Quarkonium polarization measurements

Nuclear Physics B (Proc. Suppl.) 214 (2011) 97–102 www.elsevier.com/locate/npbps Quarkonium polarization measurements 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 existing measurements of quarkonium polarization in proton-antiproton and proton-nucleus collisions are puzzling. We highlight issues which are often underestimated in the experimental analyses: the importance of the choice of the reference frame, the interplay between observed decay and production kinematics, and the consequent influence of the experimental acceptance on the comparison between experimental measurements and theoretical calculations. New measurements must provide more detailed information, such that physical conclusions can be derived without relying on model-dependent assumptions. We also describe a frame-invariant formalism which minimizes the dependence of the measurements on the experimental acceptance, facilitates the comparison with theoretical calculations, and probes systematic effects due to experimental biases. Keywords: Quarkonium, polarization, QCD 1. The experimental situation Our present understanding of quarkonium production is far from satisfactory, despite the considerable experimental efforts and the multitude of data accumulated over more than 30 years [1]. The transversemomentum dependence of the quarkonium production cross sections measured by CDF in the mid 1990’s [2] are today approximately described both in the context of fully perturbative calculations (Colour Singlet Model, CSM) [3] and in the context of calculations deriving from the data themselves the contribution of non-perturbative processes in the transition from the initially produced coloured quark pair to the observable bound state (non-relativistic QCD, NRQCD) [4]. Despite subtending completely different elementary production mechanisms, the two theoretical approaches do not differ substantially in the cross section predictions. Given this situation, differential cross sections are clearly insufficient information to ensure further progress. On the other hand, the distinct qualities and topologies of the quarkonium production processes are closely reflected in the shapes of the decay angular distributions and lead to very different expected polarizations of the quarkonia produced at high transverse momentum: transverse polarization in NRQCD [5, 6, 7] and longitudinal in the CSM [3], with respect to the momentum direction of the quarkonium state. Polariza- tion measurements have therefore the decisive role in our fundamental understanding of quarkonium production. However, the present experimental knowledge is incomplete and contradictory. The pattern measured by CDF [8] of a slightly longitudinal polarization of the inclusive prompt J/ψ is incompatible with any of the two theory approaches mentioned above, as well as with a previous measurement published by the same experiment [9]. The situation is further complicated by the intriguing lack of continuity between fixed-target and collider results, which can only be interpreted in the framework of some specific (and speculative) assumptions [10]. The Υ data from the Tevatron indicate that the Υ(1S ) is produced either unpolarized (CDF [11]) or longitudinally polarized (D0 [12]) in the helicity frame, and this discrepancy cannot be reasonably attributed to the different rapidity windows covered by the two experiments. At lower energy and transverse momentum, the E866 experiment [13] has shown yet a different polarization pattern: the Υ(2S ) and Υ(3S ) states have maximal transverse polarization, with no significant dependence on transverse or longitudinal momentum, with respect to the direction of motion of the colliding hadrons (Collins–Soper frame). Surprisingly, the Υ(1S ), whose spin and angular momentum properties are identical to those of the heavier Upsilon states, is, instead, found to be only weakly polarized. This rather 0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.03.065 98 P. Faccioli et al. / Nuclear Physics B (Proc. Suppl.) 214 (2011) 97–102 confusing situation demands a significant improvement in the accuracy and detail of the polarization measurements. Ideally, the measurements should also distinguish between the properties of directly and indirectly produced states, given that, for example, about one third of the promptly produced J/ψ mesons come from ψ′ and χc decays [14]. It is true that measurements of the quarkonium decay angular distributions are challenging, multi-dimensional kinematic problems, which require large event samples and a very high level of accuracy in the subtraction of spurious kinematic correlations induced by the detector acceptance. The complexity of the experimental problems that have to be faced in the polarization measurements is testified, for example, by the mentioned disagreements between the Tevatron results on J/ψ and Υ polarizations. However, as we have discussed and exemplified in Ref. [10], it is also true that most experiments have exploited, and presented in the published reports, only a fraction of the physical information derivable from the data. These incomplete measurements do not allow definite physical conclusions. This happens, for example, when the measurement is performed in only one polarization frame and is limited to the polar projection of the decay angular distribution, imposing genuinely model-dependent interpretations. Moreover, such a fragmentary description of the observed physical process obviously reduces the chances of detecting possible biases induced by not fully controlled systematic effects. 2. Basic concepts Because of angular momentum conservation and basic symmetries of the electromagnetic and strong interactions, a particle produced in a certain superposition of elementary mechanisms may be observed preferentially in a state belonging to a definite subset of the possible eigenstates of the angular momentum component Jz along a characteristic quantization axis. When this happens, the particle is said to be polarized. In the dilepton decay of quarkonium, the geometrical shape of the angular distribution of the two decay products (emitted back-to-back in the quarkonium rest frame) reflects the average polarization of the quarkonium state. A spherically symmetric distribution would mean that the quarkonium would be, on average, unpolarized. Anisotropic distributions signal polarized production. The measurement of the distribution requires the choice of a coordinate system, with respect to which the momentum of one of the two decay products is expressed in spherical coordinates. In inclusive quarkonium measurements, the axes of the coordinate system are fixed with respect to the physical reference provided by the directions of the two colliding beams as seen from the quarkonium rest frame. The polar angle ϑ is determined by the direction of one of the two decay products (e.g. the positive lepton) with respect to the chosen polar axis. The azimuthal angle ϕ is measured with respect to the plane containing the momenta of the colliding beams (“production plane”). The actual definition of the decay reference frame with respect to the beam directions is not unique. Measurements of the quarkonium decay distributions have used three different conventions for the orientation of the polar axis: the direction of the momentum of one of the two colliding beams (Gottfried–Jackson frame [15], GJ), the opposite of the direction of motion of the interaction point (i.e. the flight direction of the quarkonium itself in the center-of-mass of the colliding beams: helicity frame, HX) and the bisector of the angle between one beam and the opposite of the other beam (Collins–Soper frame [16], CS). The motivation of this latter definition is that, in hadronic collisions, it coincides with the direction of the relative motion of the colliding partons, when their transverse momenta are neglected. For our considerations, we will take the HX and CS frames as two extreme (physically relevant) cases, given that the GJ polar axis represents an intermediate situation. We note that these two frames differ by a rotation of 90◦ around the y axis when the quarkonium is produced at high pT and negligible longitudinal momentum (pT ≫ |pL |). All definitions become coincident in the limit of zero quarkonium pT . In this limit, moreover, for symmetry reasons any azimuthal dependence of the decay distribution is physically forbidden. The most general decay angular distribution for inclusively observed quarkonium states can be written as 1 (1 + λϑ cos2 ϑ (3 + λϑ ) +λϕ sin2 ϑ cos 2ϕ + λϑϕ sin 2ϑ cos ϕ) . W(cos ϑ, ϕ) ∝ (1) 3. The importance of the frame choice The coefficients λϑ , λϕ and λϑϕ depend on the polarization frame. To illustrate the importance of the choice of the polarization frame, we consider specific examples assuming, for simplicity, that the observation axis is perpendicular to the natural axis (δ = ±90◦ ). This case is of physical relevance since when the decaying particle is produced with small longitudinal momentum (|pL | ≪ pT , a frequent kinematic configuration in collider experiments) the CS and HX frames are actually perpendicular to one another. When δ = 90◦ , a natural 99 “transverse” polarization (λϑ = +1 and λϕ = λϑϕ = 0), for example, transforms into an observed polarization of opposite sign (but not fully “longitudinal”), λ′ϑ = −1/3, with a significant azimuthal anisotropy, λ′ϕ = 1/3. In terms of angular momentum wave functions, a state which is fully “transverse” with respect to one quantization axis (|J, Jz  = |1, ±1) is a coherent superposition of 50% “transverse” and 50% “longitudinal” components with respect to an axis rotated by 90◦ : 1 1 1 |1, +1 + |1, −1 ∓ √ |1, 0 . 2 2 2 (2) The decay distribution of such a “mixed” state is azimuthally anisotropic. The same polar anisotropy λ′ϑ = −1/3 would be measured in the presence of a mixture of at least two different processes resulting in 50% “transverse” (|J, Jz  = |1, ±1) and 50% “longitudinal” (|J, Jz  = |1, 0) natural polarization along the chosen axis. In this case, however, no azimuthal anisotropy would be observed. As a second example, we note that a fully “longitudinal” natural polarization (λϑ = −1) translates, in a frame rotated by 90◦ with respect to the natural one, into a fully “transverse” polarization (λ′ϑ = +1), accompanied by a maximal azimuthal anisotropy (λ′ϕ = −1). In terms of angular momentum, the measurement in the rotated frame is performed on a coherent admixture of states, |1, ±1 90◦ −−→ |1, 0 90◦ −−−→ 1 1 √ |1, +1 − √ |1, −1 , 2 2 (3) while a natural “transverse” polarization would originate from the statistical superposition of uncorrelated |1, +1 and |1, −1 states. The two physically very different cases of a natural transverse polarization observed in the natural frame and a natural longitudinal polarization observed in a rotated frame are experimentally indistinguishable when the azimuthal anisotropy parameter is integrated out. These examples show that a measurement (or theoretical calculation) consisting only in the determination of the polar parameter λϑ in one frame contains an ambiguity which prevents fundamental (model-independent) interpretations of the results. The polarization is only fully determined when both the polar and the azimuthal components of the decay distribution are known, or when the distribution is analyzed in at least two geometrically complementary frames. Due to their frame-dependence, the parameters λϑ , λϕ and λϑϕ can be affected by a strong explicit kinematic dependence, reflecting the change in direction of the chosen experimental axis (with respect to the “nat- λϑHX P. Faccioli et al. / Nuclear Physics B (Proc. Suppl.) 214 (2011) 97–102 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 p [GeV/c] T Figure 1: Kinematic dependence of the J/ψ polarization seen in the HX frame, for a natural polarization λϑ = +1 in the CS frame. The curves correspond to different rapidity intervals; from the solid line: |y| < 0.6 (CDF), |y| < 0.9 (ALICE), |y| < 1.8 (D0), |y| < 2.5 (ATLAS and CMS), 2 < y < 5 (LHCb). For simplicity the event populations were generated flat in rapidity. ural axis”) as a function of the quarkonium momentum. As an example, we show in Fig. 1 how a natural transverse J/ψ polarization (λϑ = +1) in the CS frame (with λϕ = λϑϕ = 0 and no intrinsic kinematic dependence) translates into different pT -dependent polarizations measured in the HX frame in different rapidity acceptance windows, representative of the acceptance ranges of several Tevatron and LHC experiments. This example shows that an “unlucky” choice of the observation frame may lead to a rather misleading representation of the experimental result. Moreover, the strong kinematic dependence induced by such a choice may mimic and/or mask the fundamental (“intrinsic”) dependencies reflecting the production mechanisms. However, not always an “optimal” quantization axis exists. This is shown in Fig. 2, where we consider, for illustration, that 60% of the J/ψ events have natural polarization λϑ = +1 in the CS frame while the remaining fraction has λϑ = +1 in the HX frame. Although the polarizations of the two event subsamples are intrinsically independent of the production kinematics, in neither frame, CS or HX, will measurements performed in different transverse and longitudinal momenta windows find “simple”, identical results. Corresponding figures for the Υ(1S ) case can be seen in Ref. [17]. 4. A frame-invariant approach It can be shown that the following combination of coefficients is independent on the polarization frame: λ̃ = λϑ + 3λϕ . 1 − λϕ (4) 100 λϑCS P. Faccioli et al. / Nuclear Physics B (Proc. Suppl.) 214 (2011) 97–102 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 p [GeV/c] λϑHX T 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 p [GeV/c] T Figure 2: Kinematic dependence of the polar anisotropy of the J/ψ decay distribution as seen in the CS (top) and HX (bottom) frames, when all the events have full transverse polarization, but 60% in the CS frame and 40% in the HX frame. The curves represent measurements in different acceptance ranges (see Fig. 1). An account of the fundamental meaning of the frameinvariance of this quantity can be found in Ref. [18]. In the special case when the observed distribution is the superposition of n “elementary” distributions of the kind 2 (i) 1 + λ(i) ϑ cos ϑ, with event weights f , with respect to n different polarization axes, λ̃ represents a weighted average of the n polarizations, insensitive to the orientations of the corresponding axes: λ̃ = n  i=1 f (i) 3+ λ(i) ϑ λ(i) ϑ n  i=1 f (i) 3 + λ(i) ϑ . (5) The determination of an invariant quantity is immune to “extrinsic” kinematic dependencies induced by the observation perspective and is, therefore, less acceptancedependent than the standard anisotropy parameters λϑ , λϕ and λϑϕ . Referring to the example shown in Fig. 2, any arbitrary choice of the experimental observation frame will always yield the value λ̃ = +1, independently of kinematics. This particular case, where all contributing processes are transversely polarized, is formally equivalent to the Lam-Tung relation [19], as discussed in Ref. [18]. The existence of frame-invariant parameters also provides a useful instrument for experimental analyses. Checking, for example, that the same value of an invariant quantity is obtained, within systematic uncertainties, in two distinct polarization frames is a nontrivial verification of the absence of unaccounted systematic effects. In fact, detector geometry and/or data selection constraints may strongly polarize the reconstructed dilepton events. Background processes also affect the measured polarization, if not well subtracted. The spurious anisotropies induced by detector effects and background do not obey the frame transformation rules characteristic of a physical J = 1 state. If not well corrected and subtracted, these effects will distort the shape of the measured decay distribution differently in different polarization frames. In particular, they will violate the frame-independent relations between the angular parameters. Any two physical polarization axes (defined in the rest frame of the meson and belonging to the production plane) may be chosen to perform these “sanity tests”. The HX and CS frames are ideal choices at high pT and mid rapidity, where they tend to be orthogonal to each other. At forward rapidity and low pT , we can maximize the significance of the test by using the CS axis and the perpendicular helicity axis [20], which coincides with the helicity axis at zero rapidity and remains orthogonal to the CS axis at nonzero rapidity. Given that λ̃ is “homogeneous” to the anisotropy parameters, the difference λ̃(B) − λ̃(A) between the results obtained in two frames provides a direct evaluation of the level of systematic errors not accounted in the analysis. 5. A few concrete examples We mention here further examples to illustrate concepts described in the previous sections. It is natural to wonder how the J/ψ polarization pattern measured by CDF in the HX frame would look like in the CS frame. Unfortunately, the measurement itself, a slight longitudinal polarization, does not suggest any educated guess on what we could assume for the unmeasured azimuthal anisotropy. For example, as shown in Fig. 3 (left), if the distribution in the HX frame were azimuthally isotropic, the measured polarization would correspond to a practically undetectable polarization in the CS frame (dashed line). However, if we take into account all physically possible values of the azimuthal anisotropy, we can only derive a broad spectrum of possible CS polarizations, approximately included between −0.5 and +1 (shaded band). This 101 λϑ λϕ P. Faccioli et al. / Nuclear Physics B (Proc. Suppl.) 214 (2011) 97–102 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 5 10 15 20 25 30 p [GeV/c] T 5 10 15 20 25 30 p [GeV/c] T Figure 3: Interpretations of the CDF J/ψ polarization measurement. Left: the measurement in the helicity frame (data points) and the range for the corresponding polarization in the CS, allowing for all possible values of the azimuthal anisotropy (shaded band). The dashed line is the CS polarization for λHX ϕ = 0. Center and right: the pT dependencies of λϑ and λϕ in the HX frame, according to a scenario where the J/ψ has always full transverse polarization, either in the CS frame or in the HX frame, with a suitable pT -dependent proportion between the two event samples. example shows how a measurement reporting only the polar anisotropy is amenable to several interpretations in fundamental terms, often corresponding to drastically different physical cases. One possible hypothesis would be that all processes are naturally polarized in the HX frame and that transverse and longitudinal polarizations are superimposed in proportions varying from approximately 2/3 transverse and 1/3 longitudinal at pT = 5 GeV/c (λϑ 0) to around 60% / 40% at pT = 20 GeV/c (λϑ −0.2). In this case, no azimuthal anisotropy should be observed in the HX frame. Alternatively, we can consider a scenario where the observed slightly longitudinal HX polarization is actually the result of a mixture of two processes, both producing J/ψ mesons with fully transverse polarizations, but one in the HX frame and the other in the CS frame. Figure 3 (middle) shows that this scenario is perfectly compatible with the CDF λϑ measurement if the proportion fHX /( fHX + fCS ) between the two sub-processes is assumed to vary linearly between 30% at pT = 5 GeV/c and 15% at pT = 20 GeV/c. The difference with respect to the previous hypothesis is that now we would 0.3, measure a significant azimuthal anisotropy, λϕ as shown in Fig. 3 (right). As an attempt to reconcile low-pT measurements with collider data, Ref. [10] described one further conjecture, in which the polarization arises naturally in the CS frame, and becomes increasingly transverse with increasing total J/ψ momentum. Again, a direct measurement of λϕ (which, in this case, should be zero in the CS frame but positive and increasing in the HX frame) would easily clarify the situation. We finally illustrate the application of the frameindependent formalism as a tool to estimate residual systematic uncertainties in experimental data analyses. Figure 4 shows a putative set of J/ψ polarization measurements performed in the CS and HX frames, versus pT . While the λϑ values seem to change significantly from one frame to the other, the two λϕ patterns are very similar. This observation should alert to a possible experimental artifact in the data analysis. We can evaluate the significance of the apparent contradiction by calculating the frame-invariant λ̃ variable in each of the two frames. For the case illustrated in Fig. 4, averaging the four represented pT bins, we see that λ̃ in the HX frame is larger than in the CS frame by 0.5 (a rather large value, considering that the decay parameters are bound between −1 and +1). In other words, an experiment obtaining such measurements would learn from this simple exercise that its determination of the decay parameters must be biased by systematic errors of roughly this magnitude. Given the puzzles and contradictions existing in the published experimental results, as recalled in Section 1, the use of a frame-invariant approach to perform self-consistency checks, which can expose unaccounted systematic effects due to detector limitations and analysis biases, constitutes a non-trivial complementary aspect of the methodologies for quarkonium polarization measurements. 6. Summary Several puzzles affect the existing measurements of quarkonium polarization. The experimental determination of the J/ψ and Υ polarizations must be improved significantly. Measurements and calculations of vector quarkonium polarization should provide results for the full dilepton decay angular distribution (a three-parameter function) and not only for the polar anisotropy parameter. Only in this way can the measurements and calculations represent unambiguous determinations of the average angular momentum composition of the produced quarkonium state in terms of the three base eigenstates, with Jz = +1, 0, −1. 102 λϑ P. Faccioli et al. / Nuclear Physics B (Proc. Suppl.) 214 (2011) 97–102 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 λϕ pT [GeV/c] 0.2 0.1 0 -0.1 spectively of the specific geometrical framework chosen by the observer. Extrinsic dependencies on kinematics and acceptances are cancelled exactly, enabling more robust comparisons with other experiments and with theory. Stripped-down analyses which only measure the polar anisotropy in a single reference frame, as often done in past experiments, give more information about the frame selected by the analyst (“is the adopted quantization direction an optimal choice?”) than about the physical properties of the produced quarkonium (“along which direction is the spin aligned, on average?”). For example, a natural longitudinal polarization will give any desired λϑ value, from −1 to +1, if observed from a suitably chosen reference frame. Lack of statistics is not a reason to “reduce the number of free parameters” if the resulting measurements become ambiguous. The forthcoming measurements of quarkonium polarization in proton-proton collisions at the LHC have the potential of providing a very important step forward in our understanding of quarkonium production, if the experiments adopt a more robust analysis framework, incorporating the ideas here presented. -0.2 References 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 pT [GeV/c] Figure 4: Example of a “gedankenexperiment” where the J/ψ polarization measurements in the CS and HX frames (empty and filled symbols, respectively) would be inconsistent with each other. Moreover, it is advisable to perform the experimental analyses in at least two different polarization frames. In fact, the self-evidence of certain signature polarization cases (e.g. a full polarization with respect to a specific axis) can be spoiled by an unfortunate choice of the reference frame, which can lead to artificial (“extrinsic”) dependencies of the results on the kinematics and on the experimental acceptance. 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