Neutrinoless double-$\beta$ decay and double Gamow-Teller transitions

The neutrinoless double-$\beta$ ($0\nu\beta\beta$) decay and the double Gamow-Teller (DGT) transition are investigated with the state-of-the-art Relativistic Configuration-interaction Density functional theory. A strong linear correlation between the nuclear matrix elements (NMEs) of the $0\nu\beta\beta$ decay and the DGT transition is demonstrated. This linear correlation is found to originate from the similarity of the leading-order term of the $0\nu\beta\beta$-decay operator and the DGT-transition one, as revealed by expanding the $0\nu\beta\beta$-decay operator in terms of the spherical harmonics. The present results provide a strong support to constrain the $0\nu\beta\beta$-decay NMEs through the double charge-exchange reactions.

The search for neutrinoless double-β (0νββ) decay is one of the top priorities in the field of nuclear and particle physics and has received worldwide attentions [1][2][3][4][5][6][7][8].The nuclear matrix element that governing the 0νββ-decay half-life is a key issue for the decay process and its evaluation is being pursued energetically by the community [9][10][11].However, the discrepancy of the NMEs between model predictions is as large as a factor around four, and this limits severely the capability to anticipate the reach of the future 0νββ-decay experiments and the extraction of the effective neutrino mass once the decay signals are observed.
Given the difficulty of evaluating the NMEs, the double Gamow-Teller (DGT) transition, a double spin-isospin flip mode, has been proposed to shed light on the values of NMEs [12][13][14][15].A good linear correlation between the NMEs of the 0νββ decay and those governing the ground-state-to-ground-state DGT transition has been found from the calculations of the configuration-interaction shell model (CISM) and the nonrelativistic density functional theory (NRDFT) [16].By studying the behavior of the NMEs as a function of the relative distance between two decaying nucleons, the correlation is attributed to the dominant shortrange character of both transitions.These results open the door to constrain the 0νββ-decay NMEs from the DGT transition.
The correlation between the 0νββ decay and the DGT transition is then investigated extensively within the framework of the quasiparticle random-phase approximation (QRPA) [17][18][19], the valence-space in-medium similarity renormalization group (VS-IMSRG), and the in-medium generator coordinate method (IM-GCM) [20].Nevertheless, all these investigations predict much weaker correlations.Moreover, the short-range dominant behavior of the DGT transition, as revealed in the CISM calculation, is not well supported in these calculations.The sensitive dependence of the NMEs on the isoscalar pairing strength in the QRPA [17][18][19] and the in-medium renormalization effect from the IMSRG evolution in the VS-IMSRG and IM-GCM calculations [20] are considered to be the possible reasons to weaken the correlation.It should be emphasized that the conclusions in the QRPA are obtained under the assumption that nuclei have spherical symmetry, while those in the VS-IMSRG and the IM-GCM are limited to the decay processes in very light nuclei.Therefore, whether and why there exist correlations between the 0νββ decay and the DGT transition are still important open questions.Exploring the correlation for nuclei that are relevant to the current and next-generation experiments based on advanced nuclear many-body approaches is highly desirable.
In this Letter, the correlation between the 0νββ decay and the DGT transition is investigated within the framework of Relativistic C onfiguration-interaction Density functional (ReCD) theory.The ReCD theory combines the advantages of the CISM [21] and the relativistic DFT [22] and allows fully microscopic and self-consistent calculations for nuclear properties within a full model space.It has been successfully applied to describe nuclear spectroscopic properties [23][24][25] and nuclear ββ decays [26].Compared to the CISM, VS-IMSRG, and IM-GCM, the ReCD theory can be applied to all the ββ-decay candidates.
Moreover, the breaking of spherical and axial symmetries, which is important for describing the 0νββ decay [26][27][28], is considered.Based on the ReCD theory, the NMEs of the DGT transition and the 0νββ decay in nuclei 48 Ca, 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, 124 Sn, 128 Te, 130 Te, and 136 Xe, which are most relevant to the 0νββ decay experiments, are evaluated.A strong linear correlation between the 0νββ decay and the DGT transition is demonstrated, providing a strong support to constrain the 0νββ-decay NMEs through the DGT transition experiments [13][14][15].Nevertheless, the present results do not support the conclusion that the linear correlation originates in the dominant short-range character of both transitions as proposed in Ref. [16].Instead, by expanding the 0νββ-decay operator in terms of the spherical harmonics, the presence of the linear correlation is suggested to be from the similarity of the leading-order term of the 0νββ-decay operator and the DGT-transition operator.
The 0νββ-decay NME can be separated into five parts, where VV, AA, AP, PP, and MM represent respectively the vector, axial-vector, axial-vector and pseudoscalar, pseudoscalar, and weak-magnetism coupling channels [29].The closure approximation that is reliable for the calculation of M 0ν is adopted to avoid the explicit calculation of the odd-odd intermediate nuclear states [30].Each term shown in Eq. ( 1) can be expressed as a "sandwich" form with a decay operator between initial and final nuclear The detailed formula of M 0ν can be found in Ref. [26].The NME of the DGT transition is defined as, where σ and τ + are respectively the spin and isospin-raising operators.
The initial and final nuclear wavefunctions in Eqs. ( 1) and ( 2) are given by the ReCD theory.They are expressed as linear combinations of the projected wavefunctions with good angular momentum I and M, Here, P I M K is the three-dimensional angular momentum projection operator [31], and |Φ κ represent a set of intrinsic multi-quasiparticle states determined by solving the triaxial relativistic Hartree-Bogoliubov (TRHB) equation [22].The set {|Φ κ } forms the intrinsic configuration space of the ReCD theory and more details about its construction can be found in Refs.[25,26].The expansion coefficients F I Kκ are obtained by diagonalizing the Hamiltonian in a shell-model space spanned by the basis { P I M K |Φ κ }, and this leads to the Hill-Wheeler equation, The energy kernel |Φ κ are evaluated by the Pfaffian algorithms [32,33].
It should be emphasised that the Hamiltonian Ĥ and the TRHB equation are derived with the same density functional.In the present work, two well-known relativistic density functionals PC-PK1 [34] and PC-F1 [35] are adopted.A finite-range separable force with strength G = 728 MeV fm 3 [36] is used to treat the pairing correlations.The intrinsic states |Φ κ are obtained by solving the TRHB equation in a set of three-dimensional harmonic oscillator bases in Cartesian coordinates with 10 major shells.Similar to our previous investigations [23][24][25][26], a sufficiently large intrinsic configuration space truncated by a quasiparticle excitation energy cutoff E cut = 5.0 MeV is adopted.Our calculations are free of adjustable parameters.M 0ν L=0 strongly correlates with M DGT and the Pearson's correlation coefficient r = 0.999.

The correlation between M 0ν
L=1 and M DGT is much weaker and the Pearson's correlation coefficient r = 0.898.Compared to M 0ν L=0 , M 0ν L with L = 2, 4, 6, • • • are suppressed, and including their contributions would not worsen the correlation between the 0νββ decay and the DGT transition.As an example, the correlation between M 0ν L=2 and M DGT is shown in Fig. 3 (c).The consideration of higher-order terms with odd-L values tend to contaminate the linear correlation.However, the calculations show that M 0ν L with odd L values are smaller than M 0ν L with even L values.These lead to the fact that the final M 0ν , obtained by summing over M 0ν L with all L components, still correlates with M DGT .
The results presented in Fig. 3 suggest that dominant leading-order term M 0ν L=0 plays a key role for the presence of the strong correlation between M 0ν and M DGT .In the following, the formula of decay operator Ô0ν L=0 corresponding to M 0ν L=0 will be derived in terms of the spherical harmonics.Then, the similarity between Ô0ν L=0 and DGT-transition operator ÔDGT will be demonstrated to explain the strong correlation between M 0ν L=0 and M DGT .Due to the fact that the AA coupling channel dominates the 0νββ-decay process and its contribution exhausts more than 95% of the total NME [37], we focus on the derivation of Ô0ν L=0 in the AA coupling channel.The decay operator of the AA coupling channel in its second-quantized form reads where d † is the proton creation operator, ĉ is the neutron annihilation operator, and indices 1, 2, 3, 4 characterize a set of spherical harmonic oscillator bases |nljm [31].The neutrino potential O AA (r 1 , r 2 ) in coordinate space is with H(q) the neutrino potential in momentum space.See more details about the decay operator in Refs.[9,11].Using the multipole expansion for plane waves e ±iq•r by the spherical harmonics, O AA (r 1 , r 2 ) can be reformulated as where j L and Y LM denote respectively the spherical Bessel function and the spherical harmonics with rank L. The neutrino potential with L = 0 in Eq. ( 7) has the following form, O AA L=0 (r 1 , r 2 ) = 1 2π 2 q 2 dqH(q)j 0 (qr 1 )j 0 (qr 2 ), (8) and its distribution as functions of nucleon coordinates r 1 and r 2 is depicted in Fig. 4.
It should be emphasized that there is no analytical expression for O AA L=0 (r 1 , r 2 ).However, as a function of two variables, O AA L=0 (r 1 , r 2 ) can be decomposed as . By integrating Eq. ( 8) over the momentum q numerically, it is found that O AA L=0 (r 1 , r 2 ) can be well approximated by Here, X 1 (r) and Y 1 (r) are smoothly decreasing functions that are larger than zero and do not have any node.The single-particle matrix elements in terms of |nljm then read approximately The contributions of n 1 l 1 |X 1 (r 1 )|n 2 l 2 and n 3 l 3 |Y 1 (r 2 )|n 4 l 4 with n 1 = n 2 or n 3 = n 4 are suppressed, as the radial nodes of the wavefunctions are different.Comparing Eq. ( 10) to the single-particle matrix elements of the DGT-transition operator, one can easily find that they share the same structure and, thus, the corresponding decay operators capture similar nuclear many-body correlations.This explains the strong correlation between M 0ν L=0 and M DGT .We have noticed that the presence of the linear correlation was attributed to the dominant short-range character both in the 0νββ decay and the DGT transition in the previous study [16].To examine such explanation, the NME distributions C α (r 12 ), defined by Therefore, the explanation for the presence of the linear correlation given in Ref. [16] is not supported by the present study.

Figure 1
Figure 1 depicts the potential energy surfaces (PESs) of the 0 + states calculated by the ReCD theory for ten ββ-decay partners that are most relevant to the current 0νββ decay experiments.The PESs based on the PC-F1 density functional are similar to those of the PC-PK1 and therefore, only the PESs obtained with the PC-PK1 are shown.The triaxial energy minima are seen for all nuclei considered here, indicating a crucial role of the triaxiality in describing the corresponding 0νββ decays and DGT transitions.With the wavefunctions that minimize the energies of the 0 + states, the NMEs for both 0νββ decay

4 FIG. 1 .FIG. 2 .FIG. 3 .
FIG. 1. (Color online) Potential energy surfaces of the 0 + states for nuclei 48 Ca, 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, 124 Sn, 128 Te, 130 Te, and 136 Xe, and their ββ-decay daughter nuclei calculated by the ReCD theory with the PC-PK1 functional.The red solid dots denote the positions of the energy minima and the neighboring contour lines are separated by 0.2 MeV.

FIG. 4 .
FIG. 4. (Color online) Distribution of the neutrino potential O AA L=0 (r 1 , r 2 ) as functions of the nucleon coordinates r 1 and r 2 .

FIG. 5 .
FIG. 5. (Color online) The NME distributions for the 0νββ decay (upper) and the DGT transition (lower) as a function of relative distance r 12 = |r 1 − r 2 | between the decaying nucleons.

dr 12 C
α (r 12 ) = M α with α specifying either 0νββ or DGT, are depicted in Fig. 5.The results for 76 Ge, 82 Se, 100 Mo, and 136 Xe are shown as examples.The 0νββ-decay NMEs mainly distribute at the range of r 12 < 3 fm, which corresponds to the so-called dominant short-range character of the 0νββ decay [16].However, the results of C DGT (r 12 ) indicate that the NMEs of the DGT transition are not short-range dominated.Despite the distinct features of C 0ν and C DGT as a function of r 12 , the good linear correlation still appears.
In summary, the NMEs of the 0νββ decay and the DGT transition in ten nuclei that are most relevant to the current and next-generation 0νββ decay experiments are investigated with the ReCD theory.A strong linear correlation between the 0νββ decay and the DGT transition is demonstrated.To understand the origin of the linear correlation, the 0νββdecay operator is expanded by using the spherical harmonics.It is found that the dominant leading-order term of the 0νββ-decay operator is very similar to the DGT-transition one, and this explains the presence of the linear correlation.The present results provide a strong support to constrain the 0νββ-decay NMEs from the double charge-exchange reactions.To examine the conclusion in Ref.[16], i.e., the linear correlation originates from the dominant short-range character, the NME distributions as a function of relative distance between two decaying nucleons are shown.The so-called short-range dominant character is observed in the 0νββ decay, but not in the DGT transitions.This work was partly supported by the National Natural Science Foundation of China (Grants No. 12141501, No. 12105004, No. 12070131001, No. 11875075, No. 11935003, and No. 11975031), the China Postdoctoral Science Foundation under Grant No. 2020M680183, the State Key Laboratory of Nuclear Physics and Technology, Peking University, and the High-performance Computing Platform of Peking University.