#### Four-lepton production at hadron colliders: aMC@NLO predictions with theoretical uncertainties

Rikkert Frederix
4
Stefano Frixione
2
3
Valentin Hirschi
3
Fabio Maltoni
0
Roberto Pittau
1
Paolo Torrielli
3
0
Centre for Cosmology, Particle Physics and Phenomenology (CP3), Universite catholique de Lou- vain
, B-1348 Louvain-la-Neuve,
Belgium
1
Departamento de Fsica Teorica y del Cosmos y CAFPE, Universidad de Granada
, Granada,
Spain
2
PH Department, TH Unit
, CERN, CH-1211 Geneva 23,
Switzerland
3
ITPP, EPFL, CH-1015 Lausanne,
Switzerland
4
Institut fur Theoretische Physik, Universitat Zurich
, Winterthurerstrasse 190, CH-8057 Zurich,
Switzerland
We use aMC@NLO to study the production of four charged leptons at the LHC, performing parton showers with both HERWIG and Pythia6. Our underlying matrix element calculation features the full next-to-leading order O(S) result and the O(S2) contribution of the gg channel, and it includes all off-shell, spin-correlation, virtualphoton-exchange, and interference effects. We present several key distributions together with the corresponding theoretical uncertainties. These are obtained through a processindependent technique that allows aMC@NLO to compute scale and PDF uncertainties in a fully automated way and at no extra CPU-time cost.
1 Introduction 2 3 4
Technical aspects of the computation
2.1 Scale and PDF uncertainties Phenomenological results Conclusions Acknowledgments
A Definition of W coefficients for NLO cross sections
Introduction
Light charged leptons constitute a particularly clean trigger for high-energy experiments. It
is thus relatively easy to measure, despite their being small, the cross sections of processes
that feature several final-state leptons. Prominent among such processes are those mediated
by a pair of electroweak vector bosons which, depending on their identities, may give rise to
a missing-energy signature as well. Vector boson pair production is interesting in at least
two respects. Firstly, it is an irreducible background to Higgs signals, in particular through
the W +W and ZZ channels which are relevant to searches for a standard model Higgs
of mass larger than about 140 GeV. While always smaller than the W +W channel, ZZ
decays may provide a cleaner signal due to the possibility of fully reconstructing the decay
products of the two Zs. Secondly, di-boson cross sections are quite sensitive to violations
of the gauge structure of the Standard Model, and hence are good probes of scenarios
where new physics is heavy and not directly accessible at the LHC, yet the couplings in
the vector boson sector are affected.
The Higgs being such an elusive particle, searches require a combination of data-driven
and theoretical methods in order to overcome irreducible backgrounds. From the theory
viewpoint, this essentially implies computations as accurate as possible, and ideally able
to realistically reproduce actual experimental events. On the other hand, modifications to
tri-linear gauge couplings will tend to manifest themselves at large transverse momenta,
and therefore this is the region where theoretical results have to be reliable. The inclusion
of next-to-leading order (NLO) QCD corrections into four-lepton cross section predictions
is a minimal, and fairly satisfactory, way to fulfill both of the two conditions above.
The aim of this paper is that of studying four-lepton hadroproduction at the NLO
accuracy in QCD, by also adding the (finite) contribution due to gg fusion (which is
formally of next-to-next-to-leading order, NNLO, yet enhanced by the gluon PDFs), and by
including the matching with parton showers according to the MC@NLO formalism [1]; this
is done by means of the framework aMC@NLO [2, 3] . Since aMC@NLO provides the
full automation of the matching procedure, and of the underlying matrix element
computations, we can give results for any four-lepton final state. In order to be definite, we have
chosen to consider the process:
pp (Z/) (Z/) +()+() ,
with , = e , . This choice is simply motivated by the fact that ZZ production as
currently implemented in the MC@NLO package [4] does not feature production spin
correlation and off-shell effects (which on the other hand are included, although in an
approximated way, in the case of W W + and W Z production in the codes of ref. [4]), and
virtual-photon contributions with their interference with the Zs. The present aMC@NLO
application remedies to these deficiencies by computing in an exact manner all relevant
matrix elements (including the contributions of singly-resonant diagrams, which are
potentially relevant to analyses in kinematical regions where one of the Z bosons is forced to
be off-shell), and by including, as was already mentioned, the gg-initiated contribution as
well which, although perturbatively suppressed, can be numerically important at the LHC
owing to its dominant parton luminosity. We remind the reader that ZZ production at
parton level and NLO accuracy in QCD has been studied for two decades now. The on-shell
calculations of refs. [5, 6] have been subsequently improved to include leptonic decays [7],
singly-resonant diagrams [8], and anomalous couplings [9]; these results have been used to
study the effects of the NLL resummation of soft gluons [10], and more recently to match
the NLO computation to parton showers according to the POWHEG formalism [11]. The
O(S2) process gg ZZ was first computed in refs. [1214]. These papers have been later
superseded by the inclusion of off-shell effects and Z/ interference [15, 16]. As far as
MC@NLO results are concerned, and on top of the phenomenological results presented
here, this paper is a first for three reasons:
The matching with Pythia6 [17] for a kinematically non-trivial process.
The use of MadLoop [18] for the computation of loop-induced processes (i.e., the
contributions of finite one-loop amplitudes squared).
The computations of scale and PDF uncertainties with a reweighting technique.
In fact, the use of Pythia for the shower phase in MC@NLO has been limited so far to
a proof-of-concept case [19], while the squares of loop amplitudes have been considered in
MadLoop only for pointwise tests [18], and not in phenomenological applications. Finally,
the reweighting procedure which we describe in this paper, and that is used to compute
the scale and PDF dependences of the cross section, allows one to determine the
corresponding uncertainties without requiring any additional computing time. We point out
that the capabilities listed in the three items above are not process dependent and are fully
automated.
This paper is organized as follows: in section 2 we discuss some of the technicalities
relevant to four-lepton production, and the procedure implemented in aMC@NLO for the
determination of scale and PDF uncertainties. In section 3 we present selected
phenomenological results, and in section 4 we draw our conclusions. Further details on scale and PDF
dependence computations are reported in appendix A.
Technical aspects of the computation
The framework of the aMC@NLO programme used for the phenomenology studies of
this paper is unchanged w.r.t. that employed in ref. [3]; in particular, the underlying
tree-level computations are performed with MadGraph v4 [20]. We remind the reader
that aMC@NLO automates all aspects of an NLO computation and of its matching with
partons showers. One-loop amplitudes are evaluated with MadLoop [18], whose core
is the OPP integrand reduction method [21] as implemented in CutTools [22]. Real
contributions and the corresponding phase-space subtractions, achieved by means of the
FKS formalism [23], as well as their combination with the one-loop and Born results and
their subsequent integration, are performed by MadFKS [24], which finally takes care of
the MC@NLO matching [1] as well.
The novel features of aMC@NLO whose results we present here are the matching
with the virtuality-ordered Pythia6 shower, based on ref. [19]; the possibility of
determining through reweighting the scale and PDF uncertainties affecting our predictions, which
we shall discuss in general in section 2.1 (with some further technical details given in
appendix A); and the use of MadLoop for the computation of the O(S2) partonic process
gg (Z/) (Z/) +()+() .
The matrix elements relevant to this process are UV- and IR-finite, and therefore the
corresponding generated events are treated as an LO sample from the viewpoint of matching
with parton showers. The amplitudes can be straightforwardly computed by MadLoop. In
the current version, MadLoop assumes that these amplitudes will have to be multiplied by
Born ones; we have extended the scope of the code, in order for it to compute amplitudes
squared.
The fact that the process in eq. (2.1) is not a genuine virtual correction, i.e. it lacks an
underlying Born, implies that event unweighting is essentially a brute-force operation (as
opposed to the normal situation where the presence of an underlying Born allows one to
pre-determine with good accuracy the peak structure of the one-loop matrix elements, thus
significantly increasing the unweighting efficiency). This results in a fairly large number of
calls to the matrix elements of eq. (2.1) per unweighted event and, given the computing
performances of the current version of MadLoop, renders it very time expensive to obtain
a good-sized unweighted-event sample (say, O(1 M)). We have therefore opted for an
alternative approach. We have generated unweighted events using the matrix elements of
the Born process
uu (Z/) (Z/) +()+() ,
multiplied by the gg parton luminosity (rather than by the uu one that would be required if
computing the Born contribution proper). The kinematical configurations so obtained have
been used to compute the matrix elements of eq. (2.1), which thus provide the correct event
weights. A similar procedure has been employed in a recent study on Higgs production via
heavy-quark loops and has been shown to work well even for more complex final states [25].
Scale and PDF uncertainties
Among the parameters which enter a short-distance cross section, renormalization ( R) and
factorization ( F ) scales and PDFs play a special role, since their variations are typically
associated with the purely theoretical uncertainty affecting observable predictions (which,
in the case of PDFs, is not quite correct one may say that PDF variations parametrize
the uncertainties not directly arising from the process under study). It is therefore
fortunate that (the bulk of) the time-consuming matrix element computations can be rendered
independent of scales and PDFs, as opposed to what happens in the case of other
parameters, e.g. particle masses. This is doable thanks to the fact that short-distance cross
sections can be written as linear combinations of scale- and PDF-dependent terms, with
coefficients independent of both scales and PDFs; it is thus possible to compute such
coefficients once and for all, and to combine them at a later stage with different scales and
PDFs at essentially zero cost from the CPU viewpoint. The crucial (and non-trivial) point
is that the noteworthy structure mentioned before is a feature not only of the parton-level
LO and NLO cross sections, but also of the MC@NLO ones. This implies that from the
conceptual point of view the same procedure for determining scale and PDF uncertainties
can be adopted in MC@NLO as in LO-based Monte Carlo simulations; for the former we
shall simply need to compute a larger number of coefficients than for the latter.
In order to illustrate what is done in aMC@NLO, we start from dealing with LO
computations, for which the notation is simpler. Here and in what follows, the expressions
for the short-distance cross sections are taken from ref. [24]. The fully-differential cross
section is:
d(LO) = f1(x1, F )f2(x2, F )d(B,n)JBj dBj ,
N
Here, dBj denotes the integration measure over the Bjorken xs, and JBj is the (possibly
trivial) corresponding jacobian factor. The quantities on the r.h.s. of eq. (2.4) are the
scattering amplitude squared M(n,0) (including coupling constants, and colour/spin average
and flux factors), a set of cuts J n(LB) which prevent phase-space singularities from appearing
at this perturbative order, an average factor N for identical final-state particles, and the
n-body phase space dn. We shall write the latter as follows:
where dn is a measure that understands the choice of 3n 4 independent integration
variables, and Kn(n) denotes the four-momentum configuration associated with a given
choice of the latter. Finally, (n) is the phase-space factor (including jacobian) that arises
from the choice of dn. One can then rewrite eq. (2.3) as follows:
d(LO) = f1(x1, F )f2(x2, F )gS2b( R)w(B,n)dBj dn ,
N
and, consistently with ref. [24], b is the power of S implicit in M(n,0). By construction,
the quantity w(B,n) defined in eq. (2.7) is independent of scales and PDFs, and is the first
example of the coefficients mentioned at the beginning of this section.
When integrating eq. (2.3), one obtains the N -event set
in the case of weighted and unweighted event generation respectively.1 Given an observable
O, the cross section in the range OLOW O < OUPP (e.g., an histogram bin) will be
at the hard-subprocess level, and
after shower. In eq. (2.11), MC(Kn;i, x1;i, x2;i) denotes the complete final-state
configuration obtained by showering the hard subprocess.
When making alternative choices for scales and PDFs, say R, F , and fi, one can
simply repeat the procedure outlined above; this is straightforward, but extremely time
consuming if the full scale and PDF uncertainties have to be determined. Alternatively,
one can compute eqs. (2.10) and (2.11) by performing the replacement
Ri = f1 (x1;i, F )f2 (x2;i, F )gS2b( R)w(B,n)(Kn;i)
The denominator here is computed using eq. (2.6), i.e. with the original choices of scales
and PDFs. As we shall show in the following, an analogous equation will hold for both the
NLO and MC@NLO cases. From eq. (2.6) we obtain:
f1 (x1;i, F )f2 (x2;i, F )gS2b( R) ,
Ri = f1(x1;i, F )f2(x2;i, F )gS2b( R)
1It should be obvious that the kinematic configurations and Bjorken xs in eq. (2.8) are not the same
for weighted- and unweighted-event generation. Nevertheless, we have used an identical notation since no
confusion is possible.
which shows explicitly that the computation of Ri does not entail any matrix-element
calculation. Therefore, after performing the bulk of the calculation, i.e. the determination
of the set in eq. (2.8), one can fill a central histogram with the weights defined in
eq. (2.9), and as many variation histograms as one likes with the weights that appear
on the r.h.s. of eq. (2.12), by simply recomputing the factors Ri for all the different scales
and PDFs needed.
The procedure outlined above is exact when applied to eq. (2.10), but it implies an
approximation in the case of eq. (2.11). This is because the quantity MC(Kn;i, x1;i, x2;i)
does contain an implicit dependence on fi, and these functions are not replaced by fi when
using eq. (2.12) (since such a replacement would imply performing the shower for each
new choice of PDFs). However, this approximation is usually a very good one (barring
perhaps the corners of the phase space, and in particular the large-rapidity regions). This
is empirically well known,2 and is due to the fact that the Sudakov form factors used in
the backward evolution of initial-state partons are sensitive to the ratios of PDFs, whose
variation is much smaller than that of their absolute values (the more so within a
PDFerror set, which is the typical application of the procedure discussed here). Furthermore, it
should be clear that both scale and PDF uncertainties are defined using well-motivated but
ultimately arbitrary conventions, and therefore are not quantities that can be computed
with arbitrary precision.
We now turn to discussing the case of NLO short-distance cross sections.
Equation (2.6) is generalized as follows:
d(NLO) = X d(NLO,) ,
d(NLO,) = f1(x(1), (F))f2(x(2), (F))W ()dBj dn+1 ,
where = E, S, C, and SC correspond to the contributions of the fully-resolved
configuration (the event), and of its soft, collinear, and soft-collinear limits (the counterevents)
respectively. The quantities W () will be written as follows:
where Q is the Ellis-Sexton scale; in eq. (2.17), this scale (which is extensively used in the
manipulation of the one-loop contribution see ref. [24] for a discussion in the context
of MadFKS) offers a convenient way to parametrize the dependence on the factorization
and renormalization scales. As the notation in eqs. (2.16) and (2.17) suggests, the values
that such scales assume in the event and counterevents may be different from each other
2 We are not aware of any quantitative statement regarding this matter. In all the cases we have
considered, both in the context of past (a)MC@NLO studies and when validating the reweighting
procedure proposed here (which involved the generation of two-lepton, four-lepton, and tt event samples), no
statistically-significant differences have been observed.
(however, they must tend to the same value when considering the relevant infrared limits).
Furthermore, eq. (2.16) takes into account the fact that, when using event projection
to write an NLO partonic cross section (see ref. [1]), the Bjorken xs of the event and
counterevents need not coincide. The last term on the r.h.s. of eq. (2.17) is the Born
contribution, which in MadFKS can be integrated simultaneously with the other terms
that enter an NLO cross section; having a soft-type kinematics, it is naturally associated
with the soft counterevent. The coefficients Wc0(), WcF(), and WcR() introduced in eq. (2.17)
are the analogues of w(B,n) defined in eq. (2.7) they are scale- and PDF-independent;
their explicit forms are given in appendix A. As far as WcB is concerned, it is equal to
w(B,n), up to a normalization factor (due to the different integration measures in eq. (2.6)
and (2.16)).
The integration of the NLO cross section leads to the set of N weighted events:3
Equation (2.10) is generalized as follows:
As is well known, for any given i the weights defined in eq. (2.19) may diverge, but the sum
in eq. (2.20) is finite for any infrared-safe observable. When changing scales and PDFs, we
can now adopt the same procedure introduced before for the LO cross sections, and define
the rescaling factors:
Ri() = f1 (x(1;i), F())f2 (x(2;i), F())
The quantities Ri() thus defined do not factorize the coefficients Wc, and hence have a
more complicated form than their LO counterpart, eq. (2.14). Nevertheless, precisely as in
the case of eq. (2.14), their computations only require the evaluation of PDFs and coupling
constants for each new choice of PDFs and scales, the idea being that the coefficients Wc
3We remind the reader that parton-level, NLO events not matched with showers cannot be unweighted
without the introduction of an arbitrary and unphysical cutoff.
are computed once and for all when integrating the partonic cross section, and are stored
for their later re-use in eq. (2.21).
We have also understood that the formulae presented above apply to the contribution
due to one given FKS pair, i.e. they are proportional to a given S function Sij . We remind
the reader that while the observable cross section is obtained by summing over all
Sfunction contributions, these are fully independent from each other, and can be effectively
treated as independent partonic processes.
We finally discuss the case of MC@NLO. The short-distance cross sections can be
written as follows [1]:
with the Monte Carlo counterterms given by:
d(MC,c) = f1(x(1MC,c), F )f2(x(2MC,c), F )gS2b+2( R)w(MC,c)dBj dn+1 .
The index c in eqs. (2.22) and (2.23) runs over all particles which are colour-connected with
the particle that branches. The latter is identified with the parent particle of the FKS pair
(i, j) which determines the S function Sij implicit in d(NLO,). This is sufficient to render
eqs. (2.22) and (2.23) locally finite, as prescribed by the MC@NLO formalism. Note that
the sum over FKS pairs is equivalent to summing over all possible MC branchings. By
integrating eqs. (2.22) and (2.23) one obtains the sets of events:
whose kinematic parts are:
EH;i = nKH;i , x(1E;i) , x(2E;i) , {x(1M;iC,c), x(2M;iC,c)}co ,
ES;i = nKS;i , {x(1;i), x(2;i)}=S,C,SC , {x(1M;iC,c), x(2M;iC,c)}co .
The final-state kinematic configurations relevant to H and S events are:
KH;i = Kn(E+)1;i ,
KS;i = Kn(S+)1;i ,
where we have used the fact that in MadFKS and aMC@NLO the phase-space
parametrization is chosen in such a way that the kinematics of the counterevents coincide (up to the
irrelevant unresolved partons; see ref. [24]). The weights that appear in eq. (2.25) are:
for weighted events, and
Ri(H) =
f1 (x(1E;i), F(E))f2 (x(2E;i), F(E))gS2b+2( R(E))
Ri(S) =
Wc0(E)(EH;i) + WcF(E)(EH;i) log
Several observations are in order here. Firstly, the weights defined in eq. (2.29) are finite,
at variance with those relevant to the computation of the parton-level NLO cross section,
eq. (2.19). This is due to the properties of the MC counterterms, and to the fact that
the kinematics of the H and S events is uniquely defined. Secondly, and as a consequence
of the former point, the quantities relevant to the counterevents are summed together in
eq. (2.32), at variance with what happens in eq. (2.21). Thirdly, in eqs. (2.31) and (2.32)
the scales entering the counterevents have been set equal to the values assumed in the
soft configurations, and those entering the MC counterterms equal to the corresponding
NLO event contributions. Many variants of these choices are possible (which is part of the
ambiguity in the definition of the scale dependence in an NLO computation), and those
given here are simply the current defaults in aMC@NLO.
Using the definitions given above, eq. (2.11) is generalized as follows:
Here, the same remark as made in the case of LO-showered predictions applies, that
concerns the PDF dependence implicit in MC(EH;i) and MC(ES;i). There is however a further
where we have assumed an emission from leg 1 to be definite, and all overall factors
irrelevant to this discussion have been neglected. The ratio of PDFs in eq. (2.35) is due to
the MC branching, and [f1(x1)Born] is the short-distance contribution. When S events are
reweighted with Ri(S) without changing the PDFs in the shower, eq. (2.35) becomes:
and therefore the cancellation of the PDFs with argument x1 does not take place any longer.
Since changing the PDFs in the shower would amount to replacing f1 with f1 in eq. (2.36),
eq. (2.36) implies a fractional difference w.r.t. the use of f1 in both the short-distance and
shower computations equal to:
subtlety in the case of MC@NLO. When showering the Born contribution to S events, one
cancels the MC counterterm contribution to H events. In the case of initial-state
emissions, we can schematically write the contribution to the H events obtained by showering
the Born as follows:
which is a number generally fairly close to one. In fact, what is discussed here is precisely
the situation one faces when running MC@NLO and adopting different PDFs in the
shortdistance calculation and shower phases. This is rather commonly done, and no significant
differences are observed w.r.t. the results obtained with the strictly correct procedure of
using the same PDFs in the two phases. While this approximation may be undesirable
when very accurate results are mandatory, it is always acceptable when computing the
PDF uncertainty, which is by definition not a precision result. Furthermore, it should be
kept in mind that the procedure currently used for determining PDF error sets is that of
employing parton-level NLO cross sections; hence, such PDFs are not guaranteed to serve
their purpose for those observables for which matched predictions differ significantly from
the NLO ones (i.e., where resummation effects are important).
Phenomenological results
In this section we present some sample results obtained with aMC@NLO, its LO
counterpart (aMC@LO), and for parton-level NLO (O(S)) and LO (O(S0)) cross sections. We
also give showered predictions for the gg O(S2) contribution. In both HERWIG [26] and
Pythia, we have switched QED showers off. We consider the production of the following
leptonic final states:
++ = e+e + ,
at the 7 TeV LHC. The only cut at the generation level is:
M (()) 30 GeV ,
which is applied to all unlike-sign lepton pairs. This cut, that serves the purpose of
avoiding the zero-virtuality divergences of diagrams with resonant photons, is obviously
not necessary in the case of eq. (3.1) for e pairs, but it is nonetheless applied there in
order to perform a comparison on equal footing between the e+e + and e+ee+e final
states. The parameters used in this paper are reported in table 1, where the values of S
have been chosen as prescribed by the PDF sets we have adopted, MSTW2008(n)lo68cl [27]
(with their associated error sets). The default renormalization and factorization scales
are set equal to the invariant mass of the four leptons,
0 = M (+()+()) .
When studying scale dependences we shall vary R and F independently, i.e. we shall
set ( R, F ) = (R 0, F 0) and consider the combinations (R, F ) = (1, 1), (1/2, 1/2),
(1/2, 1), (1, 1/2), (1, 2), (2, 1), (2, 2). The overall scale uncertainty will be the envelope of
all individual results.
We begin by presenting in table 2 the fully inclusive cross sections (bar the
invariantmass cuts of eq. (3.3)). For the NLO results and the gg contribution we also give the
scale (first error) and PDF (second error) uncertainties. The scale dependence at the
NLO is fairly small, which is in part due to the fact that the qq parton luminosity is
almost F -independent in the Bjorken-x range that gives the dominant contribution to
the cross section. This implies that at the LO (which, being O(S0), is independent of
R) the scale uncertainty is comparable or smaller, and thus not particularly meaningful:
we refrain from quoting it explicitly here. The PDF uncertainty is estimated following
the prescription given by the PDF set [27] (asymmetric Hessian); at the LO, we obtain
a fractional uncertainty similar to that of the NLO result. On the other hand, the gg
channel displays the typical scale dependence of an O(S2) LO cross section, the figures
in the table receiving the dominant contributions from the R dependence. The scale
uncertainty is largely dominant over the PDF one. Not surprisingly, the ratio between the
two leptonic channels considered in this paper is equal to two up to a very small correction
(only slightly larger than the integration uncertainty), since differences arise in the
offresonance regions (where singly-resonant diagrams are more important than elsewhere)
whose contribution to the total integral is small. Given that the gg matrix elements do
feature only doubly-resonant diagrams, we have not computed the result relevant to the
pp e+e +
pp e+ee+e
12.90+00..2237((21..18%%))+00..2262((21..07%%))
6.43+00..1133((22..10%%))+00..1110((11..76%%))
0.566+00..111682((2208..86%%))+00..001124((22..15%%))
Table 2. Total cross sections for e+e + and e+ee+e production at the LHC (S = 7 TeV),
for the production channels considered in this paper, and within the cuts given in eq. (3.3). The first
and second errors affecting the results are the scale and PDF uncertainties (also given as fractions
of the central values). See the text for details.
e+ee+e channel. Finally, we point out that the entries of table 2 have been compared
with their analogues obtained from MCFM (also for the non-central scale values, which is
a check of the reweighting technique discussed in section 2.1), and complete agreement has
been found.
As was mentioned above, the scale uncertainties reported in table 2 are the envelopes
of the results obtained with the seven combinations of ( R, F ) considered here. It is
obvious that this is not equivalent to variations in the range 0/2 R, F 2 0 if the
dependence on either scale is not monotonic. While the study of continuous variations
is clearly impossible, by means of reweighting one can probe the relevant scale ranges
with more than three values for each scale, at a very modest CPU cost. In fact, each
choice of ( R, F ) corresponds to filling an extra set of histograms (each set containing all
observables of interest), with the same kinematics as for the central set and with rescaled
weights, which is a negligible fraction of the generation-plus-shower computing time.
We now turn to discussing the case of differential distributions, and we start by
considering the O(S0) and O(S) contributions. We shall later address the (generally very small)
differences between HERWIG and Pythia6, and here we shall use only HERWIG in order
to be definite. The aim of figures 13 is that of assessing the perturbative behaviour of our
predictions. In the main frame, these figures present the results for the e+e + channel
obtained with aMC@NLO (solid black), NLO (dashed red), aMC@LO (solid blue), LO
(dashed magenta). The upper insets display the ratios of the aMC@LO (dot-dashed blue),
NLO (dashed red) and the LO (dotted magenta) predictions over the aMC@NLO ones.
The middle insets show the scale (dashed red) and PDF (solid black) uncertainties, both
with aMC@NLO, and given as ratios over the central values. Finally, the lower insets
present the ratios
with O any of the observables considered, as computed by aMC@NLO.
Figure 1 displays two observables constructed with the sum of the four-momenta of the
four leptons, the invariant mass (left panel) and the transverse momentum (right panel).
Figure 1. Cross section per bin for the four-lepton invariant mass (left panel) and transverse
momentum (right panel), as predicted by aMC@NLO (solid black), aMC@LO (dot-dashed blue),
and at the (parton-level) NLO (dashed red) and LO (dotted magenta). The upper insets show
the ratios of the aMC@LO (dot-dashed blue), parton-level NLO (dashed red), and LO (dotted
magenta) predictions over the aMC@NLO ones; the middle insets the aMC@NLO scale (dashed
red) and PDF (black solid) fractional uncertainties; and the lower insets the ratio of the two leptonic
channels, eq. (3.5). See the text for details.
These have very different behaviours w.r.t. the extra radiation provided by the parton
shower, with the former being (almost) completely insensitive to it, and the latter (almost)
maximally sensitive to it. In fact, the predictions for the invariant mass are basically
independent of the shower, with NLO (LO) being equal to aMC@NLO (aMC@LO) over
the whole range considered. The NLO corrections amount largely to an overall rescaling,
with a very minimal tendency to harden the spectrum. The four-lepton pT , on the other
hand, is a well known example of an observable whose distribution at the parton-level LO
is a delta function (in this case, at pT = 0). Radiation, be it through either showering or
hard emission provided by real matrix elements in the NLO computation, fills the phase
space with radically different characteristics, aMC@LO being meaningful at small pT and
NLO parton level at large pT aMC@NLO correctly interpolates between the two. The
different behaviours under extra radiation of the two observables shown in figure 1 is
reflected in the scale uncertainty: while in the case of the invariant mass the band becomes
very marginally wider towards large M (e+e + ) values, the corresponding effect is
dramatic in the case of the transverse momentum. This is easy to understand from the
purely perturbative point of view, and is due to the fact that, in spite of being O(S)
for any pT > 0, the transverse momentum in this range is effectively an LO observable
(the NLO effects being confined to pT = 0). The matching with shower blurs this picture,
and in particular it gives rise to the counterintuitive result where the scale dependence
increases, rather than decreasing, when moving towards large pT [19]. Finally, the lower
insets of figure 1 display the ratio defined in eq. (3.5) which, in agreement with the results
Figure 2. Cross section per bin for the inclusive pT of the positively-charged leptons (left panel),
and the inclusive pT of the same-charge lepton pairs (right panel), both with Z-id cuts, as predicted
by aMC@NLO (solid black), aMC@LO (dot-dashed blue), and at the (parton-level) NLO (dashed
red) and LO (dotted magenta). The upper insets show the ratios of the aMC@LO (dot-dashed
blue), parton-level NLO (dashed red), and LO (dotted magenta) predictions over the aMC@NLO
ones; the middle insets the aMC@NLO scale (dashed red) and PDF (black solid) fractional
uncertainties; and the lower insets the ratio of the two leptonic channels, eq. (3.5). See the text for
details.
of table 2, is equal to one half in the whole kinematic ranges considered. The only exception
is the small invariant mass region, where off-resonance effects become relevant.
A gauge-invariant way to suppress off-resonance effects, and to select doubly-resonant
contributions, is that of imposing:
M (+) mZ 10 GeV
on all equal-flavour lepton pairs; we call the cut of eq. (3.6) the Z-id cut. Lepton pairs that
pass the Z-id cut are called Z-id matched, and can be roughly seen as coming from the
decay of a (generally off-shell) Z boson. While in the case of the e+e + channel there
is only one way to choose two same-flavour lepton pairs, there are two different pairings in
e+ee+e production. In the case both of these pairings result in lepton pairs that fulfill
eq. (3.6), we choose that with the smallest pair invariant mass, and assign the Z-id matched
pairs according to this choice; in practice, this is a rare event. By imposing the Z-id cuts
the M (+()+()) distribution falls steeply below threshold and gets no contributions
below 160 GeV.
In figure 2 we present two transverse momentum distributions, relevant to the
positivelycharged leptons (left panel), and to same-charge lepton pairs (right panel); hence, there are
two entries in each histogram for any given event. These results are obtained by applying
the Z-id cuts, but we have in fact verified that without such cuts we obtain exactly the
same patterns. In the case of the pT of the individual lepton, the aMC@NLO (aMC@LO)
prediction is fairly close to the NLO (LO) one, but tends to be slightly harder, owing to
the extra radiation generated by the shower. This effect is more pronounced at the LO
than at the NLO, which is the sign of a behaviour consistent with perturbation theory
expectations. In fact, at the LO all hadronic transverse momentum is provided by the
shower, while at the NLO this is not the case; therefore, at the NLO the shower will have
less necessity to correct the prediction obtained at the parton level, a tendency which is
naturally embedded in a matching prescription such as aMC@NLO. The scale dependence
is quite small over the whole range in pT , but tends to grow larger towards larger pT . This
effect has the same origin as that observed in the right panel of figure 1, but it is much more
moderate than there. This is due to the fact that in the present case the whole range in pT
is associated with complete NLO corrections. The PDF uncertainty is seen to be similar to
or slightly smaller than that due to scale variation; parton densities are well determined in
the x range probed here. Finally, there is no difference between the two leptonic channels
for this observable; as already mentioned above, this conclusion is independent of whether
one applies the Z-id cuts. The pT of the lepton pairs shown in the right panel of figure 2
follows the same pattern as the one we have just discussed, but the differences between
the various predictions are larger in this case. In particular, aMC@LO is closer to NLO
than to LO, which is a consequence of the more important role played by extra radiation in
this case (as one expects, the present one being a correlation between two particles rather
than a single-inclusive observable). Again, the closeness of NLO and aMC@NLO results
shows the desired perturbative behaviour. The more significant impact of extra radiation
on this variable is reflected in the slightly larger scale dependence at large pT s w.r.t. what
happens for the transverse momentum of the individual leptons discussed before. The two
leptonic channels agree well, also when removing the Z-id cuts.
Figure 3 shows two observables constructed after applying the Z-id cuts, namely the
pseudorapidity of lepton pairs with opposite charge which are also Z-id matched (left
panel; this is then the pseudorapidity of would-be Z bosons), and the azimuthal distance
between leptons of opposite charge which are not Z-id matched (right panel; thus, these
are leptons emerging from different would-be Z bosons). As in the case of figure 2, there
are two entries in each histogram for any given event. These two observables are dominated
by small transverse momenta, and therefore it is not surprising that, at both O(S0) and
O(S), the predictions are quite independent of whether a shower is generated or not.
Slight differences can be seen in the case of the distribution, which is indeed known to
be more sensitive than pseudorapidity to extra radiation. The small-pT dominance ensures
that scale and PDF uncertainties are flat over the whole kinematic ranges, and of the order
of those relevant to total cross section.
We now discuss the impact of the O(S2) gg channel on our predictions. The argument
for considering such a channel, despite its being of the same perturbative order as all other
NNLO contributions which cannot be included, is the dominance of its parton luminosity
over those of the qq and qg channels. This dominance grows stronger with decreasing
final-state invariant masses, and hence the O(S2) versus NLO comparison is significantly
influenced by the cut in eq. (3.3) by lowering such a cut, the relative importance of the
gg contribution will grow bigger than the 5%-ish reported in table 2. We also discuss in the
Figure 3. Cross section per bin for the inclusive of the opposite-charge, Z-id matched lepton pairs
(left panel), and the inclusive distance of the opposite-charge, non-Z-id matched lepton pairs
(right panel), as predicted by aMC@NLO (solid black), aMC@LO (dot-dashed blue), and at the
(parton-level) NLO (dashed red) and LO (dotted magenta). The upper insets show the ratios of the
aMC@LO (dot-dashed blue), parton-level NLO (dashed red), and LO (dotted magenta) predictions
over the aMC@NLO ones; the middle insets the aMC@NLO scale (dashed red) and PDF (black
solid) fractional uncertainties; and the lower insets the ratio of the two leptonic channels, eq. (3.5).
See the text for details.
following the differences that arise when matching our calculation to Pythia6 rather than
to HERWIG. We remind the reader that, depending on input parameters, Pythia is rather
effective in producing radiation in the whole kinematically-accessible phase space. This is
not particularly useful in the context of a matched computation, where hard radiation
is provided (in a way fully consistent with perturbation theory) by the underlying
realemission matrix elements. Therefore, we have set the maximum virtuality in Pythia
equal to the four-lepton invariant mass. For consistency, this setting has been used also
when showering the gg-initiated contribution.
Figures 4, 5 and 6 present the same observables as figures 1, 2 and 3 respectively. In
the main frame, we show the aMC@NLO predictions plus the gg contribution (including
shower), as resulting from HERWIG (solid black) and Pythia (dashed blue) we shall
call these predictions aMC@NLO+gg for brevity. The dashed-plus-symbol histograms are
the gg contributions (open black boxes: HERWIG; open blue circles: Pythia), rescaled in
order to fit on the same scale as aMC@NLO+gg; the rescaling factors are equal either to 10
or 20, as indicated in the figures they are chosen so as to render the comparisons of shapes
as transparent as possible. The upper insets display the ratios for the aMC@NLO+gg
HERWIG predictions over the aMC@NLO+gg Pythia ones (solid black) and the gg
HERWIG predictions over the gg Pythia ones (open blue boxes). The middle insets show
the fractional scale (dashed red) and PDF (black solid) uncertainties of the gg contribution
alone (i.e., the extrema divided by the central gg prediction); those shown are computed
Figure 4. Cross section per bin for the four-lepton invariant mass (left panel) and transverse
momentum (right panel), for aMC@NLO+gg HERWIG (solid black) and Pythia (dashed blue)
results. The rescaled gg contributions with HERWIG (open black boxes) and Pythia (open blue
circles) are shown separately. Upper insets: ratios of aMC@NLO+gg HERWIG predictions over
the aMC@NLO+gg Pythia ones (solid black) and of the gg HERWIG predictions over the gg
Pythia ones (open blue boxes). Middle insets: scale (dashed red) and PDF (solid black) fractional
uncertainties. Lower insets: aMC@NLO/(aMC@NLO+gg) with HERWIG (solid black) and
Pythia (dashed blue).
with HERWIG, but Pythia gives identical results. The lower insets present the ratios
aMC@NLO/(aMC@NLO+gg) for HERWIG (solid black) and Pythia (dashed blue).
The HERWIG and Pythia results for the four-lepton invariant mass shown in the
left panel of figure 4 are identical in the case of the gg contribution, and practically so in
the case of aMC@NLO. This is because both event generators fix the invariant mass of
the final-state system when doing initial-state showers; hence, any difference in the
fourlepton invariant mass can only arise from the real-emission contribution to aMC@NLO,
where the final-state system does not coincide with the four leptons. The gg channel
has a softer distribution than the qq + qg ones, since by moving towards large masses one
probes larger values of the Bjorken xs, where quark parton luminosities are relatively more
important than at threshold. In the small-invariant-mass region, the gg contribution does
not exhibit any peak structure, owing to the lack of singly-resonant diagrams. Ultimately,
the only non-negligible effects are around the peak region, where they are larger than
5%. The four-lepton transverse momentum is marginally softer with Pythia6 than with
HERWIG, an effect that is more pronounced in the gg contribution (because of a larger
amount of radiation there w.r.t. the case of the qq and qg channels, radiation being on
average softer in Pythia6 than in HERWIG). On the other hand, this larger amount of
radiation implies that the gg channel is significantly harder than aMC@NLO, and is about
10% of the aMC@NLO+gg result at pT = O(100 GeV). The scale uncertainty shows a
behaviour typical of an LO computation, and is vastly different from that of figure 1; it
is constant to a good approximation, which is due to a decreasing R dependence with
increasing pT , compensated by an increasing F dependence.
As was already pointed out in the case of figure 2, the observables presented in figure 5
are quite insensitive to extra radiation, and especially so for the single-inclusive lepton pT .
Therefore, the similarity between the predictions obtained with HERWIG and Pythia
has to be expected. To a very good extent, the gg contribution has the same shape as the
aMC@NLO one. Things are marginally different in the case of the pair transverse
momentum presented in the right panel; however, any difference between HERWIG and Pythia
is not significant, being much smaller than the scale uncertainty of the aMC@NLO
contribution alone (shown in figure 2). Likewise, the shape change induced in the aMC@NLO
prediction by the gg contribution is very minor.
The small-pT dominance for the observables displayed in figure 6 implies that again the
HERWIG and Pythia results are very similar. As far as the gg contribution is concerned,
its shape is almost identical to that of aMC@NLO in the case of . On the other hand,
the lepton-pair pseudorapidity turns out to be significantly more central than that of the
qq + qg channels; the impact on the overall aMC@NLO+gg shape is however modest for
the invariant-mass cuts considered here.
Figure 6. Cross section per bin for the inclusive of the opposite-charge, Z-id matched lepton
pairs (left panel), and the inclusive distance of the opposite-charge, non-Z-id matched lepton
pairs (right panel), for aMC@NLO+gg HERWIG (solid black) and Pythia (dashed blue)
results. The rescaled gg contributions with HERWIG (open black boxes) and Pythia (open blue
circles) are shown separately. Upper insets: ratios of aMC@NLO+gg HERWIG predictions over
the aMC@NLO+gg Pythia ones (solid black) and of the gg HERWIG predictions over the gg
Pythia ones (open blue boxes). Middle insets: scale (dashed red) and PDF (solid black)
fractional uncertainties. Lower insets: aMC@NLO/(aMC@NLO+gg) with HERWIG (solid black)
and Pythia (dashed blue).
Conclusions
In this paper, we have used the aMC@NLO framework to study the hadroproduction of
four leptons, by considering, in particular, the process mediated by the exchange of two
Z or vector bosons. Di-bosons final states are relevant in both Higgs searches and
newphysics studies, where precise predictions are an important ingredient to cope with the
overwhelming irreducible Standard Model backgrounds. We have included in our
calculation all contributions which are expected to be important for accurate phenomenological
analyses, such as singly-resonant diagrams, exact spin correlations and off-shell effects,
Z/ interference, and gg-initiated subprocesses. Our computation being fully automated,
any four-lepton final states can be studied in the very same way, including those that
feature W/Z interference effects such as e+eee.
In any realistic study, precise calculations are as important as the determination
of the associated theoretical uncertainties. For this reason, we have also given in this
work practical applications of the fully automatic procedures implemented to this aim in
aMC@NLO, namely:
i) The matching of the hard process to both Pythia6 and HERWIG, paving the way
to systematic studies of the influence of different showers on any observable with
MC@NLO;4
ii) A framework for studying both scale and PDF uncertainties by a simple (and
CPUwise costless) reweighting procedure of the unweighted-event samples.
As for item i), this paper provides the first working example of matching with Pythia6
for a kinematically non-trivial process. As for item ii), we stress that our procedure allows
one to obtain precise results with very limited statistics. In fact, the central results and all
its variations are correlated, since the same events (up to the weights) are used. This is
manifest in the smoothness of the scale and PDF fractional uncertainty plots presented in
this paper, which may be compared with, for example, those presented in ref. [19], which
had been obtained by rerunning the code.
As far as our phenomenological results are concerned, they can be summarized as
follows. The impact of NLO corrections on total rates is +40%; for differential distributions,
they generally correspond to an overall rescaling, but there are cases where they give
nontrivial kinematics effects. The uncertainties due to PDFs are of the order of 2%. The scale
dependence we obtain by varying R and F independently is also of the order of 2% for
the qq/qg channels, and of the order of 20% in the case of the gg-initiated subprocess.
Parton-level NLO and fully showered NLO distributions are generally in good agreement,
except in the very few cases in which the former is not expected to give sensible predictions.
These findings are rather independent of possible Z-identification cuts applied to enhance
doubly-resonant contributions. Generally speaking, tiny differences are observed between
HERWIG and Pythia, with the exception of distributions dominated by resummation
effects, where they are more pronounced. The contribution of the gg channel is of the order of
5% of the total with a cut of 30 GeV applied to the invariant mass of unlike-sign lepton pairs.
We have explicitly verified that the inclusion of the gg H ZZ contribution
(not considered for producing the results presented here) is straightforward, and does not
increase significantly the running time. Therefore signal/background interference in Higgs
production with four-lepton decay channels at the leading order can be studied [13, 14, 28,
29]. Clearly, for this to be phenomenologically sensible, NLO corrections to the gg H
signal should to be included as well, which could be achieved in several ways, with different
levels of approximation and automation [2931].
As a last remark, we point out that ready-to-shower four-lepton NLO (i.e.,
resulting from qq/qg-initiated processes) event files are available at the aMC@NLO web page
http://amcatnlo.cern.ch, for all possible (massless) lepton flavour combinations.5
4The matching with Herwig++ and Pythia8 will become available as well.
5Processes that feature two W propagators interfere at the NLO with tW production. We have excluded
the latter contribution by setting the b-quark PDF equal to zero.
This research has been supported by the Swiss National Science Foundation (SNF) under
contracts 200020-138206 and 200020-129513, by the IAP Program, BELSPO P6/11-P, the
IISN convention 4.4511.10, by the Spanish Ministry of education under contract
PR20100285, and in part by the US National Science Foundation under Grant No. NSF
PHY0551164. 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 during the completion of this paper.
Definition of W coefficients for NLO cross sections
We collect in this section the expressions of the coefficients that enter eq. (2.17). These
are taken from ref. [24], whose equation (n) will be denoted by eq. (I.n) here; the reader
can find fuller details in that paper. The master equation for the determination of all the
coefficients W () is eq. (I.C.14).
Event
Only the (n + 1)-body cross section contributes to the event. Therefore, from
eq. (I.4.29) and eq. (I.6.7) one gets:
WcF(E) = 0 ,
WcR(E) = 0 ,
wreal(Kn+1) =
ij = (1 yij)i2M(n+1,0) Sij
with (n+1) = (n+1)/i. The quantity defined in eq. (A.5) is the combination of the
real-emission matrix element, prefactors, and the phase-space factor that is routinely
used in the FKS subtraction. We point out that according to our conventions the
matrix elements include the coupling constant, hence eq. (A.4) is independent of
gS, since we understand that the factorization scale R there is computed according
to the kinematic configuration Kn+1. The coupling-constant dependence has been
factored out in eq. (2.17).
Collinear counterevent
The coefficient W (C) receives contributions from the (n + 1)-body and the degenerate
(n + 1)-body cross sections. The latter can be read from eq. (I.4.41) (to be definite,
1
g2b+2( (RC))
S
WcF(C) = gS2b+21( (RC)) S(2(RC)) P (I01)iI1 (1 i) 1i
WcR(C) = 0 .
we assume that the FKS sister is the parton coming from the left). One obtains:
The normalization factor F in eqs. (A.6) and (A.7) is due to the fact that the
degenerate (n + 1)-body contribution is integrated together with the pure (n +
1)body one, but is originally defined with a quasi-n-body measure (see eq. (I.4.41)).
Soft counterevent, n-body contribution, and Born
The coefficient W (S) receives contributions from the (n + 1)-body and the n-body
cross sections. The latter is the sum of the Born (eq. (I.4.4)), collinear (eq. (I.4.5)),
soft (eq. (I.4.12)) and finite virtual (eq. (I.4.14)) contributions, plus the last term on
the r.h.s. of eq. (I.C.14), which we shall call the RGE term here. The expression for
the finite part of the virtual contribution that includes the full scale dependence must
be read from eq. (I.C.13). In ref. [24] we have used the Ellis-Sexton scale wherever
possible, which now renders it easy to extract the factorization and renormalization
scale dependences more specifically, these dependences are present only in one
term in eq. (I.4.5), in one term in eq. (I.C.13), and in the RGE term. This implies:
Q ( = Q) M(n,0)
1
g2b+2( (RS))
S
WcF(S) = gS2b+21( (RS)) M(n,0) Kn(S+)1 N
((I1) + 2C(I1) log cut + (I2) + 2C(I2) log cut) ,
(B) is the analogue of that introduced in eq. (A.6), and
The normalization factor F
its form can be found e.g. in eq. (I.6.14) (note that it includes an Sij factor). In
eq. (A.9), by setting = Q one eliminates the first term on the r.h.s. of eq. (I.4.6);
this is included here in eq. (A.10).
Soft-collinear counterevent
Finally, the results for W (SC) can be obtained by computing the soft limit of W (C),
and by taking into account the fact that the original equations contained plus
distributions. One obtains:
P (I01)iI1 (1)
WcF(SC) =
WcR(SC) = 0 .
1
g2b+2( (RSC))
S
WcR(S) =
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