#### The Z boson spin observables as messengers of new physics

Eur. Phys. J. C
The Z boson spin observables as messengers of new physics
J. A. Aguilar-Saavedra 2
J. Bernabéu 0 1
V. A. Mitsou 0
A. Segarra 0 1
0 Instituto de Física Corpuscular, CSIC-Universitat de València , 46890 Paterna , Spain
1 Departament de Física Teòrica, Universitat de València , 46100 Burjassot , Spain
2 Departamento de Física Teórica y del Cosmos, Universidad de Granada , 18071 Granada , Spain
We demonstrate that the eight multipole parameters describing the spin state of the Z boson are able to disentangle known Z production mechanisms and signals from new physics at the LHC. They can be extracted from appropriate asymmetries in the angular distribution of lepton pairs from the Z boson decay. The power of this analysis is illustrated by (1) the production of Z boson plus jets; (2) Z boson plus missing transverse energy; (3) W and Z bosons originating from the two-body decay of a heavy resonance.
1 Introduction
The successful operation of the large hadron collider (LHC)
has allowed one to accumulate a wealth of collision data in the
search of new physics in the ATLAS and CMS experiments,
at centre-of-mass (CM) energies of 7, 8 and 13 TeV. With
the ever-increasing statistics, measurements beyond simple
event counts are possible, which provide further insight into
the standard model (SM) processes and possible new physics.
Of particular interest are polarisation measurements, possible
for particles with a short lifetime, through analyses of the
angular distributions of their decay products.
For spin-1/2 fermions there are three independent spin
observables, which can be conveniently taken as the
expectation values of the spin operators in three orthogonal
directions. At the LEP experiments, this program was exploited
for τ leptons [
1
] and, for general e+e− colliders, proposed
for heavy quarks [
2–5
]. But for spin-1 vector bosons the
number of independent spin observables is eight, requiring a
more elaborate discussion. There have been various studies
of spin observables or decay angular distributions for
vector bosons, often focusing on specific processes [
6, 7
]. In
previous work [
8
] some of us have provided a full
modelindependent analysis of the W boson spin observables. For
the Z boson the framework is quite similar, the main
difference being that while for the W boson the couplings to leptons
violate parity maximally, for the Z boson they do not. This
difference can be taken into account by the introduction of
an additional coupling factor relating the observed angular
distributions to the Z boson spin observables, a factor which
we may call the polarisation analyser. This study has been
applied to an e+e− collider [
9
].
The purpose of this paper is to demonstrate that the
information content in the eight multipole parameters of the Z
boson, polarisations and alignments, is able to clearly
discriminate among different production mechanisms, acting as
messengers of the physics involved in the process. This is
particularly important at the LHC, taking into account the
hadronic environment. In Sect. 2 we write down the relation
between Z boson spin observables and the parameters of the
decay angular distributions. Beyond the application to Drell–
Yan Z plus jets events at LHC, considered in Sect. 3, we
move to the study of the production of a Z boson plus missing
transverse energy (MET) in Sect. 4, where the discrimination
between the SM production mechanism and that for extended
models is apparent. In Sect. 5 we discuss in detail the
predictions for the eight spin observables of Z (or W ) bosons
produced from the two-body decay of a heavy resonance of
spin 0, 1/2 or 1. Our conclusions are presented in Sect. 6.
2 Formalism
The spin state of Z bosons is described, as for any other
spin1 massive particle, by a 3 × 3 density matrix ρ, Hermitian
with unit trace and positive semidefinite. We follow closely
the analysis and notation introduced [
8
] for the analysis of the
W boson spin.1 If we fix a coordinate system (x , y , z ) in
the Z boson rest frame, we can write the spin density matrix
as
1 An equivalent description of the formalism for Z boson spin
observables was later made in Ref. [
9
].
1 1
ρ = 3 1 + 2
1
with S±1 = ∓ √12 (S1 ± i S2), S0 = S3 the spin operators in
the spherical basis and TM five rank 2 irreducible tensors,
2
M=−2
SM ∗ SM +
TM ∗TM ,
(1)
Here, at variance with W boson decays,2 we have two
possible helicity combinations (λ1, λ2) = (±1/2, ∓1/2). The
differential decay width reads
S±1 S0 + S0 S±1 ,
S+1 S−1 + S−1 S+1 + 2S02 .
Their expectation values SM and TM are the multipole
parameters corresponding to the three polarisation and five
alignment components. The second term in Eq. (1) can be
rewritten using the spin operators in the Cartesian basis, and
the third one defining the Hermitian operators
1 1
A1 = 2 (T1 − T−1), A2 = 2i (T1 + T−1),
1 1
B1 = 2 (T2 + T−2), B2 = 2i (T2 − T−2).
Therefore, the Z boson density matrix elements,
parameterised in terms of expectation values of observables, read
1 1 1
ρ±1±1 = 3 ± 2 S3 + √6
1 1
ρ±10 = 2√2 [ S1 ∓ i S2 ] ∓ √2
T0 ,
ρ1 −1 =
B1 − i B2 ,
[ A1 ∓ i A2 ] ,
and ρm m = ρm∗m . The angular distribution of the Z boson
decay products in its rest frame is determined by ρ. Let
us restrict ourselves to leptonic decays Z → + −, with
= e or = μ. Using the helicity formalism of Jacob and
Wick [
10
], the amplitude for the decay of a Z boson with
third spin component m giving − with helicity λ1 and +
with helicity λ2 is written as
Mmλ1λ2 = bλ1λ2 Dm1∗λ(φ∗, θ ∗, 0),
with (θ ∗, φ∗) the polar and azimuthal angles of the −
momentum in the Z boson rest frame, λ = λ1 − λ2 and
D mj m (α, β, γ ) = e−iαm e−iγ m dmj m (β)
the so-called Wigner D functions [
11
]; bλ1λ2 are constants,
and all the dependence of the amplitude on the angular
variables of the final state products is given by the D functions.
with λ = λ1−λ2 = ±1. The constants b1/2 −1/2 and b−1/2 1/2
are, respectively, proportional to the right- and left-handed
couplings of the Z boson to the charged leptons gR , gL ,
The angular distribution of the Z boson decay products can
easily be obtained from the distribution for W ± decays [
8
]
by noting that the only terms that change sign when
replacing λ by −λ are those proportional to Sk , k = 1, 2, 3. By
introducing the polarisation analyser,
(gL )2 − (gR )2 1 − 4sW2
η = (gL )2 + (gR )2 = 1 − 4sW2 + 8sW4 ,
with sW the sine of the weak mixing angle, we get
1
d 3
d cos θ ∗dφ∗ = 8π
(7)
(8)
(9)
1
× 2 (1 + cos2 θ ∗) − η S3 cos θ ∗
1 1
+ 6 − √6 T0 1 − 3 cos2 θ ∗
−η S1 cos φ∗ sin θ ∗−η S2 sin φ∗ sin θ ∗
− A1 cos φ∗ sin 2θ ∗− A2 sin φ∗ sin 2θ ∗
+ B1 cos 2φ∗ sin2 θ ∗
+ B2 sin 2φ∗ sin2 θ ∗ .
(10)
The angular asymmetries introduced in Ref. [
8
] for the
measurement of W boson spin observables can be
straightforwardly used for the Z boson as well, with the appropriate
replacements. We have
1
x
AFB =
y
AFB =
z
AFB =
z
AEC =
1
1
1
(cos φ∗ > 0) −
(sin φ∗ > 0) −
(cos θ ∗ > 0) −
3
(cos φ∗ < 0) = − 4 η S1 ,
3
(sin φ∗ < 0) = − 4 η S2 ,
3
(cos θ ∗ < 0) = − 4 η S3 ,
1
| cos θ ∗| > 2 −
1
| cos θ ∗| < 2
2 For the W boson the left-handed interaction fixes (λ1, λ2) =
(±1/2, ∓1/2) for W ± → ±ν decays, that is, there is a single helicity
combination in each case.
(cos φ∗ cos θ ∗ > 0) −
(cos φ∗ cos θ ∗ < 0)
(sin φ∗ cos θ ∗ > 0) −
(sin φ∗ cos θ ∗ < 0)
4 Z boson plus MET production
AFxB,z =
AFyB,z =
A1φ =
A2φ =
3 3
= 8 2
1
2
= − π
1
2
= − π
1
1
A1 ,
A2 ,
(cos 2φ∗ > 0) −
(sin 2φ∗ > 0) −
2
(cos 2φ∗ < 0) = π
2
(sin 2φ∗ < 0) = π
B1 ,
B2 .
(11)
As seen, these asymmetries separate out the eight multipole
parameters one by one.
3 Drell–Yan production
The angular distribution (10) has been investigated by the
CDF [
12
], CMS [
13
] and ATLAS [
14
] Collaborations in
Drell–Yan Z production in hadron collisions, using the
Collins–Soper coordinate system [
15
]. The doubly
differential angular distribution is parameterised using unknown
coefficients labelled A0– A7. The density matrix analysis of
the Z boson decay provides an interpretation of the measured
coefficients in terms of Z boson spin observables,
2
A0 = 3 − 2
A2 = 4 B1 ,
2
3 T0 ,
A3 = −2η S1 ,
A1 = −2 A1 ,
A4 = −2η S3 ,
A6 = −2 A2 ,
A5 = 2 B2 ,
A7 = −2η S2 .
Experiments have measured these coefficients
differentially, as a function of the transverse momentum and rapidity
of the Z boson. An interpretation in terms of Z spin
observables, apart from providing more insight into the nature of
the physical observables measured, provides a rationale for
the smallness of A3, A4 and A7, since they are
proportional to the small polarisation analyser η 0.14. The
most recent measurement by the ATLAS Collaboration [
14
]
exhibits a noticeable deviation in A2 with respect to
nextto-leading order [
16
] and next-to-next-to-leading order [
17
]
SM predictions. Nevertheless, the CMS Collaboration finds
agreement with the multi-leg SM prediction from
MadGraph5_aMC@NLO [
18
]. This coefficient corresponds to
the rank-two alignment B1 .
We also point out that – besides the method commonly
used to extract the angular coefficients A0−7 based on
inte(12)
gration with suitable weight functions [
7
] – the asymmetries
in (11) provide an alternative way for their determination.
Whether this simpler method also gives more precise results
depends on the systematic uncertainties in each case, and a
detailed analysis is compulsory to draw any conclusion.
The production at the LHC of final states containing a
sameflavour opposite-sign lepton (electron or muon) pair with
invariant mass around the Z boson mass, possibly jets, and
large MET is very relevant in the search by ATLAS [
19
] and
CMS [
20
] of Supersymmetry (SUSY) signals and collider
production of Dark Matter [
21
]. Besides the simple event
counting, the use of spin observables in these Z + MET
searches provides an additional handle to test the SM
predictions and uncover possible effects of new physics.
Similarly to the previous section, the full angular
distribution (10) can be measured differentially as a function of
the MET in these final states. The leading SM processes
yielding Z plus missing energy are (1) Z Z production, with
Z Z → + −νν¯ ; (2) W Z production, with Z → + −,
W → ν, and the additional charged lepton undetected,
because of having a small transverse momentum or large
rapidity. We have used MadGraph5_aMC@NLO to
simulate these processes at the tree level, followed by
hadronisation by Pythia [
22
], in order to estimate the SM prediction
for the Z + MET final state in pp collisions at a CM energy
of 13 TeV. We restrict our analysis to events happening at the
Z peak, with the two charged leptons in an invariant mass
window of 88–94 GeV.
We set our reference system in the Z boson rest frame
with the zˆ axis in its momentum direction, while the other
two axes are left unspecified – so the non-diagonal elements
of the density matrix vanish, leaving only S3 and T0 as
observables. The calculated values of S3 and T0 , as a
function of the lower cut on MET, are presented in Fig. 1.
The bands represent the Monte Carlo statistical uncertainty
of our results. Notice that our simulation only includes signal
events, so the meaning of these MET cuts is dynamical, i.e.
the inclusion of a MET threshold leads to a different
production mechanism of the Z boson. For the SM predictions
(blue and orange bands), the large dependence of these two
observables on the MET cut makes their measurement very
interesting to test the SM, as well as providing a reference
for beyond the standard model (BSM) searches.
To illustrate the power of the declared strategy, we
compare with the expected values of these observables in a SUSY
dark matter model with the gravitino G˜ as lightest
supersymmetric particle (LSP) and the lightest neutralino χ˜10 as
next-to-LSP. We consider a massless gravitino and a lightest
neutralino χ10 whose mass is around 100 GeV. For simplicity,
0 0
we assume the direct electroweak production of a pair χ˜1 χ˜1
from quark–antiquark, as shown in Fig. 2. The simulation
is performed within the MadGraph5_aMC@NLO
framework utilising the gravitino implemented in FeynRules
output in the Universal FeynRules Output (UFO) [
23
]. The BSM
values of T0 and S3 do not depend on the MET
threshold. The former is T0 = 1/√6 due to angular momentum
conservation, as discussed in the next section (with j = 1/2,
j = 3/2). The latter is fixed for a particular process, but may
change in the presence of an additional production channel.
Since in both diagrams in Fig. 2 we have the same
incoming and outgoing particles, qq¯ → χ10χ10 → Z G˜ Z G˜ , the
kinematical distributions, including the missing transverse
momentum, should be the same. Therefore, the change in
the MET threshold should not affect the relative contribution
of each diagram, and S3 should be independent of the MET
cut.
The consideration of Z boson plus MET production in
SUSY models with gravitinos is motivated by previous
work [
24
], where it was shown that Gauge Mediation
models could lead to a privileged new production mechanism of
Z bosons. In this model, the production mechanism of Z
bosons plus MET is χ˜ 0 → Z G˜ , which produces Z bosons
with the diagonal spin parameters shown in Fig. 1 (green and
red bands).
5 Z bosons from heavy particle decays
Let us consider that a Z boson is produced in the two-body
decay of some spin- j particle A, yielding also a spin- j
particle B as decay product,
where λ1, λ2 are the helicities of Z and B in the rest frame
of the decaying particle A, and m its third spin component.
Let us fix a (xˆ, yˆ, zˆ) coordinate system in the rest frame of
A. The amplitude for the decay can be written as
Mmλ1λ2 = aλ1λ2 D mj∗λ(φ, θ , 0),
with θ and φ the polar and azimuthal angle of the Z boson
momentum, in full analogy to the Z boson decay discussed
in Sect. 2. Restricting ourselves to particles up to spin 3/2,
we have six combinations for j and j , collected in Table 1.
We assume that spin 3/2 particles are massless, so their only
possible helicities are ±3/2; furthermore, the decaying
particle A cannot have spin 3/2 in this case. For each pair j, j ,
angular momentum conservation implies that only a subset
of aλ1λ2 are non-zero; these combinations are also given in
Table 1.
Angular momentum conservation restricts the form of the
Z boson spin density matrix, whose elements also depend
on the angles θ and φ. Differential measurements of the Z
spin observables as functions of θ and φ are possible with
sufficient statistics, but for simplicity we consider here the
integrated density matrix, using for the Z boson rest frame the
(xˆ , yˆ , zˆ ) coordinate system implied by the standard boost
from the rest frame of A, with the zˆ axis in the Z helicity
direction.3 Experimentally, this would require the
measurement of the momentum of the B particle too, in order to
reconstruct the momentum of A and choose a coordinate
3 This boost is given by a rotation R(φ, θ, 0) in the Euler
parameterisation, followed by a pure boost in the Z momentum direction to set it
at rest.
system in its rest frame. (In the previous section we have not
made such assumption.) With this setup, integrating over θ
and φ does not generally give a diagonal ρ, and a dependence
on the values of S3 and T0 for the particle A, denoted
here as S A and T0A , respectively, is retained. This
sub3
tle effect is due to the fact that, when performing the above
specified standard boost from the rest frame of A, the yˆ axis
is always in the x y plane (see for example Ref. [
25
] for a
detailed discussion) irrespectively of the values of θ and φ,
and the integration over these two angles is not equivalent to
considering an isotropic distribution of the xˆ and yˆ axes.
The predictions for the different combinations of j, j are
as follows.
• j = 1/2, j = 3/2. The density matrix has all entries
vanishing except ρ11 and ρ−1−1, implying S1 = S2 =
0, A1 = A2 = 0, B1 = B2 = 0. The only
nontrivial observables are
S3 = |a1 3/2|2 − |a−1 −3/2|2 /N ,
√
T0 = 1/ 6.
Here and below, N is the sum of the moduli squared of
all the non-zero amplitudes in Table 1 for the case under
consideration. These results were referred to in Sect. 4 of
this paper.
• j = 1/2, j = 1/2. Here ρ1−1 = 0, therefore B1 =
B2 = 0. The remaining observables can be non-zero,
1 − 3 |a0 1/2|2 + |a0 −1/2|2 /N ,
S3 = |a1 1/2|2 − |a−1 −1/2|2 /N ,
1
T0 = √6
π
S1 = − √2
π
S2 = − √2
π
A1 = − √
2 2
π
A2 = − √
2 2
S3A Re[a−1 −1/2 a0∗−1/2 + a1 1/2 a0∗1/2]/N ,
S3A Im[a−1 −1/2 a0∗−1/2 − a1 1/2 a0∗1/2]/N ,
S3A Re[a−1 −1/2 a0∗−1/2 − a1 1/2 a0∗1/2]/N ,
S3A Im[a−1 −1/2 a0∗−1/2 + a1 1/2 a0∗1/2]/N .
• j = 1, j = 0, 1. In these two cases all the density matrix
elements and Z boson spin observables are generally
different from zero. We can write them as
S3 = C1 − C−1 /N ,
1
T0 = √ [1 − 3C0/N ] ,
6
3π
S1 = − 8
3π
S2 = − 8
S3A Re[C−10 + C10]/N ,
S3A Im[C−10 − C10]/N ,
T0A Re C2,
T0A Im C2,
where we have abbreviated products of amplitudes aλ1λ2
for j = 0 ( j = 1) as
C1 = |a1 0|2 ( + |a1 1|2),
C−1 = |a−1 0|2 ( + |a−1 −1|2),
C0 = |a0 0|2 ( + |a0 1|2 + |a0 −1|2),
• j = 0, j = 1. The Z spin density matrix is diagonal,
implying S1 = S2 = 0, A1 = A2 = 0, B1 =
B2 = 0. The diagonal spin observables are
S3 = |a1 1|2 − |a−1 −1|2 /N ,
• j = 0, j = 0. This case is particularly interesting,
implying a full longitudinal Z (helicity λ = 0) or,
equivalently, a p-wave (orbital angular momentum l = 1) state.
The only non-zero density matrix element is ρ00 = 1,
therefore S1 = S2 = S3 = 0, A1 = A2 = 0,
B1 = B2 = 0 and
It would be the situation in the decay A(0−) → Z +
h(0+) in the Higgs sector of SUSY and left–right models.
This analysis can be applied to current and future heavy
resonance searches. If a new resonance is discovered, the use
of spin observables in the Z leptonic decay products might
shed light into its nature. If any spin observable is measured to
have a non-trivial value, or T0 = −2/√6, the j = 0, j = 0
hypothesis is discarded. If, additionally, a non-zero value is
found for any off-diagonal spin observable, the j = 0, j = 1
hypothesis can be discarded too. Furthermore, a non-trivial
value for B1,2 > can only be explained by a j = 1 parent.
However, discriminating the two j = 1 possibilities requires
an analysis of the decay of the extra particle B.
To conclude this section we remark that, as anticipated,
the classification and relation of spin observables with decay
amplitudes obtained also apply for a W boson, namely to the
decay
A( j, m) →
W (1, λ1) B( j , λ2)
(21)
because we have not used any other property than the spin.
In particular, one example of j = 1/2, j = 1/2 decays is
given by t → W b, studied in Ref. [
8
].
6 Conclusions
With the wealth of collision data being accumulated by LHC
experiments, the measurement of the eight Z boson spin
observables becomes feasible from the angular distribution of
its leptonic decay. We have proved in this work the
discriminating power of these polarisation and alignment observables
for identifying the production mechanism of the Z boson,
with apparent different values for known processes in the
SM and for extended models. These observable quantities
thus play the role of messengers from the hidden physics
leading to the Z boson production.
Interesting physical processes include the Drell–Yan Z
boson production, for which we have given the physical
interpretation of the parameters of the lepton angular
distribution measured by ATLAS and CMS, pointing out the
alternative method of extracting the Z boson spin
observables by means of selected asymmetries. When we move to
processes able to generate large missing transverse energy,
the SM reference values for the Z boson longitudinal
polarisation S3 and alignment T0 present a characteristic rapid
variation above 100 GeV of MET, contrary to the values and
behaviour obtained from SUSY models with new sources of
Z boson production like the decay of neutralinos to
gravitinos. For two-body decays of a heavy particle involving
a Z boson (or W boson) in its decay products, we have
demonstrated that different spin assignments of the parent
and daughter particles lead to specific zeros and values of
the Z boson polarisations and alignments. The use of these
observables will increasingly become an invaluable
interesting handle to test the SM predictions and look for new
physics.
Acknowledgements This research has been supported by MINECO
Projects FPA 2016-78220-C3-1-P, FPA 2015-65652-C4-1-R, FPA
2014-54459-P, FPA 2013-47836-C3-2-P (including ERDF), Junta
de Andalucía Project FQM-101, Generalitat Valenciana Project GV
PROMETEO II 2013-017, Severo Ochoa Excellence Centre Project
SEV 2014-0398, and European Commission through the contract
PITNGA-2012-316704 (HIGGSTOOLS). VAM acknowledges the support
by Spanish National Research Council (CSIC) under the CT
Incorporation Program 201650I002, and AS acknowledges the MECD support
through the FPU14/04678 Grant.
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Funded by SCOAP3.
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