Nuclear Physics B (Proc. Suppl.) 214 (2011) 107–109
www.elsevier.com/locate/npbps
A simple and robust method to measure χc and χb polarizations
Pietro Facciolia , Carlos Lourençob , João Seixasa,c , Hermine K. Wöhria
a Laboratório
de Instrumentação e Fı́sica Experimental de Partı́culas (LIP), Lisbon, Portugal
Organization for Nuclear Research (CERN), Geneva, Switzerland
Department, Instituto Superior Técnico (IST), Lisbon, Portugal
b European
c Physics
Abstract
The polarizations of the χ states produced in high energy collisions can be derived from the dilepton decay distributions of the daughter J/ψ or Υ mesons, with a reduced dependence of the measurement on the details of photon
reconstruction and simulation, and eliminating the dependence of the polarization measurement on the actual details
of the multipole structure of the radiative transition.
Keywords: Quarkonium, polarization, radiative decays
1. Introduction
Existing J/ψ and Υ polarization measurements make
no distinction between directly produced states and
those resulting from the decay of higher-mass states.
The J/ψ and Υ mesons coming from χ decays have in
principle very different polarizations with respect to the
directly produced ones. Therefore, the measurement of
the polarization of the χc and χb states, accompanied by
the knowledge of how these states transmit their polarization to the daughter J/ψ and Υ, is an important ingredient in the understanding of quarkonium production.
We have studied the angular distributions of the radiative decay χcJ (χbJ ) → J/ψ(Υ) γ (“photon” distribution)
and of the consecutive decay J/ψ(Υ) → ℓ+ ℓ− (“dilepton” distribution). We discuss here the sensitivity of
these observable distributions to the angular momentum
composition (“polarization”) of the decaying χ meson
and their additional dependence on the orbital angular
momentum of the photon. As a result of the study, we
propose a convenient way of measuring χ polarizations
in high-energy experiments.
2. Photon distribution
We start by considering the measurement of the photon distribution in the χ rest frame. We assume that
the χ is produced in a single “subprocess” as a given
m=+J
superposition of Jz eigenstates: |χ = m=−J
bm |J, m.
Notations for axes and angles for the photon distribution are shown in Fig. 1a. We indicate with g1 , g2 and
(only for the χ2 ) g3 , the partial amplitudes (normalized
so that i g2i = 1) corresponding to the elements of total angular momentum: electric dipole, E1, magnetic
quadrupole, M2, and electric octupole, E3, transitions.
The expected hierarchy g3 < g2 < g1 is confirmed by
χc measurements [2], albeit with seriously inconsistent
results for the χc1 , while no experimental information
exists for the χb . We restrict here our considerations
to the case of the χ2 decay. Analogous methodological
conclusions can be obtained for the χ1 case. A comprehensive report is in preparation. The photon angular
distribution is
γ
(2)
2
4
W J=2
∝ 1 + λ(1)
Θ cos Θ + λΘ cos Θ
(1)
(2)
2
4
+ λ(1)
Φ sin Θ cos 2Φ + λΦ sin Θ cos 2Φ
(1)
4
+ λ(3)
Φ sin Θ cos 4Φ + λΘΦ sin 2Θ cos Φ
(3)
2
2
+ λ(2)
ΘΦ sin Θ sin 2Θ cos Φ + λΘΦ sin Θ sin 2Θ cos 3Φ,
where
3
2
2
5
λ(1)
Θ = − D [2(1 + ∆1 )|b0 | + (1 − 3 ∆1 − 3 ∆2 )
× (|b+1 |2 + |b−1 |2 ) − (2 + 13 ∆1 − 35 ∆2 )(|b+2 |2 + |b−2 |2 )],
5
λ(2)
Θ = 6D (∆1 + ∆2 )
× [6|b0 |2 − 4(|b+1 |2 + |b−1 |2 ) + |b+2 |2 + |b−2 |2 ],
√
2
∗
λ(1)
Φ = D Re[ 6(1 + ∆1 ) b0 (b+2 + b−2 )
+ (3 − 2∆1 − 5∆2 ) b∗+1 b−1 ],
√ ∗
5
∗
λ(2)
Φ = − 3D (∆1 + ∆2 ) Re[ 6 b0 (b+2 + b−2 ) − 4 b+1 b−1 ],
0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.nuclphysbps.2011.03.067
108
P. Faccioli et al. / Nuclear Physics B (Proc. Suppl.) 214 (2011) 107–109
Figure 1: Definition of axes and decay angles for χcJ (χbJ ) → J/ψ(Υ) γ (a) and for J/ψ(Υ) → ℓ+ ℓ− (two options: b and c).
5
(∆1 + ∆2 ) Re(b∗+2 b−2 ),
3D
√
1
∗
2
5
λ(1)
ΘΦ = D Re[ 6(1 − 3 ∆1 − 3 ∆2 ) b0 (b+1 − b−1 )
∗
∗
+ (6 + 83 ∆1 − 10
3 ∆2 ) (b+2 b+1 − b−2 b−1 )],
√
5
∗
λ(2)
ΘΦ = 3D (∆1 + ∆2 ) Re[ 6 b0 (b+1 − b−1 )
− (b∗+2 b+1 − b∗−2 b−1 )],
5
∗
∗
λ(3)
ΘΦ = 3D (∆1 + ∆2 ) Re(b+2 b−1 − b−2 b+1 ),
+ λϑϕ sin 2ϑ cos ϕ.
λ(3)
Φ =
with
D = (10 + ∆1 − ∆2 )|b0 |2 + (9 − ∆2 )(|b+1 |2 + |b−1 |2 )
+ (6 − 21 ∆1 + 32 ∆2 )(|b+2 |2 + |b−2 |2 ),
√
√
∆1 = 4 g22 + 6 5 g2 g3 − 2 5 g1 g2 − 2g23 + 14 g1 g3 ,
√
√
∆2 = 4 g22 + 4 5 g2 g3 + 2 5 g1 g2 + 3g23 + 4 g1 g3 .
The dependence of this distribution on the angular
momentum configuration of the χ is very sensitive to
the contribution of the higher photon multipoles. Fig. 2a
shows, for example, that λ(1)
Θ can deviate significantly
from the value expected in the E1-dominance case, increasing by 30 − 70%, depending on the χ2 polarization
state, at the average value of g2 measured for the χc2 [2]
(assuming g3 = 0).
3. Lepton distribution
We now consider the measurement of the dilepton
distribution, which has, in any polarization frame, the
well-known form of the parity-conserving two-body decay angular distribution of a vector state:
+ −
ℓ ℓ
W J=2
∝ 1 + λϑ cos2 ϑ + λϕ sin2 ϑ cos 2ϕ
(2)
The traditional choice of axes, adopted in calculations [1] and measurements [2] of the full decay angular
distribution for χc mesons produced at low laboratory
momentum, is represented in Fig. 1b, where the J/ψ / Υ
polarization axis is the J/ψ / Υ direction in the χ rest
frame. With respect to this system of axes any measurement will always find, for example, λϑ = −1/3 plus M2
corrections for the χ1 and +1/13 plus M3 and E3 corrections for the χ2 , independently of what the χ polarization state is. This choice of axes is, therefore, suitable
for measuring the contribution of the higher-order multipoles, but it does not provide any information on the
polarization and, therefore, on the production mechanism of the χ. We propose here an alternative definition
of the J/ψ / Υ polarization frame, enabling the determination of the χ polarization in high-momentum experiments without the need of measuring the full photondilepton kinematic correlations. This definition, shown
in Fig. 1c, “clones” the χ polarization frame, defined in
the χ rest frame, into the J/ψ / Υ rest frame, taking the
x , y , z axes to be parallel to the x, y, z axes. The definition of the x, y, z axes and, therefore, of x , y , z , uses the
momenta of the colliding hadrons as seen in the χ rest
frame, so that it requires, in general, the knowledge of
the photon momentum. However, for sufficiently high
dilepton momentum, the χ and J/ψ / Υ rest frames coincide and the x , y , z axes can be approximately defined
using only momenta seen in the J/ψ / Υ rest frame. For
example, if the χ polarization axis (z) is defined along
the bisector of the beam momenta in the χ rest frame
(Collins–Soper frame [3]), the corresponding z axis is
approximated by the bisector of the beam momenta in
the J/ψ / Υ rest frame. In the case of λϑ , for instance,
P. Faccioli et al. / Nuclear Physics B (Proc. Suppl.) 214 (2011) 107–109
χ2 o ψ/b J
a)
ψ/b o ℓ+ℓ
χc2 world
average
109
+ 2(3 + δ)(|b+2 |2 + |b−2 |2 ),
δ = 2 g22 + 75 g23 .
These formulas suggest two remarks. First, with this
frame definition the dilepton distribution contains as
much information as the photon distribution regarding
the χ polarization state. The two distributions are even
identical when higher-order multipoles are neglected, as
can be recognized by comparing Eq. 1 with Eq. 3 (in this
(2)
(3)
(2)
(3)
case, in particular, λ(2)
Θ = λΦ = λΦ = λΘΦ = λΘΦ = 0).
(1)
For example, λϑ = λΘ = +1, −1/3 and −3/5, respectively for pure |2, ±2, |2, ±1 and |2, 0 χ states with
g2 = g3 = 0. Second, the dependence of the dilepton
distribution on the higher photon multipoles is negligible, as shown in Fig. 2b for λϑ .
4. Summary
b)
g2 (%)
Figure 2: Dependence of the parameters λ(1)
Θ (a) and λϑ (b) of the
χc2 (χb2 ) photon and dilepton distributions on the M2 contribution.
the relative error, |∆λϑ /λϑ |, induced by this approximation is O[(∆m/p)2 ], where ∆m is the χ – J/ψ / Υ mass
difference and p the total laboratory momentum of the
dilepton. For p > 4 GeV/c this error is smaller than 1%.
Therefore, for not-too-small momentum the frame definition we propose coincides with the frame defined in
the measurement of the polarization of inclusively produced J/ψ / Υ mesons (Collins–Soper or helicity, for example).
The dilepton decay distribution of a J/ψ / Υ originatm=+J
bm |J, m has the following
ing from the χ2 state m=−J
coefficients in the frame defined in Fig. 1c:
3
λϑ = − (1 − δ)[2|b0 |2 + |b+1 |2 + |b−1 |2
(3)
D
2
2
− 2(|b+2 | + |b−2 | )],
√
2
λϕ = (1 − δ) Re[ 6 b∗0 (b+2 + b−2 ) + 3 b∗+1 b−1 ],
D
√
1
λϑϕ = (1 − δ) Re[ 6 b∗0 (b+1 − b−1 )
D
+ 6 (b∗+2 b+1 − b∗−2 b−1 )],
with
D = 2(5 − δ)|b0 |2 + (9 − δ)(|b+1 |2 + |b−1 |2 )
We have shown that the χ polarizations can be
measured (for not-too-low-momentum experiments) directly from the angular distribution of the dilepton decay in the J/ψ / Υ rest frame, with respect to the same
kind of system of axes (Collins–Soper, helicity, etc.) as
adopted in inclusive J/ψ / Υ measurements.
In fact, the dilepton distribution in the J/ψ / Υ rest
frame is a clone of the photon distribution in the χ
rest frame, stripped of the contribution of the higher
multipoles of photon radiation. This gives a significant
advantage, given that such contributions can have a very
large impact in the measurement and that a simultaneous determination of χ polarization and of the multipole
parameters is scarcely feasible at hadron colliders.
Furthermore, this method does not use the photon
identification to reconstruct the event-by-event decay
topology. The photon momentum is used only on a statistical basis, to apply an invariant-mass cut isolating the
corresponding sample of indirectly produced J/ψ or Υ
mesons. It is not necessary, for example, to determine
the photon acceptance/efficiency as a function of the decay angles.
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