Rimantas LAZAUSKAS, 07/07/2009

Elastic p-3H scattering below the n-3He threshold

Rimantas LAZAUSKAS, 07/07/2009

Why 4N scattering?

Few-nucleon systems:Few-nucleon systems: present the most natural testground for the inter-nucleon interaction models

Present interaction models (>70 years of research):o Phenomenological (INOY,ISUJ,..)o Meson exchange theory (Nijm, AV18, CD-Bonn,…)o Chiral perturbation theory (Idaho, Epelbaum,…)

2N is made to work with perfect fits (2/data~1.) and several parameters 3N problems begin, requires one more fit to fix energies (3N-Force), no effective remedies for some scattering observables Beyond 3N: the most accessible testground A=3-12 bound states, but not easy to find uncorelated observables

Rimantas LAZAUSKAS, 07/07/2009

Why 4N scattering?

-2 -1 0 1 23725

3750

3755

3760

dd

dnpdnn

dpp

...4He

p3H

n3Hep3He

n3H

ppppnppp

nnppnnnp

nnnn

m (

MeV

)

Tz

Ab-initio calculations very limited beyond A=4 system: first very

promising calculations exist using GFMC (K. Nollett et al.), NCSM (S. Quaglioni on

friday) and RGM (previous talk) require benchmarking. A=4 scattering below break-up threshold became accessible and well established using ab-initio methods: FY, AGS (A. Deltuva et al.), HH

(M. Viviani et al.) as well as RGM (H.M. Hofmann et al.)

Very rich dynamics of 4N

0.1 1 10

1

10 n-1H

n-2H

n-3H

(b

)

En (MeV)

Rimantas LAZAUSKAS, 07/07/2009

Why 4N scattering?

4 4 1 4 20 12 12,3 12 23,1 23,4 13,2 13,4 13,24 23,14

2 10 12 12,34 12 34,1 34,2 34,1

12

26

ˆ( )

ˆ( )

E H V K V K K K K H H

E H V H V K K H

( , , )ˆ ˆ ˆ( , , ) ( , , )LST

LST

F x y zx y z L x y z S T

x y z

Basis is infinite

Solution is searched by decomposing FY components in the partial wave basis

( , , )LSTF x y z Radial parts of the amplitudes are developed in the spline basis, converting differential equations into linear algebra problem. Extensions are made to incorporate 3NF and Coulomb interaction

Faddeev-Yakubovski (FY) equations in configuration space

Reality is finite. ConvergenceConvergence

32

1

4

32

1

4x x

y yz z

412,3K 12,34H

Rimantas LAZAUSKAS, 07/07/2009

4He continuum

3725

3750

3755

...

dd

dnp

...

4He

p3H - n3He

nnpp

m (

Me

V)

No Coulomb

Becomes Becomes a bound a bound state!!!state!!!0-

3725

3750

3755

3760

...

(2+,0)

(1-,1)(0-,1)

(1-,0)

(0-,0)(2-,0)

(1-,1)(2-,1)

(0+,0)

dd

dnp

...

4He

p3H

n3He

nnpp

m (

MeV

)

B=]0.04-0.26[ MeV

Rimantas LAZAUSKAS, 07/07/2009

p-3H scattering/convergence

INOY potential jmax=max(jx,jy,jz)

32

1

4

32

1

4jx jx

jy jyjz lz

412,3K 12,34H

0.76 MeVn+3He

p+3H0+

Rimantas LAZAUSKAS, 07/07/2009

p-3H scattering lengths

2

202

]20,115[a

E keV

0.76 MeVn+3He

p+3H0+

24 25 26 27 28 29-40

-35

-30

-25

-20

-15

I-N3LO+UIX

a0 (

fm)

B4He (MeV)

AV18

I-N3LO

INOY

ISUJ

AV18+UIX

Rimantas LAZAUSKAS, 07/07/2009

p-3H scattering below n-3He threshold

0 250 500 750 10000

200

400

600

800

1000

AV18+UIX N3LO+UIX* N3LO AV18

d

/d

(mb/

sr)

Ep (keV)

INOY ISUJ

Exp: Balashko et al., Sov. Phys. JETP 9 (1959) 137; Jarmie et al., Phys. Rev 130 (1963)1987; Ennis et al., Phys. Rev. 95 (1954) 772

c.m.=120°

Rimantas LAZAUSKAS, 07/07/2009

p-3H scattering below n-3He threshold

0 250 500 750 10000

200

400

600

800

1000

AV18+UIX(S,P) AV18+UIX N3LO+UIX* N3LO AV18

d

/d

(mb/

sr)

Ep (keV)

INOY (S,P) INOY ISUJ

Exp: Balashko et al., Sov. Phys. JETP 9 (1959) 137; Jarmie et al., Phys. Rev 130 (1963)1987; Ennis et al., Phys. Rev. 95 (1954) 772

c.m.=120°

Rimantas LAZAUSKAS, 07/07/2009

p-3H scattering below n-3He threshold

0 45 90 135 1800

500

1000

1500

2000

2500

c.m.

(deg)

d/d

(mb/

sr)

INOY(S,P)/INOY/ISUJ AV18+UIX(S,P)/I-N3LO+UIX I-N3LO AV18

0 45 90 135 1800

100

200

300

400

500

600

c.m.

(deg)

d/d

(mb/

sr)

AV18+UIX(S+P) INOY(S,P) INOY/ISUJ AV18+UIX/I-N3LO+UIX* I-N3LO AV18

Ep=300 keV Ep=900 keV

Exp: Y. G. Balashko et al., Journ. Izv. Rossiiskoi Akademii Nauk, Ser. Fiz. 28 (1964) 1124; Proc. Nuclear Physics Congress, Paris (1964) 255

Rimantas LAZAUSKAS, 07/07/2009

Conclusion

Accurate theoretical calculations for 4N scattering are now available

Continuum of 4He turns to be a rich laboratory to test NN & NNN interactions: revealing some discrepencies already for 0+ state

More experiments in 4N sector and in particular for low energies will be very appreciated

Acknowledgements: The numerical calculations have beenperformed at IDRIS (CNRS, France). We thank the staff members of the IDRIScomputer center for their constant help.