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. References [1] M.G. Olsson and C.J. Suchyta, Phys. Rev. D 34, 2043 (1986); F.L. Ridener, K.J. Sebastian and H. Grotch, Phys. Rev. D 45, 3173 (1992). [2] T.A. Armstrong et al. (E760 Coll.), Phys. Rev. D 48, 3037 (1993); M. Ambrogiani et al. (E835 Coll.), Phys. Rev. D 65, 052002 (2002); M. Oreglia et al. (Crystal Ball Coll.), Phys. Rev. D 25, 2259 (1982); M. Artuso et al. (CLEO Coll.), Phys. Rev. D 80, 112003 (2009). [3] J.C. Collins and D.E. Soper, Phys. Rev. D 16, 2219 (1977).