Quenching of nuclear matrix elements for $0\nu \beta \beta$ decay by chiral two-body currents

We examine the leading effects of two-body weak currents from chiral effective field theory on the matrix elements governing neutrinoless double-$\beta$ decay. In the closure approximation these effects are generated by the product of a one-body current with a two-body current, yielding both two- and three-body operators. When the three-body operators are considered without approximation, they quench matrix elements by about 10%, less than suggested by prior work, which neglected portions of the operators. The two-body operators, when treated in the standard way, can produce somewhat larger quenching. In a consistent effective field theory, however, these two-body effects become divergent and must be renormalized by a contact operator, the coefficient of which we cannot determine at present.

Figure 1

$0\nu \beta \beta$ decay with electron lines omitted. Diagram (a) shows the leading contribution in which the one-body current acts twice, turning two neutrons (thick solid red lines) into two protons (dotted blue lines) via the exchange of a Majorana neutrino. Diagram (b) shows the action of the pion-exchange two-body current at one vertex; the thin solid black line on the left represents either a proton or a neutron. In diagram (c) the contact current replaces the pion-exchange current.

Figure 2

Goldstone-diagram contributions to the $0\nu \beta \beta$ matrix element from the two-body current in symmetric nuclear matter. The thick solid red lines represent neutrons, the dotted blue lines represent protons, the thin black lines represent either neutrons or protons, and the wiggly black lines represent the exchanged neutrino. Upward-pointing arrows signify particles and downward-pointing arrows holes. The diagrams (a) and (b) in the top row represent the contributions considered in Ref. [18]. The diagrams (c)–(f) in the bottom row have not been considered before.

Figure 3

Relative effects on the $0\nu \beta \beta$ matrix element from the three-body-operator parts of diagrams involving chiral two-body currents [as shown in Figs. 1 and 1, and Eq. (8) with several sets of coefficients $c_3,c_4$, and as a function of $c_D$ for ${}^{76}\mathrm{Ge}$. The solid line represents the full results, the dashed line represents the approximate results when three-body operators are discarded after normal ordering with respect to the intercore, and the dotted line represents the results when the normal ordering is with respect to an ensemble containing the GCM ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$ ground states. The results in the panels on the left include only contributions from the contraction of creation an annihilation operators at the same vertex in Fig. 1. See the text for details.

Figure 4

Relative effects on the $0\nu \beta \beta$ matrix element from the two-body-operator parts of diagrams involving chiral two-body currents [as shown in Figs. 1 and 1] as a function of $c_D$. The results in panel (a) are for three sets of values for $c_3$ and $c_4$. Panel (b) shows the effects of several modifications to the operator, both separately and when combined. See the text for details.