Species scale associated with Weinberg operator and bound on Majorana neutrino mass

TL;DR

This work investigates quantum gravity (swampland) constraints on Majorana neutrino masses generated by the Weinberg operator, focusing on towers of states that couple to the Higgs via the same operator. By defining a Majorana species scale $\Lambda_{\mathrm{sp},\nu}=M/\sqrt{N_{\mathrm{sp},\nu}}$ and demanding it lie below the gravitational species scale $\Lambda_{\mathrm{sp}}=M_{\rm Pl}/\sqrt{N_{\rm sp}}$, the authors derive an upper bound on the Weinberg operator scale $M$ and a corresponding lower bound on the physical neutrino mass $m_\nu = v^2/M$. Depending on whether KK or string towers dominate, the resulting lower bound on $m_\nu$ is of order $\sim 2.5\times 10^{-5}$ eV (KK) up to $\sim 10^{-4}-10^{-1}$ eV (string), still below current observational sensitivity; Festina-Lente constraints are found to be weaker. The paper also discusses the coexistence of Dirac and Majorana masses within EFT, and how these quantum gravity considerations interact with broader swampland ideas such as the AdS distance conjecture.

Abstract

When states in a tower like the Kaluza-Klein or the string tower couple to another state through the irrelevant operators of the same type, their contributions to the loop corrections of the relevant or the marginal operators are not negligible, threatening the perturbativity. This can be avoided provided the cutoff scale is lower than the species scale associated with the irrelevant operator. We apply this to towers of states associated with the neutrino which couple to the Higgs through the Weinberg operator, the dimension-5 irrelevant operator generating the Majorana neutrino mass. Requiring the `Majorana species scale', the species scale associated with the Weinberg operator, to be below the gravitational species scale, one finds the lower bound on the Majorana neutrino mass determined by the species number. The Festina-Lente bound also gives the lower bound on the Majorana neutrino mass, but it is not so stringent. Meanwhile, even if the neutrino mass is of the Dirac type at the renormalizable level, the Majorana mass term still can be written in the effective field theory action so far as the Weinberg operator is not forbidden. Even if the Majorana neutrino mass is larger than the Dirac one, so far as there are sufficient degrees of freedom with mass smaller than the scale of the cosmological constant, the observation of the Majorana nature of the neutrino may not contradict to quantum gravity constraints which rules out the neutrino mass purely given by the Majorana type.

Figures (3)

• Figure 1: $(a)$ : Propagators for $\xi_n$, $\xi_n^\dagger$ with $n\in \mathbb{Z}_{\geq 0}$ (arrow lines) and $\phi$ (dotted line). The arrow in the $\xi_n$($\xi_n^\dagger$) line toward(runaway from) the undotted(dotted) index represents the left(right)-handed spinor. The KK mass insertions in the propagators of $\xi_n$ and $\xi_n^\dagger$ are indicated by the cross. $(b)$ : Interaction vertices for the Weinberg operator and the mass of the neutrino (i.e., zero mode $\xi$) generated by the Weinberg operator. The cross indicates that the Higgs VEV is inserted.
• Figure 2: The $s$-channel diagram for the one-loop contribution of the tower states to the Higgs quartic coupling through the Weinberg operator.
• Figure 3: Feynman diagrams associated with the corrected Higgs mass up to two-loop order, generated by the Higgs quartic coupling and the Weinberg operator. The cross indicates the insertion of the Higgs VEV.