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Effective Field Theories for Electroweak Interactions in Nuclei

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Effective Field Theories for Electroweak Interactions in Nuclei 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,bla Dekens, Gandolfi, Mereghetti (LANL) 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 102 MeV: Accelerator neutrinos * ω ∼ 101 MeV: EM decay, β-decay ∼ * ω . 101 MeV: Nuclear Rates for Astrophysics ℓ′ µ e′ ,p′e P µ, Ψ f | f i q θe γ∗ √α Z√α jµ ℓ qµ = pµ p µ e − ′e = (ω, q) µ e,pµ P , Ψi e i | i 4/62 The ab initio Approach The nucleus is made of A interacting nucleons and its energy is A H = T + V = ∑ ti + ∑υij + ∑ Vijk + ... i=1 i<j i<j<k where υij and Vijk are two- and three-nucleon operators based on EXPT data fitting and fitted parameters subsume underlying QCD 1b 2b A ℓ ′ ℓ′ ρ = ∑ ρi + ∑ρij + ... , i=1 i<j q q A ℓ ℓ j = ∑ ji + ∑jij + ... i=1 i<j Two-body 2b currents essential to satisfy current conservation q j =[H, ρ ] = ti + υij + V , ρ · ijk * “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 mN + 2mπ ∆ ∼ Transition amplitude in time-ordered perturbation theory ∞ n 1 1 − Tf i = N′N′ H1 ∑ H1 NN ∗ h | Ei H + iη | i n=1 − 0 - - H0 = free π, N, ∆ Hamiltonians - H1 = interacting π, N, ∆, and external electroweak fields Hamiltonians 0 T = N′N′ T NN ∝ υij , T = N′N′ T NN;γ ∝ (A ρij,A jij) f i h | | i f i h | | i · 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 H0 between successive terms of H1 1 Tf i = N′N′ H1 NN;γ + ∑ N′N′ H1 I I H1 NN;γ + ... h | | i I h | | i Ei EI h | | i | i − The contributions to the Tf i are represented by time ordered diagrams Example: seagull pion exchange current HπNN = |I > + HγπNN N number of H ’s (vertices) N! time-ordered diagrams 1 → 7/62 Nuclear Chiral Effective Field Theory (χEFT) approach S. Weinberg, Phys. Lett. B251, 288 (1990); Nucl. Phys. B363, 3 (1991); Phys. Lett. B295, 114 (1992) * χEFT is a low-energy (Q Λχ 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/Λχ L L (0) L (1) L (2) L (n) eff = + + + ... + + ... 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 h i − h i − h i − * A theoretical error due to the truncation of the expansion can be assigned 8/62 (Na¨ıve) Power Counting Each contribution to the Tf i scales as H N πN∆ Qαi βi Q (N 1) Q3L ∏ − − − HππNN ∼ eQ ∼ eQ i=1 !× × |Ii denominators loopintegration HγπN∆ H scaling 1 | {z } |{z} | {z } αi = # of derivatives (momenta) in H1; βi = # of π’s; N = # of vertices; N 1 = # of intermediate states; − L = # of loops 1 1 0 2 0 H scaling Q Q Q Q− Q 1 ∼ × × × ∼ HπN∆ HππNN HπγN∆ 1 |{z} |{z}1 |{z} 1 1 denominators I I = I I ∼ Ei H | i ∼ 2mN (m + mN + ωπ ) | i − m mN + ωπ | i ∼ Q | i − 0 − ∆ ∆ − 1 0 2 3 Q = Q Q− Q × × * 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 HπNN HπN∆ HππNN σ gA † gA k 1 1/2 H = dxN (x) [σ ∇π (x)] τ N(x) V = i · τ Q Q− πNN F a a πNN F √ a π Z · −→ − π 2ωk ∼ × hA † hA S k 1 1/2 H = dx∆ (x) [S ∇π (x)] T N(x) V = i · T Q Q− πN∆ F a a πN∆ F √2ω a π Z · −→ − π k ∼ × gA 1.27; Fπ 186 MeV; hA 2.77 (fixed to the width of the ∆) ≃ ≃ are ‘known’∼ LECs 1 ik x πa(x) = ∑ ck,a e · + h.c. , k √2ωk ip x N(x) = ∑ bp,στ e · χστ , p,στ 10/ 62 χEFT nucleon-nucleon potential at LO k LO 0 vNN = + + ∼ Q 1 2 π vCT OPE v LO 1 Tf i = N′N′ HCT,1 NN + ∑ N′N′ HπNN I I HπNN NN h | | i I h | | i Ei EI h | | i | i − Leading order nucleon-nucleon potential in χEFT g2 σ σ LO σ σ A 1 k 2 k τ τ υNN = υCT + υπ = CS + CT 1 2 2 · 2 · 1 2 · − Fπ ωk · * Configuration space * p p σ σ σ σ τ τ τ τ υ12 = ∑υ12(r)O12; O12 = 1, 1 2, 1 2 1 2, S12 1 2 p · · · · S = 3σ rˆ σ rˆ σ σ 12 1 · 2 · − 1 · 2 11/ 62 χEFT nucleon-nucleon potential at NLO (without ∆’s) Ci NLO 2 vNN = ∼ Q renormalize CS, CT, and gA * At NLO there are 7 LEC’s, Ci, fixed so as to reproduce nucleon-nucleon scattering data (order of k data) * Ci’s multiply contact terms with 2 derivatives acting on the nucleon fields (∇N) * Loop-integrals contain ultraviolet divergences reabsorbed into gA, CS, CT , and Ci’s (for example, use dimensional regularization) * Configuration space * p p σ σ τ τ υ12 = ∑υ12(r)O12; O12 =[1, 1 2, S12,L S] [1, 1 2] 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 ); ∼ mπ TPE = Two Pion Exchange (range 1 ) ∼ 2mπ 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 12C 1010B5 8Be105 103 6 5 Li10 3 10 1 4He105 10 ) 3 3 10 5 101 10 3 10-1 10 1 0 1 2 3 4 5 10 -1 103 10 101 0 1 2 3 4 5 10-1 101 0 1 2 3 4 5 (q,Q=0) (fm 10-1 pN 0 1 2 3 4 5 -1 ρ 10 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 χEFT many-body potential: Hierarchy 2N Force 3N Force 4N Force ▲ ✵ ✭ ◗✁✂ ✮ ✄ ◆▲ ✷ ✭ ◗✁✂ ✮ ✄ ◆◆▲ ✸ ✭ ◗✁✂ ✮ ✄ +... ✸ ◆ ▲ ✹ ✭ ◗✁✂ ✮ ✄ +... +... +... Machleidt & Sammarruca - PhysicaScripta91(2016)083007 CT = Contact Term (short-range); OPE = One Pion Exchange (range 1 ); ∼ mπ TPE = Two Pion Exchange (range 1 ) ∼ 2mπ 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 χ2/datum 1; * N2LO with∼∆ 3-nucleon∼ force fits 3H binding energy and the nd scattering∼ length 17/ 62 “Phenomenological” aka “Conventional” aka “Traditional” aka “Realistic” Two- and Three- Nucleon Potentials NUCLEAR HAMILTONIAN H = X Ki + Xvij + X Vijk i i<j i<j<k Ki: Non-relativistic kinetic energy, mn-mp effects included γ π I S p Argonne v 18 : vij = vij + vij + vij + vij = P vp(rij )Oij 60 Argonne v 18 np 40 Argonne v pp • 18 spin, tensor, spin-orbit, isospin, etc., operators 18 Argonne v 18 nn • full EM and strong CD and CSB terms included 20 SAID 7/06 np (deg) δ • predominantly local operator structure 0 2 1 • ®ts Nijmegen PWA93 data with χ /d.o.f.=1.1 -20 S0 -40 Wiringa, Stoks, & Schiavilla, PRC 51 , (1995) 0 100 200 300 400 500 600 Elab (MeV) 2π 3π R Urbana & Illinois: Vijk = Vijk + Vijk + Vijk π π π π π ∆ • Urbana has standard 2π P -wave + ∆ ∆ π short-range repulsion for matter saturation π π ∆ • Illinois adds 2π S -wave + 3π rings π π to provide extra T =3/2 interaction • Illinois-7 has four parameters ®t to 23 levels in A ≤10 nuclei Pieper, Pandharipande, Wiringa, & Carlson, PRC 64 , 014001 (2001) Pieper, AIP CP 1011 , 143 (2008) Courtesy of Bob Wiringa * AV18 fitted up to 350 MeV, reproduces phase shifts up to 1 GeV * ∼ 18/ 62 Spectra of Light Nuclei M.
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