In certain cases these effects can be enhanced as is the case, e.g., in the neutral versus charged
B
(or
D
) meson lifetimes. The full SU(3) symmetry is broken by
m
s
/Λ
χ
SB
, and is known to have
typically 20
−
30% corrections.
(The same is true for its
u
spin and
d
spin SU(2) subgroups,
which act on the
ds
and
us
pairs, respectively.)
Some of the most prominent cases of isospin symmetry in the context of the CKM matrix
include relations between amplitudes involving charged and neutral pions, the determination of

V
ud

(Sec. 3.1), and the extraction of the UT angle
α
from
B
→
ππ
decays (Sec. 5.2). Simi
larly, SU(3) symmetry and chiral perturbation theory are key ingredients in determining

V
us

(Sec. 3.2.2). It also relates form factors and certain matrix elements involving pions and kaons
to one another, a relation that has many applications. Recently, the SU(3) relations between
nonleptonic decays have been extensively studied, because the
ππ
,
Kπ
, and
KK
amplitude re
lations give sensitivity to the UT angle
γ
and possibly to new physics. SU(3) has also been used
as a bound on the SMinduced deviations of the timedependent
CP
asymmetries from sin 2
β
in the penguindominated modes (see Sec. 7.2).
2.3
HeavyQuark Symmetry and HeavyQuark Effective Theory
In mesons composed of a heavy quark and a light antiquark (plus gluons and
q
q
pairs), the
energy scale of strong interactions is small compared with the heavyquark mass. The heavy
quark acts as a static pointlike color source with fixed fourvelocity, which cannot be altered
by the soft gluons responsible for confinement. Hence the configuration of the light degrees of
freedom (the socalled “brown muck”) becomes independent of the spin and flavor (mass) of the
heavy quark, which, for
N
f
heavyquark flavors, results in a SU(2
N
f
) heavyquark spinflavor
symmetry [22].
Heavyquark spinflavor symmetry has many important implications for the spectroscopy and
strong decays of
B
and
D
mesons (for a review, see e.g. [23]).
It is especially predictive for
exclusive
B
→
D
(
∗
)
ℓ
ν
semileptonic decays, which are relevant for the determination of

V
cb

CP
violation and the CKM matrix
14
(Sec. 3.2.4).
When the weak current suddenly changes the flavor (on a time scale
≪
Λ
−
1
QCD
),
momentum, and possibly the spin of the
b
quark to a
c
quark, the brown muck only notices that
the fourvelocity of the static color source has changed,
v
b
→
v
c
. Therefore, the form factors that
describe the wavefunction overlap between the initial and final mesons become independent of
the Dirac structure of weak current, and depend only on a scalar quantity,
w
=
v
b
·
v
c
. Thus all six
B
→
D
(
∗
)
ℓ
ν
form factors are related to a single IsgurWise function,
ξ
(
v
b
·
v
c
), which contains all
the lowenergy nonperturbative hadronic physics relevant for these decays. Moreover,
ξ
(1) = 1
because at zero recoil—where the
c
quark is at rest in the
b
quark’s rest frame—the configuration
of the brown muck does not change at all.