Study of bound and scattering states of few-body … of bound and scattering states of few ... A....

Study of bound and scattering states of few-bodysystems with the HH method

M. Viviani

INFN, Sezione di Pisa &Department of Physics, University of Pisa

Pisa (Italy)

Electron-Nucleus Scattering XIII, June 23-27, 2014Mini-symposium to honor Professor Sergio Rosati on his 80th Birthday

M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/06/14 1 / 32

Outline

1 The rise (and fall...) of the Few-Body Pisa Group

2 3NF effects in few-nucleon systemsp − d scatteringp − 3He scatteringp − 3H scattering

CollaboratorsS. Rosati, A. Kievsky & L.E. Marcucci - INFN & Pisa University, Pisa (Italy)

L. Girlanda - INFN & Universitá del Salento, Lecce (Italy)

R. Schiavilla, M. Piarulli, A. Baroni, F. Spadoni - Jefferson Lab. & ODU, Norfolk (VA, USA)

S. Pastore - USC, Columbia (SC, USA)


The rise (and fall...) of the Few-Body Pisa Group

The Jastrow period (1961–1972)

B. Barsella and S. Rosati, On the Effect of n-p Tensor Forces in 3HΛ, Nuovo Cimento 20,914 (1961).

J. Murphy and S. Rosati, A two-body method for the bound states of a three-body system,Nucl. Phys. 63, 625 (1965).

M. Barbi and S. Rosati, Direct numerical solution of the three-body problem, Phys. Rev.147, 730 (1966).

S. Fantoni, L. Panattoni, and S. Rosati, Calculation on nuclear 3-body and 4-body systemswith jastrow-type correlated wave functions, Nuovo Cimento 69, 80 (1970).

L. Lovitch and S. Rosati, Bound State Solution of the Two-Nucleon Schroedinger Equationwith Tensor Forces, Comp. Phys. Com. 2, 353 (1971).

Ψ = g(r12)g(r13)g(r23) δ〈Ψ|H − E |Ψ〉 = 0

“Euler” equation: for an S-state and for a central spin-independent potential v(r):

−1

M∇2g(r) +

(

v(r) + w [g, r ])

g(r) = Eg(r)

Non-linear – iterative solution


Interlude 1The Many-body period (1972–1988)

[S. Fantoni and S. Rosati, Expansion Procedure for Jastrow-Type Correlated Wave Functions,Nuovo Cimento A 10, 145 (1972)]

My first conference:Third International Conference on Recent Progress in Many-Body Theories

Odenthal-Altemberg, Germany, August 29 – September 3, 1983A wonderful 2-days trip from Pisa to Altenberg (and back) with Sergio, Stefano, and Adelchi,

crammed in Stefano’s FIAT 127


A new start....

Improvements of the wave functions and extension to realistic potentials (1989–2005)

From the CHO to the CHH expansion (selected papers)

A. Kievsky, S. Rosati and M. Viviani, Euler and Correlated Harmonic Oscillator WaveFunctions for Three-Nucleon Systems, Nucl. Phys. A 501, 503 (1989). Few-Body Systems9, 1 (1990).

M. Viviani, A. Kievsky and S. Rosati, Correlated Hyperspherical Harmonic Calculations forThree- and Four-Body Systems, Nuovo Cimento A 105, 1473 (1992).

A. Kievsky, M. Viviani and S. Rosati, The Three-Nucleon Bound-State with Realistic Softand Hard Core Potentials, Nucl. Phys. A 551, 241 (1993).

A. Kievsky, M. Viviani and S. Rosati, Variational Calculations for Scattering States inThree-Nucleon Systems, Few-Body Sys. Suppl. 7, 278 (1994).

M. Viviani, A. Kievsky and S. Rosati, Calculation of the Alpha-Particle Ground-State,Few-Body Systems 18, 25 (1995).

A. Kievsky, L. E. Marcucci, S. Rosati and M. Viviani, High-Precision Calculation of theTriton Ground State within the Hyperspherical Harmonics Method, Few-Body Systems 22,1 (1997).

M. Viviani, S. Rosati and A. Kievsky, Neutron-3H and Proton-3He Zero Energy Scattering,Phys. Rev. Lett. 81, 1580 (1998).


The goal: reach the accuracy achieved by other groups (Friar and Coll., Glöckle and Coll.,Kamimura and Coll.,. . .) + be able to treat realistic interactions, also hard-core potentials

full inclusion of the Coulomb interaction –Ay puzzle in p − d elastic scattering

Ψ =∑

i=1,3

Nc∑

α=1

fα(rjk )g(rij )g(rik )

[

Ylα (xi )YLα (yi )]

Λα

[

(sj sk )Sαsi

]

Σα

JJz

[

(tj tk )Tαti]

TαFα(xi , yi )

Fα(xi , yi ) expanded first in either HO or HH basisCHH=correlated Hyperspherical harmonics expansion (“PHH” if g = 1)

Method B (MeV) T (MeV) Ps′ (%) PD (%) PP (%)AV18 potential

PHH −7.624 46.727 1.293 8.510 0.066Witala et al. (2003) −7.621 45.73 1.291 8.510 0.066

Hamada-Johnston (hard-core) potentialCHH −7.06 72.95 1.46 10.18 0.09

Delves & Hennel (1971) −6.5 9.00


Interlude 2

The first Few-Body conference:12th International Conference on Few-Body Problems in Physics (FBXII)

Vancouver, B.C., Canada, July 2-8, 1989


Many ideas....

Some of the ideas produced by Sergio during the many discussions we had in those years....


One-point integration....

Idea: to integrate one needs to compute the integrand in just one point....

0 1 2 3 4 5x

0

0.2

0.4

0.6

0.8

1

x0

I =∫ b

adx f (x) = (b − a) ∗ f (x0)

It would be very useful in case ofmultidimensional integration...

I =∫

dx1 . . .

∫

dxn f (x1, . . . , xn) = Volume ∗ f (x1, . . . , xn)

Unfortunately we could not succeed...


The last period

Extension of the method to non-local momentum space potentials (2005–present)no correlation factor – the interactions are softer & easy to trasform the wave function from

coordinate to momentum space (and viceversa)

(selected papers)

M. Viviani, A. Kievsky and S. Rosati, Calculation of the Alpha–Particle Ground State withinthe Hyperspherical Harmonic Basis, Phys. Rev. C 71, 024006 (2005)

M. Viviani, L. E. Marcucci, S. Rosati, A. Kievsky and L. Girlanda, Variational Calculation onA=3 and 4 Nuclei with Non-Local Potentials, Few-Body Systems 39, 159 (2006)

A. Kievsky, S. Rosati, M. Viviani, L. E. Marcucci, L. Girlanda, A High-Precision VariationalApproach to Three- and Four-Nucleon Bound and Zero-Energy Scattering States, J. Phys.G: Nucl. Part. Phys. 35, 063101 (2008)

M. Viviani, A. Deltuva, R. Lazauskas, J. Carbonell, A. C. Fonseca, A. Kievsky, L.E.Marcucci, and S. Rosati, Benchmark calculation of n-3H and p-3He scattering, PRC 84,054010 (2011)


Interlude 3

18th International Conference on Few-Body Problems in Physics (FBXII)Santos, SP, Brazil, August 21-26, 2006


Applications

EM and Weak transitions in light nuclei

p−, p − d , and “hep” astrophysical factor Form factors of 3H, 3He, and 4He

Parity-violation

Longitudinal polarization in ~n + 3He → p + 3H

Study of chiral effective field theory potentials and currents

0 10 20 30 40 50E

c.m.(keV)

0

0.1

0.2

0.3

0.4

0.5

S(E

c.m

.) (e

V b

)

LUNA Griffiths et al.Schmid et al.


3NF effects in few-nucleon systems

Nuclear Dynamics: the EFT approach

Chiral effective field theory (χEFT): N − π interaction “dictated” by chiral symmetry[Weinberg (1990), Bernard, Kaiser, & Meissner (1995), Ordonéz, Ray, & van Kolck (1996),Epelbaum, Meissner, & Epelbaum (1998), . . .]

Pionless effective field theory (π/EFT): low energy processes [Kaplan, Savage, & Wise(1998), van Kolck (1999), . . .]

Study of A = 3, 4 reactions using χEFT

Test of the derived NN & 3N interactions

Different accurate theoretical techniques

FY/AGS equations in momentum space [Witala et al., 1990-2014], [Deltuva &

Fonseca, 2007, 2012] FY equations in coordinate space [Lazauskas & Carbonell, 2009, 2012] HH expansion + Kohn variational principle NCSM/RG method [Navratil, Quaglioni & Coll., 2010–2014]

Ay puzzle, energy production (NIF), reactions of astrophysical interest, study of

fundamental symmetries, etc.

muon capture on d “MUSUN” [Marcucci, 2011 ], . . .


Ay ”puzzle”

NN interaction

”Old models”: Argonne V18, CD-Bonn, Nijmengen (χ2 ≈ 1)

Derived from χEFT

J-N3LO [Epelbaum and Coll, 1998-2006] N3LO500 & N3LO600 [Entem & Machleidt, 2003 & 2011]

0

0.0005

0.001

0.0015

0.002

Ay

0

0.002

0.004

0.006

0.008

0

0.005

0.01

0.015

0.02

0 30 60 90 120 150 180θ

cm [deg]

0

0.01

0.02

0.03

0.04

Ay

0 30 60 90 120 150 180θ

cm [deg]

0

0.01

0.02

0.03

0.04

0.05

0 30 60 90 120 150 180θ

cm [deg]

0

0.01

0.02

0.03

0.04

0.05

0.06

Ecm

=266 keV Ecm

= 431 keV Ecm

= 666 keV

Ecm

= 1.33 MeV Ecm

= 1.66 MeV Ecm

= 2.0 MeV

A = 3Delicate balance between 4Pj waves

0 30 60 90 120 150θ[c.m.] [deg]

0

0.1

0.2

0.3

0.4

0.5

0.6

GeorgeFisherN3LO500AV18

0 30 60 90 120 150θ[c.m.] [deg]

Fisher

0 30 60 90 120 150 180θ[c.m.] [deg]

AlleyAlley-2

Ep=2.25 MeV E

p=4 MeV E

p=5.54 MeV

A = 4Confirmed also for other NN potentials

(Deltuva & Fonseca 2007)

dashed lines: N3LO500, dotted-dashed lines: N3LO500/UIX, solid lines N3LO500/N2LO500*


Inclusion of the 3NF

Models from EFT

3n force at N2LO: 2 unknonw LEC’s cd& cE + cutoff Λ [Epelbaum et al., 2002]

N2LO500 & N2LO600: LECs fixed byL. Marcucci (see next slide)

N3LO & N4LO pion exchanges: [Krebs,Epelbaum, et al., (2012-2013)]

10 Contact terms at N4LO: [Girlanda,Kievsky, MV, (2011)]

(f)(a) (b) (c) (d) (e)

Illinois models

Fujita−Miyazawa S−wave scattering pion rings

[Pieper et al., 2001]

3π exchanges and π rings

Illinois-7: coefficients chosen toreproduce the A = 4 − 12 spectrum


New fit of cD and cE

Sensitivity to cD and cE

cD enters also in β-decay processes

[Gardestig & Phillips, 2006], [Gazit et al., 2009] dR = M

4πfπgAcD + M

3 (c3 + 2c4) +16

dR (and thus cD) can be fixed from the tritiumGamow-Teller m.e. GT EXPT = 0.955 ± 0.004

d cR D

X

Models under study:

Model Λ [MeV] cD cE B(4He) [MeV] a2(n − d) [fm]N3LO500 500 25.38 1.100N3LO500/N2LO500 500 −0.12 −0.196 28.49 0.666N3LO600/N2LO600 600 −0.26 −0.846 28.64 0.698AV18 24.22 1.275AV18/IL7 28.44 0.552Expt. 28.30 0.645 ± 0.003 ± 0.007

Expt a2(n − d): K. Schoen et al., 2003


HH expansion

Example: p −

3He elastic scattering

Ω±LS(A,B) =

√

1N

N∑

perm.=1

[

YL(yp)⊗ [φA⊗φB]S

]

JJz

(

fL(yp)GL(η, qAByp)

qAByp± i

FL(η, qAByp)

qAByp

)

|ΨLS〉 =∑

n,[K ]

aLS,[K ]|n, [K ]〉+ |Ω−LS(p,3He)〉 −

∑

L′S′

SLS,L′S′ |Ω+L′S′(p, 3He)〉

|n, [K ]〉 HH states

SLS,L′S′ = S-matrix

aLS,[K ] and SLS,L′S′ computed using the Kohn variational principle

For a review, see [J. Phys. G: Nucl. Part. Phys. 35, 063101 (2008) ]


Convergence

n −

3H at Ecm = 3 MeV, 3P2 wave

0 10 20 30 40 50K

0.001

0.01

0.1

1

δ(K

)-δ(

K-2

) [d

eg]

AV18

N3LO-Idaho

Convergence with the grand-angular quantum number


Benchmark test of 4N scattering calculations [PRC 84,054010 (2011)]

p − 3He elastic scattering

N3LO500 potential

AGS= Deltuva & FonsecaFY= Lazauskas & Carbonell

0 60 1200

100

200

300

400

500

dσ/d

Ω [m

b/sr

]

Famularo 1954Fisher 2006AGS

2.25 MeV

0 60 1200

0,2

0,4

Ay0

Fisher 2006George 2001

0 60 120θc.m. [deg]

0

0,1

0,2

A0y

Daniels 2010

0 60 120

Mcdonald 1964Fisher 2006

4.05 MeV

0 60 120

Fisher 2006

0 60 120θc.m. [deg]

Daniels 2010

0 60 120 180

Mcdonald 1964

5.54 MeV

0 60 120 180

Alley 1993

0 60 120 180θc.m. [deg]

Alley 1993Daniels 2010




N3LO500 potential


0 60 1200

100

200

300

400

500

dσ/d

Ω [m

b/sr

]

Famularo 1954Fisher 2006AGSHH

2.25 MeV

0 60 1200

0,2

0,4

Ay0


0 60 120θc.m. [deg]

0

0,1

0,2

A0y

Daniels 2010

0 60 120


4.05 MeV

0 60 120

Fisher 2006

0 60 120θc.m. [deg]

Daniels 2010

0 60 120 180

Mcdonald 1964

5.54 MeV

0 60 120 180

Alley 1993

0 60 120 180θc.m. [deg]





N3LO500 potential


0 60 1200

100

200

300

400

500

dσ/d

Ω [m

b/sr

]

Famularo 1954Fisher 2006AGSHHFY

2.25 MeV

0 60 1200

0,2

0,4

Ay0


0 60 120θc.m. [deg]

0

0,1

0,2

A0y

Daniels 2010

0 60 120


4.05 MeV

0 60 120

Fisher 2006

0 60 120θc.m. [deg]

Daniels 2010

0 60 120 180

Mcdonald 1964

5.54 MeV

0 60 120 180

Alley 1993

0 60 120 180θc.m. [deg]



Effect of the N2LO & Illinois 3NF in p − d scattering

0 30 60 90 120 150 180θ[deg]

0

0.01

0.02

0.03

0.04

0.05

0.06

Ay

NN onlyNN+3NAV18/IL7

0 30 60 90 120 150 180θ[deg]

0

0.005

0.01

0.015

0.02

0.025

0.03

iT11

p-d scattering at Ep=3 MeV

Bands: results obtained for Λ = 500 & Λ = 600 MeVNN: N3LO500 & N3LO600 — NN+3N: N3LO500/N2LO500 & N3LO600/N2LO600


Study of the N3LO & N4LO 3NF in p − d scattering

[Krebs, Epelbaum, et al., 2012,2013]

(f)(a) (b) (c) (d) (e)

(10)(6) (7)

(5)(4)(3)(2)(1)

(8) (9)

(4)(1) (2) (3)

+ many others diagrams...

First study in n − d scattering[Witala et al., 2013]

0 60 120 180Θ

cm [deg]

0.0

0.1

0.2

Ay E

n=14.1 MeV

d(n,n)d

green band: J-N3LO NN interaction onlymagenta band: + N3LO 3NF (no 2π contact

terms)

Very complicate implementation!


3N contact terms at N4LO (1)

[Girlanda, Kievsky, MV, PRC 84, 014001 (2011)]

(N†O1N)(N†O2N)(N†O3N)

Two of the operators O = spatial derivatives

146 possible combinations, which can be reduced to 10 using Fierz transformation andother constraints

O1O2O3

1− 3←→∇ 1 ·

←→∇ 2[1, τ1 · τ2, τ1 · τ3]

4− 6←→∇ 1 ·

−→σ 1←→∇ 2 ·

−→σ 2[1, τ1 · τ2, τ1 · τ3]

7− 9←→∇ 1 ·

−→σ 2←→∇ 2 ·

−→σ 1[1, τ1 · τ2, τ1 · τ3]

10− 12←→∇ 1 ·

←→∇ 2−→σ 1 ·

−→σ 2[1, τ1 · τ2, τ1 · τ3]

13− 16←→∇ 1 ·

−→σ 1←→∇ 2 ·

−→σ 3[1, τ1 · τ2, τ1 · τ3, τ2 · τ3]

17− 20←→∇ 1 ·

−→σ 3←→∇ 2 ·

−→σ 1[1, τ1 · τ2, τ1 · τ3, τ2 · τ3]

21− 24←→∇ 1 ·

←→∇ 2−→σ 1 ·

−→σ 3[1, τ1 · τ2, τ1 · τ3, τ2 · τ3]

25←→∇ 1 ×

←→∇ 2 ·

−→σ 1[τ1 × τ2 · τ3]

26←→∇ 1 ×

←→∇ 2 ·

−→σ 3[τ1 × τ2 · τ3]

27←→∇ 1 ·

←→∇ 2−→σ 1 ×

−→σ 2 ·−→σ 3[τ1 × τ2 · τ3]

. . . . . .

146←→∇ 1 ·

−→σ 2←→∇ 1 ×

−→σ 1 ·−→σ 3[τ1 × τ2 · τ3]


3N contact terms at N4LO (2)

Final form of the new 3NF terms

V =∑

i 6=j 6=k

(E1 + E2τi · τj + E3σi · σj + E4τi · τjσi · σj )

[

Z ′′0 (rij ) + 2Z ′0(rij )

rij

]

Z0(rik )

+(E5 + E6τi · τj )Sij

[

Z ′′0 (rij )−Z ′0(rij )

rij

]

Z0(rik )

+(E7 + E8τi · τk )(L · S)ijZ ′0(rij )

rijZ0(rik )

+(E9 + E10τj · τk )σj · rijσk · rik Z ′0(rij )Z′0(rik )

Local (using a cutoff of the type F (k2j ; Λ)F (k2

k ; Λ))

Z0(r ; Λ) =∫

dp(2π)3

eip·rF (p2; Λ)

At N4LO there are new 10 LEC’s


Fit of the LEC’s – Preliminary study

“Minimal” model: AV18 + 3NF only from 3N contact terms

Terms E3, E5, E7

100

200

300

400

50 100 150θ (degrees)

dσ/d

Ω

-0.04

-0.02

0

0.02

0 50 100 150θ (degrees)

T20

-0.01

0

0.01

0.02

0.03

0 50 100 150θ (degrees)

T20

-0.04

-0.03

-0.02

-0.01

0

0 50 100 150θ (degrees)

T22

0

0.01

0.02

0.03

0 50 100 150θ (degrees)

i T11

0

0.02

0.04

0.06

0 50 100 150θ (degrees)

Ay

Λ=200 MeVχ2/d.o.f = 4a2 = 0.652 fmB(3H) = 8.483 MeV

cE=0.654E3=-0.780 E5=-0.143 E7=1.523

Terms E5, E7, E10

100

200

300

400

50 100 150θ (degrees)

dσ/d

Ω

-0.04

-0.02

0

0.02

0 50 100 150θ (degrees)

T20

-0.01

0

0.01

0.02

0.03

0 50 100 150θ (degrees)

T20

-0.04

-0.03

-0.02

-0.01

0

0 50 100 150θ (degrees)

T22

0

0.01

0.02

0.03

0 50 100 150θ (degrees)

i T11

0

0.02

0.04

0.06

0 50 100 150θ (degrees)

Ay

Λ=300 MeVχ2/d.o.f = 3.5a2 = 0.615 fmB(3H) = 8.483 MeV

cE=0.542E5=-0.262 E7=1.756 E10=-0.649


3NF effect in p −

3He scattering (1)

p − 3He phase-shift - Comparison PSA/Theory (PSA: [Daniels et al., 2010])– N2LO 3NF only –

-60

-50

-40

-30

-20

phas

e-sh

ift [

deg]

NN onlyNN+3NTUNL 2010

2 3 4 5 6E

p [MeV]

-70

-60

-50

-40

-30

phas

e-sh

ift [

deg]

2 3 4 5 6E

p [MeV]

0

1

2

3

phas

e-sh

ift [

deg]

10

20

30

phas

e-sh

ift [

deg]

1S

0

3S

1ε(1

+)

3P

0


3NF effect in p −

3He scattering (1)


-60

-50

-40

-30

-20

phas

e-sh

ift [

deg]

NN onlyNN+3NTUNL 2010AV18/IL7

2 3 4 5 6E

p [MeV]

-70

-60

-50

-40

-30

phas

e-sh

ift [

deg]

2 3 4 5 6E

p [MeV]

0

1

2

3

phas

e-sh

ift [

deg]

10

20

30

phas

e-sh

ift [

deg]

1S

0

3S

1ε(1

+)

3P

0


3NF effect in p −

3He scattering (1)


-60

-50

-40

-30

-20

phas

e-sh

ift [

deg]

NN onlyNN+3NTUNL 2010N3LO500/N2LO500N3LO600/N2LO600

2 3 4 5 6E

p [MeV]

-70

-60

-50

-40

-30

phas

e-sh

ift [

deg]

2 3 4 5 6E

p [MeV]

0

1

2

3

phas

e-sh

ift [

deg]

10

20

30

phas

e-sh

ift [

deg]

1S

0

3S

1ε(1

+)

3P

0


3NF effect in p −

3He scattering (2)

p − 3He phase-shift - Comparison PSA/TheoryPSA: [Daniels et al., 2010]

10

20

30

40

phas

e-sh

ift [

deg]

NN onlyNN+3NTUNL 2010

2 3 4 5 6E

p [MeV]

10

20

30

40

50

phas

e-sh

ift [

deg]

2 3 4 5 6E

p [MeV]

10

20

30

40

50

60

phas

e-sh

ift [

deg]

8

12

16

20

phas

e-sh

ift [

deg]

1P

1

3P

1

3P

2

ε(1-)


3NF effect in p −

3He scattering (2)


10

20

30

40

phas

e-sh

ift [

deg]

NN onlyNN+3NTUNL 2010AV18/IL7

2 3 4 5 6E

p [MeV]

10

20

30

40

50

phas

e-sh

ift [

deg]

2 3 4 5 6E

p [MeV]

10

20

30

40

50

60

phas

e-sh

ift [

deg]

8

12

16

20

phas

e-sh

ift [

deg]

1P

1

3P

1

3P

2

ε(1-)


3NF effect in p −

3He scattering (2)


10

20

30

40

phas

e-sh

ift [

deg]

NN onlyNN+3NTUNL 2010N3LO500/N2LO500N3LO600/N2LO600

2 3 4 5 6E

p [MeV]

10

20

30

40

50

phas

e-sh

ift [

deg]

2 3 4 5 6E

p [MeV]

10

20

30

40

50

60

phas

e-sh

ift [

deg]

8

12

16

20

phas

e-sh

ift [

deg]

1P

1

3P

1

3P

2

ε(1-)


p −

3He observables at Ep = 5.54 MeV (1)

θc.m.

[deg]0

100

200

300

400

dσ/dΩ [mb/sr]

NN bandNN+3N band

θc.m.

[deg]0

0.1

0.2

0.3

0.4

0.5

Ay

θc.m.

[deg]-0.1

0

0.1

0.2 A0y


p −


θc.m.

[deg]0

100

200

300

400

dσ/dΩ [mb/sr]

NN bandNN+3N bandAV18/IL7

θc.m.

[deg]0

0.1

0.2

0.3

0.4

0.5

Ay

θc.m.

[deg]-0.1

0

0.1

0.2 A0y


p −


θc.m.

[deg]0

100

200

300

400

dσ/dΩ [mb/sr]

NN bandNN+3N bandN3LO500/N2LO500N3LO600/N2LO600

θc.m.

[deg]0

0.1

0.2

0.3

0.4

0.5

Ay

θc.m.

[deg]-0.1

0

0.1

0.2 A0y


p −


0 30 60 90 120 150 180θ

c.m. [deg]

0

0.1

0.2A

yy

0 30 60 90 120 150 180θ

c.m. [deg]

0

0.1

0.2A

xx

0 30 60 90 120 150 180θ

c.m. [deg]

-0.2

-0.1

0

0.1

0.2

Axz


S = 0 & S = 1 n −

3He scattering lenght [fm]

Int. Method a0 (fm) a1 (fm)AV18 HH 7.69 − i5.70 3.56 − i0.0077

RGM 7.79 − i4.98 3.47 − i0.0066FY 7.71 − i5.25 3.43 − i0.0082

N3LO500 HH 7.57 − i4.97 3.46 − i0.0048FY 3.56 − i0.0070AGS 7.82 − i4.51 3.47 − i0.0068

N3LO500/N2LO500* HH 7.61 − i4.32 3.37 − i0.0042N3LO500/N2LO500 HH 7.67 − i3.99 3.38 − i0.0050N3LO600/N2LO600 HH 7.92 − i4.95 3.38 − i0.0049Exp.[ILL-1] 7.370(58)− i4.448(5) 3.278(53)− i0.001(2)Exp.[ILL-2] 7.46(2) 3.36(1)Exp.[NIST] 7.57(3) 3.48(2)

RGM: [Hofmann & Hale, PRC 77, 044002(2008)]FY: [Lazauskas et al., PRC 83, 034006 (2011)]AGS: [Deltuva, priv. comm.]ILL-1: [Zimmer et al., EPJA 4, 1 (2002)]ILL-2: [Ketter et al., EPJA 27, 243 (2006)]NIST: [Huffman et al., PRC 70, 014004(2004)]

From neutron interferometry at NIST[Huber et al., PRL 103, 179903 (2009)]

Re(a1 − a0)EXPT = −4.20(3) fm

Re(a1 − a0)N3LO500/N2LO500∗ = −4.24 fm

Re(a1 − a0)N3LO500/N2LO500 = −4.29 fm


Proton-3H scattering at low energies

First monopole resonance of 4He

E = −8.20 ± 0.05 MeV, W = 270 ± 50keV [Walcher, 1970]

FY: [R. Lazauskas, 2009]

Monopole resonance:

[Hiyama et al., 2004 ] [Bacca et al., 2013 ]

“Tension” with the experimentalresonance transition FF FM(q)

0 1 2 3 4

q2

[fm-2

]

0

1

2

3

4

5

|FM

|2 /4π

* 10

-4

Koebschall et al. Frosch et al. WalcherHiyama et al.AV18+UIX

NN(N3LO) +

3NF(N2LO)

0 30 60 90 120 150θ

c.m. [deg]

0

200

400

600

800

1000

dσ/d

Ω [

mb/

sr]

Balaskho (1960)N3LO500/N2LO500 (only S-waves)

0 30 60 90 120 150θ

c.m. [deg]

0 30 60 90 120 150 180θ

c.m. [deg]

Ep=0.4 MeV E

p=0.6 MeV E

p=0.99 MeV

PRELIMINARY

PRELIMINARY

p-3H elastic scattering


Work in progressfix the contact N4LO 3NF in 3N systems and see their effect in A = 4

Extension of the HH technique to treat N − d breakup (in collaboration with E. Garrido -

CSIC, Madrid (Spain)

comparison with the benchmark calc. of [Friar et al., (1995)] – PRC (in press):

→ p − d breakup: effect of the Coulomb int. Complete the study of p3H, n − 3He & d − d scattering

Extension to A > 4Extension of the HH technique to A > 4 systems (in collaboration with M. Gattobigio -

INLN, Nice (France) ) [PRA 79, 032513 (2009)], [PRC 83, 024001 (2011)]

Use of Integral Relations to compute observables (in collaboration with E. Garrido - CSIC,

Madrid (Spain) & C. Romero-Redondo - TRIUMF, Vancouver (Canada) ) – No need of the

asymptotic part [PRA 83, 022705 (2011)], [PRC 85, 014001 (2012)]


and....

Thank you Sergio!!!!

Australia, 1993, Kangaroo Island


Study of bound and scattering states of few-body … of bound and scattering states of few ... A....

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