# Study of forbidden $\beta$ decays within the realistic shell model

G. De Gregorio  Dipartimento di Matematica e Fisica, Università degli Studi
della Campania “Luigi Vanvitelli”, viale Abramo Lincoln 5 - I-81100 Caserta,
Italy  Istituto Nazionale di Fisica Nucleare,
Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
R. Mancino  Institute of Particle and Nuclear Physics, Faculty of Mathematics
and Physics, Charles University, V Holešovičkách 2, 180 00 Prague, Czech
Republic  Institut für Kernphysik (Theoriezentrum), Fachbereich Physik,
Technische Universität Darmstadt,
Schlossgartenstrasse 2 - 64298 Darmstadt, Germany  GSI Helmholtzzentrum für
Schwerionenforschung, Planckstrasse 1, 64291 Darmstadt, Germany  L. Coraggio
Dipartimento di Matematica e Fisica, Università degli Studi della Campania
“Luigi Vanvitelli”, viale Abramo Lincoln 5 - I-81100 Caserta, Italy  Istituto
Nazionale di Fisica Nucleare,
Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy
N. Itaco  Dipartimento di Matematica e Fisica, Università degli Studi della
Campania “Luigi Vanvitelli”, viale Abramo Lincoln 5 - I-81100 Caserta, Italy
Istituto Nazionale di Fisica Nucleare,
Complesso Universitario di Monte S. Angelo, Via Cintia - I-80126 Napoli, Italy

###### Abstract

For the first time, half-lives and energy spectra of forbidden $\beta$ decays
are calculated within the realistic shell model. Namely, we approach this
issue starting from a realistic nucleon-nucleon potential and deriving
effective Hamiltonians and decay operators. Our goal is to explore the
sensitivity of the shape of calculated energy spectra to the renormalization
of forbidden $\beta$-decay operators, an operation that allows to take into
account those configurations that are not explicitly included in the chosen
model space. The region that has been considered for this investigation are
nuclei outside the 78Ni core, more precisely we have studied the second-
forbidden $\beta$ decays of 94Nb and 99Tc, and fourth-forbidden $\beta$ decays
of 113Cd and 115In, that are currently of a renewed experimental interest in
terms of novel spectroscopic techniques. Our results evidence that the
introduction of a renormalized $\beta$-decay operator leads to a marked
improvement of the reproduction of experimental half-lives. As regards the
spectra of both second-forbidden and fourth-forbidden decays, we have found
that their calculated shapes are in good agreement with the observed ones,
even if scarcely responsive to the renormalization of the decay operator. We
carry out also a detailed inspection of the different components of the
calculated spectra for a deeper insight about their role in reproducing the
experimental shapes.

###### pacs:

21.60.Cs, 21.30.Fe, 27.60.+j

##  I Introduction

The understanding of the renormalization mechanisms of electroweak currents is
nowadays a cornerstone of the nuclear structure research. The attention to
this issue is motivated by the need of calculating reliable nuclear matrix
elements $M^{0\nu}$ for the $0\nu\beta\beta$ decay, and relating the inverse
half-life $\left[T^{0\nu}_{1/2}\right]^{-1}$ of such a rare process to the
neutrino effective mass.

As a matter of fact, the accurate calculation of the wave functions of parent
and grand-daughter nuclei does not ensure a trustable $M^{0\nu}$, since most
nuclear models are based on the reduction of the dimension of the Hilbert
space where the nuclear Hamiltonian is defined. Then, a sound knowledge of the
renormalization of the electroweak currents, to account for the configurations
that are not explicitly included in the components of the nuclear wave
function, is crucial to enhance the predictivity of the calculated
$M^{0\nu}$s.

The ability of nuclear models to reproduce $\beta$-decay observables is,
consequently, the better way to validate both wave functions and
renormalization procedures, and such an issue is connected to the so-called
“quenching puzzle” of the axial coupling constant, namely the need by most
nuclear structure models to resort to a reduction of $g_{A}$ to reproduce the
observables directly linked to Gamow-Teller (GT) transitions Martínez-Pinedo
et al. (1996); Barea et al. (2015); Suhonen and Kostensalo (2019). However,
this is an empirical procedure, and it cannot be generalized to any
$\beta$-decay operator that depends on the value of the axial coupling
constant.

The realistic shell model (RSM) provides a consistent approach to derive
effective Hamiltonians and decay operators, the only parameter that is
involved being the nuclear force one starts from. In such a framework, single-
particle (SP) energies and two-body matrix elements (TBMEs) of the effective
shell-model Hamiltonian $H_{\rm eff}$, as well as every matrix element of
decay operators, are derived from a realistic free nucleon-nucleon ($NN$)
potential $V_{NN}$ by way of the many-body theory Kuo and Osnes (1990); Suzuki
and Okamoto (1995). The bare matrix elements of the $NN$ potential, and of any
transition operator, are renormalized with respect to the truncation of the
full Hilbert space into the reduced shell-model (SM) model space, to take into
account the neglected degrees of freedom without resorting to any empirical
parameter Coraggio and Itaco (2020). In other words, this approach does not
apply effective charges to calculate electromagnetic transition strengths, and
empirical quenching of $g_{A}$ to reproduce the $\beta$-decay matrix elements.

We have successfully employed RSM to study the $2\nu\beta\beta$-decay of 48Ca,
76Ge, 82Se,100Mo, 130Te, and 136Xe Coraggio et al. (2017, 2019, 2022, 2024),
and then extended it to predict the nuclear matrix elements of their
$0\nu\beta\beta$-decay Coraggio et al. (2020a); Coraggio et al. (2022). Now,
in order to validate the RSM in predicting $\beta$-decay observables, in the
present work we investigate the sensitivity to the renormalization of SM
forbidden $\beta$-decay operators describing the energy spectra of the emitted
electrons.

To this end, we have considered the second-forbidden $\beta$-decays of 94Nb
and 99Tc into 94Mo and 99Ru, as well as the fourth-forbidden $\beta$-decays of
113Cd and 115In into 113In and 115Sn, respectively, and compared their
calculated log$ft$s and $\beta$-decay energy spectra as obtained with the bare
and the renormalized decay operators, as well as with the available data.

The motivations of such a choice are twofold: first, these decays have been
already the subject of a few works with a similar goal, where the effective
operators was obtained tuning the quenching factor $q$ of $g_{A}$, and the
focus was spotted on the dependence of the shape of energy spectra on the
value of $q$ Mustonen et al. (2006); Haaranen et al. (2016, 2017); Kostensalo
et al. (2021, 2023). Second, novel experimental techniques have triggered new
measurements of the energy spectra of $\beta$ decays in the region of the
$0\nu\beta\beta$ decay of 100Mo. Among them, we mention the COBRA demonstrator
Zuber (2001), that has been developed for double-$\beta$ decay experiments,
and also adopted to study the spectra of $\beta$ decays of 113Cd Goessling et
al. (2005); Bodenstein-Dresler et al. (2020); Kostensalo et al. (2021); the
ACCESS project that aims to perform precision measurements of forbidden
$\beta$-decays using cryogenic calorimeters Pagnanini et al. (2023). Another
new experimental project is ASPECT-BET (An SDD-SPECTrometer for BETa decay
studies), that has developed a new detection strategy based on silicon drift
detectors (SDD), and it should be able to perform high-precision, high-
accuracy measurements of the energy spectra of $\beta$ decays at room
temperature Biassoni et al. (2023).

This paper is organized as follows. First, in Sec. II we sketch out briefly
the derivation of the effective SM Hamiltonian and decay operators, as well as
the basic theory of the $\beta$-decay and the structure of the second- and
fourth-forbidden $\beta$-decay matrix elements.

The effective shell-model Hamiltonian and decay operators have been derived
within a model space that is spanned by four
$0f_{5/2},1p_{3/2},1p_{1/2},0g_{9/2}$ proton orbitals and five
$0g_{7/2},1d_{5/2},1d_{3/2},2s_{1/2},0h_{11/2}$ neutron orbitals outside 78Ni
core starting from the high-precision CD-Bonn $NN$ potential Machleidt (2001),
whose repulsive high-momentum components are renormalized using the $V_{{\rm
low}\mbox{-}k}$ procedure Bogner et al. (2002). This is the same framework we
have employed in our previous study of the double-$\beta$ decay of 100Mo
Coraggio et al. (2022).

The results of the shell-model calculations are discussed and compared with
the available experimental data in Sec. III. There, we check first our nuclear
wave functions by comparing the calculated low-energy spectra and $E2$
transition strengths of parent and daughter nuclei, which are involved in the
decays under consideration, with their experimental counterparts. Then, we
report the results of our theoretical log$ft$s and energy spectra of the
emitted electron and size up them to the available data. We complete our
analysis with a detailed analysis of the different components of the spectra,
and how the interplay among their contributions plays an important role in
reproducing data.

In the last section (Sec. IV), we summarize the conclusions of this study, as
well as the outlook of our current research project.

##  II Outline of the theory

###  II.1 The effective SM Hamiltonian

The procedure of the derivation of the effective SM Hamiltonian is the same as
the one followed in Ref. Coraggio et al. (2022). First, we consider the high-
precision CD-Bonn $NN$ potential Machleidt (2001), then the non-perturbative
repulsive high-momentum components are integrated out, by way of the $V_{{\rm
low}\mbox{-}k}$ unitary transformation Bogner et al. (2002); Coraggio et al.
(2009), that provides a smooth potential preserving all the two-nucleon
observables calculated with the CD-Bonn one.

The $V_{{\rm low}\mbox{-}k}$ matrix elements are chosen as the interaction
vertices of a perturbative expansion of $H_{\rm eff}$, and a detailed
description of the many-body perturbation theory approach to the nuclear
$H_{\rm eff}$ can be found in Refs. Hjorth-Jensen et al. (1995); Coraggio et
al. (2012); Coraggio and Itaco (2020), so here we only sketch briefly the
steps that have been followed to obtain $H_{\rm eff}$.

The starting point is the full nuclear Hamiltonian $H$ for $A$ interacting
nucleons, which, according to the SM ansatz, is divided into the one-body term
$H_{0}$, whose eigenvectors set up the SM basis, and the residual interaction
$H_{1}$, using the harmonic-oscillator (HO) auxiliary potential $U$:

$\displaystyle H$ $\displaystyle=$ $\displaystyle T+V_{{\rm
low}\mbox{-}k}=(T+U)+(V_{{\rm low}\mbox{-}k}-U)=$ $\displaystyle=$
$\displaystyle H_{0}+H_{1}~{}.$ (1)

The eigenvalue problem of $H$ for a many-body system, in an infinite basis of
eigenvectors of $H_{0}$, cannot be solved, then an effective Hamiltonian is
derived, which is defined in the truncated model space spanned by four proton
– $0f_{5/2},1p_{3/2},1p_{1/2},0g_{9/2}$ – and five neutron orbitals –
$0g_{7/2},1d_{5/2},1d_{3/2},2s_{1/2},0h_{11/2}$ – outside 78Ni core.

The effective Hamiltonian is derived by way of the time-dependent perturbation
theory, through the Kuo-Lee-Ratcliff folded-diagram expansion in terms of the
$\hat{Q}$-box vertex function Kuo and Osnes (1990); Hjorth-Jensen et al.
(1995); Coraggio et al. (2012):

$H^{\rm
eff}_{1}(\omega)=\hat{Q}(\epsilon_{0})-PH_{1}Q\frac{1}{\epsilon_{0}-QHQ}\omega
H^{\rm eff}_{1}(\omega)~{},$ (2)

where $\omega$ is the wave operator decoupling the model space $P$ and its
complement $Q$, and $\epsilon_{0}$ is the eigenvalue of the unperturbed
degenerate Hamiltonian $H_{0}$.

The $\hat{Q}$ box is defined as

$\hat{Q}(\epsilon)=PH_{1}P+PH_{1}Q\frac{1}{\epsilon-QHQ}QH_{1}P~{},$ (3)

and $\epsilon$ is an energy parameter called “starting energy”.

Since the exact calculation of the $\hat{Q}$ box is not possible, then the
term $1/(\epsilon-QHQ)$ is expanded as a power series

$\frac{1}{\epsilon-QHQ}=\sum_{n=0}^{\infty}\frac{1}{\epsilon-
QH_{0}Q}\left(\frac{QH_{1}Q}{\epsilon-QH_{0}Q}\right)^{n}~{},$ (4)

namely we perform an expansion of the $\hat{Q}$ box up to the third order in
perturbation theory Coraggio and Itaco (2020).

As a matter of fact, the calculation of the $\hat{Q}$ box is the start to
solve the non-linear matrix equation (2) and obtain $H_{\rm eff}$ by way of
iterative techniques such as the Kuo-Krenciglowa and Lee-Suzuki ones
Krenciglowa and Kuo (1974); Suzuki and Lee (1980), or graphical non-iterative
methods Suzuki et al. (2011).

It should be pointed out that, since the nuclei that are involved in the decay
processes under investigation are characterized by a large number of valence
nucleons, we have included contributions from induced three-body forces in the
calculation of the $\hat{Q}$ box, that involve also three valence nucleons.

Since the SM code we have employed for our calculations Shimizu et al. (2019)
cannot diagonalize a three-body $H_{\rm eff}$, we have performed a normal-
ordering decomposition of the three-body induced-force contributions arising
at second order in perturbation theory, and retained only the two-body term
that is density-dependent from the number of valence nucleons. This procedure
is presented in details in Refs. Coraggio and Itaco (2020); Coraggio et al.
(2020b), together with a discussion about the contribution of such terms to
the eigenvalues of the SM Hamiltonian.

The SM parameters, namely the SP energies and the TBMEs of the residual
interaction, are reported in the Supplemental Material sup .

###  II.2  $\beta$-decay theory

The theory of $\beta$-decay is here briefly outlined. More details can be
found in Refs. Behrens and Bühring (1971); Fahlin Strömberg (2020).

In the following we focus on the $\beta$--decay, moreover, we use natural
units ( $\hbar=c=m_{e}=1$).

The total half-life of the $\beta$ decay is expressed in terms of the $k$-th
partial decay half-life $t^{k}_{1/2}$ as follows:

$\frac{1}{T_{1/2}}=\sum_{k}\frac{1}{t^{k}_{1/2}}~{}.$ (5)

On the other hand, the partial half-life $t^{k}_{1/2}$ is related to the
dimensionless integrated shape function $\tilde{C}$ by way of the relation:

$t^{k}_{1/2}=\frac{\kappa}{\tilde{C}}~{},$ (6)

where $\kappa=6144\pm 2$ $s$ Hardy and Towner (2009).

For a given $k$-th final state, the integrated shape function $\tilde{C}$ –
whose integrand defines the $\beta$-decay energy spectrum – is written as

$\tilde{C}=\int^{w_{0}}_{1}C(w_{e})p_{e}w_{e}(w_{0}-w_{e})^{2}F(Z,w_{e})dw_{e}~{}.$
(7)

The quantities on the right-hand side of the above definition are listed as:

  * a)

$Z$ is the atomic number of the daughter nucleus, $w_{e}$ the adimensional
energy of the emitted electron, $w_{0}$ the endpoint energy – namely the
maximum electron energy for a given transition–, and $p_{e}$ the electron
momentum.

  * b)

The function $F(Z,w_{e})$ is the Fermi function which is factorized in terms
of two functions $F_{0}$ and $L_{0}$:

$F(Z,w_{e})=F_{0}(Z,w_{e})L_{0}(Z,w_{e})~{},$ (8)

where $F_{0}$ defines the effects of the Coulomb interaction between the
electron and the daughter nucleus, and $L_{0}$ accounts for the
electromagnetic finite-size effect, whose explicit expressions can be found in
Ref. Behrens and Bühring (1971).

  * c)

$C(w_{e})$ is the so-called nuclear shape function, which depends on the
nuclear matrix elements (NMEs). For allowed $\beta$ transitions, it
corresponds to the GT reduced transition probability, and in such a case does
not depend on the electron energy.

For $n$-forbidden transitions, $C(w_{e})$ depends on the electron energy, and
is expressed as

$\displaystyle
C(w_{e})=\sum_{K}\sum_{k_{e},k_{\nu}}\lambda_{k_{e}}\biggl{[}M_{K}^{2}(k_{e},k_{\nu})+m_{K}^{2}(k_{e},k_{\nu})-$
$\displaystyle\frac{2\gamma_{k_{e}}}{k_{e}w_{e}}M_{K}(k_{e},k_{\nu})m_{K}(k_{e},k_{\nu})\biggr{]}~{},$
(9)

where $K$ is the tensor rank of the forbidden $\beta$-decay operators involved
in the decay. $K$ can range from 0 to 2 for $n=1$, and from $n$ to $n+1$ for
$n>1$. The quantities $k_{e}$ and $k_{\nu}$ are the positive integers emerging
from the partial wave expansion of the leptonic wave functions. The latter,
for a given value of $K$, must satisfy either $k_{e}+k_{\nu}=K+1$ or
$k_{e}+k_{\nu}=K+2$.

The auxiliary quantities $\gamma_{k_{e}}$ and $\lambda_{k_{e}}$ are defined as

$\gamma_{k_{e}}=\sqrt{k_{e}^{2}-(\alpha
Z)^{2}},\quad\lambda_{k_{e}}=\frac{F_{k_{e}-1}(Z,w_{e})}{F_{0}(Z,w_{e})}~{},$
(10)

where $F_{k_{e}-1}(Z,w_{e})$ is the so-called generalized Fermi function (see
Ref. Behrens and Bühring (1971) for its explicit expression). The quantities
$M_{K}$ and $m_{K}$ are complicated combinations of some kinematic factors and
coefficients $F^{(N)}_{KLS}(k_{e},m,n,\rho)$, the latter being functions of
the orbital, spin, and total rank of the transition operators $L$, $S$, and
$K$, respectively, and the integers $m$, $n$ and $\rho$ depending on the
nuclear charge distribution which accounts for the influence of the nuclear
charge on the electron Behrens and Bühring (1971); Fahlin Strömberg (2020).
The index $N$ labels the order of the expansion in powers of $qR$ (where $R$
is the nuclear radius and $q$ is the momentum transfer $q=\mid
p_{e}+p_{\nu}\mid$) of the nuclear form factor $F_{KLS}$, which is defined by
the following expression

$F_{KLS}(q^{2})=\sum_{N}\frac{(-1)^{N}(qR)^{2N}(2L+1)!!}{(2N)!!(2L+2N+1)!!}F^{(N)}_{KLS}~{}.$
(11)

If we adopt the impulse approximation ($qR\ll 1$) to derive the formalism of
forbidden $\beta$-decay transitions, and also assume that bound nucleons
interact weakly as free nucleons, then we may neglect the effect of any many-
body current. Within such an approximation, it has been shown in Refs. Behrens
and Bühring (1971); Fahlin Strömberg (2020) that the form factor coefficients
${}^{V/A}F^{(N)}_{KLS}(k_{e},m,n,\rho)$ can be related to the NMEs
${}^{V/A}M^{N}_{KLS}(k_{e},m,n,\rho)$ by a phase factor:

${}^{V/A}F^{(N)}_{KLS}(k_{e},m,n,\rho)=(-1)^{K-L}~{}^{V/A}M^{N}_{KLS}(k_{e},m,n,\rho)~{},$

where the label $V/A$ indicates the separation of the coefficients
$F^{(N)}_{KLS}(k_{e},m,n,\rho)$ according to the axial and vector components
of the decay operator.

In a shell-model calculation, the NMEs can be expressed in terms of the
single-particle matrix elements of the one-body decay operator (SPMEs) and the
one-body transition densities (OBTDs), that can be obtained from the shell-
model wave functions, through the expression

${}^{V/A}M^{(N)}_{KLS}(k_{e},m,n,\rho)={\frac{1}{\hat{J_{i}}}}\sum_{\pi,\nu}~{}^{V/A}m^{(N)}_{KLS}(\pi,\nu)(k_{e},m,n,\rho)\times{\rm
OBTD}(\Psi_{f},\Psi_{i},\pi,\nu,K)~{},$ (13)

where $\hat{J_{i}}=\sqrt{(2J_{i}+1)}$, and $J_{i}$ is the angular momentum of
the initial state of the parent nucleus. The OBTDs are defined as

${\rm
OBTD}(\Psi_{f},\Psi_{i},\pi,\nu,K)=\frac{\langle\Psi_{f}\mid\mid[a^{\dagger}_{\pi}\otimes\tilde{a}_{\nu}]^{K}\mid\mid\Psi_{i}\rangle}{\hat{K}}~{},$
(14)

where $\Psi_{i}$ and $\Psi_{f}$ are the wave function of the initial and final
state, respectively, $a^{\dagger}_{\pi}$ is the particle creation operator,
and $\tilde{a}_{\nu}=(-1)^{j_{\nu}+m_{\nu}}a_{\nu-m_{\nu}}$ is the tensor
spherical form of the particle annihilation operator ($a_{\nu}$). The indices
$\pi$ and $\nu$ label the proton ($\pi$) and neutron ($\nu$) single-particle
states, and the symbol $\otimes$ denotes the angular-momentum coupling.

The SPMEs correspond to the following matrix elements:

${}^{V}m^{(N)}_{KLS}(k_{e},m,n,\rho)=\langle\phi_{\kappa_{\pi}\mu}\mid\mid\left(\frac{r}{R}\right)^{L+2N}\mathcal{I}(k_{e},m,n,\rho,r)T_{KLS}\mid\mid\phi_{\kappa_{\nu}\mu}\rangle~{},$
(15)
${}^{A}m^{(N)}_{KLS}(k_{e},m,n,\rho)=\langle\phi_{\kappa_{\pi}\mu}\mid\mid\left(\frac{r}{R}\right)^{L+2N}\mathcal{I}(k_{e},m,n,\rho,r)\gamma_{5}T_{KLS}\mid\mid\phi_{\kappa_{\nu}\mu}\rangle~{}.$
(16)

Now, it is worth to list and specify the quantities which appear in the Eqs.
(15,16) for the matrix elements of the vector and axial components of the
$\beta$-decay operator:

  * •

The functions $\mathcal{I}(k_{e},m,n,\rho,r)$ are the so-called Coulomb
factors whose explicit expression can be found in Ref. Behrens and Bühring
(1971).

  * •

The operator $T_{KLS}$ is the transition operator defined as

$T^{M}_{KLS}=\left\\{\begin{matrix}Y_{LM}\delta_{KL}&S=0,\\\
(-1)^{L-K+1}\gamma_{5}[Y_{L}\otimes\sigma]_{KM}&~{}S=1~{}.\end{matrix}\right.$
(17)

  * •

The single-particle relativistic wave functions $\phi_{\kappa\mu}$ are the
eigenfunctions of the operators $J_{z}$ and $\mathcal{K}=\beta(\sigma\cdot
L+\mathbb{I})$, and are labeled by their eigenvalues $\mu$ and $\kappa$:

$\displaystyle J_{z}\phi_{\kappa\mu}$ $\displaystyle=$
$\displaystyle\mu\phi_{\kappa\mu}~{},$
$\displaystyle\mathcal{K}\phi_{\kappa\mu}$ $\displaystyle=$
$\displaystyle\beta(\sigma\cdot
L+\mathbb{I})\phi_{\kappa\mu}=\kappa\phi_{\kappa\mu}~{}.$

It can be shown that the eigenvalue $\kappa$ is related to the total and
orbital angular momenta $j$ and $l$ through the relation

$\kappa=\left\\{\begin{matrix}j+\frac{1}{2}&\textrm{for }l=j+\frac{1}{2}\\\
-(j+\frac{1}{2})&\textrm{for }l=j-\frac{1}{2}~{}.\end{matrix}\right.$ (18)

In the Condon-Shortley (CS) phase convention, the $\phi_{\kappa\mu}$ functions
are defined as

$\phi_{\kappa\mu}=\binom{-if_{\kappa}(r)\chi_{-\kappa\mu}}{g_{\kappa}(r)\chi_{\kappa\mu}}~{},\\\
~{}$ (19)

where $\chi_{\kappa\mu}=[Y_{l}(\hat{r})\otimes\chi]^{j\mu}$, and the radial
functions $f_{\kappa}(r)$ and $g_{\kappa}(r)$ are the solutions of the radial
Dirac equations, and they are usually indicated as the small and large
components, respectively:

$\displaystyle\frac{dg_{\kappa}(r)}{dr}+\frac{\kappa+1}{r}g_{\kappa}(r)-(E+M-V(r))f_{\kappa}(r)=0~{},$
(20)
$\displaystyle\frac{df_{\kappa}(r)}{dr}-\frac{\kappa-1}{r}f_{\kappa}(r)+(E-M-V(r))g_{\kappa}(r)=0~{}.$
(21)

The matrix elements of Eqs. (15,16) can be grouped into two different types.

The first one contains the product of both small components of the initial and
final wave functions, as well as the product of both large components of the
same wave functions, which are the solutions of the coupled equations (20,21),
and usually they are dubbed in literature as the non-relativistic matrix
elements Behrens and Bühring (1971); Sadler and Behrens (1993); Zhi et al.
(2013); Fahlin Strömberg (2020). Their explicit expression is:

${}^{V}m^{(N)}_{KK0}(\pi,\nu)(k_{e},m,n,\rho)=\sqrt{2}g_{V}\biggl{[}G_{KK0}(\kappa_{\pi},\kappa_{\nu})\int_{0}^{\infty}g_{\pi}(r,\kappa_{p})\left(\frac{r}{R}\right)^{K+2N}\mathcal{I}(k_{e},m,n,\rho,r)g_{\nu}(r,\kappa_{\nu})r^{2}dr$
$\displaystyle+G_{KK0}(-\kappa_{\pi},-\kappa_{\nu})\int_{0}^{\infty}f_{\pi}(r,\kappa_{\pi})\left(\frac{r}{R}\right)^{K+2N}\mathcal{I}(k_{e},m,n,\rho,r)f_{\nu}(r,\kappa_{\nu})r^{2}dr\biggr{]}~{},$
(22)
${}^{A}m^{(N)}_{KL1}(\pi,\nu)(k_{e},m,n,\rho)=\text{sign}(K-L+\frac{1}{2})\sqrt{2}g_{A}\biggl{[}G_{KK0}(\kappa_{\pi},\kappa_{\nu})\int_{0}^{\infty}g_{\pi}(r,\kappa_{\pi})\left(\frac{r}{R}\right)^{L+2N}\mathcal{I}(k_{e},m,n,\rho,r)g_{\nu}(r,\kappa_{\nu})r^{2}dr$
$\displaystyle+G_{KK0}(-\kappa_{\pi},-\kappa_{\nu})\int_{0}^{\infty}f_{i_{\pi}}(r,\kappa_{p})\left(\frac{r}{R}\right)^{L+2N}\mathcal{I}(k_{e},m,n,\rho,r)f_{\nu}(r,\kappa_{\nu})r^{2}dr\biggr{]}~{}.$
(23)

The second group contains the product of the small and large components of the
initial and final wave functions that are the solutions of Eqs. (20,21), and
in such a case they are dubbed as the relativistic matrix elements. Now, their
expression is:

${}^{V}m^{(N)}_{KL1}(\pi,\nu)(k_{e},m,n,\rho)=\text{sign}(K-L+\frac{1}{2})\sqrt{2}g_{V}\biggl{[}G_{KL1}(\kappa_{\pi},-\kappa_{\nu})\int_{0}^{\infty}g_{\pi}(r,\kappa_{\pi})\left(\frac{r}{R}\right)^{L+2N}\mathcal{I}(k_{e},m,n,\rho,r)f_{\nu}(r,\kappa_{\nu})r^{2}dr$
$\displaystyle-
G_{KK0}(-\kappa_{\pi},\kappa_{\nu})\int_{0}^{\infty}f_{\pi}(r,\kappa_{\pi})\left(\frac{r}{R}\right)^{L+2N}\mathcal{I}(k_{e},m,n,\rho,r)g_{\nu}(r,\kappa_{\nu})r^{2}dr\biggr{]}~{},$
(24)
${}^{A}m^{(N)}_{KK0}(\pi,\nu)(k_{e},m,n,\rho)=\sqrt{2}g_{A}\biggl{[}G_{KK0}(\kappa_{\pi},-\kappa_{\nu})\int_{0}^{\infty}g_{\pi}(r,\kappa_{\pi})\left(\frac{r}{R}\right)^{K+2N}\mathcal{I}(k_{e},m,n,\rho,r)f_{\nu}(r,\kappa_{\nu})r^{2}dr$
$\displaystyle-
G_{KK0}(-\kappa_{\pi},\kappa_{\nu})\int_{0}^{\infty}f_{\pi}(r,\kappa_{\pi})\left(\frac{r}{R}\right)^{K+2N}\mathcal{I}(k_{e},m,n,\rho,r)g_{\nu}(r,\kappa_{\nu})r^{2}dr\biggr{]}~{}.$
(25)

It is worth to note that in the above equations we have introduced the
quantity

$\displaystyle G_{KLS}(\kappa_{\pi},\kappa_{\nu})=$
$\displaystyle(-1)^{j_{\pi}-j_{\nu}+l_{\pi}}\hat{S}\hat{K}\hat{j}_{\pi}\hat{j}_{\nu}\hat{l}_{\pi}\hat{l}_{\nu}\langle
l_{\pi}l_{\nu}00\lvert L0\rangle\times$ (26)
$\displaystyle\begin{Bmatrix}K&S&L\\\ j_{\pi}&\frac{1}{2}&l_{\pi}\\\
j_{\nu}&\frac{1}{2}&l_{\nu}\end{Bmatrix},$

and we remind that $L$, $S$, and $K$ are the orbital, spin, and total rank of
the transition operators, respectively, and $l_{\tau}=k_{\tau}$ if
$k_{\tau}>0$ and $l_{\tau}=\mid k_{\tau}\mid-1$ if $k_{\tau}<0$.

Until now, the nucleon wave functions are expressed in a fully relativistic
framework, being the solutions of the Dirac equation. However, within the
nuclear shell model, the nucleon wave functions are expressed as solutions of
the single-particle Schrödinger equation, introducing the auxiliary harmonic-
oscillator potential.

Such an inconsistency impacts especially on the calculation of the
relativistic matrix elements, and this problem may be tackled in two ways. The
first approach is to resort to a non-relativistic reduction of the Dirac
equation by considering the non-relativistic limit of the coupled Eqs.
(20,21), namely the kinetic energy $T$ and the auxiliary potential $V(r)$
satisfy the conditions $T=E-M_{N}\ll 2M_{N}$ and $V(r)\ll 2M_{N}$ Rose and
Osborn (1954). Within this limit, the function $g_{\kappa}$ becomes the
solution of the Schrödinger equation, and $f_{\kappa}$ is related to
$g_{\kappa}$ through the relation

$f_{\kappa}(r)=\frac{1}{2M_{N}}\left(\frac{d}{dr}+\frac{\kappa+1}{r}\right)g_{\kappa}(r)~{}.$
(27)

However, as it was observed in Ref. Behrens and Bühring (1971), whether or not
such a limit of the Dirac equation is a good approximation to calculate the
relativistic NMEs is a difficult question to answer, since to test this
approach a fully relativistic calculation should be performed and compared
with the approximated results.

Moreover, it should be noted that, if we consider the relation (27), the
radial function $f_{\kappa}(r)$ is suppressed by a factor $1/(2M_{N})$ with
respect to the function $g_{\kappa}(r)$, but the relativistic form factor,
even if it is small with respect to the non-relativistic ones, it has been
found to be relevant to determine both the shape of the energy spectrum and
the half-life of the $\beta$ decay Sadler and Behrens (1993); Kirsebom et al.
(2019); Fahlin Strömberg (2020); Kossert et al. (2022); Paulsen et al. (2023).

Its relevance, within such a reduction of the Dirac equation, is also stressed
in Ref. Kostensalo et al. (2021), where a study of the dependence of the
energy spectrum and half-life of the fourth-forbidden $\beta$ decay of 113Cd
with respect to the quenching factor $q$ of the axial coupling constant
$g_{A}$ has been carried out. As a matter of fact, the authors show that a
fitting procedure of both $q$ and of the relativistic form factor is needed to
reproduce the experimental shape of the energy spectrum as well as the
observed half-life.

An alternative approach to calculate the relativistic NMEs, is to resort to
the conserved vector current theory (CVC) Behrens and Bühring (1971), which
leads to derive a connection between the relativistic NMEs and the non-
relativistic ones, that is developed as a relation between the corresponding
form factors. An early application of this approach can be found in Ref.
Sadler and Behrens (1993).

In particular, for the four decays under investigation, since we stop at the
leading order in the expansion of $F_{KLS}(q^{2})$ ($N=0$ in Eq. (11)), the
relativistic form factors entering in Eq. (II.2) are only the ${}^{V}F_{211}$,
for the second-forbidden decay of 94Nb, 99Tc, and the ${}^{V}F_{431}$ for the
fourth-forbidden decay of 113Cd, 115In. These form factors, using the CVC
relation Behrens and Bühring (1971); Sadler and Behrens (1993), depends on
${}^{V}F_{220}$ and ${}^{V}F_{440}$, respectively, through the relations

$^{V}F_{211}=-\frac{1}{\sqrt{10}}RE_{\gamma}~{}^{V}F_{220}~{},\quad^{V}F_{431}=-\frac{1}{\sqrt{36}}RE_{\gamma}~{}^{V}F_{440}~{},$
(28)

where

$E_{\gamma}=[W_{0}+\Delta E_{C}-(M_{\nu}-M_{\pi})]~{},$ (29)

is the energy of the analogue electromagnetic transition Smith and Simms
(1970), $M_{\nu(\pi)}$ is the neutron (proton) mass, and $\Delta E_{C}$ is the
Coulomb displacement energy that can be evaluated in different ways. $\Delta
E_{C}$ can be evaluated in different ways, here we have used the results of
the fit procedure as outlined in Ref. Antony et al. (1997), obtaining $\Delta
E_{C}=11.99$, 12.38, 13.27, and 13.48 MeV for 94Nb, 99Tc, 113Cd, and 115In
decays, respectively.

It is important to stress that these CVC relations have been employed in
different studies, leading to a general improvement of the calculated spectra
Sadler and Behrens (1993); Kirsebom et al. (2019); Fahlin Strömberg (2020);
Kossert et al. (2022); Paulsen et al. (2023) and half-lives.

However, we point out that these relations are valid in the full nuclear
Hilbert space of the single-particle configurations, and it is difficult to
evaluate the impact of resorting to a truncated model space, but, on the other
hand, it should be noted that our approach relies on the derivation of
effective Hamiltonians and operators by way of the many-body theory to account
for the configurations outside the model space.

Nevertheless, the CVC relations for the relativistic form factors represent a
viable route to get an estimation of the relativistic matrix elements entering
the calculation, and/or, to understand the reliability of the non-relativistic
reduction of the relativistic matrix elements.

###  II.3 Effective shell-model decay operators

In this section, we sketch briefly the procedure to derive effective SM decay
operators $\Theta_{\rm eff}$ by way of many-body perturbation theory.

As is well known, the diagonalization of the $H_{\rm eff}$ does not produce
the true nuclear wave-functions, but their projections onto the chosen model
space $P$. Then, any decay operator $\Theta$ should be renormalized by taking
into account for the neglected degrees of freedom corresponding to the $Q$
subspace.

The derivation of effective SM operators within a perturbative approach traces
back to the pioneering period of employing $NN$ potentials in SM calculations
Mavromatis et al. (1966); Mavromatis and Zamick (1967); Federman and Zamick
(1969); Ellis and Osnes (1977); Towner and Khanna (1983); Towner (1987). We
have followed the method that has been introduced by Suzuki and Okamoto Suzuki
and Okamoto (1995), which allows a derivation of decay operators $\Theta_{\rm
eff}$ which is consistent with the one of $H_{\rm eff}$, as presented in Sec.
II.1. This is based on the perturbative expansion of a vertex function
$\hat{\Theta}$ box – analogously with the derivation of $H_{\rm eff}$ in terms
of the $\hat{Q}$ box –, whose details may be found in Refs. Suzuki and Okamoto
(1995); Coraggio and Itaco (2020).

According to such a procedure, the starting point is the perturbative
calculation of two energy-dependent vertex functions:

$\hat{\Theta}(\epsilon)=P\Theta P+P\Theta Q\frac{1}{\epsilon-QHQ}QH_{1}P~{},$
$\hat{\Theta}(\epsilon_{1};\epsilon_{2})=PH_{1}Q\frac{1}{\epsilon_{1}-QHQ}Q\Theta
Q\frac{1}{\epsilon_{2}-QHQ}QH_{1}P~{},$

and of their derivatives calculated in $\epsilon=\epsilon_{0}$, $\epsilon_{0}$
being the eigenvalue of the degenerate unperturbed Hamiltonian $H_{0}$:

$\hat{\Theta}_{m}=\frac{1}{m!}\frac{d^{m}\hat{\Theta}(\epsilon)}{d\epsilon^{m}}\biggl{|}_{\epsilon=\epsilon_{0}}~{},$
$\hat{\Theta}_{mn}=\frac{1}{m!n!}\frac{d^{m}}{d\epsilon_{1}^{m}}\frac{d^{n}}{d\epsilon_{2}^{n}}\hat{\Theta}(\epsilon_{1};\epsilon_{2})\biggl{|}_{\epsilon_{1}=\epsilon_{0},\epsilon_{2}=\epsilon_{0}}~{}$

Then, a series of operators $\chi_{n}$ is calculated:

$\displaystyle\chi_{0}$ $\displaystyle=$
$\displaystyle(\hat{\Theta}_{0}+h.c.)+\hat{\Theta}_{00}~{}~{},$ (30)
$\displaystyle\chi_{1}$ $\displaystyle=$
$\displaystyle(\hat{\Theta}_{1}\hat{Q}+h.c.)+(\hat{\Theta}_{01}\hat{Q}+h.c.)~{}~{},$
$\displaystyle\chi_{2}$ $\displaystyle=$
$\displaystyle(\hat{\Theta}_{1}\hat{Q}_{1}\hat{Q}+h.c.)+(\hat{\Theta}_{2}\hat{Q}\hat{Q}+h.c.)+$
$\displaystyle(\hat{\Theta}_{02}\hat{Q}\hat{Q}+h.c.)+\hat{Q}\hat{\Theta}_{11}\hat{Q}~{}~{}.$
$\displaystyle\cdots$

At the end, $\Theta_{\rm eff}$ is written in the following form:

$\Theta_{\rm eff}=H_{\rm
eff}\hat{Q}^{-1}(\chi_{0}+\chi_{1}+\chi_{2}+\cdots)~{},$ (32)

the $\chi_{n}$ series being arrested in our calculations at $n=2$, and the
$\hat{\Theta}$ function expanded up to third order in perturbation theory.

In Refs. Coraggio et al. (2018, 2019); Coraggio et al. (2020a) we have tackled
the issue of the convergence of the $\chi_{n}$ series and of the perturbative
expansion of the $\hat{\Theta}$ box, showing the robustness of such a
procedure.

It is worth to point out that, even if the decay operator has a one-body
structure, the shell-model effective operator has many-body components which
account for a number of valence of nucleons larger than one Ellis and Osnes
(1977); Coraggio et al. (2020a). As a matter of fact, the fourth-forbidden
$\beta$ decay of 115In into 115Sn involves 37 valence nucleons outside the
doubly-magic 78Ni, then for such a process $\Theta_{\rm eff}$ should contain
contributions up to a 37-body term.

The shell-model code KSHELL can employ transition operators with a one- and
two-body components Shimizu et al. (2019), then for the calculation of
$\beta$-decay effective operators we include just the leading terms of these
many-body contributions in the perturbative expansion of the $\hat{\Theta}$
box, namely the second-order two-body diagrams (a) and (b), that are reported
in Fig. 1.

Figure 1: Second-order two-body diagrams which are included in the
perturbative expansion of $\Theta_{\rm eff}$. The dashed line indicates the
bare second- and fourth-forbidden $\beta$-decay operators $\Theta$, the wavy
line the two-body potential $V_{{\rm low}\mbox{-}k}$. For the sake of
simplicity, for each topology only one of the diagrams which correspond to the
permutation of the external lines has been drawn.

The two topologies of second-order connected two-valence-nucleon diagrams (a)
and (b) accounts for the so-called “blocking effect”, which is necessary to
consider the Pauli exclusion principle in systems with more than one valence
nucleon Towner (1987), if the transition operator has a one-body structure. It
is worth to point out that these two-body contributions to the effective decay
operators mirror the role of induced three-body forces in the calculation of
the $\hat{Q}$ box, which we have introduced in section II.1.

In the present work, the decay operators $\Theta$ are the one-body electric-
quadrupole $E2$ transition
$q_{\scriptscriptstyle{p,n}}r^{2}Y^{2}_{m}(\hat{r})$ – the charge
$q_{\scriptscriptstyle{p,n}}$ being $e$ for protons and 0 for neutrons –, as
well as the one-body second- and fourth-forbidden $\beta$-decay operators, as
introduced in Sec. II.2.

##  III Results

In this section we present the results of our SM calculations. First, we
compare theoretical and experimental low-energy spectroscopic properties of
the parent and daughter nuclei under investigation, namely 94Nb, 94Mo, 99Tc,
99Ru, 113Cd, 113In, 115In and 115Sn. Successively, we focus to study
$\beta$-decay properties between the ground states (g.s.) of these nuclei.

We point out that, rather than compare the experimental and calculated half-
lives, we consider the quantity log$ft$ to describe the strength of a
$\beta$-transition. This is defined as the logarithm of

$ft=\frac{\kappa}{\tilde{C}}\int^{w_{0}}_{1}p_{e}w_{e}(w_{0}-w_{e})^{2}F(Z,w_{e})dw_{e}~{}.$
(33)

It is worth to stress again that all the calculations have been performed
without any empirical renormalization of $H_{\rm eff}$, as well as without
resorting to quenching factors of the axial constant $g_{A}$.

###  III.1 Spectroscopy

Table 1: Theoretical versus experimental low-energy levels of the nuclei under investigation. | $J^{\pi}$ | Eth (MeV) | EExp (MeV)
---|---|---|---
94Nb | $6^{+}$ | 0.000 | 0.000
| $3^{+}$ | 0.014 | 0.041
| $4^{+}$ | 0.014 | 0.058
| $7^{+}$ | 0.017 | 0.079
94Mo | $0^{+}$ | 0.000 | 0.000
| $2^{+}$ | 0.836 | 0.871
| $4^{+}$ | 1.450 | 1.574
99Tc | $\frac{9}{2}^{+}$ | 0.000 | 0.000
| $\frac{7}{2}^{+}$ | 0.140 | 0.140
| $\frac{1}{2}^{-}$ | 0.794 | 0.143
| $\frac{5}{2}^{+}$ | 0.812 | 0.181
99Ru | $\frac{5}{2}^{+}$ | 0.000 | 0.000
| $\frac{3}{2}^{+}$ | 0.343 | 0.096
| $\frac{3}{2}^{+}$ | 0.433 | 0.321
| $\frac{7}{2}^{+}$ | 0.588 | 0.341
113Cd | $\frac{1}{2}^{+}$ | 0.000 | 0.000
| $\frac{11}{2}^{-}$ | 0.069 | 0.263
| $\frac{3}{2}^{+}$ | 0.019 | 0.299
| $\frac{5}{2}^{+}$ | 0.401 | 0.316
113In | $\frac{9}{2}^{+}$ | 0.000 | 0.000
| $\frac{1}{2}^{-}$ | 1.375 | 0.392
| $\frac{3}{2}^{-}$ | 1.613 | 0.646
| $\frac{5}{2}^{+}$ | 1.270 | 1.024
115In | $\frac{9}{2}^{+}$ | 0.000 | 0.000
| $\frac{1}{2}^{-}$ | 1.196 | 0.336
| $\frac{3}{2}^{-}$ | 1.445 | 0.597
| $\frac{3}{2}^{+}$ | 2.444 | 0.828
115Sn | $\frac{1}{2}^{+}$ | 0.068 | 0.000
| $\frac{3}{2}^{+}$ | 0.000 | 0.497
| $\frac{7}{2}^{+}$ | 0.233 | 0.612
| $\frac{11}{2}^{-}$ | 0.396 | 0.713

In Table LABEL:spec99 we compare the low-energy spectra of the nuclei under
investigation.

The calculated energy levels are in reasonable agreement with the
corresponding experimental states with a few exceptions, while the comparison
between theory and experiment for the observed $B(E2)$ (Table 2) is less
satisfactory for some transitions. In fact, except for the 99Tc, almost all
the other calculated $B(E2)$s underestimate the experimental ones.

Table 2:  Theoretical versus experimental ens  low-energy $B(E2)$ transition strengths, in W.u., of the nuclei under investigation. Nucleus | $J_{i}\rightarrow J_{f}$ | Theory | Experiment
---|---|---|---
94Mo | $2+\rightarrow 0^{+}$ | 7.9 | 26 (4)
| $4^{+}\rightarrow 2^{+}$ | 7.7 | 16.0 (4)
99Tc | $\frac{7}{2}^{+}\rightarrow\frac{9}{2}^{+}$ | 21 | 30 (19)
| $\frac{5}{2}^{+}\rightarrow\frac{9}{2}^{+}$ | 10.3 | 15.1 (5)
99Ru | $\frac{3}{2}^{+}\rightarrow\frac{5}{2}^{+}$ | 7.4 | 50.1 (10)
113Cd | $\frac{3}{2}^{+}\rightarrow\frac{1}{2}^{+}$ | 2 | 20 (8)
| $\frac{5}{2}^{+}\rightarrow\frac{1}{2}^{+}$ | 7.0 | 0.372 (25)
113In | $\frac{5}{2}^{+}\rightarrow\frac{9}{2}^{+}$ | 7.2 | 3.9 (4)
115In | $\frac{1}{2}^{+}\rightarrow\frac{3}{2}^{+}$ | 17 | 121 (23)
115Sn | $\frac{3}{2}^{+}\rightarrow\frac{1}{2}^{+}$ | 0.1 | 2.1 (6)
| $\frac{7}{2}^{+}\rightarrow\frac{3}{2}^{+}$ | 0.010 | 0.130 (4)

Here, it is worth to notice that a sound shell-model description of these
nuclei is not an easy task for different reasons, especially in a parameter
free calculation. In fact, we start from a 78Ni core, and, therefore, the
number of valence particle is sizeable, ranging from 16 to 37. Then, the
inclusion of the contributions from three-body diagrams is only the leading
order of many-body contributions, whose role grows when increasing the number
of valence nucleons. Besides this, Cd, In and Sn isotopes are at the limit of
the proton configuration space and, therefore, $Z=50$ cross-shell excitations
may play a relevant role in the renormalization of the electric-quadrupole
transition operator Coraggio et al. (2015, 2016). Actually, the enlargement of
the proton model space, in order to account explicitly for such excitations,
is out of the present computational resources since the dimensions of the
Hamiltonians to be diagonalized could reach $\approx 10^{13}$.

###  III.2 Forbidden $\beta$ decays of 94Nb, 99Tc, 113Cd, and 115In

We focus now on the properties of the $\beta$ decay between ground states, and
it is important to start by discussing the relevance of the CVC relations (See
Eq. 28) in determining the relativistic form factors.

As shown in Table 3, in the case of 94Nb, 99Tc second-forbidden $\beta$
decays, the relativistic form factors ${}^{V}F_{211}$ obtained using the bare
operator is zero since the SPMEs of the corresponding operator (Eq. II.2) are
identically zero in the model space. Even though the renormalization procedure
gives SPMEs different from zero, the value of the ${}^{V}F_{211}$ form factors
calculated using the effective operator has an opposite sign, and it is a
factor two smaller than the form factor obtained by using the CVC relation
(${}^{V}F_{211}^{CVC}$).

Table 3: 99Tc, 113Cd, and 115In $\beta$ decay relativistic form factors determined with and without resorting to the CVC relations, and the non-relativistic form factors connected with the relativistic ones by CVC. The values are in adimensional units. 94Nb | Bare | Effective
---|---|---
${}^{V}F_{211}$ | 0.000 | 0.009
${}^{V}F_{211}^{CVC}$ | -0.031 | -0.016
${}^{V}F_{220}$ | 0.304 | 0.164
99Tc | Bare | Effective
${}^{V}F_{211}$ | 0.000 | 0.008
${}^{V}F_{211}^{CVC}$ | -0.030 | -0.017
${}^{V}F_{220}$ | 0.286 | 0.161
113Cd | Bare | Effective
${}^{V}F_{431}$ | 0.0003 | -0.008
${}^{V}F_{431}^{CVC}$ | 0.032 | 0.015
${}^{V}F_{440}$ | -0.521 | -0.237
115In | Bare | Effective
${}^{V}F_{431}$ | -0.0004 | -0.009
${}^{V}F_{431}^{CVC}$ | 0.031 | 0.017
${}^{V}F_{440}$ | -0.473 | -0.267

As regards the fourth-forbidden form factors of 113Cd, it is interesting to
note that, in this case, the bare value of the relativistic form factor
${}^{V}F_{431}$ has the same sign of ${}^{V}F_{431}^{CVC}$, but it is two
order of magnitude smaller. The effect of the renormalization is remarkable,
but, as it happens for the relativistic form factor of the 94Nb, 99Tc decays,
the final result again has an opposite sign and is a factor two smaller with
respect to the one calculated with the form factor ${}^{V}F_{431}^{CVC}$.

The same considerations may be drawn for 115In, except that, using the bare
operator, the sign of ${}^{V}F_{431}$ form factor is not consistent with the
one obtained with the ${}^{V}F_{431}^{CVC}$ one.

These results point out to a problem in determining the relativistic form
factors within the non-relativistic reduction of the Dirac equation for the
nuclei under investigations. On the above grounds, in the following
calculation of the $\beta$ decay properties we use the CVC relations for the
relativistic form factors.

Table 4:  Theoretical and experimental log$ft$ values. Data are taken from Ref. ens . | Bare | Effective | Exp.
---|---|---|---
94Nb | 11.30 | 11.58 | 11.95 (7)
99Tc | 11.580 | 11.876 | 12.325 (12)
113Cd | 21.902 | 22.493 | 23.127 (14)
115In | 21.22 | 21.64 | 22.53 (3)

We start the discussion of our results from the comparison between our
calculated values of the log$ft$s and the experimental ones, as they are
reported in Table 4. There, we have reported the log$ft$s obtained using both
the bare second- and fourth-forbidden $\beta$-decay operators and the
effective ones. It is worth stressing again that, as we have reported in Sec.
II.3, our SM effective operators consists of one- and two-body components.

As can be seen, the results that are obtained by employing the bare
$\beta$-decay operators underestimate the experimental log$ft$s, a result that
is consistent with the general consideration that nuclear models, which
operate truncation of the Hilbert space of the single-nucleon configurations,
require the introduction of a quenching factor $q$ in order to reproduce the
experimental half-lives for allowed $\beta$-decay transitions (see for example
Refs. Martínez-Pinedo et al. (1996); Barea et al. (2015)).

On the other hand, the calculations employing the SM effective operators
provide results that substantially recover the gap with respect to the
experimental log$ft$s, a result that is consistent with our previous studies
of the allowed $\beta$-decay within the realistic shell model Coraggio et al.
(2017, 2019, 2022, 2024).

In order to discuss the role of the renormalization of the $\beta$-decay
operator on the calculation of the shape of forbidden $\beta$-decay energy
spectra, we evaluate the quenching factors that are needed to tune the axial
coupling constant $g_{A}$ in order to obtain the same results we have obtained
for the log$ft$s by employing the SM effective operators. Using Eq. (7), we
obtain that the quenching factors that reproduce the values in column
“Effective” in Table 4 are $q=0.27$, 0,50, 0.22, and 0.41 for 94Nb, 99Tc,
113Cd, and 115In decays, respectively.

In Fig. 2, the calculated and experimental normalized spectra of the second-
forbidden and fourth-forbidden decays, that are under our present
investigation, are reported. The available data are drawn, with the
corresponding errors, using red dots. It should be noted that the energy
spectra are normalized in the energy region of the available data, and that
those for 99Tc decay have been extracted from Fig. (5) in Ref. Paulsen et al.
(2023). Moreover, the experimental spectra of 99Tc and 113Cd are obtained
after an unfolding procedure which takes into account the detector response
function, as discussed in Ref. Belli et al. (2007) and Refs. Paulsen et al.
(2023, 2020), respectively. Conversely, in the energy spectrum of 115In the
detector response is not decoupled, leading to a more difficult direct
comparison. However, according to the results presented in Refs. Pagnanini et
al. (2023, 2024), the detector response effects are expected to be small and,
in any case, not significant for our purposes.

As regards 94Nb decay, there are no experimental results, at present, and they
are normalized in the full range of the kinetic-energy interval.

The calculated values are labelled and drawn as follows:

  * (I)

the calculated values, obtained using the bare operators, follow the dashed
blue line;

  * (II)

the spectra calculated using the SM effective operators are drawn with a
continuous black line;

  * (III)

finally, we report also the results coming out by using the bare operator, but
quenching the axial coupling constant $g_{A}$ with the $q$ factors that
reproduce the theoretical log$ft$s using the effective decay operators.

Figure 2:  Theoretical and experimental normalized $\beta$-spectra of 94Nb
(a), 99Tc (b),113Cd (c) and 115In (d) as a function of the electron kinetic
energy $\rm T_{e}$. The theoretical spectra are calculated with the bare
operator (blue dashed line), the SM effective operator (continuous black
line), and using the quenching factors for $g_{A}$ extracted from Eq. (34)
(green dashed line) (see text for details). The red dots corresponds to
experimental values, where they are available Paulsen et al. (2023); Belli et
al. (2007, 2019); Pagnanini et al. (2023).

From the inspection of Fig. 2, we can clearly assert that the theoretical
energy spectra, calculated employing the bare and the effective $\beta$-decay
operators, are in a very good agreement with the corresponding experimental
shapes Paulsen et al. (2023); Belli et al. (2007, 2019); Pagnanini et al.
(2023), for the forbidden $\beta$-decays of 99Tc, 113Cd, and 115In. Moreover,
all of them exhibit a small sensitivity to the renormalization of the
$\beta$-decay operators, with small differences appearing only at low energy
for all the decays ($\lessapprox 100$ keV).

As regards the energy spectra obtained with the bare operator and quenching
$g_{A}$, we see that they show a pronounced disagreement with the experimental
shapes, for 99Tc, 113Cd, and 115In, mostly in the low- and intermediate-energy
intervals.

These results lead us to two main conclusions.

First, the calculated shapes of the normalized energy spectra are
substantially insensitive to the renormalization of the forbidden
$\beta$-decay operator by way of many-body perturbation theory, that is our
approach to realistic shell model, and are in a good agreement with current
data.

Second, it seems that the mere renormalization of the axial coupling constant
$g_{A}$ by a quenching factor $q$ makes it difficult to provide simultaneously
better log$ft$s and shapes of the energy spectra which reproduce the observed
behavior.

It is worth mentioning that in Refs. Haaranen et al. (2016); Kostensalo and
Suhonen (2017); Paulsen et al. (2023) the authors carried out a study of the
sensitivity of their calculated spectra upon the renormalization of the axial
coupling constant, and they found a noticeable dependence of their results on
the choice of the quenching factor $q$.

Now, in order to reach a better insight of the results, we analyze the
different components of the shape factor defined in Eq. (7). This factor can
be divided in three components, the vector, axial and vector-axial terms,
namely

$C(w_{e})=C_{V}(w_{e})+C_{A}(w_{e})+C_{VA}(w_{e}),$ (34)

where $C_{V}$ contains the coupling constant $g^{2}_{V}$, $C_{A}$ contains
$g^{2}_{A}$ and $C_{VA}$ contains $g_{V}\cdot g_{A}$.

The integrated shape functions $\tilde{C}_{k}$ are reported in Table 5, as
well as the total value $\tilde{C}$ (see Eq.(7)) for all the decays under
investigation. As can be seen, at variance with what was observed in Ref.
Haaranen et al. (2017), for all the decays $\tilde{C}_{VA}$ is positive and
therefore it is summed in phase with the vector and axial components.

Actually, such a result is a consequence of the CVC theory, since if we do not
use the CVC relations to determine the relativistic form factor, the mixed
terms $\tilde{C}_{VA}$ for the cases under investigation become negative, and
very close in absolute value to the sum of the correspondent vector and axial
components. This is similar to the results reported in Ref. Haaranen et al.
(2017), and such a feature highlights the relevance of this form factor, as it
was also discussed in other papers Sadler and Behrens (1993); Kirsebom et al.
(2019); Fahlin Strömberg (2020); Kossert et al. (2022); Paulsen et al. (2023).

Even though it is small compared to the other form factors, the relativistic
one is relevant do determine $C(w_{e})$, and, therefore, shapes and half-
lives, since, at variance with non-relativistic form factors, it enters the
quantity $M_{K}(k_{e},k_{\nu})$ of Eq. (II.2) without any suppression
coefficient (for the explicit expression of $M_{K}(k_{e},k_{\nu})$ see Table 4
in Ref. Behrens and Bühring (1971)).

The positive values of $\tilde{C}_{VA}$ explain the stability of the shape
with respect to the renormalization of the decay operators. In fact, without
using the CVC relations, as a consequence of the delicate balance of the
vector, axial and vector-axial terms, it is obtained a shape of the energy
spectrum that is very sensitive to the renormalization procedure.

Table 5: Integrated shape functions $\tilde{C}$ of the studied transitions and their vector $\tilde{C}_{V}$, axial-vector $\tilde{C}_{A}$, and mixed components $\tilde{C}_{VA}$. Parent | Op | $\tilde{C}_{V}$ | $\tilde{C}_{A}$ | $\tilde{C}_{VA}$ | $\tilde{C}$
---|---|---|---|---|---
94Nb | Bare | $5.44\times 10^{-9}$ | $1.23\times 10^{-8}$ | $1.34\times 10^{-8}$ | $3.11\times 10^{-8}$
| Effective | $1.40\times 10^{-9}$ | $9.07\times 10^{-9}$ | $5.72\times 10^{-9}$ | $1.62\times 10^{-8}$
99Tc | Bare | $3.10\times 10^{-9}$ | $5.90\times 10^{-9}$ | $7.15\times 10^{-9}$ | $1.61\times 10^{-8}$
| Effective | $8.82\times 10^{-10}$ | $4.14\times 10^{-9}$ | $3.14\times 10^{-9}$ | $8.17\times 10^{-9}$
113Cd | Bare | $1.19\times 10^{-19}$ | $3.72\times 10^{-19}$ | $2.80\times 10^{-19}$ | $7.70\times 10^{-19}$
| Effective | $2.14\times 10^{-20}$ | $1.14\times 10^{-19}$ | $6.20\times 10^{-20}$ | $1.98\times 10^{-19}$
115In | Bare | $6.14\times 10^{-19}$ | $1.93\times 10^{-18}$ | $1.15\times 10^{-18}$ | $3.69\times 10^{-18}$
| Effective | $1.75\times 10^{-19}$ | $8.68\times 10^{-19}$ | $3.72\times 10^{-19}$ | $1.42\times 10^{-18}$

##  IV Conclusions and Outlook

This work is the first attempt to describe the features of forbidden
$\beta$-decays within the framework of the realistic shell model, without
resorting to any phenomenological quenching factor for the axial and vector
coupling constants.

Such a study represents not only a validation of our theoretical framework to
assess the reliability to predict $0\nu\beta\beta$ nuclear matrix elements
Coraggio et al. (2020a), but can give also useful information for the recent
experimental studies of the electron energy spectra of forbidden
$\beta$-decays.

First, we have verified the ability of our effective Hamiltonian and
transition operators by comparing the calculated low-energy spectra and $E2$
transition strengths of parent and daughter nuclei, involved in the forbidden
$\beta$-decays under consideration, with their experimental counterparts. The
comparison of the spectroscopic data with the corresponding experimental ones
is quite satisfactory, especially if we consider the large number of valence
particles – ranging from 16 to 37 –, which characterizes the nuclear system we
have investigated.

Then, we have calculated both the half-lives and the energy spectra of the
emitted electrons of the second-forbidden $\beta$-decay of 94Nb and 99Tc, and
the fourth-forbidden $\beta$-decay of 113Cd and 115In.

As regards the outcome of our calculation of the properties of the forbidden
$\beta$-decay processes under investigation, the results may be outlined as
follows:

  * •

The exam of the theoretical log$ft$s and the experimental ones shows that the
results that are obtained with bare operators always underestimate the data, a
feature that is resembling the problem of the quenching of $g_{A}$ in the
allowed $\beta$-decay transitions. The theory moves towards experiment by
employing the theoretical effective operators, as expected.

  * •

Starting from the wave functions that are obtained through the diagonalization
of our $H_{\rm eff}$, the shape of the calculated energy spectra is rather
insensitive to the choice of the $\beta$-decay operator, bare or effective,
and in both cases the reproduction of the observed normalized energy spectra
is more than satisfactory.

  * •

The latter result seems to be unrelated to the considerations about the
calculated log$ft$s, and the comparison with data. In fact, using the bare
operator, but introducing a quenching factor of the axial constant to improve
the reproduction of the experimental log$ft$s, it results in a distortion of
the shape of the energy spectra, that affects the agreement with the observed
ones.

Wrapping up the results we have obtained, we may say that the goal to obtain,
on the same footing, a better reproduction of half-lives and the shape of the
energy spectrum of the emitted electrons in forbidden $\beta$ decays, by
employing effective decay operators, is a delicate matter. As it has been
mentioned in Sec. III.2, such an issue was met also in other studies Haaranen
et al. (2016); Kostensalo and Suhonen (2017); Paulsen et al. (2023), where the
authors showed that, without a renormalization of the $\beta$-decay operator
that is framed in the many-body theory, the reproduction of the observed
properties of forbidden $\beta$ decays cannot rely only on the quenching of
$g_{A}$, and other empirical parameters should be considered.

It is worth pointing out that such an issue does not emerge in the study of
allowed $\beta$ decays, since the calculated energy spectrum does not depend
on the nuclear matrix element of the electroweak currents, and the most
relevant observable to be tested is the half-life.

Our considerations lead to suggest that the study of forbidden $\beta$-decay
processes could be a valuable tool to refine the theoretical knowledge of the
renormalization of transition operators, and to rule out models that could be
not reliable to predict the value of nuclear matrix elements for decays, such
as in the case of the $0\nu\beta\beta$ decay.

On the above grounds, we plan to extend the present work by studying forbidden
$\beta$ decays in other mass regions, close to nuclei that are candidates for
detecting the $0\nu\beta\beta$ decay. Another interesting subject could be to
tackle the forbidden $\beta$-decay problem starting from the derivation of
electroweak currents by way of the chiral perturbation theory, which represent
the new frontier to frame the nuclear many-body problem within its underlying
fundamental theory – the QCD –, and it is currently employed to investigate GT
transitions with different nuclear approaches Gysbers et al. (2019); King et
al. (2020); Baroni et al. (2021); Gnech et al. (2021); Gnech and Schiavilla
(2022); King et al. (2023); Coraggio et al. (2024).

##  V Acknowledgments

We thank J. Kostensalo, T. Miyagi, M. Biassoni, C. Brofferio, S. Ghislandi, A.
Nava and L. Pagnanini for useful comments and discussion. G. D. G.
acknowledges the support from the EU-FESR, PON Ricerca e Innovazione 2014-2020
- DM 1062/2021. We acknowledge the CINECA award under the ISCRA initiative and
under the INFN-CINECA agreement, for the availability of high performance
computing resources and support. This work is partly supported by the Czech
Science Foundation (Czech Republic) P203-23-06439S.

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# Vid3D: Synthesis of Dynamic 3D Scenes using 2D Video Diffusion

Rishab Parthasarathy  Zack Ankner  Aaron Gokaslan

###### Abstract

A recent frontier in computer vision has been the task of 3D video generation,
which consists of generating a time-varying 3D representation of a scene. To
generate dynamic 3D scenes, current methods explicitly model 3D temporal
dynamics by jointly optimizing for consistency across both time and views of
the scene. In this paper, we instead investigate whether it is necessary to
explicitly enforce multiview consistency over time, as current approaches do,
or if it is sufficient for a model to generate 3D representations of each
timestep independently. We hence propose a model, Vid3D, that leverages 2D
video diffusion to generate 3D videos by first generating a 2D ”seed” of the
video’s temporal dynamics and then independently generating a 3D
representation for each timestep in the seed video. We evaluate Vid3D against
two state-of-the-art 3D video generation methods and find that Vid3D is
achieves comparable results despite not explicitly modeling 3D temporal
dynamics. We further ablate how the quality of Vid3D depends on the number of
views generated per frame. While we observe some degradation with fewer views,
performance degradation remains minor. Our results thus suggest that 3D
temporal knowledge may not be necessary to generate high-quality dynamic 3D
scenes, potentially enabling simpler generative algorithms for this task.

Machine Learning, ICML, Computer Vision, 3D Video, Video Generation

##  1 Introduction

A key theme throughout the evolution of computer vision has been the modeling
of new modalities that better capture the complexity of the real world (Brooks
et al., 2024; Kerbl et al., 2023b; Mildenhall et al., 2020). A less explored
and currently emerging frontier in computer vision is modelling dynamic 3D
scenes instead of static 3D scenes. In this work, we tackle the challenge of
efficiently generating high-quality 3D videos.

Recent works have proposed methods for 3D video generation that fall into one
of two paradigms. In the first paradigm, classifier guidance from 2D video
models and static 3D scene models is leveraged to optimize a dynamic 3D scene
representation along multiple axes (Zhao et al., 2023; Bahmani et al., 2023).
In the second paradigm, a 3D representation is constructed for the first frame
and then deformed over time to be consistent with a 2D rendering of the scene
(Ren et al., 2023). While these results achieve impressive results, they still
have significant limitations.

Methods that rely on classifier guidance are extremely computationally
intensive, requiring hours to generate a single 3D video. In contrast, methods
that deform an initial 3D scene representation are more efficient, but they
instead require rigid temporal structure. Furthermore, both paradigms are very
technically complex, introducing many hyperparameters that must be tuned.

Figure 1: An overview of the Vid3D method. In stage 1, we generate a 2D video
from a reference image to seed the dynamics of the scene. In stage 2, we
generate multiple views for each timestep in the 2D video. In stage 3, we
train a Gaussian splat on the collection of views from each timestep.
Ultimately, each trained Gaussian splat represents a timestep in the 3D video.

In this work, we thus investigate the common assumption that 3D temporal
dynamics need to be explicitly modelled to perform 3D video generation.
Specifically, we consider whether a model can generate high-quality 3D videos
without explicitly enforcing consistency between multiple views over time. We
propose Vid3D, a simple end-to-end pipeline for synthesizing dynamic 3D scenes
from a single reference image. We generate 3D videos in three steps: (1)
generating a 2D video outline for the 3D video, (2) independently generating
multiple views for each timestep in the seed video, and (3) using the multiple
views to generate 3D representations of each frame. We evaluate Vid3D against
two state-of-the-art 3D video generation methods, finding that it achieves
comparable quality even without explicitly modelling the 3D temporal dynamics
of the scene. We further ablate key parameters of Vid3D to gain a deeper
understanding of the method. Overall, we challenge the current assumption that
methods require explicit temporal consistency to achieve SOTA results in the
important task of 3D video generation (Ren et al., 2023; Zhao et al., 2023).

##  2 Our Approach

We hypothesize that it is possible to generate dynamic 3D scenes from a source
image without explicitly modeling 3D temporal dynamics, meaning that it is not
necessary to enforce multi-view consistency across time. To create a model
capable of generating 3D videos without modeling 3D temporal dynamics, we
factorize the task into generating the 2D temporal dynamics of the scene and
then generating 3D representations of each timestep in the 2D scene.
Specifically, we propose a three-step process for generating 3D scenes from a
source image: (1) seeding the scene dynamics with a 2D video, (2) synthesizing
multiple views of the scene for each time-step (also known as a frame) in the
seed video, and (3) generating a 3D representation of each time step based on
the generated views. We present a diagram of our Vid3D method in Figure 1.

To construct the dynamics of the scene, which we refer to as _temporal
seeding_ , we query a 2D video model with the reference image. This seeding
provides us with a dynamic rendering of the object from a singular view. We
next transform the obtained 2D representation of the dynamic scene into a 3D
representation. As we do not enforce multi-view consistency across time, we
generate the 3D representation of each frame in our 2D seed independently of
all other frames. To generate 3D representations of each 2D frame we follow
the methodology proposed by Chen et al. (2024). For each seed frame, we
generate multiple views of the target object by querying a 2D video model
finetuned for multi-view generation on a curated subset of the Objaverse
dataset (Deitke et al., 2023). For each frame, we then train a Gaussian Splat
(Kerbl et al., 2023a) on the collection of generated views. The sequence of
Gaussian Splats ultimately defines a 3D video.

###  2.1 Model Choices

We use the Stable Video Diffusion model (Blattmann et al., 2023) to generate
2D temporal seeds. The model generates seeds that consist of 25 frames. The
seed generations are conditioned on a ”motion score” that controls the degree
of motion in the video; our main results use a motion score of 120. To
generate multiple views for each timestep in the seed video, we use a version
of Stable Video Diffusion fine-tuned for the multiview generation task on
Objaverse (Chen et al., 2024; Voleti et al., 2024; Deitke et al., 2023). We
select this model over Zero 1-to-3 as it generates higher quality multi-views
over a number of quantitative and qualitative metrics including human
evaluation (Liu et al., 2023; Chen et al., 2024; Voleti et al., 2024). By
default, this model generates 18 views that are uniformly sampled from a full
orbit with zero elevation along the azimuth. To obtain 3D representations from
the collection of views, we train Gaussian splats (Ren et al., 2023)
consisting of 100K splats that are optimized for 4K steps.

Figure 2: An example of multiple 2D renderings of 3D videos generated by our
method. The 3D videos are rendered from two different camera views (y-axis)
through time (x-axis). We observe consistency between different camera views
for the same timestep as well as plausible dynamics across time for the same
view.

###  2.2 Evaluation Setup

To evaluate Vid3D we use the benchmark proposed by Zhao et al. (2023), which
is composed of 24 high-quality reference images. This benchmark has been
adopted by recent 3D video papers as the standard evaluation framework (Zhao
et al., 2023; Ren et al., 2023). Following their evaluation procedure, to
measure the quality of each 3D video, we render 10 2D videos from ten
different uniformly sampled camera angles. As a quantitative evaluation
metric, we report the CLIP-I score, which is defined as the average cosine
similarity between the CLIP-features (Radford et al., 2021) of the reference
image and each frame in each 2D video rendering. Specifically, we use the
CLIP-ViT-Base with a patch size of 32 as the image encoder. As CLIP features
represent an embedding of an image in a semantically meaningful latent space,
this metric captures the average semantic similarity (and hence model
fidelity) across both time and camera angle. We compare Vid3D to both
Animate124 and DreamGaussian4D which represent the state-of-the-art for 3D
video generation.

##  3 Results and Discussion

###  3.1 Main Results

We provide 2D video renderings from two 3D videos generated by Vid3D in Figure
2. For each 3D video, we render 2D videos from two distinct camera views. For
both reference images, the renderings demonstrate subject consistency with the
original frame as well as interesting and plausible dynamic details such as
the cat raising and lowering its paws or the monkey pedaling the bike. In
addition to temporal consistency across time for the same view, there is
spatial consistency for different views from the same timestep; the subject
and the content agree across multiple views. While the capacity of Vid3D for
general 3D scene rendering is impressive, there are still weaknesses to the
model. As can be seen in the cat rendering, rendering quality degrades moving
from views aligned with the reference image to alternative camera views. We
believe this decrease in quality is not due to multi-view information being
dropped between timesteps, suggesting that the quality could be improved with
higher quality multi-view synthesis models.

Table 1: CLIP-I score for Vid3D compared to Animate124 and DreamGaussian4D,
showing that our model does not need 3D temporal dynamics to yield competitive
results.
Model | CLIP-I
---|---
Animate124 | 0.8544
DreamGaussian4D | 0.9227
Vid3D (Ours) | 0.8946

We also quantitatively compare our proposed Vid3D method to both Animate124
(Zhao et al., 2023) and DreamGaussian4D (Ren et al., 2023) in Table 1. While
Vid3D achieves a higher CLIP-I score than Animate124, it underperforms the
CLIP-I score of DreamGaussian4D. Although Vid3D does not achieve a new state-
of-the-art score, that Vid3D’s quantitative performance is competitive with
state-of-the-art 3D video baselines implies that 3D temporal dynamics may not
be fully necessary for generating dynamic 3D scenes. Instead, it is possible
to rely solely on 2D video model priors to ensure multi-view consistency
through time, which with further tuning, could achieve even greater
performance.

We present further qualitative results and an analysis of failure modes in
Appendix D.

###  3.2 Varying Number of Views

Table 2: CLIP-I score values for Vid3D for different numbers of views. This
result shows that reducing the number of views from 18 to 9 does not
significantly degrade performance, while further reduction does.
Number of views | CLIP-I
---|---
3 | 0.8532
9 | 0.8879
18 (Baseline) | 0.8946

We ablate how the number of views produced per timestep during the multi-view
synthesis process affects the quality of the resulting 3D generations. The
number of views generated determines the information content available when
training Gaussian splats and more views provides denser coverage of the scene.
While for capturing views of real objects, i.e., object views captured by
sensors, more views corresponds to higher quality 3D representations, our work
synthesizes views from a 2D video model and as such we investigate the effect
of view number on quality. We report the CLIP-I score for varying view numbers
in Table 2. We find that the CLIP-I score strictly decreases as the number of
views decreases. Furthermore, the decrease in CLIP-I score grows larger at
lower view numbers; there is only a drop of 0.0067 from 18 to 9 frames, but a
drop of 0.0347 from 9 to 3 frames. As compared to the baselines, even at 3
frames Vid3D is comparable to Animate124, only performing worse by 0.0012.

Figure 3: Rendering of singular frame from 3D videos generated using the same
reference image but trained with a varying number of synthesized views. There
are no perceivable differences between 18 and 9 views, but there is
significant degradation and noise using 3 views.

We provide a qualitative example of single timestep renderings from Gaussian
splats trained from varying number of views in Figure 3. There is no
perceivable degradation in quality when comparing 18 views to 9 views, but
there is a significant loss of quality for the model trained with 3 views.
Specifically, at 3 views all high frequency detail is lost and the boundary of
the object is poorly defined.

##  4 Conclusion

In this work, we systematically analyze an alternative to recent dynamic 3D
scene generation algorithms and challenge the assumption that 3D video
generation methods must explicitly model 3D temporal dynamic. We propose
Vid3D, a simple method for 3D video generation that generates 3D
representations for each timestep in the video independently of all other
timesteps. We find that this simpler methodology still holds up to the state-
of-the-art baselines of Animate124 and DreamGaussian4D. Vid3D achieves a
CLIP-I score of 0.8946 as compared to baseline scores of 0.8544 and 0.9227 for
Animate124 and DreamGaussian4D respectively. We ablate the effect that the
number of views generated for each timestep has on 3D scene quality, finding
that scene quality only significantly degrades for a very low number of views.
Overall, Vid3D is a simple alternative to current 3D video generation methods
and it achieves results consistent with the state-of-the-art even though it
does not explicitly model 3D temporal dynamics.

## Impact Statement

This paper presents work whose goal is to advance the field of Machine
Learning. With easier generation of dynamic 3D scenes, embodied intelligence
models can be trained more easily. This could result in societal benefits such
as better trained robotics and prosthetics. However, we also elect to also
point out the potential effects of this research on the creation of malicious
videos like deepfakes. We encourage further work along preventing this
malicious behavior for greater alignment of future models.

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##  Appendix A Implementation

We open-source our implementation here, which provides training scripts for
running Vid3D on an 8xA100 node.

##  Appendix B Related Work

Efficiently generating 3D scenes from 2D inputs has been a long-standing goal
in computer vision (Chang et al., 2015). To automate and thus scale the
building of static 3D scenes, researchers have recently leveraged advances in
generative 2D video models to generate 3D scenes by querying a video model for
multiple poses of said scene (Chen et al., 2024; Han et al., 2024). However,
these existing methods only generate static 3D data, meaning that there is no
temporal component.

#### Domain-Specific Datasets

Some existing highly domain-specific 3D video datasets (i.e. human bodies,
human faces, etc.) have also been manually curated using a large number of
sensors (Chatzitofis et al., 2020; Reimat et al., 2021; Yoon et al., 2021;
Pagés et al., 2021). However, the use of these dynamic 3D scenes is limited
due to their domain-specific nature. Additionally, it is hard to scale their
hardware-intensive video capture technology to produce more scenes.

#### Sim2Real

Outside of 3D and 4D video datasets, other works focus on developing sim2real
worlds, where robots learn real-world features in simulation (Zhang et al.,
2022; Du et al., 2021). These works develop worlds where robots can interact
in 3D space over time, but these 4D worlds require physics and temporal
correlations to be designed by hand, hence not being able to generate dynamic
scenes directly from a sample image.

#### Dynamic 3D Scenes

Finally, a number of papers have focused on synthesizing dynamic 3D scenes
either by using classifier guidance (Bahmani et al., 2023; Zhao et al., 2023)
or by training models that learn how to deform 3D objects (Ren et al., 2023).
As stated before, these methods either suffer from high computational cost
taking hours to render a single scene, or from rigid assumptions about the
temporal structure of 3D scenes, limiting these methods’ applicability for
modeling arbitrary dynamic 3D scenes. Our work instead leverages the strong
generalization capabilities of 2D video models to flexibly generate 3D scenes
without the need to train complex auxiliary models.

##  Appendix C Varying Scene Motion

Table 3: CLIP-I score values for Vid3D for different temporal seed motion
scores. This result shows that there is a slight loss in quality for scenes
with more motion.
Motion Score | CLIP-I
---|---
120 (Baseline) | 0.8946
160 | 0.8893
200 | 0.8897

We ablate how the degree of motion in the scene’s temporal seed (i.e., 2D
video outline) affects the quality of the resulting 3D generations. The model
we use for temporal seeding, Stable Video Diffusion, generates videos
conditioned on a ”motion score”, which corresponds to the time and spatial
average of the optical flow maps of the video. We thus vary the degree of
motion in temporal seeds by varying the motion score.

We quantitatively measure the impact of varied motion in terms of CLIP-I score
in Table 3. While the initial increase of conditioned motion score from 120 to
160 degrades the CLIP-I score of the renderings by 0.0053, there is almost no
difference between the CLIP-I metrics for motion scores of 160 and 200. This
result suggests that while there is some initial loss in quality due to
increased motion, past a given motion score the quality of Vid3D is relatively
robust to added motion.

Figure 4: Rendering of singular frame from 3D videos generated by the same
reference image but with different amount of motion synthesized. As desired,
the higher motion score video has higher variability, along with similar
rendering quality to the lower motion score, demonstrating robustness to
motion.

We provide a qualitative example of varying the motion score in Figure 4. For
each motion score, we render a 2D video from the 3D video using a fixed camera
view. We observe greater variability in poses between frames for the higher
motion score. Furthermore, the added motion is achieved without a noticeable
drop in rendering quality.

##  Appendix D Qualitative Analysis

Figure 5: A qualitative comparison of Animate124, DreamGaussian4D, and Vid3D
for a seed image of an astronaut riding a horse. Here, Vid3D both creates
accurate representations from multiple angles, but also does not recolor the
horse like Animate124 or have worse renders from a non-reference view like
DreamGaussian4D. Figure 6: A qualitative comparison of Animate124,
DreamGaussian4D, and Vid3D for a seed image of a dancing panda. Here, although
Vid3D includes the hallucination of a orange ball (discussed in Figure 10), in
comparison to DreamGaussian4D, Vid3D does not hallucinate only one ear to the
panda. Figure 7: A qualitative comparison of Animate124, DreamGaussian4D, and
Vid3D for a seed image of a space shuttle. Here, the quality of the exhaust is
higher in Vid3D, with the exhaust recoloring the rocket in Animate124 and the
space shuttle being deformed in the side angle in DreamGaussian4D. Figure 8: A
qualitative comparison of Animate124, DreamGaussian4D, and Vid3D a seed image
of a tiger playing the guitar. Here, we notice that our method has higher
coherence to the source image than Animate124, keeping the same style and not
distorting the shape of the tiger like DreamGaussian4D.

We compare the performance qualitatively of Vid3D to DreamGaussian4D and
Animate124 on four examples (Ren et al., 2023; Zhao et al., 2023) in Figure 5,
Figure 6, Figure 7, and Figure 8. We note that only four examples are chosen
because Animate124 requires 10 hours of textual inversion and 7 hours of
classifier guidance in order to produce a singular video, in comparison to the
ten minutes from our method.

We observe that Vid3D has high quality generated 3D videos on multiple axes.
First, the video hallucinates less than Animate124, which frequently changes
the color and appearance of objects, such as in the space shuttle and the
astronaut, along with higher resolution. Second, DreamGaussian4D has high
quality reference-view renders but struggles on alternate views, such as those
for the space shuttle. On the other hand, Vid3D achieves high performance over
all axes.

Figure 9: Further examples of high quality generations from Vid3D,
demonstrating generalization over multiple timesteps and views.

In Figure 9, we present further examples of generations from Vid3D, which
demonstrate strong coherence both temporally and over multiple views. This
demonstrates that Vid3D can learn strong 3D temporal priors even though it
solely relies on 2D temporal priors and 2D video diffusion models.

###  D.1 Failure Modes

Figure 10: A depiction of various failure modes for Vid3D. The first is poor
render quality, where the skeleton has its facial features blurred due to the
failure of the 3D modeling to appropriately represent it. The second is
hallucination, where a orange ball appears near the panda. The final is the
Janus problem, where the same object appears twice, which is a known problem
in 3D generation and once again appears (Armandpour et al., 2023).

We address various failure modes of Vid3D in Figure 10. The first is a lack of
quality in the generation of multi-views, which leads to issues like the face
of the skeleton not being properly rendered. We plan to hopefully address this
by potentially training our own multi-view generation model. The next issue is
hallucination, where new objects, like the orange ball are added to existing
scenes. This issue arises because of the flexibility of our algorithm–we do
not simply deform the scene. However, this means that occassionally, small
aberrations like the orange ball may appear. Finally, we encounter the Janus
problem common in 3D generation (Armandpour et al., 2023). For example, in the
third frame of the bike, the bike appears twice, which is an issue that can
only be addressed through better training of multi-view models.

These issues are all common issues faced by 3D generation models, and do not
take away from the value of this experiment. We thus still maintain that 3D
temporal knowledge may not be necessary for 3D video generation.