1

Few-body systems as neutron targets

A. Fix (Tomsk polytechnic university)

2

• 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

3

• 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

4

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)

5

Results for π– photoproduction (total cross section)

σn (MAID2003)

Full theory

Total cross section

σn(free) ≈ σn(bound)a

• Reason: Transparency of the target

6

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

7

Important corrections

• Fermi motion

• NN interaction

• Pauli blocking

• Meson rescattering

• Other two-nucleon mechanisms (MEC, pion absorption on

nucleon pairs, etc.)

8

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

9

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)

10

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(

11

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

12

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)

13

NN FSI in near-threshold region

FSI included:

Large initial nucleon momentum not required

3body

SM

NN FSI

γd→ηX )(σ)(σ

SM

FSI

20

01

5

3

7

40]MeV[ EE th

η

η

η´

Enhancement effect is larger for η´

Very difficult to extract σn

14

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

15

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

16

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

17

Absorption of pions

• Large exchanged momentum short-range nature of the absorption mechanism

fm 5.01~ p

ra

221mTT MeV/c 360

21

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

18

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

19

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

20

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

21

Simple method to estimate FSI effect

• If closure 1ff

)()(FSIclosureSM dNN

)()()( SMFSI ddNN np

A

B

sm

FSI

πd

A ≈ B

is used

22

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

23

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

24

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

25

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

26

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 )

27

Pauli exclusion

• Important at forward meson angles (small relative momentum of recoil nucleons)

• Effect: Decreases cross section ppd

pn

Allowed

Excluded