Does The $\triangle I ={1/2}$ Rule Hold In D And B $\to \pi \pi$ Decays?

Abstract

Although two pion decays of $K$, $D$ and $B$ have similar isospin structures, there are dramatic differences in the ratios $R_K$ and $R_{D,B}$ of amplitudes from $\Delta I =3/2$ and $\Delta I =1/2$ interactions. In $K\to \pi\pi$ decays there is the famous $\Delta I =1/2$ rule with $R_K \approx 1/22$, whereas in $B(D) \to \pi\pi$ decays the ratios $R_{D,B}$ are of order one and therefore there is no such a rule. In this work we study decay amplitudes in $B(D)\to \pi\pi$ using QCD factorization calculations paying particular attention to discrepancies between data and theoretical estimates. Since isospin does not play a special role in factorization calculations, no $\Delta I =1/2$ rule is expected. We find that theoretical results on the size of the amplitudes are in qualitative agreement with data. However the phases for the amplitudes are very different. We show that the effects of re-scattering between the two pions in the final state can play a crucial rule in understanding the differences between $B(D)\to \pi\pi$ and $K\to \pi\pi$ decays. We also comment on the role of isospin analysis which applies to the study of CP violation in $B\to \pi \pi$ decays.