Fractionally Charged Particles at the Energy Frontier: The SM Gauge Group and One-Form Global Symmetry

We are grateful to the editor and to the referees for their close reading and thoughtful, constructive comments. We have made minor updates to the text to address their questions, as we summarize below.

Indeed, we had thought about electroweak precision constraints and concluded they did not impose any bound but should have included a bit more discussion.

• For light new species $m_X \ll M_Z$, the limits from anomalous $Z$ decays, such as extra $Z \to$ invisible, provide the strongest constraint. The only benchmark case for which such a small mass is allowed by collider constraints is for $Y=1/6$, and as we had already mentioned such a particle can live in the current uncertainty on $M_Z$.

• For heavier $m_X$, our setup -- a single fractionally charged particle -- has all of the requirements for an analysis via oblique parameters (specifically, $X$ only couples to SM gauge bosons at tree level). In more modern terms, this setup is a "universal theory" (one where S,T,U are actual observables). See Wells & Zhang 2016 (1510.08462).

• In our simple setup, many of the oblique parameters are automatically zero: $T$ = 0 (and the same for its derivative $U$), as these measure violations of custodial symmetry and our setup involves no interactions other than gauge interactions. It's $T$ which is sensitive to modifications to the ratio of $M_W, M_Z$, so this means there is no constraint from the $W$ mass measurements. $S = 0$ as well: For scenarios where $X$ only has $U(1)$ charge this is obvious, however it is true even when $X$ has $SU(2)$ interactions. The reason is that $S$ is sensitive to EWSB, given that it involves the two point of a single $SU(2)$ current, while our $X$ is completely blind to EWSB (again stemming from the fact that it has no interactions with $H$).

• The only oblique parameters which are nonzero are $W$ and $Y$ (which are both insensitive to custodial or $SU(2)$ breaking). To determine the bounding power of these parameters, we can use the results of Cynolter & Lendvai 2008 (0804.4080), rescaling the expressions by a factor of $J(J+1)/(3/4)$ in $W$ (for $X$ in the $J$ $SU(2)$ irrep) or $Y^2_X$ in $Y$ (for $X$ with hypercharge $X$), and compare to the LEP2 bounds they quote. In both cases ($W$ and $Y$) we find there are no meaningful constraints. Specifically, for particles charged only under $U(1)$, the rescaled formulae from Cynolter & Lendvai place bounds well below $M_Z$ and are thus meaningless. For particles with $SU(2)$ charge, $W$ places bounds that are within the realm of validity of an oblique analysis, but the bounds are weaker than the collider constraints, at least for the benchmarks we consider.

• To derive more accurate bounds for $m_X < m_Z$, we would need to reinterpret LEP2 $e^+e^- \to \bar f f$ data in terms of the full one-loop two point function (rather than just Taylor expanding it as a function of $q^2/m^2_X$). Given that the $Z$ decay constraint already applies in this regime and is superseded by collider bounds in all scenarios with non-zero $SU(2)$ charge, we do not pursue this direction further here.

• It is worth mentioning that, in more complicated scenarios where there are 2+ fractionally charged particles, if Yukawa interactions between these states and the Higgs is allowed (e.g. if one is an $SU(2)$ doublet, one is a singlet, and their hypercharge differs by $1/2$), then we expect to find $S, T != 0$. Laying out the constraints from EWPO on these slightly-non-minimal but well-motivated models of multiple fractionally charged particles would be an interesting target for future investigation.

- Added a paragraph at the end of Section 3.0 noting that electroweak precision observables do not offer constraints on the benchmark models we consider.
- Deleted a clause in the caption to Table 2 as suggested by Referee 2.
- Added a footnote at the end of Section 6.3 on measuring irrational charges vs. rational charges with large denominators.