[PPT] - A covariant approach to the partial wave analysis of the hadron PowerPoint Presentation

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Petersburg Nuclear Physics Institute

A covariant approach to the partial wave analysis of the hadron reactions

A. Sarantsev

HISKP (Bonn, Germany), PNPI (Gatchina, Russia)

GSI, December 14, 2016

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Bonn-Gatchina partial wave analysis group:

A. Anisovich, E. Klempt, V. Nikonov, A. Sarantsev, U. Thoma

http://pwa.hiskp.uni-bonn.de/

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Search for baryon states

1. Analysis of single and double meson photoproduction reactions.

γp → πN, ηN, KΛ, KΣ, ππN, πηN, ωp, K∗Λ,

CB-ELSA, CLAS, MAMI, GRAAL, LEPS.

2. Analysis of single and double meson production in pion-induced reactions.

πN → πN, ηN, KΛ, KΣ, ππN.

Search for meson states

1. Analysis of the p¯

p annihilation at rest and ππ interaction data.

2. Analysis of the p¯

p annihilation in flight into two and tree meson final state.

3. Analysis of the BES III data on J/Ψ decays (in collaboration with JINR Dubna).

Analysis of NN interaction

1. Analysis of single and double meson production NN → πNN and (Wasa, PNPI,

HADES)

2. Analysis of hyperon production NN → KΛp (WASA, HADES)

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Energy dependent approach

In many cases an unambiguous partial wave decomposition at fixed energies is

impossible. Then the energy and angular parts should be analyzed together:

A(s, t) = ∑

ββ′n

Aββ′

n

(s)Q(β)+

µ1...µnF µ1...µn ν1...νn Q(β′) ν1...νn

Aββ′

n

(s) - the partial wave amplitude with total spin J = n for bosons and J = n + 1/2 for fermions.

1. A. V. Anisovich, V. V. Anisovich, V. N. Markov, M. A. Matveev and A. V. Sarantsev, J. Phys. G 28, 15 (2002)
2. A. Anisovich, E. Klempt, A. Sarantsev and U. Thoma, Eur. Phys. J. A 24, 111 (2005)
3. A. V. Anisovich and A. V. Sarantsev, Eur. Phys. J. A 30, 427 (2006)
4. A. V. Anisovich, V. V. Anisovich, E. Klempt, V. A. Nikonov and A. V. Sarantsev, Eur. Phys. J. A 34, 129 (2007).
1. C. Zemach, Phys. Rev. 140, B97 (1965); 140, B109 (1965).
2. S.U.Chung, Phys. Rev. D 57, 431 (1998).
3. B. S. Zou and D. V. Bugg, Eur. Phys. J. A 16, 537 (2003)

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Partial wave amplitude:

transition amplitude with fixed initial and final states Quantum numbers: mesons IG JP C, baryons: IJP , decay LS basis: 2S+1LJ

IG1

1 JP1C1 1

+IG2

2 JP2C2 2

(2S+1LJ ) → IGJP C → I′

1 G′

1J′

1 P ′

1C′ 1 +I′

2 G′

2J′

2 P ′

2C′ 2

(

2S′+1L′ J

) G = G1G2 G = G′

1G′ 2

P = P1P2(−1)L P = P ′

1P ′ 2(−1)L′

|I1 − I2| < I < I1 + I2 |I′

1 − I′ 2| < I < I′ 1 + I′ 2

|J1 − J2| < S < J1 + J2 |J′

1 − J′ 2| < S′ < J′ 1 + J′ 2

|S − L| < J < S + L |S′ − L′| < J < S′ + L′ A(s, t) = Vµ1...µn(S, L) P µ1...µn

ν1...νn V ′ ν1...νn(S′, L′)A(s)

n = J mesons n = J − 1/2 baryons

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Boson projection operators

The wave function of boson with J = n:

Ψµ1···µn = 1 √ 2εuµ1···µneipx ∫ Ψµ(x)Ψ∗(x)d4x = αpµ = 0 = ⇒ pµΨµ = 0 ∫ Ψµν(x)Ψ∗(x)d4x = β ( gµν − pµpν

p2

) = 0 = ⇒ gµνΨµν = 0

Properties of uµ1···µn:

p2uµ1µ2...µn = m2uµ1µ2...µn pµiuµ1µ2...µn = 0 gµiµjuµ1µ2...µn = 0 uµ1...µi...µj...µn = uµ1...µj...µi...µn

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

In momentum representation:

P µ1µ2...µn

ν1ν2...νn = (−1)nOµ1µ2...µn ν1ν2...νn = 2n+1

∑

i=1

u(i)

µ1µ2...µnu(i)∗ ν1ν2...νn

O = 1 Oµ

ν

= g⊥

µν = gµν − pµpν

p2 Oµ1µ2

ν1ν2

= 1 2(g⊥

µ1ν1g⊥ µ2ν2 + g⊥ µ1ν2g⊥ µ2ν1) − 1

3g⊥

µ1µ2g⊥ ν1ν2

Recurrent expression for the boson projector operator

Oµ1...µL

ν1...νL = 1

L2  

L

∑

i,j=1

g⊥

µiνjOµ1...µi−1µi+1...µL ν1...νj−1νj+1...νL −

4 (2L − 1)(2L − 3)

L

∑

i<j,k<m

g⊥

µiµjg⊥ νkνmOµ1...µi−1µi+1...µj−1µj+1...µL ν1...νk−1νk+1...νm−1νm+1...νL

 

Normalization condition:

Oµ1...µL

ν1...νL Oν1...νL α1...αL = Oµ1...µL α1...αL

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Orbital momentum operator

The angular momentum operator is constructed from momenta of particles k1, k2 and metric tensor gµν. For L = 0 this operator is a constant: X0 = 1 The L = 1 operator is a vector X(1)

µ

, constructed from: kµ = 1

2(k1µ − k2µ) and

Pµ = (k1µ + k2µ). Orthogonality: ∫ d4k 4π X(1)

µ1 X(0) =

∫ d4k 4π X(n)

µ1...µnX(n−1) µ2...µn = ξPµ1 = 0

Then:

X(1)

µ Pµ = 0

X(n)

µ1...µnPµj = 0

and:

X(1)

µ

= k⊥

µ = kνg⊥ νµ;

g⊥

νµ =

( gνµ − PνPν p2 ) ;

in c.m.s k⊥ = (0,⃗

k)

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Recurrent expression for the orbital momentum operators X(n)

µ1...µn

X(n)

µ1...µn = 2n−1

n2

n

∑

i=1

k⊥

µiX(n−1) µ1...µi−1µi+1...µn − 2k2 ⊥

n2

n

∑

i,j=1 i<j

gµiµjX(n−2)

µ1...µi−1µi+1...µj−1µj+1...µn

Taking into account the traceless property of X(n) we have:

X(n)

µ1...µnX(n) µ1...µn = α(n)(k2 ⊥)n

α(n) =

n

∏

i=1

2i − 1 i = (2n − 1)!! n! .

From the recursive procedure one can get the following expression for the operator X(n):

X(n)

µ1...µn = α(n)

[ k⊥

µ1k⊥ µ2 . . . k⊥ µn −

k2

⊥

2n − 1 ( g⊥

µ1µ2k⊥ µ3 . . . k⊥ µn + · · ·

) + k4

⊥

(2n − 1)(2n − 3) ( g⊥

µ1µ2g⊥ µ3µ4k⊥ µ5 · · · kµ4 + · · ·

) + · · · ] .

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Scattering of two spinless particles Denote relative momenta of particles before and after interaction as q and k,

correspondingly. The structure of partial–wave amplitude with orbital momentum

L = J is determined by convolution of operators X(L)(k) and X(L)(q): AL = BWL(s)X(L)

µ1...µL(k)Oµ1...µL ν1...νL X(L) ν1...νL(q) = BWL(s)X(L) µ1...µL(k)X(L) µ1...µL(q)

BWL(s) depends on the total energy squared only.

The convolution X(L)

µ1...µL(k)X(L) µ1...µL(q) can be written in terms of Legendre

polynomials PL(z):

X(L)

µ1...µL(k)X(L) µ1...µL(q) = α(L)

(√ k2

⊥

√ q2

⊥

)L PL(z) , z = (k⊥q⊥) √ k2

⊥

√ q2

⊥

α(L) =

L

∏

n=1

2n − 1 n

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The ¯

pp → 3π0 reaction A(S − wave) =

N

∑

L=0

X(L)

µ1...µL(k⊥ 23)X(L) µ1...µL(k⊥ 1 )AL(s23)

200 400 600 800 1000 1200 )

2

) (GeV π π (

2

M 0.5 1 1.5 2 2.5 3 )

2

) (GeV π π (

2

M 0.5 1 1.5 2 2.5 3

in liquid hydrogen π 3 → p p in liquid hydrogen π 3 → p p

200 400 600 800 1000 1200 )

2

) (GeV π π (

2

M 0.5 1 1.5 2 2.5 3 )

2

) (GeV π π (

2

M 0.5 1 1.5 2 2.5 3

in liquid hydrogen π 3 → p p in liquid hydrogen π 3 → p p

200 400 600 800 1000 1200 1400 1600 1800 )

2

) (GeV π π (

2

M 0.5 1 1.5 2 2.5 3 )

2

) (GeV π π (

2

M 0.5 1 1.5 2 2.5 3 in gaseous hydrogen π 3 → p p in gaseous hydrogen π 3 → p p 200 400 600 800 1000 1200 1400 1600 1800 )

2

) (GeV π π (

2

M 0.5 1 1.5 2 2.5 3 )

2

) (GeV π π (

2

M 0.5 1 1.5 2 2.5 3 in gaseous hydrogen π 3 → p p in gaseous hydrogen π 3 → p p

pp

− - 3π0 Liquid target

20000 40000 1 2 3 M2 (ππ) GeV2 2500 5000 7500 10000 x 10

1
0.5

0.5 1 Cos θ for 0.00<Mππ<0.90 GeV 500 1000 1500 x 10 2

1
0.5

0.5 1 Cos θ for 0.95<Mππ<1.05 GeV 500 1000 1500 x 10 2

1
0.5

0.5 1 Cos θ for 1.20<Mππ<1.40 GeV 500 1000 1500 x 10 2

1
0.5

0.5 1 Cos θ for 1.45<Mππ<1.55 GeV 500 1000 1500 2000 x 10 2

1
0.5

0.5 1 Cos θ for 1.55<Mππ<1.65 GeV

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

J/Ψ decay into three pseudoscalar mesons

A = VαAα ∑ V ∗

µ V ν = g⊥ µν = gµν − PµPν

P 2 Aα = εαηβµXην2...νJ (k23)Xβν2...νJ (k⊥

1 )PµAJ(s23)

k⊥

1µ = k1νg⊥ µν

c) d)

)

2

K ) ( G e V π s (

2 3 4 5 6

)

2

K) (GeV π s(

2 3 4 5 6 100 200 300 400 500

c)

)

2

K ) ( G e V π s (

2 3 4 5 6

)

2

K) (GeV π s(

2 3 4 5 6 100 200 300 400 500

d)

e) f)

20 40 60 80 100 120 140

)

2

K) (GeV π s(

2 3 4 5 6

)

2

K) (GeV π s(

2 3 4 5 6

e)

20 40 60 80 100 120 140

)

2

K) (GeV π s(

2 3 4 5 6

)

2

K) (GeV π s(

2 3 4 5 6

f)

Data Fit

M(Κ+Κ−)

0.5 1 1.5 1 2

×103

M(π0Κ−)

0.5 1 1.5 1.5 2 2.5

×103

M(π0Κ+)

0.5 1 1.5 1.5 2 2.5

×103

cos θ(π0)

1 2

1
0.5

0.5 1

×103

cos θ(Κ+)

1 2

1
0.5

0.5 1

×103

cos θ(Κ−)

1 2

1
0.5

0.5 1

×103

cos θ(Κ+Κ−)

1 2 3

1
0.5

0.5 1

×103

cos θ(π0Κ−)

1 2 3

1
0.5

0.5 1

×103

cos θ(π0Κ+)

1 2 3

1
0.5

0.5 1

×103

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Structure of the fermion propagator

The orthogonality condition has a different form in a fermion case:

∫ Ψµ(x)Ψ∗(x)d4x = A pµ + B γµ

where A and B are matrices in spinor space. It means that we have an additional condition:

γµΨµ = 0 γµuµ = 0

Thus in momentum space we have:

(ˆ p − m)uµ1...µn = 0 pµiuµ1...µn = 0 uµ1...µi...µj...µn = uµ1...µj...µi...µn gµiµjuµ1...µn = 0 γµiΨµ1...µn = 0

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

These properties define the structure of the fermion projection operator P µ1...µn

ν1...νn :

Gµ1...µn

ν1...νn = (−1)n

2m m + ˆ p m2 − p2 P µ1...µn

ν1...νn

P µ1...µn

ν1...νn = Oµ1...µn α1...αnT α1...αn β1...βn Oβ1...βn ν1...νn

T α1...αn

β1...βn = n + 1

2n+1 ( gα1β1 − n n+1σα1β1 )

n

∏

i=2

gαiβi

where

σαiαj = 1 2(γαiγαj − γαjγαi)

For particle with spin 3/2 it has form:

P µ

ν = 1

2 ( g⊥

µν − γ⊥ µ γ⊥ ν /3

)

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

πN interaction

States with J = L − 1/2 are called ’-’ states (1/2+, 3/2−,5/2+,. . .) and states with

J = L + 1/2 are called ’+’ states (1/2−, 3/2+,5/2−,. . .). ˜ N +

µ1...µn = X(n) µ1...µn

˜ N −

µ1...µn = iγνγ5X(n+1) νµ1...µn

A = ¯ u(k1)N ±

µ1...µLF µ1...µL−1 ν1...νL−1 N ± ν1...νLu(q1)BW ± L (s) − − − − → c.m.s. ω∗ [G(s, t) + H(s, t)i(⃗

σ⃗ n)] ω′ G(s, t) = ∑

L

[ (L+1)F +

L (s) − LF − L (s)

] PL(z) , H(s, t) = ∑

L

[ F +

L (s) + F − L (s)

] P ′

L(z) .

F +

L

= (−1)L+1(|⃗ k||⃗ q|)L√χiχf α(L) 2L+1BW +

L (s) ,

F −

L

= (−1)L(|⃗ k||⃗ q|)L√χiχf α(L) L BW −

L (s) .

χi = mi + ki0 α(L) =

L

∏

l=1

2l − 1 l = (2L − 1)!! L! .

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The fit of the the π−p → KΛ reaction

Full experiment for πN → KΛ: differential cross section, analyzing power, rotation parameter. A clear evidence for resonances which are hardly seen (or not seen) in the elastic reactions: N(1710)P11,

N(1900)P13,

The total cross section for the reaction π−p → K0Λ and contributions from leading partial waves.

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Amplitude for the πN transition into channels πN, ηN, KΛ and KΣ:

AπN = ω∗ [G(s, t) + H(s, t)i(⃗ σ⃗ n)] ω′ ⃗ nj = εµνj qµkν |⃗ k||⃗ q| . G(s, t) = ∑

L

[ (L+1)F +

L (s) + LF − L (s)

] PL(z) , H(s, t) = ∑

L

[ F +

L (s) − F − L (s)

] P ′

L(z) .

z = cos Θ, the angle of the final meson in c.m.s. |A|2 = 1 2Tr [A∗

πNAπN] = |G(s, t)|2+|H(s, t)|2(1−z2)

and the recoil asymmetry can be calculated as:

P = Tr [A∗

πNσ2AπN]

2|A|2 cos ϕ = sin Θ2Im (H∗(s, t)G(s, t)) |A|2 .

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Near threshold, only contributions from S and P -waves are expected. For the S2I,2J and P2I,2J amplitudes we have

S2I,1; G= F +

0 ;

H = 0; |A|2 = |F +

0 |2

(1)

P2I,1; G= F −

1 z; H = −F − 1 ;

|A|2 = |F −

1 |2

(2)

P2I,3; G= 2F +

1 z; H = F + 1 ;

|A|2 = |F +

1 |2(3z2+1)

where the indices (2I, 2J) remind of the isospin I and the spin J of the partial waves. The recoil asymmetry vanishes unless different amplitudes interfere. S2I,1+P2I,1 : P |A|2

sin Θ = −2Im(F + 0 F −∗ 1

) |A|2 =|F +

0 |2+|F − 1 |2+2zRe(F +∗

F −

1 )

S2I,1+P2I,3 : P |A|2

sin Θ = 2Im(F + 0 F +∗ 1

) |A|2 =|F +

0 |2+|F + 1 |2(3z2+1)+4zRe(F +∗

F +

1 )

P2I,1+P2I,3 : P |A|2

sin Θ =6zIm(F +∗ 1

F −

1 )

|A|2 =|F +

1 −F − 1 |2+z2 (

3|F +

1 |2−2Re(F +∗ 1

F −

1 )

) . where |A|2 represents the angular distribution and P |A|2/ sin Θ an observable proportional to the recoil polarization parameter P .

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The fit of the the π−p → KΛ reaction (differential cross section)

0.1 1632

dσ/dΩ, mb/sr

1661 1670 0.1 1676 1678 1681 0.1 1684 1686 1687 0.1 1689 1693 1698 0.1 1701 1707 1743 0.05 1758 1792 1825 0.05 1847 1879 1909 0.05 1938 1966 1999 0.05 2027 2059 2104 0.5

0.5

cos θcm

0.05 2159 0.5

0.5

2183 0.5

0.5

0.05 0.1

Wcm=1633

P dσ/dΩ /sinθ, mb/sr

Wcm=1661 Wcm=1683

0.05 0.1

Wcm=1694 Wcm=1724 Wcm=1758

0.05 0.1

Wcm=1792 0.5

0.5

Wcm=1825 0.5

0.5

cosθcm

Wcm=1847 0.5

0.5

19

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The fit of the the π−p → KΛ reaction

1

1 1633

P

1661 1683

1

1 1694 1724 1758

1

1 1792 1825 1847

1

1 1879 1909 1938

1

1 1966 1999 2027 0.5

0.5

cos θcm

1

1 2059 0.5

0.5

2104 0.5

0.5
5
2.5

2.5 5 W=1851

β, rad

W=1940 W=2030

5
2.5

2.5 5 W=2062 W=2107 0.5

0.5

W=2159 0.5

0.5

cos θcm

5
2.5

2.5 5 W=2261 0.5

0.5

20

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Energy independent analysis of the π−p → KΛ data

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 0.00 0.05 0.10 0.15 0.20 0.25

11

S

magnitude

W(GeV)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

11

P

magnitude

W(GeV)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 0.00 0.02 0.04 0.06 0.08 0.10 0.12

13

P

magnitude

W(GeV)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

150
100
50

50 100 150

11

P

W(GeV) phase (deg)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

150
100
50

50 100 150

13

P

W(GeV) phase (deg)

However this is not a unique solution

1.65 1.7 1.75 1.8 1.85 0.00 0.05 0.10 0.15 0.20 0.25

11

S

magnitude

W(GeV) 1.65 1.7 1.75 1.8 1.85 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

11

P

magnitude

W(GeV)

1.65 1.7 1.75 1.8 1.85 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

13

P

magnitude

W(GeV) 1.65 1.7 1.75 1.8 1.85

150
100
50

50 100 150

11

P

W(GeV) phase (deg)

1.65 1.7 1.75 1.8 1.85

150
100
50

50 100 150

13

P

W(GeV) phase (deg)

21

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The fit of the the π+p → K+Σ+ reaction with BG2011-02

0.1 0.2 0.3 0.4 0.5 0.6 1700 1800 1900 2000 2100 2200

3/2 1/2− 3/2 3/2+ 3/2 7/2+

M(π+p), MeV σtot, mb π+p→ K+Σ+ 0.05 1822

dσ/dΩ, mb/sr

1845 1870 0.05 1891 1926 1939 0.05 1970 1985 2019 0.05 2031 2059 2074 0.05 2106 2118 0.5

0.5

2147 0.5

0.5

cos θcm

0.05 2158 0.5

0.5
1

1 1822

P

1845 1870

1

1 1891 1926 1939

1

1 1970 1985 2019

1

1 2031 2059 2074

1

1 2106 2118 0.5

0.5

2147 0.5

0.5

cos θcm

1

1 2158 0.5

0.5

22

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Status is confirmed. The new properties are defined

Citation: J. Beringer et al. (Particle Data Group), PR D86, 010001 (2012) and 2013 partial update for the 2014 edition (URL: http://pdg.lbl.gov)

N(1710) 1/2+

I(JP) =

1 2( 1 2+) Status: ∗∗∗

The latest GWU analysis (ARNDT 06) finds no evidence for this resonance.

N(1710) BREIT-WIGNER MASS N(1710) BREIT-WIGNER MASS N(1710) BREIT-WIGNER MASS N(1710) BREIT-WIGNER MASS

VALUE (MeV) DOCUMENT ID TECN COMMENT

1680 to 1740 (≈ 1710) OUR ESTIMATE 1680 to 1740 (≈ 1710) OUR ESTIMATE 1680 to 1740 (≈ 1710) OUR ESTIMATE 1680 to 1740 (≈ 1710) OUR ESTIMATE 1710±20 ANISOVICH 12A DPWA Multichannel 1700±50 CUTKOSKY 80 IPWA πN → πN 1723± 9 HOEHLER 79 IPWA πN → πN

• • We do not use the following data for averages, fits, limits, etc. • • •

1662± 7 SHRESTHA 12A DPWA Multichannel 1725±25 ANISOVICH 10 DPWA Multichannel 1729±16 1 BATINIC 10 DPWA πN → N π, N η 1752± 3 PENNER 02C DPWA Multichannel 1699±65 VRANA 00 DPWA Multichannel 1720±10 ARNDT 96 IPWA γ N → πN 1717±28 MANLEY 92 IPWA πN → πN & N ππ 1706 CUTKOSKY 90 IPWA πN → πN 1730 SAXON 80 DPWA π− p → ΛK0 1720 2 LONGACRE 77 IPWA πN → N ππ 1710 3 LONGACRE 75 IPWA πN → N ππ

N(1710) BREIT-WIGNER WIDTH N(1710) BREIT-WIGNER WIDTH N(1710) BREIT-WIGNER WIDTH N(1710) BREIT-WIGNER WIDTH

VALUE (MeV) DOCUMENT ID TECN COMMENT

50 to 250 (≈ 100) OUR ESTIMATE 50 to 250 (≈ 100) OUR ESTIMATE 50 to 250 (≈ 100) OUR ESTIMATE 50 to 250 (≈ 100) OUR ESTIMATE 200± 18 ANISOVICH 12A DPWA Multichannel 93± 30 CUTKOSKY 90 IPWA πN → πN 90± 30 CUTKOSKY 80 IPWA πN → πN 120± 15 HOEHLER 79 IPWA πN → πN

Citation: J. Beringer et al. (Particle Data Group), PR D86, 010001 (2012) and 2013 partial update for the 2014 edition (URL: http://pdg.lbl.gov)

∆(1920) 3/2+

I(JP) =

3 2( 3 2+) Status: ∗∗∗

The latest GWU analysis (ARNDT 06) finds no evidence for this resonance.

∆(1920) BREIT-WIGNER MASS ∆(1920) BREIT-WIGNER MASS ∆(1920) BREIT-WIGNER MASS ∆(1920) BREIT-WIGNER MASS

VALUE (MeV) DOCUMENT ID TECN COMMENT

1900 to 1970 (≈ 1920) OUR ESTIMATE 1900 to 1970 (≈ 1920) OUR ESTIMATE 1900 to 1970 (≈ 1920) OUR ESTIMATE 1900 to 1970 (≈ 1920) OUR ESTIMATE 1900 ± 30 ANISOVICH 12A DPWA Multichannel 1920 ± 80 CUTKOSKY 80 IPWA πN → πN 1868 ± 10 HOEHLER 79 IPWA πN → πN

• • We do not use the following data for averages, fits, limits, etc. • • •

2146 ± 32 SHRESTHA 12A DPWA Multichannel 1990 ± 35 HORN 08A DPWA Multichannel 2057 ± 1 PENNER 02C DPWA Multichannel 1889 ±100 VRANA 00 DPWA Multichannel 2014 ± 16 MANLEY 92 IPWA πN → πN & N ππ 1840 ± 40 CANDLIN 84 DPWA π+ p → Σ+ K+ 1955.0± 13.0 1 CHEW 80 BPWA π+ p → π+ p 2065.0+ 13.6 − 12.9 1 CHEW 80 BPWA π+ p → π+ p

∆(1920) BREIT-WIGNER WIDTH ∆(1920) BREIT-WIGNER WIDTH ∆(1920) BREIT-WIGNER WIDTH ∆(1920) BREIT-WIGNER WIDTH

VALUE (MeV) DOCUMENT ID TECN COMMENT

180 to 300 (≈ 260) OUR ESTIMATE 180 to 300 (≈ 260) OUR ESTIMATE 180 to 300 (≈ 260) OUR ESTIMATE 180 to 300 (≈ 260) OUR ESTIMATE 310 ± 60 ANISOVICH 12A DPWA Multichannel 300 ±100 CUTKOSKY 80 IPWA πN → πN 220 ± 80 HOEHLER 79 IPWA πN → πN

23

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Gauge invariant γN vertices

Photon has quantum numbers JP C = 1−−, proton 1/2+. Then in S-wave two states can be formed is 1/2− and 3/2−. Then P-wave 1/2+, 3/2+ and 1/2+, 3/2+, 5/2+.

V (1+)µ

α1...αn = γ⊥⊥ µ

iγ5X(n)

α1...αn ,

V (1−)µ

α1...αn = γξγ⊥⊥ µ

X(n+1)

ξα1...αn ,

V (2+)µ

α1...αn = γνiγ5X(n+1) να1...αng⊥⊥ µαn ,

V (2−)µ

α1...αn = X(n−1) α2...αng⊥⊥ α1µ

V (3+)µ

α1...αn = ˆ

kiγ5X(n)

α1...αnZµ ,

V (3−)µ

α1...αn = ˆ

kγχX(n+1)

χα1...αnZµ , .

Zµ = ( (Pkγ)kγ

µ − (kγ)2Pµ

) γ⊥⊥

µ

= γνg⊥⊥

µν

g⊥⊥

νµ =

( gνµ − PνPν P 2 − k⊥

ν k⊥ ν

k2

⊥

)

24

SLIDE 25

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

General structure of the single–meson electro-production amplitude:

Jµ =iF1˜ σµ+F2(⃗ σ⃗ q)εµijσikj |⃗ k||⃗ q| +iF3 (⃗ σ⃗ k) |⃗ k||⃗ q| ˜ qµ+iF4 (⃗ σ⃗ q) ⃗ q2 ˜ qµ + iF5 (⃗ σ⃗ k) |⃗ k|2 kµ+iF6 (⃗ σ⃗ q) |⃗ q||⃗ k| kµ µ = 1, 2, 3, F1(z) =

∞

∑

L=0

[LM +

L +E+ L ]P ′ L+1(z)+[(L+1)M − L +E− L ]P ′ L−1(z),

F2(z) =

∞

∑

L=1

[(L + 1)M +

L + LM − L ]P ′ L(z) ,

F3(z) =

∞

∑

L=1

[E+

L − M + L ]P ′′ L+1(z) + [E− L + M − L ]P ′′ L−1(z) ,

F4(z) =

∞

∑

L=2

[M +

L − E+ L − M − L − E− L ]P ′′ L(z) ,

F5(z) =

∞

∑

L=0

[(L + 1)S+

L P ′ L+1(z) − LS− L P ′ L−1(z)] ,

F6(z) =

∞

∑

L=1

[LS−

L − (L + 1)S+ L ]P ′ L(z)

25

SLIDE 26

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

For the positive states J = L + 1/2 (L=n):

Ji+

µ

= εµ¯ u(qN)X(n)

α1...αn(q⊥)F α1...αn β1...βn V (i+)µ β1...βn(k⊥)u(kN)

F1+

1

= λn P ′

n+1

F2+

1

= 0 F3+

1

= 0 F1+

2

= λn P ′

n

F2+

2

= − λn

n P ′ n

F3+

2

= 0 F1+

3

= 0 F2+

3

= λn

n P ′′ n+1

F3+

3

= 0 F1+

4

= 0 F2+

4

= λn

n P ′′ n

F3+

4

= 0 F1+

5

= 0 F2+

5

= 0 F3+

5

= +ξn P ′

n+1

F1+

6

= 0 F2+

6

= 0 F3+

6

= −ξn P ′

n

λn = αn 2n + 1(|⃗ k||⃗ q|)nχiχf χi,f = √ mi,f + k0i,f ξn = (kγ)2 αn 2n + 1(|⃗ k||⃗ q|)nχiχf

No singularities and the correct behavior at Q2 → 0.

26

SLIDE 27

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Meson Photoproduction experiments

GRAAL (Grenoble): Polarized beam. Ideal for the beam asymmetry and double

polarization observables for hyperon final states.

γp → π0p, ηp, KΛ, KΣ, π0π0p, π0ηp, ωp, γn → π0n, ηn.

CLAS (JLAB): High statistic, very good detector of charged particles:

γp → π−n, KΛ, KΣ, π+π−p, ωp. As missing mass data γp → π0p, ηp.

Data on deuterium target. Energy is up to W=2.5 GeV. Analysis: EBAC, SAID and recently Bonn-Gatchina.

MAMI (Mainz): High statistic, very good detector of neutral particles: (Crystal Ball):

γp → π0p, KΛ, KΣ, π0π0p, π0ηp, γn → ηn, π0n, π0π0n, π0ηn.

Energy is only up to W=1.85 GeV. Analysis: MAID and Bonn-Gatchina.

CB-ELSA (Bonn): Moderate statistic, very good detector of neutral particles:

(Crystal Barrel): γp → π0p, ηp, π0π0p, π0ηp, ωp, γn → ηn, π0n. Energy is up to W=2.3 GeV. Analysis: Bonn-Gatchina.

Independent analysis groups: J¨

ulich (M.Doering), OSAKA (T. Sato), Giessen (V. Shklyar), M. Manley (Kent Uni)

27

SLIDE 28

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The main task: search for new baryon resonances
Polarization data are sensitive to weak signals
Double polarization data are available Cx, Cz, Ox, Oz, E, G, H .
Double polarization observables (assuming XZ is the reaction plane)

Photon Target Recoil Target + Recoil

− − − − x′ y′ z′ x′ x′ z′ z′ − x y z − − − x z x z

unpol.

σ0 T P Tx′ −Lx′ Tz′ Lz′

lin.pol.

−Σ H −P −G Ox′ −T Oz′ −Lz′ Tz′ −Lx′ −Tx′

circ.pol.

F −E −Cx′ −Cz′

28

SLIDE 29

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

New MAMI data on γp → π0p

10 20 30 40 50 60 70 80 90 100 1300 1400 1500 1600 1700 1800

3/2 3/2 + 1/2 3/2 − 1/2 5/2 + 3/2 7/2 +

σtot, µb MAMI 2015 W, MeV

2.5 5

1

1

1478

2.5 5

1

1

1480.75

2.5 5

1

1

1483.49

2.5 5

1

1

1488.96

2.5 5

1

1

1491.69

2.5 5

1

1

1494.4

2.5 5

1

1

1497.11

2.5 5

1

1

1499.81

2.5 5

1

1

1502.51

2.5 5

1

1

1505.14

2.5 5

1

1

1507.88

2.5 5

1

1

1510.55

2.5 5

1

1

1513.22

2.5 5

1

1

1515.88

2.5 5

1

1

1518.53

2.5 5

1

1

1521.18

2.5 5

1

1

1523.81

2.5 5

1

1

1529.07

2.5 5

1

1

1531.68

2.5 5

1

1

1534.31

2.5 5

1

1

1536.84

2.5 5

1

1

1539.5

2.5 5

1

1

1542.08

2.5 5

1

1

1544.66

2.5 5

1

1

1547.24

cos θ 29

SLIDE 30

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The γp → KΛ reaction (CLAS 2009)

0.5 1 1.5 2 2.5 3 1600 1800 2000 2200 2400 1/2 1/2 − 1/2 1/2 + 1/2 3/2 + 1/2 5/2 +

M(γp), MeV σtot, µb

0.2 1625

dσ/dΩ, µb/sr

1635 1645 1655 1665 0.25 1675 1685 1695 1705 1715 0.25 1725 1735 1745 1755 1765 0.25 1775 1785 1795 1805 1815 0.25 1825 1835 1845 1855 1865 0.25 1875 1885 1895 1905 1915 0.25 1925 1935 1945 1965 1975 0.25 1985 1995 2005 2015 2025 0.25 2035 2045 2055 2065 2075 0.25 2085 2095 2105 2115 2125 0.2 2135 2145 2155 2165 2175 0.2 2185 2195 2205 2215 2225 0.2 2235 2245 2255 2265 2275 0.2 2285 2295 2305 2315 2325 0.2 2335

0.5 0 0.5

2345

0.5 0 0.5

2355

0.5 0 0.5

2365

0.5 0 0.5

2375

0.5 0 0.5

cos θcm

New S11 state with mass 1890 ± 10 MeV and width 90 ± 10 MeV improves description

f the data.

30

SLIDE 31

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The Ox, Oz and T (GRAAL) observables from γp → KΛ

description is notably improved with S11(1890)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

31

SLIDE 32

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The γp → K+Λ: Cx, Cz (CLAS) and beam asymmetry (GRAAL) BG2011-02 M (dashed) BG2013-02 (solid)

1678

Cx Cz

2

2

1733 1787 1838

2

2

1889 1939 1987

2

2

2035 2081 2126

2

2

2169 2212 2255

2

2

2296 2338 2377

2

2

0.5

0.5

2416

0.5

0.5

2454

0.5

0.5

1649

Σ

0.5

0.5

1676 1702 1728

0.5

0.5

1754 1781 1808

0.5

0.5

1833 1859

0.5

0.5

1883

0.5

0.5

0.5

0.5

1906

0.5

0.5

cos θK

32

SLIDE 33

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

New resonances are found. One of them has 3* and was proposed to be defined as 4* state

Citation: J. Beringer et al. (Particle Data Group), PR D86, 010001 (2012) and 2013 partial update for the 2014 edition (URL: http://pdg.lbl.gov)

N(1895) 1/2−

I(JP) =

1 2( 1 2−) Status: ∗∗

OMITTED FROM SUMMARY TABLE

The latest GWU analysis (ARNDT 06) finds no evidence for this resonance.

N(1895) BREIT-WIGNER MASS N(1895) BREIT-WIGNER MASS N(1895) BREIT-WIGNER MASS N(1895) BREIT-WIGNER MASS

VALUE (MeV) DOCUMENT ID TECN COMMENT

≈ 2090 OUR ESTIMATE ≈ 2090 OUR ESTIMATE ≈ 2090 OUR ESTIMATE ≈ 2090 OUR ESTIMATE 1895±15 ANISOVICH 12A DPWA Multichannel 2180±80 CUTKOSKY 80 IPWA πN → πN 1880±20 HOEHLER 79 IPWA πN → πN

• • We do not use the following data for averages, fits, limits, etc. • • •

1910±15 SHRESTHA 12A DPWA Multichannel 1812±25 BATINIC 10 DPWA πN → N π, N η 1822±43 VRANA 00 DPWA Multichannel 1897±50+30 − 2 PLOETZKE 98 SPEC γ p → p η′(958) 1928±59 MANLEY 92 IPWA πN → πN & N ππ

N(1895) BREIT-WIGNER WIDTH N(1895) BREIT-WIGNER WIDTH N(1895) BREIT-WIGNER WIDTH N(1895) BREIT-WIGNER WIDTH

VALUE (MeV) DOCUMENT ID TECN COMMENT

90+ 30 − 15 ANISOVICH 12A DPWA Multichannel 350±100 CUTKOSKY 80 IPWA πN → πN 95± 30 HOEHLER 79 IPWA πN → πN

Citation: J. Beringer et al. (Particle Data Group), PR D86, 010001 (2012) and 2013 partial update for the 2014 edition (URL: http://pdg.lbl.gov)

N(1900) 3/2+

I(JP) =

1 2( 3 2+) Status: ∗∗∗

The latest GWU analysis (ARNDT 06) finds no evidence for this resonance.

N(1900) BREIT-WIGNER MASS N(1900) BREIT-WIGNER MASS N(1900) BREIT-WIGNER MASS N(1900) BREIT-WIGNER MASS

VALUE (MeV) DOCUMENT ID TECN COMMENT

≈ 1900 OUR ESTIMATE ≈ 1900 OUR ESTIMATE ≈ 1900 OUR ESTIMATE ≈ 1900 OUR ESTIMATE 1905±30 ANISOVICH 12A DPWA Multichannel 1915±60 NIKONOV 08 DPWA Multichannel

• • We do not use the following data for averages, fits, limits, etc. • • •

1900± 8 SHRESTHA 12A DPWA Multichannel 1951±53 PENNER 02C DPWA Multichannel 1879±17 MANLEY 92 IPWA πN → πN & N ππ

N(1900) BREIT-WIGNER WIDTH N(1900) BREIT-WIGNER WIDTH N(1900) BREIT-WIGNER WIDTH N(1900) BREIT-WIGNER WIDTH

VALUE (MeV) DOCUMENT ID TECN COMMENT

∼ 250 OUR ESTIMATE ∼ 250 OUR ESTIMATE ∼ 250 OUR ESTIMATE ∼ 250 OUR ESTIMATE 250+120 − 50 ANISOVICH 12A DPWA Multichannel 180± 40 NIKONOV 08 DPWA Multichannel

• • We do not use the following data for averages, fits, limits, etc. • • •

101± 15 SHRESTHA 12A DPWA Multichannel 622± 42 PENNER 02C DPWA Multichannel 498± 78 MANLEY 92 IPWA πN → πN & N ππ

33

SLIDE 34

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Parameterization of the partial wave amplitude

A1i = K1j(I − iρK)−1

ji

and

Kij = ∑

α

gα

i gα j

M 2

α − s + fij(s)

fij = f (1)

ij + f (2) ij

√s s − sij .

where fij is non-resonant transition part. For the small coupled initial state, e.g. photoproduction:

Ak = Pj(I − iρK)−1

jk

The vector of the initial interaction has the form:

Pj = ∑

α

Λαgα

j

M 2

α − s + Fj(s)

Here Fj is non-resonant production of the final state j.

34

SLIDE 35

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Three body phase volume:

ρ3(s) =

(√s−m1)2

∫

(m2+m3)2

ds23 π ρ(s, √s23, m1) MRΓR

tot

(M 2

R−s23)2+(MRΓR tot)2 ,

MRΓR

tot = ρ(s23, m2, m3)g2(s23) ,

0.8
0.6
0.4
0.2

0.2 1.4 1.6 1.8 2 2.2 2.4

1 1′ 2 2′ 3 3′ 4 4′ scut

Re s, GeV2 Im s, GeV2

0.4
0.3
0.2
0.1

0.1 0.2 1 1.2 1.4 1.6 1.8 2

MR

2 - i MR ΓR

(√s

-m1)2

(m2+m3)2 A A B B

Re s23, GeV2 Im s23, GeV2

35

SLIDE 36

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

N/D based (D-matrix) analysis of the data

J m J K m δJK

π η K π η K

Djm = Djk ∑

α

Bkm

α (s)

1 Mm − s + δjm M 2

j − s

ˆ D = ˆ κ(I − ˆ Bˆ κ)−1 ˆ κ = diag ( 1 M 2

1 − s,

1 M 2

2 − s, . . . ,

1 M 2

N − s, R1, R2 . . .

) ˆ Bij = ∑

α

Bij

α =

∑

α

∫ ds′ π g(R)i

α

ρα(s′, m1α, m2α)g(L)j

α

s′ − s − i0

36

SLIDE 37

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

In the present fits we calculate the elements of the Bij

α using one subtraction taken at

the channel threshold Mα = (m1α + m2α):

Bij

α (s) = Bij α (M 2 α) + (s − M 2 α) ∞

∫

m2

a

ds′ π g(R)i

α

ρα(s′, m1α, m2α)g(L)j

α

(s′ − s − i0)(s′ − M 2

α)

.

In this case the expression for elements of the ˆ

B matrix can be rewritten as: Bij

α (s) = g(R)i a

  bα + (s − M 2

α) ∞

∫

m2

a

ds′ π ρα(s′, m1α, m2α) (s′ − s − i0)(s′ − M 2

α)

   g(L)j

β

= g(R)i

a

Bαg(L)j

β

and D-matrix method equivalent to the K-matrix method with loop diagram with real part taken into account:

A = ˆ K(I − ˆ B ˆ K)−1 Bαβ = δαβBα

37

SLIDE 38

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Pole position of the resonances

20 40 60 80 100 120 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

√s

, MeV

δ0

0, deg

a)

K-matrix D-matrix

σ-meson

420-i 395 407-i 281

f0(980)

1014-i 31 1015-i 36

f0(1300)

1302-i 140 1307-i 137

f0(1500)

1487-i 58 1487-i 60

f0(1750)

1738-i 152 1781-i 140

Re M Im M

−250 500 1500 1000

445+16

−8 − i272+9 −13

I.Caprini, G.Colangelo, and H.Leutwyler, Phys.Rev.Lett.96, 132001 (2006)

457+14

−15 − i279+11 −7

R.Garcia-Martin, R.Kaminski, J.R.Pelaez, J.Ruiz de Elvira, and F.J.Yndurain

38

SLIDE 39

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Pole parameters of the S11 states

N(1535)S11 N(1650)S11 N(1890)S11

K-matrix D-matrix K-matrix D-matrix K-matrix D-matrix

Mpole

1501±4 1494 1647±6 1651 1900±15 1905

Γpole

134±11 116 103±8 95

90+30

−15

106 Elastic residue 31±4 25 24±3 23 1±1 1.5 Phase

(29±5)o
38o
(75±12)o
62o

– – ResπN→Nη 28±3 25 15±3 15 4±2 5 Phase

(76±8)o
69o

(132±10)o 140 (40±20)o 42o ResπN→∆π 7±4 4 11±3 12 – – Phase (147±17)o 157o

(30±20)o
40

– –

A1/2 ( GeV− 1

2 )

0.116±0.010 0.107 0.033±0.007 0.029 0.012±0.006 0.010 Phase (7±6)o 1o

(9±15)o

0o 120±50o 150o

39

SLIDE 40

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Minimization methods

1. The two body final states πN, γN → πN, ηN, KΛ, KΣ, ωN, K∗Λ: χ2 method.

For n measured bins we minimize

χ2 =

n

∑

j

(σj(PWA) − σj(exp))2 (∆σj(exp))2

Present solution χ2 = 48710 for 31180 points. χ2/NF = 1.6

2. Reactions with three or more final states are analyzed with logarithm likelihood
method. πN, γN → ππN, πηN, ωp, K∗Λ. The minimization function:

f = −

N(data)

∑

j

ln σj(PWA)

N(rec MC)

∑

m

σm(PWA)

This method allows us to take into account all correlations in many dimensional phase space. Above 500 000 data events are taken in the fit.

40

SLIDE 41

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Resonance amplitudes for meson photoproduction

)

1

u(k )

2

(k ∈

3

q )

1

(q u

π π p → π

2

R →

1

R → p γ

2

R

(L, S) )

2

R π

, S

2

R π

(L

2

q

1

R General form of the angular dependent part of the amplitude:

¯ u(q1) ˜ Nα1...αn(R2 →µN)F α1...αn

β1...βn (q1 + q2) ˜

N (j)β1...βn

γ1...γm

(R1 →µR2) F γ1...γm

ξ1...ξm (P)V (i)µ ξ1...ξm(R1 →γN)u(k1)εµ

F µ1...µL

ν1...νL (p) = (m+ˆ

p)Oµ1...µL

α1...αL

L + 1 2L+1 ( g⊥

α1β1 −

L L+1σα1β1 )

L

∏

i=2

gαiβiOβ1...βL

ν1...νL

σαiαj = 1 2(γαiγαj − γαjγαi)

41

SLIDE 42

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

DATA BG2011-2014 added in BG2015-2016

πN → πN ampl.

SAID or Hoehler energy fixed

γp → πN

dσ dΩ , Σ, T, P, E, G, H dσ dΩ E, G, T, P, H, F (CB-ELSA, CLAS, MAMI)

γn → πN

dσ dΩ, Σ, T, P dσ dΩ (MAMI), Σ (CLAS)

γn → Λn, Σ−p

dσ

dΩ

γn → ηn

dσ dΩ , Σ dσ dΩ (MAMI)

γp → ηp

dσ dΩ , Σ

T, P, H, E, F γp → η′p

dσ dΩ , Σ

γp → K+Λ

dσ dΩ , Σ, P, T, Cx, Cz, Ox′, Oz′

Σ, P, T, Ox, Oz (CLAS) γp → K+Σ0

dσ dΩ , Σ, P, Cx, Cz

Σ, P, T, Ox, Oz (CLAS) γp → K0Σ+

dσ dΩ, Σ, P

π−p → ηn

dσ dΩ

π−p → K0Λ

dσ dΩ , P, β

π−p → K0Σ0

dσ dΩ , P (K0Σ0) dσ dΩ (K+Σ−)

π+p → K+Σ+

dσ dΩ , P, β

π−p → π0π0n

dσ dΩ (Crystal Ball)

π−p → π+π−n

dσ dΩ (HADES)

π−p → π−π0p

dσ dΩ (HADES)

γp → π0π0p

dσ dΩ , Σ, E, Ic, Is

T, P, H, F, Px, Py γp → π0ηp

dσ dΩ , Σ, Ic, Is

γp → π+π−p

dσ dΩ , Ic, Is (CLAS)

γp → ωp

dσ dΩ, Σ, ρ0 ij, ρ1 ij, ρ2 ij, E, G (CB-ELSA)

42

SLIDE 43

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The description of the γp → π0π0p and γp → π+π−p data (preliminary)

1.2 1.4 1.6 1.8 2 2.2 1000 2000 3000 4000 5000 6000 7000 8000 9000

M23

Total

p) π M (

Total

p) π M (

Total

p) π M (

Total

p) π M (

Total

p) π M (

Total

p) π M (

Total

0.4 0.6 0.8 1 1.2 1.4 1000 2000 3000 4000 5000 6000

M12

Total

) π π M (

Total

) π π M (

Total

) π π M (

Total

) π π M (

Total

) π π M (

Total

) π π M (

Total

1
0.5

0.5 1 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

Z1

Total

) π ( Θ cos

Total

) π ( Θ cos

Total

) π ( Θ cos

Total

) π ( Θ cos

Total

) π ( Θ cos

Total

) π ( Θ cos

Total

1
0.5

0.5 1 500 1000 1500 2000 2500

Z3

Total

(p) Θ cos

Total

(p) Θ cos

Total

(p) Θ cos

Total

(p) Θ cos

Total

(p) Θ cos

Total

(p) Θ cos

Total

1
0.5

0.5 1 200 400 600 800 1000 1200 1400 1600 1800 2000

Z23

Total

p) π ( Θ cos

Total

p) π ( Θ cos

Total

p) π ( Θ cos

Total

p) π ( Θ cos

Total

p) π ( Θ cos

Total

p) π ( Θ cos

Total

1
0.5

0.5 1 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

Z12

Total

) π π ( Θ cos

Total

) π π ( Θ cos

Total

) π π ( Θ cos

Total

) π π ( Θ cos

Total

) π π ( Θ cos

Total

) π π ( Θ cos

Total 1.2 1.4 1.6 1.8 2 2.2 500 1000 1500 2000 2500 3000 3500 4000 4500 M23 Total

)

π M (p Total

)

π M (p Total

)

π M (p Total

)

π M (p Total

)

π M (p Total

)

π M (p Total 1.2 1.4 1.6 1.8 2 2.2 1000 2000 3000 4000 5000 6000 7000 M13 Total +) π M (p Total +) π M (p Total +) π M (p Total +) π M (p Total +) π M (p Total +) π M (p Total 0.4 0.6 0.8 1 1.2 1.4 1000 2000 3000 4000 5000 6000 M12 Total

)

π + π M ( Total

)

π + π M ( Total

)

π + π M ( Total

)

π + π M ( Total

)

π + π M ( Total

)

π + π M ( Total

1
0.5

0.5 1 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Z1 Total +) π ( Θ cos Total +) π ( Θ cos Total +) π ( Θ cos Total +) π ( Θ cos Total +) π ( Θ cos Total +) π ( Θ cos Total

1
0.5

0.5 1 500 1000 1500 2000 2500 Z2 Total

)

π ( Θ cos Total

)

π ( Θ cos Total

)

π ( Θ cos Total

)

π ( Θ cos Total

)

π ( Θ cos Total

)

π ( Θ cos Total

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 Z3 Total (p) Θ cos Total (p) Θ cos Total (p) Θ cos Total (p) Θ cos Total (p) Θ cos Total (p) Θ cos Total

1
0.5

0.5 1 200 400 600 800 1000 1200 1400 1600 1800 Z23 Total

p)

π ( Θ cos Total

p)

π ( Θ cos Total

p)

π ( Θ cos Total

p)

π ( Θ cos Total

p)

π ( Θ cos Total

p)

π ( Θ cos Total

1
0.5

0.5 1 200 400 600 800 1000 1200 1400 1600 1800 2000 Z13 Total +) π (p Θ cos Total +) π (p Θ cos Total +) π (p Θ cos Total +) π (p Θ cos Total +) π (p Θ cos Total +) π (p Θ cos Total

1
0.5

0.5 1 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Z12 Total

)

π + π ( Θ cos Total

)

π + π ( Θ cos Total

)

π + π ( Θ cos Total

)

π + π ( Θ cos Total

)

π + π ( Θ cos Total

)

π + π ( Θ cos Total

43

SLIDE 44

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The description of the γp → π0π0p and γp → π+π−p data for 1700 MeV< W,1800 MeV (preliminary)

1.1 1.2 1.3 1.4 1.5 1.6 1.7 200 400 600 800 1000 1200 1400 1600 1800 2000

M23M(123)-1.72

1.660<M<1.780

p) π M (

1.660<M<1.780

p) π M (

1.660<M<1.780

p) π M (

1.660<M<1.780

p) π M (

1.660<M<1.780

p) π M (

1.660<M<1.780

p) π M (

1.660<M<1.780

0.3 0.4 0.5 0.6 0.7 0.8 0.9 200 400 600 800 1000 1200 1400 1600 1800

M12M(123)-1.72

1.660<M<1.780

) π π M (

1.660<M<1.780

) π π M (

1.660<M<1.780

) π π M (

1.660<M<1.780

) π π M (

1.660<M<1.780

) π π M (

1.660<M<1.780

) π π M (

1.660<M<1.780

1
0.5

0.5 1 400 500 600 700 800 900 1000 1100 1200 1300

Z1M(123)-1.72

1.660<M<1.780

) π ( Θ cos

1.660<M<1.780

) π ( Θ cos

1.660<M<1.780

) π ( Θ cos

1.660<M<1.780

) π ( Θ cos

1.660<M<1.780

) π ( Θ cos

1.660<M<1.780

) π ( Θ cos

1.660<M<1.780

1
0.5

0.5 1 200 400 600 800 1000 1200 1400

Z3M(123)-1.72

1.660<M<1.780

(p) Θ cos

1.660<M<1.780

(p) Θ cos

1.660<M<1.780

(p) Θ cos

1.660<M<1.780

(p) Θ cos

1.660<M<1.780

(p) Θ cos

1.660<M<1.780

(p) Θ cos

1.660<M<1.780

1
0.5

0.5 1 400 500 600 700 800 900 1000 1100 1200

Z23M(123)-1.72

1.660<M<1.780

p) π ( Θ cos

1.660<M<1.780

p) π ( Θ cos

1.660<M<1.780

p) π ( Θ cos

1.660<M<1.780

p) π ( Θ cos

1.660<M<1.780

p) π ( Θ cos

1.660<M<1.780

p) π ( Θ cos

1.660<M<1.780

1
0.5

0.5 1 400 600 800 1000 1200

Z12M(123)-1.72

1.660<M<1.780

) π π ( Θ cos

1.660<M<1.780

) π π ( Θ cos

1.660<M<1.780

) π π ( Θ cos

1.660<M<1.780

) π π ( Θ cos

1.660<M<1.780

) π π ( Θ cos

1.660<M<1.780

) π π ( Θ cos

1.660<M<1.780 1.1 1.2 1.3 1.4 1.5 1.6 1.7 200 400 600 800 1000 1200 1400 M23M(123)-1.75 1.70<M<1.80

)

π M (p 1.70<M<1.80

)

π M (p 1.70<M<1.80

)

π M (p 1.70<M<1.80

)

π M (p 1.70<M<1.80

)

π M (p 1.70<M<1.80

)

π M (p 1.70<M<1.80 1.1 1.2 1.3 1.4 1.5 1.6 1.7 200 400 600 800 1000 1200 1400 1600 1800 2000 M13M(123)-1.75 1.70<M<1.80 +) π M (p 1.70<M<1.80 +) π M (p 1.70<M<1.80 +) π M (p 1.70<M<1.80 +) π M (p 1.70<M<1.80 +) π M (p 1.70<M<1.80 +) π M (p 1.70<M<1.80 0.3 0.4 0.5 0.6 0.7 0.8 0.9 200 400 600 800 1000 1200 M12M(123)-1.75 1.70<M<1.80

)

π + π M ( 1.70<M<1.80

)

π + π M ( 1.70<M<1.80

)

π + π M ( 1.70<M<1.80

)

π + π M ( 1.70<M<1.80

)

π + π M ( 1.70<M<1.80

)

π + π M ( 1.70<M<1.80

1
0.5

0.5 1 100 200 300 400 500 600 Z1M(123)-1.75 1.70<M<1.80 +) π ( Θ cos 1.70<M<1.80 +) π ( Θ cos 1.70<M<1.80 +) π ( Θ cos 1.70<M<1.80 +) π ( Θ cos 1.70<M<1.80 +) π ( Θ cos 1.70<M<1.80 +) π ( Θ cos 1.70<M<1.80

1
0.5

0.5 1 100 200 300 400 500 600 700 800 Z2M(123)-1.75 1.70<M<1.80

)

π ( Θ cos 1.70<M<1.80

)

π ( Θ cos 1.70<M<1.80

)

π ( Θ cos 1.70<M<1.80

)

π ( Θ cos 1.70<M<1.80

)

π ( Θ cos 1.70<M<1.80

)

π ( Θ cos 1.70<M<1.80

1
0.5

0.5 1 100 200 300 400 500 600 700 800 900 Z3M(123)-1.75 1.70<M<1.80 (p) Θ cos 1.70<M<1.80 (p) Θ cos 1.70<M<1.80 (p) Θ cos 1.70<M<1.80 (p) Θ cos 1.70<M<1.80 (p) Θ cos 1.70<M<1.80 (p) Θ cos 1.70<M<1.80

1
0.5

0.5 1 200 300 400 500 600 700 800 Z23M(123)-1.75 1.70<M<1.80

p)

π ( Θ cos 1.70<M<1.80

p)

π ( Θ cos 1.70<M<1.80

p)

π ( Θ cos 1.70<M<1.80

p)

π ( Θ cos 1.70<M<1.80

p)

π ( Θ cos 1.70<M<1.80

p)

π ( Θ cos 1.70<M<1.80

1
0.5

0.5 1 200 300 400 500 600 700 Z13M(123)-1.75 1.70<M<1.80 +) π (p Θ cos 1.70<M<1.80 +) π (p Θ cos 1.70<M<1.80 +) π (p Θ cos 1.70<M<1.80 +) π (p Θ cos 1.70<M<1.80 +) π (p Θ cos 1.70<M<1.80 +) π (p Θ cos 1.70<M<1.80

1
0.5

0.5 1 100 200 300 400 500 600 700 800 900 Z12M(123)-1.75 1.70<M<1.80

)

π + π ( Θ cos 1.70<M<1.80

)

π + π ( Θ cos 1.70<M<1.80

)

π + π ( Θ cos 1.70<M<1.80

)

π + π ( Θ cos 1.70<M<1.80

)

π + π ( Θ cos 1.70<M<1.80

)

π + π ( Θ cos 1.70<M<1.80

44

SLIDE 45

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Ic and Is for γp → π+π−p from CLAS (Preliminary)

Courtesy of V. Crede, Florida State U

Is Is

1
0.5

0.5 1 Is 1400-1450 cosθ (-1 -0.8) 1400-1450 cosθ (-0.8 -0.6) 1400-1450 cosθ (-0.6 -0.4) 1400-1450 cosθ (-0.4 -0.2) 1400-1450 cosθ (-0.2 0)

1
0.5

0.5 1 1400-1450 cosθ (0 0.2) 1400-1450 cosθ (0.2 0.4) 1400-1450 cosθ (0.4 0.6) 1400-1450 cosθ (0.6 0.8) 1400-1450 cosθ (0.8 1)

1
0.5

0.5 1 1450-1500 cosθ (-1 -0.8) 1450-1500 cosθ (-0.8 -0.6) 1450-1500 cosθ (-0.6 -0.4) 1450-1500 cosθ (-0.4 -0.2) 1450-1500 cosθ (-0.2 0)

1
0.5

0.5 1

90

90 1450-1500 cosθ (0 0.2)

90

90 1450-1500 cosθ (0.2 0.4)

90

90 1450-1500 cosθ (0.4 0.6)

90

90 1450-1500 cosθ (0.6 0.8)

90

90 1450-1500 cosθ (0.8 1) φ(π+)

1
0.5

0.5 1 Is 1500-1550 cosθ (-1 -0.8) 1500-1550 cosθ (-0.8 -0.6) 1500-1550 cosθ (-0.6 -0.4) 1500-1550 cosθ (-0.4 -0.2) 1500-1550 cosθ (-0.2 0)

1
0.5

0.5 1 1500-1550 cosθ (0 0.2) 1500-1550 cosθ (0.2 0.4) 1500-1550 cosθ (0.4 0.6) 1500-1550 cosθ (0.6 0.8) 1500-1550 cosθ (0.8 1)

1
0.5

0.5 1 1550-1600 cosθ (-1 -0.8) 1550-1600 cosθ (-0.8 -0.6) 1550-1600 cosθ (-0.6 -0.4) 1550-1600 cosθ (-0.4 -0.2) 1550-1600 cosθ (-0.2 0)

1
0.5

0.5 1

90

90 1550-1600 cosθ (0 0.2)

90

90 1550-1600 cosθ (0.2 0.4)

90

90 1550-1600 cosθ (0.4 0.6)

90

90 1550-1600 cosθ (0.6 0.8)

90

90 1550-1600 cosθ (0.8 1) φ(π+)

1
0.5

0.5 1 Is 1600-1650 cosθ (-1 -0.8) 1600-1650 cosθ (-0.8 -0.6) 1600-1650 cosθ (-0.6 -0.4) 1600-1650 cosθ (-0.4 -0.2) 1600-1650 cosθ (-0.2 0)

1
0.5

0.5 1 1600-1650 cosθ (0 0.2) 1600-1650 cosθ (0.2 0.4) 1600-1650 cosθ (0.4 0.6) 1600-1650 cosθ (0.6 0.8) 1600-1650 cosθ (0.8 1)

1
0.5

0.5 1 1650-1700 cosθ (-1 -0.8) 1650-1700 cosθ (-0.8 -0.6) 1650-1700 cosθ (-0.6 -0.4) 1650-1700 cosθ (-0.4 -0.2) 1650-1700 cosθ (-0.2 0)

1
0.5

0.5 1

90

90 1650-1700 cosθ (0 0.2)

90

90 1650-1700 cosθ (0.2 0.4)

90

90 1650-1700 cosθ (0.4 0.6)

90

90 1650-1700 cosθ (0.6 0.8)

90

90 1650-1700 cosθ (0.8 1) φ(π+)

1
0.5

0.5 1 Is 1700-1750 cosθ (-1 -0.8) 1700-1750 cosθ (-0.8 -0.6) 1700-1750 cosθ (-0.6 -0.4) 1700-1750 cosθ (-0.4 -0.2) 1700-1750 cosθ (-0.2 0)

1
0.5

0.5 1 1700-1750 cosθ (0 0.2) 1700-1750 cosθ (0.2 0.4) 1700-1750 cosθ (0.4 0.6) 1700-1750 cosθ (0.6 0.8) 1700-1750 cosθ (0.8 1)

1
0.5

0.5 1 1750-1800 cosθ (-1 -0.8) 1750-1800 cosθ (-0.8 -0.6) 1750-1800 cosθ (-0.6 -0.4) 1750-1800 cosθ (-0.4 -0.2) 1750-1800 cosθ (-0.2 0)

1
0.5

0.5 1

90

90 1750-1800 cosθ (0 0.2)

90

90 1750-1800 cosθ (0.2 0.4)

90

90 1750-1800 cosθ (0.4 0.6)

90

90 1750-1800 cosθ (0.6 0.8)

90

90 1750-1800 cosθ (0.8 1) φ(π+)

45

SLIDE 46

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The total cross section from the

π−p → π+π−n and π−p → 2π0n data

1.46 1.48 1.5 1.52 1.54 1 2 3 4 5 6 7

Graph

1/2 1/2 + all 1/2 3/2 - all s-chan D(1232)-pi s-chan N-sigma s-chan N(940)-rho

Graph

1.3 1.35 1.4 1.45 1.5 1.55 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

Graph

1/21/2+ all 1/23/2- all 1/21/2+ D(1232)-pi 1/21/2+ N-sigma

Graph

46

SLIDE 47

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Crystal Ball data on π−p → π0π0n

655 (MeV/c) 691 (MeV/c) 748 (MeV/c)

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 500 1000 1500 2000 2500 3000 3500 4000 4500

M230655 0655 p) π M ( 0655 p) π M ( 0655 p) π M ( 0655 p) π M ( 0655 p) π M ( 0655 p) π M ( 0655

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 500 1000 1500 2000 2500 3000 3500 4000M120655

0655 ) π π M ( 0655 ) π π M ( 0655 ) π π M ( 0655 ) π π M ( 0655 ) π π M ( 0655 ) π π M ( 0655

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500Z10655

0655 ) π ( Θ cos 0655 ) π ( Θ cos 0655 ) π ( Θ cos 0655 ) π ( Θ cos 0655 ) π ( Θ cos 0655 ) π ( Θ cos 0655

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500 4000

Z30655 0655 (p) Θ cos 0655 (p) Θ cos 0655 (p) Θ cos 0655 (p) Θ cos 0655 (p) Θ cos 0655 (p) Θ cos 0655

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500 4000

Z230655 0655 p) π ( Θ cos 0655 p) π ( Θ cos 0655 p) π ( Θ cos 0655 p) π ( Θ cos 0655 p) π ( Θ cos 0655 p) π ( Θ cos 0655

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500 4000

Z120655 0655 ) π π ( Θ cos 0655 ) π π ( Θ cos 0655 ) π π ( Θ cos 0655 ) π π ( Θ cos 0655 ) π π ( Θ cos 0655 ) π π ( Θ cos 0655

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1000 2000 3000 4000 5000

M230691 0691 p) π M ( 0691 p) π M ( 0691 p) π M ( 0691 p) π M ( 0691 p) π M ( 0691 p) π M ( 0691

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 500 1000 1500 2000 2500 3000 3500 4000M120691

0691 ) π π M ( 0691 ) π π M ( 0691 ) π π M ( 0691 ) π π M ( 0691 ) π π M ( 0691 ) π π M ( 0691

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500 Z10691

0691 ) π ( Θ cos 0691 ) π ( Θ cos 0691 ) π ( Θ cos 0691 ) π ( Θ cos 0691 ) π ( Θ cos 0691 ) π ( Θ cos 0691

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500 4000

Z30691 0691 (p) Θ cos 0691 (p) Θ cos 0691 (p) Θ cos 0691 (p) Θ cos 0691 (p) Θ cos 0691 (p) Θ cos 0691

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500 4000

Z230691 0691 p) π ( Θ cos 0691 p) π ( Θ cos 0691 p) π ( Θ cos 0691 p) π ( Θ cos 0691 p) π ( Θ cos 0691 p) π ( Θ cos 0691

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500 4000 4500Z120691

0691 ) π π ( Θ cos 0691 ) π π ( Θ cos 0691 ) π π ( Θ cos 0691 ) π π ( Θ cos 0691 ) π π ( Θ cos 0691 ) π π ( Θ cos 0691

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1000 2000 3000 4000 5000

M230748 0748 p) π M ( 0748 p) π M ( 0748 p) π M ( 0748 p) π M ( 0748 p) π M ( 0748 p) π M ( 0748

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 500 1000 1500 2000 2500 3000 3500

M120748 0748 ) π π M ( 0748 ) π π M ( 0748 ) π π M ( 0748 ) π π M ( 0748 ) π π M ( 0748 ) π π M ( 0748

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500Z10748

0748 ) π ( Θ cos 0748 ) π ( Θ cos 0748 ) π ( Θ cos 0748 ) π ( Θ cos 0748 ) π ( Θ cos 0748 ) π ( Θ cos 0748

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500

Z30748 0748 (p) Θ cos 0748 (p) Θ cos 0748 (p) Θ cos 0748 (p) Θ cos 0748 (p) Θ cos 0748 (p) Θ cos 0748

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500 4000Z230748

0748 p) π ( Θ cos 0748 p) π ( Θ cos 0748 p) π ( Θ cos 0748 p) π ( Θ cos 0748 p) π ( Θ cos 0748 p) π ( Θ cos 0748

1
0.5

0.5 1 500 1000 1500 2000 2500 3000 3500 4000

Z120748 0748 ) π π ( Θ cos 0748 ) π π ( Θ cos 0748 ) π π ( Θ cos 0748 ) π π ( Θ cos 0748 ) π π ( Θ cos 0748 ) π π ( Θ cos 0748

—— 1/2 1/2+ —— 1/2 3/2-

47

SLIDE 48

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Meson production in NN collision

A = ( ¯ u(p′

1)V S′,L′ µ1...µJ(k′ ⊥)uc(−p′ 2)

) Oµ1...µn

ν1...νn

( ¯ uc(−p2)GJ,P

ν1...νJu(p1)

) Apw(s) .

Let us consider the transition from the NN state with JP into a pseudoscalar meson with momentum k1 and NN system with momenta k2, k3 in state with S′, L′, j, P ′ . For P = P ′(−1)J+j+1 we have the following vertex:

GJ,P

µ1...µJ = V S′,L′ µ1...µkνk+1...νj(k23)Oνk+1...νjµk+1µJ α1...αJ+j−2k

(k2+k3)X(L)

α1...αJ+j−2k(k⊥ 1 ),

Here k = 0, . . . j and L = J +j−2k ≥ 0. For P = P ′(−1)J+j:

GJ,P

µ1...µJ

= εµ1αβηV S′,L′

αµ2...µkνk+1...νj(k23)Oνk+1...νjµk+1...µJ α1...αJ+j−2k+1

(k2+k3) × X(L)

βα1...αJ+j−2k+1(k⊥ 1 )Pη,

Here k = 1, . . . j and L = J +j−2k + 1 ≥ 0.

48

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Partial wave analysis of HADES pp → K+Λp data

No P11(1710) No S11(1650) Both are included

M(KΛ) Total

100 200 300 400 500 600 700 800 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

M(KΛ) Total

100 200 300 400 500 600 700 800 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

M(KΛ) Total

100 200 300 400 500 600 700 800 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

S11(1/2−) P11(1/2+) P13(3/2+)

49

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Partial wave contributions to π−p → KΛ and pp → K+Λp

M(KΛ) Total

100 200 300 400 500 600 700 800 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

S11(1/2−) P11(1/2+) P13(3/2+) πN + γN pp → K+Λp P11(1710) 1690 ± 10 1692 ± 9 168 ± 27 170 ± 20 S11(1895) 1891 ± 7 1907 ± 15 84 ± 22 100+40

−15

P13(1900) 1906 ± 19 1910 ± 30 290 ± 55 280 ± 50

For HADES pp → K+Λp only systematic errors are given.

50

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The following data sets were analyzed in the framework of event-by-event maximum likelihood approach:

n

Reaction

pbeam Ndata

Origin 1

pp → π0pp

1683 MeV/c 1094 Gatchina 2

pp → π0pp

1581 MeV/c 903 Gatchina 3

pp → π0pp

1536 MeV/c 1319 Gatchina 4

pp → π0pp

1485 MeV/c 997 Gatchina 5

pp → π0pp

1437 MeV/c 918 Gatchina 6

pp → π0pp

1389 MeV/c 996 Gatchina 7

pp → π0pp

1341 MeV/c 883 Gatchina 8

pp → π0pp

1279 MeV/c 621 Gatchina 9

pp → π0pp

1217 MeV/c 544 Gatchina 10

np → π−pp

1-1.9 GeV/c 8210 Gatchina 11

pp → π0pp

950 MeV/c 154972 T¨ ubingen 12

pp → π+pn

2032 MeV/c 7902 T¨ ubingen 13

pp → π0pp σtot 1217-1683 MeV

9 Gatchina

51

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Parameterization

dσ = (2π)4|A|2 4|⃗ k|√s dΦ3(P, q1, q2, q3) , A = ∑

α

Aα

tr(s)Qin µ1...µJ(SLJ)A2b(i, S2L2J2)(si)Qfin µ1...µJ(i, S2L2J2S′L′J) .

Angular-spin momentum operators Qµ1...µJ(SLJ) are given in

A. V. Anisovich et. al Eur.Phys.J. A34 (2007) 129.

Aα

tr(s) = aα 1 + aα 3

√s s − aα

4

eiaα

2 ,

Decay modes: ∆(1232)N, P11(1440)N and π(NN). In NN channel amplitude was parameterized with generalized Watson-Migdal formula:

Aβ

2b(si) =

√si 1 − 1

2rβq2aβ pp + iqaβ ppq2L/F(q, rβ, L)

,

52

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Description of pp → ppπ0:

cos(π)

pp→ppπ0 (Gatchina 1437 MeV)

25 50 75 100

1
0.5

0.5 1

cos(p)

25 50 75 100

1
0.5

0.5 1

Mπp, GeV

50 100 1.1 1.2 1.3

Mpp, GeV

50 100 1.9 2 2.1

cos(π)h in πp

20 40 60

1
0.5

0.5 1

cos(p)h in pp

20 40 60

1
0.5

0.5 1

cos(π)GJ in πp

20 40 60 80

1
0.5

0.5 1

cos(p)GJ in pp

25 50 75 100

1
0.5

0.5 1

cos(π)

pp→ppπ0 (Gatchina 1485 MeV)

25 50 75 100

1
0.5

0.5 1

cos(p)

25 50 75 100

1
0.5

0.5 1

Mπp, GeV

50 100 1.1 1.2 1.3

Mpp, GeV

50 100 1.9 2 2.1

cos(π)h in πp

20 40 60

1
0.5

0.5 1

cos(p)h in pp

20 40 60

1
0.5

0.5 1

cos(π)GJ in πp

20 40 60 80

1
0.5

0.5 1

cos(p)GJ in pp

25 50 75 100

1
0.5

0.5 1

cos(π)

pp→ppπ0 (Gatchina 1536 MeV)

50 100

1
0.5

0.5 1

cos(p)

50 100 150

1
0.5

0.5 1

Mπp, GeV

50 100 150 1.1 1.2 1.3

Mpp, GeV

50 100 150 1.9 2 2.1

cos(π)h in πp

25 50 75 100

1
0.5

0.5 1

cos(p)h in pp

25 50 75 100

1
0.5

0.5 1

cos(π)GJ in πp

50 100

1
0.5

0.5 1

cos(p)GJ in pp

50 100 150

1
0.5

0.5 1

53

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Description of the HADES data pp → ppπ0 and pp → pnπ+:

cosθπ

1 2 3

1
0.5

0.5 1

a) dσ/d cosθ, mb/0.1 a) dσ/d cosθ, mb/0.1 cosθp

2 4 6

1
0.5

0.5 1

b) b) Mπp (GeV)

5 10 15 20 1 1.2 1.4

c) dσ/dM, mb/(0.025 GeV) c) dσ/dM, mb/(0.025 GeV) Mpp (GeV)

5 10 15 1.8 2 2.2

d) d) cosθπ in (h)πp

1 2 3

1
0.5

0.5 1

e) dσ/d cosθ, mb/0.1 e) dσ/d cosθ, mb/0.1 cosθp in (h)pp

1 2

1
0.5

0.5 1

f) f) cosθπ in (GJ)πp

1 2 3

1
0.5

0.5 1

g) dσ/d cosθ, mb/0.1 g) dσ/d cosθ, mb/0.1 cosθp in (GJ)pp

2 4 6

1
0.5

0.5 1

h) h)

cosθπ

5 10 15

1

1

a) dσ/d cosθ, mb/0.1 a) dσ/d cosθ, mb/0.1 cosθp

5 10 15 20

1

1

b) b) cosθn

5 10 15 20

1

1

c) c) Mπp (GeV)

50 100 1 1.2 1.4

d) dσ/dM, mb/(0.025 GeV) d) dσ/dM, mb/(0.025 GeV) Mπn (GeV)

20 40 60 1 1.2 1.4

e) e) Mpn (GeV)

20 40 60 80 1.8 2 2.2

f) f) cosθπ in (h)πp

2.5 5 7.5 10

1

1

g) dσ/d cosθ, mb/0.1 g) dσ/d cosθ, mb/0.1 cosθπ in (h)πn

5 10 15

1

1

h) h) cosθn in (h)pn

2.5 5 7.5 10

1

1

i) i) cosθπ in (GJ)πp

5 10

1

1

j) dσ/d cosθ, mb/0.1 j) dσ/d cosθ, mb/0.1 cosθπ in (GJ)πn

5 10 15

1

1

k) k) cosθn in (GJ)pn

10 20

1

1

l) l)

54

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

np → π−pp

20 40 60

N entries/0.125 pn<1.2

25 50 75 100 100 200 300

1.2 - 1.4

200 400 50 100 150

1.4 - 1.6

100 200 300 400

cos(π)

50 100 150 200

1
0.5

0.5 1

1.6 - 1.77 cos(p)

200 400 600

1
0.5

0.5 1 50 100 150 1.1 1.15 1.2

N entries/10 Mev

pn<1.2

20 40 60 80 1.9 1.95 2 200 400 600 800 1.1 1.15 1.2 1.25

N entries/15 Mev

1.2 - 1.4

100 200 300 400 1.9 1.95 2 2.05 200 400 1.1 1.2 1.3

N entries/19 Mev

1.4 - 1.6

100 200 1.9 2 2.1

M(πp), GeV

200 400 600 1.1 1.2 1.3 1.4

N entries/23 Mev

1.6 - 1.77

M(pp), GeV

100 200 300 1.9 2 2.1 2.2

Dashed lines - I = 1, dotted lines - I = 0

55

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The cross section for pion production in nucleon-nucleon collision with I = 1 is well

known. However there are very poor data about I = 0 cross section.

σ(I = 1) = σ(pp → ppπ0)

pp→ppπ

1 2 3 4 5 6 900 1000 1100 1200 1300 1400 1500 1600

pbeam, MeV σtot, mb

σ(I = 0) = 3[2σ(np → ppπ−)

−σ(pp → ppπ0)]

2
1

1 2 3 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

p(n), GeV/c

σI=0 , mb

56

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Reggezied exchanges:

)

t

(k ρ

)

2

(q π

)

1

p(k

)

1

p(q )

2

(k γ

)

u

N(k

)

2

(q π

)

1

p(k

)

1

p(q )

2

(k γ

The amplitude for t-channel exchange:

A = g1(t)g2(t)R(ξ, ν, t) = g1(t)g2(t)1 + ξexp(−iπα(t)) sin(πα(t)) ( ν ν0 )α(t) ν = 1 2(s − u).

Here α(t) is the reggion trajectory, and ξ is its signature:

R(+, ν, t) = e−i π

2 α(t)

sin( π

2 α(t))Γ

(

α(t) 2

) ( ν ν0 )α(t) , R(−, ν, t) = ie−i π

2 α(t)

cos( π

2 α(t))Γ

(

α(t) 2

+ 1

2

) ( ν ν0 )α(t) .

57

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

t,u-exchange subtraction procedure

= Σ − Σ

s,p

58

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

t,u-exchange subtraction procedure

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1700 1800 1900 2000 2100 2200 1/2 1/2 − 1/2 1/2 + 1/2 3/2 +

M(πp), MeV σtot, mb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1700 1800 1900 2000 2100 2200 1/2 1/2 − 1/2 1/2 + 1/2 3/2 +

M(πp), MeV σtot, mb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1700 1800 1900 2000 2100 2200 1/2 1/2 − 1/2 1/2 + 1/2 3/2 +

M(πp), MeV σtot, mb

59

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The new MAMI data on helicity 1/2 and 3/2 cross section

[-1.00,-0.60]

p 1/2

σ

[-0.60,-0.20]

p 1/2

σ

[-0.20,0.20]

p 1/2

σ

[0.20,0.60]

p 1/2

σ

[0.60,1.00]

p 1/2

σ

[-1.00,-0.60]

p 3/2

σ

[-0.60,-0.20]

p 3/2

σ

[-0.20,0.20]

p 3/2

σ

[0.20,0.60]

p 3/2

σ

[0.60,1.00]

p 3/2

σ

MAID BnGa

[-1.00,-0.60]

n 1/2

σ

[-0.60,-0.20]

n 1/2

σ

[-0.20,0.20]

n 1/2

σ

[0.20,0.60]

n 1/2

σ

[0.60,1.00]

n 1/2

σ

MAID BnGa (a) BnGa (b) BnGa (c)

[-1.00,-0.60]

n 3/2

σ

[-0.60,-0.20]

n 3/2

σ

[-0.20,0.20]

n 3/2

σ

[0.20,0.60]

n 3/2

σ

[0.60,1.00]

n 3/2

σ 1600 1800 1600 1800 1600 1800 1600 1800 1600 1800 1 2 3 0.5 1 2 0.5

W [MeV] b/sr] µ [ Ω /d σ d

p 1/2

σ

MAID BnGa

p 3/2

σ

n 1/2

σ

MAID BnGa (a) BnGa (b) BnGa (c)

n 3/2

σ

1500 1600 1700 1800 1900 20

2

2 4 10 20 2 4

W [MeV] b] µ [ σ

n 1/2

σ 5 ×

n 3/2

σ A

1

A

2

A

MAID BnGa (a) BnGa (b) BnGa (c) 3

A

1500 1600 1700 1800 1500 1600 1700 1800 1900 1 2 3 4 5

1
0.5

0.5

0.5

0.5 1

1.5
1
0.5

0.5 1

W [MeV] b/sr] µ [

i

A

60

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The solution with the fitted γn → KΛ, KΣ data

σ1/2 (MAMI 2016) γn→ηn(d)

5 10 15 20 25 1500 1550 1600 1650 1700 1750 1800 1850

M(γn), MeV

σ3/2 (MAMI 2016) γn→ηn(d)

0.5 1 1.5 2 2.5 3 3.5 4 1500 1550 1600 1650 1700 1750 1800 1850

M(γn), MeV

——– Prediction ———– Fit

61

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

The solution with the fitted γn → KΛ, KΣ data

σ1/2 (MAMI 2016) γn→ηn(d)

1 2 1504 1514 1524 1534 1547 1 1562 1577 1592 1607 1622 0.5 1 1637 1652 1667 1682 1697 0.5 1712 1727 1742 1757 1772 0.25 0.5 1787 1802

1

1 1817

cos θ

1

1 1832

cos θ

1

1 1847

cos θ

0.2

1

1 1862

cos θ

1

1 1882

cos θ

σtot=(σ1/2+σ3/2)/2 (γn→ηn (d)) MAMI 2016

0.5 1 1504 1514 1524 1534 1547 0.5 1562 1577 1592 1607 1622 0.25 0.5 1637 1652 1667 1682 1697 0.2 0.4 1712 1727 1742 1757 1772 0.2 1787 1802

1

1 1817

cos θ

1

1 1832

cos θ

1

1 1847

cos θ

0.1 0.2

1

1 1862

cos θ

1

1 1882

cos θ

——– Prediction ———– Fit

62

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Vector mesons in the final state: Density matrices

dσ dΩω dΩdec = dσ dΩω W(cos Θdec, Φdec) γp → pω(π+π−π0) W(cos Θ, Φ) = 3 4π (1 2(1 − ρ00) + 1 2(3ρ00 − 1) cos2 Θ− √ 2Reρ10 sin 2Θ cos Φ − ρ1−1 sin2 Θ cos 2Φ ) . cos Θ, Φ direction of the vector n = εijkmpπ+

j

pπ−

k pπ0 m in the ω rest frame.

γp → pω(γπ0) W(cos Θ, Φ) = 3 8π (1 2(1 + cos2 Θ) + 1 2(1 − 3 cos2 Θ)ρ00+ √ 2Reρ10 sin(2Θ) cos Φ + ρ1−1 sin2 Θ cos 2Φ ) . cos Θ, Φ angles of photon from ω decay in the ω rest frame

63

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Fit of the density matrices γp → pω (CB-ELSA) (A.Wilson)

ρ0

00

ρ0

1−1

0.5 1 1756

ρ00

1783 1809 1835 1860 1885 0.5 1 1910 1934 1959 1982 2006 2029 0.5 1 2052 2075 2097 2120 2142 2163 0.5 1 2185 0.5

0.5

2206 0.5

0.5

2228 0.5

0.5

2248 0.5

0.5

2269 0.5

0.5

2290 0.5

0.5

cos θω

0.5
0.25

0.25 0.5 1756

ρ1-1

1783 1809 1835 1860 1885

0.5
0.25

0.25 0.5 1910 1934 1959 1982 2006 2029

0.5
0.25

0.25 0.5 2052 2075 2097 2120 2142 2163

0.5
0.25

0.25 0.5 2185 0.5

0.5

2206 0.5

0.5

2228 0.5

0.5

2248 0.5

0.5

2269 0.5

0.5

2290 0.5

0.5

cos θω

64

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A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14

Photoproduction of vector mesons. γp → pω (A.Wilson)

2 4 6 8 10 12 14 1800 1900 2000 2100 2200 P-exchange 1/2 3/2 + 1/2 3/2 − 1/2 5/2 +

σtot (γp → ωp), µb M(γp), MeV

Strong contribution from the P13

partial wave: interference of P13(1700) and

P13(1900) states.

A confirmation of the F15(2000) state.
A structure in the D13 partial wave in the

region 2100 MeV.

No large contributions either from 7/2+ or

7/2− states are found

65