#### The complete NLO corrections to dijet hadroproduction

Received: January
The complete NLO corrections to dijet hadroproduction
R. Frederix 1 2 4 7 8 9
S. Frixione 1 2 4 8 9
V. Hirschi 1 2 4 6 8 9
D. Pagani 1 2 3 4 7 8 9
H.-S. Shao 1 2 4 5 8 9
M. Zaro 0 1 2 4 8 9
g 1 2 4 8 9
Open Access 1 2 4 8 9
c The Authors. 1 2 4 8 9
0 Sorbonne Universites, UPMC Univ. Paris 06
1 2575 Sand Hill Road, Menlo Park, CA 94025-7090 , U.S.A
2 Via Dodecaneso 33, I-16146, Genoa , Italy
3 Centre for Cosmology , Particle Physics and Phenomenology (CP3)
4 James-Franck-Str. 1, D-85748 Garching , Germany
5 TH Department, CERN , CH-1211 Geneva 23 , Switzerland
6 SLAC, National Accelerator Laboratory
7 Physik Department T31, Technische Universitat Munchen
8 UMR 7589, LPTHE , F-75005, Paris , France
9 Universite Catholique de Louvain , B-1348 Louvain-la-Neuve , Belgium
We study the production of jets in hadronic collisions, by computing all contriand next-to-leading order results, respectively, for single-inclusive and dijet observables in a perturbative expansion that includes both QCD and electroweak e ects. the leading and next-to-leading order contributions largely respect the relative hierarchy established by the respective coupling-constant combinations.
hadroproduction; n m; with n + m = 2 and n + m = 3; These correspond to leading S
1 Introduction 2 3 4
Calculation setup
De nition of jets
Photon-jet cross sections
Jet production is a very common occurrence at high-energy hadron colliders; for example,
s 1, there are
several tens of thousands events per second that contain at least one jet with transverse
momentum larger than 100 GeV. Such an abundance allows experiments to carry out
measurements a ected by very small statistical uncertainties, and thus to probe all corners
of the phase space in a multi-di erential manner. At the same time, it constitutes a severe
problem for new-physics searches characterised by jet nal states, with the signal possibly
swamped by Standard Model (SM) backgrounds. This also applies to the easiest of cases,
that of a dijet signature (which is present in many beyond-the-SM scenarios, such as those
that feature heavy vector bosons, excited quarks, axigluons, Randall-Sundrum gravitons,
and so forth | see e.g. ref. [1] for a review of experimental searches that focus on the
dijet-mass spectrum), whose peak structure can be diluted by QCD e ects or be di cult
to study if at the border of the kinematically accessible region. A well known example of
the latter situation was the high-pT excess reported by CDF [2] in inclusive jet events, that
triggered a lot of interest owing to its being a possible evidence of quark compositness,
but that was ultimately entirely due to an SM e ect. In particular, the PDFs used for
computing the SM predictions to which the data had been compared were insu ciently
constrained in the x region that dominated high-pT jet production, and the uncertainties
associated with their determination were unknown.
The case of the large transverse momentum excess at CDF typi es the necessity of
computing jet cross sections at the highest possible accuracy in the SM. The largest of such
cross sections is the dijet one (which also gives the dominant contribution to single-inclusive
rates); we shall exclusively deal with it in this paper. Next-to-leading order (NLO) QCD
results for inclusive and two-jet distributions have been available since the early 1990's [3{
6]. The rst complete next-to-NLO (NNLO) QCD predictions have appeared only very
recently [7]. As a rule of thumb based on the values of the respective coupling constants,
NNLO QCD e ects (O( S4)) have the same numerical impact as the so-called NLO ones
in the electroweak (EW) theory (O( S2 )). Partial pure-weak contributions to the latter
had been computed in refs. [8, 9], and the complete weak results published in ref. [10].
The rationale for ignoring the NLO EW corrections of electromagnetic origin, which to the
best of our knowledge have not been calculated so far, is the possible enhancement of weak
contributions due to the growth of logarithmic terms of Sudakov origin in certain regions
of the phase space associated with large scales [11{14], in particular at high transverse
momenta. Incidentally, such Sudakov e ects can also be responsible for large violations of the
natural hierarchy of QCD and EW corrections, with NLO EW ones becoming signi cantly
larger than their NNLO QCD counterparts and competitive with the NLO QCD results.
Motivated by the previous considerations, in this paper we present the computation
of all the leading and next-to-leading order contributions to the dijet cross section in
a mixed QCD-EW coupling scenario. In other words, we compute all the terms in the
perturbative series that factorise the coupling-constant combinations S
n m, with n+m = 2
O( S2 ) electromagnetic contribution, and the two NLO terms of O( S 2) and O( 3). Our
computations are carried out in the MadGraph5 aMC@NLO framework [15] (MG5 aMC
henceforth), and are completely automated; this work therefore constitutes a further step
in the validation of the MG5 aMC code, in a case that requires the subtraction of QED
infrared singularities which is signi cantly more involved than that studied in ref. [16]. We
also take the opportunity to discuss issues that arise when one de nes jets in the presence
of nal-state photon and leptons.
This paper is organised as follows. In section 2 we outline the contents of our
computation and the general features of the framework in which it is performed. The problem of the
de nition of jets in the context of higher-order EW calculations is discussed in section 3.
Phenomenological results for the LHC Run II are given in section 4. Finally, we present
our conclusions in section 5.
Calculation setup
A generic observable in two-jet hadroproduction can be written as follows:
at the LO and NLO respectively. The notation we adopt throughout this paper is fully
analogous to that of refs. [15{17]. We refer the reader, in particular, to ref. [17] for a detailed
discussion on the physical meaning of the terms that appear in eqs. (2.1) and (2.2), and
the relevant terminology; we limit ourselves to recalling here that what is conventionally
denoted by NLO QCD and NLO EW corrections can be identi ed with
In our computation, jj
(LO) receives contributions from all Feynman diagrams relevant
to tree-level four-point Green functions with external massless SM particles | namely,
light quarks (including bottoms, since we work with
ve light avours), gluons, photons,
and leptons.1 As far as jj
(NLO) is concerned, all one-loop four-point and tree-level ve-point
functions with massless external legs contribute. Note that this implies that while both
real and virtual photons enter NLO corrections, W 's and Z's only appear as internal
particles. Thus, what has been called HBR (for Heavy Boson Radiation) in refs. [16, 17],
that is the contribution from tree-level diagrams that correspond to the real emission of
or a Z (and, in principle, one might consider top-quark emissions, too) from a
Born-level con guration, is not included in our results (incidentally, this is also the reason
why in the present case
NLO2 ). In fact, in order to consider HBR cross
sections, one would need either to possibly cluster a heavy vector boson together with
other massless particles when reconstructing jets (an option which is not appealing from a
physics viewpoint, given the procedure followed by experiments), or to rst decay any W
and Z into a pair of quarks or leptons. Having said that, we point out that MG5 aMC can
be used to simulate HBR contributions to dijet observables, and that the corresponding
calculations are fully independent of those performed here.
All of the computations of the matrix elements mentioned above, the renormalisation
procedure, and the subtraction of the real-emission infrared singularities (IR) are handled
automatically by MG5 aMC (with a still-private version of the code).
We remind the
reader that MG5 aMC makes use of the FKS method [18, 19] (automated in the module
MadFKS [20, 21]) for dealing with IR singularities. The computations of one-loop
amplitudes are carried out by switching dynamically between two integral-reduction techniques,
OPP [22] or Laurent-series expansion [23], and TIR [24{26]. These have been automated
in the module MadLoop [27], which in turn exploits CutTools [28], Ninja [29, 30], or
IREGI [31], together with an in-house implementation of the OpenLoops optimisation [32].
Two remarks are in order here. Firstly, there is no element in the MG5 aMC code that
has been customised to compute dijet observables, in keeping with the general strategy
that underpins the code. Secondly, although the papers cited above mostly treat explicitly
the case of QCD corrections, MG5 aMC has been constructed for being capable to handle
other theories as well. For what concerns the subtraction of real-emission singularities,
the QED case descends from the QCD one, with the most signi cant complications in the
context of automation due to bookkeeping (which understands the necessity of retaining
independent control of the various
k;q terms). The underlying strategy has been outlined
in section 2.4.1 of ref. [15]; the necessary extensions to the code were chie y carried out
for the work of ref. [16], and further validated for the present paper. As far as one-loop
computations are concerned, MadLoop has been completely overhauled in ref. [15] (see in
particular sects. 2.4.2 and 4.3 there), and it is since then that it is able to evaluate virtual
amplitudes in theories other than QCD.
Finally, we point out that our simulations are entirely based on a Monte Carlo
integration of the short-distance subtracted cross sections, that results in (weighted) events and
1The reasons for this choice will be discussed in section 3.
their associated counterevents. In particular, we do not use any factorised formulae for the
one-loop EW logarithmic corrections (see e.g. ref. [13]).
De nition of jets
The prescription for the computation of a jet cross section, possibly in association with
other objects, is unambiguous in perturbative QCD: jets are composed of massless coloured
particles (quarks and gluons), and this determines the nature of parton-level processes.
Things become more complicated as soon as one considers the rst subleading higher-order
correction, i.e. the electromagnetic one at the NLO. Among other things, this entails
the contribution of diagrams with an extra (w.r.t. the underlying Born con guration) real
photon in the nal state. In order to have an IR- nite cross section, such a photon must
be recombined (at least in a suitable subset of the phase space) with nearby QCD partons
to form a jet.
However, this raises an issue when the jet is made of a photon and a
gluon: IR safety demands (consider the soft-gluon limit) that there be an associated Born
con guration in which a jet coincides with a photon. In other words, Born-level amplitudes
must feature both QCD partons and photons (which in turn implies that one cannot limit
oneself to considering only the leading, pure-QCD, Born contribution).
This does not really pose any problem: one must simply enlarge the set of particles
that can form jets at the level of short-distance cross sections (both at the leading and
at higher orders), and include photons on top of light quarks and gluons; the resulting
objects are called democratic jets.2 The fact that a jet might be predominantly a
nonhadronic quantity is not surprising in a realistic experimental environment; for example,
in certain LHC analyses a jet is a spray of collimated particles with up to 99% of its
energy of electromagnetic origin, of which up to 90% can be carried by a single photon (see
e.g. refs. [34{46] for a list of recent ATLAS and CMS papers approved as publications in
the context of jet physics). Having said that, xed-order perturbation theory is somehow
pathological, precisely because a jet can coincide with a photon. Although, as we shall
show later, this situation is numerically unimportant, it has motivated the introduction
of procedures with the aim of getting rid of jets whose energy content is dominated by a
photon | in this paper, we shall call such objects photon jets. Recent examples can be
found in refs. [47{50], that deal with NLO EW corrections to vector boson production in
association with jets. The common feature of these procedures is the use of the photon
energy (or of a related quantity, such as the transverse momentum), which is necessary to
de ne the photon hardness, and thus its relative contribution to that of the jet the photon
belongs to.
Unfortunately, the photon energy is an ill-de ned perturbative concept, starting from
the third-leading NLO correction (i.e.
NLO3 in the case of dijet production). This can
be easily seen by considering a Born-level diagram with a nal-state photon, and the
realones) it is necessary to include massless leptons as well in the jet-clustering procedure. As far as we know,
the term \democratic" applied to jets in a similar context has been used for the rst time in ref. [33].
emission diagram obtained from the former by means of a
q k q limit, one sees that the photon energy is not an IR-safe quantity.
! qq splitting: by taking the
In order to use photon degrees of freedom in an IR-safe way, the photon must be a
physical nal-state object (in other words, \taggable" or \observable"). For this to happen,
the following rule must be obeyed:
Photons can be considered as observable objects only if emerging from a
fragmentation process. A photon that appears in a Feynman diagram has not been fragmented,
and thus cannot be tagged.
A taggable photon is quite analogous to e.g. a pion, which is described in perturbative
QCD by means of a (non-perturbative) fragmentation process. As such, we shall have
fragmentation functions that account for the long-distance process:
where i is any massless particle that can fragment into a photon, and z the fraction of
the longitudinal momentum of i carried by the photon.
Thus, the particle i may be
itself a photon, which is the most signi cant di erence between the photon and the pion
cases (since no pion can appear at the short-distance level). In particular, owing to the
elementary nature of the photon, one will necessarily have [51]:
D( )(z) = (A + B
( )(z) a regular function at z ! 1. We point out that the O( 0) (1
in eq. (3.2) is all one needs in the context of QCD computations that feature
photons:3 in that case, the di erence between taggable photons and short-distance photons
is irrelevant (and indeed it is not necessary to introduce it). We also remark that it is
perfectly acceptable to have a process with both taggable and short-distance photons in
nal state; the degrees of freedom of the latter must be integrated over (as e.g. in a
jet- nding algorithm), while this is not necessary (but still possible) for the former ones.
The scheme outlined above allows one to de ne a photon jet regardless of the
perturbative order in
one is working at: for example, a photon jet is any jet that
contains a taggable photon with energy E such that E
zcutEj , with Ej the jet energy
and zcut a pre-de ned constant. However, in the context of a jet analysis what one is really
interested in is a \hadronic" jet, i.e. a jet in which the content of EM energy is smaller,
not larger, than a given threshold (we shall call these jets anti-tagged jets in this paper).
This poses two problems. Firstly, a photon can be anti-tagged not only if E
< zcutEj , but
also if it simply escapes detection (which, for a xed-order theoretical calculation, is the
case where the jet is made of quarks and gluons only, i.e. one in which there is no photon).
Secondly, the anti-tagging condition creates a practical problem, because fragmentation
functions can only be measured (if at all) for su ciently large z's.
3Such a term corresponds to what is usually called the direct contribution in pQCD calculations.
where the dots on the r.h.s. generically denote power-suppressed terms. In words: parton
i fragments into any \hadrons", which will be eventually clustered into a jet (note that
parton i can be dressed by the perturbative radiation of other massless particles | these
are understood in the notation of eq. (3.3)). In the rightmost side of eq. (3.3), the sum
over parton-to-hadron fragmentation functions is split into the sum of a term that features
all hadrons di erent from the photon, and of a parton-to-photon term. By neglecting the
power-suppressed terms we re-write eq. (3.3) as follows:
i =
X Dh(i)(z) + D(i)(z) (zcut
z) + D(i)(z) (z
i.e. we introduce tagging and anti-tagging conditions, which we can do because the photon
emerges from a fragmentation process, and thus is taggable. Thence:
A possible solution to these problems employs again the idea of photon jet. The starting
point is the following identity (which is the hadron-parton-duality unitary condition):
i =
X Dh(i)(z) + : : : =
X Dh(i)(z) + D(i)(z) + : : : :
with X any set of objects that have to be found in the nal state on top of n jets
(importantly, taggable photons may appear in such a set). The rst term on the r.h.s. of eq. (3.6)
is the democratic jet cross section; no taggable photons are present, except those possibly
in X. Each of the n cross sections that appear in the second term on the r.h.s. of eq. (3.6)
is constructed by using the same short-distance processes as those that contribute to the
rst term, and by fragmenting k
nal-state quarks, gluons, and photons in all possible
ways; n jets are nally reconstructed. All n + 1 terms on the r.h.s. of eq. (3.6) are
and IR safe, and can be computed independently of each other in perturbation theory.
What has been done so far for photons can essentially be repeated in the case of
massless leptons. The main di erence is that a fermion line cannot be made to disappear
by splitting, and this implies that there is a way to tag a lepton that is not viable in the case
X Dh(i)(z) + D(i)(z) (zcut
z) = i
The l.h.s. of eq. (3.5) is what we want: the anti-tag jet contribution. Unfortunately, neither
of the terms that appear there can be reliably computed (for all z's). Conversely, the
r.h.s. of that equation is just ne: the two terms there correspond to the fully-democratic
jet cross section and to the photon-tagged one. If eq. (3.5) is iterated over all possible
nal-state partons, one ends up by de ning in a natural manner the anti-tag jet cross
section as the democratic cross section, minus all tagged-photon cross sections, with the
number of photons ranging from one to the maximum number of jets compatible with the
perturbative order considered. In formulae, this can be expressed as follows:
d X(a;nntjitag) = d X(d;enmj)
of photons. Still, IR safety requires that such a tagging is performed on an object which
is not the (short-distance) lepton itself, but its dressed version: this is nothing but a jet,
typically constructed with a small aperture, that contains one lepton and whatever extra
radiation surrounds it. Alternatively, one can follow the same procedure as for photons,
namely introduce parton-to-lepton fragmentation functions. Either way, one arrives at the
idea of taggable leptons, which can be employed to de ne lepton jets; the anti-tag jet cross
section in the l.h.s. of eq. (3.6) is then de ned by inserting on the r.h.s. subtraction terms
relevant to the lepton-jet cross sections.4
The procedure outlined so far puts QCD and QED on a rather similar footing. In
particular, this implies that as far as EW corrections are concerned all computations can
be conveniently performed in an MS-like scheme (such as the G
(mZ ) ones). We
point out that this procedure naturally leads to the prescription usually adopted in NLO
EW computations (see e.g. ref. [53]) that associates a factor (0) to each external
(shortdistance) photon: such a factor results from the RG evolution of the photon-to-photon
fragmentation function, whose (1
Photon-jet cross sections
We now return to eq. (3.6) in order to de ne the photon-jet cross sections that appear in
the second term on the r.h.s. of that equation, for the case of dijet hadroproduction we are
interested in. As was discussed above, a construction valid for all the
necessarily entails the use of fragmentation functions, whose knowledge is presently far
from being satisfactory (bar perhaps for the quark-to-photon one).
Therefore, we have to adopt a pragmatic solution; this amounts to de ning the
photonjet cross sections only for those O( Sn m) terms for which the introduction of a
fragmentation function can be bypassed; for the other terms, the photon-jet cross sections will be set
equal to zero, and thus our anti-tag dijet cross section will coincide with the democratic
one.5 We do this in the following way. The photon-jet cross sections are de ned by
using the isolated-photon cross sections for one and two photons, constructed identically to
what one usually does in perturbative QCD, and whose
nal states are suitably clustered
into jets (as we shall specify later). This implies that the relevant perturbative orders are
the following:
for the one- and two-isolated-photon cross sections respectively. This is implicitly
equivalent to setting the photon-to-photon fragmentation function equal to (1
z), i.e. to
neglecting the contribution to it due to higher-order QED e ects. The cross sections that
4The use of the physical lepton masses leads to alternative approaches (see e.g. ref. [52]). These typically
feature large-logarithmic terms, that expose their IR sensitivity and necessitate a careful treatment; we
believe that they are best avoided in the context of jet analyses and lepton-jet rejection.
5We always cluster leptons democratically, which is fully justi ed by the fact that their contributions
are very subleading, and numerically completely negligible.
correspond to eqs. (3.7) and (3.8) could still depend on quark-to-photon and
gluon-tophoton fragmentation functions; in order to avoid this, we choose to work with the smooth
isolation prescription of ref. [54], which sets their contributions identically equal to zero.
More in details, we have implemented the following procedure:
1. nd jets democratically;
2. nd isolated photons; they are de ned following ref. [54] (using transverse momenta),
= 1;
3. loop over those photons: if a photon belongs to a jet, and it carries more than 90%
of the pT of that jet, then ag the jet as a candidate photon jet;
4. candidate photon jets are considered as proper photon jets if and only if:
there is exactly one isolated photon, and one computes either
there are exactly two isolated photons, and one computes either LO3 or
5. each photon jet gives an entry to the histograms relevant to single-inclusive
observables. For dijet correlations, there is an histogram entry for each pair of jets, at least
one of which is a photon jet.6
There are many possible variants to items 1{5 above, but we believe that all those
that are consistent with the general ideas outlined before will give very similar numerical
results. The most important thing to bear in mind is that, regardless of the speci c choices
made for the isolation procedure, one is guaranteed to get rid of those con gurations where
a photon jet coincides with a photon, which is the semi-pathological situation, peculiar of
xed-order calculations, that one typically would like to avoid.
We point out that, with the choices made here, each photon jet will coincide with a
democratic jet (while the opposite is obviously not true). Therefore, item 5 implies a local
and exact cancellation of the photon-jet contributions, if all the computations relevant to
the cross sections on the r.h.s. of eq. (3.6) are performed simultaneously (i.e. during the
same run), which is what we do. This not only improves the numerical stability of the
results, but also resembles very closely any possible experimental procedure that would
reject jets with too high a content of EM energy.
We now turn to presenting our predictions for a variety of single-inclusive and dijet
observables that result from pp collisions at a center of mass energy of 13 TeV (LHC Run II). We
show di erent linear combinations of these quantities. Jets are de ned by means of the kT
6We point out that dijet correlations can be constructed by using a subset of all possible two-jet pairings,
and we choose in section 4 to consider only observables de ned by means of the two hardest jets.
results relevant to democratic jets, but also explicitly assess the e ect of removing photon
jets, as discussed in section 3. For all of the observables considered here the contribution
of forward jets is discarded, by imposing the constraint:
We work in the ve- avour scheme (5FS) where all quarks, including the b, are massless;
electrons, muons, and taus, collectively called leptons, are massless as well, while the vector
boson masses and widths have been set as follows:
mW = 80:419 GeV ;
mZ = 91:188 GeV ;
W = 2:09291 GeV ;
Z = 2:50479 GeV :
The CKM matrix is taken to be diagonal, and the complex-mass scheme [57, 58] is
employed throughout. The PDFs are those of the NNPDF2.3QED set [59], extracted from
LHAPDF6 [60] with number 244600; these are associated with
S(mZ ) = 0:118 :
We work in the G
EW scheme, where:
= 1:16639 10 5
= 132:507 :
The central values of the renormalisation ( R) and factorisation ( F ) scales are both
0 =
1 X pT (i) ;
where the sum runs over all nal-state particles. The theoretical uncertainties due to the
F dependencies have been evaluated by varying these scales independently in
and by taking the envelope of the resulting predictions. The scale dependence of
ignored, and the systematics associated with the variations in eq. (4.7) is evaluated by
means of the exact reweighting technique introduced in ref. [61].
Reweighting is also
employed for the computation of PDF uncertainties, with individual weights combined
according to the NNPDF methodology [62]. We report the 68% CL symmetric interval
(that is the one that contains only 68 replicas out of a total of a hundred; this is done in
order to avoid the problem of outliers, which is severe in this case owing to the photon
PDF [59]). Finally, we note that the NNPDF2.3 set adopts a variable- avour-number
scheme. For scales larger than the top mass, this scheme is equivalent to the six- avour
one (6FS). Since the hard matrix elements are evaluated in the 5FS, the impact of the
sixth avour has to be removed from the running of S and from the DGLAP evolution of
the PDFs. This corresponds to adding to the NLO 6FS-PDF cross section the following
Here, n(gi;k) and b(i;k) are the number of initial-state gluons and the power of S in
respectively, with k numbering the individual partonic channels that contribute to
The interested reader can
nd more details in ref. [63] or in section IV.2.2 of ref. [64].
In order to determine which transverse-momentum cuts are sensible in an NLO
computation, we follow the procedure of ref. [65] and present in
gure 1 the total dijet cross
section as a function of , according to the following de nition:
( ) =
60 GeV +
60 GeV ;
with p(j1) and p(Tj2) the transverse momentum of the hardest and second-hardest jet,
re
T
spectively. In other words,
measures the asymmetry between the pT cuts imposed on the
two hardest jets, having assumed the transverse momentum of the second-hardest jet to be
larger than 60 GeV. Such a value is arbitrary, and is chosen as typical of LHC jet analyses;
we point out that its impact on the pattern of the dijet cross section dependence upon
negligible (within a reasonable range). There are ve curves in the main frame of gure 1.
The red histogram overlaid with full diamonds represents the
blue one corresponds to the sum of all of the LO contributions, jj
LO1 contribution, while the
(LO). The green histogram
overlaid with open boxes is the sum
NLO1 , i.e. of the leading terms (pure QCD) at
the LO and NLO; the black histogram is the sum of all of the LO and NLO contribution,
(NLO), and is denoted by \all orders". Finally, the brown curve represents the
sum of all LO and NLO contributions, bar the pure QCD ones ( LO1 and
NLO1 ); in order
for it to t into the frame of the gure, this histogram has been rescaled by a factor of 103.
In the region where the latter curve is displayed with a dashed pattern, the cross section
is negative, and thus what is represented is its absolute value; this convention will be used
throughout this section. The lower panel in gure 1 presents the ratios of the results shown
in the main frame, over the
LO1 prediction.
As is explained in detail in ref. [65], the dijet cross section behaves in a pathological
manner for small
values at the NLO, owing to the presence of large log
terms. Given
the de nition in eq. (4.9), one would expect a monotonically increasing rate for
This is indeed the behaviour of the LO results (red-with-diamonds and blue histograms),
while the NLO ones actually decrease as
decide which value of
is appropriate in order to carry out sensible NLO computations.
Inspection of the plot suggests to set
& 20 GeV | for such values, the three NLO
predictions are still monotonically growing. In order to be de nite, we shall thus impose
80 GeV ;
in our simulations for dijet correlations, while for single-inclusive distributions we impose
60 GeV :
There are a couple of further observations relevant to gure 1. Firstly, the full LO and NLO
results (blue and black histograms, respectively) are extremely close (but not identical,
although that is hard to see directly from the plot) to their leading, pure-QCD, counterparts
(red-with-diamonds and green-with-boxes histograms, respectively). This is the well-known
fact that EW contributions are negligible as far as dijet rates are concerned, their e ects
being manifest only in certain phase-space regions characterised by large scales and that
contribute little to total cross sections. Secondly, it appears that the impact of log
terms is larger when the pure-QCD contributions are not included (the peak of the brown
histogram occurs at a much larger
value than that relevant to the two other NLO results).
This suggests that a conservative choice of
(similar to or even more stringent than that
of eq. (4.10)) is recommended where EW e ects are particularly prominent.
We now turn our attention to di erential observables. We shall present six of them in
gures 2{14, with two gures for each observable (plus one relevant to the direct comparison
of pT results in di erent rapidity ranges, gure 10). The patterns in the layout of the plots
are the same for all of the observables; thus, we shall explain their meaning by using the
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
be de nite.
There are three panels in gure 2. The upper one presents the absolute values of the
three LO and the four NLO contributions to the cross section, as well as their sum; as
was previously mentioned, a solid (dashed) pattern indicates that the corresponding result
is positive (negative). The three LO results are displayed as histograms overlaid with
symbols: red with full diamonds for
LO1 , green with open boxes for
LO2 , and brown
with open circles for
LO3 . The four NLO results are associated with plain histograms:
NLO1 , purple for
NLO2 , yellow for
NLO3 , and cyan for
NLO4 ; the sum of all
contributions is represented by the black histogram. The middle inset presents the ratios
of the results shown in the upper inset, over the all-orders prediction; in other words, these
are the fractional contributions of the
NLOi terms to the most accurate result
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
obtained from our simulations. The patterns employed in the middle inset are identical
to those of the upper inset. Finally, the bottom inset presents the relative theoretical
uncertainty of the all-orders result, in two di erent ways: the light gray band corresponds
to the hard-scale and PDF systematics (with the two summed linearly), while the dark
gray band shows the hard-scale uncertainty only (see eq. (4.7) and thereabouts).
The main frame of gure 3 presents various linear combinations of the results shown
in gure 2, in the form of ratios over the leading LO prediction,
LO1 . In particular, we
have de ned the quantities:
which are displayed as a brown histogram overlaid with full triangles ( 1), a red histogram
1 =
2 =
3 =
overlaid with full diamonds ( 2), and a green histogram overlaid with open boxes ( 3),
respectively. We also show the sum 2 + 3 as a blue histogram, and the sum 1 + 2 + 3
as a black histogram. Finally, we report for reference the two uncertainty bands already
shown in gure 2. In view of the de nition of
NLOi , the physical meaning of the
various curves presented in
gure 3 is the following. 1 is equal to KQCD
1, with KQCD
the K factor associated with a pure-QCD computation. 2 measures the relative impact of
the two Born contributions which are non pure-QCD. 3 is equal to KQCD
1, with KQCD
the K factor associated with NLO contributions that are not pure QCD.7 Thus, 1 + 2 + 3
shows the e ect on the best (i.e. the all-orders one) prediction of all contributions di erent
from the dominant Born one ( LO1 ), while the comparison between 1 and 2 + 3 allows an
immediate understanding of how much of that is due to either pure-QCD NLO corrections,
or to other LO and NLO contributions.
The lower panel of gure 3 displays two results, both of which are ratios of all-orders
predictions obtained with speci c conditions over the all-orders default prediction. The red
histogram overlaid with full circles corresponds to setting to zero the photon PDF, while
the green histogram corresponds to removing the photon-jet contributions.
The predictions for the single-inclusive jet transverse momentum shown in gures 2
and 3 are dominated by the leading contributions at both the LO and the NLO for piTncl .
2 TeV. The impact of non-QCD contributions is essentially negligible up to those values,
well within the scale uncertainty band. As is clear from
gure 3, speci cally from the
comparison of 2
, 3, and 2 + 3, this is chie y due to the very large cancellation that
occurs between the LOi and the NLOi terms (i
2) | note, from
gure 2, that this is not
only true for the sums of such terms, but to some extent also for them individually, since the
NLO ones are negative either in all or in a large part of the pT range considered. Eventually,
the LO cross sections grow faster in absolute value than their NLO counterparts. Thus, the
sum of all results minus the leading LO term
LO1 is indistinguishable from
2 TeV, but then starts to di er signi cantly from it, to the extent that
NLO1 contributes to
less than 50% to the sum for those transverse momenta at the upper end of the range probed
in our plots, pincl & 4:5 TeV. When one moves towards such large piTncl's, one sees that the
T
NLO scale uncertainty remains moderate, while that due to the PDFs grows rapidly, owing
to the poor constraining power of the data currently used in PDF ts on the corresponding
equal to about 3% of the total PDF uncertainty at piTncl ' 2:6 TeV, and to about 22% at
' 4:6 TeV), but is never the dominant e ect. From
the contributions that depend on the photon PDF is negligible for piTncl . 3:5 TeV, while
it becomes substantial for larger values of the transverse momentum. Needless to say, the
validity of this observation is restricted to the PDF used in the present simulations. The
photon component in the NNPDF2.3QED set is mainly constrained by LHC Drell-Yan data
via a reweighting procedure. This results in a signi cant photon density at large x that,
however, is associated with a sizeable uncertainty. Other approaches, which rely either on
assumptions on the functional form at some initial scale [66{68], or on a direct extraction
gure 3 we see that the impact of
7As was already said, and for the sake of consistency, this K factor is de ned by using the pure-QCD
Born LO1 in the denominator.
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Figure 4. Single-inclusive transverse momentum, for jyj
from proton structure functions [69], suggest that its central value is much smaller than
the NNPDF2.3 one at large x and rather precisely determined (in the recent sets), thus
e ectively lying close to the lower limit of the NNPDF2.3QED uncertainty band.
We also remark that the removal of the photon-jet cross sections has a negligible impact
in the whole transverse momentum range considered. It does a ect the individual LOi and
2 contributions, especially LO2 where it can be as large as 30%; however, this
occurs mostly for piTncl . 0:5 TeV, where non-QCD terms can be safely ignored.
The single-inclusive transverse momentum is again shown in gures 4 and 5, subject to
the constraint jyj
1 (in other words, each jet that gives a contribution to these histograms
must satisfy a small-rapidity constraint). The patterns in these gures are very similar to
those of gures 2 and 3, respectively, owing to the dominance of central jets in the case
inclusive over the whole rapidity range.
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Figure 5. Single-inclusive transverse momentum, for jyj
Things slightly change when one considers the rapidity intervals 1 < jyj
2:8, whose cases are presented in
gures 6 and 7, and in
gures 8 and 9,
atter when the rapidity is increased; conversely, 3 decreases, somehow more rapidly.
The net e ect is that the amount of cancellation between the LO and NLO cross sections
is smaller the farther away one moves from central rapidities in this range of relatively
small piTncl's, so that the overall EW e ects, that decrease the pure-QCD cross sections, are
stronger the larger the rapidities. This is seen more clearly in
gure 10, where the results
, 3, and 2 + 3, already shown in gures 5, 7, and 9, are presented together (as red,
green, and blue histograms; the jyj
2:8 predictions are
displayed as solid, dashed, and short-dashed histograms, respectively), by using a smaller
y-axis scale w.r.t. those of the original plots. For larger transverse momenta the trend
changes, with the positive LO contributions eventually becoming larger than their NLO
counterparts (in absolute value). Thus, the 2 + 3 prediction crosses zero at piTncl
1, and at piTncl
2:5 TeV for 1 < jyj
conclusion in the range 2 < jyj
2 (the statistics is insu cient to draw any
Figure 6. Single-inclusive transverse momentum, for 1 < jyj
We conclude that, as far as the single-inclusive transverse momentum is concerned, the
impact of LO and NLO contributions beyond the leading ones do depend on the rapidity
range considered, and tends to decrease (increase) the pure-QCD results when moving
away from the central region for small (large) piTncl; in all cases, the absolute values of the
overall e ects are relatively small. This pattern is due to a variety of reasons; in particular,
one may mention the fact that, the larger the rapidity, the more di cult it is to reach the
high-pT region where EW e ects are known to be more prominent, but also the fact that
the extent of the cancellation between LO and NLO results is di cult to be predicted a
priori. In any case, such a pattern must be taken into account in the context of PDF ts
that aim to include EW corrections, and that need to consider di erent rapidity ranges in
order to constrain more e ectively the small-x region.
Our predictions for the invariant mass of the hardest-jet pair are given in gures 11
and 12 (note that some of the histograms have been rescaled in the latter gure, in order
Figure 7. Single-inclusive transverse momentum, for 1 < jyj
to make them more clearly visible in the layout). NLO corrections are dominated by the
pure-QCD ones
NLO1 , that turn negative around M12 ' 1 TeV.8 EW e ects tend to
decrease the cross section further, with the second-leading NLO corrections
negative and larger in absolute value than the second-leading LO term
overall impact on the physical cross section is rather small, and in particular smaller than
the hard-scale uncertainty. As was observed in ref. [10], even for mass values of several
TeV's one is not fully in the Sudakov region, and thus EW contributions tend to follow
the hierarchy established by the couplings, without major logarithmic enhancements. We
also observe a very small impact of the removal of the photon jets. In this regard, the
same comments as for the single-inclusive transverse momentum apply here. By removing
photon-jet cross sections from
LO2 , that term is halved at invariant masses smaller than
0.5 TeV; however, as can be seen from
gure 11, in that region its contribution to the
NLO2 being
LO2 . However, the
all-orders rate is in practice negligible.
We nally show, in gures 13 and 14, the rapidity separation between the two hardest
jets (again, some of the histograms have been rescaled in the latter plot to improve its
8There is a visible numerical instability that a ects the large-mass predictions of
signi cant cancellations between the real-emission and virtual contributions to that mass region.
2<|yj|<2.8
Figure 8. Single-inclusive transverse momentum, for 2 < jyj
readability). This observable is dominated by low-pT con gurations, and as a consequence
of that subleading terms, both at the LO and the NLO, are numerically extremely small,
and completely swamped by hard-scale uncertainties. Leading NLO corrections are large,
but almost at in the whole range considered. As in the previous cases, the removal of
photon jets is irrelevant to the all-orders result, while being important up to the largest
rapidity separations in particular for
Conclusions
In this paper we have studied the hadroproduction of dijets, and considered all of the LO
and NLO contributions of QCD and EW origin to the corresponding cross section,
presented as single-inclusive distributions and two-jet correlations for pp collisions at 13 TeV.
Figure 9. Single-inclusive transverse momentum, for 2 < jyj
By doing so, we have computed for the rst time three subleading NLO corrections: the
O( S2 ) electromagnetic one (our results include the contributions due to real-photon
emissions), and the O( S 2) and O( 3) EW ones. The calculations have been performed in
the automated MadGraph5 aMC@NLO framework, which is thus extensively tested in a
mixed-coupling scenario that features both EW and QCD loop corrections, and both QCD
and QED real-emission subtractions.
When all subleading NLO corrections are computed, it is necessary to be
particularly careful in the case one wants to not take into account jets that are predominantly
of electromagnetic origin. Although from the phenomenological viewpoint we do not
consider this operation to have a compelling motivation, we have outlined an IR-safe scheme
through which this result can be achieved. Its exact implementation requires the use of
fragmentation functions, whose determination from data is either poor or not available at
present.9 For the sake of this paper, we have adopted a more pragmatic strategy, which
9However, the necessary ingredients for a technically-viable computation that leads to IR- nite cross
sections can all be derived from purely perturbative information.
pincl [GeV]
Single-inclusive transverse momentum; 2 and 3 predictions for the three rapidity
regions already considered in gures 5, 7, and 9.
is a (perturbative) approximation of the more general scheme, that does not employ the
fragmentation functions. We have shown that the removal of EW-dominated jets has a
negligible impact at the level of observable di erential rates, and one can thus safely work
with democratic jets, in which all massless particles (quarks, gluons, photons, and leptons)
are treated on equal footing.
In general, contributions that are expected to be subleading according to the
couplingconstant combination they feature turn out to be indeed numerically subleading, with
pureQCD e ects being dominant everywhere, except in the very-high transverse momentum
region of the single-inclusive jet pT . In other words, within the LO and NLO cross sections,
nd that the hierarchy naively established on the basis of the couplings is largely
respected, but we also remark that, in a signi cant fraction of the phase space,
LO2 . For all observables considered here, there are large cancellations between
the LO and NLO subleading terms, which is one of the major motivations for computing
them all in a consistent manner.
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
M12 [GeV]
δ3=(NLO2+NLO3+NLO4)/LO1 102×δ3
Acknowledgments
This work is supported in part by the ERC grant 291377 \LHCtheory: Theoretical
predictions and analyses of LHC physics: advancing the precision frontier". SF thanks the
CERN TH division for hospitality during the course of this work. SF is indebted to Stefano
Catani for many stimulating discussions, and for comments on the manuscript; he is also
grateful to Fabio Cossutti, Hannes Jung, Andrew Larkoski, and Wouter Waalewijn for
comments on di erent aspects of their work. The work of MZ is supported by the European
Union's Horizon 2020 research and innovation programme under the Marie
SklodovskaCurie grant agreement No 660171 and in part by the ILP LABEX (ANR-10-LABX-63), in
turn supported by French state funds managed by the ANR within the \Investissements
d'Avenir" programme under reference ANR-11-IDEX-0004-02. RF and DP are supported
by the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja
Award Project \Event Simulation for the Large Hadron Collider at High Precision". The
work of VH is supported by the Swiss National Science Foundation (SNSF) with grant
PBELP2 146525.
Open Access.
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.
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