Double hard scattering without double counting
Double hard scattering without double counting
Markus Diehl 0 1 2 5
Jonathan R. Gaunt 0 1 2 3
Kay Sch¨onwald 0 1 2 4
Open Access 0 1 2
c The Authors. 0 1 2
0 Platanenallee 6 , 15738 Zeuthen , Germany
1 De Boelelaan 1081 , 1081 HV Amsterdam , The Netherlands
2 Notkestraße 85 , 22607 Hamburg , Germany
3 Nikhef Theory Group and VU University Amsterdam
4 Deutsches Elektronen-Synchrotron DESY
5 Deutsches Elektronen-Synchroton DESY
Double parton scattering in proton-proton collisions includes kinematic regions in which two partons inside a proton originate from the perturbative splitting of a single parton. This leads to a double counting problem between single and double hard scattering. We present a solution to this problem, which allows for the definition of double parton distributions as operator matrix elements in a proton, and which can be used at higher orders in perturbation theory. We show how the evaluation of double hard scattering in this framework can provide a rough estimate for the size of the higher-order contributions to single hard scattering that are affected by double counting. In a numeric study, we identify situations in which these higher-order contributions must be explicitly calculated and included if one wants to attain an accuracy at which double hard scattering becomes relevant, and other situations where such contributions may be neglected.
1 Introduction 2 3 5
A Fourier integrals
Leading order analysis: collinear factorisation
Squared box graph
Leading order analysis: TMD factorisation
Subtraction formalism in momentum space
Subtraction terms at higher orders
Double DGLAP evolution
Resummation of large logarithms
Dependence on the cutoff scale
Approximation of the intrinsic distribution
Collinear DPDs in momentum space
Comparison with other work
The approach of Manohar and Waalewijn
The approaches of Blok et al. and of Ryskin and Snigirev
Simplified analytic estimates
Collinear parton luminosities
Production of two scalars
Setting the scene
Contributions to the cross section: power behaviour and logarithms
Short-distance limit of DPDs
Collinear DPDs: splitting contribution
Collinear DPDs: all contributions
A scheme to regulate DPS and avoid double counting
The precise description of high-energy proton-proton collisions in QCD is imperative for
maximising the physics potential of the LHC and of possible future hadron colliders. An
important issue in this context is to understand the mechanism of double parton scattering
(DPS), in which two pairs of partons undergo a hard scattering in one and the same
proton-proton collision. In many situations DPS is suppressed compared with single parton
scattering (SPS), but this suppression generically becomes weaker with increasing collision
energy. For specific kinematics or specific final states, DPS can become comparable to or
even larger than SPS. An overview of recent experimental and theoretical activities in this
area can for instance be found in [1, 2].
Consider a DPS process pp → Y1 + Y2 + X, where Y1 and Y2 are observed particles or
groups of particles produced in two separate hard scattering processes, whilst X denotes
all unobserved particles in the final state. A cross section formula has been put forward
long ago in the framework of collinear factorisation, where the transverse momenta q
and q2 of Y1 and Y2 are integrated over . A corresponding expression has been given
in [4, 5] for transverse momentum dependent (TMD) factorisation, where q1 and q2 are
small compared with the hard scales in Y1 and Y2.
Inside a proton, the two partons that take part in the two hard scatters can originate
from the perturbative splitting of one parton. The relevance of this splitting mechanism
for the evolution equations of double parton distributions (DPDs) has been realised long
ago [6, 7] and studied more recently in [8–10]. However, it was only noted in [4, 5] that
the same mechanism dominates DPDs in the limit of small transverse distance between
the two partons, and that the splitting contribution leads to infinities when inserted into
the DPS cross section formula. These infinities are closely connected with double counting
between DPS and SPS in particular Feynman graphs, a problem that had been pointed
out earlier in the context of multi-jet production .
Different ways of dealing with this issue have been proposed in [12–15] and . As
discussed later in this paper, we find that these proposals have shortcomings either of
theoretical or of practical nature. In the present work we present an alternative scheme for
computing the cross section in a consistent way, including both DPS and SPS (as well as
other contributions). Our scheme allows for a nonperturbative definition of DPDs in terms
of operator matrix elements, and it is suitable for pushing the limit of theoretical accuracy
Our paper is structured as follows. In section 2 we recall the theoretical framework for
describing DPS and specify the theoretical problems mentioned above. The short-distance
behaviour of DPDs is discussed in section 3, since it is essential for the scheme we propose
examples of its application at next-to-leading order (NLO) in section 5. Collinear DPDs
evolve according to DGLAP equations, and in section 6 we discuss several consequences
of this scale evolution. Our scheme is naturally formulated with DPDs that depend on
the transverse distance y between the two partons, but we show in section 7 how one may
instead use DPDs depending on the transverse momentum conjugate to y. This allows
us to compare our results with those of [12, 13] and [14, 15], which we do in section 8.
Whilst the focus of the present work is theoretical rather than phenomenological, we give
in section 9 some quantitative illustrations of our scheme, obtained with a relatively simple
ansatz for the DPDs. We summarise our findings in section 10. Some Fourier integrals
required in the main text are given in an appendix.
Setting the scene
In this section we recall theoretical issues originating from the perturbative splitting
mechanism in double parton distributions, namely the appearance of ultraviolet divergences in
the naive cross section for double parton scattering, the problem of double counting
between DPS and single parton scattering, and the treatment of the so-called 2v1 (two versus
one) contributions to DPS.1 We also give some basic definitions and results. Throughout
Since the perturbative splitting mechanism in DPDs leads to issues in the ultraviolet
(UV) region, renormalisation plays a crucial role in our context. We work in dimensional
Fbare(xi, zi, y) = 2p+
with twist-two operators
Oi (y, zi) = q¯ y − 2 zi
for quark distributions, where q denotes the bare field. W (z) is a Wilson line from z to
different quark polarisations. Analogous definitions hold for distributions involving gluons,
with quark fields replaced by gluon field strength operators. For ease of writing, we omit
colour labels on the operators and distributions throughout this work, bearing in mind
that different colour couplings are possible for the four parton fields in (2.1).
In the process of deriving factorisation, one finds that the proton matrix element
in (2.1) needs to be multiplied by a combination of so-called soft factors, which are vacuum
expectation values of products of Wilson lines. More information for the case of single
parton distributions can be found in chapter 13.7 of  and in the recent overview .
A brief account for DPDs is given in section 2.1 of , and a more detailed discussion will
regulates rapidity divergences. A scheme in which a soft factor appears explicitly in the
cross section formula was presented in . Since the treatment of soft gluons does not
1We follow here the nomenclature of . The 2v1 contribution is referred to as 4 × 2 in , as 3 → 4
in [12, 13] (where four-jet production is considered), and as 1 × 2 in [14, 15].
affect parton splitting in any special way, we will not discuss it further in the present paper.
As a final step one performs UV renormalisation, which we assume to be done in the
MS scheme. The DPDs obtained from (2.1) are appropriate for TMD factorisation, with
the transverse positions zi being Fourier conjugate to the transverse parton momenta that
determine the final state kinematics. These distributions renormalise multiplicatively, with
one renormalisation factor for each product of a quark field (or gluon field strength) with
the Wilson line at the same transverse position and one factor for each pair of Wilson lines
at the same transverse position in the soft factors. Denoting the product of these factors
a renormalisation group equation which is a straightforward generalisation of the one for
TMDs (given e.g. in ).
traviolet divergences in the operators O1 and O2, and in the associated soft factors. The
where Zi renormalises the operators associated with parton i and where the convolution
products are in the momentum fractions xi. In the colour singlet channel, where both
operators Oi in (2.1) are colour singlets, the soft factors reduce to unity and one obtains
the renormalised twist-two operators that appear in single parton densities.
Since the operators associated with partons 1 and 2 renormalise independently (both
for the TMD and the collinear case) one may choose different renormalisation scales µ 1
and µ 2 in each of them. This is useful when the two hard subprocesses in double parton
scattering have widely different hard scales. In particular, one can then approach the
kinematics of the so-called underlying event, with a very hard scattering at scale µ 1 and
additional jet production at a much lower scale µ 2 (of course µ 2 needs to remain in the
perturbative region for our factorisation approach to be justified).
With different scales µ 1, µ 2 in the collinear DPDs, we have a homogeneous evolution
in µ 1 and its analogue for µ 2. For colour singlet DPDs, the kernels Pa1b1 on the r.h.s.
are the usual DGLAP splitting functions for single parton densities. In colour non-singlet
channels, both the DPDs and the splitting kernels have an additional dependence on the
gluons, respectively. Note that the polarisations of the two partons can be correlated with
each other, even in an unpolarised proton.
To simplify our presentation, we will consider the production pp → V1 + V2 + X of two
for which DPS factorisation can be established; in the case of TMD factorisation this
requires that the produced particles are colour singlets. We denote the four-momenta of
proton with momentum p (p¯) to move in the positive (negative) 3 direction. Furthermore
i e−Yi ,
with s = (p + p¯)2. For the phase space of each gauge boson we have
dxi dx¯i d2qi =
The “naive” cross section formulae (not taking into account the UV problems discussed
dx1 dx2 dx¯1 dx¯2 d2q1 d2q2
Z (d22πz)12 (d22πz)22 d2y e−i(q1z1+q2z2) Fb1b2 (x¯i, zi, y; µ i) Fa1a2 (xi, zi, y; µ i)
for TMD factorisation and
Z 1−x2 dx′1 Z 1−x′1 dx′2 Z 1−x¯2 dx¯′1 Z 1−x¯′1 dx¯′2
for collinear factorisation. The combinatorial factor C is 1 if the observed final states of the
hard scatters are different and 2 if they are identical. For simplicity we will consider the case
of , there are further contributions with DPDs that describe the interference of different
parton species. They can be discussed in full analogy to the contributions given in (2.6)
or (2.7), and we do not treat them explicitly in the present work for ease of notation.
different in the two factorisation frameworks, but in both cases they lead to a dependence
on the factorisation scale µ i that cancels against the µ i dependence of the DPDs, up to
As was pointed out in , the framework discussed so far suffers from problems in
the region of small transverse distances between the two partons in a DPD. The leading
behaviour of the collinear distributions F (xi, y) at small y can be computed from the
When inserted in the factorisation formula (2.7) this gives a quadratically
divergent integral at small y, which clearly signals an inappropriate treatment of the
ultraviolet region. As we will review in section 3.1, the short-distance behaviour of the
distributions F (xi, zi, y) is less singular but still leads to logarithmic divergences in the
TMD factorisation formula (2.6).
Instead of using DPDs depending on transverse positions, one may Fourier transform
them to transverse momentum space, integrating
(q1 − q2) .
d2−2ǫz1 d2−2ǫz2 e−i(z1k1+z2k2)
for TMDs and collinear distributions. For collinear distributions, this transformation must
additional subtraction as we will review in section 7. Rather than being associated with
the individual operators O1 and O2, this subtraction is related to the singularity arising
when the transverse distance y between the two operators goes to zero. It leads to an
that this extra µ dependence does not cancel in the cross section when (2.7) is rewritten
does not remove all UV divergences at the cross section level. The singularity of F (xi, y)
It is easy to identify the origin of the UV divergences just discussed. Both in the y and
thus left the region in which the approximations leading to the DPS cross section formulae
in ). Outside this region, the DPS approximations are not only unjustified, but they
give divergent integrals in the cross section.
This points to another problem, namely that of double counting contributions between
SPS and DPS. To see this, let us analyse the graph in figure 1a. Since the transverse boson
2 are approximately back to back (up to effects from the transverse
momenta of the incoming gluons) it is convenient to introduce the combination
(indicated by the boxes). (b) A 2v1 contribution to DPS, with a perturbative splitting DPD in
only one proton (indicated by the box). (c) A 2v2 contribution to DPS. Here and in the following it
is understood that partons emerging from the oval blobs are approximately collinear to their parent
proton. A line for the final state cut across the blobs and the produced vector bosons (wavy lines)
is not shown for simplicity.
momenta in the loops are all of order Qi. This carries the quark lines far off shell, so
that this contribution is naturally associated with SPS, with g + g → V1V2 as the hard
subprocess. A leading contribution is also obtained from the region where all transverse
quark momenta are much smaller than Qi. This region is naturally described as DPS,
with two disconnected hard scattering processes qq¯ → V1 and qq¯ → V2 and double parton
distributions with perturbative g → qq¯ splittings, indicated by the boxes in the graph.
We denote this as a 1v1 (1 versus 1) contribution to DPS, emphasising its close relation
A double counting problem for this graph obviously arises if one takes the loop integrals
in the SPS cross section over all transverse quark momenta (including the DPS region), and
likewise if one integrates the DPDs cross section over the full range of transverse positions,
which is equivalent to integrating over all transverse momenta in the quark loops. The
problem persists if one integrates the cross section over q.
Let us now turn to the graph in figure 1b and consider the cross section integrated
have transverse momenta of order q and are far off shell. The proper description of this
region is in terms of a hard scattering qq¯ + g → V1V2, convoluted with a collinear
twisttwo distribution (i.e. an ordinary PDF) at the top and a collinear twist-four distribution
at the bottom of the graph. For brevity we refer to this as the “twist-four contribution”
mations for DPS are appropriate. We call this the 2v1 contribution to DPS, recalling that
there is a qq¯+ g subprocess in the graph. Both small and large q give leading contributions
to the integrated cross section, and in a naive calculation adding up the twist-four term
and the DPS term has again a double counting problem, as well as divergences in each
contribution. The naive DPS cross section has a logarithmic divergence at small y, which
is seen by inserting the 1/y2 splitting behaviour of only one DPD in (2.7). In turn, the
hard scattering cross section in the twist-four term contains a collinear divergence in the
form of an integral behaving like dq2/q2 at q → 0, as we will show in section 4.1.2.
Clearly, one needs a consistent scheme for computing the overall cross section, without
double counting and without divergences in individual terms. An intuitive approach for
evaluating DPS is to separate the “perturbative splitting” part of a DPD from its “intrinsic”
nonperturbative part.2 This has been pursued independently by Blok et al. [12, 13] and
by Ryskin and Snigirev [14, 15]. Taking the intrinsic DPD for each proton, one obtains
the 2v2 part of DPS, which does not contain any perturbative splitting and is shown
in figure 1c. The splitting part of the DPD is explicitly computed in terms of a single
parton distribution function (PDF) and a perturbative kernel. This is multiplied with an
intrinsic DPD to compute the 2v1 term. Finally, the product of two splitting DPDs is used
to compute the 1v1 contribution in the approach of [14, 15], where an ultraviolet cutoff
must be imposed to regulate the quadratic divergence we mentioned earlier. By contrast,
the authors of [12, 13] advocate to omit this term and replace it entirely with the SPS
contribution to the cross section.
We are, however, not able to give a field theoretic definition of the “intrinsic” or
“nonperturbative” part of a DPD. The consideration of Feynman graphs in the preceding
arguments is instructive, but a satisfactory definition should only appeal to perturbation
theory in regions where it is applicable. We regard a nonperturbative definition of DPDs as
indispensable for a systematic theory approach, for instance for deriving evolution equations
and other general properties.
The setup we propose in this work defines DPDs as operator matrix elements as
described above, containing both splitting and intrinsic contributions. UV divergences in the
DPS cross section are avoided by introducing (smooth or hard) cutoffs in the integrations
over transverse distances. The double counting problems are treated within the subtraction
formalism used in standard factorisation theorems, described in detail in sections 10.1 and
10.7 of  and briefly recalled in section 4 here. The subtraction terms that avoid double
counting also remove the above mentioned collinear divergence in the twist-four term. A
distinction between “splitting” and “intrinsic” contributions to a DPD will be made in
the limit of small transverse distances, where it can be formulated in terms of an operator
product expansion (see section 3.3), and when making a model ansatz for DPDs at large
distances, which is of course necessary for phenomenology.
Contributions to the cross section: power behaviour and logarithms
In preparation for later sections, we now recall some results for the power behaviour of
different contributions to the cross section, referring to section 2.4 of  for a derivation.
We also recall which logarithms appear in the lowest order 1v1 and 1v2 graphs. As already
stated, we take the process pp → V1 + V2 + X as a concrete example, but the discussion
readily generalises to other cases.
using TMD factorisation. Here and in the following we write Qi to denote the generic size
of Q1 and Q2, and likewise for qi. The transverse momenta qi may be of nonperturbative
2The intrinsic part of a DPD may be studied using quark models [23–30], at least in the valence region,
or it may be related to the product of two PDFs if correlations between the two partons are neglected.
associated with DPS but the loop on the r.h.s. does not. (b) Interference between DPS in the
amplitude and SPS in the complex conjugate amplitude. (c) 1v1 graph in the region where the
quark loop on the left is collinear whilst the one on the right is hard. This corresponds to the
region of SPS/DPS interference, with the boxes indicating DPDs with perturbative splitting for the
parton pair in the amplitude.
transverse momentum dependent distributions in terms of collinear ones , but we shall
not discuss this here. The leading power behaviour of the cross section is
∼ Qi4 qi2
to the leading behaviour, namely DPS, SPS, and the interference between SPS and DPS.
Corresponding graphs are shown in figures 1c, 2a and 2b, respectively.
As discussed in the previous section, certain graphs contribute both to DPS and to
SPS, depending on the kinematics of their internal lines. The 1v1 graph in figure 1 also has
leading regions in which one of the loops is hard and the other is collinear. These regions
contribute to the SPS/DPS interference, as shown in figure 2c.
The double counting
problem thus concerns both SPS, DPS and their interference. Note that the SPS graph in
figure 2a contributes to the SPS/DPS interference but not to DPS.
Both the amplitude and its conjugate in the 1v1 graph contains a loop integral that
ference depends on how exactly one handles the double counting problem. We come back
to this issue in section 4.2.
Let us now turn to the cross section integrated over q1 and q2, which can be described
given by the SPS mechanism alone. Several mechanisms contribute at the power suppressed
1. DPS, which is suppressed because it can only populate the region qi ≪ Qi rather than the full phase space up to qi ∼ Qi,
for the other. (b) A graph with twist-three distributions for both protons.
2. the interference between SPS and DPS, which is suppressed for the same reason,
3. hard scattering with a twist-four distribution for one proton and a twist-two
distribution for the other. Example graphs are figure 3a, as well as figure 1b with the
4. hard scattering with twist-three distributions for both protons. An example graph is
The rationale for considering such contributions is that — whilst being power suppressed
compared with SPS — they may be enhanced by higher parton luminosities at small
momentum fractions x, or by coupling constants in the relevant hard scattering subprocesses.
corrections would be a formidable task, and it is not even established whether factorisation
(in particular the cancellation of Glauber gluons) holds at that level.
Notice that in collinear factorisation, the SPS/DPS interference term involves collinear
twist three distributions for both protons, because the SPS mechanism forces the two
partons in the interfering DPS amplitude to be at same transverse position (see section 2.4.1
in ). In this sense, mechanism 2 in the above list may be regarded as a special case of
mechanism 4, with a disconnected hard scattering in the amplitude or its conjugate (see
A full treatment of contributions with twist-three or twist-four distributions is beyond
the scope of this paper. We remark however that twist-n operators contain n or less than
n parton fields, and that different operators are related by the equations of motion. For a
detailed discussion we refer to [31–33]. Twist-n operators with n parton fields were called
“‘quasipartonic” in  and involve only the “good” parton fields in the parlance of
lightcone quantisation . These are exactly the fields appearing in the definitions of
multiparton distributions, so that graphs with a double counting issue between higher-twist hard
scattering and DPS (or the SPS/DPS interference) involve only quasipartonic operators.
The matrix elements of quasipartonic twist-three operators in an unpolarised target
satisfy the important selection rule that the helicities carried by the parton lines must
balance. This excludes three-gluon operators since three helicities ±1 cannot add up to
zero. For quark-antiquark-gluon operators it forces the quark and antiquark fields to have
index j is contracted with the polarisation index of the gluon. As for non-quasipartonic
twist-three distributions in an unpolarised target, one finds that they are absent in the
pure gluon sector , whereas the corresponding quark-antiquark distributions are again
chiral-odd . Since chiral-odd distributions cannot be generated by gluon ladder graphs,
they lack the small x enhancement that is one of the motivations to keep higher twist
contributions in the cross section. We will therefore not discuss them further in this work. Note
that corresponding selection rules do not hold for TMD correlators, where an imbalance
in the helicities of the parton fields can be compensated by orbital angular momentum.
Let us finally recall the appearance of DPS logarithms in collinear factorisation. The
∼ 1/q2 in the region Λ ≪ q ≪ Qi, which
counting between DPS and the twist-four mechanism is resolved, this logarithm can appear
in different contributions to the cross section. We will discuss this in section 4.1.2.
Short-distance limit of DPDs
In this section, we analyse the behaviour of DPDs in the limit where the transverse distance
between partons becomes small compared with the scale of nonperturbative interactions.
In this region, the splitting of one parton into two becomes dominant. Generalising results
steps when constructing a factorisation formula for the cross section.
A useful choice of position variables for describing the parton splitting mechanism is
with Fourier conjugate momenta3
y± = y ± 2
(z1 − z2) ,
K = k1 + k2 .
The relation between DPDs in position and momentum space reads
d2−2ǫK d2−2ǫk+ d2−2ǫk− eiZK+i(y+k−−y−k+) F (xi, k±, K)
distance between the two partons on the left (right) hand side of the final state cut in the
DPD. Correspondingly, the transverse momentum difference between the partons on the
occur at y± → 0 or k± → ∞.
ments. Here and in the following, the line for the final-state cut of the spectator partons is not
The perturbative splitting contribution Fspl,pt to transverse-momentum dependent
DPDs in momentum space has been calculated at leading order in section 5.2.2 of .
Generalising these results to D = 4 − 2ǫ dimensions, we have
Fa1a2,spl,pt(xi, k±, K) =
kj− kj+ (2µ )2ǫ
where j, j′ are transverse Lorentz indices and fa0 (x1 + x2, K) is an unpolarised
singleparton TMD.4 The ellipsis denotes a term that involves a TMD for polarised partons in
an unpolarised proton and depends on K but not on k±. In position space we then get
Fa1a2,spl,pt(xi, y±, Z) =
using the Fourier integral (A.2), where the term denoted by an ellipsis depends on Z but
not on y±. It is understood that for transverse quark or linear gluon polarisation, both
Fa1a2 and the kernel T carry additional transverse Lorentz indices. fa0 (x1 + x2, Z) is the
Fourier transform of fa0 (x1+x2, K). The form (3.4) gives the leading behaviour of the DPD
d2y d2z1 d2z2 e−i(q1z1+q2z2) =
d2Z d2y+ d2y− e−i(q1+q2)Z e−iq(y+−y−)
with q defined in (2.10). Performing the angular integration in
4Compared with section 5.2 of , the kernel T jj′ used here has the opposite order of indices jj′ and
includes a colour factor, e.g. TF = 1/2 for the colour singlet distribution 1Fg→qq¯.
are finite for y → ∞ due to the oscillations of the Bessel functions.
with additional partons radiated into the final state as shown in figure 5a, as well as virtual
this contribution is outside the scope of the present work, so that we will limit our analysis
of TMD factorisation to perturbative splitting at LO.
To compute the DPD cross section, we must also consider the case where only one of
a perturbative splitting only on one side of the final-state cut, as illustrated in figure 5b.
We will not discuss the detailed expression of the DPD in this regime, but give its general
structure. Setting D = 4 for simplicity, we have
momentum dependent twist-three distribution, constructed from the hadronic matrix
ele(Z − y+)
(y − Z)
three parton fields.
as shown in figure 5c. We have
defined in analogy. In the second step of (3.9) we have used translation invariance and
shifted the parton fields to the same position as in the corresponding DPD (see figure 4).
equations parton helicities are taken fixed on the l.h.s. and must be appropriately summed
over on the r.h.s. Note the difference between this notation and the labels ai, which denote
parton species and polarisation (none, longitudinal, transverse or linear) and thus refer to
a pair of parton legs. The notation with ai is hence not suitable for distributions with
Notice that a quark and antiquark produced by perturbative splitting have opposite
heliciwith perturbative splitting only to the right of the final-state cut. The blob denotes a distribution
Inserting (3.10) into (3.8) we obtain
yj+ y2j− Uαj0→α1α2 (xi) Uαj′0→α1α2 (xi) ∗ fα0 (x1 + x2, Z) .
Taking appropriate linear combinations of parton helicities, we recover the form of
Fa1a2,spl,pt in (3.5) at ǫ = 0.
Collinear DPDs: splitting contribution
distributions for two unpolarised or two longitudinally polarised partons, so that the DPDs
do not carry any transverse Lorentz indices. The lowest order splitting has been computed
in . For 4 − 2ǫ dimensions, one finds the general form
Fa1a2,spl,pt(x1, x2, y; µ )
µ 2ǫ Γ2(1 − ǫ) fa0 (x1 + x2; µ ) αs(µ )
The kernel for the splitting g → qq¯ reads for instance
Pg→qq¯ (u, ǫ) =
f u2 + (1 − u)2 − ǫ
1 − ǫ
with a factor f = 1 for the colour singlet and f = −1/√N 2
− 1 for the colour octet DPD.
Going beyond leading order, one can deduce the general form of the perturbative
splitting contribution using dimensional analysis and boost invariance. For colour singlet
distributions one finds
formulae for hard scattering processes. Both f and V on the right-hand side are understood
V is a double series
The µ (and thus on dimensional grounds the y) dependence of V follows from the fact that
terms with n > m in (3.15) are due to the subtractions of ultraviolet or collinear
divergences. At lowest order, the hard splitting graphs are disconnected (with no partons across
are 1/ǫn−1, we find
in the physical limit ǫ = 0.
For colour nonsinglet DPDs one must regulate rapidity divergences, which complicates
the preceding result. Taking e.g. Wilson lines along non-lightlike paths introduces
additional vectors and changes the analysis of boost properties of the kernel. We will not pursue
DPDs with transverse quark or linear gluon polarisation carry transverse Lorentz
indices. Their perturbative splitting expressions thus have a tensor structure containing
additional factors of yj /y compared with the formulae above. At leading order one readily
Collinear DPDs: all contributions
Let us now study the small y behaviour of collinear DPDs in more general terms. We start
by writing the relation between unrenormalised DPDs in position and momentum space as
twist-four distribution, independent of any transverse variable. For small y, the second
Fy→0(y) = Fspl,pt(y) + Ftw3,pt(y) + Fint,pt(y) ,
xi and ui denote longitudinal momentum fractions.
One may also derive the expansion (3.18) from the operator product expansion, without
taking recourse to the transverse momentum representation (3.17). In the definition of
fields are at transverse position 0. These operators have twist 2, 3, 4 for the first, second
and third term in (3.18), respectively.
The spitting contribution Fspl,pt is given by graphs as in figures 4 and 5a and has
already been discussed in the previous subsection. The term Ftw3,pt originates from two
types of graphs. The first type involves a single perturbative splitting and a quasipartonic
collinear twist-three distribution as shown in figure 5b. The second type has two splittings
as in figure 4 and a twist-three distribution with one “good” and one “bad” parton field.
Given the helicity constraints discussed in section 2.1, collinear twist-three distributions in
an unpolarised proton involve a quark and antiquark with opposite chirality (and possibly
an extra gluon). As announced earlier, we discard twist-three terms in the following, since
they are expected to become unimportant at small momentum fractions x1, x2.
Finally, the term Fint,pt contains contributions without any perturbative splitting; we
hence refer to it as the “intrinsic” part of the DPD. It can be written as
Fint,pt(x1, x2, y; µ ) = G(x1, x2, x2, x1; µ ) + C(· · ·, y; µ ) ⊗ G(· · ·; µ ) + . . .
where G is a quasipartonic collinear twist-four distribution and C a perturbative splitting
momentum fractions indicated by · · · (cf. figure 6a). The first term in (3.19) corresponds to
the first term in (3.17) and is the only contribution that does not involve a hard splitting at
all. The ellipsis denotes terms with non-quasipartonic twist-four distributions containing
three or two parton fields, together with one or two parton splittings. While having the
same power behaviour in y, one may expect that at small x1, x2 these terms become less
important than the terms with quasipartonic twist-four distributions, which should roughly
grow as fast as the square of two parton densities.
We now take a closer look at the second term in (3.19). The kernel C can be determined
interactions, i.e. interactions connecting partons 1 and 2 as in figure 6a and b, contribute
to C. The ladder graph in figure 6c is independent of y and thus gives identical contributions
to the matrix elements Fint,pt and G. As a consequence it does not contribute to C.
At this point we can comment on the scale evolution of the different terms in (3.19).
The l.h.s. evolves according to the homogeneous double DGLAP equation for DPDs, which
describes “diagonal” interactions, either between the partons with final momentum fraction
x1 or between those with final momentum fraction x2. By contrast, the evolution of
G(x1, x2, x2, x1; µ ) contains both diagonal and non-diagonal ladder interactions . The
non-diagonal interactions in the evolution must thus be cancelled by the µ dependence of
function C, which indeed contains just non-diagonal interactions as just discussed.
We finally emphasise that an unambiguous decomposition of F (y) into splitting,
intrinsic and twist-three parts is only possible in the limit of small y. If y is of hadronic size,
neither the operator product expansion nor the notion of perturbative parton splitting
make sense. One may however use the short-distance decomposition (3.18) as a starting
point for a model parameterisation of DPDs in the full y range. We describe a simple
version of this strategy in section 9.
A scheme to regulate DPS and avoid double counting
In this section, we present a scheme that regulates the DPS cross section and solves the
double counting problem between DPS and SPS, as well as between DPS and the twist-four
contribution (figure 1b). Before doing so, we discuss some general considerations that
motivate our scheme.
bing double parton scattering.
The following properties are in our opinion desirable for any theoretical setup
descri1. It should permit a field theoretical definition of DPDs, without recourse to
perturbation theory. This is the same standard as for the ordinary parton distributions in SPS
processes. In particular, it allows one to derive general properties and to investigate
these functions using nonperturbative methods, for instance lattice calculations.
One may object that so far not even ordinary PDFs can be computed to a precision
sufficient for phenomenology. However, important progress has been made in the
area of lattice computations, and more can be expected for the future. Furthermore,
whereas ordinary PDFs are being extracted with increasing precision from
experiment, it is hard to imagine a similar scenario for DPDs, because of their sheer
number and because DPS processes are much harder to measure and analyse than most
processes from which PDFs are extracted. In such a situation, even semi-quantitative
guidance from nonperturbative calculations (such as the relevance of correlations of
different types) is highly valuable.
As already discussed in section 2, the requirement of a nonperturbative definition
prevents us from separating the “perturbative splitting” contribution of a DPD in a
2. To pave the way for increased theoretical precision, the scheme should permit a
formulation at higher orders in perturbation theory. Furthermore, the complexity of
the required higher order calculations should be manageable in practice.
3. For collinear factorisation, one would like to use as much as possible existing
higherorder results for SPS processes, namely partonic cross sections and splitting functions.
This means that the scheme should not modify the collinear subtractions to be made
in hard scattering kernels, nor the validity of standard DGLAP evolution for DPDs
in the colour singlet channel.
4. For TMD factorisation, it is desirable not to modify Collins-Soper evolution and the
handling of rapidity divergences. This again allows one to re-use calculations done
for SPS, although rapidity evolution for DPS is necessarily more complicated due to
the complexities caused by colour [5, 21].
5. One would like to keep procedures as similar as possible for collinear and TMD factorisation. This will in particular facilitate the computation of DPS processes at perturbatively large transverse momenta in terms of collinear DPDs , adapting the well known procedure for single Drell-Yan production .
In principle one can use dimensional regularisation to handle the UV divergences that
are induced in the DPS cross section by the perturbative splitting mechanism, as is done
with the UV divergences that arise in simpler situations such as single hard scattering.
However, contrary to that case, the UV divergences discussed in section 2 arise not at the
level of individual DPDs but only when two DPDs are multiplied together and integrated
over y. This means that if one treats these divergences in dimensional regularisation, only
possibility was explored in . However, DPDs and their products remain nonperturbative
functions at large y, which according to present knowledge cannot be reduced to simpler
quantities in a model independent way. In practice, one therefore needs to model or
parameterise them at some starting scale. This is more involved for the product of DPDs
than for DPDs themselves, as is the practical implementation of scale evolution. We will
come back to this scheme in section 8.
Ultraviolet regularisation. We define the regularised DPS cross section by
multiplying the integrand in the DPS formula (2.6) for measured transverse momenta with
Collinear and transverse-momentum dependent DPDs are defined as specified in
section 2, without any modifications. Constructed from operator matrix elements, they
contain both splitting and non-splitting contributions. They quantify specific
properties of the proton and have a simple physical interpretation, with the same limitations
as single parton densities. (We recall that a literal density interpretation of PDFs and
TMDs is hindered by the presence of Wilson lines and of ultraviolet renormalisation.)
Double counting subtraction. To treat the double counting between DPS and other
contributions, we adapt the recursive subtraction formalism of Collins, which we
briefly sketch now (details are given in sections 10.1 and 10.7 of ). Consider a
In each term one integrates over all loop momenta. The operator TR applies
approximations designed to work in momentum region R. Subtraction terms avoid double
counting the contributions from smaller regions R′ (regions that are contained in R).
In these subtraction terms one applies the approximations designed for R and those
approximation in the region R and in all smaller regions, and P
approximation of the graph in all relevant regions. All approximations discussed here
are valid up to power corrections.
In our context, we have graphs in which a set of collinear partons split into partons
that can be either collinear (as in DPS) or hard (as in SPS). A slight adaptation of
the above formalism is required since we compute DPS using DPDs and a regulating
collinear splitting region R′ then corresponds to large y and the corresponding hard
region R to small y, but we keep the ordering of regions R′ < R from momentum
space when implementing (4.1). We will show in section 4.3 that our use of
subtractions in position space is equivalent to the one in momentum space up to power
The subtraction terms for the DPS region turn out to have a very simple form. They
can be obtained by replacing the DPDs in the UV regularised DPS cross section with
their appropriate limits for small y± in TMD factorisation and for small y in collinear
factorisation. Details will be given in sections 4.1, 4.2 and 5.
Criteria 1 and 5 above are obviously satisfied in this scheme. Regarding criterion 2, we
note that the higher-order calculations required for the double counting subtraction terms
are for the short-distance limit of DPDs, which involve much simpler Feynman graphs than
the full scattering process.
fication of the definition of DPDs and thus respects criteria 3 and 4. In particular, the
collinear DPDs F (xi, y) in transverse position space follow the homogeneous DGLAP
evolution equation (2.3). Since for colour singlet DPDs, the evolution kernels are the familiar
DGLAP kernels, the associated scale dependence in the cross section cancels by
construction against the one of the hard cross sections computed with the same collinear subtraction
to the UV renormalisation scale µ in DPDs, but we find it useful to keep it separate
For the single PDFs fa(x, µ 0), we take the MSTW2008LO distributions . In (9.9) we
have multiplied the PDFs by a function of the xi that does not affect the DPD at small
for different parton species. For this we use a simplified version of the model in section 4.1
of , taking the width h to be x independent (corresponding to h(x1, x2) of  evaluated
ha1a2 = ha1 + ha2
hg = 2.33 GeV−2 ,
hq = hq¯ = 3.53 GeV−2 .
For the splitting piece of the DPD we generalise our ansatz in (9.3), choosing an
initialisation scale that goes to b0/y at small y but freezes to a constant value b0/ymax when y
exceeds a value ymax that marks the transition between perturbative and nonperturbative
evaluated at too low scales. A suitable choice of scale is
µ y =
in the recent TMD study . Using the same parton dependent Gaussian damping as
in (9.13), we have
To obtain the splitting and intrinsic DPDs at scale µ , as in (9.8), the input forms
just discussed must be evolved, starting from µ 0 for Fint and from µ y for Fspl, according
to the homogeneous double DGLAP equations. For this we use a modified version of the
code developed in . The modified code works on a grid in the xi, µ and µ y directions
(the grid of the original code is in xi and µ only). The grid points in the xi directions
spaced in log µ or log µ y. The integrals appearing in the double DGLAP equations are
computed from points in the xi grids using Newton-Cotes rules (for details see appendix
A of ), and evolution from one point in the µ grid to the next is carried out using the
with 60 points in the µ direction, spanning µ 0 < µ <
170 GeV, and with 60 points in the
µ y direction, spanning b0/ymax < µ y < 340 GeV. According to the studies made in , this
suggests an error on the level of a few per cent in the DPD values obtained after evolution,
which is tolerable in this first study. The DPDs computed on this grid are used together
with an interpolation code to produce numerical values for the investigations below.
production of a W boson pair). Taking the collider energy to be √
s = 14 TeV, we set x1
and x¯1 in (9.1) to correspond to central production of the first system and x2 and x¯2 to
correspond to the production of the second system with rapidity Y (all rapidities refer to
the pp centre of mass). This gives
x1 = x¯1 = 5.7 × 10−3 ,
x2 = 5.7 × 10−3 exp(Y ) ,
x¯2 = 5.7 × 10−3 exp(−Y ) .
parton combination appears e.g. in ZZ production, the second is important in four-jet
production, and the last appears in W +W +. For ease of language, we will refer to these
parton combinations as the uu¯, gg and ud¯ channels, respectively. We split the overall
luminosity into contributions from 1v1 (Fspl × Fspl), from 2v1 (Fspl × Fint + Fint × Fspl)
value of 80 GeV, in order to see how DPS alone is affected by variation of this cutoff. For
expected from basic considerations, see (9.4)), so this appears as a dashed line in each plot.
For the 1v1 contribution of the gg and uu¯ channels, we also plot a band generated by
naive formula (9.6), where DPD evolution is neglected. Any discrepancy between this band
and the actual 1v1 band is therefore due to evolution effects. We do not plot this band for
the ud¯ channel: there is no LO splitting process giving ud¯, so that the scale variation (and
central value) from (9.6) is zero in this case.
We immediately notice in figures 12a and b that the 1v1 contribution is generally much
obtain a sensible prediction in these cases, one must include the SPS corrections up to the
order that includes the double box graph in figure 1a, so that the associated subtraction
significant evolution effects. We shall explore the reasons for these differences below.
with Q1 = Q2 = 80 GeV at √
Figure 12. Double parton scattering luminosities La1a2b1b2 (Y ) for the production of two systems
The parton combinations a1a2 b1b2 are uu¯u¯u + u¯uuu¯ (a), gggg (b) and ud¯d¯u + d¯uud¯ (c).
dependence. This is because, as already mentioned, there is no LO splitting directly giving
ud¯ (generation of a ud¯ pair requires at least two steps, such as u → u + g → u + d + d¯).
the order that contains the lowest-order DPS-type loop (in both amplitude and conjugate).
rather than one-step splittings are required in both protons), and the corresponding SPS
calculation will not be available for some time.
In the gg channel, the 2v1 and 2v2 contributions are of similar magnitude and shape.
This is consistent with what has been found in previous phenomenological studies [13, 51]
at similar scales and x values. In the uu¯ channel, the magnitude of 2v1 and 2v2 is broadly
the same but their shape is somewhat different. In the ud¯ channel, the shape of the two
contributions is similar, but 2v1 lies well below 2v2 (and is close to 1v1).
by adding the remaining fixed-order terms in the cross section, even without considering
defined as in (9.1) except that the DPDs are replaced by the fixed-order splitting expression
evaluated at scale µ ,
Fa1a2,spl,fo(x1, x2, y; µ ) =
For a given parton combination a1a2 b1b2, one can directly subtract L1v1,pt from the DPS
L1v1,pt is equal to that of the full cross section. On the other hand, the overall magnitude
needs to add the SPS contribution to obtain the physical cross section.
is indeed reduced compared to LDPS, with the reduction being much stronger for uu¯ than
for gg. The latter is consistent with our previous observation that evolution effects are
weaker in the uu¯ channel at central rapidities: if evolution effects are weak, Fspl and Fspl,fo
allows us to sharpen the argument we already made in the discussion of figure 12: not only
traction luminosities, shown for the uu¯ (a) and gg (b) channels.
Kinematic conditions are as in
may even serve as a rough estimate for the size of these terms.
To better understand the behaviour we have seen in the 1v1 luminosities for the uu¯
and gg channels, we now take a closer look at how evolution affects the y dependence of
also plot the ug DPD, which mixes with uu¯ and gg under evolution. For comparison we
from which Fspl(y; µ ) is obtained by evolution. The distributions are plotted in the range
b0/(160 GeV) < y < b0/(40 GeV), i.e. in the range over which the lower integration limit
damping factor in our DPD model is irrelevant, so that Fspl,ini and Fspl,fo only differ by the
For the initial conditions Fspl,ini we note that the uu¯ and ug distributions are of similar
size; the former is initialised by a larger PDF (fg instead of fu) but has a smaller splitting
coefficient P (1/2) as we noted before (9.6). By contrast, the gg distribution is much bigger;
here both the initialising PDF and the splitting coefficient are large.
is more significant for gg and ug than for uu¯, i.e. that DPD evolution has a much stronger
effect on the former two channels. This is to be expected at small x, since the 1/v behaviour
The parton combinations shown are uu¯ (a), gg (b) and ug (c). The vertical grey lines correspond
to y = b0/(160 GeV), y = b0/(80 GeV) and y = b0/(40 GeV).
of the splitting kernels Pgg(v) and Pgq(v) at small v favours the radiation of a gluon with
much smaller momentum than its parent parton, whereas the kernels Pqq(v) and Pqg(v)
giving a quark stay finite for v → 0.
An interesting point to note is that, whilst for gg and ug the curves for Fspl are more
shallow than for Fspl,ini, in the case of uu¯ the Fspl curve is actually steeper. The latter
is surprising since — based on the experience with single PDFs at small x — one may
expect that forward evolution for y > b0/(80 GeV) would always increase a DPD, and
backward evolution for y < b0/(80 GeV) would always decrease it. This indeed happens in
the gg and ug channels, whilst forward evolution results in a decrease of the uu¯ DPD. The
reason for this is that Fug,spl and Fgu¯,spl, which directly feed into Fuu¯,spl during evolution,
are comparatively small. In the case of PDFs, fg is much larger than fu and hence can
drive its small-x evolution although the splitting function Pqg does not favour the radiation
of low-momentum quarks. The evolution of the gg and ug DPDs is driven by the large
distribution Fgg,spl and enhanced by the Pgg splitting function.
Note that in all three channels, Fspl has a smaller slope in y than Fspl,fo. This implies
For uu¯, the evolution from µ y to µ turns out to have much the same quantitative effect
as adjusting the scale in the fixed order expression from µ y to µ , such that Fspl and Fspl,fo
of the 1v1 uu¯ contribution in figure 12a coincides almost exactly with the naive prediction
LDPS, which we indeed see in figure 13a.
In the gg channel, the modification of the y slope by evolution is significant. In
decrease. This was already anticipated in [14, 15], where studies in the double leading
logarithm approximation were performed.
Here we investigate this effect using our full LO DGLAP set-up. We consider the gg
channel with all x values set equal and study the DPS luminosity as a function of the
to 7 GeV < µ <
varies with x. We make one plot with √
2800 GeV) and another one with higher collider energy √
s = 100 TeV
s = 14 TeV and 5 × 10−4 < x < 0.2 (corresponding
and 10−4 < x < 0.02 (corresponding to 10 GeV < µ <
2000 GeV). In our numerical code
this requires a new DPD grid with larger µ and µ y ranges, which was generated using 60
points in the µ and µ y directions (as in the original grid). The resulting 1v1 luminosity is
comparison we also show the 2v1 and 2v2 luminosities.
central value as x (and µ ) decreases. At the lowest x values it is much smaller than the
variation becomes larger: for √
dominate over 1v1. For given µ , this effect is more pronounced for √
the left of each plot, where x and µ become small, the 2v2 and 2v1 contributions begin to
s = 100 TeV than for
kinematics of figure 15.
is significantly shallower in y than the fixed-order y−2 form. For √
s = 14 TeV we find that
given at the top of the plot.
a behaviour like y−0.5 is reached around 5 × 10−4. For √
s = 100 TeV the y behaviour is like
as the subtraction term. This is confirmed in figure 16, which is analogous to figure 13b
7.1 × 10−4, corresponding to Q1 = Q2 = 10 GeV at √
s = 14 TeV. Notice the suppressed zero on
DPS-type loop, one may in this situation justifiably make predictions that include the
DPS piece but omit the order of SPS just specified, as well as the associated subtraction
term. This is encouraging, for instance in the context of four-jet production, where the
computation of the relevant SPS order (namely NNLO) is well beyond the current state of
the art. Note that lower orders in SPS should be computed and included, if possible.
than the fixed-order form y−2.
splitting DPD in perturbation theory and must rely on a model. Likewise, the 2v1 part of
Another kinematic regime where the 1v1 contribution to LDPS becomes large compared
in figure 12, but only in the uu¯ and not in the gg channel. In both channels, the effect
becomes more pronounced once the rapidity separation of the hard systems is increased
beyond 4. To illustrate this, we make plots similar to figure 12 but now with one hard
0), such that a given value of Y corresponds to a rapidity separation of 2Y . The results are
reduced towards the right hand side of the plots, becoming hardly visible in the uu¯ channel.
Also notable is the fact that for large Y , the 2v2 contribution in this channel exceeds 1v1,
which strongly decreases between Y = 0 and 2.
splitting DPDs with one large x and one small x parton. From the point of view of small x
logarithms, it is preferable to generate such a configuration by having the 1 → 2 splitting
with Q1 = Q2 = 80 GeV at √
Figure 17. Double parton scattering luminosities La1a2b1b2 (Y ) for the production of two systems
s = 14 TeV, one with rapidity Y and the other with rapidity −Y .
The parton combinations a1a2 b1b2 are uu¯u¯u + u¯uuu¯ (a) and gggg (b).
at large x, generating directly the large x parton plus a gluon with smaller x, the latter
of which splits in a number of stages into smaller x gluons, eventually yielding the small
x parton. This increasingly happens with increasing y, since the “evolution distance”
between the initial and final scales, µ y and µ , is increased. Thus, evolution again flattens
The effect is particularly drastic in the uu¯ channel, because the lowest-order splitting
g → uu¯ is inefficient at generating a pair with very different x values (as Pg→qq¯(v) goes to a
constant at small v). Therefore, the repeated splitting mechanism described in the previous
paragraph is strongly preferred, even though its last step is penalised by the lack of small-v
the curve for uu¯ crossing zero rather close to this point. In the former case, the
lowestorder 1 → 2 splitting process is forbidden, whereas in the latter it is numerically almost
irrelevant. The situation for x1 ≫ x2 is the same.
To end this section, we study polarised distributions and luminosities. We limit
ourselves to the splitting part Fspl of the DPD and to the 1v1 contribution to LDPS. In fact,
we have little guidance for modelling the intrinsic part Fint, where a product ansatz as
in (9.9) makes no sense. We note that in , different ansa¨tze were made for polarised
DPDs, and the effects of evolution on these distributions were investigated. It was found
that, in many cases, evolution to higher scales leads to a suppression of the polarised DPD
with respect to its unpolarised counterpart.
which is needed for the point Y = 4 in figure 17a.
We initialise the polarised Fspl at µ y using the expression in (9.13) with the unpolarised
1 → 2 splitting functions replaced by their polarised counterparts (given in appendix
These distributions are then evolved to the scale µ using the appropriate
polarised double DGLAP equations (the required polarised splitting functions are collected
in appendix A of ). The settings we used for the evolution code are the same as specified
in section 9.2.1.
Q1 = Q2 = 80 GeV at √
We consider the same scenario as in figure 12, i.e. the production of two systems with
and now include polarised 1v1 contributions. In figure 19a, we reproduce figure 12a but
transverse quark polarisation, respectively. These polarised luminosities appear in the DPS
also appear but are not shown here). To avoid cluttering the plots, we omit the unpolarised
2v1 luminosity, reminding the reader that it is of the same magnitude as 2v2. In figure 19b
we repeat the exercise for the pure gluon channel, including the 1v1 luminosities for the
evolution has a rather weak effect on all three DPDs for central rapidities. This close
(and the unpolarised 2v1 band omitted).
the other two for larger Y , because Pg→qq¯ > |Pg→δq δq¯| for v 6= 1/2.
somewhat below the one for gg. This is mostly driven by the differences in initialisation
respectively. The remaining difference is due to evolution effects, which increase gg rather
Overall, we see that the polarised 1v1 luminosities can be of the same order of
magnitude as the unpolarised ones, so that one must in general take into account all possible
polarisation combinations in the DPS contribution, together with the SPS term and
subtraction (where the subtraction will also contain both unpolarised and polarised contributions).
Production of two scalars
is of similar size as the corresponding perturbative order of the SPS cross section. The
process we investigate is an artificial one, chosen for ease of computation. We consider
one light quark flavour; including further light flavours just multiplies all following results
(SPS and subtraction) by nf . We will not compare the subtraction term to the full SPS
hence be meaningfully considered by itself.
In keeping with the notation of the paper, let us denote the mass of each produced
p1 − q2/qm2ax
p1 − β2 and the gg → φφ matrix element squared, |Agg→φφ|2, is
averaged over the spin and colour of the incoming gluons. The momentum fractions x and x¯
element squared given in  by making the replacement
The right hand side of (9.18) is given in  for general quark mass mq. We evaluate this
expression numerically for very small mq, using analytic mq → 0 approximations where
this is necessary to avoid numerical instabilities.
ambiguity. The DPS subtraction term is given by
x(1+β), 2 x(1−β), y RFb1b2,spl,pt 2
where we sum over all possible colour, spin, and quark number interference/correlation
same expression, but with the quark number diagonal distributions replaced by the quark
number interference ones (see section 2 of ).
The splitting kernels for the relevant spin combinations are 
v2 + (1 − v)2 , 1
Pg→δqδq¯ (v) = − δjj′ v(1 − v) (9.20)
term with interference DPDs I gives the same contribution as that with F , since the
corresponding diagrams are simply related by reversing the direction of fermion flow in
one of the two quark loops, and this does not change the expression for the diagram. The
squared subprocess amplitudes, including an average over colour in the initial state, read
where j, j′ are the indices for transverse quark polarisation and where we have used the
spin projection operators given in equation (2.90) of .
Inserting (9.20) and (9.21) into (9.19), we obtain
Z ∞ dy2
Nc2 − 1 0
Note that both (9.16) and (9.22) contain a product of gluon PDFs evaluated at the same
x values, g(x) g(x¯). For the comparison, we can divide the common PDF factor out of
the two expressions, in order to avoid having to use an explicit parameterisation. We also
divide out various factors appearing in both expressions and compare the quantity
low q approximation to the matrix element squared outside the region where the
approxwhich correspond to the high energy limit and the threshold limit, respectively. It is to
be expected that the agreement is especially bad at these points, since in the subtraction
term we effectively assume that the integration over the squared transverse momentum q2
this assumption becomes a poor one: near threshold the phase space in q2 shrinks to zero,
whilst in the high energy limit, q2 can go up to size sˆ ≫ Q2.
Consistently incorporating the perturbative splitting of one parton into two is a highly
nontrivial problem for the theoretical description of double parton scattering. DPS graphs
in which such splittings occur in both protons (1v1 graphs) overlap with loop corrections
to single parton scattering. Another type of graph, typically referred to as 2v1, in which
one parton pair arises from a perturbative splitting, and the other pair is an “intrinsic”
one already existing at the nonperturbative level, overlaps with twist-four contributions to
the cross section. Finally there is an overlap between DPS contributions where a splitting
occurs in both protons only in the amplitude or its conjugate, and SPS/DPS
We have presented a scheme to compute DPS and to consistently merge its
contribution to the cross section with SPS and the other terms just mentioned. The scheme works
in a similar manner for collinear and for TMD factorisation. Ultraviolet divergences that
arise from perturbative splitting in a naive treatment of DPS are regulated by a cutoff
DPDs, which are defined via operator matrix elements in close analogy to single-parton
distributions. In collinear factorisation, these DPDs hence evolve according to a
homogeneous DGLAP equation, whilst their TMD counterparts satisfy a generalisation of the
renormalisation group equations for single-parton TMDs. No modification of hard
scattering cross sections computed for standard collinear or TMD factorisation is necessary
in our scheme. Collins-Soper type equations describe the rapidity evolution of
transversemomentum dependent DPDs and of collinear DPDs in colour non-singlet channels .
The problem of double counting between DPS and other contributions — notably
between DPS and SPS — is solved by subtraction terms as specified in (4.16) and (4.22),
propriate short-distance limits. This paves the way for using the scheme at higher
orpared with the full hard scattering process at the corresponding order. With a suitable
choice of starting conditions and scales, specified in section 6.1, the DPS part of the
cross section correctly resums DGLAP logarithms that are not included in the fixed order
malised in transverse momentum space and satisfy an inhomogeneous DGLAP equation
rather than a homogeneous one. This relation has the form of a perturbative matching
equation, see (7.15), and is somewhat similar to the matching between PDFs defined in
different schemes such as the MS and the DIS scheme. The momentum space
representation also allows us to show that for the 2v1 contribution to DPS our scheme is equivalent
to the ones in [12, 13] and in [14, 15] to leading logarithmic accuracy.
For collinear DPDs, one can make a model ansatz consisting of two terms which in
the distribution. With such an ansatz, the DPS cross section naturally splits into 1v1,
2v1 and 2v2 terms, where 2v2 refers to contributions in which the parton pairs from both
protons are intrinsic. A crucial question is how large DPS is compared with SPS at the
perturbative order where graphs contribute to both mechanisms. This is especially acute
in collinear factorisation, where DPS is power suppressed with respect to SPS. Note that
only in very few channels (notably pair production of electroweak gauge bosons) SPS
calculations are available at the required order. We argue in section 6.2 that in our scheme
PDFs, overall coupling constants and kinematic region (small y, corresponding to large
transverse momenta and virtualities of internal lines). An alternative estimate is provided
small y. For the hypothetical process of scalar boson pair production from two gluons, we
have shown that the latter estimate works well within about a factor of two, provided that
boson pair rest frame is close to 0 or 1.
We constructed explicit (collinear) DPD input forms using the model ansatz just
described, restricting ourselves to three quark flavours for simplicity and ease of
These inputs were then numerically evolved to other scales using a code that
implements the homogeneous double DGLAP equation. We used the resultant DPDs to
compute so-called DPS luminosities (DPS cross sections omitting the process-dependent
hard parts) and plotted these under various conditions. We observed that generically, the
1v1 contributions to the luminosity (both unpolarised and polarised) are comparable to
This demonstrates that, when including DPS in a cross section calculation, one must in
with the associated subtraction term (with unpolarised and polarised partons). Otherwise,
one would have an uncertainty on the overall cross section that is as large as, or larger than
indeed comparable to the central value of the associated double counting subtraction term,
so that either of them may be used as an estimate for the SPS contribution.
luminosity is considerably smaller than the central value. As we argued above, one may
contribution is reduced are when √
when the flavour indices in both DPDs are ud¯ (the luminosity with this parton flavour
combination appears in W +W + production). The suppression of the DPS-like double box
s becomes very large compared to the hard scales Qi,
or when the rapidity separation between the produced hard systems is large. Both of these
scenarios involve small x values in the DPDs — in the first, both x values in each DPD
are small, whilst in the second, one x value in each DPD becomes much smaller than the
other. Such processes and kinematic regions are the most promising ones to make useful
calculations and measurements for DPS. In fact, several measurements investigating DPS
have already been made in kinematics with Qi ≪
s or with large rapidity differences. It
will be interesting in future work to make more complete and comprehensive predictions
for such processes and kinematic regions in our framework, including for instance the full
flavour dependence and contributions from all partonic channels for a considered final state.
Fourier transform of a fractional power.
As a corollary we obtain
using kj eiyk = −i∂/(∂yj) eiyk.
d2−2ǫk eiyk k
We now compute the integrals in (4.19), which are also needed
Z ∞ du
2F3 1, 1; 2, 2, 2; − 4
where b0 is defined in (4.3) and 2F3(1, 1; 2, 2, 2; z) is a generalised hypergeometric function,
proceed as follows:
Z ∞ du
Z ∞ du
where in the second case we have used integration by parts. Both (A.5) and (A.6) behave
Connection between the Fourier-Bessel transform and a cutoff. Consider the
Z ∞ dy
The following argument is similar to the derivation given in appendix B of . The integral
on the r.h.s. is readily performed and gives
Z ∞ du
J2(ur)h1 − e−u2/4i =
1 − e−r2
dv logn v =
The expression on the l.h.s. of (A.7) can be rewritten as
binomial series, we obtain
n + 1 k=0
are zero) and replaced the remaining ones by coefficients dk whose values are not important
here. Comparison with (A.8) gives the desired result (A.7).
We gratefully acknowledge discussions with V. Braun, K. Golec-Biernat, D. Boer,
A. Sch¨afer and F. Tackmann. Special thanks are due to T. Kasemets for valuable remarks
on the manuscript. J.G. acknowledges financial support from the European Community
under the FP7 Ideas program QWORK (contract 320389). Two of us (M.D. and J.G.)
thank the Erwin Schr¨odinger International Institute for Mathematics and Physics (ESI)
for hospitality during the programme “Challenges and Concepts for Field Theory and
Applications in the Era of LHC Run-2”, when portions of this work were completed. The
Feynman diagrams in this paper were produced with JaxoDraw [56, 57].
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Interactions at the Large Hadron Collider, arXiv:1506.05829.
International Workshop on Multiple Partonic Interactions at the LHC,
Collisions, Nuovo Cim. A 70 (1982) 215 [INSPIRE].
Phys. Lett. B 698 (2011) 389 [arXiv:1102.3081] [INSPIRE].
QCD, JHEP 03 (2012) 089 [Erratum ibid. 1603 (2016) 001] [arXiv:1111.0910] [INSPIRE].
Phys. Lett. B 84 (1979) 266 [INSPIRE].
QCD, Phys. Lett. B 113 (1982) 325 [INSPIRE].
perturbative QCD, Phys. Rev. D 68 (2003) 114012 [hep-ph/0304172] [INSPIRE].
Evolution and Momentum and Quark Number Sum Rules, JHEP 03 (2010) 005
Phys. Lett. B 697 (2011) 482 [arXiv:1011.6586] [INSPIRE].
multi-parton collisions, Eur. Phys. J. C 74 (2014) 2926 [arXiv:1306.3763] [INSPIRE].
Phys. Rev. D 83 (2011) 114047 [arXiv:1103.3495] [INSPIRE].
of perturbative QCD, Phys. Rev. D 86 (2012) 014018 [arXiv:1203.2330] [INSPIRE].
Phys. Lett. B 713 (2012) 196 [arXiv:1202.5034] [INSPIRE].
JHEP 01 (2013) 042 [arXiv:1207.0480] [INSPIRE].
Eur. Phys. J. A 52 (2016) 153 [arXiv:1509.04766] [INSPIRE].
exchange in the double Drell-Yan process, JHEP 01 (2016) 076 [arXiv:1510.08696]
factorisation, evolution and matching, preprint in preparation DESY-2017-014,
Model, Phys. Rev. D 87 (2013) 034009 [arXiv:1211.3132] [INSPIRE].
models, Phys. Rev. D 87 (2013) 114021 [arXiv:1302.6462] [INSPIRE].
quark models: a Light Front approach to the valence sector, JHEP 12 (2014) 028
Distributions: Perturbative and Non-Perturbative effects, JHEP 10 (2016) 063
distribution function, Phys. Rev. D 95 (2017) 034040 [arXiv:1611.04793] [INSPIRE].
simple model, Few Body Syst. 55 (2014) 381 [arXiv:1310.8419] [INSPIRE].
Parton Distributions, Few Body Syst. 57 (2016) 405 [arXiv:1602.00254] [INSPIRE].
dressed quark model, Phys. Rev. D 94 (2016) 074029 [arXiv:1606.05686] [INSPIRE].
of the nucleon. Proceedings, International School of Nucleon Structure, 1st Course, Erice,
 P.J. Mulders and J. Rodrigues, Transverse momentum dependence in gluon distribution and
fragmentation functions, Phys. Rev. D 63 (2001) 094021 [hep-ph/0009343] [INSPIRE].
Pair and W and Z Boson Production, Nucl. Phys. B 250 (1985) 199 [INSPIRE].
boson pair production in gluon fusion including interference effects with off-shell Higgs at the
 A.V. Belitsky and A.V. Radyushkin, Unraveling hadron structure with generalized parton
 M. Diehl and J.R. Gaunt, Double parton scattering in the ultraviolet: addressing the double
counting problem, in Proceedings, 7th International Workshop on Multiple Partonic
Interactions at the LHC (MPI@LHC 2015): Miramare, Trieste, Italy, November 23–27,
 M. Diehl and J.R. Gaunt, Double parton scattering in the ultraviolet: addressing the double
 Z. Ligeti, I.W. Stewart and F.J. Tackmann, Treating the b quark distribution function with
 R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart, Thrust at N3LL with
Power Corrections and a Precision Global Fit for alphas(mZ),
 I.W. Stewart, F.J. Tackmann, J.R. Walsh and S. Zuberi, Jet pT resummation in Higgs
 M. Mekhfi and X. Artru, Sudakov Suppression of Color Correlations in Multiparton
transverse-momentum-dependent parton densities and the Collins-Soper evolution kernel,
Phys. Rev. D 91 (2015) 074020 [arXiv:1412.3820] [INSPIRE].
contributions to double parton scattering production of two quarkonia, two Higgs bosons and
cc¯cc¯, Phys. Rev. D 90 (2014) 054017 [arXiv:1407.5821] [INSPIRE].
JHEP 05 (2013) 150 [arXiv:1303.0842] [INSPIRE].
JHEP 01 (2013) 121 [arXiv:1210.5434] [INSPIRE].
Nucl. Phys. B 309 (1988) 282 [INSPIRE].
descriptions of semi-inclusive processes at low and high transverse momentum,
JHEP 08 (2008) 023 [arXiv:0803.0227] [INSPIRE].
drawing Feynman diagrams. Version 2.0 release notes,
 R. Astalos et al., Proceedings of the Sixth International Workshop on Multiple Partonic  H. Jung , D. Treleani , M. Strikman and N. van Buuren eds. Proceedings of the 7th  N. Paver and D. Treleani , Multi-Quark Scattering and Large pT Jet Production in Hadronic  M. Diehl and A. Sch ¨afer, Theoretical considerations on multiparton interactions in QCD,  M. Diehl , D. Ostermeier and A. Sch¨afer, Elements of a theory for multiparton interactions in  R. Kirschner , Generalized Lipatov-Altarelli-Parisi Equations and Jet Calculus Rules ,  V.P. Shelest , A.M. Snigirev and G.M. Zinovev , The Multiparton Distribution Equations in  A.M. Snigirev , Double parton distributions in the leading logarithm approximation of  J.R. Gaunt and W.J. Stirling, Double Parton Distributions Incorporating Perturbative QCD  F.A. Ceccopieri , An update on the evolution of double parton distributions ,  M. Cacciari , G.P. Salam and S. Sapeta , On the characterisation of the underlying event ,  B. Blok , Yu. Dokshitser, L. Frankfurt and M. Strikman , pQCD physics of multiparton  M.G. Ryskin and A.M. Snigirev , A fresh look at double parton scattering ,  M.G. Ryskin and A.M. Snigirev , Double parton scattering in double logarithm approximation  A.V. Manohar and W.J. Waalewijn , What is Double Parton Scattering?,  J.R. Gaunt , Single Perturbative Splitting Diagrams in Double Parton Scattering,  J. Collins , Foundations of perturbative QCD , Cambridge University Press, ( 2013 ).
 T.C. Rogers, An Overview of Transverse Momentum Dependent Factorization and Evolution,  M. Diehl, J.R. Gaunt, D. Ostermeier, P. Plo¨ßl and A. Sch¨afer, Cancellation of Glauber gluon  M. Buffing, M. Diehl and T. Kasemets, Transverse momentum in double parton scattering:  A.V. Manohar and W.J. Waalewijn, A QCD Analysis of Double Parton Scattering: Color Correlations, Interference Effects and Evolution, Phys. Rev. D 85 (2012) 114009  H.-M. Chang, A.V. Manohar and W.J. Waalewijn, Double Parton Correlations in the Bag  M. Rinaldi, S. Scopetta and V. Vento, Double parton correlations in constituent quark  M. Rinaldi, S. Scopetta, M. Traini and V. Vento, Double parton correlations and constituent  M. Rinaldi, S. Scopetta, M.C. Traini and V. Vento, Correlations in Double Parton  M. Rinaldi and F.A. Ceccopieri, Relativistic effects in model calculations of double parton  W. Broniowski and E. Ruiz Arriola, Valence double parton distributions of the nucleon in a  W. Broniowski, E. Ruiz Arriola and K. Golec-Biernat, Generalized Valon Model for Double  T. Kasemets and A. Mukherjee, quark-gluon double parton distributions in the light-front  V.M. Braun, A.N. Manashov and J. Rohrwild, Baryon Operators of Higher Twist in QCD and Nucleon Distribution Amplitudes, Nucl. Phys. B 807 (2009) 89 [arXiv:0806.2531]  V.M. Braun, A.N. Manashov and J. Rohrwild, Renormalization of Twist-Four Operators in  Y. Ji and A.V. Belitsky, Renormalization of twist-four operators in light-cone gauge,  A.P. Bukhvostov, G.V. Frolov, L.N. Lipatov and E.A. Kuraev, Evolution Equations for  R.L. Jaffe, Spin, twist and hadron structure in deep inelastic processes, in The spin structure  R.L. Jaffe and X.-D. Ji, Chiral odd parton distributions and Drell-Yan processes,  J.C. Collins, D.E. Soper and G.F. Sterman, Transverse Momentum Distribution in Drell-Yan  M. Diehl, T. Kasemets and S. Keane, Correlations in double parton distributions: effects of  F. Caola, M. Dowling, K. Melnikov, R. R¨ontsch and L. Tancredi, QCD corrections to vector  A.D. Martin, W.J. Stirling, R.S. Thorne and G. Watt, Parton distributions for the LHC, Eur. Phys. J. C 63 (2009) 189 [arXiv:0901.0002] [INSPIRE].
 J.C. Collins and D.E. Soper , Back-To-Back Jets: Fourier Transform from B to K-Transverse, Nucl. Phys . B 197 ( 1982 ) 446 [INSPIRE].
 J. Collins and T. Rogers , Understanding the large-distance behavior of  J.R. Gaunt , R. Maciula and A. Szczurek , Conventional versus single-ladder-splitting  M. Diehl and T. Kasemets , Positivity bounds on double parton distributions ,  T. Kasemets and M. Diehl , Angular correlations in the double Drell-Yan process ,  E.W.N. Glover and J.J. van der Bij, Higgs Boson pair production via gluon fusion ,  A. Bacchetta , D. Boer , M. Diehl and P.J. Mulders , Matches and mismatches in the  D. Binosi and L. Theussl , JaxoDraw: A graphical user interface for drawing Feynman diagrams , Comput. Phys. Commun . 161 ( 2004 ) 76 [hep-ph/0309015] [INSPIRE].
 D. Binosi , J. Collins , C. Kaufhold and L. Theussl , JaxoDraw: A graphical user interface for