A review of asymmetry measurements in vector meson photoproduction experiments GlueX-doc-3076 Curtis A. Meyer Carnegie Mellon University August 2016 Abstract In vector meson photoproduction, the spin observables of the vector mesons are represented by their density matrix. The elements, the spin-density matrix elements, or SDMEs are the physical observables. With polarized and unpolarized photon beams, it is possible to measure 11 of the 17 real values that describe the complex SDMEs. Historically, many experiments have reported various asymmetries, where it appears that many of these measurements were trying to measure a combinations of a small number of SDMEs. In this report, we survey the various asymmetries found in the literature and discuss. when possible, how they are related to the SDMEs. Finally, we provide a summary of existing data on ρ0 meson photoproduction using linearly-polarized photons. 1 Introduction In the photoproduction of vector mesons, it is possible for the vector meson to be polarized, even in the case of unpolarized photon beams. The spin-state of the vector meson is described by its density matrix, and spin-density matrix elements (SDMEs) provide information on both the underlying production mechanism, as well the decay distribution, W (cos θ; φ), of vector meson. We measure the SDMEs through the decay distribution. The decay distribution depends not only on the SDMEs, but also on the decay mode of the vector mesons. In particular, purely hadronic, radiative, and lepton decays may have different decay distributions. Throughout this note, we use Wh(cos θ; φ) to describe the purely hadronic decay, while Wr(cos θ; φ) describes the radiative decays. From experiments utilizing linearly-polarized photon beams, a number of different asymmetries have been reported. As we will see, some of these are easily related to some SDMEs, while for others, the relation is not so clear. The nomenclature used for these asymmetries is also confusing. We will survey the history of various asymmetries reported for vector mesons, and to the extent possible, show how they relate to the SDMEs. 1 2 Spin Observables in Vector Meson Photoproduction 2.1 Production Amplitudes We can characterize the photoproduction of vector mesons by an amplitude T , which connects the density matrix for the initial photon beam, ρ(γ), the the density matrix of the produces vector meson, ρ(V ). Following Schilling from reference [1], we can write ρ(V ) = T ρ(γ) T y : This can be written in the center-of-mass frame helicity representation [2] as 1 X y 0 0 0 ρλV λ (V ) = TλV λ ,λγ λN ρλγ λγ T 0 0 0 ; V N λV λN ,λγ λN N 0 0 λN λγ λN λγ 0 where the λx represent the helicity of the incoming (N) and outgoing (N ) nucleon, the photon spin-density matrix elements, (γ, γ0), and the vector meson spin-density matrix elements (V , V 0). The term N is a normalization factor given as 1 X 2 N = T 0 ; λV λN ,λγ λN 2 0 λV λN λγ λN which for center-of-mass momentum of the incoming photon, k, can be related to the unpolarized differential cross section as dσ 1 2π 2 = N : dΩ 2 k Thus, measuring the differential cross section and the density matrix elements of the vector meson photoproduction allows us to study the production processes. Experimentally, we can access elements of the vector meson density matrix through the decay angular distributions of the daughter products of the vector meson as discussed on Section 2.3. 2.2 Spin Density Matrix Elements We can now include the photon polarization into our description of the vector meson density matrices. We start by writing down the photon polarization vectors, Pγ for linearly- and circularly polarized photons. In the case of linearly-polarized photons, the photon polarization is given as Pγ = Pγ (− cos 2Φ; − sin 2Φ; 0) (1) where Pγ is the degree of linear polarization (between 0 and 1), and Φ is the angle between the polarization vector of the photon and the production plane of the vector meson. For circularly- polarized photons, Pγ = Pγ (0; 0; ±1) (2) where Pγ is again the degree of polarization, and the ±1 corresponds to the helicity of the photon, λγ = +1 or λγ = −1. If we now consider the three components of the photon polarization (1 and 2 2 for linearly polarized and 3 for circularly polarized), we can now write the vector meson density matrix as the sum 3 0 X α α ρ(V ) = ρV + Pγ ρ ; (3) α=1 where the ρα form a decomposition of the total density matrix, based on photon polarization. The ρα are related to the amplitudes, T , as 0 1 X ∗ ρ 0 (V ) = Tλ λ0 ,λ λ T 0 0 0 (4) λV λV V N γ N λV λN ,λγ λN 2N 0 λN λγ λN 1 1 X ∗ ρ 0 (V ) = Tλ λ0 ;−λ λ T 0 0 0 (5) λV λV V N γ N λV λN ,λγ λN 2N 0 λN λγ λN 2 i X ∗ ρ 0 (V ) = λγ Tλ λ0 ;−λ λ T 0 0 0 (6) λV λV V N γ N λV λN ,λγ λN 2N 0 λN λγ λN 3 i X ∗ ρ 0 (V ) = λγ Tλ λ0 ,λ λ T 0 0 0 (7) λV λV V N γ N λV λN ,λγ λN 2N 0 λN λγ λN Utilizing symmetry properties, including the fact that the density matrices are Hermitian, it is possible to write the ρα in terms of their spin-density matrix elements (SDMEs) as described by Schilling [1]. The spin-density matrices can be expressed in terms of 17 real numbers, of which 11 are measurable. These 17 real numbers are given below. 0 1 0 0 0 0 1 2 1 − ρ00 Reρ10 + iImρ10 Reρ1−1 0 B 0 0 0 C ρ = @ ρ00 − Reρ10 − iImρ10 A 1 0 2 1 − ρ00 0 1 1 1 1 1 ρ11 Reρ10 + iImρ10 Reρ1−1 1 B 1 1 1 C ρ = @ ρ00 − Reρ10 − iImρ10 A 1 ρ11 0 2 2 2 2 1 ρ11 Reρ10 + iImρ10 iImρ1−1 2 B 2 2 C ρ = @ 0 Reρ10 − iImρ10 A 2 −ρ11 0 3 3 3 3 1 ρ11 Reρ10 + iImρ10 iImρ1−1 3 B 3 3 C ρ = @ 0 Reρ10 − iImρ10 A 3 −ρ11 0 0 1 0 Because the trace of ρ is 1, we can define ρ11 = 2 (1 − ρ00). As we will see in Section 2.3, with unpolarized photon beams, we can measure three of the four real values describing the the ρ0 SDMEs. With circularly-polarized photons, we can measure the same three ρ0 SDMEs plus two of the four real values from the ρ3 SDMEs.With linearly-polarized photons, the same ρ0 elements are accessible. In addition, we can measure four of the five real elements of the ρ1 SDMEs, and two of the four form ρ2. 3 2.3 Decay Distributions of Vector Mesons In photoproduction experiments, the decay angular distribution for vector mesons can be expressed α in terms of the SDMEs, ρij. We start by writing this as a sum 3 0 X α α W (cos θ; φ) = W (cos θ; φ) + Pγ W (cos θ; φ) ; (8) α=1 α where Pγ is the polarization vector of the photon given by equation 1 and 2. For hadronic decays 0 + − + − 0 + − α of the vector mesons (ρ ! π π , ! ! π π π , φ ! K K and φ ! ρπ) the Wh are given as 3 1 1 W 0(cos θ; φ, ρ0) = 1 − ρ0 + 3ρ0 − 1 cos2 θ h 4π 2 00 2 00 p 0 0 2 i − 2 Reρ10 sin 2θ cos φ − ρ1−1 sin θ cos 2φ (9) 3 h p i W 1(cos θ; φ, ρ1) = ρ1 sin2 θ + ρ1 cos2 θ − 2 Reρ1 sin 2θ cos φ − ρ1 sin2 θ cos 2φ (10) h 4π 11 00 10 1−1 3 hp i W 2(cos θ; φ, ρ2) = 2 Imρ2 sin 2θ sin φ + Imρ2 sin2 θ sin 2φ (11) h 4π 10 1−1 3 hp i W 3(cos θ; φ, ρ3) = 2 Imρ3 sin 2θ sin φ + Imρ3 sin2 θ sin 2φ (12) h 4π 10 1−1 The angles θ and φ denote the direction of one of the daughter particles of the vector meson decay, nominally chosen in the helicity system, as described below. However the results are true for any system that can be reached by a rotation. For three-body decays, such as the ! ! π+π−π0, the angles describe the orientation of the normal to the decay plane of the vector meson. We note that the trace of ρ is invariant under these rotations. The above formula account for the hadronic decays of vector mesons. There are also non- negligible radiative decays of vector mesons. The largest is ! ! π0γ, but others include ρ ! πγ, ∗ α φ ! ηγ and K ! Kγ. We include here Wr for radiative decays, as taken from references [3, 4]. 2 3 There is an overall sign difference between the two references for Wr and Wr . We present the 2 3 sign from reference [4]. There is also a discrepancy in the φ dependence of Wr and Wr , where reference [4] have sin 2θ cos φ and reference [3] have sin 2θ sin φ. We believe the latter is correct. 3 h W 0(cos θ; φ, ρ0) = 1 − ρ0 sin2 θ − ρ0 cos2 θ r 8π 11 00 p 0 0 2 i + 2 Reρ10 sin 2θ cos φ + ρ1−1 sin θ cos 2φ (13) 3 h p W 1(cos θ; φ, ρ1) = 2ρ1 + ρ1 − ρ1 sin2 θ + 2 Reρ1 sin 2θ cos φ r 8π 11 00 11 10 1 2 i +ρ1−1 sin θ cos 2φ (14) 3 hp i W 2(cos θ; φ, ρ2) = − 2 Imρ2 sin 2θ sin φ + Imρ2 sin2 θ sin 2φ (15) r 8π 10 1−1 3 hp i W 3(cos θ; φ, ρ3) = − 2 Imρ3 sin 2θ sin φ + Imρ3 sin2 θ sin 2φ (16) r 8π 10 1−1 2.4 Coordinate Systems We now introduce the three coordinate systems in which SDMEs are reported in the literature.