Triple parton scatterings in high-energy proton-proton collisions

A generic expression to compute triple parton scattering (TPS) cross sections in high-energy proton-proton (pp) collisions is presented as a function of the corresponding single-parton cross sections and the transverse parton distribution in the proton encoded in an effective parameter $\sigma_{\rm eff,TPS}$. The value of $\sigma_{\rm eff,TPS}$ is closely related to the similar effective cross section that characterizes double-parton scatterings, and amounts to $\sigma_{\rm eff,TPS} = 12.5 \pm 4.5$ mb. Estimates for triple charm ($\rm c\overline{c}$) and bottom ($\rm b\overline{b}$) production in pp collisions at LHC and FCC energies are presented based on next-to-next-to-leading order perturbative calculations for single $\rm c\overline{c},\rm b\overline{b}$ cross sections. At $\sqrt{s}\approx$ 100 TeV, about 15% of the pp collisions produce three $\rm c\overline{c}$ pairs from three different parton-parton scatterings.

The existence of multiparton interactions (MPI) in high-energy hadronic collisions is a natural consequence of the finite size of hadrons, the fast increase of the parton flux at small parton longitudinal momentum fractions (Bjorken x), and the requirement of unitarization of the cross sections in perturbative quantum chromodynamics (pQCD) [1]. The field of MPI has attracted an increasing interest in the last years motivated, among other reasons, by the accumulated experimental evidence of double parton scattering (DPS) processes concurrently producing two independently-identified hard particles in the same proton-(anti)proton (pp, pp) collision, at Tevatron and LHC energies [2][3][4][5]. Multiple hard parton interaction rates depend chiefly on the degree of overlap between the matter distributions of the colliding hadrons [6], and provide valuable information on the poorly known transverse parton profile of the proton, the unknown energy evolution of the parton density as a function of impact parameter (b), as well as on the role of many-parton correlations in the hadronic wave functions [7].
In this paper, we present for the first time a quantitative estimate of the cross section for observing three separate hard interactions in a pp collision (triple parton scattering, TPS), a possibility considered earlier [8][9][10], but for which no simple formula of the expected rates based on the underlying single-parton scattering (SPS) cross sections existed so far. A good understanding of TPS is not only useful to improve our knowledge of the 3D parton structure of the proton, but is also of relevance for a realistic characterization of backgrounds for rare final-states with multiple heavy-particles in searches of new physics. As we show below, TPS can represent a non-negligible fraction of some particular final states at future hadron colliders such as FCC, projected to deliver several 100 fb −1 /year integrated luminosities in pp collisions at a c.m. energy of √ s = 100 TeV [11].
We review first the theoretical expression for TPS cross sections in a generic hadron-hadron collision, expressed as a convolution of SPS cross sections and generalized parton densities (dependent on x, Q 2 and b), and derive a simple factorized expression for the TPS cross sections as a triple product of SPS cross sections normalized by an effective cross section σ eff,TPS characterizing the transverse area of triple partonic interactions. From simple geometric considerations, we show how σ eff,TPS is closely connected to the effective cross section σ eff,DPS already measured in DPS. From the existing measurements of σ eff,DPS and realistic proton profiles, we derive the numerical value of σ eff,TPS , which proves very robust with respect to any particular choice of the underlying transverse parton density. As a concrete numerical example, we provide estimates for triple charm (cc) and bottom (bb) cross sections in pp collisions at the LHC and FCC, based on next-to-next-to-leading-order (NNLO) calculations of the corresponding SPS cross sections.
In a generic hadron-hadron collision, the inclusive TPS cross section from three independent parton subprocesses (hh → abc) can be written [8][9][10] as a convolution of generalized parton distribution functions (PDFs) and elementary cross sections summed over all involved partons (1) 3 ) are the triple parton distribution functions, depending on the momentum fractions x 1 , x 2 , x 3 at transverse positions b 1 , b 2 , b 3 of the three partons i, j, k, producing final states a, b, c at energy scales Q 1 , Q 2 , Q 3 , with subprocess cross sectionsσ il a ,σ jm b ,σ kn c . The combinatorial prefactor m/3! takes into the different cases of (indistinguishable or not) final states: 3 ) encode all the parton structure information of relevance for TPS, and are typically assumed to be decomposable in terms of longi-tudinal and transverse components where f (b 1 ) describes the transverse parton density, often assumed to be a universal function for all types of partons, from which the corresponding hadron-hadron overlap function is derived: Making the further assumption that the longitudinal components reduce to the product of inde- , the cross section of TPS can be expressed in the simple generic form i.e., as a triple product of independent single inclusive cross sections normalized by the square of an effective TPS cross section which is closely related to the similar quantity determined to be σ eff,DPS 15 ± 5 mb in DPS measurements at Tevatron and the LHC [5,[12][13][14][15]. Whereas any TPS cross section can always be expressed as the triple product (4), with σ eff,TPS encoding all unknowns about the TPS dynamics, the geometrical interpretation of this latter quantity given by Eq. (6) relies on the assumption of the aforementioned simplifying yet economical assumptions on the factorization of longitudinal and transverse degrees of freedom, the absence of multiparton correlations, and flavor (gluon, quark)independent transverse parton profiles. In addition, there is no a priori reason to take the transverse hadron profile f (b), and therefore also σ eff,TPS and σ eff,DPS , as a constant with collision energy. As a matter of fact, DPS studies [12,13,16] indicate that the experimentallyextracted σ eff, DPS 15 mb values are about a factor of two smaller (i.e., the DPS cross sections are about twice larger) than expected from Eq. (7) for a "standard" proton geometric profile. The concurrent measurement of σ eff,TPS and σ eff,DPS in different colliding systems and for different final-states can help clarify all these open issues, shedding light on the 3D partonic structure of the proton and its evolution as a function of energy.
To estimate the value of σ eff,TPS , let's consider first a simplistic class of analytical models for the overlap function of a given hadronic AA collision normalized to d 2 b T AA (b) = A 2 , where exponents n = 0, 1, ∞ give respectively a flat, hard-sphere-like, and Gaussian-like distribution in the interval [0, 2 R A ]. This generic expression applies to any hadronic system, including collisions of nuclei with nucleon number A [17], but we will mostly focus below on the proton case for which A = 1, and R A = r p is its characteristic transverse "radius". From Eq. (6) one obtains: Instead of an expression which depends on R A , it's more convenient to express the effective TPS cross sections as a function of the experimentally determined σ eff,DPS parameter. From [17] = A 2 T AA (0) n + 1 2n + 1 one obtains, via Eqs. (6)- (7), the relationship between σ eff,TPS and σ eff,DPS : which for a proton-proton collision reads: For the distributions given by (8) normalized to one, where Γ(2/m) is the gamma function. The exponent m depends on the MC "tune" obtained from fits to the measured pp "underlying event" and various DPS cross sections [13]. It varies between a pure Gaussian (m = 2) to more peaked exponentiallike (m = 0.7, 1) distributions. From the corresponding integrals of the square and cube of T (b): and we obtain, via Eqs. (6)- (7), which is again independent of the exact numerical value of the proton "radius" r p , but depends on the overall shape of its transverse profile characterized by the exponent m. For typical pythia m = 0.7, 1, 2 exponents tuned from experimental data [13], one obtains σ eff,TPS = [0.66, 0.75, 0.87] × σ eff,DPS respectively. An alternative phenomenologically-motivated description of the proton profile is given by the dipole fit of the twogluon form factor in the momentum representation [19] F 2g (q) = 1/(q 2 /m 2 where the gluon mass m g parameter characterizes its transverse momentum q distribution, and the transverse density is obtained from its Fourier-transform: Such a proton profile is used in other standard pp MCs, such as herwig [16]. The corresponding DPS and TPS effective cross sections read [10]: and Numerically integrating the latter and combining it with (17), we obtain σ eff,TPS = 0.83 × σ eff,DPS , quite close to the value obtained for the Gaussian pp overlap function. We note that in order to reproduce the experimentally measured σ eff, DPS 15 mb, one should fix the characteristic proton "size" to r p 0.11, 0.24, 0.49 fm for pp overlaps of the form given by Eq. (12) with m = 0.7, 1, 2; to r p 0.35, 0.40, 0.46 fm for exponents n = 0, 1, 2 as defined in Eq. (8); and to r g = 1/m g 0.13 fm in the case of the two-gluon dipole fit (16). Despite the wide range of effective proton radius parameter derived, one of the main conclusion of this study is the robustness of the σ eff,TPS σ eff,DPS result. Indeed, the average and standard deviation of all typical parton transverse profiles considered here, yields σ eff,TPS = k × σ eff,DPS , with k = 0.82 ± 0.11 , (18) i.e., for the typical σ eff,DPS 15±5 values extracted from a wide range of DPS measurements at Tevatron [12] and LHC [5,[12][13][14][15], we finally obtain σ eff,TPS = 12.5 ± 4.5 mb.
The experimental observation of triple parton scatterings in pp collisions requires perturbatively-calculable processes with not too high energy (mass) scales so that their SPS cross section are not much smaller than O(1 µb) since, otherwise, their corresponding TPS cross section (which goes as the cube of the SPS values) are extremely reduced. Indeed, according to Eq. (4) with the data-driven estimate (19), a triple hard process pp → a a a, with SPS cross sections σ SPS pp→a ≈ 1 µb, has a very small TPS cross section σ TPS pp→a a a ≈ 1 fb, already without accounting for extra reducing factors from decay branching ratios and experimental acceptances and reconstruction inefficiencies. Promising processes to probe TPS, with not too small pQCD cross sections, are inclusive charm pp → cc + X, and bottom pp → bb + X production. These processes are dominated by gluon-gluon fusion, gg → QQ, and at high energies receive contributions from scatterings at very small x, for which one can expect a non-negligible probability of DPS [20][21][22] and TPS in their total inclusive production.
The TPS heavy-quark cross sections are computed via Eq. (4) for m = 1, i.e. σ TPS pp→QQ = (σ SPS pp→QQ ) 3 /(6σ 2 eff,TPS ) with σ eff,TPS given by (19), and σ SPS pp→QQ calculated via Eq. (5) at NNLO accuracy using a modified version [23] of the Top++ (v2.0) code [24], with N f = 3, 4 light flavors, heavy-quark pole masses at m c,b = 1.67, 4.66 GeV, default renormalization and factorization scales µ R = µ F = 2 m c,b , and using the ABMP16 PDFs [25]. The NNLO calculations increase the total SPS heavy-quark cross sections by up to 20% at LHC energies compared to NLO results [26,27], reaching a better agreement with the experimental data, and featuring much reduced scale uncertainties (±50%, ±15% for cc,bb) [23]. Figure 1 shows the resulting total SPS and TPS cross sections for charm and bottom production over √ s = 35 GeV-100 TeV, and Table I collects a few values with associated uncertainties for nominal LHC and FCC energies. The TPS cross sections are small but rise fast with √ s, as the cube of the SPS cross sections. Although triple-bb cross sections remain quite small and reach only about 1% of the inclusive bottom cross section at the FCC ( √ s = 100 TeV), triple-cc production from three independent parton scatterings amounts to 5% of the inclusive charm yields at the LHC ( √ s = 14 TeV) and to more than half of the total charm cross section at the FCC. Since the total pp inelastic cross section at √ s = 100 TeV is σ pp 105 mb [28], charm-anticharm triplets are expected to be produced in about 15% of the pp collisions at these energies.  7.1 ± 3.5sc ± 0.3 pdf 25.0 ± 16.0 ± 1.3 pdf σ(cc cc cc + X) 0.39 ± 0.28tot 16.7 ± 11.8tot σ(bb + X) 0.56 ± 0.09sc ± 0.01 pdf 2.8 ± 0.6sc ± 0.1 pdf σ(bb bb bb + X) (0.19 ± 0.12tot) 10 −3 (24 ± 17tot) 10 −3 The study presented here demonstrates that tripleparton scatterings are a non-negligible source of pertur-bative particle production in pp collisions at increasingly higher energies. The formulas derived here allow one to estimate the TPS yields for any final-state of interest. Using accurate NNLO predictions for single heavy-quark production, we have shown that the production cross section of three cc-pairs from three separate parton interactions is, in principle, observable at the LHC, and approaches the total charm cross section at √ s ≈ 100 TeV.