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 : 𝑜 𝑙 = 𝑫/𝑙^4 for 𝑙 → ∞ ℏ 2 𝑒 3 𝑙 ℏ 2 𝑙 2 𝑜 𝜏 𝒍 − 𝑫 𝑈 + 𝑉 = 4𝜌𝑛𝑏 𝑫 + ෍ 2𝜌 3 𝑙 4 2𝑛 𝜏 And many more… S. Tan, Ann. Phys. (N.Y.) 323, 2952 (2008); Ann. Phys. (N.Y.) 323, 2971 (2008); Ann. Phys. (N.Y.) 323, 2987 (2008)

 The basic fa factoriz ization tion assumption: 1 − 1 𝑠 𝑗𝑘 →0 𝜔 × 𝐵 𝑺 𝑗𝑘 , 𝒔 𝑙 𝑙≠𝑗,𝑘 𝑠 𝑏 𝑗𝑘 𝑜 𝑙 = 1 𝑙 4 × 𝑫

 The basic fa factoriz ization tion assumption: 1 − 1 𝑠 𝑗𝑘 →0 𝜔 × 𝐵 𝑺 𝑗𝑘 , 𝒔 𝑙 𝑙≠𝑗,𝑘 𝑠 𝑏 𝑗𝑘 𝑜 𝑙 = 1 𝑙 4 × 𝑫 NOT FOR NUCLEAR PHYSICS 𝑠 0 ≪ 𝑒, 𝑏

1 − 1 𝑠 𝑗𝑘 →0 𝜔 × 𝐵 𝑺 𝑗𝑘 , 𝒔 𝑙 𝑙≠𝑗,𝑘 𝑠 𝑏 𝑗𝑘 𝑠 𝑗𝑘 →0 𝜔 𝜒 𝑗𝑘 𝑠 × 𝐵 𝑺 𝑗𝑘 , 𝒔 𝑙 𝑙≠𝑗,𝑘 𝑗𝑘

1 − 1 𝑠 𝑗𝑘 →0 𝜔 × 𝐵 𝑺 𝑗𝑘 , 𝒔 𝑙 𝑙≠𝑗,𝑘 𝑠 𝑏 𝑗𝑘 𝑠 𝑗𝑘 →0 𝜔 𝜒 𝑗𝑘 𝑠 × 𝐵 𝑺 𝑗𝑘 , 𝒔 𝑙 𝑙≠𝑗,𝑘 𝑗𝑘 𝑠 𝑗𝑘→0 ෍ 𝛽 𝒔 𝑗𝑘 × 𝐵 𝑗𝑘 𝛽 (𝑺 𝑗𝑘 , 𝒔 𝑙 𝑙≠𝑗,𝑘 ) 𝜔 𝜒 𝑗𝑘 𝛽 The pair kind Channels 𝛽 “ univ nivers ersal al “ 𝑗𝑘 ∈ {𝑞𝑞, 𝑜𝑜, 𝑞𝑜} = (ℓ 2 𝑇 2 )𝑘 2 𝑛 2 function

One-body body moment entum um distribut ibution ion - 𝒐 𝑶 (𝒍) – The probability to find a proton/neutron with momentum 𝑙 Two-bod ody y momentum entum distr tribut bution ion - 𝑮 𝑶𝑶 (𝒍) – The probability to find an NN pairs with relative momentum k

One-body body moment entum um distribut ibution ion - 𝒐 𝑶 (𝒍) – The probability to find a proton/neutron with momentum 𝑙 Two-bod ody y momentum entum distr tribut bution ion - 𝑮 𝑶𝑶 (𝒍) – The probability to find an NN pairs with relative momentum k 𝑠 𝑗𝑘→0 ෍ 𝛽 𝒔 𝑗𝑘 × 𝐵 𝑗𝑘 𝛽 (𝑺 𝑗𝑘 , 𝒔 𝑙 𝑙≠𝑗,𝑘 ) 𝜔 𝜒 𝑗𝑘 𝛽 k→∞ σ 𝛽,𝛾 ෤ 𝛽𝛾 𝛽𝛾 𝛽 † 𝒍 ෤ 𝛽 † 𝒍 ෤ 2𝐷 𝑞𝑞 𝐷 𝑞𝑜 𝛾 𝛾 𝑜 𝑞 𝒍 𝜒 𝑞𝑞 𝜒 𝑞𝑞 𝒍 16𝜌 2 + ෤ 𝜒 𝑞𝑜 𝜒 𝑞𝑜 𝒍 16𝜌 2

One-body body moment entum um distribut ibution ion - 𝒐 𝑶 (𝒍) – The probability to find a proton/neutron with momentum 𝑙 Two-bod ody y momentum entum distr tribut bution ion - 𝑮 𝑶𝑶 (𝒍) – The probability to find an NN pairs with relative momentum k 𝑠 𝑗𝑘→0 ෍ 𝛽 𝒔 𝑗𝑘 × 𝐵 𝑗𝑘 𝛽 (𝑺 𝑗𝑘 , 𝒔 𝑙 𝑙≠𝑗,𝑘 ) 𝜔 𝜒 𝑗𝑘 𝛽 k→∞ σ 𝛽,𝛾 ෤ 𝛽𝛾 𝛽𝛾 𝛽 † 𝒍 ෤ 𝛽 † 𝒍 ෤ 2𝐷 𝑞𝑞 𝐷 𝑞𝑜 𝛾 𝛾 𝑜 𝑞 𝒍 𝜒 𝑞𝑞 𝜒 𝑞𝑞 𝒍 16𝜌 2 + ෤ 𝜒 𝑞𝑜 𝜒 𝑞𝑜 𝒍 16𝜌 2 𝛽𝛾 𝐷 𝑗𝑘 k→∞ ෍ 𝛽 † 𝒍 ෤ 𝛾 𝒍 𝐺 𝑗𝑘 𝒍 𝜒 𝑗𝑘 ෤ 𝜒 𝑗𝑘 16𝜌 2 𝛽,𝛾

 As a result we get the asymptotic relation: 𝑜 𝑞 𝒍 → 𝐺 𝑞𝑜 𝒍 + 2𝐺 𝑞𝑞 (𝒍)

 As a result we get the asymptotic relation: 𝑜 𝑞 𝒍 → 𝐺 𝑞𝑜 𝒍 + 2𝐺 𝑞𝑞 (𝒍) Using the variational Monte Carlo data (VMC) Wiringa et al. Phys. Rev. C 89, 024305 (2014)

 Assuming only two wo signi gnifi fican ant channels nnels: 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  We get: 2 + 𝐷 𝑞𝑜 2 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 𝑒 0 𝐺 𝑞𝑜 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 0 𝜒 𝑜𝑜 0 2 𝐺 𝑜𝑜 𝑙 𝑙→∞ 𝐷 𝑜𝑜 𝑙

 Assuming only two wo signi gnifi fican ant channels nnels: 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  We get: 2 + 𝐷 𝑞𝑜 2 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 𝑒 0 𝐺 𝑞𝑜 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 0 𝜒 𝑜𝑜 0 2 𝐺 𝑜𝑜 𝑙 𝑙→∞ 𝐷 𝑜𝑜 𝑙 Zero-energy The VMC solution of the data two-body system (AV18)

2 + 𝐷 𝑞𝑜 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 2 𝑒 0 𝐺 𝑞𝑜 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 0 𝜒 𝑜𝑜 0 2 𝐺 𝑜𝑜 𝑙 𝑙→∞ 𝐷 𝑜𝑜 𝑙 Momentum space 10 B

2 + 𝐷 𝑞𝑜 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 2 𝑒 0 𝐺 𝑞𝑜 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 0 𝜒 𝑜𝑜 0 2 𝐺 𝑜𝑜 𝑙 𝑙→∞ 𝐷 𝑜𝑜 𝑙 Momentum space Coordinate space 10 B 10 B

2 + 𝐷 𝑞𝑜 2 + 2𝐷 𝑞𝑞 2 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 0 𝜒 𝑞𝑞 𝑒 0 0 𝑜 𝑞 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 𝑙 Universal functions - Calculated for the two-body system

2 + 𝐷 𝑞𝑜 2 + 2𝐷 𝑞𝑞 2 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 0 𝜒 𝑞𝑞 𝑒 0 0 𝑜 𝑞 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 𝑙 Fitted to 𝐺 𝑗𝑘 (𝑙) for 𝑙 > 4 fm −1

2 + 𝐷 𝑞𝑜 2 + 2𝐷 𝑞𝑞 2 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 0 𝜒 𝑞𝑞 𝑒 0 0 𝑜 𝑞 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 𝑙 The VMC data

2 + 𝐷 𝑞𝑜 2 + 2𝐷 𝑞𝑞 2 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 0 𝜒 𝑞𝑞 𝑒 0 0 𝑜 𝑞 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 𝑙 𝑜 𝑞 (𝑙) 4 He

2 + 𝐷 𝑞𝑜 2 + 2𝐷 𝑞𝑞 2 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 0 𝜒 𝑞𝑞 𝑒 0 0 𝑜 𝑞 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 𝑙 𝑜 𝑞 (𝑙) 𝑞𝑞/𝑜𝑞 4 He 4 He

2 + 𝐷 𝑞𝑜 2 + 2𝐷 𝑞𝑞 2 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 0 𝜒 𝑞𝑞 𝑒 0 0 𝑜 𝑞 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 𝑙 𝑜 𝑞 (𝑙) 𝑞𝑞/𝑜𝑞 12 C 12 C

2 + 𝐷 𝑞𝑜 2 + 2𝐷 𝑞𝑞 2 𝑒 𝜒 𝑞𝑜 0 𝜒 𝑞𝑜 0 𝜒 𝑞𝑞 𝑒 0 0 𝑜 𝑞 𝑙 𝑙→∞ 𝐷 𝑞𝑜 𝑙 𝑙 𝑙 ∞ 𝜒 𝑗𝑘 𝛽 2 𝑒 3 𝑙 = 1 Normalization: ׬ 𝑙 𝐺 ∞ %𝑇𝑆𝐷 ≡ 1 𝑜 𝑞 𝒍 𝑒 3 𝑙 = 1 𝑒 + 𝐷 𝑞𝑜 0 + 2𝐷 𝑜𝑜 0 𝑎 න 𝑎 𝐷 𝑞𝑜 𝐿 𝐺

4 He Total number of pairs: pp – 1 np-4

4 He Total number of pairs: pp – 1 np-4 𝟏 /𝒂 (%) 𝟏 /𝒂 (%) 𝒆 /𝐚 (%) 𝑫 𝒒𝒒 𝑫 𝒒𝒐 𝑫 𝒒𝒐 k-space 𝟏. 𝟕𝟔 ± 𝟏. 𝟏𝟒 𝟏. 𝟕𝟘 ± 𝟏. 𝟏𝟒 𝟐𝟑. 𝟒 ± 𝟏. 𝟐 Non-combinatorial Neutron-proton isospin symmetry dominance (T=1)

4 He Total number of pairs: pp – 1 np-4 𝟏 /𝒂 (%) 𝟏 /𝒂 (%) 𝒆 /𝐚 (%) %SRCs 𝑫 𝒒𝒒 𝑫 𝒒𝒐 𝑫 𝒒𝒐 k-space 14.3 % 0.65 ± 0.03 0.69 ± 0.03 12.3 ± 0.1

4 He Total number of pairs: pp – 1 np-4 𝟏 /𝒂 (%) 𝟏 /𝒂 (%) 𝒆 /𝐚 (%) %SRCs 𝑫 𝒒𝒒 𝑫 𝒒𝒐 𝑫 𝒒𝒐 k-space 14.3 % 0.65 ± 0.03 0.69 ± 0.03 12.3 ± 0.1 r-space 13.3% 0.567 ± 0.004 11.61 ± 0.03 Similar results are obtained for all the available nuclei in the VMC data

 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 Two-body momentum coordinate distribution for density for 𝒍 > 𝟓 𝐠𝐧 −𝟐 𝒔 < 𝟐 𝐠𝐧 Extracting the contacts Full details on SRCs for 𝒍 > 𝒍 𝑮

Two-body Two-body momentum coordinate distribution for density for 𝒍 > 𝟓 𝐠𝐧 −𝟐 𝒔 < 𝟐 𝐠𝐧 np dominance & pp/np Extracting the contacts Isospin symmetry %SRCs Main (𝑀, 𝑇, 𝐾, 𝑈) Full details channels on SRCs for 𝒍 > 𝒍 𝑮 1B momentum distribution