INTRODUCTION

The Standard Model (SM) of elementary particles perfectly describes relevant experimental data. Nevertheless, it is certain that it should be expanded in order to solve problems inherent to the SM (too many free parameters, the hierarchy problem, the origin of neutrino masses, strong CP violation) as well as to explain the nature of Dark Matter and Dark Energy and to construct predictive quantum gravity to say the least. In the course of the SM expansion, new heavy particles are usually introduced.

If these particles are electrically charged they should be produced in the γγ-fusion, \(\gamma {{\gamma }^{{(*)}}} \to {{\chi }^{ + }}{{\chi }^{ - }}\), and the cross section of this reaction is determined by the values of the electric charge and the mass of \(\chi \). One popular example of \({{\chi }^{ \pm }}\) is chargino, which is a mixture of superpartners of charged Higgs and \({{W}^{ \pm }}\) bosons.Footnote 1 In what follows we consider protons accelerated at high energy pp-colliders as a source of colliding photons.

We consider two reactions: ultraperipheral collisions when both protons remain intact and can be used for event tagging with the help of forward spectrometers, and the semi-exclusive process when only one proton survives while the second disintegrates. We calculate their cross sections for the planned colliders: HE-LHC (collision energy 27 TeV), SPPC (70 TeV) and FCC (100 TeV) and compare them with what is obtainable at the LHC (13 TeV).

In Section 2 necessary formulas are presented. Numerical results are given in Section 3. In Section 4 we conclude.

FORMULAS FOR THE CROSS SECTIONS

One of the necessary ingredients of the calculation is the cross section of the \(\gamma {{\gamma }^{{(*)}}} \to {{\chi }^{ + }}{{\chi }^{ - }}\) reaction. Due to the elastic form factor, virtuality of the photon emitted by the survived proton \(Q_{1}^{2} \equiv - q_{1}^{2}\), where \({{q}_{1}}\) is the photon 4-momentum, is bounded by approximately (200 MeV)2 (see Appendix A in [3]). Since the contribution of longitudinally polarized photon is proportional to \(Q_{1}^{2}{\text{/}}{{W}^{2}}\) where W is the invariant mass of the produced pair, it can be safely neglected and only the transverse polarization of this photon should be taken into account. However, in the case of the disintegrating proton both transversal and longitudinal polarizations should be accounted for. Formulas for the cross section of the massive fermions pair production in the collision of a real and a virtual photons are presented in [4] (see Eq. (E3) in Appendix E):

$$\begin{gathered} {{\sigma }_{{TS}}} = \frac{{4\pi {{\alpha }^{2}}}}{{{{W}^{2}}}}\frac{1}{{{{{\left( {1 + \frac{{Q_{2}^{2}}}{{{{W}^{2}}}}} \right)}}^{3}}}}\frac{{4Q_{2}^{2}}}{{{{W}^{2}}}} \\ \times \;\left( {\sqrt {1 - \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} - \frac{{2m_{\chi }^{2}}}{{{{W}^{2}}}}\ln \frac{{1 + \sqrt {1 - \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} }}{{1 - \sqrt {1 - \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} }}} \right), \\ \end{gathered} $$
(1)
$$\begin{gathered} {{\sigma }_{{TT}}} = \frac{{4\pi {{\alpha }^{2}}}}{{{{W}^{2}}}}\frac{1}{{{{{\left( {1 + \frac{{Q_{2}^{2}}}{{{{W}^{2}}}}} \right)}}^{3}}}}\left( {\mathop {\left( {1 + \frac{{Q_{2}^{4}}}{{{{W}^{4}}}} + \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}} - \frac{{8m_{\chi }^{4}}}{{{{W}^{4}}}}} \right)}\limits_{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}} } \right. \\ \left. { \times \;\ln \frac{{1 + \sqrt {1 - \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} }}{{1 - \sqrt {1 - \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} }} - \left( {{{{\left( {1 - \frac{{Q_{2}^{2}}}{{{{W}^{2}}}}} \right)}}^{2}} + \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} \right)\sqrt {1 - \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} } \right), \\ \end{gathered} $$
(2)

where \({{\sigma }_{{TS}}}\) is the cross section when the photon emitted by the disintegrating proton is polarized longitudinally, \({{\sigma }_{{TT}}}\) is that when it is polarized transversally, α is the fine structure constant, \({{m}_{\chi }}\) is the mass of \({{\chi }^{ \pm }}\), W is the invariant mass of the produced pair, \(Q_{2}^{2}\) is the virtuality of the photon emitted by the disintegrating proton.

In what follows we need the total cross section:

$$\begin{gathered} {{\sigma }_{{\gamma \gamma * \to {{\chi }^{ + }}{{\chi }^{ - }}}}}({{W}^{2}},Q_{2}^{2},m_{\chi }^{2}) \equiv {{\sigma }_{{TS}}} + {{\sigma }_{{TT}}} \\ = \frac{{4\pi {{\alpha }^{2}}}}{{{{W}^{2}}}}\frac{1}{{{{{\left( {1 + \frac{{Q_{2}^{2}}}{{{{W}^{2}}}}} \right)}}^{3}}}} \\ \times \;\left( {\left( {1 + \frac{{Q_{2}^{4}}}{{{{W}^{4}}}} + \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}} - \frac{{8m_{\chi }^{4}}}{{{{W}^{4}}}} - \frac{{8m_{\chi }^{2}Q_{2}^{2}}}{{{{W}^{4}}}}} \right)} \right. \\ \times \;\ln \frac{{1 + \sqrt {1 - \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} }}{{1 - \sqrt {1 - \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} }} \\ \left. { - \;\left( {1 - \frac{{6Q_{2}^{2}}}{{{{W}^{2}}}} + \frac{{Q_{2}^{4}}}{{{{W}^{4}}}} + \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} \right)\sqrt {1 - \frac{{4m_{\chi }^{2}}}{{{{W}^{2}}}}} } \right). \\ \end{gathered} $$
(3)

The cross section of \({{\chi }^{ + }}{{\chi }^{ - }}\) pair production in the fusion of photons emitted by a disintegrating and an elastically scattered protons equals (see Eqs. (18)–(21) and (41) in [5])

$$\frac{{{\text{d}}{{\sigma }_{{pp \to p{{\chi }^{ + }}{{\chi }^{ - }}X}}}}}{{{\text{d}}W}} = \frac{{4\alpha W}}{\pi }\sum\limits_q Q_{q}^{2}$$
$$\begin{gathered} \times \int\limits_{\frac{{{{W}^{4}}}}{{36{{\gamma }^{2}}s}}}^{s - {{W}^{2}}} \frac{{{{\sigma }_{{\gamma \gamma * \to {{\chi }^{ + }}{{\chi }^{ - }}}}}({{W}^{2}},Q_{2}^{2},m_{\chi }^{2})}}{{({{W}^{2}} + Q_{2}^{2})Q_{2}^{4}}}{\text{d}}Q_{2}^{2} \\ \times \;\int\limits_{\frac{{{{W}^{2}} + Q_{2}^{2}}}{s}\max \left( {1,\frac{{{{m}_{p}}}}{{3\sqrt {Q_{2}^{2}} }}} \right)}^1 {\text{d}}x{{f}_{q}}(x,Q_{2}^{2}) \\ \times \;\int\limits_{\frac{1}{2}\ln \left( {\frac{{{{W}^{2}} + Q_{2}^{2}}}{{{{x}^{2}}s}}\max \left( {1,\frac{{m_{p}^{2}}}{{9Q_{2}^{2}}}} \right)} \right)}^{\frac{1}{2}\ln \frac{s}{{{{W}^{2}} + Q_{2}^{2}}}} {{\omega }_{1}}{{n}_{p}}({{\omega }_{1}})[Q_{2}^{2} - {{\left( {{{\omega }_{2}}{\text{/}}3x\gamma } \right)}^{2}}]{\kern 1pt} {\text{d}}y, \\ \end{gathered} $$
(4)

where q is a quark inside the disintegrating proton, \({{Q}_{q}}\) is its charge, s is the square of the invariant mass of the colliding protons, \({{m}_{p}}\) is the proton mass, γ = \(\sqrt s {\text{/}}\left( {2{{m}_{p}}} \right)\) is the Lorentz factor of the proton, \({{f}_{q}}(x,Q_{2}^{2})\) is the parton distribution function (PDF) for quark q, \({{n}_{p}}\left( \omega \right)\) is the equivalent photon spectrum of proton (see Eqs. (4) and (5) in [6]), \(y = (1{\text{/}}2){\text{ln}}({{\omega }_{1}}{\text{/}}{{\omega }_{2}})\) is the pair rapidity, \({{\omega }_{1}}\) and \({{\omega }_{2}}\) are photons energies, ω1 = \(\sqrt {{{W}^{2}} + Q_{2}^{2}} {{e}^{y}}{\text{/}}2\), \({{\omega }_{2}} = \sqrt {{{W}^{2}} + Q_{2}^{2}} {\kern 1pt} {\kern 1pt} {{e}^{{ - y}}}{\text{/}}2\).

The accuracy of (4) is at the level of 20% which is sufficient for our estimates. More details can be found in [5] including the discussion of uncertainties due to PDFs and the impact of low-Q2 physics. The results of this paper were obtained with the help of MMHT2014nnlo68cl PDF set [7] provided by LHAPDF [8].

For the quasielastic process \(pp \to p{{\chi }^{ + }}{{\chi }^{ - }}p\) we have (see Eqs. (2.15) and (2.16) in [9])

$$\begin{gathered} \frac{{{\text{d}}{{\sigma }_{{pp \to p{{\chi }^{ + }}{{\chi }^{ - }}p}}}}}{{{\text{d}}W}} = {{\sigma }_{{\gamma \gamma * \to {{\chi }^{ + }}{{\chi }^{ - }}}}}({{W}^{2}},Q_{2}^{2} = 0,m_{\chi }^{2}) \\ \times \;\frac{W}{2}\int\limits_{ - \infty }^\infty {{n}_{p}}\left( {\frac{W}{2}{{e}^{y}}} \right){{n}_{p}}\left( {\frac{W}{2}{{e}^{{ - y}}}} \right){\text{d}}y. \\ \end{gathered} $$
(5)

PAIR PRODUCTION AT FUTURE pp COLLIDERS

In this section we consider pair production of the charged fermions \({{\chi }^{ + }}{{\chi }^{ - }}\) in the photon fusion in semi-exclusive reactions and ultraperipheral collisions at the planned pp colliders:

• HE-LHC (energy \(\sqrt s = 27\) TeV, luminosity \(\mathcal{L} = \) 16 × 1034 cm–2 s–1) [10];

• SPPC (energy \(\sqrt s = 70\) TeV, luminosity \(\mathcal{L} = \) 12 × 1034 cm–2 s–1) [11];

• FCC-hh (energy \(\sqrt s = 100\) TeV, luminosity \(\mathcal{L} = \) 5 × 1034 cm–2 s–1) [12].

To get an integrated luminosity in one year of operation, one should multiply these luminosities by 107 s (the following results are obtained assuming this duration of collecting data at peak luminosity). For comparison we present results for the LHC with \(\sqrt s = \) 13 TeV and 140 fb–1 of integrated luminosity collected by the ATLAS and the CMS collaborations in the years 2016–2018 (\(140 {\text{ f}}{{{\text{b}}}^{{ - 1}}} = 1.4 \times {{10}^{{41}}} {\text{ c}}{{{\text{m}}}^{{ - 2}}}\)).

Differential cross sections \({\text{d}}\sigma {\text{/d}}W\) for \(\sqrt s = 13\), 27, 70, 100 TeV are presented in Figs. 1–4. Each figure contains plots for \(\chi \) masses 100, 200, 400, 800 GeV as well as for muon pair production. By increasing energy from 13 to 100 TeV we get an order of magnitude enhancement in the differential cross sections for W = 200–300 GeV, while for W = 3–4 TeV we get more than two orders of magnitude. Therefore, at future colliders we can access now unavailable regions of large W. It is not that valuable for light particles since the most of the cross section comes from the region near the threshold, while it is crucial for the heavy ones. When considering the number of events for future colliders one should also keep in mind expected significant enhancement in the luminosity.

Fig. 1.
figure 1

(Color online) Differential cross sections given by Eqs. (solid lines) (4) and (dashed lines) (5) for \(\sqrt s = 13\) TeV and mχ = (from top to bottom) 0.106, 100, 200, 400, 800 GeV.

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Fig. 2.
figure 2

(Color online) Same as in Fig. 1, but for \(\sqrt s = \) 27 TeV.

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Fig. 3.
figure 3

(Color online) Same as in Fig. 1, but for \(\sqrt s = \) 70 TeV.

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Fig. 4.
figure 4

(Color online) Same as in Fig. 1, but for \(\sqrt s = \) 100 TeV.

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Total cross sections are collected in Table 1. In the case of muons we integrate over \(W > 12\) GeV since this lower bound was implemented in [13] in order to suppress the background. Cross sections of muon pair production for \(\sqrt s = 13\) TeV and \(W > 12\) GeV were already calculated in [14]: 203 and 60 pb for the inelastic and the quasielastic cross sections correspondingly. The small differences are due to additional approximations made in [14] (the most notable of these is neglecting the Pauli form factor contribution in the equivalent photons spectrum of proton).

Table 1. Total cross sections (in femtobarn) for \({{\chi }^{ + }}{{\chi }^{ - }}\) pair production in ultraperipheral collisions \(pp \to p{{\chi }^{ + }}{{\chi }^{ - }}p\) (UPC; integral of (5)) and in the inelastic process \(pp \to p{{\chi }^{ + }}{{\chi }^{ - }}X\) (SE; integral of (4)). The column with \({{m}_{\chi }} = 0.106\) GeV corresponds to muon pair production with the threshold \(W > 12\) GeV
Full size table

Though the obtained cross sections are model independent, the search strategy is determined by the decay modes and lifetimes of these new particles which are model dependent. In the case when they are long-lived (i.e., pass through the detector) we can use the method presented in [15]. The main idea behind this technique is to reconstruct complete kinematics of the process by measuring momenta of the particles produced in the main detector in combination with energy losses of each of the protons provided by the forward spectrometers. Let us note that for \({{m}_{\chi }} = \) 100 GeV in the case of the LHC the requirement of both protons being registered by the forward detectors diminishes the cross section by a factor of 3.5 (see Eqs. (2.9) and (2.11) in [15]).

It was shown in [15] that for the quasielastic processes the background can be almost eliminated with the help of proton tagging. So the discovery potential is mostly defined by the number of events. Though the designs of the future experiments are different (and therefore selection criteria will also be different), we can compare total numbers of produced pairs assuming similar efficiencies of event selection. These numbers are presented in Table 2.

Table 2. Total number of events for \({{\chi }^{ + }}{{\chi }^{ - }}\) pair production in ultraperipheral collisions \(pp \to p{{\chi }^{ + }}{{\chi }^{ - }}p\) (UPC; integral of (5)) and in the inelastic process \(pp \to p{{\chi }^{ + }}{{\chi }^{ - }}X\) (SE; integral of (4)). The column with \({{m}_{\chi }} = 0.106\) GeV corresponds to muon pair production with the threshold W > 12 GeV. While for the LHC we take the available integrated luminosity 140 fb–1, for the future colliders we assume one year of operation (107 s) at expected luminosity
Full size table

In [15] it was shown that heavy charged fermions with the mass up to almost 200 GeV can be found at \(3\sigma \) level in the process \(pp \to p{{\chi }^{ + }}{{\chi }^{ - }}p\) with the help of LHC Run-2 data. We see from Table 2 that we should have a similar number of events for mχ = 800 GeV at the SPPC. Therefore it is possible to push the model independent lower bound on heavy charged fermions mass to about 800 GeV, or discover these new particles. The SPPC has the greatest potential due to larger expected luminosity.

CONCLUSIONS

Cross sections for pair production of heavy charged particles in both inelastic \(pp \to p{{\chi }^{ + }}{{\chi }^{ - }}X\) and quasielastic \(pp \to p{{\chi }^{ + }}{{\chi }^{ - }}p\) processes were calculated for the future pp colliders. Total numbers of events were estimated based on the expected luminosity of these experiments. The SPPC has the greatest potential and can find heavy charged fermions with masses up to about 800 GeV in one year of operation. Let us stress that there are many more semi-exclusive events than quasielastic ones.

The main advantage of the considered processes is the possibility to detect survived proton(s) which provides effective means for background suppression. Nowadays, when the detectors for these colliders are intensively discussed, we would like to emphasize the importance of forward spectrometers that could provide unique model independent methods for Beyond Standard Model searches.

Numerical results were obtained with the help of libepa library [16].