Fixed-$t$ analyticity as a constraint in single-energy partial-wave analyses of meson photoproduction reactions

Partial-wave amplitudes of meson photoproduction reactions are an important source of information in baryon spectroscopy. We investigate a new approach in single-energy partial-wave analyses of these reactions. Instead of using a constraint to theoretical models in order to achieve solutions which are continuous in energy, we enforce the analyticity of the amplitudes at fixed values of the Mandelstam variable $t$. We present an iterative procedure with successive fixed-$t$ amplitude analyses which constrain the single-energy partial-wave analyses and apply this method to the $\gamma p\to \eta p$ reaction. We use pseudodata, generated by the EtaMAID model, to test the method and to analyze ambiguities. Finally, we present an analytically constrained partial-wave analysis using experimental data for four polarization observables recently measured at MAMI and GRAAL in the energy range from threshold to $\sqrt{s}=1.85$ GeV.

Figure 1

The Mandelstam plane for $\eta$ photoproduction on the nucleon. The red solid curves are the boundaries of the physical region from $\theta =0$ to $\theta ={180}^{\circ }$ and the red dashed line shows $\theta ={90}^{\circ }$. The tilted vertical lines from left to right are the thresholds for $\pi N,\eta N,\eta \prime N$ production, respectively. The (blue) dotted line shows the $u$-channel threshold $u={(m_N+m_{\pi })}^2$. The horizontal lines denote the $t$ values $-0.2,-0.5{\mathrm{GeV}}^2$, where we will show our fixed-$t$ analyses. The magenta parts give the part inside the physical region, where the cyan parts have to be evaluated in the unphysical region. The threshold values for $\gamma ,\eta$ in $W$ are $W_{\mathrm{thr}}=1.486$ GeV ($t=-0.2{\mathrm{GeV}}^2$) and $W_{\mathrm{thr}}=1.554$ GeV ($t=-0.5{\mathrm{GeV}}^2$).

Figure 2

Kinematics of photoproduction and frames for polarization. The frame $\{x,y,z\}$ is used for target polarization $\{P_x,P_y,P_z\}$, whereas the recoil polarization $\{P_{x^{\prime }},P_{y^{\prime }},P_{z^{\prime }}\}$ is defined in the frame $\{x^{\prime },y^{\prime },z^{\prime }\}$, which is rotated around $y^{\prime }=y$ by the polar angle $\theta$. The azimuthal angle $\phi$ is defined in the $\{x,y\}$ plane and is zero in the projection shown in the figure.

Figure 3

Analytic structure of invariant amplitudes for a fixed-$t$ value in the complex ${\nu }^2$ plane.

Figure 4

Iterative minimization scheme which achieves point-to-point continuity in energy using fixed-$t$ analyticity as a constraint. (IA, invariant amplitudes; FT AA, fixed-$t$ amplitude analysis; SE PWA, single-energy partial wave analysis).

Figure 5

Result of unconstrained single-energy fit described in the text. The blue and red points show the real and imaginary parts of the multipoles obtained in the fit compared to the “true” multipoles from the underlying EtaMAID-2015 model (blue and red solid lines).

Figure 6

Fixed-$t$ amplitude analysis of a complete set of pseudodata at $t=-0.2{\mathrm{GeV}}^2$ (top) and $t=-0.5{\mathrm{GeV}}^2$ (bottom). The minimization is performed according to Eq. (47). The data points are pseudodata with a precision of $0.1\%$. The lines are obtained from the Pietarinen expansion with fitted coefficients.

Figure 7

Single-energy partial-wave analysis of a complete set of pseudodata at $W=1543$ MeV (top) and $W=1770$ MeV (bottom). The minimization is performed according to Eq. (49). The data points are pseudodata with a precision of $0.1\%$. The lines are obtained from the CGLN amplitudes with fitted multipoles up to $L_{\max }=5$.

Figure 8

Helicity amplitudes $H_k(W,cos\theta )$ after final (second) iteration at $W=1543\mathrm{MeV}$ (left) and $W=1660\mathrm{MeV}$ (right). Real and imaginary parts of the helicity amplitudes (blue and red dots) are obtained from independent fixed-$t$ AA at different $t$ values in previous (first) iteration. The full lines are the helicity amplitudes from final iteration in SE PWA.

Figure 9

Real (blue) and imaginary (red) parts of electric and magnetic multipoles up to $L=4$. The points are the result of the analytically constrained single-energy fit to the pseudodata and are compared to the multipoles of the underlying EtaMAID-2015 model, shown as solid lines.

Figure 10

Fixed-$t$ amplitude analysis of four observables of pseudodata at $t=-0.2{\mathrm{GeV}}^2$ (left) and $t=-0.5{\mathrm{GeV}}^2$ (right). The data points are pseudodata with a precision of $0.1\%$. The lines are obtained from the Pietarinen expansion with fitted coefficients.

Figure 11

Single-energy partial wave analysis of a set with four observables of pseudodata at $W=1543$ MeV (left) and $W=1770$ MeV (right). The data points are pseudodata with a precision of $0.1\%$. The lines are the results of the fixed-$t$ constrained single-energy fit describes in the text.

Figure 12

Predictions for polarization observables from groups of type $\mathcal{S},\mathcal{BT}$, and $\mathcal{BR}$ that are not fitted; shown at $W=1770$ MeV (top) and $W=1900$ MeV (bottom). The data points are pseudodata with a precision of $0.1\%$ (red points). The lines are obtained in the fit with four observables ${\sigma }_0,\check{\mathrm{\Sigma }},\check{T}$, and $\check{F}$, from the CGLN amplitudes with fitted multipoles up to $L_{\max }=5$.

Figure 13

Real (blue) and imaginary (red) parts of electric and magnetic multipoles up to $L=4$ after the final iteration. The points are the result of the analytically constrained single-energy fit to the pseudodata and are compared to the multipoles of the underlying EtaMAID-2015 model, solution I, shown as solid lines.

Figure 14

Example of our interpolated fixed-$t$ database for $t=-0.2{\mathrm{GeV}}^2$ and $t=-0.5{\mathrm{GeV}}^2$. The threshold values of the physical region are $W_{\mathrm{thr}}=1486$ MeV for $t=-0.2{\mathrm{GeV}}^2$ and $W_{\mathrm{thr}}=1554$ MeV for $t=-0.5{\mathrm{GeV}}^2$.

Figure 15

Invariant amplitudes $B_1,B_2,B_6$, and $B_8$ as functions of the total c.m. energy $W$ at $t=-0.2{\mathrm{GeV}}^2$ (left) and $-0.5{\mathrm{GeV}}^2$ (right). The blue and red curves show real and imaginary parts of the amplitudes of solutions II (dashed) and III (solid) respectively.

Figure 16

Helicity amplitudes $H_1, H_2, H_3$, and $H_4$ as functions of the total c.m. energy $W$ at $t=-0.2{\mathrm{GeV}}^2$ (left) and $t=-0.5{\mathrm{GeV}}^2$ (right). The blue and red curves show real and imaginary parts of the amplitudes of solutions II (dashed) and III (solid) respectively.

Figure 17

Pietarinen fit of the interpolated data at $t=-0.2{\mathrm{GeV}}^2$ and $t=-0.5{\mathrm{GeV}}^2$. The dashed (black) and solid (blue) curves are the results starting with solutions II and III, respectively, and are on top of each other.

Figure 18

Single-energy fit to the experimental data at $E=818$ MeV ($W=1554$ MeV) (top left) and $E=898$ MeV ($W=1602$ MeV) (top right) and at $E=1191$ MeV ($W=1765$ MeV) (bottom left) and $E=1335$ MeV ($W=1840$ MeV) (bottom right), using the analytical constraint of solutions II and III, drawn as dashed and solid lines, respectively.

Figure 19

Helicity amplitudes obtained by FT AA (in first iteration) using solutions II (left) and III (right) as constraints at fixed energies $W=1602$ MeV (top) and $W=1840$ MeV (bottom). The blue and red points show the result of the fixed-$t$ analysis and are independent from point to point. The blue and red solid lines are obtained from the single-energy partial wave analysis and are continuous over the angular range. At the higher energy, the kinematical range is restricted due to the restrictions in the $t$ value, $-1.0\le t\le 0$.

Figure 20

Evolution of the helicity amplitudes at fixed energies $W=1602$ MeV (top) and $W=1840$ MeV (bottom) from the initial (left panel) to the final step (right panel) after three iterations using solutions II (dashed) and III (solid) as a constraint. The real parts of the amplitudes are shown in blue and the imaginary parts in red.

Figure 21

Evolution of the $S$-, $P$-, and $D$-wave multipoles from the initial solutions II (full lines) to the final step (blue dots) after three iterations.

Figure 22

Evolution of the $S$-, $P$-, and $D$-wave multipoles from the initial solutions III (full lines) to the final step (red dots) after three iterations.

Figure 23

Real and imaginary parts of the $S$-, $P$-, and $D$-wave multipoles obtained in the final step after three iterations using analytical constraints from helicity amplitudes obtained from initial solutions II (blue) and III (red).

Figure 24

Predictions from solutions II (dashed) and III (solid) for polarization observables that are not fitted $\{P,G,C_x,O_x\}$ (left) and $\{E,H,C_z,O_z\}$ (right) at two energies, $W=1554$ MeV (top) and $W=1765$ MeV (bottom). Together with our current experimental database $\{{\sigma }_0,\mathrm{\Sigma },T,F\}$, each additional group of four observables would make a complete experiment.