32 Decay Amplitudes: Effective Hamiltonian

The diagrammatic approach discussed in the previous chapter is useful to get an order-of-magnitude estimate of the importance of the various contributions to each amplitude, and to find approximate relations among the amplitudes for different decays. However, rigorous computations require a correct treatment of the QCD corrections, and this is best accomplished using the operator product expansion (OPE) already discussed in the context of kaon decays—see Chapter 20. The OPE allows the construction of a low-energy effective Hamiltonian as a sum of local operators multiplied by Wilson coefficients. The effective Hamiltonian is the sum of pieces, each of them of the type

with a definite CKM factor $VCKM$⁠. The Wilson coefficients $Cn(μ)$ depend on the renormalization scale $μ$ and on the renormalization scheme. These dependences should cancel in the physical amplitude

with the renormalization-scale and renormalization-scheme dependences of the operator matrix elements $〈f∣Qn∣i〉(μ)$⁠. The cancellations may involve simultaneously many Wilson coefficients and matrix elements. The scale $μ∼5GeV$ separates the high-energy QCD corrections, which are treated perturbatively and included in the Wilson coefficients, from the low-energy non-perturbative confinement effects implicitly contained in the operator matrix elements. However, upon change of $μ$ or of the renormalization scheme, contributions previously included in the matrix elements of some operator may move into the Wilson coefficients or into the matrix elements of other operators. One uses the renormalization group to run the Wilson coefficients from high to low energies. We refer the reader to the review articles by Buchalla et al. (1996) and by Buras and Fleischer (1998), where detailed discussions of the calculations can be found, together with extensive references to the original literature.

The relevant diagrams are analogous to those depicted in the figures of Chapters 20 and 31. The operators that one needs are

The operators in eqns (32.3)–(32.5) are entirely analogous to the ones introduced in Chapter 20 for the decays of the kaons. Still, as the decays of $b¯$ may involve quarks with different flavours in the final state, one needs to introduce more operators, and a more complex notation. Thus, k stands for a light down-type quark, either d or s; while $α$ and $β$ stand for either u or c (out of the current-current operators in eqn 32.3, those with $α=β$ correspond to $ΔC=ΔU=0$ decays, while the $∣ΔC∣=∣ΔU∣=1$ decays involve the operators with $α≠β$⁠). Furthermore, x and y are colour indices, just as in Chapter 20. In the sums, q runs over the four quarks u, d, s, and c; the electromagnetic-penguin operators involve the electric charges of the quarks: $eq=2/3$ for $q=u$ and $q=c$⁠, and $eq=−1/3$ for $q=d$ and $q=s$⁠. The Dirac structures V and A refer to the vector and axial-vector currents, respectively; $V±A$ stands for $γμ(1±γ5)$⁠.

The magnetic-penguin operators in eqn (32.6) correspond to the effect of the non-negligible mass of the bottom quark in penguin diagrams (see Fig. 32.1); in practice, only the external-mass effects on gluonic and photonic penguins need to be considered. These operators are important, for example, for the decay $b→sγ$⁠. This topic is beyond the scope of this book, and we shall drop these operators henceforth. We refer the reader to the reviews by Buchalla et al. (1996) and by Buras and Fleischer (1998).

An important feature of the Wilson coefficients is that they do not depend on the flavour k of the down-type quark; that is, the Wilson coefficients are the same for $k=d$ and $k=s$⁠.

As an example, consider the effective Hamiltonian responsible for $ΔC=ΔU=0$ decays,

We have used the definition

Unitarity implies $λuk+λck+λtk=0$⁠.

The Wilson coefficients are renormalization-scheme- and renormalization-scale-dependent. For instance, Buras and Fleischer (1998) give, in the $MS¯$ renormalization scheme with anti-commuting $γ5$⁠, for $μ=m¯b(mb),mt=170GeV$⁠, and with $ΛQCD(5)=225MeV$⁠,

in the next-to-leading-logarithmic approximation. Here, $α$ is the electromagnetic fine-structure constant. Due to the large value of $C9$⁠, electroweak-penguin operators need to be considered in certain decays where tree-level diagrams are either heavily CKM-suppressed or altogether absent.

Let us define (Buras and Fleischer 1998)

We then rewrite the effective Hamiltonian in eqn (32.7) as

This expression shows explicitly that, in the standard model, one may use the unitarity of the CKM matrix to rewrite the decay amplitudes in terms of only two independent weak phases.

Although the computations of the Wilson coefficients have become very elaborate, one still faces huge uncertainties in the estimates of the matrix elements of the operators. In B decays, the standard procedure has been to assume factorization, splitting the matrix element of each four-fermion operator into the product of matrix elements of two quark bilinears. This is achieved by inserting the vacuum state in all possible ways. The resulting matrix elements of quark bilinears are parametrized with form factors, one for each momentum structure consistent with Lorentz invariance and parity. These form factors are directly determined from experiment whenever possible; most often, they must be calculated within a model. Many recent articles use the relativistic BSW model (Bauer et al. 1985, 1987) to evaluate the matrix elements. In this approach, the matrix elements are written in terms of physical quantities and are, therefore, renormalization-scheme- and renormalization-scale-independent. Thus, they do not satisfy the requirements in the previous section.

Admittedly, the computation of the matrix elements is an awkward, unreliable step of the computation of the amplitudes, and any results must be taken as mere indications. Should the matrix elements be determined from the lattice, one will be able to get definite predictions.

In this section we study how the effective Hamiltonian transforms under CP, and the corresponding relations among the decay amplitudes that may be derived.

We use the CP transformation of the quark field operators

or else we may use directly the results in Table 3.1 for the transformation properties of quark bilinears. We find

Therefore, as the Wilson coefficients are real,

Let us assume that the final states that we are considering in these $ΔC=0=ΔU$ decays only include particles of definite flavour—thus excluding the presence of mesons like $KS$⁠. In that case, the $b¯q→qα¯αk¯$ decay leads into a final state f whose quark content is given exclusively by $qα¯αk¯$⁠, with no other component. Similarly, the $bq¯→q¯αα¯k$ transition leads into a final state $f¯$ whose quark content is given exclusively by $q¯αα¯k$⁠. Under these circumstances, $〈f∣T∣Bq0〉$ is obtained directly from the effective Hamiltonian. It is important to note that this is not always the case. If the final state contains particles that are not in a flavour eigenstate, such as $KS$ or $Dfcp$⁠, then one must include in the calculation the transformation from the flavour-eigenstate basis into the basis of the experimentally observed particles. We shall encounter this problem when we calculate the CP-violating asymmetry in $Bd0→J/ψKS$⁠. For the moment, we ignore that possibility.

Using the CP transformations of the final and initial states,

we find

to be compared with

When f is a CP eigenstate, $f¯=f,eiξf=ηf=±1$⁠, and eqns (32.16) and (32.17) yield

We see that the ratio reduces to a pure phase if the decay is such that, in the sums in eqn (32.18), only the operators with a certain up-type quark $α$ and a certain down-type quark k contribute.

The effective Hamiltonian relevant for these decays is

This Hamiltonian leads to four amplitudes,

As we have seen in § 31.3.2, for the $Sf≠0≠Df$ decays, i.e., for $q≠k$⁠, only the final states with either a $KS$ or a $KL$ are common to $Bq0$ and $Bq0¯$⁠. On the contrary, all final states reached by $Sf=0=Df$ decays—when $q=k$—are common. In this case the amplitudes in eqn (32.20) describe the decays into two final states: f with quark content $qc¯uq¯$⁠; and $f¯$ with quark content $qu¯cq¯$⁠. Those amplitudes therefore correspond to $Af,Af¯,A¯f¯$⁠, and $A¯f$⁠, respectively. Using the CP-transformation properties of the operators and kets, we find

However, for decays into non-CP eigenstates this does not give us any information useful for the calculation of $λf$⁠; indeed, $λf=(q/p)(A¯f/Af)$ while, for instance, the first eqn (32.21) relates $Af$ with $A¯f¯$⁠.

As for CP eigenstates, the results of § 31.3.2 show that all $ΔC≠0≠ΔU$ decays into CP eigenstates involve $Dfcp$⁠. (Also, the CP eigenstates to be reached by $ΔC≠0≠ΔU$ and $Sf≠0≠Df$ decays are $DfcpKL,S$⁠.) These are defined as the combination of $D0$ and $D0¯$ which decays into the CP eigenstate $fcp$⁠. Then, the decay amplitudes $〈fcp∣T∣D0〉$ and $〈fcp∣T∣D0¯〉$ must also be included in the computation—see § 34.4 and 34.5.