Momentum Distributions and Short-Range Correlationsin Few-Nucleon Systems Using Realistic Interactions

Laura E. Marcucci & Michele Viviani

University of Pisa & INFN-Pisa

Results obtained in collaboration with:

R. Machleidt and F. Sammarruca (Univ. of Idaho - Moscow - USA)

L.E. Marcucci et al., Phys. Rev. C 99, 034003 (2019)R. Cruz-Torres et al., arXiv:1902.06358

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 1 / 15

Motivations and outline

Structure of nuclei → strong force (strong repulsion at short distances andstrong attraction at moderate distances)

Nucleus = dilute system ⇒ collection of hard objects in a mean field

Limitations → nucleons = waves that strongly overlap ⇒ NN SRCs

Strong experimental program (JLab)

See O. Hen’s talk

Theoretical study of single-N and two-N MDs

Study of the model-dependence (AV18 vs. CDBonn)

Study with χEFT potentials

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 2 / 15

Definitions for A = 3

Single-nucleon and two-nucleon momentum distributions

nn/p(Kc.m.) =

∫dK̂c.m.

∫d k̂relΨ

†(krel ,Kc.m.)Pn/pΨ(krel ,Kc.m.)

nnp/pp(krel ,Kc.m.) =

∫d k̂rel

∫dK̂c.m.Ψ

†(krel ,Kc.m.)Pnp/ppΨ(krel ,Kc.m.)

nnp/pp(krel ,Kc.m. = 0) → back− to− back (BB)

nnp/pp(krel ) = 4π

∫ K+c.m.0

K2c.m. dKc.m.nnp/pp(krel ,Kc.m.)

The HH method in a nutshell

Ψ(krel ,Kc.m.) =∑µ

cµφµ

cµ −→ Rayleigh-Ritz variational principle∑µ′

cµ′ 〈φµ|H − E |φµ′ 〉 = 0

φµ both in r- and p-space (just a Fourier Transform)

K

−Kc.m.

c.m.

k rel

large krelsmall Kc.m. compared to kF⇒ SRCs

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 3 / 15

Single-N MDs in 3He: phenomenological potentials

0 1 2 3 4 5

k [fm-1

]

10-6

10-5

10-4

10-3

10-2

10-1

100

np

/n(k

) [f

m3]

proton - AV18

neutron - AV18proton - AV18/UIX

neutron - AV18/UIXproton - CDBonn

neutron - CDBonnproton - CDBonn/TM

neutron - CDBonn/TM

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 4 / 15

Two-N MDs in 3He: phenomenological potentials

0 1 2 3 4 5

krel

[fm-1

]

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

nn

p/p

p(k

rel,

Kc.

m. =

0)

[fm

6]

np AV18

pp AV18

np AV18/UIX

pp AV18/UIX

np CDBonn

pp CDBonn

np CDBonn/TM

pp CDBonn/TM

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 5 / 15

Conclusions I and SRCs

Large model-dependence (see also F. Sammarruca on d1)! → 6= VNN -tensorVery small TNI contribution

NBBN1N2 = 4π

∫ ∞0

nN1N2 (krel ,Kc.m. = 0)k2rel dkrel

NSRC ,BBN1N2= 4π

∫ ∞k−rel

nN1N2 (krel ,Kc.m. = 0)k2rel dkrel

NSRCN1N2 (k−rel ) = 4π

∫ ∞k−rel

nN1N2 (krel )k2rel dkrel

k−rel = 1.5 fm−1

NN1N2 = 4π

∫ ∞0

nN1N2 (krel )k2rel dkrel ≡ N

SRCN1N2

(k−rel = 0)

N1N2 = np N1N2 = pp

NBB NSRC,BB NSRC (k−rel

) N NBB NSRC,BB NSRC (k−rel

) N

AV18 6.922 0.241 0.093 1.997 2.194 0.009 0.026 0.998AV18/UIX 5.751 0.210 0.106 1.997 1.897 0.009 0.031 0.999CDBonn 6.552 0.171 0.060 1.997 2.078 0.005 0.012 0.999

CDBonn/TM 5.931 0.157 0.063 1.998 1.924 0.005 0.014 0.998

AV18 - d1 0.042

CDBonn - d1 0.032

1 F. Sammarruca, Phys. Rev. C 92, 044003 (2015)

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 6 / 15

Single-N MDs in 3He: chiral potentials (I)

0 1 2 3 4 5

k [fm-1

]

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

nn(k

) [f

m3]

LONLON2LON3LON4LO

(a)

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 7 / 15

Single-N MDs in 3He: chiral potentials (II)

0 1 2 3 4 5

k [fm-1

]

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

np(k

) [f

m3]

LONLON2LON3LON4LO

(a)

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 8 / 15

Single-N MDs in 3He: chiral potentials (III)

0 1 2 3 4 510

-1010-910-810-710-610-510-410-310-210-1100101102

0 1 2 3 4 510

-1010-910-810-710-610-510-410-310-210-1100101102

0 1 2 3 4 510

-1010-910-810-710-610-510-410-310-210-1100101102

np(k

) [f

m3]

VNN

Λ=450 MeV

VNN

Λ=500 MeV

VNN

Λ=550 MeV

VNN

+VNNN

Λ=450 MeV

VNN

+VNNN

Λ=500 MeV

VNN

+VNNN

Λ=550 MeV

0 1 2 3 4 5

krel

[fm-1

]

10-1010

-910-810-710-610-510-410-310-210-1100101102

0 1 2 3 4 5

krel

[fm-1

]

10-1010

-910-810-710-610-510-410-310-210-1100101102

LO NLO

N2LO N3LO

N4LO

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 9 / 15

np3He/np3H ∼ σ3He(e,e′p)/σ3H(e,e′p) −→ JLab

R. Cruz-Torres et al., arXiv:1902.06358

| [MeV/c]miss

p|0 100 200 300 400 500

|)m

iss

p(|H

(e,e

'p)

3σ/H

e(e,

e'p)

3σ 1

2

3

Momentum Density (VMC)

AV18+UX

LO E1 (1.0 fm)2N (1.0 fm)τLO E2N

Spectral Function

CDA & Kaptari

Benhar & Pandharipande

CDBonn/TMNLO

LO2NLO3NLO4N

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 10 / 15

Two-N MDs in 3He: chiral potentials (I)

0 1 2 3 4 5

krel

[fm-1

]

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

nnp(k

rel,

Kc.

m. =

0)

[fm

6]

LONLON2LON3LON4LO

VNN

with Λ=500 MeV

0 0.5 10

10

20

30

40

Order-by-order convergence

0 1 2 3 4 5

krel

[fm-1

]

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

nnp(k

rel,

Kc.

m. =

0)

[fm

6]

N2LON3LO-IN3LO-IIN4LO-IN4LO-II

VNN

+VNNN

with Λ=500 MeV

0 0.5 1 1.5 20

10

20

30

40

asymptotic w.f. at LO 6= NLO, N2LO . . .Small TNI contributions

N3LO 6= N4LO for krel ≥ 2.2 fm−1

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 11 / 15

Two-N MDs in 3He: chiral potentials (I)

0 1 2 3 4 510

-1010-910-810-710-610-510-410-310-210-1100101102

0 1 2 3 4 510

-1010-910-810-710-610-510-410-310-210-1100101102

0 1 2 3 4 510

-1010-910-810-710-610-510-410-310-210-1100101102

nnp(k

rel, K

c.m

. =

0)

[fm

6]

VNN

Λ=450 MeV

VNN

Λ=500 MeV

VNN

Λ=550 MeV

VNN

+VNNN

Λ=450 MeV

VNN

+VNNN

Λ=500 MeV

VNN

+VNNN

Λ=550 MeV

0 1 2 3 4 5

krel

[fm-1

]

10-1010

-910-810-710-610-510-410-310-210-1100101102

0 1 2 3 4 5

krel

[fm-1

]

10-1010

-910-810-710-610-510-410-310-210-1100101102

LO NLO

N2LO N3LO

N4LO

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 12 / 15

Conclusion II and SRCs (NSRCnp )

cutoff-dependence only for krel ≥ 2.2− 2.5 fm−1

TNI contribution visible only for krel ≥ 3.0− 3.5 fm−1

NSRCnp (k−rel ) = 4π

∫ ∞k−rel

nnp(krel )k2rel dkrel with k

−rel = 1.5 fm

−1

deuteronΛ = 450 MeV Λ = 500 MeV Λ = 550 MeV Λ = 500 MeV

LO 0.089 0.112 0.126 0.046NLO 0.016 0.024 0.037 0.015

N2LO 0.029 0.038 0.045 0.026N2LO/N2LO 0.030 0.040 0.050

N3LO 0.039 0.039 0.044 0.024N3LO/N2LO-II 0.042 0.043 0.050

N4LO 0.038 0.042 0.047 0.024N4LO/N2LO-II 0.040 0.045 0.051

CDBonn/TM ⇒ 0.063 (no TNI: 0.060)AV18/UIX ⇒ 0.106 (no TNI: 0.093)⇒ small Λ-dependence ⇒ small contributions from krel ≥ 2.2 fm−1 region

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 13 / 15

Outlook

Perform a systematic study of d , 3H, 3He and 4He

Apply the same formalism to density distributions

Give inputs for the Generalized Contact Formalism

ρNN,αA (r) → CNN,αA × |φ

αNN(r)|2

nNN,αA (k) → CNN,αA × |φ

αNN(k)|2

See O. Hen’s talk for references

⇒ Set the code for nNN,αA (k)

⇒ nST3He(k)

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 14 / 15

0 1 2 3 4 5

k [fm-1

]

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

nS

T

3H

e(k

) [f

m3]

HH+AV18/UIX ST=00; N=0.010

HH+AV18/UIX ST=01; N=1.351

HH+AV18/UIX ST=10; N=1.490

HH+AV18/UIX ST=11; N=0.149

VMC+AV18/UIX ST=00; N=0.010

VMC+AV18/UIX ST=01; N=1.358

VMC+AV18/UIX ST=10; N=1.490

VMC+AV18/UIX ST=11; N=0.142

PRELIMINARY

L.E. Marcucci (Univ. Pisa & INFN-Pisa) MDs and SRCs in Few-Nucleon Systems INPC2019 August 2, 2019 15 / 15