# Detailed study of decay properties of 7He nucleus states in ab initio
approach

D. M. Rodkin  Dukhov Research Institute for Automatics, 127055, Moscow, Russia
Moscow Institute of Physics and Technology, 141701 Dolgoprudny, Moscow Region,
Russia  Yu. M. Tchuvil’sky  Skobeltsyn Institute of Nuclear Physics, Lomonosov
Moscow State University, 119991 Moscow, Russia

###### Abstract

The energies and decay widths of the states of the exotic 7He nucleus are
studied in an ab initio approach. The spectrum of these states is calculated
using No-Core Shell Model and corresponding extrapolation procedure. Well-
proven on a large amount of data the Daejeon16 potential is used in the
calculations. The previously developed NCSM-based approach, which includes a
method for constructing the basis of orthogonalized functions of cluster
channels and a procedure for matching the cluster form factors obtained within
this method with the corresponding asymptotic wave functions, is applied to
compute the decay widths of the levels. The possibilities of the approach for
calculating the partial decay widths of nuclear states into various channels
strongly differing in type of fragmentation, spin, angular momentum of
relative motion, and amplitude are demonstrated. The results obtained are
compared with the results of other microscopic calculations.

##  I Introduction

In recent decades, research related to low-energy nuclear physics has shown
two basic trends.

In experimental research, there has been a rapid transition from the study of
stable and neighboring long-lived nuclei to the study of exotic, including
nucleon-unstable systems. The most important factor in the development of such
a research is the upgrading of existing and constructing of new radioactive
beam facilities. Programs for upgrading existing facilities are being
implemented in RIKEN (creation of RIBF), GANIL (SPIRAL2), GSI (FAIR), NSCL
(FRIB). Building of new centres for the study of radioactive isotopes: HIAF
and RAON is in progress.

In theoretical research, largely due to the rapid progress of computer
technologies, an increasing place is occupied by high-precision microscopic
approaches, in particular ab initio (from first principles) methods of
describing nuclear systems.

An important place among ab initio methods is taken by: different versions of
No-Core Shell Model (NCSM) ncsm1 ; ncsm2 ; ncsm3 ; dyt1 ; dr , Gamov Shell
Model (GSM) gamov1 ; gamov2 ; gamov3 , Green’s functions Monte Carlo Method
gfmc1 ; gfmc2 ; gfmc3 ; gfmc4 , the Coupled Cluster Method ccm1 ; ccm2 ,
Hyperspherical Harmonic Approach hsh1 ; hsh2 , In-Medium Similarity
Renormalization Group imsrg , Self Consistent Green’s Function Method scgf1 ;
scgf2 ; scgf3 , and Lattice Effective Field Theory for Multi-nucleon Systems
left1 ; left2 . These methods are based on realistic two-nucleon (NN) and
three-nucleon (NNN) potentials. These potentials could be derived from Chiral
Effective Field Theory dj16 ; machleidt1 ; machleidt2  or from nucleon
scattering data by the use of $J$-matrix inverse scattering method jisp1 .

At present, there is a process of a sharp increase in the attention of various
theoretical groups to the problems of describing unbound nuclear states within
the framework of ab initio approaches or methods approaching them in terms of
their theoretical level. This trend is clearly reflected in review johns .

In view of the mentioned above, the choice of the object of research and the
methodology for solving the problem posed in this work seems to be timely.

It should be noted that ab initio approaches focused on the discussed problem
are already presented in the literature. Among them the methods which combine
NCSM and Resonating Group Model (RGM) namely No-Core Shell Model / Resonating
Group Model (NCSM/RGM) ncsmrg1  and No-Core Shell Model with Continuum (NCSMC)
ncsmc0 ; ncsmc1 ; ncsmc2 ; ncsmc3 ; ncsmc4 ; ncsmc5 ; ncsmc6  seem to be the
most versatile. To describe resonances (including those undergoing
multiparticle decay) using calculations of scattering phase shifts the NCSM-
based SS-HORSE method was proposed sshorse1 ; sshorse2 . Some nuclear
resonances can be also studied with the use of mentioned above GSM gamov3 .

A significant place among the methods of high-precision description of unbound
states of nuclei is occupied by ab initio microscopic approaches, which are
not, in strict definition, ab initio, but accurately take into account almost
all dynamic properties of unbound systems. Such approaches are, in particular,
the Microscopic Cluster Model (MDM) desc1 ; desc2 ; desc3 , the version of RGM
kv  which exploits realistic NN-potentials, and the Fermionic Molecular
Dynamics (FMD) neff1 ; neff2 ; neff3 . Admittedly, all mentioned above
approaches aimed at describing unbound states can be applied to a very limited
number of nuclear resonances compare to the list of bound states whose total
binding energies (TBEs) and electromagnetic properties are described by
conventional ab initio methods, NCSM, for example.

In our previous papers our4 ; our5 ; our6 , an ab initio approach was
developed. It allows one to calculate the asymptotic characteristics of the
real decay of unbound and virtual splitting of bound states of nuclei namely
the total and partial decay widths and the asymptotic normalization
coefficients (ANCs) of certain channels, respectively. This approach includes
NCSM as a basic building block.

Both in the previous works and in the present one, at the first stage the
A-nucleon Schrödinger equation with realistic NN-potentials is solved using
the basis of totally antisymmetric A-nucleon oscillator wave functions (WFs).
M-scheme, in which one-nucleon functions are characterized by the angular
momentum $j_{i}$ and its projection $m_{i}$, is exploited. The discussed basis
is built to be complete up to the maximal total number of oscillator quanta
$N^{max}_{tot}$. The size of the used basis in the M-scheme in our
calculations is usually in the range $10^{8}\div 10^{9}$ components. The
binding energies and the WFs of the ground, excited, and resonance states of
nuclei are computed. In the current paper we use well-proven in calculating
the spectra of nuclei with mass A$\leq$16, their sizes and other
characteristics the Daejeon16 potential dj16 , which is based on Chiral
Effective Field Theory. Next, so-called Cluster Channel Orthogonalized
Functions Method (CCOFM) our1 ; our2 ; our3  is applied. The procedure, as a
whole, looks as follows. A basis of the orthogonalized cluster-channel WFs is
built by this method, and the eigenfunctions of using Hamiltonian computed in
the NCSM are projected onto the functions of this basis. At the third stage,
the functions obtained within the projecting procedure – so called cluster
form factors (CFFs) – are matched with the asymptotic WFs of the corresponding
channels.

The most important problem determining the prospects for the application of
the discussed approach as a whole is that the range of distances where
solutions of the Schrödinger equation are correctly described by NCSM
A-nucleon WFs expands proportionally to the size of the classically allowed
area of the harmonic-oscillator potential and therefore proportionally to
$[N_{max}^{tot}]^{1/2}$. So this range is somewhat limited. Does the CFF
obtained in these calculations reproduce the shape of the asymptotic WF at
inter-cluster distances where the nuclear interaction is negligible? The
results of our studies presented in Refs. our5 ; our6  give the answer yes to
this question for nuclei 7Li and 8Be which are close to 7He in mass, and not
only for one-nucleon, but even for cluster channels.

We also point out that the methodological scheme of this work includes a new
element in comparison with those discussed above. For a more precise
determination of the energies of the levels calculated in NCSM, the
extrapolation procedure presented in Ref. extrapA  is used.

In Sections II and III, we give a brief outline of the just described
methodology.

In this paper we present the results of its application to the detail
theoretical study of characteristics of 7He nucleus spectrum. The choice of
this object is determined by the following motivations. Despite the fact that
the 7He system has no bound states, there are experimental data on some of its
states. The prospects for a more detailed experimental study of this object
appear to be good. We have successful experience in studying nuclei of the
same and close mass.The literature contains theoretical works devoted to ab
initio studies of this object, which makes it possible to compare the results
of the approaches used in them with the results of this one.

##  II Outline of methodology of the approach

###  II.1 Total binding energy extrapolation procedure

In our previous works our4 ; our5 ; our6 , it was shown that even a not too
large deviation of the calculated resonance energy from the experimental one
leads to a noticeable change in the calculated value of the decay width. For
this reason, the use of experimental resonance energies in calculating decay
widths is preferable. In the framework of theoretical study of exotic nuclei
such as 7He, it is much more difficult to rely on experimental results due to
their incompleteness, unreliability or absence. So, the requirements for the
accuracy of calculating the level energies for this kind of problems are
stringent. At present, despite broad capabilities of the shell model
calculations, for neutron-rich nuclei such as 6He and 7He they are not fully
converged. The only way out of this situation is the use of one of the
extrapolation procedures. There are three widely known techniques, namely:
deep learning extrapolation tool based on the artificial neural networks (ANN)
extrap , or five-parameter ”Extrapolation A5” extrapA . or three-parameter
”Extrapolation B” extrapB  methods. The ”Extrapolation B” method adopts a
three-parameter extrapolation function that contains a term that is
exponential in total number of quanta above the minimum harmonic oscillator
energy configuration (cut-off parameter) $N^{*max}_{tot}$. The ”Extrapolation
A5” method adopts a five-parameter extrapolation function that contains a term
proportional to ($exp(-{N^{*max}_{tot}}^{1/2})$ in addition to the single
exponential term $exp(-N^{*max}_{tot}$) used in the ”Extrapolation B” method.
According to paper extrap  both these methods give comparable results with ANN
deep learning tool, at the same time they are much easier. ”Extrapolation A5”
method demonstrates slightly smaller deviation values, so in our work we used
this method for obtaining total binding energies of 7He and 6He states in
”infinite” shell-model basis. The extrapolation function proposed in Ref.
extrapA  depends on the five free parameters: $E_{\infty},a,c,d,k_{\infty}$
and also on parameters $b=\sqrt{{\hbar/}{m\omega}}$,
$\Lambda_{i}=b^{-1}\sqrt{2(N^{*max}_{tot,i}+3/2)}$,
$L_{i}=b\sqrt{2(N^{*max}_{tot,i}+3/2)}$,
$L_{t}=L_{i}+0.54437b(L_{i=0}/b)^{-1/3}$. It is written in the following form:

$E_{state}(N^{*max}_{tot,i},\hbar\omega)=E_{\infty}+a\cdot
exp(-c\Lambda_{i}^{2})+d\cdot exp(-2k_{\infty}L_{t})$ (1)

The values of free parameters are determined for each level independently by
fitting this function to theoretically calculated total binding energies. For
the 6He nucleus states we fit these parameters in the $\hbar\omega$ range from
10 to 25 MeV with 2.5 MeV step and $N^{*max}_{tot,i}$ = 4, 6, 8, 10, 12, 14.
For the 7He nucleus states the $\hbar\omega$ range is from 2.5 MeV to 22.5 MeV
with 2.5 MeV step and $N^{*max}_{tot,i}$ = 5, 7, 9, 11, 13. Changes in energy
values resulting from this fit are presented bellow in tables I – III.

###  II.2 Cluster channel orthogonalized basis constructing

The method of construction of the basis of the WFs of cluster channels is
described in detail in our previous works our1 ; our2 ; our3 ; our4 ; our5 ;
our6 , so here we demonstrate only its key points.

Let us consider translationally invariant A-nucleon WFs of arbitrary two-
fragment decay channel basis corresponding to the separation A = A1 \+ A2

$\Psi^{\\{k_{1},k_{2}\\}}_{{A\,},nlm}=\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A\,_{1}}\Psi^{\\{k_{2}\\}}_{A\,_{2}}\varphi_{nlm}(\vec{\rho})\\}_{J_{c},M_{J_{c}},M_{J}T},$
(2)

where $\hat{A}$ is the antisymmetrizer, $\Psi^{\\{k_{i}\\}}_{A\,_{i}}$ is a
translationally invariant internal WF of the fragment labelled by a set of
quantum numbers $\\{k_{i}\\}$; $\varphi_{nlm}(\vec{\rho})$ is the WF of the
relative motion. The channel WF as a whole is labelled by the set of quantum
numbers $c_{\kappa}$ that includes
$\\{k_{1}\\},\\{k_{2}\\},l,J_{c},M_{J_{c}},M_{J},T$, where $J_{c}$ is the
channel spin. The basic idea of the method is to represent each function of
the cluster basis as a linear combination of the functions of the M-scheme. To
do that function (2) is multiplied by the function of the center of mass (CM)
zero vibrations $\Phi_{000}(\vec{R})$. Then the transformation of WFs caused
by changing from $\vec{R},\vec{\rho}$ to $\vec{R}_{1},\vec{R}_{2}$ coordinates
– different-mass Talmi-Moshinsky-Smirnov transformation talmi-moshinsky  – is
performed and WF (2) takes the form

$\begin{array}[]{rcl}\Phi_{000}(\vec{R})\Psi^{\\{k_{1},k_{2}\\}}_{{A\,},nlm}=\sum\limits_{N_{i},L_{i},M_{i}}{\left\langle{{\begin{array}[]{*{20}c}{000}\\\
{nlm}\\\
\end{array}}}\mathrel{\left|{\vphantom{{\begin{array}[]{*{20}c}{000}\\\
{nlm}\\\ \end{array}}{\begin{array}[]{*{20}c}{N_{1},L_{1},M_{1}}\\\
{N_{2},L_{2},M_{2}}\\\
\end{array}}}}\right.\kern-1.2pt}{{\begin{array}[]{*{20}c}{N_{1},L_{1},M_{1}}\\\
{N_{2},L_{2},M_{2}}\\\ \end{array}}}\right\rangle}\\\\[12.0pt]
\hat{A}\\{\Phi_{N_{1},L_{1},M_{1}}^{A_{1}}(\vec{R}_{1})\Psi^{\\{k_{1}\\}}_{A\,_{1}}\Phi_{N_{2},L_{2},M_{2}}^{A_{2}}(\vec{R}_{2})\Psi^{\\{k_{2}\\}}_{A\,_{2}}\\}_{J_{c},M_{J_{c}},M_{J}T}.\end{array}$
(3)

The key technical procedure of the method is to transform each of the two
products of the internal WF of the fragment by the function of non-zero
oscillations of its CM into a superposition of Slater determinants (SDs)

$\Phi_{N_{i},L_{i},M_{i}}^{A_{i}}(\vec{R}_{i})\Psi^{\\{k_{i}\\}}_{A\,_{i}}=\sum\limits_{k}{X_{N_{i},L_{i},M_{i}}^{A_{i}(k)}\Psi_{A\,_{i}(k)}^{SD}}.$
(4)

It is the possibility of implementing this procedure that imposes restrictions
on the list of cluster channels available for research within the framework of
this method. Quantity $X_{N_{i},L_{i},M_{i}}^{A_{i}(k)}$ is called a cluster
coefficient (CC). Technique of these objects is presented in detail in nem .
Various approaches to calculating these coefficients, convenient in various
special cases, are presented in papers ar ; sa1 ; sa2 ; sa3 ; sa4  In the
present work we use the formalism based on the method of the second
quantization of the oscillator quanta described in details in our5 ; our6 . As
a result of these transformations, the antisymmetrized product of functions in
the right-hand side of expression (3) also turns out to be a superposition of
SDs.

It should be noted that WFs of cluster-channel basis terms (2) of one and the
same channel $c_{\kappa}$ characterized by the pair of internal functions
$\Psi^{\\{k_{1}\\}}_{A_{1}}$, $\Psi^{\\{k_{2}\\}}_{A_{2}}$ and extra quantum
numbers $l,J_{c},J,M_{J},T$ (i. e. the vector coupling is meant here) briefly
denoted as $\Psi^{c_{\kappa}}_{{A\,},n}$ are non-normalized due to the
properties of the antisymmetrization operator and, with rare exceptions, non-
orthogonal. Creation of orthonormalized basis functions of a separate channel
$c_{\kappa}$ is performed by the diagonalization of the overlap kernel matrix

$||N_{nn^{\prime}}||\equiv\langle\Psi^{c_{\kappa}}_{{A\,},n^{\prime}}|\Psi^{c_{\kappa}}_{{A\,},n}\rangle=$
$\langle\Phi_{00}(\vec{R})\Psi^{\\{k_{1}\\}}_{A_{1}}\,\Psi^{\\{k_{2}\\}}_{A_{2}}\,\varphi_{nl}(\rho)|\hat{A}^{2}|\Psi^{\\{k_{1}\\}}_{A_{1}}\,\Psi^{\\{k_{2}\\}}_{A_{2}}\,\varphi_{n^{\prime}l}(\rho)\Phi_{00}(\vec{R})\rangle.$
(5)

The eigenvalues and eigenvectors of this overlap kernel are the same in the
shell-model and translationally-invariant representations and can be written
as:

$\varepsilon_{\kappa,k}=\langle\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A_{1}}\,\Psi^{\\{k_{2}\\}}_{A_{2}}\,f_{l}^{k}(\rho)\\}|\hat{1}|\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A_{1}}\,\Psi^{\\{k_{2}\\}}_{A_{2}}\,f_{l}^{k}(\rho)\\}\rangle;$
(6)

$f_{l}^{k}(\rho)=\sum\limits_{n}{B_{nl}^{k}\varphi_{nl}(\rho)}.$ (7)

On the other hand, the WFs of the orthonormalized channel basis $c_{\kappa}$

$\Psi^{SD,c_{\kappa}}_{{A\,},kl}=\varepsilon^{-1/2}_{\kappa,k}|\Phi_{00}(\vec{R})\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A_{1}}\,\Psi^{\\{k_{2}\\}}_{A_{2}}\,f_{l}^{k}(\rho)\\}\rangle.$
(8)

turn out to be represented in the form of the superposition of the SDs. The
basis of such functions is complete in the sense that a function of this
channel

$\Psi^{SD,c_{\kappa}}=\Phi_{00}(\vec{R})\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A\,_{1}}\Psi^{\\{k_{2}\\}}_{A\,_{2}}\Phi(\rho)Y_{l}(\Omega)\\}_{J_{c},M_{J}T}$
(9)

including arbitrary WF $\Phi(\rho)$ can be represented as a superposition of
such WFs.

These functions can be introduced into the basis of the NCSM calculations for
its expansion towards large $N^{tot}_{max}$ or used to find the cluster
characteristics of nuclear states calculated in the NCSM basis, both extended
due to their inclusion, and the traditional one.

###  II.3 Cluster characteristics of in ab initio calculations

In this work, the energies of the levels of the initial and resulting nuclear
states are calculated in the usual M-scheme, and the basis described above is
used to calculate the cluster characteristics of decay channels: the cluster
form factor and the spectroscopic factor (SF).

The CFF $\Phi^{c_{\kappa}}_{A}(\rho)$ describes the relative motion of
subsystems in A-nucleon configuration space and is defined by the following
overlap

$\Phi^{c_{\kappa}}_{A}(\rho)=$
$\langle\Psi_{A}|\hat{N}^{-1/2}\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A\,_{1}}\Psi^{\\{k_{2}\\}}_{A\,_{2}}\frac{\delta(\rho-\rho^{\prime})}{\rho^{\prime
2}}Y_{l}(\Omega)\\}_{J_{c},M_{J}T}\rangle,$ (10)

where $\Psi_{A}$ is the WF of the initial nucleus – the NCSM solution of the
A-nucleon Schrödinger equation, and $\hat{N}$ is the norm operator of the
generalized function of the cluster channel which takes the form:

$\hat{N}=\langle\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A\,_{1}}\Psi^{\\{k_{2}\\}}_{A\,_{2}}\frac{\delta(\rho-\rho^{\prime})}{\rho^{\prime
2}}Y_{l}(\Omega)\\}_{J_{c},M_{J}T}|\times$
$|\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A\,_{1}}\Psi^{\\{k_{2}\\}}_{A\,_{2}}\frac{\delta(\rho-\rho^{\prime})}{\rho^{\prime
2}}Y_{l}(\Omega)\\}_{J_{c},M_{J}T}\rangle.$ (11)

Representation of the generalized function of the relative motion in the form
of an expansion in terms of oscillator functions

$[\delta(\rho-\rho^{\prime})/\rho^{\prime
2}]Y_{lm}(\Omega)=\sum\limits_{n}\varphi_{nlm}(\vec{\rho})\varphi_{nlm}(\vec{\rho}^{\prime})$
(12)

first, reduces the norm operator to the overlap kernel matrix (5) and, second,
makes it possible to write the CFF in the form

$\Phi^{c_{\kappa}}_{A}(\rho)=\sum\limits_{k}\varepsilon^{-1/2}_{\kappa,k}\langle\Psi_{A}|\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A_{1}}\,\Psi^{\\{k_{2}\\}}_{A_{2}}\,f_{l}^{k}(\rho^{\prime})\\}\rangle
f_{l}^{k}(\rho).$ (13)

After that the CCF can be expressed in the form of an expansion in the
oscillator basis using above presented techniques:

$\Phi^{c_{\kappa}}_{A}(\rho)=\sum\limits_{k}\varepsilon^{-1/2}_{\kappa,k}\sum\limits_{n,n^{\prime}}B_{nl}^{k}B_{n^{\prime}l}^{k}C_{AA_{1}A_{2}}^{n^{\prime}l}\varphi_{nl}(\rho)$
(14)

The coefficient contained in this expression has the form

$C_{AA_{1}A_{2}}^{nl}=\langle\hat{A}\\{\Psi^{\\{k_{1}\\}}_{A_{1}}\Psi^{\\{k_{2}\\}}_{A_{2}}\varphi_{nl}(\rho)\\}|\Psi_{A}\rangle=$
$\langle\Psi^{SD,c_{\kappa}}_{A,nl}|\Phi_{000}(R)|\Psi_{A}\rangle=\langle\Psi^{SD,c_{\kappa}}_{A,nl}|\Psi^{SM}_{A}\rangle.$
(15)

This coefficient is traditionally called the spectroscopic amplitude (SA). A
number of very diverse methods of its calculation depending on the masses of
the initial nuclei and fragments were described in nem ; sa1 ; sa2 ; sa3 . All
of them are based on the CCs formalism.

The SF is defined as the norm of CFF, for the discussed channel $c_{\kappa}$it
can be written as

$S_{l}^{c_{\kappa}}=\int{|\Phi^{c_{\kappa}}_{A}(\rho)|^{2}}\rho^{2}d\rho=$
$\sum\limits_{k}{\varepsilon_{k}^{-1}\sum\limits_{nn^{\prime}}{C_{AA_{1}A_{2}}^{nl}}C_{AA_{1}A_{2}}^{n^{\prime}l}B_{nl}^{k}}\;B_{n^{\prime}l}^{k}.$
(16)

It should be noted that an alternative method for calculating CFFs (and
obviously SF, if necessary) that does not use the formalism of cluster
coefficients is presented in the literature cffn .

The definitions of the CFF (10) and SF (16) are completely equivalent to those
proposed in newsf1  (the so-called “new” spectroscopic factor and CFF as
opposed to ”old” ones). In contrast to the traditional definition, the new CFF
and SF characterize the total contribution of orthonormalized cluster
components to the solution of the Schrödinger equation describing an A-nucleon
system. Reasons for the necessity of its use to describe decays and reactions
can be found in newsf2 ; newsf3 . In vt1 ; vt2 , it was shown that the correct
definition eliminates a sharp contradiction between theoretically calculated
values of the cross sections for reactions of knock-out and transfer of
$\alpha$ clusters and experimental data.

Here it is pertinent to make an essential remark. For the decay of the 7He
nucleus discussed here, only neutron channels are relevant. In contrast to
cluster channels, the ”old” definition is still often used for form factors
(FFs), SAs (14), and SFs of single-nucleon channels. This definition ignores
the necessary renormalization of any, including one-nucleon FF by the norm
kernel. This is due to a well-established tradition and the fact that the
numerical differences in the results of calculations in the ”old” and ”new”
definitions are usually not large for one-nucleon channels. We have to stress
that only “new” version is used in the current work and we consider this to be
the only correct approach.

###  II.4 Application of CFFs for studies of decay characteristics of nuclear
states

Both SFs and CFFs are the objects used in theoretical studies on nuclear
decays and reactions. Evidently CFF is more informative characteristic of a
cluster channel. In the current work the obtained CFFs are exploited for
computing the widths of resonances. The norms of these values – SFs – are used
to distinguish the main channels against the background of a multitude of
other ones, existence of which practically do not affect the results of
experiments.

As in our previous works our5 ; our6 , we use the procedure of matching the
CFF with the asymptotic WF of the corresponding channel. The results of our
studies presented in these works demonstrate that the CFF in its new
definition allows matching with the asymptotic WF at relatively small
distances, where the nuclear interaction is negligibly weak, but exchange
effects generated by the antisymmetry of the total channel WF and manifested
through the exchange terms of the overlap kernel are still not negligible.
”Switch off” the effects of antisymmetrization – the vanishing of the matrix
elements of the permutation operators included in the antisymmetrizer – occurs
at larger distances. This is precisely what constitutes the advantages of the
”new” definition as applied to the study of nuclear decays. This property is
very important for dealing with NCSM CFFs.

So in the discussed approach, a direct matching procedure described in
classical textbooks of quantum mechanics is applied to calculate the widths of
narrow resonances. For such resonances or, more precisely, for those of them
whose small width is due to a low penetrability of the potential barrier, we
used a very compact procedure proposed in matchingpoint . This procedure is
applicable because for such resonances there is sufficiently wide range of
distances in which the nuclear attraction is already switched off and at the
same time the potential barrier is high enough. At any inner point $\rho_{in}$
of this area, the relationship between the regular and irregular solutions of
the two-body Schrödinger equation in the Wentzel-Kramers-Brillouin (WKB)
approximation has the form

$F_{l}(\rho_{in})/G_{l}(\rho_{in})=P_{l}(\rho_{in})\ll 1,$ (17)

where $P_{l}(\rho_{in})$ is the penetrability of the part of the potential
barrier that is located between the point $\rho_{in}$ and the outer turning
point. The smallness of this penetrability is the condition of applicability
of the approximation in which the contribution of the regular solution can be
neglected. To determine the position of the matching point $\rho_{m}$ of the
CFF and irregular WF in this range, we use the condition of equality of the
logarithmic derivatives

$\frac{d\Phi_{A}^{c_{\kappa}}(\rho)/d\rho}{\Phi_{A}^{c_{\kappa}}(\rho)}=\frac{dG_{l}(\rho)/d\rho}{{G_{l}(\rho)}}.$
(18)

Comparison of the values of the CFF and function $G_{l}(\rho)$ in the matching
point allows one to determine the amplitude of the channel WF in the
asymptotic region which takes the form $\beta G_{l}(\rho)$, where

$\beta=\Phi_{A}^{c_{\kappa}}(\rho_{m})/G_{l}(\rho_{m}).$ (19)

As a result, the decay width is given by the expression

$\Gamma=\frac{{\hbar^{2}}}{{\mu
k}}\left[\frac{{\Phi_{A}^{c_{\kappa}}(\rho_{m})}}{{G_{l}(\rho_{m})}}\right]^{2}.$
(20)

To make the list of the properties of the states of a certain nucleus broader,
large-width resonances are considered too. If the resonance is wide and so the
penetrability $P_{l}(\rho_{in})$ (17) is not small the width of this resonance
is calculated using the simple version of the conventional R-matrix theory:

$\Gamma=\frac{\hbar^{2}}{\mu
k_{0}}(F^{2}_{l}(\rho_{m})+G^{2}_{l}(\rho_{m}))^{-1}(\Phi_{A}^{c_{\kappa}}(\rho_{m}))^{2}.$
(21)

Naturally the use of this version leads to reduction in accuracy of
calculation results. However, this technique seems natural for describing
experimental decay widths, since when extracting their values from the cross
sections of resonance processes, different versions of the R-matrix theory of
nuclear reactions are also used. Besides that, the accuracy of the data,
concerning large-width resonances, both decay widths and excitation energies,
extracted from various experiments, is also very limited. It is important to
note that, in contrast to calculating the total fragmentation width of any
resonant state, finding the partial widths of its decay into various channels
requires calculating the amplitudes of channel WFs in the asymptotic region.

In this paper the proposed approach is utilized to study the spectrum of
resonance states of the 7He nucleus and the partial decay widths of these
states.

An important point is that the Daejeon16 potential dj16  is exploited as a
model of NN-interaction in the current work. It is built using the N3LO
limitation of Chiral Effective Field Theory ceft1  softened by Similarity
Renormalization Group (SRG) transformation srg1 . This potential is designed
to calculate all kinds of characteristics of nuclei with the masses $A\leq
16.$ It was tested in the framework of large-scale computations of the total
binding energies, excitation energies, radii, moments of nuclear states and
the reduced probabilities of electromagnetic transitions. These tests
demonstrated that such characteristics are, in general, reproduced well.
Besides that this choice is supported by our previous studies of the
asymptotic characteristics of the cluster channels of light nuclei, in which
other NN-potentials were also involved in the analysis. The results of these
studies are presented in the Refs. our1 ; our3 ; our4 .

##  III Results of the calculations and discussion

In this work, we calculate the total binding energies (TBEs), excitation
energies, as well as decay energies and widths of the levels of 7He system;
and also calculate TBEs of the lower levels of the fragment 6He nucleus
necessary for that. The NCSM calculations were carried out with the use of
newest version of Bigstick shell model code bigstick . The basis is limited by
the value of cut-off parameter $N^{*max}_{tot}$=13 i. e. maximal total number
of oscillator quanta N${}^{max}_{tot}$ =16. For a limited basis, the optimal
value of the oscillator parameter turns out to be $\hbar\omega$=12.5 MeV.

Let us consider, first of all, the calculated values of the total binding
energy and the excitation energy of the 6He nucleus. They are presented in
Tab. I. Despite the fact that the ground state of the 6He nucleus is a classic
example of a nuclear system with a two-neutron halo, the experimental value of
its TBE ($E_{exp.}$) is well reproduced in the shell-model calculations on a
limited basis (values of such a type are denoted by symbol $E^{lim.}_{th.}$).
The extrapolated TBE value $E^{extr.}_{th.}$ deviates from the experimental
result even somewhat more significantly, but, in any case, the deviation does
not exceed 0.4%. The magnitude of the absolute deviation which is equal to 128
keV also appears to be satisfactory. This is not the case for the first
excited state, for which the deviation of the TBE calculation result from the
experimental one is 273 keV. This fact, and especially the significant
overestimation of the excitation energy for this level, demonstrates the need
to use an extrapolation procedure. As for the second excited state of the 6He
nucleus, its gigantic width ($\sim$ 12 MeV), obviously, does not allow
simultaneously evaluating its excitation energy from the experiment with an
accuracy better than several MeV.

Table 1: TBEs and excitation energies (MeV) of the lowest 6He nucleus states ($T$=1). $J^{\pi}$ |  $E_{exp.}$ exp7he  | $E^{lim.}_{th.}$ | $E^{extr.}_{th.}$ | $E^{*}_{exp.}$ | $E^{{}^{*}lim.}_{th.}$ | $E^{{}^{*}extr.}_{th.}$
---|---|---|---|---|---|---
0${}^{+}_{1}$ | 29.269 | 29.239 | 29.397 | 0 | 0 | 0
2${}^{+}_{1}$ | 27.472 | 27.199 | 27.533 | 1.797 | 2.040 | 1.864
2${}^{+}_{2}$ | 23.7 | 24.161 | 25.856 | 5.6 | 5.087 | 3.541

An even more expressive pattern is observed for the lower resonance level
3/2${}^{-}_{1}$ of the 7He nucleus and the higher resonance 5/2${}^{-}_{1}$,
for which experimental data have been obtained so far. For the former one the
calculations on the limited basis underestimate TBEs by 200 keV; for the
latter case, this underestimation exceeds 1 MeV. In contrast, the extrapolated
results are in good agreement with the measurement ones. The differences
between the results obtained on the limited basis and the extrapolation
results for other levels that have not yet been discovered also exceed 1 MeV.

In Tab. II and its caption, the energies of neutron decay of 7He nucleus into
the 6Hegs channels presented in published papers are also given, both
calculated and obtained in experiments. The experimental data are better
described by the GSM gamov3 , although the energy higher lying level is
reproduced rather well in all approaches. Our data correlate fairly well with
the data obtained within the framework of the NCSMC ncsmc1 . It is interesting
to compare the resonance energies which are calculated in the current work
with the same values obtained in the framework of approach SS-HORSE sshorse2 ,
since this approach, first, is also based on NCSM and, second, this work also
uses the NN-potential of Daejeon16. This comparison shows that the use of the
extrapolation procedure lowers the values of the resonance energies more than
the simultaneous use of a higher value of the cut-off parameter
$N^{*max}_{tot}$ and the inclusion of the continuous spectrum in the
calculations performed by SS-HORSE method.

Table 2: TBEs (MeV) and resonance energies (keV) of 7He nucleus states ($T$=3/2). a) – $E^{exp.}_{n}$=430 keV exp1 ; b) – $E^{exp.}_{n}$=3360 keV exp7he . $J^{\pi}$ |  $E_{exp.}$ exp7he  | $E^{lim.}_{th.}$ | $E^{extr.}_{th.}$ | $E^{extr.}_{n}$ |  $E_{n}$ncsmc1  |  $E_{n}$ gamov3  |  $E_{n}$sshorse2
---|---|---|---|---|---|---|---
3/2${}^{-}_{1}$ | 28.83a | 28.625 | 28.850 | 547 | 710 | 390 | 240
1/2${}^{+}_{1}$ | — | 26.057 | 27.701 | 1696 | — | — | —
1/2${}^{-}_{1}$ | — | 25.864 | 27.079 | 2318 | 2390 | — | 2700
5/2${}^{-}_{1}$ | 25.91b | 24.743 | 25.960 | 3437 | 3130 | 3470 | 3630
3/2${}^{+}_{1}$ | — | 24.115 | 25.905 | 3492 | — | — | 4100
5/2${}^{+}_{1}$ | — | 23.937 | 25.833 | 3564 | — | — | 4200
3/2${}^{-}_{2}$ | — | 23.966 | 25.455 | 3921 | — | — | —

The results of our calculations of the decay widths of 7He nucleus resonances
are presented in Tab. III. A wide list of decay channels was analysed,
including decays into channels in which fragment 6He is in 2${}^{+}_{1}$ or
2${}^{+}_{2}$ excited state. In some cases the partial widths of decay into
channels containing the former state are dominant, as can be seen from Tab.
III. Because of the gigantic decay width of the latter state, the results of
calculations of the characteristics of the channels in which it is contained
are physically of little interest. By including the channels containing this
state in the analysis, we, first, demonstrate the capabilities of the
developed method and, second, we follow the tradition laid down in Ref. ncsmc1
.

Table 3: Resonance energies, decay widths of open channels (keV), and channel
spectroscopic factors of 7He nucleus states. Numeric subscript denotes the
value $N^{*max}_{tot}$.
a) – for the experimental value of 3/2${}^{-}_{1}$ state resonance energy 430 keV computed value $\Gamma$ =250 keV; b) – there are other experimental results: 2.0 MeV exp2  and 10.0 MeV exp3 . $J^{\pi}$ (7He) |  $J^{\pi}$ (6He) | $E^{lim.}_{n}$ | $E^{extr.}_{n}$ | l(S) | SF | $\Gamma^{lim.}$ | $\Gamma^{extr.}_{11}$ | $\Gamma^{extr.}_{13}$ |  $\Gamma_{tot}$ ncsmc1  |  $\Gamma_{tot}$ gamov3  |  $\Gamma_{tot}$ sshorse2  | $\Gamma^{exp.}_{tot}$
---|---|---|---|---|---|---|---|---|---|---|---|---
3/2${}^{-}_{1}$ | 0${}^{+}_{1}$ | 614 | 547 | 1(1/2) | 0.730 | 387 | 336 | 334a | 300 | 178 | 110 | 182
1/2${}^{+}_{1}$ | 0${}^{+}_{1}$ | 3184 | 1696 | 0(1/2) | 0.844 | 3670 | 2670 | 2670 | — | — | — | —
1/2${}^{-}_{1}$ | 0${}^{+}_{1}$ | 3375 | 2318 | 1(1/2) | 0.814 | 2440 | 1940 | 1850 | 2890 | — | 4300 | 750b
| 2${}^{+}_{1}$ | 1335 | 454 |  | 1(3/2)
---
3(5/2)
| 0.509
---
0.21$\cdot 10^{-3}$
| 812
---
—–
| 253
---
0.47 eV
| 221
---
0.46 eV
— | — | — | —
5/2${}^{-}_{1}$ | 0${}^{+}_{1}$ | 4496 | 3437 | 3(1/2) | $0.37\cdot 10^{-3}$ | 110 eV | 56 eV | 52 eV | — | — | — | —
| 2${}^{+}_{1}$ | 2606 | 1573 |  | 1(3/2)
---
1(5/2)
| 0.420
---
0.758
| 1366
---
1783
| 980
---
1200
| 881
---
1060
1070 | 2300 | 1360 | 1990
3/2${}^{+}_{1}$ | 0${}^{+}_{1}$ | 5124 | 3492 | 2(1/2) | 0.041 | 125 | 84.4 | 83.4 | — | — | — | —
| 2${}^{+}_{1}$ | 3084 | 1628 |  | 0(3/2)
---
2(3/2)
| 0.752
---
0.051
| 3430
---
58
| 2590
---
22.3
| 2490
---
17.9
— | — | 4400 | —
5/2${}^{+}_{1}$ | 0${}^{+}_{1}$ | 5302 | 3564 | 2(1/2) | 0.126 | 382 | 285 | 258 | — | — | — | —
| 2${}^{+}_{1}$ | 3262 | 1700 |  | 0(5/2)
---
2(5/2)
| 0.704
---
0.019
| 3100
---
29
| 2210
---
13.7
| 2240
---
11.1
— | — | 5000 | —
| 2${}^{+}_{2}$ | 224 | 23 |  | 0(5/2)
---
2(3/2)
| 0.001
---
0.012
| 9.3
---
38.6 eV
| 3.33
---
0.15 eV
| 2.98
---
0.14 eV
— | — | — | —
3/2${}^{-}_{2}$ | 0${}^{+}_{1}$ | 5273 | 3921 | 1(1/2) | 0.069 | 276 | 249 | 229 | — | — | — | —
| 2${}^{+}_{1}$ | 3233 | 2057 |  | 1(3/2)
---
1(5/2)
| 0.190
---
0.470
| 556
---
1486
| 408
---
1150
| 399
---
1060
— | — | — | —
| 2${}^{+}_{2}$ | 196 | 380 | 1(3/2) | 0.268 | 44 | 126 | 108 | — | — | — | —

It is natural to start the discussion by assessing the accuracy of the method
used. Comparison of the results contained in the 3rd and 4th and,
respectively, in the 7th and 9th columns of Tab. III, demonstrates a strong
dependence of the decay width of a certain state on its energy. Roughly
assessing the tendency, one may say that the relative change in the decay
energy leads to a proportional change in the width. From this point of view,
the relative error in computing the decay energy value of a particular state
is reproduced when calculating its width. The relative decay energy errors of
ab initio computations on a limited basis for highly excited levels, are quite
large because decay energy values themselves are the difference between TBEs.
Therefore, the introduction of approaches that refine the energy values, in
particular the extrapolation method, is extremely important.

The size of the used basis affects the behavior of the CFF curves, which also
exerts some influence on the magnitude of the decay width, however this
influence is somewhat weaker in comparison with the effect produced by the
change in the decay energy. The scale of the change in the decay widths with a
change in the size of the basis from $N^{*max}_{tot}$ = 11 to
$N^{*max}_{tot}$= 13 is illustrated by columns 8 and 9 of Tab. III. The
computation results show that convergence has actually been achieved for a
significant part of the examples. Changes in decay widths for other examples
are in the region of 10%. Only in two cases does this change exceed 20%. There
is no doubt that this accuracy is sufficient for planning experiments aimed at
studying the spectra of exotic nuclei.

Comparison of the obtained results with experimental data leads to the
following conclusions. TBEs of the levels of 7He nucleus, values of which were
measured, are very well reproduced in the framework of the calculations.
Therefore the first source of the discrepancy between the calculated and
measured values of the decay width of the ground state of the 7He nucleus is
the overestimation of the binding energy of the 6He nucleus. This
overestimation, evidently, has nothing to do with the choice of the parameters
of the NCSM basis and, very likely, with the peculiarities of the
extrapolation procedure. Its causes are, most likely, the properties of the
Daejeon16 potential. The second source of the 30% overestimation of the decay
width, revealed by its calculation using the experimental decay energy, is not
clear, since all the conditions listed above for correct matching in this
example are satisfied, and the convergence of the result with respect to
$N^{*max}_{tot}$ is achieved. In principle, it can be assumed that this
discrepancy is generated by the procedure of extracting the width from the
experiment. Indeed, there are a variety of procedures used by evaluators to
determine the decay energy and width, and these procedures contain various
variation parameters. Thirty percent differences in the results of processing
the same experiments can be found in the same databases (see, for example,
exp7he ). On the other hand the GSM calculations gamov3  are in a good
agreement with the experimental data, while other theoretical methods show
overestimation or underestimation of 3/2${}^{-}_{1}$ resonance width. So, in
our opinion, the issue remains open.

The calculated values of the decay energy of state 5/2${}^{-}_{1}$ which is
equal to 3437 keV and its total width 1941 keV are in a very good agreement
with the experimental data. A good agreement was also achieved in the
calculations based on the GSM gamov3 .

The differences between the predicted values of the energies and decay widths
of the 7He nucleus states obtained by us and the results of other authors are
large enough. That makes future experiments exciting.

Table 4: Channel spectroscopic factors of closed and strongly suppressed decay channels of 7He nucleus states. Symbol * indicates closed channels $J^{\pi}$ (7He) |  $J^{\pi}$ (6He) | l(S) | SF
---|---|---|---
3/2${}^{-}_{1}$ | 2${}^{+*}_{1}$ |  | 1(3/2)
---
1(5/2)
3(3/2)
3(5/2)
| 0.691
---
0.254
0.0006
0.0046
| 2${}^{+*}_{2}$ |  | 1(3/2)
---
1(5/2)
3(3/2)
3(5/2)
| 0.338
---
0.0015
0.0003
0.00243
1/2${}^{+}_{1}$ | 2${}^{+*}_{1}$ |  | 2(3/2)
---
2(5/2)
| 0.0258
---
0.0198
| 2${}^{+*}_{2}$ |  | 2(3/2)
---
2(5/2)
| 0.0117
---
0.0017
1/2${}^{-}_{1}$ | 2${}^{+*}_{2}$ |  | 1(3/2)
---
3(5/2)
| 0.562
---
0.0004
5/2${}^{-}_{1}$ | 2${}^{+}_{1}$ |  | 3(3/2)
---
3(5/2)
| 0.0098
---
0.0036
| 2${}^{+*}_{2}$ |  | 1(3/2)
---
1(5/2)
3(3/2)
3(5/2)
| 0.641
---
0.604
0.0008
0.0001
3/2${}^{+}_{1}$ | 2${}^{+}_{1}$ | 2(5/2) | 0.0052
| 2${}^{+*}_{2}$ |  | 0(3/2)
---
2(3/2)
2(5/2)
| 0.124
---
0.014
0.0005
5/2${}^{+}_{1}$ | 2${}^{+}_{1}$ |  | 2(3/2)
---
4(3/2)
4(5/2)
| 0.0052
---
0.0006
0.0000
| 2${}^{+}_{2}$ |  | 2(5/2)
---
4(3/2)
4(5/2)
| 0.0079
---
0.0001
0.0002
3/2${}^{-}_{2}$ | 2${}^{+}_{1}$ |  | 1(5/2)
---
3(3/2)
3(5/2)
| 0.0025
---
0.0064
0.0002
| 2${}^{+}_{2}$ |  | 1(5/2)
---
3(3/2)
3(5/2)
| 0.0025
---
0.0064
0.0002

Approaching the conclusion, it should be emphasized that one of the main goals
of this work is to support experiments aimed at studying spectrum of 7He that
are carried out or planned. We believe that the data presented in Tab. III can
be used as preliminary for the analysis of the decay properties of these
states. For the same purposes, we consider it useful to supplement the tabular
data presented above with one more table. Tab. IV contains SFs of many
channels not discussed above. The data presented in the table indicate that in
order to analyse the spectrum of the nucleus, one can restrict oneself to the
channels, the parameters of which are presented in tab. III, since the other
channels are either closed or their SFs are small.

##  IV Conclusions

The main results of this work are the following.

I. Based on the successful application of the Cluster Channel Orthogonalized
Functions Method for describing 7Li and 8Be nuclei, total binding energies
(TBEs), excitation energies, as well as decay energies and widths of the
levels of 7He nucleus are calculated in this NCSM-based ab initio scheme.
Extrapolation A5 method is involved for obtaining TBEs with better precision.

II. The approach used allows one to solve multichannel problems, which made it
possible for the first time to calculate in an ab initio scheme not only the
total but the partial widths of decay of the 7He nucleus into many exit
channels.

III. The studies carried out demonstrate good prospects of the method used for
the theoretical study of neutron-rich nuclei, in particular, for predicting
the results of planned experiments.

IV. The prospects for using this approach in the field of studies of the
interaction of neutrons with light nuclei seem to be even wider.

##  V Acknowledgments

We are grateful to A. M. Shirokov, A. I. Mazur and I. A. Mazur for fruitful
discussions and providing us the Daejeon16 NN-potential matrixes, as well as
to C. W. Johnson for supporting our efforts to introduce high-performance code
Bigstick for our NCSM calculations.

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# HDR-VDP-3: A multi-metric for predicting image differences, quality and
contrast distortions in high dynamic range and regular content

Rafał K. Mantiuk
University of Cambridge
<EMAIL_ADDRESS>
Dounia Hammou
University of Cambridge
<EMAIL_ADDRESS>
Param Hanji
University of Cambridge
<EMAIL_ADDRESS>

###### Abstract

High-Dynamic-Range Visual-Difference-Predictor version 3, or HDR-VDP-3, is a
visual metric that can fulfill several tasks, such as full-reference
image/video quality assessment, prediction of visual differences between a
pair of images, or prediction of contrast distortions. Here we present a high-
level overview of the metric, position it with respect to related work,
explain the main differences compared to version 2.2, and describe how the
metric was adapted for the HDR Video Quality Measurement Grand Challenge 2023.

_Keywords_ Image Metric $\cdot$ High Dynamic Range

##  1 Introduction

High-Dynamic-Range Visual-Difference-Predictor version 3, or HDR-VDP-3 is an
image metric that can address multiple applications: the prediction of image
quality, visible differences, and contrast distortions. If we want to optimize
for image quality, for example, by selecting the right resolution and
compression configuration for video streaming, we want to use _a full-
reference quality metric_ that compares a distorted image/video (e.g., decoded
frame) to its reference and assesses the overall magnitude of introduced
distortion. However, if we want to ensure that the introduced distortions are
invisible, for example, for visually lossless compression, we may want to use
_a visibility metric_ , which predicts the probability of detecting
differences in each part of the image. In other cases, we may want to test an
image processing algorithm that modifies an image but should not introduce
disturbing artifacts. An example is tone mapping, where an image tone mapped
for a low-dynamic range display must be different from the input high-dynamic-
range image, but it should preserve the general visibility of the contrast.
For such applications, we want to use _a contrast distortion metric_. HDR-
VDP-3 addresses all three applications using the same core visual model but
uses different parameters and final processing stages to provide predictions
for each application.

This short paper is not meant to be a complete description of the metric but a
rather high-level overview with references to the relevant papers, which
provide further details. HDR-VDP-3 has the same processing pipeline as HDR-
VDP-2, which is explained in detail in (Mantiuk et al., 2011). In this short
paper, we first position HDR-VDP-3 with respect to other metrics (Section 2),
explain the list of tasks it can perform (Section 3), give a high-level
overview (Section 4), itemize the differences with respect to HDR-VDP-2
(Section 5) and finally explain how the metric was adapted to perform quality
assessment for the WACV HDR Video Quality Measurement Grand Challenge (Section
6).

##  2 Related metrics

HDR-VDP-3 is the third major iteration of the metric, which was originally
inspired by seminal works on visibility and detection metrics by Daly 1993,
Lubin 1995, and Watson 2000. The original HDR-VDP-1 (Mantiuk et al., 2005) was
an extension of the VDP by Daly 1993. The extension incorporated changes
allowing to compare high dynamic range images: the models of glare,
photoreceptor response (Mantiuk et al., 2006) (precursor of the PQ function
later used for HDR coding), and contrast sensitivity, which adapts to local
luminance. Similar to the VDP, this metric focused on predicting visible
difference maps, and it did not provide single-value quality predictions. This
was addressed in HDR-VDP-2 (Mantiuk et al., 2011), which was a major redesign
of the original VDP metric: it incorporated separate pathways for rod and cone
vision, replaced cortex transform with steerable pyramids (for performance and
accuracy), incorporated a new contrast masking model with intra- and inter-
channel masking, and provided the predictions of both visual difference maps
and image quality. But probably the most significant difference was that HDR-
VDP-2 was extensively recalibrated and tested on a large range of basic
psychophysical detection and discrimination data. When HDR-VDP-2 was released,
its quality predictions could be calibrated only on standard dynamic range
image datasets (TID2008 and LIVE) as no HDR quality datasets were available.
This was rectified in HDR-VDP-2.2 (Narwaria et al., 2015), which used two new
HDR datasets in addition to TID2008 and CSIQ to recalibrate quality
predictions.

A few important works led to the development of HDR-VDP-3. First, new
components were added to simulate the effect of aging on the visual system
(Mantiuk and Ramponi, 2018): the age-dependent model of glare, crystalline
lens aging and senile miosis (reduced pupil dilation in an older eye). Second,
a series of new measurements on an HDR display let us model the effect of
adaptation to local luminance (Vangorp et al., 2015). Finally, our effort to
combine multiple HDR and SDR datasets and bring them to the same quality scale
(Perez-Ortiz et al., 2020) let us recalibrate the metric on the largest HDR
image quality dataset (of over 4000 images) — UPIQ (Mikhailiuk et al., 2022).
Other major changes are discussed in Section 5.

A critical component of the metric is the perceptually uniform encoding of
luminance. Such encoding was shown to be an effective method of representing
and compressing HDR video (Mantiuk et al., 2004, 2006), and its refined
version was later standardized as a Perceptual Quantizer (SMPTE ST 2084)
(Miller et al., 2013). However, we have also demonstrated that such
perceptually uniform (PU) encoding can be used to adapt existing SDR quality
metrics to HDR images (Aydın et al., 2008; Mantiuk and Azimi, 2021).

Parallel to the work on visibility and quality predictions, we also worked on
predicting contrast distortions caused by tone-mapping (Aydin et al., 2008). A
modern implementation of this metric is one of the "tasks" of HDR-VDP-3
(Section 3).

Both us (Wolski et al., 2018; Ye et al., 2019a) and others (Banterle et al.,
2020) made an attempt to replace existing HDR-VDP-2 and HDR-VDP-3 metrics with
deep-learning architectures. Neural networks bring the advantage of
potentially faster processing speeds, no-reference predictions (Banterle et
al., 2020), higher accuracy, and easier re-calibration. Although deep-learning
metrics show promising results in selected applications, such as visually
lossless coding (Ye et al., 2019b), they are not explainable and often suffer
from over-fitting, as image and quality datasets are typically small in size,
and the measurements tend to be noisy.

More recently, we released Foveated Video VDP (FovVideoVDP) (Mantiuk et al.,
2021), a metric intended to predict quality in video, assuming a gaze point
(foveated viewing) or assuming that the user can look everywhere (as in
traditional video quality metrics). FovVideoVDP is a simplified version of
HDR-VDP-3, which adds temporal processing (sustained and transient visual
channels) and a contrast sensitivity function that accounts for the distance
from the gaze location (eccentricity). The new implementation runs on a GPU
(both in Matlab and Python/PyTorch) and offers much faster processing speeds.
However, because the new metric has not been calibrated on regular (non-
foveated) videos and contains multiple simplifications, it provides slightly
worse accuracy of predictions.

##  3 HDR-VDP-3 tasks

HDR-VDP-3 acts as a predictor of different quantities, depending on the "task"
parameter. The tasks include:

  * quality

— the prediction of a single-valued perceived degradation of quality measured
in the units of Just-Objectionable-Differences (JOD) (Perez-Ortiz et al.,
2020, Sec. IVb), which are related to the mean-opinion-scores. This task was
calibrated using UPIQ dataset (Mikhailiuk et al., 2022). The highest quality
is assumed to be 10. The degradation of 1 JOD unit corresponds to 75% of the
population noticing the difference between the pair of images. A further
explanation of the JOD scale can be found at
https://github.com/gfxdisp/FovVideoVDP#predicted-quality-scores and in (Perez-
Ortiz et al., 2020).

  * side-by-side

— the prediction of visible difference maps for pairs of test and reference
images presented side-by-side. The values in the map represent the proportion
of the population that is likely to notice the difference in a particular part
of the image. This task was calibrated using the datasets from (Wolski et al.,
2018; Ye et al., 2019a).

  * flicker

— similar to the "side-by-side" task, but for test and reference images that
are flipped (swapped) every 0.5 sec. This task was fine-tuned on a smaller
dataset, measured in a similar manner as in (Wolski et al., 2018)
(unpublished).

  * detection

— the detection task predicts the probability of detecting the difference
between two images (single-valued) and was calibrated on the same datasets as
HDR-VDP-2 — basic psychophysical detection and discrimination data for Gabor
patches, sinusoidal gratings, and discs. This task should provide better
accuracy for simple stimuli, but potentially lower accuracy for complex
images.

  * civdm

— a contrast distortion metric, which is a modern implementation of the
dynamic-range-independent visual quality assessment (Aydin et al., 2008). It
predicts the maps which indicate in which image parts the contrast will be
lost and in which parts it will be (over-)enhanced. The predictions are
different from those of (Aydin et al., 2008) as the original contrast-
independent metric was based on HDR-VDP-1.

##  4 HDR-VDP-3 overview

Figure 1: The processing diagram of the HDR-VDP-3. The metric requires the
images to be calibrated in the absolute radiance or luminance quantity emitted
from a display (HDR or SDR). The physical image representation (radiance map)
is processed by the optical and retinal pathway which simulates the eye’s
optics and photoreceptor responses. The resulting retinal images are then
decomposed into multiple scales, each isolating a band of spatial frequencies
and orientations. The core component of the metric is the model of contrast
masking and (neural) contrast sensitivity, which predicts the visibility of
the differences between a pair of images. The multi-band representation from
those stages is then fed to one of the different "heads", responsible for the
prediction of visibility, quality and contrast distortions.

A high-level overview of the metric is shown in Figure 1. As any full-
reference metric, HDR-VDP-3 takes as input a pair of test and reference
images. However, those images need to be calibrated in absolute radiometric
(or photometric) units by the _display model_ since HDR-VDP-3 relies on models
of low-level human vision, which operate on photometric units. The _retinal
and optical pathway_ simulates optics of the eye (glare), age-adaptive lens
opacity, pupil, photoreceptor (cones and rods) spectral response, local
adaptation and luminance masking. The resulting retinal images are then
decomposed into multiple bands of spatial frequencies and orientations. The
most important part of the metric is the model of neural contrast sensitivity
and contrast masking, which predicts the ability of the visual system to
detect and discriminate patterns. The result of that stage is then passed to
one of the three "heads" of the metric: one that predicts visibility maps, one
that predicts single-values quality and one that predicts contrast distortion
maps. The description of those stages is beyond the scope of this short paper,
but further details can be found in (Mantiuk et al., 2011; Aydin et al., 2008;
Mantiuk and Ramponi, 2018).

##  5 Differences with respect to HDR-VDP-2

Compared to version 2.2 of the metric, HDR-VDP-3 contains the following major
changes:

  * •

It requires specifying a prediction task, as explained in Section 3.

  * •

It includes a contrast distortion metric, which is a modern implementation of
(Aydin et al., 2008).

  * •

The contrast sensitivity function was refitted to newer data (from (Kim et
al., 2013)).

  * •

The model of glare (MTF) can be disabled or switched to the CIE99 Glare Spread
Function (Vos and van den Berg, 1999).

  * •

The metric now accounts for the age-related effects, as described in (Mantiuk
and Ramponi, 2018).

  * •

The metric includes a model of local adaptation from (Vangorp et al., 2015).

  * •

The tasks "side-by-side" and "flicker" have been calibrated on large datasets
from (Wolski et al., 2018; Ye et al., 2019a).

  * •

The task "quality" has been recalibrated using a new UPIQ dataset (Mikhailiuk
et al., 2022) with over 4000 SDR and HDR images, all scaled in JOD units.

  * •

The code now includes multiple examples of how to use the metric in different
scenarios.

  * •

The code has been reorganised and tested to run on a recent version of Matlab
(2022a) but also GNU Octave.

  * •

The code runs on a GPU (CUDA) in Matlab.

##  6 The submission for WACV HDR Video Quality Measurement Grand Challenge

HDR-VDP-3 was submitted to the WACV HDR Video Quality Measurement Grand
Challenge 2023. The organizers of the challenge provided a new HDR video
quality dataset — LIVE HDR (Shang et al., 2022).

For the challenge, we used the _quality_ task of HDR-VDP-3. We did not
consider the temporal aspect of the videos. The metric was applied separately
on selected frames, and the scores were averaged to obtain the final video
quality score. The FFmpeg program (Tomar, 2006) was used to decode every 30th
frame in the video and stored as PNG files. Furthermore, a display model
consisting of the inverse PQ function (Miller et al., 2013) was used to
transform the display-encoded pixel values into radiometric (or photometric)
units as:

$L(x,y)=PQ^{-1}(I(x,y))+E_{amb}\frac{k_{refl}}{\pi}\,,$ (1)

where $I(x,y)$ is the PQ-encoded RGB frame, $L(x,y)$ is the frame in absolute
linear RGB (BT.2020) units (photometric), $E_{amb}$ is the room ambient
illumination, and $k_{refl}$ is the display reflectivity. The room ambient
illumination and the display reflectivity were set to $200$ and $0.005$
respectively. These terms model the effect of ambient light, which was
reported for the experiment.

It should be mentioned that the HDR-VDP-3 _quality_ task has not been
(re-)calibrated on the LIVE HDR training dataset. We used the original model,
calibrated on the UPIQ dataset. Regardless, the metric was able to correlate
well with the mean-opinion scores.

Although the metric was originally implemented in Matlab, we adapted the code
so that it can be run in GNU Octave. The snippet of the code used to run the
metric can be found in `examples/hdr_video_pq_eotf.m` in release 3.0.7 of the
metric.

## ACKNOWLEDGMENTS

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement N∘ 725253–EyeCode).

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