Erratum: Gravitational echoes of lepton number symmetry breaking with light and ultralight Majorons

We rectify a typographical error in equation (3.14) from the original version of this article, pertaining to the effective Higgs-Majoron coupling, λ(0) JJh 1 , expressed in the mass eigenbasis. Notably, the same error persisted in the computational routines used in our work. This oversight resulted in a discrepancy in the numerical calculation of the invisible Higgs decay branching fraction, Br(h 1 → JJ), affecting a subset of the sampled points. While the main conclusions remain unaltered, we provide an updated version of our results.


Correction in equation (3.14)
The effective triple Higgs-Majoron coupling in eq.(3.14) of the original version of this work contains a misplaced v h parameter that should only multiply the first term inside the square brackets rather than the entire expression.The correct form reads as

Updated numerical results
The identified typographical error was also present in the computational routines developed for this study, causing its impact to propagate into the numerical calculations.Specifically, the effect on the invisible Higgs decay branching ratio is noticeable.The originally randomly sampled points, which were intended to satisfy 10 −15 < Br(h 1 → JJ) < 0.18 as specified in table 3 of the original article, now correspond to a range that includes values below Br(h 1 → JJ) < 1.Instead of conducting a new and computationally expensive scan on the parameter space, we have filtered our dataset to require Br(h 1 → JJ) < 0.18.
In figures 1 to 4 and 6, we present the updated results in comparison to the original figures.While the allowed parameter space region is slightly narrower and features lower statistics due to the imposed filter, there are no further qualitative deviations to note regarding the original discussion, which remains valid.
-1 - The impact of the correction presented in this erratum is most noticeable in figure 5. Specifically, the horizontal band with κ λ ≈ 1, present in all six panels of figure 6 in the original manuscript, is no longer viable.This band corresponds to scenarios with a large invisible Higgs decay branching fraction resulting from a large λ JJh 1 , which is experimentally excluded.Fortunately, all such points fall well below the sensitivity reach of LISA, DECIGO, or BBO, and therefore, no discussion about them was conducted in the original version of this article.The allowed range of the scalar mixing angle has also slightly reduced to |sin α h | ≲ 0.2.

JCAP03(2024)E01
In summary, the updated results do not impact neither the discussion in the original work nor its key conclusions.

Figure 1 .
Figure 1.Scatter plots showing the strength (top panels) and inverse duration (bottom panel) of the phase transition in terms of the peak frequency and peak energy density amplitude of the SGWB.The PISCs for LISA, BBO and DECIGO are represented in red dashed, green dot-dashed and blue dotted curves respectively [1].The wall velocity v w is also indicated.

Figure 2 .Figure 3 .
Figure 2. SNR plots showing order parameters of the EW (left) and lepton number symmetry (right) phase transitions.The coloured isolines represent the SNR at LISA while the grey dahsed ones denote the shock formation time.The grey shaded are represents the area where the sound waves treatment is mostly reliable.These plots were produced using the public software PTPlot [2].

Figure 4 .Figure 5 .
Figure 4. Scatter plots showing the correlation between the strength of the Higgs self interaction v 2 h δ0 2Λ 2 and the effective portal couplings v 2 h δ2 2Λ 2 (both left panels), v 2 σ δ4 2Λ 2 (top-right) and λ σh (bottom-right).The SGWB peak amplitude is shown in the colour scale of the top and bottom-right panels, while in the bottom-right we show the trilinear Higgs coupling modifier.