††institutetext: Department of Physics and Astronomy, University of Notre
Dame, South Bend, IN, 46556 USA

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

Seth Koren<EMAIL_ADDRESS>Adam Martin<EMAIL_ADDRESS>
###### Abstract

The observed Standard Model is consistent with the existence of vector-like
species with electric charge a multiple of $e/6$. The discovery of a
fractionally charged particle would provide nonperturbative information about
Standard Model physics, and furthermore rule out some or all of the minimal
theories of unification. We discuss the phenomenology of such particles and
focus particularly on current LHC constraints, for which we reinterpret
various searches to bound a variety of fractionally charged representations.
We emphasize that in some circumstances the collider bounds are surprisingly
low or nonexistent, which highlights the discovery potential for these species
which have distinctive signatures and important implications. We additionally
offer pedagogical discussions of the representation theory of gauge groups
with different global structures, and separately of the modern framework of
Generalized Global Symmetries, either of which serves to underscore the
bottom-up importance of these searches.

##  1 Background and Motivation

#### The Fundamental Charge Quantum of QED

What is the fundamental quantum of electric charge in the infrared quantum
electrodynamics of our universe? This is an important particle physics
question which is as yet unresolved. The Bayesian prior of high energy theory
orthodoxy expects the answer to be $e$, the electric charge of the electron.
If the Standard Model fields are ever unified in $SU(5)$ or $SO(10)$, this is
necessarily true.111For a reminder of the experimental and theoretical reasons
which would point one toward this preference, see Witten’s beautiful 2002
Heinrich Hertz Lecture ‘Quest for Unification’ Witten:2002ei .

But a lesson one could contemplate from recent decades of Beyond the Standard
Model physics is that grand theories about the ultraviolet which we have come
to love seem not to be realized in quite the way we thought. We have not
produced superparticles, nor directly detected dark matter, nor found exotic
kaon decays, nor observed an electron electric dipole moment. And we have not
seen protons decay. We should indeed always be questioning which of our
cherished principles to cling to, and which to consider counterfactually.

Notably, with less ambitious unification schemes we can have a smaller quantum
of electric charge. As examples, in Pati-Salam theories (where we do not have
full gauge coupling unification) the fundamental infrared charge can be $e/2$,
and in theories of trinification (where we must add additional fermions) the
quantum can be $e/3$. If the Standard Model matter never organizes into one of
these minimal unified theories, then the fundamental quantum of charge can be
$e/6$.222Early work on extended models of unification which feature
fractionally charged particles includes Li:1981un ; Goldberg:1981jt ;
Goldberg:1982af ; Hong:1982ub ; Dong:1982sa ; Yamamoto:1982sk ;
Frampton:1982gc ; Kang:1981nr , and early discussions of the appearance of
fractionally charged particles in string theories include Wen:1985qj ;
Athanasiu:1988uj ; Antoniadis:1989zy ; Schellekens:1989qb . In more exotic
scenarios that would even more generally challenge our usual UV paradigms, the
charge could be even smaller.

The core message of our work is that particles with $\mathcal{O}(1)$ electric
charges are an important probe of ultraviolet physics which have a universal
infrared understanding. And it is not unreasonable to believe that they could
exist near the electroweak scale to be found at the energy frontier. After
all, we have only recently uncovered the full chiral spectrum of the Standard
Model; it is certainly possible that this matter content cannot tell apart
different UV scenarios but that our discovery of the least-massive vector-like
states will distinguish them further.

One may be misled into thinking that the question of the smallest charge of
quantum electrodynamics is ultimately a question about normalization, and
should not make much difference physically. It is true that the perturbative
physics of QED is not modified in any case. But the nonperturbative physics is
modified, as we will discuss in detail below.

And while the nonperturbative physics of the Standard Model is difficult to
access with only the SM degrees of freedom, the discovery of a new particle
can reveal nonperturbative aspects of the Standard Model physics. We learn
that the allowed charges of magnetic monopoles, the spectrum of fractional
instantons, and the possible Aharonov-Bohm phases are all modified. And as we
have just said, the possibilities for the Standard Model species to unify in
the ultraviolet depend crucially on this nonperturbative physics. This means
that determining the fundamental charge quantum of QED could falsify large
classes of models of grand unification, or potentially all of them.

#### From QCD to QED

Do not be confused by the charges of the quarks—by quantum electrodynamics we
mean a long-distance theory far below the scale of confinement where the
degrees of freedom are leptons and hadrons. The particular pattern of Yang-
Mills representations we see borne out in the Standard Model unavoidably
implies that all colorless hadrons have charge quantized in units of $e$, the
electron’s charge.

We can see this with a quick representation-theoretic argument, and we’ll
understand what’s happening more generally in Section 6. Let us begin with the
Standard Model having flowed to energies below electroweak symmetry breaking.
At these energies it is sensible to speak of quarks as Dirac fermions, as in
Table 1. Of the known colored particles, each quark $\psi_{i}^{a}$ in the
fundamental $\mathbf{3}$ representation has electric charge $q_{i}$ which
obeys $3q_{i}=2\ (\text{mod }3)$, and their antiparticles the
$\bar{\mathbf{3}}$ anti-fundamental $\bar{\psi}_{jb}$ necessarily have
$3q_{j}=1\ (\text{mod }3)$. The gluons in the adjoint $8$ are of course
electrically uncharged.

| $u_{i}$ | $d_{i}$ | $g$
---|---|---|---
$SU(3)_{C}$ | $\mathbf{3}$ | $\mathbf{3}$ | $\mathbf{8}$
$U(1)_{\rm EM}$ | $\frac{2}{3}$ | $-\frac{1}{3}$ | $0$
Table 1: Colored particles in the Standard Model after electroweak symmetry
breaking. $i$ is the generation index and here we use Dirac fermions. The
charge is given in units of $e$.

The only invariant tensors of $SU(3)_{C}$ are $\delta^{a}_{b}$,
$\varepsilon^{abc}$, and $\varepsilon_{abc}$, and we seek to build composite
operators which are colorless. Working $(\text{mod }3)$, we see
$\delta^{a}_{b}$ pairs a $1$ with a $2$, and the Levi-Civita symbol composes
three of the same charge—either way resulting in an electric charge $\sum
3q_{i}=0\ (\text{mod }3)$. Dividing through by three, this is exactly the
condition that every hadron has electric charge an integer multiple of $e$.
For an arbitrarily complex bound state, ultimately color indices can only be
contracted in these ways, and the same argument applies.

So with the particles of the Standard Model, there are no asymptotic states
with fractional charge. But it is not clear from this argument whether this
fundamentally must be the case, or whether this relationship might be broken
once we discover new BSM particles. Indeed we do not know the answer, which
ultimately must be settled by empirical data. We can understand the issue
systematically and gauge-invariantly as being a question about a certain
generalized symmetry which infrared physics may or may not have.

#### Generalized Global Symmetries

While the local, perturbative physics is not modified by the charge quantum,
the nonperturbative physics certainly is. A useful strategy to understand
these aspects systematically is by enlarging our notion of symmetries to
include symmetries of extended operators that appear in our field theories,
such as Wilson and ’t Hooft loops. Symmetries that act on such one-dimensional
line operators are known as ‘one-form symmetries’—to be contrasted with
symmetries that act on local, point operators which are called ‘zero-form
symmetries’.

From the modern field theory perspective, which such one-dimensional gauge-
invariant operators exist is part of the data needed to define a quantum field
theory Seiberg:2010qd ; Banks:2010zn ; Aharony:2013hda ; Gaiotto:2014kfa ;
Kapustin:2014gua ; Tong:2017oea . As a basic picture one can think of these
operators as accessing the response of the system to a probe particle in a
particular representation in the limit where the probe particle is infinitely
massive so that it has a well-defined worldline. Note that we do not specify
that the worldline must be a geodesic, or even timelike. With a spacelike
worldline, one is familiar with using a Wilson loop operator
$\exp(i\oint_{\gamma}A)$ to understand the Aharonov-Bohm effect where we think
about adiabatically moving an electron on the spatial path $\gamma$ around a
solenoid (or possibly a cosmic string).

As such, to fully understand the quantum field theory describing the particles
of the Standard Model, we must also analyze the symmetries of the one-
dimensional gauge-invariant operators we can write down, whether in the
electroweak phase or at lower energies. In the full Standard Model the
different ‘global structures of the gauge group’ (to be reviewed below) are
exactly the question of whether the Standard Model has a discrete group of
electric one-form global symmetries, or whether (some of) these electric one-
form symmetries should actually be gauged to instead produce extra magnetic
one-form global symmetries. This trade-off is as could be expected from Dirac
quantization.

Furthermore, this generalized symmetry language will provide a unifying,
general understanding of what we learn from experimentally probing the
existence of fractionally charged particles at the energy frontier. The
question of the charge quantum of quantum electrodynamics can be rephrased
universally in terms of emergent global electric one-form symmetry. We will
introduce these concepts pedagogically in Section 7.

Such one-form symmetries are data about the field theory which are in some
sense nonperturbative. That is, they are needed to have a more refined
understanding of the Yang-Mills theory which goes past what minimal coupling,
a Lagrangian procedure which only knows about local fields, depends upon. The
Lagrangian depends only on perturbative data which are local in field space.
In order to learn information about the global structure of the field space,
we must have data which allow us to probe paths in field space, not just
points. This is why there is new understanding to be gained by thinking about
extended operators in our QFTs.333Of course it is also natural to think about
maps of higher-dimensional manifolds into field space, and one may indeed talk
about $n$-dimensional operators and $n$-form symmetries, but in this work we
will only use the concepts of Wilson and ’t Hooft lines and their $1$-form
symmetries.

#### The Energy Frontier

As we have motivated above, searches for fractionally charged particles are
some of the highest stakes experimental probes we have at the energy frontier.
The observation of a particle with electric charge $e/6$, be it fundamental or
hadronic, fermionic or bosonic, would unequivocally falsify all minimal grand
unified theories. Perhaps no other single new particle discovery could teach
us so much about the far ultraviolet of our universe, so it is well worth
devoting experimental effort to searching for such particles.

Great energy frontier searches sensitive to fractionally charged particles
have been undertaken in recent years by CMS (e.g. CMS:2024eyx ) and ATLAS
(e.g. ATLAS:2019gqq ) but efforts have mainly been focused on SUSY-motivated
scenarios. To the extent that we can design searches sensitive to the electric
charges, fractionally charged particles can provide extremely distinctive
signatures, since as discussed above there are strictly no particles with
these properties in the Standard Model. We take here a first step toward a
more general paradigm by reinterpreting existing searches for various
benchmark SM quantum numbers which result in fractionally charged states.

We discuss the production cross-sections in Section 2 and give analytic
expressions in Appendix A for general representations. There is a rich variety
of phenomenologies of fractionally charged particles produced at the energy
frontier depending on their quantum numbers, which we discuss roughly in
Section 3, emphasizing where further dedicated theoretical or experimental
study is needed to have a better handle on their signatures. In Section 4 we
place bounds by reinterpreting various searches we find to be sensitive to
fractionally charged particles with caveats for reasonable assumptions we have
had to make as phenomenologists in the process. The constraints we find are
summarized schematically in Figure 4, and the reader should be struck by the
laxity of the bounds for certain combinations of quantum numbers.

Given the enormous amount these searches could teach us about the universe
quite generally, it is well worth both theorists and experimentalists
revisiting the possibilities for these searches, optimizing them for electric
charges at least down to $e/6$, and thinking about possible new strategies for
detection.

#### Previous Work on SM Global Structure

Recent motivation for thinking about fractionally charged particles comes from
discussions of the ‘global structure’ of the Standard Model gauge group, as we
will introduce pedagogically in Section 6. The basic point is that various
distinct gauge groups can nonetheless share the same structure close to the
identity, which is all that is probed by minimal coupling. Nonetheless the
representation theory for these different gauge groups is modified. And
indeed, the Standard Model gauge group has just such an ambiguity, being

$G_{\rm SM_{n}}\equiv\left(SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}\right)/\mathbb{Z}_{n},$ (1)

for $n=1,2,3,6$ (where ‘$\mathbb{Z}_{1}$’ is slang for $\mathds{1}$). We do
not yet know which is realized in nature, but $G_{\rm SM_{n}}$ allows
particles of infrared electric charge $ne/6$, and so the discovery of a
particle with charge $q<e$ will distinguish between them.

The different possibilities for the global structure of the Standard Model
gauge group were laid out first by Hucks Hucks:1990nw . The impact on the
allowed line operators was studied recently by Tong Tong:2017oea , where it
was made clear that with access to only the Standard Model degrees of freedom
the different theories cannot be distinguished on flat space. The consequences
of the global structure on a space of nontrivial topology have been explored
in depth in Anber:2021upc . Recently multiple groups have investigated how the
discovery of an axion and the careful measurement of its couplings to
different gauge groups also provides constraints on the global structure
Reece:2023iqn ; Choi:2023pdp ; Cordova:2023her . This essentially promotes the
discussion in Tong:2017oea  about the range of the SM theta angles to a new
dynamical probe—as we likewise here emphasize that a discovery of a new
fractionally charged particle directly probes the allowed line operators by
upgrading them to dynamical particles.

Some complementary perspectives on fractionally charged particles have
recently appeared as well. In Alonso:2024pmq  the authors focus on a
classification of representations consistent with general fractional charges
and global structures. In particular the case where the quantum of hypercharge
is smaller than expected in the SM is treated in full depth, which we will
comment on only briefly below. In Li:2024nuo  the authors focus on the effects
of fractionally charged particles in the Standard Model Effective Field Theory
(SMEFT). Indeed fractionally charged particles are an interesting case of
SMEFT operators being generated only at loop level, since they transform non-
trivially under gauge rotations for which all SM particles are neutral, which
implies that they must couple in pairs to SM matter. But resultingly the
ability of SMEFT to investigate the existence of fractionally charged
particles is quite limited, and we will see the energy frontier is our best
probe. In some sense this is necessarily true from the generalized symmetry
perspective because the emergent symmetry one finds below the mass of the
lightest fractionally charged particle is a global one-form symmetry under
which Wilson loops are charged, but local fields are strictly blind to.444This
‘in principle’ statement is a bit too quick. There is no ‘smoking gun’ in
SMEFT for the existence of fractionally charged particles, as some integer-
charged particles can turn on all the same operators in full generality. But
we anyway always must interpret some SMEFT deviation in terms of models that
only add a few new particles, as you cannot directly reverse the
renormalization group flow.

##  2 LHC Production

The primary phenomenological goal of this paper is to revisit collider bounds
on fractionally charged particles, fleshing out their different signatures and
how they are dictated by the particle’s quantum numbers. The scenario we focus
on is a single new Dirac fermion or complex scalar, denoted by $\psi$, sitting
in an ‘exotic’ representation of the SM gauge group such that the electric
charge of $\psi$ is some multiple $k\in\mathbb{Z}$ of $e/6$ (excluding
$k=0,\pm 6,\dots$ obviously).555 As $\psi$ is necessarily electrically
charged, it cannot be a Majorana fermion or a real scalar. In Table 2 we show
some example non-Abelian representations and which hypercharge they must have
to produce only integer electric charges in the far IR. Away from this choice,
the electric charge will be fractional, in a multiple of $e/6$. As we will
derive in Section 6, these are well-motivated to consider from the structure
of the Standard Model.

$SU(3)_{C}$ | $SU(2)_{L}$ | $6Y\text{ mod }6$
---|---|---
– | – | $0$
– | $\mathbf{2}$ | $3$
– | $\mathbf{3}$ | $0$
$\mathbf{3}$ | – | $4$
$\mathbf{6}$ | – | $2$
$\mathbf{8}$ | – | $0$
$\mathbf{3}$ | $\mathbf{2}$ | $1$
$\mathbf{6}$ | $\mathbf{2}$ | $5$
$\mathbf{8}$ | $\mathbf{3}$ | $0$
Table 2: For a given representation of $SU(2)_{L}$ and $SU(3)_{C}$,
fractionally charged particles are avoided only with this assignment of
hypercharge, up to the addition of an integer. Here we list the requirements
for some sample representations, but a full explanation of the structure is
given in Section 6 and in particular for the Standard Model in and below
Equation 25.

We will label $\psi$ by its full quantum numbers and its electric charge when
necessary, $\psi_{(SU(3),SU(2),Y),Q}$, though when the context is clear we
will drop subscripts other than the charge. We denote the electric charge in
fractions of $e$ throughout, e.g. using $Q=1/3$ for $e/3$. For color singlet
$\psi_{Q}$, the electric charge is given by the usual combination of
$\tau_{3}$ and $Y$, while for colored $\psi$ the charge of the outgoing states
is more subtle as $\psi_{Q}$ will combine with SM matter to form color
singlet, exotically charged ‘hadrons’.

We assume the only interactions $\psi_{Q}$ has are gauge interactions dictated
by its quantum numbers. As mentioned above, interactions involving a single
$\psi_{Q}$ and SM matter are forbidden, and we will ignore interactions
between pairs of $\psi$ (really $\bar{\psi}_{Q}\psi_{Q}$, etc.) and the SM,
such as $H^{\dagger}H\bar{\psi}_{Q}\psi_{Q}$. For fermionic $\psi$, all such
interactions are non-renormalizable, while for scalar $\psi_{Q}$ the Higgs
portal term is marginal (as is the quartic interaction
$(\psi^{\dagger}_{Q}\psi_{Q})^{2})$. Nevertheless, we will neglect this
possibility as we expect it to play little role in the collider phenomenology
for reasonable values of the couplings. For this initial study, we will also
largely ignore the possibility of multiple exotically charged states. For
certain quantum number assignments, it is possible to arrange for more
renormalizable interactions between the exotic and SM sectors, such as
$H\bar{\psi}_{Q}\psi^{\prime}_{Q^{\prime}}$ when one of
$\psi_{Q},\psi^{\prime}_{Q^{\prime}}$ is an $SU(2)$ doublet and
$Y_{\psi^{\prime}}-Y_{\psi}=1/2$. 666More exotic terms, such as
$\phi_{Q^{\prime}}\psi_{Q}f$ (where we have used $\phi_{Q^{\prime}}$ for an
exotic scalar in this context, $\psi_{Q}$ for a fermion, and $f$ a SM fermion)
are also possible, either with or without flavor structure. Multi-exotic
interactions could lead to interesting phenomenology, but are beyond the scope
of this paper.

Within this setup, $\psi_{Q}$ must be pair produced at colliders via its gauge
interactions. The dominant production mechanism depends on whether or not the
particles carry $SU(3)$ quantum numbers, irrespective of the spin of
$\psi_{Q}$. For color singlets, the particles are produced in Drell-Yan
$\bar{q}q\to\bar{\psi}_{Q}\psi_{Q}$ via $\hat{s}$ channel photon and $Z$. If
$\psi_{Q}$ is an $SU(2)$ singlet, the entire cross section is proportional to
$Q^{2}_{\psi}$, while the cross section for $\psi_{Q}$ in larger $SU(2)$
multiplets will contain pieces proportional to $(\tau_{3})_{\psi}$, the
entries of the diagonal $SU(2)$ generator appropriate for $\psi_{Q}$s
representation. When $(\tau_{3})_{\psi}\neq 0$, these terms typically dominate
the cross section as each power of $Q_{\psi}$ (which we have assumed to be
$<1$) comes with a factor of $\sin^{2}{\theta_{W}}\sim 1/4$. For $\psi_{Q}$ in
non-trivial $SU(2)$ representations, there is also a charged current
production mode, $\bar{q}q^{\prime}\to\bar{\psi}_{Q}\psi_{Q\pm 1}+c.c.$ via
$\hat{s}$ channel $W^{\pm}$.

If $\psi_{Q}$ carries $SU(3)$ quantum numbers, QCD production
$gg\to\bar{\psi}_{Q}\psi_{Q},\bar{q}q\to\bar{\psi}_{Q}\psi_{Q}$ becomes the
dominant mechanism. Of these, $gg$ is the larger channel when $\psi_{Q}$ is
light, but $\bar{q}q$ takes over for heavier $\psi_{Q}$. The crossing point
depends somewhat on the representation and spin of $\psi_{Q}$ but is
$\mathcal{O}(1\,\text{TeV})$ for a Dirac fermion color triplet.

The partonic cross sections for $pp\to\bar{\psi}_{Q}\psi_{Q}$ production are
compiled in Appendix A for both fermionic and scalar $\psi_{Q}$. For now, we
opt for analytic expressions over adding new particles to Monte Carlo programs
such as MadGraph Alwall:2014hca . In part, this is because we are focused on
pair production where the expressions are still simple, but the analytic
expressions also allow us to consider exotic color representations (such as a
decouplet) which are not easily implemented in MadGraph. Throughout this paper
we will only consider lowest order calculations, as our goal is to roughly
illustrate the current bounds rather than focus on a particular search or
$\psi_{Q}$.

Folding parton distribution functions into the partonic cross sections
(Appendix A), we find the LHC proton level cross sections
$pp\to\bar{\psi}\psi$. We use NNPDF3.0nlo parton distribution functions
Ball_2015 ; Hartland_2013  with $\alpha_{s}=0.118$,
factorization/renormalization scales of
$\hat{\mu}_{F}=\hat{\mu}_{R}=\sqrt{\hat{s}}$ and assume a collider center of
mass energy of $13\,\text{TeV}$. We have also imposed the parton-level cut
$|\eta_{\psi}|<2.5$ so that these particles appear in the tracker volume.

The proton level cross sections for some illustrative $\psi_{Q}$ are shown
below in Figs. 1 and 2 below. In Fig. 1 we show the cross section for $SU(2)$
singlet $\psi_{Q}$, either charged only under hypercharge (left panel), or
under several different color representations (right panel). Figure 2 shows
the cross sections for color singlet $\psi_{Q}$ sitting in non-trivial $SU(2)$
representations, both via neutral current (left panel) and charged current
(right panel).

Figure 1: Left panel: Lowest order pair production cross section for
$\psi_{Q}$ charged solely under hypercharge, $Q=1/6$ (blue), $Q=1/3$ (red),
$Q=1/2$ (green), $Q=2/3$ (brown). Right panel: lowest order LHC cross section
for colored $\psi_{Q}$ as a function of $M_{\psi}$ (only QCD interactions are
considered). For a fixed mass, the cross section increases with the size of
the representation: red (triplet), green (sextet), blue (octet), brown
(decouplet) and orange ($15$-plet (Dynkin label (21)). In both panels we
assume a center of mass energy $\sqrt{s}=13\,\text{TeV}$ and use solid lines
are for Dirac fermions and dashed lines for charged scalars.

Figure 2: Cross sections for $\psi_{Q}$ under different $SU(2)$
representations, all with $Y=1/3$. The red line shows the cross section for
the $(\tau_{3})_{\psi}=1/2$ component of a $SU(2)$ doublet, while the green
shows the $(\tau_{3})_{\psi}=1$ component of an $SU(2)$ triplet. As in Fig. 1,
solid lines are for Dirac fermions while dashed are charged scalars. The blue
lines (dot dashed for fermions, dotted for scalars) repeat the $SU(2)$
singlet, $Y=1/3$ curves from Fig. 1 for comparison. Changing the hypercharge,
the curves for the doublet and triplet cases would barely move, as the cross
section is dominated by the $SU(2)$ portion. Right panel: Charged current
cross section (via $W^{+}$) for doublets, triplets, and $SU(2)$ singlet for
comparison

The cross sections $\psi_{Q}$ charged only under hypercharge are quite small,
$\mathcal{O}(1\,\text{pb}\times Q^{2}_{\psi})$ for a fermionic $\psi_{Q}$ and
$M_{\psi}=100\,\text{GeV}$ and falling precipitously as $M_{\psi}$ increases
to $\mathcal{O}(2\,\text{fb}\times Q^{2}_{\psi})$ at
$M_{\psi}=500\,\text{GeV}$. Charging $\psi_{Q}$ under $SU(3)$, the cross
section jumps by orders of magnitude, $\sigma(pp\to\bar{\psi}_{Q}\psi_{Q})\sim
3\,\text{pb}\,(60\,\text{pb})$ for a $500\,\text{GeV}$ color triplet fermion
(color octet). The cross section for color singlet, $SU(2)$ charged $\psi_{Q}$
sits between these two, $\mathcal{O}(5\,\text{fb})$ for Drell-Yan production
of either state in a $500\,\text{GeV}$ doublet $\psi_{Q}$, and
$\mathcal{O}(10\,\text{fb})\ (\mathcal{O}(5\,\text{fb}))$ for charged current
production via $W+(W^{-})$. For other $SU(2)$ representations, both types of
cross section grow with the size of the multiplet; labelling the $SU(2)$ part
of the $\psi_{Q}$ state as $|I_{0},i_{3}\rangle$, Drell-Yan $\propto
i^{2}_{3}$, while the charged current is
$\propto(I_{0}(I_{0}+1)-i_{3}(i_{3}+1))$. The LHC cross section for a few
different $SU(2)$ multiplets (both Drell-Yan and charged current pieces) are
shown in the right panel of Fig. 2.

For fixed quantum numbers, the cross sections for fermionic $\psi_{Q}$ are
larger than their scalar counterparts by roughly an order of magnitude. This
difference stems from the fact that fermions contain more degrees of freedom
and that angular momentum conservation demands the amplitude to produce a pair
of scalars from a pair of massless quarks/gluons is proportional to the final
state velocity and therefore suppressed close to threshold.

##  3 Collider Signatures of Fractionally Charged Particles

To explore how $\psi_{Q}$ can be bounded at the LHC, we turn to the
experiments. There are a few searches for fractionally charged particles at
the LHC in the literature. The searches assume the fractionally charged
particle is stable (or metastable), and rely on anomalously low $dE/dx$ in the
tracking system and odd time-of-flight measurements to distinguish from
background. The predominant energy loss mechanism of charged particles is via
the electromagnetic interaction. For a range of quasi-relativistic velocities,
this loss is described by the Bethe-Bloch equation. In this range, $dE/dx$ is
independent of the particle’s mass, but it is proportional to $Q^{2}$.

The CMS analysis CMS:2024eyx  is the most recent and most easily translated to
the scenarios we envision. In Ref. CMS:2024eyx , events were triggered using
information in the muon system, then investigated for tracks with anomalously
low $dE/dx$. Events are required to have either one or two tracks, and the
number of tracker hits with low ionization is used to discriminate signal from
SM background. The CMS technique is optimal for $Q\sim 2/3$; particles with
higher electric charge leave fewer low $dE/dx$ signals, while the analysis
efficiency for lower charge states drops precipitously as lower charge leads
to fewer tracker hits and therefore smaller signal/noise which inhibits track
reconstruction. For $Q\simeq 1/3$, the efficiency is so poor that the bound
drops to the minimum considered signal mass, $50\,\text{GeV}$.

A second reference we rely on is an ATLAS analysis for long lived
gluinos/stops/sbottoms, Ref ATLAS:2019gqq  (other searches, either for stable
particles or optimized for metastable variations, can be found in Ref.
CMS:2016kce ; ATLAS:2023zxo ). Upon hadronization, gluinos/stops/sbottoms all
form ‘R-hadrons’ with integer charge, with the fraction with charge $\pm 1$
playing the largest role in the analysis. This search relies on large missing
energy and/or the muon system for triggering. Given that R-hadrons are
strongly interacting particles, the usage of the missing energy trigger may
seem out of place. However, heavy exotic hadrons deposit negligible energy in
the calorimeter, so if they are not picked up by the muon system because they
are neutral (either truly neutral, as in charge zero R-hadrons, or effectively
neutral for $\psi_{Q}$ hadrons with small $Q$), most of their energy will
escape undetected. Of course, in order for this undetected energy to register
as missing energy in an event, it must be balanced by something visible,
either a charged exotic hadron or initial state radiation.

Regardless of how they are triggered, retained events with at least one
energetic track are further scrutinized, using time-of-flight information (as
determined from tracker info, muon system, or both) to separate signal from
background. Because this analysis is designed for $|Q|=1$ particles, it is not
easily adapted to fractional charges much less than one. However, it is useful
for estimating bounds when $Q\gtrsim 2/3$, where the CMS search loses
sensitivity.

While Ref. CMS:2024eyx ; ATLAS:2019gqq  are most relevant for our purposes,
we’ll see that $\psi$ in some corners of parameter space are best bounded by
LHC searches unrelated to fractionally charged or stable particles, such as
the invisible width of the $Z$ ParticleDataGroup:2022pth , monojet-style
searches ATLAS:2021kxv  that look for unbalanced, energetic jets, and
disappearing tracks searches CMS:2023mny  that look for tracks which end
suddenly. We will introduce more details of these searches when we encounter a
scenario where they are needed.

The steps needed to go from a $pp\to\bar{\psi}_{Q}\psi_{Q}$ cross section to a
bound, and exactly which bound is best, differ greatly depending on how $\psi$
is charged under the SM groups. In the next subsections, we explore some of
the options.

###  3.1 Solely $U(1)_{Y}$ charges

This is the simplest scenario, as $Q_{\psi}=Y$, so there is no hadronization
or $SU(2)$ partners to worry about. This scenario is also the closest to the
signal model used by CMS. The only difference is that CMS assumes a particle
which only couples to the photon, while we include couplings both to photon
and $Z$ as dictated by $Y$. As a result, we find slightly different masses
corresponding to the quoted cross section limits.

###  3.2 $SU(3)_{C}$ charges

Colored $\psi_{Q}$ particles will quickly hadronize after being produced at
the LHC. And if $\psi$ does not have the hypercharge demanded in Table 2, then
all of the hadrons containing one $\psi_{Q}$ will be fractionally charged.
Hadronization with the light quarks of the Standard Model will result in a
variety of fractional charges for hadrons containing $\psi$. These will differ
in electric charge by units of $e$, depending upon how many up-type versus
down-type quarks are included.

At least as a first pass at reinterpreting the CMS search for colored
representations, we follow the Lund string model Andersson:1983ia  as used in
Pythia Sjostrand:2006za  with application to R-hadrons Fairbairn:2006gg . In
this model, the $\psi_{Q}$, $\bar{\psi}_{Q}$ sit at the endpoints of color
strings which fragment. When the strings break, colored remnants join up with
$\psi_{Q}$ to form color singlet handrons.

For color triplets, the strings break into quark-antiquark or diquark-
antidiquark pairs. The three light quarks are taken to arise democratically in
string breaking, modulo a phase space factor for the strange: $(u:d:s\sim
1:1:0.3)$; the diquark fraction is suppressed by an amount set by data
DELPHI:1996sen ; OPAL:1995cgp . Following this model Fairbairn:2006gg ;
Adersberger:2018bky , triplet $\psi_{Q}$ form mesons with
$\bar{u},\bar{d},\bar{s}$ and the abundance of the ‘down-type’ mesons compared
to ‘up-type mesons’ is 60:40. $\psi_{Q}$ baryons arise less frequently, $\sim
10\%$ of the time, with the light quark composition roughly following the same
$(u:d:s)$ ratio as in $\psi_{Q}$ mesons.

Color octets are treated as if they connect to two strings, one giving a
quark/antidiquark and the other an antiquark/diquark – which then combine with
the octet to form a color singlet. The flavor composition for the gluino
R-hadron case can be found in Ref. Fairbairn:2006gg  and is well approximated
by taking each quark/antiquark as independent and with the same $(u:d:s)$
ratio as above. For our scenario, the only difference is the charge of the
hadrons will be shifted by whatever fractional charge $\psi_{Q}$
carries.777Color octets can also bind with gluons (a string breaks to $gg$,
with one $g$ binding to $\psi_{Q}$ and the other binding to remaining string
fragments). Reference Fairbairn:2006gg  takes to be $O(10\%)$ of $\psi$-gluon
bound states, though in our case these states will retain whatever electric
charge $\psi$ carries (and therefore interact with the tracker/muon system),
while in the gluino case this fraction is invisible.

For more exotic color representations, there is no $R$-hadron literature to
borrow from, so we make the assumption that the bound state involving the
fewest constituents are the most likely to form, and use the same $(u:d:s)$
ratio to determine the flavor (and therefore charge) of the hadrons.

The type of interpretation outlined above ignores the possibility that exotic
hadrons change their electric charge via hadronic interactions as the traverse
the calorimenter. For our purposes, this means that we assume the muon system
triggering works out as it would in the color singlet case. Charge flipping
has been modeled somewhat for R-hadrons Fairbairn:2006gg ; deBoer:2007ii ,
which we could export to exotic color triplets or octets. However, the
behavior of the bound states depends on their composition (baryonic vs.
mesonic, and involving quarks vs. antiquarks), and varies depending on the
phenomenological model used, so we will neglect it for this initial study. For
all exotic hadrons, we ignore the mass splitting between the different exotic
states and assume that the excited (higher spin) bound states immediately
decay to the lowest bound state.

When using these simple hadronization rules to determine the charge of exotic
hadrons, we often find some fraction of the bound states have charge $\sim 1$,
e.g. $5/6$ from a color triplet $\psi$ with $Y=1/2$ (a $\psi_{Q}\bar{d}$
meson), or $7/6$ from a color octet with $Y=1/6$, (a $\psi u\bar{d}$). The
proximity of these charges to $\pm 1$ makes the technique in CMS ineffective.
To determine bounds in this scenario, we will instead reinterpret R-hadron
searches from Ref ATLAS:2019gqq , making the assumption that the R-hadron
bounds are driven by the $Q=\pm 1$ ‘meson’ (i.e. $(\psi_{R}\bar{q})$888We use
a subscript $R$ for the heavy gluino/stop/sbottom in a $R$-hadron. for
R-hadrons from color triplet $\psi_{R}$ or $\psi_{R}q\bar{q}$ for color octet
$\psi_{R}$) bound states and that the experiments are not sensitive to the
difference between $Q\simeq\pm 1$ and $\pm 1$. For color representations not
studied in R-hadron analysis, we will set bounds by equating (cross section
$\times$ fraction of events with at least one exotic hadrons with near integer
charge) = R-hadron cross section $\times$ fraction of events with at least one
$\pm 1$ charge R-hadrons. We note that there are searches for exotic, multiply
charged particles, but these searches begin at $Q=\pm 2$ ATLAS:2023zxo .

This sort of reasoning will allow us to roughly reinterpret tracker based
searches for some colored representations, but we emphasize that for detailed
constraints dedicated simulations of hadronization and detector response for
these fractionally charged representations should be done.

###  3.3 $SU(2)_{L}$ charges

When $\psi_{Q}$ sits in a non-trivial $SU(2)$ representation, it splits upon
EWSB into a multiplet of $(2I+1)$ states, for representation $I$, with
components separated by $|\Delta Q|=1$. At tree-level, and in the absence of
operators such as $H^{\dagger}H\bar{\psi}_{Q}\psi_{Q}$ as we have assumed, the
components of $\psi_{Q}$ are mass-degenerate. Loops of $W/Z$ bosons break this
degeneracy, introducing a splitting of
$\alpha_{em}m_{W}/\pi\sim\mathcal{O}(100)$ MeV, though with a degree of
variation depending on the exact quantum numbers of $\psi_{Q}$. For a
multiplet with hypercharge $Y$ containing a state with charge
$Q=(\tau_{3})_{\psi}+Y$ and a state with charge
$Q^{\prime}=(\tau^{\prime}_{3})_{\psi}+Y$ the one-loop mass difference between
the two is Cirelli:2005uq ; Yamada:2009ve :

$\displaystyle
M_{Q^{\prime}}-M_{Q}=\frac{\alpha_{2}M}{4\pi}\Big{\\{}(\tau^{\prime
2}_{3}-\tau^{2}_{3})\,\Big{[}f\left(\frac{m_{W}}{M_{\psi}}\right)-c^{2}_{W}\,f\left(\frac{m_{Z}}{M_{\psi}}\right)\Big{]}+2\,(\tau^{\prime}_{3}-\tau_{3})\,Y\,s^{2}_{W}\,f\left(\frac{m_{Z}}{M_{\psi}}\right)\Big{\\}}$
(2)

where $M_{\psi}$ is the tree-level mass,
$c_{W}=\cos\theta_{W},s_{W}=\sin\theta_{W}$, and

$\displaystyle f(r)=\begin{cases}+r\left[2r^{3}\ln
r-2r+\left(r^{2}-4\right)^{1/2}\left(r^{2}+2\right)\ln(r^{2}-2-r\sqrt{r^{2}-4})\right]/2&\text{
for a fermion }\\\ -r\left[2r^{3}\ln
r-kr+\left(r^{2}-4\right)^{3/2}\ln(r^{2}-2-r\sqrt{r^{2}-4})\right]/4&\text{
for a scalar}\end{cases}$

In the majority of cases, the state with smaller $|Q|$ is the lightest. For
$M_{\psi}\gg m_{W},m_{Z}$ and using $m_{Z}=m_{W}/c_{W}$, we see that the mass
splitting asymptotes to

$\displaystyle\Delta M\simeq
160\,\text{MeV}\times(\tau^{\prime}_{3}-\tau_{3})(\tau^{\prime}_{3}+\tau_{3}+2Y+2\,Y/\cos{\theta_{W}}).$
(3)

While there can clearly be cancellations, the general trend is that the
splitting grows with the hypercharge of the multiplet and the
$\tau^{\prime}_{3}$ value of the excited state.101010Note that for
$Y=0,|\tau^{\prime}_{3}|=|\tau_{3}|$ the mass splitting vanishes. For $\psi$ a
Weyl fermion in the $n$-dim representation, $\bar{\psi}\varepsilon^{n}$
transforms the same way ($\varepsilon^{n}$ is $n$ copies of the $SU(2)_{L}$
Levi-Civita), and there is an $SU(2)$ flavor symmetry between them. After
$SU(2)_{L}$ symmetry-breaking this flavor symmetry disallows any mass
splitting between the fermions of the same charge.

This multiplet structure has several implications for how $\psi_{Q}$ appears
at the LHC.

  * •

Even if one component of $\psi$ has $Q\lesssim 1/3$ – where the CMS search has
limited sensitivity – it will always be accompanied by a component with larger
charge. For example, a $SU(2)$ doublet with $Y=1/3$ has one state with
$Q=-1/6$, but also a state with $Q=5/6$.

  * •

The phenomenology of the heavier, larger charge state depends crucially on its
lifetime (and therefore crucially on $\psi$’s quantum numbers, which dictate
the mass splitting). For mass splittings $>m_{\pi}$, the two-body decay
$\psi_{Q+1}\to\psi_{Q}+\pi^{+}$ dominates, while for smaller splitting
$\psi_{Q+1}$ mostly decays to $\psi_{Q}+e\,\bar{\nu}_{e}$ (three-body), with a
small branching fraction to $\psi_{Q}+\mu\,\bar{\nu}_{\mu}$. The decay length
for an illustrative set of $SU(2)$ and $Y$ choices are shown below in Fig. 3.

Figure 3: Decay length for the excited state(s) in an $SU(2)$ doublet $\psi$
(left panel) and $SU(2)$ triplet $\psi$ (right panel). In the left panel the
blue line shows the choice $Y=1/3$ ($Q=-1/6,Q^{\prime}=5/6$) while the green
and red show $Y=2/3$ ($Q=1/6$, $Q^{\prime}=7/6$) and $Y=1/6$
($Q=-1/3,Q^{\prime}=2/3$) respectively. In all cases the $\tau_{3}$ component
of the multiplet has the lowest (magnitude) charge. The solid lines are the
results for fermionic $\psi$ while scalar $\psi$ are dashed. In the right
panel, the red lines show $Y=1/6$ while blue show $Y=2/3$. There are more
lines as there are more possible decays. The solid (dashed) red shows the
decay length for $Q=7/6$ to $Q=1/6$ decay, while the dotted (dot-dashed) show
$Q=-5/6$ to $Q=1/6$. Unlike the case when $Y=0$, the lifetimes of the
$\tau_{3}=+1$ and $\tau_{3}=-1$ components are not equal. For the blue lines,
the choice $Y=2/3$ means the $\tau_{3}=-1$ component has the smallest $|Q|$
and is the lightest. Therefore, the solid (dashed) lines show the decay of
$Q=5/3$ to $Q=2/3$ while the dotted (dot-dashed) show the decay of $Q=2/3$ to
$Q=-1/3$.

The decay lengths asymptote at large $M_{\psi}/m_{W}$, as expected from the
mass splitting formulae, while at smaller $M_{\psi}/m_{W}$ there are
significant differences for fermion vs. scalar $\psi$ and cusps where the two-
body decay to $\psi_{Q}+\pi^{\pm}$ turns on or off. 111111The one exception to
the general mass splitting trend is the red dot-dashed line in the right panel
of Fig. 3, the mass difference between the $Q=2/3$ and $Q=-1/3$ components of
a scalar $SU(2)$ triplet with $Y=2/3$, which decreases for larger $M_{\psi}$
(leading to longer decay lengths). This is due to the fact that, while Eq. (2)
generically increases the mass of the larger $|Q|$ state, there are
exceptions. For example, for an $SU(2)$ triplet and $Y=1/3$, the lightest
state is the $Q=-2/3$ component rather than the $Q=1/3$ component. The
proximity of $Y=2/3$ to $Y=1/3$, where the ‘inverted mass’ situation occurs,
leads to the different behavior of the mass splitting as a function of
$M_{\psi}$.

For the selection of charges in Fig 3, none of the excited states would be
considered prompt. Several choices, such as the $Q=7/6$, $SU(2)$ doublet state
(green in the left panel of Fig. 3), or the $Q=5/3$, $SU(2)$ triplet state
(blue in the right panel of Fig. 3) have decay lengths of $O(\text{cm})$ and
would lead to displaced vertices or kinked tracks. A second category of
excited states, such as the $Q=2/3$ state in a $SU(2)$ doublet with $Y=1/6$ or
the $Q=2/3$ state in an $SU(2)$ triplet with $Y=2/3$ have accidentally small
mass splitting from the lightest state in their respective multiplet, and are
therefore effectively stable on collider scales. The roughly bi-modal
distribution of decay lengths can be traced back to whether or not the higher
charge state can decay to the lower charge state by emitting a pion.

Of course, we can have $\psi_{Q}$ in non-trivial representations of both
$SU(3)$ and $SU(2)$, in which case the phenomenology becomes even richer, as
each $SU(2)$ component will undergo hadronization, leading to a zoo of
fractionally charged bound states with a variety of lifetimes.

##  4 Reinterpreted LHC Bounds for Assorted Representations

In this section we show a sampling of the LHC bounds on different exotic
$\psi_{Q}$ by reinterpreting a variety of searches. Given the huge number of
scenarios with fractionally charged $\psi_{Q}$, we obviously cannot explore
them all here. The goal of this benchmark study is to show roughly where
things stand, identify different signal classes and detection strategies, and
point out challenges and hidden assumptions in current searches.

  * •

As our first benchmark, we take $\psi$ to be a color and $SU(2)$ singlet with
$Y=Q$ a multiple of $1/6$ (obviously avoiding multiples that result in integer
charge). This benchmark maps directly onto the CMS search in Ref. CMS:2024eyx
. Using the quoted cross section numbers to bound fermionic (scalar)
$\psi_{Q}$: $Q=1/6$ – no LHC bound, $Q=1/3$
$M_{\psi}>88\,\text{GeV}\,\,(45\,\text{GeV})$, $Q=1/2$
$M_{\psi}>610\,\text{GeV}\,\,(340\,\text{GeV})$, $Q=2/3$
$M_{\psi}>650\,\text{GeV}\,\,(370\,\text{GeV})$. It is worth mentioning that
the bounds for the lower charge regime, $|Q|=1/3$, have loosened substantially
in Ref. CMS:2024eyx  compared to previous iterations, Ref.CMS:2012xi ;
CMS:2013czn . The loosening of the bounds can be traced to a mismodeling in
the efficiency of the muon trigger for low charge CMS:2024eyx .

For the lower charge scenarios, we must look to other searches for bounds. One
obvious place to look is the invisible $Z$ partial width. If we require
$\Gamma(Z\to\bar{\psi}_{Q}\psi_{Q})\leq 1.5\,\text{MeV}$, the 1-sigma
uncertainty on the invisible width ParticleDataGroup:2022pth , fermionic
$\psi_{Q}$ with $|Q|=1/6$ are ruled out except right at $\sim m_{Z}/2$ where
the phase space suppression is severe. However, if relax the constraint to
$2\times$ this uncertainty, the bound disappears. For scalar $\psi_{Q}$,
$|Q|=1/6$ there is no bound even if we impose the stronger condition of
$\Gamma(Z\to\bar{\psi}_{Q}\psi_{Q})\leq 1.5\,\text{MeV}$.

We can also approximate $|Q|\lesssim 1/3$ as invisible and constrain these
scenarios using monojet style analyses $pp\to\not{E}_{T}+j$ ATLAS:2021kxv ,
with the $\psi_{Q}$ playing the role of the missing energy. Reference
ATLAS:2021kxv  quotes model independent cross section limits on
$pp\to\not{E}_{T}+j$ in bins beginning with $\sigma_{lim}<736\,\text{fb}$ for
$p_{T,j}>200\,\text{GeV}$. Requiring such an energetic jet suppresses the
cross section by $\mathcal{O}(200-500)$ depending on $M_{\psi}$ (larger
suppression for lighter $\psi_{Q}$) 121212We derive this factor by running
$pp\to\tau^{+}\tau^{-}(+j)$ in MadGraph and varying the mass of the
$\tau$.,131313The large $p_{T,j}$ values are needed to suppress the
irreducible background from $Z(\bar{\nu}\nu)+j$. The suppression this causes
for our signal is much less than in dark matter models where
$pp\to\not{E}_{T}+j$ proceeds through a contact interaction, as the latter
grows with the energy.. For fermionic $\psi_{Q}$, the monojet analysis places
a bound of only $\sim\text{few}$ GeV, while for scalar $\psi_{Q}$ the cross
section is so low there is no LHC bound even for massless $\psi_{Q}$.

Light ($\sim\text{few}$ GeV), fractionally charged $\psi_{Q}$ could also be
similarly to millicharged matter, a topic of intense work and interest
recently Antel:2023hkf ; depending on the exact mass and charge, such
scenarios are ruled out by fixed target experiments, rare meson decay, star
cooling, etc. See e.g. Ref. Lanfranchi:2020crw ; Gan:2023jbs  for a summary of
limits on millicharged matter. The most relevant bound for the range of masses
and charges we are interested in comes from the SLAC anomalous single photon
$e^{+}e^{-}\to\gamma X$ search, which rules out fermionic $\psi_{Q}$ lighter
than $10\,\text{GeV}$ for $Q>0.08$ Davidson:2000hf ; Prinz:1998ua ;
Essig:2013lka . We know of no reinterpretation of this experiment in terms of
a fractionally charged, complex scalar, but assume the mass bound will be in
the same ballpark.

Next, let us keep the hypercharge and $SU(2)$ assignments the same but take
$\psi_{Q}$ to be a color triplet. As we change the hypercharge assignment, we
change the charge of the exotic hadrons that form, and the hadron charge
determines how strict the bound is. For example:

  * •

$Y=0$: following the argument in Sec. 3.2 above, $\psi$ forms exotic mesons
with $|Q|=2/3$ 40% of the time, and $|Q|=1/3$ 60% of the time. The $|Q|=2/3$
limits from CMS are much more stringent, so equating the cross section for the
production of at least one $|Q|=2/3$ particle – $((0.4)^{2}+2\times 0.4\times
0.6)\times\sigma(pp\to\bar{\psi}_{Q}\psi_{Q})=0.64\,\times\sigma(pp\to\psi\psi)$
to the CMS $|Q|=2/3$ bound, we find masses less than $1.8\,\text{TeV}$
($1.4\,\text{TeV}$) are excluded for fermionic (scalar) $\psi$. Note that
$Y=1/3$ results in hadrons with the same $|Q|$ and therefore is subject to the
same bounds.

  * •

$Y=1/6$: For this choice, all $\psi_{Q}\bar{q}$ bound states have $|Q|=1/2$.
From the CMS bound, we find masses less than $1.9\,\text{TeV}$
($1.5\,\text{TeV}$) are excluded for fermion (scalar) $\psi_{Q}$.

For our next two examples, we consider more exotic color representations, and
for convenience define $d_{i}$ which is either a down or strange quark:

  * •

Color octet with $Y=1/6$: $\psi_{(8,0,1/6),1/6}$. Within our framework, this
state leads to hadrons with charge $Q=1/6\,(\psi
u\bar{u},\psi\bar{d}_{i}d_{j},\psi g$) 55% of the time, and $Q=7/6$
$(\psi\bar{d}_{i}u)$ or $Q=-5/6$ $(\psi d_{i}\bar{u})$ each 22% of the time.
As the CMS search is insensitive to $|Q|\lesssim 1/3$ or $\sim 1$, this is a
scenario where we turn to stable R-hadron searches ATLAS:2019gqq  to place
bounds. From this breakdown, we see that $67\%$ of events contain at least one
$|Q|\sim 1$ hadron. Equating $0.2\times\sigma(pp\to\bar{\psi}\psi)$ to the
gluino $R$-hadron cross section bound of $\sim 1\,\text{fb}$, we find masses
less than $2.0\,\text{TeV}$ ($1.65\,\text{TeV}$) are excluded for fermion
(scalar) $\psi_{Q}$. In applying the R-hadron bounds, we are assuming the
$Q=1/6$ can be treated as neutral for the purposes of missing energy triggers.

  * •

A color sextet with $Y=0$: $\psi_{(6,0,0),0}$ After hadronization, this yields
states with charge $Q=-4/3$ $(\psi\bar{u}\bar{u})$, $Q=2/3$
$(\psi\bar{d}_{i}\bar{d}_{j})$ and $|Q|=1/3$ $(\psi\bar{u}\bar{d}_{i}+c.c.)$
with fractions $\leavevmode\nobreak\ \sim 20\%:30\%:50\%$. The strongest bound
comes from the $|Q|=2/3$ fraction. The fraction of events with at least one
$|Q|=2/3$ particle is $\sim 50\%$, and equating
$0.5\times\sigma(pp\to\bar{\psi}\psi)$ to the CMS $|Q|=2/3$ limit, we find
masses less than $2.2\,\text{TeV}$ ($1.8\,\text{TeV}$) are excluded.

Finally, we consider benchmark color singlet $\psi$ in non-trivial $SU(2)$
representations. We pick from the examples used in the decay length plot, Fig.
3:

  * •

An $SU(2)$ doublet with $Y=2/3$, leading to one state with $Q=1/6$
($\psi_{(0,2,2/3),1/6}$) and one with $Q=7/6$ ($\psi_{(0,2,2/3),7/6}$). The
$Q=7/6$ state decays within $\mathcal{O}(\rm cm)$, leaving a disappearing
track signature. In the context of the CMS search, the $Q=7/6$ state just adds
to the cross section for $Q=1/6$ production, but as CMS is not sensitive to
$Q=1/6$ this gives no bound. Limits on the invisible $Z$ decay width bound
$M_{\psi}\gtrsim 45\,\text{GeV}$ for either spin $\psi_{Q}$. Additionally, as
the LHC production cross section is much larger than the $SU(2)$ singlet case,
it is possible to bound this $\psi_{Q}$ using monojet searches. The total
production for $|Q|=1/6$ is the sum of the Drell-Yan cross sections for
$|Q|=1/6$ and $|Q|=7/6$ along with the charged current production
$pp\to\bar{\psi}_{1/6}\psi_{7/6}+c.c.$. Adding these and comparing to the
$95\%$ CL allowed cross section for $p_{T,j}>200\,\text{GeV}$, we find a
monojet bound of $\sim 50\,\text{GeV}$ (fermionic). However, we can place a
stronger bound by utilizing the disappearing track signal from the $|Q|=7/6$
state. In a disappearing track search, events triggered with large missing
energy are investigated for tracks which end, signaling the decay of a charged
state into a nearly degenerate neutral state. This search strategy has been
applied to the scenario of nearly degenerate higgsinos (electroweak doublets
with $Y=0$), placing a bound of 190 GeV. Applying this strategy to the
scenario here, one issue is that the mass splitting between $Q=7/6$ and
$Q=1/6$ is larger than the higgsino case. For an electroweak doublet, the mass
splitting in Eq. 3 is $\propto Y$, and $Y=2/3$ is larger than the higgsino
value of $Y=1/2$. As a result, the lifetime of the excited state is shorter,
leading to shorter tracks and a less efficient search. Taking the difference
in lifetime into account and applying the cross section bound from Ref.
CMS:2023mny , we find the current scenario is excluded for $\psi_{Q}$ masses
below 115 GeV (70 GeV).

  * •

An $SU(2)$ doublet with $Y=1/6$. The only difference compared to the case
above is that the states now have charge $Q=-1/3$ and $Q=2/3$, with the
$Q=2/3$ slightly heavier. However, as $Y$ is smaller, so is the mass
splitting, to the point that for $Y=1/6$ the mass splitting drops below
$m_{\pi}$. As a result, the lifetime of the excited state is significantly
longer than in the previous case, $\mathcal{O}(20\rm m)$, and we can consider
it to be collider stable. We can therefore bound this scenario by ignoring the
$Q=-1/3$ component and equating the total cross section for $Q=2/3$
production,
$pp\to\bar{\psi}_{2/3}\psi_{2/3}+pp\to\bar{\psi}_{-1/3}\psi_{2/3}+c.c$ to the
$|Q|=2/3$ limit from CMS CMS:2024eyx . We find masses below $1.1\,\text{TeV}$
($750\,\text{GeV}$) are ruled out.

  * •

An $SU(2)$ triplet with $Y=2/3$, leading to states with $Q=-1/3,Q=2/3,Q=5/3$.
The $Q=5/3$ decays rapidly to the $Q=2/3$, which then flies $\mathcal{O}(\rm
cm)$ before decaying to the $Q=-1/3$. Only the $Q=-1/3$ particle survives to
the muon system, so if we rely on the fractionally charged bound the limits
are low; summing Drell-Yan production of all three charged states along with
their charged current counterparts and applying the limit from Ref.
CMS:2024eyx , we find limits of $M_{\psi}>350\,\text{GeV}$ (200 GeV). The
lifetime of the $Q=2/3$ is long enough that one expects it should leave a
trace in disappearing track searches. The limit from Ref. CMS:2023mny  on
nearly degenerate electroweak triplets (a wino) is $650\,\text{GeV}$, though
extrapolating this to the present scenario is not straightforward as the
efficiency for the $Q=2/3$ will be worse than the wino. Not only is the
electric charge smaller, but the $Q=2/3$ to $Q=-1/3$ mass splitting is larger
(and thus its lifetime shorter) than in the charged to neutral wino case, and
the sensitivity in Ref. CMS:2023mny  falls precipitously with mass splitting.
Part of this lack in sensitivity can be compensated by a larger cross section,
since we can lump the production of $Q=5/3$ and $Q=2/3$ together as the
effective disappearing track signal. However, we find this enhancement is
insufficient. The bounds fall so quickly for larger mass splittings that we
estimate limits from disappearing track searches are $<100$ GeV, worse than
the fractional charge bounds relying on $|Q|=1/3$.

The bounds from these benchmark scenarios are illustrated below in Fig. 4, and
we can use our experience with those setups to extrapolate to other multiplets
to some extent. For $\psi_{Q}$ charged solely under hypercharge, bounds come
from the CMS dedicated fractionally charged search. The fractionally charged
bounds are maximzed near $Q=2/3$; for larger charge, the technique fails and
is superseded by time-of-flight based searches, while for smaller charge the
sensitivity drops precipitously. Monojet style searches are an interesting
avenue to explore, but these perform best for heavier $\psi_{Q}$ – where the
cross section is even lower – or contact interactions from a heavy mediator
(which do not apply to our setup). For colored $\psi_{Q}$, the large
production cross section pushes the current limits much higher, roughly
$1.8\,\text{TeV}$ for color triplets fermions. The bounds increase with the
size of the $SU(3)$ representation and, at least at the level of our study,
are fairly insensitive to the hypercharge of $\psi_{Q}$.

Comparing the above numbers we see that the scenarios we can recast into the
CMS fractional charge search have slightly stronger limits than those we
interpret as $R$-hadrons, as fractional charge signatures have an additional
handle – low $dE/dx$ – to separate signal from background. We see the most
variability in the bounds for color singlet, $SU(2)$ charged $\psi_{Q}$, as
the signatures in the detector depend strongly on the charges and lifetimes of
all the states in the multiplet. If the excited states are short lived, they
add to the cross section for the lowest $|Q|$ state, but this boost can be
insufficient to strongly bound the scenario if the lightest state has $|Q|\leq
1/3$. Disappearing track searches, which target the decay of the excited
state, can provide another handle, though we find they are hampered by the
fact that excited state lifetimes for fractionally charged scenarios are
typically shorter than in scenarios familiar from supersymmetry (e.g. $Y=1/2$
for pure higgsino or $Y=0$ for wino). If the excited state happens to be long-
lived, the bounds to jump significantly, as the higher charge state gives us
another handle on the setup. The $SU(2)$ charged scenarios are also the most
complicated, as the number of processes one needs to consider (Drell-Yan for
each component, charged current between pairs of components) grows with the
size of the multiplet.

Figure 4: Graphic illustrating the mass bounds for the benchmark fractionally
Dirac fermions, the details of which are discussed in the text. The bounds for
fractionally charged complex scalars are lower than the fermionic case by
$\sim 20\%$.

We emphasize that all of these bounds are just an estimate. We have ignored
higher order QCD corrections, which for inclusive cross sections are
encapsulated into a $K$ factor that is typically $\sim 1-2$. More
significantly, we have assumed that the triggering efficiency – either in the
muon system efficiency or using $\not{E}_{T}$ – for fractionally charged
particles with other (non-hypercharged) quantum numbers (or much larger mass)
is not significantly different than in Ref. CMS:2024eyx .

We conclude this section with some items worth thinking about in order to
maintain a robust collider search program for fractionally charged particles.

  * •

The LHC is an evolving apparatus, with many detector upgrades planned for the
high luminosity phase. Some ways these upgrades will affect searches for
fractionally charged particles include:

    * –

The ability to trigger using tracker information alone (at both ATLAS and CMS)
may help increase sensitivity in regions where the CMS analysis is limited by
the muon trigger efficiency. It is worth noting that the upgraded outer
portion of the tracker will be upgraded to a digital device to facilitate the
high data transfer rate needed for track triggering. However, this comes with
the price that ionization energy on the individual hits is no longer kept.
Multiple hits are combined together into a single output, so there will be
less granular $dE/dx$ information. Exactly how much this impacts the analysis
strategy for fractionally charged particles in Ref. CMS:2024eyx  has not yet
been studied.

    * –

The introduction of a timing layer in CMS between the tracker and ECAL will
improve time-of-flight measurements, enhancing signal discrimination based on
velocity or displaced vertices CMS:2667167 ; Liu:2018wte ; DiPetrillo:2022qsb
.

    * –

Reference Vannerom:2693948  explored how the low $dE/dx$ search could be
improved, especially for low $Q$, by moving from a muon trigger to a
$\not{E}_{T}$ trigger. Detector upgrades are expected to increase the
efficiency for lower $\not{E}_{T}$ events ATLAS:2024xna , which should help
this approach further.

  * •

The bounds above primarily rely on tracker information, using other systems
only to trigger. More precise bounds, or perhaps even novel signals, could be
achieved by improved modeling of the interaction of colored, fractionally
charged particles as they traverse the detector. Current models are limited to
heavy color triplets/octets that are lumped into hadrons with integer charge,
and even within this subset there are considerable differences among models in
the charge vs. neutral and meson vs. baryon fractions as a function of
distance traversed Adersberger:2018bky ; Kraan:2004tz ; deBoer:2007ii .

  * •

Some improvement in the most challenging cases is already underway from the
milliQan experiment, which is forecasted to probe up to $45$ GeV for a fermion
of charge $e/6$ using LHC Run 3 data milliQan:2021lne .

  * •

Some percentage of $\psi_{Q}$ produced at the LHC will stop inside the
detector as a result of their energy loss to the detector material. The
fraction that stop depends on the mass of $\psi_{Q}$, its charge, and its
color representation. The stopped, stable $\psi_{Q}$ may form atomic or
nuclear bound states which will have a fractional charge that cannot be
screened by Standard Model material. It is not clear to us whether there might
be discovery potential in looking for later trajectories being subtly affected
by this small persistent electric charge localized somewhere in the detector.
If nothing else, it may be interesting to attempt to search disused detector
parts for embedded fractional charges.

##  5 Cosmology

Not only would the discovery of a fractionally charged particle tell us an
enormous amount about ultraviolet particle physics—it would also tell us a
huge amount about the early universe. So for completeness we offer a brief
discussion here.

Since the lightest fractionally charged particle is necessarily stable, strong
constraints on the relic abundance of particles with $\mathcal{O}(1)$ electric
charges are present. Our understanding thereof is mainly from the fantastic
Dunsky, Hall, Harigaya papers Dunsky:2018mqs ; Dunsky:2019api  as we briefly
summarize in Section 5.1.

These imply that such a species could only ever have been in thermal
equilibrium with the Standard Model if there were large Boltzmann factor
suppression. That is, discovering a fractionally charged particle of mass
$M_{\psi}$ gives an upper bound on the reheating temperature $T_{\rm
reheat}\lesssim M_{\psi}/r$. In Section 5.2 we give some basic estimates of
$r$ depending on both the details of reheating and the quantum numbers of
$\psi$.

This means that just as such an energy frontier discovery would falsify some
of our grand models of ultraviolet physics, it would also falsify the high-
scale inflation models that have been proposed in these frameworks. Of course
all we know experimentally is that there was a Standard Model plasma in a
radiation era at the temperatures of Big Bang Nucleosynthesis $T_{\rm
reheat}\gtrsim T_{\rm BBN}$, but there need not have been an era of much
hotter temperature Hannestad:2004px ; deSalas:2015glj ; Bhattiprolu:2022sdd ;
Gan:2023jbs .

###  5.1 Abundance Constraints

There have been various lab-based searches for fractionally charged particles
in which a sample of some material is tested for fractional charge. Indeed the
ensuing constraints on fractional charges present in the sample are very
strong, but extrapolating to a constraint on the relic abundance is fraught
with difficulties. The dust in our proto-planetary disk originated in an
earlier generation of stars that underwent supernovae, and that which formed
the Earth has undergone billions of years of geological activity. That is to
say, tracking the evolution of heavy particles from an initial relic abundance
through this non-trivial evolution requires great care. Some of these issues
are discussed further in Dunsky:2019api ; DeLuca:2018mzn .

However, in fact there is a better source of constraints on the relic
abundance from the flux of fractionally charged particles on the Earth. In
general, virialized dark matter which strongly interacts with SM particles is
unable to reach underground direct detection experiments that are shielded by
the Earth’s atmosphere and meters of rock (see e.g. Starkman:1990nj ;
DeLuca:2018mzn ; Digman:2019wdm ). However, the electric charges of the states
we are considering mean that there is necessarily a component which gets
boosted by supernova shocks, as impressively understood in Dunsky:2018mqs .
Indeed for the GeV - TeV mass range of interest at the energy frontier and the
$\mathcal{O}(1)$ electric charges of our states, a relic abundance of such
particles collapses into the Milky Way disk as it forms along with the
baryons, thermalizes with the ISM, and undergoes Fermi acceleration from
supernova shock waves. These accelerated particles appear on Earth in the form
of cosmic rays, and their large boosts would allow them to penetrate the Earth
down to deep underground detectors, providing strict upper limits on such a
flux. In the range of parameter space of interest to us, the strictest bounds
come from experiments like IceCube IceCube:2011cwc , searches for lightly
ionizing particles like MAJORANA Majorana:2018gib  and MACRO MACRO:2004iiu ,
and searches for magnetic monopoles like ICRR Kajino:1984ug  and Baksan
Alekseev:1982yr . These constraints are extremely strong, giving upper bounds
on the relic abundance $10^{-10}-10^{-16}$ as a fraction of the dark matter
abundance, depending on the exact charge and mass.

###  5.2 Thermal Plasma Production

The bounds on the relic abundance can roughly be translated into an upper
bound on the reheating temperature $T_{\rm reheat}\lesssim M_{\psi}/r$ where
$M_{\psi}$ is the mass of the lightest fractionally charged particle. If we
assume instantaneous reheating of all species with SM quantum numbers to a
temperature $T_{\rm reheat}\ll M_{\psi}$, we get a Boltzmann suppressed
equilibrium abundance of $\psi_{Q}$:

$\displaystyle n_{\psi}=g_{\psi}\left(\frac{M_{\psi}\,T_{\rm
reheat}}{2\pi}\right)^{3/2}\exp\left(-M_{\psi}/T_{\rm reheat}\right),$ (4)

where $g_{\psi}$ is the number of degrees of freedom of $\psi_{Q}$. This gives
a relic abundance relative to dark matter of

$\displaystyle\frac{\Omega_{\psi}}{\Omega_{DM}}=\left(\frac{M_{\psi}\,n_{\psi}}{\rho_{DM}}\right)\left(\frac{s_{0}}{s_{*}}\right)$
(5)

where $s_{0}$ is the entropy today, $s_{*}$ is the entropy at $T_{\rm reheat}$
and $\rho_{DM}$ is the average dark matter energy density. Given a bound on
$\Omega_{\psi}/\Omega_{DM}$, we can translate Eq. 5 into a bound on
$r=M_{\psi}/T_{\rm reheat}$. If we impose $\Omega_{\psi}/\Omega_{DM}\leq
10^{-16}$, the most stringent bound in the parameter space of interest
according to Ref. Dunsky:2018mqs , this translates to

$r\sim 65,$ (6)

with only weak dependence on $M_{\psi}$. If the $n_{\psi}$ produced were large
we should include the effects of annihilations like for a standard freeze-out,
as done in Langacker:2011db , but since the allowed regime is so small we can
ignore this process.

Above we assumed $\psi_{Q}$ is instantaneously in equilibrium at $T_{\rm
reheat}$. As a test of how sensitive the $r$ value derived is to our
assumptions of the reheating process, we can imagine an extreme scenario where
only the SM matter is reheated at $T_{\rm reheat}$ (which may be more or less
contrived depending on the quantum numbers of $\psi$). In this case, an
abundance of $\psi_{Q}$ is built up via freeze-in, generated from collisions
among energetic SM particles on the Boltzmann tails of their equilibrium
distributions. The frozen in abundance of $\psi$ can be estimated using the
results of Ref. Cosme:2023xpa . Specifically, if we assume a threshold cross
section times relative velocity of
$\sigma_{SM\,SM\to\bar{\psi}_{Q}\psi_{Q}}v\sim\frac{c_{eff}}{16\pi
M^{2}_{\psi}}$, where $c_{eff}$ is a combination of couplings and factors
counting degrees of freedom (both initial and final), we find

$\displaystyle\frac{\Omega_{\psi}}{\Omega_{DM}}\sim\frac{135\sqrt{5/2}\,M_{pl}\,c_{eff}\,e^{-2/r}(2/r+1)\,s_{0}}{256\,\pi^{7}\,g^{3/2}_{*}\,\rho_{DM}}.$
(7)

For QCD production (assuming six SM fermion flavors and ignoring all SM
masses) , $c_{eff}\sim 75$, while production of $\psi_{Q}$ charged only under
hypercharge has $c_{eff}\sim 0.1\,Y^{2}_{\psi}$. Plugging in numbers, the
freeze-in case decreases $r$ by $\mathcal{O}(15)$ relative to the case of
directly reheating $\psi$, with only some mild dependence on the value
$c_{eff}$.

We note that the strong bound on $\Omega_{\psi}/\Omega_{DM}$ we have taken
above may be loosened slightly for certain quantum numbers of $\psi$. In
particular, there do exist colored representations for which all hadrons
formed with SM partons have fractional electric charge, but which also have
bound states with zero electric charge, such as $Q\sim(3,X)_{0}$ where
$X=1,3,\cdots$. Triply-exotic $(QQQ)$ bound states (for $Q$ a fermion) are
neutral “dark” baryons and one could investigate them as a component of DM,
much as in the “colored DM” story DeLuca:2018mzn ; Gross:2018zha ;
Mitridate:2017izz . However, there is a severe danger posed by the existence
of mixed bounds states such as $(Q\bar{q})$ (for $\bar{q}$ a SM quark) which
have fractional charges, so must have extremely suppressed relic abundances as
discussed above. As understood for colored DM, the QCD phase transition
automatically gives some suppression of the fractionally charged abundance,
since $H(\Lambda_{QCD})\ll\Lambda^{-1}_{QCD}$. Then after the QCD phase
transition, many scatterings occur among the mixed bound states, which
depletes their abundance in favor of the much more tightly bound $(QQQ)$ by
some orders of magnitude, $\Omega_{Q\bar{q}}\sim 10^{-4}\Omega_{QQQ}$. This
leads to a less stringent restriction on $T_{\rm reheat}/M_{\psi}$ than in a
case without electrically-neutral bound states by about $\mathcal{O}(10)$.

##  6 Global Structure of Gauge Theory

In this section we give a basic review of some group and representation theory
and its appearance in gauge theories. Our focus is on conceptual understanding
moreso than technical detail. The key point is to understand the differences
between symmetry groups which are identical for infinitesimal symmetry
transformations near the identity (they have the same Lie algebra) but differ
for large symmetry transformations (they have different Lie groups as the
result of non-trivial ‘global structure’). This will allow us to appreciate
the distinct possibilities for the gauge group of the Standard Model. Some
pedagogical references for the group theory are Stillwell:2008a ; Hall:2015xtd
.

###  6.1 Abelian Warmup: $\mathbb{R}$ vs. $U(1)$

Often in particle physics we are interested in continuous symmetry groups
which have a notion of infinitesimal transformations which are close to the
trivial, identity transformation. The earliest such example in a field theory
(and indeed the farthest infrared example) is the theory of electromagnetism.

#### As Groups

When we consider a gauge field theory based on a symmetry group, the gauge
bosons correspond to the generators of the group. Electromagnetism has only
one photon, so we are interested in groups with only one generator. In fact,
the photon corresponds to the generator of $U(1)_{\rm EM}$ gauge
transformations, a global element of which we can represent as

$U(\theta)=e^{i\theta Q},$ (8)

a circle’s worth of transformations which compose by complex multiplication
$U(\theta)U(\eta)=e^{i(\theta+\eta)Q}$ with $\theta,\eta\in[0,2\pi)$. But
alternatively we may view this as a mapping of $\theta\in\mathbb{R}$ onto the
unit circle. Indeed, if we look nearby the identity transformation we cannot
tell $U(1)$ from $\mathbb{R}$

$U(\theta)\simeq 1+i\theta Q,$ (9)

where we have expanded for small $\theta$. Then we could alternatively think
about just defining the group operation

$U(\theta)U(\eta)\equiv 1+i(\theta+\eta)Q.$ (10)

This is a group which is not compact—$\theta$ has no finite period now; the
group is just $\mathbb{R}$ equipped with addition. While $U(1)$ and
$\mathbb{R}$ differ as Lie groups, they share the same Lie algebra.

Thinking in the other direction, if we had begun with $\mathbb{R}$ with the
group operation of addition, we could see the relation to $U(1)$ by
considering the quotient group $\mathbb{R}/\mathbb{Z}\simeq U(1)$. That is, we
may view $U(1)$ as coming from an $\mathbb{R}$ group where we have imposed the
additional equivalence relation $\theta\sim\theta+2\pi\mathbb{Z}$—two elements
of the group are now identified if they differ by an integer (the factor of
$2\pi$ is a normalization convention of the period). We diagram this structure
in Figure 5, and of course this is exactly what the exponential map above
does.

Figure 5: The group $U(1)$ constructed by quotienting $\mathbb{R}/\mathbb{Z}$.
We can think about the quotient projecting the real line down to the circle
such that every integer maps to the identity element.

Thinking about the physics, the perturbative, low-energy dynamics of the
vector gauge bosons depend only on the gauge transformations which are close
to the identity. That is, Maxwell’s equations and the covariant derivative
depend only on the Lie algebra of the gauge group. Yet the two theories differ
in important ways, as we discuss presently.

#### Electric Representations:

In fact, there are nonperturbative aspects of physics which do depend on the
global properties of the gauge group, and the closest at hand is simply the
representation theory. In physics our objects transform in representations of
the relevant symmetry groups, and the representation theory of groups with
different global structures may differ.

The question in the one-dimensional case is: Which charges should be allowed?
A field $\psi(x)$ with charge $q$ transforms under a $U(\theta)$
transformation as $\psi(x)\rightarrow\psi(x)\exp(iq\theta)$. If the group is
$\mathbb{R}$, then any charge $q\in\mathbb{R}$ is fine. But if the gauge group
is $U(1)$, then $U(2\pi)\equiv\mathds{1}$, a rotation around the full circle
is equivalent to an identity transformation. Each field must be trivially
mapped back to itself by an identity transformation, but a field of general
charge $q$ transforms to $\psi(x)\exp(2\pi qi)$. The requirement $\exp(2\pi
qi)\equiv 1$ implies that for a $U(1)$ group we must have $q\in\mathbb{Z}$ and
charge is quantized.

Thus, we see that the representation theory depends crucially on the global
structure of the group, rather than just its local structure near the
identity. Turned around, this means that by discovering particles with
particular representations, you can learn about the global structure. If you
discover two particles $\psi,\chi$ with relatively irrational charges
$q_{\psi}/q_{\chi}\notin\mathbb{Q}$ then the gauge group must be $\mathbb{R}$
instead of $U(1)$. Note that you only need to discover two because for any
real number can be approximated arbitrarily closely by a sequence of
$aq_{\psi}+bq_{\chi}$ for $a,b\in\mathbb{Z}$.141414We note for fun that this
fact was used to intriguing effect in the ‘irrational axion’ of Banks:1991mb .

#### Magnetic Representations:

Gauge theories may also allow representations which carry magnetic, rather
than electric charge. In the low energy theory of electromagnetism, these are
the familiar Dirac monopoles. Of course it is simple enough to postulate a
monopole magnetic field

$\vec{B}=\frac{g}{4\pi}\frac{\hat{r}}{r^{2}},$ (11)

but in a quantum mechanical theory (where Aharonov-Bohm teaches us we really
must talk about the potential $A^{\mu}$) such configurations connect to rich,
deep physics. See e.g. Preskill’s classic Preskill:1984gd  for an in-depth
introduction.

Figure 6: The Dirac monopole as the limit where a semi-infinite solenoid
becomes the Dirac string.

The problem is that when we define the magnetic field in terms of the vector
potential, $\vec{B}=\nabla\times\vec{A}$, the absence of magnetic monopoles in
the Maxwell equations follows necessarily,
$\nabla\cdot\vec{B}=\nabla\cdot(\nabla\times\vec{A})\equiv 0$ because the
divergence of a curl is identically zero. In the relativistic theory this is
often referred to as the ‘Bianchi identity’,
$\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}F_{\rho\sigma}=0$.

As Dirac understood, their construction in the low-energy theory of the gauge
field $A^{\mu}(x)$ requires a singular line in the electromagnetic field in
some direction from the monopole off to infinity known as a ‘Dirac string’.
This is on display in his

$A_{\rm Dirac}(x)=\frac{g}{4\pi r}\tan\frac{\theta}{2}\hat{\phi},$ (12)

in polar coordinates with $\phi$ the azimuthal angle and $\theta$ the polar
angle. This indeed gives rise to the monopole magnetic field above, but this
potential is singular from $r=0$ out to all $r$ along the line $\theta=\pi$.
This is not a deficiency of Dirac; any function $A(x)$ which produces this
magnetic field will unavoidably have such a singular line, which we call a
‘Dirac string’. An isolated singularity at $r=0$ appears also of course in the
electric field of an elementary charged particle—this can essentially be
ignored in the low-energy theory and relativistic quantum field theory teaches
us how to deal with it using renormalization. But a line-like singularity can
lead to physical effects which we do not want and must avoid, as follows.

One can think of a monopole so constructed as being one end of an infinitely-
thin solenoid where the other end has been sent off to infinity.151515It is
not clear to us who first discussed the Dirac string in this language, though
Dirac’s paper Dirac:1931kp  invites this interpretation easily enough. We
refer to Felsager Felsager:1981iy  for one construction, Franklin:2019znr  for
some explicit formulae, and Beche:2013wua  for an experiment at creating an
approximate monopole in the lab by taking just such a limit. The magnetic flux
$g$ of the monopole flows into it from infinity through the solenoid, creating
a monopole magnetic field at its end. The famous Dirac quantization condition
arises from requiring that the Dirac string is truly unphysical, so that we
can really view the solution as just the point monopole. Given an electrically
charged particle with charge $e$ and dragging it in a closed path around the
would-be Dirac string of a monopole with magnetic charge $g$, the charge picks
up an Aharonov-Bohm phase

$\exp\left(ie\oint_{\gamma}\vec{A}\cdot\text{d}\vec{s}\right)=\exp\left(ie\iint(\vec{\nabla}\times\vec{A})\text{d}^{2}x\right)=\exp{ieg},$
(13)

which is a physical phase we could measure in an interference experiment.
Then, in order for the Dirac string to truly be unphysical, the charge $g$ of
a fundamental monopole must satisfy

$eg=2\pi n,\quad n\in\mathbb{Z}.$ (14)

The smallest-charge monopole is found for $n=\pm 1$, and of course the most
stringent requirement is from the electrically-charged particle with the least
charge. That is, if $q_{\rm min}$ satisfies Eqn 14, then so will every
multiple of $q_{\rm min}$, so we have implicitly used this normalization of
$e$ in writing that equation.

Alternatively to this construction (and more than 40 years later) Wu and Yang
showed that magnetic monopoles can be described in a manifestly singularity-
free language by using some concepts from topology Wu:1975es . In fact
historically it is these ideas that have sparked theoretical physicists’
enduring fascination with topology in field theory, but let us try only to
appreciate some elementary points.

From this point of view, the unphysical Dirac string appears in the naive
description because there is no way to express the vector potential
$A^{\mu}(x)$ globally as a function for all $x$. In topological language we
must instead think of fields as sections of certain fiber bundles, but
elementarily we can imagine we must describe the gauge field using two
functions $A^{\mu}_{\rm N/S}(x)$ with an overlapping range of validity.
Thinking in spherical coordinates, $A^{\mu}_{\rm N}(x)$ is defined for polar
angles $\theta\in\left[0,(\pi+\delta)/2\right)$ and $A^{\mu}_{\rm S}(x)$ is
defined on the ‘southern hermisphere’
$\theta\in\left((\pi-\delta)/2,\pi\right]$ where the small $\delta$ addition
to the domains ensures that these two descriptions overlap on a small ring
around the equator. They have the explicit expressions

$\displaystyle A_{\rm N}(x)$ $\displaystyle=\frac{g}{4\pi
r\sin\theta}\left(1-\cos\theta\right)\hat{\phi}$ (15) $\displaystyle A_{\rm
S}(x)$ $\displaystyle=\frac{-g}{4\pi
r\sin\theta}\left(1+\cos\theta\right)\hat{\phi}.$ (16)

If we have two overlapping descriptions on the equator they must surely
somehow match, and this is possible despite them being different functions
locally because there is an underlying $U(1)$ gauge redundancy. That is these
functions describe the same physics on the equator if they agree up to a
$U(1)$ gauge transformation, which we can see as

$\text{On overlap:
}A^{\mu}_{N}(x)=A_{S}^{\mu}(x)-ie^{-i\alpha(x)}\partial^{\mu}e^{i\alpha(x)},\quad\alpha(x)=g\frac{\phi}{2\pi}k$
(17)

Figure 7: The local descriptions $A_{N/S}^{\mu}(x)$ of the vector potential in
their separate patches, and the transition function on their overlap.

Then morally speaking the different monopole solutions are classified by the
value of this gauge transformation on a path around the equator
$U(\phi):\phi\rightarrow U(1)$ as $\phi=0..2\pi$ with $U(0)=U(2\pi)$. In fact
the collection of such paths is familiar in algebraic topology as the
‘fundamental group’ $\pi_{1}(G)$ of a space $G$. In the case of a $U(1)$
group, $\pi_{1}(G)=\mathbb{Z}$ tells us that there are magnetic monopoles
labeled by any integer charge.

In contrast, in the case of an $\mathbb{R}$ gauge group there is no way to
draw a closed path in $\mathbb{R}$ which cannot be shrunk down to a single
point, so $\pi_{1}(G)=\mathds{1}$ is trivial and this group does not have any
magnetic monopoles. One may have intuited this already from the Dirac
quantization condition and the results above about electric representations.
Since in an $\mathbb{R}$ gauge group the electric charge can be an arbitrarily
small real number, the Dirac quantization cannot be satisfied for any magnetic
charges.

###  6.2 Global Structure of Non-Abelian Groups

#### Case Study 1: $SU(N)$ vs. $SU(N)/\mathbb{Z}_{N}$

Recall that the group $SU(N)$ consists of $N\times N$ complex matrices which
are unitary ($V^{\dagger}V=1$) and special ($\det V=1$). The structure of
infinitesimal transformations in $SU(N)$ is generated by traceless hermitian
$N\times N$ matrices

$U(\theta^{a})=\mathds{1}^{i}_{j}+i\theta^{a}\left(T^{a}\right)^{i}_{j}$ (18)

where $a=1..N^{2}-1$. These $T^{a}$ generate the Lie algebra of $SU(N)$ in a
way that generalizes the familiar Pauli matrices of $SU(2)$. The group $SU(N)$
is non-Abelian but it has a nontrivial ‘center’ $\mathbb{Z}_{N}$, where the
center of a group is the subgroup of elements which commute with all others,

$\mathbb{Z}_{N}\subset SU(N):\left\\{\exp\left(\frac{2\pi
k}{N}i\right)\mathds{1}_{N}\right\\}_{k=0..N-1},$ (19)

which is generated by the element
$\omega_{N}=\exp\left(\frac{2\pi}{N}i\right)\mathds{1}_{N}$. We can sensibly
form the quotient group $SU(N)/\mathbb{Z}_{N}$ where we ‘mod out’ by the
center subgroup. This group can be thought of as $SU(N)$ with the equivalence
relation $\omega_{N}\sim\mathds{1}_{N}$ imposed. But this does not change the
structure of transformations near the identity; the Lie algebra remains the
same.

In the quotient group any two elements of $SU(N)$ which differ by a center
element are now identified. In particular, each element of the center is now
identical to $\mathds{1}_{N}$. Thinking now about the representation theory,
this means that such elements must necessarily act trivially on each field.

If we think about the familiar $SU(N)$ representations, this is not the case
for all of them. Consider a field $\psi^{a}$ in the fundamental representation
of $SU(N)$, which transforms generally as
$\psi^{a}\rightarrow\psi^{a}V_{a}^{b}$. Then in particular under an
$(\omega_{N})^{b}_{a}$ transformation it picks up an $N^{\text{th}}$ root of
unity phase. In $SU(N)$ this is as it should be, but this is nonsensical for a
representation of $SU(N)/\mathbb{Z}_{N}$, in which this element was literally
the identity—then the fundamental representation of $SU(N)$ is not an allowed
representation of $SU(N)/\mathbb{Z}_{N}$!

The field theory of $SU(N)/\mathbb{Z}_{N}$ is a theory of adjoint fields,
including of course the gauge bosons which are necessarily present. An adjoint
representation can be thought of as the product of a fundamental and
antifundamental with the trace removed, with the math
$N\otimes\bar{N}=(N^{2}-1)\oplus 1$. With equal number of fundamental and
antifundamental indices,
$A^{a}_{c}\rightarrow(V^{\dagger})^{c}_{d}A^{a}_{c}(V)_{a}^{b}$ is easily seen
to be invariant under a center transformation. The $SU(N)/\mathbb{Z}_{N}$
theory allows arbitrary matter which is in either the adjoint or irreps which
can be built from it and the Levi-Civita symbol $\varepsilon_{a_{1}\dots
a_{n}}$.

The global structure also here crucially changes the topological properties of
the gauge group, just as did the quotient in the Abelian case. We can see this
again in the allowed magnetic representations, which are controlled by the
fundamental group $\pi_{1}(G)$. This can be thought of elementarily as simply
the group of topologically equivalent maps of circles into $G$,
$\pi_{1}(G)\simeq\\{\phi:S^{1}\rightarrow G\\}$. The question is what sorts of
closed loops we can draw in $G$. For $SU(N)$ it is a fact that
$\pi_{1}(SU(N))=1$ and there are no magnetic monopoles. But now let us
consider the following diagonal generator of $SU(N)$

$T^{N^{2}-1}=\begin{pmatrix}1&&&\\\ &1&&\\\ &&\ddots&\\\
&&&-(N-1)\end{pmatrix},$ (20)

which is a hermitian, traceless matrix you can think of as the generalization
of the Pauli $\sigma_{3}$ to $SU(N)$. Of course close to the identity we can
think of an infinitesimal transformation in this direction
$\theta^{a}=\delta^{a}_{N^{2}-1}\theta$,

$U(\theta^{a})=\mathds{1}+i\theta T^{N^{2}-1}+\mathcal{O}(\theta^{2}),$ (21)

just as in $SU(N)$. But now in $SU(N)/\mathbb{Z}_{N}$ we will see something
interesting if we go a large distance in this direction, say $\theta=2\pi/N$.
The higher order terms form into the exponential

$U(\theta^{a})=\exp\left(i\frac{2\pi}{N}T^{N^{2}-1}\right)=\exp\left(i\frac{2\pi}{N}\right)\mathds{1}$
(22)

and because $-(N-1)=1\text{ (mod) }N$ we see that by following a path along
the $T^{N^{2}-1}$ direction we have ended up at an element of the
$\mathbb{Z}_{N}$ center. In $SU(N)$ there’s nothing special to say about this,
but in $SU(N)/\mathbb{Z}_{N}$ this means that you can go far out along this
direction and end up back at the origin! So now there is a map
$\phi:[0,2\pi)\mapsto G$ where $\phi(\theta)=U(\theta/N)$ and this gives us
one-dimensional loops around $SU(N)/\mathbb{Z}_{N}$.

This means that in addition to the electric representations discussed above,
$SU(N)/\mathbb{Z}_{N}$ also has magnetic representations. In this case there
are not monopoles of any integer charge as in $\pi_{1}(U(1))=\mathbb{Z}$ but
rather only $N$ distinct closed loops
$\pi_{1}(SU(N)/\mathbb{Z}_{N})=\mathbb{Z}_{N}$ and so only $N$ distinct
monopoles. If you wind $N$ times around $SU(N)/\mathbb{Z}$ you end up with a
path that can be deformed into lying only in $SU(N)$, where it can be shrunk
to a point.

The familiar example of this is $SU(2)$ which has
$\pi_{1}\left(SU(2)\right)=1$, and you will recall is only locally isomorphic
to the rotation group $SO(3)$, while globally double-covering it. Then the
quotient group $SU(2)/\mathbb{Z}_{2}\cong SO(3)$ is isomorphic to 3D rotations
and has
$\pi_{1}\left(SU(2)/\mathbb{Z}_{2}\right)=\pi_{1}\left(SO(3)\right)=\mathbb{Z}_{2}$.
The fact that looping $N$ times around $SU(N)/\mathbb{Z}_{N}$ returns you to
the identity is nothing more than ‘Dirac’s belt trick’—in 3D space taking the
belt buckle on a loop in $\phi:[0,2\pi)\mapsto SO(3)$ puts it in a
topologically twisted sector yet going around twice returns it to the
identity.

#### Case Study 2: $SU(N)\times U(1)$ vs. $U(N)$

In the case of a product group there may be a more subtle choice of global
structure which interrelates the allowed representations of the group factors.
In fact $U(N)\cong\left(SU(N)\times U(1)\right)/\mathbb{Z}_{N}$ differs in its
global structure from $SU(N)\times U(1)$, though the fact that they are
equivalent locally is often used when analyzing perturbative physics.

In this case the quotienting is done by a diagonal combination of the
$\mathbb{Z}_{N}$ center subgroups of the two factors, and identifies them with
each other $\exp\frac{2\pi i}{N}\mathds{1}_{N}\sim\exp\frac{2\pi i}{N}Q$. This
means that every field must be invariant under the diagonal combination of
rotations, $\exp\frac{2\pi i}{N}\mathds{1}_{N}\times\exp\frac{-2\pi
i}{N}Q\equiv 1$.

There is in general for $SU(N)$ representations a notion of ‘$N$-ality’ which
simply tracks how the field transforms under a $\mathbb{Z}_{N}$ center
transformation. A fundamental has $N$-ality of 1, as we saw above, and in the
$SU(N)/\mathbb{Z}_{N}$ theory the representation theory required $N$-ality of
$0\text{ (mod) }N$. Here in the $U(N)$ theory the quotient instead correlates
the $N$-ality of the representations with the Abelian charge. A fundamental
must have a charge under $Q$ which is $1\text{ (mod) }N$ such that it is
invariant under the quotiented subgroup. Since every representation may be
constructed by taking tensor products of fundamental and anti-fundamental
representations, this informs us of the charge $Q\text{ (mod) }N$ which each
$SU(N)$ representation must have in order to be an allowed representation of
$\left(SU(N)\times U(1)\right)/\mathbb{Z}_{N}$. The two-index $\phi^{ab}$
either symmetric or anti-symmetric irrep comes from $N\otimes N=N(N-1)/2\oplus
N(N+1)/2$ so must have $U(1)$ charge $2\text{ (mod }N)$. The adjoint
$\phi^{a}_{b}$ is built from $N\otimes\bar{N}=N^{2}-1\oplus 1$ so must have
$U(1)$ charge $0\text{ (mod }N)$, and so on.

Now what of the magnetic representations? Early physics work in this direction
includes Corrigan:1976wk ; Goddard:1976qe ; Daniel:1979yz ; Dokos:1979vu ;
Wen:1985qj , in which much further detail may be found. In $SU(N)\times U(1)$
the two factors are separate, and $\pi_{1}(SU(N))=1$ does not have monopoles
while $\pi_{1}(U(1))=\mathbb{Z}$ gives the simple monopoles familar from the
Abelian case above.

Turning to $\left(SU(N)\times U(1)\right)/\mathbb{Z}_{N}$, the structure is a
bit subtle. The fundamental group $\pi_{1}(U(N))=\mathbb{Z}$ tells us we have
distinct monopoles for any integer, but in this case the spectrum of monopoles
is skewed away from just being the $\mathbb{Z}$-valued monopoles of the
Abelian group. Let us picture the different classes of closed paths. Of course
one thing we can do is simply go all the way around $U(1)$ as
$U(\phi)=\exp(i\phi Q)$ and wrap around the $U(1)$ direction to get a monopole
with only $U(1)$ magnetic flux.

However, now if we go a fraction of $k/N$ around the circle, the quotient
combined with our understanding of the $SU(N)/\mathbb{Z}_{N}$ case above tells
us $\exp\left(i\frac{2\pi k}{N}T^{N^{2}-1}\right)\sim\exp\left(i\frac{2\pi
k}{N}Q\right)$. Then we can return to the origin not by continuing around the
$U(1)$ direction, but by taking a path along $T^{N^{1}-1}$ in $SU(N)$ that
when we get close to the origin looks like $U(\theta)=\mathds{1}+i\theta
T^{N^{2}-1}$.

So this case is something of a mixture of the two we have seen before. There
are $k\in\mathbb{Z}$ magnetic monopoles, but they now have both Abelian and
non-Abelian magnetic fluxes for $k\neq 0\text{ (mod N)}$. It is only in the
case $k\in N\mathbb{Z}$ for which they have $U(1)$ magnetic flux only.

###  6.3 The Standard Models

| $Q_{i}$ | $\overline{u}_{i}$ | $\overline{d}_{i}$ | $L_{i}$ | $\overline{e}_{i}$ | $H$
---|---|---|---|---|---|---
$SU(3)_{C}$ | $\mathbf{3}$ | $\overline{\mathbf{3}}$ | $\overline{\mathbf{3}}$ | – | – | –
$SU(2)_{L}$ | $\mathbf{2}$ | – | – | $\mathbf{2}$ | – | $\mathbf{2}$
$U(1)_{Y}$ | $+1$ | $-4$ | $+2$ | $-3$ | $+6$ | $-3$
Table 3: Representations of the Standard Model fields under the subgroups of
the gauge symmetries, switching notation from the earlier sections in which we
used Dirac fermions and the standard convention for the normalization of
hypercharge. Herein we speak of Weyl fermions—as appropriate for the Standard
Model in the unbroken phase—and henceforth we normalize $U(1)_{Y}$ so the
least-charged particle has unit charge. This will make various statements
simpler to see.

The case of the Standard Model is not much more difficult than the above
examples we have discussed. As you know, the Standard Model is a Yang-Mills
theory with a certain continuous gauge group which near the identity includes
factors of $SU(3)_{C}$, $SU(2)_{L}$, and $U(1)_{Y}$. The perturbative physics
of these theories, including the spectrum of gauge bosons, is controlled by
the local structure of gauge transformations which are close to the identity
transformation. Thinking just of the symmetry group, we may write a general
such infinitesimal group element as

$U(\theta_{1},\theta^{i}_{2},\theta^{a}_{3})=\mathds{1}+i\theta_{1}Y+i\theta^{i}_{2}T^{i}_{2}+i\theta^{a}_{3}T^{a}_{3},$
(23)

where $\theta_{1,2,3}$ parametrize the transformations in the hypercharge,
weak, and strong directions, and $T_{3},T_{2},Y$ are the generators of the
respective subalgebra. Thinking about the SM as a Yang-Mills theory we wish to
upgrade this invariance from global to local transformations which depend on
spacetime position $\theta_{i}\mapsto\theta_{i}(x)$. Then as is familiar we
must introduce vector gauge bosons in the adjoint representation and couple
them to our charged fields.

The transformations close to the identity explore only the Lie algebraic
structure, and in fact are not sensitive to the ‘global structure’ of the
gauge group. This is what we see in the covariant derivative to minimally
couple charged particles to a gauge field

$D_{\mu}=\partial_{\mu}-ig_{1}Q_{Y}B_{\mu}-ig_{2}T^{\alpha}_{R_{2}}W^{\alpha}_{\mu}-ig_{3}T^{a}_{R_{3}}G^{a}_{\mu},$
(24)

which explores only the local structure of the gauge group, just as the
position derivative explores only the local structure of the spacetime
manifold. That means we are only experimentally sure of this local
information, and in fact there are multiple possible Lie groups which have
this same Lie algebra.

The four different possibilities are

$G_{\rm SM_{n}}\equiv\left(SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}\right)/\mathbb{Z}_{n},$ (25)

where $n=1,2,3,6$ and we use the slang term $\mathbb{Z}_{1}\equiv\mathds{1}$
for convenience. As far as we are aware, this was first laid out
systematically in a little-known 1990 solo paper by a UCSB grad student
Hucks:1990nw  but has been well-publicized in recent years Tong:2017oea . The
options with $n>1$ can be viewed as quotient groups of $G_{\rm SM_{1}}$ where
we quotient out certain diagonal center transformations as follows.

In the group $G_{\rm SM_{2}}=G_{\rm SM_{1}}/\mathbb{Z}_{2}$, we impose an
equivalence relationship between the $\mathbb{Z}_{2}$ center subgroups of
$SU(2)_{L}$ and of $U(1)_{Y}$. That is, $(-1)\mathds{1}_{2}\sim\exp(i\pi Y)$,
working now in the normalization that the least-hypercharged particle has unit
charge (see Table 3). In the group $G_{\rm SM_{3}}=G_{\rm
SM_{1}}/\mathbb{Z}_{3}$, we impose an equivalence relationship between the
$\mathbb{Z}_{3}$ center subgroups of $SU(3)_{C}$ and of $U(1)_{Y}$. That is,
$\exp(2\pi i/3)\mathds{1}_{3}\sim\exp(i2\pi Y/3)$. In the group $G_{\rm
SM_{6}}$ we impose both of these quotients simultaneously.

| $Q_{i}$ | $\overline{u}_{i}$ | $\overline{d}_{i}$ | $L_{i}$ | $\overline{e}_{i}$ | $H$
---|---|---|---|---|---|---
$\mathbb{Z}_{3}\subset SU(3)_{C}$ | $e^{i2\pi/3}$ | $e^{-i2\pi/3}$ | $e^{-i2\pi/3}$ | $1$ | $1$ | $1$
$\mathbb{Z}_{2}\subset SU(2)_{L}$ | $-1$ | $1$ | $1$ | $-1$ | $1$ | $-1$
$\mathbb{Z}_{3}\subset U(1)_{Y}$ | $e^{i2\pi/3}$ | $e^{-i2\pi/3}$ | $e^{-i2\pi/3}$ | $1$ | $1$ | $1$
$\mathbb{Z}_{2}\subset U(1)_{Y}$ | $-1$ | $1$ | $1$ | $-1$ | $1$ | $-1$
Table 4: How each SM field transforms under a center transformation by the
generator of each noted subgroup.

Of course we can always consider these as abstract quotient groups, as in the
constructions of the previous sections. But we have also observed the
particles of the SM, which transform in a variety of representations. To see
if we can legitimately consider these other possibility for global structure,
we must check that the representation theory of any of these options actually
allows for the needed particles.161616One must additionally check that each of
these versions of the Standard Model is free of global anomalies, which is
indeed true as discussed in Davighi:2019rcd ; Wan:2019gqr ; Davighi:2020bvi ;
Davighi:2020uab ; Wang:2020xyo ; Wang:2021ayd . Indeed, it does work, as may
be checked easily from the data in Table 3. In the case of the
$\mathbb{Z}_{2}$ quotient, we see that the fields which are $SU(2)$ doublets
all have odd hypercharge, and the fields which are $SU(2)$ singlets all have
even hypercharge (and the $SU(2)$ triplet $W^{a}$ of course has zero
hypercharge) which means that indeed none of the fields are charged under this
diagonal $\mathbb{Z}_{2}$ center transformation. The $\mathbb{Z}_{3}$ subgroup
may be checked just as easily and the conclusion is the same, meaning that
indeed there is a four-fold ambiguity in the global structure of the gauge
group of the SM.

It is useful also to note that a particular global structure may be demanded
by the UV embedding of the SM in a unified gauge group. Either of $SO(10)$ or
$SU(5)$ demand the $\mathbb{Z}_{6}$ quotient. Less stringently, Pati-Salam
$SU(4)_{C}\times SU(2)_{L}\times SU(2)_{R}$ requires the $\mathbb{Z}_{3}$
quotient and trinification $SU(3)_{C}\times SU(3)_{L}\times SU(3)_{R}$ needs
the $\mathbb{Z}_{2}$ quotient.

Given an embedding of the SM gauge algebra in a UV theory, we can see the
global structure demanded simply by examining the decomposition of the
fundamental irreps of the UV under this breaking, and asking which center
elements they are invariant under. For example, the embedding of the SM in
$SU(5)$ is such that the fundamental decomposes as
$5\rightarrow(3,1)_{+2}\oplus(1,2)_{-3}$, and we see manifestly that these are
invariant under the $\mathbb{Z}_{6}$ center. Since all irreps of $SU(5)$ can
be found in tensor products of $5$ and $\bar{5}$, the embedding of the SM in
$SU(5)$ produces only representations which are invariant under the
$\mathbb{Z}_{6}$. More formally, of course, one can find group theoretically
that it really is $SU(3)\times SU(2)\times U(1)/\mathbb{Z}_{6}$ which is
actually a subgroup of $SU(5)$, as has been known since 1980 at latest
Scott:1979zu .

From the above argument, it is clear that finding a representation which is
charged under the $\mathbb{Z}_{6}$ center falsifies the embedding into
$SU(5)$. More generally, discovering a particle with electric charge $e/6$
(either at colliders or elsewhere) would rule out all the minimal unified
models of the universe.171717Notably this statement only applies for the
minimal theories of so-called ‘vertical’ unification; that is theories which
consolidate one generation of SM fermions into fewer irreps. Unification among
generations may be compatible with the existence of any of these fractionally
charged particles. Obviously so when the horizontal gauge group is factorized
from the Standard Model gauge group e.g. Cordova:2022fhg , but even in non-
factorized cases such as color-flavor unification Cordova:2024ypu . A new
particle with charge $e/2$ would tell us we can have Pati-Salam but it cannot
be further embedded into $SO(10)$, and a new particle of charge $e/3$ would
allow a unified theory like trinification but rule out its embedding in
$E_{6}$.

#### Some Additional Possibilities:

Thinking just as low-energy effective field theorists, there are a couple
further possibilities are useful to note. For one, it is conceivable that the
hypercharge assignments we have in Table 3 are not actually in terms of the
charge quantum. That is, we could discover a particle which has hypercharge
$1/N$ that of the left-handed quark doublet field $Q$. This would rule out all
the UV unification models we normally think about, but is possible. In terms
of thinking about the global structure of the SM gauge group, this would
effectively tell us that the $U(1)_{Y}$ circle is actually a factor of $N$
‘larger’ than we had thought. Correspondingly the magnetic monopole charges
are a factor of $N$ larger as a result of Dirac quantization. Recently
Alonso:2024pmq  has fully classified which such possibilities are consistent
with the various SM quotients. It would be interesting to understand which of
these could still be consistent with new unification models.

Most exotically, we can think about $\mathbb{R}_{Y}$, in which irrrational
charges are allowed. At a generic point in some constraint plot of
fractionally charged particles, one can have this in mind as the alternate
hypothesis that is being tested. It is true that we expect theories of quantum
gravity do not contain non-compact gauge groups like $\mathbb{R}$ (see e.g.
Banks:2010zn ; Harlow:2018tng ), but it is not obvious there is anything wrong
with them strictly as quantum field theories. Flipped around, we can say that
searches for irrationally fractionally charged particles are testing deep
principles of UV physics. These ideas are also subject to precision tests of
atom neutrality for example using interferometry Arvanitaki:2007gj ;
Bressi:2011yfa ; Unnikrishnan:2004 .

Finally we mention that these are not the only possible ambiguities in the
gauge group of the Standard Model. In Koren:2022axd  (App. B.1) we introduced
the SM+, in which the SM is extended by gauging
$\mathbb{Z}_{N_{c}N_{g}}^{B-N_{c}L}\times\mathbb{Z}_{N_{g}}^{L}$, which is the
Standard Model’s anomaly-free, generation-independent, global zero-form
symmetry. This entails no modification of the local dynamics, but ensures
absolute proton stability. We will further explore these and related
possibilities in future work Koren:2024a .

##  7 Generalized Global Symmetries

As particle physicists we are often used to field theories at weak coupling,
but it can be useful as field theorists to develop tools to analyze field
theories which work at arbitrary coupling. And away from perturbative limits,
one can be forced to reckon with the fact that gauge ‘symmetries’ are merely
redundancies, and sometimes the same physics can be understood in terms of
gauge theories with different groups. So it is useful to focus on global
symmetries, which do have physical content that is independent of any choice
of description.

In the framework of Generalized Global Symmetries, symmetries correspond to
the existence of certain operators which have topological correlation
functions. These are known as ‘symmetry defect operators’ (SDOs), can be
thought of as implementing the global symmetry transformation by ‘acting on’
(or ‘sweeping past’) the charged objects, and beautifully generalize familiar
notions like Noether charges and Gauss’ law.

In the following we will aim to describe relevant basic ideas of Generalized
Global Symmetries in an elementary fashion intending to convey some conceptual
lessons. For further detail, generalization, and technicalities we refer to
the seminal Gaiotto:2014kfa  and to some reviews aimed to be accessible for
particle physicists Brennan:2023mmt ; Gomes:2023ahz ; Bhardwaj:2023kri
.181818We note also introductions and reviews a bit further afield such as
Sharpe:2015mja ; Kong:2022cpy ; Schafer-Nameki:2023jdn ; Luo:2023ive ;
Shao:2023gho ; Loaiza-Brito:2024hvx ; Borsten:2024gox . But we will eschew any
topic whose introduction would require cohomology, as well as many interesting
GGS possibilities broader than the basics we require. Ideas and technology
from GGS are gradually being utilized in (or towards) particle physics
applications, for example Cordova:2022qtz ; Cordova:2022fhg ; Cordova:2024ypu
; Brennan:2020ehu ; Cordova:2022ieu ; Cordova:2023ent ; Cordova:2023her ;
Brennan:2023kpw ; Brennan:2023tae ; vanBeest:2023dbu ; vanBeest:2023mbs ;
Davighi:2019rcd ; Davighi:2020kok ; Davighi:2024zip ; Anber:2021upc ;
Cheung:2024ypq ; Hinterbichler:2022agn ; Hinterbichler:2024cxn ;
Hidaka:2020izy ; Hidaka:2021mml ; Hidaka:2021kkf ; Yokokura:2022alv ;
Choi:2022jqy ; Choi:2022rfe ; Choi:2022fgx ; Choi:2023pdp ;
Heidenreich:2023pbi ; Fan:2021ntg ; McNamara:2022lrw ; Asadi:2022vys ;
Reece:2023iqn ; Aloni:2024jpb ; Wan:2019gqr ; Wang:2020xyo ; Wang:2021ayd ;
Wang:2021vki ; Wang:2021hob ; Wang:2022eag ; Putrov:2023jqi ; Wang:2023tbj .

#### Familiar (Zero-Form) Noether Charges

A familiar symmetry which acts on local fields (so the charged operators are
zero-dimensional) has an associated Noether charge. In the case of a
continuous symmetry (for simplicity, $U(1)_{X}$) we may build this out of a
Noether current $J^{\mu}$ which obeys the conservation equation
$\partial_{\mu}J^{\mu}=0$. From this current we can build a family of
topological, unitary operators by exponentiating its integral over any three-
manifold $\Sigma_{3}$,

$U_{\alpha}(\Sigma_{3})=\exp\left(i\alpha\int_{\Sigma_{3}}J^{\mu}\epsilon_{\mu\nu\rho\sigma}dx^{\nu}dx^{\rho}dx^{\sigma}\right),$
(26)

where $\epsilon_{\mu\nu\rho\sigma}J^{\mu}\equiv\star J$ is the Hodge dual. We
refrain from the index-free notation of differential forms, but mention that
the benefit thereof is to emphasize that the metric tensor is not needed to
define these operators—they are supposed to be topological, after all.

The familiar Noether charge restricts $\Sigma_{3}$ to be all of space at a
given time, and the topological invariance of the charge is then the fact that
it can be moved forward or back in time and the charge remains the same. But
this more covariant set of operators is well-defined for any $\Sigma_{3}$, and
the conservation $\partial_{\mu}J^{\mu}=0$ implies that any deformations of
this surface do not change the correlation functions of
$U_{\alpha}(\Sigma_{3})$. Let us discuss further how to think about this,
drawing from Brennan:2023mmt  among others.

We consider smoothly deforming $\Sigma_{3}$ to $\Sigma^{\prime}_{3}$, where
for now we assume doing so does not cross any charged operators. That is, the
spacetime volume in between these is a four-manifold $\Sigma_{4}$ bounded by
these two three-surfaces,
$\partial\Sigma_{4}=\Sigma_{3}\bigcup\Sigma^{\prime}_{3}$, and $\Sigma_{4}$
does not have any charged operators in it. We compute the product of an SDO on
$\Sigma_{3}$ implementing a rotation by $\alpha$ and an SDO on
$\Sigma^{\prime}_{3}$ implementing a rotation by $-\alpha$ using the
generalized Stokes’ theorem

$\displaystyle U_{\alpha}(\Sigma_{3})U_{-\alpha}(\Sigma^{\prime}_{3})$
$\displaystyle=\exp\left(i\alpha\int_{\Sigma_{3}}J^{\mu}\epsilon_{\mu\nu\rho\sigma}dx^{\nu}dx^{\rho}dx^{\sigma}-i\alpha\int_{\Sigma^{\prime}_{3}}J^{\mu}\epsilon_{\mu\nu\rho\sigma}dx^{\nu}dx^{\rho}dx^{\sigma}\right)$
(27)
$\displaystyle=\exp\left(i\alpha\int_{\Sigma_{4}}\partial_{\mu}J^{\mu}d^{4}x\right)=1.$
(28)

Where we have used current conservation to find the volume integral vanishes
and we get $1$ on the right-hand side. Since these SDOs are unitary operators,
we learn $U_{\alpha}(\Sigma_{3})\simeq U_{\alpha}(\Sigma^{\prime}_{3})$. That
is correlation functions containing an insertion of $U_{\alpha}(\Sigma_{3})$
are invariant under deforming $\Sigma_{3}$, so the SDOs are topological as we
said above.

Now, the above equations assumed that there are no charged particle in the
volume $\Sigma_{4}$ between the initial and final surfaces. How do the SDOs
behave when we move the surface $\Sigma_{3}$ past a local field $\psi(y)$
charged under $U(1)_{X}$?

Recall that the Ward identity encodes how the conservation of a symmetry
current jibes with the existence of operators sourcing that current. That is,
we must upgrade the classical $\partial_{\mu}J^{\mu}(x)=0$ to an operator
equation which tells us what to do with a charged field $\psi(y)$. One derives
the consequences of the symmetry in the quantum mechanical theory by
performing a symmetry transformation for a general correlation function
calculated by a path integral, demanding the action is invariant under the
symmetry, and observing the consequences for the charged operators—for example
in Section 14.8 of Schwartz Schwartz:2014sze . In the Abelian case we have
simply

$\partial_{\mu}J^{\mu}(x)\psi(y)=\delta^{(4)}(x-y)q_{\psi}\psi(y).$ (29)

This tells us that while $\partial_{\mu}J^{\mu}(x)=0$ away from other
operators, there are important contact terms when this symmetry current hits
an operator charged under this symmetry. One should properly view such
statements as taking place inside arbitrary correlation functions separated
from other local operators,

$\langle\dots\partial_{\mu}J^{\mu}(x)\psi(y)\dots\rangle=\delta^{(4)}(x-y)q_{\psi}\langle\dots\psi(y)\dots\rangle,$
(30)

where the ‘$\dots$’ is a stand-in for any other operators away from $x,y$. The
action of the Ward identity will be crucial in understanding the use of the
symmetry defect operators.

Now let us repeat the computation above of deforming $\Sigma_{3}$ to
$\Sigma^{\prime}_{3}$ but now in the case where doing so does cross a charged
operator. A simple case has $\Sigma_{3}$ as a hypersphere $S^{3}$ and the
local operator $\psi(x)$ at a point $x$ which is inside $\Sigma_{3}$. We
consider then shrinking $\Sigma_{3}$ down
$\Sigma_{3}\rightarrow\Sigma^{\prime}_{3}$ so $x$ is now outside of this
surface, as in Figure 8, and then acting with the inverse SDO. Overall this
acts on $\psi(x)$ as

$U_{\alpha}(\Sigma_{3})\psi(x)U_{-\alpha}(\Sigma^{\prime}_{3})=\exp\left(i\alpha\int_{\Sigma_{4}}\partial_{\mu}J^{\mu}d^{4}x\right)\psi(x)=\psi(x)e^{i\alpha
q_{\psi}}.$ (31)

Where we have used the Ward identity and the fact that $x\in\Sigma_{4}$, and
we refer to Brennan:2023mmt  for further detail. We note also that if no other
charged operators were in $\Sigma_{3}$ to begin with, then conceptually we can
skip this second step of acting with $U_{-\alpha}(\Sigma^{\prime}_{3})$ and
just imagine shrinking $\Sigma_{3}$ all the way down to a point after it
passes $x$.

We can state the result more generally by saying that these SDOs act by
‘linking’, and writing
$U_{\alpha}(\Sigma_{3})\psi(x)U_{-\alpha}(\Sigma^{\prime}_{3})=\psi(x)e^{i\alpha
q_{\psi}\text{Link}(\Sigma_{3},x)}$. In the situation we have described, the
‘Linking number’ $\text{Link}(\Sigma_{3},x))=1$. The ‘Linking number’ is a
topological invariant of a configuration in $d$ spacetime dimensions between a
$p$-dim submanifold $\Sigma_{p}$ and a $d-p-1$-dim submanifold
$\Sigma_{d-p-1}$. This action by linking keeps track of the charge inside the
SDO when we move a charged operator from the interior to the exterior or vice-
versa. To gain some intuition, it is useful to think about the case $d=3$
(say, 3-space at some fixed time), where it’s easy to visualize that a $p=0$
point is either inside or outside a $d-p-1=2$ sphere, and a $p=1$ loop can be
linked with another $d-p-1=1$ loop. 191919We note for fun that general linking
numbers can be defined by certain topological quantum field theories
Horowitz:1989km .

Figure 8: A local operator $\psi(x)$ charged under a $U(1)$ zero-form symmetry
and the action of a symmetry defect operator $U_{\alpha}(\Sigma_{3})$ on it by
linking as described in the text. One dimension is suppressed.

#### Discrete Symmetries

We note also that a useful aspect of this formalism is a unified language for
both continuous and discrete symmetries. A discrete $\mathbb{Z}_{N}$ symmetry
doesn’t have an associated current because the Noether procedure requires a
notion of infinitesimal transformation. However, there are still well-defined
SDOs that we can write down and have the expected properties when they act on
charged operators,

$U_{\frac{2\pi k}{N}}(\Sigma_{3})\psi(x)=\psi(x)\exp\left(i\frac{2\pi
k}{N}q_{\psi}\text{Link}(\Sigma_{3},x)\right).$ (32)

This suffices as a definition in the case of a discrete symmetry by describing
how $U(\Sigma_{3})$ behaves in arbitrary correlation functions. Of course it
may be useful—and depending on the scenario it may be more-or-less easy—to
realize the SDO as the integral over $\Sigma_{3}$ of some local operator.
Sometimes we are thinking about a $\mathbb{Z}_{N}$ subgroup of what is (or
began as) a $U(1)$ symmetry, and we can realize $U_{\alpha}(\Sigma)$ as an
integral over a current with the angle restricted to $\mathbb{Z}_{N}$. This is
effectively an operator which measures a global charge (mod $N$), and will be
the relevant case for us below with the electric one-form symmetry of
electromagnetism.

In other cases when the symmetry is really intrinsically $\mathbb{Z}_{N}$, it
is sometimes useful to introduce an auxiliary $U(1)$-valued field and then
project out its dynamics. This becomes an invaluable technique when one wants
to understand discrete gauge theories, and we refer to Brennan:2023mmt  for an
expansive discussion of this topic.

###  7.1 One-form Symmetries

Yang-Mills theories have long been appreciated to include some gauge-invariant
one-dimensional operators known as Wilson loops and ’t Hooft loops. These are
not local operators because they are defined on a 1-dimensional path $\gamma$
through spacetime which is either a closed loop or an infinite
line.202020Which are closed loops on the one-point compactification of
spacetime. Physically a Wilson loop can be seen as the effect of a massive
particle of charge $q$ traversing the path $\gamma$, and in the limit where
the mass is taken to infinity these Wilson loops capture fully their physical
effects. In the Abelian case, the Wilson loop simply integrates the vector
potential along this path as

$W_{q}(\gamma)\equiv\exp iq\int_{\gamma}A_{\mu}dx^{\mu}.$ (33)

In the general non-Abelian case the Wilson loops are instead labeled by a
representation over which we take the trace $W_{R}(\gamma)\equiv\text{Tr}\exp
i\int_{\gamma}A^{a}_{\mu}T_{R}^{a}dx^{\mu}$. The ’t Hooft loops are defined
analogously for magnetic representations but with the electromagnetic dual
vector potential $A\mapsto\tilde{A}$.212121For completeness we recall that the
dual potential is related to the vector potential in the following nonlocal
way. The field strength is
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and its Hodge dual
is $\tilde{F}_{\mu\nu}=\varepsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$. This
dual field strength is related to the dual potential as
$\tilde{F}_{\mu\nu}=\partial_{\mu}\tilde{A}_{\nu}-\partial_{\nu}\tilde{A}_{\mu}$.

Now the question of which representations our theory allows can be understood
field theoretically and gauge-invariantly by examining these line operators
and the possible symmetries they might enjoy, which are called one-form
symmetries since they act on one-dimensional operators.

We recall Gauss’ law in electromagnetism where you think about integrating the
electric field over some closed 2-dimensional spatial manifold $\Sigma_{2}$
and finding some notion of an enclosed charge $Q_{\rm
encl}=\int_{\Sigma_{2}}\vec{E}\cdot d\vec{A}$. But we can more clearly and
more covariantly think about this by recognizing the generalized symmetry
structure behind Gauss’ law: The Gaussian surface computes a Noether charge
for a one-form symmetry!

Figure 9: The familiar form of Gauss’ law on a timeslice (left) and the more
covariant interpretation of the Gaussian surface as a symmetry defect operator
$U_{\alpha}(\Sigma_{2})$ acting on Wilson lines charged under a global one-
form symmetry.

Pure electromagnetism in fact has both an electric one-form symmetry and a
magnetic one-form symmetry. The photon equation of motion and the Bianchi
identity reveal the conserved two-index currents which generate these one-form
symmetries,

$\partial_{\mu}F^{\mu\nu}=0,\quad\partial_{\mu}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}=0.$
(34)

The familiar Gaussian surface can in fact be covariantly upgraded and
exponentiated to realize SDOs supported on any two-dimensional surface
$\Sigma_{2}$

$U_{\alpha}(\Sigma_{2})=\exp\left(i\alpha\int_{\Sigma_{2}}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}dS_{\mu\nu}\right).$
(35)

The SDOs are topological except when they cross Wilson lines and their
correlation functions are controlled by

$U_{\alpha}(\Sigma_{2})W_{q}(\gamma)=W_{q}(\gamma)\exp\left(i\alpha
q\text{Link}(\Sigma_{2},\gamma)\right).$ (36)

This is just the analogue of what we observed above for zero-form symmetries.
Now we can talk about the allowed representations in terms of the electric
one-form symmetries of the Wilson lines of the theory. Analogously to the
argument in terms of gauge transformations, if the electric one-form symmetry
is compact ($U(1)$ or a $\mathbb{Z}_{N}$ subgroup) then there is a
transformation by $\alpha=2\pi$ which should act as the identity

$U_{2\pi}(\Sigma_{2})W_{q}(\gamma)\equiv W_{q}(\gamma)$ (37)

and this is seen by the above equation to imply $q\in\mathbb{Z}$, since the
linking number is an integer. On the other hand it is conceivable that the
electric one-form symmetry is $\mathbb{R}$, though with the same difficulties
discussed above that this is thought not to occur in a theory of quantum
gravity.

###  7.2 One-form Symmetry-Breaking

There is an important qualitative difference between 0-form and $(n>0)$-form
symmetries when it comes to their breaking. For a zero-form symmetry, the
charged operators are zero-dimensional local operators—precisely the sort
which can appear in a Lagrangian density governing the local dynamics of a
theory. This means that such symmetries may be explicitly broken by adding a
charged operator to the Lagrangian. For a familiar example, if we add a
Majorana mass for neutrinos $\mathcal{L}+=(\tilde{H}L)(\tilde{H}L)/\Lambda$
then we explicitly break the zero-form global $U(1)_{L}$ lepton number
symmetry.

On the other hand, for a higher-form symmetry the charged objects are extended
operators. These don’t appear in the Lagrangian, and indeed no deformation of
the Lagrangian with additional operators can break a higher-form symmetries.
Rather, these symmetries can only break if, as you go to high energies, you
see that the charged extended operators are realized as dynamical objects in a
more-fundamental theory. For example, when you see that (some of) the Wilson
lines of electromagnetism are in fact in our universe completed into dynamical
charged particles like electrons and protons.

A useful qualitative picture to have of this breaking is of the ‘endability’
of the Wilson lines Rudelius:2020orz ; Heidenreich:2021xpr . For simplicity we
consider an Abelian gauge symmetry where the Wilson lines are labeled by a
charge, but the translation to general representations of non-Abelian
symmetries is obvious. Consider an ‘open’ Wilson line

$\displaystyle W_{q}(\gamma;x,y)$
$\displaystyle=\exp\left(iq\int^{y}_{x}A\right),$ (38) $\displaystyle
A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\lambda\quad\Rightarrow\quad
W_{q}(\gamma;x,y)$ $\displaystyle\rightarrow
e^{iq\lambda(y)}W_{q}(\gamma;x,y)e^{-iq\lambda(x)},$ (39)

which implies that in the infrared the only gauge invariant line operators are
closed loops or infinite lines. This is also why is possible for the SDOs
$U(\Sigma_{2})$ to have topological correlation functions with the Wilson
lines—if $\Sigma_{2}$ is linked with $\gamma$, it cannot be unlinked by any
smooth deformation. Indeed this is the definition of a topological invariant,
and this is what breaks when we go to higher energies and see dynamical
charged matter.

Figure 10: Bilocal line operator one can write cutting a Wilson loop. Such a
possibility explicitly breaks any symmetries acting on the Wilson loop because
e.g. an SDO on $\Sigma_{2}$ cannot have non-trivial topological correlation
functions any longer when it can smoothly ‘slide off’ the Wilson line.

When we have access to the electron, we can write a gauge-invariant, bilocal
line operator

$\bar{\psi}(y)W_{q}(\gamma;x,y)\psi(x),$ (40)

which ends on matter fields of charge $q$. Now it is easy to see why the
appearance of the dynamical electron breaks the electric one-form symmetries
which acted on the integer-charged Wilson lines in the far infrared.

In Figure 10 we depict a time-like Wilson line beginning and ending on a
charged fermion, and a Gaussian surface on a time-slice which would measure
the charge of the Wilson line. But the surface $\Sigma_{2}$ can be smoothly
deformed up or down the Wilson line and ‘slide off’ the end, where it can be
shrunk to a point. Then the correlation functions of $\Sigma_{2}$ cannot any
longer be topological and depend only on data like
$\text{Link}(\Sigma_{2},\gamma)$ because this topological linking is no longer
well-defined. So the appearance of the dynamical $\psi$ field means that any
one-form symmetry under which $W_{q}(\gamma)$ is charged must necessarily be
broken. Of course this holds true also for a Wilson line of charge $nq$,
$n\in\mathbb{Z}$, which can end on $n$ of these charged fields. But if the
charge $q$ of $\psi$ is not the minimum electric charge, there will still be
Wilson lines that are not ‘endable’, and so there may remain an electric one-
form symmetry.222222The case of an $\mathbb{R}$ gauge theory has some slight
subtleties in the language one must use to discuss one-form symmetry-breaking,
as discussed in Section 6 of Heidenreich:2021xpr . We now discuss this
possibility in more detail, specializing to QED.

###  7.3 Standard Model One-Form Symmetry

As suggested by the preceding sections, in the full theory of the Standard
Model the different global structures correspond to different one-form
symmetries. But in fact the latter statement is more general. The existence of
a heavy fractionally charged particle implies the existence of an emergent
electric one-form symmetry at low energies. We can understand any example
universally at low energies where this matches on to an electric one-form
symmetry of QED. We reserve a discussion of the electric one-form symmetry in
the electroweak phase for Appendix B.

At energies far below the electron mass $E\ll m_{e}$, none of the Wilson lines
of electromagnetism can be ‘cut’ or ‘screened’ by dynamical matter, and there
is a $U(1)^{(1)}_{e}$ electric one-form symmetry corresponding to
$\theta\in[0,2\pi)$. This is responsible for Gauss’ law.

When we approach energies of order the electron mass $E\gtrsim m_{e}$, the
continuous electric one-form symmetry is necessarily broken. In terms of our
Gaussian surface SDOs, the statement is that for general $\theta$, these
surfaces will no longer be topological. As shown in Cordova:2022rer  we can
interpret this violation of the topological invariance of the Gaussian surface
as the electric charge being ‘screened’ and relate it to the running of the
fine-structure constant $\alpha$. And indeed we have long appreciated that at
these high energies, charges are screened by virtual electron-positron loops.
The Uehling potential Uehling:1935uj  describing the one-loop photon vacuum
polarization tells us the corrected form of the charge $q(r)$ one measures for
a Wilson line operator of charge $q$ using a Gaussian sphere of radius $r$,

$q(r\gg
m_{e})=q\left(1+e^{2}\frac{e^{-2m_{e}r}}{\sqrt{64\pi^{3}m_{e}r}}+\dots\right),\qquad
q(r\ll m_{e})=q\left(1-e^{2}\frac{\log{m_{e}r}}{6\pi^{2}}+\dots\right),$ (41)

where we have given the asymptotic forms. Indeed at energies below the
electron mass the electric one-form symmetry becomes good exponentially
rapidly $dq(r)/dr\approx 0$, while above the electron mass the electric one-
form symmetry is clearly broken as the Gaussian surface is far from
topological. The question is whether the electron can screen all charges, or
whether there may remain some unbroken electric one-form symmetry corresponds
to fractional charges which the electron cannot screen.

The Gaussian surface in Eqn 35 is normalized such that the electron has $q=1$,
and

$U_{\theta}(\Sigma_{2})W_{\gamma}(q)=W_{\gamma}(q)\exp\left(i\theta
q\text{Link}(\Sigma_{2},\gamma)\right).$ (42)

Clearly $U_{2\pi}(\Sigma_{2})$ acts trivially on the electron, and on every
particle with charge a multiple of the electron’s. But if there is remaining
discrete electric one-form symmetry at energies above the electron’s mass,
then there are some Wilson lines with $0<q<1$ in units of the electron charge.
Correspondingly, some $U_{\theta}(\Sigma_{2})$ which act trivially on all
Wilson lines of SM representations act non-trivially only on these new Wilson
lines, and so remain topological at $E>m_{e}$. The SM gauge group with the
quotient $\mathbb{Z}_{n}$ has discrete electric one-form symmetry
$\mathbb{Z}_{6/n}$ above the electron’s mass.

If instead there is no remaining electric one-form symmetry above the
electron’s mass, as in the case where the SM is embedded in $SU(5)$ in the UV,
then every Wilson line has $q\in\mathbb{Z}$. So if we consider $\theta=2\pi$
then the Gaussian surface will act trivially on any operator, and there are no
nontrivial $U_{\theta}(\Sigma_{2})$ which remain topological.

So the language of generalized global symmetry conceptually unifies the low-
energy experimental signatures by focusing on the symmetry-breaking. In
Section 6 above we saw that the SM gauge group could have different global
structures. Or it could be that the left-handed quarks $Q_{i}$ do not actually
have the minimum of hypercharge and there is a less-charged particle. Or the
hypercharge gauge group could even be $\mathbb{R}_{Y}$. In any of these cases,
the signature in the far infrared where experimentalists work is simply the
existence of fractionally charged particles, and we have a unifying statement
of what we may learn from such searches as follows

By discovering a particle with fractional electric charge $q_{\psi}$ and mass
$m_{\psi}$ we learn the SM has an emergent electric one-form symmetry at $E\ll
m_{\psi}$. If $q_{\psi}=n/N$ (in units of the electron charge $e$) with
$\text{gcd}(n,N)=1$ then the SM has emergent $\mathbb{Z}_{N}^{(1)}$ electric
one-form symmetry. The unbroken one-form symmetry is measured by the Gaussian
surfaces $U_{k}(\Sigma_{2})=\exp\left(i2\pi k\int_{\Sigma_{2}}F\right),$ (43)
with $\theta=2\pi k$, $k=1..N$. And in the case where
$q_{\psi}\notin\mathbb{Q}$ then the one-form symmetry is $\mathbb{Z}^{(1)}$,
and each $k\in\mathbb{Z}$ makes for a distinct Gaussian surface.

The fact that these Gaussian surfaces remain topological continues to mean
that these fractional charges cannot be screened by matter at lower energies.
That is, if we surround a heavy fractional charge with a conductor made out of
Standard Model particles, it will be unable to prevent a nonzero electric
field in its volume.

#### Magnetic monopoles

The low-energy theory of QED also has a magnetic one-form symmetry as seen by
the existence of ’t Hooft lines and the non-existence of any magnetic
monopoles to cut them in the infrared theory. Just as the electric one-form
symmetry of the far infrared is always $U(1)^{(1)}$, the magnetic one-form
symmetry group is also $U(1)^{(1)}$. But the existence of a discrete electric
one-form symmetry above the electron mass controls how the charge of the ’t
Hooft lines is related to the electron’s electric charge. That is, with no
electric one-form symmetry, Dirac quantization implies the fundamental
magnetic charge is $g=2\pi/e$. With $\mathbb{Z}_{N}$ worth of electric one-
form symmetry, the quantum of magnetic flux is instead $g=2\pi N/e$.

##  8 Conclusions

In this work we have called attention to the interesting physics of
fractionally-charged particles from both the theoretical and observational
perspectives. We have seen that their existence may be tied to the structure