \int \limits _{\max (-{\hat{y}}, {\tilde{y}})}^{\min ({\hat{y}}, {\tilde{Y}})} n_p \left( \tfrac{W}{2} \, \textrm{e}^y \right) \, n_p \left( \tfrac{W}{2} \, \textrm{e}^{-y} \right) \, \textrm{d} y, \end{aligned}$$
(12)

where FD stands for the “forward detector”,

$$\begin{aligned} \begin{aligned} {\tilde{y}}&= \ln \max \left( \frac{2 \omega _{1, \text {min}}}{W}, \frac{W}{2 \omega _{2, \text {max}}} \right) , \\ {\tilde{Y}}&= \ln \min \left( \frac{2 \omega _{1, \text {max}}}{W}, \frac{W}{2 \omega _{2, \text {min}}} \right) , \end{aligned} \end{aligned}$$
(13)

\(\omega _{1, \text {min}}\), \(\omega _{1, \text {max}}\), \(\omega _{2, \text {min}}\), \(\omega _{2, \text {max}}\) are the limits on energy losses for each of the protons.Footnote 3 Calculation with

$$\begin{aligned} \begin{aligned} \omega _{1, \text {min}}&= \omega _\text {min},&\omega _{2, \text {min}}&= 0, \\ \omega _{1, \text {max}}&= \omega _\text {max},&\omega _{2, \text {max}}&= \infty \end{aligned} \end{aligned}$$
(14)

will result in the fiducial cross section with the first proton hitting the forward detector. Calculation with

$$\begin{aligned} \begin{aligned} \omega _{1, \text {min}}&= \omega _\text {min},&\omega _{2, \text {min}}&= \omega _\text {min}, \\ \omega _{1, \text {max}}&= \omega _\text {max},&\omega _{2, \text {max}}&= \omega _\text {max} \end{aligned} \end{aligned}$$
(15)

yields the fiducial cross section with both protons hitting the forward detectors. To calculate the cross section measured in [8], one has to multiply the former by 2 and subtract the latter to avoid double counting:

(16)

In both measurements in [8], the selected region of invariant masses of lepton pairs was \(W > 20\) GeV with the region \(70~\text {GeV}< W < 105~\text {GeV}\) excluded to suppress the background from Z decays. Collecting together all of the phase space constraints relevant to the exclusive process (see Ref. [94] in [8]), we get:

  • \(20~\text {GeV}< W < 70~\text {GeV}\) or \(W > 105\) GeV.

  • \(0.035< \xi < 0.08\) which is equivalent to \(227~\text {GeV}< \omega < 520~\text {GeV}\).

  • For muons:

    • \({\hat{p}}_T = 15\) GeV, \(\hat{\eta }= 2.4\).

    • \(\sigma _{\text {fid.,}[8]}(pp \rightarrow p + \mu ^+ \mu ^- + p) = 8.6\) fb.

  • For electrons:

    • \({\hat{p}}_T = 18\) GeV, \(\hat{\eta }= 2.47\).

    • \(\sigma _{\text {fid.,}[8]}(pp \rightarrow p + e^+ e^- + p) = 10.1\) fb.

3 Fiducial cross section for the reaction \(pp \rightarrow p + \ell ^+ \ell ^- + X\)

In this section we calculate the cross section for lepton pair production with one of the protons scattered elastically and detected by the forward detector, while the other proton disintegrates. Following the parton model, we consider this process as a two-photon lepton pair production in a collision of a proton and a quark, summed over all quarks:

$$\begin{aligned}{} & {} \sigma (pp \rightarrow p + \ell ^+ \ell ^- + X) \nonumber \\{} & {} \quad = \sum \limits _q \sigma (pq \rightarrow p + \ell ^+ \ell ^- + q). \end{aligned}$$
(17)

One of the Feynman diagrams of the \(p q \rightarrow p + \ell ^+ \ell ^- + q\) reaction is presented in Fig. 1.

Fig. 1
figure 1

Lepton pair production in semiexclusive reaction

Full size image

In the laboratory system we have the following expressions for the momenta of the colliding particles:

$$\begin{aligned} p_1 = (E, 0, 0, E), ~ p_2 = (xE, 0, 0, -xE), \end{aligned}$$
(18)

where x is the fraction of the momentum of the disintegrating proton carried by the quark. In the following we will integrate over x from a value much less than 1 to 1, but the numerically important values of x are of the order of 1/3, so we have neglected here the masses of the proton \(m_p\) and the quark \(m_q\): \(E / m_p = \gamma \approx 6.93 \cdot 10^3 \gg 1\), \(xE / m_q \gg 1\) even for the (sea) b quark.

The momentum of the photon emitted by the proton,

$$\begin{aligned} q_1 = \left( \omega _1,\overrightarrow{ \textbf{q}}_{1 \perp }, \frac{\omega _1 \gamma }{\sqrt{\gamma ^2 - 1}} \right) , \end{aligned}$$
(19)

where \(\overrightarrow{ \textbf{q}}_{1 \perp }\) is its transversal momentum. The 4-momentum squared,

$$\begin{aligned} Q_1^2 \equiv -q_1^2 \approx q_{1 \perp }^2 + (\omega _1 / \gamma )^2, \end{aligned}$$
(20)

is limited by the condition that the proton survives: \(Q_1^2 \lesssim \hat{q}^2 \approx (0.2~\text {GeV})^2\) [9], so the first photon is approximately real: \(Q_1^2 \ll W^2\). Therefore, the equivalent photon approximation can be used here.

For the second photon, we also apply the equivalent photon approximation, but with a correction.Footnote 4 Since the quark is bound within the proton, we use the constituent quark mass \(m_q = m_p/3 \approx 300~\text {MeV}\) to describe its properties.Footnote 5 The momentum of the second photon

$$\begin{aligned} q_2 \approx \left( \omega _2, \overrightarrow{\textbf{q}}_{2 \perp }, -\frac{\omega _2 \gamma _q}{\sqrt{\gamma _q^2 - 1}} \right) , \end{aligned}$$
(21)

and its square is

$$\begin{aligned} Q_2^2 \equiv -q_2^2 \approx q_{2 \perp }^2 + (\omega _2 / \gamma _q)^2, \end{aligned}$$
(22)

where \(\gamma _q = E_q / m_q = 3 x E_p / m_p = 3 x \gamma \). In the ATLAS measurement the following upper bound was effectively imposed on \(q_{2 \perp }\): \(|\overrightarrow{\textbf{q}}_{1 \perp } + \overrightarrow{\textbf{q}}_{2 \perp } | \sim q_{2 \perp } < {\hat{p}}_T^{\ell \ell } = 5\) GeV.

The square of the invariant mass of the lepton pair \(W^2 = (q_1 + q_2)^2 \approx 2 q_1 q_2 + q_2^2 \approx 4 \omega _1 \omega _2 - Q_2^2\). For \(W \sim 100\) GeV, \(Q_2^2 \ll W^2\), and \(\omega _2 \approx W^2 / 4 \omega _1\). Here \(\omega _1\) is bounded by the requirement that the proton hits the forward detector (11). Therefore, \(\omega _2\) is of the order of a few GeV — the lepton pair will have large rapidity towards the detector that registers the proton. We assume \(\omega _1 \ll E\) and \(\omega _2 \ll xE\) in the following.

To derive the cross section for the \(pq \rightarrow p + \ell ^+ \ell ^- + q\) reaction, we follow closely Ref. [14] and begin with its Eq. (5.1) (multiplied by the parton distribution function \(f_q(x, Q_2^2)\)):

$$\begin{aligned}{} & {} \textrm{d} \sigma (pq \rightarrow p + \ell ^+ \ell ^- + q) \nonumber \\{} & {} \quad = \frac{Q_q^2 (4 \pi \alpha )^2}{q_1^2 q_2^2} \rho _1^{\mu \nu } \rho _2^{\alpha \beta } M_{\mu \alpha } M^*_{\nu \beta } \nonumber \\{} & {} \quad \times \frac{(2 \pi )^4 \delta ^{(4)}(q_1 + q_2 - k_1 - k_2) \textrm{d} \Gamma }{4 \sqrt{(p_1 p_2)^2 - p_1^2 p_2^2}} \nonumber \\{} & {} \quad \times \frac{\textrm{d}^3 p'_1}{(2 \pi )^3 2 E'_1} \frac{\textrm{d}^3 p'_2}{(2 \pi )^3 2 E'_2} \times f_q(x, Q_2^2) \textrm{d} x, \end{aligned}$$
(23)

where \(Q_q\) is the electric charge of quark q, \(\rho _1^{\mu \nu }\) and \(\rho _2^{\alpha \beta }\) are the photon density matrices for the photons emitted by the proton or the quark respectively, \(M_{\mu \alpha }\) is the amplitude for the \(\gamma ^* \gamma ^* \rightarrow \ell ^+ \ell ^-\) process, \(\textrm{d} \Gamma \) is the phase space element of the lepton pair, \(E'_1\) and \(E'_2\) are the energies of the proton and the quark after the collision.

Since \(p'_i = p_i - q_i\), we rewrite the phase space element of the proton and the quark as follows:

$$\begin{aligned} \frac{\textrm{d}^3 p'_1}{E'_1} \cdot \frac{\textrm{d}^3 p'_2}{E'_2} \approx \frac{\textrm{d}^2 q_{1 \perp } \textrm{d} \omega _1}{E} \cdot \frac{\textrm{d}^2 q_{2 \perp } \textrm{d} \omega _2}{xE}. \end{aligned}$$
(24)

The photon density matrix for the photon emitted by the proton is

$$\begin{aligned}{} & {} \rho _1^{\mu \nu } = -\frac{1}{2 q_1^2} {{\,\textrm{Tr}\,}}\left\{ ({\hat{p}}'_1 + m_p) \left( F_1(Q_1^2) \gamma ^\mu + F_2(Q_1^2) \frac{\sigma ^{\mu \alpha } q_1^\alpha }{2 m_p} \right) \nonumber \right. \\{} & {} \quad \times \left. ({\hat{p}}_1 + m_p) \left( F_1(Q_1^2) \gamma ^\nu - F_2(Q_1^2) \frac{\sigma ^{\nu \alpha } q_1^\alpha }{2 m_p} \right) \right\} , \end{aligned}$$
(25)

where \(F_1(Q_1^2)\) and \(F_2(Q_1^2)\) are the Dirac and Pauli form factors. The Sachs form factors (3) are their linear combinations:

$$\begin{aligned} \begin{aligned} G_E(Q_1^2)&= F_1(Q_1^2) - \frac{Q_1^2}{4 m_p^2} F_2(Q_1^2), \\ G_M(Q_1^2)&= F_1(Q_1^2) + F_2(Q_1^2). \end{aligned} \end{aligned}$$
(26)

Equation (25) simplifies to

$$\begin{aligned} \rho _1^{\mu \nu }= & {} -\left( g^{\mu \nu } - \frac{q_1^\mu q_1^\nu }{q_1^2} \right) G_M^2(Q_1^2) \nonumber \\{} & {} - \frac{(2 p_1 - q_1)^\mu (2 p_1 - q_1)^\nu }{q_1^2} D(Q_1^2), \end{aligned}$$
(27)

where \(D(Q_1^2)\) is defined in Eq. (2).

The photon density matrix for the photon emitted by the quark is

$$\begin{aligned} \rho _2^{\mu \nu }= & {} -\frac{1}{2 q_2^2} {{\,\textrm{Tr}\,}}\left\{ {\hat{p}}'_2 \gamma ^\mu {\hat{p}}_2 \gamma ^\nu \right\} = -\left( g^{\mu \nu } -\frac{q_2^\mu q_2^\nu }{q_2^2} \right) \nonumber \\{} & {} \quad - \frac{(2 p_2 - q_2)^\mu (2 p_2 - q_2)^\nu }{q_2^2}. \end{aligned}$$
(28)

It is convenient to consider the lepton pair production in the basis of virtual photon helicity states. In the center of mass system of the colliding photons, let \(q_1 = (\tilde{\omega }_1, 0, 0, {\tilde{q}})\), \(q_2 = (\tilde{\omega }_2, 0, 0, -{\tilde{q}})\). The standard set of orthonormal 4-vectors orthogonal to \(q_1\) and \(q_2\) is

$$\begin{aligned} \begin{aligned}&e_1^+ = \frac{1}{\sqrt{2}} (0, -1, -i, 0), e_1^- = \frac{1}{\sqrt{2}} (0, 1, -i, 0), \\&\quad e_1^0 =\frac{i}{\sqrt{-q_1^2}} ({\tilde{q}}, 0, 0, \tilde{\omega }_1),\\&e_2^+ = \frac{1}{\sqrt{2}} (0, 1, -i, 0), e_2^- = \frac{1}{\sqrt{2}} (0, -1, -i, 0),\\&\quad e_2^0 = \frac{i}{\sqrt{-q_2^2}} (-{\tilde{q}}, 0, 0, \tilde{\omega }_2). \end{aligned} \end{aligned}$$
(29)

Due to the conservation of vector current, the covariant density matrices \(\rho _i^{\mu \nu }\) satisfy \(q_1^\mu \rho _1^{\mu \nu } = q_2^\mu \rho _2^{\mu \nu } = 0\). Thus, we can write

$$\begin{aligned} \rho _i^{\mu \nu }&= \sum \limits _{a,b} (e_i^{a \mu })^* e_i^{b \nu } \rho _i^{ab},\end{aligned}$$
(30)
$$\begin{aligned} \rho _i^{ab}&= (-1)^{a+b} e_i^{a \mu } (e_i^{b \nu })^* \rho _i^{\mu \nu }, \end{aligned}$$
(31)

where \(a, b \in \{ \pm 1, 0 \}\), and \(\rho _i^{ab}\) are the density matrices in the helicity representation. The amplitudes of the lepton pair production in the helicity basis \(M_{ab}\) comply to the equation

$$\begin{aligned} \rho _1^{\mu \nu } \rho _2^{\alpha \beta } M_{\mu \alpha } M^*_{\nu \beta } = (-1)^{a + b + c + d} \rho _1^{ab} \rho _2^{cd} M_{ac} M^*_{bd}. \end{aligned}$$
(32)

In this expression, non-diagonal terms (those with \(a \ne b\) or \(c \ne d\)) originate from the interference and cancel out when integrated over azimuthal angles of the proton and the quark in the final state [14, 15]. Contributions to the cross section for lepton pair production by longitudinally polarized photons are proportional to \(Q_1^2 / W^2 \le {\hat{q}}^2 / W^2 \ll 1\) and \(Q_2^2 / W^2 \le (p_T^{\ell \ell })^2 / W^2 \ll 1\) and are neglected in the following. Thus, we can rewrite (23) in the helicity representation:

$$\begin{aligned}{} & {} \textrm{d} \sigma (p q \rightarrow p + \ell ^+ \ell ^- + q) \nonumber \\{} & {} \quad = Q_q^2 (4 \pi \alpha )^2 \left( \rho _1^{++} \rho _2^{++} |M_{++} |^2 + \rho _1^{++} \rho _2^{--} |M_{+-} |^2 \right. \nonumber \\{} & {} \qquad + \rho _1^{--} \rho _2^{++} |M_{-+} |^2 \left. + {} \rho _1^{--} \rho _2^{--} |M_{--} |^2 \right) \nonumber \\{} & {} \qquad \times \frac{(2 \pi )^4 \delta ^{(4)}(q_1 + q_2 - k_1 - k_2) \textrm{d} \Gamma }{4 p_1 p_2}\nonumber \\{} & {} \qquad \times \frac{\textrm{d}^2 q_{1 \perp } \textrm{d} \omega _1}{(2 \pi )^3 q_1^2 E} \cdot \frac{\textrm{d}^2 q_{2 \perp } \textrm{d} \omega _2}{(2 \pi )^3 q_2^2 xE} \cdot f_q(x, Q_2^2) \textrm{d} x. \end{aligned}$$
(33)

Substitution of (27), (28) to (31) yields the following expressions

$$\begin{aligned} \begin{aligned} \rho _1^{++} = \rho _1^{--}&= G_M^2(Q_1^2) + \frac{2 p_{1 \perp }^2}{q_1^2} D(Q_1^2), \\ \rho _2^{++} = \rho _2^{--}&= 1 + \frac{2 p_{2 \perp }^2}{q_2^2}, \end{aligned} \end{aligned}$$
(34)

where \(p_{i \perp }\) is the component of the momentum \(p_i\) orthogonal to \(q_i\) in the c.m.s. of the colliding photons. To calculate it, we follow [14] and introduce the symmetrical tensor

$$\begin{aligned}{} & {} R^{\mu \nu }(q_1, q_2) = -g^{\mu \nu } \nonumber \\{} & {} \quad + \frac{ q_1 q_2 \cdot (q_1^\mu q_2^\nu + q_1^\nu q_2^\mu ) - q_1^2 q_2^\mu q_2^\nu - q_2^2 q_1^\mu q_1^\nu }{ (q_1 q_2)^2 - q_1^2 q_2^2 }. \end{aligned}$$
(35)

This tensor is the metric tensor of the subspace orthogonal to \(q_1\) and \(q_2\), and has the following properties:

$$\begin{aligned} q_i^\mu R^{\mu \nu } = 0, ~ R^{\alpha \beta } R^{\beta \gamma } = -R^{\alpha \gamma }, ~ R^{\alpha \beta } R^{\alpha \beta } = 2. \end{aligned}$$
(36)

Then \(p_{i, \perp }^\mu = -R^{\mu \nu } p_i^{\nu }\), andFootnote 6

$$\begin{aligned} \begin{aligned}&\rho _1^{++} = \rho _1^{--} = G_M^2(Q_1^2) \\&\quad + D(Q_1^2) \left[ \frac{2 m_p^2}{q_1^2} + \frac{1}{2} \left( \frac{(2 p_1 q_2 - q_1 q_2)^2}{(q_1 q_2)^2 - q_1^2 q_2^2} - 1 \right) \right] ,\\&\quad \rho _2^{++} = \rho _2^{--} = \frac{1}{2} + \frac{2 m_q^2}{q_2^2} + \frac{1}{2} \cdot \frac{(2 p_2 q_1 - q_1 q_2)^2}{(q_1 q_2)^2 - q_1^2 q_2^2}. \end{aligned} \nonumber \\ \end{aligned}$$
(37)

With the help of Eqs. (18)–(22), under the assumptions \(q_1^2 q_2^2 \ll (q_1 q_2)^2\), \(\omega _1 \ll E\), \(\omega _2 \ll x E\), these expressions simplify to

$$\begin{aligned} \begin{aligned} \rho _1^{++} = \rho _1^{--}&\approx G_M^2(Q_1^2) + 2 D(Q_1^2) \left[ \frac{m_p^2}{q_1^2} + \left( \frac{E}{\omega _1} \right) ^2 \right] \\&\approx D(Q_1^2) \cdot \frac{2 E^2 q_{1 \perp }^2}{\omega _1^2 Q_1^2}, \\ \rho _2^{++} = \rho _2^{--}&\approx 1 + 2 \left[ \frac{m_q^2}{q_2^2} + \left( \frac{x E}{\omega _2} \right) ^2 \right] \approx \frac{2 x^2 E^2 q_{2 \perp }^2}{\omega _2^2 Q_2^2}. \end{aligned} \end{aligned}$$
(38)

Substitution of Eqs. (38) to (33) yields

$$\begin{aligned}{} & {} \textrm{d} \sigma (pq \rightarrow p + \ell ^+ \ell ^- + q) \approx \left( \frac{2 Q_q \alpha }{\pi } \right) ^2 \nonumber \\{} & {} \quad \frac{q_1 q_2}{p_1 p_2} x E^2 \sigma (\gamma \gamma \rightarrow \ell ^+ \ell ^-) D(Q_1^2) \frac{q_{1 \perp }^3 \textrm{d} q_{1 \perp }}{Q_1^4} \frac{\textrm{d} \omega _1}{\omega _1^2}\nonumber \\{} & {} \quad \times \frac{q_{2 \perp }^3 \textrm{d} q_{2 \perp }}{Q_2^4} \frac{\textrm{d} \omega _2}{\omega _2^2} \cdot f_q(x, Q_2^2) \textrm{d} x,\nonumber \\ \end{aligned}$$
(39)

where

$$\begin{aligned}{} & {} \sigma (\gamma \gamma \rightarrow \ell ^+ \ell ^-) = \int \frac{1}{4} [ |M_{++} |^2 + |M_{+-} |^2 \nonumber \\{} & {} \quad + |M_{-+} |^2 + |M_{--} |^2 ] \frac{(2 \pi )^4 \delta ^{(4)}(q_1 + q_2 - k_1 - k_2) \textrm{d} \Gamma }{4 q_1 q_2},\nonumber \\ \end{aligned}$$
(40)

is the cross for lepton pair production in a collision of two real unpolarized photons. Integration over \(q_{1 \perp }\) yields the equivalent photon spectrum of proton (1). To derive the fiducial cross section, we change the integration variables: \(\textrm{d} \omega _1 \textrm{d} \omega _2 = \frac{W}{2} \textrm{d} W \textrm{d} y\). Then

$$\begin{aligned}{} & {} \textrm{d} \sigma (pq \rightarrow p + \ell ^+ \ell ^- + q) \approx \frac{2 Q_q^2 \alpha }{\pi } \; n_p \left( \frac{W}{2} \textrm{e}^y \right) \nonumber \\{} & {} \quad \,\sigma (\gamma \gamma \rightarrow \ell ^+ \ell ^-) \; \textrm{e}^y \; \frac{q_{2 \perp }^3 \textrm{d} q_{2 \perp }}{Q_2^4} \textrm{d} W \textrm{d} y \; f_q(x, Q_2^2) \textrm{d} x. \end{aligned}$$
(41)

Introducing the function

$$\begin{aligned} n_q(\omega ) = \frac{2 Q_q^2 \alpha }{\pi \omega } \int \limits _{\omega / E}^1 \textrm{d} x \int \limits _0^{p_T^{\ell \ell }} \textrm{d} q_{2 \perp } \, \frac{q_{2 \perp }^3}{Q_2^4} f_q(x, Q_2^2), \end{aligned}$$
(42)

which can be loosely interpreted as the equivalent photon spectrum of quark q (cf. (1)), we derive the equation for the fiducial cross section similar to (7), (12):

$$\begin{aligned}{} & {} \frac{\textrm{d} \sigma _\text {fid.}(pp \rightarrow p + \ell ^+ \ell ^- + X)}{\textrm{d} W} \nonumber \\{} & {} \quad = \sum \limits _q \int \limits _{\max \left( {\hat{p}}_T, \frac{W}{2 \cosh \hat{\eta }} \right) }^ {W / 2} \textrm{d} p_T \frac{\textrm{d} \sigma (\gamma \gamma \rightarrow \ell ^+ \ell ^-)}{\textrm{d} p_T}\nonumber \\{} & {} \qquad \times \frac{W}{2} \int \limits _{\max (-{\hat{y}}, {\tilde{y}})}^{\min ({\hat{y}}, {\tilde{Y}})} \textrm{d} y \, n_p \left( \frac{W}{2} e^y \right) \, n_q \left( \frac{W}{2} e^{-y} \right) . \end{aligned}$$
(43)

Similar to (16), to calculate the cross section measured in [8], this value should be multiplied by 2 to take into account that either of the protons can hit the detector:

(44)

Using the parton distribution functions MSHT20nnlo_as118 [16] provided by the LHAPDF [17] library, we get the cross sections

$$\begin{aligned} \sigma _{\text {fid.,}[8]}(pp \rightarrow p + \mu ^+ \mu ^- + X)&= 9.6~\text {fb}, \end{aligned}$$
(45)
$$\begin{aligned} \sigma _{\text {fid.,}[8]}(pp \rightarrow p + e^+ e^- + X)&= 11.4~\text {fb}. \end{aligned}$$
(46)

In order to estimate the accuracy of these cross sections, we have calculated them with the shifted arguments of parton density functions:

$$\begin{aligned} f_q(x, Q_2^2/2): \quad&\left\{ \begin{aligned} \sigma (pp \rightarrow p + \mu ^+ \mu ^- + X)&= 7.7~\text {fb}, \\ \sigma (pp \rightarrow p + e^+ e^- + X)&= 9.1~\text {fb}, \\ \end{aligned} \right. \end{aligned}$$
(47)
$$\begin{aligned} f_q(x, 2 Q_2^2): \quad&\left\{ \begin{aligned} \sigma (pp \rightarrow p + \mu ^+ \mu ^- + X)&= 11.5~\text {fb}, \\ \sigma (pp \rightarrow p + e^+ e^- + X)&= 13.6~\text {fb}. \\ \end{aligned} \right. \end{aligned}$$
(48)

We have checked that the contribution from the region of small \(Q^2\) where the PDFs are known with less accuracy is small. To do that we have calculated the inelastic contribution with much stronger cut on the transverse momentum: \(Q^2 \approx q_\perp ^2 < 1~\text {GeV}^2\). We get 2.2 fb for the production of muons and 2.6 fb for the production of electrons. Therefore, it is about 20% of the inelastic contribution, and of the order of what we have estimated for the PDFs uncertainty. We add this as a separate source of the uncertainty of the inelastic contribution.

We must also stress here that PDFs alone do not describe all inelastic contribution and there is a contribution to proton structure functions from resonance phenomena and other effects, see [13, 18, 19]. From Figure 18 of Ref. [13] and Table I of Ref. [18] we can see that the contribution of these effects is non-vanishing but at the level of 10–15% from elastic contribution. We do not aim at such level of precision and take into account just major contributions.

4 Conclusions

Let us compare our results with experimental data from [8]:

$$\begin{aligned} \sigma _{\mu \mu + p}^\text {exp.}&= 7.2 \pm 1.6~\text {(stat.)} \pm 0.9~\text {(syst.)} \pm 0.2~\text {(lumi.)} ~\text {fb}, \end{aligned}$$
(49)
$$\begin{aligned} \sigma _{ee + p}^\text {exp.}&= 11.0 \pm 2.6~\text {(stat.)} \pm 1.2~\text {(syst.)} \pm 0.3~\text {(lumi.)} ~\text {fb}. \end{aligned}$$
(50)

Summing up the cross sections calculated in Sects. 2 and 3, we get:

$$\begin{aligned} \tilde{\sigma }_{\mu \mu + p}^\text {theor.}&= 18 \pm 3~\text {fb}, \end{aligned}$$
(51)
$$\begin{aligned} \tilde{\sigma }_{e e + p}^\text {theor.}&= 22 \pm 3~\text {fb}, \end{aligned}$$
(52)

where the uncertainty values were obtained by comparing Eqs. (47), (48) and (45), (46) and include the uncertainty of the contribution from the region of small \(Q^2\), see the paragraph after (48). Also was taken into account the very small contribution from quark mass uncertainty in (21), see footnote 5 at page 4.

The so-called survival factor takes into account the diminishing of the cross sections due to breaking of both protons occurring when the protons scatter with small impact parameter b. The survival factor decreases when the invariant mass of the lepton pair grows and for the elastic cross section at \(W \sim 100\) GeV it approximately equals to 0.9 according to Fig. 3 from [2].

Table 1 in [8] contains results for the cross section obtained by Monte Carlo simulation. From the first two lines of this Table it follows that the survival factor (the probability for the colliding protons to avoid strong interactions) decreases the cross section by approximately factor 1.5. Here the values of cross sections for \(S_\text {surv} = 1\) are the combined results of LPAIR and HERWIG event generators, and the results for the survival factor from papers [5, 6] are used. Results of SUPERCHIC4 code [20] which takes the survival factor into account are within one standard deviation, see the third line in Table 1.

In the case of the elastic process, according to [2] the cross section is approximately 10% less, while in the case of elastic-inelastic scattering the cross section is approximately 50% less, see the lower left panel of Fig. 28 from Ref. [18].

We see that when the survival factor is taken into account the derived formulae are in agreement with experimental data at the level of 2–3 standard deviations. In the case of elastic process, the survival factor is calculated in [2] and can be easily taken into account without resorting to Monte Carlo simulations. Whether calculations of survival factor for semiexclusive process can be performed in a similar way is an interesting subject for further study.

Monte Carlo codes for photon–photon processes in UPC of protons and/or nuclei are presented in the recent paper [21].

Our calculations were performed with the help of libepa [22].

We are grateful to the referee for emphasizing the necessity of analysis of the contribution of low \(Q^2\) domain to the inelastic cross section.