New graphical criterion for the selection of complete sets of polarization observables and its application to single-meson photoproduction as well as electroproduction

This paper combines the graph-theoretical ideas behind Moravcsik's theorem with a completely analytic derivation of discrete phase ambiguities, recently published by Nakayama. The result is a new graphical procedure for the derivation of certain types of complete sets of observables for an amplitude-extraction problem with $N$ helicity amplitudes. The procedure is applied to pseudoscalar meson photoproduction ($N=4$ amplitudes) and electroproduction ($N=6$ amplitudes), yielding complete sets with minimal length of $2N$ observables. For the case of electroproduction, this is the first time an extensive list of minimal complete sets is published. Furthermore, the generalization of the proposed procedure to processes with a larger number of amplitudes, i.e., $N>6$ amplitudes, is sketched. The generalized procedure is outlined for the next more complicated example of two-meson photoproduction ($N=8$ amplitudes).

Figure 1

The general consistency relation (5) is illustrated for the example of an amplitude-extraction problem with $N=4$ amplitudes $b_1,...,b_4$. The relation one deduces geometrically from the given diagram is ${\varphi }_{12}+{\varphi }_{23}+{\varphi }_{34}+{\varphi }_{41}=2\pi$, which up to addition of $2\pi$ is equivalent to ${\varphi }_{12}+{\varphi }_{23}+{\varphi }_{34}+{\varphi }_{41}=0$.

Figure 2

The meaning of the angle $\zeta \equiv {\zeta }_{1+,2+}^n$ is illustrated. This picture directly corresponds to the definitions (B4) and (B5) in Appendix pp2. The modulus of the value for each observable $({\mathcal{O}}_{1+}^n,{\mathcal{O}}_{2+}^n)$ is limited by the unpolarized differential cross section ${\sigma }_0$ of the considered process.

Figure 3

The three possible start topologies for pseudoscalar meson photoproduction ($N=4$ amplitudes) are drawn here. Each graph is drawn with a particular direction, which is intimately connected to our way of writing the corresponding consistency relation, i.e., Eqs. (15, 16, 17) (cf. comments made below Eq. (5) in Sec. 2). Each topology corresponds to the relative phases from a particular combination of two shape classes (cf. Table 1), as is indicated above the graphs (cf. discussion in the main text).

Figure 4

The first example for a fully complete graph according to Theorem 1 from Sec. 2 is shown. This graph can be inferred from the selection of observables (18). The dashed single-lined arrows indicate the fact that the selection A.1 has been applied for the two observables belonging to shape class $c$, with corresponding relative phases $\{{\varphi }_{12},{\varphi }_{34}\}$. The dashed double-lined arrows indicate the fact that a selection of type B has been applied for the two observables from the shape class $b$, with corresponding relative phases $\{{\varphi }_{14},{\varphi }_{23}\}$. The $\zeta$-sign arrows have been drawn into the dashed double-lined arrows, according to the selection B.2 taken from shape class $b$ [cf. Eq. (12)].

Figure 5

The second example for a fully complete graph according to Theorem 1 from Sec. 2 is shown, which can be inferred from the selection of observables (27). The dashed double-lined arrows indicate a selection of type B from the shape class $b$ (relative phases $\{{\varphi }_{14},{\varphi }_{23}\}$), while the dotted double-lined arrows represent a B-type selection from shape class $a$ (relative phases $\{{\varphi }_{13},{\varphi }_{24}\}$). The $\zeta$-sign arrows have been drawn into the respective double-lined arrows according to the selection (27).

Figure 6

The third example-graph mentioned in the main text is shown here, which can be inferred from the set of observables (32). The solid and dashed double-lined arrows have the same meaning as in Fig. 4. The $\zeta$-sign arrows, which have been drawn inside of the double-lined arrows, correspond to the selection (32). This graph is not fully complete according to Theorem 1 from Sec. 2.

Figure 7

These five types of graphs have been derived from the first (i.e., boxlike) topology I shown in Fig. 3. They correspond to different combinations of selections of type A and B taken from the shape classes $b$ (relative phases $\{{\varphi }_{14},{\varphi }_{23}\}$) and $c$ (relative phases $\{{\varphi }_{12},{\varphi }_{34}\}$), as described in Sec. 2. At least one pair of double-lined arrows has to be contained in the graphs, i.e., at least one selection of type B has to have been made. Graphs resulting solely from selections of type A have not been shown (there exist 4 of such graphs), since they cannot yield fully complete sets (cf. Theorem 2 from Appendix pp1).

Figure 8

These four graphs are directly deduced from graph type (I.i) shown in Fig. 7, by inserting all possible combinations of $\zeta$-sign arrows into the double-lined arrows. The two graphs (I.i.2) and (I.i.3) satisfy all the completeness conditions posed by Theorem 1 from Sec. 2, while the graphs (I.i.1) and (I.i.4) violate these criteria.

Figure 9

This set of 16 graphs is directly deduced from graph type (I.v) shown in Fig. 7, by inserting all possible combinations of $\zeta$-sign arrows into the double-lined arrows. From these 16 graphs, only four violate the completeness criterion posed in Theorem 1 of Sec. 2, namely: (I.v.1), (I.v.4), (I.v.13), and (I.v.16).

Figure 10

The 8 possible start topologies for pseudoscalar meson electroproduction ($N=6$ amplitudes), which allow for a selection of three pairs of observables from three different shape classes, are shown here. The direction of translation is indicated for each graph as well. This direction is intimately connected to our convention of writing the corresponding consistency relation, i.e., Eqs. (39) to (46) (see also the comments made below Eq. (5) in Sec. 2). Each of the shown topologies corresponds to the relative phases from a particular combination of three shape classes for electroproduction (cf. Table 3). The combinations of shape classes are also indicated above the graphs (cf. discussion in the main text).

Figure 11

The first example for a graph in electroproduction ($N=6$ amplitudes) is shown. This graph is fully complete according to Theorem 1 from Sec. 2. It can be inferred from the selection of observables (47). The dashed single-lined arrows indicate the selection of type A.1 for the shape class $e$, while the solid single-lined arrows represent the selection of type A.2 for the shape class $g$. The solid double-lined arrows indicate the fact that a selection of type B has been applied for the two observables from the shape class $a$. The $\zeta$-sign arrows have been drawn into the solid double-lined arrows according to the selection (47).

Figure 12

The second example graph for electroproduction is shown. This graph violates the completeness criterion posed in Theorem 1 from Sec. 2. It can be inferred from the selection of observables (48). The solid single-lined arrows indicate the selection of type A.2 for the shape class $g$. The solid double-lined arrows indicate the fact that a selection of type B has been applied for the two observables from the shape class $a$, while the dashed double-lined arrows represent the same fact for the relative phases belonging to shape class $e$. The $\zeta$-sign arrows have been drawn into the double-lined arrows according to the selection (48).

Figure 13

These nineteen types of graphs arise from the first topology with direction shown in Fig. 10. They correspond to different combinations of selections of types A and B (cf. Sec. 2) for the three pairs of relative phases belonging to the shape classes $(a,e,g)$ (i.e., pairs of relative phases $\{{\varphi }_{13},{\varphi }_{24}\}, \{{\varphi }_{36},{\varphi }_{45}\}$ and $\{{\varphi }_{15},{\varphi }_{26}\}$, respectively), with at least one pair of double-lined arrows, i.e., at least one selection of type B. The solid double-lined arrows mark relative phases belonging to the shape class $a$, dashed double-lined arrows mark relative phases from the shape class $e$ and the dotted double-lined arrows refer to the shape class $g$. Graphs resulting solely from selections of type A are not shown in this plot (there exist eight such graphs), since they cannot yield fully complete sets according to Theorem 2 from Appendix pp1.

Figure 14

The diagrams show three connected graphs which meet all the criteria posed by Theorem 2. These graphs with 4 vertices correspond to the well-known example case of pseudoscalar meson photoproduction, which is a problem with $N=4$ amplitudes. Green dashed lines denote the imaginary part of a bilinear amplitude product, while the real part of such a product is represented by a blue solid line. These graphs have been taken over identically from Ref. [69].