End Matter

Appendix C The Vacuum-Insertion Approximation in $K^{0} - \bar{K^{0}}$ Mixing

https://doi.org/10.1093/oso/9780198503996.005.0003

Pages

469–475

Published:

July 1999

Appendix C

C.1 Introduction

When we consider processes involving hadrons we often face the difficult task of evaluating matrix elements of operators between hadronic states. The difficulty is associated with the fact that the operators are written in terms of quarks while the physical asymptotic states are hadrons. Unfortunately, there is no reliable, first-principles calculation of the hadronization mechanism and, therefore, of the matrix elements of operators. For some leptonic or semileptonic decays, approximate symmetries allow trustworthy matrix-element computations to be performed. This is the case of chiral symmetry for hadrons with light quarks, in flavour-SU(3) multiplets; and heavy-quark symmetry for hadrons with a b or a c quark—the corrections being significant in the latter case. This is not possible however in the case of matrix elements for nonleptonic decays. Those matrix elements are usually evaluated using specific phenomenological models such as the BSW model (Bauer et al. 1985). Ultimately, they should be computed using lattice methods.

However, we may get an estimate of some matrix elements of operators quartic in the quark fields by using the vacuum-insertion approximation (VIA). The idea of the VIA is to separate the matrix element of a quartic operator into the product of two matrix elements of operators bilinear in the quark fields. The separation is not performed through a complete set of intermediate states—a partition of unity; rather, only the vacuum state is inserted. The assumption of the VIA is that a reasonable estimate, correct at least in order of magnitude, may be achieved by using only the vacuum as intermediate state. Of course, the corrections to the VIA must be estimated by using some other method for computing the matrix element.

Here we shall illustrate the VIA by evaluating a number of matrix elements relevant for neutral-kaon mixing (McWilliams and Shanker 1980). The matrix elements that we shall compute are $〈 \bar{K^{0}} ∣ {(\bar{s} d)}^{2} ∣ K^{0} 〉, 〈 \bar{K^{0}} ∣ {(\bar{s} γ_{5} d)}^{2} ∣ K^{0} 〉, 〈 \bar{K^{0}} ∣ (\bar{s} γ^{μ} d) ∣ K^{0} 〉,$ and $〈 \bar{K^{0}} ∣ (\bar{s} γ^{μ} γ_{5} d) (\bar{s} γ_{μ} γ_{5} d) ∣ K^{0} 〉 .$ We shall also compute $〈 \bar{K^{0}} ∣ (\bar{s} Γ^{μ} d) (\bar{s} Γ_{μ} d) ∣ K^{0} 〉$ ⁠, where $Γ^{μ} \equiv γ^{μ} (1 - γ_{5}) / 2$ ⁠. The operators in these matrix elements are always the product of two colour-singlet quark bilinears, with definite transformation properties under the Lorentz group.

C.2 Creation and destruction operators

The quark fields are linear combinations of destruction operators and creation operators. Those operators act on the initial and final hadron states in all possible ways. Destruction operators act to the right and creation operators act to the left. We must therefore reorder the operators, bringing the creation operators to the left and the destruction operators to the right. When we do this, there is a minus sign for each transposition, because fermionic operators anticommute.¹⁰⁹

Without loss of generality, we consider d to be the sum of a single operator $d^{-}$ which destroys d quarks and a single operator $d^{+}$ which creates $\bar{d}$ antiquarks. Similarly, $\bar{s}$ is the sum of an operator ${\bar{s}}^{-}$ which destroys $\bar{s}$ antiquarks and an operator ${\bar{s}}^{+}$ which creates s quarks. In the initial state $K^{0}$ one has a d quark and an $\bar{s}$ antiquark. In the final state $\bar{K^{0}}$ one has a $\bar{d}$ antiquark and an s quark. In the operator which effects the transition one must therefore have one destruction and one creation operator for each flavour. Thus,

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} d) (\bar{s} d) | K^{0} 〉 = 〈 \bar{K^{0}} | ({\bar{s}}^{+} d^{+}) ({\bar{s}}^{-} d^{-}) + ({\bar{s}}^{+} d^{-}) ({\bar{s}}^{-} d^{+}) \\ + ({\bar{s}}^{-} d^{+}) ({\bar{s}}^{+} d^{-}) + ({\bar{s}}^{-} d^{-}) ({\bar{s}}^{+} d^{+}) | K^{0} 〉 \\ = 2 〈 \bar{K^{0}} | ({\bar{s}}^{+} d^{+}) ({\bar{s}}^{-} d^{-}) + ({\bar{s}}^{+} d^{-}) ({\bar{s}}^{-} d^{+}) | K^{0} 〉 . \end{array}

(C.1)

In the same way,

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} γ_{5} d) (\bar{s} γ_{5} d) | K^{0} 〉 = 2 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{5} d^{-}) | K^{0} 〉 \\ + 2 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ_{5} d^{-}) ({\bar{s}}^{-} γ_{5} d^{+}) | K^{0} 〉, \end{array}

(C.2)

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} γ^{μ} d) (\bar{s} γ_{μ} d) | K^{0} 〉 = 2 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} d^{+}) ({\bar{s}}^{-} γ_{μ} d^{-}) | K^{0} 〉 \\ + 2 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} d^{-}) ({\bar{s}}^{-} γ_{μ} d^{+}) | K^{0} 〉, \end{array}

(C.3)

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} γ^{μ} γ_{5} d) (\bar{s} γ_{μ} γ_{5} d) | K^{0} 〉 = 2 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{μ} γ_{5} d^{-}) | K^{0} 〉 \\ + 2 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} γ_{5} d^{-}) ({\bar{s}}^{-} γ_{μ} γ_{5} d^{+}) | K^{0} 〉, \end{array}

(C.4)

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} Γ^{μ} d) (\bar{s} Γ_{μ} d) | K^{0} 〉 = 2 〈 \bar{K^{0}} | ({\bar{s}}^{+} Γ^{μ} d^{+}) ({\bar{s}}^{-} Γ_{μ} d^{-}) | K^{0} 〉 \\ + 2 〈 \bar{K^{0}} | ({\bar{s}}^{+} Γ^{μ} d^{-}) ({\bar{s}}^{-} Γ_{μ} d^{+}) | K^{0} 〉 . \end{array}

(C.5)

C.3 Fierz transformations

C.3.1 Dirac matrices

The Fierz reshuffling theorem for the Dirac matrices states that

{(Γ^{i})}_{αβ} {(Γ_{i})}_{γξ} = \sum_{j} F_{ij} {(Γ^{j})}_{αξ} {(Γ_{j})}_{γβ},

(C.6)

where the covariant Dirac matrices $Γ^{i}$ are $1, γ^{μ}, σ^{μν}, γ^{μ} γ_{5}$ ⁠, and $γ_{5}$ ⁠, respectively, and the matrix of indices F is

F = \frac{1}{8} (\begin{array}{r} 2 & 2 & 1 & - 2 & 2 \\ 8 & - 4 & 0 & - 4 & - 8 \\ 24 & 0 & - 4 & 0 & 24 \\ - 8 & - 4 & 0 & - 4 & 8 \\ 2 & - 2 & 1 & 2 & 2 \end{array}) .

(C.7)

From the basic Fierz identity

\begin{matrix} δ_{α β} δ_{γ ξ} = \frac{1}{4} δ_{α ξ} δ_{γ β} + \frac{1}{4} {(γ_{5})}_{α ξ} {(γ_{5})}_{γ β} + \frac{1}{8} {(σ^{μ ν})}_{α ξ} {(σ_{μ ν})}_{γ β} \\ + \frac{1}{4} {(γ^{μ})}_{α ξ} {(γ_{μ})}_{γ β} - \frac{1}{4} {(γ^{μ} γ_{5})}_{α ξ} {(γ_{μ} γ_{5})}_{γ β} \end{matrix}

(C.8)

one may derive various other Fierz identities for Dirac matrices, in particular

{(Γ^{μ})}_{αβ} {(Γ_{μ})}_{γξ} = - {(Γ^{μ})}_{αξ} {(Γ_{μ})}_{γβ} .

(C.9)

C.3.2 Gell-Mann matrices

The basic Fierz identity for the Gell-Mann matrices $λ^{a}$ is

δ_{wx} δ_{yz} = \frac{1}{3} δ_{wz} δ_{yx} + \frac{1}{2} \sum_{a = 1}^{8} λ_{wz}^{a} λ_{yx}^{a} .

(C.10)

C.3.3 Applications in the VIA

The Fierz transformations are needed in the context of the VIA because the second terms in the right-hand sides of eqns (C.1)–(C.5) contain creation operators paired with destruction operators in a manner which is inappropriate to insert the vacuum state. One must therefore change the order of the operators.

Our aim is to reorder a combination of the type $({\bar{s}}^{+} Γ^{i} d^{-}) ({\bar{s}}^{-} Γ_{i} d^{+})$ ⁠. Let us display explicitly the colour indices w, x, y, and z, and the Dirac indices $α, β, γ,$ and $ξ$ ⁠. We have

\begin{array}{l} ({\bar{s}}_{w α}^{+} d_{x β}^{-} {\bar{s}}_{y γ}^{-} d_{z ξ}^{+}) [{(Γ^{i})}_{α β} {(Γ_{i})}_{γ ξ}] (δ_{w x} δ_{y z}) \\ = (- {\bar{s}}_{w α}^{+} d_{z ξ}^{+} {\bar{s}}_{y γ}^{-} d_{x β}^{-}) [\sum_{j} F_{i j} {(Γ^{j})}_{α ξ} {(Γ_{j})}_{γ β}] (\frac{1}{3} δ_{w z} δ_{y x} + \frac{1}{2} \sum_{a = 1}^{8} λ_{w z}^{a} λ_{y x}^{a}) . \end{array}

(C.11)

The colour structure of the $λ^{a} \otimes λ^{a}$ term in eqn (C.11) forbids the insertion of a colour-singlet state like the vacuum. One assumes that this term may be neglected, and one obtains

({\bar{s}}^{+} Γ^{i} d^{-}) ({\bar{s}}^{-} Γ_{i} d^{+}) = - \frac{1}{3} \sum_{j} F_{ij} ({\bar{s}}^{+} Γ^{j} d^{+}) ({\bar{s}}^{-} Γ_{j} d^{-}) .

(C.12)

Applying this formula to eqns (C.1)–(C.5), one gets

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} d) (\bar{s} d) | K^{0} 〉 = \frac{11}{6} 〈 \bar{K^{0}} | ({\bar{s}}^{+} d^{+}) ({\bar{s}}^{-} d^{-}) | K^{0} 〉 \\ - \frac{1}{6} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{5} d^{-}) | K^{0} 〉 \\ - \frac{1}{12} 〈 \bar{K^{0}} | ({\bar{s}}^{+} σ^{μ ν} d^{+}) ({\bar{s}}^{-} σ_{μ ν} d^{-}) | K^{0} 〉 \\ - \frac{1}{6} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} d^{+}) ({\bar{s}}^{-} γ_{μ} d^{-}) | K^{0} 〉 \\ + \frac{1}{6} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{μ} γ_{5} d^{-}) | K^{0} 〉, \end{array}

(C.13)

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} γ_{5} d) (\bar{s} γ_{5} d) | K^{0} 〉 = - \frac{1}{6} 〈 \bar{K^{0}} | ({\bar{s}}^{+} d^{+}) ({\bar{s}}^{-} d^{-}) | K^{0} 〉 \\ + \frac{11}{6} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{5} d^{-}) | K^{0} 〉 \\ - \frac{1}{12} 〈 \bar{K^{0}} | ({\bar{s}}^{+} σ^{μ ν} d^{+}) ({\bar{s}}^{-} σ_{μ ν} d^{-}) | K^{0} 〉 \\ + \frac{1}{6} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} d^{+}) ({\bar{s}}^{-} γ_{μ} d^{-}) | K^{0} 〉 \\ - \frac{1}{6} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{μ} γ_{5} d^{-}) | K^{0} 〉, \end{array}

(C.14)

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} γ^{μ} d) (\bar{s} γ_{μ} d) | K^{0} 〉 = - \frac{2}{3} 〈 \bar{K^{0}} | ({\bar{s}}^{+} d^{+}) ({\bar{s}}^{-} d^{-}) | K^{0} 〉 \\ + \frac{2}{3} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{5} d^{-}) | K^{0} 〉 \\ + \frac{7}{3} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} d^{+}) ({\bar{s}}^{-} γ_{μ} d^{-}) | K^{0} 〉 \\ + \frac{1}{3} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{μ} γ_{5} d^{-}) | K^{0} 〉, \end{array}

(C.15)

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} γ^{μ} γ_{5} d) (\bar{s} γ_{μ} γ_{5} d) | K^{0} 〉 = \frac{2}{3} 〈 \bar{K^{0}} | ({\bar{s}}^{+} d^{+}) ({\bar{s}}^{-} d^{-}) | K^{0} 〉 \\ - \frac{2}{3} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{5} d^{-}) | K^{0} 〉 \\ + \frac{1}{3} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} d^{+}) ({\bar{s}}^{-} γ_{μ} d^{-}) | K^{0} 〉 \\ + \frac{7}{3} 〈 \bar{K^{0}} | ({\bar{s}}^{+} γ^{μ} γ_{5} d^{+}) ({\bar{s}}^{-} γ_{μ} γ_{5} d^{-}) | K^{0} 〉, \end{array}

(C.16)

〈 \bar{K^{0}} ∣ (\bar{s} Γ^{μ} d) (\bar{s} Γ_{μ} d) ∣ K^{0} 〉 = \frac{8}{3} 〈 \bar{K^{0}} ∣ ({\bar{s}}^{+} Γ^{μ} d^{+}) ({\bar{s}}^{-} Γ_{μ} d^{-}) ∣ K^{0} 〉 .

(C.17)

C.4 Vacuum insertion

We are now ready to insert the vacuum state, assuming for instance that

\begin{array}{l} 〈 \bar{K^{0}} ∣ ({\bar{s}}^{+} d^{+}) ({\bar{s}}^{-} d^{-}) ∣ K^{0} 〉 = 〈 \bar{K^{0}} ∣ ({\bar{s}}^{+} d^{+}) ∣ 0 〉 〈 0 ∣ ({\bar{s}}^{-} d^{-}) ∣ K^{0} 〉 \\ = 〈 \bar{K^{0}} ∣ (\bar{s} d) ∣ 0 〉 〈 0 ∣ (\bar{s} d) ∣ K^{0} 〉 . \end{array}

(C.18)

This is the crucial approximation, together with that of neglecting the $λ^{a} \otimes λ^{a}$ term in the right-hand side of eqn (C.11).

At this juncture one may use a few simplifications. Firstly, the kaon state vectors carry four-momentum $p^{μ} = (E, \vec{p})$ ⁠. As it is impossible to construct an antisymmetric tensor with two indices out of $g^{μν}, ϵ^{μνρσ}$ ⁠, and $p^{μ}$ alone, one concludes that

〈 0 ∣ \bar{s} σ^{μν} d ∣ K^{0} (E, \vec{p}) 〉 = 〈 \bar{K^{0}} (E, \vec{p}) ∣ \bar{s} σ^{μν} d ∣ 0 〉 = 0 .

(C.19)

Secondly, we have seen in § 4.6 that, in the convention in which the s and d quarks transform under parity¹¹⁰ with the same phase $β_{p}$ ⁠,

\begin{array}{c} P d (t, \vec{r}) P^{†} = e^{i β_{p}} γ_{0} d (t, - \vec{r}), \\ P \bar{s} (t, \vec{r}) P^{†} = e^{- i β_{p}} \bar{s} (t, - \vec{r}) γ_{0}, \end{array}

(C.20)

the neutral kaons have negative parity:

\begin{array}{c} P ∣ K^{0} 〉 = − ∣ K^{0} 〉, \\ ∣ \bar{K^{0}} 〉 P^{†} = − 〈 \bar{K^{0}} ∣ . \end{array}

(C.21)

The four-momentum $p^{μ}$ transforms to $p_{μ}$ under parity. Because of eqns (C.20), the operator $\bar{s} γ^{μ} d$ transforms to $\bar{s} γ_{μ} d$ ⁠. Therefore, taking into account eqns (C.21),

〈 0 ∣ \bar{s} γ^{μ} d ∣ K^{0} (E, \vec{p}) 〉 = 〈 \bar{K^{0}} (E, \vec{p}) ∣ \bar{s} γ^{μ} d ∣ 0 〉 = 0 .

(C.22)

In the same way,

〈 0 ∣ \bar{s} d ∣ K^{0} (E, \vec{p}) 〉 = 〈 \bar{K^{0}} (E, \vec{p}) ∣ \bar{s} d ∣ 0 〉 = 0 .

(C.23)

On the other hand, $\bar{s} γ^{μ} γ_{5} d \to - \bar{s} γ_{μ} γ_{5} d$ under parity, and therefore

〈 0 ∣ \bar{s} γ^{μ} γ_{5} d ∣ K^{0} (E, \vec{p}) 〉 = - e^{iφ} p^{μ} f_{K},

(C.24)

where $f_{K}$ is a real positive parameter and $exp (iφ)$ is an arbitrary phase, which is usually set to be either $+ i$ or $- i$ ⁠. Similarly, the matrix elements of $\bar{s} γ_{5} d$ do not vanish.

Thus, if one defines (McWilliams and Shanker 1980)

\begin{array}{l} V_{1} \equiv 〈 \bar{K^{0}} ∣ \bar{s} γ_{5} d ∣ 0 〉 〈 0 ∣ \bar{s} γ_{5} d ∣ K^{0} 〉, \\ V_{2} \equiv 〈 \bar{K^{0}} ∣ \bar{s} γ^{μ} γ_{5} d ∣ 0 〉 〈 0 ∣ \bar{s} γ_{μ} γ_{5} d ∣ K^{0} 〉, \end{array}

(C.25)

one directly obtains from eqns (C.13)–(C.17) the following results:

〈 \bar{K^{0}} ∣ (\bar{s} d) (\bar{s} d) ∣ K^{0} 〉 = - \frac{1}{6} V_{1} + \frac{1}{6} V_{2},

(C.26)

〈 \bar{K^{0}} ∣ (\bar{s} γ_{5} d) (\bar{s} γ_{5} d) ∣ K^{0} 〉 = \frac{11}{6} V_{1} - \frac{1}{6} V_{2},

(C.27)

〈 \bar{K^{0}} ∣ (\bar{s} γ^{μ} d) (\bar{s} γ_{μ} d) ∣ K^{0} 〉 = \frac{2}{3} V_{1} + \frac{1}{3} V_{2},

(C.28)

〈 \bar{K^{0}} ∣ (\bar{s} γ^{μ} γ_{5} d) (\bar{s} γ_{μ} γ_{5} d) ∣ K^{0} 〉 = - \frac{2}{3} V_{1} + \frac{7}{3} V_{2},

(C.29)

〈 \bar{K^{0}} ∣ (\bar{s} Γ^{μ} d) (\bar{s} Γ_{μ} d) ∣ K^{0} 〉 = \frac{2}{3} V_{2} .

(C.30)

C.5 Use of symmetries

Let us now use the CP symmetry of the strong interaction to relate the matrix element in eqn (C.24) to $〈 \bar{K^{0}} (E, \vec{p}) ∣ \bar{s} γ^{μ} γ_{5} d ∣ 0 〉$ ⁠. CP acts in the following way:

\begin{array}{l} 〈 0 | (C P)^{†} = 〈 0 |, \\ C P | K^{0} (E, \vec{p}) 〉 = e^{i ξ_{K}} | \bar{K^{0}} (E, - \vec{p}) 〉, \\ (C P) \bar{s} {(C P)}^{†} = - e^{- i ξ_{s}} s^{T} C^{- 1} γ^{0}, \\ (C P) d {(C P)}^{†} = e^{i ξ_{d}} γ^{0} C {\bar{d}}^{T} . \end{array}

(C.31)

Therefore,

\begin{array}{l} - e^{iφ} p^{μ} f_{K} = 〈 0 ∣ \bar{s} γ^{μ} γ_{5} d ∣ K^{0} (E, \vec{p}) 〉 \\ = - e^{i (ξ_{K} + ξ_{d} - ξ_{s})} 〈 0 ∣ \bar{d} γ_{μ} γ_{5} s ∣ \bar{K^{0}} (E, - \vec{p}) 〉 . \end{array}

(C.32)

Complex-conjugating eqn (C.32) and changing $\vec{p}$ into $- \vec{p}$ leads to

e^{i (ξ_{K} + ξ_{d} - ξ_{s})} e^{- iφ} p_{μ} f_{K} = 〈 \bar{K^{0}} (E, \vec{p}) ∣ \bar{s} γ_{μ} γ_{5} d ∣ 0 〉 .

(C.33)

Therefore,

V_{2} = - e^{i (ξ_{K} + ξ_{d} - ξ_{s})} \frac{f_{K}^{2} m_{K}^{2}}{2 m_{K}} .

(C.34)

In this equation we have inserted a denominator $2 m_{K}$ to correct for the relativistic normalization of states, in which there are 2E particles per unit of volume. (A similar factor shall be used in the next subsection when we extract the value of $f_{K}$ from experiment.) As we consider the kaons to be in their rest frame we use $E = m_{K}$ ⁠.

Contracting eqn (C.33) with $p^{μ}$ and using the equations of motion for the s and d quarks, one obtains

e^{i (ξ_{K} + ξ_{d} - ξ_{s})} e^{- iφ} m_{K}^{2} f_{K} = (m_{s} + m_{d}) 〈 \bar{K^{0}} (E, \vec{p}) ∣ \bar{s} γ_{5} d ∣ 0 〉,

(C.35)

where $m_{s}$ and $m_{d}$ are the current masses of the strange quark and down quark, respectively. Similarly, from eqn (C.24)

(m_{s} + m_{d}) 〈 0 ∣ \bar{s} γ_{5} d ∣ K^{0} (E, \vec{p}) 〉 = + e^{iφ} m_{K}^{2} f_{K} .

(C.36)

Therefore,

V_{1} = + e^{i (ξ_{K} + ξ_{d} - ξ_{s})} \frac{f_{K}^{2} m_{K}^{4}}{2 m_{K} {(m_{s} + m_{d})}^{2}} .

(C.37)

We are now able to write down the final results for the matrix elements in the VIA:

\begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} d) (\bar{s} d) | K^{0} 〉 = e^{i (ξ_{K} + ξ_{d} - ξ_{s})} \frac{f_{K}^{2} m_{K}}{12} [- 1 - \frac{m_{K}^{2}}{{(m_{s} + m_{d})}^{2}}], \\ 〈 \bar{K^{0}} | (\bar{s} γ_{5} d) (\bar{s} γ_{5} d) | K^{0} 〉 = e^{i (ξ_{K} + ξ_{d} - ξ_{s})} \frac{f_{K}^{2} m_{K}}{12} [1 + 11 \frac{m_{K}^{2}}{{(m_{s} + m_{d})}^{2}}], \\ \begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} γ^{μ} d) (\bar{s} γ_{μ} d) | K^{0} 〉 = e^{i (ξ_{K} + ξ_{d} - ξ_{s})} \frac{f_{K}^{2} m_{K}}{6} [- 1 + 2 \frac{m_{K}^{2}}{{(m_{s} + m_{d})}^{2}}], \\ \begin{array}{l} 〈 \bar{K^{0}} | (\bar{s} γ^{μ} γ_{5} d) (\bar{s} γ_{μ} γ_{5} d) | K^{0} 〉 = e^{i (ξ_{K} + ξ_{d} - ξ_{s})} \frac{f_{K}^{2} m_{K}}{6} [- 7 - 2 \frac{m_{K}^{2}}{{(m_{s} + m_{d})}^{2}}], \\ 〈 \bar{K^{0}} | (\bar{s} Γ^{μ} d) (\bar{s} Γ_{μ} d) | K^{0} 〉 = - e^{i (ξ_{K} + ξ_{d} - ξ_{s})} \frac{f_{K}^{2} m_{K}}{3} . \end{array} \end{array} \end{array}

(C.38)

Usually, a CP transformation in which $ξ_{d} = ξ_{s} = 0$ is assumed, while $ξ_{K}$ is chosen to be either 0 or $π$ ⁠. We display all the phases explicitly so that overall rephasing-invariance becomes evident.

C.6 $f_{K}$

Sometimes one finds in the literature the quantity ${\tilde{f}}_{K} = f_{K} / \sqrt{2}$ being denoted by $f_{K}$ ⁠. In order to avoid confusion, we explicitly give the value of $f_{K}$ for the notation we use. From

〈 0 ∣ \bar{s} γ^{μ} γ_{L} d ∣ K^{0} (E, \vec{p}) 〉 = e^{iφ} p^{μ} \frac{f_{K}}{2}

(C.39)

one derives, by means of isospin symmetry,

〈 0 ∣ \bar{s} γ^{μ} γ_{L} u ∣ K^{+} (E, \vec{p}) 〉 = e^{iφ} p^{μ} \frac{f_{K}}{2} .

(C.40)

We normalize the ket $∣ K^{+} 〉$ according to the conventional relativistic normalization of states $〈 K^{+} (E, \vec{p}) ∣ K^{+} (E, \vec{p}) 〉 = 2 EV$ ⁠, which means that there are 2E particles per unit of volume. Using eqn (C.40) one gets

\begin{matrix} Γ (K^{+} \to μ^{+} ν) + Γ (K^{+} \to μ^{+} ν γ) = \frac{G_{F}^{2} {|V_{u s}|}^{2}}{8 π} f_{K}^{2} m_{μ}^{2} m_{K} {(1 - \frac{m_{μ}^{2}}{m_{K}^{2}})}^{2} \\ \times [1 + O (α)] . \end{matrix}

(C.41)

The measured decay widths then yield (Particle Data Group 1996, p. 319) $f_{K} = 159.8 \pm 1.4 \pm 0.44 MeV$ ⁠, where the two error bars arise, respectively, from $∣ V_{us} ∣$ and from the $O (α)$ corrections. Thus, $f_{K} \approx 160 MeV$ ⁠.

Notes

¹⁰⁹

This is so even when the fermionic operators have different quantum numbers, like creation and destruction operators for the s and d quarks and their respective antiquarks.

¹¹⁰

We are allowed to use parity symmetry—and later we shall be using CP symmetry and isospin symmetry too—because we are computing strong-interaction matrix elements. Weak interactions, which are not invariant under those symmetries, have been taken care of in the process of computing Feynman diagrams and thereby deriving an effective Hamiltonian, as was done for instance in Appendix B.

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Contents

Appendix C The Vacuum-Insertion Approximation in $K^{0} - \bar{K^{0}}$ Mixing

Appendix C

C.1 Introduction

C.2 Creation and destruction operators

C.3 Fierz transformations

C.3.1 Dirac matrices

C.3.2 Gell-Mann matrices

C.3.3 Applications in the VIA

C.4 Vacuum insertion

C.5 Use of symmetries

C.6 $f_{K}$

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Contents

Appendix C The Vacuum-Insertion Approximation in K0−K0¯ Mixing

Appendix C

C.1 Introduction

C.2 Creation and destruction operators

C.3 Fierz transformations

C.3.1 Dirac matrices

C.3.2 Gell-Mann matrices

C.3.3 Applications in the VIA

C.4 Vacuum insertion

C.5 Use of symmetries

C.6 fK

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Appendix C The Vacuum-Insertion Approximation in $K^{0} - \bar{K^{0}}$ Mixing

C.6 $f_{K}$