1.  Review of QCD  Introduction to HQET  Applications  Conclusion Paper: M.Neubert PRPL 245,256(1994) Yoon yeowoong(윤여웅) Yonsei Univ. 2004.04.30

2. Introduction to HQET - Review of QCD Bjorken scaling : structure function only depend on. (1969) → Point-like structure inside proton, Asymtotic freedom → Non-Abelian gauge field theory. Yang, Mills → Asymtotic freedom in Non-Abelian gauge field theory. t ’ Hooft(1973) → Gell-Mann propose extra symmetry of non-Abelian color symmetry(1972) QCD was born → Quark confinement( Only colorless states are physically observable) is explained in QCD by infrared divergences due to the massless gluons High Energy probe Asymtotic freedom Color charge Confinement 1 fermi Barrier Distance from the bare quark color chage

3. Introduction to HQET - Review of QCD Summary of Non Abelian Gauge theory SU(3)

4. Introduction to HQET - physical picture Heavy Quark : m Q > Λ QCD Heavy Quark limit : m Q →∞ Heavy Quark + light quark system Q q q Q “ Brown muck ” light quark q cannot see the quantum numbers of Heavy Quark Comptom wavelength of Q : λ Q ~ To resolve the quantum number of Heavy quark, need a hard probe with

5. Introduction to HQET - physical picture The configuration light Degree of freedoms with different heavy quark flavor, spin system of hadron does not change if the velocity of heave quark is same. Heavy Quark velocity ≒ Meson velocity Momentum transfer ~ Λ QCD ⇒ velocity change ~ Λ QCD /m Q ~ 0 We can regard heavy quark velocity as conserved quantity v v Therefore this picture gives spin – flavor symmetry in QCD under m Q →∞ limit. N h heavy quark flavor → SU(2N h ) spin-flavor symmetry group It provide the relations between the properties of hadrons with different flavor and spin of heavy quark. Such as B, D, B *,D *, Λ b Λ c

6. Introduction to HQET - details with elementary field theory Heavy quark momentum almost on-shell Divide quark field by large and small component respectively QCD Lagrangian where

7. Introduction to HQET - details with elementary field theory On a classical level, DOF of H v can be eliminated by EOM of QCD Variation of Lagrangian with respect to Considering order of 1/m Q (n=0) And using the relation

8. Introduction to HQET - details with elementary field theory Inserting gluon field strength tensor It can be shown by and, Then the effective Lagrangian of order 1/m Q is Kinetic term From residual momentum k P Q =m Q v+k h v =e im Q v · x P + Q v Chromo-magnetic momentum interaction (Halzen Ex6.2)

9. Introduction to HQET - details with elementary field theory Now we consider heavy quark limit m Q →∞ 1. It has spin symmetry Associated group is SU(2) symmetry group under which L eff is invariant An infinitesimal SU(2) transformation On-Shell condition satisfied

10. Introduction to HQET - details with elementary field theory 2. It has flavor symmetry When there are N h heavy quark flavor Because this Lagrangian do not contain heavy quark mass, It is invariant under rotations in flavor space Combined with spin symmetry the effective Lagrangian belong to SU(2N h ) symmetry group.

11. Introduction to HQET - details with elementary field theory Now consider Feynman rules Feynman propagator, and vertex factor can be derived by effective Lagrangian Propagator Vertex It can be also derived by taking the appropriate limit of the QCD Feynman rules

12. For the heavy quark gluon vertex Introduction to HQET - details with elementary field theory Using the relation Therefore vertex factor in Heavy quark limit become

13. Application - Spectroscopy Strong Interaction dynamics is independent of the spin and mass of the heavy quark by heavy quark symmetry. Therefore hadronic states can be classified by the quantum number of the light DOF such as flavor, spin, parity, etc. Spin-flavor symmetry in HQET predict some relations of properties of hadron states, typically mass spectrum of different Hadrons states Meson Constituent Quarks JP Dc, (u or d)0 - D*c, (u or d)1 - D1D1 1 + D2*D2* 2 + DsDs c, s0 - Ds*Ds* 1 - Meson Constituent Quarks JP Bb, (u or d)0 - B*b, (u or d)1 - B1B1 ?? B2*B2* ?? BsBs b, s0 - Bs*Bs* 1 -

14. Application - Spectroscopy 1. Ground state mesons Experimentally degenerate states Need a hyperfine correction of order 1/m Q Quite small as expected So we can expect

15. Application - Spectroscopy 2. Excited state mesons degenerate states It is small mass splitting supporting our assertion One can expect also 3. Excitation energy

16. Application - Weak decay form factors Physical picture of weak decay Hadronic matrix element parameterized by several form factors.

17. Application - Weak decay form factors Q q Kinematical picture Q’Q’ q Maximum q 2 =(m M ’ -m M ) 2 ; minimum w=1Zero recoil Q q Q’Q’ Q q Minimum q 2 =0 ; maximum w Q’Q’

18. Application - Weak decay form factors Typical hadronic matrix element M.Wirbel ZPHY C29,637(1985)

19. Now in HQET Application - Weak decay form factors Why not Using flavor symmetry Is called Isgur-Wise function Normalized at zero recoil as For equal velocityis conserved current explained by following

20. Application - Weak decay form factors Using spin symmetry In the rest frame of the final state meson

21. HQETTypical Application - Weak decay form factors Summarize parameterization

22. Application - Weak decay form factors Relations between form factors and Isgur-Wise function.

23. Application - Weak decay form factors

24. Renormalization group equation Study hard ! Model independent V cb Inclusive decay with HQET Conclusion - more study