Partial wave analysis of eta meson photoproduction using fixed-t - PowerPoint PPT Presentation

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 1

Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard Overview • Motivation and goals • Theory and formalism • η MAID model • Results and conclusion 2

Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard Resonance spectrum in photoproduction E W/GeV /GeV γ 2 ** * * ** 2 ** ** ** ** * **** 1.5 *** **** ** ** ηΝ **** ** ** MAMI � energy range ** * ωΝ **** *** **** **** **** *** 1 **** **** ΚΛ / ΚΣ *** **** **** 1.5 ηΝ **** 0.5 ∆ **** π Ν N 1 **** 0 1/2+ 5/2+ 1/2- 3/2- 5/2- 3/2+ 7/2+ 3

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 K + Σ o new data from K o Σ + 10 P 11 (1710) A2/MAMI, 2017 (PRL) 15 0.4 ω S 11 (1535) D 15 (1675) η ‘ σ [µ b ] σ [µ b ] 10 D 13 (1520) 0.2 5 P 13 (1720) S 11 (1650) bg Regg F 15 (1680) bg D 13 (1700) 0 0 1.5 1.6 1.7 1.8 1.5 1.6 1.7 1.8 1 1.5 1.6 1.7 1.8 1.9 2 W [ GeV ] W [ GeV ] W [ GeV ] 4

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 ( η MAID). • By construction η MAID 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. 5

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 ( p i ) → η ( q ) + p ( p f ) (1) Variables in brackets denote the 4-momenta of the participating particles. k - photon, p i , p f - target and recoil proton, i and f denote initial and final states q - meson ( η ) Mandelstam variables: ( p i + k ) 2 = ( q + p f ) 2 , s = ( q − k ) 2 = ( p f − p i ) 2 , t = ( p i − q ) 2 = ( p i − q ) 2 . u = (2) s + t + u = 2 m p + m η √ s - total energy, t - momentum transfer squared from the photon to the meson . 6

Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard Mandelstam plane for γp → ηp Π N Η N Η 'N 0.0 t � GeV 2 � Crossing symmetrical variable. � 0.5 ν = s − u (3) 4 m p � 1.0 Crossing EVEN f ( − ν ) = f ( ν ) Crossing ODD f ( − ν ) = − f ( ν ) � 1.5 0.0 0.5 1.0 1.5 Ν � GeV � 7

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: 4 i u ( p i ) = − 4 πW ∑ A i ( ν, t ) ε µ M µ χ † t γ,η = ¯ u ( p f ) f F χ i , (4) M N i =1 − 1 M µ 2 iγ 5 ( γ µ k − kγ µ ) , = 1 ( ) P µ k · ( q − 1 2 k ) − ( q − 1 2 k ) µ k · P M µ = 2 iγ 5 , 2 − iγ 5 ( γ µ k · q − kq µ ) , M µ = 3 − 2 iγ 5 ( γ µ k · P − kP µ ) − 2 M N M µ M µ = 1 , (5) 4 where k = k µ γ µ , P µ = ( p µ i + p µ f ) / 2 . σ × ˆ σ · ˆ F = i ( ⃗ σ · ˆ ϵ ) F 1 + ( ⃗ σ · ˆ q ) ( ⃗ k ) · ˆ ϵ F 2 + i (ˆ ϵ · ˆ q ) ( ⃗ k ) F 3 + i (ˆ ϵ · ˆ q )( ⃗ σ · ˆ q ) F 4 (6) where ϵ µ = ( ϵ 0 ,⃗ ϵ · ⃗ ϵ ) and ⃗ k = 0 . Both types of amplitudes are used further. 8

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. ℓ max ∑ [( ℓM ℓ + + E ℓ + ) P ′ ℓ +1 ( x ) + (( ℓ + 1) M ℓ − + E ℓ − ) P ′ F 1 ( W, x ) = ℓ − 1 ( x )] , ℓ =0 ℓ max ∑ [( ℓ + 1) M ℓ + + ℓM ℓ − ] P ′ F 2 ( W, x ) = ℓ ( x ) , ℓ =1 ℓ max ∑ [( E ℓ + − M ℓ + ) P ′′ ℓ +1 ( x ) + ( E ℓ − + M ℓ − ) P ′′ F 3 ( W, x ) = ℓ − 1 ( x )] , ℓ =1 ℓ max ∑ [ M ℓ + − E ℓ + − M ℓ − − E ℓ − ] P ′′ F 4 ( W, x ) = ℓ ( x ) . (7) ℓ =2 where ℓ is an orbital angular momentum of the ηN system, x = cos θ is the cosine of the scattering angle, M ℓ ± , E ℓ ± are multipoles, ” ± ” → J = ℓ ± 1 / 2 . Consider example: E 0+ : ℓ = 0 , J = 0 + 1 / 2 , P = − ( − 1) ℓ = − 1 9

Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard Some important resonances in η photoproduction ℓ J P J = ℓ ± 1 / 2 Resonance Multipole N ( mass ) J P N (1535) 1 / 2 − E 0+ 1 / 2 = 0 + 1 / 2 0 1/2 - N (1650) 1 / 2 − E 0+ 1 / 2 = 0 + 1 / 2 0 1/2 - N (1440) 1 / 2 + M 1 − 1 / 2 = 1 − 1 / 2 1 1/2 + N (1710) 1 / 2 + M 1 − 1 / 2 = 1 − 1 / 2 1 1/2 + N (1720) 3 / 2 + E 1+ , M 1+ 3 / 2 = 1 + 1 / 2 1 3/2 + N (1520) 3 / 2 − E 2 − , M 2 − 3 / 2 = 2 − 1 / 2 2 3/2 - N (1700) 3 / 2 − E 2 − , M 2 − 3 / 2 = 2 − 1 / 2 2 3/2 - N (1675) 5 / 2 − E 2+ , M 2+ 5 / 2 = 2 + 1 / 2 2 5/2 - N (1680) 5 / 2 + E 3 − , M 3 − 5 / 2 = 3 − 1 / 2 3 5/2 + 10

Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard Constituents of the η MAID isobar model γ η γ η γ η ρ, ω N ∗ p p = p p γ γ η p p p p p p 11

Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard Resonance parametrization of the η MAID isobar model Multipoles M ℓ ± ( E ℓ ± , M ℓ ± ) have Breit Wigner form: M R Γ tot ( W ) M ℓ ± ( W ) = ¯ M ℓ ± f γN ( W ) R − W 2 − iM R Γ tot ( W ) f ηN ( W ) C ηN , (8) M 2 ¯ M ℓ ± is related to the photo decay amplitudes listed in PDG. Γ tot ( W ) is the energy dependent width. f γN ( W ) and f ηN ( W ) are vertex functions. 12

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 A2MAMI ● K + Σ o CBELSA/TAPS-09 10 ● K o Σ + ω η ‘ σ [µ b ] 1 -1 10 1.6 1.8 2 2.2 2.4 2.6 2.8 W [ GeV ] η MAID2003: solid - full model, dashed - background ( ρ + ω + born ), η MAID2015: solid - full model, dashed - background ( ρ + ω + born ) 13

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 ��!��������!��%�������� ���� ��� %��������� 14

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 � ���������� ����������������� η MAID2003, η MAID2015 T - polarized target, F - polarized beam and target. 15

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: ∫ ∞ dν ′ ν ′ Im A i ( ν ′ , t ) ( ν, t ) + 2 A pole Re A i ( ν, t ) = π P , for i = 1 , 2 , 4 (9) ν ′ 2 − ν 2 i ν thr ( t ) Crossing odd: ∫ ∞ dν ′ Im A i ( ν ′ , t ) ( ν, t ) + 2 ν A pole Re A i ( ν, t ) = , for i = 3 π P (10) ν ′ 2 − ν 2 i ν thr ( t ) 16

Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard Real part of A1 ReA1 � GeV � 2 � , t � � 0.5GeV 2 ReA1 0.0 � 0.5 � 1.0 Re � Η MAID � � 1.5 Re � Η MAID � calculated from the Im � Η MAID � � 2.0 � 2.5 � 3.0 W � MeV � 1600 1800 2000 2200 2400 17

Partial wave analysis of eta meson photoproduction using fixed-t dispersion relations 30’th August 2017, Boppard Integrating regions Π N Η N Η 'N ImA1 � GeV � 2 � , t � � 0.5GeV 2 0.0 ImA1 Π N Η N Η 'N t � GeV 2 � � 0.5 10 Η MAID Η MAID � 1.0 Η MAID 5 � 1.5 W � MeV � 0.0 0.5 1.0 1.5 1400 1600 1800 2000 Ν � GeV � 18