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

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

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Preface PWA method Fit results Conclusions Backup slides PWA with full rank density matrix of the + and 0 0 systems at VES setup Igor Kachaev, Dmitry Ryabchikov for VES group, Protvino, Russia Institute for High

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Preface PWA method Fit results Conclusions Backup slides

PWA with full rank density matrix of the π+π−π− and π−π0π0 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

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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 GeV2/c2

  • 20 · 106 events in π−π0π0 (leading statistics in the world)
  • 30 · 106 events in π+π−π− (next to leading)
  • Analysis is done for |t′| = 0–0.03–0.15–0.30–0.80 GeV2/c2
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Raw data

GeV/c 0.5 1 1.5 2 2.5 3 10000 20000 30000 40000 50000 60000 70000 80000 90000

) π Mass(3

2

/c

2

GeV 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3

10

4

10

5

10

6

10

T prime

) GeV/c π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 For |t'| < 0.03

Acceptance

0.4 0.45 0.5 0.55 0.6 500 1000 1500 2000

π K->3

Here and below: Blue line - π+π−π− Red line - π−π0π0

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PWA method. Partial waves

p p π- π- π+ π- X R(ππ)

PWA amplitudes are constructed using isobar model, sequential decay via ππ subsystem. Wave has quantum numbers JPLMη R where JP is spin-parity for 3π system, Mη is its projection of spin and naturality, R is the known resonance in ππ system, L is orbital momentum in Rπ decay. IG = 1− is implicit for 3π charged states.

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Method details

  • Amplitudes use d-functions (Hansen, Illinois PWA)
  • Amplitudes are relativistic (Chung, Filippini)
  • resonanses are relativistic Breit-Wigners

R = f0(980), ε(1300), f0(1500), ρ(770), f2(1270), ρ3(1690) To describe ππ S-wave we use modified Au, Morgan, Pennnington M-solution with f0(980) withdrawn. We name it ε(1300)

  • Notation for some known 3π resonances:

a1(1260) — 1+S0+ρ(770), a2(1320) — 2+D1+ρ(770), π2(1670) — 2−S0+f2(1270), π(1800) — 0−S0+f0(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.

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Extended likelihood function

ln L =

Nev

  • e=1

ln

Nw

  • i,j=1

Ck(i)Rm(i)m(j)C∗

k(j)Mi(τe)M∗ j (τe)

− Nev

Nw

  • i,j=1

Ck(i)Rm(i)m(j)C∗

k(j)

  • ε(τ)Mi(τ)M∗

j (τ) dτ

  • Nev — number of events, Nw — number of waves
  • M(τe) — 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(3π), . . . — phase space variables
  • ε(τ) — acceptance of the setup
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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 R =

d

  • k=1

ek ∗ Vk ∗ V+

k

where ek is k-th eigenvalue Vk is k-th eigenvector Let e1 ≫ e2 > . . . > ed > 0. Leading term RL is coherent part of density matrix and RS is the rest (incoherent part). This decomposition is stable w.r.t. variations of R matrix elements. R = RL + RS, RL = e1 ∗ V1 ∗ V+

1 ,

RS =

d

  • k=2

ek ∗ Vk ∗ V+

k

Experience shows that resonances tend to concentrate in RL.

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Isospin relations

If we neglect phase space factors R = σ(π−π0π0) σ(π+π−π−) = 1 for waves with ρ(770), ρ3(1690) 1/2 for waves with f0(...), f2(1270) All waves coupled to π0π0 have factor 1/2 To simplify comparison, they are scaled 2x. To compensate for some losses, all π−π0π0 are scaled by 1.25 This factor is obtained comparing signals of a2(1320) Blue line - π+π−π− Red line - π−π0π0

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Largest waves, |t′| < 0.03 GeV2/c2

) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 200 400 600 800 1000 1200 1400 1600 1800

3

10 × 1+S0+ RHO(770)

Entries 100

(1260)

1

a

1+S0+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 50 100 150 200 250 300

3

10 × 0-S0+ EPSMX Entries 9095186

(1300) π (1800) π

0-S0+ EPSMX ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 20 40 60 80 100 120 140 160 180 200

3

10 × 2-S0+ F2(1270) Entries 3827642

(1670)

2

π (2100)

2

π

2-S0+ F2(1270) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 20 40 60 80 100 120 140 160 180

3

10 × 0-P0+ RHO(770) Entries 4348903 0-P0+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 100 200 300 400 500 600 700

3

10 × 1+P0+ EPSMX Entries 8020903 1+P0+ EPSMX ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 10000 20000 30000 40000 50000 60000 70000 0-S0+ F0(975) Entries 1708639

(1800) π

0-S0+ F0(975) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 20 40 60 80 100

3

10 × 2-P0+ RHO(770) Entries 2576145 2-P0+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 10000 20000 30000 40000 50000 60000 2-F0+ RHO(770) Entries 1555797 2-F0+ RHO(770)

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Minor waves — 1, |t′| < 0.03 GeV2/c2

) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 5000 10000 15000 20000 25000 30000 2+D1+ RHO(770)

Entries 347211

|t'| < 0.03 (1320)

2

a (1700) ?

2

a

2+D1+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 20 40 60 80 100 120

3

10 × 2+D1+ RHO(770) Entries 1019707

|t'| = 0.03..0.15 (1320)

2

a

2+D1+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 10000 20000 30000 40000 50000 60000 70000 2-D0+ EPSMX Entries 2261640 2-D0+ EPSMX ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2000 4000 6000 8000 10000 12000 14000 16000 2-D0+ F0(975) Entries 327941

(1670) ?

2

π

2-D0+ F0(975) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 5000 10000 15000 20000 25000 2-D0+ F2(1270) Entries 444637

(1670)

2

π

2-D0+ F2(1270) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 3+D0+ RHO(770) Entries 614174 3+D0+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 3+P0+ F2(1270) Entries 453530 3+P0+ F2(1270) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 3+S0+ RHO3 Entries 191753

(1875) ?

3

a

3+S0+ RHO3

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Minor waves — 2

) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 1+P0+ F0(975)

Entries 249681 |t'| < 0.03 (1420) ?

1

a

1+P0+ F0(975) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 1+P0+ F0(975) Entries 132167

|t'| = 0.03..0.15 (1420) ?

1

a

1+P0+ F0(975) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 1000 2000 3000 4000 5000 6000 7000 8000 9000 0-S0+ F0(1500) Entries 137809

|t'| < 0.03 (1800) ? π

0-S0+ F0(1500) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 500 1000 1500 2000 2500 3000 3500 4000 4500 4+G1+ RHO(770) Entries 117050

|t'| = 0.03..0.15 (2050)

4

a

4+G1+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 500 1000 1500 2000 2500 3000 3500 4+G1+ RHO(770) Entries 75374

|t'| = 0.15..0.30 (2050)

4

a

4+G1+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 500 1000 1500 2000 2500 4+F1+ F2(1270) Entries 64438

|t'| = 0.03..0.15 (2050)

4

a

4+F1+ F2(1270) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 1000 2000 3000 4000 5000 6000 1-P0- RHO(770) Entries 75782

|t'| = 0.03..0.15

1-P0- RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2000 4000 6000 8000 10000 12000 1-P1- RHO(770) Entries 204921

|t'| = 0.03..0.15

1-P1- RHO(770)

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Exotic (non q¯ q) wave JPC = 1−+

Results for |t′| = 0–0.03–0.15–0.30–0.80 GeV2/c2 Full density matrix

) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 5000 10000 15000 20000 25000 30000 1-P1+ RHO(770)

Entries 511121 1-P1+ RHO(770)

) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 1-P1+ RHO(770) Entries 442687 1-P1+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 1000 2000 3000 4000 5000 6000 7000 1-P1+ RHO(770) Entries 155386 1-P1+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 500 1000 1500 2000 2500 3000 3500 4000 1-P1+ RHO(770)

Entries 97848 1-P1+ RHO(770)

Largest Eigenvalue

) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 200 400 600 800 1000 1200 1400 1600 1800 2000 1-P1+ RHO(770)

Entries 31799 1-P1+ RHO(770)

) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 1000 2000 3000 4000 5000 1-P1+ RHO(770) Entries 119892 1-P1+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 500 1000 1500 2000 2500 3000 1-P1+ RHO(770)

Entries 50083 1-P1+ RHO(770)

) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 200 400 600 800 1000 1200 1-P1+ RHO(770)

Entries 22219 1-P1+ RHO(770)

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Conclusions

  • Mass-independent PWA is done for π+π−π− and π−π0π0
  • data. A lot of waves look alike in both reactions, some are
  • not. Naive isospin relations can be violated by the difference

in phase space and in interference on Dalitz plot for I = 0 isobars.

  • Unestablished decay modes π(1800) → f0(1500)π,

a3(1875) → ρ3(1690)π are much better seen in π+π−π−. The same is true for 1+S f0(980)π bump at M = 1.4 GeV/c2, a1(1420). State a2(1700) is not seen in both reactions.

  • State a2(1320) is in good agreement in both systems for all

t′. For a4(2050) this is true for |t′| > 0.03 GeV2/c2. We have not enough statictics for low t′.

  • Waves with JPC = 1−+ are small, in largest eigenvalue are

even smaller, albeit in agreement for |t′| > 0.03 GeV2/c2. In negative naturality in π−π0π0 they are severely suppressed.

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Backup slides

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Wave set used in the analysis

JPC Waves FLAT 0−+ 0−S0+ ε 0−S0+f0(975) 0−S0+f0(1500) 0−P0+ ρ 1++ 1+S0+ ρ 1+P0+ ε 1+D0+ ρ 1+P0+f0(975) 1+P0+f2 1+S1+ ρ 1+P1+ ε 1+S1− ρ 1−+ 1−P1+ ρ 1−P0− ρ 1−P1− ρ 2−+ 2−S0+f2 2−P0+ ρ 2−D0+ ε 2−F0+ ρ 2−D0+f0(975) 2−D0+f2 2−S1+f2 2−P1+ ρ 2−F1+ ρ 2−D1+ ε 2−D1+f2 2++ 2+D1+ ρ 2+P1+f2 2+D0− ρ 2+D1− ρ 3++ 3+S0+ ρ3 3+P0+f2 3+D0+ ρ 4−+ 4−F0+ ρ 4++ 4+F0+f2 4+G0+ ρ

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Largest waves, |t′| < 0.03 GeV2/c2, Largest EigenValue

) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 200 400 600 800 1000 1200 1400 1600 1800

3

10 × 1+S0+ RHO(770)

Entries 100

(1260)

1

a

1+S0+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 50 100 150 200 250

3

10 × 0-S0+ EPSMX Entries 7521969

(1300) π (1800) π

0-S0+ EPSMX ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 20 40 60 80 100 120 140 160 180

3

10 × 2-S0+ F2(1270) Entries 2981739

(1670)

2

π (2100)

2

π

2-S0+ F2(1270) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 20 40 60 80 100 120 140 160 180

3

10 × 0-P0+ RHO(770) Entries 3301021 0-P0+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 100 200 300 400 500 600 700

3

10 × 1+P0+ EPSMX Entries 6008642 1+P0+ EPSMX ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 10000 20000 30000 40000 50000 0-S0+ F0(975) Entries 1104545

(1300) π (1800) π

0-S0+ F0(975) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 5000 10000 15000 20000 25000 30000 35000 40000 2-P0+ RHO(770) Entries 1161937 2-P0+ RHO(770) ) π M(3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 10000 20000 30000 40000 50000 60000 2-F0+ RHO(770) Entries 1137914 2-F0+ RHO(770)