Determining dominant partial waves in photoproduction via moment analysis

Abstract. Important insights into the excitation spectra of baryons are provided by measurements of polarization observables in reactions that involve particles with spin. The photoproduction of a single pseudoscalar meson constitutes an example-reaction that has been under intense investigation recently. We present the basic method of moment-analysis for pseudoscalar meson photoproduction, in which just the angular distributions are analyzed. Using this method, the total angular momentum quantum number of the dominant partial waves contributing in the data can be extracted quickly. Furthermore, the Legendre-coefficients extracted from the angular distributions show interesting composition-patterns in terms of multipoles and allow for instructive comparisons to models. In this contribution, recent results for moment analyses of polarization data for the photoproduction of pions and eta-mesons are shown.

In baryon spectroscopy, the extraction of resonance parameters from scattering data represents the most important central problem. A standard-approach to the solution of this problem, which has taken center stage over the last 50 years, is given by fits of so-called energy-dependent (ED) partial-wave analysis (PWA-) models. Well-known examples for such approaches are the Bonn-Gatchina (BnGa-) model [1], the Jülich-Bonn (JüBo-) model [2,3], the MAID-analysis [4] and the SAID-PWA [5]. A comprehensive overview of these examples is given in reference [6]. In each case, a reaction-theory is constructed, which satisfies the well-established theoretical S -Matrix principles [7] (analyticity, unitarity and crossing) with varying degrees of rigor. Within the context of such a model, resonances are extracted as poles on the second Riemann-sheet of the scattering amplitude. Another approach to analyze scattering observables is represented by single-energy (SE) fits, or truncated partialwave analyses (TPWAs). The simplest possible example is given by 2 → 2-scattering of spinless particles. The only possible observable, i.e. the differential cross section σ 0 , is defined in terms of the amplitude as σ 0 = |A(W, θ)| 2 . The infinite partial-wave series for the amplitude reads (2 + 1)A P (cos θ). (1) In case one truncates this series at some maximal angular momentum quantum number max , one obtains the following expansion for the cross section (see reference [8]) σ 0 (W, θ) = q k 2 max n=0 a σ 0 n (W)P n (cos θ), (2) * e-mail: wunderlich@hiskp.uni-bonn.de Here, for the scalar example, the coupling-coefficients C n k in the bilinear equation (3) are well-known and given by [8,9] C n k = , 0; k, 0|n, 0 2 (2 + 1) (2k + 1) (2n + 1) .
Here, the , m; , m |n, M are the usual Clebsch-Gordan coefficients.
The central problem in a SE fit is then to solve for the realand imaginary parts of generally phase-constrained 1 partial waves. Thus, one first extracts the Legendre-moments using equation (2). Then, one solves the bilinear equations (3) and it is this second step in which multiple discrete ambiguities [9,10] can, and probably will, occur. Such ambiguities require additional theoretical input on some subset of the partial waves in order to be resolved. However, one could also remain, in a first instance, with the moment-analysis using (2) and try to use it in order to learn as much about the present dataset as possible. This latter idea can be generalized to more complicated reactions involving particles with spin without a lot of additional effort. The photoproduction of a single pseudoscalar meson is one example for such a reaction. We write the process in the most general form as γN −→ ϕB, where ϕ is a pseudoscalar meson and B a recoil-baryon. The following formalism can thus be applied to the most commonly met example of pion-photoproduction ϕB = πN, but also other reactions such as eta-photoproduction ϕB = ηN or the production of kaons ϕB = KΛ, . . . may be studied. The photoproduction reaction is described using 4 1 Some kind of constraint has to be introduced on the overall phase of the partial waves due to the bilinear nature of the equations (3). Table 1: We list here the parameters needed to evaluate the angular parametrizations for the 16 polarization observables of pseudoscalar meson photoproduction given in equations (5) and (6). Such a Table was first published by Tiator [15] (see also references [9,14]).
complex spin-amplitudes, for instance CGLN-amplitudes {F i (W, θ), i = 1, . . . , 4} [11], accompanied by 16 polarization observables [12,13]. The observables are defined in terms of the amplitudes as bilinear hermitean forms and they divide into the subsets of group S observables {σ 0 , Σ, T, P} and furthermore three classes of beam-target (BT ), beam-recoil (BR) and target-recoil (T R) observables with four quantities each. The CGLN-amplitudes can be expanded into a well-known partial-wave series using electric and magnetic multipoles E ± , M ± [11,12]. In case one truncates the multipole-series at some max , the observables acquire the following standard-form [9,15], which is tantamount to the equations (2,3) in the scalar case:Ω where we have adopted a notation by Chiang and Tabakin [13] to denote the observablesΩ α and the index α runs as α = 1, . . . , 16. The P β α n (cos θ) are associated Legendre polynomials. The multipoles which are present in a certain truncation-order define the vector A set of parameters which defines the photoproduction moment-expansion for all 16 observables is given in Table 1 (cf. reference [14]). Tiator [15] first published similarly formalized expansions for an expansion into powers of cos(θ). The matrices defining the coefficients (6) as bilinear forms are hermitean and have dimensions (4 max ) × (4 max ), for max ≥ 1. We have to report that during the course of the thesis [9], no closed expression for these matrices has been found, which would be analogous to the result (4) from the scalar reactions. However, they can be calculated numerically for each relevant truncation order max . The results for group S and BT observables in the order max = 5 have been collected in the appendices of [9]. As an example, the matrix CĚ 0 is shown in Table 2 for the truncation order max = 2.
It is a fact that once the matrices CΩ α k are evaluated, one observes that in all cases the matrixes decompose into blocks of interference-terms among multipoles of certain definite orbital angular momentum quantum number . Therefore, we introduce a short-notation [14] for these interferenceblocks, using the spectroscopic notation of S , P, D, . . . for the angular momenta = 0, 1, 2, . . .. For example, a contribution coming from an interference-block between S -and P-wave multipoles would be just written as S , P . Each Legendre-moment generally contains multiple such blocks and therefore, several of the short-forms for interference-blocks are added together in order to describe the partial wave decomposition of a particular moment (see Table 2). The question whether or not specific minimal subsets of the bilinear forms (6) allow for an unambiguous amplitude extraction in so-called complete experiments has been addressed in the classic work by Omelaenko [16] and in the more recent references [9,17]. The formalism outline above allows for moment-analyses in pseudoscalar meson photoproduction. The following two main aspects are important: (I) max -analysis: The parametrization (5) of the angular distribution is fitted for different max . The χ 2 /ndf is compared for different fits, ascending from the lowest possible order. In case the goodness of fit is unsatisfactory, one has to increase max . Once a good fit is obtained, the resulting max gives, in a lot of cases, already quite a good estimate for the maximal angular momentum detectable in the data. Plots of χ 2 /ndf vs. energy show 'bumps' whenever new important contributions from higher angular momenta enter the data (cf. Figures 3,4, top).
(II) Model-comparisons: The fitted Legendre-moments a max Ωα k can be compared to the right hand side of equation (6), i.e. to the definitions of the moments as bilinear forms in terms of multipoles. The latter can be evaluated using multipoles M stemming from an ED model. These comparisons can be performed switching on/off  The matrix CĚ 0 which defines the coefficient (a 2 )Ě 0 for an expansion ofĚ up to max = 2 is shown. Every matrix element defines a particular multipole-interference term. As a generic feature, the matrices CΩ α n decompose into blocks of interference terms between multipoles with definite orbital angular momentum quantum number (cf. equations (6) and (7)). Therefore, a short-notation for interference-terms is introduced, using the spectroscopic notation of S , P, D, . . . for = 0, 1, 2, . . .. certain model partial waves. In this way, sometimes one obtains valuable information on which partial wave interferences are important. Furthermore, one can try to interpret the results of such comparisons in view of important physical effects visible in the data.
We commence with the discussion of results obtained from new data for π 0 -and η-photoproduction [18][19][20]. In case of π 0 -photoproduction, we consider a dataset for the observableĚ recently measured by the A2collaboration at MAMI, and which has been first published in the PhD-thesis [18]. The formula (5) reads, when adapted to the observableĚ, as follows (see also Table 1) This expression has been fitted to the data for different max and an order of max = 4 was sufficient to yield a satisfactory fit over the whole measured energy-region. The plot of χ 2 /ndf vs. energy, as well as several exemplary angular distributions, are shown in Figure 3. Furthermore, we show plots of the two selected Legendremoments aĚ 0 and aĚ 2 in Figure 1 (see also the thesis [18]). In these plots, the energy-region has been zoomed in, in order to facilitate a more detailed view of the fit-results for the moments. Furthermore, a model-comparison is shown for the Bonn-Gatchina solution BnGa 2017_02, which has been used to re-evaluate both of the moments using different truncation orders. We see that for aĚ 0 below the pη-threshold, i.e. for roughly W < 1500 MeV, BnGa S -and P-waves are largely sufficient to describe the fit-results for the moment. Very close to the pη-threshold, a small correction due to D-waves is needed. Then, for all energies above the pη-threshold, the BnGa F-waves are required to correct the continuous curve such that it coincides with the fitted values for the Legendre-moment. One feature of the shown result is striking: exactly at the pη-threshold, the fitted values show a pronounces, sudden change in direction, or 'cusp'. Furthermore, analytic S -Matrix theory requires amplitudes to have branch-point singularities precisely at the energies corresponding to the opening of thresholds [7,21]. Such singularities lead to cusp-like behaviours and they are basically a direct consequence of the combination of the principles of analyticity and unitarity. We conclude that this particular polarization measurement is precise enough to show such sophisticated theoretical effects directly in the data, or more precisely in the extracted Legendre-moments. Furthermore, we see in the comparison-plot that for aĚ 0 , the cusp-effect enters due to a D, D interference-term. The same effect can be seen in the plot for aĚ 2 on the right of Figure 1. The extracted values for the Legendremoment show a pronounced cusp right at the pη-threshold. Furthermore, the cusp enters into the BnGa model-curves once the D-waves are included. Here, this cusp-effect is due to the S , D -and D, D interference-blocks. However, contrary to the results for the moment aĚ 0 , (small) F-wave corrections are important already below the pη-threshold. For the process of η-photoproduction, we consider a dataset for the beam-asymmetryΣ, which has been recently measured by the CBELSA/TAPScollaboration [18][19][20]. When using the general expression (5) for the observableΣ, one obtains the following angular parametrizatioň The resulting plots of χ 2 /ndf vs. energy, as well as some fitted angular distributions, can be seen in Figure 4. Over the whole fitted energy-region, again a truncation-order of max = 4 is found to yield a satisfactory fit quality.
As an example for an extracted Legendre-moment, we consider here the quantity aΣ 4 . A plot of the extracted values, as well as a comparison to the Bonn-Gatchina solution BnGa 2014_02, are shown in Figure 2a. The fitted Legendre-moment, which is consistent with zero in the lower energy region, shows a pronounced rise, which looks like a cusp-like structure, right at the pη -threshold. Furthermore, the solution BnGa 2014_02 cannot describe this structure. As can be seen in some exemplary angular distributions shown in Figure 2a, the rise of the moment aΣ 4 to values significantly larger than zero is in correspondence with the appearance of a backward-peak in the  Figure 1: Shown are the two Legendre-moments aĚ 0 (left) and aĚ 2 (right), extracted from a recent dataset for the observablě E measured in the reaction γp → π 0 p by the A2-collaboration at MAMI [18]. The coefficients have been extracted using equation (9) in the truncation-order max = 4. The fitted values for the moments are shown as blue dots. Continuous curves show the respective Legendre-moment, evaluated using multipoles from the recent Bonn-Gatchina solution BnGa 2017_02, up to the finite truncation orders: max = 1 (green curve), max = 2 (blue), max = 3 (red) and max = 4 (black). The composition of the respective Legendre moment as a bilinear form in the multipoles is written in the short-notation on the right of the respective plot. Threshold-energies of important photoproduction final-states are illustrated by dash-dotted vertical lines. (color online) measured angular distribution forΣ. This backward-peak needs the modulation aΣ 4 P 2 4 (cos θ) in order to be described. This will be discussed in more detail in upcoming publications (for instance [19]).
For a comparison of the extracted values for the moment aΣ 4 to a more recent BnGa-solution, see Figure 2b. There, it is seen that the more recent solution BnGa 2017_02 can describe the cusp a lot better and that furthermore, the cusp-structure enters the model-curve once the G-waves are included, via an S , G interference-block. For a more involved discussion of the physical meaning of this cusp-effect in η-photoproduction data, see upcoming publications [19]. We conclude that in a new era of polarizationmeasurements, the precision of the data has become good enough to be directly sensitive to singularities in the amplitudes as dictated by S -matrix theory. This puts new increased demands to the construction and fit of ED models. Furthermore, moment-analysis is the ideal method to make sensitivities to such singularities visible.
p r e l i m i n a r y (a) The Legendre-moments aΣ 4 , extracted from a recent dataset for the observableΣ measured in the reaction γp → ηp by the CBELSA/TAPS-collaboration [18][19][20] is shown at the top. The coefficient has been extracted using equation (10) [18][19][20]. p r e l i m i n a r y (a) Figure 3: The doublepolarization observableĚ data from the A2-collaboration at MAMI [18], with only statistical error, was fitted using associated Legendre polynomials according to the Legendre moment expansions (5,9) and truncating the expansion at max = 1, . . . , 4. p r e l i m i n a r y (a) Figure 4: The beam-asymmetry Σ data from the CBELSA/TAPScollaboration [18,19], with only statistical error, was fitted using the expansion into Legendre moments given in equations (5,10) and truncating the expansion at max = 1, . . .