On the Incorporation of the D Meson into the CKM Matrix

1. Introduction

Explorations of the neutrino sector, both experimental and theoretical, have revealed that the physical states associated with the electron, muon, and tau flavours appear as characteristic mixtures of mass eigenstates. This understanding emerged from the observation, reported in 1998 by Takaaki Kajita’s team [1], that neutrinos change flavour during propagation through an oscillatory phenomenon. This observation confirmed a bold prediction made by Bruno Pontecorvo [2,3] in 1957, inspired by a similar type of oscillation between the neutral kaon and its antiparticle, predicted in 1955 by Abraham Pais and Oreste Piccioni [4]. That prediction was experimentally confirmed in 1959 by groups at Berkeley and elsewhere, establishing the existence of two distinct neutral-kaon lifetimes [5].

The phenomenon of neutral-meson oscillation, which also occurs in the neutral D- and B-meson systems, contributed to the development in 1973 of the 3×3 CKM matrix by Makoto Kobayashi and Toshihide Maskawa [6]. The CKM matrix extended the 2×2 mixing matrix introduced by Nicola Cabibbo in 1963 [7]. In the Cabibbo framework, the down-type $d^{\prime }$ and $s^{\prime }$ quarks are mixed quantum states of d and s, providing a mathematical description of the observed behaviour of the neutral kaon.

Within the Standard Model of particle physics, the CKM matrix is incorporated into the theoretical framework of Feynman box diagrams. These diagrams account for the possible interaction paths connecting the relevant quark states, including contributions from off-shell virtual particles [8]. This framework, however, has a notable shortcoming: while the neutral kaon and the two neutral B-meson systems fit naturally into it, the neutral D-meson system does not [9].

In this article, it will be shown that the neutral D meson can be given a natural place in the CKM matrix by adopting a theoretical approach based on the principles of the Structural Model of particle physics developed by the author [10].

2. The Structural Neutral Meson System

The neutral meson is considered here as a two-body oscillator system. The two bodies are the quark and antiquark that constitute a bound state, maintained in equilibrium between the attractive force associated with the monopole property of the quark and the repulsive force associated with its non-angular dipole moment. In the Structural Model of physics, the bosonic force associated with this non-angular dipole moment is identified as the weak interaction [10].

A symmetrical configuration, such as a pion, has a single mode: interchanging the particle and antiparticle states returns an identical system. This is different for an asymmetrical configuration, such as a kaon. Interchanging the particle and antiparticle states produces a two-body oscillator that can operate in two different modes. Each of these modes can be regarded as a resonator. The system can therefore be described as an assembly of two resonators. An operational mechanism that induces the interchange of particle and antiparticle states can then be interpreted as a coupling mechanism between the two resonators.

Such a system has useful analogies with electrical $LC$ oscillators or mechanical mass-spring oscillators, as shown in Figure 1.

If $k_1=k_2=k$, $m_1=m_2=m$ and $k_c\ne 0;k_c<<k$ the system resonates in two different modes, with an offset around the central angular frequency ${\omega }_0$. This offset is observable as a beat frequency $f_{beat}$.

$$2\pi f_{beat}=\sqrt{{\displaystyle \frac{k+2k_c}{m}}}-\sqrt{{\displaystyle \frac{k}{m}}}\approx {\displaystyle \frac{k_c}{k}}{\omega }_0;{\omega }_0=\sqrt{{\displaystyle \frac{k}{m}}}.$$

(1)

Let us now return to the neutral meson. The process that drives the interchange is the weak interaction, mediated by the weak-interaction boson W in a virtual state. According to the principles of the Structural Model, a weak-interaction boson is capable of reversing the polarity of the non-angular dipole moment of a quark [10]. In other words, the weak interaction may change an up-isospin quark into a down-isospin quark, and vice versa. This is evident in the transformation of u or d quarks into d or u quarks, as occurs in beta decay. However, a transformation of a u quark into an s quark may also occur. Such a transformation is shown, for example, in the creation of the s quark through pion excitation.

Within this framework, the interaction between $d\overline{s}$ and $s\overline{d}$ can occur along different routes, each with a different probability. This is illustrated in the first row of figurte 2. Tentatively, it is assumed here that the quark hierarchy contains, beyond the bottom quark $\left(b\right)$ sector, another up-quark. Such a quark is taken for granted in the Standard Model because of the existence of the top quark. In the Structural Model, however, the natural candidate would have a mass of only approximately three times that of b is incapable of hadronization, making it non-existent [10,11].

In the Structural Model, the top quark appears instead as the relativistic state of the $u/d$ quark and as the quark that, together with the antitop quark, allows the W boson to be conceived as a meson. As illustrated in the figure, if such a quark—whether identified with the top quark or regarded as a hypothetical quark—is accepted, the probabilities associated with the three routes sum to unity, with decreasing amplitudes along the row.

Figure 2. transient probabilities in the neutral kaon system

Figure 2. transient probabilities in the neutral kaon system

The lower row of the figure gives the format of the transient probabilities that enable the transformation of the kaon state $s\overline{d}$ into the antikaon state $\overline{s}d$. This format is given by the squared product of transient amplitudes. In terms of the mass-spring analogy, the kaon mass can be regarded as occurring in two different modes around a central value $m_K$ , with an offset $\Delta m_K$. One may expect this mass offset to be proportional to the mass $m_s$ of the s quark and to the sum of the transient probabilities involved, weighted according to their mass contributions. In general, this can be expressed as

$$\begin{array}{cc}\hfill \Delta m_K^{\prime }& =A{\displaystyle \frac{2m_s^{\prime }}{m_W^{\prime 2}}}\left\{{\left(V_{ud}{V}_{us}\right)}^2{m}_u^{\prime 2}+{\left(V_{cd}{V}_{cs}\right)}^2{m}_c^{\prime 2}+{\left(V_{td}{V}_{ts}\right)}^2{m}_t^{\prime 2}\right\}\hfill \\ \hfill & \approx 2A{m}_s^{\prime }{\left(V_{cd}{V}_{cs}\right)}^2{\displaystyle \frac{m_c^{\prime 2}}{m_W^{\prime 2}}}.\hfill \end{array}.$$

(2)

In this expression, the proportionality factor A is dimensionless, and the constituent masses $m_u$, etc., are normalized to the known rest mass of the weak-interaction boson W. In principle, of course, the various contributions compete with one another in relative strength. The approximation shown here will be explained later in the text. The formula is heuristic, but it is based on a physical line of reasoning. Its validity must ultimately be demonstrated by comparison with experimental evidence.

Applying the same approach to the neutral bottom B mesons, we can construct Figure 3.

Let us now consider the transformation of a quark into an antiquark more closely. Within the neutral kaon meson, the s quark transforms into d through the up-quarks u,c and t, respectively, via the transient amplitudes $V_{ud}$,$V_{cd}$ and $V_{td}$. The squared amplitudes represent the corresponding probabilities. Consequently, the sum of the squared amplitudes must equal unity. On the other hand, the antiquark $\overline{d}$ transforms into $\overline{s}$ through $V_{us}$,$V_{cs}$ and $V_{ts}$, respectively. Hence,

$$\begin{array}{cc}\hfill V_{ud}^2+V_{cd}^2+V_{td}^2& =1,\hfill \\ \hfill V_{us}^2+V_{cs}^2+V_{ts}^2& =1.\hfill \end{array}$$

(3)

A similar relationship applies to the neutral mesons in the bottom sector. The resulting relationships can be represented by a single 3 × 3 probability matrix $M_{prob}$ in which the sum of the elements in each row and each column is equal to 1. Thus,

$$M_{prob}=\left[\begin{array}{ccc}{V}_{ud}^2& V_{us}^2& V_{ub}^2\\ V_{cd}^2& V_{cs}^2& V_{cb}^2\\ V_{td}^2& V_{ts}^2& V_{tb}^2\end{array}\right].$$

(4)

The amplitude matrix derived from this probability matrix resembles the $CKM$ (Cabibbo-Kobayashi-Maskawa) matrix, which will be discussed later in this text. Generalizing Equation (2), we now obtain

$$\begin{array}{cc}\hfill \Delta m_K^{\prime }& =A{\displaystyle \frac{2m_s^{\prime }}{m_W^{\prime 2}}}\left\{{\left(V_{ud}{V}_{us}\right)}^2{m}_u^{\prime 2}+{\left(V_{cd}{V}_{cs}\right)}^2{m}_c^{\prime 2}+{\left(V_{td}{V}_{ts}\right)}^2{m}_t^{\prime 2}\right\}\hfill \\ \hfill & \approx 2A{m}_s^{\prime }{\left(V_{cd}{V}_{cs}\right)}^2{\displaystyle \frac{m_c^{\prime 2}}{m_W^{\prime 2}}},\hfill \\ \hfill \Delta m_{B_s}^{\prime }& =A{\displaystyle \frac{2m_b^{\prime }}{m_W^{\prime 2}}}\left\{{\left(V_{us}{V}_{ub}\right)}^2{m}_u^{\prime 2}+{\left(V_{cs}{V}_{cb}\right)}^2{m}_c^{\prime 2}+{\left(V_{ts}{V}_{tb}\right)}^2{m}_t^{\prime 2}\right\}\hfill \\ \hfill & \approx 2A{m}_b^{\prime }{\left(V_{ts}{V}_{tb}\right)}^2{\displaystyle \frac{m_t^{\prime 2}}{m_W^{\prime 2}}},\hfill \\ \hfill \Delta m_{B_d}^{\prime }& =A{\displaystyle \frac{2m_b^{\prime }}{m_W^{\prime 2}}}\left\{{\left(V_{ub}{V}_{ud}\right)}^2{m}_u^{\prime 2}+{\left(V_{cb}{V}_{cd}\right)}^2{m}_c^{\prime 2}+{\left(V_{tb}{V}_{td}\right)}^2{m}_t^{\prime 2}\right\}\hfill \\ \hfill & \approx 2A{m}_b^{\prime }{\left(V_{tb}{V}_{td}\right)}^2{\displaystyle \frac{m_t^{\prime 2}}{m_W^{\prime 2}}}.\hfill \end{array}.$$

(5)

By relating the upper two equations of this set, we obtain

$${\displaystyle \frac{\Delta m_{B_d}^{\prime }}{\Delta m_K^{\prime }}}\approx {\displaystyle \frac{m_b^{\prime }}{m_s^{\prime }}}{\displaystyle \frac{{\left(V_{tb}{V}_{td}\right)}^2}{{\left(V_{cd}{V}_{cs}\right)}^2}}{\displaystyle \frac{{m_t^{\prime }}^2}{{m_c^{\prime }}^2}}.$$

(6)

Similarly, relating the lower two equations gives

$${\displaystyle \frac{\Delta {m^{\prime }}_{B_s}}{\Delta {m^{\prime }}_{B_d}}}\approx {\displaystyle \frac{V_{ts}^2}{V_{td}^2}}.$$

(7)

Let us next attempt to determine the values of the probability matrix from empirical values of the beat-frequency ratios. These empirical values are given in Figure 4. The energy values of the constituent quarks have been derived within the Structural Model of particle physics from first principles.

Because nine unknown parameters have to be determined, nine independent equations are required. The probability matrix in Equation (4) provides six independent equations. Equations (6) and (7) provide two additional equations, leaving one equation still to be determined.

The missing equation can be obtained from the neutral D meson. This meson belongs to a different class. The class is complementary in the sense that the roles of the up-quarks and down-quarks are reversed. Figure 5 illustrates the proposed transformation of the neutral D meson into its antiparticle. Note that the probability amplitudes and their relationships can conveniently be extracted from the probability matrix in Equation (4). Thus, for the neutral D meson, we obtain

$$\Delta m_D^{\prime }=A{\displaystyle \frac{2m_c^{\prime }}{m_W^{\prime 2}}}\left\{{\left(V_{cd}{V}_{ud}\right)}^2{m_d^{\prime }}^2+{\left(V_{cs}{V}_{us}\right)}^2{m_s^{\prime }}^2+{\left(V_{cb}{V}_{ub}\right)}^2{m_b^{\prime }}^2\right\}.$$

(8)

Because of the competing contributions, this expression cannot be simplified. Compared with the results obtained for the other neutral mesons, it appears at first that the top quark plays no role here. This would indeed be the case if the top quark were an isospin-up quark. In the Structural Model of particle physics, however, the top quark is regarded as the relativistic state of the $u/d$ quark. Evidence for this interpretation is presented in [10]. It therefore has the same isospin properties, possessing both the up-quark and down-quark properties. Because of its down-quark property, the top quark must be included in the transient path $c\overline{u}\leftrightarrow \overline{c}u$. See Figure 6. In doing so, the top quark behaves as the d quark, with a weighting factor determined by the top-quark contribution. Equation (8) must therefore be modified to

$$\Delta m_D^{\prime }=A{\displaystyle \frac{2m_c^{\prime }}{m_W^{\prime 2}}}\left\{{\left(V_{cd}{V}_{ud}\right)}^2{m}_d^{\prime 2}+{\left(V_{cs}{V}_{us}\right)}^2{m}_s^{\prime 2}+{\left(V_{cb}{V}_{ub}\right)}^2{m}_b^{\prime 2}+{\left(V_{cd}{V}_{td}\right)}^2{m}_t^{\prime 2}\right\}.$$

(9)

Relating (9) with the upper equation in (5), we have

$${\displaystyle \frac{\Delta {m^{\prime }}_D}{\Delta {m^{\prime }}_K}}=$$

$${\displaystyle \frac{{m^{\prime }}_c}{{\left(V_{cd}{V}_{cs}\right)}^2{m^{\prime }}_s}}\{{\left(V_{ud}{V}_{cd}{\displaystyle \frac{{m^{\prime }}_d}{{m^{\prime }}_c}}\right)}^2+{\left(V_{us}{V}_{cs}{\displaystyle \frac{{m^{\prime }}_s}{{m^{\prime }}_c}}\right)}^2+{\left(V_{ub}{V}_{cb}{\displaystyle \frac{{m^{\prime }}_b}{{m^{\prime }}_c}}\right)}^2+{\left(V_{td}{V}_{cd}{\displaystyle \frac{{m^{\prime }}_t}{{m^{\prime }}_c}}\right)}^2\}.$$

(10)

With this additional requirement, it is now possible, using elementary algebra, to calculate an amplitude matrix that satisfies all the requirements defined above. The result is

$$\left[\begin{array}{ccc}0.974& 0.225& 0.006\\ 0.225& 0.974& 0.0352\\ 0.006& 0.0352& 0.999\end{array}\right].$$

(11)

From this result the validity of the assumptions in (5) can be readily validated.

The obtained amplitude matrix shows a surprisingly close fit with the CKMmatrix currently formulated within the Standard Model. The latter is

$$\left[\begin{array}{ccc}0.974& 0.224& 0.0038\\ 0.221& 0.975& 0.0415\\ 0.0086& 0.0410& 0.9991\end{array}\right]$$

(12)

.

The way in which the Standard Model 3 × 3 matrix evolved is different from the approach presented here. The result obtained in this text is based on a heuristic formulation of the coupling mechanism between the two modes of the neutral meson. Its advantage is that the neutral D meson has a natural place within the system. Even more: the incorporation of the D meson is a prerequisite for deriving the elements of the probability matrix as a necessary and sufficient set of basic equations.

The history of the 3 × 3 CKM matrix is quite different. It was conceived in 1973 by Makoto Kobayashi and Toshihide Maskawa [6], building on the 2 × 2 matrix introduced in 1963 by Nicola Cabibbo. Cabibbo’s formulation formalized the underlying unitary weak-interaction relationship between the d quark and the s quark. Whereas Cabibbo related the d quark and the s quark by hypothesizing that they could occur as a unitary mixture defined by

$$\left[\begin{array}{c}{d}^{\prime }\\ s^{\prime }\end{array}\right]=\left[\begin{array}{cc}{V}_{ud}& V_{us}\\ V_{cd}& V_{cs}\end{array}\right]\left[\begin{array}{c}d\\ s\end{array}\right]=\left[\begin{array}{cc}cos{\vartheta }_c& sin{\vartheta }_c\\ -sin{\vartheta }_c& cos{\vartheta }_c\end{array}\right]\left[\begin{array}{c}d\\ s\end{array}\right].$$

(13)

Kobayashi and Maskawa subsequently incorporated the later-discovered quarks into a framework that permits unitary mixing among all quark generations, such that

$$\left[\begin{array}{c}{d}^{\prime }\\ s^{\prime }\\ b^{\prime }\end{array}\right]=\left[\begin{array}{ccc}{V}_{ud}& V_{us}& V_{ub}\\ V_{cd}& V_{cs}& V_{cb}\\ V_{td}& V_{ts}& V_{tb}\end{array}\right]\left[\begin{array}{c}d\\ s\\ b\end{array}\right].$$

(14)

Note: The magnitude of the Cabibbo angle is a measure of the relative strength of the probability transitions between u and d, and between $u/d$ and s.

This formalism marked the beginning of intensive experimental efforts to establish the numerical values of the matrix elements that are known today. In parallel, a theoretical framework was developed to provide mathematical support for the observed numerical values. This framework, based on the application of Feynman’s path-integral approach, includes the possible interaction paths between the $d,s,b$ states, including contributions from off-shell virtual states. The result, however, has a significant shortcoming from the perspective adopted in this text: it does not incorporate the neutral D meson into the framework.

The two probability matrices shown in Equations (11) and (12) are not sufficient to define unique unitary matrices. Both share the property that the sums of the squared elements over each row and each column are equal to 1. They can therefore be regarded as generalizations of Cabibbo’s 2 × 2 matrix, with its single mixing angle ${\theta }_c$, to a product of three rotation matrices $M=R_1\otimes R_2\otimes R_3$, each with its own mixing angle:

$$M=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos{\vartheta }_{23}& sin{\vartheta }_{23}\\ 0& -sin{\vartheta }_{23}& cos{\vartheta }_{23}\end{array}\right]\left[\begin{array}{ccc}cos{\vartheta }_{13}& 0& sin{\vartheta }_{13}\\ 0& 1& 0\\ -sin{\vartheta }_{13}& 0& cos{\vartheta }_{13}\end{array}\right]\left[\begin{array}{ccc}cos{\vartheta }_{12}& sin{\vartheta }_{12}& 0\\ -sin{\vartheta }_{12}& cos{\vartheta }_{12}& 0\\ 0& 0& 1\end{array}\right].$$

(15)

The results discussed so far allow the mixing angles to be calculated from the amplitude matrix. The mixing angles associated with Equations (10) and (11), respectively, are

$$\begin{array}{cc}\hfill & {\vartheta }_{12}=13.0^{\circ };{\vartheta }_{13}=0.{34}^{\circ };{\vartheta }_{23}=2.{02}^{\circ },\hfill \\ \hfill & {\vartheta }_{12}=13.0^{\circ };{\vartheta }_{13}=0.{22}^{\circ };{\vartheta }_{23}=2.{38}^{\circ }.\hfill \end{array}.$$

(16)

An amplitude matrix with real coefficients is compatible with $CP$ conservation, because complex conjugation leaves the real coefficients unchanged. Unitarity can nevertheless be preserved if, in addition to the rotation matrices, one or more phase angles are introduced. For example, this can be achieved by an additional matrix

$$\left[\begin{array}{ccc}0& 0& e^{-i\delta }\\ 0& 1& 0\\ e^{i\delta }& 0& 0\end{array}\right].$$

(17)

The introduction of such phase angles makes the coefficients of the probability matrix complex while leaving their squared absolute values unchanged.

3. CP Violation

The freedom to introduce a phase parameter into the unitary neutral-meson system makes it possible to account for the unexpected phenomenon of charge-parity $\left(CP\right)$ violation in neutral-meson systems. $CP$ violation was first observed in 1964 by Christenson, Cronin, Fitch, and Turlay through the decay behaviour of neutral kaons [12].

To understand this phenomenon, it is instructive to regard the meson as a two-body oscillator, as is done in the Structural Model of particle physics. In this model, a quark and an antiquark form an anharmonic quantum-mechanical oscillator through an equilibrium between the strong interaction associated with the quark’s monopole property and the weak interaction associated with its non-angular dipole property. The two constituents of this oscillator may move either in phase or out of phase with respect to each other. The resulting states are symmetric and antisymmetric superpositions of the two constituent configurations:

$$\begin{array}{cc}\hfill {\psi }_{K_S}& ={\psi }_{d\overline{s}}+{\psi }_{s\overline{d}},\hfill \\ \hfill {\psi }_{K_L}& ={\psi }_{d\overline{s}}-{\psi }_{s\overline{d}}.\hfill \end{array}$$

(18)

or, equivalently,

$$\begin{array}{cc}\hfill K_S& =K_1={\displaystyle \frac{K_0+{\overline{K}}_0}{\sqrt{2}}},\hfill \\ \hfill K_L& =K_2={\displaystyle \frac{K_0-{\overline{K}}_0}{\sqrt{2}}}.\hfill \end{array}$$

(19)

(Note that superposition as such is different from mixing due to coupling discussed so far. Mixing may occur, though, under superposition)

The $CP$ properties of these two superpostions are different. Whereas $CP|K_S\rangle =1$ has even symmetry, $CP|K_L\rangle =-1$ has odd symmetry. Under decay, the $CP$ character of the initial state is expected to be preserved. Consequently, the hadronic decay of $K_S$ should result in a two-pion state, $\left({\pi }^{+}{\pi }^{-}\right)$ or $\left({\pi }^0{\pi }^0\right)$, whereas the hadronic decay of $K_L$ should result in a three-pion state, $\left({\pi }^{+}{\pi }^{-}{\pi }^0\right)$ or $\left({\pi }^0{\pi }^0{\pi }^0\right)$. This is indeed what is normally observed.

The fact that the decay associated with $K_L$ takes considerably longer than that associated with $K_S$ is also understandable within this picture, because the available phase space and the corresponding mass gaps of the two decay paths are different. Against this background, the observation in 1964 of $K_L\to \pi \pi$ decay events was particularly remarkable and was interpreted as evidence for a violation of $CP$ symmetry.

Although the observation of CP violation came as a surprise, there is no fundamental reason why interactions between quarks should be restricted to real transition amplitudes. Within the Structural Model of particle physics, complex transition amplitudes can even be anticipated. The reason is that reversal of the polarity of the non-angular dipole moment of the quarks, which in the Structural Model is identified with the origin of the weak interaction, represents only one component of the quark interaction. A second component arises from the interaction associated with the regular angular dipole moment.

As analysed in the Structural Model, this angular dipole moment gives rise to a bosonic spin-spin interaction between quarks. This is a quantum-mechanical process governed by coefficients constructed from Pauli matrices, and these coefficients naturally introduce complex quantities into the transition amplitudes. A complete first-principles treatment of these effects would, however, be rather cumbersome. A practical approach is therefore to introduce an additional phase-offset matrix into the product of the three rotation matrices derived from measurements of the beat frequencies of the four neutral-meson configurations.

The remaining problem is to determine a numerical value for the phase parameter. Whereas the numerical values of the transition amplitudes can be obtained from appropriate measurements of the beat frequencies, determining the phase parameter is less straightforward. A detailed discussion of this issue is beyond the scope of the present article. It is discussed in the literature describing the theory underlying the $CKM$ matrix [13].

4. Predictive Power

The probability matrix (11) derived in this article is correct by construction. When applied to the four neutral-meson systems, it reproduces the observed numerical relationships between their beat frequencies. This agreement, however, is obtained by assigning empirical data to the parameters of a heuristic model constructed on physical grounds. Such a construction is meaningful only if the resulting model has predictive power for physical phenomena other than those used to determine its parameters.

In this respect, the Structural Model approach is not fundamentally different from the canonical $CKM$ matrix shown in (12). The striking difference lies in the way in which the two matrices are derived. The Structural Model approach is based on a direct and comparatively simple structural interpretation. By contrast, the canonical $CKM$ matrix has undergone continuous refinement over the years in order to improve its predictive accuracy. This refinement has required the introduction of a number of corrective factors and phenomenological parameters, including bag parameters, decay constants, loop functions, $QCD$ correction factors, and related quantities [13].

Moreover, the canonical framework encounters the additional difficulty that the neutral D meson does not fit naturally into the theoretical framework underlying the conventional $CKM$ description. As argued in this article, this discrepancy can be traced to the different interpretations of the top quark in the Structural Model of particle physics and in the Standard Model of particle physics.

5. Conclusions

Whereas the conventional theory of the $CKM$ matrix within the Standard Model encounters difficulties in accommodating the neutral D meson within the same structural framework, the Structural Model of particle physics provides a different perspective. It permits the incorporation of this meson by recognizing non-canonical relationships between the weak interaction associated with the non-angular dipole moment of quarks, the weak-interaction boson W, represented in the Structural Model as a meson composed of top quarks, and the isospin property attributed to the top quark as a consequence of its relativistic state as a $u/d$ quark.

This structural interpretation suggests that the relationships between the different neutral-meson systems need not be identical to those assumed in the canonical Standard Model description. Instead, they may reflect the underlying structural properties of the quarks and their associated dipole interactions.