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(2) When is a quark heavy. strongcouplingconstanti. where r. small coupling for Q. Nao jO. 100MeV. 2. Naco. useperturbative QCD. Large coupling for Q2Cta non perturbative. Iacp. 3200MeV MB is the scale separating long from short distance QCD. We call a quark Q heavy if. In nature. u. d s. light. c. ma Aacp. b. heavy. t w. tooheavy decaystooquickly.
(3) Heavy quarksymmetry strict mo Consider. a. hadron. limit. Hq containing aheavyquark Qand lightdegrees of freedom lightquarks gluons. heavyquark. golley Mi. meeeeheeled. tightquarks. Mleeeeee Iggy. brownmuck. t. gf.EE. m wellhead a. mermaid. This is a horribly complex system Simplification Consider the limit. or more. ma 9. precisely AQCD ma. in this limit the Compton wavelengths. O. to Imo and. Trac satisfy. Xa Csx the light d o f cannot resolve the exact value of xo and thus ma.
(4) Theheavyquark also carries a spin magnetic moment. ma. s. m. In the strict heavyquark limit the light d. 0. ma. of. are. D. blind to the. mass andspin of the heavyquark. SUKNa spin flavoursymmetry Relates different hadrons Hn v. in nature Na 2. Hz v with the same velocity but differentmomenta. CET from the point of view of the. Consider the decay b. d af. t O. t eo H. light d af reassemble around. He. ight d o f s ee t. light. msn.it. ifeng.cqgnqanj.it. t. c. heavyquarkrecoilsfrom. velocity u to v in general He can be an excited state. Probability amplitude for Hc to be a groundstate D meson f V.V's 7 Nothinghappens from the p o. g. r. 7. v. of the light d af. w. isguv wise function.
(5) The matrix elements. can. all be described by a singlefunction. w. welearned. The strict ma thanks to the. But. a. limit leads togreatsimplification. heavyquark spin flavoursymmetry. How good is the approximation ma. For instance. Iacp me. n. 0,15 is not THATsmall. Togetprecise predictions. we have toquantify our. ignorance. Systematic expansion around theheavyquark limit.
(6) HQ ET. Heavyquarkeffectivetheory. stolen from M Neubert arXiv 9610266. A systematic expansionaroundthe heavyquark limit in terms of powers of NCD Ma. isobtainedbythe HQET. Resultsof HQETcomputations are doubleexpansions in terms of. QC ma. power corrections. and. radiative corrections. The effectiveLagrangian Consider a heavyquark Q boundin a colour neutral hadron We can. write its momentum as OLAND Q large on shell part. W. small. deviation. fromitsmassshell.
(7) Heavyquarkpropagator. in. projector. P. T independent. of M Q. projects onto thepositive frequency upper components of. Spiner. RF. v. tf. Y. Pi. goo. o8g8g fog. Only the uppercomponentspropagate in theheavyquark limit PQ. x. QQ to. Ia. P Qx I 1. 1. 0 t. Action of the heavyquarkfield on a heavyquarkstate. Qlx. Qp. e. P. 40. x ji mantle Oy. multiplybothsides by a phase eima eima. 0,111apps e. i. to. independent of ma. define ma independentheavyquark field. Offa. a. Divac.
(8) its smallcomponents are givenby. eima necessary for effects at 0. Lagrangian. Ot. Vma. LO i. take. plug in and obtain. hu. x. PtVr Pt. Pthr x Vm. This Lagrangian corresponds to the heavyquark limit. Explicitly independent of classical e am. mo and the heavyquarkspin. NV. OfTna.
(9) Including. Imo. corrections. take. plug in. fall. find. where. Dj DM. use e am. v Dum. such that. QCDLagrangian. D vs O. to. eliminate the small component Hu. then expand in Yma. now we can use. I. to. find.
(10) we. then write. small perturbation. free Lagrangian. with. breaks. flavour symmetry. EffectiveStates andcurrents. Relativisticnormalization of States. heavy hadron. depend on my. We want a mass independentnormalizationcondition. relation betweenStatesdefined in QCD and HQET. breaks heavyspinAND. flavoursymmetry.
(11) for thespinors. 0. Img. Introduce two effective fields c 7 hi. b. a thesame. Effectivecurrent for b QCD. jn clr.gs. t. Ceti. r. j .HQETi. h. eliminate thesmallcomponentfields in finding. Xa plugthis intothe current c p. LOobtained by c Ih b. hub. b.
(12) Hadronic matrixelements We want toevaluate hadronicmatrixelements. need a covariantrepresentation of States. introduce annihilatedby. thevectorcurrent. annihilated by the axialvectorcurrent. underheavy quarkspinrotations transformation behaviour of a singleheavyquark. underLorentzrotations. transforms like theproduct ofspinors. hug. Feynman rules Heavyquark part of decay amplitude x a. Iv. uh. contained in Mtv. Decayamplitude. x. MTM 4 4 Dirac matrix. indices have to be contracted.
(13) Lorentz covariance andparity. and cyclicity Of the trace. use. guv wise function. consequently. S nl 7. Forgroundstate B and D. mesons. a. Using which we will compute the inclusive B at the LO in the exercises. Keefe decay rate.
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(15) Inclusive B. decays.
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