#### Triple-parton scatterings in proton–nucleus collisions at high energies

Eur. Phys. J. C
Triple-parton scatterings in proton-nucleus collisions at high energies
David d'Enterria david.d' 2
Alexander M. Snigirev 0 1
0 Bogoliubov Laboratory of Theoretical Physics , JINR, 141980 Dubna , Russia
1 Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University , 119991 Moscow , Russia
2 EP Department , CERN, 1211 Geneva , Switzerland
A generic expression to compute triple-parton scattering (TPS) cross sections in high-energy protonnucleus (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 about 9 A 2000 in pPb as compared to those in proton-nucleon collisions at the same center-of-mass energy. Estimates for triple charm (cc) and bottom (bb) production in pPb collisions at LHC and FCC energies are presented based on nextto-next-to-leading-order calculations for cc and bb singleparton cross sections. At √snn = 8.8 TeV, about 10% of the pPb events have three cc pairs produced in separate partonic interactions. At √snn = 63 TeV, the pPb cross sections for triple- J /ψ and triple-bb are O(1-10 mb). In the most energetic collisions of cosmic rays in the upper atmosphere, equivalent to √snn ≈ 400 TeV, the TPS cc cross section equals the total p-Air inelastic cross section.
1 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
( p2T + m2 3 GeV) in independent multiparton
interactions (MPIs) in high-energy proton–(anti)proton (pp, pp¯)
collisions [
1–5
]. Many experimental final states – involving the
concurrent production of heavy-quarks, quarkonia, jets, and
gauge bosons – have been found to be consistent with
double parton scatterings (DPS) processes at Tevatron (see e.g.
[
6
]) and the LHC (see e.g. [
7–9
] 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 (1) the
badly known 3D parton profile of the proton, (2) the unknown
energy evolution of the parton density as a function of impact
parameter (b), and (3) the role of many-parton correlations in
the hadronic wave functions [
10
]. In our previous work [
11–
14
], we highlighted the importance of studying DPS also in
proton–nucleus (pA) and nucleus–nucleus (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–
21
].
The possibility of triple-parton scatterings (TPS) in
hadronic collisions has also been considered in the literature
[
22–26
], and estimates of their expected cross sections have
been recently provided for pp collisions [
27
]. 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 sections normalized by the square of an
effective cross section σeff,tps, characterizing the transverse
area of triple-partonic interactions, that is closely related
to the DPS-equivalent σeff,dps parameter [
27
]. The paper is
organized as follows. In Sect. 2, we review the theoretical
expression for TPS cross sections in generic hadron–hadron
collisions, first expressed as a convolution of SPS cross
sections and generalized parton densities dependent on parton
fractional momentum x , virtuality Q2, and impact parameter
b, and then in their factorized form as a function of σeff,tps.
In Sect. 3, a generic expression for TPS cross sections in pA
collisions is presented based on realistic parametrizations of
the nuclear transverse profile. As concrete numerical
examples, Sect. 4 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) [
28
] energies, as well as in
protonair collisions at the highest energies observed so far, based
on next-to-next-to-leading-order (NNLO) calculations of the
corresponding SPS cross sections. The main conclusions are
summarized in Sect. 5.
2 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 [
22–27,29
]
(2)
(3)
σhthps→abc
m
= 3
! i, j,k,l,m,n
Γhi jk (x1, x2, x3; b1, b2, b3; Q12, Q22, Q32)
×σˆail (x1, x1, Q12)σˆbjm (x2, x2, Q22)σˆckn(x3, x3, Q32)
×Γhlmn(x1, x2, x3; b1 − b, b2 − b, b3 − b; Q21, Q22, Q32)
×d x1d x2d x3d x1d x2d x3d2b1d2b2d2b3d2b. (1)
Here, Γhi jk (x1, x2, x3; b1, b2, b3; Q2, Q2, Q23) are the
triple1 2
parton distribution functions, depending on the momentum
fractions x1, x2, x3 at transverse positions b1, b2, b3 of the
three partons i , j , k, producing final states a, b, c at energy
scales Q1, Q2, Q3, with subprocess cross sections σˆail , σˆbjm ,
σˆckn. The combinatorial prefactor m/3! takes into account
the different cases of (indistinguishable or not) final states:
m = 1 if a = b = c; m = 3 if a = b, or a = c, or b = c;
and m = 6 if a, b, c are different. The triple-parton
distribution functions Γ i jk (x1, x2, x3; b1, b2, b3; Q2, Q2, Q2)
h 1 2 3
encode all the parton structure information of relevance for
TPS.
Without any loss of generality, any TPS cross section can
always be expressed in a very economical form as a triple
product of single inclusive cross sections,
Dhi (x1; Q12) σˆaik (x1, x1) Dhk (x1; Q12)dx1dx1,
σhshps→a =
i,k
σhthps→abc =
normalized by the square of an effective TPS cross section,
namely
m
3!
σhshps→a · σhshps→b · σhshps→c ,
2
σeff,tps
(5)
(6)
(7)
(8)
(9)
−1
−1
,
,
where σeff,tps encodes all the unknowns related to the
generalized PDFs and their correlations in space, color, flavor,
pT,... The value of σeff,tps can be estimated theoretically
making a few common approximations. First, the triple-PDF are
commonly assumed to be factorizable in terms of
longitudinal and transverse components, i.e.
Γhi jk (x1, x2, x3; b1, b2, b3; Q12, Q22, Q32)
= Dhijk (x1, x2, x3; Q12, Q22, Q32) f (b1) f (b2) f (b3), (4)
where f (b1) describes the transverse parton density of the
hadron, often considered a universal function for all types of
partons, from which the corresponding hadron–hadron
overlap function is derived:
T (b) =
f (b1) f (b1 − b)d2b1 .
Making the further assumption that the longitudinal
components reduce to the product of independent single PDF,
Dhijk (x1, x2, x3; Q21, Q22, Q23) = Dhi (x1; Q21)Dhj (x2; Q22)
Dhk (x3; Q23), the effective TPS cross section has a very
simple geometric interpretation as the inverse of the cube of the
integral of the hadron–hadron overlap function:
2
σeff,tps =
d2b T 3(b)
which is closely related to the similar quantity,
σeff,dps =
d2b T 2(b)
determined in DPS measurements. In the proton–proton case,
making use of Eqs. (5), (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:
σeff,tps = (0.82 ± 0.11) · σeff,dps ,
which, for the typical σeff,dps = 15 ± 5 mb values extracted
from a broad range of DPS measurements at Tevatron [
6
] and
LHC [
4,6–9
], translates into
σeff,tps = 12.5 ± 4.5 mb .
This data-driven numerical value allows for the computation
of any TPS cross section in pp collisions via Eq. (3), once
the corresponding SPS cross section, Eq. (2), is known.
In the next section, we extend and exploit these results for
the pA case. We note that, since we start off with a
numerical value for σeff,dps that is directly obtained from data,
our approach effectively takes into account effects which
go beyond pure geometrical quantities computed in terms
of transverse overlap of parton densities. Indeed, the fact
that the experimentally extracted σeff,dps = 15 ± 5 mb
values are about a factor of 2 smaller (i.e., that the DPS cross
sections are about twice larger) than expected from Eq. (7)
for a “naive” proton profile, is indicative of the presence of
“beyond mean-field” effects, including e.g. parton
correlations in momentum, flavor, color, x ,... Perturbative partonic
correlations due to 1-to-2 parton splittings have been
discussed for the DPS case in [
30, 31
] in pp collisions and in
[21] for pA collisions. The first attempt to estimate those for
TPS in pp collisions was done in [
26
]. As we will see later,
however, any such potential effects are of less relevance in
pA than in pp collisions, since the majority of the TPS yields
for the former arise from interactions among different
“target” nucleons for which any potential partonic correlations
are of smaller magnitude than in the pp case.
3 Triple-parton-scattering cross sections in
proton–nucleus collisions
The first theoretical analysis of double- and triple-parton
scattering cross sections in pA collisions was done in [
15, 16
],
where large enhancements of the DPS and TPS yields were
anticipated. Here we focus on the derivation of a compact
“pocket formula” for TPS scattering in pA collisions,
particularly useful for phenomenological and numerical
applications. The starting point of our approach is to remember
that the parton flux in pA compared to pp is enhanced by
the nucleon number A and, modulo shadowing effects in the
nuclear PDF [32], 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 [
33
],
σpsAps→abc = σpsNps→abc
d2b TA(b) = A · σpsNps→abc .
(10)
Here TA(b) = fA(√r2 + z2)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 [
34
]) over the longitudinal
direction with respect to the impact parameter b between the
colliding proton and nucleus, normalized to d2b TA(b) =
A. In order to obtain a TPS “pocket formula” of the form
of Eq. (3) for pA collisions, we follow the approach
developed in our previous work for the DPS case [
11–14
]. The
TPS pA cross section is thus obtained from the sum of three
contributions:
– A “pure TPS” cross section, given by Eq. (3) for pN
collisions scaled by A, namely:
σptAps→,1abc = A · σptNps→abc .
(11)
– A second contribution, involving interactions of partons
from two different nucleons in the nucleus, depending on
the square of TA,
FpA =
A − 1
A
d2b TA2(b) ,
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 TA,
σptAps→,3abc = σptNps→abc · σeff,tps · CpA, with
2
CpA =
(A − 1)(A − 2)
A2
d2b T3A(b) .
The factor (A−1)(A−2)/A2 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):
σ tps
pA→abc = A σ tps
pN→abc
1 + 3
2
σeff,tps FpA
σeff,dps A
2 CpA
+ σeff,tps A
which is enhanced by the factor in parentheses compared to
the corresponding TPS cross section in pN collisions scaled
by A. The analytical (unintegrated) expression for TPS cross
sections in proton–nucleus collisions was first derived in
[
15
], but our compact expression (16) provides a more
useful formula to easily derive numerical TPS estimates for
any pA system for phenomenological purposes. The value
of this enhancement 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 RA = 6.36 fm, and surface thickness a = 0.54 fm)
[
34
]. The first integral is identical to the overlap function
at zero impact parameter for the corresponding AA
collision, FpA = (A − 1)/A TAA(0) = 30.25 mb−1 [
11–14
]. The
second one can be obtained by means of a Glauber Monte
Carlo (MC) [33] and amounts to CpA = 4.75 mb−2. From the
relationship (8) between the effective DPS and TPS cross
sections, and the experimental σeff,dps = 15 ± 5 mb value
[
4, 6–9
], we can finally determine the relative importance for
pPb of the three TPS terms of Eq. (16): σpA→abc : σptAps→,2abc :
tps,1
σptAps→,3abc = 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
interac(12)
(13)
(14)
(15)
,
(16)
tions in three different Pb nucleons. The fact that, for heavy
nuclei, the second and third terms of Eq. (16) are much larger
than the first one was first pointed out in Ref. [
15
], and we
provide here an exact numerical evaluation for the pPb case.
The sum of the three contributions in Eq. (16) amounts to 9.1,
namely the TPS cross sections in pPb are about nine times
larger than the naive expectation based on A-scaling of the
corresponding pN TPS cross sections, Eq. (11). We note that
in the DPS case the equivalent pA enhancement factor was
[1 + σeff,dpsFpA/A] 3 [
11–14
]. The final formula for TPS
in the proton–nucleus case reads
σptAps→abc =
m
6
σpsNps→a · σpsNps→b · σpsNps→c ,
2
σeff,tps,pA
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:
2
σeff,tps,pA =
A/σe2ff,tps + 2.46 FpA/σeff,tps + CpA
=
A/156. + FpA[mb−1]/5. + CpA[mb−2] −1 ,
(17)
−1
(18)
where the latter equality is obtained using Eqs. (8)–(9). The
effective TPS cross section in the pPb case amounts thereby
to σeff,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
2
as a function of the proton and nucleus radii: σeff,tps,pA =
3/4 σe2ff,dps/{A[1+9/2A (rp/RA)2 +4A2 (rp/RA)4]}, which
amounts to σeff,tps,pA 0.28 mb (fixing r p so as to σeff,dps =
15 mb), in perfect agreement with our more accurate estimate
above.
4 Triple cc and bb production cross sections in pA
collisions
As concrete numerical examples of our calculations,
following our previous similar pp study [
27
], we compute the charm
(pPb → cc + X) and bottom (pPb → bb + X) TPS cross
sections first at the LHC and FCC center-of-mass (c.m.)
energies, and then also in proton-air collisions of relevance for
ultra-high-energy cosmic-ray collisions in the atmosphere.
These processes are dominated by gluon–gluon scattering
gg → qq at low parton fractional momentum x , for which
the DPS and TPS mechanisms have a growing contribution to
the total inclusive production at increasing c.m. energy. This
expectation has been discussed for the DPS case in [
35
], and
we extend those studies to the TPS case here. The TPS
heavyquark cross sections are computed via Eq. (17) for m = 1, i.e.
σ tps )3/(6 σe2ff,tps,pA) with σeff,tps,pA
gipvPebn→cbcy,bb(18=), (aσnpdsNps→σscpcs,bb
pN→cc,bb is calculated via Eq. (2) at
NNLO accuracy using a modified version [
36
] of the Top++
(v2.0) code [
37
]. Top++ is run with Nf = 3, 4 light flavors,
charm and bottom pole masses set to mc,b = 1.67, 4.66 GeV
[
38
], default renormalization and factorization scales set to
μR = μF = 2 mc,b, and using the NNLO ABMP6 PDF of the
proton [
39
] and the nuclear PDF modification factors of the
Pb nucleus given by EPS09-NLO [
32
]. 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 μR and μF within a factor of 2. In the
pp case, such a theoretical NNLO setup yields SPS
heavyquark cross sections which are larger by up to 20% compared
to the NLO [
40,41
] predictions at the LHC, reaching a better
agreement with the experimental data, and showing a much
reduced scale uncertainty (±50%, ±15% for cc, bb) [36].
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 σpsNps→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 (σpsPpbs,→nnclco = 650 ± 290sc ± 60pdf mb)
agrees well with the ALICE total D-meson measurement
[
42
] extrapolated using [
40
] to a total charm cross section
(σpaPlbic→e cc = 640 ± 60stat +60
−110 syst mb, Fig. 1 left).
Table 1 collects the heavy-quark cross sections and
associated uncertainties predicted in pPb collisions at the
nominal LHC and FCC c.m. energies. The large SPS cc cross
section at the LHC (∼1 b) results in triple-cc cross
sections from independent parton scatterings amounting to about
20% of the inclusive charm yields. Since the total inelastic
pPb cross sections are σpPb = 2.2, 2.4 b at √snn = 8.8 and
63 TeV [
33
], charm TPS takes place in about 10% of the pPb
events at 8.8 TeV. At the FCC, the theoretical TPS charm
cross section even overcomes the inclusive charm one. Such
a seemingly “unphysical” result indicates that quadruple,
quintuple,... parton–parton scatterings are expected to
produce extra cc pairs with non-negligible probability in pPb at
√snn = 63 TeV. We recall that inclusive cross sections can
be related to the factorial moments of the multiplicity
distribution and, thus, SPS, DPS, TPS... cross sections are not
bounded by the inelastic cross section [
43
]. The huge TPS cc
cross sections at the FCC will make triple- J /ψ production
ALICE (pPb→ cc), 5.02 TeV
10
1
10−1
indicate scale, PDF (and σeff,tps, in the TPS case) uncertainties added
in quadrature. The pPb → cc + X charm data point on the left plot has
been derived from [
42
]
Fig. 2 Charm (left) and bottom (right) cross sections in p–Air
collisions as a function of nucleon-nucleon c.m. energy, in single-parton
(solid line) and triple-parton (dashed line) parton scatterings, compared
to the total inelastic p–Air cross section (dotted line). Bands around
curves indicate scale, PDF (and σeff,tps, in the TPS case) uncertainties
added in quadrature
observable. Indeed, the SPS J /ψ cross section corresponds
to a few percent of the cc one [
11–14
], which translates1 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
1 We note that the large production of double quarkonia seen in the
data seem to indicate values of σeff,dps smaller than the (15 ± 5 mb)
world-average used here [44]. Therefore, one would also expect that
the corresponding TPS yields for J /ψ could well be above our 1 mb
order-of-magnitude estimate.
reach ∼10 mb (i.e. 3% of the total inclusive bottom cross
section) at the FCC.
Figure 1 shows pPb cross sections over √snn ≈ 40 GeV–
100 TeV for SPS (solid bands), TPS (dashed bands) 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 √snn ≈ 25 TeV, the
total charm and inelastic pPb cross sections are equal
implying that the average number of charm pairs produced in pPb
collisions is larger than one. 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(1020 eV) cosmic rays, at the
socalled “GZK cutoff” [
45
], with N and O nuclei at rest in
the upper atmosphere. To study the amount of triple
heavyquark production produced in such collisions at equivalent
c.m. energies of √snn ≈ 430 TeV, we show in Fig. 2
similar curves as those in Fig. 1 but for the p–Air case. The
TPS cross sections have been obtained using Eq. (17) with
the same cubic power of the SPS pN cross sections
computed with the Top++ (ABMP6+EPS09) setup, but
normalized now to an effective TPS p–Air cross section amounting
to σeff,tps,pA = 2.2 ± 0.4 mb obtained from Eq. (18) using:
A = 14.3 (from a 78%–21% air mixture of 14N and 16O),
FpA = 0.51 mb−1, and CpA = 0.016 mb−2, the latter two
values being obtained via a Glauber MC [
33
]. Around the
GZK cutoff, the cross sections for inclusive as well as TPS
charm production equal the total inelastic proton-air cross
section, σpAir ≈ 0.61 b, indicating that the average number
of cc-pairs produced in p–Air collisions is larger than one. 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 [
46
] which,
so far, do not include any heavy-quark production. Given that
the measurements [
47
] observe unexplained excesses in the
number of muons compared to the model predictions, and
that charmed and bottom mesons feed less the non-muonic
component of the air-shower, it is worth to explore the impact
of properly including all such multiple heavy-quark
production in the MC generators used in high-energy cosmic-ray
physics.
5 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 σeff,tps,pA
parameter characterizing the transverse densities of partons
in the proton and nucleus. 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,
the average number of cc-pairs produced in p–Air collisions
is larger than one. 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
and cosmic-ray physics at very high energies.
Acknowledgements Discussions with A.P. Kryukov and M.A.
Malyshev on TPS, and with M. Cacciari, M. Czakon, A. Mitov and G. Salam
on NNLO heavy-quark calculations are gratefully acknowledged.
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Funded by SCOAP3.
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