Nuclear RG perspective on SRC and EMC physics Dick Furnstahl Department of Physics Ohio State University MIT Workshop on SRC and EMC Physics December, 2016 Collaborators: S. Bogner (MSU), K. Hebeler (TU Darmstadt), S. K¨ onig (TU Darmstadt), S. More (MSU)
Large Q 2 scattering at different RG decoupling scales What is this vertex? e’ k � k q = k − k � e ν = E k − E k � q p � N 1 Q 2 = − q 2 Q 2 N p � x B = 2 2 m N ν A A ! 2 Subedi et al., Science 320, 1476 (2008) Higinbotham, arXiv:1010.4433 a) 3 SRC interpretation: r( 4 He/ 3 He) 2.5 2 NN interaction can scatter 1.5 q 1 states with p 1 , p 2 � k F 4 b) p � r( 12 C/ 3 He) to intermediate states with 3 1 p 1 p � 2 which are p � 1 , p � 2 2 � k F p 2 1 knocked out by the photon 6 c) r( 56 Fe/ 3 He) 4 How to explain cross sections in terms of 2 low-momentum interactions? 1 1.25 1.5 1.75 2 2.25 2.5 2.75 1 . 4 < Q 2 < 2 . 6 GeV 2 x B Vertex depends on the resolution! Egiyan et al. PRL 96, 1082501 (2006) SRC explanation relies on high-momentum nucleons in structure
Large Q 2 scattering at different RG decoupling scales What is this vertex? e’ k � q = k − k � k e q ν = E k − E k � p � N 1 Q 2 = − q 2 Q 2 N p � x B = 2 2 m N ν A A ! 2 Subedi et al., Science 320, 1476 (2008) Higinbotham, arXiv:1010.4433 3 a) SRC interpretation: r( 4 He/ 3 He) 2.5 2 NN interaction can scatter 1.5 q 1 states with p 1 , p 2 � k F 4 b) p � r( 12 C/ 3 He) to intermediate states with 3 p 1 1 p � 2 which are 2 p � 1 , p � 2 � k F p 2 1 knocked out by the photon 6 c) r( 56 Fe/ 3 He) 4 How to explain cross sections in terms of 2 low-momentum interactions? 1 1.25 1.5 1.75 2 2.25 2.5 2.75 1 . 4 < Q 2 < 2 . 6 GeV 2 x B Vertex depends on the resolution! Egiyan et al. PRL 96, 1082501 (2006) RG evolution changes physics interpretation but not cross section!
Ab initio calculations: The nuclear structure hockey stick Realis'c: BEs within 5% and starts from NN + 3NFs Gaute Hagen, DNP 2016 Why has the reach of precision structure calculations increased? Application of effective field theory (EFT) and renormalization group (RG) methods = ⇒ low-resolution (“softened”) potentials Explosion of many-body methods: GFMC/AFDMC, (IT-)NCSM, coupled cluster, lattice EFT, IM-SRG, SCGF , UMOA, MBPT, . . .
Uses of the renormalization group (RG) [cf. S. Weinberg (1981)] Improving perturbation theory; e.g., in QCD calculations Mismatch of energy scales can generate large logarithms Shift between couplings and loop integrals to reduce logs Identifying universality in critical phenomena Filter out short-distance degrees of freedom Simplifying calculations of nuclear structure/reactions Make nuclear physics look more like quantum chemistry!
Uses of the renormalization group (RG) [cf. S. Weinberg (1981)] Improving perturbation theory; e.g., in QCD calculations Mismatch of energy scales can generate large logarithms Shift between couplings and loop integrals to reduce logs Identifying universality in critical phenomena Filter out short-distance degrees of freedom Simplifying calculations of nuclear structure/reactions Make nuclear physics look more like quantum chemistry! Coupling of low- k /high- k modes: non-perturbative, strong correlations, . . . Remedy: Use RG to decouple modes = ⇒ low resolution � k | V AV18 | k ′ � AV18, Bonn, Reid93
Uses of the renormalization group (RG) [cf. S. Weinberg (1981)] Improving perturbation theory; e.g., in QCD calculations Mismatch of energy scales can generate large logarithms Shift between couplings and loop integrals to reduce logs Identifying universality in critical phenomena Filter out short-distance degrees of freedom Simplifying calculations of nuclear structure/reactions Make nuclear physics look more like quantum chemistry! V low k : lower cutoff Λ i in k , k ′ via dT ( k , k ′ ; k 2 ) / d Λ = 0 SRG: drive H toward diagonal with flow equation dH s / ds = [[ G s , H s ] , H s ] Continuous unitary transforms “ V low k ” Similarity RG (cf. running couplings)
Uses of the renormalization group (RG) [cf. S. Weinberg (1981)] Improving perturbation theory; e.g., in QCD calculations Mismatch of energy scales can generate large logarithms Shift between couplings and loop integrals to reduce logs Identifying universality in critical phenomena Filter out short-distance degrees of freedom Simplifying calculations of nuclear structure/reactions Make nuclear physics look more like quantum chemistry! V low k : lower cutoff Λ i in k , k ′ via dT ( k , k ′ ; k 2 ) / d Λ = 0 SRG: drive H toward diagonal with flow equation dH s / ds = [[ G s , H s ] , H s ] Continuous unitary transforms Block diagonal SRG Similarity RG (cf. running couplings)
Uses of the renormalization group (RG) [cf. S. Weinberg (1981)] Improving perturbation theory; e.g., in QCD calculations Mismatch of energy scales can generate large logarithms Shift between couplings and loop integrals to reduce logs Identifying universality in critical phenomena Filter out short-distance degrees of freedom Simplifying calculations of nuclear structure/reactions Make nuclear physics look more like quantum chemistry! AV18: Decoupling naturally visualized in momentum space for G s = T Phase-shift equivalent! Width of diagonal given by λ 2 = 1 / √ s What does this look like in coordinate space?
Uses of the renormalization group (RG) [cf. S. Weinberg (1981)] Improving perturbation theory; e.g., in QCD calculations Mismatch of energy scales can generate large logarithms Shift between couplings and loop integrals to reduce logs Identifying universality in critical phenomena Filter out short-distance degrees of freedom Simplifying calculations of nuclear structure/reactions Make nuclear physics look more like quantum chemistry! N 3 LO: (500 MeV) Decoupling naturally visualized in momentum space for G s = T Phase-shift equivalent! Width of diagonal given by λ 2 = 1 / √ s What does this look like in coordinate space?
Visualizing the softening of NN interactions � d 3 r ′ V λ ( r , r ′ ) Project non-local NN potential: V λ ( r ) = Roughly gives action of potential on long-wavelength nucleons Central part (S-wave) [Note: The V λ ’s are all phase equivalent!] Tensor part (S-D mixing) [graphs from K. Wendt et al., PRC (2012)] = ⇒ Flow to universal potentials!
Compare changing a cutoff in an EFT to RG decoupling (Local) field theory version in perturbation theory (diagrams) ∆Λ c Loops (sums over intermediate states) ⇐ ⇒ LECs � � d + = 0 d Λ c � �� � � �� � � Λ c d 3 q C 0 (Λ c ) ∝ Λ c C 0 MC 0 2 π 2 + ··· ( 2 π ) 3 k 2 − q 2 + i ǫ ⇒ Taylor expansion in k 2 Momentum-dependent vertices = This implements an operator product expansion! Claim: V low k RG and SRG decoupling work analogously SRG (“T” generator) “ V low k ”
Approach to universality (fate of high- q physics!) Run NN to lower λ via SRG = ⇒ ≈ Universal low- k V NN Off-Diagonal V λ ( k , 0 ) k ′ < λ 1.0 −1 λ = 5.0 fm V λ 0.5 1 S 0 = C 0 + · · · ⇒ 0.0 q ≫ λ V λ (k,0) [fm] −0.5 V λ −1.0 k < λ 550/600 [E/G/M] 600/700 [E/G/M] q ≫ λ (or Λ ) intermediate states −1.5 500 [E/M] 600 [E/M] = ⇒ change is ≈ contact terms: C 0 δ 3 ( x − x ′ ) + · · · −2.0 2 C 0 ( ψ † ψ ) 2 + · · · ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 [cf. L eft = · · · + 1 − 1 ] k [fm Similar pattern with phenomenological potentials (e.g., AV18) � λ ( q ′ , k ′ ) for k , k ′ < λ, q , q ′ ≫ λ Factorization: ∆ V λ ( k , k ′ ) = U λ ( k , q ) V λ ( q , q ′ ) U † � U λ → K · Q Q ( q ) V λ ( q , q ′ ) Q ( q ′ )] K ( k ′ ) with K ( k ) ≈ 1! − → K ( k )[
Approach to universality (fate of high- q physics!) Run NN to lower λ via SRG = ⇒ ≈ Universal low- k V NN Off-Diagonal V λ ( k , 0 ) k ′ < λ 1.0 −1 λ = 4.0 fm V λ 0.5 1 S 0 = C 0 + · · · ⇒ 0.0 q ≫ λ V λ (k,0) [fm] −0.5 V λ −1.0 k < λ 550/600 [E/G/M] 600/700 [E/G/M] q ≫ λ (or Λ ) intermediate states −1.5 500 [E/M] 600 [E/M] = ⇒ change is ≈ contact terms: C 0 δ 3 ( x − x ′ ) + · · · −2.0 2 C 0 ( ψ † ψ ) 2 + · · · ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 [cf. L eft = · · · + 1 − 1 ] k [fm Similar pattern with phenomenological potentials (e.g., AV18) � λ ( q ′ , k ′ ) for k , k ′ < λ, q , q ′ ≫ λ Factorization: ∆ V λ ( k , k ′ ) = U λ ( k , q ) V λ ( q , q ′ ) U † � U λ → K · Q Q ( q ) V λ ( q , q ′ ) Q ( q ′ )] K ( k ′ ) with K ( k ) ≈ 1! − → K ( k )[
Approach to universality (fate of high- q physics!) Run NN to lower λ via SRG = ⇒ ≈ Universal low- k V NN Off-Diagonal V λ ( k , 0 ) k ′ < λ 1.0 −1 λ = 3.0 fm V λ 0.5 1 S 0 = C 0 + · · · ⇒ 0.0 q ≫ λ V λ (k,0) [fm] −0.5 V λ −1.0 k < λ 550/600 [E/G/M] 600/700 [E/G/M] q ≫ λ (or Λ ) intermediate states −1.5 500 [E/M] 600 [E/M] = ⇒ change is ≈ contact terms: C 0 δ 3 ( x − x ′ ) + · · · −2.0 2 C 0 ( ψ † ψ ) 2 + · · · ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 [cf. L eft = · · · + 1 − 1 ] k [fm Similar pattern with phenomenological potentials (e.g., AV18) � λ ( q ′ , k ′ ) for k , k ′ < λ, q , q ′ ≫ λ Factorization: ∆ V λ ( k , k ′ ) = U λ ( k , q ) V λ ( q , q ′ ) U † � U λ → K · Q Q ( q ) V λ ( q , q ′ ) Q ( q ′ )] K ( k ′ ) with K ( k ) ≈ 1! − → K ( k )[
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