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Lecture III

Lecture III. Factorization approaches SCET. Outlines. Introduction B !  form factor Nonleptonic decays Charming penguin Summary. Introduction. An effective theory by integrating out high energy (E) modes. Effective degrees of freedom: collinear fields, soft fields,…

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Lecture III

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  1. Lecture III Factorization approaches SCET

  2. Outlines • Introduction • B! form factor • Nonleptonic decays • Charming penguin • Summary

  3. Introduction • An effective theory by integrating out high energy (E) modes. • Effective degrees of freedom: collinear fields, soft fields,… • Express an amplitude in 1/E in terms of the effective operators. • The Wilson coefficients of these operators are the hard kernels. • The (nonlocal) matrix element of the operators are DAs (or form factors). • Convenient for deriving factorization theorem.

  4. B! form factor • Kinematics • Soft spectator in B, r» • If p2» mb, pg2=(p2-r)2=-2r¢ p2»O(mb) • Then the internal quark is off-shell by (mbv+k+pg)2-mb2»O(mb2 ) • SCET is careful in matching at different scales.

  5. Matching • Demonstrate the matching in SCET • Full theory! SCETI: integrating out the lines off-shell by mb2 Power scaling mb-3/2 from HQET Wilson coeff at SCETI W 2 b C()J(0)()!C()() J(0) T(0)J(1)(0) g mb2 mb 1/mb suppressed current

  6. Jet=Wilson coeff at SCETII =hard kernel in PQCD • SCETI! SCETII: integrating out the lines off-shell by mb J(0,)O() 2 ! T(0)J(0,)M()B() Power scaling mb-1/2 Two terms have the same power scaling.

  7. Comparison • The fundamental inputs, B meson transition forjm factors, are treated differently in different approaches. • fNF is not calculable, so FB is not in QCDF. No matching at mb. Just input it from sum rules. • FNF is factorizable in kT factorization theorem, so FB is in PQCD. No matching at mb. Input B from sum rules, and compute FB. • FB contains both fNF and fF, so it is a mixture in SCET. They are determined from the fit to the B! data.

  8. Nonleptonic decays • SCET can be applied to nonleptonic decays. The result for B! MM’ is

  9. Charming penguin • SCET gives another example that the leading amplitude in a nonleptonic decay does not need to be in the BSW form. • At leading power, no alrge source of strong phases in SCET (no annihilation) . • Long-distance charming penguin is then introduced, parameterized as Acc.

  10. Decay amplitude • SCET factotrization formula for B! M1M2 Wilson coeff Color-allowed Color-suppressed factorizable Wilson coeff

  11. Comment on charming penguin • Charming penguin is factorizable at leading power (see BBNS). • Compute one-loop correction to the charm loop, and see no IR divergence. • No need for additional nonperturbative parameter at leading power. • IR divergence could appear at next-to-leading power. • Then annihilation should be also formulated into SCET.

  12. Fit to data • Do not compute the jet function J(s(mb)) • Determine complex Acc, real B, real JB=s dz JB(z) from the B! data, Absorb JB+,  from somewhere

  13. Results • Small FB • Acc dominates penguin contribution • Predict Why is P so large?

  14. Amplitude parameterization • I can get the same “prediction” using T, C, P, assuming C to be real, same as in SCET---4 parameters with 4 inputs. • The 00 amplitude is fixed by the isospin relation. • A stringent test will be B! K modes. Need more parameters. +-: T+P p200: P-C p2+0: T+C

  15. Amplitude topologies

  16. Summary (Beneke at CKM 2005) SCET QCDF/PQCD QCDF/SCET PQCD

  17. Summary • QCDF, PQCD, SCET go beyond FA. • They have different assumptions, whose verification or falsification may not be easy. • They all have interesting phenomenological applications. • Huge uncertainty from QCDF is annoying. Input from time-like form factor for annihilation? • NLO correction in PQCD needs to be checked. • SCET should be applied to explore heavy quark decay dynamics more.

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