Effective Field Theories for Electroweak Interactions in Nuclei - PowerPoint PPT Presentation

Effective Field Theories for Electroweak Interactions in Nuclei Saori Pastore Winter Workshop on Neutrino-Nucleus Interactions FNAL - Batavia IL - November 2017 WITH nnn Schiavilla (ODU and JLab) & Carlson, Cirigliano, Dekens, Gandolfi, Mereghetti (LANL) bla Piarulli, Pieper, Wiringa (ANL) & Baroni (U. of SC) & Girlanda (Salento U. and INFN) Kiewsky, Marcucci, Viviani (Pisa U. and INFN) REFERENCES nnn PRC78(2008)064002 - PRC80(2009)034004 - PRL105(2010)232502 - PRC84(2011)024001 - PRC87(2013)014006 - PRC87(2013)035503 - PRL111(2013)062502 - PRC90(2014)024321 - JPhysG41(2014)123002 - PRC(2016)015501 - arXiv:1709.03592 & arXiv:1710.05026 1 / 62

A special request from Andreas S Kronfeld & Maria del Pilar Coloma Escribano * Cover similar range of topics as in the NuSTEC School * - two- and three-nucleon pion exchange interactions - realistic models of two- and three-nucleon interactions - realistic models of many-body nuclear electroweak currents - short-range structure of nuclei, nuclear correlations, and quasi-elastic scattering with emphasis on how the nuclear-physics concepts are grounded in quantum field theory 2 / 62

The Microscopic (aka ab initio ) Description of Nuclei ℓ ′ q ℓ GOAL Develop a comprehensive theory that describes quantitatively and predictably all nuclear structure and reactions * The ab initio Approach* In the ab initio Approach one assumes that all nuclear phenomena can be explained in terms of (or emerge from) interactions between nucleons, and interactions between nucleons and external electroweak probes (electrons, photons, neutrinos, DM, ... ) 3 / 62

Electroweak Reactions * ω ∼ 10 2 MeV: Accelerator neutrinos * ω ∼ 10 1 MeV: EM decay, β -decay * ω � 10 1 MeV: Nuclear Rates for Astrophysics ℓ ′ e ′ , p ′ µ P µ e f , | Ψ f � θ e γ ∗ q Z √ α √ α j µ ℓ q µ = p µ e − p ′ µ e = ( ω, q ) P µ e , p µ i , | Ψ i � e 4 / 62

The ab initio Approach The nucleus is made of A interacting nucleons and its energy is A υ ij + ∑ ∑ t i + ∑ H = T + V = V ijk + ... i = 1 i < j i < j < k where υ ij and V ijk are two- and three-nucleon operators based on EXPT data fitting and fitted parameters subsume underlying QCD 1b 2b A ∑ ρ i + ∑ ℓ ′ ρ = ρ ij + ... , ℓ ′ i = 1 i < j q q A ℓ ∑ j i + ∑ ℓ j = j ij + ... i = 1 i < j Two-body 2b currents essential to satisfy current conservation � � q · j = [ H , ρ ] = t i + υ ij + V ijk , ρ * “Longitudinal” component fixed by current conservation * “Transverse” component “model dependent” 5 / 62

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 ) ∗ Note no pions in the initial or final states, i.e. , pion-production not accounted in the theory 6 / 62

Transition amplitude in time-ordered perturbation theory Insert complete sets of eigenstates of H 0 between successive terms of H 1 1 T f i = � N ′ N ′ | H 1 | NN ; γ � + ∑ � N ′ N ′ | H 1 | I � � I | H 1 | NN ; γ � + ... E i − E I | I � The contributions to the T f i are represented by time ordered diagrams Example: seagull pion exchange current H π NN | I > = + H γπ NN N number of H 1 ’s (vertices) → N! time-ordered diagrams 7 / 62

Nuclear Chiral Effective Field Theory ( χ EFT) approach S. Weinberg, Phys. Lett. B251 , 288 (1990); Nucl. Phys. B363 , 3 (1991); Phys. Lett. B295 , 114 (1992) * χ EFT is a low-energy ( Q ≪ Λ χ ∼ 1 GeV) approximation of QCD * It provides effective Lagrangians describing π ’s, N ’s, ∆ ’s, ... π interactions that are expanded in powers n of a perturbative Q Q (small) parameter Q / Λ χ L eff = L ( 0 ) + L ( 1 ) + L ( 2 ) + ... + L ( n ) + ... N N * The coefficients of the expansion, Low Energy Constants (LECs), are unknown and need to be fixed by comparison with exp data, or take them from LQCD * The systematic expansion in Q naturally has the feature � O � 1 − body > � O � 2 − body > � O � 3 − body * A theoretical error due to the truncation of the expansion can be assigned 8 / 62

(Na¨ ıve) Power Counting Each contribution to the T f i scales as � � H π N∆ N Q α i − β i × Q − ( N − 1 ) Q 3 L ∏ × H ππ NN ∼ eQ ∼ eQ � �� � ���� i = 1 | I � denominators loopintegration � �� � H γπ N∆ H 1 scaling α i = # of derivatives (momenta) in H 1 ; β i = # of π ’s; N = # of vertices; N − 1 = # of intermediate states; L = # of loops × Q − 2 ∼ Q 0 H 1 scaling ∼ Q 1 × Q 1 × Q 0 ���� ���� ���� H π N ∆ H ππ NN H πγ N ∆ 1 1 1 | I �∼ 1 denominators ∼ | I � ∼ 2 m N − ( m ∆ + m N + ω π ) | I � = − Q | I � E i − H 0 m ∆ − m N + ω π Q 1 = Q 0 × Q − 2 × Q 3 * This power counting also follows from considering Feynman diagrams, where loop integrations are in 4D 9 / 62

π , N and ∆ Strong Vertices ∼ Q ∼ Q ∼ Q k , a H π NN H π N∆ H ππ NN � σ · k H π NN = g A V π NN = − i g A τ a ∼ Q 1 × Q − 1 / 2 d x N † ( x ) [ σ · ∇ π a ( x )] τ a N ( x ) − → √ 2 ω k F π F π � H π N ∆ = h A V π N ∆ = − i h A S · k T a ∼ Q 1 × Q − 1 / 2 d x ∆ † ( x ) [ S · ∇ π a ( x )] T a N ( x ) − → √ 2 ω k F π F π g A ≃ 1 . 27; F π ≃ 186 MeV; h A ∼ 2 . 77 (fixed to the width of the ∆ ) are ‘known’ LECs 1 � c k , a e i k · x + h . c . � π a ( x ) = ∑ √ 2 ω k , k ∑ b p , στ e i p · x χ στ , N ( x ) = p , στ 10 / 62

χ EFT nucleon-nucleon potential at LO k v LO = Q 0 + + ∼ NN 1 2 v CT v π OPE 1 f i = � N ′ N ′ | H CT , 1 | NN � + ∑ � N ′ N ′ | H π NN | I � T LO � I | H π NN | NN � E i − E I | I � Leading order nucleon-nucleon potential in χ EFT υ CT + υ π = C S + C T σ 1 · σ 2 − g 2 σ 1 · k σ 2 · k υ LO A = τ 1 · τ 2 NN F 2 ω 2 π k * Configuration space * υ p 12 ( r ) O p υ 12 = ∑ 12 ; O 12 = 1 , σ 1 · σ 2 , σ 1 · σ 2 τ 1 · τ 2 , S 12 τ 1 · τ 2 p S 12 = 3 σ 1 · ˆ r σ 2 · ˆ r − σ 1 · σ 2 11 / 62

χ EFT nucleon-nucleon potential at NLO (without ∆ ’s) C i v NLO ∼ Q 2 = NN renormalize C S , C T , and g A * At NLO there are 7 LEC’s, C i , fixed so as to reproduce nucleon-nucleon scattering data (order of k data) * C i ’s multiply contact terms with 2 derivatives acting on the nucleon fields ( ∇ N ) * Loop-integrals contain ultraviolet divergences reabsorbed into g A , C S , C T , and C i ’s (for example, use dimensional regularization) * Configuration space * υ 12 = ∑ υ p 12 ( r ) O p O 12 = [ 1 , σ 1 · σ 2 , S 12 , L · S ] ⊗ [ 1 , τ 1 · τ 2 ] 12 ; p 12 / 62

Nucleon-nucleon potential Aoki et al. Comput.Sci.Disc.1(2008)015009 CT = Contact Term (short-range); OPE = One Pion Exchange (range ∼ 1 m π ); TPE = Two Pion Exchange (range ∼ 1 2 m π ) 13 / 62

Nucleon-Nucleon Potential and the Deuteron M = ± 1 M = 0 Carlson and Schiavilla Rev.Mod.Phys.70(1998)743 14 / 62

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 10 1 0 1 2 3 4 5 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 JLab, Subedi et al. Science320(2008)1475 Nuclear properties are strongly affected by two-nucleon interactions! 15 / 62

χ EFT many-body potential: Hierarchy 2N Force 3N Force 4N Force ▲ � ✵ ✭ ◗✁✂ ✮ ✄ ◆ ▲ � ✷ ✭ ◗✁✂ ✮ ✄ ◆◆ ▲ � ✸ ✭ ◗✁✂ ✮ ✄ +... ✸ ◆ ▲ � ✹ ✭ ◗✁✂ ✮ ✄ +... +... +... Machleidt & Sammarruca - PhysicaScripta91(2016)083007 CT = Contact Term (short-range); 1 OPE = One Pion Exchange (range ∼ m π ); 1 TPE = Two Pion Exchange (range ∼ 2 m π ) 16 / 62

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 17 / 62