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3-Body Decays of D and B Mesons

Brian Meadows

University of Cincinnati

- Dalitz plot amplitude analyses
- The tools we use
- Recent results from BaBar
- Ds+-++
- Ds+K- K++
- Ds+Ks+-
- D0-D0 Mixing measurements
- CP Violation measurements from D decays
- A way forward and D+K-+- ?
- Summary

Brian Meadows, U. Cincinnati

- These decays are dominated by resonance structures
- Analysis of their phase and amplitude structure can provide information on:
- Presence of CP Violation in D decays
- D0 Mixing parameters
- Scalar (and other) light quark resonances
- CP Violation parameters (e.g. CKM °)
- Decay couplings
- S-waves are both ubiquitous and “useful”
- A suitable parametrization is to be found
- Interference with P –wave resolves ambiguity in the sign of cos2¯ in BdÃK* analysis
- Likewise, in K+K- system. Resolves sign of cos2¯s in BsÃÁ analysis
- As larger data samples are examined, and rarer phenomena sought, more care will need to be taken with the analysis details

BK+¼-¼0

DK-¼+¼0

Same Model

as B Decay

Brian Meadows, U. Cincinnati

Analyses fall into two categories:

- Model Dependent - assume isobar (quasi-two body) decay
- Using Breit-Wigner forms for resonances (BWM)
- K-matrix form for resonance and background
- Model Independent (or quasi-independent)
- Moments analysis (in restricted parts of the phase space)
- Quasi Model independent partial wave analysis (QMIPWA) for a single wave.

Attempts to extend this to more than one wave are, so far, questionable and difficult to make work.

Other hybrid schemes are also used.

Brian Meadows, U. Cincinnati

Isobar (Quasi-two Body) Model

- Decays (e.g. D+K-¼+¼+) have amplitudes F(s), s=sK-¼+,are related to final state scattering amplitude T(s)by:

Ff (s)=Tfk (s)Pk (s)

Intermediate states

- Weak decay/fragmentation:
- I-spinnot conserved
- kscattering on +during
- fragmentation can impart
- an overall phase

D +

p+

T

P

k

Scattering:

kf

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 overall phase shift.

Behaviour of P(s) is unknown.

Brian Meadows, U. Cincinnati

(L=0)

D form

factor

spin

factor

R form

factor

Mass-Dependent

width

Resonance

Mass

Breit-Wigner Model “BWM”- The BWM ignores re-scattering, and problems of double-counting:
- Amplitude is sum over channels {ij}and the isobarsR in each:
- In the BWM each resonance “Rij” (mass mR, width R) has mass dependence:

2

{12}

{23}

{13}

“NR”

1

1

1

2

2

2

3

3

3

1

3

2

Ak

Lots of problems with this theoretically – especially in S- wave

Brian Meadows, U. Cincinnati

K is real

is phase space factor.

Define production

vector for complex

couplings gk

D +

p+

Tkf

k

Pk

f

p -

p+

K-Matrix Model- Designed to overcome problem

in BWM of broad, overlapping

resonances in a partial wave

- Usual recipe:
- Usually only applied to one partial wave in one channel

I.J.R. Aitchison,

NP A189, 417 (1972)

T (s) is not

unitary in BWM

For each K-matrix

partial wave

Same as BWM but exclude isobars in KM waves

Brian Meadows, U. Cincinnati

PWA (Moments) Analysis

B0K+¼-¼0

s(K+0) (GeV/c2)2

s(K+-) (GeV/c2)2

- Works only where just ONE channel has significant isobar contributions

Would NOT work here due to 0, K *-,K2*-in other channels

Would

work

here

K2*

- Slice selected regions of Dalitz plot into invariant mass strips.
- In each invariant mass strip:
- Compute moments <YL0> = YL0(cosµ) for all events in bin
- Solve for partial waves S, P, ... Using the relationship:

- Isobars from other two channels contribute many higher “moments”.

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

PWA (Moments) Analysis

Moments with L > 2 are consistent with zero (for m < 1.2 GeV/c2) Lmax = 2

- First used on 3-body D0 decays by Babar

PRD 72, 052008 (2005)

- Plot m(K+K-), weighting events by factors YL0(cos )/ to obtain “moments <YL0(m)>”

K+K-CMS

Helicity angle

in K+K- channel

Brian Meadows, U. Cincinnati

Moments Analysis

Moments<YL0> for L=0, 1 and 2 are related to S- and P-wave amplitudes S and P, with relative phase SP by

S

|P|2

S*P

P-wave phase assumed to be dominated by meson Breit-Wigner form

Same technique also applied to K+K0 channel

Brian Meadows, U. Cincinnati

The E791 collaboration introduced the QMIPWA, to measure the K-+S-wave from D+K-++ decays in a way that required no knowledge of its isobar structure.

The BWM for the S-wave was replaced by a spline defined by complex numbers (fit parameters) defined at discrete invariant masses.

While the moments method can decompose multiple partial waves in small regions, the E791 method takes on other channels over the whole Dalitz plot as a kind of background, describing them by BWM isobars.

E791 Quasi-Model-IndependentPartial Wave Analysis (QMIPWA)E791 Phys.Rev. D 73, 032004 (2006)

The D+K-+p+ channel is characterized by presence of a large S-wave evidenced by clear asymmetry in the K*(890) helicity distribution

D+K-++

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Brian Meadows, U. Cincinnati

Partial Wave expansion in angular momentum L of K -+channel(s) from D+ K-a+b+ decaysE791 Quasi-Model-IndependentPartial Wave Analysis (QMIPWA)

E791 Phys.Rev. D 73, 032004 (2006)

Bose

Symmetrization

Spin Factor

Decay amplitude :

S-wave (L = 0): ReplaceBWMby discrete pointscne in

P-orD-wave: Define as inBWM

Parameters (cn , n) provide quasi-model independent estimate

of total S- wave (sum of both I- spins).

BUT S-wave values do depend onmodels forP- and D- waves.

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Brian Meadows, U. Cincinnati

Fitting Procedures

BWM, K matrix and QMIPWA fits maximize log-likelihood

with respect to the parameters (coefficients d, , m0’s, 0’s, etc)

Ps,Pb are PDF’s for signal ( ) and background, respectively

The PDF’s are normalized to unity over the Dalitz Plot area.

fs,fb are fractions of signal and background, respectively

Fraction of events attributed to any isobar or wave taken as

To evaluate a fit, we compute the 2 (sum of squares of normalized residuals over 2D bins in the Dalitz plot) divided by , the number of d.o.f

Ds-->K+K-+ - Recent Result from Babar

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Previous Analyses of Ds+K+K-π+

PLB 351:591-600 (1995)

PRL 100:161804 (2008)

arXiv:1011.4190v1 [hep-ex]

- This channel represents the “standard” for measurements of Ds BF’s.
- Prevous DP analyses by:
- E687 with 300 events
- CLEO-c (14,400 evs)
- This analysis (96000 evs)

Brian Meadows, U. Cincinnati

10

14

Ds+K+K-π+ Event Selection

~101K Events

96% purity

Background regions

From ±6s ±10s

Event Selection

- BaBar tracking and particle ID
- Select D*(2112)+Ds*to reduce background

Dm=|m(K+K-p+g)–m(K+K-p+)-mPDG|< 2

- Cut on likelihood ratio based on
- Momentum of Ds* in event CMS
- c2 difference due to Ds flight length
- Signed decay distance

Backgrounds:

- Remove D*+p+D0 (K-p+π0)
- Remaining backrounds mostly combinatorial with some D+K-p+π+
- Model DP for these from sidebands.

Brian Meadows, U. Cincinnati

10

15

Dalitz Plot Overview

f(1020)

K*0(892)

- f(1020) and K*(892) are clearly visible
- Before making a fit to the full Dalitz Plot, PWA analyses were attempted in the two box regions
- Parameterization of low mass K+K-S-wave was possible
- No evidence for D-wave in K-p+ channel
- No structure in K-p+ threshold region could be parameterized
- Probably due to presence of f0(1370) in the box.

Brian Meadows, U. Cincinnati

10

16

K+K- Partial Wave Analysis

θKK

π+

Ds+

<Y00>

<Y10>

<Y20>

Subtract background, correct for efficiency and phase-space factor at each m (K+K-)

Weight each event by spherical harmonics to get coefficients <Yk0> of cosKK distribution.

Brian Meadows, U. Cincinnati

10

17

PWA for low mass K+K-S-Wave

Ambiguity in ÁSP

Resolved by

direction

of Á(1020)

phase motion

Extract |S| and |P| magnitudes and relative phase fSP as a function of m(K+K-)

Fit distributions to a Flatte form for f0(980) for S-wave and an RBW for (1020) for the P-wave (solid curves)

Most of the phase motion comes from P-wave

S-wave phase obtained by subtracting fitted P-wave phase.

Brian Meadows, U. Cincinnati

10

18

P-/S-wave Ratio in f(1020) Region

Ds+fπ+ often used as normalization mode

Presence of S-wave must be taken into account

As a service, calculate S- and P-wave contributions as fractions of overall 3-body rate for different cuts on m(K+K-)

These can be used to make normalization to the well-measured DsK+K- π + branching fraction instead.

Brian Meadows, U. Cincinnati

10

19

K+K-S-wave Comparison

_

_

L. Maiani, A. D. Polosa and V. Riquer,

Phys. Lett. B651, 129 (2007)

K+K- S-wave shape compared with charm analyses of D0K+K-K0 and D0K+K-0

The K0K+ mode would be dominated by a0(980)

For Ds, expect large f0(980)

Similarity in form is evident for all

Evidence that they are both 4-quark states ?

Brian Meadows, U. Cincinnati

10

20

PWA Analysis in K-p+ System

- Proceeding as for the K-K+ system, evidence exists for strong K*(890) and for S-wave interference.
- |S|2 indicates that this wave is small
- This makes it difficult to measure S-wave phase
- ALSO, |S|2 goes negative
- This indicates that other (f0?) amplitudes play a role in this region.

Brian Meadows, U. Cincinnati

Fit to the whole Dalitz Plot

The Model:

- S-wave K+K- channel –

Use empirical parameterization off0(980) from fit to K+K- PWA.

- S-wave K-p + channel –

Use BWM forK0*(1430) with no non-resonant component.

- P- and D-waves for K+K- and K-p + channels

Modeled with BWM’s

The Fit:

- K*(892) mass and width are floated parameters
- Unbinned maximum likelihood fit with complex coefficients floating
- c2 computed by adaptive binning algorithm

Brian Meadows, U. Cincinnati

22

Dalitz Plot Results

Decay dominated by vector resonances

K*(892) width is 5 MeV lower than PDG ‘08 (consistent with CLEO-c)

Contributions from K*1(1410), K2(1430), κ(800), f0(1500), f2(1270), f2’(1525), NR - all consistent with zero

Adding LASS non-resonant part of S-wave K-p+ system makes fit worse

Brian Meadows, U. Cincinnati

10

26

Ds +-+ - Recent Result from Babar

384 fb-1

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Dalitz Plot Preliminaries

13K events

f0(980)

f0(980)

Sample Selection:

- Again, use Ds(2112)Ds
- Cut on likelihood ratio based on
- CMS momentum of Ds
- c2 probability difference due to Ds flight length
- Signed decay distance
- Dalitz plot is symmetrized (each event plotted twice)
- Efficiency and backgrounds modeled from MC simulated uniformly over the Dalitz plot

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

10

28

Decay Amplitude Modeling

Magnitude

f0(980)

Phase (radians)

The analysis closely follows the QMIPWA of D+K-++ decays by the E791 collaboration

The p+p-S-wave is described by a spline defined at 29 invariant mass values by complex numbers (58 parameters in the fit)

P- and D-waves are modeled as linear combinations of BWM contributions with complex coefficients that are also determined in the fit

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

10

29

Fit Results - Dalitz Plot m(p-p+)Moments

p-

p+

µ

p+

Agreement

between data

and model is

generally good

Unknown

reflection?

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

10

31

Dalitz Plot Results

BaBar

3

2

1

4

2/DOF =

437/(422-64)

P(2)=1.2%

Derived from sum of

S-wave isobar fractions

1Phys.Rev.D 79 032003,2009

2 Phys.Lett.B407:79-91,1997

3 Phys.Rev.Lett.86:765-769,2001

4 Phys.Lett.B585:200-212,2004

Interesting to compare different modeling of S-wave from E791 (BWM model) and FOCUS (K Matrix formalism)

S-wave component is Large

Significant D-wave f2(1270)

Brian Meadows, U. Cincinnati

10

32

S-wave Comparison

The f0(980) behavior is evident and as expected

Activity at f0(1370) and f0(1520) consistent with E791 fit

This wave is small in f0(600) region

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

10

33

Other Observations

_

*L. Maiani, A. D. Polosa, and V. Riquer, Phys. Lett.B651, 129 (2007)

Line shapes for K+K-, K+K0and+-S-wave (both f0(980) and a0(980) modes) are in striking agreement.

Does this suggest these are 4-quark states*?

Presence of signals in Ds decay suggests either W-annihilation process or significant re-scattering

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

34

Ds Ks-+ - Recent Result from Babar

468 fb-1

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

D0 Ks+-

A large, clean sample (540K events in the signal at 98.5% purity) was identified as D0 or D0 from charge of slow pion from D*D0s+ decays

For this, we choose events where

m=m(Ks+-s)-m(Ks+-)¼145.5 Mev/c2.

Backgrounds mostly from mis-reconstructed D’s or combinatorial sources.

Other, readily identifiable backgrounds were from D0K0K0 or from D0+-+- decays.

Self-conjugate channel is used in measurements of CKM phase AND D0-D0 mixing parameters

_

_

_

D0 Ks+-

Time-integrated

population

s(Ks+) (GeV/c2)2

s(Ks-) (GeV/c2)2

- The Dalitz plot has rich structure:
- Cabibbo-favoured (CF) “right sign” (RS)K*-(890) – vertical band
- “Wrong sign” (WS)K*(890), horizontal band,is upside-down” wrt S-wave background as expected for Doubly Cabibbo Suppressed (DCS) decay
- The P-wave+-system includes (770) and states at higher mass,
- S-wave +- structures including f0(980) and possible higher mass isobars and overall S-wave background.

- Also in the +- system is the K0 from D0K0K0.

Brian Meadows, U. Cincinnati

A Decay model for D0 Ks+-

Decays include isobars in all three channels

TheS-wave +- system is parameterized using the K-matrix form derived from much +- data*

For theS-wave K0§ systems, the form introduced by the LASS collaboration is used.

For P- and D-waves the BWM is used.

Time-integrated Fit Results

* Anisovitch, Sarantev, EPJ:A16, 229 (2003)

- Sum of fractions 103%

- WS
- Decays
- << RS

Mass K *(890)- 893.7 § 0.1

K *(890)- 46.7 § 0.1

Mass K0(1430)- 1422 § 2

K0(1430)- 247 § 3

* D. Aston, et.al., NP:B296, 493 (1988)

BUT 2/ = 10,429/8,585 = 1.21

Very poor 2 probability !

D0 Mixing and D0 Ks+-

Mixing

D0

_

- This (self-conjugate) channel provides information from the time-dependence of on D0-D0 mixing parameters xD, yD and q/p.
- Each point on the DP can be reached either by direct decay OR by mixing followed by decay
- The plot shows the mean decay time as function of position in the plot.

Note, for theWS K*band, where mixing is a significant contributor and mean decay time is long, this time is BELOW the average.

This results in a mean WS decay time that depends on the time-dependent quantity

Time-dependence of determines xD, yD and q/p

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

With so much data, the “best” fit has very small probability !

Variations in the model lead to no improvement. E.g.:

Several other models also yield comparable 2/ values

Which model (if any) is right ? Which can be ignored ?

Obviously, this impacts seriously on how much we can learn about LQ spectroscopy from HQ decays like these.

A Significant ProblemAll fitsareBAD – some are just more so than others !

The Problem with Significance

Each model also produces a different set of values for D0 mixing parameters xD, yD , |q/p| and Arg{q/p}.

Creates an Irreducible model uncertainty (IMU) ~10-4 in xD and yD .

This channel is also used to measure CKM (see Tim Gershon’s talk). In this case IMU ~ §40 in .

Neither effect is comparable to statistical uncertainty yet, but LHCb and SuperB/Belle 2 will alter that.

The more data we have, the less we realize we know !!

- Either we need a better way to model
- CLEO-c, BES3, SuperB
- Or we must measure amplitudes in a more model-independent way.

CPV Tests – D0-+0

384 fb-1

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

CPV in D0-+0 and K-K+0

Potential for extended search within these 3-body modes:

CPV is unlikely to be in all channels – perhaps in one

Search each channel - e.g. D0 0 + 0

Each channel can be normalized to whole Dalitz plot.

Systematic uncertainties from s+ tagging or from production asymmetries become 2nd order effects

CPV signalled by different D0 and D0 phase behavior

Dalitz plot for these 3-body final states yields information on phase behaviour between channels.

BaBar used three search strategies

Two model-independent searches for CPV in exclusive parts of DP.

A model-dependent search based on fit to the DP distributions

Two Model-Independent Searches for CPV in D0-+0 and K-K+0

_

Dalitz Plots for D 0 and for D 0 are

normalized and differences plotted

Unbiassed frequentist test 16.6%

conf. level there is no difference.

_

Legendre polynomial moments

up to order 8 for D 0 and for D 0 are normalized and compared, in each channel.

Unbiassed frequentist test indicates

23-66% conf. level there are no differences in the various channels.

[+-]+ 0

channel

[+0]+ -

channel

Model-dependent Search for CPV in D0-+0 and K-K+0

384 fb-1

_

Dalitz plots for D0 and for D0 were

fitted to isobar model expansions of

interfering amplitudes in each channel.

Differences (+ for D0 and – for D0) in magnitudes a and phasesfor each of theNR=15 isobars were insignificantly small.

_

Towards Model-Independence - D+ K-++

Much interest in this S-wave dominant channel

In particular, E791 observed a need for “(800)-like” scalar to explain low mass K-+ data

Three QMIPWA fits have now been made

Agreement is quite good – especially the phase in the elastic region.

Notably, 2/ are all of acceptable quality

CLEO-c claim that I=2 (++) amplitude is also required.

A physics issue arising from these fits is why the S-wave phase does not match the LASS I=1/2 data in the elastic region (Watson theorem)

1Phys. Rev. D73, 032004 (2006)

2 Phys.Rev.D78:052001 (2008)

3Phys.Lett.B681:14-21 (2009)

Isobar Model and D+K-++

+or e+e

s

c

K-

_

_

+

d

d

- A physics issue arising from the QMIPWA fits is why the S-wave phase does not match LASS I=1/2 data in the elastic region (Watson theorem). Possible reasons are
- The QMIPWA measures the total K-+ wave that can include an I=3/2 component
- FOCUS has checked for evidence of this, but is inconclusive1.
- The model for the P-wave does not match the LASS I=1/2 data either. The S-wave phase is measured relative to this.
- Perhaps the isobar model, does not work well in this hadronic final state, and three-body scattering is involved.
- We note that recent evidence from D+K-+e+e decays show good agreement with the LASS I=1/2 data
- Suggests that I=3/2 production

is small if external spectator

diagram dominates

1Phys.Lett.B653:1-11,2007

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

K-+ S-wave Phase from D+K -+e+ºe

New data on from Babar extract S-wave phase information in 5-variable fit

Use P-wave model:

K*(890) + K*(1410)

Preliminary

arXiv:1012.1810 (347.5 fb-1)

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

A Way Forward ?

It seems important to improve on parameterization of the reference waves.

For D+K-++, the P-wave should be modelled on LASS data rather than using the RBW model, for example

Is there a way to make a QMIPWA for more than one wave at a time?

Probably, but only if another wave provides a good reference phase

This could work in cases with Bose symmetrization

In their D+K-++, CLEOc claim to have done this, BUT it is difficult to understand what they used as a reference phase.

Summary

Dalitz plots for D decays hold much information on hadron and flavor physics, but we are getting close to the limit of what we can extract from them

It is clear that, with the large data samples expected from LHCb, BES3 and the new B factories, that better models, or new model-independent techniques will be required

Analyses of decays to 4-body states could add much, but they have yet to be tried

This is an area where theorists and experimentalists need to work together.

BW Isobar Model Analysis of D0 KsK-K+

BaBar: Phys.Rev.Lett. 105 081803 (2010)

79,900 Events, 468.5 fb-1.

Isobar Model Parameters:

a0(980)+

and a0(980) Parameters:

a0/f0(980)0

- Primary goal was to determine CKM and D0 mixing parameters
- No distinction between a0/f0(980) was attempted.

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Earlier Moments Analysis of D0 KsK-K+

S

|P|2

S*P

BaBar: Phys.Rev.D72:052008,2005

12,500 Events, 91.5 fb-1.

_

Low KKinvariantmass parts of Dalitz plot allow PWA of S- and P-waves:

|S|2

_

(in KK CMS)

where

Shapes of K+K- and K+K0 are much the same !

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Brian Meadows, U. Cincinnati

S-waves in Heavy Flavor Decays

- The S- wave is both ubiquitous and “useful”
- Interference in hadronic final states through Dalitz plot analyses plays a major role in studying much that is new in flavor physics:
- CKM
- D0-D0 mixing
- Sign of cos2, etc….
- 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 (694-841)-i(300-400) MeV/c2 cites many of the relevant references:

D. V. Bugg, J. Phys. G 34, 151 (2007).

- General belief is that P-andD-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.

Brian Meadows, U. Cincinnati

S-waves in Heavy Flavor Decays

- The S- wave is 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….
- 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 (694-841)-i(300-400) MeV/c2 cites many of the relevant references:

D. V. Bugg, J. Phys. G 34, 151 (2007).

- General belief is that P-andD-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.

Brian Meadows, U. Cincinnati

3-Body Decays of D and B Mesons

- These decays provide information on
- D0-D0 mixing
- CKM
- Sign of cos2, etc….
- Scalar mesons
- General belief is that P-andD-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.

Brian Meadows, U. Cincinnati

L = 0

Phase 0

|T |

Phase degrees

|T |

M (K-p+) (GeV/c2)

M (K-p+) (GeV/c2)

What is Known about K p Scattering ?SLAC/LASS experiment E135:K -p K -p+n (11 GeV/c)

NPB 296, 493 (1988)

+++ Total S-wave

+++I = 1/2

+++I = 3/2

I =3/2 Phase 0

K +p K +p+n

K -p K –p-D++

I- spins are separated using I=3/2 phases from

K +p K +p+n andK -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)

Brian Meadows, U. Cincinnati

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”

Brian Meadows, U. Cincinnati

B. Hyams, et al, NP B64, 134 (1973)

KK

Threshold

Re T

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.

Brian Meadows, U. Cincinnati

02 (degrees)

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 - d02is negative.

Fit assumes amplitude to be unitary:

W. Hoogland, et al, NP B69, 266-278 (1974)

N. Durusoy, et al, PL B45, 517-520 (1973)

Reasonable assumption

up to r±r± threshold

Brian Meadows, U. Cincinnati

Study D Decay Channels withLarge S-wave Component

D + K -++ (shown to right)

Prominent feature:

- Strong asymmetry inK*(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 S–Prelative 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+ -++

Brian Meadows, U. Cincinnati

k(800) in BWM Fit to D+ K-p+p+

E791: E. Aitala, et al, PRL 89 121801 (2002)

Withoutk(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.

Phase 0

Fraction %

S~89 %

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 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

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

Brian Meadows, U. Cincinnati

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-1+2+ decays

Bose

Symmetrization

Decay amplitude :

S-wave (L = 0): ReplaceBWMby discrete pointscne in

P-orD-wave: Define as inBWM

Parameters (cn, n) provide quasi-model independent estimate

of total S- wave (sum of both I- spins).

(S-wave values do depend onP- and D- wave models).

Brian Meadows, U. Cincinnati

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?

Compare QMIPWA with LASS for S-wave|F0 (s) |

arg{F0(s)}

E791

LASS

Not obvious if Watson theorem is broken in these decays ?

Brian Meadows, U. Cincinnati

Watson Theorem Applicability or I = 3/2 ?

FOCUS / Pennington: D K-++arXiv:0705.2248v1 [hep-ex] 2007

K-matrix fit using LASS Data

ForI=1/2production vector:

Includes separateI=3/2wave

Bigimprovement in 2.

LASS I=1/2

phase

S- wave phase(deg.)

TotalK-+

S-wave

I =1/2K-+

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 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 (%):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 BWMfit.

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

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 …

QIMPWAfit

BWMfit

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)

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)

- 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 I=2 wave over-constraint?

BUT – only float C(s) for one wave at a time.

Brian Meadows, U. Cincinnati

Data from CLEO c: D -++

arXiv:0704.3965v2 [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.B585:200-212,2004

E. Aitala, et al, PRL 89 121801 (2002)

CLEO c

Brian Meadows, U. Cincinnati

Complex Pole for :

J. Oller: PRD 71, 054030 (2005)

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

arXiv:0704.3965v2 [hep-ex] Jul 20, 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:

- Full recipe includes both weak and strong mixing between and f0(980)

– 7 parameters in all

arXiv:0704.3965v2, 2007

Weakly mixed

Poles and f0(980)

Unitary

. . .+ usual BW terms for f0 (1350) and f0 (1500)

%

%

%

Excellent fit:

Brian Meadows, U. Cincinnati

CLEO c: D -++

arXiv:0704.3965v2 [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

Brian Meadows, U. Cincinnati

This suggests that EITHER

Both S-wave charges + and 0 are from a0(980)

there is no f0(980)

OR

The the a0 and f0(980) have the same line shape

However, if there is no f0(980), then it is difficult to understand why a 3f0(1370) signal is required.

Perhaps, indeed

The a0(980) and f0(980) have the same line shape

(but different couplings to KK, , ) ?

Brian Meadows, U. Cincinnati

Ds -++

f0(980)0

BaBar: Phys.Rev.D79:032003,2009

13,000 Events, 384 fb-1.

Isobar Model Parameters:

and a0(980) Parameters:

- Primary goal was to determine CKM and D0 mixing parameters
- No distinction between a0/f0(980) was attempted.

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Brian Meadows, U. Cincinnati

Ds -++

BaBar: Phys.Rev.D79:032003,2009

13,000 Events, 384 fb-1.

- S-wave amplitude extracted using E791-style MIPWA analysis
- P-wave parametrized as linear combination of RBW’s

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Brian Meadows, U. Cincinnati

Ds K –K ++

f0(980)0

BaBar: Phys.Rev.D79:032003,2009

13,000 Events, 384 fb-1.

Isobar Model Parameters:

and a0(980) Parameters:

- Primary goal was to determine CKM and D0 mixing parameters
- No distinction between a0/f0(980) was attempted.

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Brian Meadows, U. Cincinnati

Moments Analysis in D+ K-K++

K++channel has no resonances

Remove meson in K+K+channel

Allows Legendre polynomial moments analysis inK-+channel free from cross-channel:

where

Focus: hep-ex/0612032v1 (2007)

6400 Events before cut.

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

(in K –+ CMS)

|S|2

|P|2

S*P

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Brian Meadows, U. Cincinnati

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 > 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

Brian Meadows, U. Cincinnati

S- Wave in D+ +K+-

Phys.Lett.B621:72-80,2005

- The FOCUS experiment first pointed out the FB asymmetry in K-p+ system in these decays arising from interference with the S-wave. This was observed to follow the LASS behaviour.

Clearly an interesting way to probe the

K- +S- wave

Brian Meadows, U. Cincinnati

K-+ S-wave Phase from D+K -+e+ºe and D+K -++ Decays

- New data on from Babar extract S-wave phase information in 5-variable fit
- Use P-wave model:

K*(890) [+ K*(1410)]

Brian Meadows, U. Cincinnati

K-K+ S-wave from Ds+K -K+e+e

- Babar data see some evidence for a small S-wave background under the .
- S-wave amplitude information could not be extracted with

Amplitude Analysis Workshop, Trento, Jan 26, 2011

Brian Meadows, U. Cincinnati

Brian Meadows, U. Cincinnati

Some Kp S-wave MeasurementsCompared to LASS Amplitude

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.

Adapted from W.M. Dunwoodie, Workshop on 3-Body Charmless B Decays, LPHNE, Paris, Feb. 1-3, 2006

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.

Brian Meadows, U. Cincinnati

Back Up Slides

Brian Meadows, U. Cincinnati

Efficiency

Efficiency

Efficiency

Taken from flat-generated MC

Fitted to 2-dimensional third order polynomial

c2/dof = 1133/1140

Projections onto m(K-p+) and m(K+K-) look good

Brian Meadows, U. Cincinnati

10

88

Charged (800) ?

Babar: D0 K-K+0

Tried three recipes for K±0S-wave:

- 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

?

11,278 ± 110 events (98% purity)

?

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:

allows Legendre polynomial moments analysis inK-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)

Brian Meadows, U. Cincinnati

Red curves are ±1 bounds on BWM fit.

Black curves are ±1 bounds on QMIPWA fit.

Completely flexibleS-wave changesP-&D-waves.

Compare QMIPWA with BWM Fitarg{F(s)}

S

P

D

E791 Phys.Rev. D 73, 032004 (2006)

(S-wave values do depend on P- and D- wave models).

Brian Meadows, U. Cincinnati

E791 Require s(500) in D+ p-p+p+

E. Aitala, et al, PRL 86:770-774 (2001)

Withouts(500):

- NR ~ 40% dominates
- r (1400) > 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.

Phase 0

Fraction %

With

S~116 %

No

c2/d.o.f. = 0.90 (76 %)

This caught the attention of our theorist friends !

Brian Meadows, U. Cincinnati

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) toKthreshold usingPT

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

Brian Meadows, U. Cincinnati

FOCUS / Pennington: D K-++

arXiv:0705.2248v1 [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

k=1K ; k=2K’

Same as pole

in K-matrix

Brian Meadows, U. Cincinnati

… Is Watson Theorem Broken ?

- E791 concludes:

“If the data are mostlyI= 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.

Brian Meadows, U. Cincinnati

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