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Low Mass S -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

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Low mass s wave k and systems
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

Brian Meadows, U. Cincinnati

S waves in heavy flavour physics
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  (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-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

What is known about k p scattering

L = 0

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
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.


  • 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

P s wave scattering i 0

Im T

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



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



M(pp) (MeV/c2

No evidence for s(500) – essentially no data below 500 MeV/c2 either.

Brian Meadows, U. Cincinnati

P s wave scattering i 2 from n achasov and g shestakov prd 67 243 2005


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

How this should apply to 3 body d decays
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-spinnot conserved

  • kscattering on +during

  • fragmentation can impart

  • an overall phase

D +










 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.

Brian Meadows, U. Cincinnati

Conventional approach breit wigner model bwm
Conventional Approach – Breit-Wigner Model “BWM”

  • The “isobar model” ignores all this, and problems of double-counting:

  • Amplitude for channel {ij}with angular momentumL:

  • In the BWM each resonance “R” (mass mR, width R) described as:

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


















NR - constant


D form




R form


Brian Meadows, U. Cincinnati

Study d decay channels with large s wave component
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



M 2(K -+)


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
k(800) in BWM Fit to D+ K-p+p+

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


  • NR ~ 90%

  • Sum of fractions 130%

  • Very Poor fit (10-5 %)


  • 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 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 independent partial wave analysis qmipwa
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): ReplaceBWMby discrete pointscne in

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

Compare qmipwa with lass for s wave

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) |




Not obvious if Watson theorem is broken in these decays ?

Brian Meadows, U. Cincinnati

Watson theorem breaking vs i 3 2
Watson Theorem Breaking vs. 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


S- wave phase(deg.)



I =1/2K-+


Large Data sample:

52,460 § 245 events (96.4% purity)

s 1/2 (GeV/c2)


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
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 k1
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 …



Brian Meadows, U. Cincinnati

Total s wave from d k decays
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 k2
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

New data from cleo c d
New 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)


Brian Meadows, U. Cincinnati

Complex pole for
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
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)


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




Excellent fit:

Brian Meadows, U. Cincinnati

Cleo c d
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

Moments analysis in d k k


Moments Analysis in D+ K-K++

Focus: hep-ex/0612032v1 (2007)

  • K++channel has no resonances

  • Remove  meson in K+K+channel

  • Allows Legendre polynomial moments analysis inK-+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)




Brian Meadows, U. Cincinnati

S wave in b j k
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
S- Wave in D+ +K+-

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


… and friends?

Clearly an interesting way to probe the

K- +S- wave

Brian Meadows, U. Cincinnati

Some k p s wave measurements compared to lass amplitude
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.

(More coming for LP07)

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

Brian Meadows, U. Cincinnati


  • 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

Charged 800
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 d 0 k k 0
    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)


    • |S| consistent with either

    • a0(980) or f0(980) lineshapes.

    Babar: 385 fb-1: PRC-RC 76, 011102 (2007)

    Brian Meadows, U. Cincinnati

    Compare qmipwa with bwm fit

    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 Fit





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

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


    • NR ~ 40% dominates

    • r (1400) > r (770) !!

    • Very Poor fit (10-5 %)


    • 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
    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) toK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:


    T- pole is at: 1.408 – i 0.011 GeV/c2

    Brian Meadows, U. Cincinnati

    Focus pennington d k1
    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
    … 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