1. CPV in three-body decays: the Dalitz plot analysis
DIF06
LNF - February 28 –March 3
Sandra Malvezzi
INFN Milano

2. Outline The power of the Dalitz plot analysis
CPV and Dalitz plot
Recent applications of the Dalitz technique in the beauty sector
Results
Problems/complications
Some guidance from charm
D mesons and FSI
A pioneering anlysis in D ppp
Conclusions

3. Dalitz plot in the last few years
SPIRES search for “Dalitz and date after 1999”
91 entries
after 2004
29 entries
Experiments:
FOCUS, E791, CLEO, BaBar-Belle
From D to B decays
From decay dynamics to CPV to New Physics
new millennium

4. Dalitz plot: the revenge
The experimentalist’s struggle!
“When the going gets tough, the tough get going”
for the younger in the auditurium:
the analysis is certainly complex but not impossible
if you survive, you might understand how QM works!

5. The power of the Dalitz plot
Dalitz plot analysis allows for determination of a complete set of decay parameters, i.e. amplitudes and phases
CP is a matter of phase
Exploit interference and make use of formalisms with explicit CKM phases.
B  rp a angle
B  D(*)K (*) g angle
...promising

6. CPV and Dalitz plot Promising and complementary approach
Independent measurements to over determine the unitarity triangle provide a non-trivial test of the Standard Model.
Comparing the results in various channels and via different analysis techniques will allow us to find possible inconsistency...
the way to New Physics.

7. Some pilot Dalitz-plot analyses in the beauty sector
Results and complications

8. B  rp
A theoretically clean way to extract a is via a time-dependent Dalitz plot analysis of B  rp
Snyder - Quinn formalism Phys. Rev. D48, 2139 (1993)
from the operative point of view B  p+p-p0 (all charge combinations) with all possible resonant structures and interferences.
A full Dalitz analysis from BaBar
a = ( ± 6)°
213 ML BB hep-ex/ (ICHEP04)
A “partial’’ Dalitz analysis from Belle
Selecting distinct bands in the p+p-p0 Dalitz Plot
a = (102 ± 11 ± 15)°
152 ML BB hep-ex/
Phys. Rev. Lett. 94, (2005)
Measuring a is more difficult than measuring b
Decay amplitudes modulated by Vub rather than Vcb making the overall rates small.....( )
Gluonic penguin rates are of the same order as the trees causing large theoreticanl uncertainties in cleanly extracting a from asymmetry measurements.

9. B  rr (not Dalitz)
This decay has recently received attention: small theoretical uncertainty
Potentially highly complicated
Three possible helicity states for the decay
Helicity 0 is CP-even
Helicity ±1 are not CP eigenstates
BaBar
a=(100 ± 13)° fL = ±
232 ML BB hep-ex/ Phys. Rev. Lett. 95, (2005)
Belle
a= (88 ± 17)° fL = ± 0.030
275 ML BB hep-ex/
This decay has received much attention because it allows one to determine alpha with small theoreticaluncertainty (penguin amplitude is small). Alpha is determined by measuring the decay time distributions of B and Bbar. We need to know the rho polarization.
The analysis assumes a 100% longitudinal polarization of the mode. If it is not true an angular analysis would be required which needs more data ....

10. Some complications to go from Bppp to B  rp from Bpppp to B  rr
means selecting and filtering the desired states among the possible contributions, e.g. sp, f0(980)p, sr, ss, rpp...
How to deal with the underlying strong dynamics effects?
The pp S-wave is characterized by broad, overlapping states: unitarity is not explicitly guaranteed by a simple sum of Breit -Wigner (BW) functions
Independently of the nature of s (genuine resonance or a strong dynamics structure), it is not a simple BW
f0(980) is a Flatté-like function, coupling to KK and pp

11. BDK Possibility of observing CP violation in BDK decays
B+ D(*)K(*)+ can produce neutral D mesons of both flavors
D0 and D0 mesons can decay into a common final state u b u
K(*)+ s
D(*)0 c b c B+
The measurement is based on two key observations: Do Dobar can decay to a common final state and B can produce D mesons of both flavors with a relative phase between the two interfering amplitudes which is the sum of strong and weak phases that can be extracted
From measurements of charge conjugate B decay modes.
B+ to DK+ are given by M1 propto Vcb*Vus and M2 Vub*Vcs. M1 and M2 interferes as the D0 D0bar decay to the same final state. s B+
D(*)0
K(*)+ u u u u
Relative phase q+=d+g is the sum of strong and weak interaction phases
q-=d-g for charge conjugate mode

12. Dalitz plot and the g angle
Dalitz plot analysis to extract g
Originally: interference of Cabibbo-favored D0  K+p-p0 and doubly Cabibbo-suppressed D0  K+p-p0
Recently: interference D0, D0  KSp+p- (both CF decays)
Belle ML BB
g=(64 ±15)° for B± DK ± ( 137 – 139 events )
g=(75 ±25)° for B± D*K ± ( events )
combined samples
The idea of using Dalitz plot analysis to extract gamma dates back to 1997 (ADS)
The technique uses the interference of the ...
hep-ex

13. Dalitz plot and the g angle (II)
BaBar ML BB
Phys. Rev. Lett. 95 (2005)
A model for D0 decay is needed
Dominating source of systematic error
hep-ex/
Method proposed by Belle was implemented by BaBar as well

14. Some complications Model assumptions ....
Set of 15 two-body amplitudes
(Kp)p K*(892)p, K*(1430)p, K2*(1430)p, K*(1680)p
(plus doubly Cabibbo-suppressed partners for each of these states)
Ks(pp) Ksr, Ksv, Ksf0(980), Ksf2(1270), Ksf0(1370), KSs1, KSs2
s1 and s2 are “ad hoc” resonances introduced to describe excess of events at pp threshold and at 1.1 GeV2
Ms1 = 539 ± 9 MeV Gs1= 453 ± 16 MeV
Ms2 = 1048 ± 7 MeV Gs1= 109 ± 11 MeV

15. A word of caution Some questions
Do we really understand the systematics?
Are we confident of controlling strong dynamics effects in the analysis?
Where can we look for directions?
Charm: we have already come across parametrization and formalism issues
Low and intermediate energy processes
Hadron spectroscopy
Scattering
A lot of science and in the decades (light quarks)

16. A way to proceed ... BaBar The right track to pursue ... promising!
Implemented the K-matrix formalism to describe the pp S-wave component in D0, D0 KSp+p-
Benefiting from charm expertise and work
FOCUS three-pion Dalitz plot analysis
No ad “ad hoc” resonances needed
tried to quote a preliminary, reliable, systematic error on the g angle: 3° (hep-ex/ )
The right track to pursue ... promising!

17. What is the K-matrix?
E.P.Wigner,
Phys. Rev. 70 (1946) 15
S.U. Chung et al.
Ann. Physik 4 (1995) 404
It follows from the S-matrix and, because of S-matrix unitarity, it is real
Vice versa, any real K-matrix will generate a unitary S-matrix
This is the real advantage of the K-matrix approach:
It (drastically) simplifies the formalization of any scattering problem since the unitarity of S is automatically respected.

18. For a single-pole problem, far away from any threshold,
a K-matrix amplitude reduces to the standard BW formula
The two descriptions are equivalent
In all the other cases, the BW representation is no longer valid
The most severe problem is that it does not respect unitarity
Add BW
Add BW
Add K
The Unitarity circle
Adding BWs a la “traditional Isobar Model”
Breaks Unitarity
Heavily modify the phase motion!
At this point, on the basis of a pretty solid theory, it is very easy to understand when we can employ the traditional Isobar Model and when not.
Add K

19. FOCUS D+ p +p +p - analysis
Yield D+ = 1527 51
S/N D+ = 3.64
Sideband
Signal
PLB 585 (2004) 200

20. K-matrix fit results C.L fit 7.7 % Reasonable fit with no retuning
Low mass projection
High mass projection
decay channel
phase (deg)
fit fractions (%)
Reasonable fit with no retuning
of the A&S K-matrix.
No new ingredients (resonances),
not present in the scattering, required !

21. Isobar analysis of D+ p +p +p would instead require
An “ad hoc” scalar meson: s
m = ± 27.0 MeV/c
G = ± 65.5 MeV/c
With s
C.L. ~ 7.5%
C.L. ~ 10-6
Without s

22. FOCUS D s+  p +p +p - analysis
Yield Ds+ = 1475 50
S/N Ds+ = 3.41
Sideband
Signal
Observe:
f0(980)
f2(1270)
f0(1500)

23. K-matrix fit results C.L fit 3 %
Low mass projection
High mass projection
decay channel
phase (deg)
fit fractions (%)
C.L fit 3 %
No three-body non-resonant contribution

24. The effort continues, grows and matures....

25. B   DK*
Statistical accuracy of the g extraction can be improved by adding excited K states to the analysis
Belle
B   DK* (hep-ex/ )
253 fb signal candidates B   DK* g = ( 112  35  9  11  8 )°
BaBar
B   DK* and B   D(*)K* (hep-ex/ )
g = ( 67  28  13  11 )°
Beside the usual problems with formalism and additional complication from NR
Before averaging we should understand clearly the systematics of each channel.
(D Ksp+p-)
non-resonant B   DKSp

26. Dalitz Analysis of B  Khh
Belle hep-ex/
140 fb-1 B+ K+p+p- and B+ K+K+K-
357 fb-1 B0 K0p+p-
Already mentioned complications due to pp states
KK final state can come from f0(980), f0(1300), f0(1500) – coupled-channel parametrization
CP asymmetry is predicted very small in B+ K*0(892) p+
window to NP
Kp model is needed.

27. Dalitz Analysis of B  hhh
BaBar
210 fb-1 B± p±p±p hep-ex/
Phys. Rev. D72, (2005)
205.4 fb-1 B± K±p±p hep-ex/
Phys. Rev. D72, (2005)
230 fb-1 B0 K+K-KS hep-ex/
B in three pions would allow to extract angle gamma from interference
Investigate sigma.
B to Kpp has also the complication of the Kp resonant structures (LASS parametrization)

28. Dalitz plot and B  fKs Promising way to search for New Physics
A reliable SM prediction exists for
sin2b(Bd J/yKs)  sin2b(Bd fKs)
BaBar/Belle average for 2005
sin2b(Bd J/yKs) = ± 0.032
sin2b(Bd fKs) =
= 0.50 ± – BaBar
= 0.44 ± 0.27 ± Belle
How do other resonant (e.g. f0(980)) and non-resonant KK components underneath f affect the measurement?
It is mandatory to measure various contributions and related interference via a Dalitz plot analysis.
530 events in kkk.

29. First set of conclusions
Dalitz plot analysis represents a powerful, unique and promising tool to study CP violation in the beauty sector
The analysis is challenging but there are no shortcuts to perform precise studies (New Physics)
There is a new vigorous effort to perform amplitude analyses
more robust formalism implemented
many different channels analysed
beauty community can benefit from charm experience and expertise
but need to go on..

30. Beauty and charm relationship...
B rp
B ppp D ppp
B D(*)K(*)
Kspp
 Kpp0
B  Kpp D Kpp
from charm we can learn something for beauty .... but not only ...

31. CPV in charm
In the SM, the D system is not as sensitive to CP as the K and B mesons.
The small effects predicted could leave open a window onto NP
Charm is unique (I. Bigi):
non-Standard-Model effects might exhibit very different patterns for the up and down classes of quarks
Charm decays are the only up-type decays that afford a probe of such physics
Important to measure it!
Asymmetry in decay rates are already measured, also
in three-body decays
Alternative approaches are worth being exploited ...
(DKK p )

32. Dalitz plot analysis and CPV in the charm sector
FOCUS D+K+K– p+ (ICHEP 02)
BaBar D0 K0K+K– hep-ex/050702
Phys. Rev. D72, (2005)
CLEO
D0  p+ p- p0 hep-ex/
Phys. Rev. D70, (2005)
D0  KS p+ p- hep-ex/
Phys.Rev. D70, (2004)
No statistically significant asymmetries reported ...
improve accuracy!

33. D+K–K+p+ is (would be) a good candidate
Yield D+ = 7106 92
1.7
1.8
1.9
2.0
GeV
2.1
1250
1750
250
500
750
1000
1500
2000
2250
Two amplitudes (spectator CSD - penguin)
Good yield and S/N ratio
Strong phases present
D+ , Ds KK 1
1.5 2
2.5 3
3.5
m(KK)(GeV)
m(Kp)(GeV)
0.2
0.4
0.6
0.8
1.2
1.4
1.6
1.8

34. Simple idea ... look at D+/D–
Measure coefficient and phase for each amplitude
Look for possible local asymmetry in D+/D– parametrs
Complications in the final state (KK) (Kp) treatment
f0(980)/a0(980) coupled-channel lineshape
Higher mass f0(1370)-f0(1500) ...
Broad K*0(1430) ...
q = d + f
Measured phase:
q = d - f
CP conjugate
CP conserving
d = d
CP violating
f = -f

35. D+/D- split samples Fit based on BW formalism ICHEP2002
preliminary and tentative
No CPV but a more reliable parametrization needed
Start studying scattering S-matrix (K-matrix)
ICHEP2002
Coefficients: D±, D+, D-
Phases: D±, D+, D-

36. Hadronic physics The other perspective
The hadronic physics challenge ...
very clean samples of HF decays offer an unprecedented opportunity to investigate light meson physics
enriching, testing and finding consistency with the already available measurements from low-intermediate energy experiments ...
BES, BaBar, Belle, Cleo-c have (and/or) will have clean, high-statistics samples to provide phase-shift behaviour, measuring resonance parameters ... etc. ...

37. Conclusions high-precision studies and NP search
Dalitz plot analysis will definitely keep us company over the next few years
Some complications have already emerged
expecially in the charm field
others (unexpected) will only become clearer when we delve deeper into the beauty sector
Bs will be a new chapter (hep-ph/ Bs  Kpp, Bs  KKp)
There will be a lot of work for both theorists and experimentalists
Synergy invaluable!
The are no shortcuts toward ambitious and
high-precision studies and NP search