# In certain cases these effects can be enhanced as is

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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 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 , , 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 SM-induced deviations of the time-dependent CP asymmetries from sin 2 β in the penguin-dominated modes (see Sec. 7.2). 2.3 Heavy-Quark Symmetry and Heavy-Quark 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 heavy-quark mass. The heavy quark acts as a static point-like color source with fixed four-velocity, which cannot be altered by the soft gluons responsible for confinement. Hence the configuration of the light degrees of freedom (the so-called “brown muck”) becomes independent of the spin and flavor (mass) of the heavy quark, which, for N f heavy-quark flavors, results in a SU(2 N f ) heavy-quark spin-flavor symmetry [22]. Heavy-quark spin-flavor 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 ( ) 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 four-velocity of the static color source has changed, v b v c . Therefore, the form factors that describe the wave-function 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 ( ) ν form factors are related to a single Isgur-Wise function, ξ ( v b · v c ), which contains all the low-energy 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.