SciPost Physics

Submission

Fractionally Charged Particles at the Energy Frontier:
The SM Gauge Group and One-Form Global Symmetry
Seth Koren⋆ and Adam Martin†
Department of Physics and Astronomy, University of Notre Dame, South Bend, IN, 46556 USA⋆ skoren@nd.edu ,† amarti41@nd.edu

Abstract
The observed Standard Model is consistent with the existence of vector-like species withelectric 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 wereinterpret various searches to bound a variety of fractionally charged representations.
We emphasize that in some circumstances the collider bounds are surprisingly low ornonexistent, 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 ofwhich serves to underscore the bottom-up importance of these searches.
Copyright attribution to authors.
This work is a submission to SciPost Physics.
License information to appear upon publication.
Publication information to appear upon publication.

Received Date
Accepted Date
Published Date

1

2

Contents

3

1 Introduction

2

4

2 LHC Production

6

8

3 Collider Signatures of Fractionally Charged Particles
3.1 Solely U(1)Y charges
3.2 SU(3)C charges
3.3 SU(2) L charges

8
9
10
11

9

4 Reinterpreted LHC Bounds for Assorted Representations

13

5 Cosmology
5.1 Abundance Constraints
5.2 Thermal Plasma Production

17
17
18

6 Global Structure of Gauge Theory
6.1 Abelian Warmup: R vs. U(1)6.2 Global Structure of Non-Abelian Groups

19
19
23

5
6
7

10
11
12

13
14
15

1

SciPost Physics

6.3

16

Submission

The Standard Models

26

20

7 Generalized Global Symmetries
7.1 One-form Symmetries
7.2 One-form Symmetry-Breaking
7.3 Standard Model One-Form Symmetry

28
31
32
34

21

8 Conclusions

35

22

A Fractionally Charged Particle Partonic Cross Sections

36

23

B Electroweak Phase One-Form Symmetry

37

24

References

38

17
18
19

25
26

27

28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53

1

Introduction

The Fundamental Charge Quantum of QED What is the fundamental quantum of electriccharge 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.1
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 tobe 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 whichof our cherished principles to cling to, and which to consider counterfactually.
Notably, with less ambitious unification schemes we can have a smaller quantum of electriccharge. 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 wemust add additional fermions) the quantum can be e/3. If the Standard Model matter neverorganizes into one of these minimal unified theories, then the fundamental quantum of charge
can be e/6.2 In more exotic scenarios that would even more generally challenge our usual UVparadigms, the charge could be even smaller.
The core message of our work is that particles with 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 dif1
For 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’ [1].
2
Early work on extended models of unification which feature fractionally charged particles includes [2–9], andearly discussions of the appearance of fractionally charged particles in string theories include [10–13].

2

SciPost Physics

54
55
56
57
58
59
60
61
62
63

64
65
66
67
68
69
70
71
72
73
74
75

Submission

ference 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 aspectsof the Standard Model physics. We learn that the allowed charges of magnetic monopoles, thespectrum 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 thefundamental 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 wesee 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 understandwhat’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
ψai in the fundamental 3 representation has electric charge qi which obeys 3qi = 2 (mod 3),
and their antiparticles the 3̄ anti-fundamental ψ̄ j b necessarily have 3q j = 1 (mod 3). The
gluons in the adjoint 8 are of course electrically uncharged.

ui

di

g

SU(3)C

3

3

8

U(1)EM

2
3

− 31

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.
76
77
78
79
80
81
82
83
84
85
86
87

88
89
90
91
92

The only invariant tensors of SU(3)C are δ ab , ϵ abc , and ϵabc , and we seek to build composite
operators which are colorless. Working (mod 3), we see δ ab pairs a 1 with a 2, and the LeviCivita
symbol composes three of the same charge—either way resulting in an electric chargeP
3qi = 0 (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 fractionalcharge. But it is not clear from this argument whether this fundamentally must be the case, orwhether 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 understandthe 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 understandthese aspects systematically is by enlarging our notion of symmetries to include symmetries ofextended 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’—to3

SciPost Physics

93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124

125
126
127
128
129
130
131
132
133
134
135
136
137
138

Submission

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 [14–19]. As abasic picture one can think of these operators as accessing the response of the system to aprobe 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
H
a Wilson loop operator exp(i γ A) to understand the Aharonov-Bohm effect where we think
about adiabatically moving an electron on the spatial path γ 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 fullStandard Model the different ‘global structures of the gauge group’ (to be reviewed below) areexactly 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 fieldspace, 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.3The 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, fermionicor bosonic, would unequivocally falsify all minimal grand unified theories. Perhaps no othersingle 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. [20]) and ATLAS (e.g. [21]) but efforts have mainly been
focused on SUSY-motivated scenarios. To the extent that we can design searches sensitive tothe 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.
3
Of 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 conceptsof Wilson and ’t Hooft lines and their 1-form symmetries.

4

SciPost Physics

139
140
141
142
143
144
145
146
147
148
149
150
151

152
153
154
155
156
157
158

159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185

Submission

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 fractionallycharged 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 placebounds 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 inthe 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 thesesearches, 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 fractionallycharged 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 isprobed 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
GSMn ≡ (SU(3)C × SU(2) L × U(1)Y ) /Zn ,(1)for n = 1, 2, 3, 6 (where ‘Z1 ’ is slang for 1). We do not yet know which is realized in nature,
but GSMn 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 werelaid out first by Hucks [22]. The impact on the allowed line operators was studied recently
by Tong [19], 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 [23].
Recently multiple groups have investigated how the discovery of an axion and the carefulmeasurement of its couplings to different gauge groups also provides constraints on the global
structure [24–26]. This essentially promotes the discussion in [19] about the range of the SMtheta angles to a new dynamical probe—as we likewise here emphasize that a discovery ofa new fractionally charged particle directly probes the allowed line operators by upgradingthem to dynamical particles.
Some complementary perspectives on fractionally charged particles have recently appeared
as well. In [27] 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 [28] 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 thatthey must couple in pairs to SM matter. But resultingly the ability of SMEFT to investigate theexistence 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 fractionallycharged particle is a global one-form symmetry under which Wilson loops are charged, but

5

SciPost Physics

Submission

SU(3)C
–
–
–
3
6
8
3
6
8

SU(2) L
–
2
3
–
–
–
2
2
3

6Y mod 6
0
3
0
4
2
0
1
5
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.

186

local fields are strictly blind to.4

187

2

188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209

LHC Production

The primary phenomenological goal of this paper is to revisit collider bounds on fractionallycharged 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 ψ, sitting in an ‘exotic’ representation of the SM gauge group such thatthe electric charge of ψ is some multiple k ∈ Z of e/6 (excluding k = 0, ±6, . . . obviously).5
In Table 2 we show some example non-Abelian representations and which hypercharge theymust have to produce only integer electric charges in the far IR. Away from this choice, theelectric 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.
We will label ψ by its full quantum numbers and its electric charge when necessary, ψ(SU(3),SU(2),Y ),Q ,though when the context is clear we will drop subscripts other than the charge. We denotethe electric charge in fractions of e throughout, e.g. using Q = 1/3 for e/3. For color singlet
ψQ , the electric charge is given by the usual combination of τ3 and Y , while for colored ψ thecharge of the outgoing states is more subtle as ψQ will combine with SM matter to form color
singlet, exotically charged ‘hadrons’.
We assume the only interactions ψQ has are gauge interactions dictated by its quantum
numbers. As mentioned above, interactions involving a single ψQ and SM matter are forbidden, and we will ignore interactions between pairs of ψ (really ψ̄Q ψQ , etc.) and the SM, suchas H † H ψ̄Q ψQ . For fermionic ψ, all such interactions are non-renormalizable, while for scalarψQ the Higgs portal term is marginal (as is the quartic interaction (ψQ† ψQ )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 the4This ‘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.
5
As ψ is necessarily electrically charged, it cannot be a Majorana fermion or a real scalar.

6

SciPost Physics

210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257

Submission

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 ψ̄Q ψQ′ ′ when one of ψQ , ψQ′ ′ is an SU(2) doublet and Yψ′ − Yψ = 1/2. 6 Multi-exotic
interactions could lead to interesting phenomenology, but are beyond the scope of this paper.
Within this setup, ψQ 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 ψQ . For color singlets, the particles are produced in
Drell-Yan q̄q → ψ̄Q ψQ via ŝ channel photon and Z. If ψQ is an SU(2) singlet, the entire crosssection is proportional to Q2ψ , while the cross section for ψQ in larger SU(2) multiplets willcontain pieces proportional to (τ3 )ψ , the entries of the diagonal SU(2) generator appropriate
for ψQ s representation. When (τ3 )ψ ̸= 0, these terms typically dominate the cross section aseach power of Q ψ (which we have assumed to be < 1) comes with a factor of sin2 θW ∼ 1/4.
For ψQ in non-trivial SU(2) representations, there is also a charged current production mode,q̄q′ → ψ̄Q ψQ±1 + c.c. via ŝ channel W ± .
If ψQ carries SU(3) quantum numbers, QCD production g g → ψ̄Q ψQ , q̄q → ψ̄Q ψQ becomes the dominant mechanism. Of these, g g is the larger channel when ψQ is light, but q̄q
takes over for heavier ψQ . The crossing point depends somewhat on the representation andspin of ψQ but is O(1 TeV) for a Dirac fermion color triplet.
The partonic cross sections for pp → ψ̄Q ψQ production are compiled in Appendix A for
both fermionic and scalar ψQ . For now, we opt for analytic expressions over adding newparticles to Monte Carlo programs such as MadGraph [29]. In part, this is because we arefocused 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 noteasily 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 aparticular search or ψQ .
Folding parton distribution functions into the partonic cross sections (Appendix A), we
find the LHC proton level cross sections pp → ψ̄ψ. We use NNPDF3.0nlo parton distributionpfunctions [30, 31] with αs = 0.118, factorization/renormalization scales of µ̂ F = µ̂R = ŝ andassume a collider center of mass energy of 13 TeV. We have also imposed the parton-level cut|ηψ | < 2.5 so that these particles appear in the tracker volume.
The proton level cross sections for some illustrative ψQ are shown below in Figs. 1 and
2 below. In Fig. 1 we show the cross section for SU(2) singlet ψQ , either charged only under
hypercharge (left panel), or under several different color representations (right panel). Figure
2 shows the cross sections for color singlet ψQ sitting in non-trivial SU(2) representations, both
via neutral current (left panel) and charged current (right panel).
The cross sections ψQ charged only under hypercharge are quite small, O(1 pb × Q2ψ ) for afermionic ψQ and Mψ = 100 GeV and falling precipitously as Mψ increases to O(2 fb × Q2ψ ) atMψ = 500 GeV. Charging ψQ under SU(3), the cross section jumps by orders of magnitude,
σ(pp → ψ̄Q ψQ ) ∼ 3 pb (60 pb) for a 500 GeV color triplet fermion (color octet). The crosssection for color singlet, SU(2) charged ψQ sits between these two, O(5 fb) for Drell-Yan production of either state in a 500 GeV doublet ψQ , and O(10 fb) (O(5 fb)) for charged current
production via W + (W − ). For other SU(2) representations, both types of cross section growwith the size of the multiplet; labelling the SU(2) part of the ψQ state as |I0 , i3 〉, Drell-Yan∝ i32 , while the charged current is ∝ (I0 (I0 + 1) − i3 (i3 + 1)). The LHC cross section for a few
different SU(2) multiplets (both Drell-Yan and charged current pieces) are shown in the rightpanel of Fig. 2.
For fixed quantum numbers, the cross sections for fermionic ψQ are larger than theirMore exotic terms, such as φQ′ ψQ f (where we have used φQ′ for an exotic scalar in this context, ψQ for afermion, and f a SM fermion) are also possible, either with or without flavor structure.
6

7

SciPost Physics

Submission

10100

σ(pp ψ QψQ) [pb]σ(pp ψ QψQ) [pb]

1
0.100
0.010
0.001
10-4

1

0.01

10-4
100

200

300

400
500
Mψ [GeV]

600

700

500

800

1000

1500
Mψ [GeV]

2000

2500

10

10

0.100

0.100

dσ/ds [pb]σ(pp ψ QψQ)(pb)]Figure 1: Left panel: Lowest order pair production cross section for ψQ 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 ψQ as a functionof Mψ (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 asp
sume a center of mass energy s = 13 TeV and use solid lines are for Dirac fermionsand dashed lines for charged scalars.

0.001

0.001

100

200

300

400
500
Mψ [GeV]

600

700

800

200

400

600

800

1000

mψ [GeV]Figure 2: Cross sections for ψQ under different SU(2) representations, all with
Y = 1/3. The red line shows the cross section for the (τ3 )ψ = 1/2 componentof a SU(2) doublet, while the green shows the (τ3 )ψ = 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, thecurves for the doublet and triplet cases would barely move, as the cross section isdominated by the SU(2) portion. Right panel: Charged current cross section (via
W + ) for doublets, triplets, and SU(2) singlet for comparison

261

scalar counterparts by roughly an order of magnitude. This difference stems from the factthat 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.

262

3

258
259
260

263
264

Collider Signatures of Fractionally Charged Particles

To explore how ψQ 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 the8SciPost Physics

Submission

304

fractionally charged particle is stable (or metastable), and rely on anomalously low d E/d x 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, d E/d x is independent of the particle’s mass, but it is proportional to Q2 .
The CMS analysis [20] is the most recent and most easily translated to the scenarios weenvision. In Ref. [20], events were triggered using information in the muon system, then investigated for tracks with anomalously low d E/d x. 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 ∼ 2/3; particles with higher
electric charge leave fewer low d E/d x 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 ≃ 1/3, the efficiency is so poor thatthe bound drops to the minimum considered signal mass, 50 GeV.
A second reference we rely on is an ATLAS analysis for long lived gluinos/stops/sbottoms,
Ref [21] (other searches, either for stable particles or optimized for metastable variations, canbe found in Ref. [32, 33]). Upon hadronization, gluinos/stops/sbottoms all form ‘R-hadrons’
with integer charge, with the fraction with charge ±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 seemout of place. However, heavy exotic hadrons deposit negligible energy in the calorimeter, so ifthey 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 ψQ hadrons with small Q), most of theirenergy 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 arefurther 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 ≳ 2/3, where the CMS search loses sensitivity.
While Ref. [20, 21] are most relevant for our purposes, we’ll see that ψ in some corners ofparameter space are best bounded by LHC searches unrelated to fractionally charged or stableparticles, such as the invisible width of the Z [34], monojet-style searches [35] that look for
unbalanced, energetic jets, and disappearing tracks searches [36] 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 → ψ̄Q ψQ cross section to a bound, and exactly whichbound is best, differ greatly depending on how ψ is charged under the SM groups. In the next
subsections, we explore some of the options.

305

3.1

265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303

306
307
308
309
310

Solely U(1)Y charges

This is the simplest scenario, as Q ψ = Y , so there is no hadronization or SU(2) partners toworry 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.

9

SciPost Physics

311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355

3.2

Submission

SU(3)C charges

Colored ψQ particles will quickly hadronize after being produced at the LHC. And if ψ does
not have the hypercharge demanded in Table 2, then all of the hadrons containing one ψQ will
be fractionally charged. Hadronization with the light quarks of the Standard Model will resultin a variety of fractional charges for hadrons containing ψ. 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, wefollow the Lund string model [37] as used in Pythia [38] with application to R-hadrons [39].
In this model, the ψQ , ψ̄Q sit at the endpoints of color strings which fragment. When thestrings break, colored remnants join up with ψQ 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 spacefactor for the strange: (u : d : s ∼ 1 : 1 : 0.3); the diquark fraction is suppressed by an amountset by data [40, 41]. Following this model [39, 42], triplet ψQ form mesons with ū, d̄, s̄ andthe abundance of the ‘down-type’ mesons compared to ‘up-type mesons’ is 60:40. ψQ baryons
arise less frequently, ∼ 10% of the time, with the light quark composition roughly following
the same (u : d : s) ratio as in ψQ 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. [39] and is wellapproximated 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 shiftedby whatever fractional charge ψQ carries.7
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 thecolor singlet case. Charge flipping has been modeled somewhat for R-hadrons [39, 43], 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 initialstudy. 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 boundstate.
When using these simple hadronization rules to determine the charge of exotic hadrons,
we often find some fraction of the bound states have charge ∼ 1, e.g. 5/6 from a color triplet ψwith Y = 1/2 (a ψQ d̄ meson), or 7/6 from a color octet with Y = 1/6, (a ψud̄). The proximityof these charges to ±1 makes the technique in CMS ineffective. To determine bounds in this
scenario, we will instead reinterpret R-hadron searches from Ref [21], making the assumptionthat the R-hadron bounds are driven by the Q = ±1 ‘meson’ (i.e. (ψR q̄)8 for R-hadrons fromcolor triplet ψR or ψR qq̄ for color octet ψR ) bound states and that the experiments are notsensitive to the difference between Q ≃ ±1 and ±1. For color representations not studied
Color octets can also bind with gluons (a string breaks to g g, with one g binding to ψQ and the other binding
to remaining string fragments). Reference [39] takes to be O(10%) of ψ-gluon bound states, though in our casethese states will retain whatever electric charge ψ carries (and therefore interact with the tracker/muon system),
while in the gluino case this fraction is invisible.
8
We use a subscript R for the heavy gluino/stop/sbottom in a R-hadron.
7

10

SciPost Physics

Submission363

in R-hadron analysis, we will set bounds by equating (cross section × fraction of events withat least one exotic hadrons with near integer charge) = R-hadron cross section × fraction of
events with at least one ±1 charge R-hadrons. We note that there are searches for exotic,
multiply charged particles, but these searches begin at Q = ±2 [33].
This sort of reasoning will allow us to roughly reinterpret tracker based searches for somecolored representations, but we emphasize that for detailed constraints dedicated simulationsof hadronization and detector response for these fractionally charged representations should
be done.

364

3.3

356
357
358
359
360
361
362

365
366
367
368
369
370
371
372

373

SU(2) L charges

When ψQ 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 |∆Q| = 1. At tree-level,
and in the absence of operators such as H † H ψ̄Q ψQ as we have assumed, the components of
ψQ are mass-degenerate. Loops of W /Z bosons break this degeneracy, introducing a splitting of αem mW /π ∼ O(100) MeV, though with a degree of variation depending on the exact
quantum numbers of ψQ . For a multiplet with hypercharge Y containing a state with charge
Q = (τ3 )ψ + Y and a state with charge Q′ = (τ′3 )ψ + Y the one-loop mass difference between
the two is [44, 45]:







α2 M ¦ ′2
mW
mZ ©
mZ 
2
2
2
′
MQ′ − MQ =(τ3 − τ3 ) f− cW f+ 2 (τ3 − τ3 ) Y sW f
(2)
4π
Mψ
Mψ
Mψ
where Mψ is the tree-level mass, cW = cos θW , sW = sin θW , and(f (r) =

374
375



1/2 2p+r 2r 3 ln r − 2r + r 2 − 4
r + 2 ln(r 2 − 2 − r r 2 − 4) /2


3/2
p
−r 2r 3 ln r − kr + r 2 − 4ln(r 2 − 2 − r r 2 − 4) /4

377
378

for a scalar9

In the majority of cases, the state with smaller |Q| is the lightest. For Mψ ≫ mW , m Z and usingm Z = mW /cW , we see that the mass splitting asymptotes to∆M ≃ 160 MeV × (τ′3 − τ3 )(τ′3 + τ3 + 2Y + 2 Y / cos θW ).

376

for a fermion(3)

While there can clearly be cancellations, the general trend is that the splitting grows with thehypercharge of the multiplet and the τ′3 value of the excited state.10
This multiplet structure has several implications for how ψQ appears at the LHC.
• Even if one component of ψ has Q ≲ 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.

379
380
381

• The phenomenology of the heavier, larger charge state depends crucially on its lifetime(and therefore crucially on ψ’s quantum numbers, which dictate the mass splitting). For
mass splittings > mπ , the two-body decay ψQ+1 → ψQ +π+ dominates, while for smaller
splitting ψQ+1 mostly decays to ψQ + e ν̄e (three-body), with a small branching fraction
to ψQ + µ ν̄µ . The decay length for an illustrative set of SU(2) and Y choices are shown
below in Fig. 3. The decay lengths asymptote at large Mψ /mW , as expected from the

382
383
384
385
386
387

The factor k is UV divergent but can be absorbed by counterterms for the mass and ψQ quartic
Note that for Y = 0, |τ′3 | = |τ3 | the mass splitting vanishes. For ψ a Weyl fermion in the n-dim representation,
ψ̄ϵ n transforms the same way (ϵ 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 thefermions of the same charge.
9

10

11

SciPost Physics

Submission

104

104
1000

1000

100
cτ [cm]cτ [cm]

100
10

10
1
1

50

0.1

100

150

200
250
Mψ [GeV]

300

350

400

50

100

150

200
250
Mψ [GeV]

300

350

400

Figure 3: Decay length for the excited state(s) in an SU(2) doublet ψ (left panel)and SU(2) triplet ψ (right panel). In the left panel the blue line shows the choice
Y = 1/3 (Q = −1/6, Q′ = 5/6) while the green and red show Y = 2/3 (Q = 1/6,
Q′ = 7/6) and Y = 1/6 (Q = −1/3, Q′ = 2/3) respectively. In all cases the τ3component of the multiplet has the lowest (magnitude) charge. The solid lines are
the results for fermionic ψ while scalar ψ are dashed. In the right panel, the redlines 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 τ3 = +1 and τ3 = −1 components are notequal. For the blue lines, the choice Y = 2/3 means the τ3 = −1 component has thesmallest |Q| and is the lightest. Therefore, the solid (dashed) lines show the decayof Q = 5/3 to Q = 2/3 while the dotted (dot-dashed) show the decay of Q = 2/3 to
Q = −1/3.
mass splitting formulae, while at smaller Mψ /mW there are significant differences forfermion vs. scalar ψ and cusps where the two-body decay to ψQ + π± turns on or off. 11

388
389

390
391
392
393
394
395
396
397
398
399
400
401

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 ofO(cm) and would lead to displaced vertices or kinked tracks. A second category of excitedstates, 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 intheir 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 ψQ 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.
11

The one exception to the general mass splitting trend is the red dot-dashed line in the right panel of Fig. 3, themass 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 Mψ (leading to longer decay lengths). This is due to the fact that, while Eq. (2) genericallyincreases 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 afunction of Mψ .

12

SciPost Physics

402

403
404
405
406
407

4

Reinterpreted LHC Bounds for Assorted Representations

In this section we show a sampling of the LHC bounds on different exotic ψQ by reinterpretinga variety of searches. Given the huge number of scenarios with fractionally charged ψQ , weobviously 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 ψ 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. [20]. Using the quoted cross section numbers to bound fermionic (scalar) ψQ : Q = 1/6 – no LHC bound, Q = 1/3
Mψ > 88 GeV (45 GeV), Q = 1/2 Mψ > 610 GeV (340 GeV), Q = 2/3 Mψ > 650 GeV (370 GeV).
It is worth mentioning that the bounds for the lower charge regime, |Q| = 1/3, haveloosened substantially in Ref. [20] compared to previous iterations, Ref. [46, 47]. The
loosening of the bounds can be traced to a mismodeling in the efficiency of the muon
trigger for low charge [20].

408
409
410
411
412
413
414
415
416

417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440

Submission

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 Γ (Z → ψ̄Q ψQ ) ≤ 1.5 MeV, the 1-sigma
uncertainty on the invisible width [34], fermionic ψQ with |Q| = 1/6 are ruled out except rightat ∼ m Z /2 where the phase space suppression is severe. However, if relax the constraint to
2× this uncertainty, the bound disappears. For scalar ψQ , |Q| = 1/6 there is no bound even if
we impose the stronger condition of Γ (Z → ψ̄Q ψQ ) ≤ 1.5 MeV.
We can also approximate |Q| ≲ 1/3 as invisible and constrain these scenarios using monojet/ T + j [35], with the ψQ playing the role of the missing energy. Reference
style analyses pp → E/ T + j in bins beginning with[35] quotes model independent cross section limits on pp → E
σl im < 736 fb for p T, j > 200 GeV. Requiring such an energetic jet suppresses the cross sectionby O(200−500) depending on Mψ (larger suppression for lighter ψQ ) 12,13 . For fermionic ψQ ,the monojet analysis places a bound of only ∼ few GeV, while for scalar ψQ the cross section
is so low there is no LHC bound even for massless ψQ .
Light (∼ few GeV), fractionally charged ψQ could also be similarly to millicharged matter, a
topic of intense work and interest recently [48]; depending on the exact mass and charge, such
scenarios are ruled out by fixed target experiments, rare meson decay, star cooling, etc. Seee.g. Ref. [49,50] for a summary of limits on millicharged matter. The most relevant bound forthe range of masses and charges we are interested in comes from the SLAC anomalous single
photon e+ e− → γX search, which rules out fermionic ψQ lighter than 10 GeV for Q > 0.08 [51–53]. 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 ψQ to be acolor 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, ψ 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 aremuch more stringent, so equating the cross section for the production of at least one|Q| = 2/3 particle – ((0.4)2 + 2 × 0.4 × 0.6) × σ(pp → ψ̄Q ψQ ) = 0.64 × σ(pp → ψψ)

441
442
443
444

We derive this factor by running pp → τ+ τ− (+ j) in MadGraph and varying the mass of the τ.
The large p T, j values are needed to suppress the irreducible background from Z(ν̄ν) + j. The suppression this/ T + j proceeds through a contact
causes for our signal is much less than in dark matter models where pp → E
interaction, as the latter grows with the energy.
12

13

13

SciPost Physics

445
446
447

448
449
450
451
452
453
454
455
456
457
458
459
460

461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489

490
491

Submission

to the CMS |Q| = 2/3 bound, we find masses less than 1.8 TeV (1.4 TeV) are excluded
for fermionic (scalar) ψ. Note that Y = 1/3 results in hadrons with the same |Q| andtherefore is subject to the same bounds.
• Y = 1/6: For this choice, all ψQ q̄ bound states have |Q| = 1/2. From the CMS bound,
we find masses less than 1.9 TeV (1.5 TeV) are excluded for fermion (scalar) ψQ .
For our next two examples, we consider more exotic color representations, and for convenience
define di which is either a down or strange quark:• Color octet with Y = 1/6: ψ(8,0,1/6),1/6 . Within our framework, this state leads tohadrons with charge Q = 1/6 (ψuū, ψd̄i d j , ψg) 55% of the time, and Q = 7/6 (ψd̄i u) or
Q = −5/6 (ψdi ū) each 22% of the time. As the CMS search is insensitive to |Q| ≲ 1/3 or∼ 1, this is a scenario where we turn to stable R-hadron searches [21] to place bounds.
From this breakdown, we see that 67% of events contain at least one |Q| ∼ 1 hadron.
Equating 0.2 × σ(pp → ψ̄ψ) to the gluino R-hadron cross section bound of ∼ 1 fb, wefind masses less than 2.0 TeV (1.65 TeV) are excluded for fermion (scalar) ψQ . 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: ψ(6,0,0),0 After hadronization, this yields states with charge
Q = −4/3 (ψūū), Q = 2/3 (ψd̄i d̄ j ) and |Q| = 1/3 (ψūd̄i +c.c.) with fractions ∼ 20% : 30% : 50%.
The strongest bound comes from the |Q| = 2/3 fraction. The fraction of events with atleast one |Q| = 2/3 particle is ∼ 50%, and equating 0.5 × σ(pp → ψ̄ψ) to the CMS|Q| = 2/3 limit, we find masses less than 2.2 TeV (1.8 TeV) are excluded.
Finally, we consider benchmark color singlet ψ 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 (ψ(0,2,2/3),1/6 ) and
one with Q = 7/6 (ψ(0,2,2/3),7/6 ). The Q = 7/6 state decays within O(cm), leaving adisappearing track signature. In the context of the CMS search, the Q = 7/6 state justadds 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 Mψ ≳ 45 GeV for
either spin ψQ . Additionally, as the LHC production cross section is much larger thanthe SU(2) singlet case, it is possible to bound this ψQ 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 → ψ̄1/6 ψ7/6 + c.c.. Adding
these and comparing to the 95% CL allowed cross section for p T, j > 200 GeV, we finda monojet bound of ∼ 50 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 searchstrategy 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 thanthe higgsino case. For an electroweak doublet, the mass splitting in Eq. 3 is ∝ Y , andY = 2/3 is larger than the higgsino value of Y = 1/2. As a result, the lifetime of theexcited state is shorter, leading to shorter tracks and a less efficient search. Taking thedifference in lifetime into account and applying the cross section bound from Ref. [36],we find the current scenario is excluded for ψQ masses below 115 GeV (70 GeV).
• An SU(2) doublet with Y = 1/6. The only difference compared to the case above is thatthe states now have charge Q = −1/3 and Q = 2/3, with the Q = 2/3 slightly heavier.
14

SciPost Physics

492
493
494
495
496
497
498

499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516

517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540

Submission

However, as Y is smaller, so is the mass splitting, to the point that for Y = 1/6 the mass
splitting drops below mπ . As a result, the lifetime of the excited state is significantly
longer than in the previous case, O(20m), 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 → ψ̄2/3 ψ2/3 + pp → ψ̄−1/3 ψ2/3 + c.c
to the |Q| = 2/3 limit from CMS [20]. We find masses below 1.1 TeV (750 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 O(cm) before decaying to
the Q = −1/3. Only the Q = −1/3 particle survives to the muon system, so if we relyon 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 thelimit from Ref. [20], we find limits of Mψ > 350 GeV (200 GeV). The lifetime of theQ = 2/3 is long enough that one expects it should leave a trace in disappearing track
searches. The limit from Ref. [36] on nearly degenerate electroweak triplets (a wino) is
650 GeV, though extrapolating this to the present scenario is not straightforward as theefficiency 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. [36] falls
precipitously with mass splitting. Part of this lack in sensitivity can be compensatedby 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 canuse our experience with those setups to extrapolate to other multiplets to some extent. For ψQ
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, thetechnique fails and is superseded by time-of-flight based searches, while for smaller charge thesensitivity drops precipitously. Monojet style searches are an interesting avenue to explore,
but these perform best for heavier ψQ – where the cross section is even lower – or contact
interactions from a heavy mediator (which do not apply to our setup). For colored ψQ , thelarge production cross section pushes the current limits much higher, roughly 1.8 TeV for color
triplets fermions. The bounds increase with the size of the SU(3) representation and, at leastat the level of our study, are fairly insensitive to the hypercharge of ψQ .
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 d E/d x – to separate signal frombackground. We see the most variability in the bounds for color singlet, SU(2) charged ψQ ,as the signatures in the detector depend strongly on the charges and lifetimes of all the statesin the multiplet. If the excited states are short lived, they add to the cross section for thelowest |Q| state, but this boost can be insufficient to strongly bound the scenario if the lightest state has |Q| ≤ 1/3. Disappearing track searches, which target the decay of the excitedstate, can provide another handle, though we find they are hampered by the fact that excitedstate lifetimes for fractionally charged scenarios are typically shorter than in scenarios familiarfrom 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
15

SciPost Physics

541
542

Submission

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.
(1,1)1/6 (1,1)1/3(1,1)2/3
U(1)Y only(3,1)0(1,2)2/3

0 TeV(3,1)1/6(8,1)1/6 (6,1)0(1,2)1/6(1,3)2/3
0.5 TeV

SU(3)c charged

SU(2)L + U(1)Y charged
1.5 TeV

1 TeV

2 TeV

Figure 4: Graphic illustrating the mass bounds for the benchmark fractionally Diracfermions, the details of which are discussed in the text. The bounds for fractionallycharged complex scalars are lower than the fermionic case by ∼ 20%.
543
544
545
546
547
548
549
550
551
552
553

554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569

570
571
572

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 ∼ 1 − 2. More significantly, we have assumed that the triggering efficiency – either/ T – for fractionally charged particles with other (nonin the muon system efficiency or using E
hypercharged) quantum numbers (or much larger mass) is not significantly different than in
Ref. [20].
We conclude this section with some items worth thinking about in order to maintain arobust collider search program for fractionally charged particles.
• The LHC is an evolving apparatus, with many detector upgrades planned for the highluminosity 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 asingle output, so there will be less granular d E/d x information. Exactly how much
this impacts the analysis strategy for fractionally charged particles in Ref. [20] 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 [54–56].
– Reference [57] explored how the low d E/d x search could be improved, especially/ T trigger. Detector upgrades arefor low Q, by moving from a muon trigger to a E/ T events [58], which should help thisexpected to increase the efficiency for lower E
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
16

SciPost Physics

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 afunction of distance traversed [42, 43, 59].

573
574
575
576

• 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 [60].

577
578
579

• Some percentage of ψQ 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 massof ψQ , its charge, and its color representation. The stopped, stable ψQ may form atomic
or nuclear bound states which will have a fractional charge that cannot be screenedby 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 persistentelectric charge localized somewhere in the detector. If nothing else, it may be interesting
to attempt to search disused detector parts for embedded fractional charges.

580
581
582
583
584
585
586
587

588

Submission

5

Cosmology605

Not only would the discovery of a fractionally charged particle tell us an enormous amountabout 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 O(1) electric charges are present. Our understanding
thereof is mainly from the fantastic Dunsky, Hall, Harigaya papers [61, 62] 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 afractionally charged particle of mass Mψ gives an upper bound on the reheating temperature
Treheat ≲ Mψ /r. In Section 5.2 we give some basic estimates of r depending on both the detailsof reheating and the quantum numbers of ψ.
This means that just as such an energy frontier discovery would falsify some of our grandmodels of ultraviolet physics, it would also falsify the high-scale inflation models that havebeen proposed in these frameworks. Of course all we know experimentally is that there wasa Standard Model plasma in a radiation era at the temperatures of Big Bang Nucleosynthesis
Treheat ≳ TBBN , but there need not have been an era of much hotter temperature [50, 63–65].

606

5.1

589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604

607
608
609
610
611
612
613
614
615
616

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 fractionalcharges present in the sample are very strong, but extrapolating to a constraint on the relicabundance 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 [62, 66].
However, in fact there is a better source of constraints on the relic abundance from the fluxof fractionally charged particles on the Earth. In general, virialized dark matter which strongly

17

SciPost Physics

Submission

631

interacts with SM particles is unable to reach underground direct detection experiments thatare shielded by the Earth’s atmosphere and meters of rock (see e.g. [66–68]). However, theelectric charges of the states we are considering mean that there is necessarily a component
which gets boosted by supernova shocks, as impressively understood in [61]. Indeed for the
GeV - TeV mass range of interest at the energy frontier and the O(1) electric charges of ourstates, a relic abundance of such particles collapses into the Milky Way disk as it forms alongwith 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 theirlarge 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 [69], searches for lightly ionizing
particles like MAJORANA [70] and MACRO [71], and searches for magnetic monopoles like
ICRR [72] and Baksan [73]. 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.

632

5.2

617
618
619
620
621
622
623
624
625
626
627
628
629
630

633
634
635
636

Thermal Plasma Production

The bounds on the relic abundance can roughly be translated into an upper bound on thereheating temperature Treheat ≲ Mψ /r where Mψ is the mass of the lightest fractionally charged
particle. If we assume instantaneous reheating of all species with SM quantum numbers to atemperature Treheat ≪ Mψ , we get a Boltzmann suppressed equilibrium abundance of ψQ :nψ = gψ

637
638

639
640
641
642

M T
3/2
ψ reheat2πexp −Mψ /Treheat ,where gψ is the number of degrees of freedom of ψQ . This gives a relic abundance relative todark matter ofM n  
Ωψ
s0
ψ ψ
=
(5)
Ω DM
ρ DM
s∗
where s0 is the entropy today, s∗ is the entropy at Treheat and ρ DM is the average dark matter energy density. Given a bound on Ωψ /Ω DM , we can translate Eq. 5 into a bound on
r = Mψ /Treheat . If we impose Ωψ /Ω DM ≤ 10−16 , the most stringent bound in the parameter space of interest according to Ref. [61], this translates tor ∼ 65,

643
644
645
646
647
648
649
650
651
652
653

(4)

(6)

with only weak dependence on Mψ . If the nψ produced were large we should include theeffects of annihilations like for a standard freeze-out, as done in [74], but since the allowed
regime is so small we can ignore this process.
Above we assumed ψQ is instantaneously in equilibrium at Treheat . 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 Treheat (which may be more or less contrived
depending on the quantum numbers of ψ). In this case, an abundance of ψQ 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 ψ can be estimated using theresults of Ref. [75]. Specifically, if we assume a threshold cross section times relative velocity
ce f fof σS M S M →ψ̄Q ψQ v ∼ 16πM
2 , where c e f f is a combination of couplings and factors countingψ654

degrees of freedom (both initial and final), we findpΩψ
135 5/2 M pl ce f f e−2/r (2/r + 1) s0
∼
.
3/2
Ω DM
256 π7 g∗ ρ DM
18

(7)

SciPost Physics

Submission

672

For QCD production (assuming six SM fermion flavors and ignoring all SM masses) , ce f f ∼ 75,
while production of ψQ charged only under hypercharge has ce f f ∼ 0.1 Yψ2 . Plugging in numbers, the freeze-in case decreases r by O(15) relative to the case of directly reheating ψ, with
only some mild dependence on the value ce f f .
We note that the strong bound on Ωψ /Ω DM we have taken above may be loosened slightly
for certain quantum numbers of ψ. In particular, there do exist colored representations forwhich all hadrons formed with SM partons have fractional electric charge, but which also havebound states with zero electric charge, such as Q ∼ (3, X )0 where X = 1, 3, · · · . 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 [66, 76, 77]. However, there
is a severe danger posed by the existence of mixed bounds states such as (Qq̄) (for q̄ 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−1gives some suppression of the fractionally charged abundance, since H(ΛQC D ) ≪ ΛQC
D . Thenafter 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 ofmagnitude, ΩQq̄ ∼ 10−4 ΩQQQ . This leads to a less stringent restriction on Treheat /Mψ than in acase without electrically-neutral bound states by about O(10).

673

6

655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671

Global Structure of Gauge Theory

681

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 technicaldetail. 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 Liealgebra) but differ for large symmetry transformations (they have different Lie groups as theresult 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 [78, 79].

682

6.1

674
675
676
677
678
679
680

683
684
685
686

687
688
689
690

Abelian Warmup: R vs. U(1)Often in particle physics we are interested in continuous symmetry groups which have a notionof 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 theoryof electromagnetism.
As Groups When we consider a gauge field theory based on a symmetry group, the gaugebosons 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 thegenerator of U(1)EM gauge transformations, a global element of which we can represent as
U(θ ) = e iθQ ,

691
692
693

(8)

a circle’s worth of transformations which compose by complex multiplication U(θ )U(η) = e i(θ +η)Q
with θ , η ∈ [0, 2π). But alternatively we may view this as a mapping of θ ∈ R onto the unit
circle. Indeed, if we look nearby the identity transformation we cannot tell U(1) from R
U(θ ) ≃ 1 + iθQ,

19

(9)

SciPost Physics

694
695

696
697
698
699
700
701
702
703

Submission

where we have expanded for small θ . Then we could alternatively think about just defining
the group operation
U(θ )U(η) ≡ 1 + i(θ + η)Q.
(10)
This is a group which is not compact—θ has no finite period now; the group is just R equippedwith addition. While U(1) and R differ as Lie groups, they share the same Lie algebra.
Thinking in the other direction, if we had begun with R with the group operation of addition, we could see the relation to U(1) by considering the quotient group R/Z ≃ U(1). That is,
we may view U(1) as coming from an R group where we have imposed the additional equivalence relation θ ∼ θ + 2πZ—two elements of the group are now identified if they differ by
an integer (the factor of 2π is a normalization convention of the period). We diagram thisstructure in Figure 5, and of course this is exactly what the exponential map above does.

Figure 5: The group U(1) constructed by quotienting R/Z. We can think about thequotient projecting the real line down to the circle such that every integer maps tothe identity element.
704
705
706
707

708
709
710
711
712
713
714
715
716
717

Thinking about the physics, the perturbative, low-energy dynamics of the vector gaugebosons 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 dodepend 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
ψ(x) with charge q transforms under a U(θ ) transformation as ψ(x) → ψ(x) exp(iqθ ). If thegroup is R, then any charge q ∈ R is fine. But if the gauge group is U(1), then U(2π) ≡ 1, arotation 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

20

SciPost Physics

718
719
720
721
722
723
724
725
726

727
728
729
730

731
732
733

Submission

transforms to ψ(x) exp(2πqi). The requirement exp(2πqi) ≡ 1 implies that for a U(1) group
we must have q ∈ 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 meansthat by discovering particles with particular representations, you can learn about the global
structure. If you discover two particles ψ, χ with relatively irrational charges qψ /qχ ∈/ Q then
the gauge group must be 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 aqψ + bqχ fora, b ∈ Z.14
Magnetic Representations: Gauge theories may also allow representations which carry magnetic, rather than electric charge. In the low energy theory of electromagnetism, these are thefamiliar Dirac monopoles. Of course it is simple enough to postulate a monopole magnetic
field
g r̂
⃗=
B
,
(11)
4π r 2but in a quantum mechanical theory (where Aharonov-Bohm teaches us we really must talkabout the potential Aµ ) such configurations connect to rich, deep physics. See e.g. Preskill’sclassic [81] for an in-depth introduction.

Figure 6: The Dirac monopole as the limit where a semi-infinite solenoid becomesthe Dirac string.
734
735
736
737

The problem is that when we define the magnetic field in terms of the vector potential,⃗⃗ the absence of magnetic monopoles in the Maxwell equations follows necessarily,
B = ∇ × A,
⃗ = ∇ · (∇ × A)
⃗ ≡ 0 because the divergence of a curl is identically zero. In the relativistic∇·Btheory this is often referred to as the ‘Bianchi identity’, εµνρσ ∂ν Fρσ = 0.
14

We note for fun that this fact was used to intriguing effect in the ‘irrational axion’ of [80].

21SciPost Physics

738
739
740

Submission

As Dirac understood, their construction in the low-energy theory of the gauge field Aµ (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
ADirac (x) =

741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756

g
θ
tan φ̂,
4πr
2

(12)

in polar coordinates with φ the azimuthal angle and θ the polar angle. This indeed gives riseto the monopole magnetic field above, but this potential is singular from r = 0 out to all r
along the line θ = π. This is not a deficiency of Dirac; any function A(x) which produces thismagnetic 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 leadto 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.15 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 wouldbe Dirac string of a monopole with magnetic charge g, the charge picks up an Aharonov-Bohm
phase

 I
 ZZ

2
⃗
⃗
⃗
s = exp ie(∇ × A)d x = exp ie g,(13)
exp ie A · d⃗
γ

757
758

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
e g = 2πn,

759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774

n ∈ Z.

(14)

The smallest-charge monopole is found for n = ±1, and of course the most stringent requirement is from the electrically-charged particle with the least charge. That is, if qmin satisfies
Eqn 14, then so will every multiple of qmin , 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 thatmagnetic monopoles can be described in a manifestly singularity-free language by using someconcepts from topology [86]. In fact historically it is these ideas that have sparked theoreticalphysicists’ enduring fascination with topology in field theory, but let us try only to appreciatesome 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 Aµ (x) globally as a function for all x. In
topological language we must instead think of fields as sections of certain fiber bundles, but eleµmentarily we can imagine we must describe the gauge field using two functions AN/S(x) with anµoverlapping range of validity. Thinking in spherical coordinates, AN (x) is defined for polar anµ
gles θ ∈ [0, (π + δ)/2) and AS (x) is defined on the ‘southern hermisphere’ θ ∈ ((π − δ)/2, π]
where the small δ addition to the domains ensures that these two descriptions overlap on a
It is not clear to us who first discussed the Dirac string in this language, though Dirac’s paper [82] invites
this interpretation easily enough. We refer to Felsager [83] for one construction, [84] for some explicit formulae,
and [85] for an experiment at creating an approximate monopole in the lab by taking just such a limit.
15

22

SciPost Physics

775

Submission

small ring around the equator. They have the explicit expressionsg(1 − cos θ ) φ̂
4πr sin θ−gAS(x) =
(1 + cos θ ) φ̂.
4πr sin θAN (x) =

776
777
778
779

(15)
(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 ifthey agree up to a U(1) gauge transformation, which we can see asµµ

On overlap: AN (x) = AS (x) − ie−iα(x) ∂ µ e iα(x) ,α(x) = g

φ
k
2π

(17)

µ

Figure 7: The local descriptions AN /S (x) of the vector potential in their separate
patches, and the transition function on their overlap.

790

Then morally speaking the different monopole solutions are classified by the value of thisgauge transformation on a path around the equator U(φ) : φ → U(1) as φ = 0..2π with
U(0) = U(2π). In fact the collection of such paths is familiar in algebraic topology as the‘fundamental group’ π1 (G) of a space G. In the case of a U(1) group, π1 (G) = Z tells us that
there are magnetic monopoles labeled by any integer charge.
In contrast, in the case of an R gauge group there is no way to draw a closed path in R whichcannot be shrunk down to a single point, so π1 (G) = 1 is trivial and this group does not haveany magnetic monopoles. One may have intuited this already from the Dirac quantization
condition and the results above about electric representations. Since in an R gauge groupthe electric charge can be an arbitrarily small real number, the Dirac quantization cannot besatisfied for any magnetic charges.

791

6.2

780
781
782
783
784
785
786
787
788
789

792
793
794

Global Structure of Non-Abelian Groups

Case Study 1: SU(N ) vs. SU(N )/ZN Recall that the group SU(N ) consists of N ×N complex
matrices which are unitary (V † V = 1) and special (det V = 1). The structure of infinitesimaltransformations in SU(N ) is generated by traceless hermitian N × N matrices
U(θ a ) = 1ij + iθ a (T a )ij

795
796
797
798

(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’ ZN , where the center of a group is the subgroup of elements which commute with allothers,§ ª
2πk
ZN ⊂ SU(N ) : exp
i 1N
,
(19)
N
k=0..N −1
23

SciPost Physics

799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826

Submission

which is generated by the element ωN = exp 2π
N i 1N . We can sensibly form the quotientgroup SU(N )/ZN where we ‘mod out’ by the center subgroup. This group can be thoughtof as SU(N ) with the equivalence relation ωN ∼ 1N imposed. But this does not change thestructure 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 1N . Thinking now
about the representation theory, this means that such elements must necessarily act triviallyon each field.
If we think about the familiar SU(N ) representations, this is not the case for all of them.
Consider a field ψa in the fundamental representation of SU(N ), which transforms generally
as ψa → ψa Vab . Then in particular under an (ωN )ab transformation it picks up an N th root ofunity phase. In SU(N ) this is as it should be, but this is nonsensical for a representation of
SU(N )/ZN , in which this element was literally the identity—then the fundamental representation of SU(N ) is not an allowed representation of SU(N )/ZN !
The field theory of SU(N )/ZN is a theory of adjoint fields, including of course the gaugebosons 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 ⊗N̄ = (N 2 −1)⊕1.
With equal number of fundamental and antifundamental indices, Aac → (V † )cd Aac (V )ab is easily
seen to be invariant under a center transformation. The SU(N )/ZN theory allows arbitrary
matter which is in either the adjoint or irreps which can be built from it and the Levi-Civita
symbol ϵa1 ...an .
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 allowedmagnetic representations, which are controlled by the fundamental group π1 (G). This can bethought of elementarily as simply the group of topologically equivalent maps of circles intoG, π1 (G) ≃ {φ : S 1 → G}. The question is what sorts of closed loops we can draw in G. For
SU(N ) it is a fact that π1 (SU(N )) = 1 and there are no magnetic monopoles. But now let us
consider the following diagonal generator of SU(N )


1
1


2
,
T N −1 = 
(20)
.


..
−(N − 1)

827
828
829

which is a hermitian, traceless matrix you can think of as the generalization of the Pauli σ3 toSU(N ). Of course close to the identity we can think of an infinitesimal transformation in thisdirection θ a = δNa 2 −1 θ ,2

U(θ a ) = 1 + iθ T N −1 + O(θ 2 ),

830
831

(21)

just as in SU(N ). But now in SU(N )/ZN we will see something interesting if we go a large
distance in this direction, say θ = 2π/N . The higher order terms form into the exponential
2π N 2 −1
2πaU(θ ) = exp iT1(22)= exp i
N
N
2

832
833
834
835
836
837
838

and because −(N − 1) = 1 (mod) N we see that by following a path along the T N −1 direction
we have ended up at an element of the ZN center. In SU(N ) there’s nothing special to say
about this, but in SU(N )/ZN this means that you can go far out along this direction and end
up back at the origin! So now there is a map φ : [0, 2π) 7→ G where φ(θ ) = U(θ /N ) and thisgives us one-dimensional loops around SU(N )/ZN .
This means that in addition to the electric representations discussed above, SU(N )/ZN
also has magnetic representations. In this case there are not monopoles of any integer charge
24

SciPost Physics

839
840
841
842
843
844
845
846
847

848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884

Submission

as in π1 (U(1)) = Z but rather only N distinct closed loops π1 (SU(N )/ZN ) = ZN and so only
N distinct monopoles. If you wind N times around SU(N )/Z you end up with a path that canbe 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 π1 (SU(2)) = 1, and you will recall is only
locally isomorphic to the rotation group SO(3), while globally double-covering it. Then thequotient group SU(2)/Z2 ∼= SO(3) is isomorphic to 3D rotations and has π1 (SU(2)/Z2 ) = π1 (SO(3)) = Z2 .
The fact that looping N times around SU(N )/ZN returns you to the identity is nothing morethan ‘Dirac’s belt trick’—in 3D space taking the belt buckle on a loop in φ : [0, 2π) 7→ SO(3)puts it in a topologically twisted sector yet going around twice returns it to the identity.
Case Study 2: SU(N ) × 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 groupfactors. In fact U(N ) ∼= (SU(N ) × U(1)) /ZN differs in its global structure from SU(N ) × 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 ZN center subgroups of
2πi
the two factors, and identifies them with each other exp 2πi
N 1N ∼ exp N Q. This means that ev−2πi
ery field must be invariant under the diagonal combination of rotations, exp 2πi
N 1N ×exp N Q ≡ 1.
There is in general for SU(N ) representations a notion of ‘N -ality’ which simply tracks
how the field transforms under a ZN center transformation. A fundamental has N -ality of 1,
as we saw above, and in the SU(N )/ZN theory the representation theory required N -ality of
0 (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 (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 (mod) N which each SU(N ) representation must havein order to be an allowed representation of (SU(N ) × U(1)) /ZN . The two-index φ ab either
symmetric or anti-symmetric irrep comes from N ⊗ N = N (N − 1)/2 ⊕ N (N + 1)/2 so musthave U(1) charge 2 (mod N ). The adjoint φ ba is built from N ⊗ N̄ = N 2 − 1 ⊕ 1 so must haveU(1) charge 0 (mod N ), and so on.
Now what of the magnetic representations? Early physics work in this direction includes[10, 87–90], in which much further detail may be found. In SU(N ) × U(1) the two factors are
separate, and π1 (SU(N )) = 1 does not have monopoles while π1 (U(1)) = Z gives the simple
monopoles familar from the Abelian case above.
Turning to (SU(N ) × U(1)) /ZN , the structure is a bit subtle. The fundamental group
π1 (U(N )) = Z tells us we have distinct monopoles for any integer, but in this case the spectrumof monopoles is skewed away from just being the 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(φ) = exp(iφQ) and wrap around the U(1) direction to get amonopole with only U(1) magnetic flux.
However, now if we go a fraction of k/N around the circle,
the quotient
combined with
N 2 −1
our understanding of the SU(N )/ZN case above tells us exp i 2πk
T
∼
exp
i 2πk
N
N Q . Then
we can return to the origin not by continuing around the U(1) direction, but by taking a path12
along T N −1 in SU(N ) that when we get close to the origin looks like U(θ ) = 1 + iθ T N −1 .
So this case is something of a mixture of the two we have seen before. There are k ∈ Z
magnetic monopoles, but they now have both Abelian and non-Abelian magnetic fluxes fork ̸= 0 (mod N). It is only in the case k ∈ N Z for which they have U(1) magnetic flux only.

25

SciPost Physics

Submission

Qi

ui

di

Li

ei

H

SU(3)C

3

3

3

–

–

–

SU(2) L

2

–

–

2

–

2

U(1)Y

+1

−4

+2

−3

+6

−3

Table 3: Representations of the Standard Model fields under the subgroups of thegauge 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.

885
886
887
888
889
890
891
892

893
894
895
896
897
898
899
900

6.3

The Standard Models

The case of the Standard Model is not much more difficult than the above examples we havediscussed. 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 elementas(23)
U(θ1 , θ2i , θ3a ) = 1 + iθ1 Y + iθ2i T2i + iθ3a T3a ,where θ1,2,3 parametrize the transformations in the hypercharge, weak, and strong directions,and T3 , T2 , 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 θi 7→ θi (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 infact are not sensitive to the ‘global structure’ of the gauge group. This is what we see in thecovariant derivative to minimally couple charged particles to a gauge field
Dµ = ∂µ − i g1Q Y Bµ − i g2 TRα Wµα − i g3 TRa Gµa ,
2

901
902
903
904
905

3

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 whichhave this same Lie algebra.
The four different possibilities are
GSMn ≡ (SU(3)C × SU(2) L × U(1)Y ) /Zn ,

906
907
908
909
910
911
912
913
914

(24)

(25)

where n = 1, 2, 3, 6 and we use the slang term Z1 ≡ 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[22] but has been well-publicized in recent years [19]. The options with n > 1 can be viewed
as quotient groups of GSM1 where we quotient out certain diagonal center transformations asfollows.
In the group GSM2 = GSM1 /Z2 , we impose an equivalence relationship between the Z2
center subgroups of SU(2) L and of U(1)Y . That is, (−1)12 ∼ exp(iπY ), working now in thenormalization that the least-hypercharged particle has unit charge (see Table 3). In the group
GSM3 = GSM1 /Z3 , we impose an equivalence relationship between the Z3 center subgroups of26

SciPost Physics

915
916

Submission

SU(3)C and of U(1)Y . That is, exp(2πi/3)13 ∼ exp(i2πY /3). In the group GSM6 we impose
both of these quotients simultaneously.

Z3 ⊂ SU(3)C

e

Qi

ui

di

Li

ei

H

i2π/3

−i2π/3−i2π/3

1

1

1

e

e

Z2 ⊂ SU(2) L

−1

1

1

−1

1

−1

Z3 ⊂ U(1)Ye i2π/3e−i2π/3e−i2π/3

1

1

1

Z2 ⊂ 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.
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946

Of course we can always consider these as abstract quotient groups, as in the constructionsof the previous sections. But we have also observed the particles of the SM, which transformin 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.16 Indeed, it does work, as may be checked easily from thedata in Table 3. In the case of the Z2 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 Z2 center transformation. The Z3 subgroup may bechecked just as easily and the conclusion is the same, meaning that indeed there is a four-foldambiguity 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 Z6
quotient. Less stringently, Pati-Salam SU(4)C × SU(2) L × SU(2)R requires the Z3 quotient andtrinification SU(3)C × SU(3) L × SU(3)R needs the Z2 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 → (3, 1)+2 ⊕(1, 2)−3 ,and we see manifestly that these are invariant under the Z6 center. Since all irreps of SU(5)can be found in tensor products of 5 and 5̄, the embedding of the SM in SU(5) produces only
representations which are invariant under the Z6 . More formally, of course, one can find grouptheoretically that it really is SU(3) × SU(2) × U(1)/Z6 which is actually a subgroup of SU(5),as has been known since 1980 at latest [97].
From the above argument, it is clear that finding a representation which is charged under
the Z6 center falsifies the embedding into SU(5). More generally, discovering a particle withelectric charge e/6 (either at colliders or elsewhere) would rule out all the minimal unified
models of the universe.17 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 E6 .
16

One must additionally check that each of these versions of the Standard Model is free of global anomalies,
which is indeed true as discussed in [91–96].
17
Notably 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 becompatible 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. [98], but even in non-factorized cases suchas color-flavor unification [99].

27

SciPost Physics

Submission

971

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, thiswould effectively tell us that the U(1)Y circle is actually a factor of N ‘larger’ than we hadthought. Correspondingly the magnetic monopole charges are a factor of N larger as a resultof Dirac quantization. Recently [27] 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 RY , in which irrrational charges are allowed. At ageneric point in some constraint plot of fractionally charged particles, one can have this inmind as the alternate hypothesis that is being tested. It is true that we expect theories ofquantum gravity do not contain non-compact gauge groups like R (see e.g. [15, 100]), but it
is not obvious there is anything wrong with them strictly as quantum field theories. Flippedaround, 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 forexample using interferometry [101–103].
Finally we mention that these are not the only possible ambiguities in the gauge groupof the Standard Model. In [104] (App. B.1) we introduced the SM+ , in which the SM is
B−N L
extended by gauging ZN N c × ZNL , which is the Standard Model’s anomaly-free, generationc gg
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 infuture work [105].

972

7

947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970

973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990

Generalized Global Symmetries

As particle physicists we are often used to field theories at weak coupling, but it can be usefulas 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 termsof 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 [17] and to some reviews aimed to beaccessible for particle physicists [106–108].18 But we will eschew any topic whose introduction would require cohomology, as well as many interesting GGS possibilities broader than thebasics we require. Ideas and technology from GGS are gradually being utilized in (or towards)
particle physics applications, for example [23–26, 91, 92, 95, 96, 98, 99, 116–145].
18

We note also introductions and reviews a bit further afield such as [109–115].

28

SciPost Physics

991
992
993
994
995

Submission

Familiar (Zero-Form) Noether Charges A familiar symmetry which acts on local fields (sothe 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 µwhich obeys the conservation equation ∂µ J µ = 0. From this current we can build a family of
topological, unitary operators by exponentiating its integral over any three-manifold Σ3 , Z
Uα (Σ3 ) = exp iα

996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009


= exp iα

Σ3

Z
Σ4

1010

1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024

1025
1026
1027
1028

(26)

where εµνρσ J µ ≡ ⋆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 Σ3 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 forany Σ3 , and the conservation ∂µ J µ = 0 implies that any deformations of this surface do notchange the correlation functions of Uα (Σ3 ). Let us discuss further how to think about this,drawing from [106] among others.
We consider smoothly deforming Σ3 to Σ′3 , where for now we assume doing so does notcross any charged operators. That is, the spacetime volume
in between these is a four-manifoldSΣ4 bounded by these two three-surfaces, ∂ Σ4 = Σ3 Σ′3 , and Σ4 does not have any charged
operators in it. We compute the product of an SDO on Σ3 implementing a rotation by α andan SDO on Σ′3 implementing a rotation by −α using the generalized Stokes’ theorem ZZUα (Σ3 )U−α (Σ′3 ) = exp iα

1011

Σ3

J µ εµνρσ d x ν d x ρ d x σ ,J µ εµνρσ d x ν d x ρ d x σ − iαµ 4∂µ J d xΣ′3

J µ εµνρσ d x ν d x ρ d x σ

(27)


= 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 Uα (Σ3 ) ≃ Uα (Σ′3 ).
That is correlation functions containing an insertion of Uα (Σ3 ) are invariant under deforming
Σ3 , so the SDOs are topological as we said above.
Now, the above equations assumed that there are no charged particle in the volume Σ4
between the initial and final surfaces. How do the SDOs behave when we move the surface
Σ3 past a local field ψ( 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∂µ J µ (x) = 0 to an operator equation which tells us what to do with a charged field ψ( 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 forthe charged operators—for example in Section 14.8 of Schwartz [146]. In the Abelian case
we have simply∂µ J µ (x)ψ( y) = δ(4) (x − y)qψ ψ( y).
(29)This tells us that while ∂µ J µ (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,〈. . . ∂µ J µ (x)ψ( y) . . . 〉 = δ(4) (x − y)qψ 〈. . . ψ( y) . . . 〉,
29

(30)

SciPost Physics

1029
1030
1031
1032
1033
1034
1035

Submission

where the ‘. . . ’ is a stand-in for any other operators away from x, y. The action of the Wardidentity will be crucial in understanding the use of the symmetry defect operators.
Now let us repeat the computation above of deforming Σ3 to Σ′3 but now in the case where
doing so does cross a charged operator. A simple case has Σ3 as a hypersphere S 3 and the local
operator ψ(x) at a point x which is inside Σ3 . We consider then shrinking Σ3 down Σ3 → Σ′3
so x is now outside of this surface, as in Figure 8, and then acting with the inverse SDO. Overall
this acts on ψ(x) as ZUα (Σ3 )ψ(x)U−α (Σ′3 ) = exp iα

1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048

Σ4

∂µ J µ d 4 x ψ(x) = ψ(x)e iαqψ .

(31)Where we have used the Ward identity and the fact that x ∈ Σ4 , and we refer to [106] forfurther detail. We note also that if no other charged operators were in Σ3 to begin with, thenconceptually we can skip this second step of acting with U−α (Σ′3 ) and just imagine shrinking
Σ3 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’, andwriting Uα (Σ3 )ψ(x)U−α (Σ′3 ) = ψ(x)e iαqψ Link(Σ3 ,x) . In the situation we have described, the‘Linking number’ Link(Σ3 , x)) = 1. The ‘Linking number’ is a topological invariant of a configuration in d spacetime dimensions between a p-dim submanifold Σ p and a d − p − 1-dim
submanifold Σd−p−1 . This action by linking keeps track of the charge inside the SDO when wemove 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 tovisualize 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. 19

Figure 8: A local operator ψ(x) charged under a U(1) zero-form symmetry and theaction of a symmetry defect operator Uα (Σ3 ) on it by linking as described in the text.
One dimension is suppressed.

1049
1050
1051
1052

Discrete Symmetries We note also that a useful aspect of this formalism is a unified language for both continuous and discrete symmetries. A discrete ZN symmetry doesn’t havean 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 the19
We note for fun that general linking numbers can be defined by certain topological quantum field theories[147].

30

SciPost Physics

1053

Submission

expected properties when they act on charged operators,

2πk
U 2πk (Σ3 )ψ(x) = ψ(x) exp iqψ Link(Σ3 , x) .
N
N

(32)

1064

This suffices as a definition in the case of a discrete symmetry by describing how U(Σ3 ) behavesin 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 Σ3 of some local operator.
Sometimes we are thinking about a ZN subgroup of what is (or began as) a U(1) symmetry,
and we can realize Uα (Σ) as an integral over a current with the angle restricted to ZN . Thisis effectively an operator which measures a global charge (mod N ), and will be the relevantcase for us below with the electric one-form symmetry of electromagnetism.
In other cases when the symmetry is really intrinsically ZN , it is sometimes useful to introduce an auxiliary U(1)-valued field and then project out its dynamics. This becomes aninvaluable technique when one wants to understand discrete gauge theories, and we referto [106] for an expansive discussion of this topic.

1065

7.1

1054
1055
1056
1057
1058
1059
1060
1061
1062
1063

1066
1067
1068
1069
1070
1071
1072

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 γ through spacetime which is either a closed loop
or an infinite line.20 Physically a Wilson loop can be seen as the effect of a massive particleof charge q traversing the path γ, 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 asZ
Wq (γ) ≡ exp iq

γ
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088

Aµ d x µ .

(33)

In the general non-Abelian case the Wilson loops
R are instead labeled by a representation
over which we take the trace WR (γ) ≡ Tr exp i γ Aaµ TRa d x µ . The ’t Hooft loops are defined
analogously for magnetic representations but with the electromagnetic dual vector potential
A 7→ Ã.21
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 electricfield over some closedR 2-dimensional spatial manifold Σ2 and finding some notion of an en⃗ But we can more clearly and more covariantly think about thisclosed charge Q encl = Σ E⃗ ·d A.
2
by recognizing the generalized symmetry structure behind Gauss’ law: The Gaussian surface
computes a Noether charge for a one-form symmetry!
Pure electromagnetism in fact has both an electric one-form symmetry and a magnetic oneform symmetry. The photon equation of motion and the Bianchi identity reveal the conserved
two-index currents which generate these one-form symmetries,∂µ F µν = 0,∂µ εµνρσ Fρσ = 0.

20

(34)

Which are closed loops on the one-point compactification of spacetime.
For completeness we recall that the dual potential is related to the vector potential in the following nonlocal
way. The field strength is Fµν = ∂µ Aν − ∂ν Aµ , and its Hodge dual is F̃µν = ϵµνρσ F ρσ . This dual field strength isrelated to the dual potential as F̃µν = ∂µ Ãν − ∂ν Ãµ .
21

31

SciPost Physics

Submission

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 Uα (Σ2 ) acting
on Wilson lines charged under a global one-form symmetry.

1089
1090

The familiar Gaussian surface can in fact be covariantly upgraded and exponentiated to realize
SDOs supported on any two-dimensional surface Σ2 ZUα (Σ2 ) = exp iα

1091
1092

1093
1094
1095
1096
1097

Σ2

εµνρσ Fρσ dSµν .

(35)

The SDOs are topological except when they cross Wilson lines and their correlation functions
are controlled by
Uα (Σ2 )Wq (γ) = Wq (γ) exp (iαqLink(Σ2 , γ)) .
(36)
This is just the analogue of what we observed above for zero-form symmetries. Now we can talkabout the allowed representations in terms of the electric one-form symmetries of the Wilsonlines of the theory. Analogously to the argument in terms of gauge transformations, if theelectric one-form symmetry is compact (U(1) or a ZN subgroup) then there is a transformation
by α = 2π which should act as the identity
U2π (Σ2 )Wq (γ) ≡ Wq (γ)

(37)

1101

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

1102

7.2

1098
1099
1100

1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113

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 zerodimensional 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 L+= (H̃ L)(H̃ L)/Λ 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 withadditional 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
32

SciPost Physics

1114
1115
1116
1117
1118
1119
1120

Submission

as dynamical objects in a more-fundamental theory. For example, when you see that (someof) the Wilson lines of electromagnetism are in fact in our universe completed into dynamicalcharged particles like electrons and protons.
A useful qualitative picture to have of this breaking is of the ‘endability’ of the Wilsonlines [148, 149]. For simplicity we consider an Abelian gauge symmetry where the Wilsonlines are labeled by a charge, but the translation to general representations of non-Abelian
symmetries is obvious. Consider an ‘open’ Wilson line Z y Wq (γ; x, y) = exp iq

A ,

(38)

x

Aµ → Aµ + ∂µ λ
1121
1122
1123
1124
1125

⇒

Wq (γ; x, y) → e iqλ( y) Wq (γ; x, y)e−iqλ(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(Σ2 ) to have topological correlation
functions with the Wilson lines—if Σ2 is linked with γ, 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 onΣ2 cannot have non-trivial topological correlation functions any longer when it cansmoothly ‘slide off’ the Wilson line.
1126

When we have access to the electron, we can write a gauge-invariant, bilocal line operatorψ̄( y)Wq (γ; x, y)ψ(x),

1127
1128
1129
1130
1131
1132
1133
1134

(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 Σ2 can be smoothly deformed up or down the Wilson line and ‘slide off’ theend, where it can be shrunk to a point. Then the correlation functions of Σ2 cannot any longer
be topological and depend only on data like Link(Σ2 , γ) because this topological linking is no33

SciPost Physics

Submission

1140

longer well-defined. So the appearance of the dynamical ψ field means that any one-form
symmetry under which Wq (γ) is charged must necessarily be broken. Of course this holds truealso for a Wilson line of charge nq, n ∈ Z, which can end on n of these charged fields. But if
the charge q of ψ 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.22 We now discuss thispossibility in more detail, specializing to QED.

1141

7.3

1135
1136
1137
1138
1139

1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160

1161
1162
1163
1164
1165
1166
1167

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 existenceof 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 forAppendix B.
At energies far below the electron mass E ≪ me , 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 θ ∈ [0, 2π). This is responsible for Gauss’ law.
When we approach energies of order the electron mass E ≳ me , the continuous electric oneform symmetry is necessarily broken. In terms of our Gaussian surface SDOs, the statement
is that for general θ , these surfaces will no longer be topological. As shown in [150] we caninterpret this violation of the topological invariance of the Gaussian surface as the electriccharge being ‘screened’ and relate it to the running of the fine-structure constant α. And
indeed we have long appreciated that at these high energies, charges are screened by virtualelectron-positron loops. The Uehling potential [151] 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,e−2me r2
2 log me rq(r ≫ me ) = q 1 + e p+ ... ,q(r ≪ me ) = q 1 − e+ . . . , (41)6π264π3 me rwhere we have given the asymptotic forms. Indeed at energies below the electron mass theelectric one-form symmetry becomes good exponentially rapidly dq(r)/d r ≈ 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, andUθ (Σ2 )Wγ (q) = Wγ (q) exp (iθ qLink(Σ2 , γ)) .

1168
1169
1170
1171
1172
1173
1174

(42)

Clearly U2π (Σ2 ) acts trivially on the electron, and on every particle with charge a multipleof 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 theelectron charge. Correspondingly, some Uθ (Σ2 ) 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 > me . The SM gauge group with the quotient Zn has discrete electric one-form symmetry
Z6/n above the electron’s mass.
22

The case of an 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 [149].

34

SciPost Physics

1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185

Submission

If instead there is no remaining electric one-form symmetry above the electron’s mass, asin the case where the SM is embedded in SU(5) in the UV, then every Wilson line has q ∈ Z. So
if we consider θ = 2π then the Gaussian surface will act trivially on any operator, and there
are no nontrivial Uθ (Σ2 ) 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 RY . In any of these cases, the signaturein 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

1186

By discovering a particle with fractional electric charge qψ and mass mψ we learn the SM has
an emergent electric one-form symmetry at E ≪ mψ . If qψ = n/N (in units of the electron
(1)

1187

charge e) with gcd(n, N ) = 1 then the SM has emergent ZN electric one-form symmetry. The
unbroken one-form symmetry is measured by the Gaussian surfacesZ
Uk (Σ2 ) = exp i2πk

F ,

(43)

Σ2

1188
1189
1190
1191
1192

with θ = 2πk, k = 1..N . And in the case where qψ ∈/ Q then the one-form symmetry is Z(1) ,and each k ∈ Z makes for a distinct Gaussian surface.
The fact that these Gaussian surfaces remain topological continues to mean that thesefractional charges cannot be screened by matter at lower energies. That is, if we surrounda 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.

1200

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 infraredis always U(1)(1) , the magnetic one-form symmetry group is also U(1)(1) . But the existenceof a discrete electric one-form symmetry above the electron mass controls how the charge ofthe ’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 = 2π/e. With ZN
worth of electric one-form symmetry, the quantum of magnetic flux is instead g = 2πN /e.

1201

8

1193
1194
1195
1196
1197
1198
1199

1202
1203
1204
1205
1206
1207
1208
1209
1210
1211

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 of the Standard Model as a quotient group, and correspondingly
their discovery would probe nonperturbative aspects of SM physics which could rule out minimal unification schemes from the infrared. More generally, the language of Generalized Global
Symmetries provides an interpretation of the existence of heavy, fractionally-charged states interms of an emergent symmetry possessed by the observed Standard Model.
On the empirical front, we have reinterpreted various LHC searches to derive energy frontier constraints on fractionally-charged particles for a variety of Standard Model representations. In some cases they possess signatures which are well-covered by existing searches

35

SciPost Physics

Submission

1216(modulo subtleties in particle-detector interactions which we have ignored and deserve further attention), but in other cases the constraints on these exotic, electrically-charged particles
from energy frontier searches are weak or nonexistent. Further exploration of possible experimental strategies is clearly warranted to ensure a robust observational program for thesestriking new particles which could teach us an enormous amount about the universe.

1217

Acknowledgements

1212
1213
1214
1215

1218
1219
1220

We thank Clay Córdova, Antonio Delgado, and Sungwoo Hong for valuable conversations andSteven Lowette for helpful comments on searches for fractionally charged particles. We are
grateful to Sungwoo Hong for comments on a draft of this manuscript.

1222

Funding information The work of SK and AM was supported in part by the National Science
Foundation under Grant Number PHY2112540.

1223

A

1221

1224
1225
1226
1227
1228

Fractionally Charged Particle Partonic Cross Sections

In this appendix we summarize the partonic cross sections for ψQ pair production. The expressions are organized by the spin of ψQ and whether or not ψQ is charged under SU(3).
We begin with color singlets. For a fermionic ψQ with charge Q ψ = (τ3 )ψ +Y , where (τ3 )ψ
is the eigenvalue of the third generator of SU(2) appropriate for ψQ ’s SU(2) representation,
we find:


4
4 2 2
2
2
2
dimψ  8e Q q Q ψ 2Mψ + 2Mψ (ŝ − t̂ − û) + t̂ + û
d σ̂ EW (q̄q → ψ̄Q ψQ )(44)=
192 πŝ2
ŝ2
d t̂

2
2
  2
  2

2
4g Z4 2Mψŝ x L x R q2L + qR2 + Mψ− t̂
q2L x R2 + qR2 x 2L + Mψ− û
q2L x 2L + qR2 x R2+
2
Γ Z2 m2Z + m2Z − ŝ
8 e2 g Z2 Q q Q ψ m2Z − ŝ
4
2


−
Mψ
(q L + qR )(x L + x R ) + Mψ(ŝ(q L + qR )(x L + x R )ŝ m4Z + m2Z Γ Z2 − 2 ŝ + ŝ2− 2 t̂ (q L x R + qR x L ) − 2 û (q L x L + qR x R )) + t̂ 2 (q L x R + qR x L ) + û2 (q L x L + qR x R )1229

d σ̂ EW (q̄q′ → ψ̄Q ψQ′ ′ )
d t̂

=

4
2
2
2
dimψ e4 (I(I + 1) − i3 (i3 ± 1))  t̂ + û + 2 Mψ (ŝ − t̂ − û) + 2 Mψ 

192 πŝ2 sin4 θW

2(m2W − ŝ)2 + ΓW
m2W

(45)
1230
1231
1232
1233
1234
1235
1236
1237
1238

Here ŝ, t̂, û are the partonic Mandelstam variables, g Z = e/ cos θW , q L , qR = τ3 −Q q sin2 θW andx L , x R = (τψ )3 − Q ψ sin2 θW factors for ψQ . The quark factors Q q , q L , qR depend on whether
up-type or down-type quarks initiate the collision, while Q ψ , x L , x R depend on which SU(2)representation and hypercharge ψQ carries. If ψQ is an SU(2) singlet, x L , x R ∝ Q ψ so theentire partonic cross section scales as Q2ψ . Note ψQ must have vectorial charge assignment,
meaning x L = x R . The factor of dimψ is the size of ψQ ’s SU(3) representation, should wewant to know the electroweak production in that case; dimψ = 1 when ψQ is a color singlet.
The second expression, σ̂ EW (q̄q′ → ψ̄Q ψQ′ ′ ), shows the charged current production crosssection for ψQ in a SU(2) multiplet of size I(I + 1). For production via W + , i3 = (τψ )3 for36

SciPost Physics

1239
1240
1241

Submission

the lower charge state within the ψ multiplet and we take the + sign, i3 (i3 + 1), while for W −production we take the minus sign and i3 = (τψ )3 for the higher charge ψ state.
Keeping the representation the same but switching to scalar ψQ , the expressions become:


4 2 2
2
2
2
dimψ  2e Q q Q ψ ŝ − ( t̂ − û) − 4 Mψ ŝd σ̂ EW (q̄q → ψ̄Q ψQ )=+2
192 π ŝ2
ŝ
d t̂


2
g Z4 x 2L (q2L + qR2 ) ŝ2 − ( t̂ − û)2 − 4 Mψ
ŝ
+
(46)
Γ Z2 m2Z + (m2Z − ŝ)2
22 g Z2 e2 Q q Q ψ x 2L (q L + qR )(m2Z − ŝ) ŝ2 − ( t̂ − û)2 − 4 Mψŝ−ŝ(Γ Z2 m2Z + (m2Z − ŝ)2 )1242

d σ̂ EW (q̄q′ → ψ̄Q ψQ′ ′ )
d t̂
1243
1244

1245
1246
1247

=



2
2
2
dimψ e4 (I(I + 1) − i3 (i3 ± 1)) ŝ − ( t̂ − û) − 4 Mψ ŝ
2(m2W − ŝ)2 + ΓW
m2W

768 π ŝ2 sin4 θW(47)

If ψ carries SU(3) quantum numbers, QCD production g g → ψ̄Q ψQ , q̄q → ψ̄Q ψQ becomes
the dominant mechanism. For fermionic ψ at leading order, we have


4
6
2
d σ̂(g g → ψ̄Q ψQ ) π α2s C2 (ψ) ¦ 18 2Mψ − 3Mψ ( t̂ + û) + 6Mψ t̂ û − t̂ û( t̂ + û)



=
−
(48)
2
2
64 ŝ2d t̂
ŝ Mψ− t̂ Mψ
− û





2
2
2
2
 9C2 (ψ) Mψ
2Mψ
− t̂ Mψ(3 − 2C2 (ψ)) 4Mψ
− û
− ŝ



+ dimψ
+
−
2
2
ŝ2
Mψ
− t̂ Mψ
− û




4
4
2
2
2C2 (ψ) Mψ
+ Mψ
( t̂ + 3û) − t̂ û ©
2C2 (ψ) Mψ
+ Mψ
(3 t̂ + û) − t̂ û
−

2

2
2
2
Mψ
− t̂
Mψ
− û


4
2
2
2
2
d σ̂QC D (q̄q → ψ̄Q ψQ ) π αs dimψ C2 (ψ) 2Mψ + 2Mψ (ŝ − t̂ − û) + t̂ + û
=
.
(49)
9 ŝ4
d t̂
Here dimψ is the size of the ψ SU(3) representation, C2 (ψ) is the appropriate quadratic
Casimir, and we have used dimG C(ψ) = dimψ C2 (ψ) to remove all instances of the index
C(ψ) and clean up the formulae. For scalar ψQ , the analogous expressions are:


2
2
2
2
d σ̂QC D (q̄q → ψ̄Q ψQ ) π αs dimψ C2 (ψ) ŝ − ( t̂ − û) − 4 Mψ ŝ
=.
(50)
36 ŝ4d t̂

1248

d σ̂(g g → ψ̄Q ψQ )
d t̂

=

¦

× C2 (ψ)

1249

1250
1251
1252

B

π α2s dimψ C2 (ψ)
128 ŝ212ŝ2 (Mψ− t̂)2+4 2
68( t̂ 2 û2 + Mψ( t̂ + û2 ) − 4 Mψ( t̂ + û) + 5 Mψ)12ŝ2 (Mψ− û)2+2(C2 (ψ) − 1)22ŝ2 (Mψ− t̂)(Mψ
− û)

©

(51)

Electroweak Phase One-Form Symmetry

We have focused on the electric one-form symmetry in the U(1)QED phase of the SM, but let
us turn briefly to the TeV-scale phase, noting that a more technical discussion may be foundin [23].
37

SciPost Physics

Submission

1281

An electric one-form symmetry in the far IR matches on to some electric one-form symmetryof the SM, so the general statement is that there are some Wilson lines which are not endable
by the SM matter. The one-form symmetry has rank 1, so we need only one new Wilson line
to generate any that is allowed but not realized by the SM matter. We may think of Wilsonlines as fusing via the composition of representations.
Then we can always for simplicity choose an SU(3) × SU(2) singlet representation withsome hypercharge. In the cases of the ‘global structure’ we can think of these as Wilson lines in
the representation R = (1, 1, q) with q = 1, 2, 3 for Z6/q electric one-form symmetry. More generally, sticking with this normalization where the left-handed quark doublet has hyperchargeq = 1, some q = k/N where gcd(k, N ) = 1 has ZN electric one-form symmetry and q ∈/ Q hasZ.
By combining these and Wilson lines in the knoown SM representations one can build thecolored or weakly charged representations that give rise to fractionally charged particles aswell. However, it is a more subtle task to write down the symmetry defect operators as theintegral of some sort of current, since the centers of SU(3)C , SU(2) L are intrinsically discrete.
But we know these two-dimensional SDOs measure certain combinations of the non-Abeliancenter symmetry fluxes and the hypercharge flux. The SM fields do not carry these combinations of charges and so these SDOs act trivially upon them.
In general such operators are known as Gukov-Witten operators [152, 153]. For detailed
calculations involved the generalized symmetries it may be useful to introduce auxiliary fields
to write the SDOs in a local-looking form, but this goes beyond our remit. For this purpose
one would likely wish to begin with the SU(3)C × SU(2) L × U(1)Y theory and view the extra
ZN electric one-form symmetry as deriving from gauging the ZN discrete magnetic symmetryof this theory.
The magnetic one-form symmetry of the Standard Model remains group-theoretically U(1)no matter the choice of global structure, but the hypermagnetic monopoles may possess also
discrete color- and weak- magnetic fluxes in the case where the global structure is non-trivial.
We refer to [19,96] for further detail. Note if we have RY there are no magnetic representations
at all, so no magnetic one-form symmetry.

1282

References

1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280

1283
1284
1285

1286
1287

1288
1289
1290

1291
1292
1293

1294
1295

[1] E. Witten, Quest for unification, In 10th International Conference on Supersymmetry
and Unification of Fundamental Interactions (SUSY02), pp. 604–610 (2002), hep-ph/
0207124.
[2] L. F. Li and F. Wilczek, Price of Fractionally Charged Particles in a Unified Model, Phys.
Lett. B 107, 64 (1981), doi:10.1016/0370-2693(81)91148-5.
[3] H. Goldberg, T. W. Kephart and M. T. Vaughn, Fractionally Charged Color Singlet Fermions in a Grand Unified Theory,
Phys. Rev. Lett. 47, 1429 (1981),
doi:10.1103/PhysRevLett.47.1429.
[4] H. Goldberg, Fractionally Charged Heavy Leptons: Cosmological Implications of Their
Existence, and a Prediction of Their Abundance, Phys. Rev. Lett. 48, 1518 (1982),
doi:10.1103/PhysRevLett.48.1518.
[5] D. K. Hong, J. Kim, J. E. Kim and K. S. Soh, Fractionally Charged Lepton And Low Mass
Magnetic Monopole, Phys. Rev. D 27, 1651 (1983), doi:10.1103/PhysRevD.27.1651.

38

SciPost Physics

1296
1297
1298

1299
1300

1301
1302

1303
1304

1305
1306

1307
1308
1309

1310
1311
1312

1313
1314

1315
1316

1317
1318

1319
1320

1321
1322

1323
1324

1325
1326

1327
1328

1329
1330
1331

1332
1333

1334
1335

Submission[6] F.-X. Dong, T.-S. Tu, P.-Y. Xue and X.-J. Zhou, SU(8) GUT Model With Fractionally Charged
Color Singlet Fermions And Low Mass Magnetic Monopoles, Phys. Lett. B 119, 121 (1982),
doi:10.1016/0370-2693(82)90258-1.
[7] K. Yamamoto, Fractionally Charged Leptons in the SO(14) GUT, Phys. Lett. B 120, 157(1983), doi:10.1016/0370-2693(83)90643-3.
[8] P. H. Frampton and T. W. Kephart, Fractionally Charged Particles as Evidence for Supersymmetry, Phys. Rev. Lett. 49, 1310 (1982), doi:10.1103/PhysRevLett.49.1310.
[9] K. Kang and I.-G. Koh, Hypercharge Generators in SU(7) Grand Unification Models, Phys.
Rev. D 25, 1700 (1982), doi:10.1103/PhysRevD.25.1700.
[10] X.-G. Wen and E. Witten, Electric and Magnetic Charges in Superstring Models, Nucl.
Phys. B 261, 651 (1985), doi:10.1016/0550-3213(85)90592-9.
[11] G. G. Athanasiu, J. J. Atick, M. Dine and W. Fischler, Remarks on Wilson Lines, Modular
Invariance and Possible String Relics in Calabi-yau Compactifications, Phys. Lett. B 214,55 (1988), doi:10.1016/0370-2693(88)90451-0.
[12] I. Antoniadis, J. R. Ellis, J. S. Hagelin and D. V. Nanopoulos, The Flipped SU(5)x U(1) String Model Revamped, Phys. Lett. B 231, 65 (1989), doi:10.1016/03702693(89)90115-9.
[13] A. N. Schellekens, Electric Charge Quantization in String Theory, Phys. Lett. B 237, 363(1990), doi:10.1016/0370-2693(90)91190-M.
[14] N. Seiberg, Modifying the Sum Over Topological Sectors and Constraints on Supergravity,
JHEP 07, 070 (2010), doi:10.1007/JHEP07(2010)070, 1005.0002.
[15] T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev.
D 83, 084019 (2011), doi:10.1103/PhysRevD.83.084019, 1011.5120.
[16] O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional
gauge theories, JHEP 08, 115 (2013), doi:10.1007/JHEP08(2013)115, 1305.0318.
[17] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP
02, 172 (2015), doi:10.1007/JHEP02(2015)172, 1412.5148.
[18] A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04, 001(2014), doi:10.1007/JHEP04(2014)001, 1401.0740.
[19] D. Tong,
Line Operators in the Standard Model,
doi:10.1007/JHEP07(2017)104, 1705.01853.

JHEP 07, 104 (2017),[20] A. Hayrapetyan et al., Search for fractionally charged particles in proton-proton collisionsp
at s = 13 TeV (2024), 2402.09932.
[21] M. Aaboud et al., Search for heavy charged long-lived particles in the ATLAS detector inp
36.1 fb−1 of proton-proton collision data at s = 13 TeV, Phys. Rev. D 99(9), 092007(2019), doi:10.1103/PhysRevD.99.092007, 1902.01636.
[22] J. Hucks, Global structure of the standard model, anomalies, and charge quantization,
Phys. Rev. D 43, 2709 (1991), doi:10.1103/PhysRevD.43.2709.
[23] M. M. Anber and E. Poppitz, Nonperturbative effects in the Standard Model with gauged
1-form symmetry, JHEP 12, 055 (2021), doi:10.1007/JHEP12(2021)055, 2110.02981.
39

SciPost Physics

1336
1337

1338
1339
1340

1341
1342

1343
1344

1345
1346

1347
1348
1349
1350

1351
1352
1353

1354
1355
1356

1357
1358
1359

1360
1361
1362

1363
1364

1365
1366
1367

1368
1369
1370

1371
1372

1373
1374

1375
1376
1377

Submission[24] M. Reece, Axion-gauge coupling quantization with a twist, JHEP 10, 116 (2023),
doi:10.1007/JHEP10(2023)116, 2309.03939.
[25] Y. Choi, M. Forslund, H. T. Lam and S.-H. Shao, Quantization of Axion-Gauge Couplings and Noninvertible Higher Symmetries, Phys. Rev. Lett. 132(12), 121601 (2024),
doi:10.1103/PhysRevLett.132.121601, 2309.03937.
[26] C. Cordova, S. Hong and L.-T. Wang, Axion Domain Walls, Small Instantons, and NonInvertible Symmetry Breaking (2023), 2309.05636.
[27] R. Alonso, D. Dimakou and M. West, Fractional-charge hadrons and leptons to tell theStandard Model group apart (2024), 2404.03438.
[28] H.-L. Li and L.-X. Xu, The Standard Model Gauge Group, SMEFT, and Generalized Symmetries (2024), 2404.04229.
[29] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao,T. Stelzer, P. Torrielli and M. Zaro, The automated computation of tree-level and next-toleading order differential cross sections, and their matching to parton shower simulations,
JHEP 07, 079 (2014), doi:10.1007/JHEP07(2014)079, 1405.0301.
[30] R. D. Ball, V. Bertone, S. Carrazza, C. S. Deans, L. Del Debbio, S. Forte, A. Guffanti, N. P.
Hartland, J. I. Latorre and et al., Parton distributions for the lhc run ii, Journal of HighEnergy Physics 2015(4) (2015), doi:10.1007/jhep04(2015)040.
[31] N. P. Hartland and E. R. Nocera,
Nuclear Physics B - Proceedings
doi:10.1016/j.nuclphysbps.2012.11.013.

A mathematica interface to nnpdfs,
Supplements 234,
54–57 (2013),[32] V. Khachatryan et al., Search for long-lived charged particles in proton-proton collisions atp
s = 13 TeV, Phys. Rev. D 94(11), 112004 (2016), doi:10.1103/PhysRevD.94.112004,
1609.08382.
[33] G. Aad et al., Search for heavy long-lived multi-charged particles in the full LHC Run 2 pp
collision data at s=13 TeV using the ATLAS detector, Phys. Lett. B 847, 138316 (2023),
doi:10.1016/j.physletb.2023.138316, 2303.13613.
[34] R. L. Workman et al., Review of Particle Physics,
doi:10.1093/ptep/ptac097.

PTEP 2022, 083C01 (2022),[35] G. Aad et al., Search for new phenomena in events with an energetic jet and missingptransverse momentum in pp collisions at s =13 TeV with the ATLAS detector, Phys. Rev.
D 103(11), 112006 (2021), doi:10.1103/PhysRevD.103.112006, 2102.10874.
[36] A. Hayrapetyan et al., Search for supersymmetry in final states with disappearing
tracks in proton-proton collisions at s=13 TeV, Phys. Rev. D 109(7), 072007 (2024),
doi:10.1103/PhysRevD.109.072007, 2309.16823.
[37] B. Andersson, G. Gustafson, G. Ingelman and T. Sjostrand, Parton Fragmentation andString Dynamics, Phys. Rept. 97, 31 (1983), doi:10.1016/0370-1573(83)90080-7.
[38] T. Sjostrand, S. Mrenna and P. Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05,
026 (2006), doi:10.1088/1126-6708/2006/05/026, hep-ph/0603175.
[39] M. Fairbairn, A. C. Kraan, D. A. Milstead, T. Sjostrand, P. Z. Skands and
T. Sloan,
Stable Massive Particles at Colliders,Phys. Rept. 438, 1 (2007),
doi:10.1016/j.physrep.2006.10.002, hep-ph/0611040.
40

SciPost Physics

1378
1379

1380
1381

1382
1383

1384
1385
1386

1387
1388

1389
1390
1391

1392
1393
1394

1395
1396
1397

1398
1399

1400
1401
1402

1403
1404

1405
1406

1407
1408

1409
1410

1411
1412

1413
1414
1415

1416
1417
1418

Submission[40] P. Abreu et al., Tuning and test of fragmentation models based on identified particles andprecision event shape data, Z. Phys. C 73, 11 (1996), doi:10.1007/s002880050295.
[41] G. Alexander et al., A Comparison of b and (uds) quark jets to gluon jets, Z. Phys. C 69,
543 (1996), doi:10.1007/s002880050059.
[42] M. Adersberger, Searches for Heavy Charged Long-Lived Particles with the ATLAS detector,
Ph.D. thesis, Munich U., doi:10.5282/edoc.23448 (2018).
[43] Y. R. de Boer, A. B. Kaidalov, D. A. Milstead and O. I. Piskounova, Interactions of Heavy
Hadrons using Regge Phenomenology and the Quark Gluon String Model, J. Phys. G 35,
075009 (2008), doi:10.1088/0954-3899/35/7/075009, 0710.3930.
[44] M. Cirelli, N. Fornengo and A. Strumia, Minimal dark matter, Nucl. Phys. B 753, 178(2006), doi:10.1016/j.nuclphysb.2006.07.012, hep-ph/0512090.
[45] Y. Yamada,Electroweak two-loop contribution to the mass splitting withina new heavy SU(2)(L) fermion multiplet,
Phys. Lett. B 682, 435 (2010),
doi:10.1016/j.physletb.2009.11.044, 0906.5207.
p[46] S. Chatrchyan et al., Search for Fractionally Charged Particles in pp Collisions at s = 7
TeV, Phys. Rev. D 87(9), 092008 (2013), doi:10.1103/PhysRevD.87.092008, [Erratum:
Phys.Rev.D 106, 099903 (2022)], 1210.2311.
p[47] S. Chatrchyan et al., Searches for Long-Lived Charged Particles in pp Collisions at s=7
and 8 TeV, JHEP 07, 122 (2013), doi:10.1007/JHEP07(2013)122, [Erratum: JHEP
11, 149 (2022)], 1305.0491.
[48] C. Antel et al., Feebly-interacting particles: FIPs 2022 Workshop Report, Eur. Phys. J. C
83(12), 1122 (2023), doi:10.1140/epjc/s10052-023-12168-5, 2305.01715.
[49] G. Lanfranchi, M. Pospelov and P. Schuster, The Search for Feebly Interacting Particles,
Ann. Rev. Nucl. Part. Sci. 71, 279 (2021), doi:10.1146/annurev-nucl-102419-055056,
2011.02157.
[50] X. Gan and Y.-D. Tsai, Cosmic Millicharge Background and Reheating Probes (2023),
2308.07951.
[51] S. Davidson, S. Hannestad and G. Raffelt, Updated bounds on millicharged particles,
JHEP 05, 003 (2000), doi:10.1088/1126-6708/2000/05/003, hep-ph/0001179.
[52] A. A. Prinz et al., Search for millicharged particles at SLAC, Phys. Rev. Lett. 81, 1175
(1998), doi:10.1103/PhysRevLett.81.1175, hep-ex/9804008.
[53] R. Essig et al., Working Group Report: New Light Weakly Coupled Particles, In Snowmass
2013: Snowmass on the Mississippi (2013), 1311.0029.
[54] C. CMS, A MIP Timing Detector for the CMS Phase-2 Upgrade, Tech. rep., CERN, Geneva(2019).
[55] J. Liu, Z. Liu and L.-T. Wang,
Enhancing Long-Lived Particles Searches at the
LHC with Precision Timing Information, Phys. Rev. Lett. 122(13), 131801 (2019),
doi:10.1103/PhysRevLett.122.131801, 1805.05957.
[56] K. F. Di Petrillo, J. N. Farr, C. Guo, T. R. Holmes, J. Nelson and K. Pachal, Optimizing
trigger-level track reconstruction for sensitivity to exotic signatures, JHEP 02, 034 (2023),
doi:10.1007/JHEP02(2023)034, 2211.05720.
41

SciPost Physics

1419
1420

1421
1422

1423
1424

1425
1426
1427

1428
1429

1430
1431

1432
1433

1434
1435
1436

1437
1438
1439

1440
1441

1442
1443
1444

1445
1446
1447
1448

1449
1450
1451

1452
1453
1454

1455
1456

1457
1458
1459

Submission[57] D. Vannerom, Search for new physics in the dark sector with the CMS detector Frominvisible to low charge particles, Ph.D. thesis, Brussels U. (2019).
[58] G. Aad et al., The ATLAS Trigger System for LHC Run 3 and Trigger performance in 2022
(2024), 2401.06630.
[59] A. C. Kraan, Interactions of heavy stable hadronizing particles, Eur. Phys. J. C 37, 91(2004), doi:10.1140/epjc/s2004-01946-6, hep-ex/0404001.
[60] A. Ball et al.,Sensitivity to millicharged particles in future proton-proton collisions at the LHC with the milliQan detector, Phys. Rev. D 104(3), 032002 (2021),
doi:10.1103/PhysRevD.104.032002, 2104.07151.
[61] D. Dunsky, L. J. Hall and K. Harigaya, CHAMP Cosmic Rays, JCAP 07, 015 (2019),
doi:10.1088/1475-7516/2019/07/015, 1812.11116.
[62] D. Dunsky, L. J. Hall and K. Harigaya, Higgs Parity, Strong CP, and Dark Matter, JHEP
07, 016 (2019), doi:10.1007/JHEP07(2019)016, 1902.07726.
[63] S. Hannestad, What is the lowest possible reheating temperature?, Phys. Rev. D 70,
043506 (2004), doi:10.1103/PhysRevD.70.043506, astro-ph/0403291.
[64] P. F. de Salas, M. Lattanzi, G. Mangano, G. Miele, S. Pastor and O. Pisanti, Bounds
on very low reheating scenarios after Planck, Phys. Rev. D 92(12), 123534 (2015),
doi:10.1103/PhysRevD.92.123534, 1511.00672.
[65] P. N. Bhattiprolu, G. Elor, R. McGehee and A. Pierce, Freezing-in hadrophilic dark matterat low reheating temperatures, JHEP 01, 128 (2023), doi:10.1007/JHEP01(2023)128,
2210.15653.
[66] V. De Luca, A. Mitridate, M. Redi, J. Smirnov and A. Strumia, Colored Dark Matter,
Phys. Rev. D 97(11), 115024 (2018), doi:10.1103/PhysRevD.97.115024, 1801.01135.
[67] G. D. Starkman, A. Gould, R. Esmailzadeh and S. Dimopoulos,
Opening theWindow on Strongly Interacting Dark Matter,
Phys. Rev. D 41, 3594 (1990),
doi:10.1103/PhysRevD.41.3594.
[68] M. C. Digman, C. V. Cappiello, J. F. Beacom, C. M. Hirata and A. H. G. Peter, Not
as big as a barn: Upper bounds on dark matter-nucleus cross sections, Phys. Rev. D100(6), 063013 (2019), doi:10.1103/PhysRevD.100.063013, [Erratum: Phys.Rev.D
106, 089902 (2022)], 1907.10618.
[69] R. Abbasi et al., IceCube Sensitivity for Low-Energy Neutrinos from Nearby Supernovae,
Astron. Astrophys. 535, A109 (2011), doi:10.1051/0004-6361/201117810e, [Erratum: Astron.Astrophys. 563, C1 (2014)], 1108.0171.
[70] S. I. Alvis et al., First Limit on the Direct Detection of Lightly Ionizing Particles for Electric
Charge as Low as e/1000 with the Majorana Demonstrator, Phys. Rev. Lett. 120(21),211804 (2018), doi:10.1103/PhysRevLett.120.211804, 1801.10145.
[71] M. Ambrosio et al., Final search for lightly ionizing particles with the MACRO detector(2004), hep-ex/0402006.
[72] F. Kajino, S. Matsuno, T. Kitamura, T. Aoki, Y. K. Yuan, K. Mitsui, Y. Ohashi and A. Okada,
New Limit On The Flux Of Slowly Moving Magnetic Monopoles, J. Phys. G 10, 447 (1984),
doi:10.1088/0305-4616/10/4/007.
42

SciPost Physics

1460
1461
1462

1463
1464
1465

1466
1467

1468
1469
1470

1471
1472
1473

1474
1475

1476
1477

1478
1479
1480

1481
1482

1483
1484

1485
1486

1487
1488

1489
1490

1491
1492

1493
1494

1495
1496

1497
1498

Submission[73] E. N. Alekseev, M. M. Boliev, A. E. Chudakov, B. A. Makoev, S. P. Mikheev and Y. V.
Stenkin, Search For Superheavy Magnetic Monopoles At The Baksan Underground Telescope, Lett. Nuovo Cim. 35, 413 (1982), doi:10.1007/BF02754761.
[74] P. Langacker and G. Steigman, Requiem for an FCHAMP? Fractionally CHArged, Massive
Particle, Phys. Rev. D 84, 065040 (2011), doi:10.1103/PhysRevD.84.065040, 1107.
3131.
[75] C. Cosme, F. Costa and O. Lebedev, Freeze-in at stronger coupling, Phys. Rev. D 109(7),075038 (2024), doi:10.1103/PhysRevD.109.075038, 2306.13061.
[76] C. Gross, A. Mitridate, M. Redi, J. Smirnov and A. Strumia, Cosmological Abundance ofColored Relics, Phys. Rev. D 99(1), 016024 (2019), doi:10.1103/PhysRevD.99.016024,
1811.08418.
[77] A. Mitridate, M. Redi, J. Smirnov and A. Strumia, Cosmological Implications of Dark
Matter Bound States, JCAP 05, 006 (2017), doi:10.1088/1475-7516/2017/05/006,
1702.01141.
[78] J. Stillwell, Naive Lie Theory, Undergraduate Texts in Mathematics. Springer,
doi:10.1007/978-0-387-78214-0 (2008).
[79] B. C. Hall, Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics.
Springer, doi:10.1007/978-3-319-13467-3 (2015).
[80] T. Banks, M. Dine and N. Seiberg, Irrational axions as a solution of the strong CP
problem in an eternal universe, Phys. Lett. B 273, 105 (1991), doi:10.1016/03702693(91)90561-4, hep-th/9109040.
[81] J. Preskill,
Magnetic Monopoles,
Ann. Rev. Nucl. Part. Sci. 34, 461 (1984),
doi:10.1146/annurev.ns.34.120184.002333.
[82] P. A. M. Dirac, Quantised singularities in the electromagnetic field,, Proc. Roy. Soc. Lond.
A 133(821), 60 (1931), doi:10.1098/rspa.1931.0130.
[83] B. Felsager, Geometry, Particles, and Fields, Graduate Texts in Contemporary Physics.
Univ.Pr., Odense, doi:10.1007/978-1-4612-0631-6 (1981).
[84] J. Franklin and G. Goel, Finite and half-infinite solenoids and the Aharonov-Bohm effect,
Am. J. Phys. 87(11), 862 (2019), doi:10.1119/1.5124814.
[85] A. Béché, R. Van Boxem, G. Van Tendeloo and J. Verbeeck, Magnetic monopole field
exposed by electrons, Nature Phys. 10, 26 (2014), doi:10.1038/nphys2816, 1305.0570.
[86] T. T. Wu and C. N. Yang, Concept of Nonintegrable Phase Factors and Global Formulationof Gauge Fields, Phys. Rev. D 12, 3845 (1975), doi:10.1103/PhysRevD.12.3845.
[87] E. Corrigan and D. I. Olive, Color and Magnetic Monopoles, Nucl. Phys. B 110, 237(1976), doi:10.1016/0550-3213(76)90525-3.
[88] P. Goddard, J. Nuyts and D. I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys.
B 125, 1 (1977), doi:10.1016/0550-3213(77)90221-8.
[89] M. Daniel, G. Lazarides and Q. Shafi, SU(5) Monopoles, Magnetic Symmetry and Confinement, Nucl. Phys. B 170, 156 (1980), doi:10.1016/0550-3213(80)90483-6.

43

SciPost Physics

1499
1500

1501
1502

1503
1504
1505

1506
1507

1508
1509

1510
1511

1512
1513
1514

1515
1516

1517
1518

1519
1520

1521
1522
1523

1524
1525
1526

1527
1528
1529

1530
1531

1532
1533

1534
1535

1536
1537

Submission[90] C. P. Dokos and T. N. Tomaras, Monopoles and Dyons in the SU(5) Model, Phys. Rev. D
21, 2940 (1980), doi:10.1103/PhysRevD.21.2940.
[91] J. Davighi, B. Gripaios and N. Lohitsiri, Global anomalies in the Standard Model(s) and
Beyond, JHEP 07, 232 (2020), doi:10.1007/JHEP07(2020)232, 1910.11277.
[92] Z. Wan and J. Wang, Beyond Standard Models and Grand Unifications: Anomalies,
Topological Terms, and Dynamical Constraints via Cobordisms, JHEP 07, 062 (2020),
doi:10.1007/JHEP07(2020)062, 1910.14668.
[93] J. Davighi and N. Lohitsiri, Anomaly interplay in U(2) gauge theories, JHEP 05, 098(2020), doi:10.1007/JHEP05(2020)098, 2001.07731.
[94] J. Davighi and N. Lohitsiri, The algebra of anomaly interplay, SciPost Phys. 10(3), 074(2021), doi:10.21468/SciPostPhys.10.3.074, 2011.10102.
[95] J. Wang, Anomaly and Cobordism Constraints Beyond the Standard Model: Topological
Force (2020), 2006.16996.
[96] J. Wang, Z. Wan and Y.-Z. You, Cobordism and deformation class of the standard model,
Phys. Rev. D 106(4), L041701 (2022), doi:10.1103/PhysRevD.106.L041701, 2112.
14765.
[97] D. M. Scott, Monopoles in a Grand Unified Theory Based on SU(5), Nucl. Phys. B 171,95 (1980), doi:10.1016/0550-3213(80)90361-2.
[98] C. Córdova, S. Hong, S. Koren and K. Ohmori, Neutrino Masses from Generalized Symmetry Breaking (2022), 2211.07639.
[99] C. Cordova, S. Hong and S. Koren, Non-Invertible Peccei-Quinn Symmetry and the Massless Quark Solution to the Strong CP Problem (2024), 2402.12453.
[100] D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity,
Commun. Math. Phys. 383(3), 1669 (2021), doi:10.1007/s00220-021-04040-y, 1810.
05338.
[101] A. Arvanitaki, S. Dimopoulos, A. A. Geraci, J. Hogan and M. Kasevich, Testing Atomand Neutron Neutrality with Atom Interferometry, Phys. Rev. Lett. 100, 120407 (2008),
doi:10.1103/PhysRevLett.100.120407, 0711.4636.
[102] G. Bressi, G. Carugno, F. Della Valle, G. Galeazzi, G. Ruoso and G. Sartori, Testing
the neutrality of matter by acoustic means in a spherical resonator, Phys. Rev. A 83(5),052101 (2011), doi:10.1103/PhysRevA.83.052101, 1102.2766.
[103] C. S. Unnikrishnan and G. T. Gillies, The electrical neutrality of atoms and of bulk matter,
Metrologia 41(5), S125 (2004), doi:10.1088/0026-1394/41/5/S03.
[104] S. Koren, Cosmological Lithium Solution from Discrete Gauged B-L, Phys. Rev. Lett.
131(9), 091003 (2023), doi:10.1103/PhysRevLett.131.091003, 2204.01750.
[105] S. Hong and S. Koren,
Inpreparation.

Which Gauge Group for the Standard Model Fermions?

[106] T. D. Brennan and S. Hong, Introduction to Generalized Global Symmetries in QFT and
Particle Physics (2023), 2306.00912.

44

SciPost Physics

1538
1539

1540
1541
1542

1543
1544

1545
1546

1547
1548

1549
1550

1551
1552

1553
1554

1555
1556

1557
1558

1559
1560

1561
1562

1563
1564

Submission[107] P. R. S. Gomes, An introduction to higher-form symmetries, SciPost Phys. Lect. Notes 74,
1 (2023), doi:10.21468/SciPostPhysLectNotes.74, 2303.01817.
[108] L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. W. Gould, A. Platschorreand H. Tillim, Lectures on generalized symmetries, Phys. Rept. 1051, 1 (2024),
doi:10.1016/j.physrep.2023.11.002, 2307.07547.
[109] E. Sharpe, Notes on generalized global symmetries in QFT, Fortsch. Phys. 63, 659 (2015),
doi:10.1002/prop.201500048, 1508.04770.
[110] L. Kong and Z.-H. Zhang, An invitation to topological orders and category theory (2022),
2205.05565.
[111] S. Schafer-Nameki, ICTP lectures on (non-)invertible generalized symmetries, Phys. Rept.
1063, 1 (2024), doi:10.1016/j.physrep.2024.01.007, 2305.18296.
[112] R. Luo, Q.-R. Wang and Y.-N. Wang, Lecture notes on generalized symmetries and applications, Phys. Rept. 1065, 1 (2024), doi:10.1016/j.physrep.2024.02.002, 2307.09215.
[113] S.-H. Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries(2023), 2308.00747.
[114] O. Loaiza-Brito and V. M. López-Ramos, A mini-course on Generalized Symmetries and a
relation to Cobordisms and K-theory (2024), 2404.03599.
[115] L. Borsten, M. J. Farahani, B. Jurco, H. Kim, J. Narozny, D. Rist, C. Saemann and M. Wolf,
Higher Gauge Theory (2024), 2401.05275.
[116] C. Cordova and S. Koren, Higher Flavor Symmetries in the Standard Model, Annalen
Phys. 535(8), 2300031 (2023), doi:10.1002/andp.202300031, 2212.13193.
[117] T. D. Brennan and C. Córdova, Axions, higher-groups, and emergent symmetry, JHEP
02, 145 (2022), doi:10.1007/JHEP02(2022)145, 2011.09600.
[118] C. Cordova and K. Ohmori, Noninvertible Chiral Symmetry and Exponential Hierarchies,
Phys. Rev. X 13(1), 011034 (2023), doi:10.1103/PhysRevX.13.011034, 2205.06243.
[119] C. Córdova and K. Ohmori, Quantum Duality in Electromagnetism and the Fine-Structure
Constant (2023), 2307.12927.

1566[120] T. D. Brennan, S. Hong and L.-T. Wang, Coupling a Cosmic String to a TQFT, JHEP 03,
145 (2024), doi:10.1007/JHEP03(2024)145, 2302.00777.

1567[121] T. D. Brennan, A New Solution to the Callan Rubakov Effect (2023), 2309.00680.

1565

1568
1569

1570
1571

1572
1573

1574
1575

[122] M. van Beest, P. Boyle Smith, D. Delmastro, Z. Komargodski and D. Tong, Monopoles,
Scattering, and Generalized Symmetries (2023), 2306.07318.
[123] M. van Beest, P. Boyle Smith, D. Delmastro, R. Mouland and D. Tong, Fermion-Monopole
Scattering in the Standard Model (2023), 2312.17746.
[124] J. Davighi and N. Lohitsiri, Omega vs. pi, and 6d anomaly cancellation, JHEP 05, 267(2021), doi:10.1007/JHEP05(2021)267, 2012.11693.
[125] J. Davighi, A. Greljo and N. Selimovic, Topological Portal to the Dark Sector (2024),
2401.09528.

45

SciPost Physics

1576
1577

1578
1579
1580

1581
1582

1583
1584
1585

1586
1587
1588

Submission[126] C. Cheung, M. Derda, J.-H. Kim, V. Nevoa, I. Rothstein and N. Shah, Generalized Symmetry in Dynamical Gravity (2024), 2403.01837.
[127] K. Hinterbichler, D. M. Hofman, A. Joyce and G. Mathys, Gravity as a gapless phaseand biform symmetries, JHEP 02, 151 (2023), doi:10.1007/JHEP02(2023)151, 2205.
12272.
[128] K. Hinterbichler, A. Joyce and G. Mathys, Impossible Symmetries and Conformal Gravity(2024), 2403.03256.
[129] Y. Hidaka, M. Nitta and R. Yokokura, Global 3-group symmetry and ’t Hooft anomalies inaxion electrodynamics, JHEP 01, 173 (2021), doi:10.1007/JHEP01(2021)173, 2009.
14368.
[130] Y. Hidaka, M. Nitta and R. Yokokura, Topological axion electrodynamics and 4-groupsymmetry, Phys. Lett. B 823, 136762 (2021), doi:10.1016/j.physletb.2021.136762,
2107.08753.

1591[131] Y. Hidaka, M. Nitta and R. Yokokura,
Global 4-group symmetry and ’t Hooft
anomalies in topological axion electrodynamics, PTEP 2022(4), 04A109 (2022),
doi:10.1093/ptep/ptab150, 2108.12564.

1592[132] R. Yokokura, Non-invertible symmetries in axion electrodynamics (2022), 2212.05001.

1589
1590

1593
1594
1595

1596
1597

1598
1599

1600
1601

[133] Y. Choi, H. T. Lam and S.-H. Shao,
Noninvertible Global Symmetries in the Standard Model,
Phys. Rev. Lett. 129(16), 161601 (2022),
doi:10.1103/PhysRevLett.129.161601, 2205.05086.
[134] Y. Choi, H. T. Lam and S.-H. Shao, Non-invertible Time-reversal Symmetry (2022), 2208.
04331.
[135] Y. Choi, H. T. Lam and S.-H. Shao, Non-invertible Gauss law and axions, JHEP 09, 067(2023), doi:10.1007/JHEP09(2023)067, 2212.04499.
[136] B. Heidenreich, J. McNamara and M. Reece, Non-standard axion electrodynamics and thedual Witten effect, JHEP 01, 120 (2024), doi:10.1007/JHEP01(2024)120, 2309.07951.

1604[137] J. Fan, K. Fraser, M. Reece and J. Stout, Axion Mass from Magnetic Monopole Loops,
Phys. Rev. Lett. 127(13), 131602 (2021), doi:10.1103/PhysRevLett.127.131602, 2105.
09950.

1605[138] J. McNamara and M. Reece, Reflections on Parity Breaking (2022), 2212.00039.

1602
1603

1606
1607
1608

1609
1610

1611
1612

1613
1614
1615

[139] P. Asadi, S. Homiller, Q. Lu and M. Reece, Chiral Nelson-Barr models: Quality andcosmology, Phys. Rev. D 107(11), 115012 (2023), doi:10.1103/PhysRevD.107.115012,
2212.03882.
[140] D. Aloni, E. García-Valdecasas, M. Reece and M. Suzuki, Spontaneously Broken (−1)Form U(1) Symmetries (2024), 2402.00117.
[141] J. Wang and Y.-Z. You, Gauge Enhanced Quantum Criticality Between Grand Unifications:Categorical Higher Symmetry Retraction (2021), 2111.10369.
[142] J. Wang and Y.-Z. You, Gauge enhanced quantum criticality beyond the standard
model, Phys. Rev. D 106(2), 025013 (2022), doi:10.1103/PhysRevD.106.025013,
2106.16248.
46

SciPost Physics

1616
1617
1618

1619
1620

1621
1622

1623
1624

1625
1626
1627

1628
1629
1630

1631
1632
1633

1634
1635
1636

1637
1638

1639
1640

1641
1642

Submission[143] J. Wang, Z. Wan and Y.-Z. You, Proton stability: From the standard model to beyond grandunification, Phys. Rev. D 106(2), 025016 (2022), doi:10.1103/PhysRevD.106.025016,
2204.08393.
[144] P. Putrov and J. Wang, Categorical Symmetry of the Standard Model from Gravitational
Anomaly (2023), 2302.14862.
[145] J. Wang, Family Puzzle, Framing Topology, c− = 24 and 3(E8)1 Conformal Field Theories:
48/16 = 45/15 = 24/8 =3 (2023), 2312.14928.
[146] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University
Press, ISBN 978-1-107-03473-0, 978-1-107-03473-0 (2014).
[147] G. T. Horowitz and M. Srednicki, A Quantum Field Theoretic Description of Linking Numbers and Their Generalization, Commun. Math. Phys. 130, 83 (1990),
doi:10.1007/BF02099875.
[148] T. Rudelius and S.-H. Shao, Topological Operators and Completeness of Spectrum in
Discrete Gauge Theories, JHEP 12, 172 (2020), doi:10.1007/JHEP12(2020)172,
2006.10052.
[149] B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius and I. Valenzuela,
Non-invertible global symmetries and completeness of the spectrum, JHEP 09, 203 (2021),
doi:10.1007/JHEP09(2021)203, 2104.07036.
[150] C. Cordova, K. Ohmori and T. Rudelius, Generalized symmetry breaking scales and weakgravity conjectures, JHEP 11, 154 (2022), doi:10.1007/JHEP11(2022)154, 2202.
05866.
[151] E. A. Uehling, Polarization effects in the positron theory, Phys. Rev. 48, 55 (1935),
doi:10.1103/PhysRev.48.55.
[152] S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands
Program (2006), hep-th/0612073.
[153] S. Gukov and E. Witten, Rigid Surface Operators, Adv. Theor. Math. Phys. 14(1), 87(2010), doi:10.4310/ATMP.2010.v14.n1.a3, 0804.1561.

47