Transcript Low MassS-wave K and  Systems Brian Meadows University of Cincinnati  S- waves in heavy flavour physics ?  What is known about S- wave.

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
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
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  (476–628)− i (226–346) and of  (694841)-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.
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
What is Known about K Scattering ?
SLAC/LASS experiment E135: K -p  K -+n (11 GeV/c)
NPB 296, 493 (1988)
+++
L=0
L=0
|T |
+++
Total S-wave
I = 1/2
I = 3/2
I =3/2 Phase 0
|T |
Phase
degrees
Phase
0
+++
K +p  K ++n
M (K -+) (GeV/c2)
M (K -+) (GeV/c2)
K -p  K –-D++
I- spins are separated using I=3/2 phases from
K +p  K ++n and K -p  K –-D++ (13 GeV/c)
M (K§ §) (GeV/c2)
No evidence for (800) – yet ~no data below 825 MeV/c2 either.
Charm 2007, Ithaca, NY, 8/06/2007
Estabrooks, et al, NP B133, 490 (1978)
Brian Meadows, U. Cincinnati
Effective Range Parametrization (LASS)
NPB 296, 493 (1988)

Scattering amplitude is unitary (elastic) up to K’ threshold
(for even L):
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”
Strictly, only valid below ~1460 MeV/c2.
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
 S-wave Scattering (I = 0)
Excellent Data from - p  - + n
G. Greyer, et al, NP B75, 189-245 (1975)
(several analyses - including other reactions)
B. Hyams, et al, NP B64, 134 (1973)
00 (degrees)
I=0
Im T
Au, Morgan,
Pennington, PR D35,
1633-1664 (1987)
c PT
KK
Threshold
KK
Threshold
M() (MeV/c2
No evidence for (500) – essentially no data below 500 MeV/c2 either.
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
Re T
 S-wave Scattering (I = 2)
0 2
(degrees)
from N. Achasov and G. Shestakov, PRD 67, 243 (2005)
02
Data included in fit:
+ p  + + n (12.5 GeV/c)
W. Hoogland, et al, NP B69,
266-278 (1974)
+ d  - - ppspec
(9 GeV/c)
N. Durusoy, et al, PL B45, 517520 (1973)
NOTE - 02 is negative.
Fit assumes amplitude to be unitary:
Reasonable assumption
up to r§r§ threshold
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
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
D+
• k scattering on
during
fragmentation can impart
an overall phase
Q
+
+
k T
f
K-
+
Scattering:
kf
 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.
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
Conventional Approach –
Breit-Wigner Model “BWM”

The “isobar model” ignores all this, and problems of double-counting:
“NR”
{12}
1
1
2
3

3
2
{13}
1
2
{23}
3
2
1
Amplitude for channel {i j} with angular momentum L:
NR - constant
(L=0)
R form
factor
D form
factor
spin
factor

In the BWM each resonance “R” (mass mR, width R) described as:

Lots of problems with this theoretically – especially in S- wave
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
3
2
D +  K -++ (shown to right)
M 2(K -+)
Prominent feature:
 Strong asymmetry in K*(892) bands
 F-B asymmetry vs. K*(892) BreitWigner phase (inset) is zero at 560.
 (Differs from LASS where this is
zero at 135.50
E791
Asymmetry
Study D Decay Channels with
Large S-wave Component
BW
 Interference with large S– wave
component.
 Shift in S–P relative phase wrt
elastic scattering by -79.50
M 2(K -+)
0
Another channel with similar features w.r.t. the r (770) is D+  -++
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
(800) in BWM Fit to D+  K-++
E791: E. Aitala, et al, PRL 89 121801 (2002)
Without (800):



Fraction %
0
NR ~ 90%
Sum of fractions 130%
Very Poor fit (10-5 %)
BUT

Phase
S~89 %
Inclusion of  makes
K0*(1430) parameters
differ greatly from PDG
or LASS values.
M1430 = 1459 § 7 § 12 MeV/c2
1430 = 175 § 12 § 12 MeV/c2
M = 797 § 19 § 42 MeV/c2
 = 410 § 43 § 85 MeV/c2
Similarly, (500) is required in D+  -++
E791: E. Aitala, et al, PRL 86:770-774 (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
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
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 - arXiv:0705.2248v1 [hep-ex] 2007
CLEO c - arXiv:0707.3060v1 [hep-ex] 2007
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
E791 Quasi-Model-Independent
Partial 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).
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
Compare QMIPWA with LASS for S-wave
arg{F0(s)}
|F0 (s) |
E791
LASS


S-wave phase for E791 is shifted by –750 wrt LASS.
Energy dependence compatible above ~1100 MeV/c2.
 Parameters for K*0(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 ?
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
Watson Theorem Breaking vs. I = 3/2 ?
S- wave phase (deg.)
FOCUS / Pennington: D  K-++
arXiv:0705.2248v1 [hep-ex] 2007
K-matrix fit using LASS Data
For I=1/2 production vector:
LASS I=1/2
phase
Includes separate I=3/2 wave
 Big improvement in c2.
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:
I=½ phase does agree well with LASS as
S- wave fractions (%): I=1/2:
207.25 § 24.45 § 1.81 § 12.23
I=3/2: 40.50 § 9.63 § 0.55 § 3.15
stat.
syst.
Model
P- and D- wave fractions & phases ~same as BWM fit.
Charm 2007, Ithaca, NY, 8/06/2007
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 ?
Brian Meadows, U. Cincinnati
CLEO c: D  K-++
arXiv:0707.3060v1 [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
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
CLEO c: D  K-++
arXiv:0707.3060v1 [hep-ex] Jul 20, 2007


BWM fit is also significantly improved by adding I=2 ++
amplitude – repairs poor fit to ++ inv. mass spectrum.
Best fit uses a modification of E791 QMIPWA method …
BWM fit
Charm 2007, Ithaca, NY, 8/06/2007
QIMPWA fit
Brian Meadows, U. Cincinnati
Total S- wave from D+  K-++ Decays
• General agreement
is good
• All differ from LASS
(blue curves, 2nd row)
CLEO c (Solid line)
arXiv:0707.3060v1, 2007
E791 (Error bars)
Phys.Rev.D73:032004, 2006
FOCUS (Range)
arXiv:0705.2248v1, 2007
M(K- +) (GeV/c2)
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
CLEO c: D  K-++
arXiv:0707.3060v1 [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)
Interpolation table
(26 complex values)

Breit-Wigner type
of propagator:
K-+ S- wave – K0*(1430)
K-+ P- wave – K*(890)
D- wave
– K2*(1420)
++ S- wave – R = 0
Total of ~ 170 parameters:
BUT – only float C(s) for one wave at a time.
Charm 2007, Ithaca, NY, 8/06/2007
• Is final fit converged. (Errors?)
• Is solution unique?
• Is I=2 wave over-constraint?
Brian Meadows, U. Cincinnati
New Data from CLEO c: D  -++
arXiv:0704.3965v2 [hep-ex] Jul 20, 2007

Use 281 pb-1 sample (3770):
•


~4,086 events including
background.
Had to remove large slice in
m+- invariant mass
corresponding to
D+  Ks+
FOCUS: Phys.Lett.B585:200-212,2004
E. Aitala, et al, PRL 89 121801 (2002)
General morpholgy similar to
E791 and FOCUS


BWM fits
Standard BWM fit requires a 
amplitude much the same
Introduced several variations in
S- wave parametrization:
…………………..
Charm 2007, Ithaca, NY, 8/06/2007
CLEO c
Brian Meadows, U. Cincinnati
Complex Pole for :
J. Oller: PRD 71, 054030 (2005)

Replace S- wave Breit-Wigner
for  by complex pole:
arXiv:0704.3965v2 [hep-ex] Jul 20, 2007

Best fit:
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
Linear  Model inspired Production Model
Black, et al. PRD 64, 014031 (2001), J. Schecter et al., Int.J.Mod.Phys. A20, 6149 (2005)
Replace S- wave  and f0 (980) by weakly mixed
complex poles:
Unitary
arXiv:0704.3965v2, 2007
Weakly mixed
Poles  and f0(980)
. . . + usual BW terms for f0 (1350) and f0 (1500)

%
%
Full recipe includes both weak and strong
mixing between  and f0(980)
– 7 parameters in all
%
Excellent fit:
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
CLEO c: D  -++
arXiv:0704.3965v2 [hep-ex] Jul 20, 2007


A fourth, “custom model” for Swave (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
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
Moments Analysis in D+  K-K++



Focus: hep-ex/0612032v1 (2007)
K++ channel has no resonances
Remove  meson in
K+K+
6400 Events before  cut.
channel
Allows Legendre polynomial
moments analysis in K-+ channel
free from cross-channel:
|S| similar to LASS
Phase was not computed, but appears
to be shifted ~900 wrt LASS.
|S|2
Charm 2007, Ithaca, NY, 8/06/2007
(in K –
where
S*P
Brian Meadows, U. Cincinnati
+
CMS)
|P|2
S- Wave in B  J/ K+


Similar analysis (more complex due to
vector nature of J/) on K- + system
Mass dependence of S- and P-wave
relative phase in K-+ system was used to
determine sign: cos 2 > 0
A clear choice agrees with the LASS data
with overall shift + radians.
Clearly an interesting
way to probe the
K- + S- wave
89 fb-1
PRD 71: 032005 (2005)
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
S- Wave in

D+
+
   K+-
FB asymmetry in K- + system in these
decays observed by FOCUS to follow
closely the LASS behaviour.
Phys.Lett.B621:72-80,2005
… and friends?
Charm 2007, Ithaca, NY, 8/06/2007
Clearly an interesting
way to probe the
K- + S- wave
Brian Meadows, U. Cincinnati
Some K S-wave Measurements
Compared to LASS Amplitude
| Amplitude |
| Amplitude |
m(K ) < 1 GeV
m(K ) > 1 GeV
~0
Unknown;
(M/p) | ALASS |
used in fit
Similar to LASS
B0  J/ K+ -
~ + 180
Poorly defined ;
to be updated
Similar to LASS
B+  K+ - r+
~ ± 180
Unknown
Unknown
D0  K- K+ 0
~ - 90
Similar to LASS
Similar to LASS
Very different ;
significant rise
Similar to LASS
get ~ same
K0*(1430)
mass and width
Decay Process
Use of LASS S- wave
parametrization or
determination of
relative S-P phase in
various Dalitz plot
analyses leads to a
confusing picture.
B+

K+
-
+
D+  K- + +
More channels are
needed to understand
any pattern.
S – P
Meas. – LASS
( deg. )
~ - 75
toward threshold
D+  K- K+ +
~ - 90
Similar to LASS
Similar to LASS
D+  K- + l
~0
Similar to LASS
Similar to LASS
(More coming for LP07)
Adapted from W.M. Dunwoodie, Workshop on 3-Body
Charmless B Decays, LPHNE, Paris, Feb. 1-3, 2006
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
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.
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
Back Up Slides
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
Charged (800) ?
Babar: D0  K-K+0
?
11,278 § 110 events (98% purity)
Tried three recipes for K§0 S-wave:
1.
2.
3.


LASS parametrization
E791 fit
*
NR and BW’s 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
Charm 2007, Ithaca, NY, 8/06/2007
Not consistent
With “”
385 fb-1: PRC-RC 76, 011102 (2007)
Brian Meadows, U. Cincinnati
Partial Wave Analysis in D0  K-K+0

Region under  meson is ~free
from cross channel signals:
p- s
allows Legendre polynomial
moments analysis in K-K+ channel:
(Cannot do this is K channels)
|S|
(in K –K
where
+
CMS)
|S| consistent with either
a0(980) or f0(980) lineshapes.
Charm 2007, Ithaca, NY, 8/06/2007
Babar: 385 fb-1: PRC-RC 76, 011102 (2007)
Brian Meadows, U. Cincinnati
|P|
Compare QMIPWA with BWM Fit
arg{F(s)}



Red curves are §1
bounds on BWM fit.
Black curves are §1
bounds on QMIPWA fit.
Completely flexible Swave changes P- & Dwaves.
S
P
D
E791 Phys.Rev. D 73, 032004 (2006)
(S- wave values do depend on P- and D- wave models).
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
E791 Require (500) in D+  -++
E. Aitala, et al, PRL 86:770-774 (2001)
Without (500):



Fraction %
With 
NR ~ 40% dominates
r (1400) > r (770) !!
Very Poor fit (10-5 %)
S~116 %
Observations:


Phase
NR and  phases differ by ~
1800
Inclusion of  makes
K0*(1430) parameters differ
greatly from PDG or LASS
values.
No 
c2/d.o.f. = 0.90 (76 %)
This caught the attention of our theorist friends !
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
0
FOCUS / Pennington: D  K-++
arXiv:0705.2248v1 [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 cPT
I= 1/2: 2-channels (K and K’ ) one pole (K
I= 3/2: 1-channel (K only) no poles

*
1430)
This defines the D+ decay amplitudes for each I- spin:
where
T- pole is at: 1.408 – i 0.011 GeV/c2
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
FOCUS / Pennington: D  K-++
arXiv:0705.2248v1 [hep-ex] May 15, 2007

Amplitude used in fit:
I- spin 1/2 and 3/2
K-+ S-wave

Usual BWM model for
P- and D- waves
P- vectors are of form:
k=1 K ; k=2 K’
Same as pole
in K-matrix
that can have s-dependent phase except far from pole.
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati
… 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 /  = 0.99) in which the
S-wave phase variation is constrained to follow the LASS shape up
to K’ threshold.
Charm 2007, Ithaca, NY, 8/06/2007
Brian Meadows, U. Cincinnati