TABLE IV: Limits on the effective neutrino mass 〈 m$_{ν}$ 〉 (in eV) corresponding to different theoretical model calculations of nuclear matrix elements obtained for T$_{1}$$_{/}$$_{2}$(0 νββ ) > 1 . 1 · 10 24 y in case of $^{100}$Mo and T$_{1}$$_{/}$$_{2}$ (0 νββ ) > 3 . 6 · 10 23 y in case of $^{82}$Se. Nuclear matrix elements $^{100}$Mo $^{82}$Se Shell model [23] - < 2 . 43 QRPA [24],[25] < 0 . 58 − 0 . 75 < 1 . 12 − 1 . 38 QRPA [26] < 0 . 45 − 0 . 93 < 0 . 89 − 1 . 61 IBM-2 [27] < 0 . 49 − 0 . 55 < 1 . 03 − 1 . 19 PHFB [28] < 0 . 70 The 0 νχ $^{0}$ββ decay requires the existence of a Majoron. It is a massless Goldstone boson that arises due to a global breakdown of ( B - L ) symmetry, where B and L are, respectively, the baryon and the lepton number. The Majoron, if it exists, could play a significant role in the history of the early Universe and in the evolution of stars. A 2 β -decay model that involves the emission of two Majorons was proposed within supersymmetric theories and several other models of the Majoron were proposed in the 1990s (see review [29] and references therein). The possible two electrons energy spectra for different 0 νχ $^{0}$ββ decay modes of $^{100}$Mo are shown in Fig. 4. Here n is the spectral index, which defines the shape of the spectrum. For example, for an ordinary Majoron n = 1, for 2 ν decay n = 5, in the case of a bulk Majoron n = 2 and for the process with two Majoron emission n = 3 or 7. No evidence for 0 νχ $^{0}$ββ decay was found for all seven isotopes. The limits for $^{100}$Mo, $^{82}$Se, $^{150}$Nd and $^{96}$Zr are presented in Table 5. In particular, strong limits on ”ordinary” Majoron (spectral index 1) decay of $^{100}$Mo ( T$_{1}$$_{/}$$_{2}$ > 2 . 7 · 10 22 y) and $^{82}$Se ( T$_{1}$$_{/}$$_{2}$ > 1 . 5 · 10 22 y) have been obtained. Corresponding bounds on the Majoron-neutrino coupling constant are < g$_{ee}$ >< (0 . 35 − 0 . 85) · 10 $_{−}$4 and < (0 . 6 − 1 . 9) · 10 $^{$_{−}$4}$, respectively (using nuclear matrix elements from [23–28]). C. Search for double beta decay with Majoron emission 9