Amplitude- and truncated partial-wave analyses combined: A novel, almost theory-independent single-channel method for extracting photoproduction multipoles directly from measured data
AAmplitude- and truncated partial-wave analyses combined: A novel, almosttheory-independent single-channel method for extracting photoproduction multipolesdirectly from measured data ∗ A. ˇSvarc , , Y. Wunderlich , and L. Tiator Rudjer Boˇskovi´c Institute, Bijeniˇcka cesta 54, P.O. Box 180, 10002 Zagreb, Croatia Tesla Biotech, Mandlova 7,10000 Zagreb, Croatia Helmholtz-Institut f¨ur Strahlen- und Kernphysik der Universit¨at Bonn, 53115 Bonn, Germany and Institut f¨ur Kernphysik, Universit¨at Mainz, D-55099 Mainz, Germany (Dated: August 5, 2020)Amplitude- and truncated partial-wave analyses are combined into a single procedure and a novel,almost theory-independent single-channel method for extracting multipoles directly from measureddata is developed. In practice, we have created a two-step procedure which is fitted to the samedata base: in the first step we perform an energy independent amplitude analysis where continuityis achieved by constraining the amplitude phase, and the result of this first step is then taken asa constraint for the second step where a constrained, energy independent, truncated partial-waveanalysis is done. The method is tested on the world collection of data for η photoproduction, andthe obtained fit-results are very good. The sensitivity to different possible choices of amplitudephase is investigated and it is demonstrated that the present data base is insensitive to notablephase changes, due to an incomplete database. New measurements are recommended to remedy theproblem. PACS numbers: PACS numbers: 13.60.Le, 14.20.Gk, 11.80.Et
I. INTRODUCTION
Finding a connection between QCD and experiment is a conditio sine qua non for establishing whether a particulardescription of the effects of non-perturbative QCD is close to being correct or not, and a lot of effort has in lastdecades been put into doing it via comparing resonance spectra. While on the QCD side, a resonance spectrum isstandardly predicted by lattice QCD and various QCD-inspired phenomenological models, on the experimental sideit is standardly extracted by identifying poles of the scattering matrix [1]. However, as resonances/poles must havedefinite quantum numbers, finding pole structure of experimental data must necessarily go through a partial wavedecomposition where the angular dependence at a fixed energy is represented by a decomposition over the complete setof Legendre polynomials which then define proper eigenvalues of the angular momentum operator. Combining goodquantum numbers of angular momenta with the known spins of the reacting particles, resonance quantum numbers arefully defined. However, one should be aware that observables which are measured are most generally given in termsof amplitudes, and not partial waves, and to obtain partial waves one has to invest some extra work. Unfortunately,in that process the single-channel partial wave decomposition turned out to be rather non-unique. For decades, it hasbeen known that in the single-channel case, even a complete set of observables is invariant with respect to the phaserotation of all reaction amplitudes by the same arbitrary real function of energy and angle (continuum ambiguity)[2–4], and this free rotation either causes a rearrangement of strength between real and imaginary parts of amplitudesand partial waves for energy dependent phase-rotation functions, or it even mixes partial waves for angular dependentphase-rotation functions. These effects lead to unacceptable discontinuities in amplitudes and partial waves, and havebeen extensively discussed in refs [5, 6]. The main conclusion is that at least one of the reaction amplitude phasesmust be forced to be continuous in energy and angle in order to restore a continuous, unique solution. The openquestion is how to accomplish this task with minimal model dependence. With all these issues at hand, finding anoptimal method for extracting partial waves with minimal reference to a particular theoretical model for fixing thephase turns out to be of utmost importance.A direct consequence of the continuum ambiguity is that an unconstrained single-channel, single energy partialwave analysis (SE PWA), in the sense that there is absolutely no correlation among SE PWA solutions at neighboring ∗ Corresponding author: [email protected] a r X i v : . [ nu c l - t h ] A ug energies, must be discontinuous. This is the consequence of the fact that if a phase is unspecified at an isolated energy,then the free fit chooses a random phase value as there is an infinite number of phases which give an absolutely identicalset of observables. So, the variation of the phase between neighboring energies may be random and discontinuous. Ifthe variation of the phase between neighboring energies is discontinuous, the redistribution of strength between realand imaginary part at each energy will be random, so the partial wave must be discontinuous too. The standardway of achieving the continuity was to implement it on the level of partial waves, so one resolved to constrain partialwaves directly to values originating from some particular theoretical model. In this case , the model dependence isstrong. First ideas to use more general principles of analyticity for imposing the continuity instead of referring to aparticular model were introduced in the mid 1980s by the Karlsruhe-Helsinki group for Pion-Nucleon ( πN ) elasticscattering in the form of fixed- t analyticity [7]. In this case instead of demanding the proximity of fitted partial wavesto some model values, the continuity is imposed on the level of reaction amplitudes by requiring fixed- t analyticity. Inother words, the group was fitting the world collection of data requiring that the reaction amplitude for a fixed- t havea certain analytic, hence continuous form. In this way, any specific dependence on a particular model was reducedto the level of discussing what is the correct analytic structure of the reaction amplitudes, and this form is fairlywell defined by the branch-points of the analyzed reaction. Unfortunately, imposing analyticity in the Mandelstam- t variable opened quite some additional issues, and this is extensively discussed in [8].The aim of this work is to show that invoking analyticity in the Mandelstam- t variable is not really needed and raisesunwanted complications; the required continuity can be obtained by amplitude analysis (AA) in the Mandelstam- s variable, and the continuity is imposed by requiring the proper analyticity of amplitude phases only. So, everythingis done in the Mandelstam-s variable, and by this the analysis becomes much simpler. To our knowledge, this isthe first time that amplitude analysis and truncated partial wave analysis are joined into one compact, self-sustainedanalysis-scheme.Let us stress that the continuum ambiguity problem, and all problems of the continuity of the phase related to itare typical and inherent for single-channel analyses where unitarity is violated in the sense that there exists the lossof probability-flux into other channels. Unitarity equations become inequalities, and the free phase arises. However,in full coupled-channel formalisms where unitarity is at the end restored by summing up the flux in all channels, theinvariance to phase rotations disappears as the phase is fixed, and uniqueness is automatically restored. However,this work analyzes only the single-channel case.Let us also warn the reader about another aspect of PWA: the number of partial waves involved. As the partialwave decomposition is an expansion over the complete set of Legendre polynomials, it is inherently infinite, but inpractice it must be finite, so all we can talk about is a truncated PWA (TPWA). A lot of effort has recently beenput into analyzing the features of a TPWA for pseudoscalar meson photoproduction [9, 10]. A theoretical model waschosen, all observables were generated from this model with a fixed angular-momentum cutoff (cid:96) max , and a completeset of observables generated this way was taken as intput to TPWA. In that way the outcome of TPWA is known inadvance, and a lot of conclusion on the symmetries and inter-relation among the thus formed pseudo-observables havebeen drawn. Unfortunately, as this is an idealized case, most of these conclusions are not applicable for our practicalpurposes. Those pseudo-observables generated from a model by default possess explicit properties like unrealisticallyhigh precision, continuity in energy and angle, various inter-dependence among observables due to finite truncationorder, etc., which our real data do not necessarily have. This is in particular pronounced if the truncation order istoo low. Therefore, we have to be very careful in our analysis of real data to take the truncation order high enoughto avoid introducing additional, nonexisting symmetries into the analysis which may raise quite some problems. If weare careful enough, our obtained partial waves are not exact, but indeed are a good representation of the amplitudeanalysis representing the process.The paper is organized as follows: the main text goes directly into media res by proposing the new fit-methodand showing applications to polarization data in η photoproduction in a detailed way. Discussions on the necessarybackground knowledge concerning the photoproduction formalism, as well as a more elaborate mathematical discussionon the motivation of the proposed novel analysis-scheme, have been relegated to the appendices. In this way, we canpresent our main results quickly and concisely, while the interested reader can read the more elaborate mathematicaldiscussion in parallel. II. THE NOVEL APPROACH TO SINGLE-CHANNEL PWA AND APPLICATION TO η PHOTOPRODUCTION
The main intention of our proposed scheme is to obtain a continuous set of partial waves, directly from experimentaldata, with minimal involvement of theoretical models.The 0-th step of our procedure is to perform an unconstrained single energy partial wave analysis (SE PWA); namelyto fit the available set of measured data with a chosen number of partial waves at each available energy independently(at each energy the fit is independent of the neighboring energy). We know that such a process, due to the continuumambiguities on the level of reaction amplitudes, must produce a set of partial waves that are discontinuous in energy,even for a complete set of pseudo-data with very high precision. However, this procedure gives the best possible fitto the data with the chosen number of partial waves, and directly measures the consistency of the data. So, thisgives us a benchmark-set of values for the goodness-of-fit parameter chi squared - which we call χ ( W ) (’unc.’ for’unconstrained’) - and any method of enforcing continuity of partial waves must be as close as possible to this set, butcan never be better. Achieving the continuity of partial waves is, however, a demanding task. For the case of veryprecise pseudo-data, it has been shown that the task is still relatively simple: it is enough to impose the continuityonly in one amplitude phase to achieve the goal that the SE PWA becomes continuous [5]. However, for the real datawe unfortunately have a serious problem. The existing set of observables is incomplete, and errors are realistic, so thesituation changes drastically. Simple methods of imposing continuity on one phase only do not work anymore.The standard way to impose continuity in discontinuous SE PWA is the penalty function methodology. The ideais to require that the solution one obtains by fitting the data at one isolated energy SIMULTANEOUSLY reproducesthe data AND is also close to some continuous function. So, out of an in principle infinite number of solutions atisolated energy for the unconstrained SE PWA one picks only those which are also close to a predetermined continuouspenalization factor. Of course, the solution will depend on the size of the penalization coefficient. The smaller thecoefficient, the more the solution will tend to reproduce the fitted data; it will be more discontinuous and it will lesssatisfy the penalization function. On the other hand, if one increases the penalization coefficient, the more the fitwill reproduce the penalization function and be continuous, but it will less describe the fitted data. In the final limitof extremely low penalization coefficient, the fit will ideally describe the data and be discontinuous, and in the finallimit of extremely big penalization coefficient the fit will be continuous, perfectly describe the penalization function,and definitely disagree with the fitted data. The optimum lies somewhere in-between.The first, most standard approach found in the literature was to penalize partial waves. We require that the fittedpartial waves reproduce the observable O and are at the same time close to some partial waves taken from a theoreticalmodel. So, for one observable we may at one fixed energy W write (for a literature-example of a penalization-schemewhich acts on the level of partial waves, though not quite in the same way as in the definition given below, see forinstance ref. [11]): χ ( W ) = N data (cid:88) i =1 w i (cid:2) O exp. i ( W, Θ i ) − O th. i ( M fit ( W, Θ i )) (cid:3) + λ pen. N data (cid:88) i =1 (cid:12)(cid:12) M fit ( W, Θ i ) − M th. ( W, Θ i ) (cid:12)(cid:12) (1)where M def. = {M , M , M , ..., M j } is the generic notation for the set of all multipoles, w i is the statistical weight and j is the number of partial waves(multipoles). Here, M fit are fitting parameters and M th. are continuous functions taken from a particular theoreticalmodel.In this case the procedure is strongly model-dependent.A possibility to make the penalization function independent of a particular model was first formulated in theKarlsruhe-Helsinki (KH) π N-elastic PWA by G. H¨ohler and collaborators in the mid 1980s [7]. Instead of usingpartial waves which are inherently model dependent, the penalization function was chosen to be constructed fromreaction amplitudes which can be in principle directly linked to experimental data with only analyticity requirementsimposed in the amplitude reconstruction procedure. So, the equation (1) was changed to: χ ( W ) = χ ( W ) + χ ( W ) χ ( W ) = N data (cid:88) i =1 w i (cid:2) O exp. i ( W, Θ i ) − O th. i ( M fit ( W, Θ i )) (cid:3) χ ( W ) = λ pen. N data (cid:88) i =1 N amp (cid:88) k =1 (cid:12)(cid:12) A k ( M fit ( W, Θ i )) − A pen. k ( W, Θ i ) (cid:12)(cid:12) (2)where A k is the generic name for any kind of reaction amplitudes (invariant, helicity, transversity) . The amplitudes A k ( M fit ( W, Θ i )) are discontinuous ones obtained from fitted multipoles, and the amplitudes A pen. k ( W, Θ i ) are contin-uous ones obtained in the penalization procedure. In this way, one is now responding to two challenges: to get reactionamplitudes which fit the data, and also to make them continuous. In Karlsruhe-Helsinki case, this was accomplishedby implementing fixed- t analyticity and fitting the data base for fixed- t with reaction amplitudes whose analyticity isachieved by using the Pietarinen expansion, and using the obtained, continuous reaction amplitudes as penalizationfunctions A pen. k ( W, Θ i ). So, the first step of the KH fixed- t approach was to create the data base O ( W ) | t =fixed usingthe measured data base O (cos θ ) | W =fixed , and then to fit them with a manifestly analytic representation of reactionamplitudes for a fixed- t . Then, the second step was to perform a penalized PWA defined by Eq. (2) in a fixed- W representation where the penalizing function A pen. k ( W, Θ i ) was obtained in the first step in a fixed- t representation.In that way a stabilized SE PWA was performed.This approach was revived very recently for SE PWA of η photoproduction by the Main-Tuzla-Zagreb collab-oration [8], and analyzed in details. The basic result of that paper is that this fixed- t method works very reliably, butis rather complicated. First it required the creation of a completely new data base O ( W ) | t =fixed from the measureddata base O (cos θ ) | W =fixed , which introduced a certain model dependence connected with the interpolation, andsecond it involved quite some problems with the importance of the unphysical regions.Therefore, we propose an alternative.We also use Eq. (2), but the penalizing function A pen. k ( W, Θ i ) is generated by the amplitude analysis in thesame, fixed- W representation, and not in the fixed- t one. This simplifies the procedure significantly, and avoids quitesome theoretical assumptions on the behaviour in the fixed- t representation.We also propose a 2-step process: Step 1:
Complete experiment analysis/amplitude analysis (CEA/AA) of experimental data in a fixed- W repre-sentation to generate the penalizing function A pen. k ( W, Θ i ) Step 2:
Penalized TPWA, using Eq.(2) with the penalty function from
Step 1 .One has to observe one very important fact:The ”main event” happens in
Step 2 ; Step 1 serves only to impose continuity of
Step 2 . Therefore, the reac-tion amplitudes obtained in
Step 1 need not absolutely reproduce the data, it is important that they are close to theexperiment, and that they are continuous. The best agreement is then achieved in
Step 2 . Of course, finding theoptimal value of the penalization coefficient λ pen. is of utter importance. Step 1: CEA/AA
In Appendix A, we give the formalism of pseudoscalar meson photoproduction, and in Appendix B we discussthe details of CEA/AA. Out of detailed presentation of the problem we stress the most important fact: that theunconstrained CEA/AA is non-unique and discontinuous because of the continuum ambiguity. In this step, therefore,we have to achieve two goals: to find the amplitudes which achieve the best possible agreement with the data andare continuous at the same time. As we do not have a complete set of data of infinite precision at our disposal,it is by definition impossible to obtain the unique solution. We can only get the solution with errors generated byexperiment; in other words these errors originate only in the uncertainty of data, and not in continuum ambiguityeffects.
Step In their case they have chosen to use invariant amplitudes.
Observe that
Step 1 is because of part ’b.’ a model dependent step, but this model dependence will be additionallyreduced in
Step 2 . Namely, in
Step 2 we fit the data with partial waves directly, so the phases of all reaction amplitudesare changed correcting the fact that the penalization which gives continuity is model dependent. We may safely saythat the approach which is proposed here forces the phase of the final solution to be in-between the exact and the pe-nalizing solution and to be continuous at the same time. The situation may further improve by iteration, i.e. to repeat
Step 1 with input from
Step 2 as it was done in KH approach for fixed- t , but the final result so far does not require that. a. Obtaining absolute values For obtaining absolute values it is extremely useful to use the TRANSVERSITY REPRESENTATION. Namely,in the transversity representation for η photoproduction, all four absolute values | b i | are determined by a set offour observables given by the unpolarized differential cross section dσ/d Ω def = σ , the beam asymmetry Σ, thetarget-asymmetry T , and the recoil-polarization asymmetry P (cf. Table II in appendix A): σ = ρ (cid:0) | b | + | b | + | b | + | b | ) (cid:1) ˆΣ = ρ (cid:0) −| b | − | b | + | b | + | b | ) (cid:1) ˆ T = ρ (cid:0) | b | − | b | − | b | + | b | ) (cid:1) ˆ P = ρ (cid:0) −| b | + | b | − | b | + | b | ) (cid:1) (3)where O σ def = ˆ O and ρ is defined in appendices. Therefore, having all four observables with sufficient precision and inadequate number of angular points would enable us, up to discrete ambiguities, the unique extraction of the absolutevalues | b i | in SE PWA. By adequate programming (taking into account similarity of solutions at neighboring energies,one can eliminate discontinuities due to discrete ambiguities. All remaining discontinuities will be of experimentalorigin. b. Determining phases Up to this moment, our model is completely energy and angle independent, and depends only on experimen-tal data. However, results are still not continuous. Introducing analytic phases in this step produces continuity. Fora single pole amplitude, the phase is smooth (in the vicinity of the pole the phase just quickly transverses through π/ b amplitude. Other phases are very similar.So, our first solution Sol dσ/d
Ω, Σ, T and P are phase independent, and only the double-polarization observablesof type beam-target- ( BT ), beam-recoil- ( BR ) and target-recoil- ( T R ) are, we may hope that this dependence isweak. Using the fact that phases are analytic function offers us the possibility to test the size of this dependence.First, we confirm that all transversity amplitude phases indeed are analytic functions. To do this, we fit all fourphases with a 2-dimensional Pietarinen expansion, a method which has not been formulated up to now. The methodis based on Pietarinen expansion technique [14], but extended to two variables: energy and angle (more preciselycosine of the angle x := cos θ ). Namely, in energy dimension we use standard Pietarinen expansion, but each of thecoefficients also depends on the angle. Similar as for the energy part, for the angular dependence we also assumeFIG. 1: (Color online) The normalized transversity amplitude (phase) e iϕ := b / | b | from the BG2014-2 solution isshown.the expansion over a complete set of functions, in this case we use Legendre polynomials. So, one gets an analyticfunction which is analytic in energy and angle and with the analyticity we control. P T ( W, θ ) = (cid:80) Nk =0 c k ( x ) Z ( W ) k (cid:12)(cid:12)(cid:12)(cid:80) Nk =0 c k ( x ) Z ( W ) k (cid:12)(cid:12)(cid:12) Z ( W ) = α − √ W − Wα + √ W − Wc k ( x ) = M (cid:88) l =0 c k,l P l ( x ) x = cos θ (4) M and N are small numbers (angular momentum index M is always around 3, and energy index can vary from 4for very simple energy analyticity to 20 for a fairly complicated one), P l ( x ) are Legendre polynomials and α is thePietarinen range parameter. Then we make a 2-dimensional fit to the four normalized transversity amplitudes andget the coefficients α , W , and c k,l for all four absolute values.If we are able to fit the phases with such an expansion, the phases have to be analytic functions.We use a very simple Pietarinen expansion, with only one branch-point at the η photoproduction threshold andonly 4 terms in the angular expansion. However, we see that the analytic structure of the fitted phase is rathercomplicated in energy, and we need as much as N = 20 energy terms to obtain a decent fit. The result is shown inFig. 2We again show the result only for b amplitude:The analyticity of the phases of b − b offers us the possibility of testing the sensitivity of our method to the phase.Instead of the very physical phases b − b of the BG2014-2 model and illustrated for b in Figs.(1,2), we shall use aphase with much simpler analyticity, and which is generated by a 2D fit to the BG2014-02 phases with N = 4 termsonly! This phase is again shown as an illustration for the b amplitude in Fig. 3.So, our second solution Sol
Sol
Sol
Sol
Step 1 by using the original and smoothed analytic phases which are generated bythe phases from the theoretical Bonn-Gatchina model (used phases are the best fit of BG2014-02 input with 2-DPietarinen expansion given in Eq. 4).FIG. 2: (Color online) The normalized transversity amplitude (phase) e iϕ from the BG2014-02 solution (Discretesymbols) and a 2-dimensional Pietarinen fit (2D plane) are shown.FIG. 3: (Color online) The normalized transversity amplitude (phase) e iϕ from the smoothed BG2014-02 solutionis shown. Step 2 : TPWA
We perform a standard penalized TPWA defined by Eq. (2) with (cid:96) max = 5. The only issue is finding anoptimal value for the penalty-function coefficient λ pen. . This issue will be discussed further below.We, however, have to discuss two features of TPWA: the threshold behaviour and the data base. Threshold behaviour:
We know that in the vicinity of a threshold, partial waves have to behave like q ( W ) L where q ( W ) is the ab-solute value of the meson’s cm momentum, and in our procedure that has not been enforced up to now in any way. Step | b i | are concerned, so no restrictions are coming throughthe penalty function. The TPWA itself also does not require that our result obeys that rule. So, we have to imposethat threshold-behaviour somehow.A very natural way to do it is, again, via penalty function technique, and we follow the method recommendedby the KH group in ref. [7].The logic is the following: we add another penalty function to our total χ ( W ), which is to be minimized: χ ( W ) = λ thr. (cid:96) max (cid:88) l =1 |M l ± | F thr. ( W, b, l ) · l F thr. ( W, b, l ) = b · lq ( W ) · e − q ( W )0 . b M l ± = { E l ± , M l ± } (5)where F thr. ( W, b, l ) is a phenomenological function instead of the theoretical function R l of ref. [7], which is connectedwith convergence radius of the PWA expansion. We have used this function for the value b = m N where m N is thenucleon mass, and λ thr. = 2. The function F thr. ( W, b, l ) · l behaves like q ( W ) − l for small q ( W ), and vanishes for big q ( W ), so it scales down all multipoles with low q ( W ), and leaves those unchanged that have a big q ( W ). In this way, the q ( W ) l power law is automatically enforced for low q ( W ). Data base:
The data selection is particularly important as we want to be as close as possible to a complete set of ob-servables. At this moment , we take all available measured data, and we take them without any renormalization,exactly as they are published. In Table I we give our data base.TABLE I: Experimental data from A2@MAMI, GRAAL, and CBELSA/TAPS used in our PWA. Data fromCBELSA/TAPS are taken at the center of the energy bin.
Obs.
N E lab [MeV] N E θ cm [deg] N θ Reference σ − −
162 20 A2@MAMI(2010) [15]Σ 150 724 − −
160 10 GRAAL(2007) [16] T
144 725 − −
156 12 A2@MAMI(2016) [17] F
144 725 − −
156 12 A2@MAMI(2016) [17] E
64 750 − −
151 8 CBELSA/TAPS(2020) [18] P
66 725 −
908 6 41 −
156 11 CBELSA/TAPS(2020) [18] G
48 750 − −
153 8 CBELSA/TAPS(2020) [18] H
66 725 −
908 6 41 −
156 11 CBELSA/TAPS(2020) [18]
This selected set of data is somewhat specific, and deserves our special attention. The set is dominated by the verydense and very precise σ -data from A2@MAMI, while other spin observables are measured only at 6-15 energies,and much less angles. So the question arises how these sparse spin data will be combined with very precise results on σ . We shall solve this problem via interpolation. We generate two sets of interpolated data: Set 1 :We use all σ data, and all spin data are interpolated. So the whole minimization is performed on a set whichconsists of σ data + observables interpolated at energies and angles where σ is measured. These data are markedlight grey. Observe that all data are very dense, but in practice the only factually measured data are the σ values, allother data are obtained by interpolation from the measured values. This set of data is somewhat model dependent,and serves only as an indication. This set will be used in Step 1 . Set 2 :We use only part of the σ data at energies where at least one additional spin observables is exactly measured.This set is not so dense in energy, but the model dependence is reduced. We denote the results corresponding to thisset with red discrete symbols. This set will be used in Step 2 . III. RESULTS AND DISCUSSION
First we made an unconstrained fit to produce the bench-mark χ ( W ) function which represents the lowestpossible χ value for any PW fit, and consequently indicates how consistent the data base actually is. However, let usremind the reader that the resulting partial waves for such a fit are random and discontinuous. Then, we performedtwo fits using the 2-step analysis-scheme introduced above, with the results being denoted as Sol
Sol b i -phases defined in section II. For both solutions, we adjusted the following value forthe penalty coefficient: λ pen. = 10. This value represents the lower boundary of the following roughly estimatedoptimal window of penalty-coefficient values around λ pen. (cid:39) , . . . ,
50. This window of values has been determinedby ’sweetspot’-fitting techniques similar in spirit, but not exactly equal, to those proposed in sections II and III ofreference [19]. The coefficient for the threshold-penalty (5) was set to λ thr. = 2.At this moment it is essential to show the difference among χ / ndf for the unconstrained solution, as well as χ / ndf for Sol
Sol ndf is number of degrees of freedom). We show it in Fig.4.
U n c o n s t r a i n e d S o l 1 S o l 2 chi2 / ndf
W c m
FIG. 4: (Color online) A comparison of χ / ndf for the unconstrained solution, as well as χ / ndf for Sol
Sol χ / ndf from the unconstrained solution has by far the smallest values, however multipoles for this solutionare discontinuous. The χ / ndf for both solutions Sol
Sol χ / ndf for Sol
Sol
2, solutions with two different phases, is barely distinguishable! χ /ndf for Sol
1, the solution with phase directly taken over from a very good, multichannel ED model BG2014-2, is systemati-cally better than χ /ndf for Sol . Somewhat more pronounced differences canbe seen in the energy range 1600 MeV ≤ W ≤ K Λ, K Σ,...) where the phase is expected to have notable structure.Therefore, we in Figs. 5 and 6 show the lowest multipoles for
Sol 1 and
Sol 2 and corresponding predictions ofBG2014-2 theoretical solution.As it was to be expected, differences are noticeable, but not big. In spite of small differences in χ /ndf for Sol
Sol
2, the obtained multipoles are not identical. There are several points where this is not the case, but this is just the reflection of the fact that the phase from BG2014-2 model is stillonly a model and not a genuine phase, so there is a possibility that smoothed phase is accidentally better. Sol , solution with BG 2014-2 phase. - - [ MeV ] R e E + [ m F m ] - - - - [ MeV ] I m E + [ m F m ] - - [ MeV ] R e M - [ m F m ] - - [ MeV ] I mM - [ m F m ] - - - - - [ MeV ] R e E + [ m F m ] - - - - [ MeV ] I m E + [ m F m ] - - - - - - - [ MeV ] R e M + [ m F m ] - - [ MeV ] I mM + [ m F m ] - - [ MeV ] R e E - [ m F m ] - - - - [ MeV ] I m E - [ m F m ] - [ MeV ] R e M - [ m F m ] - - - - [ MeV ] I mM - [ m F m ] - - - [ MeV ] R e E + [ m F m ] - - - [ MeV ] I m E + [ m F m ] - - [ MeV ] R e M + [ m F m ] - - [ MeV ] I mM + [ m F m ] - - - [ MeV ] R e E - [ m F m ] - - - - [ MeV ] I m E - [ m F m ] - - [ MeV ] R e M - [ m F m ] - - - [ MeV ] I mM - [ m F m ] FIG. 5: (Color online) Multipoles for L=0, 1 ,2 and 3 partial waves for Sol 1. Grey discrete symbols correspond toSet 1, and red discrete symbols correspond to Set 2. Orange full line is BG2014-2 solution for comparison.2
Sol , solution with smoothed BG2014-2 phase. - - [ MeV ] R e E + [ m F m ] - - - - [ MeV ] I m E + [ m F m ] - - [ MeV ] R e M - [ m F m ] - - [ MeV ] I mM - [ m F m ] - - - - [ MeV ] R e E + [ m F m ] - - - - [ MeV ] I m E + [ m F m ] - - - - - - - [ MeV ] R e M + [ m F m ] - - - [ MeV ] I mM + [ m F m ] - - - [ MeV ] R e E - [ m F m ] - - - - - [ MeV ] I m E - [ m F m ] - [ MeV ] R e M - [ m F m ] - - - - [ MeV ] I mM - [ m F m ] - - - [ MeV ] R e E + [ m F m ] - - [ MeV ] I m E + [ m F m ] - - [ MeV ] R e M + [ m F m ] - [ MeV ] I mM + [ m F m ] - - - [ MeV ] R e E - [ m F m ] - - - - [ MeV ] I m E - [ m F m ] - - [ MeV ] R e M - [ m F m ] - - - [ MeV ] I mM - [ m F m ] FIG. 6: (Color online) Multipoles for L=0, 1, 2 and 3 partial waves for Sol 2. Grey discrete symbols correspond toSet 1, and red discrete symbols correspond to Set 2. Orange full line is BG2014-2 solution for comparison.4In Fig. 7 we repeat the plot of χ /ndf for the whole process for Sol χ /N data for the whole fit, and χ /N data for individual observables. The χ /N data for individual observables isextremely important as it gives one the internal consistency of used data base. We do not show the similar figure for Sol [ MeV ] χ ^ _ da t a / nd f [ MeV ] χ ^ _ da t a / N da t a [ MeV ] χ ^ / N da t a σ [ MeV ] χ ^ / N da t a Σ [ MeV ] χ ^ / N da t a T [ MeV ] χ ^ / N da t a F [ MeV ] χ ^ / N da t a E [ MeV ] χ ^ / N da t a H [ MeV ] χ ^ / N da t a P [ MeV ] χ ^ / N da t a G FIG. 7: (Color online) χ /N data for Sol 1.5However, the fits to the data for both solutions Sol
Sol
Sol
Sol σ (discrete symbols) with results of Sol 1 (red full line)and BG2014-2 fit (blue dashed line) at representative energies.6FIG. 9: (Color online) Comparison of experimental data for Σ (discrete symbols) with results of Sol 1 (red full line)and BG2014-2 fit (blue dashed line) at representative energies.7FIG. 10: (Color online) Comparison of experimental data for T (discrete symbols) with results of Sol 1 (red full line)and BG2014-2 fit (blue dashed line) at representative energies.8FIG. 11: (Color online) Comparison of experimental data for F (discrete symbols) with results of Sol 1 (red fullline) and BG2014-2 fit (blue dashed line) at representative energies.9FIG. 12: (Color online) Comparison of experimental data for E (discrete symbols) with results of Sol 1 (red fullline) and BG2014-2 fit (blue dashed line) at measured energies.FIG. 13: (Color online) Comparison of experimental data for P (discrete symbols) with results of Sol 1 (red fullline) and BG2014-2 fit (blue dashed line) at measured energies.0FIG. 14: (Color online) Comparison of experimental data for G (discrete symbols) with results of Sol 1 (red fullline) and BG2014-2 fit (blue dashed line) at measured energies.FIG. 15: (Color online) Comparison of experimental data for H (discrete symbols) with results of Sol 1 (red fullline) and BG2014-2 fit (blue dashed line) at measured energies.So, we obtained almost identical fits of all observables in the present data base (indistinguishable when plotted, butdifferent below drawing precision when a detailed comparison of numbers is made) with two visibly different sets ofmultipoles!1If the data base were more complete, the two sets of χ / ndf would be different between Sol
Sol
2, andthe we could refine
Step t analyisis .So, improving the precision of existing experiments and measuring additional observables to get missing phases isdefinitely needed to distinguish between the present solutions. On the basis of physics arguments we definitely claimthat Sol
Sol W representation.Furthermore, from Figs. 5 and 6 we see: • The obtained multipoles are fairly smooth and do not significantly deviate from the BG2014-02 predictions inthe sense that there is no qualitative difference between the two sets of multipoles. They are of the same sign,they have similar shape, they have comparable structure. However, one sees that both solutions
Sol 1 and
Sol2 have notably more structure than the energy-dependent BG2014-02 model, and that is to be expected asBG2014-2 is a multi-channel model, and does not ideally fit the data in one particular channel. • One does see some apparent discontinuities at certain energies in certain multipoles (i.e. jump in Im E at 1687MeV), but this is the result of inconsistencies of the data, and not of the inability of the proposed analysis-scheme to enforce continuity. Namely, as it has been shown in a former publication [5], forcing the phase tobe a continuous function is always resulting with continuity for the complete set of observables measured withsufficient precision ( in reference [5], this has been shown for pseudo-data with infinite precision). So, if suddendiscontinuities appear, they should be solely attributed to the inconsistency in the data itself. • As the low-energy behavior of the multipoles is constrained by the penalty function technique to the q L be-haviour, some low energy structure in the multipoles (mostly structures below 1550 MeV) may result from thiseffect. However, it is clearly visible that low-energy structures are more pronounced for the Sol 2 which isobtained with the smoothed nonphysical set of phases. This should and will be discussed at length in futureresearch when the pole structure will be analyzed using the Laurent+Pietarinen formalism [14]. • We see that both sets of multipoles corresponding to different choices of interpolating techniques (Set 1 lightgray and Set-2 red) are in fair agreement.From Fig. 7 we see: • The values of χ / ndf and χ /N data are extremely good but notably non-uniform throughout the analyzedenergy range. This indicates certainly inconsistencies in data set as it will be discussed later. • The distributions of χ /N data -values for particular observables notably differ.* It is uniform and very good (close to 0.5) in the complete energy range for σ .* It is very good and close to 0.5 in most of the energy range for Σ, T , F and G , but each of the observablesshow energy ranges where this quantity suddenly increases: ◦ For F it rises from an average value in the ranges 1600-1650 MeV and 1750-1840 MeV; much more forthe second range. ◦ For T it jumps only slightly at lower and higher energies. ◦ For G it also jumps in the ranges 1600-1650 Mev and 1750-1840 MeV.* It is uniform in the whole energy range, but somewhat worse than typical for E .* It is somewhat worse for P in the available energy range 1500-1650 MeV.* It notably worse for H in the complete measured energy range 1500-1650 MeV. • The quantities in these figures indicate that there exist certain inconsistencies among measured data in certainenergy ranges. In particular, H seems to deviate in the complete measured range and F seem to be problematicat higher energies . We could fit the theoretical BG2014-02 phases with 2D Pietarinen expansion with at least N = 20 terms, and then make a global,energy dependent fit of all observables fixing the absolute values of reaction amplitudes to the values of the present fit with only fourobservables, and using the Pietarinen expansion coefficients as fitting parameters for improving phases. Then we would go to Step
Sol 1 and the theoretical BG014-02 model for all experimental datafrom Table I. We conclude that the quality of fit for
Sol 1 is much better than the one of BG2014-02, and this isnot surprising as this is a fit, and BG2014-02 is an energy-dependent microscopic model. In addition, we have madesome tests, and we strongly suspect that the agreement with the data given in Fig. 7 cannot be better even for thefree fit. So, this solution is very close to the best result one can achieve. However, analyzing the details of thesefigures one can also trace the angular- and energy-ranges which are problematic and have either big dissipation orbig uncertainty, and we can very confidently predict where a particular observable is expected to be. The need fornew measurements is automatically suggested. Immediately, we may recommend that H and F should be remeasuredtowards the end of the measured energy range. In addition, the energy range of the P - and H -observable is muchsmaller, so we recommend to extend the energy range to at least 1800 MeV. IV. SUMMARY AND CONCLUSIONS
In this paper we have presented a new data analysis scheme for single-channel pseudoscalar meson photoproduction.It combines the amplitude analysis CEA/AA of a complete experiment with the truncated partial wave analysisTPWA of an idealistic case, where all higher partial waves that cannot be fitted would be completely negligible.The strength of our scheme is its simplicity and minimal reference to any particular theoretical model. But itis also robust enough that it can always extend the lack of data by additional theoretical constraints.The possible weaknesses of the scheme are that it requires a lot of experimental data, and that they should bemeasured with considerable reliability. The main opportunities of the method are that it enables the direct extractionof resonance parameters via Laurent-Pietarinen formalism [14], and at the same time gives a direct possibility tocheck the consistency of measured data sets. The scheme also allows to test the importance of certain observable tothe final result.The proposed fit-method yields a continuous and reliable set of partial waves without experiencing a strong in-fluence to any theoretical model.The new variable P , measured by the Bonn group [18], is extremely important as it helps to pin-down the ab-solute values of the transversity amplitudes in Step 1 .The present data set is insufficient to uniquely determine the reaction amplitude phase, so as an example wegenerate two solutions with almost identical quality of the fit to the data, but with notably different partial waves.More measurements are needed if one wants to better specify the pole structure of partial wave solutions.Fitting the relative phase with present data base is futile. New measurements of well selected observables canimprove the analysis a lot. With them the analysis scheme can be extended to include fitting the relative phase too,so a unique solution could be generated.The method offers the possibility to directly analyze the internal consistency of different data sets, avoiding theinfluence of different theoretical models.The separate analysis of χ /N data for 8 polarization observables in Fig. 7 suggest that certain observables should beremeasured in certain energy ranges, and Figs 8-15 imply the ranges where the consistent data are expected to be.We believe that the central result of our work consists of the fact that applying CEA/AA in practical data analysesis a very important technique, which should be employed more and more in the future. The problem of the CEA/AAhas been mostly studied as an isolated mathematical problem in the past, yielding the well-known complete sets of8 observables. However, the CEA/AA is also quickly applied to real data and it is a numerically quite well-behavedprocedure, due to the fact that it involes only 4 complex numbers for all energies. The real power of the CEA/AAresults emerges once they are combined with the TPWA. There, they have a great constraining power and make theTPWA, an analysis which is known to be very badly behaved on its own for the higher (cid:96) max , a lot more stable. Usingthe CEA/AA in such a constructive way, we have been able to derive SE PWA solutions for η photoproduction, whichhave quite controlled and small discontinuities in their energy dependence, even for the ’small’ multipoles (i.e. allmultipoles other than E ).3 Appendix A: Photoproduction formalism
In the following, we collect all aspects of the general photoproduction formalism needed for this work. We considera 2 → target nucleon in the initial state and a pseudoscalar meson anda spin- baryon in the final state: γ ( p γ ; m γ ) + N ( P i ; m s i ) −→ ϕ ( p ϕ ) + B (cid:0) P f ; m s f (cid:1) . (A1)In this expression, the 4-momenta as well as the variables necessary to label the spin-states have been indicatedfor each particle. For the reaction of η photoproduction studied in this work, the pseudoscalar ϕ is the η and therecoil-baryon B is the nucleon N . However, other combinations are also possible.In the following, we collect the customary definitions for the Mandelstam variables s , t and u . Using 4-momentumconservation, p γ + P i = p ϕ + P f , each of these variables can be written in two equivalent forms: s = ( p γ + P i ) = ( p ϕ + P f ) , (A2) t = ( p γ − p ϕ ) = ( P f − P i ) , (A3) u = ( p γ − P f ) = ( P i − p ϕ ) . (A4)Since all particles in the initial- and final state of the reaction (A1) are assumed to be on the mass-shell, the wholereaction can be described by two independent kinematic invariants. The latter are often chosen to be the pair ( s, t ).In case center-of-mass (CMS) coordinates are adopted, the following relations can be established between ( s, t ) andthe center-of-mass energy W and scattering angle θ of the reaction s = W , (A5) t = m ϕ − k (cid:113) m ϕ + q + 2 kq cos θ. (A6)Here, k and q are the absolute values of the CMS 3-momenta for the photon and the meson, respectively. Both ofthese variables can be expressed in terms of W and the masses of the initial- and final state particles. Therefore, it isseen that the reaction can be described equivalently in terms of ( W, θ ). Furthermore, the phase-space factor for theconsidered 2 → ρ = q/k .The spins of the particles in the initial- and final-state of photoproduction (A1) imply a general decomposition forthe reaction amplitude. This decomposition has been found by Chew, Goldberger, Low and Nambu (CGLN) [20] andit reads: F = χ † m sf (cid:16) i(cid:126)σ · ˆ (cid:15) F + (cid:126)σ · ˆ q (cid:126)σ · ˆ k × ˆ (cid:15) F + i(cid:126)σ · ˆ k ˆ q · ˆ (cid:15) F + i(cid:126)σ · ˆ q ˆ q · ˆ (cid:15) F (cid:17) χ m si . (A7)Here, ˆ k and ˆ q are normalized CMS 3-momenta, ˆ (cid:15) is the normalized photon polarization-vector and χ m si , χ m sf arePauli-spinors. The complex amplitudes F , . . . , F depend on ( W, θ ) and are called CGLN-amplitudes. Once this setof 4 amplitudes is determined, the full dynamics of the process is known.The axis of spin-quantization chosen for the initial-state nucleon and the final-state baryon in the decomposition (A7)coincides with the ˆ z -axis in the CMS. However, other choices are also feasible, which then lead to different butequivalent systems composed of 4 spin-amplitudes. For instance, it is possible to introduce so-called transversityamplitudes b , . . . , b by rotating the spin-quantization axis to the direction normal to the so-called reaction-plane.The latter is defined as the plane spanned by the CMS 3-momenta (cid:126)k and (cid:126)q . Using the conventions employed implicitlyin the work of Chiang and Tabakin [21], one arrives at the following set of linear and invertible relations betweentransversity- and CGLN-amplitudes: b ( W, θ ) = − b ( W, θ ) − √ θ (cid:104) F ( W, θ ) e − i θ + F ( W, θ ) e i θ (cid:105) , (A8) b ( W, θ ) = − b ( W, θ ) + 1 √ θ (cid:104) F ( W, θ ) e i θ + F ( W, θ ) e − i θ (cid:105) , (A9) b ( W, θ ) = i √ (cid:104) F ( W, θ ) e − i θ − F ( W, θ ) e i θ (cid:105) , (A10) b ( W, θ ) = i √ (cid:104) F ( W, θ ) e i θ − F ( W, θ ) e − i θ (cid:105) . (A11)4The transversity basis greatly simplifies the definitions of polarization observables (see further below) and is thereforegenerally used as a starting point for the discussion of complete-experiment problems (see appendices ?? and ?? ).Due to these mathematical advantages, this basis is also used in the discussion in the main text (section II).In order to extract information on the properties of resonances, one has to analyze partial waves. In this work, weadopt the well-known expansion of the CGLN-amplitudes into electric and magnetic multipoles, which reads [20, 22] F ( W, θ ) = ∞ (cid:88) (cid:96) =0 (cid:110) [ (cid:96)M (cid:96) + ( W ) + E (cid:96) + ( W )] P (cid:48) (cid:96) +1 (cos θ )+ [( (cid:96) + 1) M (cid:96) − ( W ) + E (cid:96) − ( W )] P (cid:48) (cid:96) − (cos θ ) (cid:111) , (A12) F ( W, θ ) = ∞ (cid:88) (cid:96) =1 [( (cid:96) + 1) M (cid:96) + ( W ) + (cid:96)M (cid:96) − ( W )] P (cid:48) (cid:96) (cos θ ) , (A13) F ( W, θ ) = ∞ (cid:88) (cid:96) =1 (cid:110) [ E (cid:96) + ( W ) − M (cid:96) + ( W )] P (cid:48)(cid:48) (cid:96) +1 (cos θ )+ [ E (cid:96) − ( W ) + M (cid:96) − ( W )] P (cid:48)(cid:48) (cid:96) − (cos θ ) (cid:9) , (A14) F ( W, θ ) = ∞ (cid:88) (cid:96) =2 [ M (cid:96) + ( W ) − E (cid:96) + ( W ) − M (cid:96) − ( W ) − E (cid:96) − ( W )] P (cid:48)(cid:48) (cid:96) (cos θ ) . (A15)The multipoles can be assigned to definite conserved spin-parity quantum numbers J P . In particular, resonances withspin J = (cid:12)(cid:12) (cid:96) ± (cid:12)(cid:12) couple to the multipoles E (cid:96) ± and M (cid:96) ± .The multipole expansion of the CGLN-amplitudes F i is formally inverted by the following well-known set ofprojection-integrals [10, 23]: M (cid:96) + = 12 ( (cid:96) + 1) (cid:90) − dx (cid:20) F P (cid:96) ( x ) − F P (cid:96) +1 ( x ) − F P (cid:96) − ( x ) − P (cid:96) +1 ( x )2 (cid:96) + 1 (cid:21) , (A16) E (cid:96) + = 12 ( (cid:96) + 1) (cid:90) − dx (cid:20) F P (cid:96) ( x ) − F P (cid:96) +1 ( x ) + (cid:96)F P (cid:96) − ( x ) − P (cid:96) +1 ( x )2 (cid:96) + 1+ ( (cid:96) + 1) F P (cid:96) ( x ) − P (cid:96) +2 ( x )2 (cid:96) + 3 (cid:21) , (A17) M (cid:96) − = 12 (cid:96) (cid:90) − dx (cid:20) − F P (cid:96) ( x ) + F P (cid:96) − ( x ) + F P (cid:96) − ( x ) − P (cid:96) +1 ( x )2 (cid:96) + 1 (cid:21) , (A18) E (cid:96) − = 12 (cid:96) (cid:90) − dx (cid:20) F P (cid:96) ( x ) − F P (cid:96) − ( x ) − ( (cid:96) + 1) F P (cid:96) − ( x ) − P (cid:96) +1 ( x )2 (cid:96) + 1 − (cid:96)F P (cid:96) − ( x ) − P (cid:96) ( x )2 (cid:96) − (cid:21) . (A19)In these projection-equations, one has x = cos θ . Polarization observables in pseudoscalar meson photoproductionare generically defined as dimensionless asymmetries among differential cross sections for different beam-, target- andrecoil polarization states O = β (cid:104)(cid:0) dσd Ω (cid:1) ( B ,T ,R ) − (cid:0) dσd Ω (cid:1) ( B ,T ,R ) (cid:105) σ . (A20)The factor β has been introduced in the work by Sandorfi et al. [22] for consistency and it takes the value β = forobservables which involve only beam- and target polarization and β = 1 for quantities with recoil polarization. Theunpolarized cross section σ always assumes the form of the sum of the two polarization configurations: σ = β (cid:34)(cid:18) dσd Ω (cid:19) ( B ,T ,R ) + (cid:18) dσd Ω (cid:19) ( B ,T ,R ) (cid:35) . (A21)The dimensioned asymmetry σ O is often called a profile function [10, 21, 24] and it is distinguished by a hat-markon the O : ˆ O = β (cid:34)(cid:18) dσd Ω (cid:19) ( B ,T ,R ) − (cid:18) dσd Ω (cid:19) ( B ,T ,R ) (cid:35) . (A22)5 Observable Group σ = (cid:0) | b | + | b | + | b | + | b | (cid:1) ˆΣ = (cid:0) − | b | − | b | + | b | + | b | (cid:1) S ˆ T = (cid:0) | b | − | b | − | b | + | b | (cid:1) ˆ P = (cid:0) − | b | + | b | − | b | + | b | (cid:1) ˆ E = Re [ − b ∗ b − b ∗ b ] = − | b | | b | cos φ − | b | | b | cos φ ˆ F = Im [ b ∗ b − b ∗ b ] = | b | | b | sin φ − | b | | b | sin φ BT ˆ G = Im [ − b ∗ b − b ∗ b ] = − | b | | b | sin φ − | b | | b | sin φ ˆ H = Re [ b ∗ b − b ∗ b ] = | b | | b | cos φ − | b | | b | cos φ ˆ C x (cid:48) = Im [ − b ∗ b + b ∗ b ] = − | b | | b | sin φ + | b | | b | sin φ ˆ C z (cid:48) = Re [ − b ∗ b − b ∗ b ] = − | b | | b | cos φ − | b | | b | cos φ BR ˆ O x (cid:48) = Re [ − b ∗ b + b ∗ b ] = − | b | | b | cos φ + | b | | b | cos φ ˆ O z (cid:48) = Im [ b ∗ b + b ∗ b ] = | b | | b | sin φ + | b | | b | sin φ ˆ L x (cid:48) = Im [ − b ∗ b − b ∗ b ] = − | b | | b | sin φ − | b | | b | sin φ ˆ L z (cid:48) = Re [ − b ∗ b − b ∗ b ] = − | b | | b | cos φ − | b | | b | cos φ T R ˆ T x (cid:48) = Re [ b ∗ b − b ∗ b ] = | b | | b | cos φ − | b | | b | cos φ ˆ T z (cid:48) = Im [ − b ∗ b + b ∗ b ] = − | b | | b | sin φ + | b | | b | sin φ TABLE II: The definitions of the 16 polarization observables of pseudoscalar meson photoproduction in terms oftransversity amplitudes b i (cf. ref. [21]) are collected here. Expressions are given both in terms of moduli andrelative phases of the amplitudes and in terms of real- and imaginary parts of bilinear products of amplitudes.Furthermore, the phase-space factor ρ has been suppressed in the given expressions. The four different groups ofobservables are indicated as well. The sign-conventions for the observables are consistent with reference [10].For the photoproduction of a single pseudoscalar meson, there exist in total 16 polarization observables [22], whichinclude also the unpolarized cross section σ and which can be further divided into the four groups of single-spinobservables ( S ), beam-target- ( BT ), beam-recoil- ( BR ) and target-recoil ( T R ) observables [21, 25]. Each group iscomposed of 4 observables. When expressed in terms of the transversity amplitudes b i , the 16 observables take thefollowing shape ˆ O α ( W, θ ) = 12 (cid:88) i,j =1 b ∗ i ( W, θ ) Γ αij b j ( W, θ ) , α = 1 , . . . , . (A23)The matrices Γ α represent a complete and orthogonal set of 4 × O α are bilinear hermitean forms in the transversity amplitudes. The matriceshave been listed by Chiang and Tabakin [21] and they can also be found in the appendices of the works [10, 24].Their algebraic properties imply useful quadratic constraints among the observables ˆ O α known as the (generalized) Fierz identities [21]. A listing of the 16 quantities (A23) is expressed in terms of moduli | b i | and relative phases φ ij := φ i − φ j of the transversity amplitudes in Table II.6 Appendix B: CEA/AA and TPWA
This appendix compiles the definitions and mathematical details of both, the complete-experiment analysis/amplitudeanalysis (CEA/AA) and the truncated partial-wave analysis (TPWA). Then, both methods are compared and theanalysis-method proposed in this work emerges as a synergy of the two.The CEA/AA represents the method to extract the 4 spin-amplitudes, for instance the transversity amplitudes b , . . . , b , from a subset of the 16 observables collected in Table II. Due to the structure of the expressions A23 assums span over bilinear amplitude products b ∗ i b j , the amplitudes can only be extracted uniquely up to one unknownoverall phase [5, 21] which is a real function that can depend on the full reaction-kinematics, i.e. on ( W, θ ). Thefinal goal of the analysis is to obtain 4 amplitudes in the complex plane, with 4 uniquely defined moduli and 3relative-phase angles. This ’rigid’ amplitude arrangement is however free to rotate as a full entity in the complexplane with energy and angle dependent phase. See Figure 16 for an illustration.
ReIm ˜ b ˜ b ˜ b ˜ b φ φ φ ReIm φ ( W , θ ) b b b b φ φ φ FIG. 16: (Color online) The photoproduction amplitudes in the transversity-basis are shown as an arrangement of 4complex numbers. The schematic is taken over from reference [10]. The plots serve to illustrate the possiblesolutions of the CEA/AA. Left: The reduced amplitudes ˜ b i , defined by the phase-contraint Im (cid:104) ˜ b (cid:105) = 0 , Re (cid:104) ˜ b (cid:105) ≥ true solution for the actualtransversity amplitudes b i is shown, and it is obtained from the ˜ b i via a rotation by the overall phase φ ( W, θ ) shownin red.The choice of variables in terms of which to parameterize the amplitude arrangement is in principle not unique forthe CEA/AA. Since one has 4 complex amplitudes and 1 unknown overall phase, the number of independent realvariables in the choice always has to amount to 8 − b i plus threesuitably chosen relative-phases, for instance | b | , | b | , | b | , | b | , φ , φ , φ . (B1)However, in numeric data analyses, the parametrization in terms of moduli and relative phases can lead to difficultiescaused by the logarithmic singularity which enters the procedure once complex exponentials have to be inverted.Alternatively, one can also think about parametrizing the CEA/AA in terms of the phase-rotation functions e iφ jk , i.e.to use the set of variables | b | , | b | , | b | , | b | , e iφ = | b || b | b b , e iφ = | b || b | b b , e iφ = | b || b | b b . (B2)This removes the difficulty of having to invert exponentials. However, this is bought at the disadvantage of havingincreased the number of real degrees of freedom artificially, since the functions e iφ jk have both a real- and an imaginarypart. Still, parametrizations in terms of phase-rotation functions are used in the main text (section II) in order to’smoothen’ phase-information coming from a PWA-model.7Whatever choice one makes to parametrize the amplitudes, the CEA/AA is always a numerical (or algebraic)procedure which takes place at one isolated point in ( W, θ ) individually. This means that in case one wishes toperform the CEA/AA for a collection of observables over a wider kinematic region, the kinematic binning of all theseobservables has to be brought to a match over this common region. The situation is illustrated in Figure 17. - - All observablescos θ W [ G e V ] FIG. 17: A schematic illustration for a specific binning of points in phase-space (blue blue polygons) for theCEA/AA. The kinematic binning has to agree for all observables. The CEA/AA then acts on each pointindividually (illustrated by the red boxes) and therefore all the observables involved in the analysis have to bebrought to the same binning. (Color Online)Consequently, the result of the CEA/AA, i.e. the 7 variables parametrizing the transversity amplitudes with afixed overall phase, is also returned as a set of ’discrete data’ in complex-space. In other words, the standardCEA/AA without any constraints returns a discrete but not necessarily continuous set of points. The direct con-sequence is that partial waves with physical meaning cannot be extracted [26] without further imposing an overallphase provided by a theoretical model.The TPWA denotes the procedure of extracting a finite set of photoproduction multipoles from experimental data byintroducing continuous angular decomposition of amplitudes over Legendre polynomials. In practice, the multipole-expansion defined by equations (A12) to (A15) is truncated at some finite angular-momentum (cid:96) max . Insertingthis truncation into the definitions of the 16 polarization observables shown in Table II yields the mathematicalparametrization lying at the heart of the analysis.The TPWA parametrization can be expressed in a concise form. Choosing to express the emerging angular de-pendence of the polarization observables ˆ O α in terms of associated Legendre polynomials, one arrives at the followingform (cf [10, 27–29]): ˆ O α ( W, θ ) = qk (cid:96) max + β α + γ α (cid:88) n = β α ( a L ) ˆ O α n ( W ) P β α n (cos θ ) , α = 1 , . . . , , (B3)( a L ) ˆ O α n ( W ) = (cid:104)M (cid:96) max ( W ) | ( C L ) ˆ O α n |M (cid:96) max ( W ) (cid:105) . (B4)The Legendre coefficients ( a L ) ˆ O α n become bilinear hermitean forms defined by a certain set of matrices ( C L ) ˆ O α n (suchmatrices are given explicitly, for the group S - and BT -observables, in the appendix of reference [10]). The multipolesare organized into the 4 (cid:96) max -dimensional complex vector |M (cid:96) max (cid:105) according to the convention |M (cid:96) max (cid:105) = [ E , E , M , M − , E , E − , . . . , M (cid:96) max − ] T . (B5)The quantities β α and γ α in equations (B3) and (B4) are constants which define the precise form of the TPWA foreach observable. These constants can be found for instance in references [10, 29].8For the TPWA, all observables have to be prepared with a common energy-binning. However, since this methodfor extracting amplitudes actually parametrizes the angular dependence continuously (cf. equation (B3)), the angularbinnings of the observables can be different. The TPWA then returns a continuous function in angle for each of thediscrete energy-bins. However, continuity in energy is another matter, and has been discussed elsewhere [5]. Thekinematic situation is illustrated in Figure 18. - - Observable 1cos θ W [ G e V ] - - Observable 2cos θ FIG. 18: The plots show schematic illustrations for the kinematic situation in the TPWA. Figures are shown for twodifferent hypothetical observables. Further observables are not shown, but surely present in the TPWA. Thekinematic binning for all the datapoints (blue polygons) does not have to fully agree between all observables.However, the energy-binning has to be the same for all datasets. The TPWA (represented by red solid lines)introduces a continuous dependence on the angular variable cos θ . Compare this to Figure 17. (Color Online)Note that the CEA/AA and the TPWA are not equivalent procedures and will not lead to identical results. Thisbecomes especially apparent once one compares the complete sets of observables [21, 30], i.e. minimal subsets of allpolarization observables which allow for an unambiguous extraction of the complex amplitudes (or multipoles), validfor both analysis-procedures. The differences among and the most important characteristics of the CEA/AA and theTPWA are listed in the following: • CEA/AA:i.) Kinematic regime: the CEA/AA takes place at individual points in the 2-dimensional space (
W, θ ) spannedby the energy W and scattering angle θ .ii.) In the CEA/AA, one has 4 complex amplitudes while one overall phase φ ( W, θ ) is not known. This resultsin 8 − S (cf. Table II), one has to select 4 double-polarization observableswhich must not belong to the same group. This becomes apparent once one considers for instance the 4 BT -observables listed in Table II: all 4 observables only contain information on two relative phases, φ and φ . Thus, even in case all BT -observables were measured, at least one connecting relative phase, forinstance φ , remains unknown, which results in a continuous ambiguity. Therefore, at least 2 observablesmust be chosen from a third group, e.g. the BR -observables. Further rules for the selection of completesets can be found in references [21, 31].Recent studies [32–35] demonstrate the fact that the completeness of the minimal complete sets of 8 is lostonce measurement-errors of realistic sizes are introduced. Then, in order to recover a unique solution forthe amplitudes, the considered complete set of 8 has to be enlarged.iv.) In case a mathematically in complete set of observables has been selected for the CEA/AA, in most casesthis results in only an additional 2-fold discrete ambiguity (in case the 4 double-polarization measurementsare not taken from the same group).v.) In a world without measurement uncertainties, the CEA/AA yields an exact representation of the photo-production T -matrix (up to one overall phase). This is accomplished by extracting 4 complex numbers,9independently of the considered energy-region. The phase of one of the 4 complex numbers has to beconstrained, e.g. by demanding this number to be real and positive. • TPWA:i.) Kinematic regime: the TPWA is performed at an individual point in W , but over a whole distribution inthe angular variable θ (or cos θ ).ii.) In the TPWA, one has 4 (cid:96) max complex multipoles while one energy-dependent overall phase φ ( W ) is notknown. This results in 8 (cid:96) max − φ ( W ) is fixed in some way.iii.) A mathematical complete set for the TPWA is given by minimally 4 observables [9, 10]. However, thesecomplete sets of 4 can only be found in numerical simulations. On the other hand, an algebraic solution-theory exists for complete sets composed of 5 carefully chosen observables [10, 28, 30, 36]. The minimalmathematical complete sets mentioned here loose their validity once measurement-errors of realistic sizesare introduced (cf. section 5.5 of reference [10]) and then have to be enlarged in order to facilitate a uniquesolution.iv.) In case a mathematically in complete set of observables has been selected for the TPWA, one obtainsan exact 2-fold discrete ambiguity called double ambiguity [10, 30, 36], but also a number of (possible)approximate accidental ambiguities exists, which scales as 4 (cid:96) max − (cid:96) max > approximation of the photoproduction T -matrix for any finite (cid:96) max .For higher energies, one generally has to choose a higher truncation order (cid:96) max , which can result in anincreased numerical instability. ⇒ In this work, we combine both analysis-procedures and use the amplitudes resulting from a CEA/AA in orderto resolve the instability-problems of the TPWA, which exist mainly for higher truncation orders.As it is explained in details in the main text (section II), a CEA/AA with smooth, analytic phases originating froma theoretical model is used as a penalty function in the two-step process in order to ensure the continuity and toincrease the stability of the TPWA. In this way, the advantages of both methods have been combined, and a synergyis created which produces a reliable, and very precise description of the data, while at the same time additionaltheoretical requirements like a good threshold behaviour are obeyed.0 [1] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019 update.[2] D. Atkinson, P.W. Johnson and R.L. Warnock, Commun. mat. Phys. , 221 (1973).[3] J.E. Bowcock and H. Burkhard, Rep. Prog. Phys. , 1099 (1975).[4] D. Atkinson and I.S. Stefanescu, Commun. Math. Phys. , 291 (1985).[5] A. ˇSvarc, Y. Wunderlich, H. Osmanovi´c, M. Hadˇzimehmedovi´c, R. Omerovi´c, J. Stahov, V. Kashevarov, K. Nikonov, M.Ostrick, L. Tiator, and R. Workman, Phys. Rev. C 97 , 054611 (2018).[6] Y. Wunderlich, A. varc, R. L. Workman, L. Tiator and R. Beck, Phys. Rev. C , no.6, 065202 (2017)doi:10.1103/PhysRevC.96.065202 [arXiv:1708.06840 [nucl-th]].[7] G. H¨ohler, Pion Nucleon Scattering , Part 2, Landolt-Bornstein: Elastic and Charge Exchange Scattering of ElementaryParticles, Vol. 9b (Springer-Verlag, Berlin, 1983).[8] H. Osmanovi´c, M. Hadˇzimehmedovi´c, R. Omerovi´c, J. Stahov, V. Kashevarov, K. Nikonov, M. Ostrick, L. Tiator, and A.ˇSvarc, Phys. Rev.
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