K -  + Scattering from D Decays

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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

University of Cincinnati

{12}

{23}

{13}

1

1

1

2

2

2

3

3

3

1

3

• 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

NR

2

NRConstant

R form

factor

D form

factor

spin

factor

~138 %

c2/d.o.f. = 2.7

Flat “NR” term does not give

good description of data.

Phys.Rev.Lett.89:121801,2002

“Traditional”  Model for S-wave - E791

~89 %

c2/d.o.f. = 0.73

(95 %)

Probability

Mk = 797 § 19 § 42 MeV/c2

Gk = 410 § 43 § 85 MeV/c2

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

• 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

ML(p,q)

Watson Theorem holds that, up to elastic limit

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

for elastic scattering.

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.

MIPWA
• 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.

MIPWA
• 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

E791

15,079 signal events

94% purity

2/NDF = 272/277 (48%)

S

Phys.Rev.D73:032004,2006

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

S

P

K1*(1410)

D

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

However:

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

S

Elastic limit

Kh’ threshold

(1454 MeV/c2)

P

D

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
• K-p+p+ invariant mass vs. likelihood (L)

(NOTE log scale).

• Some purities:
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
• 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
• 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
• Probable origin
• PDF2b

= g(MK) x Gauss (M2K)

Lost

Efficiency
• Efficiency (%) over the Dalitz plot for various laboratory momentum ranges.
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

2/NDF = 20.1x103 / 15.6x103

- a very poor fit

Partial Waves from  Model Fit

Phase

Magnitude

NOTE – no K*(1410)

Width of lines

represents 1

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 data)

2/NDF = 1007/574

– better, but still a poor fit

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

2/NDF = 18.8x103 / 15.5x103

– Better, but still a very poor fit

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
• SIMPLEX
• 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

|S|

S phase

S

P phase

|P|

P

Antimo’s Result
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

Output

Input

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
• 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

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

S

-2lnL

P

S

etc

# Function Calls

Mag

-2lnL

Phase

Mag

etc

# 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 !!

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?
• Regions exist where the P-wave is much smaller than the S-wave

This makes phase measurements more difficult to make

Summary
• 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.