Lecture III Factorization approaches SCET Outlines Introduction B
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, … • 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.
B! form factor • Kinematics • Soft spectator in B, r» • If p 2» mb, pg 2=(p 2 -r)2=-2 r¢ p 2» O(mb ) • Then the internal quark is off-shell by (mbv+k+pg)2 -mb 2» O(mb 2 ) • SCET is careful in matching at different scales.
Matching • Demonstrate the matching in SCET • Full theory! SCETI: integrating out the lines off-shell by mb 2 Power scaling mb-3/2 from HQET Wilson coeff at SCETI 2 C( )J(0)( )! C( ) W b J(0) g mb T( 0)J(1)( 0) mb 2 1/mb suppressed current
• SCETI! SCETII: integrating out the lines off -shell by mb Jet=Wilson coeff at SCET =hard kernel in PQCD J( 0, )O( ) 2 ! T( 0)J( 0, ) M( ) B( ) Power scaling mb-1/2 Two terms have the same power scaling. II
Comparison • The fundamental inputs, B meson transition forjm factors, are treated differently in different approaches. • f. NF is not calculable, so FB is not in QCDF. No matching at mb. Just input it from sum rules. • FNF is factorizable in k. T factorization theorem, so FB is in PQCD. No matching at mb. Input B from sum rules, and compute FB. • FB contains both f. NF and f. F, so it is a mixture in SCET. They are determined from the fit to the B! data.
Nonleptonic decays • SCET can be applied to nonleptonic decays. The result for B! MM’ is
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.
Decay amplitude • SCET factotrization formula for B! M 1 M 2 Wilson coeff Color-allowed Color-suppressed factorizable Wilson coeff
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-toleading power. • Then annihilation should be also formulated into SCET.
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
Results • Small FB • Acc dominates penguin contribution Why is P so large? • Predict
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 p 2 0 0: p 2 + 0: T+C P-C
Amplitude topologies
Summary (Beneke at CKM 2005) SCET QCDF/PQCD QCDF/SCET PQCD
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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