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K -  + Scattering from D Decays. Brian Meadows University of Cincinnati. {12}. {23}. {13}. 1. 1. 1. 2. 2. 2. 3. 3. 3. 1. 3. “Traditional” Dalitz Plot Analysis.

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k scattering from d decays

K-+ Scattering from D Decays

Brian Meadows

University of Cincinnati

traditional dalitz plot analysis















“Traditional” Dalitz Plot Analysis
  • The “isobar model”, with relativistic Breit-Wigner (RBW) resonant terms, is 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




R form


D form




traditional e791 dd k k p p
Traditional E791 DD+!KK-p+p+

~138 %

c2/d.o.f. = 2.7

Flat “NR” term does not give

good description of data.


traditional model for s wave e791
“Traditional”  Model for S-wave - E791

~89 %

c2/d.o.f. = 0.73

(95 %)


Mk = 797 § 19 § 42 MeV/c2

Gk = 410 § 43 § 85 MeV/c2

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

e791 wmd model independent partial wave analysis mipwa
E791 (WMD) “Model-Independent”Partial-Wave Analysis (MIPWA)
  • Make partial-wave expansion of decay amplitude in angular momentum of K-+ system produced

D form-factor

  • “Partial Wave:”
  • Describes invariant mass dependence of K-+ system
  • -> Related to K-+
  • scattering


Watson Theorem holds that, up to elastic limit

(K’ threshold for S-wave) K phases same as

for elastic scattering.

watson theorem
Watson Theorem
  • The process P   + c can be thought of as

Borrowed from M. Pennington (hep-ph/0608016)

  • The only channel open below elastic limit is elastic scattering, so  phase is same as for elastic scattering.
  • BUT the interaction between c and P introduces overall phase

This might also depend on energy, in which case Watson theorem will not apply.



means on

mass shell.

  • Define S–wave amplitude at discrete points sK=sj. Interpolate elsewhere.

 model-independent - two parameters (ccj, j) per point

  • P- and D-waves are defined by known K* resonances

and act as analyzers for the S-wave.

  • Phases are relative to K*(890) resonance.
  • Un-binned maximum likelihood fit:
    • Use 40 (cj, j) points for S
    • Float complex coefficients of KK*(1680) and K2*(1430) resonances
      • 4 parameters (d1680, 1680) and (d1430, 1430)

! 40 x 2 + 4 = 84 free parameters.

mipwa e791 mass distributions
MIPWA – E791 Mass Distributions


15,079 signal events

94% purity

2/NDF = 272/277 (48%)



watson theorem a direct test
Phases for S-, P- and D-waves are compared with measurements from LASS.

S-wave phase S for E791 is shifted by -750 wrt LASS

fs energy dependence differs below 1100 MeV/c2.

P-wave phase does not match very well above K*(892)

Probably artifact of model used

Lower arrow is at  threshold

Upper arrow is at effective limit of elastic scattering observed by LASS.

Watson Theorem - a direct test ?

Elastic limit Kh’ threshold





watson theorem enforced for s wave
A good fit was also made by constraining the shape of the S-wave phase to agree with that from K-+ scattering


S-wave phase S for E791 still shifted by -750 wrt LASS

fP match is even worse above K*(892)

fD phase also shifts.

Watson Theorem Enforced for S-wave


Elastic limit

Kh’ threshold

(1454 MeV/c2)



the babar sample of d k
The BaBar Sample of D+K-++
  • Skim carried out byRolf Andreassen.
  • A likelihood is based on PDFS (signal - MC) and PDFB (background - data sidebands) for each of the following quantities:
    • SignedD+ decay length l/sl
    • c2 probability for vertex
    • PLAB for D+
  • Likelihood is product:

Skim all with L>2

rolf andreassen s skim
Rolf Andreassen’s Skim
  • K-p+p+ invariant mass vs. likelihood (L)

(NOTE log scale).

  • Some purities:
d k dalitz plot
D+ K-++ Dalitz Plot
  • Plot includes 500K events
    • of which 13K are background.
  • Obviously large S-wave content

Interferes with K*(890) (and anything else in P-wave).

  • Some D-wave also present
max likelihood fit
Max. Likelihood Fit
  • Likelihood function covers 3-dimensions:
    • sK1, sK2 and also the reconstructed 3-body mass MK
    • Factorize MK dependence:
  • All events used in signal as well as sidebands have a D+ mass constraint.
    • Makes it possible to overlay Dalitz plot for sideband data directly on signal
    • Greatly simplifies computation of efficiency.
  • is efficiency

Subscript s is signal

Subscript b is background

background model
Background Model
  • K-p+p+ invariant mass distribution from sample with L > 3
  • Dalitz plot distributions in lower side-band, signal region and upper side-band (log. Scale)
  • Used directly as input to background function.

PDF1b - bin-by-bin interpolation

second background
Second Background
  • Probable origin
  • PDF2b

= g(MK) x Gauss (M2K)


  • Efficiency (%) over the Dalitz plot for various laboratory momentum ranges.
efficiency vs p lab
Efficiency vs. pLAB
  • Efficiency (%) vs laboratory momentum.
  • Lab. momentum for Data (black).
    • Lab. momentum for reconstructed, signal MC (red).

 No need to use efficiency as function of pLAB

traditional model for s wave babar
“Traditional”  Model for S-wave - BaBar

2/NDF = 20.1x103 / 15.6x103

- a very poor fit

partial waves from model fit
Partial Waves from  Model Fit



NOTE – no K*(1410)

Width of lines

represents 1

e791 s wave fit on babar data
E791 S-Wave Fit (on BaBar data)
  • S-wave is spline with 30 equally spaced points
  • P-wave is as in  model fit, with complex coefficients floated.
  • D-wave also as in  model fit – complex coefficient floated.
e791 s wave fit on babar data1
E791 S-Wave Fit (on BaBar data)

2/NDF = 1007/574

– better, but still a poor fit

the p wave problem
The P-wave Problem
  • Problem is – the S-wave solution depends on assumptions made about the reference P-wave.
  • Models:
    • Add K1*(1410) RBW - this crowds the wave - SHOWN HERE
    • Could use K-matrix to avoid this – TO BE TRIED
    • Use LASS phases up to elastic limit (~110 MeV/c2) – TO BE TRIED

BUT these all just transfer the ignorance.

  • Parametrize as table of complex values (spline) as for S-wave:
    • Sn(s) = splinen(s) - spline defined by n points.
    • Pm(s) = RBW[K*(890)] x splinem(s)
  • Not much progress on this yet.

 Uniqueness problem ? – can it be done at all?

add k 1410 to p wave babar
Add K*(1410) to P-wave - BaBar

2/NDF = 18.8x103 / 15.5x103

– Better, but still a very poor fit

spline for p wave fit procedures
Spline for P-wave - Fit Procedures
  • Make E791 fit
    • Sn(s) from a table of n points.
  • Fix S-wave and fit P-wave same way
    • Pm(s) from a table of n points.
  • Fix P-wave and re-fit S-wave
  • Repeat cycle several times
    • Errors from likelihood scan
  • Alternatives
    • Cycle Magnitudes (both waves) and phases (both waves).
    • Use Binned likelihood – WORKS WELL
    • 2 fit. – WORKS LESS WELL

Antimo’s procedure

antimo s result


S phase


P phase



Antimo’s Result
some mc tests
Some MC Tests
  • Generate toy MC corresponding to  model fit to data
    • No background
  • Look for self-consistency between fit and generated quantities
kappa model test
Kappa Model Test



mc test s wave only
MC Test – S-wave Only
  • S-wave is fitted tospline with 40 equally spaced points
  • P-wave is fixed as in  model fit (but defined as a spline).
  • D-wave complex coefficient floated.
mc test p wave only
MC Test – P-wave Only
  • S-wave is fixed as in  model fit.
  • P-wave is fitted tospline with 40 equally spaced points
  • D-wave complex coefficient floated.
mc test migrad s p cycles
MC Test – Migrad S-P Cycles

Cycling does work, but convergence seems far away even after 16 cycles!






# Function Calls

mc test magn phase cycles






# Function Calls

MC Test –Magn./Phase Cycles

Cycling does work, but convergence seems far away even after 16 cycles!

but float s and p waves together
BUT – Float S- and P-waves Together !!

Maybe it is not possible tofindboth S- and P-wave amplitudes without a definite form for one of them.

mipwa for both s and p waves
MIPWA for Both S- and P-waves?
  • Regions exist where the P-wave is much smaller than the S-wave

This makes phase measurements more difficult to make

  • So far, we have gone as far as E791 did, but the next steps need some more work.
  • We conclude that the S-wave can be well determined if the P-wave is known
  • Understanding the P-wave is a challenging problem.
  • New ideas how to parametrize the P-wave need be considered.