# Systematic shell-model analysis of $2\nu\beta\beta$ decay of 76Ge and 96Zr
to the ground and excited states of 76Se and 96Mo

Deepak Patel  Department of Physics, Indian Institute of Technology Roorkee,
Roorkee 247667, India  Praveen C. Srivastava  Corresponding author:
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Roorkee, Roorkee 247667, India  Jouni Suhonen  University of
Jyvaskyla, Department of Physics, P.O. Box 35 (YFL), FI-40014, University of
Jyvaskyla, Finland  International Centre for Advanced Training and Research in
Physics (CIFRA),P.O. Box MG12, 077125 Bucharest-Măgurele, Romania

###### Abstract

In this work, we have studied the $2\nu\beta\beta$ decay of 76Ge and 96Zr
isotopes utilizing large-scale shell-model calculations. The GWBXG effective
interaction has been employed in the calculation of $2\nu\beta\beta$-decay
nuclear matrix elements (NMEs). We have tested the effective interaction by
comparing the predicted spectroscopic properties, such as energy spectra and
transition probabilities, with the available experimental data. The variation
of cumulative NMEs with respect to the $1^{+}$ state energies of the
intermediate nucleus is also studied, corresponding to $0^{+}_{\rm
g.s}\rightarrow 0^{+}_{\rm g.s}$, $0^{+}_{\rm g.s}\rightarrow 0^{+}_{2}$, and
$0^{+}_{\rm g.s}\rightarrow 2^{+}_{1}$ transitions between the parent and
granddaughter nuclei. The extracted half-lives using the shell-model predicted
NMEs show good agreement with the recent experimental data. The comparison of
the shell-model predicted NMEs with previous NMEs available in the literature
is discussed. Also, the computed branching ratios for the $2\nu\beta\beta$
decay of 76Ge and both the $2\nu\beta\beta$ and single-$\beta$ decay of 96Zr
are reported.

###### pacs:

21.60.Cs, 23.20.-g, 23.20.Lv, 27.60.+j

##  I Introduction

Double beta decay (DBD) is a rare second-order weak-interaction process
wherein a nucleus undergoes a simultaneous conversion of two neutrons into two
protons, or vice versa Deppisch ; GERDA ; Agostini ; Ackerman ; Adams ;
Elliott . This phenomenon, first proposed by Goeppert-Mayer in 1935 as a
nuclear disintegration Mayer , plays a significant role in nuclear and
particle physics. DBD occurs primarily in isotopes where single beta decay is
energetically forbidden or highly suppressed. This process can occur through
two decay modes: two-neutrino double beta decay ($2\nu\beta\beta$) and
neutrino-less double beta decay ($0\nu\beta\beta$). In the realm of double
beta decay studies, $2\nu\beta\beta$ decay offers a valuable assessment of the
standard model. It also confirms the validity of the weak nuclear forces and
the existence of neutrinos as weakly interacting particles. The
double-$\beta$-minus mode of this decay process, relevant for the present
studies, can be written as: $(A,Z)\rightarrow(A,Z+2)+2e^{-}+2\bar{\nu}_{e}$.
In previous years, the $2\nu\beta\beta$-decay process has been observed in
different mass regions of the nuclear chart for several nuclei. The study of
$2\nu\beta\beta$ decay mode also provides natural background components in
$0\nu\beta\beta$ measurements Chandra ; Ejiri . In the case of the
$0\nu\beta\beta$ decay process, which is not observed yet, neutrino emission
does not occur, thus violating lepton-number conservation Rodriguez ; Shimizu1
. Determination of the half-lives of $2\nu\beta\beta$ and $0\nu\beta\beta$
decaying nuclei imparts crucial details about the decay rates and lifetimes of
nuclei involved in the nucleosynthesis phenomenon Zuber . In the present
study, we consider only the $2\nu\beta\beta$ decay process.

Several experiments have been performed for the accurate half-life estimation
of the $2\nu\beta\beta$-decaying nuclei by different groups in the last few
decades. In the NEMO-3 experiment, Arnold $et$ $al.$ Arnold  measured the
half-life for $2\nu\beta\beta$ decay of 48Ca to the ground state (g.s.) of
48Ti as
$T_{1/2}^{2\nu}=[6.4^{+0.7}_{-0.6}\text{(stat)}^{+1.2}_{-0.9}\text{(syst)}]\times
10^{19}$ yr. Recently, Agostini $et$ $al.$ Agostini  extracted the half-life
of 76Ge for $2\nu\beta\beta$ decay to the g.s. of 76Se as $(2.022\pm
0.018_{\text{stat}}\pm 0.038_{\text{syst}})\times 10^{21}$ yr from the
GERmanium Detector Array (GERDA). From another NEMO-3 experiment, Arnold $et$
$al.$ Arnold1  studied the $2\nu\beta\beta$ decay in 82Se to the g.s. of 82Kr;
they have measured the half-life as $[9.39\pm 0.17\text{(stat)}\pm
0.58\text{(syst)}]\times 10^{19}$ yr and extracted the corresponding nuclear
matrix element $|M_{2\nu}|=0.0498\pm 0.0016$ using the value $g_{\rm A}=1.27$
for the weak axial coupling. In the $A=90$ mass region, 96Zr is a well-known
candidate of $2\nu\beta\beta$ decay. Using the NEMO-3 detector, the obtained
half-life of 96Zr in $2\nu\beta\beta$ decay to the g.s. of 96Mo is
$T_{1/2}^{2\nu}=[2.35\pm 0.14\text{(stat)}\pm 0.16\text{(syst)}]\times
10^{19}$ yr and the extracted NME using the above half-life and $g_{\rm
A}=1.25$ is $M_{2\nu}=0.049\pm 0.002$ Argyriades . Another candidate of
$2\nu\beta\beta$ decay, 100Mo has one of the largest decay energies,
$Q_{\beta\beta}=3034.36(17)$ keV NNDC_qcal  and the shortest half-life
Barabash . Recently, Armengaud $et$ $al.$ Armengaud  determined the
g.s.-to-g.s. decay half-life of 100Mo very precisely as
$T_{1/2}^{2\nu}=[7.12^{+0.18}_{-0.14}\text{(stat)}\pm\text{(syst)}]\times
10^{18}$ yr using the CUPID-Mo detection technology. In the Aurora experiment,
the half-life of the $2\nu\beta\beta$ transition of 116Cd to the $0^{+}_{\rm
g.s.}$ of 116Sn was measured with the highest accuracy as
$T_{1/2}^{2\nu}=(2.63^{+0.11}_{-0.12})\times 10^{19}$ yr Barabash2 . In
another work, the half-lives of two other $2\nu\beta\beta$ decaying
candidates, 128,130Te, were proposed as $(2.41\pm 0.39)\times 10^{24}$ and
$(9.0\pm 1.4)\times 10^{20}$ yr, respectively, using geological Te specimens
Meshik . In the NEXT Collaboration, Novella $et$ $al.$ Novella  extracted the
$2\nu\beta\beta$ half-life of 136Xe as
$2.34^{+0.80}_{-0.46}(\text{stat})^{+0.30}_{-0.17}(\text{syst})\times 10^{21}$
yr for the g.s. transition. After the nucleus 48Ca, 150Nd has the second
largest $Q_{\beta\beta}$ value (3371.38 keV NNDC_qcal ), making it an
intriguing nucleus for studying both $2\nu\beta\beta$ and $0\nu\beta\beta$
decay. Recently, Arnold $et$ $al.$ Arnold2  reported findings on the search
for $0\nu\beta\beta$ decay and measured the $2\nu\beta\beta$ decay half-life
of 150Nd as $T_{1/2}^{2\nu}=[9.34\pm
0.22\text{(stat)}^{+0.62}_{-0.60}\text{(syst)}]\times 10^{18}$ yr for the
g.s.-to-g.s transition in the NEMO-3 Collaboration.

The cumulative nuclear matrix elements $(M_{2\nu})$ for $2\nu\beta\beta$ decay
should show a significant contribution from the Gamow-Teller giant resonance
(GTGR) region. Nuclear models like proton-neutron quasi-particle random-phase
approximation (pnQRPA) take the GTGR region into account in a realistic way
Suhonen1 ; Terasaki . However, for the nuclear shell model the high energy in
the GTGR region remains a challenge and could only partly be described by two
earlier shell-model calculations in the case of 48Ca Kostensalo1 ; Horoi . In
the past, due to huge dimensions, it was difficult to perform large-scale
shell-model calculations for $2\nu\beta\beta$ decay studies in the medium to
heavier mass region. At present, it is possible to perform shell-model
calculations in extended valence spaces, including the relevant shell-model
configurations, due to significant progress in configuration mixing using
novel approaches and new advancements in the computational facilities. All
this could help us describe the GTGR region realistically within the shell-
model framework in the near future.

We have studied the $2\nu\beta\beta$ decay in several nuclei in our previous
work Patel1 , excluding 76Ge and 96Zr. Now, motivated by the recent
experimental data Agostini ; Finch ; Pritychenko  on the
$2\nu\beta\beta$-decaying nuclei 76Ge and 96Zr, we have performed large-scale
shell-model calculations for studying the behavior of cumulative NMEs with
respect to the $1^{+}$ state energies in the intermediate nucleus and
extracting the $2\nu\beta\beta$ half-lives of these two nuclei (using Eqs. 1
and 2 below). The contributions of the $1^{+}$ states of the intermediate
nucleus have been included up to the saturation level of NMEs. The present
work represents the most comprehensive set of shell-model results thus far for
the various $\beta$ and/or $2\nu\beta\beta$ transitions in 76Ge and 96Zr.

This paper is organized as follows: Sec. II provides a concise introduction to
the theoretical framework used in our calculations. Sec. III presents the
outcomes of the shell-model analyses and discussion regarding the energy
spectra, transition probabilities, NMEs, extracted half-lives, and branching
ratios. Finally, we summarize our findings and draw conclusions in Sec. IV.

##  II Theoretical Framework

###  II.1 Half-life

We can express the half-life for the $2\nu\beta\beta$ decay as follows

$(T_{1/2}^{2\nu})^{-1}=G^{2\nu}g_{\rm A}^{4}|M_{2\nu}|^{2}.$ (1)

Here, $G^{2\nu}$ denotes the phase-space factor Neacsu ; Stoica ; Kotila ;
Kotila1  and $g_{\rm A}$ is the axial-vector coupling strength Suhonen .
$M_{2\nu}$ is the nuclear matrix element (NME) for $2\nu\beta\beta$ decay and
first derived by Tomoda $et$ $al.$ Tomoda . We can express it by the following
expression Kostensalo

$M_{2\nu}=\sum_{N}\frac{\langle
J^{+}||\sigma\tau^{\pm}||1_{N}^{+}\rangle\langle
1_{N}^{+}||\sigma\tau^{\pm}||0_{\rm
g.s.}^{(i)}\rangle}{\sqrt{J+1}([\frac{1}{2}Q_{\beta\beta}+E(1^{+}_{N})-M_{i}]/m_{e}+1)^{k}},$
(2)

where $m_{e}$ represents the rest mass of the electron; $E(1^{+}_{N})-M_{i}$
corresponds the energy difference between the $N^{th}$ intermediate $1^{+}$
state and the ground state (g.s.) of the initial nucleus; $0_{\rm g.s.}^{(i)}$
stands for the ground state of the initial nucleus; $\sigma$ is the Pauli spin
operator; $\tau^{-}(\tau^{+})$ is the isospin lowering (raising) operator.
$Q_{\beta\beta}$ ($Q$-value) is the energy released in the decay. $\langle
J^{+}||\sigma\tau^{\pm}||1_{N}^{+}\rangle$ and $\langle
1_{N}^{+}||\sigma\tau^{\pm}||0_{\rm g.s.}^{(i)}\rangle$ denote the reduced
Gamow-Teller (GT) matrix elements and $J$ represents the spin state of the
final nucleus. Here, $k=1$ for $J=0$ and $k=3$ for $J=2$ Doi . The nuclear
energy and decay scheme for $2\nu\beta\beta$-minus decay is shown in Fig. 1.

Figure 1:  Nuclear level and decay scheme for $2\nu\beta\beta$-minus decay.

###  II.2 Shell-model Hamiltonian

We can express the nuclear shell-model Hamiltonian in terms of single-particle
energies and two-nucleon interactions as Patel2

$H=T+V=\sum_{\alpha}\epsilon_{\alpha}\hat{N}_{\alpha}+\sum_{\alpha\leq\beta,\gamma\leq\delta,J,M}\langle
j_{\alpha}j_{\beta}|V|j_{\gamma}j_{\delta}\rangle_{J}\times\\\
O_{\alpha,\beta}^{\dagger}(JM)O_{\gamma,\delta}(JM),$ (3)

where $\alpha=\\{nljt\\}$ stand for the single-particle orbitals with $n$
being the number of nodes of the wave function, $l$ the orbital angular
momentum, $j$ the total angular momentum, and $t=1/2$ the isospin. The
$\epsilon_{\alpha}$ represent the corresponding single-particle energies.
$\hat{N}_{\alpha}=\sum_{m_{\alpha}}c^{\dagger}_{\alpha,m_{\alpha}}c_{\alpha,m_{\alpha}}$
denotes the particle-number operator. The symbol $\langle
j_{\alpha}j_{\beta}|V|j_{\gamma}j_{\delta}\rangle_{J}$ is the two-body
interaction matrix element.
$O_{\alpha,\beta}^{\dagger}(JM)(O_{\gamma,\delta}(JM))$ corresponds to the
fermion pair creation (annihilation) operator and is written as

$O_{\alpha,\beta}^{\dagger}(JM)=\frac{1}{\sqrt{1+\delta_{\alpha\beta}}}\sum_{m_{\alpha},m_{\beta}}\langle
j_{\alpha}m_{\alpha}j_{\beta}m_{\beta}|JM\rangle\times\\\
c^{\dagger}_{\alpha,m_{\alpha}}c^{\dagger}_{\beta,m_{\beta}}.$ (4)

We have employed the GWBXG effective shell-model Hamiltonian in our
calculations, where its mean-field part consists of the $0f_{5/2}$,
$1p_{3/2}$, $1p_{1/2}$, and $0g_{9/2}$ proton orbitals, as well as $1p_{1/2}$,
$0g_{9/2}$, $0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, and $2s_{1/2}$ neutron
orbitals. This interaction is a composition of different interactions. The
initial 974 two-body matrix elements (TBMEs) are derived from the bare
$G$-matrix of the H7B potential Hosaka . The bare $G$-matrix is not reasonable
due to the space truncation, and the interaction should be renormalized by
considering the core-polarization effects. Here, the present $G$-matrix
effective interaction is tuned by further modification in matrix elements
using fitted interactions: The 65 TBMEs for proton orbitals are replaced with
the effective values reported in Ref. Ji . The TBMEs connecting the
$\pi(1p_{1/2},0g_{9/2})$ and the $\nu(1d_{5/2},2s_{1/2})$ orbitals are
replaced by those from the Ref. Gloeckner . Further, Serduke $et$ $al.$
Serduke  replaced the TBMEs between the $\pi(1p_{1/2},0g_{9/2})$ and the
$\nu(1p_{1/2},0g_{9/2})$ orbitals.

It is difficult to calculate all possible $1^{+}$ states of intermediate
nuclei in the full model space due to the involved huge dimensions. Thus, we
have employed truncation on neutron orbitals above $N=50$ for calculating the
$2\nu\beta\beta$-decay NME in 76Ge. Here, we have performed $1p-1h$ excitation
across $N=50$ to the $0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, and $2s_{1/2}$
neutron orbitals. For the calculation of the $2\nu\beta\beta$-decay NME in
96Zr, we have applied the truncation, where the considered proton and neutron
partitions belong to the
$\pi[(0f_{5/2}1p_{3/2})^{0-10}(1p_{1/2}0g_{9/2})^{0-4}]$ and
$\nu[(1p_{1/2}0g_{9/2})^{12-12}(0g_{7/2}1d_{5/2}1d_{3/2}2s_{1/2})^{0-20}]$
configurations, respectively. In both calculations, we have considered 5000
eigenvalues for $1^{+}$ states in 76As and 96Nb. The shell-model codes
NuShellX Nushellx  and KSHELL Kshell  have been used in the diagonalization of
the shell-model Hamiltonian matrices.

##  III Results and Discussion

In this section, we present the results obtained from shell-model
calculations. First, we discuss the theoretical and experimental energy
spectra and the corresponding quadrupole-reduced transition probabilities
($B(E2)$) for the parent, and granddaughter nuclei of interest in the
$2\nu\beta\beta$-decay study. We also show the variation of the shell-model
calculated cumulative NME as a function of the excitation energy of the
$1^{+}$ states of the intermediate nucleus. Using the final predicted NMEs, we
extract the $2\nu\beta\beta$\- decay half-lives of 76Ge and 96Zr. We also
compare our calculated NMEs with the previously available NMEs in the
literature. Lastly, we estimate the branching ratios for the $2\nu\beta\beta$
decay of 76Ge and for both the single-$\beta$ and $2\nu\beta\beta$ decay modes
of 96Zr.

Figure 2:  Comparison between experimental and shell-model calculated low-
lying states in 76Ge and 76Se.

###  III.1 Energy spectra and electromagnetic properties

Table 1: Comparison of theoretical and experimental $B(E2)$ strengths (in $e^{2}\textrm{fm}^{4}$) NNDC_NUDAT  in the parent and grand-daughter nuclei of $2\nu\beta\beta$ decay. The effective charges are taken as $e_{\pi}=1.6e,e_{\nu}=0.8e$ Honma . Isotope | $J_{i}^{\pi}\rightarrow J_{f}^{\pi}$ | Theory | Expt.
---|---|---|---
76Ge |  $2^{+}_{1}$$\rightarrow$$0^{+}_{1}$ | 330.7 | 554.5(191)
|  $4^{+}_{1}$$\rightarrow$$2^{+}_{1}$ | 435.4 | 726.6(1721)
76Se |  $2^{+}_{1}$$\rightarrow$$0^{+}_{1}$ | 553.3 | 841.4(191)
|  $4^{+}_{1}$$\rightarrow$$2^{+}_{1}$ | 811.3 | 1357.7(382)
|  $6^{+}_{1}$$\rightarrow$$4^{+}_{1}$ | 921.9 | 1300.3(1530)
|  $8^{+}_{1}$$\rightarrow$$6^{+}_{1}$ | 961.4 | 1568.0(3251)
|  $10^{+}_{1}$$\rightarrow$$8^{+}_{1}$ | 1031.2 | 994.4(2103)
96Zr |  $2^{+}_{1}$$\rightarrow$$0^{+}_{1}$ | 62.2 | 60.1(78)
|  $3^{+}_{1}$$\rightarrow$$2^{+}_{1}$ | 0.1 | 2.6(+78-26)
|  $8^{+}_{1}$$\rightarrow$$6^{+}_{4}$ | 39.6 | 36.0(29)
96Mo |  $2^{+}_{1}$$\rightarrow$$0^{+}_{1}$ | 266.6 | 540.5(104)
|  $4^{+}_{1}$$\rightarrow$$2^{+}_{1}$ | 307.6 | 1070.5(1828)
|  $3^{+}_{1}$$\rightarrow$$2^{+}_{1}$ | 1.0 | $<33.9$
|  $6^{+}_{1}$$\rightarrow$$4^{+}_{1}$ | 243.8 | $<75.7\times 10^{2}$

In Fig. 2, we depict the comparison between the shell-model and experimental
low-energy states of the 76Ge and 76Se isotopes. It is clear from the spectra
that our calculated states are in quite good agreement with the experimental
data for both isotopes. In 76Ge, the shell-model predicted positive-parity
yrast states (g.s. band) up to $8^{+}_{1}$ are characterized by mainly
[$\pi(f_{5/2}^{4})\otimes\nu(g_{9/2}^{6})$] configuration, whereas the
$10^{+}_{1}$ state stems from
$[\pi(f_{5/2}^{2}p_{3/2}^{2})\otimes\nu(g_{9/2}^{6})\approx 25.3\%]$ and
[$\pi(f_{5/2}^{4})\otimes\nu(g_{9/2}^{6})\approx 21.8\%$] configurations.
Here, the large energy gap between the $8^{+}_{1}$ and $10^{+}_{1}$ states
compared to the other two consecutive yrast states may be caused by the
excitation of two protons from the $\pi(f_{5/2})$ orbital to the
$\pi(p_{3/2})$ orbital in the dominant configuration of the $10^{+}_{1}$
state. Contrariwise, the yrast $0^{+}_{\rm g.s.}-10^{+}_{1}$ states (g.s.
band) in 76Se are described by same dominant configuration
$\pi(f_{5/2}^{2}p_{3/2}^{2}g_{9/2}^{2})\otimes\nu(g_{9/2}^{4})$. The excited
states in the yrast band of 76Ge and 76Se show a pronounced collective
behavior. The experimental data for $B(E2)$ transitions in the yrast band of
76Ge are available only for $2^{+}_{1}-0^{+}_{1}$ and $4^{+}_{1}-2^{+}_{1}$
transitions. As reported in Table 1, except for the $10^{+}_{1}-8^{+}_{1}$
transition in 76Se, our calculated $B(E2)$ transitions for both isotopes are
slightly smaller but consistent with the experimental data.

Figure 3:  Comparison between experimental and shell-model calculated low-
lying states in 96Zr and 96Mo.

Fig. 3 shows a comparison between the shell-model predicted and the
experimental low-lying energy states in 96Zr and 96Mo. The $0^{+}_{\rm g.s.}$
in 96Zr arises primarily due to the pure
$\pi(g_{9/2}^{2})\otimes\nu(d_{5/2}^{6})$ configuration, whereas the
$0^{+}_{2}$ and $2^{+}_{1}$ states stem from the proton-aligned configuration
$\pi(g_{9/2}^{2})$ and the collective contribution
$\nu(d_{5/2}^{4}g_{7/2}^{2})$ of neutron orbitals, although the wavefunctions
of both states are quite fragmented. Experimentally, the nucleus 96Zr exhibits
a small $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$ transitition strength,
verifying the similar value obtained in our calculation. Structure change
between the involved initial and final states may cause this small $B(E2)$
value. Similarly, we obtain different structures in the largest configuration
of the $8^{+}_{1}$ and $6^{+}_{4}$ states, namely
$[\pi(g_{9/2}^{2})\otimes\nu(g_{7/2}^{2}d_{5/2}^{4})]$ and
$[\pi(p_{1/2}^{2})\otimes\nu(g_{7/2}^{1}d_{5/2}^{5})]$, due to which we record
a small $B(E2;8^{+}_{1}\rightarrow 6^{+}_{4})$ value, similar to the
experimental one (see Table 1). In the case of 96Mo, the g.s. sequence
$0^{+}_{1}-8^{+}_{1}$ shows similar structure with the dominant configuration
$\pi(g_{9/2}^{2})\otimes\nu(d_{5/2}^{4})$ predicted by the shell model. Thus,
as reported in Table 1, the obtained $B(E2)$ values are not weak for the
$2^{+}_{1}\rightarrow 0^{+}_{1}$, $4^{+}_{1}\rightarrow 2^{+}_{1}$, and
$6^{+}_{1}\rightarrow 4^{+}_{1}$ transitions but still they are not even half
of the experimental values. The choice of effective charges might be a reason
for relatively smaller $B(E2)$ values, but this is outside the scope of the
present study. However, overall, the computed $B(E2)$ values of Table 1 are in
quite a good agreement with the experimental data.

After analyzing these results for energy spectra and transition probabilities,
we are confident that the present shell-model Hamiltonian can be safely
employed in calculating the NMEs ($M_{2\nu}$) for our $2\nu\beta\beta$-decay
study.

Figure 4:  Cumulative $2\nu\beta\beta$ NMEs ($M_{2\nu}$) for 76Ge as functions of excitation energy of the intermediate $1^{+}$ state in 76As. Table 2: Shell-model calculated $2\nu\beta\beta$ NMEs and the extracted half-lives. The phase-space factors $(G^{2\nu})$ are taken from Ref. Neacsu . Transition | $|M_{2\nu}|$ |  $G^{2\nu}$ (yr-1) | $g_{\rm A}^{\rm eff}$ | Calculated $T_{1/2}^{2\nu}$ (yr) |  Experimental/Recommended value of $T_{1/2}^{2\nu}$ (yr)
---|---|---|---|---|---
${}^{76}\text{Ge}(0^{+}_{\rm g.s.})\rightarrow$${}^{76}\text{Se}(0^{+}_{\rm g.s.})$ | 0.2285 | 4.51$\times 10^{-20}$ | 0.80 | 1.037$\times 10^{21}$ |  $(2.022\pm{0.018}_{\text{stat}}\pm{0.038}_{\text{syst}})\times 10^{21}$ Agostini
${}^{76}\text{Ge}(0^{+}_{\rm g.s.})\rightarrow$${}^{76}\text{Se}(0^{+}_{2})$ | 0.2414 | 6.4$\times 10^{-23}$ |  | 6.546$\times 10^{23}$ |  $>6.2\times 10^{21}$ Klimenko
${}^{76}\text{Ge}(0^{+}_{\rm g.s.})\rightarrow$${}^{76}\text{Se}(2^{+}_{1})$ | 0.0045 | 4.0$\times 10^{-22}$ |  | 3.014$\times 10^{26}$ |  $>1.1\times 10^{21}$ Barabash1
${}^{96}\text{Zr}(0^{+}_{\rm g.s.})\rightarrow$${}^{96}\text{Mo}(0^{+}_{\rm g.s.})$ | 0.0599 | 642.0$\times 10^{-20}$ | 1.00 | 4.341$\times 10^{19}$ |  $(2.35\pm 0.14(\text{stat})\pm 0.16(\text{syst}))\times 10^{19}$ Argyriades
${}^{96}\text{Zr}(0^{+}_{\rm g.s.})\rightarrow$${}^{96}\text{Mo}(0^{+}_{2})$ | 0.0159 | 1633.8$\times 10^{-22}$ |  | 2.421$\times 10^{22}$ |  $>3.1\times 10^{20}$ Finch
${}^{96}\text{Zr}(0^{+}_{\rm g.s.})\rightarrow$${}^{96}\text{Mo}(2^{+}_{1})$ | 7.4$\times 10^{-4}$ | 730.8$\times 10^{-21}$ |  | 2.499$\times 10^{24}$ |  $>4.1\times 10^{19}$ Arpesella

###  III.2 Cumulative nuclear matrix elements and the extracted half-lives

Here, we discuss the shell-model calculated results for NMEs and half-lives
for the $2\nu\beta\beta$ decays of 76Ge and 96Zr. The calculated NMEs and
half-lives are reported in Table 2. We have taken the $Q_{\beta\beta}$ and
$Q_{\beta}$ values from Ref. NNDC_qcal . In the case of $0^{+}_{\rm
g.s.}\rightarrow 0^{+}_{2}$ and $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$
transitions, the $Q_{\beta\beta}$ values are taken as the difference of the
$Q_{\beta\beta}$ value and the experimental energies of $0^{+}_{2}$ and
$2^{+}_{1}$ states, respectively. The variation of cumulative NMEs as a
function of $1^{+}$ state energies of the intermediate nucleus is depicted in
Figs. 4-5.

Table 3: Contribution in the final NME from the first $1^{+}$ state of the intermediate nucleus. Transitions | Value | Percentage (%)
---|---|---
${}^{76}\text{Ge}(0^{+}_{\rm g.s.})\rightarrow$${}^{76}\text{Se}(0^{+}_{\rm g.s.})$ | 0.1275 | 55.8
${}^{76}\text{Ge}(0^{+}_{\rm g.s.})\rightarrow$${}^{76}\text{Se}(0^{+}_{2})$ | 0.0630 | 26.1
${}^{76}\text{Ge}(0^{+}_{\rm g.s.})\rightarrow$${}^{76}\text{Se}(2^{+}_{1})$ | 0.0043 | 95.6
${}^{96}\text{Zr}(0^{+}_{\rm g.s.})\rightarrow$${}^{96}\text{Mo}(0^{+}_{\rm g.s.})$ | 0.0596 | 99.5
${}^{96}\text{Zr}(0^{+}_{\rm g.s.})\rightarrow$${}^{96}\text{Mo}(0^{+}_{2})$ | 0.0032 | 20.1

76Ge: A careful and precise calculation of NMEs ($M_{2\nu}$) holds significant
importance in the theoretical analysis of the $2\nu\beta\beta$ decay of a
nucleus. In the case of 76Ge, we have calculated $M_{2\nu}$ for three
transitions: $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$, $0^{+}_{\rm
g.s.}\rightarrow 0^{+}_{2}$, and $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$. In
the present calculation, the excitation energies of $1^{+}$ states in 76As
were shifted such that the lowest-lying $1^{+}$ state is at the experimental
energy of 0.044 MeV. As reported in Table 3, the first $1^{+}$ state of the
intermediate nucleus contributes 55.8% of the final NME (0.2285), yielding a
value of 0.1275 for the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$
transition. In this case, we can see from Fig. 4 that the cumulative NME peaks
around 4.35 MeV, with a value of $0.2671$. As mentioned earlier, the pnQRPA
calculations realistically incorporate the contribution of the GTGR region to
the NMEs Suhonen1 ; Terasaki . But, to the best of our knowledge, previous
shell-model calculations were able to describe this region partly in the case
of only 48Ca Kostensalo1 ; Horoi . In our case for the $0^{+}_{\rm
g.s.}\rightarrow 0^{+}_{\rm g.s.}$ transition, the peak is found at the
$85^{th}$ intermediate $1^{+}$ state; however, this energy (4.35 MeV) is
notably below the energy of the GT resonance ($\approx 11$ MeV) Thies .
Comparison of our results with those of the large-scale shell-model
calculations of Kostensalo et al. Kostensalo  shows that the calculation of
Kostensalo  does not show a prominent peak at this energy, and only a small
bump at around 5 MeV of excitation can be discerned. After around 12.75 MeV,
the cumulative NMEs for the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$
transition start to saturate and show almost constant behavior with a final
value of 0.2285. Similarly, for the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$
and $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$ transitions, the cumulative NMEs
nearly saturate after around 11.69 and 7.55 MeV with the final values 0.2414
and 0.0045, respectively. Using Eq. (1) and utilizing the final value of NME
(0.2285) for the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$ transition,
together with $g_{\rm A}^{\rm eff}=0.8$ Suhonen , we have extracted the half-
life of 76Ge as $1.037\times 10^{21}$ yr, which shows good agreement with the
experimental half-life Agostini  (see Table 2). In the case of $0^{+}_{\rm
g.s.}\rightarrow 0^{+}_{2}$, and $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$
transitions, the cumulative NMEs show a very similar behavior as recorded in
the shell-model calculation of Kostensalo . The corresponding calculated half-
lives are $6.546\times 10^{23}$ and $3.014\times 10^{26}$ yr, respectively.
Both estimates are notably higher than the experimental lower limits
$6.2\times 10^{21}$ Klimenko  and $1.1\times 10^{21}$ yr Barabash1 . These
results may be useful for experimentalists in estimating the sensitivities of
their experimental set-ups to these $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$,
and $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$ transitions in the future.

Figure 5:  Cumulative $2\nu\beta\beta$ NMEs ($M_{2\nu}$) for 96Zr as functions
of excitation energy of the intermediate $1^{+}$ state in 96Nb.

96Zr: Because of the large $Q$-value (3356.03 keV), 96Mo has several excited
states through which 96Zr could undergo $2\nu\beta\beta$ decay. Here, we have
taken three transitions $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$,
$0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$, and $0^{+}_{\rm g.s.}\rightarrow
2^{+}_{1}$ from the parent to granddaughter nucleus for calculating the
$M_{2\nu}$ and extracting the $2\nu\beta\beta$-decay half-lives, like in the
case of 76Ge. There is no $1^{+}$ state having been experimentally confirmed
for 96Nb. Thus, shell-model computed excitation energies are used to calculate
the NMEs. In our calculation, we found that the contribution in the cumulative
NME from the first $1^{+}$ state of the intermediate nucleus for the
$0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$ transition is 0.0596,
accounting for 99.5% of the final value (0.0599). Thus, it demonstrates a
clear single-state dominance (SSD), consistent with the previous study
Kostensalo1 . On the contrary, the $1^{+}_{1}$ state of 96Nb contributes 20.1%
to the final NME (0.0159) for the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$
transition. Interestingly, the $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$
transition exhibits SSD with a value 0.00077 of the cumulative NME, which is
slightly higher than the final NME (0.00074). Therefore, we have not reported
its contributive percentage in Table 3.

The extracted half-lives of 96Zr using the final NMEs for the $0^{+}_{\rm
g.s.}\rightarrow 0^{+}_{\rm g.s.}$, $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$,
and $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$ transitions are $4.341\times
10^{19}$, $2.421\times 10^{22}$, and $2.499\times 10^{24}$ yr, respectively.
Here, we have taken $g_{\rm A}^{\rm eff}=1.00$ Arnold3 . However, utilizing
$g_{\rm A}^{\rm eff}=1.04$ as proposed in Ref. Kostensalo1 , we would obtain a
half-life for the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$ transition
slightly closer to the experimental value. The computed half-lives for the
$0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$ and $0^{+}_{\rm g.s.}\rightarrow
2^{+}_{1}$ transitions are notably higher than the lower limit of experimental
values, which could be useful in estimating experimental sensitivities to
these transitions in the future.

Table 4: Comparison between our calculated and previously obtained absolute values of NMEs. Here SM denotes the nuclear shell model, IBM the interacting boson model, HFB the Hartree-Fock-Bogoliubov theory, and (R)QRPA the (renormalized) quasiparticle random-phase approximation. For more values of NMEs of older calculations, see Suh1998 . Transitions | $|$ NME $|$
---|---
| Present work | SM Kostensalo  | IBM Nomura  | HFB Dhiman  | RQRPA Toi1997  | QRPA Stoica1
${}^{76}\text{Ge}(0^{+}_{\rm g.s.})\rightarrow$${}^{76}\text{Se}(0^{+}_{\rm g.s.})$ | 0.2285 | 0.168 | 0.034 | 0.879 | 0.074 | -
${}^{76}\text{Ge}(0^{+}_{\rm g.s.})\rightarrow$${}^{76}\text{Se}(0^{+}_{2})$ | 0.2414 | 0.121 | 0.078 | - | 0.130 | -
${}^{76}\text{Ge}(0^{+}_{\rm g.s.})\rightarrow$${}^{76}\text{Se}(2^{+}_{1})$ | 0.0045 | 0.0012 | - | 1.04 | 0.003 | -
${}^{96}\text{Zr}(0^{+}_{\rm g.s.})\rightarrow$${}^{96}\text{Mo}(0^{+}_{\rm g.s.})$ | 0.0599 | - | 0.154 | - | 0.036 | 0.022
${}^{96}\text{Zr}(0^{+}_{\rm g.s.})\rightarrow$${}^{96}\text{Mo}(0^{+}_{2})$ | 0.0159 | - | 0.063 | - | 0.028 | 0.012
${}^{96}\text{Zr}(0^{+}_{\rm g.s.})\rightarrow$${}^{96}\text{Mo}(2^{+}_{1})$ | 7.4$\times 10^{-4}$ | - | - | - | 0.010 | 8.1$\times 10^{-5}$

###  III.3 Comparison of NMEs

Here, we discuss the comparison between our shell-model predicted and earlier
suggested NMEs using different nuclear models, as reported in Table 4. We
notice that the previously determined NMEs for 76Ge exhibit either a notably
small value (0.034) Nomura  or significantly large value (0.879) Dhiman  with
respect to correctly reproducing the experimental half-life for the
$0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s}$ transition. The NME (0.168)
obtained in Ref. Kostensalo  is approximately 1.36 times smaller than our
calculated NME, and the one of Toi1997  (0.074) is about half of the one of
Kostensalo . We have allowed $1p-1h$ excitations across the neutron-shell
closure $N=50$, which is not considered in the previous study Kostensalo . It
is a possible cause for the relatively larger value of the NME in our case. We
have accounted for the contribution of a huge number of excited $1^{+}$ states
in the intermediate nucleus 76As, hence, it is reasonable to involve also the
contribution of those $1^{+}$ states which are arising due to the $1p-1h$
excitations across the $N=50$ shell gap. Notably, the final NME for the
$0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$ transition is larger than for the
ground-state transition. Similar results concerning the calculated NMEs for
the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$ and $0^{+}_{\rm
g.s.}\rightarrow 0^{+}_{2}$ transitions were also found in previous studies,
utilizing the interacting boson model (IBM) Nomura  and the renormalized
quasiparticle random-phase approximation (RQRPA) Toi1997  for 76Ge, as also
the quasiparticle random-phase approximation (QRPA) Jokiniemi  for 136Xe.

Previously, Šimkovic $et$ $al.$ Simkovic  proposed that the NME for the
$0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$ transition in $2\nu\beta\beta$
decay of 76Ge is suppressed because of the different deformations of the
parent and granddaughter nucleus. We can deduce the deformation parameter
$(\beta_{2})$ from the intrinsic quadrupole moment ($Q_{0}$). The $\beta_{2}$
value is related to the $Q_{0}$-moment by the following expression Kilgallon

$\beta_{2}=\frac{\sqrt{5\pi}}{3ZR^{2}}Q_{0}.$ (5)

Here, for simplicity, we have taken $R$ as $1.2\times A^{1/3}$ fm. Initially,
we determined the $Q_{0}$-moment using the spectroscopic quadrupole moment,
$Q_{s}(2^{+}_{1})=-28.714$ efm2 in 76Ge obtained from our shell-model
calculation, employing the formula $Q_{0}=-(7/2)Q_{s}$ Simkovic  which arises
from the expression Kilgallon

$Q_{s}=\frac{3K^{2}-I(I+1)}{(I+1)(2I+3)}Q_{0},$ (6)

by taking $K=0$ for $I^{\pi}=2^{+}$. Substituting the $Q_{0}$ value into Eq.
(5), we find the $\beta_{2}$ value to be approximately 0.161. Similarly,
utilizing the calculated $Q_{s}(2^{+}_{1})$ value (37.142 efm2) for 76Se from
our shell-model calculation, we obtain the value $\beta_{2}=-0.195$. Previous
$\beta_{2}$ values computed for 76Ge (0.157) and 76Se (-0.244) within
relativistic mean field (RMF) theory Lalazissis  also support our findings. As
suggested in Ref. Simkovic , the behavior of final NME can be predicted by the
value $\Delta\beta_{2}=|\beta_{2}(^{76}\text{Ge})-\beta_{2}(^{76}\text{Se})|$
and the suppression of the $2\nu\beta\beta$-decay NME becomes stronger with an
increase of $\Delta\beta_{2}$. Different deformation (prolate for 76Ge and
oblate for 76Se) yields a larger $\Delta\beta_{2}$, which may be a possible
factor for the suppression of the NME for the $0^{+}_{\rm g.s.}\rightarrow
0^{+}_{\rm g.s.}$ transition in our case (see Fig. 4).

In the case of 96Zr, our computed value of the NME, 0.0599, for the
$0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$ transition, is slightly smaller
than the 0.0747 one of Kostensalo1 . Looking at Table 4, the previously
determined NME using the IBM is larger than those obtained in the other theory
frameworks. In fact, the present shell-model results for this transition are
in good agreement with the RQRPA results of Toivanen et al. Toi1997  and QRPA
results of Stoica Stoica1 . For the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$
transition a similar situation prevails, the IBM result deviating notably from
the rest. Concerning the $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$ transition,
the RQRPA Toi1997  gives a much larger NME than the present calculation and
the QRPA model of Stoica Stoica1 . Our NME for the $0^{+}_{\rm
g.s.}\rightarrow 2^{+}_{1}$ transition agrees qualitatively with the one of
Stoica1  although is one order of magnitude larger than it. Here it has to be
pointed out that we have completely blocked neutron excitations across the
$N=50$ shell closure in order to make the calculation of the NMEs for 96Zr
feasible. Further, restricting the occupancy of the $\pi(0g_{9/2})$ orbital to
a maximum of four protons may cause inaccuracies in predicting the wave
functions of the states of interest. In the future, one could achieve more
precise NMEs if it was be possible to include additional proton excitations in
$\pi(0g_{9/2})$ orbital and the contribution of the $\nu(1p_{1/2})$ and
$\nu(0g_{9/2})$ orbitals.

###  III.4 Branching ratios

The nucleus 76Ge can decay via $2\nu\beta\beta$ decay into the g.s. and
different excited states of 76Se. But, the nucleus 96Zr can decay via
$2\nu\beta\beta$ decay into the g.s. and different excited states of 96Mo and
also it can decay via single-$\beta$ transitions into the lowest $4^{+}$,
$5^{+}$, and $6^{+}$ states of 96Nb. Here, we discuss the shell-model
predicted branching ratios for the $2\nu\beta\beta$ and/or single-$\beta$
transitions for the 76Ge and 96Zr decays.

In the case of 76Ge, we first calculated the total half-life for the
$0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$, $0^{+}_{\rm g.s.}\rightarrow
0^{+}_{2}$, and $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$ transitions of
$2\nu\beta\beta$ decay using the formula Suh2007

$\frac{1}{T_{1/2}^{\rm total}}=\sum_{k}\frac{1}{T_{1/2}^{k}},$ (7)

where $k$ refers to the final state of the decay. Subsequently, we have
calculated the branching ratios ($B.R.$), $(B.R.)^{k}=T_{1/2}^{\rm
total}/T_{1/2}^{k}$ Suh2007  for all three transitions. It is clear from the
computed $B.R.$ that the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$
transition is dominant having the largest $B.R.$ of 99.84%. The $B.R.$ is
notably smaller for the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$ transition
(15.81$\times 10^{-2}$%), and negligible for the $0^{+}_{\rm g.s.}\rightarrow
2^{+}_{1}$ transition (3.43$\times 10^{-4}$%).

Our estimated half-lives for the fourth-forbidden non-unique, fourth-forbidden
unique, and sixth-forbidden non-unique $\beta$ transitions from the
$0^{+}_{\rm g.s.}$ of 96Zr to the $4^{+}_{1}$, $5^{+}_{1}$, and $6^{+}_{1}$
states of 96Nb (corresponding to $g_{\rm A}=g_{\rm V}=1$) are $7.5\times
10^{22}$, $1.4\times 10^{20}$, and $1.9\times 10^{29}$ yr, respectively. These
are quite similar to the half-lives $7.5\times 10^{22}$, $1.1\times 10^{20}$,
and $1.6\times 10^{29}$ yr obtained in Ref. Alanssari . Our results predict
that the $2\nu\beta\beta$ decay branching ratio (76.33%) is larger than the
$\beta$ decay branching ratio (23.67%) in 96Zr, consistently with the earlier
shell-model study Kostensalo  where the $2\nu\beta\beta$ branching ratio was
81.6%. In the QRPA study of Hei2007  the $\beta$ decay was dominated by the
$0^{+}_{\rm g.s.}\rightarrow 5^{+}_{1}$ transition with a half-life of
$2.4\times 10^{20}$ yr, in agreement with the present study. In the
$2\nu\beta\beta$ branch of 96Zr, the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm
g.s.}$ transition is dominant with a 76.19% contribution, while the
contributions from $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$ (13.66$\times
10^{-2}$%), and $0^{+}_{\rm g.s.}\rightarrow 2^{+}_{1}$ (1.32$\times
10^{-3}$%) transitions are almost negligible. In the case of the
single-$\beta$ branch, the $0^{+}_{\rm g.s.}\rightarrow 5^{+}_{1}$ transition
dominates with 23.63% contribution, consistently with the previous study
Kostensalo . We found significantly smaller branching ratios for the
$0^{+}_{\rm g.s.}\rightarrow 4^{+}_{1}$ (0.04%) and $0^{+}_{\rm
g.s.}\rightarrow 6^{+}_{1}$ (0.17$\times 10^{-7}$%) transitions, again
consistently with the studies of Kostensalo et al. Kostensalo  and Heiskanen
et al. Hei2007 .

##  IV Summary and Conclusions

In the present study, we have investigated the $2\nu\beta\beta$ decay in 76Ge
and 96Zr isotopes by using the GWBXG shell-model effective Hamiltonian.
Firstly, we validated the applicability of this Hamiltonian by computing low-
lying energy levels and reduced transition probabilities in both parent and
granddaughter nuclei. Our findings indicate that the present shell-model
interaction is suitable for calculating NMEs for $2\nu\beta\beta$ decay. To
ensure the convergence of cumulative NMEs, we computed 5000 $1^{+}$ states in
76As and 96Nb. Utilizing the calculated NMEs, we extracted the half-lives of
76Ge and 96Zr for the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s}$,
$0^{+}_{\rm g.s.}\rightarrow 0^{+}_{2}$, and $0^{+}_{\rm g.s.}\rightarrow
2^{+}_{1}$ transitions through $2\nu\beta\beta$ decay. We further investigated
the variation of cumulative NMEs with respect to the energies of $1^{+}$
states of the intermediate nucleus for all three transitions. Our analyses
reveal that the present shell-model Hamiltonian partially describes the
contribution of the GTGR region in the case of 76Ge for the $0^{+}_{\rm
g.s.}\rightarrow 0^{+}_{\rm g.s.}$ transition, which was not obtained in the
previous shell-model studies. Our calculations show single-state dominance
(SSD) for the $0^{+}_{\rm g.s.}\rightarrow 0^{+}_{\rm g.s.}$ transition in
96Zr. The computed half-lives turned out to be in good agreement with the
experimental data. Additionally, we compared the predicted NMEs with those
obtained from different nuclear models in previous works. The shell-model
predicted branching ratios for the $2\nu\beta\beta$ decay of 76Ge, and both
the single-$\beta$ and $2\nu\beta\beta$ decay of 96Zr are also analyzed.

## ACKNOWLEDGEMENTS

We acknowledge financial support from SERB (India), Grant No. CRG/2022/005167.
Additionally, D.P. acknowledges financial support from MHRD, the Government of
India. We would like to thank the National Supercomputing Mission (NSM) for
providing computing resources of “PARAM Ganga” at the IIT Roorkee, implemented
by C-DAC and supported by MeitY and DST, Government of India.

## References

  * (1) F. F. Deppisch, L. Graf, and F. Šimkovic, Searching for New Physics in Two-Neutrino Double Beta Decay, Phys. Rev. Lett. 125, 171801 (2020).
  * (2) The GERDA Collaboration, Background-free search for neutrinoless double-$\beta$ decay of 76Ge with GERDA, Nature 544, 47 (2017).
  * (3) M. Agostini, A. Alexander, G. R. Araujo, A. M. Bakalyarov, M. Balata $et$ $al.$ (GERDA Collaboration), Final Results of GERDA on the Two-Neutrino Double-$\beta$ Decay Half-Life of 76Ge, Phys. Rev. Lett. 131, 142501 (2023).
  * (4) N. Ackerman, B. Aharmim, M. Auger, D. J. Auty, P. S. Barbeau $et$ $al.$ (EXO Collaboration), Observation of Two-Neutrino Double-Beta Decay in 136Xe with the EXO-200 Detector, Phys. Rev. Lett. 107, 212501 (2011).
  * (5) D. Q. Adams, C. Alduino, K. Alfonso, F. T. Avignone III, O. Azzolini $et$ $al.$, Measurement of the $2\nu\beta\beta$ Decay Half-Life of 130Te with CUORE, Phys. Rev. Lett. 126, 171801 (2021).
  * (6) S. R. Elliott, A. A. Hahn, and M. K. Moe, Direct evidence for two-neutrino double-beta decay in 82Se, Phys. Rev. Lett. 59, 2020 (1987).
  * (7) M. Goeppert-Mayer, Double Beta-Disintegration, Phys. Rev. 48, 512 (1935).
  * (8) R. Chandra, J. Singh, P. K. Rath, P. K. Raina, and J. G. Hirsch, Two-neutrino double-$\beta$ decay of $94\leq A\leq 110$ nuclei for the $0^{+}\rightarrow 0^{+}$ transition, Eur. Phys. J. A 23, 223 (2005).
  * (9) H. Ejiri, J. Suhonen, and K. Zuber, Neutrino-nuclear responses for astro-neutrinos, single beta decays and double beta decays, Phys. Rep. 797 1, (2019).
  * (10) T. R. Rodrıguez and G. M.-Pinedo, Energy Density Functional Study of Nuclear Matrix Elements for Neutrinoless $\beta\beta$ Decay, Phys. Rev. Lett. 105, 252503 (2010).
  * (11) N. Shimizu, J. Menéndez, and K. Yako, Double Gamow-Teller Transitions and its Relation to Neutrinoless $\beta\beta$ Decay, Phys. Rev. Lett. 120, 142502 (2018).
  * (12) K. Zuber, Double beta decay, Contemp. Phys. 45, 491 (2004).
  * (13) R. Arnold, Augier, A. M. Bakalyarov, J. D. Baker, A. S. Barabash $et$ $al.$ (NEMO-3 Collaboration), Measurement of the double-beta decay half-life and search for the neutrinoless double-beta decay of 48Ca with the NEMO-3 detector, Phys. Rev. D 93, 112008 (2016).
  * (14) R. Arnold, C. Augier, A. S. Barabash, A. Basharina-Freshville, S. Blondel $et$ $al.$, Final results on 82Se double beta decay to the ground state of ${82}$Kr from the NEMO-3 experiment, Eur. Phys. J. C 78, 821 (2018).
  * (15) J. Argyriades, R. Arnold, C. Augier, J. Baker, A.S. Barabash, Measurement of the two neutrino double beta decay half-life of Zr-96 with the NEMO-3 detector, Nucl. Phys. A 847, 168 (2010).
  * (16) National Nuclear Data Center, Q-Value Calculator (QCalc), https://www.nndc.bnl.gov/qcalc/.
  * (17) A. S. Barabash, Precise Half-Life Values for Two-Neutrino Double-$\beta$ Decay: 2020 Review, Universe 6, 159 (2020).
  * (18) E. Armengaud, C. Augier, A. S. Barabash, F. Bellini, G. Benato $et$ $al.$, Precise measurement of $2\nu\beta\beta$ decay of 100Mo with the CUPID-Mo detection technology, Eur. Phys. J. C 80, 674 (2020).
  * (19) A.S. Barabash, P. Belli, R. Bernabei, F. Cappella, V. Caracciolo $et$ $al.$, Final results of the Aurora experiment to study $2\beta$ decay of 116Cd with enriched 116CdWO4 crystal scintillators, Phys. Rev. D 98, 092007 (2018).
  * (20) A.P. Meshik, C.M. Hohenberg, O.V. Pravdivtseva, T.J. Bernatowicz, and Y.S. Kapusta, 130Te and 128Te double beta decay half-lives, Nuclear Physics A 809, 275 (2008).
  * (21) P. Novella, M. Sorel, A. Usón, C. Adams, H. Almazán $et$ $al.$ (NEXT Collaboration), Measurement of the 136Xe two-neutrino double-$\beta$-decay half-life via direct background subtraction in NEXT, Phys. Rev. C 105, 055501 (2022).
  * (22) R. Arnold, C. Augier, J. D. Baker, A. S. Barabash, A. Basharina-Freshville $et$ $al.$ (NEMO-3 Collaboration), Measurement of the $2\nu\beta\beta$ decay half-life of 150Nd and a search for $0\nu\beta\beta$ decay processes with the full exposure from the NEMO-3 detector, Phys. Rev. D 94, 072003 (2016).
  * (23) J. Suhonen, and O. Civitarese, Probing the quenching of $g_{\rm A}$ by single and double beta decays, Phys. Lett. B 725, 153 (2013).
  * (24) J. Terasaki, and Y. Iwata, Estimation of nuclear matrix elements of double-$\beta$ decay from shell model and quasiparticle random-phase approximation, Eur. Phys. J. Plus 136, 908 (2021).
  * (25) J. Kostensalo, and J. Suhonen, Consistent large-scale shell-model analysis of the two-neutrino $\beta\beta$ and single $\beta$ branchings in 48Ca and 96Zr, Phys. Lett. B 802, 135192 (2020).
  * (26) M. Horoi, S. Stoica, and B. A. Brown, Shell-model calculations of two-neutrino double-$\beta$ decay rates of 48Ca with the GXPF1A interaction, Phys. Rev. C 75, 034303 (2007).
  * (27) D. Patel, P. C. Srivastava, V. K. B. Kota, and R. Sahu, Large-scale shell-model study of two-neutrino double beta decay of 82Se, 94Zr, 108Cd, 124Sn, 128Te, 130Te, 136Xe, and 150Nd, Nucl. Phys. A 1042, 122808 (2024).
  * (28) S. W. Finch, and W. Tornow, Search for two-neutrino double-$\beta$ decay of 96Zr to excited states of 96Mo, Phys. Rev. C 92, 045501 (2015).
  * (29) B. Pritychenko, Systematic analysis of double-beta decay half lives, Nucl. Phys. A 1033, 122628 (2023).
  * (30) A. Neacsu, and M. Horoi, An Effective Method to Accurately Calculate the Phase Space Factors for $\beta^{-}\beta^{-}$ Decay, Adv. High Energy Phys. 2016, 7486712 (2016).
  * (31) S. Stoica, and M. Mirea, New calculations for phase space factors involved in double-$\beta$ decay, Phys. Rev. C 88, 037303 (2013).
  * (32) J. Kotila, J. Barea, and F. Iachello, Phase-space factors and half-life predictions for Majoron-emitting $\beta^{-}\beta^{-}$ decay, Phys. Rev. C 91, 064310 (2015).
  * (33) J. Kotila, and F. Iachello , Phase-space factors for double-$\beta$ decay, Phys. Rev. C 85, 034316 (2012).
  * (34) J. T. Suhonen, Value of the Axial-Vector Coupling Strength in $\beta$ and $\beta\beta$ Decays: A Review, Front. Phys. 5, 55 (2017).
  * (35) T. Tomoda, Double beta decay, Rep. Prog. Phys. 54 53 (1991).
  * (36) J. Kostensalo, J. Suhonen, and K. Zuber, The first large-scale shell-model calculation of the two-neutrino double beta decay of 76Ge to the excited states in 76Se, Phys. Lett. B 831, 137170 (2022).
  * (37) M. Doi, T. Kotani, and E. Takasugi, Double Beta Decay and Majorana Neutrino, Progress of Theoretical Physics Supplement 83, 1-175 (1985).
  * (38) D. Patel, P. C. Srivastava, N. Shimizu, Systematic shell-model study of 98-130Cd isotopes and $8^{+}$ isomeric states, Nucl. Phys. A 1039, 122742 (2023).
  * (39) A. Hosaka, K.-I. Kubo, and H. Toki, $G$-matrix effective interaction with the paris potential, Nucl. Phys. A 444, 76 (1985).
  * (40) X. Ji, and B. H. Wildenthal, Effective interaction for $N=50$ isotones, Phys. Rev. C 37, 1256 (1988).
  * (41) D. H. Gloeckner, Shell-model systematics of the zirconium and niobium isotopes, Nucl. Phys. A 253, 301 (1975).
  * (42) F. J. D. Serduke, R. D. Lawson, and D.H. Gloeckner, Shell-model study of the $N=49$ isotones, Nucl. Phys. A 256, 45 (1976).
  * (43) B.A. Brown, and W.D.M. Rae, The Shell-Model Code NuShellX$@$MSU, Nucl. Data Sheets 120, 115 (2014).
  * (44) N. Shimizu, T. Mizusaki, Y. Utsuno and Y. Tsunoda, Thick-restart block Lanczos method for large-scale shell-model calculations, Comput. Phys. Comm. 244, 372 (2019).
  * (45) NuDat 3 - National Nuclear Data Center, https://www.nndc.bnl.gov/nudat3/indx_adopted.jsp.
  * (46) M. Honma, T. Otsuka, T. Mizusaki, Y. Utsuno, N. Shimizu, and M. Hjorth-Jensen, Shell-model description of magnetic dipole bands in 105Sn, RIKEN Accel. Prog. Rep. 51, 95 (2018).
  * (47) A. A. Klimenko, S. B. Osetrov, A. A. Smolnikov, and S. I. Vasiliev, Double-Beta Decay of 150Nd and 76Ge to Excited States, Czechoslovak Journal of Physics 52, 589 (2002).
  * (48) A. S. Barabash, A. V. Derbin, L. A. Popeko, and V. I. Umatov, Search for $\beta\beta$ decay of 76Ge to the excited states in 76Se, Z. Phys. A 352, 231 (1995).
  * (49) C. Arpesella, A. S. Barabash, E. Bellotti, C. Brofferio, E. Fiorini $et$ $al.$, Search for $\beta\beta$ decay of 96Zr and 150Nd to excited states of 96Mo and 150Sm, Europhys. Lett. 27, 29 (1994).
  * (50) J. H. Thies, D. Frekers, T. Adachi, M. Dozono, H. Ejiri $et$ $al.$, The $(^{3}\text{He},t)$ reaction on 76Ge, and the double-$\beta$-decay matrix element, Phys. Rev. C 86, 014304 (2012).
  * (51) R. Arnold, C. Augier, J. Baker, A. Barabash, D. Blum $et$ $al.$, Double beta decay of 96Zr, Nucl. Phys. A 658, 299 (1999).
  * (52) K. Nomura, Two-neutrino double-$\beta$ decay in the mapped interacting boson model, Phys. Rev. C 105, 044301 (2022).
  * (53) S. K. Dhiman and P. K. Raina, Two-neutrino double-beta decay matrix elements for ground and excited states of 76Ge and 82Se nuclei, Phys. Rev. C 50, R2660(R) (1994).
  * (54) J. Toivanen and J. Suhonen, Study of several double-beta-decaying nuclei using the renormalized proton-neutron quasiparticle random-phase approximation, Phys. Rev. C 55, 2314 (1997).
  * (55) L. Jokiniemi, B. Romeo, C. Brase, J. Kotila, P. Soriano, Two-neutrino $\beta\beta$ decay of 136Xe to the first excited $0^{+}$ state in 136Ba, Phys. Lett. B 838, 137689 (2023).
  * (56) F. Šimkovic, L. Pacearescu, and A. Faessler, Two-neutrino double beta decay of 76Ge within deformed QRPA, Nucl. Phys. A 733, 321 (2004).
  * (57) J. Kilgallon, M. R. Pearson, J. Billowes, P. Campbell, U. Georg $et$ $al.$, Quadrupole moments and mean square charge radii in the bismuth isotope chain, Phys. Lett. B 405, 31 (1997).
  * (58) G. A. Lalazissis, S. Raman, and P. Ring, Ground-state properties of even-even nuclei in the relativistic mean-field theory, At. Data Nucl. Data Tables 71, 1 (1999).
  * (59) S. Stoica, Two-neutrino double-beta decay half-lives of 96Zr and 100Mo to excited states of 96Mo and 100Ru, Phys. Lett. B 350, 152 (1995).
  * (60) J. Suhonen and O. Civitarese, Weak-interaction and nuclear-structure aspects of nuclear double beta decay, Phys. Rep. 300, 123 (1998).
  * (61) J. Suhonen, From Nucleons to Nucleus: Concepts of Microscopic Nuclear Theory, Springer, Berlin, 2007.
  * (62) M. Alanssari, D. Frekers, T. Eronen, L. Canete, J. Dilling $et$ $al.$, Single and Double Beta-Decay $Q$ Values among the Triplet 96Zr, 96Nb, and 96Mo, Phys. Rev. Lett. 116, 072501 (2016).
  * (63) H. Heiskanen, M. T. Mustonen, and J. Suhonen, Theoretical half-life for beta decay of 96Zr, J. Phys. G: Nucl. Part. Phys. 34, 837 (2007).


For each $s\in\mathfrak{Z}_{M}^{\sharp}$,
$\mathbb{D}^{\lambda}_{s}(b_{i|X_{s}\setminus H_{s}})$ is $L^{2}$ with respect
to $h^{(\epsilon_{i})}_{s}$ and $\omega_{\epsilon_{i},s}$.

(a3)

The Lebesgue measure of
$\mathfrak{Z}_{M}^{\triangle}\setminus\mathfrak{Z}_{M}^{\sharp}$ is $0$.

Note that the condition (a2) implies the following.

###### Lemma 7.4

Let $s\in\mathfrak{Z}_{M}^{\sharp}$. Let $\widetilde{h}^{(\epsilon_{i})}_{s}$
be a harmonic metric of $(E_{s},\mathbb{D}^{\lambda}_{s})$ adapted to
$(\mathcal{P}_{\ast}^{(\epsilon_{i})}\mathcal{V},\mathbb{D}^{\lambda})_{|X_{s}}$
such that $\det(\widetilde{h}^{(\epsilon_{i})}_{s})=h_{\det(E)|X_{s}\setminus
H_{s}}$. Let $\widetilde{b}_{i,s}$ be the automorphism of $E_{|X_{s}\setminus
H_{s}}$ determined by
$h^{(\epsilon_{i})}_{s}=\widetilde{h}^{(\epsilon_{i})}_{s}\widetilde{b}_{i,s}$.
Then, $\widetilde{b}_{i,s}$ and $\widetilde{b}_{i,s}^{-1}$ are bounded with
respect to $\widetilde{h}^{(\epsilon_{i})}_{s}$, and
$\mathbb{D}^{\lambda}(\widetilde{b}_{i,s})$ is $L^{2}$ with respect to
$\widetilde{h}^{(\epsilon_{i})}_{s}$ and $\omega_{\epsilon_{i},s}$

Proof Let $b^{\prime}_{i,s}$ be the automorphism of $E_{s}$ determined by
$\widetilde{h}^{(\epsilon_{i})}_{s}=h^{(\epsilon_{i})}_{\mathop{\rm
in}\nolimits|X_{s}\setminus H_{s}}$. Then, $b^{\prime}_{i,s}$ and
$(b^{\prime}_{i,s})^{-1}$ are bounded with respect to
$h^{(\epsilon_{i})}_{\mathop{\rm in}\nolimits|X_{s}\setminus H_{s}}$, and
$\mathbb{D}^{\lambda}(b^{\prime}_{i,s})$ is $L^{2}$ with respect to
$h^{(\epsilon_{i})}_{\mathop{\rm in}\nolimits|X_{s}\setminus H_{s}}$ and
$\omega_{\epsilon_{i},s}$, according to Proposition 4.4. Then, we obtain the
claim of the lemma.

###### Lemma 7.5

There exists a ${\mathbb{C}}^{\ast}$-invariant subset
$\mathfrak{X}_{M}^{\sharp}\subset\mathfrak{X}_{M}^{\triangle}\times_{\mathfrak{Z}_{M}^{\triangle}}\mathfrak{Z}_{M}^{\sharp}$
such that the following holds:

  * •

The measure of
$\mathfrak{X}_{M}^{\triangle}\setminus\mathfrak{X}_{M}^{\sharp}$ is $0$.

  * •

A subsequence of $h^{\triangle(\epsilon_{i})}_{|\mathfrak{X}_{M}^{\sharp}}$ is
convergent to $h^{\triangle}_{|\mathfrak{X}_{M}^{\sharp}}$ at any point of
$\mathfrak{X}_{M}^{\sharp}$.

Proof By Proposition 4.8, for any $s\in\mathfrak{Z}_{M}^{\sharp}$, the
sequence $h_{s}^{(\epsilon_{i})}$ is weakly convergent to $h_{s}$ in
$L_{1}^{2}$ locally on $X_{s}\setminus H_{s}$. We set
$b^{(\epsilon_{i})}_{s}:=h_{s}^{(\epsilon_{i})}h_{s}^{-1}$. We obtain
$\det(b_{s}^{(\epsilon_{i})})=1$, and $b^{(\epsilon_{i})}_{s}$ converges to
the identity locally on $X_{s}\setminus H_{s}$ in $L^{p}$ for any $p\geq 1$.
We set
$g^{(\epsilon_{i})}_{s}:=\bigl{|}b^{(\epsilon_{i})}_{s}\bigr{|}_{h_{s}}$ on
$X_{s}\setminus H_{s}$. We obtain the function $g^{(\epsilon_{i})}$ on
$\mathfrak{X}_{M}^{\triangle}\times_{\mathfrak{Z}_{M}^{\triangle}}\mathfrak{Z}_{M}^{\sharp}$
from $g^{(\epsilon_{i})}_{s}$ $(s\in\mathfrak{Z}_{M}^{\sharp})$. By Lemma 3.21
and Proposition 4.8, for any compact subset
$K\subset\mathfrak{X}_{M}^{\triangle}\setminus\mathfrak{H}_{M}^{\triangle}$,
the restriction of $g^{(\epsilon_{i})}$ to
$K\cap\bigl{(}\mathfrak{X}_{M}^{\triangle}\times_{\mathfrak{Z}_{M}^{\triangle}}\mathfrak{Z}_{M}^{\sharp}\bigr{)}$
are uniformly bounded. Note that the sequence
$\bigl{(}h^{\triangle(\epsilon_{i})}(h^{\triangle})^{-1}\bigr{)}_{|K\cap
X_{s}}=b_{s|K\cap X_{s}}^{(\epsilon_{i})}$ is convergent in $L^{p}$ for any
$p\geq 1$. By Fubini’s theorem and Lebesgue theorem, we obtain the
$L^{p}$-convergence of $h^{\triangle(\epsilon_{i})}(h^{\triangle})^{-1}$ to
the identity for any $p$ on
$K\cap\bigl{(}\mathfrak{X}_{M}^{\triangle}\times_{\mathfrak{Z}_{M}^{\triangle}}\mathfrak{Z}_{M}^{\sharp}\bigr{)}$.
Then, after going to a subsequence, we obtain the desired convergence.

###### Remark 7.6

If $\lambda=0$, the argument can be simplified. Indeed, by Proposition 3.31,
the curvature $R(h_{s}^{(\epsilon_{i})})$ of $h_{s}^{(\epsilon_{i})}$ are
bounded locally on $X_{s}\setminus H$. Hence, we obtain that
$h^{(\epsilon_{i})}_{s}$ is weakly convergent to $h_{s}$ in $L_{2}^{2}$. In
particular, $h_{s}^{(\epsilon_{i})}$ is convergent to $h_{s}$ in the
$C^{0}$-sense locally on $X_{s}\setminus H_{s}$.

There exists a subset
$X^{\sharp}\subset\texttt{P}^{\triangle}_{1}(\mathfrak{X}_{M}^{\sharp})$ such
that for any $P\in X^{\sharp}$, the measure of
$(\texttt{P}^{\triangle}_{1})^{-1}(P)\setminus\bigl{(}\mathfrak{X}_{M}^{\sharp}\bigr{)}$
is $0$ in $(\texttt{P}^{\triangle}_{1})^{-1}(P)$, and that the measure of
$X\setminus X^{\sharp}$ is $0$ in $X$. We obtain that the sequence
$h^{(\epsilon_{i})}_{\mathop{\rm HE}\nolimits|X^{\sharp}}$ is convergent to a
Hermitian metric $h_{\infty}$ of $E_{|X^{\sharp}}$.

###### Lemma 7.7

For any $P\in X^{\sharp}$ and
$s\in\texttt{P}_{2}\bigl{(}(\texttt{P}_{1}^{\triangle})^{-1}(P)\bigr{)}$, we
obtain $(h_{s})_{|P}=h_{\infty|P}$.

Proof For any
$s\in\texttt{P}_{2}\bigl{(}(\texttt{P}^{\triangle}_{1})^{-1}(P)\bigr{)}\cap\mathfrak{Z}_{M}^{\sharp}$,
we obtain $(h_{s})_{|P}=h_{\infty|P}$. Then, by using the continuity of
$h_{s}$ on $s$, we obtain $(h_{s})_{|P}=h_{\infty|P}$ for any
$s\in\texttt{P}_{2}\bigl{(}(\texttt{P}^{\triangle}_{1})^{-1}(P)\bigr{)}$.

###### Lemma 7.8

Let $P\in X\setminus(W_{M}\cup H)$. Then, for any
$s_{1},s_{2}\in\texttt{P}_{2}\bigl{(}(\texttt{P}^{\triangle}_{1})^{-1}(P)\bigr{)}$,
we obtain $h_{s_{1}|P}=h_{s_{2}|P}$.

Proof For any $P\in X^{\sharp}$ and for any
$s_{1},s_{2}\in\texttt{P}_{2}\bigl{(}(\texttt{P}^{\triangle}_{1})^{-1}(P)\bigr{)}$,
we obtain $h_{s_{1}|P}=h_{\infty|P}=h_{s_{2}|P}$. By the continuity of $h_{s}$
on $s$, we obtain the claim of the lemma.

Then, $h_{\infty}$ extends to a Hermitian metric of $E_{|X\setminus(H\cup
W_{M})}$ by setting $h_{\infty|P}:=(h_{s})_{|P}$ for
$s\in\texttt{P}_{2}\bigl{(}(\texttt{P}^{\triangle}_{1})^{-1}(P)\bigr{)}$.

###### Lemma 7.9

$h_{\infty}$ induces a Hermitian metric of $E_{|X\setminus(H\cup W_{M})}$ of
$C^{1}$-class. The $C^{1}$-Hermitian metric is also denoted by $h_{\infty}$.

Proof Let $P$ be any point of $X\setminus W_{M}$. Let $(U_{P},x_{1},x_{2})$ be
a holomorphic coordinate neighbourhood as in §7.2.2. By using Proposition 4.5,
we define the continuous Hermitian metric $h^{(i)}_{P}$ of $E_{|U_{P}}$ by the
condition that $h_{P|\\{x_{i}=a\\}}$ is equal to the restriction of
$h_{s_{1}+as_{\infty}}$. By the construction of $h_{\infty}$, we obtain
$h_{\infty|U_{P}}=h^{(i)}_{P}$. Hence, we obtain that $h_{\infty|U_{P}\cap
X^{\sharp}}$ induces a continuous Hermitian metric $h_{P,\infty}$ of
$E_{|U_{P}}$, and $h^{(1)}_{P}=h_{P,\infty}=h^{(2)}_{P}$ hold. Moreover, by
Proposition 4.5, any derivative of $h^{(j)}_{P}$ with respect to
$\partial_{z_{i}}$ and $\partial_{\overline{z}_{i}}$ $(i\neq j)$ are
continuous. We obtain that $h_{P,\infty}$ is $C^{1}$. Thus, we obtain the
claim of the lemma.

We obtain the operator $\mathbb{D}^{\lambda\star}_{h_{\infty}}$ from
$\mathbb{D}^{\lambda}$ and $h_{\infty}$. We define
$G(h_{\infty}):=\bigl{[}\mathbb{D}^{\lambda},\mathbb{D}^{\lambda\star}_{h_{\infty}}\bigr{]}$
as a current.

###### Lemma 7.10

$G(h_{\infty})^{(1,1)}=0$ on $X\setminus(H\cup W_{M})$.

Proof Let $P\in X\setminus(H\cup W_{M})$. Let $(U_{P},x_{1},x_{2})$ be a
holomorphic coordinate neighbourhood as in §7.2.2. We have the expression

$G(h_{\infty})^{(1,1)}=G(h_{\infty})_{11}dx_{1}\,d\overline{x}_{1}+G(h_{\infty})_{12}dx_{1}\,d\overline{x}_{2}+G(h_{\infty})_{21}dx_{2}\,d\overline{x}_{1}+G(h_{\infty})_{22}dx_{2}\,d\overline{x}_{2}.$

Because $h_{\infty|\\{x_{i}=a\\}}$ is equal to $h_{s_{i}+as_{\infty}}$, we
obtain $G(h_{\infty})_{ii}=0$ for $i=1,2$.

By considering the holomorphic coordinate system
$(w_{1},w_{2})=(x_{1}+x_{2},x_{1}-x_{2})$ and the coefficient of
$dw_{1}\,d\overline{w}_{1}$ in $G(h_{\infty})^{(1,1)}$, we obtain
$G(h_{\infty})_{12}+G(h_{\infty})_{21}=0$. By considering the holomorphic
coordinate system $(z_{1},z_{2})=(x_{1}+\sqrt{-1}x_{2},x_{1}-\sqrt{-1}x_{2})$
and the coefficient of $dz_{1}\,d\overline{z}_{1}$ in $G(h_{\infty})^{(1,1)}$,
we obtain $G(h_{\infty})_{12}-G(h_{\infty})_{21}=0$. Therefore, we obtain that
$G(h_{\infty})_{ij}=0$.

###### Lemma 7.11

We obtain $\Lambda G(h_{\infty})=0$ on $X\setminus(H\cup W_{M})$. As a result,
$h_{\infty}$ is $C^{\infty}$ on $X\setminus(H\cup W_{M})$. If moreover
$\lambda\neq 0$, then $h_{\infty}$ is a pluri-harmonic metric of
$(E,\mathbb{D}^{\lambda})_{|X\setminus H\cup W_{M}}$.

Proof The first claim immediately follows from Lemma 7.10. We obtain the
second claim by the elliptic regularity and a standard bootstrapping argument.
The last claim follows from Corollary 2.16.

###### Lemma 7.12

In the case $\lambda=0$, we obtain $\partial_{h_{\infty}}\theta=0$, i.e.,
$h_{\infty}$ is a pluri-harmonic metric of the Higgs bundle
$(E,\overline{\partial}_{E},\theta)_{|X\setminus(H\cup W_{M})}$.

Proof Let us observe that the sequence
$\partial_{h^{(\epsilon_{i})}_{HE}}-\partial_{h_{\infty}}$ is convergent to
$0$ almost everywhere on $X\setminus H$. It is enough to prove that
$\partial_{h^{(\epsilon_{i})}_{HE|X_{s}}}-\partial_{h_{s}}$ is convergent to
$0$ for $s\in\mathfrak{Z}_{M}^{\sharp}$. Let $b^{(\epsilon_{i})}_{s}$ be the
automorphism of $E_{|X_{s}\setminus H}$ which is self-adjoint with respect to
$h_{s}$ and $h_{\mathop{\rm HE}\nolimits|X_{s}}^{(\epsilon_{i})}$ determined
by $h_{\mathop{\rm
HE}\nolimits|X_{s}}^{(\epsilon_{i})}=h_{s}b^{(\epsilon_{i})}_{s}$. By
Proposition 4.8, the sequence
$(b_{s}^{(\epsilon_{i})})^{-1}\partial_{h_{s}}(b_{s}^{(\epsilon_{i})})$ is
convergent to $0$ weakly in $L^{2}$ locally on $X_{s}\setminus H$. By
Proposition 3.31, the sequence
$(b_{s}^{(\epsilon_{i})})^{-1}\partial_{h_{s}}(b_{s}^{(\epsilon_{i})})$ is
bounded in $L_{2}^{p}$ locally on $X_{s}\setminus H$ for any $p\geq 1$. Hence,
we obtain that
$(b_{s}^{(\epsilon_{i})})^{-1}\partial_{h_{s}}(b_{s}^{(\epsilon_{i})})$ is
convergent to $0$ in $L_{1}^{p}$ locally on $X_{s}\setminus H$.

As a result, $\partial_{h_{\mathop{\rm HE}\nolimits}^{(\epsilon_{i})}}\theta$
is convergent to $\partial_{h_{\infty}}\theta$ almost everywhere. Note that

$0\leq\int_{X\setminus H}\bigl{|}\partial_{h_{\mathop{\rm
HE}\nolimits}^{(\epsilon_{i})}}\theta\bigr{|}_{h_{\mathop{\rm
HE}\nolimits}^{(\epsilon_{i})},\omega_{\epsilon_{i}}}^{2}\leq\int_{X\setminus
H}\bigl{|}G(h_{\mathop{\rm
HE}\nolimits}^{(\epsilon_{i})})\bigr{|}_{h_{\mathop{\rm
HE}\nolimits}^{(\epsilon_{i})},\omega_{\epsilon_{i}}}^{2}=-8\pi^{2}\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}^{(\epsilon_{i})}_{\ast}\mathcal{V}).$

We also have $\lim_{i\to\infty}\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}^{(\epsilon_{i})}_{\ast}\mathcal{V})=0$. We have
the following convergence almost everywhere on $X\setminus H$:

$\lim_{i\to\infty}\bigl{|}\partial_{h_{\mathop{\rm
HE}\nolimits}^{(\epsilon_{i})}}\theta\bigr{|}_{h_{\mathop{\rm
HE}\nolimits}^{(\epsilon_{i})},\omega_{\epsilon_{i}}}^{2}=\bigl{|}\partial_{h_{\infty}}\theta\bigr{|}_{h_{\infty},\omega_{X}}^{2}.$

Therefore, we obtain
$\int\bigl{|}\partial_{h_{\infty}}\theta\bigr{|}^{2}_{h_{\infty},\omega_{X}}=0$
by Fatou’s lemma.

###### Lemma 7.13

$h_{\infty}$ induces a $C^{\infty}$-metric of $E$ on $X\setminus H$, and hence
it is a pluri-harmonic metric of $(E,\mathbb{D}^{\lambda})$.

Proof It is enough to prove that $h_{\infty}$ is a $C^{\infty}$-metric around
any point of $W_{M}\setminus H$. We have only to apply the argument in [48,
Lemma 5.15].

If $\lambda=0$, we obtain that $(E,\overline{\partial}_{E},\theta,h_{\infty})$
is a good wild harmonic bundle on $(X,H)$, because
$(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is a good filtered Higgs bundle. If
$\lambda\neq 0$, the associated Higgs bundle
$(E,\overline{\partial}_{E},\theta)$ with the pluri-harmonic metric
$h_{\infty}$ is a good wild harmonic bundle by [50, Proposition 13.5.1]. We
obtain a good filtered $\lambda$-flat bundle
$(\mathcal{P}_{\ast}^{h_{\infty}}E,\mathbb{D}^{\lambda})$ on $(X,H)$. We put
$H^{[2]}=\bigcup_{i\neq j}(H_{i}\cap H_{j})$. For any $P\in
H\setminus(W_{M}\cup H^{[2]})$, there exists
$s\in\mathfrak{Z}_{M}^{\triangle}$ such that $P\in X_{s}$. By the
construction, $h_{\infty|X_{s}\setminus H_{s}}=h_{s}$. Hence, we obtain
$\mathcal{P}_{\ast}^{h_{\infty}}(E)_{|X_{s}}=\mathcal{P}_{\ast}(\mathcal{V})_{|X_{s}}$.
Let $Y:=(H\cap W_{M})\cup H^{[2]}$, which is a finite subset of $H$. We obtain
that $\mathcal{P}_{\ast}^{h_{\infty}}(E)_{X\setminus
Y}\simeq\mathcal{P}_{\ast}\mathcal{V}_{|X\setminus Y}$. By Hartogs theorem, we
obtain that
$\mathcal{P}_{\ast}^{h_{\infty}}(E)\simeq\mathcal{P}_{\ast}\mathcal{V}$. Thus,
the proof of Proposition 7.2 is completed.

###  7.3 Higher dimensional case

Let us prove Theorem 7.1 in the case $\dim X\geq 3$ by an induction on $\dim
X$. Take a sufficiently large integer $M$. We set
$\mathfrak{Z}_{M}:=H^{0}(X,L^{\otimes\,M})\setminus\\{0\\}$, and let
$\mathfrak{X}_{M}\subset X\times\mathfrak{Z}_{M}$ be defined as
$\mathfrak{s}^{-1}(0)$ as in §7.2.1. For any $s\in\mathfrak{Z}_{M}$, set
$X_{s}:=s^{-1}(0)$. Let $\mathbb{P}(T^{\ast}X)$ denote the projectivization of
the cotangent bundle of $X$. If $M$ is sufficiently large, there exists a
Zariski dense open subset $\mathfrak{Z}_{M}^{\circ}\subset\mathfrak{Z}_{M}$
such that the following holds.

  * •

$\texttt{P}_{2}^{\circ}:\mathfrak{X}_{M}^{\circ}:=\mathfrak{X}_{M}\times_{\mathfrak{Z}_{M}}\mathfrak{Z}_{M}^{\circ}\longrightarrow\mathfrak{Z}_{M}^{\circ}$
is smooth.

  * •

$\mathfrak{X}_{M}^{\circ}\cup\bigl{(}H\times\mathfrak{Z}_{M}^{\circ}\bigr{)}$
is simply normal crossing. Moreover the intersections of any tuple of
irreducible components are smooth over $\mathfrak{Z}_{M}^{\circ}$.

  * •

The induced map $\texttt{P}_{1}:\mathfrak{X}_{M}^{\circ}\longrightarrow X$ is
surjective. Moreover, the induced morphism
$\widetilde{\texttt{P}}_{1}:\mathfrak{X}_{M}^{\circ}\longrightarrow\mathbb{P}(T^{\ast}X)$
is surjective.

Let $p_{i,j}$ denote the projection of
$X\times\mathfrak{Z}_{M}^{\circ}\times\mathfrak{Z}_{M}^{\circ}$ onto the
product of the $i$-th component and the $j$-th component. For $j=2,3$, let
$(\mathfrak{X}_{M}^{\circ})^{(j)}$ denote the pull back of
$\mathfrak{X}_{M}^{\circ}$ by $p_{1,j}$. There exists a Zariski dense open
subset
$\mathfrak{U}_{M}\subset\mathfrak{Z}_{M}^{\circ}\times\mathfrak{Z}_{M}^{\circ}$
such that the following holds.

  * •

Let $(\mathfrak{X}_{M}^{\circ})^{(j)}_{\mathfrak{U}_{M}}$ denote the fiber
product of $(\mathfrak{X}_{M}^{\circ})^{(j)}$ and $\mathfrak{U}_{M}$ over
$\mathfrak{Z}_{M}^{\circ}\times\mathfrak{Z}_{M}^{\circ}$. Then,
$(\mathfrak{X}_{M}^{\circ})^{(2)}_{\mathfrak{U}_{M}}\cup(\mathfrak{X}_{M}^{\circ})^{(3)}_{\mathfrak{U}_{M}}\cup(H\times\mathfrak{U}_{M})$
is simply normal crossing. Moreover, the intersection of any tuple of
irreducible components are smooth over $\mathfrak{U}$.

By the Mehta-Ramanathan type theorem (Proposition 3.8), there exists a Zariski
dense open subset $\mathfrak{U}_{M}^{\triangle}\subset\mathfrak{U}_{M}$ such
that the following holds.

  * •

For ${\boldsymbol{s}}=(s_{1},s_{2})\in\mathfrak{U}_{M}^{\triangle}$, we set
$X_{{\boldsymbol{s}}}:=X_{s_{1}}\cap X_{s_{2}}$. Then, the restriction
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|X_{{\boldsymbol{s}}}}$
is a $\mu_{L}$-stable good filtered $\lambda$-flat bundle on
$(X_{{\boldsymbol{s}}},H\cap X_{{\boldsymbol{s}}})$.

Hence, there exists a Zariski dense open subset
$\mathfrak{Z}_{M}^{\triangle}\subset\mathfrak{Z}_{M}^{\circ}$ such that the
following holds.

  * •

For any $s\in\mathfrak{Z}_{M}^{\triangle}$,
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|X_{s}}$ is a
$\mu_{L}$-stable good filtered $\lambda$-flat bundle on $(X_{s},H\cap X_{s})$.

  * •

For any $s_{1},s_{2}\in\mathfrak{Z}_{M}^{\triangle}$, there exists a Zariski
open subset $\mathfrak{V}(s_{1},s_{2})\subset\mathfrak{Z}_{M}^{\triangle}$
such that
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|X_{(s_{i},s_{3})}}$
$(i=1,2)$ are $\mu_{L}$-stable for any $s_{3}\in\mathfrak{V}(s_{1},s_{2})$.

We set
$\mathfrak{X}_{M}^{\triangle}:=\mathfrak{X}_{M}\times_{\mathfrak{Z}_{M}}\mathfrak{Z}_{M}^{\triangle}$.
Let $\texttt{P}_{2}^{\triangle}:\mathfrak{X}_{M}^{\triangle}\longrightarrow X$
denote the naturally induced morphism. Then,
$W_{M}:=X\setminus\texttt{P}_{2}^{\triangle}(\mathfrak{X}_{M}^{\triangle})$ is
a finite subset.

For any $P\in X\setminus(H\cup W_{M})$, there exists
$s\in\mathfrak{Z}_{M}^{\triangle}$ such that $P\in X_{s}$. Then,
$(\mathcal{P}_{\ast}\mathcal{V}_{s},\mathbb{D}^{\lambda}_{s}):=(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|X_{s}}$
is $\mu_{L}$-stable, and the following holds:

$\int_{X_{s}}c_{1}(\mathcal{P}_{\ast}\mathcal{V}_{s})c_{1}(L_{|X_{s}})^{\dim
X_{s}-1}=0,\quad\int_{X_{s}}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V}_{s})c_{1}(L_{|X_{s}})^{\dim
X_{s}-2}=0.$

There exists a pluri-harmonic metric $h_{s}$ of
$(E_{s},\overline{\partial}_{E_{s}},\mathbb{D}^{\lambda}_{s}):=(E,\overline{\partial}_{E},\mathbb{D}^{\lambda})_{|X_{s}\setminus
H}$ adapted to $\mathcal{P}_{\ast}\mathcal{V}_{s}$ such that
$\det(h_{s})=h_{\det(E)|X_{s}\setminus H}$. Take another
$s^{\prime}\in\mathfrak{Z}_{M}^{\triangle}$ such that $P\in X_{s^{\prime}}$.
There exists a pluri-harmonic metric $h_{s^{\prime}}$ of
$(E_{s^{\prime}},\overline{\partial}_{E_{s^{\prime}}},\mathbb{D}^{\lambda}_{s^{\prime}})$
adapted to $\mathcal{P}_{\ast}\mathcal{V}_{s^{\prime}}$ such that
$\det(h_{s^{\prime}})=h_{\det(E)|X_{s^{\prime}}\setminus H}$.

###### Lemma 7.14

$h_{s|P}=h_{s^{\prime}|P}$.

Proof Suppose that $X_{s}\cup X_{s^{\prime}}\cup H$ is simply normal crossing.
We set $X_{s,s^{\prime}}:=X_{s}\cap X_{s^{\prime}}$. It is smooth and
connected. We obtain a good filtered $\lambda$-flat bundle
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|X_{s,s^{\prime}}}$,
and $h_{s|X_{s,s^{\prime}}}$ and $h_{s^{\prime}|X_{s,s^{\prime}}}$ are adapted
to $\mathcal{P}_{\ast}\mathcal{V}_{|X_{s,s^{\prime}}}$. Let $b_{s,s^{\prime}}$
be the automorphism of $E_{|X_{s,s^{\prime}}}$ which is self-adjoint with
respect to both $h_{s|X_{s,s^{\prime}}}$ and
$h_{s^{\prime}|X_{s,s^{\prime}}}$, and determined by
$h_{s^{\prime}|X_{s,s^{\prime}}}=h_{s|X_{s,s^{\prime}}}\cdot
b_{s,s^{\prime}}$. There exists a decomposition

$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|X_{s,s^{\prime}}}=\bigoplus(\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i}),$

which is orthogonal with respect to both $h_{s|X_{s,s^{\prime}}}$ and
$h_{s^{\prime}|X_{s,s^{\prime}}}$, and $b_{s,s^{\prime}}=\bigoplus
a_{i}\mathop{\rm id}\nolimits_{\mathcal{V}_{i}}$ for some positive constants
$a_{i}$.

There exists $s_{1}\in\mathfrak{V}(s,s^{\prime})$. Then,
$(\mathcal{P}_{\ast}\mathcal{V},\theta)_{|X_{s_{1},s}}$ and
$(\mathcal{P}_{\ast}\mathcal{V},\theta)_{|X_{s_{1},s^{\prime}}}$ are
$\mu_{L}$-stable. Therefore, we have $h_{s|X_{s_{1},s}}=h_{s_{1}|X_{s_{1},s}}$
and $h_{s^{\prime}|X_{s_{1},s^{\prime}}}=h_{s_{1}|X_{s_{1},s^{\prime}}}$. We
obtain that $h_{s|X_{s_{1}}\cap X_{s}\cap
X_{s^{\prime}}}=h_{s^{\prime}|X_{s_{1}}\cap X_{s}\cap X_{s^{\prime}}}$. It
implies that $a_{i}$ are $1$, and hence $h_{s|P}=h_{s^{\prime}|P}$.

In general, there exists $s_{2}\in\mathfrak{Z}_{M}^{\triangle}$ such that (i)
$P\in X_{s_{2}}$, (ii) $X_{s}\cup X_{s_{2}}\cup H$ and $X_{s^{\prime}}\cup
X_{s_{2}}\cup H$ are simply normal crossing. By the above consideration, we
obtain $h_{s|P}=h_{s_{2}|P}=h_{s^{\prime}|P}$.

Therefore, we obtain Hermitian metrics $h_{P}$ of $E_{|P}$ $(P\in
X\setminus(H\cup W_{M}))$. By using the argument in Lemma 7.9, we can prove
that they induce a Hermitian metric $h$ of $E_{|X\setminus(H\cup W_{M})}$ of
$C^{1}$-class. We obtain $G(h)$ from $\mathbb{D}^{\lambda}$ and $h$ as a
current. Because $h_{|X_{s}}$ $(s\in\mathfrak{U}_{M}^{\triangle})$ are pluri-
harmonic metrics of $(E,\mathbb{D}^{\lambda})_{|X_{s}\setminus H}$, we obtain
that $G(h)=0$. It also implies that $h$ is a $C^{\infty}$ on $X\setminus(H\cup
W_{M})$. By using the argument in [48, Lemma 5.15], we obtain that $h$ induces
a pluri-harmonic metric of $(E,\mathbb{D}^{\lambda})$ on $X\setminus H$. Then,
as in the proof of Proposition 7.2, we can conclude that
$(E,\mathbb{D}^{\lambda},h)$ is a good wild harmonic bundle, and that
$\mathcal{P}^{h}_{\ast}(E)=\mathcal{P}_{\ast}\mathcal{V}$. Thus, we obtain
Theorem 7.1.

##  8 Homogeneity with respect to group actions

###  8.1 Preliminary

####  8.1.1 Homogeneous harmonic bundles

Let $Y$ be a complex manifold. Let $K$ be a compact Lie group. Let
$\rho:K\times Y\longrightarrow Y$ be a $K$-action on $Y$ such that
$\rho_{k}:Y\longrightarrow Y$ is holomorphic for any $k\in K$. Let
$\kappa:K\longrightarrow S^{1}$ be a homomorphism of Lie groups.

Let $(E,\overline{\partial}_{E},\theta,h)$ be a harmonic bundle on $Y$. It is
called $(K,\rho,\kappa)$-homogeneous if $(E,\overline{\partial}_{E},h)$ is
$K$-equivariant and $k^{\ast}\theta=\kappa(k)\theta$.

###### Remark 8.1

According to [64], harmonic bundles are equivalent to polarized variation of
pure twistor structure of weight $w$, for any given integer $w$. If $\kappa$
is non-trivial, as studied in [51, §3], by choosing a vector $\mathfrak{v}$ in
the Lie algebra of $K$ such that $d\kappa(\mathfrak{v})\neq 0$, we obtain the
integrability of the variation of pure twistor structure from the homogeneity
of harmonic bundles.

####  8.1.2 Homogeneous filtered Higgs sheaves and the stability condition
with respect to the action

Let $X$ be a connected complex projective manifold with a simple normal
crossing hypersurface $H$. Let $G$ be a complex reductive algebraic group. Let
$\rho:G\times Y\longrightarrow Y$ be an algebraic $G$-action on $Y$ which
preserves $H$. Let $\kappa:G\longrightarrow{\mathbb{C}}^{\ast}$ be a
homomorphism of complex algebraic groups.

Let $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ be a filtered Higgs sheaf on
$(Y,H)$. It is called $(G,\rho,\kappa)$-homogeneous if
$\mathcal{P}_{\ast}\mathcal{V}$ is $G$-equivariant and
$g^{\ast}\theta=\kappa(g)\theta$ for any $g\in G$.

Let $L$ be a $G$-equivariant ample line bundle on $X$. A
$(G,\rho,\kappa)$-homogeneous filtered Higgs sheaf
$(\mathcal{P}_{\ast}\mathcal{V},\theta)$ on $(X,H)$ is called $\mu_{L}$-stable
(resp. $\mu_{L}$-semistable) with respect to the $G$-action if the following
holds.

  * •

Let $\mathcal{V}^{\prime}$ be a $G$-invariant saturated Higgs subsheaf of
$\mathcal{V}$ such that $0<\mathop{\rm
rank}\nolimits\mathcal{V}^{\prime}<\mathop{\rm rank}\nolimits\mathcal{V}$.
Then,
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{\prime})<\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})$
(resp.
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{\prime})\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})$)
holds.

A $(G,\rho,\kappa)$-homogeneous filtered Higgs sheaf
$(\mathcal{P}_{\ast}\mathcal{V},\theta)$ on $(X,H)$ is called
$\mu_{L}$-polystable with respect to the $G$-action if it is
$\mu_{L}$-semistable with respect to the $G$-action and isomorphic to a direct
sum of $(G,\rho,\kappa)$-homogeneous filtered sheaves
$\bigoplus(\mathcal{P}_{\ast}\mathcal{V}_{i},\theta_{i})$, where each
$(\mathcal{P}_{\ast}\mathcal{V}_{i},\theta_{i})$ is $\mu_{L}$-stable with
respect to the $G$-action.

###### Lemma 8.2

$(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is $\mu_{L}$-semistable if and only
if $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is $\mu_{L}$-semistable with
respect to the $G$-action.

Proof The “only if” part is clear. Let us prove that the “if” part. Let
$\mathcal{V}_{0}\subset\mathcal{V}$ be the $\beta$-subobject as in Proposition
3.4. Because $g^{\ast}\mathcal{V}_{0}$ also has the same property, we obtain
that $\mathcal{V}_{0}$ is $G$-invariant. Then, the claim of the proposition is
clear.

The following lemma is clear.

###### Lemma 8.3

If $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is $\mu_{L}$-stable, then
$(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is $\mu_{L}$-stable with respect to
the $G$-action.

###### Lemma 8.4

If $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is $\mu_{L}$-stable with respect
to the $G$-action, then $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is
$\mu_{L}$-polystable.

Proof According to Lemma 8.2, $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is
$\mu_{L}$-semistable. Let $\mathcal{V}_{1}$ be the socle of
$(\mathcal{P}_{\ast}\mathcal{V},\theta)$ as in Proposition 3.5. Because
$g^{\ast}\mathcal{V}_{1}$ also has the same property, $\mathcal{V}_{1}$ is
$G$-invariant. Moreover,
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{1})=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})$
holds. Hence, we obtain $\mathcal{V}_{1}=\mathcal{V}$. According to
Proposition 3.5, $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is
$\mu_{L}$-polystable.

###### Remark 8.5

In general, even if $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is
$\mu_{L}$-stable with respect to the $G$-action,
$(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is not necessarily $\mu_{L}$-stable.

####  8.1.3 Actions of a complex reductive group and its compact real form

Let $X$ be a complex projective manifold equipped with an algebraic action of
a complex reductive group $G$. Let $L$ be a $G$-equivariant ample line bundle
on $X$. Let $K$ be a compact real form of $G$.

Let $(E,\overline{\partial}_{E})$ be a $G$-equivariant holomorphic vector
bundle on $X$. Then, as the restriction, we may naturally regard
$(E,\overline{\partial}_{E})$ as a $K$-equivariant holomorphic vector bundle
on $X$.

###### Lemma 8.6

The above procedure induces an equivalence between $G$-equivariant holomorphic
vector bundles and $K$-equivariant holomorphic vector bundles on $X$.

Proof Let $(E,\overline{\partial}_{E})$ be a $K$-equivariant holomorphic
vector bundle on $X$. There exists $m_{0}>0$ such that $E\otimes
L^{\otimes\,m_{0}}$ is globally generating. We set
$\mathcal{G}_{0}:=H^{0}(X,E\otimes
L^{\otimes\,m_{0}})\otimes(L^{\otimes\,m_{0}})^{-1}$. There exists a naturally
induced epimorphism of $\mathcal{O}_{X}$-modules
$\mathcal{G}_{0}\longrightarrow E$. Let $\mathcal{K}$ denote the kernel. There
exists $m_{1}>0$ such that $\mathcal{K}\otimes L^{\otimes\,m_{1}}$ is globally
generating. We set $\mathcal{G}_{1}:=H^{0}(X,\mathcal{K}\otimes
L^{\otimes\,m_{1}})\otimes(L^{\otimes\,m_{1}})^{-1}$. There exists a naturally
induced epimorphism $\mathcal{G}_{1}\longrightarrow\mathcal{K}$. Thus, we
obtain a resolution $\mathcal{G}_{1}\longrightarrow\mathcal{G}_{0}$ of $E$.
Because $E$ is $K$-equivariant, $H^{0}(X,E\otimes L^{\otimes m_{0}})$ is
naturally a $K$-representation, $\mathcal{G}_{0}$ is a $K$-equivariant
holomorphic vector bundle on $X$, and $\mathcal{G}_{0}\longrightarrow E$ is
$K$-equivariant. Hence, $\mathcal{K}$ is a $K$-equivariant holomorphic vector
bundle. Similarly $H^{0}(X,\mathcal{K}\otimes L^{\otimes m_{2}})$ is a
$K$-representation, and $\mathcal{G}_{1}$ is $K$-equivariant holomorphic
vector bundle, and $\mathcal{G}_{1}\longrightarrow\mathcal{K}_{2}$ is
$K$-equivariant.

The $K$-representations on $H^{0}(X,E\otimes L^{\otimes\,m_{1}})$ and
$H^{0}(X,\mathcal{K}\otimes L^{\otimes m_{2}})$ naturally induce
$G$-representations on $H^{0}(X,E\otimes L^{\otimes\,m_{1}})$ and
$H^{0}(X,\mathcal{K}\otimes L^{\otimes m_{2}})$. Hence, $\mathcal{G}_{i}$ are
naturally algebraic $G$-equivariant vector bundles on $X$. Moreover, the
morphism $\mathcal{G}_{1}\longrightarrow\mathcal{G}_{0}$ is $G$-equivariant
and algebraic. Hence, $E$ is a $G$-equivariant algebraic vector bundle on $X$.

###  8.2 An equivalence

####  8.2.1 Good filtered Higgs bundles associated to homogeneous good wild
Higgs bundles

Let $X$ be a connected complex projective manifold with a simple normal
crossing hypersurface $H$. Let $G$ be a complex reductive group acting on
$(X,H)$. Let $K$ be a compact real form of $G$. The actions of $G$ and $K$ on
$X$ are denoted by $\rho$. Let $\kappa:G\longrightarrow{\mathbb{C}}^{\ast}$ be
a character. The induced homomorphism $K\longrightarrow S^{1}$ is also denoted
by $\kappa$.

Let $(E,\overline{\partial}_{E},\theta,h)$ be a $(K,\rho,\kappa)$-homogeneous
harmonic bundle on $X\setminus H$ which is good wild on $(X,H)$. We obtain a
good filtered Higgs bundle $(\mathcal{P}^{h}_{\ast}E,\theta)$ on $(X,H)$.
Because each $\mathcal{P}^{h}_{{\boldsymbol{a}}}E$ is naturally a
$K$-equivariant holomorphic vector bundle on $X$, $\mathcal{P}_{\ast}^{h}E$ is
naturally $G$-equivariant by Lemma 8.6. Because
$k^{\ast}\theta=\kappa(k)\theta$ for any $k\in K$, we obtain
$g^{\ast}\theta=\kappa(g)\theta$ for any $g\in G$. Therefore,
$(\mathcal{P}_{\ast}^{h}E,\theta)$ is a $(G,\rho,\kappa)$-homogeneous good
filtered Higgs bundle on $(X,H)$.

Let $L$ be a $G$-equivariant ample line bundle on $X$.

###### Proposition 8.7

$(\mathcal{P}_{\ast}^{h}E,\theta)$ is $\mu_{L}$-polystable with respect to the
$G$-action, i.e., there exists a decomposition
$(E,\overline{\partial}_{E},\theta,h)=\bigoplus(E_{i},\overline{\partial}_{E_{i}},\theta_{i},h_{i})$
of $(G,\rho,\kappa)$-homogeneous harmonic bundles such that each
$(\mathcal{P}_{\ast}^{h_{i}}E_{i},\theta_{i})$ is $\mu_{L}$-stable with
respect to the $G$-action.

Proof Because $(\mathcal{P}^{h}_{\ast}E,\theta)$ is $\mu_{L}$-polystable, we
obtain that $(\mathcal{P}^{h}_{\ast}E,\theta)$ is $\mu_{L}$-semistable with
respect to the $G$-action. Let $\mathcal{V}_{1}\subset\mathcal{P}^{h}E$ be a
$G$-invariant saturated Higgs $\mathcal{O}_{X}(\ast H)$-submodule such that
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{1})=\mu_{L}(\mathcal{P}_{\ast}^{h}E)=0$.
Let $E_{1}$ be the Higgs subsheaf of $E$ obtained as the restriction of
$\mathcal{V}_{1}$ to $X\setminus H$. Then, by the argument in the proof of
[50, Proposition 13.6.1], we obtain that $E_{1}$ is a subbundle, and the
orthogonal complement $E_{2}:=E_{1}^{\bot}$ is also a holomorphic subbundle.
Moreover, $\theta(E_{2})\subset E_{2}\otimes\Omega^{1}_{X\setminus H}$, and
$E_{2}$ is $K$-equivariant. Hence, we obtain a decomposition
$(E,\overline{\partial}_{E},\theta,h)=(E_{1},\overline{\partial}_{E_{1}},\theta_{1},h_{1})\oplus(E_{2}\overline{\partial}_{E_{2}},\theta_{2},h_{2})$
of $(K,\rho,\kappa)$-homogeneous harmonic bundles. Then, the claim of the
proposition is clear.

####  8.2.2 Uniqueness

Let $(E,\overline{\partial}_{E},\theta,h)$ be a $(K,\rho,\kappa)$-homogeneous
harmonic bundle on $X\setminus H$ which is good wild on $(X,H)$. Let
$h^{\prime}$ be another pluri-harmonic metric of
$(E,\overline{\partial}_{E},\theta)$ such that (i) $h^{\prime}$ is
$K$-invariant, (ii)
$\mathcal{P}_{\ast}^{h^{\prime}}E=\mathcal{P}_{\ast}^{h}E$. The following is
clear from Proposition 2.22.

###### Proposition 8.8

There exists a decomposition
$(E,\overline{\partial}_{E},\theta)=\bigoplus_{i=1}^{m}(E_{i},\overline{\partial}_{E_{i}},\theta_{i})$
such that (i) the decomposition is orthogonal with respect to both $h$ and
$h^{\prime}$, (ii) there exist $a_{i}>0$ $(i=1,\ldots,m)$ such that
$h^{\prime}_{|E_{i}}=a_{i}h_{E_{i}}$, (iii) the decomposition $E=\bigoplus
E_{i}$ is preserved by the $K$-action.

####  8.2.3 Existence theorem

Let $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ be a
$(G,\rho,\kappa)$-homogeneous good filtered Higgs bundle on $(X,H)$ such that

$\int_{X}c_{1}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim
X-1}=0,\quad\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim X-2}=0.$

Let $(E,\overline{\partial}_{E},\theta)$ be the Higgs bundle on $X\setminus H$
obtained as the restriction of $(\mathcal{P}_{\ast}\mathcal{V},\theta)$.

###### Theorem 8.9

Suppose that $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is $\mu_{L}$-stable with
respect to the $G$-action. Then, there exists a $K$-invariant pluri-harmonic
metric $h$ of $(E,\overline{\partial}_{E},\theta)$ such that
$\mathcal{P}^{h}_{\ast}E=\mathcal{P}_{\ast}\mathcal{V}$. If $h^{\prime}$ is
another $K$-invariant pluri-harmonic metric of
$(E,\overline{\partial}_{E},\theta)$, there exists a positive constant $a$
such that $h^{\prime}=ah$.

Proof By Lemma 8.4, $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is
$\mu_{L}$-polystable. There exists the canonical decomposition

$(\mathcal{P}_{\ast}\mathcal{V},\theta)=\bigoplus_{i=1}^{m}(\mathcal{P}_{\ast}\mathcal{V}_{i},\theta_{i})\otimes
U_{i},$

where $(\mathcal{P}_{\ast}\mathcal{V}_{i},\theta_{i})$ are $\mu_{L}$-stable
good filtered Higgs bundles such that
$(\mathcal{P}_{\ast}\mathcal{V}_{i},\theta_{i})\not\simeq(\mathcal{P}_{\ast}\mathcal{V}_{j},\theta_{j})$
$(i\neq j)$, and $U_{i}$ are finite dimensional complex vector spaces. Let
$(E_{i},\overline{\partial}_{E_{i}},\theta_{i})$ denote the Higgs bundle
obtained as the restriction of $(\mathcal{V}_{i},\theta_{i})$ to $X\setminus
H$. There exist pluri-harmonic metrics $h_{i}$ of
$(E_{i},\overline{\partial}_{E_{i}},\theta_{i})$ adapted to the filtered
bundles $\mathcal{P}_{\ast}\mathcal{V}_{i}$. Let $h^{(0)}_{U_{i}}$ be
Hermitian metrics of $U_{i}$. We obtain a pluri-harmonic metric
$h^{(0)}=\bigoplus(h_{i}\otimes h^{(0)}_{U_{i}})$ of
$(E,\overline{\partial}_{E},\theta)$ adapted to
$\mathcal{P}_{\ast}\mathcal{V}$. By Proposition 2.22 and the uniqueness of the
canonical decomposition, we obtain the following lemma.

###### Lemma 8.10

For any pluri-harmonic metric $h^{(1)}$ of
$(E,\overline{\partial}_{E},\theta)$ adapted to
$\mathcal{P}_{\ast}\mathcal{V}$, there uniquely exist Hermitian metrics
$h^{(1)}_{U_{i}}$ of $U_{i}$ such that $h^{(1)}=\bigoplus(h_{i}\otimes
h^{(1)}_{U_{i}})$.

For any $k\in K$, we obtain a pluri-harmonic metric $k^{\ast}h^{(0)}$ of
$(E,\overline{\partial}_{E},\kappa(k)\theta)$ adapted to
$\mathcal{P}_{\ast}\mathcal{V}$. Because $|\kappa(k)|=1$, $k^{\ast}(h^{(0)})$
is also a pluri-harmonic metric of $(E,\overline{\partial}_{E},\theta)$
adapted to $\mathcal{P}_{\ast}\mathcal{V}$. Hence, there uniquely exist
Hermitian metrics $h_{U_{i}}(k)$ of $U_{i}$ such that
$k^{\ast}(h)=\bigoplus_{i=1}^{m}(h_{i}\otimes h_{U_{i}}(k))$. By using the
Haar measure $dk$ on $K$ with $\int_{K}dk=1$, we define the Hermitian metric
$h$ of $E$ as follows:

$h:=\int_{K}k^{\ast}(h^{(0)})dk=\bigoplus_{i=1}^{m}\Bigl{(}h_{i}\otimes\int_{K}h_{U_{i}}(k)dk\Bigr{)}.$

Then, $h$ is also a pluri-harmonic metric. By the construction, $h$ is
$K$-invariant.

Let $h^{\prime}$ be another $K$-invariant pluri-harmonic metric of
$(E,\overline{\partial}_{E},\theta)$ adapted to
$\mathcal{P}_{\ast}\mathcal{V}$. We obtain the decomposition
$(E,\overline{\partial}_{E},\theta)=\bigoplus(E_{i},\overline{\partial}_{E_{i}},\theta_{i})$
as in Proposition 2.22, which induces a decomposition of good filtered Higgs
bundles
$(\mathcal{P}_{\ast}\mathcal{V})=\bigoplus(\mathcal{P}_{\ast}\mathcal{V}_{i},\theta_{i})$.
Because both $h$ and $h^{\prime}$ are $K$-invariant, the decompositions are
also $K$-invariant. Hence, the decomposition
$(\mathcal{P}_{\ast}\mathcal{V})=\bigoplus(\mathcal{P}_{\ast}\mathcal{V}_{i},\theta_{i})$
is $G$-invariant. By the $\mu_{L}$-stability of
$(\mathcal{P}_{\ast}\mathcal{V})$, we obtain $m=1$, i.e., $h^{\prime}=ah$ for
$a>0$.

###### Corollary 8.11

We obtain the equivalence between the isomorphism classes of the following
objects.

  * •

$(K,\rho,\kappa)$-homogeneous good wild harmonic bundles on $(X,H)$.

  * •

$(G,\rho,\kappa)$-homogeneous good filtered Higgs bundles
$(\mathcal{P}_{\ast}\mathcal{V},\theta)$ such that (i) it is
$\mu_{L}$-polystable with respect to the $G$-action, (ii)
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})=0$, $\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim X-2}=0$.

## References

  * [1] S. Barannikov, Quantum periods. I. Semi-infinite variations of Hodge structures, Internat. Math. Res. Notices, 23, (2001), 1243–1264.
  * [2] O. Biquard, Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse), Ann. Sci. École Norm. Sup. 30 (1997), 41–96.
  * [3] O. Biquard and P. Boalch, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179–204.
  * [4] F. A. Bogomolov, Holomorphic tensors and vector bundles on projective manifolds, Math. USSR Izv. 13 (1979), 499–555.
  * [5] N. Borne, Fibrés paraboliques et champ des racines, Int. Math. Res. Not. IMRN (2007), no 16, 38 pages.
  * [6] N. Borne, Sur les représentations du groupe fondamental d’une variété privée d’un diviseur à croisements normaux simples. (French) Indiana Univ. Math. J. 58 (2009), 137–180.
  * [7] S. Cecotti and C. Vafa, Topological–anti-topological fusion, Nuclear Phys. B, 367 (1991), 359–461.
  * [8] S. Cecotti and C. Vafa, On classification of N=2 supersymmetric theories, Comm. Math. Phys. 158 (1993), 569–644
  * [9] T. Coates, H. Iritani, H.-H. Tseng, Wall-crossings in toric Gromov-Witten theory. I. Crepant examples, Geom. Topol. 13 (2009), 2675–2744.
  * [10] K. Corlette, Flat $G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361–382.
  * [11] M. Cornalba and P. Griffiths, Analytic cycles and vector bundles on noncompact algebraic varieties, Invent. Math. 28 (1975), 1–106.
  * [12] B. Collier, R. Wentworth, Conformal limits and the Białynicki-Birula stratification of the space of $\lambda$-connections. Adv. Math. 350 (2019), 1193–1225.
  * [13] R. Donagi, T. Pantev, Geometric Langlands and non-abelian Hodge theory, Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 85–116, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009.
  * [14] S. K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983), 269–277.
  * [15] S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985), 1–26.
  * [16] S. K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54, (1987), 231–247.
  * [17] S. K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987), 127–131.
  * [18] B. Dubrovin, Geometry and integrability of topological-antitopological fusion, Comm. Math. Phys., 152 (1993), 539–564.
  * [19] D. Gieseker, On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math. 101 (1979), no. 1, 77–85.
  * [20] A. Grothendieck, Techniques de construction et théorèms d’existence en géométrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki exposé 221, (1961), IHP, Paris.
  * [21] C. Hertling, $tt^{\ast}$ geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003), 77–161.
  * [22] C. Hertling and Ch. Sevenheck, Nilpotent orbits of a generalization of Hodge structures, J. Reine Angew. Math. 609 (2007), 23–80.
  * [23] C. Hertling and C. Sevenheck, Limits of families of Brieskorn lattices and compactified classifying spaces, Adv. Math. 223 (2010), 1155–1224.
  * [24] N. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59–126.
  * [25] N. Hitchin, A note on vanishing theorems; in Geometry and analysis on manifolds, 373–382, Progr. Math., 308, Birkhäuser/Springer, Cham, 2015.
  * [26] Z. Hu and P. Huang, Flat $\lambda$-Connections, Mochizuki Correspondence and Twistor Spaces, arXiv:1905.10765
  * [27] P. Huang, Non-abelian Hodge Theory and Related Topics, arXiv:1908.08348
  * [28] H. Iritani, $tt^{\ast}$-geometry in quantum cohomology, arXiv:0906.1307
  * [29] J. Iyer and C. Simpson, A relation between the parabolic Chern characters of the de Rham bundles, Math. Ann. 338, (2007), 347–383.
  * [30] J. Jost and K. Zuo, Harmonic maps of infinite energy and rigidity results for representations of fundamental groups of quasiprojective varieties, J. Differential Geom. 47 (1997), 469–503.
  * [31] M. Kashiwara, Semisimple holonomic $D$-modules, in Topological Field Theory, Primitive Forms and Related Topics, Progress in Math, vol 160, Birkhäuser, (1998), 267–271.
  * [32] S. Kobayashi, First Chern class and holomorphic tensor fields, Nagoya Math. J. 77 (1980), 5–11.
  * [33] S. Kobayashi, Curvature and stability of vector bundles, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 158–162.
  * [34] S. Kobayashi (noted by I. Enoki), Differential geometry of holomorphic vector bundles (in Japanese) Seminary Note in the University of Tokyo 41, 1982\.
  * [35] S. Kobayashi, Differential geometry of complex vector bundles, Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo, (1987).
  * [36] T. Kotake, T. Ochiai, (eds): Non-linear problems in geometry, Proceedings of the Sixth International Symposium, Division of Mathematics, The Taniguchi Foundation, 1979.
  * [37] J. Li, Hermitian-Einstein metrics and Chern number inequalities on parabolic stable bundles over Kähler manifolds, Comm. Anal. Geom. 8 (2000), 445–475.
  * [38] J. Li, M. S. Narasimhan, Hermitian-Einstein metrics on parabolic stable bundles, Acta Math. Sin. (Engl. Ser.) 15 (1999), 93–114.
  * [39] M. Lübke, Chernklassen von Hermite-Einstein-Vektorbündeln. Math. Ann. 260 (1982), 133–141.
  * [40] M. Lübke, Stability of Einstein-Hermitian vector bundles. Manuscripta Math. 42 (1983), 245–257.
  * [41] M. Lübke, A. Teleman, The universal Kobayashi-Hitchin correspondence on Hermitian manifolds, Mem. Amer. Math. Soc. 183 (2006), no. 863, vi+97 pp.
  * [42] M. Maruyama and K. Yokogawa, Moduli of parabolic stable sheaves, Math. Ann. 293, 77–99 (1992).
  * [43] V. Mehta and A. Ramanathan, Semistable sheaves on projective varieties and their restriction to curves, Math. Ann., 258 (1982), 213-224.
  * [44] V. Mehta and A. Ramanathan, Restriction of stable sheaves and representations of the fundamental group, Invent. Math., 77, (1984), 163–172.
  * [45] V. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248, (1980), 205–239
  * [46] T. Mochizuki, Kobayashi-Hitchin correspondence for tame harmoinc bundles and an application, Astérisque 309 (2006).
  * [47] T. Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules I, II, Mem. AMS. 185 (2007)
  * [48] T. Mochizuki, Kobayashi-Hitchin correspondence for tame harmonic bundles II, Geometry&Topology 13, (2009), 359–455, math.DG/0602266
  * [49] T. Mochizuki, Asymptotic behaviour of variation of pure polarized TERP structures, Publ. Res. Inst. Math. Sci. 47 (2011), 419–534.
  * [50] T. Mochizuki, Wild harmonic bundles and wild pure twistor $D$-modules, Asterisque 340, 2011.
  * [51] T. Mochizuki, Harmonic bundles and Toda lattices with opposite sign II, Comm. Math. Phys. 328 (2014), 1159–1198.
  * [52] T. Mochizuki, Wild harmonic bundles and twistor D-modules, Proceedings of the International Congress of Mathematicians–Seoul 2014. Vol. 1, 499–527, Kyung Moon Sa, Seoul, 2014.
  * [53] T. Mochizuki, Mixed twistor D-modules, Lecture Notes in Mathematics, 2125. Springer, Cham, 2015.
  * [54] T. Mochizuki, Harmonic bundles and Toda lattices with opposite sign I, to appear from RIMS Kôkyûroku Bessatsu. (the first part of “Harmonic bundles and Toda lattices with opposite sign” arXiv:1301.1718.)
  * [55] D. Mumford, Projective invariants of projective structures and applications, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) pp. 526–530.
  * [56] D. Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, 59 Princeton University Press, Princeton, N.J. 1966
  * [57] M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567.
  * [58] C. Sabbah, Harmonic metrics and connections with irregular singularities, Ann. Inst. Fourier (Grenoble) 49 (1999), 1265–1291
  * [59] C. Sabbah, Polarizable twistor $D$-modules Astérisque, 300, Société Mathématique de France, Paris, (2005).
  * [60] K. Saito and A. Takahashi, “From primitive forms to Frobenius manifolds”, In “From Hodge theory to integrability and TQFT $tt^{\ast}$-geometry”, Proc. Sympos. Pure Math., 78, Amer. Math. Soc., Providence, RI, (2008), 31–48.
  * [61] C. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and application to uniformization, J. Amer. Math. Soc. 1 (1988), 867–918.
  * [62] C. Simpson, Harmonic bundles on non-compact curves, J. Amer. Math. Soc. 3 (1990), 713–770.
  * [63] C. Simpson, Higgs bundles and local systems, Publ. I.H.E.S., 75 (1992), 5–95.
  * [64] C. Simpson, Mixed twistor structures, math.AG/9705006.
  * [65] C. Simpson, The Hodge filtration on nonabelian cohomology, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, (1997), 217–281.
  * [66] C. Simpson, Iterated destabilizing modifications for vector bundles, IN Vector bundles and complex geometry, 183–206, Contemp. Math., 522, Amer. Math. Soc., Providence, RI, 2010.
  * [67] Y. T. Siu, Techniques of Extension of Analytic Objects, Lect. Notes in Pure and Appl Math, 8, Marcel Dekker, Inc, New York, (1974)
  * [68] Y. T. Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. of Math. (2) 112 (1980), 73–111.
  * [69] B. Steer and A. Wren, The Donaldson-Hitchin-Kobayashi correspondence for parabolic bundles over orbifold surfaces, Canad. J. Math. 53 (2001), 1309–1339.
  * [70] F. Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1972), 29–48.
  * [71] F. Takemoto, Stable vector bundles on algebraic surfaces. II. Nagoya Math. J. 52 (1973), 173–195.
  * [72] K. Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83, (1982) 31–42.
  * [73] K. Uhlenbeck and S. T. Yau, On the existence of Hermitian Yang-Mills connections in stable bundles, Comm. Pure Appl. Math., 39-S (1986), 257–293.

Address
Research Institute for Mathematical Sciences, Kyoto University, Kyoto
606-8502, Japan,
<EMAIL_ADDRESS>