One-loop corrections to light neutrino masses in gauged U(1) extensions of the standard model

We consider gauged U(1) extensions of the standard model of particle physics with three right-handed sterile neutrinos and a singlet scalar. The neutrinos obtain mass via the type I seesaw mechanism. We compute the one-loop corrections to the elements of the tree-level mass matrix of the light neutrinos and show explicitly the cancellation of the gauge-dependent terms. We present a general formula for the gauge-independent, finite one-loop corrections for arbitrary number new U(1) groups, new complex scalars, and sterile neutrinos. We estimate the size of the corrections relative to the tree-level mass matrix in a particular extension, the superweak model.

Figure 1

One-loop graphs contributing to the neutrino self-energy. Top left: Goldstone boson contribution. Top right: scalar contribution. Bottom left: neutral gauge boson contribution. Bottom right: charged gauge boson contribution. Note that the $W$ boson loop does not contribute to the matrix ${\mathbf{B}}_L$.

Figure 2

Absolute values of $\sin {\theta }_G$ (top) and mass of $Z^{\prime }$ boson (bottom) in the logarithmic $(g_y^{\prime },g_z^{\prime })$ plane. For ${\theta }_G$ the contour labels $n$ correspond to value ${10}^n$. Left plots: $w=100\mathrm{GeV}$. Right plots: $w=750\mathrm{GeV}$.

Figure 3

Matrix elements ${\mathbf{F}}_{ij}$ as a function of the mass $m_{\mathrm{loop}}$ of the boson in the loop are confined to the blue band, assuming normal neutrino mass hierarchy. We have highlighted with vertical bands the relevant mass regions where the masses of the bosons in the loop lie. The scalar $s$ is required to have mass between 144 and 558 GeV requiring stability of the vacuum [40]. Left plot: $m_1^{\mathrm{tree}}=0.01\mathrm{eV}$, $m_4^{\mathrm{tree}}=30\mathrm{keV}$, $m_5^{\mathrm{tree}}\approx m_6^{\mathrm{tree}}=2.5\mathrm{GeV}$. Right plot: $m_1^{\mathrm{tree}}=0.001\mathrm{eV}$, $m_4^{\mathrm{tree}}=7.1\mathrm{keV}$, $m_5^{\mathrm{tree}}\approx m_6^{\mathrm{tree}}=3.0\mathrm{GeV}$.

Figure 4

Eigenvalues of the matrix $\mathbf{F}$ as a function of the mass of the boson in the loop $m_{\mathrm{loop}}$, assuming $m_1^{\mathrm{tree}}=0.01\mathrm{eV}$, $m_4^{\mathrm{tree}}=30\mathrm{keV}$, $m_5^{\mathrm{tree}}\approx m_6^{\mathrm{tree}}=2.5\mathrm{GeV}$, and normal neutrino mass hierarchy.