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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30th August 2017, Boppard Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations Kirill Nikonov Johannes Gutenberg

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations

Kirill Nikonov Johannes Gutenberg University, Mainz.

30’th August 2017, Boppard

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Overview

  • Motivation and goals
  • Theory and formalism
  • ηMAID model
  • Results and conclusion

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Resonance spectrum in photoproduction

1 1.5 2

W/GeV /GeV

γ

E

0.5 1 1.5 2

**** ** **** **** **** **** *** *** ** **** ** **** *** * ** **** **** ** **** **** **** **** ** **** * ***

ηΝ π Ν ΚΛ/ΚΣ ωΝ N

∆

ηΝ

MAMI energy range

1/2+ 3/2+ 5/2+ 7/2+ 1/2- 3/2- 5/2-

** * * ** ** ** **

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Total cross section data from Mainz for γp → ηp. Resonance contribution

1 10 1.5 1.6 1.7 1.8 1.9 2

W [GeV] σ [µb]

K+Σo KoΣ+ ω η‘ new data from A2/MAMI, 2017 (PRL) 5 10 15 1.5 1.6 1.7 1.8 0.2 0.4 1.5 1.6 1.7 1.8

σ [µb] W [GeV]

S11(1535) S11(1650) bg bg Regg P11(1710) D15(1675) D13(1520) P13(1720) F15(1680) D13(1700)

W [GeV]

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Motivation and goals Motivation:

  • Study nucleon resonances using isobar model (ηMAID).
  • By construction ηMAID is non analytic.
  • Apply fixed-t dispersion relations.

I.Aznauryan in [Phys.Rev. C68 (2003) 065204]

Goals:

  • Fit the new data. Resonance parameters used as fitting parameters.
  • Obtain masses, widths, branching ratios, e.t.c in an improved and less model dependent

way.

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Kinematics of γp → ηp

Consider kinematical quantities independent of the reference frame.

γ(k) + p(pi) → η(q) + p(pf)

(1) Variables in brackets denote the 4-momenta of the participating particles.

k - photon, pi , pf - target and recoil proton, i and f denote initial and final states q - meson (η)

Mandelstam variables:

s = (pi + k)2 = (q + pf)2, t = (q − k)2 = (pf − pi)2, u = (pi − q)2 = (pi − q)2.

(2)

s + t + u = 2mp + mη √s - total energy, t - momentum transfer squared from the photon to the meson.

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Mandelstam plane for γp → ηp

0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 Ν GeV t GeV2

ΠN ΗN Η'N

Crossing symmetrical variable.

ν = s − u 4mp

(3) Crossing EVEN f(−ν) = f(ν) Crossing ODD f(−ν) = −f(ν)

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Invariant and CGLN amplitudes Chew,Golberger,Low,Nambu - CGLN [Phys.Rev.106(1957) 1337-1344].

γp → ηp can be described by 4 amplitudes, the matrix element in c.m. takes the form: tγ,η = ¯ u(pf)

4

∑

i=1

Ai(ν, t) εµM µ

i u(pi) = −4πW

MN χ†

fFχi ,

(4)

M µ

1

= −1 2iγ5 (γµk − kγµ) , M µ

2

= 2iγ5 ( P µ k · (q − 1 2k) − (q − 1 2k)µ k · P ) , M µ

3

= −iγ5 (γµ k · q − kqµ) , M µ

4

= −2iγ5 (γµ k · P − kP µ) − 2MN M µ

1 ,

(5) where k = kµγµ, P µ = (pµ

i + pµ f)/2.

F = i (⃗ σ · ˆ ϵ) F1 + (⃗ σ · ˆ q) (⃗ σ × ˆ k) · ˆ ϵ F2 + i (ˆ ϵ · ˆ q) (⃗ σ · ˆ k)F3 + i(ˆ ϵ · ˆ q)(⃗ σ · ˆ q) F4

(6) where ϵµ = (ϵ0,⃗

ϵ) and ⃗ ϵ · ⃗ k = 0. Both types of amplitudes are used further.

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Truncated partial wave expansion of CGLN amplitudes

CGLN amplitudes can be expanded in terms of the partial waves and an angle.

F1(W, x) =

ℓmax

∑

ℓ=0

[(ℓMℓ+ + Eℓ+) P ′

ℓ+1(x) + ((ℓ + 1) Mℓ− + Eℓ−) P ′ ℓ−1(x)] ,

F2(W, x) =

ℓmax

∑

ℓ=1

[(ℓ + 1) Mℓ+ + ℓMℓ−] P ′

ℓ(x) ,

F3(W, x) =

ℓmax

∑

ℓ=1

[(Eℓ+ − Mℓ+) P ′′

ℓ+1(x) + (Eℓ− + Mℓ−) P ′′ ℓ−1(x)] ,

F4(W, x) =

ℓmax

∑

ℓ=2

[Mℓ+ − Eℓ+ − Mℓ− − Eℓ−] P ′′

ℓ (x) .

(7) where ℓ is an orbital angular momentum of the ηN system, x = cos θ is the cosine of the scattering angle, Mℓ±, Eℓ± are multipoles, ” ± ” → J = ℓ ± 1/2. Consider example: E0+ : ℓ = 0, J = 0 + 1/2, P = −(−1)ℓ = −1

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Some important resonances in η photoproduction Resonance

ℓ J P

Multipole

J = ℓ ± 1/2 N(mass) JP N(1535) 1/2−

1/2

  • E0+

1/2 = 0 + 1/2 N(1650) 1/2−

1/2

  • E0+

1/2 = 0 + 1/2 N(1440) 1/2+

1 1/2 +

M1− 1/2 = 1 − 1/2 N(1710) 1/2+

1 1/2 +

M1− 1/2 = 1 − 1/2 N(1720) 3/2+

1 3/2 +

E1+, M1+ 3/2 = 1 + 1/2 N(1520) 3/2−

2 3/2

  • E2−, M2−

3/2 = 2 − 1/2 N(1700) 3/2−

2 3/2

  • E2−, M2−

3/2 = 2 − 1/2 N(1675) 5/2−

2 5/2

  • E2+, M2+

5/2 = 2 + 1/2 N(1680) 5/2+

3 5/2 +

E3−, M3− 5/2 = 3 − 1/2

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Constituents of the ηMAID isobar model

γ p η p =

γ p

N∗

η p γ η ρ, ω p p p p γ p p η γ

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Resonance parametrization of the ηMAID isobar model

Multipoles Mℓ± (Eℓ±, Mℓ±) have Breit Wigner form:

Mℓ±(W) = ¯ Mℓ± fγN(W) MRΓtot(W) M 2

R − W 2 − iMRΓtot(W) fηN(W) CηN ,

(8)

¯ Mℓ± is related to the photo decay amplitudes listed in PDG. Γtot(W) is the energy dependent width. fγN(W) and fηN(W) are vertex functions.

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Fit results for the total cross section with an isobar model

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  • 1

1 10 1.6 1.8 2 2.2 2.4 2.6 2.8

K+Σo KoΣ+ ω η‘

  • A2MAMI
  • CBELSA/TAPS-09

W [GeV] σ [µb]

ηMAID2003: solid - full model, dashed - background (ρ + ω + born), ηMAID2015: solid - full model, dashed - background (ρ + ω + born)

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Fit results for the differential cross section with an isobar model

  • !!%

%

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Fit results for T and F with an isobar model

  • $

$

  • ηMAID2003, ηMAID2015

T - polarized target, F - polarized beam and target.

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Fixed-t dispersion relations

Q:How to improve? Problem of an isobar model:non analytic. BUT: Invariant amplitudes are analytic functions of complex variables, one can derive dispersion relations at a fixed value of t. Crossing even: ReAi(ν, t)

= Apole

i

(ν, t) + 2 π P ∫ ∞

νthr(t)

dν′ ν′ ImAi(ν′, t) ν′2 − ν2 ,

for i = 1, 2, 4 (9) Crossing odd: ReAi(ν, t)

= Apole

i

(ν, t) + 2ν π P ∫ ∞

νthr(t)

dν′ ImAi(ν′, t) ν′2 − ν2 ,

for i = 3 (10)

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Real part of A1

1600 1800 2000 2200 2400 WMeV 3.0 2.5 2.0 1.5 1.0 0.5 0.0 ReA1

ReA1GeV2, t 0.5GeV2

ReΗMAID ReΗMAID calculated from the ImΗMAID

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Integrating regions

0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 Ν GeV t GeV2

ΠN ΗN Η'N

1400 1600 1800 2000 WMeV 5 10 ImA1

ImA1GeV2, t 0.5GeV2

ΠN ΗN Η'N

ΗMAID ΗMAID ΗMAID

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Fitting procedure.

  • Data: dσ

dΩ,T,F (A2) and Σ (Graal) up to 1700 MeV.

  • Model: Resonances + born + ρ-meson + ω-meson.
  • 9 resonances.
  • Constrain from dispersion relations is implemented in Minuit.

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Working progress. Fit results for A1

1800 2000 2200 2400 WMeV 4 2 2 4 6 ReA1

ReA1GeV2, t 0.5GeV2

ImΗMAID ReΗMAID after fit with dispersion relations ImΗMAID after fit with dispersion relations ReΗMAID

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Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard

Conclusion

  • Systematic cross checks.
  • Multipole extraction.
  • Resonance parameters.

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