Sensitivity to ‘pion-exchange-like’ correlations GT-AA with correlations -3 2 � 10 10 He 10 Be 0.4 GT-AA without correlations -3 2 � 10 0.3 1 � 10 -3 C(q) [MeV -1 ] 0.2 C(r) [fm 8 � 10 -4 0.1 -1 ] -4 4 � 10 0 0 -4 � 10 -4 -0.1 0 2 4 6 0 200 400 600 r [fm] q [MeV] * no ‘pion-exchange-like’ correlation operators U ij * yes ‘pion-exchange-like’ correlation operators U ij * ∼ 10% increase in the matrix elements corresponds with Mereghetti & Dekens & Cirigliano & Carlson & Wiringa PRC97(2018)014606 19 / 31
Single Beta Decay Matrix Elements in A = 6–10 10 C 10 B 7 Be 7 Li(ex) 7 Be 7 Li(gs) 6 He 6 Li 3 H 3 He Ratio to EXPT gfmc 1b gfmc 1b+2b(N4LO) Chou et al. 1993 - Shell Model - 1b 1 1.1 1.2 gfmc (1b) and gfmc (1b+2b); shell model (1b) Pastore et al. PRC97(2018)022501 A. Baroni et al. PRC93(2016)015501 & PRC94(2016)024003 Based on g A ∼ 1 . 27 no quenching factor ∗ data from TUNL, Suzuki et al. PRC67(2003)044302, Chou et al. PRC47(1993)163 20 / 31
Comparison with calculations of larger nuclei J. Menendez arXiv:1712.08691 21 / 31
Comparison with calculations of larger nuclei - 1/4 F NN GT ππ GT π N F ν F NN GT AA GT ν GT ππ GT π N 1 1 Norm Norm A=10 A=10 A=12 A=12 A=48 JM A=48 JM A=76 JM A=76 JM A=76 JH A=76 JH A=136 JM A=136 JM A=136 JH A=136 JH 0 0 -1 -1 JM = Javier Menendez private communication JH = Hyv¨ arien et al. PRC91(2015)024613 * Relative size of the matrix elements is approximately the same in all nuclei * Short-range terms approximately the same in all nuclei with Mereghetti & Dekens & Cirigliano & Carlson & Wiringa PRC97(2018)014606 22 / 31
Neutrinoless Double Beta Decay: Summary and Outlook We studied correlations and many-body currents in single beta and neutrinoless double beta decays (NLDBD) in A ≤ 12 nuclei * In single beta decays the calculations based on g A ∼ 1 . 27 are in good agreement with the data and axial two-body currents provide a negligible contribution ∼ 2% * In the neutrino-scattering Quasi Elastic kinematic region electroweak two-body are found to increase calculations based on one-body operators alone * In NLDBD we tested the neutrino-exchange potentials as well as contributions of one-pion and contact- range * Lack of correlations in the wave functions produces a ∼ 10% increase in the NLDBD matrix elements 23 / 31
Summary and Outlook Two-nucleon correlations and two-body electroweak currents are crucial to explain available experimental data of both static (ground state properties) and dynamical (cross sections and rates) nuclear observables * Two-body currents can give ∼ 30 − 40% contributions and improve on theory/EXPT agreement * Calculations of β − and ββ − decay m.e.’s in A ≤ 12 indicate two-body physics (currents and correlations) is required * Short-Time-Approximation to evaluate υ -A scattering in A > 12 nuclei is in excellent agreement with exact calculations and data * We are developing a coherent picture for neutrino-nucleus interactions * 24 / 31
Factorization: Short-Time Approximation � 0 | O † R α ( q , ω ) = ∑ � � δ ω + E 0 − E f α ( q ) | f �� f | O α ( q ) | 0 � f � α ( q ) e i ( H − ω ) t O α ( q ) | 0 � dt � 0 | O † R α ( q , ω ) = At short time, expand P ( t ) = e i ( H − ω ) t and keep up to 2b-terms H ∼ ∑ t i + ∑ υ ij i < j i and O † i P ( t ) O i + O † i P ( t ) O j + O † i P ( t ) O ij + O † ij P ( t ) O ij 1b 2b ℓ ′ ℓ ′ q q ℓ ℓ WITH Carlson & Gandolfi (LANL) & Schiavilla (ODU+JLab) & Wiringa (ANL) 25 / 31
Factorization up to two-body operators: The Short-Time Approximation (STA) ℓ ′ In STA: ∼ | f > q Response functions are given by the scattering off pairs of fully interacting nucleons that propagate into a ℓ correlated pair of nucleons = ∑ � 0 | O † � � R α ( q , ω ) δ ω + E 0 − E f α ( q ) | f �� f | O α ( q ) | 0 � f ( 1 ) ( q )+ O α ( 2 ) ( q ) = 1b + 2b O α ( q ) = O α | f � ∼ | ψ p , P , J , M , L , S , T , M T ( r , R ) � = correlated two − nucleon w . f . * We retain two-body physics consistently in the nuclear interactions and electroweak currents * STA can be implemented to accommodate for more two-body physics, e.g. , pion-production induced by e and ν � p 2 P 2 � 0 | O † � � R α ( q , ω ) ∼ δ ( ω + E 0 − E f ) d Ω P d Ω p dPdp α ( q ) | p , P �� p , P | O α ( q ) | 0 � 26 / 31
The Short-Time Approximation S(e,E) 2500 2000 1500 1000 500 0 -500 0 50 0 50 100 100 150 150 e (p) MeV 200 200 E (P) MeV 250 250 300 300 Transverse “response-density” 1b + 2b for 4 He � p 2 P 2 � 0 | O † R α ( q , ω ) ∼ δ ( ω + E 0 − E f ) d Ω P d Ω p dPdp � α ( q ) | p , P �� p , P | O α ( q ) | 0 � � * Preliminary results * 27 / 31
✌ ✔ ✟ ☞ ☛ ✡ ✠ ✟ ✕ ✓ ✎ ✒ ✑ ✏ ✎ ✍ ✌ ☞ ☛ ✍ ✏ ✠ ✌ ✕ ✔ ✓ ✒ ✑ ✏ ✎ ✍ ☞ ✑ ☛ ✡ ✠ ✟ ✕ ✔ ✓ ✒ ✡ EM Moments, EM Decays and e -scattering off nuclei 9 Be( 5 / 2 - � 3 / 2 - ) B(E2) � ✁ � ✞ ✖✗ ✘✙ ✚ ✛ ✜ ✢✣ ✤✥ ✦ ✧ ★ ✩ ✪ ★ - � � ✁ � ✝ ✫ ✬ ✭ ✮ 9 Be( 5 / 2 3 / 2 - ) B(M1) ✯ ✰ ✱ ✲ ✳ ✳ ✴ ✵ ✴ ✶✷ ✸ ✹ ✷ ✺ ✻✷ ✼ ✷ ✄ ☎ ✄ ✆ 4 8 B(3 + � 2 + ) B(M1) � ✁ � ✂ 3 8 B(1 + � 2 + ) B(M1) 9 B 7 Li p � ✁ � � 9 Li 3 H 2 8 Li(3 + � 2 + ) B(M1) � ✁ � ✞ 6 Li* 10 B � ✁ � ✝ 8 Li(1 + � 1 2 + ) B(M1) µ ( µ N ) 8 Li 8 B 2 H 6 Li ✄ ☎ ✄ ✆ 10 B* 7 Be( 1 / 2 - � 3 / 2 - ) B(M1) 0 GFMC(1b) 9 C � ✁ � ✂ GFMC(1b+2b) 9 Be 7 Be 7 Li( 1 / 2 - � 3 / 2 - ) B(E2) EXPT -1 � ✁ � � 3 He n � ✁ � ✞ 7 Li( 1 / 2 - � 3 / 2 - ) B(M1) -2 � ✁ � ✝ 6 Li(0 + � 1 + ) B(M1) -3 ✄ ☎ ✄ ✆ EXPT GFMC(1b) GFMC(1b+2b) � ✁ � ✂ 0 1 2 3 � ✁ � � Ratio to experiment ✽ ✾ ✿ ❀ ❁ ❁ ❀ ❂ ❁ ❃ ❁ ❁ ❃ ❂ ❁ ❄ ❁ ❁ ❄ ❂ ❁ ❅ ❆ ❆ ❇ ❈ ❉❊ ❋ ● Electromagnetic data are explained when two-body correlations and currents are accounted for! Pastore et al. PRC87(2013)035503 - Lovato et al. PRC91(2015)062501 28 / 31
Towards a coherent and unified picture of neutrino-nucleus interactions * ω ∼ few MeV, q ∼ 0: β -decay, ββ -decays * ω � tens MeV: Nuclear Rates for Astrophysics * ω ∼ 10 2 MeV: Accelerator neutrinos, ν -nucleus scattering 29 / 31
Understand Nuclei to Understand the Cosmos ESA, XMM-Newton, Gastaldello, CFHTL Majorana Demonstrator LBNF 30 / 31
Thank you! saori.pastore at gmail.com 31 / 31
Nuclear Physics for Neutrinoless Double Beta Decay: Kinematics ⇒ ω ∼ few MeV, q ∼ 0: EM decay, β -decay, ββ -decays ⇐ ⇒ ω ∼ few MeV, q ∼ hundreds of MeVs: 0 νββ -decays ⇐ * ω ∼ 10 2 MeV: Accelerator neutrinos, ν -nucleus scattering 32 / 31
Nuclei for Accelerator Neutrinos’ Experiments LBNF T2K 12 C CCQE on Neutrino-Nucleus scattering 8 7 ℓ ′ 6 q 5 2 ] -38 cm Ankowski, SF ℓ 4 Athar, LFG+RPA σ [x 10 Benhar, SF GiBUU 3 Madrid, RMF Martini, LFG+RPA Nieves, LFG+SF+RPA 2 � ∆ m 2 RFG, M A =1 GeV � 21 L RFG, M A =1.35 GeV P ( ν µ → ν e ) = sin 2 2 θ sin 2 1 Martini, LFG+2p2h+RPA 2 E ν 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 E ν [GeV] Alvarez-Ruso arXiv:1012.3871 * Nuclei of 12 C , 40 Ar , 16 O , 56 Fe , ... * are the DUNE, MiniBoone, T2K, Miner ν a ... detectors’ active material 33 / 31
Nuclear Physics for Neutrinoless Double Beta Decay Searches ✦ ✦ ✦ ✦ ✦ ✦ ✦ Majorana Demonstrator J. Engel and J. Men´ endez - arXiv:1610.06548 0 νββ -decay τ 1 / 2 � 10 25 years (age of the universe 1 . 4 × 10 10 years) need 1 ton of material to see (if any) ∼ 5 decays per year * Decay Rate ∝ (nuclear matrix elements) 2 ×� m ββ � 2 * 2015 Long Range Plane for Nuclear Physics 34 / 31
Nuclear Structure and Dynamics * ω ∼ few MeV, q ∼ 0: EM decay, β -decay, ββ -decays * ω � tens MeV: Nuclear Rates for Astrophysics * ω ∼ 10 2 MeV: Accelerator neutrinos, ν -nucleus scattering 35 / 31
The Microscopic (or ab initio ) Description of Nuclei ℓ ′ q ℓ Develop a comprehensive theory that describes quantitatively and predictably all nuclear structure and reactions * Accurate understanding of interactions between nucleons, p ’s and n ’s * and between e ’s, ν ’s, DM , ... , with nucleons, nucleons-pairs, ... H Ψ = E Ψ Ψ ( r 1 , r 2 , ..., r A , s 1 , s 2 , ..., s A , t 1 , t 2 , ..., t A ) Erwin Schr¨ odinger 36 / 31
Nuclear Force These Days * 1930s Yukawa Potential * 1960–1990 Highly sophisticated meson exchange potentials * 1990s– Highly sophisticated Chiral Effective Field Theory based potentials π π π Hideki Yukawa Steven Weinberg * Contact terms: short-range 1 * One-pion-exchange: range ∼ m π 1 * Two-pion-exchange: range ∼ 2 m π 37 / 31
Nuclear Interactions and the role of the ∆ Courtesy of Maria Piarulli * N3LO with ∆ nucleon-nucleon interaction constructed by Piarulli et al. in PRC91(2015)024003-PRC94(2016)054007-arXiv:1707.02883 with ∆ ′ s fits ∼ 2000 ( ∼ 3000) data up 125 (200) MeV with χ 2 /datum ∼ 1; * N2LO with ∆ 3-nucleon force fits 3 H binding energy and the nd scattering length υ 12 = ∑ υ p O 12 = [ 1 , σ 1 · σ 2 , S 12 , L · S , L 2 , L 2 σ 1 · σ 2 , ( L · S ) 2 ] ⊗ [ 1 , τ 1 · τ 2 ] 12 ( r ) O 12 ; p + operators 4 terms breaking charge independence 38 / 31
Phenomenological aka Conventional aka Traditional aka Realistic Two- and Three- Nucleon Potentials Courtesy of Bob Wiringa * AV18 fitted up to 350 MeV, reproduces phase shifts up to ∼ 1 GeV * * IL7 fitted to 23 energy levels, predicts hundreds of levels * 39 / 31
Nucleon-nucleon potential Aoki et al. Comput.Sci.Disc.1(2008)015009 CT = Contact Term ∗ - short-range; 1 OPE = One Pion Exchange - range ∼ m π ; 1 TPE = Two Pion Exchange - range ∼ 2 m π ∗ in practice CT’s in r -space are coded with representations of a δ -function ( e.g. , a Gaussian function), or special functions such as Wood-Saxon functions 40 / 31
ρ , ω , σ -exchange The One Boson Exchange (OBE) Lagrangians scalar − g S 0 ¯ − g S 1 ¯ ψψφ S 0 ψτψ · � φ S 1 pseudo-scalar − ig PS 0 ¯ − ig PS 1 ¯ ψγ 5 ψφ PS 0 ψγ 5 τψ · � φ PS 1 vector − g V 0 ¯ ψγ µ ψφ V 0 µ − g V 1 ¯ ψγ µ τψ · � φ V 1 µ tensor − g T 0 − g T 1 ψσ µν ψ∂ ν φ T 0 ψσ µν τψ · ∂ ν � φ T 1 2 m T 0 ¯ 2 m T 1 ¯ µ µ slide from my 15 mins HUGS talk... 41 / 31
CD Bonn Potential g 2 g T J π Mass (MeV) I 4 π g V π ± 0 − 139.56995 1 13.6 PS 1 0 − π 0 134.9764 1 13.6 PS 1 0 − η 547.3 0 0.4 PS 0 ρ ± , ρ 0 1 − 769.9 1 0.84 6.1 V 1; T 1 1 − ω 781.94 0 20.0 0.0 V 0; T 0 0 + σ 400-1200 0 S 0 R.Machleidt, Phys.Rev. C 63 , 014001 (2001) O 12 = [ 1 , σ 1 · σ 2 , S 12 , L · S ] ⊗ [ 1 , τ 1 · τ 2 ] vs O 12 = [ 1 , σ 1 · σ 2 ] ⊗ [ 1 , τ 1 · τ 2 ] ; S 12 from2 π − exchange slide from my 15 mins HUGS... 42 / 31
Nucleon-Nucleon Potential and the Deuteron M = ± 1 M = 0 Carlson and Schiavilla Rev.Mod.Phys.70(1998)743 43 / 31
Quantum Monte Carlo Methods ℓ ′ q ℓ Solve numerically the many-body problem H Ψ = E Ψ Ψ ( r 1 , r 2 , ..., r A , s 1 , s 2 , ..., s A , t 1 , t 2 , ..., t A ) Ψ are spin-isospin vectors in 3 A dimensions with 2 A × A ! Z ! ( A − Z ) ! components 4 He : 96 6 Li : 1280 8 Li : 14336 12 C : 540572 44 / 31
Variational Monte Carlo (VMC) Minimize expectation value of H = T + AV18 + IL7 E V = � Ψ V | H | Ψ V � ≥ E 0 � Ψ V | Ψ V � using trial function � �� � S ∏ ( 1 + U ij + ∑ ∏ | Ψ V � = U ijk ) f c ( r ij ) | Φ A ( JMTT 3 ) � i < j k � = i , j i < j * single-particle Φ A ( JMTT 3 ) is fully antisymmetric and translationally invariant * central pair correlations f c ( r ) keep nucleons at favorable pair separation * pair correlation operators U ij reflect influence of υ ij (AV18) * triple correlation operators U ijk reflect the influence of V ijk (IL7) Lomnitz-Adler, Pandharipande, and Smith NPA361(1981)399 Wiringa, PRC43(1991)1585 45 / 31
Green’s function Monte Carlo (GFMC) Ψ V can be further improved by “filtering” out the remaining excited state contamination Ψ ( τ ) = exp [ − ( H − E 0 ) τ ] Ψ V = ∑ exp [ − ( E n − E 0 ) τ ] a n ψ n n Ψ ( τ → ∞ ) = a 0 ψ 0 In practice, we evaluate a “mixed” estimates � O ( τ ) � = f � Ψ ( τ ) | O | Ψ ( τ ) � i Mixed + � O ( τ ) � f ≈ � O ( τ ) � i Mixed −� O � V � Ψ ( τ ) | Ψ ( τ ) � Mixed = f � Ψ V | O | Ψ ( τ ) � i f � Ψ ( τ ) | O | Ψ V � i ; � O ( τ ) � f � O ( τ ) � i Mixed = f � Ψ V | Ψ ( τ ) � i f � Ψ ( τ ) | Ψ V � i Pudliner, Pandharipande, Carlson, Pieper, & Wiringa, PRC 56 , 1720 (1997) Wiringa, Pieper, Carlson, & Pandharipande, PRC 62 , 014001 (2000) Pieper, Wiringa, & Carlson, PRC 70 , 054325 (2004) 46 / 31
GFMC Energy calculation: An example -20 8 Be(3 + ) 8 Be(1 + ) 8 Be(4 + ) 8 Be(2 + ) -30 8 Be(gs) E( τ ) (MeV) -40 Fig. 6 (Wiringa, et al.) -50 0 0.05 0.1 0.15 0.2 τ (MeV -1 ) Wiringa et al. PRC62(2000)014001 47 / 31
Spectra of Light Nuclei Carlson et al. Rev.Mod.Phys.87(2015)1067 48 / 31
Spectra of Light Nuclei M. Piarulli et al. - arXiv:1707.02883 * one-pion-exchange physics dominates * * it is included in both chiral and “conventional” potentials * 49 / 31
Three-body forces A υ ij + ∑ ∑ t i + ∑ H = T + V = V ijk + ... i = 1 i < j i < j < k V ijk ∼ ( 0 . 2 − 0 . 9 ) υ ij ∼ ( 0 . 15 − 0 . 6 ) H υ π ∼ 0 . 83 υ ij 10 B VMC code output Ti + Vij = -38.2131 (0.1433) + Vijk = -46.7975 (0.1150) Ti = 290.3220 (1.2932) Vij =-328.5351 (1.1983) Vijk = -8.5844 (0.0892) Two-body physics dominates! 50 / 31
(Very) Incomplete List of Credits and Reading Material ∗ Pieper and Wiringa; Ann.Rev.Nucl.Part.Sci.51(2001)53 ∗ Carlson et al. ; Rev.Mod.Phys.87(2015)1067 ∗ van Kolck et al. ; PRL72(1994)1982-PRC53(1996)2086 ∗ Kaiser, Weise et al. ; NPA625(1997)758-NPA637(1998)395 ockle, Meissner ∗ ; RevModPhys81(2009)1773 and references therein ∗ Epelbaum, Gl¨ ∗ Entem and Machleidt ∗ ; PhysRept503(2011)1 and references therin * NN Potentials suited for Quantum Monte Carlo calculations * ∗ Pieper and Wiringa; Ann.Rev.Nucl.Part.Sci.51(2001)53 ∗ Gezerlis et al. and Lynn et al. ; PRL111(2013)032501,PRC90(2014)054323,PRL113(2014)192501 ; ∗ Piarulli et al. ; PRC91(2015)024003-PRC94(2016)054007-arXiv:1707.02883 51 / 31
Summary: Nuclear Interactions * The Microscopic description of Nuclei is very successful * Nuclear two-body forces are constrained by large database of nucleon-nucleon scattering data * Intermediate– and long–range components are described in terms of one- and two-pion exchange potentials * Short-range parts are described by contact terms or special functions * Due to a cancellation between kinetic and two-body contribution, three-body potentials are (small but) necessary to reach (excellent) agreement with the data * Calculated spectra of light nuclei are reproduced within 1 − 2% of expt data * Two-body one-pion-exchange contributions dominate and are crucial to explain the data 52 / 31
Neutrinos (Fundamental Symmetries) and Nuclei Topics (5 hours) * Nuclear Theory for the Neutrino Experimental Program � * Microscopic (or ab initio ) Description of Nuclei � * “Realistic” Models of Two- and Three-Nucleon Interactions � * “Realistic” Models of Many-Body Nuclear Electroweak Currents * Short-range Structure of Nuclei and Nuclear Correlations * Quasi-Elastic Electron and Neutrino Scattering off Nuclei * Validation of the theory against available data 53 / 31
Electromagnetic Probes as tool to test theoretical models e ′ , p ′ µ P µ e f , | Ψ f � θ e γ ∗ Z √ α √ α j µ q µ = p µ e − p ′ µ e = ( ω, q ) P µ e , p µ i , | Ψ i � e * coupling constant α ∼ 1 / 137 allows for a perturbative treatment of the EM interaction; single photon γ exchange suffices * calculated x-sections factorize into a part ∝ |� Ψ f | j µ | Ψ i �| 2 with j µ nuclear EM currents and a part completely specified by the electron kinematic variables * EXPT data are (in most cases) known with great accuracy providing stringent constraints on theories * For light nuclei, the many-body problem can be solved exactly or within controlled approximations 54 / 31
Nuclear Currents: One Body Component 1b A ℓ ′ ∑ ρ = ρ i + ... , i = 1 q A ∑ j = j i + ... ℓ i = 1 * Nuclear currents given by the sum of p ’s and n ’s currents, one-body currents (1b) � L p � S n � S p * Nucleonic electroweak form factors are taken from experimental data, and, in principle, from LQCD calculations where data are poor or scarce ( e.g. , nucleonic axial form factor) * A description based on 1b operators alone fails to reproduce “basic” observables (magnetic moments, np radiative capture) * corrections from two-body meson-exchange currents are required to explain, e.g. , radiative capture Riska&Brown 1972 55 / 31
Electromagnetic Nucleonic Form Factors 1.2 1 1.0 p /( µ p G D ) Price, Hanson Price 0.8 p /G D Berger, Walker 0.9 Berger Borkowski, Murphy Hanson G E Andivahis, Qattan Borkowski 0.6 G M Gayou2002, Punjabi Bosted 0.8 Sill Christy Walker 0.4 Gayou2001 Andivahis Puckett, Crawford Christy 0.7 Zhan, Paolone 0.2 Qattan Ron -1 0 1 -2 -1 0 1 10 10 10 10 10 10 10 Bermuth A-S Bartel-69 Schiavilla 1.1 0.5 Kelly Bartel-72 Zhu Esaulov Becker BHM-SC Lung Herberg BHM-pQCD 0.4 Markowitz Ostrick GKex n /( µ n G D ) Anklin-94 Passchier n /G D 1 Bruins 0.3 Rohe Anklin-98 Eden G E Gao Meyerhoff G M 0.2 Xu-2000 Madey Xu-2003 0.9 Warren Kubon 0.1 Riordan Anderson Geis Lachniet 0 0.8 -2 -1 0 -2 -1 0 10 10 10 10 10 10 2 | (GeV/c) 2 2 | (GeV/c) 2 |Q |Q Gonz´ elez-Jim´ enez Phys.Rept.524(2013)1-35 56 / 31
Nuclear Currents: Two-Body Component 1b 2b A ℓ ′ ℓ ′ ∑ ρ i + ∑ ρ = ρ ij + ... , i < j i = 1 q q A ∑ j i + ∑ = j ij + ... j ℓ ℓ i = 1 i < j * Nuclear currents given by the sum of p ’s and n ’s currents, one-body currents (1b) � L p � S n � S p * Two-body currents (2b) essential to satisfy current conservation * We use MEC (SNPA) or χ EFT currents � � q · j = [ H , ρ ] = t i + υ ij + V ijk , ρ + . . . q − ∂ρ γ ∇ · j = ∂ t classically N N 57 / 31
Electromagnetic Reactions * ω ∼ few MeV, q ∼ 0: EM-decays * ω ∼ 10 2 MeV: e -nucleus scattering A coherent and accurate picture of the way electrons interact with nuclei in a wide range of energy and momenta exists, provided that two-body correlations and two-body currents are accounted for! 58 / 31
Electromagnetic Currents from Nuclear Interactions q · j = [ H , ρ ] = � t i + υ ij + V ijk , ρ � 1) Longitudinal component fixed by current conservation 2) Plus transverse “phenomenological” terms j (1) transverse = j π ρ ω π ∆ j (2) ( v ) + + + q N N j (3) ( V ) + Villars, Myiazawa (40-ies), Chemtob, Riska, Schiavilla . . . see, e.g. , Marcucci et al. PRC 72 (2005)014001 and references therein 59 / 31
Currents from nuclear interactions Satisfactory description of a variety of nuclear em properties in A ≤ 12 2 H(p, γ ) 3 He capture 0.5 0.4 S(E) (eV b) 0.3 LUNA 0.2 Griffiths et al. Schmid et al. 0.1 0 0 10 20 30 40 50 E CM (keV) Marcucci et al. PRC 72 , 014001 (2005) 60 / 31
Currents from χ EFT - Time-Ordered-Perturbation Theory The relevant degrees of freedom of nuclear physics are bound states of QCD * non relativistic nucleons N * pions π as mediators of the nucleon-nucleon interaction * non relativistic Delta’s ∆ with m ∆ ∼ m N + 2 m π Transition amplitude in time-ordered perturbation theory � n − 1 ∞ � 1 T f i = � N ′ N ′ | H 1 | NN � ∗ ∑ E i − H 0 + i η H 1 n = 1 - - H 0 = free π , N, ∆ Hamiltonians - H 1 = interacting π , N, ∆ , and external electroweak fields Hamiltonians T f i = � N ′ N ′ | T | NN � ∝ υ ij , T f i = � N ′ N ′ | T | NN ; γ � ∝ ( A 0 ρ ij , A · j ij ) ∗ A µ = ( A 0 , A ) photon field 61 / 31
External Electromagnetic Field ∼ e Q 0 ∼ e Q 0 ∼ e Q ∼ e Q H γ CT H γπ NN H γπ N∆ H γππ “Minimal” Electromagnetic Vertices * EM H 1 obtained by minimal substitution in the π - and N-derivative couplings (same as doing p → p + e A , minimal coupling) ∇ π ∓ ( x ) → [ ∇ ∓ ie A ( x )] π ∓ ( x ) ∇ N ( x ) → [ ∇ − iee N A ( x )] N ( x ) , e N = ( 1 + τ z ) / 2 * same LECs as the Strong Vertices * * This is equivalent to say that the currents are conserved, i.e. , the continuity equation is satisfied 62 / 31
External Electromagnetic Field µ p , µ n d ′ 8 , d ′ 9 , d ′ C ′ 15 , C ′ 21 16 H (2) H γNN H CT γ, nm γπNN “Non-Minimal” Electromagnetic Vertices * EM H 1 involving the tensor field F µν = ( ∂ µ A ν − ∂ ν A µ ) LECs are not constrained by the strong interaction there are additional LECs fixed to EM observables * H γ NN obtained by non-relativistic reduction of the covariant single nucleon currents constrained to µ p = 2 . 793 n.m. and µ n = − 1 . 913 n.m. * H γπ NN involves ∇ π and ∇ N and 3 new LECs (2 of them “mimicking” ∆ ) * H CT 2 γ involves 2 new LECs * These are the so called the “transverse” currents 63 / 31
EM Currents j from Chiral Effective Field Theory : j ( − 2) ∼ eQ − 2 LO NLO : j ( − 1) ∼ eQ − 1 N 2 LO : j ( − 0) ∼ eQ 0 * Note that j π satisfies the continuity equation with υ π (can be done analytically) − g 2 σ 1 · k σ 2 · k A υ π ( k ) = τ 1 · τ 2 F 2 ω 2 π k − ie g 2 σ 2 · k 2 A j π ( k 1 , k 2 ) = ( τ 1 × τ 2 ) z σ 1 + 1 ⇋ 2 F 2 ω 2 π k 2 ie g 2 k 1 − k 2 A + ( τ 1 × τ 2 ) z σ 1 · k 1 σ 2 · k 2 F 2 ω 2 k 1 ω 2 π k 2 * LO = one-body current * 64 / 31
EM Currents j from Chiral Effective Field Theory : j ( − 2) ∼ eQ − 2 LO NLO : j ( − 1) ∼ eQ − 1 N 2 LO : j ( − 0) ∼ eQ 0 N 3 LO : j (1) ∼ eQ unknown LEC ′ s No three-body currents at this order! * Analogue expansion exists for the Time Component (Charge Operator) ρ * Two-body corrections to the one-body Charge Operator appear at N3LO Pastore et al. PRC78(2008)064002 & PRC80(2009)034004 & PRC84(2011)024001 * analogue expansion exists for the Axial nuclear current - Baroni et al. PRC93 (2016)015501 * also derived by Park+Min+Rho NPA596(1996)515, K¨ olling+Epelbaum+Krebs+Meissner PRC80(2009)045502 & PRC84(2011)054008 65 / 31
Electromagnetic LECs d V 1 , d V 2 d S , d V 1 , d V c S , c V 2 Isovector d V 2 = 4 µ ∗ h A / 9 m N ( m ∆ − m N ) and d S , d V 1 , and d V 2 could be determined by d V 1 = 0 . 25 × d V 2 πγ -production data on the nucleon assuming ∆ -resonance saturation Left with 3 LECs: Fixed in the A = 2 − 3 nucleons’ sector * Isoscalar sector: * d S and c S from EXPT µ d and µ S ( 3 H/ 3 He) * Isovector sector: * c V from EXPT npd γ xsec. or * c V from EXPT µ V ( 3 H/ 3 He) m.m. 66 / 31
Low-energy observables and ground state properties np capture x-section/ µ V of A = 3 nuclei 360 γ 3 H/ 3 He) -1.8 σ µ V ( np 340 -2 320 -2.2 n.m. mb 300 -2.4 LO NLO N2LO 280 -2.6 N3LO (no LECs) N3LO (full) 260 EXP -2.8 ��� ��� 500 600 Λ� (MeV) Λ� (MeV) Observable ∝ � Ψ f | j | Ψ i � Piarulli et al. PRC87(2013)014006 67 / 31
Deuteron magnetic form factor 0 10 N3LO /NN(N3LO), Piarulli et al. j .. N3LO /NN(N2LO), Kolling et al. j -1 m/(M d µ d )|G M | 10 -2 10 (b) -3 10 0 1 2 3 4 5 6 7 -1 ] q [fm Observable ∝ � Ψ f | j | Ψ i � PRC86(2012)047001 & PRC87(2013)014006 68 / 31
✡ ☛ ✤ ✣ ✥ ✢ ❝ ✜ ⑤ ☞ ✌ ⑤ 12 C Charge form factor ✠ ✶ ✵ ✒ ✚ ✒ ✛ ✒ ✚ ✒✔ ☎ ✟ ✶ ✵ ✒ ✚ ✒ ✒ ✒ ✓ ✔ ✕ ✖ ✗ ✘✙ ☎ ✞ ✶ ✵ ❡✍ ✎ ☎ ✝ ✶ ✵ r ✟ ✏ r ✏✑ ✞ ✏ ✟ ☎ ✆ ✶ ✵ ✵ ✶ ✷ ✸ ✹ ✲ ✄ q �✁✂ ✮ ∝ � Ψ f | ρ | Ψ i � Lovato et al. PRL111(2013)092501 69 / 31
3 He and 3 H magnetic form factors 0 10 (a) (b) -1 10 |F T / µ | -2 10 3 He 3 H -3 10 -4 10 0 (c) (d) 10 -1 10 V | S | |F T |F T -2 LO /AV18+UIX 10 j N3LO /AV18+UIX j -3 LO /NN(N3LO)+3N(N2LO) 10 j N3LO /NN(N3LO)+3N(N2LO) j -4 10 0 1 2 3 4 0 1 2 3 4 5 -1 ] -1 ] q [fm q [fm 1b/1b+2b with AV18+UIX – 1b/1b+2b with χ -potentials NN(N3LO)+3N(N2LO) Observable ∝ � Ψ f | j | Ψ i � Piarulli et al. PRC87(2013)014006 70 / 31
Magnetic Moments of Nuclei 4 � L p 3 9 B � S n 7 Li p � 9 Li S p 3 H 2 6 Li* 10 B 1 µ ( µ N ) 8 Li 8 B 2 H 6 Li 10 B* 0 GFMC(1b) 9 C GFMC(1b+2b) 9 Be 7 Be EXPT -1 3 He n -2 -3 m.m. THEO EXP 9 C -1.35(4)(7) -1.3914(5) 9 Li 3.36(4)(8) 3.4391(6) chiral truncation error based on EE et al. error algorithm, Epelbaum, Krebs, and Meissner EPJA51(2015)53 Pastore et al. PRC87(2013)035503 71 / 31
One-body magnetic densities 0.04 0.03 7 Li( 3 / 2 - ) 8 Li(2 + ) 9 Li( 3 / 2 - ) ρ µ (r) ( µ N fm -3 ) 0.02 0.01 0.00 p L -0.01 p S n S -0.02 µ (IA) -0.03 0.03 7 Be( 3 / 2 - ) 8 B(2 + ) 9 C( 3 / 2 - ) ρ µ (r) ( µ N fm -3 ) 0.02 0.01 0.00 -0.01 -0.02 -0.03 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 5 r (fm) r (fm) r (fm) 1b magnetic moment operator µ 1b = µ N ∑ [( L i + g p S i )( 1 + τ i , z ) / 2 + g n S i ( 1 − τ i , z ) / 2 ] i 72 / 31
Electromagnetic Reactions * ω ∼ few MeV, q ∼ 0: EM-decays * ω ∼ 10 2 MeV: e -nucleus scattering A coherent and accurate picture of the way electrons interact with nuclei in a wide range of energy and momenta exists, provided that two-body correlations and two-body currents are accounted for! 73 / 31
Electromagnetic Transitions in Light Nuclei * 2b electromagnetic currents bring - � 9 Be( 5 / 2 3 / 2 - ) B(E2) the THEORY in agreement with - � 9 Be( 5 / 2 3 / 2 - ) B(M1) the EXPT 8 B(3 + � 2 + ) B(M1) * ∼ 40% 2b-current contribution found in 9 C m.m. 8 B(1 + � 2 + ) B(M1) * ∼ 60 − 70% of total 2b-current 8 Li(3 + � 2 + ) B(M1) component is due to 8 Li(1 + � 2 + ) B(M1) one-pion-exchange currents 7 Be( 1 / 2 - � 3 / 2 - ) B(M1) * ∼ 20-30% 2b found in M1 transitions in 8 Be 7 Li( 1 / 2 - � 3 / 2 - ) B(E2) 7 Li( 1 / 2 - � 3 / 2 - ) B(M1) One M1 prediction: 9 Li(1 / 2 → 3 / 2)* 6 Li(0 + � 1 + ) B(M1) + a number of B(E2)s EXPT GFMC(1b) GFMC(1b+2b) *2014 TRIUMF proposal Ricard-McCutchan et al. 0 1 2 3 Ratio to experiment Pastore et al. PRC87(2013)035503 & PRC90(2014)024321, Datar et al. PRL111(2013)062502 74 / 31
Electromagnetic Reactions * ω ∼ few MeV, q ∼ 0: EM-decays * ω ∼ 10 2 MeV: e -nucleus scattering A coherent and accurate picture of the way electrons interact with nuclei in a wide range of energy and momenta exists, provided that two-body correlations and two-body currents are accounted for! 75 / 31
Back-to-back np and pp Momentum Distributions 12 C 10 B 10 5 8 Be 10 5 10 3 6 Li 10 5 10 3 10 1 4 He 10 5 10 3 ρ pN (q,Q=0) (fm 3 ) 10 1 10 5 10 -1 10 3 0 1 2 3 4 5 10 1 10 -1 10 3 0 1 2 3 4 5 10 1 10 -1 0 1 2 3 4 5 10 1 10 -1 0 1 2 3 4 5 10 -1 0 1 2 3 4 5 q (fm -1 ) Wiringa et al. - PRC89(2014)024305 Nuclear properties are strongly affected by correlations! Triple coincidence reactions A ( e , e ′ np or pp ) A − 2 measurements at JLab on 12 C indicate that at high values of relative momenta (400 − 500 MeV), ∼ 90% of the pairs are in the form of np pairs and ∼ 5% in pp pairs 76 / 31
Two-body momentum distributions: Where to find them 1-body momentum distributions http://www.phy.anl.gov/theory/research/momenta/ 2-body momentum distributions http://www.phy.anl.gov/theory/research/momenta2/ 77 / 31
Inclusive ( e , e ′ ) scattering * inclusive xsecs * d 2 σ dE ′ d Ω e ′ = σ M [ v L R L ( q , ω )+ v T R T ( q , ω )] |� f | O α ( q ) | 0 �| 2 R α ( q , ω ) = ∑ � � δ ω + E 0 − E f f Longitudinal response induced by O L = ρ Transverse response induced by O T = j * Sum Rules * Exploit integral properties of the response functions + ℓ ′ closure to avoid explicit calculation of the final states � ∞ q S ( q , τ ) = 0 d ω K ( τ , ω ) R α ( q , ω ) ℓ * Coulomb Sum Rules * � ∞ 0 d ω R α ( q , ω ) ∝ � 0 | O † S α ( q ) = α ( q ) O α ( q ) | 0 � 78 / 31
Sum Rules and the role of two-body currents 3 1−body (1+2)−body 2.5 4 He 6 Li 2 S T (q)/S L (q) 1.5 3 He 1 0.5 200 300 400 500 600 700 800 q(MeV/c) Carlson, Jourdan, Schiavilla, and Sick PRC65(2002)024002 79 / 31
Sum Rules and Two-Body Physics 3 1−body (1+2)−body • S T ( q ) ∝ � 0 | j † j | 0 � 2.5 4 He 6 Li • j = j 1 b + j 2 b 2 S T (q)/S L (q) • enhancement of the transverse 1.5 response is due to interference between 3 He 1 1b and 2b contributions AND presence of correlations in the wave function • 0.5 200 300 400 500 600 700 800 q(MeV/c) PRC65(2002)024002 � j † 1 b j 1 b � > 0 � j † 1 b j 2 b v π � ∝ � v 2 π � > 0 80 / 31
☛ ✔ ✕ ☞ ☛ ✡ ✠ ✟ ✕ ✓ ✓ ✒ ✑ ✏ ✎ ✍ ✌ ☞ ✡ ✔ ✒ ✟ ✌ ✕ ✔ ✓ ✒ ✑ ✏ ✎ ✍ ✟ ✑ ✠ ✡ ☛ ☞ ✌ ✍ ✎ ✏ ✠ Recent Developments on 12 C � ✁ � ✞ ✖✗ ✘✙ ✚ ✛ ✜ ✢✣ ✤✥ ✦ ✧ ★ ✩ ✪ ★ � ✁ � ✝ ✫ ✬ ✭ ✮ ✯ ✰ ✱ ✲ ✳ ✳ ✴ ✵ ✴ ✶✷ ✸ ✹ ✷ ✺ ✻✷ ✼ ✷ ✄ ☎ ✄ ✆ � ✁ � ✂ � ✁ � � � ✁ � ✞ � ✁ � ✝ ✄ ☎ ✄ ✆ � ✁ � ✂ � ✁ � � � ✁ � ✞ � ✁ � ✝ ✄ ☎ ✄ ✆ � ✁ � ✂ � ✁ � � ✽ ✾ ✿ ❀ ❁ ❁ ❀ ❂ ❁ ❃ ❁ ❁ ❃ ❂ ❁ ❄ ❁ ❁ ❄ ❂ ❁ ❅ ❆ ❆ ❇ ❈ ❉❊ ❋ ● q = [ 300 − 750 ] MeV ∼ 100 million core hours Lovato, Gandolfi et al. PRC91(2015)062501 + arXiv:1605.00248 Two-body correlations and currents essential to explain the data! 81 / 31
Electromagnetic Reactions * ω ∼ few MeV, q ∼ 0: EM-decays * ω ∼ 10 2 MeV: e -nucleus scattering A coherent and accurate picture of the way electrons interact with nuclei in a wide range of energy and momenta exists, provided that two-body correlations and two-body currents are accounted for! 82 / 31
✌ ✔ ✟ ☞ ☛ ✡ ✠ ✟ ✕ ✓ ✎ ✒ ✑ ✏ ✎ ✍ ✌ ☞ ☛ ✍ ✏ ✠ ✌ ✕ ✔ ✓ ✒ ✑ ✏ ✎ ✍ ☞ ✑ ☛ ✡ ✠ ✟ ✕ ✔ ✓ ✒ ✡ EM Moments, EM Decays and e -scattering off nuclei 9 Be( 5 / 2 - � 3 / 2 - ) B(E2) � ✁ � ✞ ✖✗ ✘✙ ✚ ✛ ✜ ✢✣ ✤✥ ✦ ✧ ★ ✩ ✪ ★ - � � ✁ � ✝ ✫ ✬ ✭ ✮ 9 Be( 5 / 2 3 / 2 - ) B(M1) ✯ ✰ ✱ ✲ ✳ ✳ ✴ ✵ ✴ ✶✷ ✸ ✹ ✷ ✺ ✻✷ ✼ ✷ ✄ ☎ ✄ ✆ 4 8 B(3 + � 2 + ) B(M1) � ✁ � ✂ 3 8 B(1 + � 2 + ) B(M1) 9 B 7 Li p � ✁ � � 9 Li 3 H 2 8 Li(3 + � 2 + ) B(M1) � ✁ � ✞ 6 Li* 10 B � ✁ � ✝ 8 Li(1 + � 1 2 + ) B(M1) µ ( µ N ) 8 Li 8 B 2 H 6 Li ✄ ☎ ✄ ✆ 10 B* 7 Be( 1 / 2 - � 3 / 2 - ) B(M1) 0 GFMC(1b) 9 C � ✁ � ✂ GFMC(1b+2b) 9 Be 7 Be 7 Li( 1 / 2 - � 3 / 2 - ) B(E2) EXPT -1 � ✁ � � 3 He n � ✁ � ✞ 7 Li( 1 / 2 - � 3 / 2 - ) B(M1) -2 � ✁ � ✝ 6 Li(0 + � 1 + ) B(M1) -3 ✄ ☎ ✄ ✆ EXPT GFMC(1b) GFMC(1b+2b) � ✁ � ✂ 0 1 2 3 � ✁ � � Ratio to experiment ✽ ✾ ✿ ❀ ❁ ❁ ❀ ❂ ❁ ❃ ❁ ❁ ❃ ❂ ❁ ❄ ❁ ❁ ❄ ❂ ❁ ❅ ❆ ❆ ❇ ❈ ❉❊ ❋ ● Electromagnetic data are explained when two-body correlations and currents are accounted for! Pastore et al. PRC87(2013)035503 - Lovato et al. PRC91(2015)062501 83 / 31
Two-body Currents: Summary * Two-body correlations and currents are essential to explain the data * Two-body currents provide up to ∼ 40% contributions to the magnetic moments of nuclei (ground state observable) * Two-body currents enhance the transverse response up ∼ 50% (dynamical observable) * One-pion-exchange currents provide ∼ 0 . 8 j ij 84 / 31
Neutrinos and Nuclei 85 / 31
Towards a coherent and unified picture of neutrino-nucleus interactions * ω ∼ few MeV, q ∼ 0: β -decay, ββ -decays * ω � tens MeV: Nuclear Rates for Astrophysics * ω ∼ 10 2 MeV: Accelerator neutrinos, ν -nucleus scattering 86 / 31
Neutrinos and Nuclei: Challenges and Opportunities Beta Decay Rate Neutrino-Nucleus Scattering 12 C CCQE on 8 7 6 5 2 ] -38 cm Ankowski, SF 4 Athar, LFG+RPA σ [x 10 Benhar, SF GiBUU 3 Madrid, RMF Martini, LFG+RPA Nieves, LFG+SF+RPA 2 RFG, M A =1 GeV RFG, M A =1.35 GeV 1 Martini, LFG+2p2h+RPA 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 E ν [GeV] Alvarez-Ruso arXiv:1012.3871 → g eff in 3 ≤ A ≤ 18 − A ≃ 0 . 80 g A Chou et al. PRC47(1993)163 87 / 31
Standard Beta Decay The “ g A problem” and the role of two-body correlations and two-body currents e − ν e ¯ W ± g A * Matrix Element � Ψ f | GT | Ψ i � ∝ g A and Decay Rates ∝ g 2 A * ( Z , N ) → ( Z + 1 , N − 1 )+ e + ¯ ν e 88 / 31
“Anomalies” q ∼ 0: The “ g A problem” Gamow-Teller Matrix Elements Theory vs Expt → g eff in 3 ≤ A ≤ 18 − A ≃ 0 . 80 g A Chou et al. PRC47(1993)163 Missing Physics: 1. Correlations and/or 2. Two-body currents 89 / 31
Nuclear Interactions and Axial Currents A υ ij + ∑ ∑ t i + ∑ H = T + V = V ijk + ... i = 1 i < j i < j < k so far results are available with AV18+IL7 ( A ≤ 10) and SNPA or chiral currents ( a.k.a. hybrid calculations) * c 3 and c 4 are taken them from Entem and Machleidt PRC68(2003)041001 & LO Phys.Rep.503(2011)1 * c D fitted to GT m.e. of tritium N 3 LO Baroni et al. PRC94(2016)024003 * cutoffs Λ = 500 and 600 MeV * include also N4LO 3b currents (tiny) + ... N 4 LO * derived by Park et al. in the ′ 90 used (mainly at tree-level) in many calculations A. Baroni et al. PRC93(2016)015501 * pion-pole at tree-level derived H. Krebs et al. Ann.Phy.378(2017) by Klos, Hoferichter et al. PLB(2015)B746 90 / 31
Single Beta Decay Matrix Elements in A = 6–10 10 C 10 B 7 Be 7 Li(ex) 7 Be 7 Li(gs) 6 He 6 Li 3 H 3 He Ratio to EXPT gfmc 1b gfmc 1b+2b(N4LO) Chou et al. 1993 - Shell Model - 1b 1 1.1 1.2 gfmc (1b) and gfmc (1b+2b); shell model (1b) Pastore et al. PRC97(2018)022501 A. Baroni et al. PRC93(2016)015501 & PRC94(2016)024003 Based on g A ∼ 1 . 27 no quenching factor ∗ data from TUNL, Suzuki et al. PRC67(2003)044302, Chou et al. PRC47(1993)163 91 / 31
10 B + ,1) (0 < 0.08 % 10 C 98.54(14)% E ~ 2.15 MeV + ,0) (1 + ,1) (0 E ~ 0.72 MeV + ,0) (1 + ,0) (3 10 B * In 10 B, ∆ E with same quantum numbers ∼ 1 . 5 MeV * In A = 7, ∆ E with same quantum numbers � 10 MeV 92 / 31
Nuclei for Accelerator Neutrinos’ Experiments LBNF T2K 12 C CCQE on Neutrino-Nucleus scattering 8 7 ℓ ′ 6 q 5 2 ] -38 cm Ankowski, SF ℓ 4 Athar, LFG+RPA σ [x 10 Benhar, SF GiBUU 3 Madrid, RMF Martini, LFG+RPA Nieves, LFG+SF+RPA 2 � ∆ m 2 RFG, M A =1 GeV � 21 L RFG, M A =1.35 GeV P ( ν µ → ν e ) = sin 2 2 θ sin 2 1 Martini, LFG+2p2h+RPA 2 E ν 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 E ν [GeV] Alvarez-Ruso arXiv:1012.3871 * Nuclei of 12 C , 40 Ar , 16 O , 56 Fe , ... * are the DUNE, MiniBoone, T2K, Miner ν a ... detectors’ active material 93 / 31
Nuclei for Accelerator Neutrinos’ Experiments: More in Detail Neutrino Flux Tomasz Golan Phil Rodrigues * Oscillation Probabilities depend on the initial neutrino energy E ν * Neutrinos are produced via decay-processes, E ν is unknown! � � ∆ m 2 21 L P ( ν µ → ν e ) = sin 2 2 θ sin 2 2 E ν * E ν is reconstructed from the final state observed in the detector * !! Accurate theoretical neutrino-nucleus cross sections are vital !! to E ν reconstruction 94 / 31
e − A and ν − A Scattering µ Boone 95 / 31
Inclusive ( e , ν scattering * inclusive xsecs * d 2 σ dE ′ d Ω e ′ = σ M [ v L R L ( q , ω )+ v T R T ( q , ω )] |� f | O α ( q ) | 0 �| 2 R α ( q , ω ) = ∑ � � δ ω + E 0 − E f f Longitudinal response induced by O L = ρ Transverse response induced by O T = j ... 5 nuclear responses in ν -scattering... * Sum Rules * Exploit integral properties of the response functions + ℓ ′ closure to avoid explicit calculation of the final states � ∞ q S ( q , τ ) = 0 d ω K ( τ , ω ) R α ( q , ω ) ℓ * Coulomb Sum Rules * � ∞ 0 d ω R α ( q , ω ) ∝ � 0 | O † S α ( q ) = α ( q ) O α ( q ) | 0 � 96 / 31
P ◆ ❚ ❯ ❙ ❘ ◆ ◗ ◆ ❖ ▼ ✓ ❚ ❙ ❘ ◆ ◗ ◆ ▼ ◆ ✒ ✔ P ✣ ✪ ✩ ★ ✥ ✧ ✦ ✥ ✤ ✢ ✒ ✜ ✛ ✚ ✙ ✘ ✗ ✖ ✒ ✕ ❖ ◆ ▼ ✛ ✥ ✧ ✦ ✥ ✤ ✣ ✢ ✜ ✚ ✩ ✙ ✘ ✗ ✖ ✒ ✕ ✒ ✔ ★ ✪ ◗ ◆ ◆ ❘ ❙ ❚ ▼ ◆ ❖ P ◗ ✫ ◆ ❘ ❙ ❯ ❚ P ❖ ◆ ▼ ✫ ◆ ✒ ✢ ✩ ★ ✥ ✧ ✦ ✥ ✤ ✣ ✜ ✫ ✛ ✚ ✙ ✘ ✗ ✖ ✒ ✕ ✪ ▼ ✔ ❖ ❚ ❯ ❙ ❘ ◆ ◗ ◆ P ◆ ◆ ▼ ❚ ❙ ❘ ◆ ◗ ◆ P ❖ ✒ ✓ ❖ ✢ ✒ ✖ ✗ ✘ ✙ ✚ ✛ ✜ ✣ ✒ ✤ ✥ ✦ ✧ ✥ ★ ✩ ✪ ✫ ✕ ✔ ✒ ❖ ❚ ❯ ❙ ❘ ◆ ◗ ◆ P ◆ ✓ ▼ ❚ ❙ ❘ ◆ ◗ ◆ P ✒ ✓ Recent Developments on 12 C : Inclusive QE Scattering NC Inclusive Xsec ✡ ☛ ☞ ✬ ❇ ❈ ■ ✭ ✮ ✯ ✰ Charge-Current Cross Section ✱ ✝ ✠ ✞ ✲ ✳ ✴ ✵ ✶ ✷ ❇ ❈ ❍ ✹ ✸ ✝ ✟ ✠ ✺ ✻ ✼ ✽ ✾ ✿ ❭ ❪ ❫ ❴ ❀ ❁ 12 C ❊ ❋ ● ❂ ❃ ❵❛ CCQE on ✝ ✞ ✞ ❄ ❅ ❆ ❜❝❞ ❇ ❈ ❉ ☎ ✆ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✝ ✞ ✞ ✟ ✞ ✞ ❏ ✞ ✞ ❑ ▲ ▲ 8 ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ❱ ❲ ❳❨ ❩ ❬ ✂ ✄ 7 ✁ ✂ � ✂ ✄ 6 ♣ ● ❋ ✏ ✏ ✏ ✏ ✏ ✏ ✑ ✑ ✑ ✑ ✑ ✑ ✌ ✌ ✌ ✌ ✌ ✌ ✐ ❥ ✐ ❥ ✐ ❥ ✐ ❥ ✐ ❥ ✐ ❥ ❣ ❤ ♥ ♦ ♥ 5 2 ] -38 cm ❧ ❈ ♠ Ankowski, SF ❢ ✄ 4 Athar, LFG+RPA σ [x 10 ❇ ❈ ❦ Benhar, SF ✟ ❏ ✁ ✄ ✝ ✞ ✞ ✞ ✞ ✞ ✞ ❑ ▲ ▲ GiBUU 3 ❱ ❲ ❳❨ ❩ ❬ Madrid, RMF ❡ ✄ Martini, LFG+RPA Nieves, LFG+SF+RPA 2 RFG, M A =1 GeV � ❡ ✉ RFG, M A =1.35 GeV ② ✏ ✏ ✏ ✏ 1 ✑ ✑ ✑ ✑ ✌ ✌ ✌ ✌ ✈ ✈ ✈ ✈ ❥ ❥ ❥ ❥ Martini, LFG+2p2h+RPA s t ① 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 E ν [GeV] r ✇ ✝ ✞ ✞ ✟ ✞ ✞ ❏ ✞ ✞ ❑ ▲ ▲ Alvarez-Ruso arXiv:1012.3871 ❱ ❲ ❳❨ ❩ ❬ q CHALLENGES: � ⑦ ❾ ❿ ❹ ❹ ⑨ ⑨ ⑩ ⑩ ❶ ❷ ❶ ❷ ❸ ❸ ⑥ ❼ ❽ q ❻ 1. How do we describe electroweak-scattering off ⑤ ❺ ✝ ✞ ✞ ✟ ✞ ✞ ❏ ✞ ✞ ❑ ▲ ▲ ④ A > 12 without loosing two-body physics ❱ ❲ ❳❨ ❩ ❬ ③ (correlations and two-body currents)? � � ✂ ✄ ✝ ✞ ✞ ✝ ✠ ✞ ✟ ✞ ✞ ✟ ✠ ✞ ❏ ✞ ✞ ❏ ✠ ✞ ❑ ▲ ▲ ❑ ⑧ ▲ 2. How to incorporate (more) exlusive processes? ❱ ❲ ❳ ❨ ❩ ❬ q = 750 MeV Lovato & Gandolfi et al. PRC97(2018)022502 ∼ 100 million core hours 97 / 31
Scaling properties of the Response Functions Inclusive xsec depends on a single (scaling) function of ω and q Scaling 2 nd kind: independence form A Donnelly and Sick - PRC60(1999)065502 1. Rely on observed scaling properties of inclusive xsecs, universal behavior of nucleon/ A momentum distributions, and exhibited locality of nuclear properties to build approximate response functions for A > 12 nuclei 2. From exact ab initio calculations we know that two-body correlations and two-body currents are crucial 3. Build a model that retains two-body physics 98 / 31
Factorization: Short-Time Approximation � 0 | O † R α ( q , ω ) = ∑ � � δ ω + E 0 − E f α ( q ) | f �� f | O α ( q ) | 0 � f � α ( q ) e i ( H − ω ) t O α ( q ) | 0 � dt � 0 | O † R α ( q , ω ) = At short time, expand P ( t ) = e i ( H − ω ) t and keep up to 2b-terms H ∼ ∑ t i + ∑ υ ij i < j i and O † i P ( t ) O i + O † i P ( t ) O j + O † i P ( t ) O ij + O † ij P ( t ) O ij 1b 2b ℓ ′ ℓ ′ q q ℓ ℓ WITH Carlson & Gandolfi (LANL) & Schiavilla (ODU+JLab) & Wiringa (ANL) 99 / 31
Factorization up to one body - The Plane Wave Impulse Approximation ℓ ′ In PWIA: q Response functions given by incoherent scattering off single nucleons that propagate freely in the final state ℓ (plane waves) = ∑ � 0 | O † � � R α ( q , ω ) δ ω + E 0 − E f α ( q ) | f �� f | O α ( q ) | 0 � f ( 1 ) ( q ) = 1b O α ( q ) = O α e i ( k + q ) · r = free single nucleon w . f . | f � ∼ * PWIA Longitudinal Response in terms of the p -momentum distribution n p ( k ) * ω − ( k + q ) 2 + k 2 � � � PWIA ( q , ω ) R L = d k n p ( k ) δ 2 m N 2 m N A 1 + τ i , z ( 1 ) ( q ) e i q · r i ∑ = O L e 2 i = 1 100 / 31
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