TITLE: Mass difference of $B^0$, $\overline{B^0}$-states
QUESTION [0 upvotes]: It is known that $B^0$, $\overline{B^0}$ or $K^0$, $\overline{K^0}$ particles have different masses and that they oscillate in 'an equivalent' way to neutrino oscillations.
Is it theoretically calculable, why $B^0$ and $\overline{B^0}$ have different masses and how large this mass difference actually is?
Is the reason for the mass difference the same reason why neutrinos (presumably) have different masses?

REPLY [5 votes]: Yes, the neutral mesons you mentioned do oscillate, as you mentioned, at very short distance scales, analogously to neutrino oscillations, which oscillate at distance scales of hundreds miles. (Actually, the B mesons you mentioned are the bottom-strange ones, Bs. )
There are estimates of the relevant mass difference, computed from the "box diagram" of the Standard model, and involving two Ws, and the CKM-flavor violating matrix elements. But, beware: $B^0_s$, $\overline{B}_s^0$,   $K^0$, $\overline{K}^0$ are not the ones with different masses, anymore than the $\nu_{e,\mu,\tau}$ are!
The  propagating states are linear combinations of them, with differing masses, in both cases. Nobody can calculate such mass differences for neutrinos, by contrast; those are just determined by experiment.
That is, $$|K_0\rangle = \frac{1}{\sqrt 2} \left ( |K_S\rangle+ |K_L\rangle\right ),\\
|\bar K_0\rangle = \frac{1}{\sqrt 2} \left ( |K_S\rangle - |K_L\rangle\right ), $$
where L indicates the long-lived CP odd mass eigenstate, and S the dramatically shorter-lived  CP even one. Similarly for the Bs.
For the Kaons, of mass roughly half a GeV, the relevant mass difference proportional to the oscillation frequency is
$$ \Delta m\approx 3\cdot 10^{-15}\hbox{ GeV}. $$
For the Bs, of mass roughly 5.366 GeV,
the relevant frequency/ mass difference is
$$
\Delta m\approx 1.17 \cdot 10^{-8} \hbox{MeV}\\
 \approx 18\cdot 10^{12} \hbar /s ,
$$
dubbed 18 /ps in natural units. This corresponds to oscillation lengths of  20 μm, so, a far cry from kilometers for neutrinos!
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