Partial wave analysis of the reaction $\gamma p\to p\omega$ and the search for nucleon resonances

An event-based partial wave analysis (PWA) of the reaction $\gamma p\to p\omega$ has been performed on a high-statistics dataset obtained using the CLAS at Jefferson Lab for center-of-mass energies from threshold up to 2.4 GeV. This analysis benefits from access to the world’s first high-precision spin-density matrix element measurements, available to the event-based PWA through the decay distribution of $\omega \to {\pi }^{+}{\pi }^{-}{\pi }^0$. The data confirm the dominance of the $t$-channel ${\pi }^0$ exchange amplitude in the forward direction. The dominant resonance contributions are consistent with the previously identified states $F_{15}(1680)$ and $D_{13}(1700)$ near threshold, as well as the $G_{17}(2190)$ at higher energies. Suggestive evidence for the presence of a $J^P=5/2^{+}$ state around $2$ GeV, a “missing” state, has also been found. Evidence for other states is inconclusive.

Figure 1

Feynman diagrams for the amplitudes used in our analysis. The images were produced using the JaxoDraw package [19].

Figure 2

$\Delta \ln \mathcal{L}$ vs $W$ (MeV): Example likelihood differences from $s$-channel scans. Each fit contained the locked OTL $t$-channel terms, along with a single $s$-channel wave. Shown are $\Delta \ln {\mathcal{L}}_{1/2^{+},5/2^{+}}=\ln {\mathcal{L}}_{5/2^{+}}-\ln {\mathcal{L}}_{1/2^{+}}$ (open squares) and $\Delta \ln {\mathcal{L}}_{1/2^{+},3/2^{-}}=\ln {\mathcal{L}}_{3/2^{-}}-\ln {\mathcal{L}}_{1/2^{+}}$ (closed triangles). See text for details and discussion.

Figure 3

Results from Fit I: $\sigma$ ($\mu$b) vs $W$ (MeV): Total cross sections from all of the waves included in the fit (filled squares), only $t$-channel waves (open squares), only $J^P=5/2^{+}$ waves (circles), and only $J^P=3/2^{-}$ waves (triangles). The cross sections extracted for both $s$-channel waves are consistent with near/sub-threshold resonances. The error bars are purely statistical.

Figure 4

Results from Fit I: $\Delta \varphi ={\varphi }_{3/2^{-}}-{\varphi }_{5/2^{+}}$ (radians) vs $W$ (MeV): The dashed line was fit using constant width Breit-Wigner distributions requiring the parameters to be within the limits quoted by the PDG for the four-star $F_{15}(1680)$ and three-star $D_{13}(1700)$. The solid line was fit allowing the $3/2^{-}$ parameters to vary freely, the results are listed in the text. The error bars are purely statistical.

Figure 5

$d\sigma /d\cos {\theta }_{\mathrm{c}.\mathrm{m}.}^{\omega }$ ($\mu$b) vs $\cos {\theta }_{\mathrm{c}.\mathrm{m}.}^{\omega }$: PWA results from Fit I (solid black line), compared to our measurements [11]. The individual contributions from the $J^P=3/2^{-}$ wave (red, dashed line), $J^P=5/2^{+}$ wave (blue, dashed-dotted line), and OTL $t$-channel terms (dotted black line) are also shown. The lack of data reported in the $W=1.955$ GeV bin is due to normalization issues [11].

Figure 6

${\rho }_{{\mathit{MM}}^{\text{'}}}^0$ vs $\cos {\theta }_{\mathrm{c}.\mathrm{m}.}^{\omega }$: PWA results from Fit I: ${\rho }_{00}^0$ (solid black line), ${\rho }_{1-1}^0$ [solid middle (red) line], $\mathrm{Re}{\rho }_{10}^0$ [solid bottom (blue) line], compared to our measurements [11]. The lack of a fit in the $W=1.955$ GeV bin is due to normalization issues [11].

Figure 7

Results from Fit II: $\sigma$ ($\mu$b) vs $W$ (MeV): Total cross sections from all of the waves in the fit (filled squares), only $t$-channel waves (open squares), only $J^P=5/2^{+}$ waves (circles), and only $J^P=7/2^{-}$ waves (triangles). The cross section extracted for $J^P=5/2^{+}$ is consistent with the tail of a lower mass state (as seen in Fit I). The $J^P=7/2^{-}$ cross section is indicative of a state near $2.2$ GeV. The errors are purely statistical.

Figure 8

Results from Fit II: $\Delta \varphi ={\varphi }_{7/2^{-}}-{\varphi }_{5/2^{+}}$ (radians) vs $W$ (MeV): The dot-dashed line is the phase motion expected using constant width Breit-Wigner distributions and the parameters quoted by the PDG for the $F_{15}(1680)$ and $G_{17}(2190)$. The dashed line required the $J^P=7/2^{-}$ parameters to be within the PDG limits for the $G_{17}(2190)$, while allowing the $J^P=5/2^{+}$ parameters to vary freely. The solid line used a constant width Breit-Wigner distribution for the $G_{17}(2190)$, but a two-pole single channel $K$-matrix for the $J^P=5/2^{+}$ wave. The parameters obtained from these fits are listed in the text. The error bars are purely statistical.

Figure 9

$d\sigma /d\cos {\theta }_{\mathrm{c}.\mathrm{m}.}^{\omega }$ ($\mu$b) vs $\cos {\theta }_{\mathrm{c}.\mathrm{m}.}^{\omega }$: PWA results from Fit II (solid black line), compared to our measurements [11]. The individual contributions from the $J^P=7/2^{-}$ wave (red, dashed line), $J^P=5/2^{+}$ wave (blue, dashed-dotted line), and OTL $t$-channel terms (dotted black line) are also shown.

Figure 10

${\rho }_{{\mathit{MM}}^{\text{'}}}^0$ vs $\cos {\theta }_{\mathrm{c}.\mathrm{m}.}^{\omega }$: PWA results from Fit II: ${\rho }_{00}^0$ [solid upper (at the right-hand edge) line (black)], ${\rho }_{1-1}^0$ [solid middle (at the right-hand edge) line (red)], $\mathrm{Re}{\rho }_{10}^0$ [solid lower (at the right-hand edge) line (blue)], compared to measurements [11]. See text for discussion.

Figure 11

Results from Fit III: (a) $\sigma$ ($\mu$b) vs $W$ (MeV): Total cross sections extracted from all of the waves in the fit (filled black squares), only $t$-channel waves (open squares), only $5/2^{+}$ waves (circles), only $3/2^{-}$ waves (triangles), and only $3/2^{+}$ waves (crosses). (b) $\Delta \varphi ={\varphi }_{5/2^{+}}-{\varphi }_{3/2^{+}}$ (radians) vs $W$ (MeV). (c) $\Delta \varphi ={\varphi }_{3/2^{-}}-{\varphi }_{3/2^{+}}$ (radians) vs $W$ (MeV). The solid curves show the phase motion expected assuming the $J^P=3/2^{+}$ has Breit-Wigner parameters $M_{3/2^{+}}=1875$ MeV and ${\Gamma }_{3/2^{+}}=150$ MeV while locking the $J^P=3/2^{-}$ and $J^P=5/2^{+}$ parameters to be those of the $D_{13}(1700)$ and $F_{15}(1680)$, respectively. The phase motion obtained for the $3/2^{+}$ is not consistent with a single resonant state. The dashed line on (c) represents using the $D_{13}(1700)$ Breit-Wigner parameters for the $3/2^{-}$ and a single channel two-pole $K$-matrix for the $3/2^{+}$. There is enough freedom to describe the data (see text for discussion). All error bars are purely statistical.

Figure 12

$d\sigma /d\cos {\theta }_{\mathrm{c}.\mathrm{m}.}^{\omega }$ ($\mu$b) vs $\cos {\theta }_{\mathrm{c}.\mathrm{m}.}^{\omega }$: PWA results from Fit III (solid black line), compared to our measurements [11]. The individual contributions from the $J^P=3/2^{-}$ wave (red, dashed line), $J^P=5/2^{+}$ wave (blue, dashed-dotted line), the $J^P=3/2^{+}$ wave (green, dashed-triple-dotted line), and OTL $t$-channel terms (dotted black line) are also shown. The lack of data reported in the $W=1.955$ GeV bin is due to normalization issues [11].

Figure 13

${\rho }_{{\mathit{MM}}^{\text{'}}}^0$ vs $\cos {\theta }_{\mathrm{c}.\mathrm{m}.}^{\omega }$: PWA results from Fit III: ${\rho }_{00}^0$ [solid upper (at the right-hand edge) line (black)], ${\rho }_{1-1}^0$ [solid middle (at the right-hand edge) line (red)], $\mathrm{Re}{\rho }_{10}^0$ [solid lower (at the right-hand edge) line (blue)], compared to measurements [11]. See text for discussion.

Figure 14

Fit II $+u$-channel: $\Delta \varphi ={\varphi }_{7/2^{-}}-{\varphi }_{5/2^{+}}$ (radians) vs $W$ (MeV): Phase motion obtained with and without $u$-channel terms in Fit II. Including $u$-channel terms only creates discrepancies in the phase motion where one of the $s$-channel waves has a small contribution to the cross section. The error bars on both phase motion plots are purely statistical.