Incompleteness of complete kaon photoproduction

A possible roadmap for reaching a status of complete information in $\gamma p \rightarrow K^ {+} \Lambda$ is outlined.


Introduction
Thanks to recent technological advances in producing high-quality polarized monochromatic photon beams, and in developing polarized nucleon targets, it becomes possible to measure a sufficiently large amount of single-and doublepolarization observables in pion and kaon photoproduction from the nucleon. As a result, a status of complete quantum mechanical information of meson photoproduction comes within reach. Measurements are complete whenever they enable one to determine unambiguously all amplitudes of the underlying reaction process at some specific kinematics.
October 24, 2013 0:21 WSPC/INSTRUCTION FILE RyckebuschNstar2013 2 Authors' Names meson photoproduction from the proton. The transversity amplitudes b i (i = 1, 2, 3, 4) express the transition matrix elements in terms of the p and Λ spinors (with quantization axis perpendicular to the reaction plane) and of linear photon polarizations (J x , J y ). We propose 1 to use normalized transversity amplitudes (NTA) a i ≡ bi √ i |bi| 2 to perform an amplitude analysis of the single-and doublepolarization observables. The NTA provide complete information after determining the differential cross section. The corresponding polarization observables can be expressed in terms of linear and nonlinear equations of bilinear products of the a i . For a given kinematical setting determined by the meson angle θ cm and the invariant mass W , the a i = r i e iαi are fully determined by six real numbers conveniently expressed as three real moduli r i and three real relative phases α i − α 4 . All observables are invariant under a transformation of the type α i → α i + β, with β an arbitrarily chosen overall phase. In Fig. 1 we show predictions for the NTA at W = 1900 MeV and various θ cm . The adopted model for γp → K + Λ is the Regge-plus-Resonance (RPR) approach in its most recent version RPR-2011 2,3 . The model has a Reggeized t-channel background and the s-channel resonances S 11 (1650), F 15 (1680), P 13 (1720), D 13 (1900), P 13 (1900), P 11 (1900), and F 15 (2000). The RPR approach provides a low-parameter framework with predictive power for K + and K 0 photoproduction on the proton and the neutron 4 .

2.
Extracting the moduli and phases from γp → K + Λ data An obvious advantage of using the transversity amplitudes is that linear equations connect the moduli r i of the NTA to the single-polarization observables {Σ, T, P } Accordingly, a measurement of (Σ, T, P ) at given (W, cos θ cm ) allows one to infer the moduli r i (W, cos θ cm ) of the NTA. The GRAAL collaboration provides p(γ, K + )Λ data for {Σ, T, P } at 66 (W, cos θ cm ) combinations in the ranges 1.65 W 1.91 GeV (∆W ≈ 50 MeV) and −0.81 cos θ cm 0.86 (∆ cos θ cm ≈ 0.3). Figure 2 shows the extracted r i at three θ cm intervals along with the RPR-2011 predictions. For a few kinematic points the r i could not be retrieved from the data. This occurs whenever one or more arguments of the square roots in Eq. (1) become negative due to finite experimental error bars. The RPR-2011 model offers a fair description of the W dependence of the extracted r i except for the most forward angles at W ≈ 1.85 GeV. Furthermore, the data confirm the predicted dominance of the r 2 .
Inferring the NTA phases α i from data requires measured double asymmetries. Complete sets of the first kind, which involve seven observables (e.g. Instructions for Typing Manuscripts (Paper's Title) 3 phases 6,1            r 1 r 4 sin δ 1 + r 2 r 3 sin ∆ 23 = − Cx 2 , r 1 r 4 cos δ 1 + r 2 r 3 cos ∆ 23 = + Ox 2 , r 1 r 3 cos ∆ 13 − r 2 r 4 cos δ 2 = + E 2 , r 1 r 3 sin ∆ 13 + r 2 r 4 sin δ 2 = − F 2 , where δ i ≡ α i − α 4 and ∆ ij = δ i − δ j . Solutions to the above set of nonlinear equations gives the phases (δ 1 , δ 2 , ∆ 13 , ∆ 23 ) for given moduli (r 1 , r 2 , r 3 , r 4 ). We stress that single-polarization observables are part of any complete set as they provide the information about the moduli. Double polarization observables are required to get access to the phases. In all practical situations one has δ 1 + ∆ 23 − δ 2 − ∆ 13 ≈ 0. Finite error bars introduce a bias for the choices made with regard to the reference phase (here, α 4 ) for the above equations. A consistent set of estimators δ αj i for the independent phases (insensitive to choices made with regard to the reference phase) has been proposed in Ref. 1 .
To date, the published double polarization observables for γp → K + Λ do not allow one to extract the phases of the NTA. We have conducted studies with pseudodata generated by the RPR-2011 model for γp → K + Λ. We have considered ensembles of 200 pseudo-data sets each containing samples of 50 events for the asym- (ii) imaginary solutions for the phases; (iii) incorrect solutions which stem from the fact that δ 1 + ∆ 23 − δ 2 − ∆ 13 = 0 cannot be exactly obeyed for data with finite errors. We find that the amount of incorrect and imaginary solutions is much larger for the phases than for the moduli. The frequency of finding imaginary solutions can be dramatically reduced by improving on the experimental resolution σ exp .

Conclusions
We have sketched a possible roadmap for reaching a status of complete information in pseudoscalar-meson photoproduction. We suggest that the use of transversity amplitudes is tailored to the situation that experimental information about (Σ, P, T ) is more abundant (and most often more precise) than for the double polarization observables. Linear equations connect {Σ, P, T } to the moduli r i of the NTA. An analysis of {Σ, T, P } data for γp → K + Λ from GRAAL allowed us to extract the r i in the majority of considered (W, cos θ cm ) combinations. Extracting the NTA independent phases is far more challenging as they are connected to the double asymmetries by means of nonlinear equations. It has been suggested 7 that overcomplete sets which involve more than seven polarization observables may provide a solution to tackle the problem of extracting the relative phases of the amplitudes from the data.