Theories of exclusive B meson decays

1
Theories of exclusive B meson decays

• Hsiang-nan Li
• Academia Sinica (Taiwan)
• Presented at Beijing
• Aug. 13-17, 2005

2
Titles of Lectures

• I Naïve factorization and beyond
• II QCDF and PQCD
• III SCET
• IV Selected topics in B Physics
• Will not cover SU(3), QCD sum rules,
  determination of CKM, specific modes, new
  physics,

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Lecture I

• Naïve Factorization and beyond

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Outlines

• Introduction
• Weak Hamiltonian
• Naïve factorization
• Diagnose FA
• A plausible proposal
• Summary

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Introduction

• Missions of B factories
• Constrain standard-model parameters
• Explore heavy quark dynamics
• Search for new physics
• Must handle QCD eventually for precision
  measurement
• Several theories have been developed recently,
  which go beyond the naïve factorization.
• Semileptonic B decays (B meson transition form
  factors) are inputs to the above theories.
• Predictions are then made for nonleptonic B
  decays.
• Will discuss their ideas, differences,
  applications

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sin (2?1)0.685 0.032 _at_LP 2005
Determine ?1 using the golden mode B! J/? K.
Penguin pollution 5. When reaching this
precision, need a QCD theory.
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Isospin relation
A(D0?-)
p 2A(D0?0)/ a2
60o
A(D?-)/ a1
a1, a2 the BSW parameters a20.35-0.6, much
larger than expectation. Arg(a2/a1) 60o is
generated by decay dynamics. Their understanding
requires a theory.
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Mixing-induced CP
Penguin-dominated
Tree-dominated

• 4 S?0 due to
• new physics?
• Need a theory
• for tree
• Pollution.

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Complexity
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Weak Hamiltonian
Effective theory Low-energy lt mW
IR finite difference High-energy mW
Full theory ? indep.
Sum ln(mW/?) to all orders
4-fermion operator O(?)
Wilson coefficient C(?)
Weak Hamiltonian Heff

The factorization scale ? is arbitrary, and its
dependence cancels between C(?) and O(?)
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Penguins
At O(?s) or O(?), there are also penguin diagrams
??(1-?5) ! V-A
??
??(1?5)! VA
b
s
b
s
??
q
q
q
q
2 Color flows 2TaijTakl -?ij?kl/Nc ?il?kj
QCD penguin g EW penguin g replaced by ?, Z
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Example Heff for b! s
I, j label different color flows
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Naïve factorization
The decay amplitude for B! D?
h D?HeffBi/ C(?) hD?O(?)Bi

nonperturbative
perturbative
Must deal with the hadronic matrix element.
The factorization assumption (FA) was the first
try.
Decay constant and form factor are physical. No
? dependence. To make physical prediction,
must assume C to be constant, and It is better
to be universal.
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Color flows
O1(C)
u
d
c
u
b
c
b
d
One color trace, Tr(I)Nc1 Color-suppressed
two color traces, Tr(I)Tr(I)Nc2 Color-allowed
RHS is down by 1/N_c compared to LHS
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Decay amplitudes

Class 1 Color-allowed
Class 2 Color-suppressed
a1, a2 universal parameters
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Success of FA
a1 and a2 seem to be universal! Success due to
color transparency
D
B
Lorentz contraction Small color dipole
Decoupling in space-time From the BD system
FA is expected to work well for color-allowed
modes with a light meson emitted from the weak
vertex.

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Failure of FA
a2(D?)?a2(J/? K) is not a surprise
B
D
Large correction in color-suppressed modes due to
heavy D, large color dipole
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The failure of FA implies the importance of
nonfactorizable correction to color-suppressed
modes, for which a2(mb) 0.1lt a2(J/? K),
a2(D?) In terms of Feynman diagrams,
nonfactorizable correction is not universal.
a1(mb) 1.1
(J/?)
(K)
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Nonfactorizable corrections
Generalized naïve factorization
Exp shows that Wilson coefficients are not really
universal
Due to nonfactorizable correction?
Fine tune the mode-dependent parameters to
fit data
Equivalently, effective number of colors in
Not very helpful in understanding decay
dynamics How to calculate nonfactorizable
correction?
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Diagnose FA

• FA should make some sense (color transparency).
• The assumption of constant a1, a2 is not
  successful.
• FA fails for color-suppressed modes as expected
  (small a2 nonfactorizable correction).
• Stop data fitting. How to go beyond FA?

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Scale dependence
Decay constant and form factor are physical. No ?
dependence.

• Problem of FA
• Before applying factorization, extract the ?
  dependence from the matrix element
• The question is how to calculate g(?)

C(?)h O(?)i ¼ C(?)h O iFA

• dependence in
• C(?) remains

? dependences cancel
FA
? independent
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IR cutoff and gauge dependences

• Look at the derivation of Weak Hamiltonian again
• Considering off-shell external quarks, the
  constant a is gauge dependent, which is also
  hidden into the matrix element.
• When extracting g(?), one also extracts the
  dependences on cutoff and on gauge. The scale
  dependence is just replaced by the cutoff and
  gauge dependences. Dead end?

Evaluated between external quark states
-p2 is the off-shell IR cutoff hidden into matrix
element
ln(MW2/-p2)ln(MW2/?2)ln(?2/-p2)
Absorbed into C(\mu)
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Strong phase and CP asymmetry
CP asymmetries in charmless decays can be
measured at B factories
Tree
Penguin
Interference of T and P
Data
Extraction
Theory
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Bander-Silverman-Soni mechanism
In FA, strong phase comes from the BSS mechanism
It gives a small phase. Only source? Important
source?
q

Im/ Moreover, what is the gluon invariant mass
q2? Can not compute thye strong phase
unambiguoysly.
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A plausible proposal

• Recalculate O(?s) corrections with on-shell
  quarks (Cheng, Li, Yang, May, 99)

They are gauge invariant. But ln(?2/-p2)!
1/?IR How to deal with this IR pole?
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IR divergence

• IR divergence is physical!
• Its a long-distance phenomenon, related to
  confinement, the hadronic bound state.
• All physical hadronic high-energy processes
  involve both soft and hard dynamics.

q
weak decay occurs
g
q
Soft dynamics
t1
t0
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Factorization theorem

• The idea is to absorb IR divergence into meson
  distribution amplitudes ?
• Factorization theorem
• ?f factorization scale. Its role is like ?.
  mWgt?gtmb, mbgt?fgt?
• H is the IR finite hard kernel.
• The matrix element
• A scale-independent, gauge-invariant, IR finite
  theory is possible!

Scale dependence cancel
All allowed decay topologies
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Factorization vs. factorization

• Factorization in naïve factorization means
  breaking a decay amplitude into decay constant
  and form factor.
• Factorization in factorization theorem means
  separation of soft and hard dynamics in decay
  modes.
• After 2000, factorization approach to exclusive B
  decays changed from 1st sense to 2nd.

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Summary

• FA is a simple model for nonleptonic B decays
  based on color transparency.
• Its application is limited to branching ratios of
  color-allowed modes.
• It can not describe color-suppressed modes,
  neglects nonfactorizable contributions, and has
  incomplete sources of strong phases.
• Theoretically, it is not even a correct tool due
  to scale or gauge dependence.
• A proposal for constructing a theory with the
  necessary merits has been made.