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6 The operators of the effective Hamiltonian

So far, we have presented the exact solutions of the renormalization group equations for the Wilson coefficients. In practice, it is only possible to calculate the relevant functions in perturbation theory. For illustrative purposes, we consider the calculation of the effective Hamiltonian at the leading order in QCD. The bare Hamiltonian is given in eq. (29). In the presence of QCD interactions, other operators appear in the Wilson expansion. A complete basis is given by the following operators

The q index runs over the ``active'' flavours. The above operators are generated by gluon exchanges in the Feynman diagrams of fig. 2. In particular, is generated by current--current diagrams and -- are generated by penguin diagrams. The choice of the operator basis in not unique, and different possibilities have been considered in the literature [27]. If the electromagnetic correction, are also taken into account, the operator basis enlarges to include the following operators

Below the bottom threshold, the following relation holds

 

so that there are nine independent operators. The basis is further reduced below the charm threshold by using the relations

  
Figure: One-loop corrections to the effective Hamiltonian.

All the operators considered above are dimension-six operators. In principle, two dimension-five operators

should also be included in the operator basis. The matrix elements of and , however, enter only at in chiral perturbation theory. Since the phenomenological analysis presented in the following is only valid up to terms of , we do not need to include the contribution of the dimension-five operators in the calculation of . The effect of these operators on has recently been analysed in ref. [29]. Other operators of lower dimensionality (e.g. two-fermion operators) are also potentially present. However, it can be shown that their effect can be reabsorbed in a suitable redefinition of the fermion fields and by diagonalizing the quark mass matrix at first order in [23]--[26].

In summary, the effective Hamiltonian, renormalized at a scale , can be written as

where, in order to find the Wilson coefficients to a given order in , we have to calculate eqs. (41), (45) in perturbation theory.

The explicit expressions of and , in the LLA

can be found for example in ref. [5]. In eq. (41), using and , one obtains

At this order, the matching conditions are trivial: , eq. (43), is the identity matrix; , eq. (42), has all vanishing components with the only exception of . Thus the Wilson coefficients at the leading order for are given by

with and all the other Wilson coefficients at the scale vanish.

In the next-to-leading logarithmic approximation (NLLA), one proceeds along the general scheme described above. In this case, all quantities entering in the matching procedure have to be computed at order ( for the electromagnetic case). The -function and the anomalous dimension matrix have to be computed at second order in the coupling constants. Thus, for example, the anomalous dimension matrix in the NLLA has the form

where corrections have been neglected. We will not give here any details of the NLLA calculations. They can be found in refs. [1]--[5]. At the next-to-leading order, it is necessary to solve numerically eq. (37). Table 1 contains the coefficients, calculated at the leading (LO) and at the next-to-leading (NLO) order, using the 't Hooft--Veltman (HV) and the naïve dimensional (NDR) regularization schemes, for different values of the renormalization scale . The errors in the table take into account the variation of the values of the coefficients due to MeV and GeV. Notice that the next-to-leading Wilson coefficients and operators both depend on the regularization scheme, while the effective Hamiltonian is scheme-independent up to terms . Actually the dependence of the effective Hamiltonian on the regularization scheme, due to the unknown next-to-next-to-leading terms, can be estimated and contributes to the uncertainties in the prediction of , see ref. [9].

The coefficients in table 1 have been computed independently by the Munich group [4,14]. The definition of the renormalized operators in the HV scheme used here differ from those defined in ref. [14]. This is due to the different way of taking into account the two-loop anomalous dimension of the weak current, which does not vanish in the HV calculation. One can relate the HV coefficients of table 1 () and those of ref. [14] (). The relation is

where

Once these differences in the definition of the renormalized operators and the reduction of the operator basis, eq. (49), are properly taken into account, the numerical results presented here agree with those of ref. [14].

  
Table: Wilson coefficients of the effective Hamiltonian at GeV. For , the relation (49) has been used to reduce the operator basis. We take MeV and GeV. The values of the coefficients shown here correspond to the central values of these parameters. The first error is due to the uncertainty on , the second is due to .



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Next: 7 Relevant formulae Up: Chapter 1 Section 2 Previous: 5 QCD corrections



Carlos E.Piedrafita