Few-body systems as neutron targets
description
Transcript of Few-body systems as neutron targets
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Few-body systems as neutron targets
A. Fix (Tomsk polytechnic university)
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• Deuteron parameters:
Deuteron is particularly suited as a neutron target
pn
• Photoproduction of π, η, and η´ on few-body nuclei
- Binding energy 2.224 MeV (small)
- Matter radius 1.9 fm (large)
- Asymptotic ratio D/S 0.027 (small spins of nucleons are aligned along the deuteron spin)
a
a
ns
ps
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• Schrödinger equation for the deuteron w.f.
• Asymptotic Region
0)( ))( ()( ruErVMru d
:)( 0rr
dME
• Deuteron size parameter /1dRfm 4.10 r
rCe
r
)(rV
Nucleons are on average outside the interaction range
0fm 3.4 MeV 22.2 rRE dd
a
a
,)( rCeru
,0)()( 2 ruru
pnRd > r0
r0
)(ru
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Simplest approximation to σ(γd→mX)
• Amplitude:
• Cross section: )()()( freefree EEE pnd dm R
q1
)()()( EfEfEf pnd 0rRd
)free()bound( NN ff 0Ed aa
a
p
n
γ m
• Spectator model:
a
(diffuse structure)
(weak binding)
(weak interference)
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Results for π– photoproduction (total cross section)
σn (MAID2003)
Full theory
Total cross section
σn(free) ≈ σn(bound)a
• Reason: Transparency of the target
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Validity of the spectator model• 1st condition: weak binding
,1
m
NTT 1
m
NTU
easily satisfied for not very slow mesons
• 2nd condition: short “collision” time (impulse character of reaction)
Violated because of resonance time delay ∆t = 2/Γ∆ ~ 10–23 s
a ∆R= ∆t βΔ ~ 1fm
∆(1232) region: Tπ ≈ mπ
MeV 30 NN UTTa
∆R /Rd ~ 1
spectator model is somewhat marginal in the resonance region
• 3rd condition: dominance of incoherent mechanismsViolated for for π0 σcoh ≈ 1/3 σincoh
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Important corrections
• Fermi motion
• NN interaction
• Pauli blocking
• Meson rescattering
• Other two-nucleon mechanisms (MEC, pion absorption on
nucleon pairs, etc.)
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Fermi motion
222),(),( )(
)2()( 3
3
pfpfpPpdEf npd
• Doppler shift of the photon energy kE
• Effect: smearing of the resonance structure • Preserves energy integrated σ
dEEEE
E
Eth th
)(σ 1σmax
max
γd frame γN frame
,μb 9.106σ: d μb 4.106σ n
:GeV 1max E
σn
Momentum distribution
)free(σ)bound(σ nn
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Influence on Σ asymmetry
p,p spnpd
dddd
dddd
/σ/σ
σσσσ
||
||
a Effect of FM depends on specific behaviour of elementary cross section
1.00 GeV 1.05 GeV 1.10 GeV 1.20 GeV
Data: GRAAL, 2008
S11(1535)
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GDH on neutron
• Spins of nucleons are aligned along the deuteron spin
)free(σ)bound(σ
pnγ
pn
)σ()σ(σ ,)(σ
~Imax
GDH
E
Eth
dEE
E
• Solution: Exclusion of FM through transition to γN frame
• However (free)I(bound)I GDHGDHnn
(due to Fermi motion)
Δσ
IGDH
free
bound
free
bound
γ
Δσ Δσ/Eγ
)σ(
)σ(
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SM
NN FSI
ppd π spectrum in
NN Final state interaction
Bound pp
Virtual pp
Effect: peak near high energy limit of π spectrum caused by strong NN attraction in 1S0 state
Pπ (MeV/c)
a
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NN FSI in near-threshold region
Initial nucleon momentum p > 200 MeV/c strongly exceeds typical momentum in the deuteron α=√MEd ≈ 45 MeV/c
FSI neglected:
• Leads to strong enhancement of SM cross section
strong suppression of the SM cross section
SM
NN FSIeffect
γd→ηnp
(threshold)
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NN FSI in near-threshold region
FSI included:
Large initial nucleon momentum not required
3body
SM
NN FSI
γd→ηX )(σ)(σ
SM
FSI
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01
5
3
7
40]MeV[ EE th
η
η
η´
Enhancement effect is larger for η´
Very difficult to extract σn
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Orthogonality
γd→ηnp
SM
FSI
SM
FSI FSI
SM
‹ d (2S+1LJ =3S1 ) | → | np (3S1 ) ›
• ηnp, π–pp: FSI is insignificant • π0np: FSI is important
• Reason: Orthogonality of ψd(r) and ψnp(r) in γd→ π0np
dominates at θπ → 0
equal quantum numbers
initial state final state
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FSI
SM 0 if 0 qkqIT
Orthogonality
0 ),()(ˆ dddd ErErH
0 ),()(ˆ ErErH npnp
0 dnpI
denpT rqki )( ~
In spectator model:0at 0)( )( qpdeeT d
rqkirpi
a Orthogonality relation:
γd→π0np Amplitude:
a dσ/dΩ suppressed at θπ → 0
a Orthogonality is ignored
np is a plane wave
d npand are eigenstatesof the same HamiltonianH
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200
Absorption of pions
400 600 800Photon energy [MeV]
0
2
4
6
x102
σ [μ
b]
σd (B.Krusche et al, 1999)
σp + σn (MAID 2003)
Energy integrated σ
dEEEE
E
Eth th
)(σ 1σmax
max
,μb 8.155σ d μb 5.171σσ np
πo Photoproduction
npd σσ σ
a pions are absorbed
:GeV 8.0max E
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Absorption of pions
• Large exchanged momentum short-range nature of the absorption mechanism
fm 5.01~ p
ra
221mTT MeV/c 360
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21 ppp
• Estimate of absorption effects: pion is necessarily absorbed if r ≤ ra
arrdd
0
32absP
p
n
π } ra
p1
p2
rrd ee
rr 1)(with Hulthen w.f. and ra = 0.5 fm
a
a
aa
a rrr
eee )()(22
2
)(4
)()(1
Pabs = ≈ 0.1
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2nd resonance region
?
1. Strong absorption (unlikely)
• Disagreement at Eγ ≈ 0.7 ± 0.2 GeV :
σ σ nntheor
Measurement of free
free
ddR Q
Q
p
nσσ
C.Bacci et al (1969): R ≈ 1
Data: R ≈ 1/3Data: B.Krusche et al, 1999
• Assumptions:
2.
theorydata
≈ 1.5
Full model
Then freefree R pn σσ
σp + σn
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Spectator model for η and η´ photoproduction
• Works well, especially if Fermi motion is excluded (through transition to γN)
free neutron
• Corrections to SM are insignificant (except low energies)
NN FSI large momentum transfer to NNOrthogonality small coherent component Absorption large meson mass and weak ηNN (η´ NN) coupling
Correction Why small
γn* → η´n
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Corrections to the spectator model
• Generally important for π photoproduction in the resonance region, especially for π0 where coherence effects are strong
• Insignificant for η and η´ (except trivial Fermi motion and NN FSI in the near- threshold region)
• Rather well understood a study of reactions on neutrons
is not problematic
• ? 2nd resonance region in γd → π0np reason of discrepancy is unclear
Conclusion
NN FSI absorption
? shadowing
F e r m i m o t i o n
σp+ σn
Full model
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Simple method to estimate FSI effect
• If closure 1ff
)()(FSIclosureSM dNN
)()()( SMFSI ddNN np
A
B
sm
FSI
πd
A ≈ B
is used
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Absorption of η and η´
• Two-nucleon absorption requires large momentum exchange not effective• Main abs. mechanism – transition to pions ηd→πNN, ηd→ππNN, ...
• Time delay in the resonance region ΔQ ~ 2/ΓR
strong influence of inelastic channelsExample: ηd elastic scattering in the S11(1535) regionArgand diagram for L=0 Inelasticity parameter
Three-body calculation
S11(1535)
Complex nuclei:σ(γA → ηX) ~ A2/3
(surface production)
a
a
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Meson rescattering at low energies
• Small πN scattering length dRNa fm 1.0)(
• S11(1535) near ηN threshold dRNa ~ fm 1)( Re
• Few-body models are the only proper base for ηNN, ηNNN …
a π rescattering is insignificant
N N
η
a strong ηN attraction a rescattering concept is inadequate
Very difficult to extract σn
rescattering
3-body
rescattering
+ …
3-body
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π– Photoproduction at higher energies
NKS02 947 MeVSCH74 957 MeV
NKS02 1097 MeVSCH74 1100 MeV
dσ/d
Ω [μ
b/sr
]
dσ/d
Ω [μ
b/sr
]
0
5
10
0
5
10
50 100 150 50 100 150
θ [deg] θ [deg]
• At θ ≈ 0 theorydata
≈ 2
γ d→π– p p
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Rescattering corrections a shadowing effects
p nπ
pn
Shadowing of incident photon Shadowing of produced pion
)p(σ)p(σ
411 )n(σ)d(σ
2
drdd
dd
22 fm 3.0)(40
2
drrr dd 75.0
)pn()ppd(
dd
γ
a
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γd→ηnp SM
FSI
Why is orthogonality important only for π0 ?
γd→π0np: 3S1 large, q small a effect is importantγd→ηnp: 3S1 small, q large a effect is insignificant
• 1st condition: fraction of 3S1 in final NN is large
• 2nd condition: momentum transfer q to NN is small
FSI
SM
Isovector (Tγ=1) a Id = 0 → Inp= 1 (3S1 forbidden )
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Pauli exclusion
• Important at forward meson angles (small relative momentum of recoil nucleons)
• Effect: Decreases cross section ppd
pn
Allowed
Excluded