$B_s^0\to {\eta }^{(\mathrm{\prime })}{\eta }^{(\mathrm{\prime })}$ decays in the pQCD approach

We calculate the CP averaged branching ratios and CP-violating asymmetries for $B_s^0\to \eta \eta ,\eta {\eta }^{\mathrm{\prime }}$ and ${\eta }^{\mathrm{\prime }}{\eta }^{\mathrm{\prime }}$ decays in the perturbative QCD (pQCD) approach here. The pQCD predictions for the CP-averaged branching ratios are $Br(B_s^0 \to \eta \eta) = \left (14.2^{+18.0}{-7.5}) \times 10^{-6}$,$Br(B_s^0 \to \eta \eta^\prime)= \left (12.4 ^{+18.2}{-7.0}) \times 10^{-6}$,and$Br(B_s^0 \to \eta^{\prime} \eta^{\prime}) = \left (9.2^{+15.3}_{-4.9}) \times 10^{-6}$, which agree well with those obtained by employing the QCD factorization approach and also be consistent with available experimental upper limits. The gluonic contributions are small in size: less than 7% for $B_s\to \eta \eta$ and $\eta {\eta }^{\mathrm{\prime }}$ decays, and around 18% for $B_s\to {\eta }^{\mathrm{\prime }}{\eta }^{\mathrm{\prime }}$ decay. The CP-violating asymmetries for three decays are very small: less than 3% in magnitude.