W production at LHC: lepton angular distributions and reference frames for probing hard QCD
Eur. Phys. J. C
W production at LHC: lepton angular distributions and reference frames for probing hard QCD
E. RichterWas 1
Z. Was 0
0 Institute of Nuclear Physics Polish Academy of Sciences , 31342 Kraków , Poland
1 Institute of Physics, Jagiellonian University , Ł ojasiewicza 11, 30348 Kraków , Poland
Precision tests of the Standard Model in the Strong and Electroweak sectors play a crucial role, among the physics program of LHC experiments. Because of the nature of protonproton processes, observables based on the measurement of the direction and energy of final state leptons provide the most precise probes of such processes. In the present paper, we concentrate on the angular distribution of leptons from W → ν decays in the leptonpair restframe. The vector nature of the intermediate state imposes that distributions are to a good precision described by spherical harmonics of at most second order. We argue, that contrary to general belief often expressed in the literature, the full set of angular coefficients can be measured experimentally, despite the presence of escaping detection neutrino in the final state. There is thus no principle difference with respect to the phenomenology of the Z /γ → + − DrellYan process. We show also, that with the proper choice of the reference frames, only one coefficient in this polynomial decomposition remains sizable, even in the presence of one or more high pT jets. The necessary stochastic choice of the frames relies on probabilities independent from any coupling constants. In this way, electroweak effects (dominated by the V − A nature of W couplings to fermions), can be better separated from the ones of strong interactions. The separation is convenient for the measurements interpretation.
1 Introduction
The main purpose of the LHC experiments [1,2] is to search
for effects of New Physics. This program continues after the
breakthrough discovery of the Higgs boson [3,4] and
measurement of its main properties [5]. In parallel to searches of
New Physics, see e.g. [6–8], a program of precise
measurements in the domain of Electroweak (EW) and Strong (QCD)
interactions is ongoing. This is the keystone for
establishing the Standard Model as a consistent theory. It is focused
around two main directions: searches (setting upper limits)
for anomalous couplings and precision measurements of the
Standard Model parameters. Precision measurements of the
production and decay of Z and W bosons represent the
primary group of measurements of the latter domain, see e.g.
[9–13]. The study of the differential crosssections of W
production and decays is essential for understanding open
questions related to the electroweak physics, like the origin of
the electroweak symmetry breaking or the source of the CP
violation.
Since discovery, the W boson mass and width, have been
measured to a great precision in p p¯ and pp collisions [14].
To complete physical information on the production process,
measurements pursued the boson’s decay differential
distributions too. The measurements rely on outgoing leptons of
the W → ν decays in the W boson rest frame. Because
electroweak interaction of the decay vertex are known with
much better precision than the QCD interaction of the
production, the measurement predominantly tests dynamics of
QCD imprinted in the angular distributions of outgoing
leptons.
In principle, the same standard formalism of the Drell–
Yan production Z → [15] can be applied in the case of
W → ν production [16,17]. The angular dependence of the
differential crosssection can be written again as
where the gα(θ , φ) represent harmonic polynomials of the
second order, multiplied by normalisation constants and by
dσ α which denote helicity crosssections, corresponding to
nine helicity configurations of W matrix elements. The angle
θ and φ in dΩ∗ = d cos θ dφ are the polar and azimuthal
decay angles of the charged lepton in the W restframe. The
pT , Y denote transverse momenta and rapidity of the
intermediate W boson in the laboratory frame. The zaxis of the
W rest frame can be chosen along the W momentum of the
laboratory frame (the helicity frame), or constructed from the
directions of the two beams (the Collins–Soper frame [18]).
We rewrite Eq. (1) explicitly, defining polynomials and
corresponding coefficients
d pT2 dY d cos θ dφ = 16π d pT2 dY
where dσ U +L denotes the unpolarised differential
crosssection (a convention used in several papers of the 1980s). In
case of W boson, (θ , φ) define the orientation of the charged
lepton from W → ν decay. The coefficients Ai ( pT , Y ) are
related to ratios of corresponding crosssections for
intermediate state helicity configurations. The full set of Ai
coefficients has been explicitly calculated for p p¯ → W (→
ν) + 1 j at QCD NLO in [16,17].
The first term at Born level (no jets): (1 + cos2 θ ) results
from spin 1 of the intermediate boson. The dynamics of
the production process is hidden in the angular coefficients
Ai ( pT , Y ). This allows to treat the problem in a model
independent manner. In particular, as we will see, all the hadronic
physics is described implicitly by the angular coefficients and
it decouples from the well understood leptonic and
intermediate boson physics. Let us stress, that the actual choice of
the orientation of coordinate frames represents an important
topic; we will return to this later.
The understanding of how QCD corrections affect
lepton angular distributions is important in the measurement
of the W mass (mW ), independently of whether leptonic
transverse momentum or transverse mass (mTW ) of the W
are used. In fact, the first measurements of the angular
coefficients explored this relation in the opposite way. Assuming
the mass of the W boson measured by LEP, from the fit to
transverse mass distribution of the leptonneutrino system
mW , information on the angular orientation of the outgoing
T
leptons was extracted.
The crosssection has been parametrised [19] using only
the polarangle (i.e. integrating over azimuthal angle) as
dσ
d cos θ ∼ (1 + α1 cos θ + α2 cos2 θ )
with the following relations between coefficients;
2 A4 2 − 3 A0
It has been estimated that a 1% uncertainty on α2
corresponds to a shift of the measured mW in p p¯ collision,
determined by fitting the transverse mass distribution, of
approximately 10 MeV. The α1 measures the forwardbackward
decay asymmetry.
The measurements of α2 at 1.8 TeV p p¯ collisions have
been conducted by D0 and CDF experiments and published
in [20,21]. It was based on the data collected in 1994–1995
by Fermilab’s Tevatron Run Ia. The fit to mTW was performed
in several ranges of the W boson transverse momentum.
The measurements confirmed standard model (SM)
expectations, that α2 decreases with increasing W boson
transverse momentum, which corresponds to increase of the
longitudinal component of the W boson polarisation. The ratio
of longitudinally to transversely polarised W bosons in the
Collins–Soper W rest frame increases with the W transverse
momentum at a rate of approximately 15% per 10 GeV.
With more data collected during Fermilab Tevatron Run
Ib, the measurement of the W angular coefficients was
performed using a different technique; through direct
measurement of the azimuthal angle of the charged lepton in the
Collins–Soper restframe of the W boson [22]. The strategy
of this novel measurement was documented in a separate
paper [23]. Because of the ambiguity on determining the
sign of cos θ (due to neutrino momenta escaping detection)
which was not resolved, only the measurement of the
coefficients A2 and A3 was performed and angular coefficients
were measured as function of the transverse momentum of
the W boson. The measurement was performed specifically
for the W − bosons; angular coefficients of the W + were
obtained by CP transformation of Eq. (2).
The pure V − A interactions of W ± without QCD effects,
lead for p p¯ collisions to α2 = 1.0 and α1 = 2.0, thus to
pure transversely polarised W boson. This assumes that the
W boson is produced with no transverse momenta, and
seaquarks and gluon contributions to the structure functions can
be neglected. Such a simple partonmodel could guide
intuition for the p p¯ collisions at Tevatron, but had to be revisited
for the pp collisions at LHC.
The dominance of quarkgluon initial states, along with
the V–A nature of the coupling of the W boson to fermions,
implies that at the LHC W bosons with high transverse
momenta are expected to exhibit a different polarisation as
the production mechanism is different at low pTW and high
pTW [24,25]. W bosons produced with low pTW , and
therefore moving generally along beam axis, exhibit a lefthanded
polarisation [26]. This is because the W boson couples, in
the dominant production diagram, to the lefthanded
component of valence quarks, and to the righthanded one of the sea
antiquarks. At high pTW , the situation becomes more
complex due to contributions of higherorder processes. Of
special interest, to quantify the validity of the QCD predictions,
becomes the behavior of polarisation fractions as function of
pTW . It was recently pointed out in [27], that events with high
pTW can test the absorptive part of the scattering amplitudes
and hence offer a nontrivial test of perturbative QCD at one
and higherloop levels. In all pTW ranges, the production at
LHC therefore displays new characteristics: asymmetries in
charge and momentum for W bosons and their decay leptons.
The LHC experiments pursued measurement techniques
different than Tevatron. With 7 TeV data of pp collisions, the
helicity frame and not the Collins–Soper frame was used. The
interest was not to measure Ai coefficients directly but rather
the helicity fractions, f0, fL , f R . The helicity state of the W
boson becomes a mixture of the left and right handed states,
whose proportions are respectively described with fractions
fL and f R . The f0 denotes the fraction of longitudinally
polarised W bosons, which is possible at higher transverse
momenta, due to a more complicated production mechanism.
This state is particularly interesting as it is connected to the
mass of the gauge boson [25]. The measurements [28,29]
by CMS and ATLAS experiments established that W bosons
produced in pp collisions with large transverse momenta
are predominantly lefthanded, as expected in the Standard
Model.
In the standard notation of the helicity fractions, the
following relations with Ai ’s of Eq. (2) are valid
The difference between left and righthanded fraction is
proportional to A4
Note, that even if Eq. (2) is valid for any definition of the
Wboson rest frame, the Ai ( pT , Y ) are frame dependent. The
relations Eqs. (5) and (6), hold in the helicity frame.
Very similar arguments can be made also for the case of
the Z production. However, the different characteristic of
couplings have to be considered: the coupling of Zboson to
quarks does not involve the chirality projector 21 (1 − γ 5),
but is asymmetric between left and right handedness. The
analysing power of Z leptonic decays is severely affected
by the coupling to righthanded leptons, being similar to
the coupling to left handed leptons. As a consequence the
angular coefficients fL , f R , f0 can no longer be interpreted
directly as polarisation fractions of the Z boson. The
respective matrix transformation, involving left and right couplings
of Z boson to fermions, relates them to the Z boson
polarisation fractions [30].
For the case of Z → channel the measurement of the
complete set of Ai ’s coefficients in the Collins–Soper frame
was recently performed at 8 TeV pp collisions by the CMS
Collaboration [31] and the ATLAS Collaboration [13]. The
precision of the measurement by the ATLAS Collaboration
allowed to clearly show that the violation of the LamTung
sum rule [32] i.e. A0 = A2, is much stronger than
predicted by NLO calculations. It has shown also an evidence
of A5, A6, A7 deviation from zero.
As of today, the situation with the measurement of Ai
coefficients for W → ν production in hadronic collisions is
far from satisfactory. Measuring only some coefficients like
α2 in the Collins–Soper frame or fL , f R , f0 in the
helicityframe as function of Wboson transverse momenta does not
give a complete picture on the QCD dynamics of the
production process. In the early papers [16,33], the point was made,
that measurement of the complete set of coefficients may not
be possible, due to limitations related to the reconstruction
of lepton neutrino momentum: in particular a twofold
ambiguity in the determination of the sign of cos θ defined in the
next section.
In the present paper we argue, that following the strategy
outlined in [13], one can design a measurement which allows
to measure the complete set of coefficients also in the case
of W → ν in pp collision. Then, we move to the
discussion of the reference frames used for W → ν decay and
demonstrate that the Mustraal [34] frame introduced and
detailed for LHC in [35] will be interesting in the case of
W → ν production as well.
Our paper is organized as follows. Section 2 is devoted to
the presentation of the strategy which allows to measure
complete set of the Ai ’s coefficients in case of W → ν process.
We follow this strategy and show a proof of concept for such
measurement. In Sect. 3, we discuss variants for the frames
of the θ , φ angles definition. In Sect. 4 we collect numerical
results for the Ai ’s coefficients in the case of pp → ν + 1 j
generated with QCD LO MadGraph5_aMC@NLO Monte
Carlo generator [36] and QCD NLO Powheg+MiNLO
Monte Carlo generator [37,38]. We elaborate on possible
choices of the coordinate frame orientation. We recall
arguments for introducing the Mustraal frame [35], (where
the orientation of axes is optimized thanks to matrix element
and nexttoleading logarithm calculations) and compare the
Collins–Soper and Mustraal frames. We demonstrate that,
similarly to the Z → case discussed in [35], with the help
of probabilistic choice of reference frames for each event,
the results of formula (3.4) from [34] are reproduced and
indeed only one nonzero coefficient in the decomposition
of the angular distribution is needed. Finally, in Sect. 5 we
conclude the paper.
To avoid proliferation of the figures, we generally present
those for W − → −ν only, while the corresponding ones for
W + → +ν are deferred to Appendixes A–C.
2 Angular coefficients in W →
The production of vector bosons at LHC displays new
characteristics compared to the production at Tevatron due to
proton–proton nature of the collision: asymmetries in charge
and momentum for vector bosons and their decay leptons.
Large lefthanded polarisation is expected in the transverse
plane. Contrary to the case of p p¯ collisions, the angular
coefficients in pp collisions of the W + and W − are not related
by CP transformation, due to absence of such symmetry in
the proton structure functions. Only quarks can be valence,
while both quarks and antiquarks may be nonvalence.
For the numerical results presented in this section we use
a sample of 4M events pp → τ ν + 1 j generated at QCD LO
with MadGraph5_aMC@NLO Monte Carlo [36], with
minimal cuts on the generation level, i.e. pTj > 1 GeV, and default
initialisation of other parameters.1 The purpose of presented
results is not so much to give theoretical predictions on Ai ’s
but to illustrate the proof of concept for the proposed
measurement strategy. Therefore we will not elaborate on the choices
of PDF structure functions, QCD factorisation and
normalisation scale, or EW scheme used. However, numerical results
are sensitive to particular choices. The experimental
precision of reconstruction of hadronic recoil necessary for the
neutrino momentum reconstruction can be impressive as it
is the case of W mass [39], but it is less accurate for other
measurements where such precision was not essential.
2.1 Kinematic selection
Kinematic selections need to be applied in the experimental
analysis. Limited coverage in the phasespace is due to the
need for the efficient triggering, detection and background
suppression. It inevitably reshapes angular distributions of
the outgoing leptons. The minimal set of selections, in the
context of LHC experiments, is to require (in the
laboratory frame), the transverse momenta and pseudorapidity
of charged lepton pT > 25 GeV and η  < 2.5
respectively. Typically selection to suppress background from the
multijet events, we require neutrino transverse momenta
pTν > 25 GeV and the transverse mass of the chargelepton
and neutrino system mT > 40 GeV. This set of selections
will define fiducial phasespace of the measurement.
Similar selection was used e.g. in measurement [40]. In Fig. 1
we show as an example the pseudorapidity distribution of
the charged lepton from W ± → ±ν decay, in the full
phasespace and in the fiducial phasespace as defined above.
Clearly, the distributions are different between W + → +ν
and W − → −ν processes.
1 In principle any other lepton flavour could have been used for pre
sentation of numerical results. Our choice to generate τ ν final states is
motivated by the planned extensions of the work.
Fig. 1 The pseudorapidity distributions of charged lepton in the
laboratory frame, for the full phasespace and fiducial phasespace.
Distributions for W − → τ −ν (top) and W + → τ +ν (bottom) are shown
2.2 Solving equation for neutrino momenta
For the leptonic decay mode, W bosons have the
disadvantage (with respect to the Z bosons) that the decay kinematics
cannot be completely reconstructed due to the
unobservability of the outgoing neutrino. On the other hand, we can profit
from a simplification: the electroweak interaction does not
depend on the virtuality of the intermediate state. The
transverse components of the neutrino’s momentum pν , pνy can be
x
approximated from missing transverse momentum balancing
the event. The longitudinal component pzν can only be
calculated up to a twofold ambiguity when solving the quadratic
equation on the invariant mass of the leptonneutrino system
mW , assuming its value is known.
Let us recall the corresponding simple formulas:
a = 4 · pz − 4 · E ,
pT =
Equation (7) has two solutions. Moreover solutions exist
only if Δ = (b2 − 4a · c) is positive. It requires also, that the
mass of the W boson, mW , is fixed, usually m WPDG is taken
(no smearing due to its width). The solution for the neutrino
momentum allows to calculate its energy, completing the
kinematics of massless neutrino
Some studies of the past [41], investigated if a better option
can be designed than taking one of the two pzν solutions
randomly, with equal probabilities. In particular in case that
solutions do not exist, if replacing the m WPDG by e.g. transverse
mass mTW can be beneficial. No convincing alternative was
found. Replacing m WPDG with the transverse mass was
creating spikes in shapes of angular distributions that are difficult
to control. Similar effect (i.e. spiky distortions of the
angular distributions), was caused by favoring some solutions of
the neutrino momenta e.g. by selecting the one in the most
populated regions of the multidimensional phase space, or
taking the bigger of the two, etc.
In the analysis which will be outlined below, we propose
– Use nominal PDG value for mW to solve the equation for
the neutrino momenta pν .
z
– Drop the event if Δ = (b2 − 4a · c) is negative.
– Choose with equal probabilities, one of the two solutions
for the neutrino momenta pzν . This solution will be called
a random solution.
We estimated that the loss of events due to Δ < 0 is on
the level of 10%. In the experimental analysis this loss can
be considered as a part of other events losses due to
kinematical selection cuts, like thresholds on the lepton transverse
momenta or pseudorapidity bounds due to limited detector
acceptances.
2.3 Collins–Soper restframe
For the Drell–Yan productions of the leptonpair in hadronic
collisions, the well known and broadly used Collins–Soper
reference frame [18] is defined as a restframe of the
leptonpair, with the polar and azimuthal angles constructed using
proton directions in that frame. Since the intermediate
resonance, the W  or Z  boson are produced with nonzero
transverse momentum, the directions of initial protons are
not collinear in the leptonpair rest frame. The polar axis
(zaxis) is defined in the leptonpair restframe such that it is
bisecting the angle between the momentum of one of the
protons and the reverse of the momentum for the other one. The
sign of the zaxis is defined by the sign of the leptonpair
momentum with respect to zaxis of the laboratory frame.
To complete the coordinate system the yaxis is defined as
the normal vector to the plane spanned by the two incoming
proton momenta in the W rest frame and the x axis is
chosen to set a righthanded Cartesian coordinate system with
the other two axes. Polar and azimuthal angles are calculated
with respect to the outgoing lepton and are labeled θ and
φ respectively. In the case of zero transverse momentum of
the leptonpair, the direction of the yaxis is arbitrary. Note,
that there is an ambiguity in the definition of the φ angle in
the Collins–Soper frame. The orientation of the x axis here
follows convention of [16,42,43].
For the Z → + − production, the formula for cos θ can
be expressed directly in terms of the momenta of the outgoing
leptons in the laboratory frame [44]
pz ( + −)
2 ( P1+ P2− − P1− P2+)
 pz ( + −) m( + −) m2( + −) + pT ( + −)
2
where Ei and pz,i are respectively the energy and longitudinal
momentum of the lepton (i = 1) and antilepton (i = 2) and
pz ( + −) denotes the longitudinal momentum of the lepton
system, m( + −) its invariant mass. The φ angle is calculated
as an angle of the lepton in the plane of the x and y axes in
the Collins–Soper frame. Only the fourmomenta of outgoing
leptons and incoming proton directions are used. That is why
the frame is very convenient for experimental purposes.
In case of W ± → ±ν production we follow (in principle)
the same definition of the frame. We use the convention that
the θ and φ angles define the orientation of the charged lepton,
i.e. antilepton of W + production and lepton in case of W −
production. We calculate cos θ for the chosen solution of
neutrino momenta with formula (9) and φ from the event
kinematics as well.
However, because neutrino momentum remains
unobserved, one of the two solutions to Eq. (7) has to be chosen and
m WPDG must be used instead of ±ν pair mass. Figure 2 shows
correlation plots between cos θ gen, φgen calculated using
generated neutrino momenta, and cos θ , φ calculated using
neutrino momenta from formula (7) with mW = m WPDG .
Correlations are shown when correct or wrong solution for
θ 1
s
co 0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
θ 1
s
co 0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1−1 −0.8 −0.6 −0.4 −0.2
neutrino momenta2 are selected. The cos θ − cos θ gen and
φ − φgen , can be anticorrelated in case of correct and wrong
solutions, the effect is much stronger for wrong solutions
and the cos θ variable. We observe also inevitable migrations
between bins due to the approximation m W = m WPDG used
for solving Eq. (7). The following two figures will
demonstrate how elements of reconstruction based on Eq. (7) affect
the obtained angular distributions.
Figure 3 shows cos θ and φ distributions of the charged
lepton from W → ν decays in the Collins–Soper rest frame.
We use the generated W boson mass m gWen of a given event or
the fixed PDG value m WPDG for calculating neutrino momenta
pzν , taking the correct solution for pν . We compare the two
z
results. The losses due to the nonexistence of a solution of
Eq. (7) are concentrated around cos θ = 0 but are uniformly
distributed over the full φ range.
Figure 4 demonstrates the variation of cos θ and φ
distributions for charged lepton of W − → −ν decays when
m W = m WPDG is used for solving Eq. (7) and the selection
of the fiducial regions applied. In each case, distributions
are shown for correct, wrong and random solution for pν .
z
Selection of the fiducial region enhances modulation in the
φ distribution. Corresponding distributions for W + → +ν
decay are shown in Appendix A.
Now we are ready to illustrate the effect of folding into
fiducial phasespace, 2D distributions of (cos θ , φ) are shown
in Fig. 5: (i) for events in the full phasespace, when generated
neutrino momenta are used, (ii) in the fiducial phasespace
when m W = m WPDG is used for Eq. (7) and random solution
of neutrino momenta is taken. Clearly, original shapes of
distributions are significantly distorted, but still, as we will
see later, basic information on the angular correlation of the
outgoing charged lepton and the beam direction is preserved.
In particular, it is non trivial that the information is preserved
despite approximate knowledge of the neutrino momentum.
Moreover, the information is carried by both, correct and
wrong, solutions for neutrino momenta. These observations
are essential for the analysis presented in our paper.
2.4 Templated shapes and extracting Ai ’s coefficients
The standard experimental technique to extract parameters
of complicated shapes is to perform the multidimensional
fit to distributions of experimental data using either
analytical functions or templated shapes [45]. Given what we
observed in Fig. 4 only the second options seems feasible.
The technique of templated shapes constructed from Monte
2 As correct we denote solution which is closer to the generated pν
z
value, as wrong the other one. In practice, the twofold ambiguity for
solution of Eq. (9) is present. We will return to this point later, in
Sect. 2.3. It is important for the observables we advocate in the paper.
−1−1 −0.8 −0.6 −0.4 −0.2
Fig. 2 The correlation plots of cos θ gen and φgen calculated using
generated neutrino momenta and cos θ and φ calculated using mW =
m WPDG for solving Eq. (7). The plots for correct (two top) or wrong (two
bottom) solution for neutrino momenta are shown. Correlation plots are
prepared for the full phasespace of W − → τ −ν process
−0.8 −0.6 −0.4 −0.2
6
φ
Fig. 3 The cos θ and φ distributions of charged lepton from W − →
τ −ν, in the Collins–Soper rest frame. Effect from events loss due to
nonexisting solution for the neutrino momenta, when mW = m WPDG is
used for Eq. (7) is concentrated in the central bins of the top plot
Carlo events, elaborated in [13] for the Ai ’s measurement in
Z → case, is followed here and shortly described below.
We use the Monte Carlo sample of W ± → ±ν events
and extract angular coefficients of Eq. (2) using moments
methods [17]. The first moment of a polynomial Pi (cos θ , φ),
integrated over a specific range of pT , Y is defined as follows:
−1
−11 d cos θ 02π dφ Pi (cos θ , φ)dσ (cos θ , φ)
1 d cos θ 02π dφ dσ (cos θ , φ)
Owing to the orthogonality of the spherical polynomials
of Eq. (2), the weighted average of the angular distributions
with respect to any specific polynomial, Eq. (10), isolates its
corresponding coefficient, averaged over some phasespace
region. As a consequence of Eq. (2) we obtain:
1 3 2 1
2 (1 − 3 cos2 θ ) = 20 A0 − 3 ; sin 2θ cos φ = 5 A1;
0−1
ts240
n
veE220
200
180
160
140
120
100
80
60
40
20
00
s
t
ven350
E
300
Fig. 4 The cos θ and φ distributions of charged lepton from W − →
τ −ν in the Collins–Soper rest frame. Cases of mW = m WPDG for solving
Eq. (7) where correct, wrong or random solution for pzν are taken. Top
plots are for the distributions in the full phasespace, bottom ones for
the fiducial phasespace
0−1 −0.8 −0.6 −0.4 −0.2
1 0 −1−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6co0s.8θgen1
Fig. 5 The 2D distribution of cos θ , φ of charged lepton from W − →
τ −ν. Case of the fullphase space with generated neutrino momentum,
top plot. Case of the fiducial phasespace and mW = m WPDG used in
Eq. (7) and random solution for pzν , bottom plot
We extract coefficients Ai using generated neutrino momenta
to calculate cos θ and φ. As a technical test, 2dimensional
histogram of (cos θ , φ) distribution obtained from our events
weighted with
Σii==08 Ai Pi (cos θ , φ)
where A8 = 1.0 and P8 = 1 + cos2 θ is used.
By construction, thanks to Eqs. (11) and (10), weighted
with (12) sample, feature unchanged Y , pT distribution, but
matrix element dependence of angular distribution of leptons
in lepton pair restframe is completely removed.
If averages for (10) are taken for subsamples in
appropriately narrow bins of Y and pT this feature holds precisely
for configurations of up single high pT , thus degrading
predictions of the Monte Carlo simulation results, to at worst
NLO (NLL) level. We have found that for numerical results
binning in pT alone is sufficient. Indeed flat distribution in
(cos θ , φ), where θ , φ are calculated using the generated
neutrino momentum, see top plot of Fig. 6, is obtained. This
completes our technical test and we can continue the
construc
1 0 −1−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6c0o.s8θge1n
Fig. 6 The 2D distribution of cos θ and φ of charged lepton from
W − → τ −ν. On top, distribution of the full phasespace, with
generated neutrino momentum used, and events weighted wtΣ Ai Pi . Bottom,
the same distribution is shown, but: m WPDG is used for solving Eq. (7),
randomly one of the solutions for pzν is taken and fiducial selection
is applied. The weight wtΣ Ai Pi is calculated with generated neutrino
momenta, as it should be
tion of templates. Further refinements are straightforward but
require substantially more CPU time as binning in more than
one or even more than two variables is then necessary.
We fold now events weighted with wtΣ Ai Pi into fiducial
phasespace of the measurement: for the neutrino
momentum reconstruction we use m W = m WPDG and take one of
the solutions at random, then we recalculate θ , φ angles and
finally we apply the kinematical selection of the fiducial
phasespace. Bottom plot of Fig. 6 shows how the initially
flat distribution is distorted by this folding procedure.
We can now model any desired analytical polynomial
shape of the generated full phasespace folded into
fiducial phasespace of experimental measurement. It is enough
to apply wti = Pi · wtΣ Ai Pi to our events, to model the
shape of the Pi (cos θ , φ) polynomial in the measurement
fiducial phasespace. In Fig. 7, we show 2D distributions
modeling polynomials P0(cos θ , φ) and P4(cos θ , φ) in the
full and fiducial phasespace as an example. Distributions for
P1(cos θ , φ), P2(cos θ , φ) and P3(cos θ , φ) are shown in the
Appendix C.
We can now proceed with the fit of a linear combination of
templates to distributions of the fiducial phasespace
pseudo1 0 −1−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6c0o.s8θge1n
1 0 −1−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6c0o.s8θge1n
data. We bin in pW both templates shown in Figs. 7, 21
T
and pseudodata distributions shown in Fig. 5. We perform a
multiparameter loglikelihood fit3 in each pTW bin; the only
parameters of the fit are the 8 angular coefficients Ai ( pTW ) (of
a given pTW bin). Results of the fitting procedure are shown
in Fig. 8. The black points represent fitted values of Ai ’s
with their fit error, black open circles are the generated
values of the Ai ’s (which we extracted with moments method
3 The Root framework [46] was used for this purpose. All eight Ai ( pTW )
coefficients were fitted simultaneously. Statistical errors were
calculated. Only for selected (nonzero), coefficients statistical errors (pools)
are presented in plots, such as Fig. 8. For the remaining ones they were
found to be of the same order. Non diagonal elements of correlation
matrix [47] were all smaller than 0.7.
described above). Bottom panels show difference between
fitted and true values divided by their errors (so called pulls
distributions). Pulls are small because of the samples
correlations. We confirm closure of the method, i.e. extracted
coefficients are equal to their nominal value for analysed events
sample. However, estimation of the statistical errors for all
A0 to A7 of the method, and in particular how it would
compare with the case when ν momentum would be measured
directly and full phase space was available for the
measurement is premature. Corresponding discussion should include
discussion of experimental systematic error as well.
The same procedure has been repeated for W + → +ν
and results are shown in Appendix C.
Fig. 8 Closure test on the fitting of angular coefficients Ai ’s for
W − → τ −ν. Fit is performed in the fiducial phasespace. Shown are
generated Ai ’s coefficients (open circles) and their fitted values (black
points). In the bottom panels shown are pulls (difference between
generated and fit value, divided by the statistical error of the fit). Pulls are
smaller than one could expect. This is because events of pseudodata
and templates are statistically correlated
We have also performed the fit using templates and
pseudodata distributions prepared with only correct or only
wrong solutions for the neutrino momenta.4 In both cases the
fit returned nominal values of the Ai coefficients,
confirming that both solutions of neutrino momenta carry the same
information on the angular correlations.
In this proof of concept for the proposed measurement
strategy we have not discussed possible experimental effects
like resolution of the missing transverse energy which will
be used to reconstruct neutrino transverse momenta or e.g.
background subtraction which can limit precision of the
measurement.
– We have shown, that despite the neutrino escaping
detection, one can define, under some assumptions, the
equivalent to the rest frame of leptonneutrino system and
preserve sensitivity for the complete set of angular
coefficients of the decomposition.
– Then, we have shown with a simplified version of the
method used in [13] to measure Ai coefficients in case
of Z → decays, applied in case of W → ν decays
allows for the measurement of a complete set of angular
coefficients in this case as well.
2.5 Comments on systematic errors
Once we have collected numerical results, let us address the
issues of systematic errors. The only difference of principle
for our measurements, between W and Z decays, is the
neutrino momentum reconstruction, which is by far less precise
than the measurement of the second lepton of Z .
Let us start with the consequences of the use of m WPDG
(instead of mW the invariant mass of the leptonneutrino
system) discussed in Sect. 2.2. Substantial complications arise:
the twofold ambiguity for the solution appears and for some
other events solution for the neutrino momentum does not
exist at all. On the other hand we have found the numerical
impact of this replacement for the measurement of Ai
coefficients to be rather small. This is because m WPDG − mW is
typically of the order of W width, thus an order of magnitude
smaller than momenta of its decay products. Also the use of
wrong instead of correct solution for neutrino momenta is
not introducing bias in the result for Ai coefficients.
Detector effects on the reconstruction are much larger and must be
taken into account for the experimental measurement.
We do not attempt to estimate expected experimental error,
discussion of Ref. [48], taught us how unrealistic it can be,
4 With experimental data one can use random solution for pzν only.
The other two possibilities correct and wrong are for tests and to better
understand twofold ambiguity of neutrino reconstruction, which could
be expected to be the reason for some of the Ai coefficients to remain
nonobservable.
despite the best efforts, once it is confronted with
experimental analysis [49]. It can be simply misleading.
In Footnote 3, we have pointed that the multiparameter
fit provides numerical answer for all angular coefficients
A0− A7 simultaneously. It is justified to ask if the 8 parameter
fit will provide constraints for all A0− A7. We have verified
that it is the case; all nondiagonal elements of correlation
matrix were smaller than 0.5. The evaluation of 8
dimensional correlation matrix for the fit parameters was delegated
to specialized algorithms present in the Root framework [46].
This is an essential cross check, specialized algorithms for
such purposes were developed already long time ago [47].
This encouraging result needs to be confirmed once all
experimental effects are introduced. If this holds, the fit will not
feature unacceptably strong correlations.
3 Angular coefficients and reference frames
In this paper, we concentrate on the numerical analysis of tree
level parton–parton collisions into a lepton pair and
accompanying jets, convoluted with parton distributions, but
without parton showers. Even though such approach is limited, it
provides input for general discussions. Such configurations
constitute parts of the higher order corrections, or can be seen
as the lowest order terms but for observables of tagged high
pT jets.
For the choices of the reference frames to be discussed
here, let us point out that in the limit of zero transverse
momenta, all coefficients except A4 vanish. The ( pT , Y )
dependence of the Ai coefficients differs with the choice of
the reference frame.
So far, we have introduced and discussed angular
coefficients in the Collins–Soper frame only. Let us present now
the variant of the reference frame definition we are also going
to use, i.e. the Mustraal frame.
3.1 The Mustraal reference frame
The Mustraal reference frame is also defined as a rest
frame of the lepton pair. It has been proposed and used for
the first time in the Mustraal Monte Carlo program [34]
for the parametrization of the phase space for muon pair
production at LEP. The resulting optimal frame was minimising
higher order corrections from initial state radiation to the
e+e− → Z /γ ∗ → f f¯ and was used very successfully for
the algorithms implementing genuine weak effects in the LEP
era Monte Carlo program KORALZ [50]. A slightly different
variant was successfully used in the Photos Monte Carlo
program [51] for simulating QED radiation in decays of
particles and resonances. The parametrization was useful not
only for compact representation of single photon emissions
but for multiemission configurations as well.
Recently in [35], the implementation of the Mustraal
frame has been extended to the case of pp collisions and
studied for configurations with one or two partons in the
final state accompanying Drell–Yan production of the
lepton pairs. The details of the implementation of this phase
space parametrization have been discussed in context of
Z → events and the complete algorithm how to
calculate cos θ and φ angles was given. There is no need to repeat
it here.
Let us point out that unlike the case of the Collins–Soper
frame, the Mustraal frame requires not only information
on 4momenta of outgoing leptons but also on outgoing
jets (partons). The information on jets (partons), is used to
approximate the directions and energies of incoming partons.
This is used for the calculation of weights (probabilities) with
which each event contributes to one of two possible Bornlike
configurations. Each configuration requires different cos θ ,
φ definition. This does not have to be very precise but can
introduce additional experimental systematics, and requires
attention. No dependence on coupling constants or PDF’s is
introduced in this way.
3.2 QCD and EW structure of angular correlations
The measurement of the angular distribution of leptons from
the decay of a gauge boson V → where V = W, Z or γ ∗,
produced in hadronic collisions via a Drell–Yantype process
h1 + h2 → V + X provides a detailed test of the production
mechanism, revealing its QCD and EW structure.
The predictive power of QCD is based on the
factorisation theorem [52]. It provides a framework for separating out
longdistance effects in hadronic collisions. Consequently,
it allows for a systematic prescriptions and provides tools
to calculate the shortdistance dynamics perturbatively, at
the same time allowing for the identification of the leading
nonperturbative longdistance effects which can be extracted
from experimental measurements or from numerical
calculations of Lattice QCD.
The question of the input from the Electroweak sector of
the Standard Model is important, especially for distributions
of leptons originating from the intermediate Z /γ ∗ state. We
have addressed numerical consequences of this point recently
in [53] in the context of τ lepton polarization in Drell–Yan
processes at the LHC. A wealth of publications was devoted
during last years to this issue, see e.g. [54–56]. We should
underline limitations of separating interactions into
Electroweak and QCD part. Limitations have been well known
for more than 15 years now, see e.g. [57].
Let us come back now to Eqs. (1) and (2) and discuss
the structure of crosssection decomposition into harmonic
polynomials multiplied by angular coefficients. The Ai
coefficients represent ratios of the helicity crosssections and
following the conventions and notations of [16,17], the
folσ U +L ∼ (v2 + a2)(v2 + aq2),
A0, A1, A2 ∼ 1,
v a vq aq
A3, A4 ∼ (v2 + a2)(v2 + aq2) ,
(v2 + a2)(vq aq )
A5, A6 ∼ (v2 + a2)(v2 + aq2) ,
v a (vq2 + aq )
2
A7 ∼ (v2 + a2)(v2 + aq2)
lowing coefficients constructed from couplings, appear in
Ai ’s:
where vi , ai , i = q, denote vector and axial couplings of
intermediate boson to quarks and leptons.
In case of W boson the EW sector at leading order is
simply a (V − A) coupling only. At higher order and higher
pTW the more complicated structure (potentially also more
interesting) of the multiboson couplings, if present, may be
revealed. In case of the Z /γ ∗, the sensitivity to the EW sector
is much richer from the physics point of view, in particular
for A3 and A4 coefficients, and we have discussed it recently
in [35,53].
4 Numerical results for Collins–Soper and Mustraal
frames
Let us now present numerical results for the angular
coefficients Ai and compare predictions in the Collins–Soper and
Mustraal frame for W − production. Most of results for
W + are delegated to Appendix C. We will present
numerical results for coefficients A0 to A4 only, the other ones
are consistent with zero over all kinematical range, for both
Collins–Soper and Mustraal frames with similar statistical
errors. One can conclude that for both choices of frames all
Ai coefficients are constrained. Thus, in principle
interpretation of results in language of different helicity amplitudes
of the scattering process is possible.
4.1 Results with LO simulation
We use samples of events generated with the
MadGraph5_aMC@NLO Monte Carlo [36] for Drell–Yan production of
W + 1 j with W → τ ν and 13 TeV pp collisions.
Lowest order spin amplitudes are used in this program for the
parton level process. To better populate higher pW bins
T
we merged (adjusting properly for relative normalization)
3 samples, 2 × 106 events each, generated with thresholds
of pTj > 1, 50, 100 GeV respectively. The incoming
partons distributed accordingly to PDFs (using CTEQ6L1 PDFs
[58] linked through LHAPDF v6 interface) remain precisely
CollinsSoper frame
Mustraal frame
CollinsSoper frame
Mustraal frame
 41.6
A

Fig. 9 The A0 and A4 coefficients calculated in Collins–Soper (black)
and in Mustraal (red) frames for pp → τ −ν + 1 j process generated
with MadGraph. Further plots, but for A1, A2, A3 and A0 − A2 are
delegated to Fig. 16 of Appendix B
Fig. 10 The A0 and A4 coefficients calculated in Collins–Soper frame
for pp → τ ±ν + 1 j processes generated with MadGraph. Further
plots, but for A1, A2, A3 and A0 − A2 are delegated to Fig. 17 of
Appendix B
collinear to the beams. At this level, jet (j) denotes outgoing
parton of unspecified flavour.
Figure 9, collects results for angular coefficients Ai of the
processes with W − → τ −ν in the final state. We show sets
of five angular coefficients A0 − A4 only; the remaining ones
A5 − A7 are close to zero over the full pTW range for both
definition of frames; Collins–Soper and Mustraal. For the
A3 and A4 coefficients we show their absolute value, as in
the convention we have adopted the signs depend on the sign
of the charge of the W boson (so the charge of lepton). In
case of W + production, both A3 and A4 are negative, see
Appendix C.
In both frames, at low pTW the only nonzero coefficient is
A4, and is of the same value. Similarly as we observed in case
of Drell–Yan Z → process [35], the only significantly
nonzero coefficient in the Mustraal frame at higher pW
T
remains A4, while A0, A2 and A3 are rising steeply for higher
pTW in the Collins–Soper frame.
Figures 10 and 11 show the comparison of predicted
coefficients for W + and W −, respectively in Collins–Soper and
Mustraal frames. The noticeable difference of the A4
coefficients at low pW directly reflects different compositions
T
of the structure functions in pp collisions to produce W +
and W − which enters the average over couplings shown in
Eq. (13). This difference is present for both the Collins–Soper
and the Mustraal frame. For the Collins–Soper frame we
observe also sizable A3 coefficient above pTW = 100 GeV.
As stated already in [16], the theoretical uncertainties due
to the choice of the factorisation and renormalization scales
are very small for Ai representing the cross section ratios.
Also most of the uncertainties from the choice of structure
functions and factorisation scheme cancel in the ratios.
4.2 Results with NLO simulation So far, we have discussed results for samples of fixed order tree level matrix elements and of single parton (jet) emis
pp → Wj, W → τ ν
Powheg+MiNLO MC
CollinsSoper frame
Mustraal frame
pp → Wj, W → τ ν
Powheg+MiNLO MC
CollinsSoper frame
Mustraal frame
Fig. 11 The A0 and A4 coefficients calculated in Mustraal frame
for pp → τ ±ν + 1 j processes generated with MadGraph. Further
plots, but for A1, A2, A3 and A0 − A2 are delegated to Fig. 18 of
Appendix B
Fig. 12 TheA0 and A4 coefficients calculated in Collins–Soper (black)
and in Mustraal (red) frames for pp → τ −ν + 1 j process generated
with Powheg+MiNLO. Further plots, but for A1, A2, A3 and A0 − A2
are delegated to Fig. 19 of Appendix B
sion. In general, configurations with a variable number of jets
and effects of loop corrections and parton shower of initial
state should be used to complete our studies. We have
performed this task partially only, with the help of 10M weighted
W + + j and W − + j events, with W ± → τ ±ν generated
with Powheg+MiNLO Monte Carlo, again for pp collisions
at 13 TeV and the effective EW scheme. The PowhegBox
v2 generator [37,38], augmented with MiNLO method for
choices of scales [59] and inclusion of Sudakov form
factors [60], by construction achieves NLO accuracy for
distributions involving finite nonzero transverse momenta of the
lepton system. Two jet configurations are thus present.
In Fig. 12 results for Ai ’s coefficients for W − → τ −ν
are shown, extracted using moments method [17] described
in Sect. 2.4. Comparisons of results using Mustraal
and the Collins–Soper frames feature again the pattern,
observed already for QCD LO W + 1 j events generated
with MadGraph. As predicted for Z → and higher pTZ ,
the LamTung relation A0 = A2 [32] is again violated at
QCD NLO in the Collins–Soper frame. This confirms the
robustness of our conclusions that only the Mustraal frame
retains Born like Ai coefficients for high pTW configurations.
5 Summary
The interest in the decomposition of results for measurement
of final states in Drell–Yan processes at the LHC, into
coefficients of second order spherical polynomials (for angular
distributions of leptons in the leptonpair restframe), was
recently confirmed by experimental publications for Z →
process [13,31,61] and W → ν processes [28,29].
Inspired by those measurements, we have investigated the
possibility to apply a strategy similar to [13] for W → ν
production and decay. Thus, to contest the statements which
are often made in the literature, that the neutrino escaping
detection makes measurement of the complete set of angular
coefficients not possible. We have shown a proof of concept
for the proposed strategy, for the measurement of the
complete set of Ai . To establish precision level of 1% (required
for the W mass measurement at 10 MeV precision level),
require experimental details to be included, it is thus out
of scope of the present paper. As an example, Monte Carlo
events of simulated W → ν +1 j process were analysed
and the complete set of Ai coefficients was extracted from
a fit to pseudodata distributions in the fiducial phasespace,
in agreement with the prediction for this sample calculated
with the moments method. The results were crosschecked
to hold when QCD NLO effects were taken into account.
In the second part of this paper, we have discussed the
optimal reference frame for such measurement. Two frames:
Collins–Soper and Mustraal [35] were studied. We have
presented predictions for the angular coefficients in those
frames as function of W boson transverse momenta. We have
shown that as expected, in case of the Mustraal frame,
only one coefficient remains significantly nonzero and
constant, almost up to pW = 100 GeV, where it starts
decreas
T
ing. Similarly as we argued in [35], this may help to
facilitate the interpretation of experimental results into quantities
sensitive to strong interaction effects. The longitudinal W
boson polarisation seems to appear predominantly as
kinematic consequence of the choice of reference frames even in
configurations of high pT jets.
Acknowledgements E.RW. would like to thank Daniel Froidevaux
for numerous inspiring discussions on the angular decomposition and
importance of measuring angular coefficients for W boson production
at LHC. We would like to thank W. Kotlarski for providing us with
samples generated with MadGraph which were used for numerical results
presented here. E.RW. was partially supported by the funds of Polish
National Science Center under decision UMO2014/15/B/ST2/00049.
Z.W. was partially supported by the funds of Polish National Science
Center under decision DEC2012/04/M/ST2/00240. We acknowledge
PLGrid Infrastructure of the Academic Computer Centre CYFRONET
AGH in Krakow, Poland, where majority of numerical calculations were
performed.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
In this Appendix, we collect plots corresponding to the ones
shown in Sect. 2 but for W + → +ν. Figure 13 shows cos θ
and φ distributions of charged lepton in the Collins–Soper
rest frame. We use generated W boson mass in a given event
m W = m gWen and compare to the case when fixed PDG value
m W = m WPDG for calculating neutrino momenta pzν , taking
correct solution for pzν is used. The losses due to nonexisting
0−1
Fig. 13 As Fig. 3 but for W + → τ +ν. The distributions of cos θ
and φ of charged lepton in the Collins–Soper rest frame. Effect from
events loss due to nonexisting solution for the neutrino momenta, when
mW = m WPDG is used for Eq. (7) is concentrated in the central bins of
the top plot
−0.8 −0.6 −0.4 −0.2
Fig. 14 As Fig. 4 but for W + → τ +ν events. The distributions of
cos θ and φ of charged lepton in the Collins–Soper rest frame. Cases
of mW = m WPDG for solving Eq. (7) where correct, wrong or random
solution for pzν are taken. Top plots are for the distributions in the full
phasespace, bottom ones for the fiducial phasespace
0
−0.05
0
−0.2
20.3
AA00.2
2 0.3
A0
A0.2
Fig. 17 The A1, A2, A3 and A0 − A2 coefficients calculated in Collins–
Soper frame for pp → τ ±ν +1 j processes generated with MadGraph.
Plots for A0 and A4 are given in Fig. 10
Fig. 18 The A1, A2, A3 and A0 − A2 coefficients calculated in
Mustraal frame for pp → τ ±ν + 1 j processes generated with
MadGraph. Plots for A0 and A4 are given in Fig. 11
Fig. 16 The A1, A2, A3 and A0 − A2 coefficients calculated in Collins–
Soper (black) and in Mustraal (red) frames for pp → τ −ν + 1 j
process generated with MadGraph. Plots for A0 and A4 are given in
Fig. 9
Fig. 15 As Fig. 5 but for W + → τ +ν events. The 2D distribution of
charged lepton cos θ and φ. Case of the fullphase space with generated
neutrino momentum, top plot. Case of the fiducial phasespace and
mW = m WPDG used in Eq. (7) and random solution for pzν , bottom plot
solution of Eq. (7) are concentrated at cos θ = 0 but are
uniformly distributed over the whole φ range.
Figure 14 shows variation of cos θ and φ distributions
gen or
for charged lepton from W + decays when m W = m W
m W = m WPDG are used for solving Eq. (7). In the second
case selection of the fiducial region is applied. In each case
distributions are shown for correct, wrong and random
solution for pν . One can notice, comparing with Fig. 4, that the
z
shapes of the cos θ distributions in case of W + and W − are
not mirrored when random or wrong solution for the neutrino
momenta are used.
Figure 15 shows 2D distributions in (cos θ , φ) for events
in: full phasespace with generated neutrino momenta used
and in the fiducial phasespace (also m W = m WPDG is used for
solving Eq. (7) with random solution of neutrino momenta
taken).
Appendix B: Further plots for templated shapes and
extracting Ai ’s coefficients
Let us start the Appendix B with plots for A1, A2, A3 and
A0 − A2 (Figs. 16, 17, 18, 19) supplementing the plots for
A0 and A4 presented in Sect. 4.
2 1.6
A1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.2
Fig. 19 The A1, A2, A3 and A0 − A2 coefficients calculated in Collins–
Soper (black) and in Mustraal (red) frames for pp → τ −ν + 1 j
process generated with Powheg+MiNLO. Plots for A0 and A4 are given
in Fig. 12
Figure 20 (bottom) shows (cos θ , φ) distribution of W + →
+ν events, where θ , φ are calculated using generated
neutrino momentum and events are weighted with wtΣ Ai Pi .
Bottom plot of Fig. 20 show how the initially flat distributions
are distorted by this folding procedure. Note that the shape is
not the same as in case of W − → −ν shown in Fig. 6. This
is due to different distributions e.g. the pseudorapidity of
charged lepton and thus different acceptance of the fiducial
selection.
Figure 21 collects examples of 2D distributions for
polynomials P1(cos θ , φ), P2(cos θ , φ) and, P3(cos θ , φ) in the
full and fiducial phasespace. Now for each event we
reconstruct neutrino momenta using m W = m WPDG , take randomly
one of the solutions to recalculate θ , φ angles from Eq. (7) and
to apply kinematical selection of the fiducial phasespace.
Fig. 20 As Fig. 6 but for W + → τ +ν events. The 2D distribution
of charged lepton cos θ and φ. On top distribution of the full
phasespace, with generated neutrino momentum used, and events weighted
wtΣ Ai Pi . Bottom, the same distribution is shown, but: m WPDG is used
for solving Eq. (7), randomly one of the solutions for pzν is taken and
fiducial selection is applied. The weight wtΣ Ai Pi is still calculated with
generated neutrino momenta
Figure 22 collects results of the multilikelihood fit of
W + → +ν, displayed are Ai coefficients as function of
pTW .
1 0 10.80.60.40.2 0 0.20.40.60.8 1
cosθ
1 0 10.80.60.40.2 0 0.20.40.60.8 1
cosθ
1 0 10.80.60.40.2 0 0.20.40.60.8 1
cosθ
Fig. 22 As Fig. 8 but for W +
τ +ν events. Closure test on the fitting
→
of angular coefficients Ai ’s. Fit is performed in the fiducial phasespace.
Shown are generated Ai ’s coefficients (open circles) and their fitted
values (black points). In the bottom panels shown are pulls (difference
between generated and fit value, divided by the statistical error of the
fit). Pulls are smaller than one could expect. This is because events of
pseudodata and templates are statistically correlated
Appendix C: Additional plots on Ai s coefficients
Powheg+MiNLO.
In the generated sample information on incoming and
outgoing partons flavours is stored. We will use this
informalevel initial states. Figures 23 and 24 show predictions for
MadGraph Monte Carlo for these subsamples.
Ai ’s coefficients for W +
τ +ν and events generated with
→
Figure 25 shows predictions for Ai ’s coefficients of
and for processes generated with
0 1.6
A
2 1.6
A
0 1.6
A
2 1.6
A
2 0.3
A
0
A0.2
Col insSoper frame
Mustraal frame
Col insSoper frame
Mustraal frame
Col insSoper frame
Mustraal frame
Col insSoper frame
Mustraal frame
Col insSoper frame
Mustraal frame
Col insSoper frame
Mustraal frame
1 0.2
A
1 0.2
A
 41.6
A
 1.4
1 j generated with MadGraph
+
Fig. 23 As Figs. 9 and 16 but for W + and parton level subprocess
only. The Ai coefficients calculated in Collins–Soper (black) and in
Mustraal (red) frames for selected parton level process pp(qq)
¯ →
1 j generated with MadGraph
+
Fig. 24 As Figs. 9 and 16 but for W + and parton level subprocess
only. The Ai coefficients calculated in Collins–Soper (black) and in
Mustraal (red) frames for selected parton level process pp(Gq)
¯ →
1 0.2
A
0.15
Fig. 25 As Figs. 12 and 19 but for W + → τ +ν events. The Ai
coefficients calculated in Collins–Soper (black) and in Mustraal (red)
frames for pp → τ +ν + 1 j process generated with Powheg+MiNLO
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