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2011 QCD and High Energy Interactions - Rencontres de Moriond ...

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Estimating the <strong>High</strong>er Or<strong>de</strong>r Hadronic Matrix Elements<br />

in the Heavy Quark Expansion<br />

THOMAS MANNEL<br />

Theoretische Elementarteilchenphysik, Naturwiss. Techn. Fakultät,<br />

Universität Siegen, 57068 Siegen, Germany<br />

The non-perturbative input in the heavy quark expansion relevant for precision <strong>de</strong>terminations<br />

of CKM matrix elements from heavy hadron <strong>de</strong>cays consists of certain forward matrix<br />

elements of local operators. While at low or<strong>de</strong>rs these matrix elements may be <strong>de</strong>termined<br />

from experiment, the number of in<strong>de</strong>pen<strong>de</strong>nt matrix elements at higher or<strong>de</strong>rs is way too large<br />

to extract them from data. Hence an estimate for these matrix elements from the theoretical<br />

si<strong>de</strong> is necessary. In this contribution I present a way to estimate these matrix elements in a<br />

simple way.<br />

1 Introduction<br />

The theoretical <strong>de</strong>scription of semi-leptonic <strong>de</strong>cays of heavy hadrons is in a very mature state.<br />

The major tool for reliable calculations is the Heavy Quark Expansion (HQE) <strong>and</strong> Heavy Quark<br />

Effective Theory (HQET) together with the Heavy Quark Symmetries (HQS) appearing in the<br />

heavy mass limit.<br />

The <strong>de</strong>termination of Vcb can be performed from exclusive as well as from inclusive <strong>de</strong>cays.<br />

While the inclusive <strong>de</strong>termination makes use of the HQE, the exclusive <strong>de</strong>termination uses HQS,<br />

which constrain the form factors at the non-recoil point, where the four-velocities of the initial<br />

<strong>and</strong> final state hadrons are the same.<br />

The theory for the inclusive <strong>de</strong>termination based on HQE has reached the status of a precision<br />

calculation. It is based on the computation of the total rate, which in HQE is given as a combined<br />

series in αs(mb) n , (Λ<strong>QCD</strong>/mb) m , <strong>and</strong> (Λ<strong>QCD</strong>/mc) k (Λ<strong>QCD</strong>/mb) l+3 1 . Currently the leading term<br />

m = k = 0 is known to or<strong>de</strong>r α 2 s 2,3 , the first sub-leading corrections are partially known to or<strong>de</strong>r<br />

α 2 s × (Λ<strong>QCD</strong>/mb) 2 4 , while the tree level terms are known to or<strong>de</strong>r (Λ<strong>QCD</strong>/mb) m 5 , <strong>and</strong> the term<br />

involving inverse powers of mc (Λ<strong>QCD</strong>/mc) 2 (Λ<strong>QCD</strong>/mb) 3 6 . Overall, this calculation has reached<br />

a theoretical uncertainty at the level of one percent.<br />

In or<strong>de</strong>r to explore the higher or<strong>de</strong>rs of the HQE quantitatively, it is m<strong>and</strong>atory to have a<br />

way to estimate the hadronic matrix elements appearing as the nonperturbative input. In the<br />

following I shall present a method for such an estimate.

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