CP violation in polarized B-> pi ell+ ell- and B-> rho ell+ ell- decays

We study the decay rate and the CP violating asymmetry of the exclusive B-> pi ell+ ell- and B-> rho ell+ ell- decays in the case where one of the final leptons is polarized. We calculate the contributions coming from the individual polarization states in order to identify a so-called wrong sign decay, which is a decay with a given polarization, whose width and CP asymmetry are smaller as compared to the unpolarized one. The results are presented for electron and tau leptons. We observe that in particular decay channels, one can identify a wrong sign decay which is more sensitive to new physics beyond the Standard Model.Comment: 24 pages, 10 figures; minor errors and misprints corrected, references added, version to be published in EPJ

Crucial and bicrucial permutations with respect to arithmetic monotone patterns

A pattern $\tau$ is a permutation, and an arithmetic occurrence of $\tau$ in (another) permutation $\pi=\pi_1\pi_2...\pi_n$ is a subsequence $\pi_{i_1}\pi_{i_2}...\pi_{i_m}$ of $\pi$ that is order isomorphic to $\tau$ where the numbers $i_1<i_2<...<i_m$ form an arithmetic progression. A permutation is $(k,\ell)$-crucial if it avoids arithmetically the patterns $12... k$ and $\ell(\ell-1)... 1$ but its extension to the right by any element does not avoid arithmetically these patterns. A $(k,\ell)$-crucial permutation that cannot be extended to the left without creating an arithmetic occurrence of $12... k$ or $\ell(\ell-1)... 1$ is called $(k,\ell)$-bicrucial. In this paper we prove that arbitrary long $(k,\ell)$-crucial and $(k,\ell)$-bicrucial permutations exist for any $k,\ell\geq 3$. Moreover, we show that the minimal length of a $(k,\ell)$-crucial permutation is $\max(k,\ell)(\min(k,\ell)-1)$, while the minimal length of a $(k,\ell)$-bicrucial permutation is at most $2\max(k,\ell)(\min(k,\ell)-1)$, again for $k,\ell\geq3$