On production of heavy charged particles in γγ fusion at planned pp colliders

Production of heavy fermions in ultraperipheral collisions ( pp → p + γγ + p → p + χ + χ − + p ) and the semiexclusive reaction ( pp → p + γγ ∗ + X → p + χ + χ − + X ) is considered. Differential and total cross sections for the energies of the planned pp colliders are presented.


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, γγ ( * ) → χ + χ − , and the cross section of this reaction is determined by the values of the electric charge and the mass of χ.One popular example of χ ± is chargino -a mixture of superpartners of charged Higgs and W ± bosons. 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 semiexclusive 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.
One of the necessary ingredients of the calculation is the cross section of the γγ ( * ) → χ + χ − reaction.Due to the elastic form factor, virtuality of the photon emitted by the survived proton Q 2 1 ≡ −q 2 1 , where q 1 is the photon 4-momentum, is bounded by approximately (200 MeV) 2 [3, Appendix A].Since the contribution of longitudially polarized photon is proportional to Q 2 1 /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 longitudial 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, Appendix E, Eq. (E3)]: where σ T S is the cross section when the photon emitted by the disintegrating proton is polarized longitudinally, σ T T is that when it is polarized transversally, α is the fine structure constant, m χ is the mass of χ ± , 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: The cross section of χ + χ − pair production in the fusion of photons emitted by a disintegrating and an elastically scattered protons equals [5, Eqs. ( 18)-( 21), see also Eq. ( 41)] 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 = √ s/ (2m p ) is the proton Lorenz factor, 2 ) is the parton distribution function (PDF) for quark q, n p (ω) is the equivalent photon spectrum of proton [6, Eqs. ( 4) and ( 5)], y = (1/2) ln (ω 1 /ω 2 ) is the pair rapidity, ω 1 and ω 2 are photons energies, 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-Q 2 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 → pχ + χ − p we have (see Eqs. (2.15) and (2.16) in [9]) 3 Pair production at future pp colliders In this section we consider pair production of the charged fermions χ + χ − in the photon fusion in semiexclusive reactions and ultraperipheral collisions at the planned pp colliders: • HE-LHC (energy √ s = 27 TeV, luminosity L = 16 • 10 34 cm −2 • s −1 ) [10]; • SPPC (energy √ s = 70 TeV, luminosity L = 12 • 10 34 cm −2 • s −1 ) [11]; To get an integrated luminosity in one year of operation, one should multiply these luminosities by 10 7 s (the following results are obtained assuming this duration of collecting data at peak luminosity).For comparison we present results for the LHC with √ s = 13 TeV and 140 fb −1 of integrated luminosity collected by the ATLAS and the CMS collaborations in the years 2016-2018 (140 fb −1 = 1.4 • 10 41 cm −2 ).
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   1: Total cross sections (in fb) for χ + χ − pair production in ultraperipheral collisions pp → pχ + χ − p (UPC; integral of ( 5)) and in the inelastic process pp → pχ + χ − X (SE; integral of ( 4)).The column with m χ = 0.106 GeV corresponds to muon pair production with the threshold W > 12 GeV.2: Total number of events for χ + χ − pair production in ultraperipheral collisions pp → pχ + χ − p (UPC; integral of ( 5)) and in the inelastic process pp → pχ + χ − X (SE; integral of ( 4)).The column with m χ = 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 fulture colliders we assume one year of operation (10 7 s) at expected luminosity. of muon pair production √ s = 13 TeV and W > 12 GeV were already calculated in [14]: 203 pb 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).

UPC
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.
In [15] it was shown that heavy charged fermions with the mass up to almost 200 GeV can be found at 3σ level in the process pp → pχ + χ − p with the help of LHC Run-2 data.We see from Table 2 that we should have a similar number of events for m χ = 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 → pχ + χ − X and quasielastic pp → pχ + χ − 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 semiexclusive 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].