ar X iv : h ep - p h / 06 06 23 1 v 1 2 1 Ju n 20 06 The controversy about ” 1 / m Q duality violation ” : a quark model point of view

The controversy about " 1/m Q duality violation " : a quark model point of view Abstract. Non relativistic quark models have been invoked to support the statement of a 1/m Q duality violation in semileptonic B decays. However, as we recall, we have very explicitly shown that no 1/m Q duality violation is present in totally integrated rates, in such quark models. Then : 1) it is shown that some contributions under discussion are misleadingly described as " 1/m Q duality violation " ; as they stand, they are in fact parametrically much smaller : they are O((1/m Q) 5/2) when properly referred to the total rate under discussion ; numerically this is below 10 −2 ; 2) moreover, relying on our rigourous expansion of the harmonic oscillator model around the Shifman-Voloshin limit, it is shown that even such small terms are not present at all in the total rate, and must therefore merely cancel ; 3) finally, for physical masses, by an exact treatment of this particular, but not too unrealistic model, one finds a departure from free-quark decay rate only at the 10 −3 level ; moreover, this departure is totally explained by the OPE to a high precision (with an error of 10 −5 only), provided we include the 1/(m Q) 3 terms. Further terms are strongly suppressed. On the whole, from the various detailed studies which have been made, it appears highly probable that equality between inclusive decay and OPE holds very well for total decay rates in QCD, even for a realistic spectrum and kinematics, provided the OPE expansion itself is pushed sufficiently far. In other words, not only 1/m Q duality violation cannot be present, but possible duality violation, at high orders, is small.