Precision Analysis of the $^{136}$Xe Two-Neutrino $\beta\beta$ Spectrum in KamLAND-Zen and Its Impact on the Quenching of Nuclear Matrix Elements

TL;DR

This paper provides a high-precision analysis of the $^{136}$Xe two-neutrino double beta decay spectrum, measuring nuclear matrix element ratios and discussing implications for neutrinoless decay models.

Contribution

It introduces an improved formalism for analyzing the two-neutrino $etaeta$ rate and measures the NME ratio $\xi^{2 u}_{31}$, offering new insights into NME quenching and decay mechanisms.

Findings

Measured $\xi^{2 u}_{31}$ consistent with some theoretical models.

Excluded part of QRPA predictions at 90% confidence level.

Highlighted sensitivity of NME ratios to quenching and state contributions.

Abstract

We present a precision analysis of the $^{136}$Xe two-neutrino $\beta\beta$ electron spectrum above 0.8 MeV, based on high-statistics data obtained with the KamLAND-Zen experiment. An improved formalism for the two-neutrino $\beta\beta$ rate allows us to measure the ratio of the leading and subleading $2\nu\beta\beta$ nuclear matrix elements (NMEs), $\xi^{2\nu}_{31} = -0.26^{+0.31}_{-0.25}$. Theoretical predictions from the nuclear shell model and the majority of the quasiparticle random-phase approximation (QRPA) calculations are consistent with the experimental limit. However, part of the $\xi^{2\nu}_{31}$ range allowed by the QRPA is excluded by the present measurement at the 90% confidence level. Our analysis reveals that predicted $\xi^{2\nu}_{31}$ values are sensitive to the quenching of NMEs and the competing contributions from low- and high-energy states in the intermediate nucleus. Since these aspects are also at play in neutrinoless $\beta\beta$ decay, $\xi^{2\nu}_{31}$ provides new insights towards reliable neutrinoless $\beta\beta$ NMEs.

Tables

Table 1. Table 1: Estimated systematic uncertainties used for the 136 Xe 2 ​ ν ​ β ​ β 2 𝜈 𝛽 𝛽 2\nu\beta\beta decay rate measurement.

Source	Systematic uncertainty (%)
Fiducial volume	3.0
Enrichment factor of 136Xe	0.09
Xenon mass	0.8
Detector energy scale	0.3
Detection efficiency	0.2
Total	3.1

Equations

$$(T_{1/2}^{2\nu }{)}^{-1}\simeq (g_A^{\mathrm{eff}}{)}^4|M_{GT}^{2\nu }{|}^2{G}_0^{2\nu },$$

$$(T_{1/2}^{2\nu }{)}^{-1}$$

$$=(g_A^{\mathrm{eff}}{)}^4|M_{GT-3}^{2\nu }{|}^2\frac{1}{|{\xi }_{31}^{2\nu }{|}^2}\left|G_0^{2\nu }+{\xi }_{31}^{2\nu }{G}_2^{2\nu }\right|,$$

$$M_{GT}^{2\nu }$$

$$M_{GT-3}^{2\nu }$$

Full text

Introduction.—Double-beta ($\beta\beta$) decay is a rare nuclear process. The $\beta\beta$ decay emitting two electron antineutrinos and two electrons ($2\nu\beta\beta$) is described within the standard model of the electroweak interaction. In contrast, the $\beta\beta$ mode without neutrino emission ($0\nu\beta\beta$) implies new physics, and can only occur if neutrinos are Majorana particles. While $2\nu\beta\beta$ decay has been measured in 12 isotopes Barabash2015 , an observation of $0\nu\beta\beta$ decay remains elusive. In the standard scenario, the $0\nu\beta\beta$ rate is proportional to the square of the effective Majorana neutrino mass, $m_{\beta\beta}$ Avignone2008 , allowing the establishment of definite benchmarks toward the discovery of $0\nu\beta\beta$ decay in experiments.

The $0\nu\beta\beta$ rate, however, also depends on nuclear matrix elements (NMEs) which are poorly known Engel2016 , as $0\nu\beta\beta$ NME estimates vary between the many-body approaches used to calculate them. In addition, NMEs may be affected by a possible “quenching” or, equivalently, an effective value of the axial-vector coupling ${\it g}_{A}^{\rm eff}$ in the decay. Overall, the NME uncertainty can reduce the experimental sensitivity on $m_{\beta\beta}$ by up to a factor of 5 DellxOro2016 . To mitigate this, nuclear many-body predictions need to be tested in other observables. Several nuclear structure Schiffer2008 ; Kay2009 ; Freeman2012 ; Brown2014 and Gamow-Teller (GT) properties Yako2009 ; Frekers2013 ; Shimizu2018 have been proposed as $0\nu\beta\beta$ decay probes. Because $2\nu\beta\beta$ and $0\nu\beta\beta$ decays share initial and final nuclear states, and the transition operators are similar, a reproduction of $2\nu\beta\beta$ decay is key to reliable $0\nu\beta\beta$ NME predictions. Nonetheless, few nuclear many-body methods are well suited for both $\beta\beta$ modes, because nuclei with even and odd numbers of neutrons and protons up to high excitation energies need to be described consistently. The most notable approaches are the quasiparticle random-phase approximation (QRPA)  Vogel1986 ; Engel1988 ; Suhonen1998 ; Simkovic2011 ; Simkovic2013 and the nuclear shell model Caurier1990 ; Poves1995 ; Caurier2010 ; Horoi2013 ; Coraggio2017 ; Iwata2016 .

The $2\nu\beta\beta$ rate is usually expressed as

[TABLE]

where $M_{GT}^{2\nu}$ is the $2\nu\beta\beta$ NME and $G^{2\nu}_{0}$ a known phase-space factor Kotila2012 . As a result, ${\it g}_{A}^{\rm eff}$ can be determined from the measured $T_{1/2}^{2\nu}$ once $M_{GT}^{2\nu}$ is theoretically evaluated, a strategy followed in Ref. ibm13 . While a similar approach has been used in the nuclear shell model, especially for 136Xe Caurier2011 ; Horoi2013 , it is more common to take ${\it g}_{A}^{\rm eff}$ from GT $\beta$ decay and electron-capture (EC) rates Caurier2011 ; Neacsu2015 , assuming a common quenching for all weak processes. Likewise, the QRPA can also use $\beta$ decay and EC to obtain ${\it g}_{A}^{\rm eff}$ lisi08 ; Suhonen2004 ; Suhonen2013 , even though the standard approach is to fix ${\it g}_{A}^{\rm eff}$ first, and then adjust the nuclear interaction so that $M_{GT}^{2\nu}$ describes the $2\nu\beta\beta$ half-life Rodin2006 . In this way, the nuclear shell model and QRPA typically reproduce experimental $2\nu\beta\beta$ rates and predict nonmeasured ones Caurier1990 ; Poves1995 ; Horoi2016 ; Pirinen2015 ; Perez2018 .

Recently, the $2\nu\beta\beta$ decay of several isotopes has been observed with high statistics by the NEMO-3 Arnold2015 , EXO Albert2014a , KamLAND-Zen Gando2016 , GERDA Agostini2017 , Majorana Aalseth2018 and CUORE Alfonso2015 collaborations. These achievements demand an improved theoretical description. Reference Simkovic2018 gives a more accurate expression for the $2\nu\beta\beta$ decay rate

[TABLE]

where the phase-space factor $G_{2}^{2\nu}$ has a different dependence on lepton energies than $G_{0}^{2\nu}$, and the subleading nuclear matrix element $M^{2\nu}_{GT-3}$ enters the (real-valued) ratio $\xi_{31}^{2\nu}=M_{GT-3}^{2\nu}/M_{GT}^{2\nu}$. While $M_{GT}^{2\nu}$ is sensitive to contributions from high-lying states in the intermediate odd-odd nucleus, for $M_{GT-3}$ only the lowest-energy states are relevant due to rapid suppression in the energy denominator. Consequently $\xi_{31}^{2\nu}$ probes additional, complementary physics to the $2\nu\beta\beta$ half-life. This novel observable can be determined experimentally by fitting the $2\nu\beta\beta$ electron energy spectrum to extract the leading and second order contributions in Eq. (2). Hence, the measurement of $\xi_{31}^{2\nu}$ challenges theoretical calculations and can discriminate between those that reproduce the $2\nu\beta\beta$ rate.

In this Letter, we analyze the high-statistics $2\nu\beta\beta$ decay of 136Xe with KamLAND-Zen Gando2016 and compare the measured $T_{1/2}^{2\nu}$ and $\xi_{31}^{2\nu}$ values with the predictions from the QRPA and nuclear shell model. In KamLAND-Zen, the spectral distortion due to $\xi_{31}^{2\nu}$ could be up to 8% based on the theoretical predictions. Such effect is testable with accumulated statistics of $\sim 10^{5}$ $2\nu\beta\beta$ decays. Because $0\nu\beta\beta$ NMEs also show a competition between contributions from low- and high-energy intermediate states Simkovic2011 , testing theoretical $\xi_{31}^{2\nu}$ predictions can provide new insights on $0\nu\beta\beta$ calculations, including the possible quenching of the NMEs.

Experiment and results.—The KamLAND-Zen (KamLAND Zero-Neutrino Double-Beta Decay) detector consists of 13 tons of Xe-loaded liquid scintillator (Xe-LS) contained in a 3.08-m-diameter spherical inner balloon (IB). The IB is constructed from 25-$\mu$m-thick transparent nylon film and is suspended at the center of the KamLAND detector Gando2015 ; Gando2013b . The IB is surrounded by 1 kton of liquid scintillator (LS) which acts as an active shield. The scintillation photons are viewed by 1879 photomultiplier tubes mounted on the inner surface of the containment vessel. The Xe-LS consists of 80.7% decane and 19.3% pseudocumene (1,2,4-trimethylbenzene) by volume, 2.29 g/L of the fluor PPO (2,5-diphenyloxazole), and $(2.91\pm 0.04)$% by weight of enriched xenon gas. The isotopic abundances in the enriched xenon were measured by a residual gas analyzer to be $(90.77\pm 0.08)\%$ 136Xe, $(8.96\pm 0.02)\%$ 134Xe.

We report on data collected between December 11, 2013 and October 27, 2015, which is the same data set analyzed for the $0\nu\beta\beta$ search in Ref. Gando2016 with a total live time of 534.5 days. The selection to reduce the background contributions is the same as in Ref. Gando2016 , but we apply a tightened $2\nu\beta\beta$ event selection for this work in order to avoid systematic uncertainties arising from backgrounds. The fiducial volume for the reconstructed event vertices is defined as a 1-m-radius spherical shape at the detector center, which gives a fiducial exposure for this analysis of $(126.3\pm 3.9)$ kg yr in 136Xe. We perform a likelihood fit to the binned energy spectrum of the selected candidates between 0.8 and 4.8 MeV, tightened relative to the $2\nu\beta\beta$ analysis in Ref. Gando2016 . The systematic uncertainties on the $2\nu\beta\beta$ rate are evaluated identically as in Ref. Gando2016 and are summarized in Table 1.

A detailed energy calibration is essential for the extraction of $\xi_{31}^{2\nu}$. The energy scale was determined using $\gamma$ rays from 60Co, 68Ge, and 137Cs radioactive sources, $\gamma$ rays from the capture of spallation neutrons on protons and 12C, and $\beta+\gamma$-ray emissions from 214Bi, a daughter of 222Rn (lifetime 5.5 day) that was introduced during the Xe-LS purification. Uncertainties from the nonlinear energy response due to scintillator quenching and Cherenkov light production are constrained by the calibrations. The most important calibration is the high-statistics 214Bi from the initial 222Rn distributed uniformly over the Xe-LS volume. To ensure that the calibration with 214Bi can be applied to the entire data set, we confirmed that the time variation of the energy scale is less than 0.5% based on the spectral fit to the $2\nu\beta\beta$ decays for each time period. This uncertainty is reduced relative to the previous analysis Gando2016 , and is added to the energy scale error, which is the dominant error source for the $\xi_{31}^{2\nu}$ measurement, as discussed later.

The energy spectrum of selected candidate events between 0.8 and 2.5 MeV together with the best-fit spectral decomposition is shown in Fig. 1. In the fit, the contributions from $2\nu\beta\beta$ and major backgrounds in the Xe-LS, such as 40K, 210Bi, and the 228Th-208Pb subchain of the 232Th series are free parameters and are left unconstrained. The background contribution from 110mAg, which is important for the $0\nu\beta\beta$ analysis, is also a free parameter in the fit. The contributions from the 222Rn-210Pb subchain of the 238U series, and from 11C and 10C (muon spallation products), as well as the detector energy response model parameters, are allowed to vary but are constrained by their independent estimations Gando2016 .

The $2\nu\beta\beta$ spectrum is computed with Eq. (2), convolved with the detector response function. It is characterized by two free parameters: the total $2\nu\beta\beta$ rate and the ratio of the matrix elements $\xi_{31}^{2\nu}$. We obtained a best fit of $\xi_{31}^{2\nu}=-0.26^{+0.31}_{-0.25}$ and a 90% C.L. upper limit of $\xi_{31}^{2\nu}<0.26$. The systematic uncertainty on the energy scale limits the sensitivity of the $\xi_{31}^{2\nu}$ measurement, because an energy scale shift introduces a shape distortion similar to the change generated by a nonzero $\xi_{31}^{2\nu}$. The best-fit total $2\nu\beta\beta$ rate in the Xe-LS mass is $99.7^{+1.2}_{-1.4}$ (ton day)-1. Figure 2 shows the joint confidence intervals for the $2\nu\beta\beta$ rate and $\xi_{31}^{2\nu}$, which exhibit only a slight positive correlation. It indicates that the effect on the total $2\nu\beta\beta$ rate estimate by the introduction of the second order contribution is small. The effect on the $0\nu\beta\beta$ analysis is also negligibly small. Considering the systematic uncertainties in Table 1, the $2\nu\beta\beta$ decay half-life of 136Xe is estimated to be $T_{1/2}^{2\nu}=2.23\pm 0.03({\rm stat})\pm 0.07({\rm syst})\times 10^{21}$ yr. This result is consistent with our previous result based on phase-II data, $T_{1/2}^{2\nu}=2.21\pm 0.02({\rm stat})\pm 0.07({\rm syst})\times 10^{21}$ yr Gando2016 , and with the result obtained by EXO-200, $T_{1/2}^{2\nu}=2.165\pm 0.016({\rm stat})\pm 0.059({\rm syst})\times 10^{21}$ yr Albert2014a . Our analysis neglects counts from decays to excited states in 136Ba, for which our shell model calculations predict $T_{1/2}^{2\nu}(0_{\rm gs}^{+}\rightarrow 0_{1}^{+})>10^{26}$ yr. Even a very conservative half-life of $8.7\times 10^{24}$ yr that assumes the same NME for the decay to the ground (gs) and excited $0^{+}$ states does not affect our results. This issue might need to be revisited in the case of an unexpectedly short half-life close to the present 90% C.L. lower limit of $8.3\times 10^{23}$ yr Asakura2016 . The correction to $2\nu\beta\beta$ decay represented by $\xi_{31}^{2\nu}$ impacts KamLAND-Zen analyses of spectral distortions, including extraction of half-lives to excited states as well as searches for beyond-standard-model physics, such as for Majoron emission modes. Considering $\xi_{31}^{2\nu}$ as a free parameter, we find the additional uncertainty comparable to the energy scale error. Updated spectral analyses will be presented in future publications.

Theoretical calculations.—We obtain the $2\nu\beta\beta$ decay NMEs $M_{GT}^{2\nu}$ and $M_{GT-3}^{2\nu}$ to compare calculated $\xi^{2\nu}_{31}$ values to the KamLAND-Zen limit. The NMEs are defined as Simkovic2018

[TABLE]

with energy denominator $\Delta=[E_{j}-(E_{i}+E_{f})/2]/m_{e}$. $E_{k}$ is the energy of the nuclear state $|J^{\pi}_{k}\rangle$ with total angular momentum $J$ and parity $\pi$, and $m_{e}$ is the electron mass. The labels $i$, $j$, $f$ refer to the initial, intermediate and final nuclear states, respectively, while ${\bm{\sigma}}$ is the spin and $\tau^{-}$ the isospin lowering operator.

We perform nuclear shell model calculations in the configuration space comprising the $0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, $2s_{1/2}$, and $0h_{11/2}$ single-particle orbitals for both neutrons and protons, using the shell model code NATHAN Caurier2004 . We reproduce $M_{GT}^{2\nu}=0.064$ from Ref. Caurier2011 with the GCN interaction Caurier2010 , and also use the alternative MC interaction from Ref. Qi2012 , which yields $M_{GT}^{2\nu}=0.024$. Both interactions have been used in $0\nu\beta\beta$ decay studies Menendez2009 ; Shimizu2018 . Shell model NMEs for $\beta$ and $2\nu\beta\beta$ decays are typically too large, due to a combination of missing correlations beyond the configuration space, and neglected two-body currents in the transition operator Engel2016 . This is phenomenologically corrected with a “quenching” factor $q$, or ${\it g}_{A}^{\rm eff}=q\,{\it g}_{A}$. In general, the quenching that fits GT $\beta$ decays and ECs in the same mass region is valid for $2\nu\beta\beta$ decays as well. Around 136Xe, GT transitions with GCN are best fit with $q=0.57$ Caurier2011 , and with the same adjustment the 136Xe GT strength into 136Cs Frekers2013 , available up to energy $E\lesssim 4.5$ MeV, is well reproduced by both interactions. However, the experimental $2\nu\beta\beta$ half-life suggests different quenching factors $q=0.42(0.68)$ for GCN (MC). The calculations yield $M_{GT-3}^{2\nu}=0.011(0.0025)$. We assume a common quenching for $M_{GT}^{2\nu}$ and $M_{GT-3}^{2\nu}$ because the shell model reproduces well GT strengths at low and high energies up to the GT resonance Yako2009 . This gives ratios $\xi^{2\nu}_{31}=0.17$ for GCN and $\xi^{2\nu}_{31}=0.10$ for MC, both consistent with the present experimental analysis.

We also perform $2\nu\beta\beta$ decay QRPA calculations with partial restoration of isospin symmetry Simkovic2013 . We consider a configuration space of 23 single-particle orbitals (the six lowest harmonic oscillator shells with the addition of the $0i_{13/2}$ and $0i_{11/2}$ orbitals). We take as nuclear interactions two different G matrices, based on the charge-dependent Bonn (CD-Bonn) and the Argonne V18 nucleon-nucleon potentials. We fix the isovector proton-neutron interaction imposing the restoration of isospin Simkovic2013 . Finally, we adjust the isoscalar neutron-proton interaction to reproduce the $2\nu\beta\beta$ decay half-life for different values in the range ${\it g}_{A}^{\rm eff}\leq{\it g}_{A}=1.269$. We obtain the following ranges of results: $M_{GT}^{2\nu}=(0.011,0.164)$, $M_{GT-3}^{2\nu}=(0.0031,0.019)$ and $\xi^{2\nu}_{31}=(0.11,0.29)$ for the Argonne potential; and $M_{GT}^{2\nu}=(0.011,0.157)$, $M_{GT-3}^{2\nu}=(0.0036,0.018)$ and $\xi^{2\nu}_{31}=(0.11,0.35)$ using the CD-Bonn potential. Except for the larger $\xi^{2\nu}_{31}$ values, especially with CD-Bonn, most of the QRPA predictions are consistent with the present experimental analysis.

Discussion.—Figure 3 shows the effective axial-vector coupling constant ${\it g}_{A}^{\rm eff}$ as a function of the matrix element $M^{2\nu}_{GT-3}$ for the $2\nu\beta\beta$ decay of 136Xe. A large region in the ${\it g}_{A}^{\rm eff}-M^{2\nu}_{GT-3}$ plane is excluded by the present 90% C.L. limit $\xi^{2\nu}_{31}<0.26$. The two nuclear shell model GCN and MC results, indicated by points, are consistent with the KamLAND-Zen limit. The QRPA Argonne and CD-Bonn results are presented by curves, which accommodate $0.33\leq{\it g}_{A}^{\rm eff}\leq 1.269$ values (the lower end corresponds to vanishing isoscalar interactions). Both curves are very similar, because QRPA ratios of matrix elements with the same initial and final states are weakly sensitive to the nucleon-nucleon interaction Rodin2006 . Figure 3 shows that, even though most QRPA predictions are consistent with our measurement, ${\it g}_{A}^{\rm eff}\gtrsim 1.14(1.00)$ for the Argonne (CD-Bonn) potential is excluded at 90% C.L. by the KamLAND-Zen $\xi^{2\nu}_{31}$ limit.

Figure 3 also shows that for ${\it g}_{A}^{\rm eff}\gtrsim 0.7$ the QRPA predicts larger $\xi^{2\nu}_{31}$ values than the nuclear shell model. Elsewhere, the QRPA ratios lie between those of the GCN and MC shell model interactions. Interestingly, for ${\it g}_{A}^{\rm eff}\sim 0.5$, the QRPA and shell model GCN results are close. While such relatively small ${\it g}_{A}^{\rm eff}$ values are not always considered in $2\nu\beta\beta$ QRPA calculations of 136Xe, they are favored by QRPA statistical analyses that take into account experimental EC and $\beta$ rates lisi08 ; deppisch16 .

To illustrate the origin of the differences between the theoretical calculations, Fig. 4 compares the nuclear shell model and QRPA Argonne running sums of $M_{GT}^{2\nu}$ and $M_{GT-3}^{2\nu}$ Simkovic2018 ; Caurier2011 , multiplied by the corresponding $({\it g}_{A}^{\rm eff})^{2}$. The sums run over the excitation energy of the spin-parity $1^{+}$ states in the intermediate nucleus 136Cs. The theoretical $M_{GT}^{2\nu}$ running sums differ: while the shell model converges at $E_{\rm exc}\simeq 8$ MeV, QRPA terms contribute until $E_{\rm exc}\simeq 20$ MeV. Moreover, at $E_{\rm exc}\sim 10$ MeV the accumulated QRPA $M_{GT}^{2\nu}$ exceeds the shell model significantly, with a strong ${\it g}_{A}^{\rm eff}$ sensitivity. While for ${\it g}_{A}^{\rm eff}=1.269$ the maximum of the QRPA running sum is almost four times larger than the shell model one, for ${\it g}_{A}^{\rm eff}\sim 0.5$ —not shown in Fig. 4— the difference is only about 20%, consistent with the more similar $\xi^{2\nu}_{31}$ values predicted. $E_{\rm exc}\sim 10$ MeV shell-model contributions may be too small due to missing spin-orbit partner orbitals, but the QRPA may also overestimate them. Measurements of charge-exchange reactions up to the 136Xe GT resonance, currently limited to lower energy Frekers2013 ; Puppe2011 , can clarify this picture. Above $E_{\rm exc}\gtrsim 10$ MeV, the QRPA excess with respect to the shell model is canceled. The final value, set by the $2\nu\beta\beta$ half-life, is common to all calculations.

By contrast, Fig. 4 shows that in both shell model and QRPA the lowest $1^{+}$ state component dominates the $M_{GT-3}^{2\nu}$ NME. Such contribution is more salient for the shell model GCN and QRPA ${\it g}_{A}^{\rm eff}=1.269$ calculations, which explains the larger associated $\xi^{2\nu}_{31}$ value compared to the shell model MC and QRPA ${\it g}_{A}^{\rm eff}=0.8$ results, respectively. The contrast in the $M_{GT-3}^{2\nu}$ running sum at low energies is ultimately responsible for the different $\xi^{2\nu}_{31}$ values predicted by the QRPA and nuclear shell model.

In $0\nu\beta\beta$ decay, the running sum of the NME can extend to even higher energies, because in this case there is no dependence on the energy of the intermediate states in the denominator; see Eqs. (3) and (4). Therefore, a competition between contributions from low- and high-energy states similar to $2\nu\beta\beta$ decay is expected Simkovic2011 ; Senkov2013 ; Senkov2014 . Consequently, fixing $\xi^{2\nu}_{31}$ in $2\nu\beta\beta$ decay will allow one to identify the most promising $0\nu\beta\beta$ NME predictions.

Further experimental $\xi^{2\nu}_{31}$ sensitivity improvements may distinguish between various scenarios. On the one hand, measured values of $\xi^{2\nu}_{31}\geq 0.11$ will allow QRPA calculations to fix the quenched value of ${\it g}_{A}^{\rm eff}$, reducing uncertainties in QRPA $0\nu\beta\beta$ NMEs Rodin2006 ; Ejiri2019 ; Terasaki2016 . Likewise, a measured value $\xi^{2\nu}_{31}\simeq 0.17(0.10)$ would suggest that the GCN (MC) shell model interaction, with its associated ${\it g}_{A}^{\rm eff}$ value, leads to a more reliable $0\nu\beta\beta$ NME. Because the QRPA and shell model rely on different assumptions, and for $2\nu\beta\beta$ decay they can exhibit contrasting sensitivities on ${\it g}_{A}^{\rm eff}$ —as shown in Fig. 4— a measurement of $\xi^{2\nu}_{31}$ could lead to different ${\it g}_{A}^{\rm eff}$ values for each model. Furthermore, the quenching may not be the same in $2\nu\beta\beta$ and $0\nu\beta\beta$ decays, especially in the light of the differences in the two-body Menendez2011 ; Engel2014 ; Wang2018 and contact Cirigliano2018 corrections to the two $\beta\beta$ transition operators. On the other hand, a small ratio $\xi^{2\nu}_{31}<0.11$, which cannot be accommodated in the present QRPA calculations, or a determination of $\xi^{2\nu}_{31}$ very different to the GCN and MC predictions, would demand improved theoretical developments.

Summary.—We have presented a precision analysis of the 136Xe $2\nu\beta\beta$ electron spectrum shape with the KamLAND-Zen experiment. For the first time, we set a limit on the ratio of nuclear matrix elements $\xi^{2\nu}_{31}<0.26$ (90% C.L.). The experimental limit is consistent with the predictions from the nuclear shell model and most QRPA calculations, but excludes QRPA Argonne (CD-Bonn) results for ${\it g}_{A}^{\rm eff}\gtrsim$ 1.14(1.00). The allowed theoretical values vary in the range $\xi^{2\nu}_{31}=(0.10-0.26)$, so that future $\xi^{2\nu}_{31}$ measurements will be required to further test $2\nu\beta\beta$ calculations, and select the most successful ones. The associated ${\it g}_{A}^{\rm eff}$ value, or NME quenching, would also be identified. Future experiments such as KamLAND2-Zen Inoue2013 and others with improved resolution and reduced backgrounds promise enhanced sensitivity to reach this goal. Our analysis reveals that $\xi^{2\nu}_{31}$ is sensitive to competing contributions to the NME from low- and high-energy intermediate states. Because a similar competition is also relevant for $0\nu\beta\beta$ decay, studies of this observable provide new insights for identifying reliable $0\nu\beta\beta$ NMEs.