Analysis of E852 Data at 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

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 − π 0 π 0 p (3.0M events, after cuts) Isospin relations provide powerful cross checks.

Raw 3π Mass Distributions

Raw Distributions After Cuts (no acceptance corrections) a1a1 a2a2 π2π2 π2π2 a2a2 a1a1 f 2 (1270) ρ(770) 3π mass distributions show the familiar a 1 (1260) a 2 (1320) π 2 (1670) resonances. Dalitz plots show evidence of isobar production. Charged Mode Neutral Mode f 2 (1270)

Setting up the Partial Wave Analysis Expand the “angular distribution” in each mass and t bin into partial waves: basis functions 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 Bins 800 – 2500 MeV/c 2 13 t Bins 0.08 – 0.58 GeV 2 /c 2 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 ModeNeutral Mode 2++1+(ρ)S 1++0+(ρ)S 2−+0+(f 2 )S a 2 (1320) a 1 (1260) π 2 (1670) 35 Wave Fit º 21 Wave Fit All Waves

PWA Results: A Minor Wave Charged ModeNeutral Mode 4++1+(ρ)G2++1+(ρ)S a 2 (1320) a 4 (2040) 35 Wave Fit º 21 Wave Fit 2++1+(ρ)S4++1+(ρ)G Notice the difference in scales. (About a factor of 40)

Comparison of 2−+ and Exotic 1−+ Waves 35 Wave Fit º 21 Wave Fit 1−+1+(ρ)P 1−+M−(ρ)P 2−+1+(ρ)F 1670 MeV/c 2 Charged Mode When additional 2−+ waves are not included, the intensity appears in the 1−+ waves. 2−+0+(ρ)F Neutral Mode

II.PWA Formalism and Fitting Techniques

Expanding the Intensity in Bins of M 3π and t (and s): “Decay Amplitudes” (basis states) “Production Amplitudes” (complex fit parameters) incoherent sum (spin flip, reflectivity, background, etc. ) coherent sum (J PC, isobars, etc.) Kinematics (angles, isobar masses, etc.) CHOICES: 1. How to write A(Ω)? 2. Which terms add coherently and which incoherently? 3.Which and How Many terms are included? Extensions 1.Don’t bin in M 3π and/or t. 2.Include fit parameters inside A(Ω).

A(Ω) for 3π in the Isobar Model J,M S L 1. Each wave is characterized by J PC M ε (isobar)L: J: Spin of resonance P: Parity of resonance C: C-Parity of resonance M: z projection of J ε: Reflectivity L: Orbital angular momentum of the resonance decay S: Spin of the Isobar θ 1,φ 1 : Decay angles of the resonance (Gottfried- Jackson) θ 2,φ 2 : Decay angles of the isobar (Helicity) F L (p 1 ): Barrier factor for resonance decay F S (p 2 ): Barrier factor for isobar decay BW(isobar): Breit-Wigner with isobar parameters 3. Transform to the “reflectivity” basis: 2. Add a term for identical pions.

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

The Likelihood Fit Perform a likelihood fit in every bin of mass and t: Minimize this function: This term is modified by acceptance n: observed number of events in this bin μ: expected number of events in this bin: η(Ω): Acceptance

Likelihood Fit (acceptances) N GEN : Generated MC Events (flat in angles) N ACC : Accepted MC Events N DATA : Observed Data Events “Normalization Integrals” Minor note: V is rescaled during the fit: One-time sum over 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 Gather partial Likelihood NI’s BIG speed increase when data is CACHED.

Viewing PWA Results 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 J,M S L 1. Each wave is characterized by J PC M ε (isobar)L: J: Spin of resonance P: Parity of resonance C: C-Parity of resonance M: z projection of J ε: Reflectivity L: Orbital angular momentum of the resonance decay S: Spin of the Isobar θ 1,φ 1 : Decay angles of the resonance (Gottfried- Jackson) θ 2,φ 2 : Decay angles of the isobar (Helicity) F L (p 1 ): Barrier factor for resonance decay F S (p 2 ): Barrier factor for isobar decay BW(isobar): Breit-Wigner with isobar parameters 3. Transform to the “reflectivity” basis: 2. Add a term for identical pions. What is the systematic effect of this?

Case I. I Just Want to Modify a Barrier Factor ?? The code must be more transparent. We need more flexibility in 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 πf2f2 π 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 by Adam Szczepaniak and Jo Dudek. Mass ≈ 1740 MeV/c 2 “π 2 (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 πf2f2 π 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 Multiply the “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 files easily 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