Collaborators: O. Hen, E. Piasetzki and R. Torres Zero-range - - PowerPoint PPT Presentation

Ronen Weiss and Nir Barnea Collaborators: O. Hen, E. Piasetzki and R. Torres

 Zero-range condition: 𝑠

0 ≪ 𝑏, 𝑒

 Many quantities are connected to the conta

tact ct 𝑫:

• S. Tan, Ann. Phys. (N.Y.) 323, 2952 (2008); Ann. Phys. (N.Y.)

323, 2971 (2008); Ann. Phys. (N.Y.) 323, 2987 (2008)

𝑜 𝑙 = 𝑫/𝑙^4 for 𝑙 → ∞ 𝑈 + 𝑉 = ℏ2 4𝜌𝑛𝑏 𝑫 + ෍

𝜏

𝑒3𝑙 2𝜌 3 ℏ2𝑙2 2𝑛 𝑜𝜏 𝒍 − 𝑫 𝑙4 And many more…

 The basic fa

factoriz ization tion assumption:

𝜔

𝑠𝑗𝑘→0

1 𝑠

𝑗𝑘

− 1 𝑏 × 𝐵 𝑺𝑗𝑘, 𝒔𝑙 𝑙≠𝑗,𝑘

𝑜 𝑙 = 1 𝑙4 × 𝑫

 The basic fa

factoriz ization tion assumption:

𝜔

𝑠𝑗𝑘→0

1 𝑠

𝑗𝑘

− 1 𝑏 × 𝐵 𝑺𝑗𝑘, 𝒔𝑙 𝑙≠𝑗,𝑘

𝑜 𝑙 = 1 𝑙4 × 𝑫

NOT FOR NUCLEAR PHYSICS 𝑠

0 ≪ 𝑒, 𝑏

𝜔

𝑠𝑗𝑘→0

1 𝑠

𝑗𝑘

− 1 𝑏 × 𝐵 𝑺𝑗𝑘, 𝒔𝑙 𝑙≠𝑗,𝑘 𝜔

𝑠𝑗𝑘→0

𝜒𝑗𝑘 𝑠

𝑗𝑘

× 𝐵 𝑺𝑗𝑘, 𝒔𝑙 𝑙≠𝑗,𝑘

𝜔

𝑠𝑗𝑘→0

1 𝑠

𝑗𝑘

− 1 𝑏 × 𝐵 𝑺𝑗𝑘, 𝒔𝑙 𝑙≠𝑗,𝑘 𝜔

𝑠𝑗𝑘→0

𝜒𝑗𝑘 𝑠

𝑗𝑘

× 𝐵 𝑺𝑗𝑘, 𝒔𝑙 𝑙≠𝑗,𝑘 𝜔

𝑠𝑗𝑘→0 ෍ 𝛽

𝜒𝑗𝑘

𝛽 𝒔𝑗𝑘 × 𝐵𝑗𝑘 𝛽 (𝑺𝑗𝑘, 𝒔𝑙 𝑙≠𝑗,𝑘)

Channels 𝛽 = (ℓ2𝑇2)𝑘2𝑛2 The pair kind 𝑗𝑘 ∈ {𝑞𝑞, 𝑜𝑜, 𝑞𝑜} “univ

nivers ersal al“ function

One-body body moment entum um distribut ibution ion - 𝒐𝑶(𝒍) – The probability to find a proton/neutron with momentum 𝑙 Two-bod

• dy

y momentum entum distr tribut bution ion - 𝑮𝑶𝑶(𝒍) – The probability to find an NN pairs with relative momentum k

𝜔

𝑠𝑗𝑘→0 ෍ 𝛽

𝜒𝑗𝑘

𝛽 𝒔𝑗𝑘 × 𝐵𝑗𝑘 𝛽 (𝑺𝑗𝑘, 𝒔𝑙 𝑙≠𝑗,𝑘)

One-body body moment entum um distribut ibution ion - 𝒐𝑶(𝒍) – The probability to find a proton/neutron with momentum 𝑙 𝑜𝑞 𝒍

k→∞ σ𝛽,𝛾 ෤

𝜒𝑞𝑞

𝛽 † 𝒍 ෤

𝜒𝑞𝑞

𝛾

𝒍

2𝐷𝑞𝑞

𝛽𝛾

16𝜌2 + ෤

𝜒𝑞𝑜

𝛽 † 𝒍 ෤

𝜒𝑞𝑜

𝛾

𝒍

𝐷𝑞𝑜

𝛽𝛾

16𝜌2

Two-bod

• dy

y momentum entum distr tribut bution ion - 𝑮𝑶𝑶(𝒍) – The probability to find an NN pairs with relative momentum k

𝜔

𝑠𝑗𝑘→0 ෍ 𝛽

𝜒𝑗𝑘

𝛽 𝒔𝑗𝑘 × 𝐵𝑗𝑘 𝛽 (𝑺𝑗𝑘, 𝒔𝑙 𝑙≠𝑗,𝑘)

One-body body moment entum um distribut ibution ion - 𝒐𝑶(𝒍) – The probability to find a proton/neutron with momentum 𝑙 𝑜𝑞 𝒍

k→∞ σ𝛽,𝛾 ෤

𝜒𝑞𝑞

𝛽 † 𝒍 ෤

𝜒𝑞𝑞

𝛾

𝒍

2𝐷𝑞𝑞

𝛽𝛾

16𝜌2 + ෤

𝜒𝑞𝑜

𝛽 † 𝒍 ෤

𝜒𝑞𝑜

𝛾

𝒍

𝐷𝑞𝑜

𝛽𝛾

16𝜌2

Two-bod

• dy

y momentum entum distr tribut bution ion - 𝑮𝑶𝑶(𝒍) – The probability to find an NN pairs with relative momentum k 𝐺𝑗𝑘 𝒍

k→∞ ෍ 𝛽,𝛾

෤ 𝜒𝑗𝑘

𝛽 † 𝒍 ෤

𝜒𝑗𝑘

𝛾 𝒍

𝐷𝑗𝑘

𝛽𝛾

16𝜌2

 As a result we get the asymptotic relation:

𝑜𝑞 𝒍 → 𝐺

𝑞𝑜 𝒍 + 2𝐺 𝑞𝑞(𝒍)

 As a result we get the asymptotic relation:

𝑜𝑞 𝒍 → 𝐺

𝑞𝑜 𝒍 + 2𝐺 𝑞𝑞(𝒍)

Wiringa et al. Phys. Rev. C 89, 024305 (2014)

Using the variational Monte Carlo data (VMC)

 Assuming only two

wo signi gnifi fican ant channels nnels:

 We get:

The deuteron eron channel – L=0,2; S=1; J=1; T=0 The pure e s-wave channel – L=0; S=0; J=0; T=1

𝐺

𝑞𝑜 𝑙 𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2

𝐺

𝑜𝑜 𝑙 𝑙→∞ 𝐷𝑜𝑜 0 𝜒𝑜𝑜

𝑙

2

 Assuming only two

wo signi gnifi fican ant channels nnels:

 We get:

The deuteron eron channel – L=0,2; S=1; J=1; T=0 The pure e s-wave channel – L=0; S=0; J=0; T=1

𝐺

𝑞𝑜 𝑙 𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2

𝐺

𝑜𝑜 𝑙 𝑙→∞ 𝐷𝑜𝑜 0 𝜒𝑜𝑜

𝑙

2 Zero-energy solution of the two-body system (AV18) The VMC data

𝐺

𝑞𝑜 𝑙 𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2

𝐺

𝑜𝑜 𝑙 𝑙→∞ 𝐷𝑜𝑜 0 𝜒𝑜𝑜

𝑙

2

Momentum space 10B

𝐺

𝑞𝑜 𝑙 𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2

𝐺

𝑜𝑜 𝑙 𝑙→∞ 𝐷𝑜𝑜 0 𝜒𝑜𝑜

𝑙

2

Coordinate space Momentum space 10B 10B

Universal functions - Calculated for the two-body system

𝑜𝑞 𝑙

𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2 + 2𝐷𝑞𝑞 0 𝜒𝑞𝑞

𝑙

2

Fitted to 𝐺𝑗𝑘 (𝑙) for 𝑙 > 4 fm−1

𝑜𝑞 𝑙

𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2 + 2𝐷𝑞𝑞 0 𝜒𝑞𝑞

𝑙

2

The VMC data

𝑜𝑞 𝑙

𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2 + 2𝐷𝑞𝑞 0 𝜒𝑞𝑞

𝑙

2

𝑜𝑞 𝑙

𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2 + 2𝐷𝑞𝑞 0 𝜒𝑞𝑞

𝑙

2

4He

𝑜𝑞(𝑙)

𝑜𝑞 𝑙

𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2 + 2𝐷𝑞𝑞 0 𝜒𝑞𝑞

𝑙

2

4He

𝑜𝑞(𝑙) 𝑞𝑞/𝑜𝑞

4He

12C

𝑜𝑞(𝑙) 𝑞𝑞/𝑜𝑞

𝑜𝑞 𝑙

𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2 + 2𝐷𝑞𝑞 0 𝜒𝑞𝑞

𝑙

2

12C

Normalization: ׬

𝑙𝐺 ∞ 𝜒𝑗𝑘 𝛽 2𝑒3𝑙 = 1

𝑜𝑞 𝑙

𝑙→∞ 𝐷𝑞𝑜 𝑒 𝜒𝑞𝑜 𝑒

𝑙

2 + 𝐷𝑞𝑜 0 𝜒𝑞𝑜

𝑙

2 + 2𝐷𝑞𝑞 0 𝜒𝑞𝑞

𝑙

2

%𝑇𝑆𝐷 ≡ 1 𝑎 න

𝐿𝐺 ∞

𝑜𝑞 𝒍 𝑒3𝑙 = 1 𝑎 𝐷𝑞𝑜

𝑒 + 𝐷𝑞𝑜 0 + 2𝐷𝑜𝑜

4He

Total number of pairs: pp – 1 np-4

𝑫𝒒𝒒

𝟏 /𝒂 (%)

𝑫𝒒𝒐

𝟏 /𝒂 (%)

𝑫𝒒𝒐

𝒆 /𝐚 (%)

k-space

𝟏. 𝟕𝟔 ± 𝟏. 𝟏𝟒 𝟏. 𝟕𝟘 ± 𝟏. 𝟏𝟒 𝟐𝟑. 𝟒 ± 𝟏. 𝟐

4He

Neutron-proton dominance Non-combinatorial isospin symmetry (T=1) Total number of pairs: pp – 1 np-4

𝑫𝒒𝒒

𝟏 /𝒂 (%)

𝑫𝒒𝒐

𝟏 /𝒂 (%)

𝑫𝒒𝒐

𝒆 /𝐚 (%)

%SRCs k-space

0.65 ± 0.03 0.69 ± 0.03 12.3 ± 0.1 14.3%

4He

Total number of pairs: pp – 1 np-4

𝑫𝒒𝒒

𝟏 /𝒂 (%)

𝑫𝒒𝒐

𝟏 /𝒂 (%)

𝑫𝒒𝒐

𝒆 /𝐚 (%)

%SRCs k-space

0.65 ± 0.03 0.69 ± 0.03 12.3 ± 0.1 14.3%

r-space

0.567 ± 0.004 11.61 ± 0.03 13.3%

Similar results are obtained for all the available nuclei in the VMC data

4He

Total number of pairs: pp – 1 np-4

 Moment

entum um distribut ibutions ions

• R. Weiss, B. Bazak, N. Barnea, PRC 92

92, 054311 (2015)

• M. Alvioli, CC. Degli Atti, H. Morita, PRC 94

94, 044309 (2016)

 The

e Levin vinger er constant tant

• R. Weiss, B. Bazak, N. Barnea, PRL 114

114, 012501 (2015)

• R. Weiss, B. Bazak, N. Barnea, EPJA 52

52, 92 (2016)

 Elect

ctron ron scatt ttering ering

• O. Hen et al., PRC 92

92, 045205 (2015)

 Symme

mmetry try energ rgy

• BJ. Cai, BA. Li, PRC 93

93, 014619 (2016)

 The

e Coulomb lomb sum rule le (and nd a review) iew)

• R. Weiss, E. Pazy, N. Barnea, Few-Body Systems (2016)

 The

e EMC effect t

• JW. Chen, W. Detmold, J. E. Lynn, A. Schwenk, arxiv 1607.03065 [hep-ph] (2016)

and more…

Two-body momentum distribution for 𝒍 > 𝟓 𝐠𝐧−𝟐 Full details

• n SRCs for

𝒍 > 𝒍𝑮 Extracting the contacts Two-body coordinate density for 𝒔 < 𝟐 𝐠𝐧

Two-body momentum distribution for 𝒍 > 𝟓 𝐠𝐧−𝟐 Full details

• n SRCs for

𝒍 > 𝒍𝑮 Extracting the contacts Two-body coordinate density for 𝒔 < 𝟐 𝐠𝐧

np dominance & pp/np Isospin symmetry Main (𝑀, 𝑇, 𝐾, 𝑈) channels %SRCs

1B momentum distribution