In certain cases these effects can be enhanced as is the case, e.g., in the neutral versus chargedB(orD) meson lifetimes. The full SU(3) symmetry is broken byms/ΛχSB, and is known to havetypically 20−30% corrections.(The same is true for itsu-spin andd-spin SU(2) subgroups,which act on thedsanduspairs, respectively.)Some of the most prominent cases of isospin symmetry in the context of the CKM matrixinclude relations between amplitudes involving charged and neutral pions, the determination of|Vud|(Sec. 3.1), and the extraction of the UT angleαfromB→ππdecays (Sec. 5.2). Simi-larly, SU(3) symmetry and chiral perturbation theory are key ingredients in determining|Vus|(Sec. 3.2.2). It also relates form factors and certain matrix elements involving pions and kaonsto one another, a relation that has many applications. Recently, the SU(3) relations betweennonleptonic decays have been extensively studied, because theππ,Kπ, andKKamplitude re-lations give sensitivity to the UT angleγand possibly to new physics. SU(3) has also been usedas a bound on the SM-induced deviations of the time-dependentCPasymmetries from sin 2βin the penguin-dominated modes (see Sec. 7.2).2.3Heavy-Quark Symmetry and Heavy-Quark Effective TheoryIn mesons composed of a heavy quark and a light antiquark (plus gluons andqqpairs), theenergy scale of strong interactions is small compared with the heavy-quark mass. The heavyquark acts as a static point-like color source with fixed four-velocity, which cannot be alteredby the soft gluons responsible for confinement. Hence the configuration of the light degrees offreedom (the so-called “brown muck”) becomes independent of the spin and flavor (mass) of theheavy quark, which, forNfheavy-quark flavors, results in a SU(2Nf) heavy-quark spin-flavorsymmetry .Heavy-quark spin-flavor symmetry has many important implications for the spectroscopy andstrong decays ofBandDmesons (for a review, see e.g. ).It is especially predictive forexclusiveB→D(∗)ℓνsemileptonic decays, which are relevant for the determination of|Vcb|
CPviolation and the CKM matrix14(Sec. 3.2.4).When the weak current suddenly changes the flavor (on a time scale≪Λ−1QCD),momentum, and possibly the spin of theb-quark to ac-quark, the brown muck only notices thatthe four-velocity of the static color source has changed,vb→vc. Therefore, the form factors thatdescribe the wave-function overlap between the initial and final mesons become independent ofthe Dirac structure of weak current, and depend only on a scalar quantity,w=vb·vc. Thus all sixB→D(∗)ℓνform factors are related to a single Isgur-Wise function,ξ(vb·vc), which contains allthe low-energy nonperturbative hadronic physics relevant for these decays. Moreover,ξ(1) = 1because at zero recoil—where thecquark is at rest in thebquark’s rest frame—the configurationof the brown muck does not change at all.