The Impact of Electron Phase Shifts on ββ-Decay Kinematics

Abstract

We reexamine the angular correlation between the emitted electrons in the double beta decay (DBD) of ¹⁰⁰Mo, with particular attention to the impact of electronic wave function phase shifts. In the two-neutrino mode, the angular correlation factor increases modestly compared to calculations without phase shifts. However, a more detailed analysis of the angular correlation energy distributions uncovered a striking feature: electrons are most likely emitted in the same direction when one of them is below a certain energy threshold. We show that this feature is absent in previous Standard Model (SM) predictions and that phase shifts could also influence the angular correlations predicted by new physics models in two-neutrino DBD. For the neutrinoless mode, the direction flip is also present when phase shifts are included in the calculation. However, the angular correlation factor does not change much when phase shifts are taken into account, though our analysis is limited to the light neutrino exchange as the dominant mechanism. These findings highlight the subtle yet significant role that phase shifts can play in shaping electron emission patterns, influencing both SM and new physics predictions in DBD.

1. Introduction

Double-beta decay (DBD) processes are sensitive probes to new physics beyond the Standard Model (SM) of particle physics. The SM-allowed two-neutrino double-beta ($2\nu \beta \beta$) decay stands as the rarest observed decay mode in nature, with half-lives exceeding ${10}^{18}$ years [1]. However, its SM-forbidden counterpart, the neutrinoless double-beta ($0\nu \beta \beta$) decay, remains experimentally elusive. The best lower limits on the half-life of $0\nu \beta \beta$-decay have recently reached the order of ${10}^{26}$ years for both ⁷⁶Ge [2] and ¹³⁶Xe [3,4]. The discovery of $0\nu \beta \beta$-decay would confirm lepton number violation, a key element in the theory of leptogenesis [5,6], and would establish the Majorana nature of neutrinos [7,8]. It also holds the potential to provide crucial insights into neutrino masses, their hierarchy, and CP violation in the lepton sector [8,9,10,11,12,13,14].

The eager search for the hypothetical $0\nu \beta \beta$-decay [15] translates into rich statistics for $2\nu \beta \beta$ events. This is primarily due to the role played by the $2\nu \beta \beta$-decay spectrum as a background for the expected $0\nu \beta \beta$-decay signal. The first indirect detection of $2\nu \beta \beta$-decay of ¹³⁰Te was reported in 1950 in a geochemical experiment that measured the isotopic excess of ¹³⁰Xe in ancient tellurium-ores [16]. It was not until 1987 when the $2\nu \beta \beta$-decay of ⁸²Se was observed in a direct counting experiment [17,18]. Today, the $2\nu \beta \beta$-decay has been detected in 11 different nuclei, including decays into two excited states, using direct counting, radiochemical, and geochemical methods [1,19]. There are also positive indications of the double electron capture ($2\nu \mathrm{ECEC}$) process for ¹³⁰Ba and ¹³²Ba from geochemical measurements [20,21,22], as well as for ⁷⁸Kr [23,24]. Recently, the first direct observation of the $2\nu \mathrm{ECEC}$ process in ¹²⁴Xe was reported [25,26,27]. Moreover, although less favored due to their lower Q-value compared with the $2\nu \mathrm{ECEC}$ process, the detection of double-positron emission and electron capture with coincident positron emission modes may soon be observed if coincidence trigger logic are employed [28,29].

The growing statistics for the $2\nu \beta \beta$-decay make the exploration of new physics beyond the Standard Model (BSM) possible. Three fundamental concepts have been employed in the BSM models that extend the $2\nu \beta \beta$-decay [30]: (i) non-standard interactions, (ii) violation of fundamental symmetries, and (iii) emission of new bosons or fermions from the decay. Today, these models include the exploration of right-handed neutrino interactions [31], neutrino self-interaction [32], violation of the Pauli exclusion principle [33], violation of Lorentz invariance [34,35,36,37], sterile neutrinos with masses up to the Q-value of the process [38,39], $0\nu \beta \beta$ decays with Majoron(s) emission [40,41,42,43,44,45,46,47,48,49,50], and quadruple-$\beta$ decay [51].

The current experimental constraints on various strength parameters associated with the BSM models are obtained by analyzing the shape of the summed electron energy distribution of $2\nu \beta \beta$-decay [30]. However, the most striking signatures in many BSM scenarios are expected in the angular correlation distributions between the emitted electrons [31,32,33,36,37,38,48,49]. A notable example is the direction flip in the emission of electrons when right-handed currents are included in $2\nu \beta \beta$-decay [31]. Fortunately, the concept of tracking individual electrons [52] is also actively pursued in next-generation experiments such as SuperNEMO [53] and NEXT-100 [54]. The tracking capability will provide valuable insights into the underlying mechanism of $0\nu \beta \beta$-decay [55,56,57,58,59], if observed, and will strongly enhance the sensitivity to BSM scenarios in $2\nu \beta \beta$-decay.

In this paper, we reexamine the angular correlation between the emitted electrons in the DBD of ¹⁰⁰Mo. We have employed a modified self-consistent Dirac-Hartree-Fock-Slater potential of the final positive ion ¹⁰⁰Ru to determine the continuum states electron wave functions and the corresponding phase shifts. This approach offers a realistic description of the nucleus, incorporating finite size, diffuse surface effects, and a self-consistent treatment of atomic screening. We found that the phase shifts for the electronic wave functions play a significant role in predicting the emission patterns for the outgoing electrons for both $2\nu \beta \beta$-decay and $0\nu \beta \beta$-decay. While the overall differences in angular correlation coefficients are moderate, around 7% for $2\nu \beta \beta$-decay and around 2% for $0\nu \beta \beta$-decay, a deeper scan into the angular correlation distributions reveals a notable feature: electrons are most likely emitted in the same direction when one of them is below a certain energy threshold. Our findings might affect the predictions for the electron emission patterns corresponding to different BSM scenarios in $2\nu \beta \beta$-decay. Additionally, the inclusion of electron phase shifts could influence the current tools to unravel the mechanism driving $0\nu \beta \beta$-decay. However, this will be discussed elsewhere as the current investigation is limited to the light-neutrino exchange mechanism.

2. Formalism for DBD

The differential DBD rate for a $0^{+}\to 0^{+}$ nuclear transition with respect to the angle $0\le \theta \le \pi$ between the emitted electrons can be written as [31]

$${\displaystyle \frac{d\Gamma }{d\left(cos\theta \right)}}={\displaystyle \frac{\Gamma }{2}}\left(1+Kcos\theta \right),$$

(1)

where

$$\begin{array}{c}\hfill K=-{\displaystyle \frac{\Lambda }{\Gamma }},\end{array}$$

(2)

is the angular correlation coefficient. The above expressions are valid for both $2\nu \beta \beta$-decay and $0\nu \beta \beta$-decay.

2.1. 2$\nu \beta \beta$-Decay

The $2\nu \beta \beta$-decay rates are given by [60,61],

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left\{\begin{array}{c}{\Gamma }^{2\nu }\\ {\Lambda }^{2\nu }\end{array}\right\}=& {\left(g_A\right)}^4{\left|M^{2\nu }\right|}^{2}ln\left(2\right)\left\{\begin{array}{c}{G}_0^{2\nu }+{\xi }_{31}{G}_2^{2\nu }+{\displaystyle \frac{1}{3}}{\xi }_{31}^2{G}_{22}^{2\nu }+\left({\displaystyle \frac{1}{3}}{\xi }_{31}^2+{\xi }_{51}\right)G_4^{2\nu }\\ H_0^{2\nu }+{\xi }_{31}{H}_2^{2\nu }+{\displaystyle \frac{5}{9}}{\xi }_{31}^2{H}_{22}^{2\nu }+\left({\displaystyle \frac{2}{9}}{\xi }_{31}^2+{\xi }_{51}\right)H_4^{2\nu }\end{array}\right\}\hfill \end{array}\end{array}$$

(3)

where $g_A$ is the axial coupling constant and the phase space factors (PSFs) are given by,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left\{\begin{array}{c}{G}_N^{2\nu }\\ H_N^{2\nu }\end{array}\right\}=& {\displaystyle \frac{m_e{\left(G_F\left|V_{ud}\right|m_e^2\right)}^4}{8{\pi }^{7}ln2}}{\displaystyle \frac{1}{m_e^{11}}}{\int }_{m_e}^{E_i-E_f-m_e}{p}_{e_1}{E}_{e_1}{\int }_{m_e}^{E_i-E_f-E_{e_1}}{p}_{e_2}{E}_{e_2}{\mathcal{I}}_N\left\{\begin{array}{c}{F}_{ss}\left(E_{e_1}\right)F_{ss}\left(E_{e_2}\right)\\ E_{ss}\left(E_{e_1}\right)E_{ss}\left(E_{e_2}\right)\end{array}\right\}d{E}_{e_2}d{E}_{e_1}\hfill \end{array}\end{array}$$

(4)

with $N=\{0,2,22,4\}$. Here, $G_F$ is the Fermi coupling constant, $V_{ud}$ is the first element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and $m_e$ is the mass of the electron. $E_i$, $E_f$ and $E_{e_i}$ ($E_{e_i}=\sqrt{p_{e_i}^2+m_e^2}$, $i=1,2$) are the energies of the initial nucleus, the final nucleus and electrons, respectively. The difference $E_i-E_f$ represents the energy difference between the initial and final $0^{+}$ nuclear states, which can be determined by relating it to the Q-value of the $2\nu \beta \beta$-decay, given as $Q=E_i-E_f-2m_e$. In this study, we used $Q=3.0344$ MeV [62] for the DBD of ¹⁰⁰Mo. The functions ${\mathcal{I}}_N$ are the results from the integration over the antineutrino energies and can be found in the Appendix of [61].

The quantity $M^{2\nu }$ is the nuclear matrix element (NME). If its Fermi (F) component is negligible, as the isospin is known to be a good approximation in nuclei, the Gamow-Teller (GT) component dominates the transition, i.e.,

$$M^{2\nu }=M_{GT}^{2\nu }=m_e\sum _n{\displaystyle \frac{M_n}{E_n-(E_i+E_f)/2}},$$

(5)

with

$$M_n=\langle 0_f^{+}\parallel \sum _j{\tau }_j^{+}{\sigma }_j\parallel 1_n^{+}\rangle \langle 1_n^{+}\parallel \sum _k{\tau }_k^{+}{\sigma }_k\parallel 0_i^{+}\rangle .$$

(6)

Here, $|0_i^{+}\rangle$ ($|0_f^{+}\rangle$) is the ground state of the initial (final) even-even nucleus with energy $E_i$ ($E_f$), and the summations run over all $|1_n^{+}\rangle$ states the intermediate odd-odd nucleus with energies $E_n$ and over all $j,k$ nucleons inside the nucleus. We adopt the same assumptions as in [60], where the F component of the NME for $2\nu \beta \beta$ decay is neglected. However, it should be noted that the formalism which takes into account the F component is presented in [61]. More details on the determination $2\nu \beta \beta$-decay NMEs from experiments, nuclear structure models, and phenomenological approaches can be found in [63].

The parameters ${\xi }_{31}$ and ${\xi }_{51}$ are given by

$${\xi }_{31}={\displaystyle \frac{M_{GT-3}^{2\nu }}{M_{GT-1}^{2\nu }}},{\xi }_{51}={\displaystyle \frac{M_{GT-5}^{2\nu }}{M_{GT-1}^{2\nu }}},$$

(7)

where the additional NMEs are defined as,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill M_{GT-1}^{2\nu }& \equiv M_{GT}^{2\nu }\hfill \\ \hfill M_{GT-3}^{2\nu }& =\sum _n{M}_n{\displaystyle \frac{4m_e^3}{{\left[E_n-(E_i+E_f)/2\right]}^3}},\hfill \\ \hfill M_{GT-5}^{2\nu }& =\sum _n{M}_n{\displaystyle \frac{16m_e^5}{{\left[E_n-(E_i+E_f)/2\right]}^5}}.\hfill \end{array}\end{array}$$

(8)

From the theoretical side, one can predict the ${\xi }_{31}$ and ${\xi }_{51}$ parameters by computing the NMEs within various nuclear structure models, e.g., proton-neutron quasiparticle random phase approximation (pn-QRPA) calculation with partial isospin restoration from [60] or nuclear shell model (NSM) and pn-QRPA calculations from [64]. From the experimental side, the ratios ${\xi }_{31}$ and ${\xi }_{51}$ can be treated as free parameters that control the observables of $2\nu \beta \beta$-decay. The first limit on ${\xi }_{31}$ was set by the KamLAND-Zen collaboration by analyzing the $2\nu \beta \beta$-decay of ¹³⁶Xe [64]. Additionally, the ${\xi }_{31}$ and ${\xi }_{51}$ parameters have been recently measured by the CUPID-Mo experiment [65]. It should also be noted that ${\xi }_{31}$ and ${\xi }_{51}$ parameters are fixed under the single state dominance (SSD) and higher state dominance (HSD) hypotheses in $2\nu \beta \beta$-decay [60,61].

The functions $F_{ss}$ and $E_{ss}$ are given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_{ss}\left(E_e\right)=& {\left|{\tilde{g}}_{-1}(E_e,R)\right|}^2+{\left|{\tilde{f}}_1(E_e,R)\right|}^2,\hfill \\ \hfill E_{ss}\left(E_e\right)=& 2\Re \left\{{\tilde{g}}_{-1}(E_e,R){\tilde{f}}_1^{*}(E_e,R)\right\}.\hfill \end{array}\end{array}$$

(9)

where the electron radial wave functions, ${\tilde{g}}_{\kappa }(E_e,r)$ and ${\tilde{f}}_{\kappa }(E_e,r)$, discussed in Section 2.3, are evaluated on the nuclear surface $R=1.2A^{1/3}$. Note that the functions $f_{11}^{\left(0\right)}$ and $f_{11}^{\left(1\right)}$ defined in [66] are equivalent in our notation with $F_{ss}\left(E_{e_1}\right)F_{ss}\left(E_{e_2}\right)$ and $-E_{ss}\left(E_{e_1}\right)E_{ss}\left(E_{e_2}\right)$, respectively.

One can also define the energy-dependent angular correlation distributions [31,67],

$${\alpha }^{2\nu }\left(E_e\right)=-{\displaystyle \frac{d{\Lambda }^{2\nu }/d{E}_e}{d{\Gamma }^{2\nu }/d{E}_e}},{\kappa }^{2\nu }(E_{e_1},E_{e_2})=-{\displaystyle \frac{d^2{\Lambda }^{2\nu }/\left(d{E}_{e_1}d{E}_{e_2}\right)}{d^2{\Gamma }^{2\nu }/\left(d{E}_{e_1}d{E}_{e_2}\right)}}$$

(10)

which offer information about the direction of electron emission for given energies.

2.2. 0$\nu \beta \beta$-Decay

Assuming only the light-neutrino exchange produced by left-handed currents as the dominant mechanism, the $0\nu \beta \beta$-decay rates are given by [57,66]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left\{\begin{array}{c}{\Gamma }^{0\nu }\\ {\Lambda }^{0\nu }\end{array}\right\}=& {\displaystyle \frac{{\left(G_F\left|V_{ud}\right|\right)}^4{m}_e^2}{32{\pi }^5{R}^2}}{\left(g_A\right)}^4{\left|M^{0\nu }\right|}^2{\displaystyle \frac{{\left|m_{\beta \beta }\right|}^2}{m_e^2}}{\int }_{m_e}^{E_i-E_f-m_e}{p}_{e_1}{E}_{e_1}{p}_{e_2}{E}_{e_2}\left\{\begin{array}{c}{F}_{ss}\left(E_{e_1}\right)F_{ss}\left(E_{e_2}\right)\\ E_{ss}\left(E_{e_1}\right)E_{ss}\left(E_{e_2}\right)\end{array}\right\}d{E}_{e_1}\hfill \\ \hfill =& {\left(g_A\right)}^4{\left|M^{0\nu }\right|}^2{\displaystyle \frac{{\left|m_{\beta \beta }\right|}^2}{m_e^2}}ln\left(2\right)\left\{\begin{array}{c}{G}^{0\nu }\\ H^{0\nu }\end{array}\right\},\hfill \end{array}\end{array}$$

(11)

where $E_{e_2}=E_i-E_f-E_{e_1}$, from the energy conservation, and the effective Majorana neutrino mass is given by

$$m_{\beta \beta }=\sum _{k=1}^3{U}_{ek}^2{m}_k.$$

(12)

Here, $U_{ek}$ and $m_k$, with $k=1,2,3$, are elements of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix and the values of neutrino masses, respectively. A recent and comprehensive review on the calculation of the $0\nu \beta \beta$-decay NME, $M^{0\nu }$, can be found in [68]. In this case, one angular correlation distribution can be defined, i.e.,

$${\alpha }^{0\nu }\left(E_e\right)=-{\displaystyle \frac{d{\Lambda }^{0\nu }/d{E}_e}{d{\Gamma }^{0\nu }/d{E}_e}}.$$

(13)

2.3. Electron Wave Function and Phase Shift

The partial spherical waves of the electron wave function in a potential, $V\left(r\right)$, are given by [69,70]

$${\psi }_{\kappa ,m}(E_e,\mathit{r})=\left(\begin{array}{c}{\tilde{g}}_{\kappa }(E_e,r){\Omega }_{\kappa ,m}\left(\widehat{\mathit{r}}\right)\\ i{\tilde{f}}_{\kappa }(E_e,r){\Omega }_{-\kappa ,m}\left(\widehat{\mathit{r}}\right)\end{array}\right),$$

(14)

where $\mathit{r}$ stands for the position vector of the electron with $\widehat{\mathit{r}}=\mathit{r}/r$, $\kappa$ is the relativistic quantum number and ${\Omega }_{\kappa ,m}\left(\widehat{\mathit{r}}\right)$ are the spherical spinors [71,72]. The large- and small-component radial functions, ${\tilde{g}}_{\kappa }(E_e,r)$ and ${\tilde{f}}_{\kappa }(E_e,r)$, respectively, satisfy the following system of coupled differential equations,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({\displaystyle \frac{d}{dr}}+{\displaystyle \frac{\kappa +1}{r}}\right){\tilde{g}}_{\kappa }-(E_e-V\left(r\right)+m_e){\tilde{f}}_{\kappa }& =0,\hfill \\ \hfill \left({\displaystyle \frac{d}{dr}}-{\displaystyle \frac{\kappa -1}{r}}\right){\tilde{f}}_{\kappa }+(E_e-V\left(r\right)-m_e){\tilde{g}}_{\kappa }& =0.\hfill \end{array}\end{array}$$

(15)

For electrons emitted in DBD, we consider the modified Dirac-Hartree-Fock-Slater (DHFS) potential,

$$V\left(r\right)=V_{\mathrm{nuc}}\left(r\right)+V_{\mathrm{el}}\left(r\right)+V_{\mathrm{ex}}^{\mathrm{Slater}}\left(r\right),$$

(16)

whose components have been detailed in [73,74], based on the RADIAL subroutine package [75], that we employ in the actual calculations. The electron charge density required to construct the electronic and exchange components of the potential is obtained through a self-consistent DHFS atomic structure calculation of the final positive ion with the electronic configuration of the initial neutral atom, known as sudden approximation. This approximation is avoided in [76,77,78,79], where more complex models that account for overlap correction, and shake-up and shake-off effects have been considered. In particular, while the last two effects might influence the potential $V\left(r\right)$, the conclusions presented below remain valid. Nevertheless, the potential used in our model respects the correct asymptotic condition, i.e., $\underset{r\to \infty }{lim}rV\left(r\right)=\alpha Z_{\infty }$. The charge $Z_{\infty }=Z_e{Z}_f$ can be obtained in DBD from the electron charge $Z_e=-1$ and the charge of the final positive ion, i.e., $Z_f=2$.

The functions ${\tilde{g}}_{\kappa }$ and ${\tilde{f}}_{\kappa }$ must also satisfy the boundary condition “a plane wave plus incoming spherical waves” and are normalized in such a way that [66,70]

$$\left\{\begin{array}{c}{\tilde{g}}_{\kappa }(E_e,r)\\ {\tilde{f}}_{\kappa }(E_e,r)\end{array}\right\}=exp\left(-i{\overline{\Delta }}_{\kappa }\right)\left\{\begin{array}{c}{g}_{\kappa }(E_e,r)\\ f_{\kappa }(E_e,r)\end{array}\right\}$$

(17)

with the following asymptotic behavior,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{g}_{\kappa }(E_e,r)\\ f_{\kappa }(E_e,r)\end{array}\right\}\underset{r\to \infty }{\to }{\displaystyle \frac{1}{p_{e}r}}\left\{\begin{array}{c}\sqrt{{\displaystyle \frac{E_e+m_e}{2E_e}}}sin(p_{e}r-{\ell }_{\kappa }{\displaystyle \frac{\pi }{2}}-\eta ln\left(2p_{e}r\right)+{\overline{\Delta }}_{\kappa })\\ \sqrt{{\displaystyle \frac{E_e-m_e}{2E_e}}}cos(p_{e}r-{\ell }_{\kappa }{\displaystyle \frac{\pi }{2}}-\eta ln\left(2p_{e}r\right)+{\overline{\Delta }}_{\kappa })\end{array}\right\},\end{array}\end{array}$$

(18)

where ${\overline{\Delta }}_{\kappa }={\Delta }_{\kappa }+{\delta }_{\kappa }$ is the overall phase shift. Here, ${\delta }_{\kappa }$ is the inner phase shift induced by the finite-range component of final system potential, i.e., $V_{\mathrm{f}.\mathrm{r}.}\left(r\right)=V\left(r\right)-\alpha Z_{\infty }/r$, and $\eta =\alpha Z_{\infty }{E}_e/p_e$ is the Sommerfeld parameter. The effect of the pure Coulomb potential, i.e., $V_c\left(r\right)=\alpha Z_{\infty }/r$, is accounted for by the logarithmic phase, $-\eta ln\left(2p_{e}r\right)$, and the Coulomb phase shift [75],

$${\Delta }_{\kappa }=\nu -({\gamma }_{\kappa }-{\ell }_{\kappa }-1){\displaystyle \frac{\pi }{2}}+arg\Gamma ({\gamma }_{\kappa }+i\eta )-{\mathcal{S}}_{Z_{\infty },\kappa }\pi$$

(19)

with

$$\nu =arg\left[\alpha Z_{\infty }(E_e+m_e)-i(\kappa +{\gamma }_{\kappa })p_e\right],$$

(20)

$${\gamma }_{\kappa }=\sqrt{{\kappa }^2-{\left(\alpha Z_{\infty }\right)}^2},$$

(21)

and

$${\ell }_{\kappa }=\left\{\begin{array}{cc}\left|\kappa \right|-1\hfill & \mathrm{if}\kappa <0\hfill \\ \kappa \hfill & \mathrm{if}\kappa >0\hfill \end{array}\right.,{\mathcal{S}}_{Z_{\infty },\kappa }=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\kappa <0,Z_{\infty }<0\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right..$$

(22)

The phase shifts for electrons emitted in DBD of ¹⁰⁰Mo were obtained by solving Equation (15) in the potential generated by the final ion ¹⁰⁰Ru. The numerical inner phase shift for electrons with $\kappa =-1$ is displayed in the top-left panel of Figure 1 as a function of the kinetic energy. Due to the attractive nature of the finite-range component of the DHFS potential generated by the final positive ion ¹⁰⁰Ru, the absolute phase shift must be positive. However, the numerical inner phase shift reaches negative values because the RADIAL subroutine package [75] confines its calculation within the interval $(-\pi /2,\pi /2)$. The connection between the absolute and numerical inner phase shift was established using the graphical method.

Three points, from three regions of ${\delta }_{\kappa }\left(E_e\right)$, have been highlighted in the top-left panel of Figure 1. For each fixed energy, the radial dependence of the function $r{g}_{-1}(E_e,r)$ is displayed in the other panels of Figure 1 for both the modified DHFS potential (solid line) and pure Coulomb potential (dashed line). The radial ranges are selected to ensure that all functions reach their asymptotic behavior. By comparing the radial differences between the nodes of these functions, marked by arrows, the absolute inner phase shift can be identified. We found that in the regions corresponding to the points labeled “1”, “2”, and “3”, one must add $\pi$, $2\pi$, and $3\pi$, respectively, to the numerical phase shift to obtain the absolute phase shift. The same finding holds for electrons with $\kappa =1$ as well.

The energy dependence of the total phase shift, which includes the Coulomb phase shift, is presented in the left panel of Figure 2, for electrons with $\kappa =-1$ (solid line) and $\kappa =1$ (dashed line). The right panel of Figure 2 shows the function $cos\left({\overline{\Delta }}_{-1}-{\overline{\Delta }}_{+1}\right)$ which is essential for constructing the $E_{ss}\left(E_e\right)$ function from Equation (9). Interestingly, we observe a change in the sign of the cosine function around a kinetic energy of 2 keV for the emitted electron.

3. Results and Discussion

In Figure 3 we show the quantities $F_{ss}$ and $E_{ss}$ with and without phase shifts included in their definitions. Since $F_{ss}$ is defined in terms of the modulus squared of scattering wave functions, the two cases coincide. On the other hand, the quantity $E_{ss}$ is significantly smaller in the former case for electron energies below 100 keV, becoming even negative around 2 keV.

The angular correlation functions $\alpha$ are presented in Figure 4. In both DBD modes, at electron energies below a few tens of keV, when phase shifts are taken into account, $\alpha$ becomes positive. A similar result was also reported in [80], but the implications in the electron emission patterns were not provided. In this energy range, events where the electrons are emitted with an opening angle less than $\pi /2$ are more frequent than those where the angle exceeds $\pi /2$. The most probable emission angle is 0, meaning the electrons are most likely emitted in the same direction. Collinear emission is possible in the two-neutrino mode because the antineutrinos can be emitted in the direction opposite to that of the electrons, ensuring conservation of momentum. However, in the neutrinoless mode, momentum conservation with collinear electron emission can only occur due to the recoil of the nucleus. Note that the recoil energy is not accounted for in Equations (3) and (11), hence it is important to be cautious when relying on this intuitive picture.

An interesting feature shown in Figure 4 appears at electron energies above a few tens of keV. In the two-neutrino mode, the value of $\alpha$ obtained with phase shifts differs significantly from the value without phase shifts across the entire energy range. In contrast, in the neutrinoless mode, the two curves are nearly identical, except near the boundaries of the energy range. Additionally, we observe that including phase shifts increases the relative difference between ${\alpha }^{2\nu }$ and ${\alpha }^{0\nu }$, suggesting that the angular correlation function could be a useful tool for distinguishing between neutrinoless and two-neutrino double beta decay if high enough statistics is reached.

In Figure 5 we show the ${\kappa }^{2\nu }$ function, again with and without phase shifts included. When phase shifts are included, the negative contours are shifted towards higher electron energies and are also distorted in shape with respect to the case when phase shifts are omitted. More interestingly, in the former case there exist positive contours and the physical picture is similar to the one discussed for the the ${\alpha }^{2\nu }$ function. Since ${\kappa }^{2\nu }$ is a continuous function, the existence of positive and negative contours implies the existence of two contours where ${\kappa }^{2\nu }=0$. For the ¹⁰⁰Mo case, these contours correspond to one electron having an energy close to 2 keV. In this energy regime, electrons are emitted isotropically (if nuclear recoil is neglected).

Finally, we investigate the angular correlation coefficients given in Equation (2).

Table 1. The values of $K^{2\nu }$ for the DBD of ¹⁰⁰Mo decay for multiple sets of ${\xi }_{31}$ and ${\xi }_{51}$ parameters, corresponding to the HSD and SSD hypotheses, along with experimental measurements. The values from this work are labeled as “TW” and for the other, the reference next to the values indicates the original papers. For cases where the angular correlation coefficient value is not explicitly provided, we have derived it based on the reported PSFs. The “No Screening” case refers to the standard Fermi function approximation [70], while “Realistic Screening” involves models where atomic screening is derived from either the Thomas-Fermi equation or the DHFS method. Values in the second and third columns are computed either by neglecting phase shifts or by considering them approximately, while the final column includes results that fully account for phase shifts.

	${\mathit{K}}^{2\mathit{\nu }}$ Without Phase Shifts		${\mathit{K}}^{2\mathit{\nu }}$ With Phase Shifts
	No Screening	Realistic Screening	Realistic Screening
HSD	−0.650 [32]	−0.684 [61]	−0.640 (TW)
(${\xi }_{31}=0$; ${\xi }_{51}=0$)	−0.646 [33]	−0.684 [67]
	−0.646 [61]	−0.685 (TW)
	−0.646 [81]
SSD	−0.627 [33]	−0.668 [36]	−0.627 (TW)
(${\xi }_{31}=0.368$; ${\xi }_{51}=0.135$)	−0.627 [81]	−0.669 [37]
	−0.633 [61]	−0.668 [67]
		−0.674 [61]
		−0.675 (TW)
EXP	−0.630 [61]	−0.671 [61]	−0.624 (TW)
(${\xi }_{31}=0.450$; ${\xi }_{51}=0.165$ [65])		−0.672 (TW)

presents $K^{2\nu }$ values, calculated with and without phase shifts, for various ${\xi }_{31}$ and ${\xi }_{51}$ parameter sets, corresponding to the HSD hypothesis, SSD hypothesis, and experimental measurements. For references where the angular correlation coefficient value is not explicitly provided, we have derived it based on the reported PSFs. Our results without phase shifts align with previous studies that included atomic screening, while those neglecting this effect report larger $K^{2\nu }$ values. Notably, the ${\xi }_{31}$ and ${\xi }_{51}$ values associated with the HSD hypothesis produce the smallest $K^{2\nu }$ factors, whereas the experimentally measured set [65] results in the largest. When phase shifts are included in the decay rate calculation (last column of Table 1), we obtain $K^{2\nu }$ values around 7% higher than when phase shifts are excluded. This comparison suggests that incorporating phase shifts could similarly impact results in the other studies. In the case of the $0\nu \beta \beta$-decay of ¹⁰⁰Mo, we obtain $K^{0\nu }=-0.896$ when omitting phase shifts and considering the measured ${\xi }_{31}$ and ${\xi }_{51}$. Including phase shifts in the calculation leads to $K^{0\nu }=-0.882$, an increase of about $2\%$. This difference between modes is expected, as phase shifts impact a narrower portion of the integration domain in Equation (11) compared to Equation (3). We stress that the $2\%$ figure might differ for other mechanisms driving the $0\nu \beta \beta$-decay, but we limit this work to the light neutrino exchange mechanism.

4. Conclusions

In conclusion, we have studied the effect of phase shifts on the kinematics of $\beta \beta$-decay in ¹⁰⁰Mo, examining both $0\nu$ and $2\nu$ modes. For the investigation of the angular correlation distributions, $\alpha$ and $\kappa$, we found a remarkable feature: when electron phase shifts are included in the calculations, the electrons are most likely emitted in the same direction when one is below 2 keV. A similar result was also reported in [80], but the implications in the electron emission patterns, which are examined in detail in our study, were not provided.

From a systematic study of the previously reported $K^{2\nu }$, we showed that the inclusion of phase shifts impacts the results regardless of the approximations used in the factorization of the $2\nu \beta \beta$-decay rate or the description of the atomic screening effects. We showed that correctly accounting for phase shifts leads to an increase of the angular correlation factor, K, by $7\%$ ($2\%$) in the $2\nu$ ($0\nu$) mode. As a consequence, our findings might affect the proposals to constrain different new physics parameters from the angular correlation coefficient for the electrons emitted in $2\nu \beta \beta$-decay [31,32,33,34,36,37,38,48,49]. Additionally, the phase shits might have deeper implications in the tools to distinguish between the mechanisms driving $0\nu \beta \beta$-decay [57,58,59], but this will be discussed elsewhere as the current study is limited to the light neutrino exchange mechanism.