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Effective theory approach to new physics in $b \to u$ and $b \to c$ leptonic and semileptonic decays

Recent measurements of exclusive $B^{-} \to τ^{-}\,ν$ and $B^0 \to π^{+}\,l^{-}\,\barν_l$ decays via the $b \to u\,l\,ν$ transition process differ from the standard model expectation and, if they persist in future $B$ experiments, will be a definite hint of the physics beyond the standard model. Similar hints of new physics have been observed in $b \to c$ semileptonic transition processes as well. BABAR measures the ratio of branching fractions of $B \to (D,\,D^{\ast})\,τ\,ν$ to the corresponding $B \to (D,\,D^{\ast})\,lν$, where $l$ represents either an electron or a muon, and finds $3.4σ$ discrepancy with the standard model expectation. In this context, we consider a most general effective Lagrangian for the $b \to u\,l\,ν$ and $b \to c\,l\,ν$ transition processes in the presence of new physics and perform a combined analysis of all the $b \to u$ and $b \to c$ semi-(leptonic) data to explore various new physics operators and their couplings. We consider various new physics scenarios and give predictions for the $B_c \to τν$ and $B \to πτν$ decay branching fractions. We also study the effect of these new physics parameters on the ratio of the branching ratios of $B \to πτν$ to the corresponding $B \to π\,l\,ν$ decays.