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

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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

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

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)

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

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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

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. Piarulli et al. - arXiv:1707.02883

* one-pion-exchange physics dominates * * it is included in both chiral and “conventional” potentials *

19/ 62 Chiral Potentials (Incomplete List of Credits)

van Kolck et al.; PRL72(1994)1982-PRC53(1996)2086 ∗ Kaiser, Weise et al.; NPA625(1997)758-NPA637(1998)395 ∗ Epelbaum, Gl¨ockle, Meissner ; RevModPhys81(2009)1773 and references therein ∗ ∗ Entem and Machleidt ; PhysRept503(2011)1 and references therin ∗ ∗

* Chiral Potentials suited for Quantum Monte Carlo calculations *

Gezerlis et al. PRL111(2013)032501-PRC90(2014)054323; ∗ Lynn et al. PRL113(2014)192501

Piarulli et al. PRC91(2015)024003-PRC94(2016)054007-arXiv:1707.02883 (with ∆ s) ∗ ∗ ′

* Potentials ﬁtted and used in many-body calculations

20/ 62 External Electromagnetic Field

∼ e Q0 ∼ e Q0 ∼ e Q ∼ e Q

HγπNN HγπN∆ Hγππ HγCT

“Minimal” Electromagnetic Vertices

* EM H1 obtained by minimal substitution in the π- and N-derivative couplings (same as doing p p + eA, minimal coupling) →

∇π (x) [∇ ieA(x)]π (x) ∓ → ∓ ∓ ∇N(x) [∇ ieeN A(x)]N(x) , eN =(1 + τz)/2 → − * same LECs as the Strong Vertices *

* This is equivalent to say that the currents are conserved, i.e., the continuity equation is satisﬁed

21/ 62 External Electromagnetic Field

′ ′ ′ ′ ′ µp, µn d8,d9,d21 C15,C16

(2) HγNN HγπNN HCTγ,nm

“Non-Minimal” Electromagnetic Vertices

* EM H involving the tensor ﬁeld Fµν =(∂µ Aν ∂ν Aµ ) 1 −

LECs are not constrained by the strong interaction there are additional LECs ﬁxed to EM observables

* HγNN obtained by non-relativistic reduction of the covariant single nucleon currents constrained to µp = 2.793 n.m. and µn = 1.913 n.m. − * HγπNN involves ∇π and ∇N and 3 new LECs (2 of them are “saturated” by the ∆)

* HCT2γ involves 2 new LECs

* These are the so called the “transverse” currents

22/ 62 Electromagnetic Currents from Chiral Effective Field Theory

LO : j( 2) eQ 2 − ∼ −

NLO : j( 1) eQ 1 − ∼ −

N2LO : j( 0) eQ0 − ∼

* 3 unknown Low Energy Constants: ﬁxed so as to reproduce d, 3H, and 3He magnetic moments

N3LO: j(1) eQ ∼

unknown LEC′s

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 *

23/ 62 Technicalities I: Reducible Contributions

4 interaction Hamiltonians 4! time ordered diagrams −→

q2 Reducible

q1 1 2

Irreducible direct

Irreducible crossed

1 Ψ φ + υπ φ + ... | i ≃ | i Ei H0 | i − π 1 Ψf j Ψi φf j φi + φf υ j + h.c. φi + ... h | | i ≃ h | | i h | Ei H0 | i − * Need to carefully subtract contributions generated by the iterated solution of the Schr¨odinger equation

24/ 62 Technicalities II: The Cutoff

* χEFT operators have a power law behavior in Q (Q/Λ)n 1. introduce a regulator to kill divergencies at large Q, e.g., CΛ = e− 2. pick n large enough so as to not generate spurious contributions

Q n C 1 + ... Λ ∼ − Λ 3. for each cutoff Λ re-ﬁt the LECs 4. ideally, your results should be cutoff-independent

* In rij-space this corresponds to cutting off the short-range part of the operators that make the matrix elements diverge at rij = 0

25/ 62 Convergence and cutoff dependence

np capture x-section/ µV of A = 3 nuclei bands represent nuclear model dependence [NN(N3LO)+3N(N2LO) – AV18+UIX]

360 σγ µ 3 3 -1.8 np V( H/ He) 340 -2

320 -2.2 mb 300 n.m. LO -2.4 NLO 280 N2LO N3LO (no LECs) -2.6 N3LO (full) 260 EXP -2.8

500 600 500 600 Λ (MeV) Λ (MeV)

3 3 npdγ x-section and µV ( H/ He) m.m. are within 1% and 3% of EXPT ∗ negligible dependence on the cutoff ∗ Piarulli et al. PRC87(2013)014006

26/ 62 Predictions with χEFT EM currents for the deuteron Charge and Quadrupole f.f.’s

Bands represent cutoff Λ dependence

0 2 10 10 N3LO N3LO ρ /NN(N3LO), Piarulli et al. ρ /NN(N3LO), Piarulli et al. 1 N3LO 10 N3LO ρ /NN(N2LO), Phillips ρ /NN(N2LO), Phillips -1 10 0 10 | | C Q |G |G -1 10 -2 10

-2 10 (c) (d) -3 -3 10 0 1 2 3 4 5 6 7 10 0 1 2 3 4 5 6 7 -1 -1 q [fm ] q [fm ] Calculations include nucleonic form factors taken from EXPT data ∗ ∗ J.Phys.G34(2007)365 & PRC87(2013)014006

27/ 62 Predictions with χEFT EM currents for the deuteron magnetic f.f.

Bands represent cutoff Λ dependence

0 10 N3LO j /NN(N3LO), Piarulli et al.

N3LO .. j /NN(N2LO), Kolling et al.

| -1

M 10 )|G d µ d

-2 m/(M 10

(b) -3 10 0 1 2 3 4 5 6 7 -1 q [fm ]

PRC86(2012)047001 & PRC87(2013)014006

28/ 62 Predictions with χEFT EM currents for 3He and 3H magnetic f.f.’s

0 10 (a) (b) -1 10 | µ

/ -2 T 10 |F

-3 3 3 10 He H

-4 10 0 10 (c) (d)

-1 10 | | S V T -2 LO T |F 10 j /AV18+UIX |F N3LO j /AV18+UIX -3 LO 10 j /NN(N3LO)+3N(N2LO) N3LO j /NN(N3LO)+3N(N2LO) -4 10 0 1 2 3 4 0 1 2 3 4 5 -1 -1 q [fm ] q [fm ] LO/N3LO with AV18+UIX – LO/N3LO with χ-potentials NN(N3LO)+3N(N2LO) 3He/3H m.m.’s used to ﬁx EM LECs; 10% correction from two-body currents ∗ ∼ Two-body corrections crucial to improve agreement with EXPT data ∗ Piarulli et al. PRC87(2013)014006

29/ 62 Electromagnetic Currents from Nuclear Interactions aka Standard Nuclear Physics Approach (SNPA) Currents aka Meson Exchange Currents (MEC)

q j =[H, ρ ] = ti + υij + V , ρ · ijk 1) “Longitudinal” component ﬁxed by current conservation 2) “Plus transverse” component “model dependent”

j = j(1) transverse π π ρω + j(2)(v) + ∆ + q

N N + j(3)(V )

(2) * If υij = AV18 j (υ) has the same range of applicability ( 1 GeV) as the → AV18 * ∼ Villars, Myiazawa (40-ies), Chemtob, Riska, Schiavilla . . . see, e.g., Marcucci et al. PRC72(2005)014001 and references therein

30/ 62

Chiral vs Conventional Approach

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Girlanda et al. PRL105(2010)232502

Power Counting doesn’t know about suppressions/cancellations at LO

31/ 62 Magnetic Moments and M1 Transitions

9 5 - → 3 - Be( /2 /2 ) B(E2)

4 9 5 - → 3 - Be( /2 /2 ) B(M1)

3 8B(3+ → 2+) B(M1) 7 9 Li B 8 + → + p 3 9Li B(1 2 ) B(M1) 2 H 8Li(3+ → 2+) B(M1) 6Li* 10B

) 1 8 8Li(1+ → 2+) B(M1) N Li 8B µ 2H 6Li ( 10 B* 7 1 - → 3 - µ Be( / / ) B(M1) 0 GFMC(1b) 2 2 9C GFMC(1b+2b) 9 7Li(1/ - → 3/ -) B(E2) 7Be Be 2 2 -1 EXPT 7Li(1/ - → 3/ -) B(M1) n 3He 2 2 -2 6Li(0+ → 1+) B(M1)

-3 EXPT GFMC(1b) GFMC(1b+2b) 0 1 2 3 Ratio to experiment * 2b electromagnetic currents bring the THEORY in agreement with the EXPT * 40% 2b-current contribution found in 9C m.m. ∼ * 60 70% of total 2b-current component is due to one-pion-exchange currents ∼ − * 20-30% 2b found in M1 transitions in 8Be ∼

Pastore et al. PRC87(2013)035503 & PRC90(2014)024321, Datar et al. PRL111(2013)062502

32/ 62 Error Estimate

4 EE et al. error algorithm Epelbaum, Krebs, and 3 9 Meissner EPJA51(2015)53 7Li B p 3 9Li 2 H 6Li* 10 B δ N3LO =max Q4µLO, Q3 µLO µNLO ,

) 1 8 | − | N Li 8B µ 2H 6Li 2 NLO N2LO ( 10B* Q µ µ , µ 0 GFMC(IA) | − | 9 GFMC(FULL) C 1 N2LO N3LO 7 9Be Q µ µ EXPT Be | − | -1 n 3He m p -2 Q = max π , Λ Λ -3

m.m. THEO EXP 9C -1.35(4)(7) -1.3914(5) 9Li 3.36(4)(8) 3.4391(6)

* ‘N3LO-∆’ corrections can be ’large’ * * “Conventional” and χEFT currents qualitatively in agreement, χEFT isoscalar currents provide better description exp data * Pastore et al. PRC87(2013)035503

33/ 62 Recent Developments on 12C Quantum Monte Carlo Calculations of Nuclear Responses and Sum Rules

Electromagnetic Transverse Responses

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❅ ❆❆ ❇ ❈ ❉❊❋● q =[300 750] MeV − More by Alessandro Lovato on THUR @ 2:30 pm WH3NE

Lovato et al. PRC91(2015)062501 + arXiv:1605.00248

34/ 62 Chiral Electroweak Currents (Incomplete List of Credits)

* Electromagnetic Currents *

Park, Min, and Rho et al. - NPA596(1996)515 ∗ applications to A=2–4 systems including magnetic moments and M1 properties and radiative captures by Song, Lazauskas, Park at al. Meissner, K¨olling, Epelbaum, Krebs et al. - PRC80(2009)045502 & PRC84(2011)054008 ∗ applications to A=2–4 systems including d and 3He photodisintegration by Rozpedzik et al.; d magnetic f.f. by K¨olling, Epelbaum, Phillips; radiative N d capture by Skibinski et al. (2014) − Phillips ∗ applications to deuteron static properties and f.f.’s

* Axial Currents *

Park, Min, and Rho et al. - PhysRept233(1993)341 ∗ applications to A=2–4 systems including µ-capture, pp-fusion, hep · Krebs and Epelbaum et al. - AnnalsPhys378(2017)317 ∗ Baroni et al. - PRC93(2016)015501 ∗ applications to low-energy neutrino scattering off d and Quantum Monte Carlo calculations of β-decay matrix elements in A=3-10 nuclei

35/ 62 Observations

* Chiral Effective Field Theory *

* Chiral Formulation of Nuclear Physics is extremely successful * But limited to low-energies * Inclusion of the ∆ possibly allows for applications to higher energies

* “Conventional” Formulation *

* “Conventional” Formulation of Nuclear Physics is extremely successful * But hard to be systematically improved * “Conventional” Currents (other names are Meson Exchange Currents, MEC, or Standard Nuclear Physics Approach Currents, SNAP) satisfy the continuity equation (with, e.g., the AV18) by construction (they have the same range of applicability as the AV18, i.e. 1 GeV) ∼

36/ 62 Conclusion and Outlook I

* The Microscopic picture of the nucleus based on many-body interactions and electroweak currents successfully explains the data both qualitatively and quantitatively * It explains the spectra and shapes of nuclei * It has been validated against electromagnetic observables in a wide range of energies from keV (relevant to astrophysics) to GeV (relevant to accelerator neutrino experiments) * Two-body physics, correlations and two-body currents, is essential to understand the data both for static nuclear properties (spectra, electromagnetic moments, nuclear form factors) and dynamical properties (transitions in low-lying nuclear states, nuclear responses) * We want the same coherent picture for interactions with neutrinos *

37/ 62 Nuclei and Neutrinos

* ν-A scattering “Anomalies” the QE region * “gA-problem” low-values of momentum/energy transfer * Scarce data at moderate values of momentum transfer

ℓ′

ℓ

38/ 62 “Anomalies” GeV ∼

Neutrino-Nucleus scattering 12 CCQE on C

] 5 2

cm Ankowski, SF

-38 4 Athar, LFG+RPA Benhar, SF

[x 10 GiBUU σ 3 Madrid, RMF Martini, LFG+RPA 2 Nieves, LFG+SF+RPA RFG, MA=1 GeV RFG, M =1.35 GeV 1 A Martini, LFG+2p2h+RPA

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Eν [GeV] Alvarez-Ruso arXiv:1012.3871

39/ 62 “Anomalies” q 0: The “gA problem” ∼

Gamow-Teller Matrix Elements Theory vs Expt

eff in 3 A 18 g 0.80gA ≤ ≤ −→ A ≃ Chou et al. PRC47(1993)163

40/ 62 β and 0νββ-decay −

e−

ν¯e

Berna U.

41/ 62 Nuclear Interactions and Axial Currents

The nucleus is made of A non-relativistic interacting nucleons and its energy is

A H = T + V = ∑ ti + ∑υij + ∑ Vijk + ... i=1 i

N3LO * c3 and c4 are taken them from Entem and Machleidt PRC68(2003)041001 & Phys.Rep.503(2011)1

+... N4LO * cD ﬁtted to GT m.e. of tritium beta-decay Baroni et al. PRC94(2016)024003

A. Baroni et al. PRC93(2016)015501

H. Krebs et al. Ann.Phy.378(2017)

42/ 62 Single β-decay Matrix Elements in A = 6–10

10 10 C B

7 7 Be Li(ex)

7 7 Be Li(gs)

6 6 He Li

3 3 H He

Ratio to EXPT gfmc 1b gfmc 1b+2b(N4LO) Chou et al. 1993 - Shell Model - 1b

1 1.1 1.2

Pastore et al. arXiv:1709.03592

Based on gA 1.27 no quenching factor ∼ data from TUNL, Suzuki et al. PRC67(2003)044302, Chou et al. PRC47(1993)163 ∗

43/ 62 Fundamental Physics Quests: Double Beta Decay

observation of 0νββ-decay → lepton # L = l ¯l not conserved − implications→ in matter-antimatter imbalance Majorana Demonstrator

* detectors’ active material 76Ge * 25 10 0νββ-decay τ1/2 & 10 years (age of the universe 1.4 10 years) 1 ton of material to see (if any) 5 decays per× year * also, if nuclear m.e.’s are known, absolute∼ν-masses can be extracted *

2015 Long Range Plane for Nuclear Physics

44/ 62 Momentum Dependence

GT-AA with correlations -3 10 10 2×10 0.4 He Be GT-AA without correlations

-3 2×10 0.3

-3 [MeV C(q) 1×10 ]

-1 0.2 -4 8×10 -1

0.1 -4 ] C(r) [fm 4×10

0 0

-4 -4×10 -0.1

0 2 4 6 0 200 400 600 r [fm] q [MeV]

* Peaks at 200 MeV ∼ * no ‘pion-exchange-like’ correlation operators Uij * yes ‘pion-exchange-like’ correlation operators Uij * 10% increase in the matrix elements corresponds to a ‘gA-quenching’ of 0.95 * as∼ opposed to 0.83 found in A = 10 single beta decay ∼ ∼

ν π ππ NN

Pastore, Mereghetti, Dekens, Carlson, Cirigliano, Wiringa arXiv:1710.05026 45/ 62

Recent Developments on 12C: Weak Responses

✵ ✵

12 CCQE on C

8 ①

7 ①

] 5 2 cm

Ankowski, SF ✂ ✂ -38 4 Athar, LFG+RPA Benhar, SF

[x 10 GiBUU σ 3 Madrid, RMF Martini, LFG+RPA 2 Nieves, LFG+SF+RPA

RFG, MA=1 GeV ✂

RFG, M =1.35 GeV ✵ 1 A Martini, LFG+2p2h+RPA

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Eν [GeV] ✁

Alvarez-Ruso arXiv:1012.3871 ① ✲

q =[300 750] MeV − Lovato, Gandolﬁ et al. - PRL112(2014)182502

More by Alessandro Lovato on THUR @ 2:30 pm WH3NE

46/ 62 Conclusion and Outlook II

* We are constructing a coherent picture of neutrino-nucleus interactions spanning wide rage of energy and momenta

* Many-body correlations contribute to explain the “gA-problem”

* Studies on neutrinoless double beta decay indicate a less severe “gA-problem” at moderate values of momentum transfer * Two-body physics, correlations and two-body currents, play a crucial role in the explanation of electromagnetic responses of nuclei * Microscopic Calculations of weak response indicate that two-body physics important in electroweak responses More by Alessandro Lovato on THUR @ 2:00 pm * We are addressing how to retain two-body physics in approximated Microscopic calculations of responses in A > 12 nuclei * Benchmark with existing exact calculations by Lovato et al. is excellent * All this work impacts on Fundamental Physics Quests! *

47/ 62 Outlook

ℓ′

ℓ

j† j > 0 h 1b 1bi

2 j† j vπ v > 0 h 1b 2b i ∝ h πi

48/ 62 Fundamental Physics Quests: Accelerator Neutrinos

LBNF T2K

neutrinos oscillate Neutrino-Nucleus scattering 12 CCQE on C they have→ tiny masses 8 = 7 BSM physics 6 Beyond the Standard Model ] 5 2

Simpliﬁed 2 ﬂavors picture: cm Ankowski, SF -38 4 Athar, LFG+RPA Benhar, SF

[x 10 GiBUU 2 σ 3 2 2 ∆m L Madrid, RMF P(νµ νe) = sin 2θsin Martini, LFG+RPA → 2Eν 2 Nieves, LFG+SF+RPA RFG, M =1 GeV A RFG, M =1.35 GeV 1 A * Unknown * Martini, LFG+2p2h+RPA 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ν-mass hierarchy, CP-violation, Eν [GeV] accurate mixing angles Alvarez-Ruso arXiv:1012.3871 DUNE, MiniBoone, T2K, Minerνa ... active material * 12C, 40Ar, 16O, 56Fe, ... *

49/ 62 Inclusive (e,e′) scattering: Intro to Short-Time-Approximation

* ν/e inclusive xsecs are completely speciﬁed by the response functions

* Two response functions for (e,e′) inclusive xsec

2 Rα (q,ω) = ∑δ ω + E0 Ef f Oα (q) 0 α = L,T f − |h | | i| Longitudinal response induced by OL = ρ Transverse response induced by OT = j

* Sum Rules * Exploit integral properties of the response functions + ℓ′ closure to avoid explicit calculation of the ﬁnal states ∞ q S(q,τ) = dω K(τ,ω)Rα (q,ω) 0 ℓ Z * Coulomb Sum Rules * ∞ † Sα (q) = dω Rα (q,ω) ∝ 0 Oα (q)Oα (q) 0 0 h | | i Z

50/ 62 Sum Rules and Two-body Currents: Excess Transverse Strength

T 1.5 S enhancement of the transverse 3He • 1 response is due to interference between 1b and 2b currents AND presence of 0.5 200 300 400 500 600 700 800 two-nucleon correlations q(MeV/c) •

Carlson et al. PRC65(2002)024002

j† j > 0 h 1b 1bi

2 j† j vπ v > 0 h 1b 2b i ∝ h πi

Carlson at latest INT neutrino workshop Dec. 2016

51/ 62 Recent Developments on 12C Quantum Monte Carlo Calculations of Nuclear Responses and Sum Rules

Electromagnetic Transverse Responses

✁

✞

✖✗ ✘✙ ✚

✛ ✜

✢✣ ✤✥ ✦

✧ ★ ✩✪★

✔

✫ ✬✭✮

✁ ✝

✓

✯✰✱✲✳ ✳✴✵✴ ✒

Charge-Current Cross Section ✑

✶✷ ✸ ✹✷✺ ✻✷ ✼✷

✏

✎

✌ ✄ ☎ ✄✆

12 ☞ ✍

CCQE on C ☛

✡

✠

✟ ✁ 8 ✂

7 ✁

✁ ✞

✔

✁ ✝

✓

✒

✑ ✏ ] 5 ✎

✌

✄ ☎ ✄✆

☞ ✍ cm

Ankowski, SF ☛

✡ ✠

-38 4 Athar, LFG+RPA

✟ ✁ Benhar, SF ✂

[x 10 GiBUU σ 3 Madrid, RMF

Martini, LFG+RPA ✁

✁ ✞ 2 Nieves, LFG+SF+RPA

RFG, MA=1 GeV

✔ ✁ ✝ RFG, MA=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 [GeV] ✟ ✁ ν ✂

Alvarez-Ruso arXiv:1012.3871

✁

✽ ✾ ✿ ❀ ❁ ❁ ❀ ❂ ❁ ❃ ❁ ❁ ❃ ❂ ❁ ❄ ❁❁ ❄ ❂ ❁

❅ ❆❆ ❇ ❈ ❉❊❋● q =[300 750] MeV Lovato et al. PRC91(2015)062501− + arXiv:1605.00248 100 million core hours ∼ CHALLENGE: How do we describe electroweak-scattering off A > 12 nuclei

without loosing two-body physics? 52/ 62 Factorization: Short-Time Approximation

† Rα (q,ω) = ∑δ ω + E0 Ef 0 Oα (q) f f Oα (q) 0 f − h | | ih | | i † i(H ω)t Rα (q,ω) = dt 0 Oα (q)e − Oα (q) 0 Z h | | i i(H ω)t At short time, expand P(t) = e − and keep up to 2b-terms

H ∑ti + ∑υij ∼ i i

1b 2b

ℓ ′ ℓ′

q q

ℓ ℓ

53/ 62 Factorization I: The Plane Wave Impulse Approximation - PWIA

In PWIA: ℓ′

Response functions given by incoherent scattering off q single nucleons that propagate freely in the ﬁnal state (plane waves) ℓ

† Rα (q,ω) = ∑δ ω + E0 Ef 0 Oα (q) f f Oα (q) 0 f − h | | ih | | i

(1) Oα (q) = Oα (q) = 1b i(k+q) r f e · = free single nucleon w.f. | i ∼ * PWIA Longitudinal Response in terms of the p-momentum distribution np(k) *

2 2 PWIA (k + q) k RL (q,ω) = dk np(k)δ ω + − 2m 2m Z N N

54/ 62 Factorization II: The Short-Time Approximation - STA

In STA: ℓ′ f > ∼ | Response functions are given by the scattering off q pairs of fully interacting nucleons that propagate into a ℓ correlated pair of nucleons

† Rα (q,ω) = ∑δ ω + E0 Ef 0 Oα (q) f f Oα (q) 0 f − h | | ih | | i

(1) (2) Oα (q) = Oα (q) + Oα (q) = 1b + 2b

f ψp P J M L S T M (r,R) = correlated two nucleon w.f. | i ∼ | , , , , , , , T i −

* We retain two-body physics consistently in the nuclear interactions and electroweak currents * Rα (q,ω) requires only direct calculation of g.s. 0 w.f.’s * * STA can be implemented to accommodate for more two-body| i physics, e.g., pion-production induced by e and ν

55/ 62 The Short-Time Approximation

In STA: ℓ′ f > ∼ | Response functions are given by the scattering off q pairs of fully interacting nucleons that propagate into a ℓ correlated pair of nucleons

† Rα (q,ω) = ∑δ ω + E0 Ef 0 Oα (q) f f Oα (q) 0 f − h | | ih | | i (1) (2) Oα (q) = Oα (q) + Oα (q) = 1b + 2b

ip r iP R f is a function of p and P e.g. for free propagator e · e · | i ∼ f ψp P J M L S T M (r,R) = correlated two nucleon w.f. | i ∼ | , , , , , , , T i −

2 2 † Rα (q,ω) δ ω + E E dΩP dΩp dPdp p P 0 O (q) p,P p,P Oα (q) 0 ∼ 0 − f h | α | ih | | i Z h i

56/ 62 The Short-Time Approximation

transverse 1b transverse interference 1b*2b

1b - 600 Mev 1b*2b - 600 MeV

30 25 350 20 300 15 250 200 10 150 5 100 0 50 -5 0 -50

5 5 4 4 3 0 relative momentum p 1 3 2 0 1 3 2 1 2 2 relative momentum P 0 5 4 3 1 total momentum P relative momentum p 4 5 0

300 core hours with 1b + 2b for 4He

*Preliminary results*

57/ 62 1b vs 1b+2b

Longitudinal @ q=500 MeV - 1b vs 1b+2b 0.014 gfmc-Lovato 1b+2b STA 1b STA 1b+2b 0.012

0.01

0.008

0.006 R 1/MeV

0.004

0.002

0 0 100 200 300 400 500 600 omega MeV Transverse @ q=500 MeV - 1b vs 1b+2b 0.03 gfmc-Lovato 1b+2b STA 1b STA 1b+2b 0.025

0.02

0.015 R 1/MeV 0.01

0.005

0 0 100 200 300 400 500 600 omega MeV

58/ 62 The Short-Time Approximation

6 4 He World's data 5 LIT, Bacca et al. (2009) AV18+UIX GFMC, Lovato et al. (2015) STA, Pastore et al. PRELIMINARY PWIA ]

-3 4 10 -1 3 [MeV L

R 2

0 0 100 200 300 400 500 ω [MeV]

Longitudinal Response function at q = 500 MeV

59/ 62 The Short-Time Approximation

0.03 4 He GFMC Longitudinal, Lovato et al. (2015) 0.025 STA Longitudinal, PRELIMINARY AV18+UIX GFMC Transverse, Lovato et al. (2015) STA Transverse, PRELIMINARY ]

-1 0.02 [MeV

2 0.015 p /G

L/T 0.01 R

0.005

0 0 100 200 300 400 500 ω [MeV]

Longitudinal vs Transverse Response Function at q = 500 MeV

60/ 62 Conclusion and Outlook III

* We are constructing a coherent picture of neutrino-nucleus interactions spanning wide rage of energy and momenta

* Many-body correlations contribute to explain the “gA-problem”

* Studies on neutrinoless double beta decay indicate a less severe “gA-problem” at moderate values of momentum transfer * Two-body physics, correlations and two-body currents, play a crucial role in the explanation of electromagnetic responses of nuclei * Microscopic Calculations of weak response indicate that two-body physics important in electroweak responses More by Alessandro Lovato on THUR @ 2:00 pm * We are addressing how to retain two-body physics in approximated Microscopic calculations of responses in A > 12 nuclei with the Short Time Approximation * Benchmark with existing exact calculations by Lovato et al. is excellent * All this work impacts on Fundamental Physics Quests! *

61/ 62 Outlook

ℓ′

ℓ

j† j > 0 h 1b 1bi

2 j† j vπ v > 0 h 1b 2b i ∝ h πi

62/ 62 Institute for Nuclear Theory (INT) Program - Seattle - Summer 2018

Fundamental Physics with Electroweak Probes of Light Nuclei June 12 - July 13, 2018 S. Bacca, R. J. Hill, S. Pastore, D. Phillips

Contacts http://www.int.washington.edu/ [email protected] [email protected]

63/ 62