1Helmholtz Institut fur Strahlen- und Kernphysik ... · 1Helmholtz Institut fur Strahlen- und Kernphysik, Universit at Bonn, Germany For photoproduction reactions with nal states

Minimal complete sets for two pseudoscalar meson photoproduction

Philipp Kroenert,1 Yannick Wunderlich,1 Farah Afzal,1 and Annika Thiel1

1Helmholtz Institut für Strahlen- und Kernphysik, Universität Bonn, Germany

For photoproduction reactions with final states consisting of two pseudoscalar mesons and a spin-1/2 baryon, 8 complex amplitudes need to be determined uniquely. A modified version of Moravcsik’stheorem is employed for these reactions, resulting in slightly over-complete sets of polarizationobservables that are able to determine the amplitudes uniquely. Further steps were taken to reducethe found sets to minimal complete sets. As a final result, multiple minimal complete sets withoutany remaining ambiguities are presented for the first time. These sets consist of 2N = 16 observables,containing only one triple polarization observable.

I. INTRODUCTION

The interrelation between experiment and theory iswhat drives science. In the field of hadron spectroscopythese are the measurement of cross sections or polariza-tion observables and its counterpart Quantum Chromo-dynamics. The latter describes the transition of the ini-tial to the final state via a transition matrix T . Thismatrix comprises the employed model predictions to de-scribe a certain process. Via so-called formation experi-ments (i.e. γp → ππp) it is possible to study the emer-gence of resonant states (such as ∆(1232), N(1440)1/2+,N(1520)3/2−, etc. [1]).

These states can be analyzed via partial wave analysis(i.e. BnGa [1], MAID [2]), determining the matrix ele-ments of T and comparing it to the model prediction.However, as polarization observables depend on bilinearproducts of the complex amplitudes [3–5], mathemati-cal ambiguities arise [6]. Nevertheless, it is still possibleto determine unique solutions by employing a completeexperiment analysis [7].

Such a complete experiment analysis was performedanalytically by Chiang and Tabakin in 1997 [6] for singlepseudoscalar meson photoproduction. A detailed proofcomprising all the relevant cases was published recentlyby Nakayama [8]. It should be noted that these completeexperiments are an idealization for data with no uncer-tainty [9]. Although the process of single pseudoscalarmeson photoproduction can be fully described by onlyfour complex amplitudes [10], the calculations are nontrivial and cumbersome [8] and furthermore, quite in-volved ambiguity-structures can arise.

Within this paper, the determination of complete setsof observables is studied for the reaction of two pseu-doscalar meson photoproduction. The process can bedescribed by N = 8 complex amplitudes and thus allowsfor 64 measurable polarization observables [5], which arefour times as many observables as in the case of singlepseudoscalar meson photoproduction [6]. This results inan exponential increase of complexity, for what reasonthe algebraic techniques presented in [8] are no longerappropriate, although possible (see Section VII B). Newmethods should be employed, in order to allow for an eas-ier access to the problem of complete sets for reactionswith N > 4.

There is an already existing work on this subject byArenhövel and Fix [11] from 2014. On the one hand, theyused the inverse function theorem to derive complete setsof 16 observables. The downside of this method is thatthe resulting sets might locally be free of ambiguities,but not globally. On the other hand, they used a graphtheoretical approach, where a complex amplitude is rep-resented as a node and the bilinear product as a connec-tion between certain nodes. This method yields completesets with 25 observables. It was then shown how to fur-ther reduce such a set to 15 observables. Although, theyfound sets without triple polarization observables, therestill remain quadratic ambiguities.

To overcome these difficulties arising from remainingdiscrete mathematical ambiguities, Moravcsik’s theorem[12] is employed within this paper. This theorem allowsfor the extraction of complete sets of observables for anarbitrary number of amplitudes. Furthermore, due to itsgraph-theoretical foundation, the whole algorithm can beautomated [13].

The paper is structured in the following way: Start-ing point is a short recap of Moravcsik’s theorem andits modification in Section II. Section III introduces the64 polarization observables for two pseudoscalar mesonphotoproduction. Section IV elaborates on the difficul-ties of their experimental determination and gives an ex-tensive overview of already performed measurements intwo pseudoscalar meson photoproduction. Within Sec-tion V the actual application of Moravcsik’s theorem isdescribed and illustrated with an example. The entireanalysis results in 5964 unique, but slightly over-completesets of observables. Their characteristics are discussed inSection VI. In Section VII it is described how to trans-form the slightly over-complete sets into minimal ones(i.e. into sets containing 2N = 16 observables). Basedon these sets as well as the already performed measure-ments (see Tables IV, V and VII) the most promisingminimal complete set is presented in Section VIII. Theresults are summarized in Section IX.

arX

iv:2

009.

0435

6v4

[nu

cl-t

h] 2

7 O

ct 2

020

2

∼sin(ϕ12)

∼sin(ϕ14)∼sin(ϕ27)

∼sin(ϕ34)

∼sin(ϕ78)

∼cos(ϕ36)∼sin(ϕ56)

∼sin(ϕ58)

1

2

4

7

3 6

5

8

FIG. 1. Illustration of a cycle graph with eight nodes (enumer-ated points) and edges (solid and dashed lines). Each noderepresents one complex amplitude, whereas each edge con-necting two nodes represents either the real (solid) or imagi-nary (dashed) part of the bilinear product of those nodes. Therespective correlation to the relative phase φij is indicated.

II. MORAVCSIK’S THEOREM

The main points of Moravcsik’s paper [12] shall be re-capped in a concise form. The basic assumption of thetheorem is that the moduli of the N complex amplitudesti are known, together with the real and imaginary partsof the bilinear products tit

∗j . Furthermore, each complex

amplitude ti is treated as a node of a graph whereas anedge is the real/imaginary part of the bilinear amplitudeproduct tit

∗j connecting both nodes. An illustration is

shown in Fig. 1. Such a graph is said to correspond to acomplete set of observables if it fulfills the following tworequirements:

1. it is a connected graph

2. it has an odd number of edges which corresponds toan imaginary part of a particular bilinear product,i.e. ∼ Im(tit∗j )

The first condition is related to the “consistency relation”of the relative phases:

φ12 + φ23 + . . .+ φN1 = 0, (1)

which implies a summation of relative phases between allneighboring amplitudes ti [8]. Equation (1) has to hold inevery case, whether the considered set is fully complete,or not. The second condition is responsible for resolvingthe discrete ambiguities since it holds

Im(tit∗j ) = |ti||tj | · sin(φij), (2)

and that sine itself produces an ambiguity due to its pe-riodicity:

φij → (φij , π − φij). (3)

It turns out that any odd number of such “sine-type”ambiguities resolves the discrete ambiguities, due to thesummand π. The generalization to any odd number isthe actual modification to Moravcsik’s theorem. A proofof the original version of the theorem can be found in [12]and a quite detailed proof of the modified version of thetheorem is given in reference [13].

The following analysis focuses on cycle graphs, i.e. con-nected graphs where each node has degree two. As ex-plained in the paper [12]: “from the point of view of elim-inating discrete ambiguities” these graph types are “themost economical” ones. Thus, only the minimal numberof N bilinear products is needed in order to eliminate alldiscrete ambiguities, one for each edge.

III. POLARIZATION OBSERVABLES

The derivation of the 64 polarization observables of twopseudoscalar meson photoproduction was first publishedby Roberts and Oed [5]. The observables were definedin a ”helicity and hybrid helicity-transversity basis” [5].For the latter, the photon spin is still quantized along itsdirection of motion. For the sake of comparability, thehybrid basis shall be adopted in this paper. However,in order to work out the connection between the real(imaginary) part of the bilinear products and the relativephases φij , it is advantageous to rename the amplitudes:

b+1 → t1 b+2 → t2 b

+3 → t3 b

+4 → t4 (4)

b−1 → t5 b−2 → t6 b

−3 → t7 b

−4 → t8 (5)

Within reference [5], the observables are ordered accord-ing to the polarization of the photon beam which is re-quired to measure the respective observable. This order-ing scheme is advantageous from an experimental pointof view, unfortunately it is inappropriate when studyingambiguities. Therefore, the observables are regroupedaccording to their mathematical structure, which yieldseight groups. While the first group consists of observ-ables solely described by the squared moduli of the am-plitudes ti, any other group comprises equal amounts ofobservables containing only cos- or sin-terms. The result-ing expressions for the observables are listed in Tables Iand II.

For the purpose of an easier calculation, 64 Γ-matricesare introduced, which can be solely described by theidentity matrix, as well as the three Pauli matrices(listed in Table IX). Allowing to calculate the respec-tive observable by the bilinear form 〈t|Γ|t〉 with |t〉 :=(t1, t2, t3, t4, t5, t6, t7, t8), similar as in [6]. As expected,the Γ-matrices for each group share the same matrixstructure. Naturally, they form an orthogonal basis, arehermitian and unitary. Indeed, the matrices fulfill thesame properties as presented in [6] (with an adapted pref-actor in the orthogonality relation).

3

TABLE I. Definitions of the 64 polarization observables for two pseudoscalar meson photoproduction in hybrid helicity-transversity form. Here, φij denotes the relative phase between the complex amplitudes ti and tj . The notation used inthe original paper of Roberts and Oed [5] is also shown. The observables are classified into eight groups according to theirunderlying mathematical structure. The vector |t〉 has the form (t1, t2, t3, t4, t5, t6, t7, t8) and the shape of the Γ-matrices isoutlined in Table IX.

Observable Definition in terms of polar coordinates / 2 Bilinear form Roberts/Oed

OI1 2 · (|t1|2 + |t2|2 + |t3|2 + |t4|2 − |t5|2 − |t6|2 − |t7|2 − |t8|2) 〈t|ΓI1|t〉 I�

OI2 2 · (|t1|2 + |t2|2 − |t3|2 − |t4|2 + |t5|2 + |t6|2 − |t7|2 − |t8|2) 〈t|ΓI2|t〉 PyOI3 2 · (|t1|2 − |t2|2 + |t3|2 − |t4|2 + |t5|2 − |t6|2 + |t7|2 − |t8|2) 〈t|ΓI3|t〉 Py’OI4 2 · (|t1|2 − |t2|2 − |t3|2 + |t4|2 − |t5|2 + |t6|2 + |t7|2 − |t8|2) 〈t|ΓI4|t〉 O�yy’OI5 2 · (|t1|2 − |t2|2 − |t3|2 + |t4|2 + |t5|2 − |t6|2 − |t7|2 + |t8|2) 〈t|ΓI5|t〉 Oyy’OI6 2 · (|t1|2 − |t2|2 + |t3|2 − |t4|2 − |t5|2 + |t6|2 − |t7|2 + |t8|2) 〈t|ΓI6|t〉 P�y’OI7 2 · (|t1|2 + |t2|2 − |t3|2 − |t4|2 − |t5|2 − |t6|2 + |t7|2 + |t8|2) 〈t|ΓI7|t〉 P�yOI8 2 · (|t1|2 + |t2|2 + |t3|2 + |t4|2 + |t5|2 + |t6|2 + |t7|2 + |t8|2) 〈t|ΓI8|t〉 I0

OIIc1 |t1| |t3| cos(φ13) + |t2| |t4| cos(φ24) + |t5| |t7| cos(φ57) + |t6| |t8| cos(φ68) 〈t|ΓIIc1|t〉 −PzOIIc2 |t1| |t3| cos(φ13) + |t2| |t4| cos(φ24)− |t5| |t7| cos(φ57)− |t6| |t8| cos(φ68) 〈t|ΓIIc2|t〉 −P�zOIIc3 |t1| |t3| cos(φ13)− |t2| |t4| cos(φ24) + |t5| |t7| cos(φ57)− |t6| |t8| cos(φ68) 〈t|ΓIIc3|t〉 −Ozy’OIIc4 |t1| |t3| cos(φ13)− |t2| |t4| cos(φ24)− |t5| |t7| cos(φ57) + |t6| |t8| cos(φ68) 〈t|ΓIIc4|t〉 −O�zy’OIIs1 |t1| |t3| sin(φ13) + |t2| |t4| sin(φ24) + |t5| |t7| sin(φ57) + |t6| |t8| sin(φ68) 〈t|ΓIIs1|t〉 −PxOIIs2 |t1| |t3| sin(φ13) + |t2| |t4| sin(φ24)− |t5| |t7| sin(φ57)− |t6| |t8| sin(φ68) 〈t|ΓIIs2|t〉 −P�xOIIs3 |t1| |t3| sin(φ13)− |t2| |t4| sin(φ24) + |t5| |t7| sin(φ57)− |t6| |t8| sin(φ68) 〈t|ΓIIs3|t〉 −Oxy’OIIs4 |t1| |t3| sin(φ13)− |t2| |t4| sin(φ24)− |t5| |t7| sin(φ57) + |t6| |t8| sin(φ68) 〈t|ΓIIs4|t〉 −O�xy’

OIIIc1 |t1| |t2| cos(φ12) + |t3| |t4| cos(φ34) + |t5| |t6| cos(φ56) + |t7| |t8| cos(φ78) 〈t|ΓIIIc1 |t〉 −Pz’OIIIc2 |t1| |t2| cos(φ12) + |t3| |t4| cos(φ34)− |t5| |t6| cos(φ56)− |t7| |t8| cos(φ78) 〈t|ΓIIIc2 |t〉 −P�z’OIIIc3 |t1| |t2| cos(φ12)− |t3| |t4| cos(φ34) + |t5| |t6| cos(φ56)− |t7| |t8| cos(φ78) 〈t|ΓIIIc3 |t〉 −Oyz’OIIIc4 |t1| |t2| cos(φ12)− |t3| |t4| cos(φ34)− |t5| |t6| cos(φ56) + |t7| |t8| cos(φ78) 〈t|ΓIIIc4 |t〉 −O�yz’OIIIs1 |t1| |t2| sin(φ12) + |t3| |t4| sin(φ34) + |t5| |t6| sin(φ56) + |t7| |t8| sin(φ78) 〈t|ΓIIIs1 |t〉 Px’OIIIs2 |t1| |t2| sin(φ12) + |t3| |t4| sin(φ34)− |t5| |t6| sin(φ56)− |t7| |t8| sin(φ78) 〈t|ΓIIIs2 |t〉 P�x’OIIIs3 |t1| |t2| sin(φ12)− |t3| |t4| sin(φ34) + |t5| |t6| sin(φ56)− |t7| |t8| sin(φ78) 〈t|ΓIIIs3 |t〉 Oyx’OIIIs4 |t1| |t2| sin(φ12)− |t3| |t4| sin(φ34)− |t5| |t6| sin(φ56) + |t7| |t8| sin(φ78) 〈t|ΓIIIs4 |t〉 O�yx’

OIVc1 |t1| |t4| cos(φ14) + |t2| |t3| cos(φ23) + |t5| |t8| cos(φ58) + |t6| |t7| cos(φ67) 〈t|ΓIVc1 |t〉 Ozz’OIVc2 |t1| |t4| cos(φ14) + |t2| |t3| cos(φ23)− |t5| |t8| cos(φ58)− |t6| |t7| cos(φ67) 〈t|ΓIVc2 |t〉 O�zz’OIVc3 |t1| |t4| cos(φ14)− |t2| |t3| cos(φ23) + |t5| |t8| cos(φ58)− |t6| |t7| cos(φ67) 〈t|ΓIVc3 |t〉 Oxx’OIVc4 |t1| |t4| cos(φ14)− |t2| |t3| cos(φ23)− |t5| |t8| cos(φ58) + |t6| |t7| cos(φ67) 〈t|ΓIVc4 |t〉 O�xx’OIVs1 |t1| |t4| sin(φ14) + |t2| |t3| sin(φ23) + |t5| |t8| sin(φ58) + |t6| |t7| sin(φ67) 〈t|ΓIVs1 |t〉 Oxz’OIVs2 |t1| |t4| sin(φ14) + |t2| |t3| sin(φ23)− |t5| |t8| sin(φ58)− |t6| |t7| sin(φ67) 〈t|ΓIVs2 |t〉 O�xz’OIVs3 |t1| |t4| sin(φ14)− |t2| |t3| sin(φ23) + |t5| |t8| sin(φ58)− |t6| |t7| sin(φ67) 〈t|ΓIVs3 |t〉 −Ozx’OIVs4 |t1| |t4| sin(φ14)− |t2| |t3| sin(φ23)− |t5| |t8| sin(φ58) + |t6| |t7| sin(φ67) 〈t|ΓIVs4 |t〉 −O�zx’

4

TABLE II. (Continuation of Table I) Definitions of the 64 polarization observables for two pseudoscalar meson photoproductionin hybrid helicity-transversity form. Here, φij denotes the relative phase between the complex amplitudes ti and tj . The notationused in the original paper of Roberts and Oed [5] is also shown. The observables are classified into eight groups according totheir underlying mathematical structure. The vector |t〉 has the form (t1, t2, t3, t4, t5, t6, t7, t8) and the shape of the Γ-matricesis outlined in Table IX.

Observable Definition in terms of polar coordinates / 2 Bilinear form Roberts/Oed

OVc1 |t1| |t5| cos(φ15) + |t2| |t6| cos(φ26) + |t3| |t7| cos(φ37) + |t4| |t8| cos(φ48) 〈t|ΓVc1|t〉 −Ic

OVc2 |t1| |t5| cos(φ15) + |t2| |t6| cos(φ26)− |t3| |t7| cos(φ37)− |t4| |t8| cos(φ48) 〈t|ΓVc2|t〉 −PcyOVc3 |t1| |t5| cos(φ15)− |t2| |t6| cos(φ26) + |t3| |t7| cos(φ37)− |t4| |t8| cos(φ48) 〈t|ΓVc3|t〉 −Pcy’OVc4 |t1| |t5| cos(φ15)− |t2| |t6| cos(φ26)− |t3| |t7| cos(φ37) + |t4| |t8| cos(φ48) 〈t|ΓVc4|t〉 −Ocyy’OVs1 |t1| |t5| sin(φ15) + |t2| |t6| sin(φ26) + |t3| |t7| sin(φ37) + |t4| |t8| sin(φ48) 〈t|ΓVs1|t〉 −Is

OVs2 |t1| |t5| sin(φ15) + |t2| |t6| sin(φ26)− |t3| |t7| sin(φ37)− |t4| |t8| sin(φ48) 〈t|ΓVs2|t〉 −PsyOVs3 |t1| |t5| sin(φ15)− |t2| |t6| sin(φ26) + |t3| |t7| sin(φ37)− |t4| |t8| sin(φ48) 〈t|ΓVs3|t〉 −Psy’OVs4 |t1| |t5| sin(φ15)− |t2| |t6| sin(φ26)− |t3| |t7| sin(φ37) + |t4| |t8| sin(φ48) 〈t|ΓVs4|t〉 −Osyy’

OVIc1 |t1| |t7| cos(φ17) + |t2| |t8| cos(φ28) + |t3| |t5| cos(φ35) + |t4| |t6| cos(φ46) 〈t|ΓVIc1 |t〉 PczOVIc2 |t1| |t7| cos(φ17) + |t2| |t8| cos(φ28)− |t3| |t5| cos(φ35)− |t4| |t6| cos(φ46) 〈t|ΓVIc2 |t〉 −PsxOVIc3 |t1| |t7| cos(φ17)− |t2| |t8| cos(φ28) + |t3| |t5| cos(φ35)− |t4| |t6| cos(φ46) 〈t|ΓVIc3 |t〉 Oczy’OVIc4 |t1| |t7| cos(φ17)− |t2| |t8| cos(φ28)− |t3| |t5| cos(φ35) + |t4| |t6| cos(φ46) 〈t|ΓVIc4 |t〉 −Osxy’OVIs1 |t1| |t7| sin(φ17) + |t2| |t8| sin(φ28) + |t3| |t5| sin(φ35) + |t4| |t6| sin(φ46) 〈t|ΓVIs1 |t〉 PszOVIs2 |t1| |t7| sin(φ17) + |t2| |t8| sin(φ28)− |t3| |t5| sin(φ35)− |t4| |t6| sin(φ46) 〈t|ΓVIs2 |t〉 PcxOVIs3 |t1| |t7| sin(φ17)− |t2| |t8| sin(φ28) + |t3| |t5| sin(φ35)− |t4| |t6| sin(φ46) 〈t|ΓVIs3 |t〉 Oszy’OVIs4 |t1| |t7| sin(φ17)− |t2| |t8| sin(φ28)− |t3| |t5| sin(φ35) + |t4| |t6| sin(φ46) 〈t|ΓVIs4 |t〉 Ocxy’

OVIIc1 |t1| |t6| cos(φ16) + |t2| |t5| cos(φ25) + |t3| |t8| cos(φ38) + |t4| |t7| cos(φ47) 〈t|ΓVIIc1 |t〉 Pcz’OVIIc2 |t1| |t6| cos(φ16) + |t2| |t5| cos(φ25)− |t3| |t8| cos(φ38)− |t4| |t7| cos(φ47) 〈t|ΓVIIc2 |t〉 Ocyz’OVIIc3 |t1| |t6| cos(φ16)− |t2| |t5| cos(φ25) + |t3| |t8| cos(φ38)− |t4| |t7| cos(φ47) 〈t|ΓVIIc3 |t〉 Psx’OVIIc4 |t1| |t6| cos(φ16)− |t2| |t5| cos(φ25)− |t3| |t8| cos(φ38) + |t4| |t7| cos(φ47) 〈t|ΓVIIc4 |t〉 Osyx’OVIIs1 |t1| |t6| sin(φ16) + |t2| |t5| sin(φ25) + |t3| |t8| sin(φ38) + |t4| |t7| sin(φ47) 〈t|ΓVIIs1 |t〉 Psz’OVIIs2 |t1| |t6| sin(φ16) + |t2| |t5| sin(φ25)− |t3| |t8| sin(φ38)− |t4| |t7| sin(φ47) 〈t|ΓVIIs2 |t〉 Osyz’OVIIs3 |t1| |t6| sin(φ16)− |t2| |t5| sin(φ25) + |t3| |t8| sin(φ38)− |t4| |t7| sin(φ47) 〈t|ΓVIIs3 |t〉 −Pcx’OVIIs4 |t1| |t6| sin(φ16)− |t2| |t5| sin(φ25)− |t3| |t8| sin(φ38) + |t4| |t7| sin(φ47) 〈t|ΓVIIs4 |t〉 −Ocyx’

OVIIIc1 |t1| |t8| cos(φ18) + |t2| |t7| cos(φ27) + |t3| |t6| cos(φ36) + |t4| |t5| cos(φ45) 〈t|ΓVIIIc1 |t〉 −Oczz’OVIIIc2 |t1| |t8| cos(φ18) + |t2| |t7| cos(φ27)− |t3| |t6| cos(φ36)− |t4| |t5| cos(φ45) 〈t|ΓVIIIc2 |t〉 Osxz’OVIIIc3 |t1| |t8| cos(φ18)− |t2| |t7| cos(φ27) + |t3| |t6| cos(φ36)− |t4| |t5| cos(φ45) 〈t|ΓVIIIc3 |t〉 −Oszx’OVIIIc4 |t1| |t8| cos(φ18)− |t2| |t7| cos(φ27)− |t3| |t6| cos(φ36) + |t4| |t5| cos(φ45) 〈t|ΓVIIIc4 |t〉 −Ocxx’OVIIIs1 |t1| |t8| sin(φ18) + |t2| |t7| sin(φ27) + |t3| |t6| sin(φ36) + |t4| |t5| sin(φ45) 〈t|ΓVIIIs1 |t〉 −Oszz’OVIIIs2 |t1| |t8| sin(φ18) + |t2| |t7| sin(φ27)− |t3| |t6| sin(φ36)− |t4| |t5| sin(φ45) 〈t|ΓVIIIs2 |t〉 −Ocxz’OVIIIs3 |t1| |t8| sin(φ18)− |t2| |t7| sin(φ27) + |t3| |t6| sin(φ36)− |t4| |t5| sin(φ45) 〈t|ΓVIIIs3 |t〉 Oczx’OVIIIs4 |t1| |t8| sin(φ18)− |t2| |t7| sin(φ27)− |t3| |t6| sin(φ36) + |t4| |t5| sin(φ45) 〈t|ΓVIIIs4 |t〉 −Osxx’

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IV. DATABASE FOR TWO PSEUDOSCALARMESON PHOTOPRODUCTION

As already mentioned, 64 observables can be measuredfor two pseudoscalar meson photoproduction using thefull three-body kinematics of the reaction. These observ-ables can be organized into three groups: single, doubleand triple polarization observables, which require eitherthe use of a polarized beam (B) or a polarized target (T )or a recoil polarimeter (R) or a combination of the three.Table III gives an overview of all the observables of eachcategory. In addition to the unpolarized cross sectionI0, there are three observables in each single polarizationobservable category (B,T ,R), nine in each double polar-ization observable category (BT , BR and T R) and 27observables in the triple polarization observable category(BT R).The description of the full three-body kinematics re-quires five independent variables [14]. In this context,two planes, the reaction plane and the decay plane, areoften used [14, 15]. While the reaction plane is definedby the incoming photon and one of the outgoing parti-cles, the decay plane is spanned by the other two out-going particles. The angle between the reaction and thedecay plane is called φ∗. Integrating over φ∗ makes itpossible to treat the three-body final state as a two-bodyfinal state, resulting in a reduced number of observables.In this case, the observables correspond to observablesknown from single meson photoproduction [7], e.g. thecategory B reduces to Ic = Σ, the category T to Py = T ,the category R to Py’ = P (this observable can be alsomeasured as a double polarization observable −Pcy [5])and the category BT to Psx = H, P

sz = G, P

�x = F and

P�z = E [5].In the case of single pseudoscalar meson photoproduc-

tion, quite a lot measurements were performed to de-termine single and double polarization observables. Anextensive overview over the performed measurements, onthe basis of the SAID database [16], was brought togetherrecently by Ireland, Pasyuk and Strakovsky [17].

A similar database does not exist yet for double pseu-doscalar meson photoproduction. Thus, for the firsttime, an extensive overview of measurements of polar-ization observables for double pseudoscalar meson pho-toproduction is presented in Tables IV and V.

By far the most measurements where performed forthe reaction γp → pπ0π0, as the reaction has the lowestamount of non-resonant background amplitude contribu-tions compared to other isospin-channels [1]. The mostcommon observable is the unpolarized cross section I0,followed by the beam asymmetries Is, Ic and I�. Evena few double polarization observables in quasi two-bodykinematics were measured, i.e. E and H. Until now, notriple polarization observables were extracted, as it is ex-perimentally challenging to measure the polarization of arecoiling particle [18, 19] in addition to a polarized beamand a polarized target.

TABLE III. The 64 observables are grouped into eight cate-gories according to the polarization needed to measure theseobservables (beam (B), target (T ) and recoil (R)). The no-tation used in the original paper of Roberts and Oed [5] isused for the observables. The observable I0 corresponds tothe unpolarized cross section.

Category Subcategory Observables

I0

B Bl Is, Ic

B� I�

T Px,Py,Pz

R Px’,Py’,Pz’

BT BlT Psx,P

sy,P

sz,P

cx,P

cy,P

cz

B�T P�x ,P�y ,P�z

BR BlR Psx’,P

sy’,P

sz’,P

cx’,P

cy’,P

cz’

B�R P�x’,P�y’,P

�z’

T R Oxx’,Oxy’,Oxz’,Oyx’,Oyy’,Oyz’,Ozx’Ozy’,Ozz’

BT R

BlT R Osxx’,Osxy’,Osxz’,Osyx’,Osyy’,Osyz’,Oszx’Oszy’,Oszz’Ocxx’,Ocxy’,Ocxz’,Ocyx’,Ocyy’,Ocyz’,Oczx’Oczy’,Oczz’

B�T R O�xx’,O�xy’,O

�xz’,O

�yx’,O

�yy’,O

�yz’,O

�zx′

O�zy’,O�zz’

6

TABLE IV. A collection of polarization observable measurements for two pseudoscalar meson photoproduction.

Observable Energy range Elabγ Facility Reference

γp→ pπ0π0

I0 309− 792 MeV TAPS at MAMI Härter et al. [20]I0 309− 820 MeV TAPS at MAMI Wolf et al. [21]I0 200− 820 MeV TAPS at MAMI Kleber et al. [22]I0 300− 425 MeV TAPS at MAMI Kotulla et al. [23]I0 309− 800 MeV CB/TAPS at MAMI Zehr et al. [24]I0 309− 1400 MeV CB/TAPS at MAMI Kashevarov et al. [25]I0 432− 1374 MeV CB/TAPS at MAMI Dieterle et al. [26]I0 400− 800 MeV DAPHNE at MAMI Braghieri et al. [27]I0 400− 800 MeV DAPHNE at MAMI Ahrens et al. [28]I0 309− 820 MeV A2+TAPS at MAMI, CB-ELSA Sarantsev et al. [29]I0 400− 1300 MeV CB-ELSA Thoma et al. [30]I0 ∼ 750− 2500 MeV CB-ELSA/TAPS at ELSA Thiel et al. [31]

I0,Σ 600− 2500 MeV CB-ELSA/TAPS at ELSA Sokhoyan et al. [32]I0 650− 1500 MeV GRAAL Assafiri et al. [33]Σ 650− 1450 MeV GRAAL Assafiri et al. [33]Σ 650− 1450 MeV CB-ELSA Thoma et al. [30]I� 560− 810 MeV CB/TAPS at MAMI Krambrich et al. [34]I� ∼ 600− 1400 MeV CB/TAPS at MAMI Oberle et al. [35]I� 550− 820 MeV CB/TAPS at MAMI Zehr et al. [24]

E, σ1/2, σ3/2 ∼ 431− 1455 MeV CB/TAPS at MAMI Dieterle et al. [36]Px,Py, T,H, P 650− 2600 MeV CB-ELSA/TAPS at ELSA Seifen et al. [14]

Ic, Is 970− 1650 MeV CB-ELSA/TAPS at ELSA Sokhoyan et al. [32]

γp→ pπ+π−

I0 400− 800 MeV DAPHNE at MAMI Braghieri et al. [27]I0 400− 800 MeV DAPHNE at MAMI Ahrens et al. [37]I0 370− 940 MeV LNF Carbonara et al. [38]I0 800− 1100 MeV NKS at LNS Hirose et al. [39]I0 500− 4800 MeV CEA Crouch et al. [40]I0 ∼ 560− 2560 MeV SAPHIR at ELSA Wu et al. [41]I� 575− 815 MeV TAPS at MAMI Krambrich et al. [34]I� 502− 2350 MeV CLAS at JLAB Strauch et al. [42]I� 1100− 5400 MeV CLAS at JLAB Badui et al. [43]

7

TABLE V. (Continuation of Table IV) A collection of polarization observable measurements for two pseudoscalar mesonphotoproduction.

Observable Energy range Elabγ Facility Reference

γp→ pπ0η

I0 ∼ 930− 2500 MeV CBELSA/TAPS at ELSA Gutz et al. [15]I0 ∼ 1070− 2860 MeV CBELSA at ELSA Horn et al. [44]I0 950− 1400 MeV CB/TAPS at MAMI Kashevarov et al. [45]I0 1000− 1150 MeV GeV-γ at LNS Nakabayashi et al. [46]

I0,Σ ∼ 930− 1500 MeV GRAAL Ajaka et al. [47]Σ 970− 1650 MeV CBELSA/TAPS at ELSA Gutz et al. [48]Σ ∼ 1070− 1550 MeV CBELSA/TAPS at ELSA Gutz et al. [15]

Ic, Is 970− 1650 MeV CBELSA/TAPS at ELSA Gutz et al. [49]Ic, Is ∼ 1081− 1550 MeV CBELSA/TAPS at ELSA Gutz et al. [15]

γp→ nπ+π0

I0 300− 820 MeV TAPS at MAMI Langgärtner et al. [50]I0 ∼ 325− 800 MeV CB/TAPS at MAMI Zehr et al. [24]I0 400− 800 MeV DAPHNE at MAMI Braghieri et al. [27]I0 400− 800 MeV DAPHNE at MAMI Ahrens et al. [51]I� 520− 820 MeV CB/TAPS at MAMI Krambrich et al. [34]I� ∼ 550− 820 MeV CB/TAPS at MAMI Zehr et al. [24]

γn→ nπ0π0

I0,Σ ∼ 600− 1500 MeV GRAAL Ajaka et al. [52]I0 ∼ 430− 1371 MeV CB/TAPS at MAMI Dieterle et al. [26]

γn→ pπ−π0

I0 ∼ 370− 940 MeV LNF Carbonara et al. [38]I0 ∼ 450− 800 MeV DAPHNE at MAMI Zabrodin et al. [53]I0 ∼ 500− 800 MeV DAPHNE at MAMI Zabrodin et al. [54]

γn→ nπ+π−

I0 370− 940 MeV LNF Carbonara et al. [38]

γp→ pK+K−

I� 1100− 5400 MeV CLAS at JLAB Badui et al. [43]

8

1

2

8

3

4 5

6

7

1

2

3

4 5

6

7

8

1

2

4

3

8

7

5

6

FIG. 2. Examples of graph topologies. Only three out of 2520unique cycle graphs with 8 nodes are shown.

V. APPROACH

The results of Section II imply the following steps: Oneconstructs all unique graph topologies with N nodes us-ing combinatorial methods. A few examples of possi-ble graphs are shown in Fig. 2. The number of uniquetopologies is solely determined by the number of nodes(or edges), i.e. for N ≥ 3 it is N !/(2N) [13].

In a second step, all possible edge configurations whichyield a complete set of observables are constructed. Anexample is shown in Fig. 1. This is done for each uniquetopology. The total number of possible edge configura-

tions can be calculated by∑Nk=1

(Nk

)for all odd k ≤ N .

The final step involves the mapping from bilinear formsto the actual observables. Referring again to the exam-ple in Fig. 1, the overall question is: Which combinationsof observables can be solely described by these bilinearproducts (given that all amplitudes are known)? Consid-ering Tables I and II, the following relations are evident:

∼ sin(φ12) 7→ {OIIIs1 ,OIIIs2 ,OIIIs3 ,OIIIs4 }, (6)∼ sin(φ34) 7→ {OIIIs1 ,OIIIs2 ,OIIIs3 ,OIIIs4 }, (7)∼ sin(φ56) 7→ {OIIIs1 ,OIIIs2 ,OIIIs3 ,OIIIs4 }, (8)∼ sin(φ78) 7→ {OIIIs1 ,OIIIs2 ,OIIIs3 ,OIIIs4 } (9)∼ sin(φ14) 7→ {OIVs1 ,OIVs2 ,OIVs3 ,OIVs4 }, (10)∼ sin(φ58) 7→ {OIVs1 ,OIVs2 ,OIVs3 ,OIVs4 }, (11)∼ sin(φ27) 7→ {OVIIIs1 ,OVIIIs2 ,OVIIIs3 ,OVIIIs4 }, (12)

∼ cos(φ36) 7→ {OVIIIc1 ,OVIIIc2 ,OVIIIc3 ,OVIIIc4 }. (13)

Thus, the complete set of observables which corresponds

to the graph configuration shown in Fig. 1 is:

{OI1,OI2,OI3,OI4,OI5,OI6,OI7,OI8,OIIIs1 ,OIIIs2 ,OIIIs3 ,OIIIs4 ,OIVs1 ,OIVs2 ,OIVs3 ,OIVs4 , (14)OVIIIs1 ,OVIIIs2 ,OVIIIs3 ,OVIIIs4 ,OVIIIc1 ,OVIIIc2 ,OVIIIc3 ,OVIIIc4 }.

In general one needs to add the observables which aresolely described by moduli, in order to fix the moduli ofthe complex amplitudes ti (see Section II). In this case,these are the group I observables, as shown in Table I.Thus, Eq. (14) accounts to a total of 24 observables.

The same result can be obtained by using the relation:

t∗j ti =1

8

64∑α=1

ΓαijOα, (15)

where α is an index running through the observableslisted in Tables I and II and the Γ−matrices as listedin Table IX. Equation (15) is derived by using Oα =∑8i,j=1 t

∗iΓ

αijtj in combination with the completeness re-

lation of the Γ-matrices∑64α=1 Γ

αaiΓ

αjb = 8 · δabδij .

VI. RESULTS FOR N=8

For N = 8 one has 2520 unique cycle graphs, eachwith 128 unique edge configurations, as explained in Sec-tion V. Hence, there exist in total 128 · 2520 = 322560edge configurations which yield a complete set of observ-ables. However, the resulting sets are not all linearlyindependent.

The whole algorithm was implemented in MATHE-MATICA [55], but can just as easily be implementedin other languages such as JULIA [56]. Filtering outthe redundant sets, one is left with 5964 unique sets ofobservables. The length of the sets varies between thetopologies as well as between different edge configura-tions. To be exact, it varies between a total of 24 and 40observables.

Without loss of generality, the further analysis focuseson the 392 distinct sets with 24 observables. A numericalanalysis was performed which showed that these sets areindeed complete. The applied algorithm is described inAppendix A.

Further characteristics of the minimal sets accordingto Moravcsik involve:

• each set inhibits at least five triple polarization ob-servables,

• the sets are constructed from four or five differentshape classes.

However, these sets are slightly over complete since eachobservable depends on more than one bilinear product.According to the current knowledge [11, 13] a truly min-imal complete set consists of 2N observables. Thus thetask remains to reduce the slightly over complete sets byeight observables while retaining the completeness.

9

VII. REDUCTION TO MINIMAL SETS OF 2N

A. Numerical calculation

The smallest complete sets, which emerge from themodified version of Moravcsik’s theorem, have a lengthof 24 (for N = 8). Eight of these observables can not beomitted, namely the group I observables, as discussed inSection V. From the remaining 16 observables one con-structs all possible subsets containing eight observables,which amounts to

(168

)= 12870 distinct sets.

In principle this is done for all sets with length of24, leading to just over five million minimal, completeset candidates. This number can be further reducedby ∼ 7.7%, by noting that sets containing only oneor two distinct observable groups (apart from groupI) do not correspond to a connected graph and thusdo not form a complete set. There are also a fewcases in which three distinct observable groups (apartfrom group I) are not able to form a connected graph,i.e. {II,III,IV}, {II,V,VI}, {II,VII,VIII}, {III,V,VII},{III,VI,VIII}, {IV,V,VIII} and {IV,VI,VII}.

However, due to the enormous number of possible can-didate sets just a minor excerpt was analyzed for thispaper. The sets of interest are checked for completenessvia a numerical analysis. The employed algorithm is de-scribed in Appendix A.

So far 4185 unique truly minimal sets of length 2N =16 have been found. There are two major differences tothe slightly over complete sets with 24 observables. Onthe one hand, all sets found are constructed from exactlyfour different shape classes. On the other hand, trulyminimal complete sets exist with a minimal number oftriple polarization observables, namely only OI4 = O�yy’from group I. Hence, this observable has to be includedin every set as explained in Section II.

In total 69 sets with only one triple polarization ob-servable have been found. All of them are shown in Ta-ble VII. Hence, these are the most promising ones whenit comes to the experimental verification of Moravcsik’stheorem.

B. Algebraic phase-fixing method

In the following, the phase-fixing approach first devel-oped by Nakayama in a treatment of single-meson pho-toproduction (i.e. for N = 4 amplitudes) [8] is adaptedto two meson photoproduction. Thus it is possible, al-though tedious, to derive minimal complete sets of ob-servables with algebraic methods.

Since the full mathematical derivation is quite exten-sive, all mathematical details are given in a supplemen-tary note [57].

One starts by combining, i.e. adding or subtracting,the observables within one shape-class, in such a waythat the result only depends on two relative phases. Bydoing this, a “decoupled” shape-class is formed. See also

TABLE VI. The 14 decoupled shape-classes IIa, IIb, . . ., VI-IIa, VIIIb are listed together with their corresponding pairsof relative phases.

IIa −→ {φ13, φ24} IIb −→ {φ57, φ68}IIIa −→ {φ12, φ34} IIIb −→ {φ56, φ78}IVa −→ {φ14, φ23} IVb −→ {φ58, φ67}Va −→ {φ15, φ26} Vb −→ {φ37, φ48}

VIa −→ {φ17, φ28} VIb −→ {φ35, φ46}VIIa −→ {φ16, φ25} VIIb −→ {φ38, φ47}

VIIIa −→ {φ18, φ27} VIIIb −→ {φ36, φ45}

1

3

5

72

4

6

8

{IIa,IIb,VIb,VIIIa}

1

2

6

7

3

4

8

5

{Va,Vb,IIIa,IVb}

FIG. 3. Examples are shown for graph topologies which im-ply the possibility for a consistency relation (cf. the relativephases listed in Table VI).

Table VI. In that way, one establishes a mathematicalsimilarity with the shape-classes in single-meson photo-production [8, 13]. For shape-class II this would be:

IIa : OIIs1 +OIIs2,OIIs3 +OIIs4,OIIc1 +OIIc2,OIIc3 +OIIc4, (16)IIb : OIIs1 −OIIs2,OIIs3 −OIIs4,OIIc1 −OIIc2,OIIc3 −OIIc4. (17)

The algebraic approach shown here works out only incase the observables are selected from very particularcombinations of four decoupled shape-classes.

More precisely, it has to be possible to establish a “con-sistency relation” (cf. Eq. (1)) among the relative phasesbelonging to all the involved decoupled shape-classes. Anecessary and sufficient condition for this can be formu-lated in terms of the graph constructed from the relativephases (cf. Section II): the latter graph has to be a cyclegraph.

There exist 40 possible combinations of four decoupledshape classes fulfilling these requirements and which havethe following general form:

{Xa,Xb,Y,Z}. (18)

Two examples are shown in Fig. 3, a complete list can befound in the supplementary material [57]. The followingderivation holds for all combinations of shape-classes ofthe form given in Eq. (18).

The following discussion focuses on the example case:

{IIa, IIb,VIb,VIIIa}. (19)

10

For the remaining 39 cases, the derivation proceeds anal-ogously.

For the example case Eq. (19), the consistency relationreads (cf. Table VI):

φ13 + φ24︸︷︷︸IIa

+φ57 + φ68︸︷︷︸IIb

= φ18 + φ27︸︷︷︸VIIIa

−φ35 − φ46︸︷︷︸VIb

. (20)

According to Nakayama [8], the “phase-fixing” procedurestarts by picking a particular combination of observablesfrom the considered combination of shape-classes, i.e.from Eq. (19). In general, one picks two observablesfrom shape-class Xa, two from Xb and one observableeach from two different shape-classes selected from the12 remaining. For the case at hand, these are two observ-ables from shape-class IIa, two from IIb, one from VIb aswell as one from VIIIa. For any selection of observableswhich has this pattern, one has to work out the remainingdiscrete phase-ambiguities which exist for the associatedrelative-phases. For each combination of possible dis-crete ambiguities, the consistency relation Eq. (20) thenhas to be evaluated. In case the consistency relations ofsuch a combination are all linearly independent [8, 13],the considered set of observables is complete.

The way forward is analogous to [8]. A detailed de-scription, on how to determine all possible discrete am-biguities and determining whether a set of consistencyrelations is linear independent or not, can be found inthe supplementary material [57].

The result for the discussed example can be found inTable VIII. The shown results come only from consider-ing the left side of Eq. (20). In general, the determinationof the discrete ambiguities of the left side is easier thanthat of the right side. So theoretically even more com-binations are possible. Unfortunately, the minimal setsshown in Table VIII contain at least three triple polar-ization observables.

VIII. IMPLICATIONS FOREXPERIMENTALISTS

The experimental verification of complete sets, in theframework of single pseudoscalar meson photoproduc-tion, was studied for example by Ireland [9] and Vrancx etal. [58]. They concluded that as soon as the data have afinite measurement uncertainty, it might not be possibleto determine unique solutions for the amplitudes in caseone uses a mathematical, minimal complete set of ob-servables. It is believed that additional obervables couldresolve the ambiguity. Thus it is important to measuremore than 16 observables [9].

Observables which are relatively easy to measure rankin the categories B, T ,BT as they do not require a re-coil polarimeter. Even though they form a group of 16observables, they do not form a complete set, which wasverified numerically in this study. The measurement ofsome well chosen observables from the categories R, BR,T R and BT R is essential. Out of the 69 possible com-plete sets, that require the measurement of only one triple

polarization observable and which are listed in Table VII,the sets 6), 12), 15), 27), 30), 41) and 47) contain theminimal number of recoil polarization observables, e.g.the set 15) contains the following observables:

{I�,Py,Py’,O�yy’,Oyy’,P�y’,P

�y , I0,

Px,Pz,Px’,Psx,P

�x ,P

cz,P

�z ,P

�x’}. (21)

There exist data for 8 of these observables (I0, I�, Px, Py,

Py’, Oyy’(= −Ic), Psx, P�z ) for the pπ

0π0 final state, albeitnot having a perfect overlap of the energy and angularranges covered by the different data sets (see Tables IVand V). The remaining 8 observables could be measuredin the future in three different experiments using a lin-early polarized photon beam with a longitudinally polar-ized target (Pz, P

cz), using a circularly polarized photon

beam and a transversely polarized target (P�x , P�y ) and

by employing a recoil polarimeter in addition to the lat-ter configuration, the observables Px’, P

�x’, P

�y’ and O

�yy’

can be obtained as well.

IX. CONCLUSION AND OUTLOOK

Within this paper, the problem of findining completesets for two pseudoscalar meson photoproduction wasstudied. For this purpose, a slightly modified versionof Moravcsik’s theorem was applied. This method iscapable of extracting complete sets of observables in atotally automated manner. The automation capability,easy accessibility as well as the adaptability for reactionswith arbitrary N are the strengths of Moravcsik’s theo-rem. However, it turns out that the resulting sets fromMoravcsik are slightly over-complete since each observ-able depends on more than one bilinear product. Forthis reason, a numerical- as well as an algebraic methodis discussed in order to reduce these sets to minimal com-plete sets containing 2N = 16 observables. The charac-teristics of the minimal sets are discussed. Finally, 69minimal complete sets containing the minimal numberof triple polarization observables, namely only one, arepresented. From these subsets in combination with theextensive overview of already performed measurementsin two pseudoscalar meson photoproduction, the mostpromising set of observables, which could be measuredin the near future, is presented.

Further studies could be performed on how Moravc-sik’s theorem should be adapted in order to yield directlyminimal complete sets. This would decrease the numeri-cal effort enormously and would make the theorem evenmore accessible.

ACKNOWLEDGMENTS

The authors would like to thank Prof. Dr. Thoma, fora fruitful discussion and constructive comments on themanuscript and Prof. Dr. Beck for his support.

11

TABLE VII. Truly minimal sets consisting of 2N = 16 observables with the minimal number of triple polarization observables,i.e. just OI4 = O�yy’. However, the group I observables are not explicitly shown, as they are common to all complete sets. Theused notation is analogous to Roberts/Oed [5].

1) Pz Px’ Pz’ Psx P

cz P

�z P

�x’ P

�z’

2) Pz Psx P

cz P

�z P

�x’ P

�z’ Oyx’ Oyz’

3) Pz Psx P

cz P

�x’ P

�z’ Oyx’ Oyz’ Ozy’

4) Px’ Pz’ Psx P

cz P

�z P

�x’ P

�z’ Ozy’

5) Psx Pcz P

�z P

�x’ P


6) Px Pz Pz’ Psx P

�x P

cz P

�z P

�z’


cz P

�z’ Oxy’ Ozy’


cz Oxy’ Oyz’ Ozy’

9) Pz’ Psx P

�x P

cz P

�z P


10) Pz’ Psx P

�x P

cz P

�z Oxy’ Oyz’ Ozy’

11) Psx P�x P

cz P

�z P

�z’ Oxy’ Oyz’ Ozy’

12) Pz Pcx P

sx P

cz P

sz P

�z P

�z’ Oyz’

13) Pz Pcx P

sx P

cz P

sz P

�z’ Oyz’ Ozy’

14) Pcx Psx P

cz P

sz P

�z P

�z’ Oyz’ Ozy’

15) Px Pz Px’ Psx P

�x P

cz P

�z P

�x’


cz P

�x’ Oxy’ Ozy’


cz Oxy’ Oyx’ Ozy’

18) Px’ Psx P

�x P

cz P

�z P


19) Px’ Psx P

�x P

cz P

�z Oxy’ Oyx’ Ozy’

20) Psx P�x P

cz P

�z P

�x’ Oxy’ Oyx’ Ozy’

21) Pz Px’ Pz’ Pcx P

sz P

�z P

�x’ P

�z’

22) Pz Pcx P

sz P

�z P

�x’ P


23) Pz Px’ Pz’ Pcx P

sz Oyx’ Oyz’ Ozy’

24) Pz Pcx P

sz P

�x’ P


25) Px’ Pz’ Pcx P

sz P

�z P

�x’ P

�z’ Ozy’

26) Pcx Psz P

�z P

�x’ P


27) Pz Pcx P

sx P

cz P

sz P

�z P

�x’ Oyx’

28) Pz Pcx P

sx P

cz P

sz P

�x’ Oyx’ Ozy’

29) Pcx Psx P

cz P

sz P

�z P

�x’ Oyx’ Ozy’

30) Px Pz Pz’ Pcx P

�x P

sz P

�z P

�z’


sz P



sz Oxy’ Oyz’ Ozy’

33) Pz’ Pcx P

�x P

sz P

�z P


34) Pz’ Pcx P

�x P

sz P

�z Oxy’ Oyz’ Ozy’

35) Pcx P�x P

sz P

�z P

�z’ Oxy’ Oyz’ Ozy’

36) Px Px’ Pz’ Psx P

�x P

cz P

�x’ P

�z’

37) Px Psx P

�x P

cz P

�x’ P


38) Px Psx P

cz P

�x’ P

�z’ Oxy’ Oyx’ Oyz’

39) Px’ Pz’ Psx P

�x P

cz P

�x’ P

�z’ Oxy’

40) Psx P�x P

cz P

�x’ P

�z’ Oxy’ Oyx’ Oyz’

41) Px Pz Px’ Pcx P

�x P

sz P

�z P

�x’


sz P



sz Oxy’ Oyx’ Ozy’

44) Px’ Pcx P

�x P

sz P

�z P


45) Px’ Pcx P

�x P

sz P

�z Oxy’ Oyx’ Ozy’

46) Pcx P�x P

sz P

�z P

�x’ Oxy’ Oyx’ Ozy’

47) Px Pcx P

sx P

�x P

cz P

sz P

�z’ Oyz’

48) Px Pcx P

sx P

cz P

sz P

�z’ Oxy’ Oyz’

49) Pcx Psx P

�x P

cz P

sz P

�z’ Oxy’ Oyz’

50) Px Pz P�x P

�z P

cx’ P

sz’ Oxz’ Ozx’

51) Px Pz Pcx’ P

sz’ Oxy’ Oxz’ Ozx’ Ozy’

52) P�x P�z P

cx’ P

sz’ Oxy’ Oxz’ Ozx’ Ozy’

53) Ic

Px P�x P

cy Oxx’ Oxz’ Ozx’ Ozz’

54) Ic

Px P�x P

cy’ Oxx’ Oxz’ Ozx’ Ozz’

55) Px P�x P

cy P

cy’ Oxx’ Oxz’ Ozx’ Ozz’

56) Ic

Px Pcy Oxx’ Oxy’ Oxz’ Ozx’ Ozz’

57) Ic

Px Pcy’ Oxx’ Oxy’ Oxz’ Ozx’ Ozz’

58) Px Pcy P

cy’ Oxx’ Oxy’ Oxz’ Ozx’ Ozz’

59) Ic

P�x Pcy Oxx’ Oxy’ Oxz’ Ozx’ Ozz’

60) P�x Pcy P

cy’ Oxx’ Oxy’ Oxz’ Ozx’ Ozz’

61) Ic

Px P�x P

cx’ P

sx’ P

cy’ P

cz’ P

sz’

62) Px P�x P

cy P

cx’ P

sx’ P

cy’ P

cz’ P

sz’

63) Ic

Px Pcy P

cx’ P

sx’ P

cz’ P

sz’ Oxy’

64) Ic

P�x Pcy P

cx’ P

sx’ P

cz’ P

sz’ Oxy’

65) Ic

P�x Pcx’ P

sx’ P

cy’ P

cz’ P

sz’ Oxy’

66) P�x Pcy P

cx’ P

sx’ P

cy’ P

cz’ P

sz’ Oxy’

67) Ic

Is Pcy Psy P

cx’ P

sz’ Oxz’ Ozx’

68) Ic

Is Pcx’ Pcy’ P

sy’ P

sz’ Oxz’ Ozx’

69) Pcy Psy P

cx’ P

cy’ P

sy’ P

sz’ Oxz’ Ozx’

12

Appendix A: Algorithm to check for completeness

The following algorithm was designed by Lothar Tia-tor [59] and was already applied in the paper [60]. Itis used to check if a set of observables is able to resolvecontinuous as well as discrete ambiguities. The startingpoint is a system of multivariate homogeneous polyno-mials f1(~t), . . . , fn(~t). The input is a vector of N com-plex amplitudes ti. Without loss of generality, the over-all phase of the complex amplitudes is fixed by requiringRe(t1) > 0 and Im(t1) = 0. In a next step an N dimen-sional solution vector ~s is formed. It consist of 2N − 1randomly chosen prime numbers within a certain range.These serve as values for the real and imaginary partsof the ti. Using prime numbers and increasing the rangefrom which they are chosen should be capable of reduc-ing the chance to land on a singularity in the solutionspace, where the “condition of equal magnitudes of rela-tive phases”[8] is met.

Finally, the polynomial system:

f1(~t) = g1,

... (A1)

fn(~t) = gn,

is constructed, where gi = fi(~s) is a scalar quantity. Thefunction NSolve from MATHEMATICA [55] is employedto solve the algebraic system for the variables t1, . . . , tN .According to the Wolfram Mathematica documentation[61]: “For systems of algebraic equations, NSolve com-putes a numerical Gröbner basis using an efficient mono-mial ordering, then uses eigensystem methods to extractnumerical roots.”.

The system of polynomials is said to be complete, ifonly one solution is found which furthermore is equivalentto ~s.

TABLE VIII. Truly minimal complete sets, consisting of2N = 16 observables, obtained for the example discussedin Section VII B. The group I observables are not explicitlyshown, as they are common to all complete sets. The usednotation is analogous to Roberts/Oed [5].

Px Pz P�x P

�z P

sz P

cx Oszz’ Ocxz’

Px Pz P�x P

�z P

sz P

cx Oczx’ Osxx’

Px Pz P�x P

�z P

sz P

cx Oczz’ Osxz’

Px Pz P�x P

�z P

sz P

cx Oszx’ Ocxx’

Px Pz P�x P

�z P

cz P

sx Oszz’ Ocxz’

Px Pz P�x P

�z P

cz P

sx Oczx’ Osxx’

Px Pz P�x P

�z P

cz P

sx Oczz’ Osxz’

Px Pz P�x P

�z P

cz P

sx Oszx’ Ocxx’

Px Pz P�x P

�z Oszy’ Ocxy’ Oszz’ Ocxz’

Px Pz P�x P

�z Oszy’ Ocxy’ Oczx’ Osxx’

Px Pz P�x P

�z Oszy’ Ocxy’ Oczz’ Osxz’

Px Pz P�x P

�z Oszy’ Ocxy’ Oszx’ Ocxx’

Px Pz P�x P

�z Oczy’ Osxy’ Oszz’ Ocxz’

Px Pz P�x P

�z Oczy’ Osxy’ Oczx’ Osxx’

Px Pz P�x P

�z Oczy’ Osxy’ Oczz’ Osxz’

Px Pz P�x P

�z Oczy’ Osxy’ Oszx’ Ocxx’

Psz Pcx Oxy’ Ozy’ O�xy’ O

�zy’ O

szz’ Ocxz’


�zy’ O

czx’ Osxx’


�zy’ O

czz’ Osxz’


�zy’ O

szx’ Ocxx’

Pcz Psx Oxy’ Ozy’ O�xy’ O

�zy’ O

szz’ Ocxz’


�zy’ O

czx’ Osxx’


�zy’ O

czz’ Osxz’


�zy’ O

szx’ Ocxx’

Oxy’ Ozy’ O�xy’ O�zy’ O

szy’ Ocxy’ Oszz’ Ocxz’


szy’ Ocxy’ Oczx’ Osxx’


szy’ Ocxy’ Oczz’ Osxz’


szy’ Ocxy’ Oszx’ Ocxx’


czy’ Osxy’ Oszz’ Ocxz’


czy’ Osxy’ Oczx’ Osxx’


czy’ Osxy’ Oczz’ Osxz’


czy’ Osxy’ Oszx’ Ocxx’

13

TABLE IX. Definition of the 64 Γ-matrices in terms of the well known Pauli matrices in combination with the Kroneckerproduct. The gray shaded cells within the column ”Shape-class” correspond to the non-zero matrix entries.

Γ-matrices Definition / 2 Shape-class

ΓI1 2 · (σ3 ⊗ I2 ⊗ I2)ΓI2 2 · (I2 ⊗ σ3 ⊗ I2)ΓI3 2 · (I2 ⊗ I2 ⊗ σ3)ΓI4 2 · (σ3 ⊗ σ3 ⊗ σ3)ΓI5 2 · (I2 ⊗ σ3 ⊗ σ3)ΓI6 2 · (σ3 ⊗ I2 ⊗ σ3)ΓI7 2 · (σ3 ⊗ σ3 ⊗ I2)ΓI8 2 · (I2 ⊗ I2 ⊗ I2)

ΓIIc1 I2 ⊗ σ1 ⊗ I2ΓIIc2 σ

3 ⊗ σ1 ⊗ I2ΓIIc3 I2 ⊗ σ1 ⊗ σ3

ΓIIc4 σ3 ⊗ σ1 ⊗ σ3

ΓIIs1 −I2 ⊗ σ2 ⊗ I2ΓIIs2 −σ3 ⊗ σ2 ⊗ I2ΓIIs3 −I2 ⊗ σ2 ⊗ σ3

ΓIIs4 −σ3 ⊗ σ2 ⊗ σ3

ΓIIIc1 I2 ⊗ I2 ⊗ σ1

ΓIIIc2 σ3 ⊗ I2 ⊗ σ1

ΓIIIc3 I2 ⊗ σ3 ⊗ σ1

ΓIIIc4 σ3 ⊗ σ3 ⊗ σ1

ΓIIIs1 −I2 ⊗ I2 ⊗ σ2

ΓIIIs2 −σ3 ⊗ I2 ⊗ σ2

ΓIIIs3 −I2 ⊗ σ3 ⊗ σ2

ΓIIIs4 −σ3 ⊗ σ3 ⊗ σ2

ΓIVc1 I2 ⊗ σ1 ⊗ σ1

ΓIVc2 σ3 ⊗ σ1 ⊗ σ1

ΓIVc3 −I2 ⊗ σ2 ⊗ σ2

ΓIVc4 −σ3 ⊗ σ2 ⊗ σ2

ΓIVs1 −I2 ⊗ σ2 ⊗ σ1

ΓIVs2 −σ3 ⊗ σ2 ⊗ σ1

ΓIVs3 −I2 ⊗ σ1 ⊗ σ2

ΓIVs4 −σ3 ⊗ σ1 ⊗ σ2

Γ-matrices Definition / 2 Shape-class

ΓVc1 σ1 ⊗ I2 ⊗ I2

ΓVc2 σ1 ⊗ σ3 ⊗ I2

ΓVc3 σ1 ⊗ I2 ⊗ σ3

ΓVc4 σ1 ⊗ σ3 ⊗ σ3

ΓVs1 −σ2 ⊗ I2 ⊗ I2ΓVs2 −σ2 ⊗ σ3 ⊗ I2ΓVs3 −σ2 ⊗ I2 ⊗ σ3

ΓVs4 −σ2 ⊗ σ3 ⊗ σ3

ΓVIc1 σ1 ⊗ σ1 ⊗ I2

ΓVIc2 −σ2 ⊗ σ2 ⊗ I2ΓVIc3 σ

1 ⊗ σ1 ⊗ σ3

ΓVIc4 −σ2 ⊗ σ2 ⊗ σ3

ΓVIs1 −σ2 ⊗ σ1 ⊗ I2ΓVIs2 −σ1 ⊗ σ2 ⊗ I2ΓVIs3 −σ2 ⊗ σ1 ⊗ σ3

ΓVIs4 −σ1 ⊗ σ2 ⊗ σ3

ΓVIIc1 σ1 ⊗ I2 ⊗ σ1

ΓVIIc2 σ1 ⊗ σ3 ⊗ σ1

ΓVIIc3 −σ2 ⊗ I2 ⊗ σ2

ΓVIIc4 −σ2 ⊗ σ3 ⊗ σ2

ΓVIIs1 −σ2 ⊗ I2 ⊗ σ1

ΓVIIs2 −σ2 ⊗ σ3 ⊗ σ1

ΓVIIs3 −σ1 ⊗ I2 ⊗ σ2

ΓVIIs4 −σ1 ⊗ σ3 ⊗ σ2

ΓVIIIc1 σ1 ⊗ σ1 ⊗ σ1

ΓVIIIc2 −σ2 ⊗ σ2 ⊗ σ1



ΓVIIIs1 −σ2 ⊗ σ1 ⊗ σ1



ΓVIIIs4 σ2 ⊗ σ2 ⊗ σ2

14

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16

tutorial/SomeNotesOnInternalImplementation.html; Accessed 25 May 2020.

https://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.html

Minimal complete sets for two pseudoscalar meson photoproductionAbstractI IntroductionII Moravcsik's theoremIII Polarization observablesIV Database for two pseudoscalar meson photoproductionV ApproachVI Results for N=8VII Reduction to minimal sets of 2NA Numerical calculationB Algebraic phase-fixing method

VIII Implications for experimentalistsIX Conclusion and Outlook AcknowledgmentsA Algorithm to check for completeness References

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1Helmholtz Institut fur Strahlen- und Kernphysik ... · 1Helmholtz Institut fur Strahlen- und Kernphysik, Universit at Bonn, Germany For photoproduction reactions with nal states