Low MassS-wave K? and ?? Systems

1
Low MassS-wave K? and ?? Systems
Brian Meadows University of Cincinnati

• S- waves in heavy flavour physics ?
• What is known about S- wave ?-? and K -?
  scattering and how this should apply to D decays
• Measurements of S- wave component
• D ? K -??
• Other modes
• Summary

2
S-waves in Heavy Flavour physics ?

• Low mass K? and ?? S- wave systems are of
  intrinsic interest and important for
  understanding the spectroscopy of scalar mesons
  existence of low mass ? or ? states in particular
• This is not covered in this talk, though a review
  of recent theoretical and experimental efforts
  focussing on pole parameters for ? (476628)- i
  (226346) and of ? (694-841)-i(300-400) MeV/c2
  cites many of the relevant references
• D. V. Bugg, J. Phys. G 34, 151
  (2007).
• The S- wave is also both ubiquitous and useful
• Interference in hadronic final states through
  Dalitz plot analyses plays a major role in
  studying much that is new in flavour physics
• CKM ?
• D0-D0 mixing
• Sign of cos2?, etc.
• General belief is that P- and D- waves are well
  described by resonance contributions, but that
  better ways to parameterize the S- wave systems
  are required as our targets become more precise.
• This talk focusses on recent attempts to improve
  on this situation.

3
What is Known about K p Scattering ?
SLAC/LASS experiment E135 K -p ? K -pn (11
GeV/c)
NPB 296, 493 (1988)
Total S-wave
I 1/2
I 3/2
I 3/2 Phase ?0
K p ? K pn
K -p ? K p-D
I- spins are separated using I3/2 phases from
K p ? K pn and K -p ? K p-D (13 GeV/c)
M (K p) (GeV/c2)
No evidence for k(800) yet no data below 825
MeV/c2 either.
Estabrooks, et al, NP B133, 490 (1978)
4
Effective Range Parametrization (LASS)
NPB 296, 493 (1988)

• Scattering amplitude is unitary (elastic) up to
  K? threshold (for even L)
• Strictly, only valid below 1460 MeV/c2.

where

• S-wave (I 1/2)

• S-wave (I 3/2)

No resonances
One resonance M0 1435 ?0 275 MeV/c2
a scattering lengths b effective
ranges
5
?p S-wave Scattering (I 0)
Excellent Data from ?- p ? ?- ? n
G. Greyer, et al, NP B75, 189-245 (1975) (several
analyses - including other reactions)
I 0
Au, Morgan, Pennington, PR D35, 1633-1664 (1987)
d00 (degrees)
c PT
KK Threshold
M(pp) (MeV/c2
No evidence for s(500) essentially no data
below 500 MeV/c2 either.
6
?p S-wave Scattering (I 2)from N. Achasov and
G. Shestakov, PRD 67, 243 (2005)

• Data included in fit
• ? p ? ? ? n (12.5 GeV/c)
• ? d ? ?- ?- ppspec (9 GeV/c)
• NOTE - d02 is negative.

W. Hoogland, et al, NP B69, 266-278 (1974)
N. Durusoy, et al, PL B45, 517-520 (1973)

• Fit assumes amplitude to be unitary

Reasonable assumption up to rr threshold
7
How This Should Apply to 3-body D Decays

• Decays have amplitudes F(s) related to scattering
  amplitude T(s) by
• Ff (s) Tfk (s) Qk (s)

Intermediate states

• Weak decay/fragmentation
• I-spin not conserved
• k scattering on ? during
• fragmentation can impart
• an overall phase

D
p
T
Q
k
Scattering k ? f
f
K -
p
? Watson theorem Up to elastic limit (for each
L and I ) K -? phase has same dependence on s
as elastic scattering but there can be an from
overall phase shift.
Behaviour of Q(s) is unknown.
8
Conventional Approach Breit-Wigner Model BWM

• The isobar model ignores all this, and problems
  of double-counting
• Amplitude for channel i j with angular momentum
  L
• In the BWM each resonance R (mass mR, width ?R)
  described as
• Lots of problems with this theoretically
  especially in S- wave

2
12
23
13
NR
1
1
1
2
2
2
3
3
3
1
2
3
NR - constant (L0)
D form factor
spin factor
R form factor
9
Study D Decay Channels withLarge S-wave Component

• D ? K -?? (shown to right)
• Prominent feature
• Strong asymmetry in K(892) bands
• F-B asymmetry vs. K(892) Breit-Wigner phase
  (inset) is zero at 560.
• (Differs from LASS where this is zero at 135.50
• ? Interference with large S wave component.
• ? Shift in SP relative phase wrt elastic
  scattering by -79.50

E791
Asymmetry
M 2(K -?)
?BW
M 2(K -?)
Another channel with similar features w.r.t. the
?0(770) is D ? ?-??
10
k(800) in BWM Fit to D ? K-pp
E791 E. Aitala, et al, PRL 89 121801 (2002)
Fraction
Phase 0

• Without k(800)
• NR 90
• Sum of fractions 130
• Very Poor fit (10-5 )
• BUT
• Inclusion of k makes K0(1430) parameters differ
  greatly from PDG or LASS values.

S89
M1430 1459 7 12 MeV/c2 G1430 175 12
12 MeV/c2
Mk 797 19 42 MeV/c2 Gk 410 43 85
MeV/c2
Similarly, ?(500) is required in D ? ?-??
E791 E. Aitala, et al, PRL 86770-774 (2001)
c2/d.o.f. 0.73 (95 )
Can no longer describe S- wave by a single BW
resonance and constant NR term for either K -?
or for ?-? systems. ? Search for more
sophisticated ways to describe S- waves
11
New BWM Fits Agree

• NEW RESULTS from both FOCUS and CLEO c support
  similar conclusions
• ? required (destructively interferes with NR) to
  obtain acceptable fit.
• K0(1430) parameters significantly different
  from LASS.

These BW parameters are not physically meaningful
ways to describe true poles in the T- matrix.
FOCUS - arXiv0705.2248v1 hep-ex 2007
CLEO c - arXiv0707.3060v1 hep-ex 2007
12
E791 Quasi-Model-IndependentPartial Wave
Analysis (QMIPWA)
E791 Phys.Rev. D 73, 032004 (2006)

• Partial Wave expansion in angular momentum L of K
  -? channels from D ? K-?? decays

Decay amplitude S- wave (L
0) Replace BWM by discrete points cne i?n P-
or D- wave Define as in BWM Parameters
(cn, ?n) provide quasi-model independent
estimate of total S- wave (sum of both I-
spins). (S- wave values do depend on P- and D-
wave models).
13
Compare QMIPWA with LASS for S-wave
F0 (s)
argF0(s)
E791
LASS

• S-wave phase for E791 is shifted by 750 wrt
  LASS.
• Energy dependence compatible above 1100 MeV/c2.
• Parameters for K0(1430) are very similar
  unlike the BWM
• Complex form-factor for the D ? 1.0 at 1100
  MeV/c2 ?

Not obvious if Watson theorem is broken in these
decays ?
14
Watson Theorem Breaking vs. I 3/2 ?
FOCUS / Pennington D ? K-?? arXiv0705.2248v1
hep-ex 2007
K-matrix fit using LASS Data For I1/2
production vector Includes separate I3/2
wave ? Big improvement in ?2.

LASS I1/2 phase
S- wave phase (deg.)
Total K-? S- wave
I 1/2 K-? S- wave
Large Data sample 52,460 245 events (96.4
purity)
s 1/2 (GeV/c2)
Observations I½ phase does agree well with
LASS as required by Watson theorem except near
pole (1.408 GeV/c2) This possibility is built in
to the fit model Huge fractions of each I- spin
interfere destructively. What about P- wave ?
S- wave fractions () I1/2 207.25 24.45
1.81 12.23 I3/2 40.50 9.63
0.55 3.15
stat. syst. Model P- and D-
wave fractions phases same as BWM fit.
15
CLEO c D ? K-??
arXiv0707.3060v1 hep-ex Jul 20, 2007

• Very clean sample from ?(3770) data
• 67,086 events with 98.9 purity.
• BWM fit similar to E791
• ?(800) in S- wave is required (as a Breit-Wigner)
  with NR.
• K (1410) in P- wave not required

16
CLEO c D ? K-??
arXiv0707.3060v1 hep-ex Jul 20, 2007

• BWM fit is also significantly improved by adding
  I2 ?? amplitude repairs poor fit to ??
  inv. mass spectrum.
• Best fit uses a modification of E791 QMIPWA
  method

QIMPWA fit
BWM fit
17
Total S- wave from D ? K-?? Decays

• General agreement
• is good
• All differ from LASS
• (blue curves, 2nd row)

CLEO c (Solid line)
arXiv0707.3060v1, 2007
E791 (Error bars)
Phys.Rev.D73032004, 2006
FOCUS (Range)
arXiv0705.2248v1, 2007
M(K- ?) (GeV/c2)
18
CLEO c D ? K-??
arXiv0707.3060v1 hep-ex Jul 20, 2007

• QMIPWA (E791 method applied to all waves and
  channels!)
• Define wave in each channel as
• F(s) C(s) ae i? R(s)
• Total of 170 parameters

Breit-Wigner type of propagator K-? S- wave
K0(1430) K-? P- wave K(890) D- wave
K2(1420) ?? S- wave R 0
Interpolation table (26 complex values)

• Is final fit converged. (Errors?)
• Is solution unique?
• Is I2 wave over-constraint?

BUT only float C(s) for one wave at a time.
19
New Data from CLEO c D ? ?-??
arXiv0704.3965v2 hep-ex Jul 20, 2007
BWM fits

• Use 281 pb-1 sample ?(3770)
• 4,086 events including background.
• Had to remove large slice in m??- invariant mass
  corresponding to
• D ? Ks?
• General morpholgy similar to E791 and FOCUS
• Standard BWM fit requires a ? amplitude much the
  same
• Introduced several variations in S- wave
  parametrization ..

FOCUS Phys.Lett.B585200-212,2004
E. Aitala, et al, PRL 89 121801 (2002)
CLEO c
20
Complex Pole for ?
J. Oller PRD 71, 054030 (2005)

• Replace S- wave Breit-Wigner for ? by complex
  pole
• Best fit

arXiv0704.3965v2 hep-ex Jul 20, 2007
21
Linear ? Model inspired Production Model
Black, et al. PRD 64, 014031 (2001), J. Schecter
et al., Int.J.Mod.Phys. A20, 6149 (2005)
arXiv0704.3965v2, 2007

• Replace S- wave ? and f0 (980) by weakly mixed
  complex poles
• Full recipe includes both weak and strong mixing
  between ? and f0(980)
• 7 parameters in all

Weakly mixed Poles ? and f0(980)
Unitary
. . . usual BW terms for f0 (1350) and f0 (1500)



Excellent fit
22
CLEO c D ? ?-??
arXiv0704.3965v2 hep-ex Jul 20, 2007

• A fourth, custom model for S- wave (Achasov,
  et. Al., priv. comm.) also gave excellent fit
• All models tried (including BWM)
• Give essentially the same non S- wave parameters
• Provide excellent descriptions of the data

23
Moments Analysis in D ? K-K?
Focus hep-ex/0612032v1 (2007)

• K? channel has no resonances
• Remove ? meson in KK channel
• Allows Legendre polynomial moments analysis in
  K-? channel free from cross-channel

6400 Events before ? cut.

• S similar to LASS
• Phase was not computed, but appears to be shifted
  900 wrt LASS.

(in K ? CMS)
S2
P2
SP
24
S- Wave in B ? J/? K?-

• Similar analysis (more complex due to vector
  nature of J/?) on K- p system
• Mass dependence of S- and P-wave relative phase
  in K-? system was used to determine sign cos
  2b gt 0
• A clear choice agrees with the LASS data with
  overall shift p radians.

Clearly an interesting way to probe the K- ? S-
wave
PRD 71 032005 (2005)
89 fb-1
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S- Wave in D ? ?? K?-

• FB asymmetry in K- p system in these decays
  observed by FOCUS to follow closely the LASS
  behaviour.

Phys.Lett.B62172-80,2005
Clearly an interesting way to probe the K- ? S-
wave
and friends?
26
Some Kp S-wave MeasurementsCompared to LASS
Amplitude
Decay Process dS dP Meas. LASS ( deg. ) Amplitude m(K p) lt 1 GeV Amplitude m(K p) gt 1 GeV
B ? K p- p 0 Unknown (M/p) ALASS used in fit Similar to LASS
B0 ? J/y K p- 180 Poorly defined to be updated Similar to LASS
B ? K p- r 180 Unknown Unknown
D0 ? K- K p0 - 90 Similar to LASS Similar to LASS
D ? K- p p - 75 Very different significant rise toward threshold Similar to LASS get same K0(1430) mass and width
D ? K- K p - 90 Similar to LASS Similar to LASS
D ? K- p l? 0 Similar to LASS Similar to LASS
Use of LASS S- wave parametrization or
determination of relative S-P phase in various
Dalitz plot analyses leads to a confusing
picture. More channels are needed to understand
any pattern. (More coming for LP07)
Adapted from W.M. Dunwoodie, Workshop on 3-Body
Charmless B Decays, LPHNE, Paris, Feb. 1-3, 2006
27
Conclusions

• The most reliable data on S- wave scattering are
  still from LASS or CERN-Munich data.
• More information on very low mass data may be
  accessible through study of
• semi-leptonic D decays
• larger samples of B ? J/? K-(?-)? decays
• New techniques seem to yield information on the
  S- wave in various decay modes, BUT it is not yet
  obvious how to carry that over information from
  one decay to another.
• Understanding this will require a systematic
  study of many more D and B decays
• This should remain a goal before it becomes a
  limiting systematic uncertainty in other heavy
  flavour analyses.

28

• Back Up Slides

29
Charged ?(800) ?

• Babar D0 ? K-K?0
• Tried three recipes for K?0 S-wave
• LASS parametrization
• E791 fit
• NR and BWs for ? and K0(1430)
• Best fit from 1 rotated by -900.
• No need for ? nor ?-, though not excluded
• Fitted with
• M (870 30) MeV/c2,
• ? (150 20) MeV/c2

?
11,278 110 events (98 purity)
?
Not consistent With ?
385 fb-1 PRC-RC 76, 011102 (2007)
30
Partial Wave Analysis in D0 ? K-K?0

• Region under ? meson is free from cross channel
  signals
• allows Legendre polynomial moments analysis in
  K-K channel
• (Cannot do this is K? channels)

?p- ?s
S
P
(in K K CMS)
where

• S consistent with either
• a0(980) or f0(980) lineshapes.

Babar 385 fb-1 PRC-RC 76, 011102 (2007)
31
Compare QMIPWA with BWM Fit
argF(s)

• Red curves are 1? bounds on BWM fit.
• Black curves are 1? bounds on QMIPWA fit.
• Completely flexible S-wave changes P- D-waves.

S
P
D
E791 Phys.Rev. D 73, 032004 (2006)
(S- wave values do depend on P- and D- wave
models).
32
E791 Require s(500) in D ? p-pp
E. Aitala, et al, PRL 86770-774 (2001)
Fraction
Phase 0

• Without s(500)
• NR 40 dominates
• r (1400) gt r (770) !!
• Very Poor fit (10-5 )
• Observations
• NR and s phases differ by 1800
• Inclusion of k makes K0(1430) parameters differ
  greatly from PDG or LASS values.

With ?
S116
No ?
c2/d.o.f. 0.90 (76 )
This caught the attention of our theorist friends
!
33
FOCUS / Pennington D ? K-??
arXiv0705.2248v1 hep-ex May 15, 2007

• Use K-matrix formalism to separate I- spins in
  S-wave.
• The K-matrix comes from their fit to scattering
  data T(s) from LASS and Estabrooks, et al
• Extend T(s) to K? threshold using ?PT
• I 1/2 2-channels (K? and K? ) one pole (K
  1430)
• I 3/2 1-channel (K? only) no poles
• This defines the D decay amplitudes for each I-
  spin

where
T- pole is at 1.408 i 0.011 GeV/c2
34
FOCUS / Pennington D ? K-??
arXiv0705.2248v1 hep-ex May 15, 2007

• Amplitude used in fit
• P- vectors are of form
• that can have s-dependent phase except far from
  pole.

Usual BWM model for P- and D- waves
I- spin 1/2 and 3/2 K-? S-wave
k1 K? k2 K?
Same as pole in K-matrix
35
Is Watson Theorem Broken ?

• E791 concludes
• If the data are mostly I 1/2 , this
  observation indicates that the Watson theorem,
  which requires these phases to have the same
  dependence on invariant mass, may not apply to
  these decays without allowing for some
  interaction with the other pion.
• Point out that their measurement can include an I
  3/2 contribution that may influence any
  conclusion.
• Note
• They also make a perfectly satisfactory fit (c2 /
  n 0.99) in which the S-wave phase variation is
  constrained to follow the LASS shape up to Kh
  threshold.