The basic model of nuclear theoryMajor goal: Describe nuclear systems (such as atomic nuclei and infinite matter) from a microscopic point of view. In such model the nucleus’ constituents— the nucleons (N), i.e., protons (p) and neutrons (n) interact with each other in terms of many-body (primarily, two and three-body) effective interactions, and with external electroweak probes via effective currents describing the coupling of these probes to individual nucleons and many-body clusters of them

Inelastic lepton-nucleus scattering

Beta-decays and Gamow-Teller matrix elements in light-nuclei

2

into account. These two-body operators, multiplied byhadronic form factors so as to regularize their short-rangebehavior in configuration space, were then constrained toreproduce the GT matrix element contributing to tritium� decay by adjusting the poorly known N -to-� axial cou-pling constant (see Ref. [19] for a recent summary).

Yet, the calculations of Ref. [11] were based on ap-proximate VMC wave functions to describe the nuclearstates involved in the transitions. This shortcoming wasremedied in the subsequent GFMC study of Ref. [12],which, however, only retained the one-body GT opera-tor. Adding to the GFMC-calculated one-body matrixelements the VMC estimates of two-body contributionsobtained in Ref. [11] led Pervin et al. [12] to speculatethat a full GFMC calculation of these A=6–7 weak tran-sitions might be in agreement with the measured values.

The last three decades have witnessed the emergenceof chiral e↵ective field theory (�EFT) [20]. In �EFT,the symmetries of quantum chromodynamics (QCD), inparticular its approximate chiral symmetry, are used tosystematically constrain classes of Lagrangians describ-ing, at low energies, the interactions of nucleons and �isobars with pions as well as the interactions of thesehadrons with electroweak fields [21, 22]. Thus �EFTprovides a direct link between QCD and its symmetries,on one side, and the strong and electroweak interac-tions in nuclei, on the other. Germane to the subjectof the present letter are, in particular, the recent �EFTderivations up to one loop of nuclear axial currents re-ported in Refs. [23, 24]. Both these studies were based ontime-ordered perturbation theory and a power-countingscheme a la Weinberg, but adopted di↵erent prescrip-tions for isolating non-iterative terms in reducible contri-butions. There are di↵erences—the origin of which is yetunresolved—in the loop corrections associated with boxdiagrams in these two independent derivations.

The present study reports on VMC and GFMC calcu-lations of weak transitions in 6He, 7Be, and 10C, based onthe Argonne v18 (AV18) two-nucleon [25] and Illinois-7(IL7) three-nucleon [26] interactions, and axial currentsobtained either in the meson-exchange [19] or �EFT [23]frameworks mentioned earlier. The AV18+IL7 Hamilto-nian reproduces well the observed spectra of light nuclei(A=3–12), including the 12C ground- and Hoyle-stateenergies [3]. The meson-exchange model for the nuclearaxial current has been most recently reviewed in Ref. [19],where explicit expressions for the various one-body (1b)and two-body (2b) operators are also listed (including fit-ted values of the N -to-� axial coupling constant). The�EFT axial current [23, 27] consists of 1b, 2b, and three-body (3b) operators. The 1b operators read

j1b5,± = �gA

AX

i=1

⌧i,±

✓�i �

ri �i ·ri � �i r2

i

2m2

◆, (1)

where ⌧i,± = (⌧i,x ± i ⌧i,y)/2 is the standard isospin rais-

ing (+) or lowering (�) operator, and �i and �iri are,respectively, the Pauli spin matrix and momentum oper-ator of nucleon i. The 2b and 3b operators are illustrateddiagrammatically in Fig. 1 in the limit of vanishing mo-mentum transfer considered here. Referring to Fig. 1,the 2b operators are from contact [CT, panel (a)], one-pion exchange (OPE) [panels (b) and (f)], and multi-pionexchange (MPE) [panels (c)-(e) and (g)],

j2b5,± =

AX

i<j=1

hjCT

5,±(ij) + jOPE

5,± (ij) + jMPE

5,± (ij)i, (2)

and the 3b operators are from MPE [panels (h)-(i)],

j3b5,± =

AX

i<j<k=1

jMPE

5,± (ijk) . (3)

Configuration-space expressions for these 2b and 3b op-erators are reported in Ref. [27].

FIG. 1. Diagrams illustrating the (non-vanishing) contribu-tions to the 2b and 3b axial currents. Nucleons, pions, andexternal fields are denoted by solid, dashed and wavy lines,respectively. The circle in panel (b) represents the vertex im-

plied by the L(2)⇡N chiral Lagrangian [28], involving the LECs

c3 and c4. Only a single time ordering is shown; in particular,all direct- and crossed-box diagrams are accounted for. Thepower counting of the various contributions is also indicated.See text for further explanations.

The 1b operator in Eq. (1) includes the leading or-der (LO) GT term and the first non-vanishing correc-tions to it, which come in at next-to-next-to-leading or-der (N2LO) [27]. Long-range 2b corrections from OPEenter at N3LO, panel (b) in Fig. 1, involving the low-energy constants (LECs) c3 and c4 in the sub-leading

L(2)

⇡N chiral Lagrangian [28], as well as at N4LO, panel (f).In terms of the expansion parameter Q/⇤�—where Qspecifies generically the low-momentum scale and ⇤�=1GeV is the chiral-symmetry-breaking scale—they scale as(Q/⇤�)3 and (Q/⇤�)4, respectively, relative to the LO.Loop corrections from MPE, panels (c)-(e) and (g), comein at N4LO, as do 3b currents, panels (h)-(i). Finally, thecontact 2b current at N3LO, panel (a), is proportional toa LEC, denoted as z0.The short-range behavior of the 2b and 3b operators

is regularized by including a cuto↵ C⇤(k)= exp(�k4/⇤4)

Objectives: • Compute inelastic electron-nucleus scattering for

which accurate experimental data are available • Within the same formalism, study the neutral-current

and charge-current response functions that are important inputs for neutrino-nucleus scattering

• Test electroweak current operators, including two-body operators

Accomplishments: • Development of an algorithm to compute the

Euclidean response functions within GFMC • Using Maximum Entropy technique to obtain

the response functions from the Euclidean response

Basic model

Chiral 2Ninteractions

Chiral 3Ninteractions

EWKinteractions

EWK QEresponse

Outlook

GFMC calculation of EM response in 12CCarlson and Schiavilla (1992); Lovato et al. (2013–2016)

Z 1

0

d! e�⌧!

R↵�(q,!)=hi | j†↵(q) e�⌧(H�Ei) j�(q) | ii

Inversion back to R↵�(q,!) by maximum entropymethods

longitudinal transverse

yXyyy

yXyy8

yXyRy

yXyR8

yXyky

yXyk8

yXyjy

yXyj8

y 8y Ryy R8y kyy k8y jyy j8y 9yy

ԇ ЈЈ�Ӹϵ Ӵզ

ᆂ(J2o)

[־[ � ��� J2of+:6J* Ԅφս�ϵս:6J* ԄφսqQ`H/ /�i�

yXyy

yXyk

yXy9

yXye

yXy3

y 8y Ryy R8y kyy k8y jyy j8y 9yy

ԇ �Ӹϵ Ӵզ

ᆂ(J2o)

[־[ � ��� J2of+:6J* Ԅφս�ϵս:6J* ԄφսqQ`H/ /�i�

yXy

yXk

yX9

yXe

yX3

RXy

RXk

yXy yX8 RXy RX8 kXy kX8 jXy jX8 9Xy

Ԉ խԠ

Ԡ (7Kφ)

:6J* Ԅφս�ϵս1tT@h_1tT1tT@h66

Coulomb sum

• EM longitudinal (right panel) and transverse (left panel) response function of 12C for q=570 MeV/c with only one-body currents (red) and also two-body currents (black) are compared with experiment data (blue)

Objectives:

Accomplishments: • GFMC calculations of Gamow-Teller matrix

elements (right panel) for light-nuclei at lowest-order theory of one-body currents (blue) using AV18+IL7

• Including two-body currents (magenta), there is an overall improvement of the theoretical predictions compared to the experimental data (black): some issues with 10C

• Understand weak properties and transition rates of light-nuclei

• Test nuclear interactions and weak current operators, including complete two-body currents up to one loop (left panel)

• Address long-standing problem “quenching” of required from shell-calculations: this is relevant for neutrinoless double-beta decay

gA

Larger nuclei and infinite neutron matter

• Validate theoretical framework for nuclear forces and current operators in heavier nuclei, including neutron-rich nuclei important in the r-process, and infinite nuclear matter

Objectives:

Accomplishments:

Currents: many-body effective electroweak operators

Inputs for the basic model:

Hamiltonian: many-body effective interactions between the nucleons (N) H =

AX

i=1

p2i

2mi+

AX

i<j=1

vij +AX

i<j<k=1

Vijk + .....

Kinetic energy Two-body Three-body

jEW =AX

i=1

ji +AX

i<j=1

jij +AX

i<j<k=1

jijk + ....

One-body Two-body Three-body

Theoretical approaches• Phenomenological: two-body Argonne V18 (AV18) + three-body Illinois-7 (IL7)

• Chiral effective field theory (𝝌EFT): two- and three-body local chiral interactions

Objectives:

Accomplishments:

Objectives:

Spectra of light-nuclei A≤12

• Study the spectra of light-nuclei: theory confronts experiment

• Validate theoretical framework used to derive nuclear interactions

• GFMC calculations of the spectra of nuclei up to A=12 using Delta-full local chiral interactions (red) compared with the ones obtained using AV18+IL7 (cyan) and experimental data (green)

Magnetic moments and electromagnetic decays in light-nuclei

• Understand electromagnetic properties and transition rates of light-nuclei

• Test nuclear interactions and electromagnetic current operators, including complete two-body currents

Accomplishments: • GFMC calculations of magnetic moments (right panel)

and electromagnetic decays (left panel) using AV18+IL7 at lowest-order theory of one-body currents (blue) disagrees with experiment (black)

• Including two-body currents based on effective field theory (red) improves all predictions

EM transitions:

Basic model

Chiral 2Ninteractions

Chiral 3Ninteractions

EWKinteractions

EWK QEresponse

Outlook

Magnetic moments in A 10 nucleiPastore et al. (2013)

GFMC calculations use AV18/IL7 (rather than chiral)potentials with �EFT EM currentsPredictions for A > 3; about 40% of µ(9C) due tocorrections beyond LO

-3

-2

-1

0

1

2

3

4

µ (µ

N)

EXPT

GFMC(IA)GFMC(FULL)

n

p

2H

3H

3He

6Li

6Li*

7Li

7Be

8Li 8B

9Li

9Be

9B

9C

10B

10B*

Magnetic moments: • Realistic two- and three-body interactions based AV18+IL7 and 𝝌EFT have been used in AFDMC to

study properties of selected close shell nuclei up to A=16 and EoS pure neutron matter (left panel) • Realistic two- and three-body interactions based AV18+IL7 have been used in CVMC to validate the

computational approach in 16O and then used to study properties of 40Ca (right panel)

Argonne National Laboratory (ANL)

Los Alamos National Laboratory (LANL)

University of South Carolina (USC)

Old Dominion University & Jefferson Lab (ODU & Jlab)

Maria Piarulli, Alessandro Lovato, Ewing Lusk, Steven Pieper, Stefan Wild, Robert Wiringa Stefano Gandolfi, Diego Lonardoni, Saori Pastore, Joseph Carlson

Alessandro Baroni Rocco Schiavilla

1 1.1 1.2

Ratio to EXPT

10C 10B

7Be 7Li(gs)

6He 6Li

3H 3He

7Be 7Li(ex)

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

Quantum Monte Carlo methodsPhysical problem: Calculate the structure and reactions of light-nuclei including spectra, form factors, transitions, low-energy scattering and response as well as equation of state (EoS) of infinite matter. Compare the theoretical results with the experimental data

Numerical method: Use Quantum Monte Carlo (QMC) methods to solve the many-body Schrödinger equation for two- and three-body interactions

• (Cluster)Variational Monte Carlo, (C)VMC • Green’s functions Monte Carlo, GFMC • Auxiliary Field Diffusion Monte Carlo, AFDMC

H (R; s1, .., sA; t1, .., tA) = E (R; s1, .., sA; t1, .., tA)

3A coordinates in r-space Nucleon spin Nucleon isospin (p or n)

Figure by Diego Lonardoni LANL

3Quantum Monte Carlo methods

Goal: solve the many-body problem for correlated systems in a non perturbative fashion

GFMCVMC AFDMCCVMC

A

Pros:‣ Work with bare interactions. ‣ Good for strongly correlated systems. ‣ Stochastic method: errors quantifiable and systematically improvable.

Cons:‣ Some limitations in A and/or in the interaction to be used.

minimizationE ! E0E � E0

� ⇠ 1/pN

⌧ propagation AFDMC

CVMC AFDMC

(C)VMC GFMC AFDMC

light systems A 12

A ⇠ 50

A ! 1infinite matter

light to medium- heavy nuclei

12

40

The basic model of nuclear theoryThe basic model of nuclear theory: achieving a comprehensive description of the wealth of data and peculiarities exhibited by nuclear systems

Nucleon-nucleon (NN) and 3N scattering data; Spectra, properties, and transition of nuclei; Nucleonic matter equation of state; ……

Electroweak current operators: jEW =

AX

i=1

ji +AX

i<j=1

jij +AX

i<j<k=1

jijk + ....

One-body Two-body Many-body

The Basic Model! The nucleus is a system made of A interacting nucleons, its energy is given by

H = T+V =A

∑i=1

ti+∑i<jυij+ ∑

i<j<kVijk+ ...

where υij and Vijk are 2- and 3-nucleon interaction operators

! Current and charge operators describe the interaction of nuclei with externalfields. They are expanded as a sum of 1−, 2−, ... nucleon operators:

ρ =A

∑i=1

ρi+∑i<jρij+ ... , j=

A

∑i=1ji+∑

i<jjij+ ...

q+ . . .

N N

γ

! EM current operator j satisfies the current conservation relation (CCR) with thenuclear Hamiltonian, hence V, ρ , j need to be derived consistently

q · j= [H, ρ ]CCR does not constrain transverse (orthogonal to q) currents

3 / 22

N

N

N

N

!, W± , Z

The Basic Model! The nucleus is a system made of A interacting nucleons, its energy is given by

H = T+V =A

∑i=1

ti+∑i<jυij+ ∑

i<j<kVijk+ ...

where υij and Vijk are 2- and 3-nucleon interaction operators

! Current and charge operators describe the interaction of nuclei with externalfields. They are expanded as a sum of 1−, 2−, ... nucleon operators:

ρ =A

∑i=1

ρi+∑i<jρij+ ... , j=

A

∑i=1ji+∑

i<jjij+ ...

q+ . . .

N N

γ

! EM current operator j satisfies the current conservation relation (CCR) with thenuclear Hamiltonian, hence V, ρ , j need to be derived consistently

q · j= [H, ρ ]CCR does not constrain transverse (orthogonal to q) currents

3 / 22

!, W± , Z

N N

NN N

N

The Basic Model! The nucleus is a system made of A interacting nucleons, its energy is given by

H = T+V =A

∑i=1

ti+∑i<jυij+ ∑

i<j<kVijk+ ...

where υij and Vijk are 2- and 3-nucleon interaction operators

! Current and charge operators describe the interaction of nuclei with externalfields. They are expanded as a sum of 1−, 2−, ... nucleon operators:

ρ =A

∑i=1

ρi+∑i<jρij+ ... , j=

A

∑i=1ji+∑

i<jjij+ ...

q+ . . .

N N

γ

! EM current operator j satisfies the current conservation relation (CCR) with thenuclear Hamiltonian, hence V, ρ , j need to be derived consistently

q · j= [H, ρ ]CCR does not constrain transverse (orthogonal to q) currents

3 / 22

N N

NN

!, W± , Z + + + ….

th+expInputs for the basic model:

Many-body interactions between the constituents

One-body Two-body (NN) Three-body (3N)

H =AX

i=1

p2i

2mi+

AX

i<j=1

vij +AX

i<j<k=1

Vijk + .....

N

N

N

N N N

NN N

N

+ + ….

{ th+exp{

- local chiral two-body interactions contains parameters called low-energy constants (LECs) that are fixed by fitting NN scattering data

- the fits to data of some of these local interactions have been performed by using POUNDERS, a SciDAC-supported derivative-free optimization solver available in PETSc libraries

• Comparison with “unquenched” shell model calculations (green) based on the inclusion of one-body current operators: important role of correlation in the wave functions

PROS:

• Keeping track of every nucleon’s spin and isospin states: in GFMC these states are summed explicitly, while in AFDMC they are sampled. GFMC is more appropriate for light-nuclei, AFDMC for heavy nuclei and matter

• QMC does not require a basis expansion or fixed grids and works with bare interactions

CONS:

▪ADLB API: – ADLB_Put – ADLB_Reserve – ADLB_Get_reserved

▪Arguments allow construction of many algorithms – E.g. multiple parallel loops across the whole machine

▪Performance: – Scalable – supports GFMC on largest machines – Flexible – work units from milliseconds to seconds

Application ProcessesADLB Servers

Computer Science impact via SciDAC on GFMC:

ADLB and DMEM rely on MPI for interaddress space communication

Ab-initio Nuclear Structure and Nuclear Reactions Gaute Hagen

initio calculations of important scattering and reactions involving ↵-particles practical. The method is em-barrasingly parallel and the code, NLEFT is fully implemented on GPU’s and thus requires very little datamovement between device and host memory. Checkpoint/restart is implemented for fault tolerance and forrunning longer jobs.

3.3 Parallel Performance

0 10 20 30 40 50 60 7030.0

50.0

100.

300.

500.

1000

Total number of threadsTi

me/

Sam

ple

(sec

onds

)

1 rank 2 ranks 4 ranks 6 ranks10 ranks 1 rank, BG/QIdeal

A) GFMC Strong OpenMP scalingon KNL node

(a)

10 40 100 400 1000 400024

25

26

27

28

29

30

Number of nodes (6 MPI ranks/node, 10 threads/rank)

Tim

e (m

inut

es)

B) GFMC Weak MPI scaling on Theta11 Monte Carlo samples/node

(b)

256 1,024 4,096 16,384 65,536 262,1440

10

20

30

40

Number of MPI ranks

Tim

e (m

inut

es)

ActualIdeal

C) 12C − GFMC+ADLB − BG/Q Weak scaling, 2 configs/rank

(c)

2,048 4,096 8,192 16,384 32,768number of nodes

1

2

4

8

16

spee

du

p

ideal scalingmatrix constructiondiagonalization

Mira (using all 64 threads per node)

(d)

1 2 4 8 16 32 64 128 256OMP scaling

1

2

4

8

16

32

64

128

256

spee

du

p

ideal scalingmatrix constructionSpMM on 8 vectors

Intel KNL (quad-flat mode)

(e)

10000 15000 20000 25000 300001.0

1.5

2.0

2.5

3.0

3.5

4.0

number of cores

relativespeedup

(f)

(g)

0

100

200

300

400

500

0 10000 20000 30000 40000 50000 60000 70000

time for

100 u

pdate

s (s

eco

nds)

number of Titan ranks

CPU only, L = 16CPU only, L = 20

CPU and GPU, L = 16CPU and GPU, L = 20

(h)

101

102

103

104

# Nodes

101

102

103

104

Tim

e t

o s

olu

tio

n

CPU

CPU+GPU

(i)Figure 4. Scaling plots on relevant systems for all applications used in this proposal.

GFMC. Fig. 4a shows good strong OpenMP scaling on a Theta node with 1 to 10 MPI ranks running onthe node. Fig. 4b shows good weak scaling up to 3168 nodes of nodes of Theta. Six ranks with 10 threadseach were used on each node. Finally, Fig. 4c shows good weak MPI scaling on Mira using 8 ranks/nodeand 6 OpenMP threads per rank.

MFDn. Fig. 4d shows good strong scaling on up to 32,768 nodes (using 64 OMP threads per node) on Mirafor the two most time-consuming parts of MFDn. Fig. 4e shows good strong scaling up to 64 threads on aKNL node, with slight performance improvement from utilizing the hardware threads. Runs were performedin quad-flat mode, with careful placement of the eigenvectors on high-bandwidth memory (HBM).

NCSMC. Fig. 4f shows good strong scaling up to 35,000 cores on Titan using 2 ranks per node 8 threadsper rank without GPU’s for the NCSM-RGM module of NCSMC.

NUCCOR. Fig. 4g shows the strong scaling of a full application run to compute the ground-state in 48Cawith iterative triples excitations, using up to 1024 nodes on Summit on a log-log scale. We achieve morethan 4 times speedup going from an optimzed CPU-only version to a GPU version. The benchmark run was

13

GFMC is a hybrid code using both MPI and OpenMP parallelism• The Asynchronous Dynamic Load Balancing (ADLB):

• Distributed Memory (DMEM):

- library that implements a flexible and scalable scheduling and load-balancing system for work units of varying types, sizes, and priorities

- manages work sharing without the bottleneck of a single master controller

- carries out memory load balancing by storing large arrays on any node with enough memory and subsequently fetching them when needed

- manages memory on all clients. Runs as separate thread, sharing memory with application processes, so local operations are fast: great improved performance of ADLB in GFMC

▪DMEM API: – DMEM_Put – DMEM_Get – DMEM_Free – DMEM_Copy – DMEM_Update

The basic model of nuclear theory: from atomic nuclei to infinite matter