Mirror symmetry of Gamow-Teller transitions by the comparison of $^{37}$Cl($^3$He,$t$)$^{37}$Ar and $^{37}$Ca(${\beta}_+$)$^{37}$K

Under the assumption of isospin symmetry, the distributions of Gamow-Teller (GT) transition strengths $B$(GT) are identical between $^{37}$Cl $\rightarrow$ $^{37}$Ar and $^{37}$Ca $\rightarrow$ $^{37}$K transitions. The differences between the empirical $B$(GT) distributions from $^{37}$Cl($p,n$)$^{37}$Al and $^{37}$Ca $\beta$-decay caused a great deal of controversy. One of problems in the $(p,n)$ experiments was the rather poor resolution. In order to study the individual transitions for $^{37}$Cl $\rightarrow$ $^{37}$Ar, we performed a high-resolution, 30~keV (FWHM), $^{37}$Cl($^3$He,$t$)$^{37}$Ar experiment at 140~MeV/u $^3$He beam. The $B$(GT) distribution was extracted from the differential cross sections and compared with the $^{37}$Ca $\beta$-decay data. Overall, they had similar distributions. However, the fine structures were not necessary consistent.