aMC@NLO predictions for Wjj production at the Tevatron
On leave of absence from INFN, Sezione di Genova,
Centre for Cosmology, Particle Physics and Phenomenology (CP3), Universite catholique de Louvain
, B-1348 Louvain-la-Neuve,
Departamento de Fsica Teorica y del Cosmos y CAFPE, Universidad de Granada
PH Department, TH Unit
, CERN, CH-1211 Geneva 23,
ITPP, EPFL, CH-1015 Lausanne,
Institut fur Theoretische Physik, Universitat Zurich
, Winterthurerstrasse 190, CH-8057 Zurich,
We use aMC@NLO to predict the + 2-jet cross section at the NLO accuracy in QCD matched to parton shower simulations. We find that the perturbative expansion is well behaved for all the observables we study, and in particular for those relevant to the experimental analyses. We therefore conclude that NLO corrections to this process cannot be responsible for the excess of events in the dijet invariant mass observed by the CDF collaboration.
1 Introduction 2 3 4
Method and validation
W jj production at the Tevatron
Recently, CDF has reported  an excess of events in two-jet production in association
with a W boson, in the form of a broad peak centered at Mjj = 144 GeV in the dijet
invariant mass. By now, i.e. with a data set corresponding to an integrated luminosity
of 7.3 fb1, the excess has reached a statistical significance of 4.1 w.r.t. the estimated
Standard Model yield. In view of the possible implications for a BSM physics discovery,
this anomaly has attracted a lot of attention, though it has so far failed to be confirmed
by a very similar D0 analysis .
One of the major challenges in a measurement of this kind is posed by the need of
reliable predictions and simulations of the processes that contribute to the observables of
interest. In the CDF and D0 analyses, for instance, such simulations are typically performed
by means of fully exclusive Monte Carlo programs based on tree-level matrix elements.
In the case of multi-jet final states in association with weak bosons, a proper merging
procedure  between multi-parton matrix elements (which give a reliable description of
large-angle and large-energy emissions) and parton shower Monte Carlos (PSMCs) (which
give a reliable description of small-angle or small-energy emissions) is employed that allows
the generation of inclusive jet samples for all relevant multiplicities, accurate to the leading
order (LO) in perturbative QCD.
Yet, the uncertainties that affect LO predictions can be very large for rates, and smaller
but still discernible for differential distributions. This is the reason why parton-level NLO
and, when possible, NNLO computations of infrared safe observables are used.
Alternatively, and if the statistics is sufficient, control data samples are employed. For example, a
theoretical analysis based on the NLO computation of the SM yield for + 2 jets +
missing transverse energy (which with the cuts used by CDF and D0 gets contributions from,
in order of importance, W jj, Zjj, W W , tt, single-t and W Z production) has recently
appeared . It has been shown that indeed the W jj process gives by far the dominant
contribution, and that the NLO QCD corrections are small. Unfortunately, even though
more accurate from the theoretical point of view, such small-multiplicity, parton-level
calculations cannot be directly compared to experimental analyses, since this would require
events with high-multiplicity, fully-fledged hadronic final states.
In order to obtain predictions that are both accurate and employable in experimental
analyses, an NLO calculation needs to be consistently matched to a PSMC. This can be
currently achieved with the MC@NLO  or POWHEG methods [8, 9]. It is interesting to
note that out of the processes listed above for the signature + 2 jets + missing transverse
energy, only the W jj and Zjj contributions are not available in either of these frameworks.
Given that the cross section of the latter process (within the experimental cuts adopted
by CDF and D0) is smaller than that of the former by more than one order of magnitude,
it is more urgent and highly desirable to have the best possible theoretical predictions for
W jj production, which is a fairly challenging task. The complexity stems not only from
the NLO computation itself, but also from its subsequent matching with parton showers,
where the technical difficulties arise mainly from the presence of phase-space singularities
at the Born level, which need to be cut-off. While this problem has already been faced in
the POWHEG implementation of dijet  and W j/Zj  production, it is significantly
simpler in these cases: a pT cut on the recoil system (one parton in dijet, and the vector
boson in W j/Zj production) is sufficient to get rid of the divergences of the Born matrix
elements. On the other hand, W jj production features a final-state three-body (of which
two light partons) kinematic configuration already at the Born level, which renders the
cutting-off of the singularities highly non trivial. In fact, the kinematics of W jj production
is sufficiently involved to provide a proof that, if a successful matching of the NLO results
with parton showers can be achieved, the same kind of matching technique can be applied
to larger final-state multiplicities, without encountering any new problems of principle.
In this paper, we compute the NLO QCD corrections to the process pp jj,1 and,
for the first time and in a fully-automated way, consistently match them to the HERWIG
parton shower  according to the MC@NLO formalism , as implemented in the
aMC@NLO program . One-loop corrections are obtained with MadLoop , which
is based on the OPP reduction method  and on its implementation in CutTools .
All the other contributions to the parton-level NLO cross section are dealt with by
MadFKS , which is based on the FKS subtraction method , and takes care of determining
the MC counterterms needed in the MC@NLO approach. Throughout the paper, we often
refer to the W boson or to W jj production; this is only for the sake of brevity, since
we actually deal with the leptonic process mentioned before, and thus doing we fully retain
the information on production and decay spin correlations and off-shell effects.
We begin by showing that the cutting-off of Born-level singularities (which is an
arbitrary procedure) has no impact on the predictions in the kinematic regions of interest. We
also show that NLO corrections are moderate, and depend mildly on the kinematics. We
conclude by presenting our predictions for the dijet invariant mass, closely following the
Method and validation
The W jj NLO cross section receives contributions from processes with W +2-parton and
W +3-parton final states; although these diverge when independently integrated over the
1The mass of the charged lepton is set equal to zero. Furthermore, since we do not compare our
predictions to data here, it is sufficient to consider only positively-charged leptons of one flavor.
phase space, their combination into any infrared-safe observable is finite, thanks to the
KLN and factorization theorems. For this to happen, it is a crucial condition that there be
two observable jets in the final state. Although such a condition can be easily included in
the definition of the short-distance cross sections, this is not the way one follows nowadays.
A much-preferred option (and the only one which is viable when matching to PSMCs) is
that of imposing jet cuts at the very last step of the computation (the physics analysis),
since this gives one the flexibility of e.g. using several jet-finding algorithms in parallel.
It should be clear, however, that some cuts (called generation cuts henceforth) must still
be imposed at the level of short-distance cross sections, which otherwise would diverge
upon integration, as mentioned before. Generation cuts are therefore a technical trick
that allow one to work with finite quantities; the idea is that kinematic configurations
that do not pass these cuts would anyhow not contribute to the observable cross sections,
which is what permits one to discard them; in other words, cross sections are not biased
by generation cuts. We remind the reader that, in the case of cross sections involving
jets, generation cuts are imposed on the kinematics of the jets that result from applying
a jet-finding algorithm to the underlying parton configurations. In general, different jet
kinematics will be associated with real-emission contributions (that in our case have three
final-state partons), and with counterterms, virtual, and Born contributions (that have two
final-state partons here). Thanks to the infrared safety property of jet algorithms, these
differences will tend to vanish in the soft and collinear limits.
Unfortunately, it is not straightforward to prove that indeed physical observables are
unbiased, which constitutes a necessary and very strong consistency check of ones
computation. An analytic proof not being viable, one exploits the fact that generation cuts
are arbitrary. Hence, one imposes several generation cuts, and then verifies that in the
kinematic regions of interest physical observables do not depend on them. This opens the
question of how to define generation cuts, and it is obvious that a necessary condition
is that they must be looser than the loosest of the set of cuts imposed in the physics
analysis. When performing a perturbative calculation at the parton level, it is quite easy
to understand whether generation cuts are sufficiently loose. This is because generation
and analysis cuts are imposed on kinematic configurations that have the same
multiplicities and particle contents. Things are significantly more complicated when one matches
matrix-element computations with parton showers; the latter will in fact generally increase
the final-state multiplicities w.r.t. those relevant to short-distance cross sections, and the
relationship between the quantities being cut at the generation and analysis level becomes
blurred. The upshot of this is the following: when considering the matching with parton
showers, generation cuts are typically softer than those one would need if only performing
perturbative parton-level computations, and they affect larger kinematic ranges than in
the latter case.
In order to address this (among others) problem, at the LO one merges different
parton multiplicities in a way consistent with parton showers . Although a generalization
of these procedures to NLO is in its infancy , we may observe that when the merging
at the LO is restricted to processes whose multiplicities differ by one unit (e.g., the W +2
and W +3 partons samples in W jj production), then one is actually dealing with a subset
of the matrix elements used in the well-established NLO-PSMC matching procedures such
as MC@NLO (a subset, since the virtual correction are not included). Hence, one may
anticipate that unphysical effects, the reduction of whose impact necessitates a merging
procedure at the LO, are smaller in the context of matched NLO computations of a given
multiplicity. We shall later see an explicit example of this fact.
To conclude this discussion, we mention that, although there is ample freedom in
the choice of generation cuts, in practice it is convenient to employ the same jet-finding
algorithm at the matrix element level as in the physics analysis, since this renders it a bit
easier the task of applying generation cuts which are looser than the analysis ones.
As a technical aside, we point out that the MC@NLO formalism does not require
modifications in order to be applied to processes whose Born contribution is divergent,
and one simply imposes generation cuts when computing MC@NLO short-distance cross
sections, fully analogously to what is done at the LO. Using the results of ref. , it is easy
to show  that this should be done in the following way. All contributions to S events,
and the MC counterterms relevant to H events, are cut according to the corresponding
Born configuration (which thus has a W +2 parton kinematics). The contributions of the
real-emission matrix elements to H events are cut according to corresponding W +3
Our predictions are obtained with the electroweak parameters reported in table 1.
For the (N)LO computations we use the MSTW(n)lo200868cl  PDFs, which also set
the value of S(MZ). The renormalization and factorization scales are chosen equal to
HT /2, with HT = Pi pT ,i + pp2T () + M 2(). The sum here runs over all final-state
QCD partons, and all the quantities that appear in the definition of HT are computed at
the matrix-element level, i.e., before showering. We have not included the simulation of
the underlying event in our predictions.
We define jets by means of the anti-kT algorithm  with R = 0.4, as implemented in
FastJet . Generation cuts are imposed by demanding the presence of at least two jets
at the hard-subprocess level (hence, at this stage the inputs to the jet-finding algorithm
are two- or three-parton configurations). All jets thus found are required to have either
pT > 5 GeV or pT > 10 GeV. The short-distance cross sections defined with these cuts
are used to obtain unweighted events as customary in MC@NLO. Such events are then
showered by HERWIG, and the resulting hadronic final states are used to reconstruct
about sixty observables (involving leptons, jets, lepton-jet, and jet-jet correlations) for each
of the two generation pT cuts mentioned above. These observables are organized in three
Figure 1. Transverse momentum of the hardest jet (upper left plot), transverse momentum of
the third-hardest jet (upper right plot), invariant mass of the pair of the two hardest jets (lower
left plot), and distance between the two hardest jets in the plane (lower right plot), in W jj
events and as predicted by aMC@NLO (histograms), and with parton-level NLO computations
(symbols). See the text for details.
classes, each being associated with jets2 defined by imposing their transverse momenta to
be larger than 10, 25, and 50 GeV; these conditions will be called analysis cuts henceforth.
We finally check that the tighter the analysis cuts, the smaller the difference between the
results obtained with the two generation cuts.
As an example of the outcome of this exercise, we present in figure 1 the transverse
momentum of the hardest jet, the dijet invariant mass, and the R separation between the
two hardest jets. In the main frame of each plot there are six histograms: the three solid
ones correspond to generation cuts pT = 5 GeV, while the three dashed ones correspond
to generation cuts pT = 10 GeV. The upper (red), middle (blue), and lower (green) pairs
of histograms are obtained with the analysis cuts pT = 10, 25, and 50 GeV, respectively.
2We stress that such jets are now reconstructed by clustering all stable final-state hadrons that emerge
from the shower.
The lower insets display three curves, obtained by taking the ratios of the pT = 5 GeV
generation-cut results over the pT = 10 GeV generation-cut results, for the three given
analysis cuts (in other words, these are the ratios of the solid over the dashed histograms).
Fully-unbiased predictions are therefore equivalent to these ratios being equal to one in the
kinematic regions of interest.
Inspection of figure 1, and of its analogues not shown here, allows us to conclude that
the results follow the expected pattern: when one tightens the analysis cuts, the bias due to
the generation cuts is reduced, and eventually disappears. Although all observables display
this behaviour, the precise dependence on generation cuts is observable-specific; the three
cases of figure 1 have been chosen since they are representative of different situations. The
transverse momentum of the hardest jet shown in the upper plot of figure 1 is (one of)
the very observable(s) on which generation cuts are imposed. Therefore, as one moves
towards large pT s, one expects the bias due to generation cuts to decrease, regardless of
values of the pT cut used at the analysis level. This is in fact what we see. Still, a residual
dependence on generation cuts can be observed at relatively large pT s for looser analysis
cuts; this could in fact be anticipated, since the events used here are W jj ones hence,
the next-to-hardest jet will tend to have a transverse momentum as close as possible to
the analysis pT cut, and thus to the region affected by the generation bias in the case of
looser analysis cuts. The dijet invariant mass, shown in the middle plot of figure 1, tells
a slightly different story. Namely, the hard scale associated with this observable is not
in one-to-one correspondence with that used for imposing the analysis cuts, at variance
with the pT of the hardest jet discussed previously. Hence, the effects of the
generationlevel cuts are more evenly distributed across the whole kinematical range considered, as
can be best seen from the lower inset. Essentially, the bias here amounts largely to a
normalization mismatch, which disappears when tightening the analysis cuts. Finally, the
R distribution, presented in the lower part of figure 1, is representative of a case where
both shapes and normalization are biased. There is a trend towards larger biases at large
R, which is understandable since this region receives the most significant contributions
from large-rapidity regions, where the transverse momenta tend to be relatively small and
hence closer to the bias region. For all the observables considered in figure 1, we have also
computed the parton-level NLO results (with MadFKS and MadLoop), which are shown
as symbols in the plots. As can be seen from the figure, these results are quite close to the
corresponding aMC@NLO ones, and especially so when the analysis cuts are tightened.
Some differences can still be observed, due to the fact that the jets reconstructed using
aMC@NLO events are at the hadronic level, and emerge from high-multiplicity final states.
We conclude this section with some further comments on validation exercises. Firstly,
we started by testing the whole machinery in the simpler case of W j production. Although,
as was discussed before, for this process generation cuts may be imposed on pT (W ), we
have chosen to require the presence of at least one jet with a transverse momentum larger
than a given value, so as to mimic the strategy followed in the W jj case. Secondly, we have
checked that we obtain unbiased results by suitably changing the jet-cone size. Thirdly,
we have exploited the fact that the starting scale of the shower is to some extent arbitrary,
and the dependence upon its value is very much reduced in the context of an NLO-PSMC
matched computation. As was discussed in ref. , in MC@NLO the information on the
starting scale is included in the MC counterterms, and the independence of the physical
results of its value constitutes a powerful check of a correct implementation. We have
verified that this is indeed the case, by considering several different scale choices in a
neighbourhood of the partonic, Born-level, c.m. energy.
W jj production at the Tevatron
The hard events obtained with the generation cuts described above can be used to impose
the selection cuts employed by the CDF collaboration . The latter are as follows (where
with lepton we always mean the charged one):
minimal transverse energy for the lepton: ET () > 20 GeV;
minimal missing transverse energy: E/T > 25 GeV;
jet definition: JetClu algorithm with 0.75 overlap and R = 0.4;
minimal transverse jet energy: ET (j) > 30 GeV;
minimal jet pair transverse momentum: pT (j1j2) > 40 GeV;
lepton isolation: transverse hadronic energy smaller than 10% of the lepton transverse
energy in a cone of R = 0.4 around the lepton.
These cuts (and their analogues in the D0 analysis , which give very similar results in
the signal region) are tighter than the pT = 25 GeV analysis cut previously discussed.
Since the latter was seen to give unbiased results in the central rapidity regions relevant
here, we deem our approach safe. The cuts reported above (which we dub exclusive)
have also been slightly relaxed by CDF (see ), by accepting events with three jets or
more in the central and hard region this amounts to not applying the jet-veto condition
reported in the last bullet above; we call these cuts inclusive.
In addition to the aMC@NLO predictions, we have performed parton-level LO and
NLO computations. Finally, we have showered events obtained by unweighting LO matrix
elements as well. As is well known, the latter case is potentially plagued by severe
doublecounting effects which, although formally affecting perturbative coefficients of order higher
than leading, can be numerically dominant. We have indeed found that this is the case
for the cuts considered here: predictions obtained with generation cuts pT = 5 GeV and
pT = 10 GeV (and with anti-kT algorithm and R = 0.4) differ by 30% or larger for total
rates (shapes are in general better agreement), even for the analysis cut of pT = 50 GeV.
We have therefore opted for using a matched LO sample, which we have obtained with
Alpgen  interfaced to HERWIG through the MLM prescription . In order to do
this, we have generated W + n parton events, with n = 1, 2, 3. The dominant contribution
to W jj observables is due to the n = 2 sample, but that of n = 3 is not negligible. The
size of the n = 1 contribution is always small, and rapidly decreasing with dijet invariant
masses; it is thus fully safe not to consider W + 0 parton events.
In figures 2 and 3 we present our predictions for the invariant mass of the pair of the
two hardest jets with exclusive and inclusive cuts, respectively. The three histograms in
the main frames are the aMC@NLO (solid red), Alpgen+MLM (dashed blue), and NLO
parton level (green symbols) predictions. The two NLO-based results are obtained with the
pT = 10 GeV generation cuts. The Alpgen+MLM curves have been rescaled to be as close as
possible to the NLO ones, since their role is that of providing a prediction for the shapes, but
not for the rates (incidentally, this is also what is done in the experimental analyses when
control samples are not available). The upper insets show the ratios of the Alpgen+MLM
and NLO results over the aMC@NLO ones. The middle insets display the fractional scale
(dashed red) and PDF (solid black) uncertainties given by aMC@NLO, computed with
the reweighting technique described in ref. . The lower insets show the ratios of the
aMC@NLO results obtained with the two generation cuts, and imply that indeed there is
no bias due to generation cuts. We have also checked that removing the lepton isolation
cut does not change the pattern of the plots, all results moving consistently upwards by a
very small amount. The fraction of the 10M generated events passing the inclusive-analysis
cuts for the pT = 10 GeV (5 GeV) generation cuts is 0.92% (0.35%), with the fraction of
negative-weight events equal to 29% (34%). In the case of exclusive cuts, the fractions
of events passing the cuts are marginally smaller, and those of negative events the same
as those reported above. We point out that the accuracy to which the scale dependence
is determined cannot be directly inferred from these numbers, being much better than a
simple counting would suggest. In fact, as discussed in ref. , by using a reweighting
technique predictions obtained with different scale or PDF settings are correlated (i.e., they
are obtained with the same random number seeds in both the NLO and the MC runs).
By inspection of figures 2 and 3, we can conclude that the three predictions agree
rather well, and are actually strictly equivalent, when the theoretical uncertainties affecting
aMC@NLO are taken into account (i.e., it is not even necessary to consider those relevant
to Alpgen+MLM and parton-level NLO). This is quite remarkable, also in view of the fact
that the dominant contribution to the latter, the scale dependence, amounts to a mere
(+10%, 15%) effect. We have verified that such a dependence is in agreement with that
predicted by MCFM .
In spite of their being not significant for the comparison with data, it is perhaps
interesting to speculate on the tiny differences between the central aMC@NLO, Alpgen+MLM,
and NLO predictions. The total rates given by aMC@NLO and NLO are close but not
identical; this is normal, and is a consequence of the fact that the kinematical
distributions in the two computations are different, and thus differently affected by the hard cuts
considered here. More interestingly, the Mjj distribution predicted at the NLO is (very)
slightly harder than that of aMC@NLO, especially in the case of exclusive cuts. This is
best seen in the upper insets of figures 2 and 3, and is due to the fact that the fraction of
events with a third central and hard jet is larger in aMC@NLO than at the parton-level
NLO. This argument applies also to the case of inclusive cuts. In fact, by requiring the
two hardest jets to have a large invariant pair mass, and given the presence of a W boson,
one forces extra QCD radiation to be fairly soft, since relatively-hard radiation is strongly
suppressed by the damping of the PDFs at large Bjorken xs. This effectively imposes a
veto-like condition on the events, which however, at Mjj 300 GeV, is still larger than the
explicit 30 GeV one imposed by CDF; hence, NLO predictions for inclusive cuts are slightly
harder than the aMC@NLO ones, but less than in the case of exclusive cuts. We point
out that a veto on the third jet (be it explicit or effective) introduces a new mass scale in
the problem, whose ratio over Mjj may grow large. In such a situation, the resummation
of large logarithms performed by the shower constitutes an improvement over fixed-order
results. Given the level of agreement we find here, we can conclude the resummation effects
are still fairly marginal.
As far as the comparison between the central aMC@NLO and Alpgen+MLM
predictions is concerned, this is affected by the choice of the hard scales, which are different in the
two codes: in Alpgen, the transverse W -boson mass is adopted (the renormalization scale
is then effectively redefined through the reweighting of the matrix elements by S factors,
which is specific of the merging procedure ). In spite of this, the agreement between the
two results is quite good, with Alpgen+MLM being slightly harder than aMC@NLO (this
effect being of the same order or smaller than that observed with parton-level NLO results).
We have also compared Alpgen+MLM with aMC@NLO, by setting the hard scales in the
latter equal to the transverse W -boson mass.3 The ratio of these two results is shown as
open boxes in the upper inset of figures 2 and 3, whence one sees a marginal improvement
in the agreement between the two predictions w.r.t. the case corresponding to = HT /2,
which is our aMC@NLO default. We finally stress again that the MLM prescription is
crucial to get rid of double-counting effects in LO samples. While double counting is
guaranteed not to occur at the NLO in MC@NLO, it can still affect terms of O(S4) and beyond.
Although we did not see any evidence of these in the form of generation-cut dependence,
we have also heuristically extended the MLM prescription to NLO, by requiring the two
hardest jets after shower to be matched with two jets reconstructed at the hard-subprocess
3Note that, since we determine the scale dependence through the reweighting technique of ref. , we
do not need to run aMC@NLO a second time.
level (where they play the same roles as the partons in the original MLM matching). This
prescription has had no visible effect on our results. Although this is a process-dependent
conclusion, it confirms the naive expectation that NLO-PSMC matching is less prone to
theoretical systematics than its LO counterpart, and suggests that a reduction of the
dependence upon unphysical merging parameters can be achieved by extending the CKKW
or MLM procedures to the NLO.
In this paper, we have presented the automated computation of the W jj cross section to
the NLO accuracy in QCD, and its matching to parton showers according to the MC@NLO
formalism. This is the first time that a process of this complexity has been matched to an
event generator beyond the LO. We believe this is significant not only as a
phenomenological result, but also in view of the fact that it is also the first time that the MC@NLO
prescription has been applied to a process that requires the presence of cutoffs at the Born
level in order to prevent phase-space divergences from appearing. In fact, the structure
of such divergences in W jj production is sufficiently involved to provide evidence that no
new problems of principle are expected in the application of MC@NLO to processes with
even larger final-state multiplicities.
We have given predictions for the dijet invariant mass in W jj events, using the same
cuts as CDF and D0 in the signal region. Perturbative, parton-level results agree well with
those obtained after shower, and we do not observe any significant effects in the shape of
distributions due to NLO corrections, which therefore cannot be responsible for the excess
of events observed by the CDF collaboration.
We would like to thank Johan Alwall, Michelangelo Mangano and Bryan Webber for useful
discussions. S. F. is indebted to Michelangelo Mangano for his assistance in running Alpgen.
This research has been supported by the Swiss National Science Foundation (SNF) under
contract 200020-138206, by the Belgian IAP Program, BELSPO P6/11-P and the IISN
convention 4.4511.10, by the Spanish Ministry of education under contract PR2010-0285.
F.M. and R.P. thank the financial support of the MEC project FPA2008-02984 (FALCON).
R.F. and R.P. would like to thank the KITP at UCSB for the kind hospitality offered while
an important part of this work was being accomplished.
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License which permits any use, distribution and reproduction in any medium,
provided the original author(s) and source are credited.
 CDF collaboration, T. Aaltonen et al., Invariant mass distribution of jet pairs produced in
association with a W boson in pp collisions at s = 1.96 TeV,
Phys. Rev. Lett. 106 (2011) 171801 [arXiv:1104.0699] [INSPIRE].