Proton 0.01 MeV resonance width and low-energy $S$-factor of p+10B fusion

{\bf{Purpose}:} In this paper, the proton resonance width of the near-threshold resonance is calculated using two different approaches. The values of the proton width are used to calculate the low-energy $S$-factor. \ {\bf{Method:}} First, the proton resonance width is estimated using the mirror symmetry of the resonance ${}^{11}\mathrm{C}(E_x=8.70\mathrm{M}\mathrm{e}\mathrm{V};{\frac{5}{2}}^{+})$ and the mirror bound state ${}^{11}\mathrm{B}(E_x=9.272\mathrm{M}\mathrm{e}\mathrm{V};{\frac{5}{2}}^{+})$. In the second approach, this width is estimated using the $R$-matrix definition of the observable resonance width, which is expressed in terms of the $p-{}^{10}\mathrm{B}$ resonant wave function calculated in the potential approach and the spectroscopic factor. \ {\bf{Results}:} Depending on the method chosen, the calculated proton resonance width varies from $1.03\times {10}^{-19}$MeV to $2.96\times {10}^{-19}$MeV. The role of the near-threshold resonance is determined using fitting of two low-energy $S$-factors from direct measurements [C. Angulo {\it et al.}, Z. Phys. A {\bf 345}, 333 (1993)] and from the indirect Trojan horse method (THM) [ A. Cvetinović {\it et al.}, Phys. Rev. C {\bf 97}, 065801 (2018)]. Within the framework of the $R$-matrix method using the determined proton resonance widths and the modified THM parameters for six low-lying resonances, the low-energy $S$-factors were calculated and compared with the corresponding experimental $S$-factors. The closest agreement with the data is achieved with the proton resonance widths $1.0\times {10}^{-19}$ MeV when fitting the $S$-factor from the THM indirect measurements, and $1.68\times {10}^{-19}$ MeV and $2.5\times {10}^{-19}$ MeV when fitting the $S$-factor from [Angulo {\it et al}, Z. Phys. A {\bf 345}, 333 (1993)].\