K swave System from D Meson Decays to Kp p in E791

1
K-? s-wave System from D Meson Decays to K-pp
in E791

• Brian Meadows
• University of Cincinnati
• For the E791 Collaboration

2
Outline

• What is known about s-wave K-? scattering
• Results from D ! K-?? Decays
• Model Independent Partial Wave Analysis
• Role of the Watson theorem
• Summary and Discussion

3
Kp Scattering in Heavy Quark Decays

• Understanding the structure of the s-wave Kp
  system is important to many analyses, and of
  vital interest to an understanding of the
  spectroscopy of scalar mesons.
• It may be possible to learn more from the large
  amounts of data on D and B decays now available.
• The applicability of the Watson theorem can also
  be tested.
• E791 is the first to use these decays to
  investigate these issues by making a Model
  Independent Partial Wave Analysis of the s-wave
  in the decay

D ! K-pp (and cc).
4
Kp Scattering

• Most information on K-p scattering comes from
  the LASS experiment (SLAC, E135)

Data from K-p! K-pn and K-p! K0p-p NPB
296, 493 (1988)
No data from E135 below 825 MeV/c2
5
Traditional Dalitz Plot Analyses

• The isobar model, with Breit-Wigner resonant
  terms, has been widely used in studying 3-body
  decays of heavy quark mesons.
• Amplitude for channel ij
• Each resonance R (mass MR, width ?R) assumed to
  have form

NR
2
D form factor
R form factor
spin factor
NR Constant
6
E791 D ! K-pp
138
c2/d.o.f. 2.7
Flat NR term does not give good description of
data.
7
E791 D ! K-pp
89
c2/d.o.f. 0.73 (95 )
Probability
Mk 797 19 42 MeV/c2 Gk 410 43 85
MeV/c2
8
E791 D ! K-pp Dalitz Plot

• Most interesting feature
• K(892) bands dominate
• Asymmetry in K(892) bands
• ! Interference with large swave component
• Also
• Structure at 1430 MeV/c2 mostly K0(1430)
• Some K2(1420)? or K1(1410)??
• Perhaps some K1(1680)?
• So
• At least the K(892) can act as interferometer
  for swave
• Perhaps other resonances can fill in some gaps
  too.

9
Asymmetry in K(892)

• Helicity angle q in K-p system
• Asymmetry

K-
q
?
p
q
cos q p q
?
tan-1m0?0/(m02-sK?)
! ?P - ?s is -750 relative to elastic scattering
10
swave from D ! K-pp Dalitz Plot?

• Divide m2(K-?) into slices
• Find swave amplitude in each slice (two
  parameters)
• Use remainder of Dalitz plot as an interferometer
• For s-wave
• Interpolate between (ck, ?k) points
• Model P and D

S (partial wave)
11
Reference Waves

• For p- and d-waves
• Use traditional Breit-Wigner isobar model
• Unbinned maximum likelihood fit
• Use 40 (ck, ?k) points for S
• Float (d1680, ?1680) and (d1430, ?1430)
• ! 40 x 2 4 84 free parameters.

P (partial wave)
D (partial wave)
K892 defines reference phase
12
Fit E791 Data for s-wave
Phase
Magnitude

• Float P and D parameters and find S
• General appearance similar to isobar model fit
• Magnitudes at low mass differ
• Phases above K0(1430)
• Tests with many MC samples of this size (15K
  events), produced to simulate the isobar model,
  produce similar differences in 15 of the cases
• Major source of systematic uncertainty
• Contribution of reference waves in region between
  K(892) and K(1680).

S
P
D
13
Comparison with Data
S
?2/NDF 272/277 (48)
14
Comparison with Elastic Scattering (LASS)

• S is related to elastic scattering amplitude T
  obtained from LASS by
• In elastic scattering K-p ! K-p the amplitude
  is unitary
• In D decays, the Kp can come from many sources
  so we expect the magnitude to differ from sin
  d(sKp).
• If applicable to these decays, the Watson theorem
  requires phases d(sKp) for each wave to be the
  same, up to the elastic limit (1454 MeV/c2).

K.M. Watson, Phys. Rev. 88, 1163 (1952)
15
Watson Theorem - a direct test

• Phases for S, P and D waves are compared with
  those from LASS.
• s-wave phase fs for E791 is shifted by 750 wrt
  LASS.
• fs energy dependence differs below 1100 MeV/c2.
• fp does not match well between K(892) and
  K(1680) resonances
• fd match is excellent up to elastic limit.
• However, a good fit can also be made by
  constraining the s-wave phase to agree with that
  from K-? scattering.

S
Elastic limit Kh threshold (1454 MeV/c2)
P
D
16
Summary

• A new technique for analyzing the amplitude
  describing a Dalitz plot distribution is used in
  D decays to K-??.
• It can provide model independent measurements of
  the complex amplitude of the K-p s-wave system,
  provided a good model for the p- and d-waves is
  used.
• The first measurements at masses below 825
  MeV/c2, are presented.
• The Watson theorem does not apply to D! K-??
  decays.
• Techniques similar to this should play a role in
  analyses of large samples of heavy meson decays
  becoming available from B factories, CLEO-c and
  the TeVatron collider.

17
Back up Slides
18
Watson Theorem - a direct test

• Phases for S, P and D waves are compared with
  those from LASS.
• s-wave phase fs for E791 is constrained to match
  shape of, but allowed to shift by 750 wrt LASS.
• fp match between K(892) and K(1680) resonances
  gets even worse.
• fd shifts -500 wrt LASS.
• Magnitude in s-wave differs too

S
Elastic limit Kh threshold (1454 MeV/c2)
P
D
19
A Different Approach

• Instead of expanding the Dalitz plot amplitude in
  BWs (or pole terms in a K matrix) for each
  resonance, expand in partial waves.
• For a D decay, barrier factors preclude all but
  s-, p- and d- waves.
• Treat the s-wave, at least, as having completely
  unknown dependence on invariant mass.
• p- and d-waves can be expanded as resonances of
  appropriate spin

20
Does this Work?
Phase
Magnitude
S

• Simulate 150K MC events with isobar model
  parameters
• Find S for them

S
P
D
21
Milestones in Dalitz Plot Analyses

• 1993-7
• E691/687 find large non-resonant (NR) fraction in
  Decays
• D ! ?-?? and D ! K-??
• 2001
• E791 find that broad, low mass scalar isobars can
  soak up most of the NR contribution
• ! NR is not constant
• 2004
• Focus collaboration use data from K-matrix fit to
  large number of hadron interactions involving
  ??- production in analysis of
• D ! ?-??.
• ! No new broad scalars required?

22
Milestones in Dalitz Plot Analyses

• 2005
• Lots more data is on the way
• Clearly, we may be able to learn which scalar
  resonances really exist
• Other information is required from the data
• We need new, less model-dependent ways to analyze
  it.
• ! One possibility is Energy Independent Partial
  Wave Analysis (EIPWA).
• E791 is the first to try.

23
E791 D ! p-pp
No s(500)
24
E791 D ! p-pp
25
Kp Scattering

• Most information on K-p scattering comes from
  the LASS experiment (SLAC, E135)

Data from K-p! K-pn and K-p! K0p-p NPB
296, 493 (1988)
a scattering length b effective range p
momentum in CM
Parametrize s-wave (I1/2) by
26
E791 D ! K-pp
27
Does this Work?
Phase
Magnitude

• Fit the E791 data
• Fix P and D parameters at ? model
• Find S
• Fix S and D parameters at ? model
• Find P
• Fix S and P parameters at ? model
• Find D
• The method works.

S
P
S
D
28
Comparison with Data
Moments
Masses
S
?2/NDF 272/277 (48)
29
Other Solution
Phase
Magnitude

• Qualitative agreement with data
• BUT does not give acceptable c2.
• This solution violates the Wigner causality
  condition.

S
S
E. P. Wigner, Phys. Rev. 98, 145 (1955)
P
D