Uncertainty quantiﬁcation in electromagnetic observables of nuclei Bijaya Acharya12 Sonia Bacca13 Francesca Bonaiti1 Simone Salvatore Li

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Uncertainty quantiﬁcation in electromagnetic

observables of nuclei

Bijaya Acharya1,2, Sonia Bacca 1,3∗, Francesca Bonaiti1, Simone Salvatore Li

Muli 1, Joanna E. Sobczyk1

1Institut f¨

ur Kernphysik and PRISMA+Cluster of Excellence, Johannes Gutenberg

Universit ¨

at, 55128 Mainz, Germany

2Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831,

USA

3Helmholtz-Institut Mainz, Johannes Gutenberg Universit ¨

at Mainz, D-55099 Mainz,

Germany

Correspondence*:

Sonia Bacca

s.bacca@uni-mainz.de

ABSTRACT

We present strategies to quantify theoretical uncertainties in modern ab-initio calculations of

electromagnetic observables in light and medium-mass nuclei. We discuss how uncertainties

build up from various sources, such as the approximations introduced by the few- or many-body

solver and the truncation of the chiral effective ﬁeld theory expansion. We review the recent

progress encompassing a broad range of electromagnetic observables in stable and unstable

nuclei.

Keywords: uncertainty quantiﬁcation, electromagnetic processes, ab-initio theory, chiral effective ﬁeld theory

1 INTRODUCTION

Uncertainty quantiﬁcation is an emerging ﬁeld in nuclear theory. It is nowadays expected for any theoretical

calculation of nuclear observables to have a corresponding uncertainty bar, which is vital to make progress

in our understanding of strongly interacting systems through the comparison of theoretical modeling with

experimental data. While this is clearly the goal, the speciﬁc approach to uncertainty quantiﬁcation and

its sophistication level strongly depends on the used theoretical method and on the observables under

investigation. In this review, we focus on electromagnetic reactions and on how they can be calculated

with corresponding uncertainty in the so-called ab-initio methods. It is fair to say that the sub-ﬁeld of

quantiﬁcation of theoretical uncertainties is just now developing, and while there is still much to be done

there has been recent signiﬁcant progress. Here, we report on such progress, discuss its philosophy and

identify areas where improvements can be expected in the future.

In the ab-initio approach to nuclear theory [

] the goal is to explain nuclear phenomena, including

electromagnetic processes, starting from protons and neutrons as degrees of freedom and to solve the related

quantum-mechanical problem in a numerical way, either exactly or within controlled approximations. To

achieve this, one typically solves the Schr

odinger equation for a given Hamiltonian

and then computes

transition matrix elements of the electromagnetic operator

Jµ

between the eigenstates of

. Hence, before

arXiv:2210.04632v1 [nucl-th] 10 Oct 2022

Acharya et al.

discussing the approach devised to quantify uncertainty in electromagnetic observables, we deﬁne the

dynamical ingredients (Hamiltonian and currents), as well as the speciﬁc observables we want to investigate.

1.1 Hamiltonians and currents

The starting point of an ab-initio computation of a nucleus composed of

nucleons is the nuclear

Hamiltonian,

H=TK+

i<j

Vij +

i<j<k

Wijk ,(1)

where

is the intrinsic kinetic energy,

Vij

is the two-body interaction and

Wijk

is the three-body

interaction. As opposed to a phenomenological derivation of nuclear forces, effective ﬁeld theories (EFT)

offer a more systematic approach [

]. In this paper, we will use effective Hamiltonians which are derived

in chiral effective ﬁeld theory (

EFT)[

]. In this framework, the Hamiltonian is expanded in powers

(Q/Λ)

, where

is the typical low–momentum characterizing nuclear physics and

is the breakdown

scale of the effective ﬁeld theory. The various components relevant for

Vij

and

Wijk

are presented in terms

of Feynman diagrams in Figure 1, where

ν0

is the ﬁrst power entering in the counting. The unresolved short

Figure 1.

The

EFT expansion of the nuclear Hamiltonian and electromagnetic currents. The ﬁlled circles,

squares and diamond denote strong-interaction vertices with chiral dimension

0,1

and 2, respectively. The

⊗

symbols denote the electromagnetic vertices. In the literature,

ν0

is usually taken as

for the potential

and −3for the currents.

range physics is encoded in the values of the low energy constants (LECs), which are usually calibrated by

ﬁtting to experimental data. Different optimization and ﬁtting strategies have been used to calibrate the

LECs [

]. Here, we will use only a selected set of different Hamiltonians obtained from

EFT.

Furthermore, interactions with explicit

∆

degrees of freedom are becoming available [

]

and should be explored. In the present work we will present results with both chiral

∆

-full and

∆

-less

interactions.

This is a provisional ﬁle, not the ﬁnal typeset article 2

Acharya et al.

The nuclear response to external probes is described by the interaction Hamiltonian, which depends

on nuclear dynamics through the nuclear current operator. The

EFT expansion exists also for the

electromagnetic four-vector current

Jµ= (ρ, J)

, where the time-like component is the charge operator

and the space-like component is the three-vector current operator. The ﬁrst diagram entering the

EFT

expansion for

(ρ, J)

are shown in Figure 1, where we omit the diagrams that contribute to the elastic form

factors. The reader can ﬁnd more details on our implementation of the currents in Ref. [

]. While different

authors adopt different power counting schemes for the currents [

], we follow the conventions

of Ref. [22].

1.2 Electromagnetic observables

Electromagnetic probes are key tools to study nuclear structure because measured cross sections are easily

related to the few-/many-body matrix elements of electromagnetic operators via perturbation theory. Here,

we focus on electromagnetic observables that can be explained to high precision in ﬁrst order perturbation

theory, i.e., processes where one single photon is exchanged between the probe and the nucleus. This is

the case for the photoabsorption process and the electron scattering process, see Figure 2. The exchanged

photon can in general transfer energy

and momentum

. In the photonuclear process, a real photon with

ω=|q|=qis absorbed by the nucleus, while in electron scattering a virtual photon is exchanged, where

one can vary ωand qindependently.

Figure 2.

Feynman diagrams for the photoabsorption process (left), where a real photon

is exchanged,

and the electron scattering process, where a virtual photon

γ∗

is exchanged between the probe and the

nucleus (cyan blob).

In the cases of the photoabsorption and the electron-scattering process (see also Sections 3, 5), the cross

section can be written in terms of a so-called response function, which, in the inclusive unpolarized case, is

deﬁned as

R(ω, q) = Z

0fhΨf|Θ(q)|Ψ0i

2δEf−E0−ω.(2)

Here,

Θ(q)

is the electromagnetic operator, which can be directly one of the operators

(ρ, J)

or can be just

a multipole of them.

|Ψ0/f i

are the ground state and the excited states of the Hamiltonian

, respectively.

The symbol

P¯

indicates an average on the initial angular momentum projection, while the symbol

corresponds to both a sum over discrete excited states and an integral over continuum eigenstates of the

Hamiltonian. Indeed,

|Ψfi

may include not only bound excited states, but also states in the continuum

where the nucleus is broken up into fragments.

Frontiers 3

Acharya et al.

The calculation of continuum wave functions represents a challenging task especially in an inclusive

process, where one needs information on all possible fragmentation channels of the nucleus at a given

energy. To avoid the issue, one can use integral transforms, such as the Lorentz integral transform (LIT)

technique [

]. Originally used in few-body calculations, the LIT technique is based on the calculation

of the following integral of the response function R(ω, q),

L(σ, Γ, q) = Γ

πZdω R(ω, q)

(ω−σ)2+ Γ2,(3)

which can be shown to be the squared norm of the solution of a Schr

odinger-like equation calculated using

bound-state techniques. Once

L(σ, Γ)

is calculated, a numerical inversion procedure allows one to recover

R(ω, q), see Ref. [24] for details.

1.3 Numerical solvers

In order to calculate electromagnetic observables, we ﬁrst need a numerical solution of the Schr

odinger

equation. In the applications discussed in Sections 3, 4 and 5, we will use either few-body or many-body

solvers depending on the mass range Aof the addressed nuclei.

We obtain the bound-state and scattering-state wave functions for the

A= 2

problem by solving

the partial-wave Lippmann-Schwinger equations for the Hamiltonian. The response functions are then

calculated by directly evaluating the matrix elements of the electromagnetic operator in coordinate space.

To calculate few-body problems with

2< A < 8

we use hyperspherical harmonics expansions. In this

framework, one expands the

-body intrinsic wave function in terms of hyperspherical harmonics

and

hyperradial functions Rnas

Ψ =

Kmax

nmax

αnK Rn(ρr)HK(Ω) ,(4)

where

αnK

are the coefﬁcients of the expansion and where for the sake of simplicity we omit spin and

isospin degrees of freedom. Here,

ρr

is the hyperradius while

Ω

is a set of hyperangles, on which the

hyperspherical harmonics

with grandangular momentum

depend. The expansion is performed up to

a maximal value of hyperradial functions

nmax

and a maximal value of grandangular momentum

Kmax

Reaching convergence in

nmax

is typically not difﬁcult. The expansion in hyperspherical harmonics is

instead more delicate and one needs to ensure that the dependence of the calculated observables on this

truncation is under control. To accelerate convergence, an effective interaction a la Lee-Suzuki can be

introduced [

], obtaining the so-called effective interaction hyperspherical harmonics (EIHH) method,

which allows to eventually achieve sub-percentage accuracy in the

He calculations of binding energies

and electromagentic observables [

]. Hyperspherical harmonics expansions can be conveniently used

also to solve the Schr

odinger-like equation obtained when applying the LIT method described above. The

interested reader can consult, e.g., Refs. [27, 25, 24, 2, 28, 26] for more details.

For nuclei with

A≥8

we use coupled-cluster theory. In this framework, for a given Hamiltonian

one starts from a Slater determinant

|Φ0i

of single particle states and assumes an exponential ansatz to

construct the correlated many-body wave function as

|Ψ0i= exp (T)|Φ0i.(5)

This is a provisional ﬁle, not the ﬁnal typeset article 4

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UncertaintyquanticationinelectromagneticobservablesofnucleiBijayaAcharya1;2,SoniaBacca1;3,FrancescaBonaiti1,SimoneSalvatoreLiMuli1,JoannaE.Sobczyk11Institutf¨urKernphysikandPRISMA+ClusterofExcellence,JohannesGutenbergUniversit¨at,55128Mainz,Germany2PhysicsDivision,OakRidgeNationalLaboratory,OakRid...

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Uncertainty quantiﬁcation in electromagnetic observables of nuclei Bijaya Acharya12 Sonia Bacca13 Francesca Bonaiti1 Simone Salvatore Li

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