Memorino on the `1/2 vs. 3/2 Puzzle' in $\bar{B}\to l\bar{\nu }{X}_c$ -- a Year Later and a Bit Wiser

Abstract

The OPE treatment that has been so successful in describing inclusive $\bar{B}\to l\bar{\nu }{X}_c$ decays yields sum rules (in particular the Uraltsev sum rule and its higher moments) implying the dominance of the $P$ wave $j_q=3/2$ charm states in $X_c$ over their $j_q=1/2$ counterparts. This prediction is supported by other general arguments as well as quark model calculations, which illustrate the OPE results, and by preliminary lattice findings. Its failure would indicate a significant limitation in our theoretical understanding of $\bar{B}\to l\bar{\nu }{X}_c$. Some experimental issues have been clarified since a preliminary version of this note had appeared, yet the verdict on the composition of the final states {\em beyond} $D$, $D^{\ast }$ and the two narrow $j_q=3/2$ resonances remains unsettled. Establishing which hadronic configurations -- $D/D^{\ast }+\pi ,D/D^{\ast }+2\pi ,...$ -- contribute, what their quantum numbers are and their mass distributions will require considerable experimental effort. We explain the theoretical issues involved and why a better understanding of them will be of significant value. Having significant contributions from a mass continuum distribution below 2.5 GeV raises serious theoretical questions for which we have no good answer. Two lists are given, one with measurements that need to be done and one with items of theoretical homework. Some of the latter can be done by employing existing theoretical tools, whereas others need new ideas.

arXiv comment21 pages, no figures; corrected address of one of the authors and included new measurements by BaBar

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