Due to the fact thatW andZ bosons are very massive, weak meson decays (whose energyE is of the order of the meson’s mass and thus E MW,Z) proceed at very small distances of the order ofO(1/MW). Then, following the ideas of the previous section, the dynamical degrees of freedom that mediate these processes1 can be integrated out of the Standard Model (SM) Lagrangian to define the Weak Effective Hamiltonian (WEH) for the given decay. Formally, this has the structure of an Operator Product Expansion (OPE) [4–6]
Heff= GF
√2 X
i
Ci(µ)Oi, (1.1.1)
where Oi is a set of local effective operators, µ is the renormalisation scale and Ci(µ) are the corresponding Wilson coefficient functions.
In this context, the amplitude of a generic meson transition M1 → M2 described by (1.1.1) admits the following decomposition
A(M1 →M2) =hM2| Heff|M1i= GF
√2 X
i
Ci(µ)hM2| Oi(µ)|M1i ≡ GF
√2 X
i
Ci(µ)hOi(µ)i, (1.1.2)
1These include the electroweak gauge bosons (W±andZ), the Higgs field and the top quark.
5
where both the initial and final state mesonsM1 and M2 may contain more than only one meson in them (i.e. B →ππ).
Wilson coefficients contain the contributions from physical processes with characteristic scales higher than the renormalisation scale µ. These are perturbatively calculable functions as long as the scale µ is large enough. Therefore, the Ci include contributions from quantum fluctuations involving the high frequency modes of the theory that we integrated out: top quarks, W and Z bosons and the Higgs, in the SM. If we assume a particular ultraviolet completion of the SM, the new heavy degrees of freedom will also be integrated out of the Lagrangian, provided that they are sufficiently heavy, and thus will contribute to the Wilson coefficients. Consequently, Ci(µ) typically depend on mt, mW, mH and the mass scales of the heavy modes of any SM extension considered. These dependencies are identified by computing the penguin and box diagrams with fully propagating top,W,Z, Higgs and new heavy mediators exchanges. It is important to stress that also short distance QCD effects are properly taken into account in the computation of Wilson coefficients, these being the ones that introduce theµscale dependence through the running of the QCD coupling constantαs.
On the contrary, the matrix elements hOi(µ)i summarise the contributions to the amplitude A(M1 →M2) from scales lower thanµ. Then, since they involve long distance interactions, it is not possible to compute them in perturbation theory and one must resort to non-perturbative methods for their assessment. Examples of such non-perturbative tools include lattice gauge theory, the 1/Nc expansion (where Ncis the number of colours), QCD sum rules, light-cone sum rules, chiral perturbation theory, etc. In B meson decays, we will see in Chapter 3 that certain dynamical conditions that emerge from Heavy Quark Effective Theory (HQET) and Large Energy Effective Theory (LEET) allow for systematic computations of these matrix elements.
Constructing an Effective Hamiltonian involves computing its corresponding Wilson coeffi-cients and identifying its relevant local effective operators. Notice that all this can be fully done in perturbation theory. All physical information about the inner structure of the mesons in the decay is irrelevant for this construction and it will only become relevant when computing the ma-trix element of the decay, through the evaluation of the mama-trix elements discussed above. Thus, the Wilson coefficients, being objects that only depend on the underlying quark transition and the perturbative interactions they experience, are independent of the particular decay considered in the same way as the usual couplings of gauge theories are universal and process-independent parameters.
Precisely, the biggest advantage of this framework is that it allows to separate the problem of computing the amplitudeA(M1 →M2) into two distinct parts: theshort-distance (perturbative) calculation of the effective couplingsCiand thelong-distance(non-perturbative) computation of the matrix elementshOii. As we discussed above, the scaleµseparates out short-distance contributions to the amplitude from the long-distance ones. Actually, this separation is somewhat arbitrary, since there is certain freedom in assigning which contribution is absorbed into the Wilson coefficients and which other is encoded within the matrix elements. Indeed, the only effect of changing the scale µ is to reshuffle terms from the matrix elements hOii into the Wilson coefficients Ci, hence there is no information loss in shifting from one scale to another. At the same time, this has the important implication that the scale dependence of the matrix elements cancels the unphysical scale dependence of the Wilson coefficients, and the other way around, so the amplitude is a physically meaningful quantity.
1.1. Weak Effective Hamiltonian in the Standard Model 7 This completes our remarks about the main features of the WEH formalism and its building blocks. Now, from a more practical perspective, we review the standard procedure for obtaining the corresponding effective Hamiltonian to a given process (see Refs. [7–10] for more detailed discussions):
I Given a process M1 → M2, compute the amplitude A(M1 → M2) in the full theory up to certain order inαs. As a result of Feynman diagrams with one or more loops, divergences will appear during the computations, which can be treated using usual renormalisation techniques by introducing a high scaleµ0 ∼MW. At this scaleαsis small, so perturbation theory works well.
I Once the computation of the amplitude is performed in the full theory and at the high scale µ0, one can identify the effective operators{Oi}. With the operators known, the OPE for the effective Hamiltonian can be written.
I Calculate the matrix elements hOii of the operators at the high scale µ0 and at the same order in αs. Perturbation theory still applies at this scale and because QCD corrections are included, divergences will arise again. Some of them can be absorbed through field renor-malisation, however in general the resulting expressions will still be divergent. Therefore an additional multiplicative renormalisation, the so-called operator renormalisation, is nec-essary. This method typically introduces mixing between different operators in the OPE that carry the same quantum numbers, in such a way that even new operators which were not present at tree level may arise. Moreover, since these matrix elements hOii must be computed within a renormalisation scheme, this introduces a certain scheme dependence on the Wilson coefficients, so the overall scheme depend cancels at the amplitude level.
I Having computed the amplitude A(M1 → M2) in the full theory and the matrix elements hOii of the operators in the OPE, at the same order in Perturbation Theory and at the scaleµ0, the Wilson coefficientsCi(µ0) can be obtained via (1.1.1). This procedure is called matching the full theory onto the effective theory.
I Finally, renormalisation group evolution techniques must be used to obtain the value of the Wilson coefficients at the low scaleµ. As we discussed, theµ scale is arbitrary, allowing us to choose what belongs to Ci(µ) and what to hOi(µ)i. However, it is customary to choose µof the order of the mass of the decaying meson, which for B decays means µ=µb ∼mb. So µ has to be run down from µ0 ∼ MW to µb ∼ mb. Then, the matching conditions for the Wilson coefficients develop large logarithms ln(MW2 /µ2b) that break their computation in a perturbative expansion. Solving for this requires the use of Renormalisation Group (RG) Improved Perturbation Theory [8,11], which we will discuss in the following section.
As a last note, it is important to stress that the description provided by a weak effective Hamil-tonian is only valid up to the order in the 1/MW expansion used for building the OPE and to the order in αs in which the matching conditions of the effective theory are based on.
1.1.1 Renormalisation Group Evolution of Wilson coefficients
As we noticed above, the matching conditions for the Wilson coefficients present an important problem: the expansion parameter αs/π ∼0.1 (at the µb ∼ mb scale) is always accompanied by
consistent powers of ln(MW2 /µ2b), establishing (αs/π) ln(MW2 /µ2b)∼0.8 as the de facto expansion parameter and spoiling the perturbative expansion. This is actually a generic problem of renormal-isable quantum field theories with vastly different characteristic scales: perturbative expansions organise in powers of αsln(M2/µ2) rather thanαs.
In order to solve the problem of these large logarithms, they must beresummed to all orders by means of (RG)-evolution equations in Renormalisation Group Improved Perturbation Theory.
This framework provides a reorganisation of perturbation theory whereαsln(M2/µ2) works as the O(1) parameter in the expansion. Then, at leading order all terms of the form (αsln(M2/µ2))n (with n = 0, . . . ,∞) are resummed, contributing to the Wilson coefficients at order O(1). Next-to-leading order contributions are obtained by resumming terms of the form αs(αsln(M2/µ2))n, which correspondingly count asO(αs) in the expansion.
Great references where RG-Improved Perturbation Theory is discussed at length, alongside many other relevant topics for the study of Effective Theories, are Refs. [9, 12]. The discussion below borrows some of its elements from these two references.
Now we will explicitly show how large logarithms must be resumed up to leading order. Con-sider a complete set of effective operators
{Oi(µ)}, (withi= 1, . . . , n) (1.1.3) allowed by the symmetries of the theory. Since arbitrary changes of the renormalisation scale µ does not affect the physical value of the amplitude
A=X
Being the basis of operators complete, we can write the derivative of every matrix element in terms of the other matrix elements in the set,
d
dlnµhOi(µ)i=−X
j
γij(µ)hOj(µ)i, (1.1.7) whereγij is usually referred to as the anomalous dimension matrix and it measures how much the matrix element changes under scale variations. It is important to stress that no large logarithms are contained in this matrix. Clearly, if there is more than one operator in the basis, in general γij will cause operators to mix when the scale is changed. Expanding the derivative in Eq. (1.1.6) and using Eq. (1.1.7), we obtain
1.1. Weak Effective Hamiltonian in the Standard Model 9 Therefore, because the operators are linearly independent, Eq. (1.1.8) necessarily implies the fol-lowing renormalisation group equation for the Wilson coefficients
d
dµCi(µ)−X
j
Cj(µ)γji(µ) = 0, (1.1.9)
which can be rewritten in matrix form as d
dµC(µ) = ˆγT(µ)C(µ), (1.1.10) Because the anomalous dimension matrix controls the scale variation of the matrix elements and scale dependence gets introduced in them as a result of QCD corrections, ˆγ depends on the scale µonly through the running of theαs(µ) coupling constant. Then, we can change variables fromµ toαs in Eq. (1.1.10) to find
d
dαs(µ)C(µ) = ˆγT(αs(µ))
β(αs(µ)) C(µ), (1.1.11) where β ≡ dαs(µ)/dlnµ is the QCD beta function. Eqs. (1.1.10) and (1.1.11) are differential equations controlling the running of the Wilson coefficients with initial conditions set by the values of the Wilson coefficients at the high scale, i.e. C(MW).
Notice that Eq. (1.1.11) is formally identical to the Heisenberg equation of motion for time-dependent operators, with the anomalous dimension matrix playing the role of the Hamiltonian.
Therefore, the general solution to equation Eq. (1.1.11) with the aforementioned initial conditions reads
C(µ) =Tαexp
"Z αs(µ) αs(MW)
dαsˆγT(αs(µ)) β(αs(µ))
#
C(MW), (1.1.12)
with the formal definition of the exponential above given by its Taylor expansion andTα being an operator that organises the powers of ˆγT(αs) such that those terms evaluated at larger values ofαs stand to the left of those with smaller values ofαs. Keeping a well-defined ordering of the powers of ˆγT(αs) is needed since, in general, two realisations of ˆγT at differentαs will not commute.
Now we apply Eq. (1.1.12) to the simplified case of having a theory with only one Wilson coefficient (i.e. no mixing), this will explicitly show how resummation of large logarithms works.
First, we write the perturbative expansion of the relevant quantities in the RG-evolution
C(MW) = 1 +O(αs), (1.1.13)
γ(αs) =γ0αs
4π +O(α2s), (1.1.14)
β(αs) =−2αs β0αs
4π +O(α2s)
, (1.1.15)
where we have assumed the effective operator we are dealing with is such that it represents a process operating at leading order in the full theory, henceC(MW) = 1. Notice that none of these quantities are afflicted by large logarithms.
Then, we find the leading order result
C(µ) =
The resummation of the large logarithm can be made explicit by expanding the evolution factor in the equation above, Hence, in RG-Improved Perturbation Theory the characteristic large logarithms that appear in the running of Wilson coefficients are encapsulated in ratios comparing the strength of QCD interactions between the high and low scales through the QCDβ-function.
If one wants to compute similar expressions for the RG-evolution of Wilson coefficients at higher orders (i.e. next-to-leading order and beyond), it is enough to write consistent perturbative expansions forC(µ),γ(αs) and β(αs),
up to the desired order inαs and proceed as detailed above. Generalisations of this procedure to cases with more than one operator, where mixing between operators is expected, can be found in Ref. [13].