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K -  + Scattering from D +  K -  +  +. Antimo Palano INFN, Bari R. Andreassen, Brian Meadows University of Cincinnati D. Aston, J. Coleman, W. Dunwoodie, K. Suzuki, D. Leith Group B, SLAC. “Traditional” Dalitz Plot Analysis.

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k scattering from d k

K-+ Scattering from D+ K-++

Antimo Palano

INFN, Bari

R. Andreassen, Brian Meadows

University of Cincinnati

D. Aston, J. Coleman, W. Dunwoodie, K. Suzuki, D. Leith

Group B, SLAC

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

{12}

{23}

{13}

NR

1

1

1

2

2

2

3

3

3

1

3

2

NR

Constant

Form

factor

D form

factor

spin

factor

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

Conserves

spin

Watson Theorem holds that, up to elastic limit

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

for elastic scattering.

mipwa a la wmd e791
MIPWA (a la WMD / E791)

`

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

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

  • P- and D- waves are parametrized by known K* resonances
unbinned max likelihood fit
Unbinned Max. Likelihood Fit
  • Likelihood function covers 3-dimensions:
    • s(K1), s(K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.
    • Can then overlay Dalitz plot for sideband data directly on signal
    • Greatly simplifies computation of efficiency.
  • is efficiency

Subscript s is signal

Subscript b is background

e791 mipwa of k p s wave
E791 MIPWA of K-p+S-wave
  • What does this show ?

“It helps reveal a k pole hidden by an Adler zero”

- D. Bugg et al.

compare e791 k p s wave with e791
Compare E791 K-p+S-wave with E791
  • Published results are multiplied by the scalar D form-factor used in the analysis.

Features similar to previously published k model fit”

compare e791 k p s wave with lass
Compare E791 K-p+S-wave with LASS
  • Multiply LASS result by phase space factorMKp / p
  • Shift LASS phasedownward700
  • Good agreement above ~ 1150 MeV/c2 !
  • Is there an S-wave D+ (complex) form-factor ?
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

d k dalitz plot
D+ K-++ Dalitz Plot
  • Plot includes 500K events (~97% purity)
    • ~13K are background.
  • Obviously large S-wave content

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

  • Some D-wave also present

L > 3

Purity

97%

three isobar model fits to babar data
Three Isobar Model Fits to BaBar Data

This is one

2/NDF = 1.3 (NDF=15,600)

- very poor fit

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.
slide13

K*(1410)

K*(1677)

K*(1677)

K*(1410)

  • A substantial problem is that we have to assume a form for the P-wave that we know is not correct
    • It contains no K*(1410) since E791 data did not require this
    • It is a sum of BW’s – known to violate unitarity (at least).
  • So, can we find the P-wave the same way ?
spline model for p wave too
Spline Model for P-wave Too
  • Antimo tried this (see BAD 1291):
  • Fix P-and D-waves as in the  (isobar) model
    • S(s) from a table of n points.
  • Now fix S-wave and fit P-wave same way
    • P(s) from a table of n points.
  • Fix P-wave and re-fit S-wave
  • Repeat cycle several times
    • SIMPLEX
    • Errors from likelihood scan
antimo s result 2 cycles

|S|

S phase

S

P phase

|P|

P

Antimo’s Result (2 cycles)

Im S

See BAD 1293

Re S

Im P

Re P

  • It is difficult to know if this has converged
    • or if it is correct
some mc tests
Some MC Tests
  • Generate 3 toy MC samples, each corresponding to isobar model fits actually made to the BaBar data

MCA: S-wave: , K*(1430)

P-wave: K*(890), K*(1410), K*(1688)

D-wave: K*(1420)

MCB: As in MCA, but no D-wave K*(1420)

MCC: As in MCA, but no P-wave K*(1410)

  • Each sample:
    • No background
    • ~4M events
  • Look for self-consistency between fit and generated quantities.
mc test s wave only
MC Test – S-wave Only

Almost

perfect fit

  • 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.

Almost

perfect fit

Fixed

Fixed

Almost

perfect fit

Almost

perfect fit

OK !

mc test p wave only
MC Test – P-wave Only
  • S-wave is fixed as in MCA model fit.
  • P-wave is fitted tospline with 40 equally spaced points
  • D-wave complex coefficient floated.

Fixed

Fixed

Almost

perfect fit

Almost

perfect fit

Almost

perfect fit

Almost

perfect fit

OK !

migrad cycle s then p wave
Migrad (Cycle S- then P-wave)

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

S

Goes on

forever …?

-2lnL

P

S

etc

# Function Calls

mc test magn phase cycles

Mag

-2lnL

Phase

Mag

etc

# Function Calls

MC Test –Magn./Phase Cycles

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

Goes on

forever …?

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 ??

so read the directions
So – “Read the directions …”
  • Clearly this is not working!
  • It also takes far too long as it stands even on 3 GHz CPU:
    • 8 hours for unbinned fit with1M events and just one wave (30 points)
    • ~7 days for fit just shown !

Therefore:

  • Try to understand how this ought to work …
  • Then improve performance of the fit.
how does the mipwa work
How Does the MIPWA Work?
  • With structure only in ONE channel (e.g. K-+1) then the density along strips like that shown varies like

A + B cos  + C cos2 

  • So we can determine

A ~ |S|2

C ~ |P|2

and B ~ |S||P| cos (fs-fP )

i.e., we measure difference|fs-fP|

- ambiguous sign

Must knoweithersor Pto find the other phase (with ambiguity)

S and P only

cos 

mipwa for both s and p waves
MIPWA for Both S- and P-waves?

In this region, below the

K*(890) peak, we know

P well:

So we measure

Both S and P

over this range

Re{P}

?

s

Im{P}

mipwa for both s and p waves1
MIPWA for Both S- and P-waves?

Build on this region

in horizontalstrips.

Measures S and P

over this range

  • [Structure in cross channels:
  • complicates cos 

dependence on each

strip

  • but can be

measured in this

channel. ]

mipwa for both s and p waves2
MIPWA for Both S- and P-waves?

Bose symmetrize

Know S and P

everywhere except

in the “Gap”

Gap

improve the fit performance
Improve the Fit Performance
  • Unbinned fit  Binned fit
    • Choose 600x600 bins
  • Poisson probability

Define

where

is expected number of events in bink

observed number of events in bink

PDF includes signal & background

is normalization (a fit parameter).

If bins are small enough, takes care of normalization !!

improve the fit performance1
Improve the Fit Performance
  • Work with square plot
    • S(K- +1) vs. cos  (K- +2)
  • Useful features of this kind of fit:
    • Choosing 600x600 bins (in effect, only about 150K bins)
      • CPU time independent of actual number of events!
    • No need to normalize – this is part of the fit.
    • About 5 times faster for ~ 1 M events.
improve the fit performance2
Improve the Fit Performance
  • Re- parametrize P-wave:
    • Spline  BWK*1430 x spline
  • Use Re A & Im A rather than |A| & 
    • More stable where|A| is small
  • In principal, all fits can now be accomplished in a single step, floating all parameters, in a reasonable time.
start at truth
Start at “Truth”

~4 M events

TRUTH

lnL=-16000316.9, chi/NDF146319.2/145889

different start
Different Start

~4 M events

Starting model:

S-wave: k, K0(1430)

P-wave: K*(890), K1*(1677) (No K*1410)

D-wave: K2*(1420)

lnL=-16000316.9, chi/NDF146314.8/145888

further improvement in fit performance
Further Improvement in Fit Performance
  • Poor performance in 1.0-1.5 GeV/c2 region

Due in part to Representation of S-wave near K*(1430)

Also results from low statistical content of bins in region

  • Re- parametrize S-wave:

Spline  BWK*1430 x spline

  • Also require at least 10 events in each bin
from 3 different starts
From 3 Different Starts:

~4 M events

lnL=-16000287.4, chi/NDF144250.7/145900

summary
Summary
  • This kind of fit is shown to work, in principal
  • By fixing the P-wave up to about 900 MeV/c2 we can now fit S- and P-waves simultaneously.
  • It seems to add stability if we require bins to have at least 10 events in them.

Next:

  • Add background
  • Fit data
  • Systematic / bias studies
  • Finish BAD
scattering a reminder
Scattering aReminder

Im {T}

s

h/2

2d

T

In LASS (s) = 1.0 from threshold up to ~(0.9 –1.0) GeV/c2

x

f

Re {T}

But this may differ in

D or B decays

Threshold

k p s p waves from lass

S

S

K-p+S- & P-waves from LASS

|S|

|S|

K’

threshold

K*(1410)

K*(1677)

K*(1677)

K*(1410)

partial waves from model fit
Partial Waves from  Model Fit

Phase

Magnitude

NOTE – no K*(1410)

Width of lines

represents 1

omit the gap region
Omit the “Gap” Region

~4 M events

TRUTH

lnL=-16000287.1, chi/NDF146240.5/145890

different start and omit gap
Different Start (and Omit “Gap”)

~4 M events

Starting model:

S-wave: k, K0(1430)

P-wave: K*(890), K1*(1677) (No K*1410)

D-wave: K2*(1420)

lnL=-16000287.4, chi/NDF146250.7/145900

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.

FD

FR

means on

mass shell.

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)

Lost

efficiency
Efficiency
  • 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

e791 s wave fit on babar data1
E791 S-Wave Fit (on BaBar data)

2/NDF = 1007/574

– better, but still a poor fit

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