Mass difference between K0 and K0-bar and other meson-antimeson pairs

[MarekS]

The K0--K0-bar, D0--D0-bar, B0--B0-bar, Bs0--Bs0-bar systems all exhibit oscillations whose rate is proportional to their mass difference via a second order weak interaction "box" diagram.

I don't understand how their masses can differ, when they are simply C conjugates of one another. Doesn't the TCP theorem forbid a difference between the masses of a particle and respective anti-particle?

I assume the mass difference between a K0 and a K0-bar (or in the other systems) is caused not by a difference in the masses of s and s-bar or d and d-bar, but by something else. Can someone explain how this mass difference comes about?

 

[mfb]

The mass differences are not between D0 and D0bar (for example) - those are not mass eigenstates anyway. I'll keep the charm meson as example, it is similar for the other systems:

D0 and D0bar are flavour eigenstates - they have a well-defined quark content.
However, they can mix into each other. This allows to find mass eigenstates D₁, D₂. Those have different masses M₁, M₂ and lifetimes $\Gamma_1$,$\Gamma_2$.
The flavour eigenstates are now superpositions of those mass eigenstates and vice versa:

$D_1=p D^0 + q \overline{D^0}$ and $D_2=p D^0 - q \overline{D^0}$
where $|p|^2+|q|^2=1$, both are complex parameters.

Without CP violation, those mass eigenstates are CP eigenstates, and p=q.

In the charm system, it is common to define
$x=\frac{M_1-M_2}{\Gamma}$ and $y=\frac{\Gamma_1-\Gamma_2}{2\Gamma}$ where $\Gamma=\frac{\Gamma_1+\Gamma_2}{2}$ is the average lifetime.
There was a recent measurement of those values by LHCb: Observation of D0-D0bar oscillations
I think you can find references to the theory there.

 

[MarekS]

The mass differences are not between D0 and D0bar (for example) - those are not mass eigenstates anyway. I'll keep the charm meson as example, it is similar for the other systems:

D0 and D0bar are flavour eigenstates - they have a well-defined quark content.
However, they can mix into each other. This allows to find mass eigenstates D₁, D₂. Those have different masses M₁, M₂ and lifetimes $\Gamma_1$,$\Gamma_2$.
The flavour eigenstates are now superpositions of those mass eigenstates and vice versa:

$D_1=p D^0 + q \overline{D^0}$ and $D_2=p D^0 - q \overline{D^0}$
where $|p|^2+|q|^2=1$, both are complex parameters.

Without CP violation, those mass eigenstates are CP eigenstates, and p=q.

In the charm system, it is common to define
$x=\frac{M_1-M_2}{\Gamma}$ and $y=\frac{\Gamma_1-\Gamma_2}{2\Gamma}$ where $\Gamma=\frac{\Gamma_1+\Gamma_2}{2}$ is the average lifetime.
There was a recent measurement of those values by LHCb: Observation of D0-D0bar oscillations
I think you can find references to the theory there.

Thanks! What you say makes sense to me. Except that the question why then is there a mass difference between D₁ and D₂ remains. $D^0$ and $\overline{D^0}$ have the same mass and p, q are normalised. What is causing the difference in the mass of the mass eigenstates?

 

[ofirg]

K^{0} and \bar{K^{0}} (and other examples you gave) are not mass eigenstates.

The mass difference which determines the rate of oscillations in these systems is the mass difference between the two mass eigenstates of the system.

If you would write the mass matrix in the K^{0}, \bar{K^{0}} basis you would get that the diagonal terms are equal ( due to CPT, as you said) but the off diagonal term (due to K^{0}\leftrightarrow\bar{K^{0}} oscillations) would cause splitting in mass between the mass eigenstates.

The mass eigenstates are not conjugates of each other.

 

[mfb]

$$M=\begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix}$$
+CPT => $M_{11}=M_{22}$
+CP => $M_{12}=M_{21}$

The mass matrix has two different eigenvalues, their difference depends on the relative strength of M₁₂ to M₁₁.