jefferson lab - electroproduction of hypernuclei...sotona and frullani, prog. theor. phys. suppl....

Electroproduction of p-shell hypernuclei in DWIA

The 13th International Conference on Hypernuclear and Strange Particle Physics

Petr BydzovskyNuclear Physics Institute, Rez, Czech Republic

in collaboration∗ with D.J. Millener (BNL),

F. Garibaldi and G.M. Urciuoli (Rome)

Outline:IntroductionFormalism of DWIAResults: elementary elektroproduction

electroproduction on 9Be, 12C, and 16O targetsUncertainty of predicted cross sectionsSummary and outlook

∗the results will be published in an archival paper by Hall A Hypernucleus Collaboration

Petr Bydzovsky (NPI Rez) Electroproduction of hypernuclei HYP 2018, June 24-29, 2018 1 / 23

Introduction – Why do we study electroproduction of hypernuclei?

γ-ray and reaction spectroscopy of Λ hypernuclei→ information on the YN interaction, its spin-dependent part

in the reaction spectroscopy we can study higher energy states, Λp

DWIA calculations with a structure from standard Shell-model→ we learn on the effective (in medium) YN interaction

reaction mechanism in DWIA

electroproduction– a better energy resolution than in the π+ and K− induced productions;– a large momentum transfer to the hypernucleus: q > 250 MeV/c;– strong spin-flip, the highest-spin states in multiplets dominate;– the electro-magnetic part is well known and the one-photon exchangeis a good approximation → production by virtual photons – simplification;– if K+ detected → production on the proton → other hypernuclei.

suitable kinematics: – a very small electron scattering angle→ very small Q2 = −q2 and sufficiently big virtual photon flux;– kaon is detected along the photon direction→ very small kaon angle


Introduction – kinematics, cross sections

e + AZ −→ e′ + K+ + AΛ(Z−1)

laboratory frame:

Φ

θe

θ

p

p'

γ

q

p

pH

Scattering (Leptonic) Plane

Reaction (Hadronic) Plane

z

x

y

e

e

K

K

K

^

^

1

input kinematics: Ee,Ee′ , θe , θKe ,ΦK

→ ε, Γ, q = pe − p′e, Q2 = −q2, s = W 2, t,...

the unpolarized lab cross section in hypernucleus electroproduction

d3σ

dE′e dΩ′e dΩK= Γ

[dσT

dΩK+ ε

dσL

dΩK+ ε

dσTT

dΩK+√

2ε(1+ε)dσTL

dΩK

]Petr Bydzovsky (NPI Rez) Electroproduction of hypernuclei HYP 2018, June 24-29, 2018 3 / 23

Introduction – previous calculations for E94-107 in Hall ASotona and Frullani, Prog. Theor. Phys. Suppl. 117 (1994) 151

Excitation Energy (MeV)0 10 20

MeV

⋅

GeV

2sr

nb

ex

c d

Ee

dE

KΩ

de

Ωd

σd

0

2

4

-Binding Energy (MeV)-10 -8 -6 -4 -2

Cro

ss S

ecti

on

(n

b/s

r^2/G

eV

/MeV

)

0.0

0.5

1.0

1.5

12C(e, e′K+)12Λ B M. Iodice,... M. Sotona,... et al,

Phys. Rev. Lett. 99, 052501 (2007)

16O(e, e′K+)16Λ N

F.Cusanno,... M. Sotona,... et al,

Phys. Rev. Lett. 103 (2009) 202501

Binding Energy (MeV)-20 -15 -10 -5 0 5 10 15

Me

V

⋅G

eV

2s

rn

b

ex

c d

Ee

dE

KΩ

de

Ωd

σd

0

1

2

3

4

5

9Be(e, e′K+)9ΛLi G.M. Urciuoli,... M. Sotona,... et al,

Phys. Rev. C 91 (2015) 034308


Formalism of DWIA – many-particle matrix element

Calculation of dσx/dΩK for γv + A→ K+ + H∗ in DWIA

particle momenta are quite high, |~q|, | ~pK| ≈ 1 GeV/c

many-particle matrix element in lab frame

〈ΨH|Z∑

i=1

χγ χ∗K Jj(pΛ, pK, p

ip, q) |ΨA〉 = (2π)3δ(~q− ~pK − ~pH) T j

χK kaon distorted wave functionI eikonal approximation: | ~pK| ≈ 1 GeV/c and KN interaction is weakI the first-order optical potential: proton-neutron averaged KN forward

scattering amplitude and target-nucleus density (bHO)

χγ photon wave function, no Coulomb distortion of e and e’

Jj elementary-production hadronic current in two-component formalism

ΨA and ΨH nucleus and hypernucleus non relativistic wave functions


Formalism of DWIA – assumptions, approximations

translational invariance of Jj : ~q + ~p ip = ~pK + ~pΛ

but the energy conservation is violated Eγ + Ep 6= EK + EΛ,– off-shell effects are neglected;

factorization approximation: ~p ip → ~p0 = 0 in the lab frame

→ Jj(~q,~pΛ) expressed via six CGNL amplitudes– important simplification; no Fermi motion;

partial-wave decomposition: χγ χ∗K =

∑FLM YLM

FLM includes the kaon distorted wave function in eikonal

approximation – Vopt depends on σKNtot , α = Re f KN(0)/Im f KN(0),

and ρ(bHO); f KN(0) is the proton-neutron averaged forward angle amplitude

from a separable model with partial waves ` = 0, 1, ... 7 (0 ≤ Ekin ≤ 2 GeV);

elementary current in spherical form: J 1i =

∑FSiλ σ

Sλ , i , λ = ±1, 0,

– FSiλ are non-flip (S=0) and spin-flip (S=1) CGNL-like amplitudes,

harmonic oscillator basis: |α〉 = |n ` 12 j 〉


Formalism of DWIA – final formula for the amplitude

The hypernucleus production amplitude

T i =∑Jm′

1

[JJH ]C JHMHJAMAJm′

∑Sλ

∑LM

C Jm′LMSλ FS

iλ ×∑α′α

RLMn′`′n`MLSJ

`′j ′`j (ΨH ||[b+α′ ⊗ aα

]J ||ΨA)

The main ingredients

the elementary production: CGNL-like amplitudes FSiλ from an isobar

model, e.g. Saclay-Lyon and BS3 [D. Skoupil, P. B.; Phys.Rev.C97(2018)025202]

the radial integral: RLMn′`′n` =

∫∞0 dr r2 R∗n′`′FLM Rn`

– includes the kaon distortion via FLM (σKNtot , α, and bHO);

– Rn` are HO or Woods-Saxon radial wave functions of the proton and Λ

nucleus structure information: reduced OBDME (ΨH ||[b+α′ ⊗ aα

]J ||ΨA)

– Shell-model calculations by John Millener


Results – elementary electroproduction

the new E94-107 p(e, e′K+)Λ data at W = 2.2 GeV, Q2 = 0.07 GeV2 isconsistent with the old data by Brown etal, W = 2.17 GeV, Q2 = 0.18 GeV2

the calcualtions are for p(γ,K+)Λ at W = 2.2 GeV

the isobar and RPR models differ in dynamics of the resonant andbackground parts (various sets of resonances, couplings, form factors)

0 30 60 90 120 150 180

θc.m.

K (deg)

0

0,1

0,2

0,3

0,4

0,5

0,6

dσ/

dΩ

(µ

b/s

r)

CLASLEPSSAPHIRBrownSLAWJCKMH2RPR-1BS1BS3E94-107

p(γ,K+)Λ

M.E.McCracken etal, P.R.C81,025201(2010)M.Sumihama etal, P.R.C73,035214(2006)K.-H.Glander etal, Eur.Phys.J.A19,251(2004)

cross sections for p(e, e′K+)ΛW = 2.2 GeV, Q2 = 0.07 GeV2, ε = 0.7,θc.m.

K = 6, ΦK = 180

(µb/sr) SLA H2 BS1 BS3

σfull 0.309 0.091 0.402 0.303

σT 0.370 0.026 0.241 0.280

σL 0.006 0.038 0.215 0.109

σTT 0.001 0.006 -0.015 0.003

σTL 0.042 -0.023 -0.014 0.036

σfull = σT + εσL + εσTT −√

2ε(1 + ε)σTL

predictions of SLA and BS3 are about 40%below the E94-107 data


Q2 dependence of the separated cross sections in p(e, e′K+)Λ

0

100

200

300

400

σ T [

nb

/sr]

Mohring et al.

Bleckmann et al.Coman et al.

0 0.5 1 1.5 2 2.5

Q2 [(GeV/c)

2]

0

100

200

300

σ L [

nb

/sr]

BS1BS3Saclay-Lyon

Kaon-MAID

W = 1.84 GeV

θK

c.m. = 0 deg

D. Skoupil, P. B.; Phys. Rev. C97 (2018) 025202


Results – the 9Be(e, e′K+)9ΛLi spectrum

Ee = 3.77, Ee′ = 1.56 GeV, θe = 6, θKe = 6, ΦK = 180

→ Eγ = 2.21 GeV, Q2 = 0.064 GeV2, |~pK| = 1.96 GeV, θlabKγ = 1.8

structure: complete p-shell basis for the core nucleus with the interaction fit4; 1/2↔ 3/2

Woods-Saxon w.f.: Epp3/2

= −16.89 and EΛs1/2

= −8.53, MeV

kaon distortion: σKNtot = 17.90 mb, α = −0.288, bHO = 1.756 fm

-10 -9 -8 -7 -6 -5 -4 -3- Binding Energy (MeV)

0

0.5

1

1.5

dσ

/(dΩ e

dΩ

KdE

edE

b)

[nb/(

sr2G

eVM

eV)]

fitSLABS3

9

ΛLi

experiment theory with SLA

Ex crs Ex Jπ crs diff

0.00 0.59±0.15 0.00 32

+0.164

0.57 0.83±0.13 0.56 52

+1.118

sum=1.42 1.28 -10%

1.42 12

+0.353

1.45 32

+0.327

1.47 0.79±0.07 0.68 -14%

2.27 52

+0.130

2.73 72

+0.324

2.27 0.54±0.06 0.45 -17%

radiative-corrected data P.R.C91(2015)034308;

FWHM = 730 keV was used for theoretical peaks; fitted FWHM = 730±60 keV


Results – the 12C(e, e′K+)12Λ B spectrum

kinematics is the same as for the 9Be target


the full 0~ω basis of p-shell states for the core coupled to a Λ or Σ in an s or p orbit;

W.-S. w.f.: Epp3/2

= −15.96, Epp1/2

= −10.37, EΛs1/2

= −11.37, EΛp = −0.40 MeV

Excitation Energy (MeV)0 10 20

Me

V

⋅G

eV

2s

rn

b

ex

c d

Ee

dE

KΩ

de

Ωd

σd

0

2

4

0 2 4 6 8 10 12 14Excitation Energy (MeV)

0

1

2

3

4

5

dσ

/(dΩ e

dΩ

KdE

edE

exc)

[nb/(

sr2G

eVM

eV)]

fitSLABS3

12

ΛB

M.Iodice etal, Phys.Rev.Lett.99,052501(2007) new radiative-corrected data

new calculations with 820 keV for the peaks


New E94-107 data for 12ΛB Theoretical predictions

Ex FWHM Cross section Ex Jπ SLA BS3

(MeV) (MeV) (nb/sr2/GeV ) (MeV) crs diff crs diff

0.00 1− 0.640 0.524

0.116 2− 2.227 2.172

0.00±0.03 1.09±0.05 4.51±0.23(0.67) 2.87 -36% 2.70 -40%

2.587 1− 0.846 0.689

2.593 0− 0.001 0.071

2.62±0.05 0.64±0.11 0.58±0.10(0.11) 0.85 +46% 0.76 +31%

5.642 2− 0.368 0.359

5.717 1− 0.119 0.097

5.94±0.06 0.56±0.10 0.51±0.09(0.09) 0.49 -4% 0.46 -10%

10.480 2+ 0.194 0.157

10.525 1+ 0.085 0.100

11.059 2+ 0.959 0.778

11.132 3+ 1.485 1.324

11.674 1+ 0.050 0.047

10.93±0.04 1.29±0.07 4.68±0.24(0.60) 2.77 -41% 2.41 -48%

12.967 2+ 0.552 0.447

13.074 1+ 0.167 0.196

12.65±0.06 0.60±0.11 0.63±0.12(0.15) 0.72 +14% 0.64 +2%


Central kinematics of the JLab Hall A and C measurements on the Carbon targetused in the calculations (the laboratory frame)

Ei Ef θe θKe ΦK Eγ θγe Q2 ε Γ pKGeV GeV deg deg deg GeV deg GeV2 GeV

E94-107 (Hall A)

3.77 1.56 6 6 180 2.21 4.20 0.0644 0.703 0.0174 1.95–1.96

E01-011 (Hall C)∗

1.851 0.351 5.4 7.11 90 1.50 1.26 0.0058 0.365 0.0287 1.20–1.22

E05-115 (Hall C)∗

2.344 0.844 5.4 7.62 90 1.50 3.02 0.0176 0.635 0.0310 1.20–1.22

* L. Tang et al, Phys.Rev.C 90 (2014) 034320; T.Gogami PhD Thesis, private communication

kinematics differ mainly in Eγ , Q2, and ΦK ;

in E01-011, photons are almost real, Q2 ≈ 0, → photoproduction approximationis possible, see the next talk by Toshio Motoba and JPS Conf. Proc. 17, 011003 (2017).


Comparison with E01-011 data on 12C(e, e′K+)12Λ B and calculations by Motoba-san

Experimental dataa our results with SLA Motoba-san (SLA)b

BΛ Ex Cross section Jπ Ex crs sum Ex crs sum

(MeV) (MeV) (nb/sr) (MeV) (nb/sr) (MeV) (nb/sr)

1− 0.0 13.90 0.0 21.1

11.517 0.0 101.0 ± 4.2 2− 0.116 44.70 58.6 0.186 89.3 100.4

1− 2.587 17.26 2.398 48.4

8.390 3.127 33.5 ± 11.3 0− 2.593 0.04 17.3 3.062 7.7 56.1

2− 4.761 0.37 5.023 7.0

2− 5.642 7.20 6.267 11.8

5.440 6.077 26.0 ± 8.8 1− 5.717 2.44 10.0 6.389 5.0 23.8

2.882 8.635 20.5 ± 7.3

2+ 10.480 5.15 11.000 1.3

1.470 10.047 31.5 ± 7.4 1+ 10.525 2.16 7.3 11.120 8.2 9.5

2+ 11.059 25.23 11.610 53.2

0.548 10.969 87.7 ±15.4 3+ 11.132 39.08 64.4 11.081 77.6 130.7

-0.318 11.835 46.3 ±10.3 1+ 11.674 5.37 5.4 12.129 6.1

2+ 12.967 13.96 12.784 20.0

-0.849 12.366 28.5 ± 7.4 1+ 13.074 4.36 18.3 13.176 3.7 29.8aL. Tang et al, Phys. Rev. C 90, 034320 (2014), Table III.bphotoproduction calculations, T. Motoba, JPS Conf. Proc. 17, 011003 (2017), Table I.


Photo- versus electroproduction in the elementary process

The laboratory cross section at E labγ = 1.5 GeV (W = 1.92 GeV), θlabK = 7 (θc.m.K = 16.6),

Q2 = 0.0058 GeV2, ε = 0.4, and ΦK = 90

Saclay-Lyon A (SLA) BS3

[µb/sr] electroproduction photoproduction electroproduction photoproduction

σT 2.044 1.728

σL 0.008 0.034

σTT -0.080 -0.062

σTL 0.100 -0.092

σfull 2.079 2.043 1.766 1.741

σfull = σT + εσL + εσTT cos 2ΦK +√

2ε(1+ε) σTL cosΦK

the elementary cross section is larger for electroproduction than for photoproduction

the difference between photo- and electroproduction hypernucleus cross sections comesmainly from the nucleus structure and kaon distortion


More hypernucleus states

Calculated spectrum of 12ΛB

2.81

0.73

0.43

C2S

0

2125

5020

7286

7978

9272

9873

11600

3/2−

1/2−

3/2−

5/2+

3/2+

5/2+3/2+

5/2+

0116

25872593

56425717

10480105251105911132

1296713074

1−2−

1−0−

2−1−

2+1+2+3+

2+1+

∆S=0

3.078

0.525

0.372

0.755

4.212

0.853

∆S=1

1.1824.108

1.655

0.7330.237

0.5630.2482.7915.572

1.7170.520

12ΛB

11B

states included in the calculationare based only on the 0~ω coreconfiguration [s4p7]→ J−c states

|12ΛB, J−〉 : |11B, J−c 〉 ⊗ |Λ, s1/2〉

|12ΛB, J+〉 : |11B, J−c 〉 ⊗ |Λ, p1/2〉

|11B, J−c 〉 ⊗ |Λ, p3/2〉

extension via 1~ω core excitations[s4p6(sd)1] and [s3p8]→ J+

c states

|12ΛB, J+〉 : |11B, J+

c 〉 ⊗ |Λ, s1/2〉

|12ΛB, J−〉 : |11B, J+

c 〉 ⊗ |Λ, p1/2〉|11B, J+

c 〉 ⊗ |Λ, p3/2〉

figure by John Millener


Results – the 16O(e, e′K+)16Λ N spectrum from E94-107

Ee = 3.66, Ee′ = 1.45 GeV, θe = 6, θKe = 6, ΦK = 180

→ Eγ = 2.21 GeV, Q2 = 0.058 GeV2, |~pK| = 1.95− 1.97 GeV, θlabKγ = 2.1


W.-S. w.f.:Ep

p3/2= −17.82, Ep

p1/2= −11.2, EΛ

s1/2= −13.5, EΛ

p1/2= −2.3, EΛ

p3/2= −2.9 MeV

-20 -15 -10 -5 0 5 10 15- Binding Energy (MeV)

0

1

2

3

4

dσ

/(dΩ e

dΩ

KdE

edE

b)

[nb/(

sr2G

eVM

eV)]

fitSLABS3

16

ΛN

experiment theory with BS3 sum

Ex crs Ex Jπ crs diff

0.0 1.45±0.26 0.0 0− 0.134 1.52

0.023 1− 1.391 +5%

6.83 3.16±0.35 6.730 1− 0.688 2.84

6.978 2− 2.153 -10%

10.92 2.11±0.37 11.000 2+ 1.627

11.116 1+ 0.679 2.38

11.249 1+ 0.071 +13%

17.10 3.44±0.52 17.303 1+ 0.181

17.567 2+ 1.783 4.01

17.515 3+ 2.045 +17%F. Cusanno etal, P. R. L. 103 (2009) 202501

fit and theory with Voight functions


Angular dependence of the cross sections

4 5 6 7 8 9 10 11 12

θKe

L [deg]

0

1

2

3

4

dσ

/(dΩ e

dΩ

Kd

Ee)

[n

b/(

sr2G

eV)]

SLA 1+

SLA 2+

SLA 3+

BS3 1+

BS3 2+

BS3 3+

16O (e,e’K

+)16

ΛNE

e= 3.66, E

e’= 1.45 GeV

θe= 6

o, Φ

K= 180

o

17 MeV multiplet 1+ 2

+ 3

+

2+

1+

3+

different angular behavior for the 2+ and3+ states is from the nucleus structure

a weak dependence on the elementaryamplitude (3+) is due to the small Q2

information on the nucleus structure andthe elementary amplitude

4 6 8 10 12

θKe

L [deg]

0

0,5

1

1,5

2

2,5

dσL

/dΩ

K [µ

b/s

r]

TLTTTL

4 6 8 10 12

θKe

L [deg]

SLA BS3

Q2 = 0.058 GeV

2

EL

γ = 2.21 GeV

p(e,e’K+)Λ

4 5 6 7 8 9 10 11 12

θKe

Lab [deg]

0

100

200

dσ

/dΩ

K [n

b/s

r]

TLTTTLds

4 5 6 7 8 9 10 11 12

16O (e,e’K

+)16

ΛN

E = 17.515 MeV, JP = 3

+

BS3 model

SLA model


Uncertainty in the DWIA calculations

the model for elementary production, e.g. SLA, BS3, ...

the kaon wave function – the eikonal approximation– parameters of the kaon-nucleus optical potential– more precise kaon wave function (Klein-Gordon equation)

the nuclear structure – the full basis of p-shell states– a larger model-space calculation of the OBDME→ the strength is distributed among more states

single-particle wave functions in the radial integral– harmonic oscillator × Woods-Saxon wave functions– parameters of the W.-S. potential

approximations– effective factorization × full folding– off-shell effects in Jj (energy conservation on elementary level is violated)

uncertainty in input kinematics – angles, momenta→ averaging arround a central kinematics (with a weighting function)


Uncertainty from the elementary production

separated lab cross sections for 12C(e, e′K+)12ΛB in central kinematics of E01-011

Eγ = 1.5 GeV, θKγ = 7, Q2 = 0.0058 GeV2, ε = 0.4, and ΦK = 90

the full cross section: ds = (dsT + dsL + dsTT + dsTL)

Ex JP model ds dsT dsL dsTT dsTL

(MeV) (nb/sr) (nb/sr) (nb/sr) (nb/sr) (nb/sr)

0.0 1− SLA 13.90 13.45 0.01 0.44 0.00

BS3 14.04 13.08 0.01 0.95 0.00

BS1 15.03 12.60 0.01 2.41 0.00

0.116 2− SLA 44.70 44.24 0.08 0.38 0.00

BS3 35.33 35.31 0.34 -0.33 0.00

BS1 36.42 33.84 0.62 1.96 0.00

11.06 2+ SLA 25.23 24.68 0.02 0.52 0.00

BS3 22.35 21.64 0.02 0.68 0.00

BS1 22.90 19.88 0.02 2.99 0.00

11.13 3+ SLA 39.08 33.17 0.11 5.80 0.00

BS3 29.70 24.66 0.44 4.60 0.00

BS1 39.73 36.76 0.89 2.08 0.00


Estimated uncertainty of predicted cross sections for the 12C target

in Hall C kinematics

very small Q2: 0.006 x 0.06 GeV2 in Hall A experiment

electroproduction (ds) x photoproduction (dsT): < 1–15% (SLA and BS3)

using the SLA, BS3, and BS1 isobar models: < 20–30%– the models are still not well fixed at very small angles, θc.m.Kγ < 30

parameters of the K+-nucleus optical potential:

σKNtot ± 10% ⇒ 7% for the Λs states

5% for the Λp states

α = Re f KN(0)/Im f KN(0)± 10% ⇒ 1%

input kinematics:θKe ± 1 ⇒ 20–30%ΦK ± 10 ⇒ 1–5%

– due to the strong angular dependence averaging with a weighting functioncan modify predictions


Summarynew DWIA calculations of the cross sections for electroproductionof 9

ΛLi, 12Λ B, and 16

Λ N in various kinematics are in a reasonableagreement with experimental data from JLab;

different isobar models for the elementary production, SLA and BS3,can be used giving near-by values (< 30%);

the calculations mostly underpredicts (by 10 – 60 %) the crosssections for 9

ΛLi and 12Λ B (could be attributed to elementary production),

however, they overpredict by 5 – 30 % the cross sections for 16Λ N;

large uncertainty in our calculations comes from the elementaryproduction and from a possible indefinitness of kaon polar angle.

Outlookmore detailed analysis with various models of elementary productionand with various nuclear structure (e.g., full basis of 1hω states).


We wish to remember

Salvatore Frullani, Miloslav Sotona, and Francesco Cusanno

who died before the archival paper on the Hall A experimentswas completed.

Thank you for your attention!


jefferson lab - electroproduction of hypernuclei...sotona and frullani, prog. theor. phys. suppl....

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