Theories of exclusive B meson decays
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Theories of exclusive B meson decays
Hsiang-nan Li
Academia Sinica (Taiwan)
Presented at Beijing
Aug. 13-17, 2005
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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 measu
rement• 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 dec
ays.• Will discuss their ideas, differences, applications
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sin (21)=0.685± 0.032@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(D+-)/ a1
p 2A(D00)/ a2A(D0-)
a1, a2: the BSW parameters|a2|=0.35-0.6, much larger than expectation.Arg(a2/a1)» 60o is generated by decay dynamics.Their understanding requires a theory.
60o
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Mixing-induced CP
4 S0 due tonew physics?Need a theory for tree Pollution.
Penguin-dominated
Tree-dominated
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Complexity
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Weak Hamiltonian
Full theoryindep.
Effective theoryLow-energy< mW
IR finite differenceHigh-energy» mW
4-fermion operator O()
Wilson coefficient C()
Weak Hamiltonian Heff =
Sum ln(mW/) to all orders
The factorization scale is arbitrary, and its dependence cancels between C() and O()
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Penguins
g
Wb s
q q
(1-5) ! V-A
(1+5)! V+A
At O(s) or O(), there are also penguin diagrams
QCD penguin: gEW penguin: g replaced by , Z
2 Color flows: 2TaijTa
kl = -ijkl/Nc + ilkj
b sb
q
q
s
q
q
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Example: Heff for b! s
I, j label different color flows
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Naïve factorization
h D|Heff|Bi/ C() hD|O()|Bi
perturbative nonperturbative
Must deal with the hadronic matrix element.
The factorization assumption (FA) was the first try.
The decay amplitude for B! D
f
BDF
Decay constant and form factor are physical. No dependence.To make physical prediction, mustassume C to be constant, and It is better to be universal.
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Color flows
b c
du
O1(C)
two color traces, Tr(I)Tr(I)=Nc
2
Color-allowed
b
c
d
u
One color trace, Tr(I)=Nc1Color-suppressed
RHS is down by 1/N_c compared to LHS
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Decay amplitudes
Df
BF
f
BDF
a1, a2: universal parameters
Class 1:Color-allowed
Class 2:Color-suppressed
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B D
a1 and a2 seem to be universal!Success due to “color transparency”
Lorentz contractionSmall color dipole
Decoupling in space-timeFrom the BD system
FA is expected to work well for color-allowed modes witha light meson emitted from the weak vertex.
Success of FA
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Large correction in color-suppressed modes due to heavy D, large color dipole
B D
a2(D)a2(J/ K) is not a surprise
Failure of FA
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The failure of FA implies the importance of nonfactorizable correction to color-suppressed modes, for which a2(mb)» 0.1< a2(J/ K), a2(D)In terms of Feynman diagrams, nonfactorizablecorrection is not universal.
a1(mb)» 1.1
(J/)
(K)
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Generalized naïve factorization
6~2 effCC NN
Exp shows that Wilson coefficients are not really universal
Due to nonfactorizable correction?
222111 , aaaa Fine tune the mode-dependent parameters to fit data
Equivalently, effective number of colors in
CNCCa )2(1)1(2)2(1
Not very helpful in understanding decay dynamicsHow to calculate nonfactorizable correction?
Nonfactorizable corrections
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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
• 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 i|FA
Decay constant and form factor are physical. No dependence.
dependences cancel dependence in C() remains
|FAindependent
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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 betweenexternal quark states
ln(MW2/-p2)=ln(MW
2/2)+ln(2/-p2)
Absorbed into C(\mu)
-p2 is the off-shell IR cutoffhidden into matrix element
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Strong phase and CP asymmetry
CP asymmetries in charmless decays can be measured at B factories
b
W
u
g
Wb d
q q
Tree Penguin
2sinsin CPA
2)( 0 ii ePeTBA
Interference of T and P
Data
Theory
Extraction
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In FA, strong phase comes from the BSS mechanism
It gives a small phase.Only source?Important source?
Bander-Silverman-Soni mechanism
Im/
Moreover, what is the gluon invariant mass q2? Can not compute thye strong phase unambiguoysly.
q
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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!
• It’s a long-distance phenomenon, related to confinement, the hadronic bound state.
• All physical hadronic high-energy processes involve both soft and hard dynamics.
q
q
g
t=1t=0
weak decayoccurs
Soft dynamics
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Factorization theorem• The idea is to absorb IR divergence into m
eson distribution amplitudes • Factorization theorem
• f: factorization scale. Its role is like . mW
>>mb, mb>f>• H is the IR finite hard kernel.• The matrix element• A scale-independent, gauge-invariant, IR fi
nite 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.