Triple parton scatterings in proton-nucleus collisions at high energies

A generic expression to compute triple parton scattering (TPS) cross sections in high-energy proton-nucleus (pA) collisions is derived as a function of the corresponding single-parton cross sections and an effective parameter encoding the transverse parton profile of the proton. The TPS cross sections are enhanced by a factor of $9\,A\approx 2000$ in pPb compared to those in proton-nucleon collisions at the same center-of-mass energy. Estimates for triple charm ($c\overline{c}$) and bottom ($b\overline{b}$) production in pPb collisions at LHC and FCC energies are presented based on next-to-next-to-leading order (NNLO) calculations for $c\overline{c}, b\overline{b}$ single-parton cross sections. At $\sqrt{s_{NN}} = 8.8$ TeV, about 10% of the pPb events have three $c\overline{c}$ pairs produced in separate partonic interactions. At $\sqrt{s_{NN}} = 63$ TeV, the pPb cross sections for triple-J$/\psi$ and triple-$b\overline{b}$ are ${\cal O}$(1--10 mb). In the most energetic cosmic-ray collisions observed on earth, TPS $c\overline{c}$-pair cross sections equal the total p-Air inelastic cross section.


I. INTRODUCTION
The extended nature of hadronic systems and their growing parton density when probed at increasingly higher collision energies, makes it possible to produce multiple particles with large transverse momentum and/or mass ( p 2 T + m 2 3 GeV) in independent multiparton interactions (MPIs) in high-energy proton-(anti)proton (pp, pp) collisions [1][2][3][4][5]. Many experimental final-states -involving the concurrent production of heavy-quarks, quarkonia, jets, and gauge bosons-have been found consistent with double parton scatterings (DPS) processes at Tevatron (see e.g. [6]) and the LHC (see e.g. [7][8][9] for a selection of the latest results). Multiple hard parton interaction rates depend chiefly on the transverse overlap of the matter densities of the colliding hadrons, and provide valuable information on (i) the badly known 3D parton profile of the proton, (ii) the unknown energy evolution of the parton density as a function of impact parameter (b), and (iii) the role of manyparton correlations in the hadronic wave functions [10]. In our previous works [11], we highlighted the importance of studying DPS also in proton-nucleus (pA) and nucleusnucleus (AA) collisions, as a complementary means to improve our understanding of hard MPIs in pp collisions. The larger transverse parton density in a nucleus (with A nucleons) compared to that of a proton, results in enhanced DPS cross sections coming from interactions where the two partons of the nucleus belong to the same or to two different nucleons, providing thereby useful information on the underlying multiparton dynamics [11][12][13][14].
The possibility of triple parton scatterings (TPS) in hadronic collisions has also been considered in the literature [15][16][17], and estimates of their expected cross sections have been recently provided for pp collisions [18]. In this paper, we extend our latest work and derive for the first time quantitative estimates of the cross sections for observing three separate hard interactions in a pA collision through a factorized formula which depends on the underlying single-parton scattering (SPS) cross sec-tions normalized by the square of an effective cross section σ eff,tps , characterizing the transverse area of triple partonic interactions, that is closely related to the DPSequivalent σ eff,dps parameter [18]. The paper is organized as follows. In Sec. II, we review the theoretical expression for TPS cross sections in generic hadron-hadron collisions -expressed as a convolution of SPS cross sections and generalized parton densities dependent on parton fractional momentum x, virtuality Q 2 , and impact parameter b-and its factorized form as a function of σ eff,tps . In Section III, a generic expression for TPS cross sections in pA collisions is presented based on realistic parametrizations of the nuclear transverse profile. As a concrete numerical example, Section IV provides estimates for triple charm (cc) and bottom (bb) cross sections from independent parton scatterings in proton-lead (pPb) collisions at the LHC and future circular collider (FCC) [19] energies, based on next-to-next-to-leading-order (NNLO) calculations of the corresponding SPS cross sections. The main conclusions are summarized in Section V.

II. TRIPLE-PARTON-SCATTERING CROSS SECTIONS IN HADRON-HADRON COLLISIONS
In a generic hadronic collision, the inclusive TPS cross section from three independent hard parton scatterings (hh → abc) can be written as a convolution of generalized parton distribution functions (PDF) and elementary cross sections summed over all involved partons [15][16][17] σ tps 3 ) encode all the parton structure information of relevance for TPS, and are commonly assumed to be factorizable in terms of longitudinal and transverse components, i.e.
describes the transverse parton density of the hadron, often considered a universal function for all types of partons, from which the corresponding hadronhadron overlap function is derived: Making the further assumption that the longitudinal components reduce to the product of inde- , the TPS cross section can be expressed in the simple generic form i.e. as a triple product of single inclusive cross sections normalized by the square of an effective TPS cross section which is closely related to the similar quantity determined in DPS measurements. In the proton-proton case, making use of the expressions (3), (6) and (7), for a wide range of proton transverse parton profiles f (b), we found a simple relationship between the effective DPS and TPS cross sections: which, for the typical σ eff,dps = 15 ± 5 values extracted from a wide range of DPS measurements at Tevatron [6] and LHC [4,[6][7][8][9], translates into σ eff,tps = 12.5 ± 4.5 mb .
This data-driven numerical value, together with Eq. (4), allows the computation of any TPS cross section in pp collisions. In the next Section, we extend and exploit these results for the pA case.

III. TRIPLE-PARTON-SCATTERING CROSS SECTIONS IN PROTON-NUCLEUS COLLISIONS
The parton flux in pA compared to pp is enhanced by the nucleon number A and, modulo shadowing effects in the nuclear PDF [20], the single-parton cross section for any hard process is that of proton-nucleon (pN) collisions (with N = p, n including their appropriate relative fraction in the nucleus) scaled by the factor A [21], Here dz is the nuclear thickness function given by the integral of the nuclear parton density function (commonly parametrized in terms of a "Woods-Saxon" Fermi-Dirac distribution [22]) over the longitudinal direction with respect to the impact parameter b between the colliding proton and nucleus, normalized to d 2 b T A (b) = A. In order to obtain a TPS "pocket formula" of the form of Eq. (4) for pA collisions, we follow the approach developed in our previous work for the DPS case [11]. The TPS pA cross section is obtained from the sum of three contributions: • A "pure TPS" cross section, given by Eq. (4) for pN collisions scaled by A, namely: • A second contribution, involving interactions of partons from two different nucleons in the nucleus, depending on the square of T A , where the factor (A-1)/A is introduced to account for the difference between the number of nucleon pairs and the number of different nucleon pairs.
• A third term, involving interactions among partons from three different nucleons, depending on the cube of T A , σ tps,3 pA→abc = σ tps pN→abc · σ 2 eff,tps · C pA , with The factor (A-1)(A-2)/A 2 is introduced to take into account the difference between the total number of nucleon TPS and that of different nucleon TPS.
The inclusive TPS cross section for three hard parton subprocesses a, b, and c in pA collisions is thus obtained from the sum of the three terms (11), (12), and (14): which is enhanced by the factor in parentheses compared to the corresponding TPS cross section in pN collisions scaled by A. The value of this factor, as well as the relative role of each one of the three TPS components, can be obtained for pPb evaluating the integrals (13) and (15) using the standard Fermi-Dirac spatial density for the lead nucleus (A = 208, radius R A = 6.36 fm, and surface thickness a = 0.54 fm) [22]. The first integral is identical to the overlap function at zero impact parameter for the corresponding AA collision, F pA = (A − 1)/A T AA (0) = 30.25 mb −1 [11]. The second one can be obtained by means of a Glauber Monte Carlo (MC) [21] and amounts to C pA = 4.75 mb −2 .
From the relationship (8) between effective DPS and TPS cross sections, and the experimental σ eff,dps = 15 ± 5 mb value [4,[6][7][8][9], we can finally determine the relative importance for pPb of the three TPS terms of Eq. (16): σ tps,1 pA→abc : σ tps,2 pA→abc : σ tps,3 pA→abc = 1 : 4.54 : 3.56. Namely, in pPb collisions, 10% of the TPS yields come from partonic interactions within just one nucleon of the lead nucleus, 50% involve scatterings within two nucleons, and 40% come from partonic interactions in three different Pb nucleons. The sum of the three contributions in Eq. (16) amounts to 9.1, namely the TPS cross sections in pPb are nine times larger than the naive expectation based on A-scaling of the corresponding pN TPS cross sections, Eq. (11). We note that for DPS the equivalent pA enhancement factor was [1 + σ eff,dps F pA /A] 3 [11]. The final formula for TPS in proton-nucleus reads where the effective TPS pA cross section in the denominator depends on the effective pp one, and on pure geometric quantities directly derivable from the well-known nuclear transverse profile: σ 2 eff,tps,pA = A/σ 2 eff,tps + 2.46 F pA /σ eff,tps + C pA −1 where the latter equality is obtained using Eqs. (8)- (9). The effective TPS cross section in the pPb case amounts thereby to σ eff,tps,pA = 0.29 ± 0.05 mb. This value is very robust with respect to the parametrization of the underlying proton and nucleus transverse profiles. Indeed, by using simplified Gaussian proton and nucleus transverse densities, all relevant factors in Eq. (16) can be analytically calculated, and the effective TPS pA cross section can be simply written as a function of the proton and nucleus radii: σ 2 eff,tps,pA = 3/4 σ 2 eff,dps /{A[1 + 9/2A (r p /R A ) 2 + 4A 2 (r p /R A ) 4 ]}, which amounts to σ eff,tps,pA 0.28 mb (fixing r p so as to σ eff,dps = 15 mb), in perfect agreement with our more accurate estimate above.

IV. TRIPLE cc AND bb PRODUCTION CROSS SECTIONS IN pA COLLISIONS
As a concrete numerical example of our calculations, following our previous similar pp study [18], we compute the charm pPb → cc + X and bottom pPb → bb + X TPS cross sections first at the LHC and FCC, and then also those in proton-air collisions of relevance for ultrahigh-energy cosmic rays. These processes are dominated by gluon-gluon scattering gg → qq at low parton fractional momentum x, and at high energies the DPS and TPS mechanisms have a growing contribution to the total inclusive production. This expectation has been discussed for the DPS case in [23], and we extend those studies to the TPS case here. The TPS heavy-quark cross sections are computed via Eq. (17) for m = 1, i.e. σ tps pPb→cc,bb = (σ sps pN→cc,bb ) 3 /(6 σ 2 eff,tps,pA ) with σ eff,tps,pA given by (18), and σ sps pN→cc,bb calculated via Eq. (5) at NNLO accuracy using a modified version [24] of the Top++ (v2.0) code [25]. Top++ is run with N f = 3, 4 light flavors, charm and bottom pole masses set to m c,b = 1.67, 4.66 GeV, default renormalization and factorization scales set to µ R = µ F = 2 m c,b , and using the NNLO ABMP6 PDF of the proton [26] and the nuclear PDF modification factors of the Pb nucleus given by EPS09-NLO [20]. The PDF uncertainties include those from the proton and nucleus, as obtained from the corresponding 28 (30) eigenvalues of the ABMP16 (EPS09) sets, combined in quadrature. The dominant uncertainty is that linked to the theoretical scale choice, which is estimated by modifying µ R and µ F within a factor of two. In the pp case, such a theoretical NNLO setup yields SPS heavy-quark cross sections which are larger by up to 20% at the LHC compared to the NLO [27,28] predictions, reaching a better agreement with the experimental data [24], and showing a much reduced scale uncertainty (±50%, 15% for cc,bb). In the pPb case, the inclusion of EPS09 nuclear shadowing reduces moderately the total charm and bottom cross sections in pN compared to pp collisions, by about 10% (13%) and 5% (10%) at the LHC (FCC). Since the TPS pPb cross section go as the cube of σ sps pN→qq , the impact of shadowing is amplified and leads to 15-35% reductions with respect to the result obtained if one used the pp (instead of the pN) SPS cross section in Eq. (17). At √ s = 5.02 TeV, our theoretical SPS prediction (σ sps pPb→cc = 650 ± 290 sc ± 60 pdf mb) agrees well with the ALICE total D-meson measurement [29] extrapolated using [27] to a total charm cross section (σ alice pPb→cc = 640 ± 60 stat +60 −110 syst mb, Fig. 1 left). Table I collects   960 ± 450sc ± 100 pdf 3400 ± 1900sc ± 380 pdf σ(cc cc cc + X) 200 ± 140tot 8700 * ± 6200tot σ(bb + X) 72 ± 12sc ± 5 pdf 370 ± 75sc ± 30 pdf σ(bb bb bb + X) 0.084 ± 0.045tot 11 ± 7tot pPb events at 8.8 TeV. At the FCC, the theoretical TPS charm cross section even overcomes the inclusive charm one. Such an unphysical result indicates that quadruple, quintuple,... parton-parton scatterings are expected to produce extra cc pairs with non-negligible probability in pPb at √ s nn = 63 TeV. The huge TPS cc cross sections at the FCC will make triple-J/ψ production observable. Indeed, the SPS J/ψ cross section corresponds to about 5% of the cc one [11], which translates into σ(3 × J/ψ + X) ≈ 1 mb. Triple-bb cross sections remain comparatively small, in the 0.1 mb range, at the LHC but reach ∼10 mb (i.e. 3% of the total inclusive bottom cross section) at the FCC. Figure 1 shows pPb cross sections over √ s nn ≈ 40 GeV-100 TeV for SPS (solid), TPS (dashed) for charm (left) and bottom (right) production, and total inelastic (dotted curve, in both plots). The TPS cross sections are small at low energies but rise fast with √ s, as the cube of the SPS cross section evolution. Whenever the theoretical central value of the TPS cross section overcomes the inclusive charm cross section, indicative of multiple (beyond three) cc-pair production, we equalize it to the latter. Above √ s nn ≈ 25 TeV, the total charm and inelastic pPb cross sections are equal implying that all pPb interactions produce at least three charm pairs. In the bb case, such a situation only occurs at much higher c.m. energies, above 500 TeV. The most energetic hadronic collisions observed in nature occur in collisions of O(10 20 eV) cosmic rays, at the so-called "GZK cutoff" [30], with N and O nuclei at rest in the upper atmosphere. To study the amount of triple heavy-quark production produced in such collisions at equivalent c.m. energies of √ s nn ≈ 430 TeV, we show in Fig. 2 [21]. Around the GZK cutoff, the cross section for inclusive as well as TPS charm production equal the total inelastic protonair cross section, σ pAir ≈ 0.61 b, indicating that all p-Air collisions produce at least three cc-pairs in multiple partonic interactions. In the bb case, about 20% of the p-Air collisions produce bottom hadrons but only about 4% of them have TPS production. These results are clearly of relevance for the hadronic models commonly used for the simulation of the interaction of ultrarelativistic cosmic rays with the atmosphere [31] which, so far, do not include any heavy-quark production. Indeed, first, the cosmic ray data [32] feature unexplained excesses in the number of muons compared to the model predictions, and charmed and bottom hadrons feed more the muonic component of the air-shower. In addition, heavy-quark decays are a significant background of high-energy atmospheric neutrinos that need to be substracted in searches of astrophysical TeV-PeV ν's [33]. For both reasons, it is worth to explore the impact of such multiple heavyquark production in the MC generators commonly used in high-energy cosmic ray and ν astrophysics.

V. SUMMARY
We have derived for the first time estimates of the cross sections for triple parton scattering (TPS) cross sections in proton-nucleus collisions as a function of the corresponding single-parton cross sections and an effective σ eff,tps,pA parameter characterizing the transverse density of partons in the proton. Using NNLO predictions for single heavy-quark production, we have shown that three cc-pairs are produced from separate parton interactions in ∼10% of the pPb events at the LHC. At FCC energies, more rare processes such as triple-J/ψ and triple-bb production have cross sections reaching the 1-10 mb range. At even higher energies, of a few hundred TeV reachable in the highest-energy collisions of cosmic rays with the nuclei in the atmosphere, events producing three charmed hadron pairs occur in all proton-air collisions. The quantitative results presented here are of relevance for a proper description and understanding of final states with multiple hard particles in heavy-ion collider physics, and for a good control of high-energy µ and ν atmospheric fluxes in cosmic ray and neutrino astrophysics.