1. Isospin Phased Topological Approach for B-Meson Decays
C. S. Kim (Yonsei University)
김 충선 (金 忠善)
ICFP2005-Taiwan
C S Kim - ICFP2005

2. Contents Introduction  Issues & Motivations
Nonleptonic B Decays  Strong Phases & FSI
SU(2) Isospin & SU(3) Topological Approaches
Isospin Phased Topological (IPT) Method
IPT Approach for Decays
Discussions & Conclusions
with: D Chang, C Chen, H Hatanaka, CSK;
CSK, T Morozumi, S Oh, C Yu
C S Kim - ICFP2005

3. Introduction  Issues & Motivations
B-meson system
Experimental tools
Recent experimental & theoretical issues
Theoretical issues & motivations
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4. B-meson system is the best place to:
- explore aspects of strong interactions
- test electroweak interactions, parameters of SM
- find new physics, new sources of flavor & CPV
- may get insight into gravity ??
(ref: CSK etal, PRD67; K. Agashe etal, PRL93)
We have experimental tools:
- BaBar & Belle already collected O (109) B ’s
- Tevatron run II are coming
- in 2007, LHC-B and LHC will give 1012 B ’s/yr
- may get Super-B, Bs factories
- soon measure rare kaon decays
 We are lucky to live this exciting time !!
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5. Recent theoretical & experimental issues:
decays
- only T, C, (W ) contribute within SM
- B 0 (t )  to measure UT angle g (f3)
- good testing ground for factorization & FSI
 within (model independent) topological analysis
(ref: CSK et al, PLB621)
* Is C / T = 0.4~0.7, large? Is dT-dC large like p/2?
* Is FSI negligible in B decays, unlike K, p decays?
(ref: H. Cheng etal, PRD71)
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6. - P dominant > T > P EW > C > … within SM
decays
- P dominant > T > P EW > C > … within SM
* A+0 > 0  Are C/P, P EW/P larger than T/P??
* Is dT = dEW ? (ref: M. Neubert etal, PRL81)
* How about FSI (Final State Interactions)?
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7. - T dominant. How large P/T ?
decays
- T dominant. How large P/T ?
* What value Sp+p- ? Belle result = -1 ?
* Large Br(B 0  p0 p0)  Is there FSI ?
* Does FSI affect determination of a (f2) ?
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8. 4. Theoretical Issues & Motivations:
 Is FSI negligible in B decays?
* there exist a few experimental evidences of FSI ??
 If FSI non-negligible, FS re-scattering may violate SU(3)F
 Topological approach may not describe B decays correctly.
 Even so, SU(2) isospin symmetry may still holds !!
 Does isospin approach describe B decay, completely details ?
 cannot include short distance strong phases properly,
 in general, weak phases are all mixed,
 cannot relate SU(3) symmetry in different B decays.
 We would like to propose new method, which can describe
systematically, the effects of FSI, SU(3)F symmetry, etc.
C S Kim - ICFP2005

9. 2. Nonleptonic B Decays  Strong Phases & FSI
Strong phases in non-leptonic B decays
Direct CPV & strong phases
sd-strong phases & topological diagrams
ld-strong phases & final state re-scattering
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10. Strong Phases in Nonleptonic B Decays:
1. Direct CPV and Strong Phases
- Direct CP asymmetry (DCPV) in decay
where and d=d1- d2
* with non-zero d from more than 1 amplitudes
 DCPV !!
- Sources of strong phases (SPs) ?
* SPs related to quark diagrams  sd-SPs
 BSS, BBNS, pQCD, …
* SPs generated by re-scattering, FSI, etc
connected to isospin rearrangement  ld-SPs
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11. 2. sd-SPs and topological diagrams:
- connected to each quark diagram (P, T, C, A, …)
 topological quark diagram approach
- calculable or parameterized in QCDF or pQCD, …
 e.g.
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12. 3. ld-SPs and FSI, FS re-scattering:
- FSI, FS re-scattering, … (H. Cheng etal, PRD71)
* e.g.
* sd W -exchange 0, but ld W -exchange induced
- re-scattering of certainly violate
SU(3)F symmetry, in scale of m(D, K, p).
? For B  f1 f2, mB >> mf, (ms2-md2)/mb2 << 0.01
However, for f1 f2  f3 f4, no such suppression
** evidences: C/T ( ) ~ 0.5 exp(i 600)
Br( ) ~ O(10-6), …
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13. - FSI  break down of SU(3)F symmetry
 topological diagramatic approach (X)
* However, SU(2)F symmetry still hold
 isospin analysis (O)
** FS re-scattering will rearrange FS isospins
to make isospin relations still hold
with proper ld(isospin)-SPs
- Topological approach can be convoluted within
isospin analysis systematically
 Isospin Phased Topological (IPT) method
 If we ignore FSI (i.e. ld-SPs), it becomes back
to topological approach
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14. 3. SU(2) Isospin and SU(3) Topological Approaches
Topological quark diagram approach (TA)
TA flavor flow diagrams
Isospin analysis for B  D K decays
Topological approach for B  D K decays
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15. (model independent) Topological SU(3)F approach:
- associated with each topological amplitude with
each strong phase by using flavor SU(3) symmetry
(ref: D. Zeppenfend, ZPC8)
* in SU(3)F limit, derive SU(3) invariant tensor operators of effective Hamiltonian
 express the amplitude to reduced amplitude by using Wigner-Eckart theorem
 match the reduced amplitude to quark line graphs (w/ weak & strong phases)
- Topological (quark diagramatic) approach (TA)
* is digramatic (is not Feynman diagram, including sd-SPs)
i.e. T = VCKM |T | exp(i dT), is NOT absolute value |T |
* concerns only favor flow (invariant under hard QCD corrections)
 arbitrary number of gluon exchange with large scale among quarks
will not change the SU(3) invariance
* has clear weak phase structure (unlike isospin analysis)
 can be regrouped with same CKM factors, and by using unitarity
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16. TA falvor flow Diagrams: (ref: M. Gronau etal, PRD50, …)
dT=dEW ?
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17. e.g. isospin analysis for B  DK decays:
 Isospin amplitude, AkDK = [AkDK] exp( i dk ) = (VcbVus*)|AkDK| exp(i dk )
* However, in general, weak CKM phases are not factored out,
but different weak phases usually mix, so mixed with |AkDK|
 Each |AkDK| can be decompsed to combination of diagrams,
|A1DK|=T +C, |A0DK|=T -C < real and positive T, C
* Be careful that T is not |T |, but =|T | exp(i dT) in original TA
* However, in original isospin analysis,
T, C, …, are assumed to be real positive
 So, the isospin relation, A+- + A00 = A0-
i.e. (|T |) + (|C |) = |T |+|C |, holds automatically
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18. e.g. topological diagram approach for B  DK :
 each diagram has sd strong phase, e.g. T = (VcbVus*) |T | exp(i dT)
 If we combine [isospin analysis] + [topological approach],
what would happen ?
** Constraints :
(i) isospin relation should always hold as isospin symmetry holds
(ii) if we ignore topological strong phases, i.e. dT – dC = 0,
must recover to the original isospin method
(iii) if we assume negligible isospin phases, i.e. d1 - d0 = 0,
must recover to the original topological method
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19. 4. Isospin Phased Topological (IPT) Approach
Isospin phased topological approach
e.g. IPT approach for B  D K decays
Topological analysis & Isospin analysis
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20. where AI is the isospin amplitude with strong phases DI ,
Isospin Phased Topological Approach:
- 2 kinds of strong phases (SPs)
strong phases (related to each topological diagram)  sd-SPs
strong phases (related to FSI & FS re-scattering)  ld-SPs
- Convolution of topological approach with isospin analysis
with all constraints satisfied
 Isospin Phased Topological (IPT) Approach:
* infuse topological decomposition into isospin analysis
where AI is the isospin amplitude with strong phases DI ,
Dk is the amplitude from diagrammatic approach, and d’s are the corresponding strong phases. Each Dk = VCKM |Dk| .
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21. - e.g. IPT approach for B  DK :
* satisfies all constraints
(i) isospin relation holds as isospin symmetry holds,
AD+K- + AD0K0 =AD0K-
(ii) if we ignore sd-strong phases, i.e. dT – dC = 0,
recovers to the original isospin method
(iii) if we assume negligible isospin phases, i.e. d1 - d0 = 0,
recovers to the original topological method
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22. c.f. IPT approach for B  DK :
- if d1 - d0 = 0: topological diagram approach (TA)
T,C = VcbVus*|T,C | exp(i dT,C)
* weak phases factored out
* sd-SPs also factored out
- if dT – dC = 0: isospin analysis
A1,A0=VcbVus*|A1,A0|exp(i d1,0)
in general, not factored
* ld-SPs always factored
** topological decomposition of isospin components
* in original isospin analysis
A1=|A1| exp(i d1)
so, dT=dC=0 or dT-dC=0
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23. 5. IPT Approach for Decays
IPT approach for B  D K and B  D p
IPT for B  K p
IPT for B  pp
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24. (1) IPT Approach for B  DK and B  Dp
Similary for B  D p
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25. - Assume d0=d1 (for BDK), d1/2=d3/2 (for BDp)  topol’cal analysis:
Experimental Results:
- Assume d0=d1 (for BDK), d1/2=d3/2 (for BDp)  topol’cal analysis:
c2 analysis with SU(3) breaking effects, e.g ,
 6 BRs with 5 parameters (|T |, |C |, |E |, dT-C, dT-E)
c2min=0.15/1 (ref: CSK etal, PLB621)
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26. Full IPT Analysis for B D K and B D p system:
 c2 analysis with SU(3) breaking effects,
6 Brs with 6 parameters (|T |, |C |, dTC, dTE, d01, d13)
** from
 c2min=0.05/0, and the best fit values are
 |C/T | = 0.22, not 0.5
dT-dC = 0, not
 3 branches of solution:
branch A = TA
branch B = IA
branch A+B+C = IPT
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27. (2) IPT Approach for B  Kp
- topological diagram approach (TA)
- isospin analysis topological decomposition
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28. - full IPT relations:
where
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29. Experimental Results: Shown in Page 6 (ref: CSK etal, hep-ph/0505060)
- Assume a1/2 = b1/2 = a3/2 = 0  Topological Analysis:
c2 analysis with 8 observables (4 BRs, 3 Acp, SKp) and w/ Acp0+=0
7 parameters (|P |, |T |, |EW |, |C |, dT, dEW, dC), w/ |A |=0
 c2min = 0.05/1, and the best fit values (for fixed f3(g)=600)
dT not = dEW !!
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30. Full IPT Analysis for B  K p system:
- c2 analysis with 9 observables (4 BRs, 4 Acp, SKp)
10 parameters (|P |,|T |,|EW |,|C |, dT, dEW, dC, a1/2, a3/2, f3)
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31. ** Null Hypothesis Test by statistical p - value:
( is probability density function for c2 distribution)
 probability that observed value is happened under
the “null hypothesis”
** p – value < 5%, then the hypothesis is disfavored !!
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32. Interesting results from IPT Analysis:
i) We fix one input f3=600 or rT=0.21 or dT = dEW alternatively
 give good fits, c2min = 0
** dT=dEW holds, contrary to topological analysis
ii) Interestingly if we fix dT = dEW and dC = dP together,
 get good fit with parameters having SM (pQCD) values
** pQCD estimated parameter values
 our estimate: rT= , rEW= , rC=
iii) In general, we have large ld-SPs, i.e. large a1/2, a3/2
 large FSI in B  K p decays
iv) ld-SPs only  completely disfavored
sd-SPs only  very much disfavored
ld-SPs plus sd-SPs  favored with all SM parameters
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33. (3) IPT Approach for B  pp - full IPT relations: - isospin relations
 now with q02=q0-q2, PEW=0, [f1(b)=23.50]:
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34. IPT approach by c2 analysis:
- 7 parameters (|T |, rP, dP, rC, dC, q02, f3) and 6 observables:
 case(a): ld-SPs=0, i.e. q0=q2 (q02=0)
 Topological Analysis
case(b): sd-SPs=0, i.e. dP=dC=dT
 Isospin Analysis
case(c): with non-zero ld-SPs, sd-SPs, f3=650
 Isospin Phased Topological Approach
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35. where rP=|VtdVtb*/VubVud*| |P /T |=2 |P /T |, rC=|C /T |
c2 fit results:
where rP=|VtdVtb*/VubVud*| |P /T |=2 |P /T |, rC=|C /T |
 rC = (TA), rC = (IPT)
rP = (TA), rP = (IPT)
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36. (4) research still going on - issues:
1) How large is P / T, which may pollute f2(a) measurement ?
2) Does FSI increase B  p0 p0 rate ?
3) Is f2(a) measurement altered due to FSI ?
4) Does SU(3)F hold between B  K p and B  p p ?
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37. 6. Discussions & Conclusions
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38. 1. We have quite a few evidences of FSI, non-negligible
in B decays, B  K p, p p, D p, D K, ...
2. We have proposed a new approach, so called Isospin
Phased Topological method, which may break SU(3)F
if we include FSI, but still hold SU(2) isospin symmetry.
3. Topological description is convoluted within isospin
components, which can describe systematically, the
effects of FSI, SU(3)F symmetry, etc.
 If we ignore ld-SPs, it becomes back to topological
approach.
 Isospin relations still hold with FSI (i.e. with id-SPs).
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39. 4 We re-analyzed B decays, and found: a) for B  D p, DK
|C / T | = 0.22, not
dT - dC = 0, not
b) for B  K p
|PEW / P | = not
|C / P | = not
dEW = dT not large dEW - dT
c) for B  p p l
|C / T | = 0.4, not 0.8
|P / T | = not
d) in general, large (non-negligible) FSI ld-SPs
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