Lattice measurement of the Isgur - Wise functions τ 1 / 2 and τ 3 / 2

We propose a method to compute the Isgur-Wise form factors τ1/2(1) and τ3/2(1) for the decay of B mesons into orbitally excited (P wave) D∗∗ charmed mesons on the lattice in the static limit. We also present the result of an exploratory numerical simulation which shows that the signal/noise ratio allows for a more dedicated computation. We find τ1/2(1) = 0.38(5) and τ3/2(1) = 0.53(8), with yet unknown systematic errors. These preliminary numbers agree fairly well with theoretical expectation. PACS: 12.38.Gc (Lattice QCD calculations), 12.39.Hg (Heavy quark effective theory), 13.20.He (Leptonic/semileptonic decays of bottom mesons). Introduction The scalar heavy-light mesons and more generally the first orbital excitations D∗∗ have attracted attention since years and they still remain somehow mysterious. The recent discovery of a cs̄-scalar meson significantly lighter than expected has renewed the interest in these states [1, 2]. There have been several lattice studies of this spectrum [3, 4] and a recent rather complete one compares quenched and unquenched [5] computations. Recently the H∗ 0 → Hπ transition (scalar-pseudoscalar-pion) have also been considered [6, 7]. The transitions of the type B → D∗∗lν raise a serious problem. In the infinite mass limit these decays are described by the Isgur-Wise form factors τ1/2 and τ3/2 [8]. To make a long story short, a series of sum rules [9]-[14] have been derived from QCD, all indicating that τ3/2 should be significantly larger than τ1/2. These sum rules relate the τj form factors, as well as form factors related to excitations, to derivatives of the ground state Isgur-Wise function ξ and allow to bound the latter derivatives in an efficient and useful way [15]-[17]. Not only does the slope of ξ verify ρ > 3/4 but also the curvature and even higher derivatives are bound. The limit in which τ1/2 = 0 has been baptised “BPS” by Uraltsev [18]-[19] and was proven to provide intersting hints. However the theoretical prediction that τ (0) 3/2 > τ (0) 1/2 and hence that the decay B → D∗ 2 should be significantly larger than the B → D∗ 0 is not verified by experiment [2, 20]. This is the ‘1/2 > 3/2’ paradox [21]. One might incriminate the corrections to the infinite mass limit. Another possibility could be that the sum rules are fulfilled by higher excitations and that the ground state obeys an opposite hierarchy i.e. τ (0) 3/2 < τ (0) 1/2 . To answer to this question one needs to compute directly τ (0) 3/2 and τ (0) 1/2. Here we propose a lattice method to do that. We will work in the static quark limit, mb,c → ∞, with the four vectors v′ = v = (1, 0, 0, 0), and we will exhibit we will exhibit operators whose matrix elements allow to measure these form factors. This letter is meant to propose this new method and to make a feasibility study. We do not intend at this stage to provide accurate results for these form factors but merely to describe the principle of the method and to show with preliminary simulations that there is good hope to make the precision calculation. 1 Principle of the calculation We are concerned with the matrix element of an electroweak current between a pseudoscalar or vector heavy-light meson H(∗) and an orbitally excited one H∗∗. However, in the conditions of the infinite mass limit on the lattice with the heavy quarks at rest, both in the initial and final state (vμ = v ′ μ), this matrix element vanishes. The way out is to use a series of relations derived in ref. [22]. In that paper it has been shown that in the case of a matrix element which vanishes linearly in the difference v′ − v, when v′ → v, there are non-vanishing forward matrix elements (for v′ = v) involving the There is no mathematical impossibility for the sum rules to be fulfilled with an reversed hierarchy for the ground state, but it does not seem very likely and is not seen in models.