Examination of reliability of nuclear matrix elements of neutrinoless double-β decay by QRPA J. Terasaki, Institute of Experimental and Applied Physics, Czech Technical Univ. in Prague Y. Iwata, Faculty of Chemistry, Materials and Bioengineering, Kansai University The goal is to determine the effective mass of the neutrino. The double-β decay of nucleus is used for this purpose. We aim at the theoretical contribution. Possible change of two neutrons to two protons in a nucleus emitting two electrons with neutrino exchange (neutrinoless double-β decay). This decay occurs, if the neutrino（ν） is a Majorana particle (𝜈= 𝜈̅ ), and the effective neutrino mass can be determined; see the equations below. Determination of the effective neutrino mass is one of the most important subjects in modern physics. 𝜈𝑒 𝑛 𝑛 𝑝 𝑝 𝑒− 𝑒− 𝜈̅𝑒 Nucleus of (neutron number, proton number) = (𝑁, 𝑍) Nucleus (𝑁−2, 𝑍+ 2) Why nuclei? Because 𝐸(final state) < 𝐸(initial state) is necessary. Other conditions for the nuclei used in the experiments: Single beta decay is suppressed. The energy spectrum of two electrons in 2𝜈𝜈𝜈 decay can be distinguished well from that of 0𝜈𝜈𝜈. Large Q value [ ≈𝐸(initial state) −𝐸(final state)]. The parent nuclei can be produced massively with high purity. 76Ge→76Se 130Te→130Xe 136Xe→136Ba 150Nd→150Sm 48Ca→48Ti 82Se→82Kr 96Zr→96Mo 100Mo→100Ru 110Pd→110Cd 116Cd→116Sn 124Sn→124Te and more List of decays searched in the experiments 1/𝑇0𝜈(g. s. →g. s. ) = 𝑀0𝜈 2𝐺0ν𝑔𝐴 4 𝑚𝜈 𝑚𝑒 2 Half-life Nuclear matrix element Phase -space factor Effective 𝜈 mass Principle to determine effective neutrino mass 𝑚𝜈= ෍𝑈𝑒𝑒 2 𝑚𝑖 𝑖=1,2,3 Theoretical calculation Experimental measurement U : Pontecorvo–Maki– Nakagawa–Sakata matrix 𝑚𝑖: eigen mass (i =1,2,3) Axial-vector current coupling 𝑚𝑒: electron mass Phase-space factor ← Wave functions of emitted electrons Nuclear matrix element ← Nuclear wave functions Approximation is indispensable. Accurate calculation more difficult than the phase-space factor Nuclear matrix element 𝑉𝑟12, 𝐸𝑏≅ℎ+ 𝑟12, 𝐸𝑏 {linear combination of Double- Gamow- Teller and Double- Fermi operators} 𝑀(0𝜈) = ෍෍෍⟨𝑝𝑝′ 𝑉𝑟12, 𝐸𝑏𝑛𝑛′⟩⟨0f +|𝑐𝑝′ † 𝑐𝑛′|𝑏⟩ 𝑛𝑛′ 𝑝𝑝′ 𝑏 ⟨𝑏|𝑐𝑝 †𝑐𝑛|0i +⟩ The transition operator used in our calculation is Final state, ground state of nucleus(N-2,Z+2) intermediate state, nucleus (N-1,Z+1). Energy is Eb. Initial state, ground state of nucleus (N,Z) ℎ+ 𝑟12, 𝐸𝑏 : neutrino potential; energy denominator included Status and purpose of this study: The calculated nuclear matrix elements by various approximation methods and groups are distributed in a range of factor of 2‒3. The nuclear matrix element cannot be obtained by experiment. Thus, examination and improvement of the calculation are essential. Approximation to the method to obtain the nuclear wave functions in our study: Quasiparticle random-phase approximation (QRPA) How good? • Transition-strength function can be well reproduced. • Sum rule is satisfied. • Widely used in nuclear and condensed-matter physics. Nuclear excitation is described as the superposition of two-quasiparticle excitations. Under the approximation to the equation of the nuclear matrix element by 𝑉𝑟12, 𝐸𝑏≅𝑉𝑟12, 𝐸ത𝑏 𝐸ത𝑏: average energy of the intermediate states |𝑏⟩, a virtual decay path is possible to use for calculation as two-neutron removal followed by two-proton addition. The nuclear matrix element of the virtual path must be equal to that of the double-β path. This is a constraint to the effective interactions used for the approximation. The strength of the isoscalar proton-neutron pairing interaction is determined. J. Terasaki, Phys. Rev. C 91, 034318 (2015); ibid 93, 024317 (2016) Why important? This interaction is necessary for calculating the nuclear matrix element of two-neutrino double-β decay. There are exp. data of half-life to this decay for those nuclei used for the neutrinoless double-β decay exp. The two-neutrino decay operator 𝑉2ν 𝐸𝑏 cannot be replaced by 𝑉2ν 𝐸ത𝑏, thus, the double-β-path calcul