PWA with full rank density matrix of the + and 0 0 systems - PowerPoint PPT Presentation

Preface PWA method Fit results Conclusions Backup slides PWA with full rank density matrix of the π + π − π − and π − π 0 π 0 systems at VES setup Igor Kachaev, Dmitry Ryabchikov for VES group, Protvino, Russia Institute for High Energy Physics, Protvino. E-mail Igor.Katchaev@ihep.ru MESON 2016 Krakow, Poland 3 June 2016

Preface PWA method Fit results Conclusions Backup slides Preface Comparison of two final states permit us to know better: • isospin relation between these states • what is resonant and what is not • bugs in hardware, methods, programs... • PWA is model dependent. It is useful to compare one model for two systems and two model (full rank and rank 1) for the same system. We have: • full featured magnetic spectrometer with 29 GeV/c π − beam, Be target, | t ′ | = 0 . . . 1 GeV 2 /c 2 • 20 · 10 6 events in π − π 0 π 0 (leading statistics in the world) • 30 · 10 6 events in π + π − π − (next to leading) • Analysis is done for | t ′ | = 0–0.03–0.15–0.30–0.80 GeV 2 /c 2

Preface PWA method Fit results Conclusions Backup slides Raw data π Mass(3 ) T prime Acceptance 0.4 2000 90000 π K->3 6 10 1500 0.35 80000 1000 0.3 70000 500 5 10 60000 0.25 0 0.4 0.45 0.5 0.55 0.6 50000 0.2 4 40000 10 0.15 30000 0.1 20000 3 10 For |t'| < 0.03 0.05 10000 0 0 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 π GeV/c 2 2 M(3 ) GeV/c GeV /c Here and below: Blue line - π + π − π − Red line - π − π 0 π 0

Preface PWA method Fit results Conclusions Backup slides PWA method. Partial waves π + π - π - R( ππ ) X π - p p PWA amplitudes are constructed using isobar model, sequential decay via ππ subsystem. Wave has quantum numbers J P LM η R where J P is spin-parity for 3 π system, M η is its projection of spin and naturality, R is the known resonance in ππ system, L is orbital momentum in R π decay. I G = 1 − is implicit for 3 π charged states.

Preface PWA method Fit results Conclusions Backup slides Method details • Amplitudes use d-functions (Hansen, Illinois PWA) • Amplitudes are relativistic (Chung, Filippini) • resonanses are relativistic Breit-Wigners R = f 0 ( 980 ) , ε ( 1300 ) , f 0 ( 1500 ) , ρ ( 770 ) , f 2 ( 1270 ) , ρ 3 ( 1690 ) To describe ππ S -wave we use modified Au, Morgan, Pennnington M-solution with f 0 ( 980 ) withdrawn. We name it ε ( 1300 ) • Notation for some known 3 π resonances: a 1 ( 1260 ) — 1 + S 0 + ρ ( 770 ) , a 2 ( 1320 ) — 2 + D 1 + ρ ( 770 ) , π 2 ( 1670 ) — 2 − S 0 + f 2 ( 1270 ) , π ( 1800 ) — 0 − S 0 + f 0 ( 980 ) • fit parameters are elements of positive definite density matrix. No rank constraints. For small number of waves (where C � = 1, see below) we have 100% coherence.

Preface PWA method Fit results Conclusions Backup slides Extended likelihood function N ev N w � � C k ( i ) R m ( i ) m ( j ) C ∗ k ( j ) M i ( τ e ) M ∗ ln L = ln j ( τ e ) e = 1 i , j = 1 N w � � C k ( i ) R m ( i ) m ( j ) C ∗ ε ( τ ) M i ( τ ) M ∗ − N ev j ( τ ) dτ k ( j ) i , j = 1 • N ev — number of events, N w — number of waves • M ( τ e ) — amplitudes for e -th event (data) • R — positive definite density matrix (parameters) • C — coupling coeff (most of C = 1, some are parameters) • m ( i ) , k ( i ) — describes wave to C and R correspondence • τ = s , t , m ( 3 π ) , . . . — phase space variables • ε ( τ ) — acceptance of the setup

Preface PWA method Fit results Conclusions Backup slides Coherent part of density matrix Coherent part of the density matrix R is the largest part of the matrix which has rank 1 and behaves like vector of amplitudes. Let � e k is k- th eigenvalue d � e k ∗ V k ∗ V + R = where k V k is k- th eigenvector k = 1 Let e 1 ≫ e 2 > . . . > e d > 0. Leading term R L is coherent part of density matrix and R S is the rest (incoherent part). This decomposition is stable w.r.t. variations of R matrix elements. d R L = e 1 ∗ V 1 ∗ V + � e k ∗ V k ∗ V + R = R L + R S , 1 , R S = k k = 2 Experience shows that resonances tend to concentrate in R L .

Preface PWA method Fit results Conclusions Backup slides Isospin relations If we neglect phase space factors � 1 R = σ ( π − π 0 π 0 ) for waves with ρ ( 770 ) , ρ 3 ( 1690 ) σ ( π + π − π − ) = 1 / 2 for waves with f 0 ( ... ) , f 2 ( 1270 ) All waves coupled to π 0 π 0 have factor 1 / 2 To simplify comparison, they are scaled 2x. To compensate for some losses, all π − π 0 π 0 are scaled by 1.25 This factor is obtained comparing signals of a 2 ( 1320 ) Blue line - π + π − π − Red line - π − π 0 π 0

Preface PWA method Fit results Conclusions Backup slides Largest waves, | t ′ | < 0.03 GeV 2 /c 2 1+S0+ RHO(770) 1+S0+ RHO(770) 0-S0+ EPSMX 0-S0+ EPSMX 2-S0+ F2(1270) 2-S0+ F2(1270) 0-P0+ RHO(770) 0-P0+ RHO(770) × 3 × 3 × 3 × 3 10 10 10 10 Entries 100 Entries 9095186 Entries 3827642 Entries 4348903 180 1800 300 200 160 1600 π 180 (1300) π (1670) 250 2 140 1400 160 π π (1800) (2100) a (1260) 2 120 140 1200 1 200 120 100 1000 150 100 800 80 80 60 600 100 60 400 40 40 50 20 200 20 0 0 0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 π π π π M(3 ) M(3 ) M(3 ) M(3 ) 1+P0+ EPSMX 1+P0+ EPSMX 0-S0+ F0(975) 0-S0+ F0(975) 2-P0+ RHO(770) 2-P0+ RHO(770) 2-F0+ RHO(770) 2-F0+ RHO(770) × 3 × 3 10 10 700 Entries 8020903 Entries 1708639 Entries 2576145 Entries 1555797 60000 70000 100 600 50000 60000 π 80 (1800) 500 50000 40000 400 60 40000 30000 300 30000 40 20000 200 20000 20 10000 100 10000 0 0 0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 M(3 π ) M(3 π ) M(3 π ) M(3 π )

Preface PWA method Fit results Conclusions Backup slides Minor waves — 1, | t ′ | < 0.03 GeV 2 /c 2 2+D1+ RHO(770) 2+D1+ RHO(770) 2+D1+ RHO(770) 2+D1+ RHO(770) 2-D0+ EPSMX 2-D0+ EPSMX 2-D0+ F0(975) 2-D0+ F0(975) × 3 10 Entries 347211 Entries 1019707 70000 Entries 2261640 Entries 327941 120 16000 30000 π 60000 14000 (1670) ? 100 25000 2 |t'| < 0.03 |t'| = 0.03..0.15 12000 50000 a (1320) 80 a (1320) 20000 2 2 10000 40000 a (1700) ? 2 60 15000 8000 30000 6000 40 10000 20000 4000 5000 20 10000 2000 0 0 0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 π π π π M(3 ) M(3 ) M(3 ) M(3 ) 2-D0+ F2(1270) 2-D0+ F2(1270) 3+D0+ RHO(770) 3+D0+ RHO(770) 3+P0+ F2(1270) 3+P0+ F2(1270) 3+S0+ RHO3 3+S0+ RHO3 20000 Entries 444637 Entries 614174 Entries 453530 Entries 191753 25000 20000 20000 18000 π (1670) 18000 18000 2 16000 20000 16000 16000 a (1875) ? 3 14000 14000 14000 12000 15000 12000 12000 10000 10000 10000 8000 10000 8000 8000 6000 6000 6000 5000 4000 4000 4000 2000 2000 2000 0 0 0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 π π π π M(3 ) M(3 ) M(3 ) M(3 )

Preface PWA method Fit results Conclusions Backup slides Minor waves — 2 1+P0+ F0(975) 1+P0+ F0(975) 1+P0+ F0(975) 1+P0+ F0(975) 0-S0+ F0(1500) 0-S0+ F0(1500) 4+G1+ RHO(770) 4+G1+ RHO(770) 24000 10000 9000 Entries 249681 Entries 132167 Entries 137809 Entries 117050 4500 22000 9000 |t'| < 0.03 |t'| = 0.03..0.15 8000 |t'| < 0.03 |t'| = 0.03..0.15 4000 20000 8000 7000 π a (1420) ? a (1420) ? (1800) ? a (2050) 18000 3500 1 1 4 7000 16000 6000 3000 6000 14000 5000 2500 12000 5000 4000 10000 2000 4000 8000 3000 1500 3000 6000 2000 2000 1000 4000 1000 1000 500 2000 0 0 0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 π π π π M(3 ) M(3 ) M(3 ) M(3 ) 4+G1+ RHO(770) 4+G1+ RHO(770) 4+F1+ F2(1270) 4+F1+ F2(1270) 1-P0- RHO(770) 1-P0- RHO(770) 1-P1- RHO(770) 1-P1- RHO(770) 12000 Entries 75374 Entries 64438 Entries 75782 Entries 204921 3500 6000 |t'| = 0.15..0.30 |t'| = 0.03..0.15 2500 10000 3000 5000 a (2050) a (2050) |t'| = 0.03..0.15 |t'| = 0.03..0.15 4 4 2500 2000 8000 4000 2000 1500 6000 3000 1500 1000 4000 2000 1000 500 1000 2000 500 0 0 0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 π π π π M(3 ) M(3 ) M(3 ) M(3 )