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Lecture 4Hadronic heavy-quark decays
Hsiang-nan LiOct. 22, 2012
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Outlines
• Naïve factorization• QCD-improved factorization• Perturbative QCD approach• Strong phases and CP asymmetries• Puzzles in B decays
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Introduction• Why B physics? Constrain standard-model parameters CKM matrix elements, weak phases Explore heavy quark dynamics Form factors, penguins, strong phases Search for new physics SUSY, 4th generation, Z’…• Need B factories and critical comparison
between data and QCD theories. 3
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Cabbibo-Kobayashi-Maskawa Matrix
Weak phase, CP
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CP violation• Thumb only on the right---P violation• Thumbs on the right of right hand, and on
the left of left hand---CP conservation • God is fair: He gives L to particle and R to
antiparticle.• CP conserved
here? • If she loses one
arm, CP at 10-3Anti- particleparticleThousand-hand Guan Yin 5
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B transition form factors
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Form factors• Measurement of structure of a bound state
• If bound state, effective strength dependent on momentum transfer
• It is like structure functions in DIS• Are form factors calculable?
b
point particle -> constant strength
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Nonleptonic decays• Much more complicated• Involve scales mW, mb~Q, and • Are they calculable?
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Naïve factorization
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Effective Hamiltonian• First step, derive effective weak
Hamiltonian by integrating out mW
• Dynamics with scale mW is organized into Wilson coefficient Ci
• The rest dynamics lower than mW goes into 4-fermion operators Oi
• H=VCKMiCi()Oi()
• Their (factorization scale) dependence cancels. 10
Buras,Buchalla, 1995
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Diagrams at scale mW b
penguinboxalso contributingto B-Bbar mixing
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Factorization assumption A= < D|Heff|B> ~ C() <D|O()|B>
FA was proposed to deal with the hadronic matrix element (Bauer, Stech, Wirbel 85).
B
DBD FfaFfaDBA 21)(
,121 CNCCa
Color-allowed Color-suppressed
CNCCa 212
f
FBD
fD
FB
form factornot calculable
1~,3.0~ 21 CC 12
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B D
Nonfactorizable corrections negligible for color-allowed amplitudes.
Lorentz contractionsmall color dipole
decoupling in space-timefrom the BD system
Expect large corrections in color-suppressed modes due to heavy D, large color dipole.No strong phase.No systematic improvement of precision
Color transparency
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Incompleteness of FA• FA cannot be complete: form factor and
decay constant are physical, independent of . Predictions depend on via C().
• Nonfactorizable contributions must exist, especially in color-suppressed modes. They may be small in color-allowed decays, which are insensitive to .
• Power corrections, like strong phases, are crucial for CP violation.
• FA was used for decades due to slow experimental and theoretical progress.
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Go beyond FA• Great experimental (Babar, Belle) and
theoretical (factorization approaches) progress around 2000.
• Rare (including color-suppressed) decays with BR down to 10-6 and CP asymmetries can be measured precisely at B factories.
• Explore nonfactorizable and power corrections, or even new physics.
• Theorists need to go beyond FA.15
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Theoretical approaches• Hadronic B decays involve abundant QCD
dynamics of heavy quarks.• Complexity of B decays dragged
theoretical progress till year 2000, when we could really go beyond naïve factorization.
• Different approaches developed: PQCD, QCDF, SCET.
• Predictive power and data discriminate different approaches. 16
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QCD-improved factorization
Beneke, Buchalla, Neubert, Sachrajda, 1999
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QCD corrections to FA• Add gluons to FA
• Nonfactorizable corrections appear at this order
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Hard kernels • TI comes from vertex corrections
• TII comes from spectator diagramsMagnetic penguin O8g
q 1
x
soft divergences cancel in pairs 19
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QCDF • Based on collinear factorization• Compute corrections to FA, ie., the heavy-
quark limit.
• Due to end-point singularity in collinear factorization
nonperturbative
perturbative
distribution amplitudelike PDF in DIS
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B -> form factors• x runs from 0 to 1. The end-point region is
unavoidable.• Collinear factorization gives end-point
singularity. Form factors not calculable
)1()(,)(2
10 xxx
xxdx
xP
PB-xP
k-xP
k
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Breakthrough and challenge • It is amazing that higher-order corrections
to FA are calculable!• Study of two-body hadrnic B decays can
be improved systematically • But, when going to higher powers,
challenge appeared….• End-point singularity avoided by absorbing
it into form factors, but reappeared at higher powers
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End-point singularity • Singularity appears at O(1/mb), twist-3
spectator and annihilation amplitudes, parameterized as X=(1+ ei)ln(mb/)
• For QCDF to be predictive, O(1/mb) corrections are better to be small ~ FA.
• Data show important O(1/mb). Different free (,) must be chosen for B-> PP, PV, VP. 23
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Perturbative QCD approach
Keum, Li, Sanda, 2000
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kT factorization• If no end-point singularity, collinear
factorization works. If yes, small x region is important, and kT factorization is more appropriate.
• kT factorization for exclusive processes is basically the same as for inclusive process (small x physics).
• Based on collinear and kT factorizations, QCDF and PQCD have been developed for exclusive B decays.
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Smearing of end-point singularity• Including parton kT accumulated through
gluon emissions
B
x’, 0Tx, kT
1/(xmB2+kT
2)
2210
1
BT mkxdx is finite
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Factorization of form factor• Form factors are calculable, if TMD is
known
• Factorization of two-body hadronic B decays changed!
b
BF
TMD
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Three scales• Start with lowest order diagram for
B -> D• This amplitude
involves three dramatically different scales: mW, mb, and
• Radiative corrections generate large logs ln(mW/mb), ln(mb/), which must be summed to all orders to improve perturbation.
b uW-
ud
g
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Two clusters of corrections• Two clusters
• Sum ln(mW/mb) first (integrating out W boson), giving effective weak Hamiltonian, and then sum ln(mb/), giving Sudakov and RG evolutions.
• PQCD approach incorporates factorizations of effective weak Hamiltonian and IR divergences.
b uW-
ud
g
Generate ln(mW/mb)
Generate ln(mb/)
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Effective Hamiltonian• also a kind of factorization theorem.
• Radiative corrections give new operators (operator mixing) O1 -> O1,O2
• Effective Hamiltonian
Full diagram
independent
Soft (effective) diagram4-fermion operator O2
dependent
Hard part Wilson coefficientdependent
b
->
Different color flows
is renormalization scale
CKM matrix elements 30
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QCD and EM penguins• Radiative corrections also introduce QCD
and electroweak penguins
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Factorization of IR divergence• Use soft divergence for demonstration.
• Factorization formula
full diagramcurrent conservationf-independent
soft diagrameikonalizedf-dependent
hard partf-dependent
TMD wave function
hardkernel
f is factorization scale
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Three-scale factorization
• A=C(mW,) * H(t,,f) * (f,1/b)
for any gluon, isolate IR divergence first
then shrink W line to get effective Hamiltonian
Precisely speaking, it is cancelled by another gluon
hard scale t2 ~mb
can choose =f33
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Four scales actually • Strictly speaking, there are 4 scales: mW,
mb, t ~ (mb)^(1/2), .• There are 3 clusters. The
cluster of hard kernel splits into one for weak b ->u vertex, and another for hard gluon.
• The above are scales in Soft-Collinear Effective Theory
b uW-
ud
g
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Feynman diagrams
emission diagrams annihilation diagrams35
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• Physical picture: large b means large color dipole. Large dipole tends to radiate during hard scattering. No radiation in exclusive processes. Small b configuration preferred.
suppression at large bbecomes stronger at larger energy
Small x
Sudakov effect increases kT
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Strong phase and CP violation
annihilation marks most crucial difference between
QCD and PQCD
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• For (V-A)(V-A), left-handed current
• Pseudo-scalar B requires spins in opposite directions, namely, helicity conservation 1=s1. p1=(-s2 ) .(-p2 ) =2 .
• For (V-A)(V+A)=(S-P)(S+P), scalar current
momentum
bfermion flow
spin (this configuration is not allowed)
b
p1p2
Survive helicity conservation,
(V-A) and (V+A) currents
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Small annihilation?• Annihilation is of higher power• But what is the power suppression factor?• If /mB , annihilation is negligible• But it turns out to be 2m0/mB ~ 0.6 for
• Introduced by pseudoscalar-current matrix element (for twist-3 DA S)
• What is impact of sizable annihilation?
GeVmm
mmdu
5.12
0
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Bander-Silverman-Soni mechanism, strongphase source in FA
Loop linecan go on-shell
kT
loop with the weight factor from kT distribution function (TMD)
Strong phase
Principle value
B
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Sources of strong phase
Annihilation is assumed to be small, down by s/mb in QCDF, but calculable and sizable in PQCD Different sources lead to different direct CP asy.
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Large strong phase• ACP(K+-)~ -0.1 implies sizable T
~ 15o
between T and P from annihilation (PQCD, 00)
T exp(i3)
P
T exp(-i3)
T exp(i3)
P
T exp(-i3)
If T=0
Br
Br = Br
If T=0
Br = BrDirect CP
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Ali et al., 2007
QCDF prediction smaller by an order of magnitude 44
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Direct CP asymmetries, pure annihilation branching ratios,
B->VV polarizations all indicate sizable annihilation
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Puzzles in B decays
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Puzzles in B physics• much larger than
predictions• much different from • lower than • New physics or QCD effect? • If new physics, how about • If QCD, but is normal• How to resolve these puzzles?
)(),( 0000 BB
)( 0 KACP )( KACP
)( 0SKS )( sccS
)(),( 0000 BB)( 00B
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Quark amplitudes
b
u
d,s
u
Color-allowed tree T Color-suppressed tree C
QCD penguin P Electroweak penguin Pew
d,s
u
u
d,s
u
u
d,s
u
ug , Z
W
2 color traces Nc2 1 color trace Nc
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K parameterization
(C2/C4)(VusVub/VtsVtb) ~ (1/2)(5/2) ~ 52
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Explanation 1• How to understand positive ACP(K+0)?• Large PEW to rotate P (Buras et al.;
Yoshikawa; Gronau and Rosner; Ciuchini et al., Kundu and Nandi)
new physics? T exp(i3)
P
T exp(-i3)PEW
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Explanation 2• Large C to rotate T (Charng and Li; He
and McKellar). mechanism missed in naïve power counting?
• C is subleading by itself. Try NLO…. An not-yet-finished story
(T+C) exp(i3)
(T+C) exp(-i3)
T exp(i3)
P
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Summary• Naïve factorization of hadronic B decays
relies only on color transparency.• Fully utilize heavy-quark expansion.
Combine effective Hamiltonian (separation of mW and mb) and IR divergence factorization (separation of mb and ).
• Factorization theorem can explain many data, but some puzzles remain
• New physics or complicated QCD dynamics? More hard working is needed
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