If we interpret $K0$ as $s¯d$ and $K0¯$ as $sd¯$⁠, in the standard model there are two one-loop diagrams which accomplish the transition $K0→K0¯$⁠. They are box diagrams, which we have depicted in Figs. 17.2 and 17.3. The diagram of Fig. 17.3 is identical with that of Fig. 17.2 after the interchange $d1↔d2$ has been made. The diagrams have two $W±$ bosons and two up-type quarks, $α$ and $β$⁠, in the loop. The quarks $α$ and $β$ may be either u, c, or t. Any of the two gauge bosons $W±$ may be substituted by the corresponding Goldstone bosons $φ±$⁠; we have not depicted the diagrams involving the $φ±$⁠, but we include them in the computation of the box diagrams.

In order to find the effective $∣ΔS∣=2$ interaction we integrate out the ‘heavy’ degrees of freedom: the $W±$ bosons and the up-type quarks. When doing this we are not interested in the ‘light’ degrees of freedom, i.e., in the masses and momenta of the external particles in the diagram. We therefore use an approximation in which the external particles are massless $(ms=md=0)$ and their four-momenta are zero. In consequence, all internal lines carry the same fourmomentum $kμ,$ which we have to integrate over.

We compute the diagram of Fig. 17.2 in an arbitrary ’t Hooft gauge in order to check its gauge-independence. We use the following shorthands for the denominators of the various propagators:

Gauge-independence means that the final result should be independent of $ξW$⁠.

We use the shorthands in eqn (13.50) for the relevant combinations of CKM matrix elements; unitarity of the CKM matrix implies eqn (13.53). Indeed, the existence of an unitary CKM matrix was first suggested by Glashow, Iliopoulos, and Maiani (1970) in the context of a computation of the box diagram. For this reason, the use of eqn (13.53) is usually referred to as the GIM mechanism.

Taking into account that each boson $W±$ may be substituted by the corresponding Goldstone boson $φ±$⁠, we have

We notice that

As a result, the terms proportional to $k4−mα2mβ2$ in eqn (B.2) yield momentum integrals whose integrand is independent either of $mα$ or of $mβ$⁠. Equation (13.53) implies that those contributions vanish upon summation either over $α$ or over $β$⁠, respectively. We thus find that all terms with denominator $Dg$ yield vanishing contributions, and the box diagram is gauge-independent:

It is now evident that the momentum integral is finite.

As the denominator of the integrand depends on k only through $k2$⁠, the integral of $kζkη$ is equivalent to $gζη/4$ times the integral of $k2$⁠. We also use the Dirac-matrix identity

to derive

where we have used the notation in eqn (17.1). We thus obtain

We introduce Feynman parameters and perform the momentum integral, obtaining

where

Defining

we have, after integrating over the Feynman parameters x and y,

Here,

for $β≠α$⁠. For $β=α$ one should use the limit when $mβ→mα$ of the function in eqn (B.12).

We use eqn (13.53) to rewrite eqn (B.11) in such a way that the sums only run over the quarks c and t. We use the definition of the Fermi constant $GF=g2/(42mW2)$ to trade $g2$ by $GF$⁠. Finally, the approximation $mu=0$ is convenient, in order to obtain expressions which do not depend on $mu$⁠. We get

where

Here,

and the function $S0(x)$ is the limit when $y→x$ of $S0(x,y)$⁠:

We interpret the box diagram as resulting from an effective Hamiltonian. Following our conventions, the Lagrangian must be equal to the Feynman diagram divided by a factor i, just as the Feynman rule for each vertex is i times the corresponding term in the Lagrangian. Moreover, the interaction Hamiltonian is minus the interaction Lagrangian. Therefore,

We have divided $ℋeff$ by an extra 2, taking into account that $ℋeff$ is the product of two identical operators, and therefore the Feynman rule for the vertex is $−2iℋeff$⁠.

The effective Hamiltonian in eqn (B.17) yields the result

for the second box diagram effecting the $K0→K0¯$ transition. This means that $ℋeff$⁠, although derived from the computation of the diagram in Fig. 17.2 only, really gives rise to the two diagrams in Figs. 17.2 and 17.3, the values of those diagrams being given by eqns (B.13) and (B.18), respectively.

We now consider the possibility that, in some extension of the standard model, there is a physical scalar particle S which has flavour-changing neutral Yukawa interactions—see § 22.10—with the s and d quarks:

where a and b are dimensionless coupling constants. The interaction in eqn (B.19) leads to $K0→K0¯$ transitions via the tree-level diagrams in Fig. B.1.

Just as in the computation of the box diagram, we assume all the external momenta to vanish. Then, the momentum of the propagator of S is zero. Denoting by m the mass of S, we obtain for the effective Hamiltonian which gives rise to the diagrams in Fig. B.1 the result

Because of parity symmetry the matrix element of the operator $(s¯d)(s¯γ5d)$ between $〈K0¯∣$ and $∣K0〉$ is zero. The other matrix elements may be estimated using the vacuum-insertion approximation in Appendix C. One obtains the following result for the contribution of the interaction in eqn (B.19) to $M21$⁠:

Suppose there is a vector boson $Xμ$ which couples to a flavour-changing neutral current between the s and d quarks:

where a and b are dimensionless coupling constants. This interaction leads to $K0→K0¯$ transitions via the tree-level diagrams depicted in Fig. B.2. The corresponding effective Hamiltonian is

where m now is the mass of the vector boson X. The contribution to $M21$⁠, with the relevant matrix elements computed in the vacuum-insertion approximation, is