About electroweak domain walls in Majoron models

Abstract

Some time ago it was claimed in "Spontaneous Breaking of Lepton Number and Cosmological Domain Wall Problem" (Phys. Rev. Lett. 122, 151301 (2019)) that non perturbative instantons of the weak interaction $\text{SU}(2)_\text{W}$ lead to the formation of domain walls in Majoron models owing to the anomaly of the spontaneously broken global lepton number $L$ symmetry $\text{U}(1)_L$ with respect to $\text{SU}(2)_\text{W}$. We point out that it has long been known, that this effect can be completely rotated away unless there is a source of explicit $B+L$ breaking present, where $B$ denotes baryon number. We further estimate the tiny instanton induced Majoron mass from $B+L$ breaking and analyze the cosmological impact of such domain walls including possible finite temperature effects. In general this scenario does not lead to a cosmological catastrophe and we demonstrate that the tiny instanton induced mass can act as a bias term to collapse walls induced by a larger source of lepton number breaking. Alternatively this electroweak Majoron could act as dynamical dark energy.

Figures (3)

• Figure 1: Diagrammatic representation of the single instanton 't Hooft vertex with the fermion zero modes for each generation of $L\;(Q)$ in green (red) closed up by three insertions of the $B+L$ breaking effective operator in Eq. \ref{['eq:B+L']} with coefficient $c_L$. Adapted from Ref. Csaki:2023ziz.
• Figure 2: Parameter space for the production of ultra-light Majoron dark matter from domain wall decay. Here we depict a Majoron mass of $M_j=e-18eV$ together with a single scalar quadruplet $(d=4)$ with mass $M=5TeV$ contributing to the one-loop $\beta$-function of $\text{SU}(2)_\text{W}$ for the electroweak instanton induced bias term $m_j$(left) and $M_j=e-17eV$ together with a quintuplet $(d=5)$ of mass $M=20TeV$(right). On the blue line one can reproduce the dark matter relic abundance solely by domain wall decay (see Eq. \ref{['eq:OmegaBias']}). The red region is ruled out because the domain walls would either decay after they dominate the universe's energy budget (see Eq. \ref{['eq:cond1']}) or after matter-radiation-equality (see Eq. \ref{['eq:cond2']}). In the orange slice the bias term contribution $m_j$ becomes larger than $M_j$. The small gray region is excluded because the lepton number breaking scale $v_L$ would need to be larger than the cut-off $M_\text{UV}$ of our EFT. Stellar cooling bounds from the Majoron's coupling to electrons typically rule out $v_L<e8GeV$ for the Type I/III seesaw Heeck:2019guh, but this can be avoided for the Type II seesaw Choi:1989hi. For larger values of $v_L>e15GeV$ the energy density from coherent oscillations from the misalignment mechanism Preskill:1982cyAbbott:1982afDine:1982ah becomes the dominant source of the dark matter relic density. In all plots we take $|c_L|\cos{(\theta_\text{EW}+ 3 \delta_L)}^{1/3}=1$.
• Figure 3: Parameter space that reproduces the preferred range of Majoron masses and decay constants $f_j\equiv 2 v_L/3$ for the hint of thawing quintessence in the combined data from DESIDESI:2024mwxDESI:2024aqxDESI:2024kobDESI:2025zgxDESI:2025fii, the CMB surveys PlanckPlanck:2018vygPlanck:2019nip and ACTACT:2023douACT:2023kun, as well as the supernova catalogs Pantheon+Brout:2022vxf(left), Union3Rubin:2023jdq(middle) and DESY5DES:2024jxu(right) found by the analysis in Ref. DESI:2025fii. The blue bands correspond to the required rage of $m_j$ and the red bands indicate the needed range of $v_L$. In the gray region the lepton number breaking scale is larger than the UV cutoff $M_\text{UV}$ of the assumed effective field theory. In all plots we take $|c_L|\cos{(\theta_\text{EW}+ 3 \delta_L)}^{1/3}=1$ and include the effect of a single triplet $(d=3)$ scalar with a mass of $M=10TeV$ on the one loop $\beta$-function of $\text{SU}(2)_\text{W}$.