FRIB: Opening New Frontiers in Nuclear Science bands James ...quantum properties of nuclei depend on the intricate interplay of strong, weak, and electromagnetic ... R = 0.9 fm, no

Topical Collaboration on Neutrinos and Fundamental

Symmetries

No Core Shell Model (NCSM)With Consistent Electroweak Interactions and

Other Recent Developments James P. Vary, Iowa State UniversityTRIUMF Workshop, Feb 26 - Mar 1, 2019

FRIB: Opening New Frontiers in Nuclear Science

2

Achieving this goal involves developing predictive theoretical models that allow us to understand the emergent phenomena associated with small-scale many-body quantum systems of finite size. The detailed quantum properties of nuclei depend on the intricate interplay of strong, weak, and electromagnetic interactions of nucleons and ultimately their quark and gluon constituents. A predictive theoretical description of nuclear properties requires an accurate solution of the nuclear many-body quantum problem — a formidable challenge that, even with the advent of super-computers, requires simplifying model assumptions with unknown model parameters that must be constrained by experimental observations.

Fundamental to Understanding

The importance of rare isotopes to the field of low-energy nuclear science has been demonstrated by the dramatic advancement in our understanding of nuclear matter over the past twenty years. We now recognize, for example, that long-standing tenets such as magic numbers are useful approximations for stable and near stable nuclei, but they may offer little to no predictive power for rare isotopes. Recent experiments with rare isotopes have shown other deficiencies and led to new insights for model extensions, such as multi-nucleon interactions, coupling to the continuum, and the role of the tensor force in nuclei. Our current understanding has benefited from technological improvements in experimental equipment and accelerators that have expanded the range of available isotopes and allow experiments to be performed with only a few atoms. Concurrent improvements in theoretical approaches and computational science have led to a more detailed understanding and pointed us in the direction for future advances.

We are now positioned to take advantage of these developments, but are still lacking access to beams of the most critical rare isotopes. To advance our understanding further low-energy nuclear science needs timely completion of a new, more powerful experimental facility: the Facility for Rare Isotope Beams (FRIB). With FRIB, the field will have a clear path to achieve its overall scientific goals and answer the overarching questions stated above. Furthermore, FRIB will make possible the measurement of a majority of key nuclear reactions to produce a quantitative understanding of the nuclear properties and processes leading to the chemical history of the universe. FRIB will enable the U.S. nuclear science community to lead in this fast-evolving field.

Figure 1: FRIB will yield answers to fundamental questions by exploration of the nuclear landscape and help unravel the history of the universe from the first seconds of the Big Bang to the present.

Progress in Ab Initio Techniques in Nuclear Physics, Feb. 2015

Emergence of rotational bands

in light nuclei from No-Core CI calculationsPieter Maris

[email protected]

Iowa State University

SciDAC project – NUCLEI

lead PI: Joe Carlson (LANL)

http://computingnuclei.org

PetaApps award

lead PI: Jerry Draayer (LSU)

INCITE award – Computational Nuclear Structure

lead PI: James P Vary (ISU)

NERSC

Progress in Ab Initio Techniques in Nuclear Physics, Feb. 2015, TRIUMF, Vancouver – p. 1/50

Magnetic moment operator

I The low energy constants d9 are L2 are determined by fitting to data.In this work we have used d9 = 0.001 GeV2 and L2 = 0.113 GeV4.

0 50 100 150 200Nmax0.0

0.5

1.0

1.5

2.0

N

Magnetic Moment

LENPIC-N2LO

=20 MeV

Order (N)bare 0.8565064

bare + 2BC 0.8894164



0 50 100 150 200Nmax0.0

0.5

1.0

1.5

2.0

N

Magnetic Moment

LENPIC-N2LO

=20 MeV


bare + 2BC 0.8894164



0 50 100 150 200Nmax0.0

0.5

1.0

1.5

2.0

N

Magnetic Moment

LENPIC-N2LO

=20 MeV


bare + 2BC 0.8894164



0 50 100 150 200Nmax0.0

0.5

1.0

1.5

2.0

N

Magnetic Moment

LENPIC-N2LO

=20 MeV


bare + 2BC 0.8894164



0 50 100 150 200Nmax0.0

0.5

1.0

1.5

2.0

N

Magnetic Moment

LENPIC-N2LO

=20 MeV


bare + 2BC 0.8894164

No-Core Configuration Interaction calculations

Barrett, Navrátil, Vary, Ab initio no-core shell model, PPNP69, 131 (2013)

Given a Hamiltonian operator

H =!

i<j

(pi − pj)2

2mA+!

i<j

Vij +!

i<j<k

Vijk + . . .

solve the eigenvalue problem for wavefunction of A nucleons

HΨ(r1, . . . , rA) = λΨ(r1, . . . , rA)

Expand wavefunction in basis states |Ψ⟩ ="

ai|Φi⟩

Diagonalize Hamiltonian matrix Hij = ⟨Φj |H|Φi⟩

No-Core CI: all A nucleons are treated the same

Complete basis −→ exact result

In practice

truncate basis

study behavior of observables as function of truncation

Progress in Ab Initio Techniques in Nuclear Physics, Feb. 2015, TRIUMF, Vancouver – p. 2/50

Expand eigenstates in basis states

No Core Full Configuration (NCFC) – All A nucleons treated equally

Effective Nucleon Interaction(Chiral Perturbation Theory)

R. Machleidt, D. R. Entem, nucl-th/0503025

Chiral perturbation theory (χPT) allows for controlled power series expansion

Expansion parameter : QΛχ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

υ

, Q−momentum transfer,

Λχ ≈1 GeV , χ - symmetry breaking scale

Within χPT 2π-NNN Low Energy Constants (LEC) are related to the NN-interaction LECs ci.

Terms suggested within theChiral Perturbation Theory

Regularization is essential, which is also implicit within the Harmonic Oscillator (HO) wave function basis

CD CE

R. Machleidt and D.R. Entem, Phys. Rep. 503, 1 (2011);E. Epelbaum, H. Krebs, U.-G Meissner, Eur. Phys. J. A51, 53 (2015); Phys. Rev. Lett. 115, 122301 (2015)

Low Energy Nuclear Physics International Collaboration

E. Epelbaum, H. Krebs

A. Nogga

P. Maris, J. Vary

J. Golak, R. Skibinski, K. Tolponicki, H. Witala

S. Binder, A. Calci, K. Hebeler,J. Langhammer, R. Roth

R. Furnstahl

H. Kamada

Calculation of three-body forces at N3LO

Goal

Calculate matrix elements of 3NF in a partial-

wave decomposed form which is suitable for

different few- and many-body frameworks

Challenge

Due to the large number of matrix elements,

the calculation is extremely expensive.

Strategy

Develop an efficient code which allows to

treat arbitrary local 3N interactions.

(Krebs and Hebeler)

U.-G Meissner

Additional Goal: Develop consistent chiral EFT theory for electroweak operators

LENPIC NN + 3NFs at N2LO (E. Epelbaum, et al., PRC 99, 024314 (2019); arXiv:1807.02848)

f rR⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟= 1−exp − r2

R2⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟

6

Coordinate space regulator:

-3

-2

-1

0

-5 0 5 10 15

c E

cD

R = 0.9 fmR = 1.0 fm

cD/cE correlation fromfitting 3He gs Energy

nd scattering length 2a

nd σtot at 70 MeV

pd minimum of dσ/dθ at 70 MeV

nd σtot at 108 MeV

nd σtot at 135 MeV




nd σtot at 70 MeV (derived data, assuming 1% sys. err.)

pd minimum of dσ/dθ at 70 MeV (assuming 2% sys. err.)




Determination of cD at N2LO, R = 0.9 fm, no additional cutoff

0-5 0 5 10 15

cD


all data: cD = 1.7±0.8; χ2DOF = 0.7

data up to 108 MeV: cD = 2.1±0.9; χ2DOF = 0.5







Determination of cD at N2LO, R = 1.0 fm, no additional cutoff


0-5 0 5 10 15 20

cD

all data: cD = 7.2±0.7; χ2DOF = 0.7

data up to 108 MeV: cD = 7.2±0.9; χ2DOF = 0.4

R = 0.9 fm R = 1.0 fm

cD (cE from 3He gs energy correlation)

R = 0.9 fm

R = 1.0 fm

R = 0.9 fm

R = 1.0 fm

cD (cE) selected via ND scattering analyses:

0

-1

-2

-3

c E

-5 0 5 10 15cD

+ +

cD/cE from fitting ND scatt.+

4He

6He

6Li

7Li

8He

8Li

8Be

9Li

9Be

10Be

10B

11B

12B

12C

-100

-90

-80

-70

-60

-50

-40

-30

-20

Gro

un

d s

tate

en

erg

y (

MeV

)

(0+, 0)

(0+, 1)

(0+, 2)

(0+, 0)

(1+, 1)

(2+, 1)

(3/2-, 1/2)

(3/2-, 3/2)

(3/2-, 1/2)

(3/2-, 1/2)

(3+, 0)

(0+, 0)

(1+, 0)

(0+, 1) chiral EFT interaction with

semilocal regulator R = 1.0 fm

(Jp, T)

LO, NLO, and N2LO

N2LO without 3N forces

Expt. values

16O

-180

-160

-140

-120

LO, NLO, and N2LO


Expt. values

LENPIC NN + 3NFs at N2LO (E. Epelbaum, et al., PRC 99, 024314 (2019); arXiv:1807.02848)

Grey bands – chiral perturb. theory uncertaintyBlue bars – extrapolation and SRG evolution

uncertainties combined

LENPIC NN + 3NFs at N2LOE. Epelbaum, et al., PRC 99, 024314 (2019); arXiv:1807.02848

6Li

7Li

8Li

8Be

9Li

9Be

10Be

10B

11B

12B

12C

-1

0

1

2

3

4

5

6

7

8

9

Ex

cita

tio

n e

ner

gy

(M

eV)


N2LO with 3N forces

Expt. values

12C

10

12

14

1/2-

5/2-

2+

1+

2+

1+

1+

2+

1/2-

3+

2+

1+

3+

7/2-5/2

-

5/2-

2+

2+

4+

3+

4+

2+

2+

0+

1/2-

5/2-

7/2- 3/2

-

1+

2+

0+

Jp

UHU † =U[T +V ]U † = Hd , the diagonalized HH eff ≡UOLSHUOLS

† = PH effP= P[T +Veff ]PUP ≡ PUP

!UP ≡ P !UPP≡ UP

UP†UP

H eff = !UP†Hd!UP = !UP†UHU † !UP = P[T +Veff ]P

We conclude that:UOLS = !UP†USimilarly, we have effective operators for observables:Oeff ≡ !U

P†UOU † !UP = P[Oeff ]P

W P ≡ PUP

!UP ≡ P !UPP≡ W P

W P†W P

OLS Transform:Unitary transformationthat block-diagonalizesthe Hamiltonian – i.e. itintegrates out Q-spacedegrees of freedom.

H

PHeffP

QHeffQ

P Q

P

Q

Nmax

PHeffQ=0

QHeffP=0

Consistent observables

See: J.P. Vary, et al.,PRC98, 065502 (2018) arXiv:1809.00276for applications

Consider two nucleons as a model problem with V = LENPIC chiral NNsolved in the harmonic oscillator basis with ħΩ = 5, 10 and 20 MeV.Also, consider the role of an added harmonic oscillator quasipotential

Hamiltonian #1

Hamiltonian #2

Hamiltonian #3

Other observables:Root mean square radius RMagnetic dipole operator M1Electric dipole operator E1Electric quadrupole moment QElectric quadrupole transition E2Gamow-Teller GTNeutrinoless double-beta decay M(0ν2β)

Dimension of the “full space” is 400 for all results depicted here

H =T +VH =T +Uosc (!Ωbasis )+VH =T +Uosc (!Ω =10MeV)+V

J.P. Vary, et al., PRC98, 065502 (2018);arXiv:1809.00276

We initially considered a 2-body contribution within EFT to 0νββ-decay at N2LO

n

n

π-

π-

e-

e-

p

p

G. Prézeau, M. Ramsey-Musolf and P. Vogel, Phys. Rev. D 68, 034016 (2003)

Iowa State work-in-progress – take 0νββ-decay operators, term-by-term, available from V. Cirigliano, W. Dekens, J. de Vries, M.L. Graesser and E. Mereghetti, arXiv:1806.02780; calculate them in the harmonic oscillator basis and use LENPICLECs for NCSM apps with LENPIC interactions.

, x= mπ!r

This operator is used in: J.P. Vary, et al., PRC98, 065502 (2018); Jon Engel’s operator used in: ISU-UNC-MSU Collaboration benchmarkingNCSM and MR-IMSRG (paper in preparation)

ISU – UNC – MSU collaboration to benchmark 0νββ-decay matrix elements for 6He -> 6Be

5 10 15 20Nmax

5

4

3

2

1

0

1M0ν (a)

0νββ Total ( M0ν)

0 4 8 125.0

4.6

4.2M0νFM0νGTM0νTM0ν

0 5 10 15 20Nmax6

5

4

3

2

1

M0ν (b)Fermi

0 4 8 121.05

1.00

0.95

0.90

0.85

M0νF (nn)M0νF (pp)M0νF (pn)M0νF

0 5 10 15 20Nmax

-10

-8

-6

-4

M0ν (c)Gamow-Teller

0 4 8 12-4.0

-3.8

-3.6

-3.4

M0νGT(nn)M0νGT(pp)M0νGT(pn)M0νGT

5 10 15 20Nmax

-0.02

0.00

0.02

0.04

0.06

M0ν (d)Tensor

M0νT (nn)M0νT (pp)M0νT (pn)M0νT

NCSM MR-IMSRG

0 6 ! 6

Nmax

0

emax = 4

Nmax = 6

Nmax

0Nmax

6 ! 60

P. Gysbers, et al., see poster

4% diff

LENPIC collaboration (in process) – adopts momentum space regulators

Note: we are working to retain dependence on external momentum transfer

Nuclear Axial Current Operators e.g. Krebs, et al., Ann. Phys. 378, 317 (2017)

Charge radiusI The charge radius squared is defined as the secondmoment of the

charge form factor,

hr2i = dGC(k2)dk2

k2=0

. (4)

I The charge form factor is obtained from the scalar component of thecharge operator.

I At leading order (Q3) the charge operator consists of point-likenucleons,

LO = |e|X

i

1 + zi

2(~p0i ~pi ~k). (5)

I There is no contribution to the charge operator at next-to-leadingorder.

I At N2LO (Q1), there are relativistic corrections arising from the finitesize of the nucleons, [Phillips, 2003]

N2LO = |e|6hr2Esi

X

i

1 + zi

2(~p0i ~pi ~k). (6)

Charge radius

I The first two body correction appears at N3LO (Q0),

[Kölling et.al., 2011]

N3LO =|e|g2A

16F2mN

1 29

X

i<j

(( z

i + ~i · ~j)~i · ~qi~j · ~kq2i +m2

+ i $ j

)

(~p0i + ~p0j ~pi ~pj ~k). (7)

I 9 is a parameter of the unitary transformation used to renormalizethe one-pion exchange potential, and the charge operator. We haveused 9 = 0.

I There are twomore terms that contribute to the charge operator atN3LO, however they do not contribute to the charge radius.

Gamow-Teller transitionI We consider the operators at zero momentum transfer.I The leading order contribution (Q3 in power counting) obtained

from chiral EFT coincides with the impulse approximation operator,

O±GT,LO = gA

X

i

±i i(~p0i ~pi). (1)

I The first two body correction from chiral EFT appears at N2LO (Q0 inpower counting), [Krebs, et.al., 2017]

O±GT,N2LO = 1

2(2)3DX

i<j

±i i + i $ j

(~p0i + ~p0j ~pi ~pj). (2)

I The low energy constant D comes from the one pion exchange loopdiagram, and is usually expressed in terms of a more well-known lowenergy constant, cD,

D =cD

F2. (3)

I We have used cD = 1 [Navratil et.al., 2007 and the chiral symmetrybreaking scale = 700MeV.

Deep Learning: Extrapolation Tool forAb Initio Nuclear Theory, G.A. Negoita, et al., arXiv:1810.04009

......

.

1

2

1

2

3

6

7

8

1

input layer hidden layer

output layer

Nmax

hΩ-

protonrms radius

or

energy

-32.2

-32.1

-32

-31.9

-31.8

-31.7

8 10 12 14 16 18 20

Extrapolation A5Extrapolation BANNExperiment -31.995

gs e

nerg

y E gs

(MeV

)

Nmax

6Li with Daejeon16

2.2

2.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

8 10 12 14 16 18 20

Extrapolation A3ANNExperiment 2.38(3)

gs p

roto

n rm

s ra

dius

r p (fm

)

Nmax

6Li with Daejeon16

G.A. Negoita, et al., “Deep Learning: Extrapolation Tool for Computational Nuclear Physics,” submitted to PRC; arXiv: 1811.01782

-34

-32

-30

-28

-26

-24

-22

5 10 15 20 25 30 35 40 45 501.8

1.9

2

2.1

2.2

2.3

2.4

ANNNCSM

1614Nmax=12 18 70 ( Egs ) / 90 ( rp )

6Li with Daejeon16

ANN results when training data limited to Nmax 10≤

Progress:

LENPIC NN+NNN (at N2LO) paper: PRC accepted; arXiv:1807.02848

Completed studies of model 2-body systems: PRC98, 065502 (2018); arXiv: 1809:00276

Implement electroweak operators in finite nuclei:

Benchmark A=6 calculations of 0ν2β-decay with UNC & MSU groups (paper in preparation)

Postprocessor code for scalar and non-scalar observables (in testing stage)Iowa State – Notre Dame collaboration

Develop extrapolations and uncertainties with Artificial Neural Networks: A. Negoita, et al., submitted to PRC; arXiv:1810.04009

Outlook:

Expand treatment to full range of EW operators within Chiral EFTat NLO, N2LO & N3LO (studies underway)

Extend effective EW operator approach to medium weight nuclei with “Double OLS” approach (sd shell investigations underway: N. Smirnova, et al)

Iowa State UniversityMembers of NUCLEI and Topical Collaboration Teams

FacultyJ.P. Vary and P. Maris

Grad StudentsRobert BasiliWeijie Du

Mengyao Huang Matthew Lockner

Alina NegoitaSoham Pal

Shiplu Sarker

New faculty position at Iowa State in Nuclear TheorySupported, in part, by the Fundamental Interactions

Topical Collaboration Interviews in process

Breaking News

Time-Dependent Basis Function approach to Coulomb ExcitationDeuteron (JISP16) on 208Pb target

Ed (lab) = 12 MeV, Lab scattering angle of n-p CM = 70o

Peng Yin, et al., in preparation

P = probability the deuteronremains in the elastic channel as function of t.

Non-perturbative treatmentof electric dipole transitionsto continuum states.

1-P = breakup probability

W. Du, et al., PRC 97, 064620 (2018); arXiv 1804.01156

E1s from/to the gs only

E1s also allowed amongintermediate states

Documents

FRIB: Opening New Frontiers in Nuclear Science bands James ...quantum properties of nuclei depend on the intricate interplay of strong, weak, and electromagnetic ... R = 0.9 fm, no