November 30, 2004
Flavor Diagram Approach to
Hadronic B Decays
National Central University
Beauty in B physics
CPV in SM
Charm in B physics
charmed decays for some B meson properties
charmed decays for indirect CPV
charmless decays for direct CPV
global c2 fits and weak phase g
Strangeness in B physics
accumulating hints of new physics in charmless B decays
FCNC Z0 as an example
Perspective and Summary
C.W. Chiang Flavor Diagram Approach 2
Beauty in B Physics
C.W. Chiang Flavor Diagram Approach 3
Motivations for Studying B Decays
We have observed in K meson system:
indirect CPV: 1964 in KL p+ p–
direct CPV: 1999 in CERN-NA48 and FNAL-KTeV exp’ts
B factories have produced a lot of interesting results, particularly in measuring
indirect CPV in the B system
sin2b=0.725 0.037 from y KS data (ICHEP)
indicating that B physics is entering a precision era
Finally, we have recently observed direct CPV in B system (ACP(Bd p¨K§) =
0.113 § 0.019).
Most branching ratios of charmless B P P and P V are currently measured
with errors about 10%~20% (mostly based upon ~100M B anti-B pairs)
5%~10% errors on amplitudes
One hopes to have <5% errors on the amplitudes of most modes
more precise information on a and g (<10±)
C.W. Chiang Flavor Diagram Approach 4
Beauty in the B Meson
The beauty of B mesons lies in its large mass or the mass hierarchy:
mq << LQCD << mb
In the heavy quark limit, mQ 1, we discover:
Flavor symmetry: dynamics unchanged under heavy flavor
exchange (b c), corrections incorporated in powers of 1/mb–1/mc;
Spin symmetry: dynamics unchanged under heavy quark spin flips,
corrections incorporated in powers of 1/mb.
B mesons provide an ideal system for studying heavy-to-heavy
Much progress has been made in understanding heavy-to-light
transitions in recent years:
Perturbative approach: naïve factorization, generalized factorization,
QCD-improved factorization, pQCD, and SCET;
Nonperturbative approach: flavor SU(3) symmetry.
C.W. Chiang Flavor Diagram Approach 5
Within SM, CPV in the quark sector is explained using the CKM matrix,
which is unitary and complex:
3 mixing angles
1 weak phase
(Kobayashi and Maskawa, 1973)
small in magnitude and least known
but contain largest CPV phases
Both these elements can be explored using various B meson mixing and/or
Vub from tree-level decays;
Vtd from loop-induced processes.
C.W. Chiang Flavor Diagram Approach 6
The Matrix Reloaded
The CKM matrix written in terms of Wolfenstein parameters (l , A, r, and
h) becomes [to O(l 3)] [Wolfenstein, PRL 51, 1945 (1983)]
l ' 0.2264
A ' 0.801
/ e– i b / e– i g
The ultimate goal of studying B physics is not only to achieve precision
measurements of the above parameters, but also to discover evidences of
new physics and possibly its type (e.g. GUT, SUSY, XD).
One way to detect new physics is to perform consistency checks for the
sizes and phases of the CKM elements.
Even if no deviation is seen from SM in these studies, we can still obtain
useful and stringent bounds on new physics scales.
C.W. Chiang Flavor Diagram Approach 7
Vub and Vtd can be related to each other through the unitarity relation
VudVub* + VcdVcb* + VtdVtb* = 0
A triangle can be formed on a complex plane as a geometrical
representation of the above relation, where a nonzero area signifies CPV.
This triangle has three sizeable angles.
decay side oscillation side
ACP [pp,ph,rp…] a
BR(BXc,uln) DMBd and DMBs
g (f3) b (f1)
ACP [DCP K§, Kp,…] ACP (t)[J/Y KS, h’ KS, f KS(?),…]
C.W. Chiang Flavor Diagram Approach 8
Charm in B Physics
C.W. Chiang Flavor Diagram Approach 9
B mesons decay dominantly into charmed final states; used to determine
many properties: [2004 PDG]
Bd: DMd = 0.502 § 0.007 ps-1; Gd = 1.542 § 0.076 ps.
Bs: DMs > 14.5 ps-1; Gs = 1.461 § 0.057 ps.
|Vcb| determined mainly from semileptonic B D(*) transitions;
important for normalization in the UT.
Bd J/y Ks involves a tree-level, dominant subprocess b c anti-c s
with no CPV phase; Bd-anti-Bd mixing involves a factor e– 2 i b.
Time-dependent CPA gives Sy Ks =sin2b = 0.725§0.037 (WA),
consistent with constraints from other processes.
The result is clean without much ambiguity or new physics pollution
(unless contrived cancellation between mixing and decay).
C.W. Chiang Flavor Diagram Approach 10
Overall UT Fit Results
CKM Fitter Group
C.W. Chiang Flavor Diagram Approach 11
Overall UT Fit Results
(2004 Summer, ICHEP)
CKM Fitter Group
Everything simply fits together nicely!
C.W. Chiang Flavor Diagram Approach 12
Charmless is Charmful !
Although rare in comparison with charmed decays (suppressed by CKM
factors), charmless decays are actually very charmful and important
Include strangeness-conserving (DS=0) and strangeness-changing
(|DS|=1) transitions; some processes in the latter category already give
us hints about new physics.
Offers opportunities to discover direct CPV because many of them
involve more than one significant subprocesses with different weak and
B p p, p h(0), r p provide info on a, as a result of the interference
between mixing and decay;
B K p provides info on g.
B Xu l n provides info on |Vub| (theoretically hard though because of
the large charm background), thus one side of the UT.
C.W. Chiang Flavor Diagram Approach 13
Importance of Strong Phases
Strong interaction matters because what we observe are hadrons but not
the fundamental degrees of freedom in the theory.
Consider rate CP asymmetry of modes with the amplitudes
Such an asymmetry requires at least two amplitudes characterized by
distinct weak phases and strong phases.
It is of great importance to understand the patterns of FSI phases in as
wide as possible a set of decays, although what we really care about are
weak phases (signals), not really strong phases (noises).
C.W. Chiang Flavor Diagram Approach 14
Getting Strong Phases
The Bander-Silverman-Soni (BSS) type strong phase calculation only
accounts for the perturbative strong phases in penguin diagrams with
intermediate q anti-q pair being on shell.
[BSS, PRL 43, 242 (1979)]
No first-principle method for computing FSI strong phases exists
because they involve nonperturbative long-distance physics.
[see a recent try by Cheng, Chua, Soni, hep-ph/0409317]
One conventional and efficient method of obtaining strong phase
information is to directly extract from data using isospin analysis.
Flavor diagram approach offers a way to extract strong phases
associated with individual topological amps and to relate them using
flavor SU(3) symmetry.
C.W. Chiang Flavor Diagram Approach 15
Flavor Diagram Approach
This approach is intended to rely, to the greatest extent, on model
independent flavor SU(3) symmetry arguments, rather than on specific
model calculations of amplitudes.
[Zeppenfeld, ZPC 8, 77 (1981); Chau + Cheng, PRL 56, 1655 (1986); PRD 36, 137 (1987); PRD 43,
2176 (1991); Savage+Wise, PRD 39, 2246 (1989); Grinstein + Lebed, PRD 53, 6344 (1996); Gronau
et. al., PRD 50, 4529 (1994); 52, 6356 (1995); 52, 6374 (1995)]
The flavor diagram approach:
is diagrammatic (can be formulated in a formal way);
only concerns the flavor flow (arbitrary gluon exchange among
has a clearer weak phase structure (unlike isospin analysis where
different weak phases usually mix).
Very recent works in this direction include:
Chua [PRD 68, 074001 (2003)] and Luo + Rosner [PRD 67, 094017 (2003)] for baryons;
Charng + Li [PLB 594, 185 (2004) and hep-ph/0410005] for weak phase extraction; and
He + McKellar [hep-ph/0410098] for analyzing recent data.
C.W. Chiang Flavor Diagram Approach 16
All these tree-level diagrams involve the same CKM factor.
q = u,d,s
tree (external W emission) color-suppressed (internal W emission)
due to fB.
exchange (neutral mesons only) annihilation (charged mesons only)
C.W. Chiang Flavor Diagram Approach 17
Loop-Level (Penguin) Diagrams
All these tree-level diagrams also have the same CKM factor.
QCD (strong) penguin penguin annihilation flavor singlet
(internal gluon emission) (neutral mesons) (external gluon emission)
C.W. Chiang Flavor Diagram Approach 18
NLO Flavor Diagrams in Weak Interactions
Nothing forbids you from drawing one of the following
diagrams whenever you see T, C, or P in your amplitude
list. They involve two weak boson propagators.
color-suppressed EW penguin
appear together with C appear together with T
and S in decay amps and P in decay amps
C.W. Chiang Flavor Diagram Approach 19
Physical Flavor Diagrams
Treat T, C, P, E, A, S as “leading-order” amplitudes (note that only S is
of loop nature) and PEW and PCEW as “higher-order” contributions (in the
sense of weak interactions).
Physical amplitudes contain flavor diagrams both “leading order” and
“next-to-leading order” in weak interactions
t ´ T + PCEW, c ´ C + PEW, p ´ P – PCEW / 3, s ´ S – PEW / 3, a ´ A.
Moreover, P contains t-, c-, and u-quark mediated penguins, Pt,c,u. One
may use the unitarity to rewrite Pt as the sum of two parts, one having
the same weak phase as Pc and the other having the same weak phase as
Pt. This amounts to separating P into Ptc + Ptu.
Ptu involves the same CKM factor as the tree-level amplitudes. One
may thus sweep this amplitude to the tree-level amplitude category.
Note that this amplitude may be sizeable, particularly for DS = 0 decays.
For example, what many people call T or C extracted from p p decays
are actually T – Ptu and C + Ptu. Thus, “| C / T |” > 1 is possible.
C.W. Chiang Flavor Diagram Approach 20
A Simple Example
Quark contents: p+ p- p+ r-
When vector mesons
are involved, one Bd p- Bd r-
further labels the
amplitude by which
meson the spectator
quark goes into. P PV
A(p+p–) = – (T + P); p+ p+
A(p+r–) = – (TV + PV). Bd Bd
Minus sign comes p- r-
from the wave
functions of p– and r–.
C.W. Chiang Flavor Diagram Approach 21
Hierarchy in Flavor Diagrams
In our definition, the amplitudes contains CKM factors and may involve
an arbitrary number of gluon exchanges.
An educated guess tells us that the magnitudes of the amplitudes should
roughly satisfy the following hierarchical structure.
It should be emphasized that l appearing in the hierarchy is not an
expansion parameter but merely an order parameter. It simply reflects
our naïve expectation in the magnitudes of flavor amplitudes.
when going from 1 st row to 2nd row:
- tree-type amps suppressed because VudVus
- loop-type amps favored because VtdVts
C.W. Chiang Flavor Diagram Approach 22
Global Fits to Charmless Decays
Goals for the global fits:
Check if the SM offers a consistent picture for all available data;
Check the working assumption of SU(3)F;
Extract weak phase g (thus a by unitarity);
Extract strong phases; check Lipkin conjecture in V P decays;
Make predictions of unseen modes based upon current data.
Parameters involved in the fits include:
Data points used in the c2 fits include:
CP asymmetries (time-dependent and -independent).
C.W. Chiang Flavor Diagram Approach 23
Some Basic Formulas
The invariant matrix element M for a decay process B M1 M2 and the
corresponding decay width are
where p is the 3-momentum of the final state particle in the rest frame of
B. Note that M may contain polarization vector summation and average
as is the case for final states containing vector mesons.
We also assume the following SU(3)F relations (with l = 0.224):
partial SU(3)F breaking
taken into account
assuming perfect SU(3)F
symmetry for these amps
C.W. Chiang Flavor Diagram Approach 24
B V P Decays
CWC, M. Gronau, Z. Luo, J. Rosner, D. Suprun, PRD 69, 034001 (2004) [hep-ph/0307395].
C.W. Chiang Flavor Diagram Approach 25
Fitting Parameters for VP Modes
start with: related by symmetry: other quantities:
ignored small amplitudes such
as color-suppressed EW
annihilation diagrams, and
C.W. Chiang Flavor Diagram Approach 26
Fitting Parameters for VP Modes
We thus have the following parameters:
• amp sizes: | tP |, | tV |, | CP |, | CV |, | p0 P |, | p0 V |, |P0 EWP |, |P0 EWV |;
• strong phases: dP, dV, f;
• weak phase: g only (no b dependence).
• symmetric under simultaneous changes:
g p – g, dP,V p – d P,V and f – f.
C.W. Chiang Flavor Diagram Approach 27
List of Modes
In our fit, there are totally 34 observables:
13 data points
C.W. Chiang Flavor Diagram Approach 28
List of Modes
another 16 data points
13 data points
finally, include time-dependent CP asymmetries:
Sf Ks=-0.147§0.697 (S=2.11), Af Ks=0.046§0.256 (S=1.08)
(contribute constant to c2) [Browder, talk at LP03]
(provide b dependence) [BaBar, talk by Jawahery at LP03]
C.W. Chiang Flavor Diagram Approach 29
c2-g Plots (34 data points)
p’V/p’P = –1; 10 parameters
p’V/p’P = real; 11 parameters
p’V/p’P = complex; 12 parameters
All of them together
major changes in c2 at g'65± and
165± from step 1 to step 2
positions of minima almost
g=(53+15-33)± if Srp is left out, c.f.
For the three major minima, we still have:
g ' 25± and 165± disfavored by CKM Fitter and dynamics
g ' 65± favored and consistent with CKMFitter
C.W. Chiang Flavor Diagram Approach 30
physical solution prefers trivial strong phases while others give large strong phases,
consistent with perturbative calculations
C.W. Chiang Flavor Diagram Approach 31
Predictions for VP Modes
(DS=0, complex p’V/p’P)
results showing significant
differences from exp’t data
results showing significant differences
among the three fits
C.W. Chiang Flavor Diagram Approach 32
Predictions for VP Modes
(|DS|=1, complex p’V/p’P)
results showing some significant
differences among the three fits
C.W. Chiang Flavor Diagram Approach 33
Overall, fits are satisfactory (at 38% and 55% CL for 10- and 12-
parameter fits) and have solutions g = (65§6)± and (63§6)± consistent
with constraints from other processes.
Data of r p play the roles of breaking the g p – g symmetry and
stabilizing the fit results.
Note that the c2 fits allow us to extract preferred values of fitting
parameters along with their 1 s errors. Moreover, we have an idea
about how good our fits are from the CL.
Global fits prefers the Lipkin conjecture p0= – p0 P .
Only partial SU(3) breaking effect included for T amps; can verify
SU(3) for penguin amps when B K* anti-K (pV) and anti-K* K (pP )
rates are measured.
C.W. Chiang Flavor Diagram Approach 34
B P P Decays
CWC, M. Gronau, J. Rosner, PRD 68, 074012 (2003) [hep-ph/0306021];
CWC, M. Gronau, J. Rosner, D. Suprun, PRD 70, 034020 (2004) [hep-ph/0404073].
C.W. Chiang Flavor Diagram Approach 35
List of Modes
large CP asymmetries;
BR too small compared
To QCD fact. predictions
BR too large compared to
QCD fact. predictions
p K anomaly
need s’=S’-P’EW / 3
C.W. Chiang Flavor Diagram Approach 36
From bottom to top, we use 8 and 6 CKM fitter
parameters to fit 14 data points (p p
and K p) and 13 and 11 parameters to
fit 24 data points (further including
final states with h and h0).
Find g ' 54± ~ 66±, results still consistent
with but less stable than the V P case.
C.W. Chiang Flavor Diagram Approach 37
consistent with other constraints
large |C/T| ratio and non-trivial
relative strong phase; unable to
account for in perturbation
all strong phases relative to P’
required to account for the large
h 0 K BR’s
satisfactory overall fit results
C.W. Chiang Flavor Diagram Approach 38
Predictions for PP Modes
no problem in p p modes
still problematic in p K modes
C.W. Chiang Flavor Diagram Approach 39
[see also Chua, Hou and Yang, Mod. Phys. Lett. A18, 1763 (2003);
Buras et al, PRL 92, 101804 (2004); hep-ph/0402112]
Fits to p p + p K data:
requires large |C/T| ratio (> 0.5);
requires a nontrivial strong phase between C and T (» 100±).
Fits to all PP data
one needs to introduce S (singlet penguin), Ptu (t,u-mediated penguin);
one needs to introduce Stu (t,u-mediated singlet penguin) particularly for the p+h’
Robust results in our fits
magnitude of QCD penguin |P|;
relative strong phase between T and P;
sizes of electroweak penguins (color-allowed and -suppressed), consistent with
obtain g ' 60± (48± < g < 73±, 68%CL), consistent with CKM fitter and earlier analysis
for VP modes.
We do not fully trust the stability of the shallow minima.
C.W. Chiang Flavor Diagram Approach 40
Strangeness in B Physics
V. Barger, CWC, P. Langacker, H.S. Lee, PLB 580, 186 (2004) [hep-ph/0310073];
V. Barger, CWC, J. Jiang, P. Langacker, PLB 596, 229 (2004) [hep-ph/0405108];
V. Barger, CWC, P. Langacker, H.S. Lee, PLB 598, 218 (2004) [hep-ph/0406126].
C.W. Chiang Flavor Diagram Approach 41
p K Anomaly
Within the SM, the following ratios should be approximately equal
but show a 2.4s 1.9s difference.
underestimate of p 0 detection efficiency; [Gronau and Rosner, hep-ph/0402112]
new physics. [Buras et al, PRL 92, 101804 (2004); NPB 697:133 (2004), hep-ph/0410407]
One minimal explanation is that the color-allowed electroweak penguins
cause the problem isospin-violating new physics
C.W. Chiang Flavor Diagram Approach 42
Z0 Model With FCNC
The B p K decay can be a tree-level process mediated by a Z' boson if
there are FCNC couplings (possible for family non-universal charges).
We simply a general Z' model by assuming:
(i) no right-handed flavor-changing couplings,
(ii) no significant RG running effect between MZ'
and MW scales,
(iii) negligible Z' effect on the QCD penguins so that the new physics is
With these simplifications, we have 3 parameters left in the model.
carrying a new CP-violating source
C.W. Chiang Flavor Diagram Approach 43
Following the arguments in Buras et al, new physics is
coded by the parameters:
Found solutions from p p and p K data:
Correspond to our parameters:
C.W. Chiang Flavor Diagram Approach 44
Simultaneous Solution to p K and f KS
Can the p K anomaly and f Ks asymmetries be accounted for by the
same thing at the same time?
According to Buras et al, their solution [our solution (AL)] to the p K
anomaly leads to S(f Ks) greater than S(Y Ks)!
However, due to different interference patterns between O7,8 and O9,10
operators in our model, it is possible [all our other solutions (BL, ALR and
BLR)] to have S(f Ks) smaller than S(Y Ks).
C.W. Chiang Flavor Diagram Approach 45
Perspective and Summary
Flavor diagram approach provides a simple and reliable picture for
describing B decays. It is phenomenological (data driven), contains LO
and NLO amps in weak interactions but all orders in strong interactions,
includes final state interacting effects, and suffers from SU(3)F breaking
Flavor SU(3) generally fits data well, with some exceptions. Predictions
are made based upon current measurements and provide tests of the
All fits provide information on g that is consistent with other extractions.
Results from ICHEP are being analyzed by D. Suprun for his thesis.
We need to know more information about strong phases. In particular,
one should understand better about final-state rescattering effects.
We hope to see a proof of factorization in processes of interest to us, in
particular, those involving penguin diagrams.
We hope to see more affirmative evidence of new physics in B physics,
e.g., Sf KS the p K anomaly, large Bs anti-Bs mixing, etc.
C.W. Chiang Flavor Diagram Approach 46
C.W. Chiang Flavor Diagram Approach 47