$\omega N$ scattering length from $\omega$ photoproduction on the proton near the threshold

TL;DR

This study measures the $oldsymbol{ ext{omega}}$ meson-proton scattering length near threshold using photoproduction data, providing the first separate determination of real and imaginary parts, indicating a repulsive interaction.

Contribution

It presents the first separate determination of real and imaginary parts of the $oldsymbol{ ext{omega}}$-proton scattering length and effective range from photoproduction data near threshold.

Findings

The scattering length $a_{\omega p}$ indicates a repulsive force.

Total cross sections were measured at incident energies 1.09-1.15 GeV.

Small P-wave contribution does not affect the results.

Abstract

Photoproduction of the $\omega$ meson on the proton has been experimentally studied near the threshold. The total cross sections are determined at incident energies ranging from 1.09 to 1.15 GeV. The 1/2 and 3/2 spin-averaged scattering length $a_{\omega p}$ and effective range $r_{\omega p}$ between the $\omega$ meson and proton are estimated from the shape of the total cross section as a function of the incident photon energy: $a_{\omega p} = \left(-0.97^{+0.16_{\rm stat}}_{-0.16_{\rm stat}}{}^{+0.03_{\rm syst}}_{-0.00_{\rm syst}}\right)+i \left(0.07^{+0.15_{\rm stat}}_{-0.14_{\rm stat}}{}^{+0.17_{\rm syst}}_{-0.09_{\rm syst}}\right)$ fm and $r_{\omega p}=\left(+2.78^{+0.68_{\rm stat}}_{-0.54_{\rm stat}}{}^{+0.11_{\rm syst}}_{-0.13_{\rm syst}}\right)+i\left(-0.01^{+0.46_{\rm stat}}_{-0.50_{\rm stat}}{}^{+0.07_{\rm syst}}_{-0.00_{\rm syst}}\right)$ fm, resulting in a repulsive force. The real and imaginary parts for $a_{\omega p}$ and $r_{\omega p}$ are determined separately for the first time. A small $P$-wave contribution does not affect the obtained values.

Tables

Table 1. Table 1: Deduced scattering parameters a ω ​ p subscript 𝑎 𝜔 𝑝 a_{\omega p} and r ω ​ p subscript 𝑟 𝜔 𝑝 r_{\omega p} for several conditions. The second, third, fourth lines show the results for different Λ Λ \Lambda cut-off parameters. The fifth line corresponds to the result with taking the P 𝑃 P -wave contribution into account where Im ​ r ω ​ p Im subscript 𝑟 𝜔 𝑝 {\rm Im\ }r_{\omega p} is fixed at 0 fm. The sixth line represents the result with the assumption that the energy dependence of the ω 𝜔 \omega production amplitude is express by a single N ∗ superscript 𝑁 N^{*} resonance as an extreme case.

parameters	Re $a_{\omega p}$ (fm)	Im $a_{\omega p}$ (fm)	Re $r_{\omega p}$ (fm)	Im $r_{\omega p}$ (fm)
$\Lambda =0.8\mathrm{GeV}/c$	$-{0.97}_{-0.16}^{+0.16}{}_{-0.00}^{+0.03}$	$+{0.07}_{-0.14}^{+0.15}{}_{-0.09}^{+0.17}$	$+{2.78}_{-0.54}^{+0.68}{}_{-0.13}^{+0.11}$	$-{0.01}_{-0.50}^{+0.46}{}_{-0.00}^{+0.07}$
$\Lambda =0.6\mathrm{GeV}/c$	$-{1.11}_{-0.16}^{+0.14}{}_{-0.02}^{+0.03}$	$+{0.12}_{-0.17}^{+0.17}{}_{-0.11}^{+0.15}$	$+{2.78}_{-0.57}^{+0.81}{}_{-0.11}^{+0.04}$	$-{0.00}_{-0.57}^{+0.44}{}_{-0.13}^{+0.11}$
$\Lambda =1.0\mathrm{GeV}/c$	$-{0.89}_{-0.18}^{+0.16}{}_{-0.00}^{+0.01}$	$+{0.04}_{-0.12}^{+0.14}{}_{-0.08}^{+0.13}$	$+{2.78}_{-0.51}^{+0.62}{}_{-0.09}^{+0.13}$	$+{0.01}_{-0.50}^{+0.47}{}_{-0.05}^{+0.03}$
$P$-wave contribution	$-{0.96}_{-0.16}^{+0.16}{}_{-0.01}^{+0.03}$	$+{0.10}_{-0.14}^{+0.14}{}_{-0.11}^{+0.14}$	$+{2.85}_{-0.53}^{+0.77}{}_{-0.15}^{+0.10}$	$0.00$
single $N^{\ast }$ contribution	$-{0.87}_{-0.12}^{+0.15}{}_{-0.02}^{+0.04}$	$+{0.22}_{-0.12}^{+0.14}{}_{-0.11}^{+0.11}$	$+{2.69}_{-0.55}^{+0.62}{}_{-0.12}^{+0.06}$	$-{0.04}_{-0.69}^{+0.48}{}_{-0.14}^{+0.04}$

Equations

$$p\cot \delta (p)=\frac{1}{a_{\omega N}}+\frac{1}{2}{r}_{\omega N}{p}^2+O(p^4),$$

$$\frac{d\sigma }{d\Omega }=\frac{N_{\omega }(\cos \theta )}{2\pi \Delta \cos \theta N_{\gamma }{N}_{\tau }{\eta }_{\mathrm{acc}}(\cos \theta )\mathrm{BR}\left(\omega \to \gamma \gamma \gamma \right)},$$

$$\sigma =\sum 2\pi \Delta \cos \theta \frac{d\sigma }{d\Omega }.$$

$${\sigma }_0(W,M)=\frac{1}{16\pi W^2}\frac{p(W,M)}{k}{\left|T_{\gamma p\to \omega p}(W,M)\right|}^2,$$

$$\sigma (E_{\gamma })={\int }_{m_{{\pi }^0}}^{W(E_{\gamma })-m_p}{\sigma }_0(W(E_{\gamma }),M)L_{\omega }(M)dM,$$

$$T_{\gamma p\to \omega p}=V_{\gamma p\to \omega p}+T_{\omega p\to \omega p}{G}_{\omega p\to \omega p}{V}_{\gamma p\to \omega p},$$

$$\begin{array}[]{l@{}l}&\left\langle\displaystyle\vphantom{\vec{k}}\omega p(p)\left|\displaystyle\vphantom{\vec{k}}T_{\gamma p\to\omega p}\left|\displaystyle\vphantom{\vec{k}}\gamma p(k)\!\right.\right.\right\rangle\\ &=\left\langle\displaystyle\vphantom{\vec{k}}\omega p(p)\left|\displaystyle\vphantom{\vec{k}}V_{\gamma p\to\omega p}\left|\displaystyle\vphantom{\vec{k}}\gamma p(k)\!\right.\right.\right\rangle+\displaystyle\int\vphantom{\vec{k}}\left\langle\displaystyle\vphantom{\vec{k}}\omega p(p)\left|\displaystyle\vphantom{\vec{k}}T_{\omega p\to\omega p}\left|\displaystyle\vphantom{\vec{k}}\omega p(q)\!\right.\right.\right\rangle{\frac{\delta^{3}\left(\vec{q}-\vec{q}\,^{\prime}\right)}{W-H_{0}+i\epsilon}}\left\langle\displaystyle\vphantom{\vec{k}}\omega p(q^{\prime})\left|\displaystyle\vphantom{\vec{k}}V_{\gamma p\to\omega p}\left|\displaystyle\vphantom{\vec{k}}\gamma p(k)\!\right.\right.\right\rangle d\vec{q}\,d\vec{q}\,^{\prime}\\ &\simeq\displaystyle\left[1+8\pi\mu\left\langle\displaystyle\vphantom{\vec{k}}\omega p(p)\left|\displaystyle\vphantom{\vec{k}}T_{\omega p\to\omega p}\left|\displaystyle\vphantom{\vec{k}}\omega p(p)\!\right.\right.\right\rangle\int\frac{d\vec{q}}{p^{2}-q^{2}+i\mu\Gamma_{\omega}}\exp\left(-\frac{q^{2}}{\Lambda^{2}}\right)\right]\left\langle\displaystyle\vphantom{\vec{k}}\omega p(p)\left|\displaystyle\vphantom{\vec{k}}V_{\gamma p\to\omega p}\left|\displaystyle\vphantom{\vec{k}}\gamma p(k)\!\right.\right.\right\rangle,\end{array}$$

$$⟨\omega p(p)|T_{\omega p\to \omega p}|\omega p(p)⟩=-\frac{1}{{\left(2\pi \right)}^2\mu }{\left(\frac{1}{a_{\omega p}}+\frac{1}{2}{r}_{\omega p}{p}^2-ip\right)}^{-1}.$$

$${\chi }^2=\sum _{i=1}^{10}\frac{{\left({\sigma }_i-\alpha Y_i\right)}^2}{{\left(\delta {\sigma }_i^{(\mathrm{stat})}\right)}^2+{\left(\delta {\sigma }_i^{(\mathrm{syst})}\right)}^2},$$

$$V_{\gamma p\to \omega p}\propto {\left(W^2-M_{N^{\ast }}^2+i{M}_{N^{\ast }}{\Gamma }_{N^{\ast }}\right)}^{-1}$$

Full text

The structure of hadrons and dynamical hadron-mass generation are the most important subjects to be studied in the non-perturbative domain of quantum chromodynamics (QCD). The $\omega$ meson ($\omega$) is one of the best established hadrons, and it is considered to give a short-ranged repulsive central force and a strong spin-orbit force between two nucleons ($N$s) machleidt . Nevertheless, the fundamental properties of $\omega$ such as the interaction with $N$ is not known yet due to the difficulties in realizing scattering experiments. Detailed information on $\omega N$ scattering would not only reveal highly excited nucleon resonances ($N^{*}$) but also have a strong relevance to the equation of state (EoS) describing the interior of neutron stars sxn1 . Gravitational wave observations just have begun to provide information on EoS sxn2 .

The low-energy $\omega N$ scattering is characterized by the scattering length $a_{\omega N}$ and effective range $r_{\omega N}$ through an effective-range expansion of the $S$-wave phase shift $\delta(p)$:

[TABLE]

where $p$ denotes the momentum of $\omega$ in the $\omega N$ center-of-mass (CM) frame. A positive (negative) Re $a_{\omega N}$ gives attraction (repulsion), and a positive Im $a_{\omega N}$ corresponds to the absorption to another channel such as $\omega N\to\pi N$. The $r_{\omega N}$ provides the momentum dependence of the interaction. Recently, the A2 collaboration at the Mainz MAMI facility has reported $\left|a_{\omega N}\right|=0.82\pm 0.03$ fm, which is extracted from $\omega$ photoproduction on the proton ($\gamma p\to\omega p$) near the threshold assuming a vector meson dominance (VMD) model mainz . The obtained value is a combination of two independent $S$-wave scattering lengths with total spins of 1/2 and 3/2. The unknown sign of $a_{\omega N}$ leaves the naive question of whether low-energy $\omega N$ scattering is repulsive or attractive.

Theoretically estimated values of $a_{\omega N}$ are scattered in a wide range from attractive to repulsive ones. The effective Lagrangian approach based on chiral symmetry gives an attractive value of $a_{\omega N}=+1.6+i0.30$ fm omega1 . A QCD sum-rule analysis provides a weakly attractive value of $a_{\omega N}=+0.41\pm 0.05$ fm omega2 . The coupled-channel unitary approach gives repulsive values of $a_{\omega N}^{(1/2)}=-0.45+i0.31$ fm and $a_{\omega N}^{(3/2)}=-0.43+i0.15$ fm for the two total spins, giving a spin-averaged value of $a_{\omega N}=-0.44+i0.20$ fm omega3 . The coupled-channel analysis of $\omega$ production in pion and photo-induced reactions gives a very weakly repulsive value of $a_{\omega N}=-0.026+i0.28$ fm omega4 . The dynamical coupled-channel analysis resulted in $a_{\omega N}^{(1/2)}=0.0454+i0.0695$ fm and $a_{\omega N}^{(3/2)}=-0.180+i0.0597$ fm, giving a repulsive spin-averaged value111 We adopt $a_{\omega N}=(1/3)a_{\omega N}^{(1/2)}+(2/3)a_{\omega N}^{(3/2)}$ for the spin average using the convention of Lutz et al. omega3 ; omega4 . of $a_{\omega N}=-0.135+i0.0630$ fm omega5 . Neither the coupled-channel analyses nor the VMD analysis by the A2 collaboration incorporates the finite width of $\omega$ in the final state.

To determine the low-energy $\omega N$ scattering parameters $a_{\omega N}$ and $r_{\omega N}$ experimentally, we investigate the $\gamma p\to\omega p$ reaction very close to the reaction threshold. Several collaborations have already measured the total cross sections near the threshold using the $\omega\to\pi^{+}\pi^{-}\pi^{0}$ decay mode (SAPHIR saphir and CLAS clas collaborations), and the $\omega\to\pi^{0}\gamma$ decay mode (CBELSA/TAPS cbelsa and A2 mainz collaborations). Currently, the data points for the total cross section near the threshold ($E_{\gamma}\lesssim 1.2$ GeV), where the $S$-wave $\omega N$ contribution is dominant, are not enough for determining $a_{\omega N}$ and $r_{\omega N}$ from the shape of the total cross section as a function of the incident energy (excitation function) through $\omega N$ rescattering in the final-state interaction. We have measured ten data points of the total cross section at incident photon energies ranging from 1.09 to 1.15 GeV. The $\omega$ meson mainly decays in the $\omega\to\pi^{0}\pi^{+}\pi^{-}$ mode with a branching ratio of 89.2% pdg . It is, however, difficult to reproduce the background shapes in the $\pi^{0}\pi^{+}\pi^{-}$ invariant mass distributions measured with poor identification for charged particles hashimoto . Thus, we determined the cross sections using the $\omega\to\pi^{0}\gamma$ decay mode with a branching ratio of $8.40\%$. In this letter, we present $a_{\omega N}$ and $r_{\omega N}$ extracted from the shape of the excitation function for the $\gamma p\to\omega p$ reaction.

A series of meson photoproduction experiments were conducted exp using the FOREST detector forest , which was installed on the second photon beamline tag2 at the Research Center for Electron Photon Science (ELPH), Tohoku University, Japan. In the present experiments, bremsstrahlung photons were produced from 1.2-GeV circulating electrons in a synchrotron stb by inserting a carbon thread (radiator) tag2 . The photons collimated with two lead apertures of 10 and 25 mm in diameter located 4.2- and 12.9-m downstream from the radiator, respectively, were incident on a 45-mm thick liquid-hydrogen target located at the center of FOREST. The energies of the incident photons were analyzed up to 1.15 GeV by detecting the post-bremsstrahlung electrons with a photon-tagging counter, STB-Tagger II tag2 . FOREST consists of three different electromagnetic calorimeters (EMCs): 192 undoped CsI crystals, 252 lead scintillating-fiber modules, and 62 lead glasses. A plastic-scintillator hodoscope (PSH) is placed in front of each EMC to identify charged particles. FOREST covers a solid angle of $\sim{88\%}$ in total. The typical photon-tagging rate was 20 MHz, and the photon transmittance (the so-called tagging efficiency) was $\sim$53% tag2 . The trigger condition of the data acquisition (DAQ), which required for an event to have more than one final-state particles in coincidence with a photon-tagging signal forest , was the same as that in Ref. dpipi-plb . The total number of collected events in DAQ was $1.79\times 10^{9}$. The average trigger rate was 1.6 kHz, and the average DAQ efficiency was 80%.

Event selection was made for the \gamma{p}$$\to$$\pi^{0}\gamma{p}$$\to$$\gamma\gamma\gamma{p} reaction. At first, events containing three neutral particles and a charged particle were selected. The time difference between every 2 neutral EMC clusters out of 3 was required to be less than thrice that of the time resolution for the difference. The two neutral EMC clusters giving the $\gamma\gamma$ invariant mass ranging from 50 to 220 MeV were selected, and the other EMC cluster was required to have an energy higher than 200 MeV. The charged particles were detected with the forward PSH. Further selection was made by applying a kinematic fit with five constraints: energy and three-momentum conservation, and $\gamma\gamma$ invariant mass being the $\pi^{0}$ mass. The momentum of the charged particle was obtained from the time delay assuming that the charged particle had proton mass. Events for which the $\chi^{2}$ probability was higher than 0.1 were selected. When the number of combinations was more than 1 in an event, the combination with the minimum $\chi^{2}$ was adopted. Sideband-background subtraction was performed for accidental-coincidence events detected in STB-Tagger II and FOREST.

All the data for incident energies above 1.09 GeV ($E_{\gamma}=1.09$–1.15 GeV) are divided into ten bins (every bin includes 4 photon-tagging channels), and 10 angular bins of the $\pi^{0}\gamma$ emission angle $\cos\theta$ in the $\gamma p$-CM frame. The typical $\pi^{0}\gamma$ invariant mass ($M_{\pi\gamma}$) distributions are shown in Fig. 1. Each $M_{\pi\gamma}$ distribution shows a prominent peak with a centroid of $\sim 0.78$ GeV, and has a broad background contribution in the lower side. This background contribution is well reproduced by a Monte-Carlo (MC) simulation based on Geant4 geant4 for the $\gamma p\to\pi^{0}\pi^{0}p\to\gamma\gamma\gamma\gamma p$ reaction, where 1 $\gamma$ out of 4 is not detected with FOREST. In the simulation, the five-fold differential cross sections are assumed to be the same as those provided by the 2-PION-MAID calculation 2-pion-maid . The $M_{\pi\gamma}$ distributions for the $\gamma p\to\pi^{0}\pi^{0}p$ reaction are also plotted in Fig. 1 where the same analysis is applied as for the $\gamma p\to\pi^{0}\gamma p$ reaction.

The $M_{\pi\gamma}$ distributions for the $\gamma p\to\omega p\to\pi^{0}\gamma p\to\gamma\gamma\gamma p$ and $\gamma p\to\pi^{0}\pi^{0}p\to\gamma\gamma\gamma\gamma p$ reactions in the MC simulation are fitted to the measured $M_{\pi\gamma}$ distribution for each emission-angle incident-energy bin only by changing the normalization coefficients. Here, the events are generated according to the pure phase space for the $\gamma p\to\omega p$ reaction. The number of the $\omega$ produced events $N_{\omega}$ is estimated for $M_{\pi\gamma}\geq 0.76$ GeV after subtracting the background $\gamma p\to\pi^{0}\pi^{0}p$ contribution for each bin. The angular differential cross section is obtained from $N_{\omega}(\cos\theta)$ as

[TABLE]

with the incident photon flux including the DAQ efficiency correction $N_{\gamma}$, the number of target protons $N_{\tau}$, the multiplication of branching ratios for the $\omega\to\pi^{0}\gamma$ and $\pi^{0}\to\gamma\gamma$ decays ${\rm BR}(\omega\to\gamma\gamma\gamma)$, and the detector acceptance calculated in the simulation $\eta_{\rm acc}(\cos\theta)$, where $\Delta\cos\theta=0.2$. Fig. 2 shows the typical $d\sigma/d\Omega$ distributions. The systematic uncertainty of $d\sigma/d\Omega$ is also given in Fig. 2. It includes the uncertainty of event selection in the kinematic fit, that of counting $N_{\omega}$ due to the $M_{\pi\gamma}$ threshold, that of acceptance owing to the uncertainties of the $d\sigma/d\Omega$ distributions for event generation in the simulation, that of detection efficiency of protons, and that of normalization resulting from $N_{\tau}$ and $N_{\gamma}$.

Every $d\sigma/d\Omega$ distribution shows a slight increase with increase of $\cos\theta$. A finite $P$-wave amplitude must produce asymmetric behavior of the angular distribution through the interference with the $S$-wave amplitude although the $S$-wave contribution is expected to be dominant near the threshold. The measured $d\sigma/d\Omega$s in this work are somewhat lower than the world available data. The obtained $d\sigma/d\Omega$ depends on the incident-energy coverage because the cross section increases rapidly as the incident energy goes up. The bin size of the incident energy is $\sim 4.5$ MeV in our results, while that for the SAPHIR saphir data is 25 MeV, 18 MeV (CLAS clas ), and 15 MeV (A2 mainz ), respectively. In Fig. 2, the angular distribution obtained by the A2 collaboration at $E_{\gamma}\simeq 1.14$ GeV shows a shape being concave upward, suggesting a $P$-wave contribution, although any significant slope changes are not observed in this work. Apparently this deviation comes from the relative difference of centroid photon-tagging energies by a few MeV. The uncertainty of the centroid photon-tagging energies is estimated to be 0.3%, which corresponds to 3–4 MeV. A calibration difference of photon-tagging energies needs to be incorporated in the estimation of the systematic uncertainty for $a_{\omega p}$ and $r_{\omega p}$.

The total cross section $\sigma$ is obtained by integrating $d\sigma/d\Omega$s all over the ten emission-angle bins:

[TABLE]

Fig. 3 shows $\sigma$ as a function of the incident photon energy. The excitation function shows a monotonic increase, and finite yields are observed below the threshold $E_{\gamma}^{\rm thr}$ for production of $\omega$ having the centroid mass. The obtained cross sections show a systematic deviation from the world available data. The uncertainty of tagging-energy determination (0.3%) for the incident photon beam may account for the deviation.

We determine $a_{\omega p}$ and $r_{\omega p}$ from the shape of the excitation function. We evaluate the excitation function for the $\gamma p\to\omega p$ reaction using a model with final-state $\omega p$ interaction (FSI) based on the Lippmann-Schwinger equation. We assume that the $S$-wave contribution is dominant at $E_{\gamma}=1.09$–1.15 GeV. The total cross section for a fixed $\omega$ mass $M$ and $\gamma p$-CM energy $W$ can be calculated using a transition amplitude $T_{\gamma p\to\omega p}(W,M)$:

[TABLE]

where $k$ and $p$ denote the momenta of an initial- and a final-state particles, respectively, in the $\gamma p$-CM frame. The total cross section $\sigma$ as a function of $E_{\gamma}$ is obtained by averaging $\sigma_{0}(W(E_{\gamma}),M)$ over available $\omega$ masses:

[TABLE]

where the probability $L_{\omega}(M)$ stands for a Breit-Wigner function with a centroid of $M_{\omega}=782.65$ MeV and a width of $\Gamma_{\omega}=8.49$ MeV pdg .

The $T_{\gamma p\to\omega p}$ is expressed by

[TABLE]

where $T_{\omega p\to\omega p}$ stands for the $\omega p$ scattering amplitude, $G_{\omega p\to\omega p}$ denotes the $\omega p$ propagator, and $V_{\gamma p\to\omega p}$ is the production amplitude without FSI. We evaluate the matrix element for $T_{\gamma p\to\omega p}$ with on-shell approximations for $T_{\omega p\to\omega p}$ and $V_{\gamma p\to\omega p}$, and introduce a Gaussian form factor in the integration of $G_{\omega p\to\omega p}$. This leads the matrix element of $T_{\gamma p\to\omega p}$ to the equation:

[TABLE]

where $H_{0}$ stands for the free Hamiltonian for the final-state $\omega p$, and $\mu$ denotes a reduced mass between $\omega$ (with a mass of $M$) and the proton. Here, we use a cut-off parameter $\Lambda=0.8$ GeV$/c$. The $\left\langle\displaystyle\vphantom{\vec{k}}\omega p(p)\left|\displaystyle\vphantom{\vec{k}}T_{\omega p\to\omega p}\left|\displaystyle\vphantom{\vec{k}}\omega p(p)\!\right.\right.\right\rangle$ is given by $a_{\omega p}$ and $r_{\omega p}$:

[TABLE]

The $\left\langle\displaystyle\vphantom{\vec{k}}\omega p(p)\left|\displaystyle\vphantom{\vec{k}}V_{\gamma p\to\omega p}\left|\displaystyle\vphantom{\vec{k}}\gamma p(k)\!\right.\right.\right\rangle$ is assumed to be a constant value of 1 in the incident-energy region of interest.

The dashed curve (gray) in Fig. 3 shows the excitation function $\sigma^{S}_{\rm nonFSI}$ with $a_{\omega p}=0$ fm and $r_{\omega p}=0$ fm corresponding to non FSI condition, which does not reproduce the experimental data. FSI is necessary and the optimal set of $a_{\omega p}$ and $r_{\omega p}$ are determined to reproduce the experimentally obtained obtained cross section data. The $\chi^{2}$ corresponding to the reproducibility is defined as

[TABLE]

where $\sigma_{i}$, $\delta\sigma_{i}^{(\rm stat)}$, $\delta\sigma_{i}^{(\rm syst)}$, and $Y_{i}$ denote the measured total cross section, its statistical error, its systematic error, and the yield estimated in Eq. (5) by taking the coverage of incident energies into account, respectively, for the $i$-th incident-energy bin. The coefficient $\alpha$ for the overall normalization is determined to minimize $\chi^{2}$ for each parameter set. The deduced values are $a_{\omega p}=\left(-0.97^{+0.16}_{-0.16}{}^{+0.03}_{-0.00}\right)+i\left(0.07^{+0.15}_{-0.14}{}^{+0.17}_{-0.09}\right)$ fm and $r_{\omega p}=\left(+2.78^{+0.68}_{-0.54}{}^{+0.11}_{-0.13}\right)+i\left(-0.01^{+0.46}_{-0.50}{}^{+0.07}_{-0.00}\right)$ fm. The first and second errors for each parameter refer to the statistical and systematic uncertainties, respectively. The systematic uncertainty is estimated from that of the mean incident energy ($\pm 0.3\%$) for each photon-tagging bin. The solid (red) curve in Fig. 3 shows the excitation function with the optimal parameters. No significant changes are observed for the $a_{\omega p}$ and $r_{\omega p}$ parameters when we shift the incident photon energies by $\pm 0.3\%$. This is because these parameters are primarily determined by the shape of the excitation function.

The parameters may be somewhat affected by the adopted $\Lambda$. We also determine $a_{\omega p}$ and $r_{\omega p}$ for $\Lambda=0.6$ and 1.0 GeV/$c$. The obtained values are summarized in Table 1. Although $\left|{\rm Re}\,a_{\omega p}\right|$ and $\left|{\rm Im}\,a_{\omega p}\right|$ become larger with decrease of $\Lambda$, changes of $a_{\omega p}$ and $r_{\omega p}$ are not significant among the realistic $\Lambda$ values.

The asymmetric behavior of the angular distribution mainly comes from interference between $S$- and $P$-wave contributions. The dotted curve (magenta) in Fig. 3 shows the shape of the $P$-wave excitation function where $\sigma_{0}^{P}(W,M)\propto p^{3}/k$ is assumed. The finite width of $\omega$ makes the $P$-wave excitation function rather flat, and the $P$-wave contribution does not explain the gap at higher incident energies between the data and calculation without FSI. We also fit the excitation function adding a $P$-wave contribution to the experimental data by fixing ${\rm Im}\,r_{\omega p}=0$ fm, obtaining the values given in Table 1. The optimal coefficient to the $P$-wave contribution is 0, and the $P$-wave total cross section is 0 with an error of $\sigma^{P}_{\rm max}$. The dotted curve in Fig. 3 corresponds to $5\sigma^{P}_{\rm max}$. The asymmetric behavior in the angular distribution shown in Fig. 2 requires a finite $P$-wave contribution. The solid curve in Fig. 2 corresponds to a solution under the condition that the $P$-wave contribution in $\sigma$ is $\sigma^{P}_{\rm max}/5$. We can conclude that the $P$-wave contribution in $\sigma$ is negligibly small in determination of $a_{\omega p}$ and $r_{\omega p}$.

We have assumed that $V_{\gamma p\to\omega p}$ is constant since the coverage of incident energies is narrow ($E_{\gamma}=1.09$–1.15 GeV) for several overlapping $N^{*}$s with a very wide width. We deduce the scattering parameters with ${\rm Im}\,r_{\omega p}=0$ fm by assuming a single $N^{*}$ contribution $D_{13}(1700)$ as an extreme condition:

[TABLE]

where $M_{N^{*}}=1.7$ GeV and $\Gamma_{N^{*}}=0.2$ GeV pdg . The change of each parameter from the constant $V_{\gamma p\to\omega p}$ is not significant.

Fig. 4 shows the real and imaginary parts of 1/2 and 3/2 spin-averaged $a_{\omega p}$ obtained by assuming a constant $V_{\gamma p\to\omega p}$ in this work together with the previously obtained values. It is consistent with $\left|a_{\omega p}\right|=0.82\pm 0.03$ fm given by the A2 collaboration mainz . The other values correspond to the theoretical predictions. The positive ${\rm Re}\,a_{\omega p}$ value, giving an attraction, is rejected at a confidence level higher than $99.9\%$. The repulsion is found to be much stronger than the $\pi N$ ones, and no bound or virtual state is expected for $\omega N$. Slightly attractive $\omega$-nucleus ($\omega A$) interactions are reported with potential depths at normal nuclear density of $-42\pm 17\pm 20$ MeV a-metag and $-15\pm 35\pm 20$ MeV a-cbelsa from $\omega$ photoproduction from nuclei. The measurement of $\omega$ line shape shows a decrease of $\omega$ mass by $9.2\%\pm 0.2\%$ (corresponding to $\omega A$ attraction) without any in-medium broadening ozawa ; naruki . The relation between strong $\omega N$ repulsion and $\omega A$ attraction would be a subject of future discussions taking into consideration spin-dependent terms, higher partial waves, and partial restoration of chiral symmetry.

In summary, the total cross sections have been measured for the $\gamma p\to\omega p$ reaction near the threshold. The $\omega$ is identified through the $\omega\to\pi^{0}\gamma$ decay. The spin-averaged scattering length $a_{\omega p}$ and effective range $r_{\omega p}$ between the $\omega$ and proton are estimated from the excitation function at incident photon energies ranging from 1.09 to 1.15 GeV: $a_{\omega p}=\left(-0.97^{+0.16}_{-0.16}{}^{+0.03}_{-0.00}\right)+i\left(0.07^{+0.15}_{-0.14}{}^{+0.17}_{-0.09}\right)$ fm and $r_{\omega p}=\left(+2.78^{+0.68}_{-0.54}{}^{+0.11}_{-0.13}\right)+i\left(-0.01^{+0.46}_{-0.50}{}^{+0.07}_{-0.00}\right)$ fm. The real and imaginary parts for $a_{\omega p}$ and $r_{\omega p}$ are determined separately for the first time. A small $P$-wave contribution does not affect the obtained values. The positive ${\rm Re}\,a_{\omega p}$ value indicates repulsion.