Analysis of E852 Dataat Indiana University

Ryan Mitchell

PWA Workshop

Pittsburgh, PA

February 2006

Outline

I. Overview of the Indiana 3π Analysis

II. PWA Formalism and Fitting Techniques

III. Limitations of the Current Method: 3 Case Studies

1pπππpπρpXpπ

pπππpπρpXpπ000

0

2005 Indiana 3π Analysis

PWA of two charge modes using 1995 E852 data:

1. “Charged Mode”π−p→π+π−π−p (2.6M events, after cuts)

2. “Neutral Mode”π−p→π−π0π0p (3.0M events, after cuts)

2pπππpfπpXpπ

pπππpfπpXpπ00

Isospin relationsprovide powerfulcross checks.

Raw 3π Mass Distributions

Raw Distributions After Cuts(no acceptance corrections)

a1

a2

π2

π2

a2

a1

f2(1270)

ρ(770)

3π mass distributionsshow the familiar a1(1260) a2(1320) π2(1670)resonances.

Dalitz plots showevidence ofisobar production.

Charged Mode Neutral Mode

f2(1270)

Setting up the Partial Wave Analysis

k

*k,k,

*k,k,

2

kk,k,tM, )(A)(AVV)(AV )(I

Expand the “angular distribution” in each mass and t bin into partial waves:

basisfunctions

complex fit parameters

incoherent sum(reflectivity)

coherent sum(JPC, isobars, M, L)

kinematics(angles,

2π masses)

Divide the data into bins of mass and t:

67 25MeV

Mass Bins

67 Mass Bins800 – 2500 MeV/c2

13 t Bins0.08 – 0.58 GeV2/c2

13 t Bins (1 standard)

Events in a t Bin~100’s of 1000’s

Events in a Fit~1’s to 10’s of 1000’s

PWA Results: Dominant Waves

Charged Mode Neutral Mode

2++1+(ρ)S 2++1+(ρ)S

1++0+(ρ)S 1++0+(ρ)S2−+0+(f2)S 2−+0+(f2)S

a2(1320)

a1(1260)

π2(1670)

• 35 Wave Fitº 21 Wave Fit

All Waves All Waves

PWA Results: A Minor Wave

Charged Mode Neutral Mode

4++1+(ρ)G 2++1+(ρ)S

a2(1320)a4(2040)

• 35 Wave Fitº 21 Wave Fit

2++1+(ρ)S 4++1+(ρ)G

Notice the difference in scales.(About a factor of 40)

Comparison of 2−+ and Exotic 1−+ Waves

• 35 Wave Fitº 21 Wave Fit

1−+1+(ρ)P 1−+1+(ρ)P1−+M−(ρ)P 1−+M−(ρ)P

2−+1+(ρ)F 2−+1+(ρ)F

1670 MeV/c2

Charged Mode

When additional 2−+ waves are not included,the intensity appears in the 1−+ waves.

2−+0+(ρ)F 2−+0+(ρ)F

Neutral Mode

II. PWA Formalism andFitting Techniques

Expanding the Intensity in Bins of M3π and t (and s):

2

kk,k,t,M )(AV )(I

3π

“DecayAmplitudes”(basis states)

“ProductionAmplitudes”(complex fit parameters)

incoherent sum

(spin flip,reflectivity,background,

etc.)

coherentsum(JPC,

isobars,etc.)

Kinematics(angles,

isobar masses,etc.)

CHOICES:

1. How to write A(Ω)?

2. Which terms addcoherently and whichincoherently?

3. Which and How Many terms are included?

Extensions

1. Don’t bin in M3π and/or t.

2. Include fit parameters inside A(Ω).

A(Ω) for 3π in the Isobar Model

r)L(-M)(isobaJ

MJ

M(isobar)LJ(isobar)LMJ

PC

PCεPC

A1)εP(

AΘ(M)A

0M0

0M1/2

0M1/2

Θ(M)

J,M SL

1. Each wave is characterized by JPCMε(isobar)L:

J: Spin of resonanceP: Parity of resonanceC: C-Parity of resonanceM: z projection of Jε: ReflectivityL: Orbital angular momentum of the resonance decayS: Spin of the Isobarθ1,φ1: Decay angles of the resonance (Gottfried- Jackson)θ2,φ2: Decay angles of the isobar (Helicity)FL(p1): Barrier factor for resonance decayFS(p2): Barrier factor for isobar decayBW(isobar): Breit-Wigner with isobar parameters

3. Transform to the “reflectivity” basis:

isobar)(BW)p(F)p(F

)0,,(D)0,,(DJ;SL0,

1)S2(1)L2()(A

2S1L

22*011

*JM

1/21/2

M(isobar)LJPC

S

2. Add a term for identical pions.

A(Ω) for 3π in the Isobar Model (pictures of angular projections)

The Likelihood Fit

n

1i

inμ

dΩΩηΩI

)I(Ω

n!

μeL

2

kk,k,tM, )(AV )(I

dΩΩηΩIμ

n

1ii dΩΩηΩI2))ln(I(Ω22ln(L)

Perform a likelihood fit in every bin of mass and t:

Minimize this function:This term is modifiedby acceptance

n: observed number of events in this binμ: expected number of events in this bin:

η(Ω): Acceptance

Likelihood Fit (acceptances)

k α α

N

1ii

*αk,iαk,

GEN

*αk,αk,

N

1i k

2

αiαk,αk,

GEN

N

1ii

GEN

ACC

ACC

ACC

ΩAΩAN

4πVV

ΩAVN

4π

ΩIN

4πdΩΩηΩI

NGEN: Generated MC Events (flat in angles)

NACC: Accepted MC Events

NDATA: Observed Data Events

“Normalization Integrals”

Minor note:V is rescaled during the fit:

ACC

GENDATA

N

NNVV

One-time sumover MC events.

MC

Master

Distributed data

Gather partial NI’s

NI’s

Slaves calculate amplitudes “on the fly” and evaluate partial contributions to normalization integrals

1. Normalization Integrals

Master Fitted

parameters

At every iteration of minimization the master sends the current parameters, and the slaves calculate the likelihood and send the result back to the master

2. The PWA Fit

Minuit runs on master

Data

GatherpartialLikelihood

NI’s

n

1ii dΩΩηΩI2))ln(I(Ω22ln(L)

BIG speed increase when data is CACHED.

Viewing PWA Results

dΩΩAV2

Dρ12Dρ12

Shown is:

in each mass bin and in a given bin of t.

III. Limitations of the Current Method

Three Case Studies:

1. The Effects of Barrier Factors

2. Incorporating the Deck Effect

3. Normalization Integral Files

Case I. The Effect of Barrier Factors

r)L(-M)(isobaJ

MJ

M(isobar)LJ(isobar)LMJ

PC

PCεPC

A1)εP(

AΘ(M)A

0M0

0M1/2

0M1/2

Θ(M)

J,M SL

1. Each wave is characterized by JPCMε(isobar)L:

J: Spin of resonanceP: Parity of resonanceC: C-Parity of resonanceM: z projection of Jε: ReflectivityL: Orbital angular momentum of the resonance decayS: Spin of the Isobarθ1,φ1: Decay angles of the resonance (Gottfried- Jackson)θ2,φ2: Decay angles of the isobar (Helicity)FL(p1): Barrier factor for resonance decayFS(p2): Barrier factor for isobar decayBW(isobar): Breit-Wigner with isobar parameters

3. Transform to the “reflectivity” basis:

isobar)(BW)p(F)p(F

)0,,(D)0,,(DJ;SL0,

1)S2(1)L2()(A

2S1L

22*011

*JM

1/21/2

M(isobar)LJPC

S

2. Add a term for identical pions.

What is the systematiceffect of this?

Case I. I Just Want to Modify a Barrier Factor

??

The code must be moretransparent.

We need more flexibilityin defining amplitudes:

• This is the HEART of the PWA.

• PWA should not be a physics black box.

• We need quick answers to simple questions of systematics.

Limitation 1.Black Box Physics

Case II: The Deck Effect

Proton

Pomeron

π π

π

πf2π

The 2-+0+(f2)S and 2-+0+(f2)D waves in 3π.

Why are they shifted in mass?

Fit with one resonance and the Deck Effect.

Preliminary work byAdam Szczepaniakand Jo Dudek.

Mass ≈ 1740 MeV/c2

“π2(1670)”

Case II. The Deck Effect

• Non-resonant amplitudes should be part of the PWA instead of being extracted from resonant amplitudes.

Proton

Pomeron

π π

π

πf2π

Limitation 2a:No Easy Way for “Users”to Define Amplitudes.

• Amplitudes like Deck could be more effective with fit parameters inside the amplitude.

Limitation 2b:Fit Parameters Always Multiplythe “Decay Amplitude”.

Case III. Normalization Integral Files

• When the data selection cuts change, or

the form of an amplitude changes, or

a Monte Carlo file changes, then

Normalization Integral files must be regenerated.

• With many versions of normalization integral files,

Organization becomes difficult.

• In November 2005, wrong normalization integral files set back the 3π analysis several weeks.

Limitation 3:Handling multiple NI fileseasily leads to confusion.

• Needs Improvement:– Black Box Physics

Separate physics and

computer science.

– Only Production Amplitudes in PWA Fit

– Normalization Integral Files

– Documenting Fits (CMU Database?)

– Viewing Results, Viewing correlations, etc.

Summary of Indiana PWA Experience

• Good Things:– Parallel Processing

– Caching Amplitudes

OTHER TALKS:

Matt: PWA Framework

Scott: Documentation

Adam: Phenomenology