On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

182 Appendix B. Neutral Meson Mixing
and for the eigenvalues
ωL,H = H11 ∓ � H12H21 . (B.7)
One can express the mass and width difference of the mass eigenstates by the real and
imaginery part of the eigenvalues
∆m ≡ mH − mL = Re (ωH − ωL) ,
which allows for the definition of the parameters
∆Γ ≡ ΓH − ΓL = −2Im (ωH − ωL) , (B.8)
x ≡ ∆m
Γ
, y ≡ ∆Γ
2Γ
, (B.9)
which are well suited to categorize the different meson-antimeson systems in (B.1).
A large mass difference expresses itself in a large x parameter and y measures the
difference in the lifetime of the two mass eigenstates. Interestingly, all of the four
systems in (B.1) differ in these parameters [159, 241], which implies different oscillation
behaviour, as one can see from plotting the percentage of antimesons M and mesons
M in an initially pure meson beam versus the time normalized to the average lifetime
T = t/τ of the involved states,
NM(t) e−T
=
NM(0) 2
NM (t) e−T
=
NM(0) 2
[cosh(yT ) + cos(xT )] ,
[cosh(yT ) − cos(xT )] , (B.10)
see [241, Sec.2.2.4] for details. The plots are reproduced in Figure B.1. For the Kaon
system, both x ∼ 1 and y ∼ −1 are large, which corresponds to a large mass difference
as well as a considerable difference in the lifetimes of the two mass eigenstates.
The sign of y does also imply, that the heavier state lifes longer, so that one usually
identifies KL = KH and KS = Kl with the subscript denoting S = short and L =
long lifetime. The relaxation process dominates, because it takes little more than
one oscillation for all of the KS to decay, so that the beam consists entirely of KL
states after one period, as shown in the upper left panel of Figure B.1. This feature
is shared by D − ¯ D mixing and Bd − ¯ Bd mixing, which both have x ∼ 0.6 − 0.8, so
that it takes only one oscillation for most of the mesons to decay. The difference in
lifetimes is with y ∼ 10 −3 however negligible for the Bd − Bd system, so that there are
basically no particles left in the beam after one period as shown in the lower left panel
of Figure B.1. For the D − D system y ∼ 0.75, which results in a not totally depleted
beam, compare the upper right panel of Figure B.1. The situation is very different
for Bs − Bs system, for which x ∼ 27 and y ∼ 10 −2 . The system shows the biggest
mass difference of the four, but it is not possible to identify long- or short lived states
with the mass eigenstates, because their lifetimes are almost equal and the oscillation
frequency allows for several oscillations before the beam is depleted, as shown in the
lower left panel of Figure B.1.