aainstitutetext: Centre for Particle Theory, Department of Mathematical
Sciences, Durham University, South Road, Durham DH1 3LE, UKbbinstitutetext:
Department of Physics, University of Toronto, 60 St George St., Toronto, ON
M5S 1A7, Canada

# Nonperturbative effects in the Standard Model with gauged 1-form symmetry

Mohamed M. Anber b Erich Poppitz<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

W

e study the Standard Model with gauged $\mathbb{Z}_{N=2,3,6}^{(1)}$ subgroups
of its $\mathbb{Z}_{6}^{(1)}$ 1-form global symmetry, making the gauge group
$SU(3)\times SU(2)\times U(1)\over\mathbb{Z}_{N}$. We show that, on a finite
${\mathbb{T}}^{3}$, there are self-dual instantons of fractional topological
charge. They mediate baryon- and lepton-number violating processes. We compare
their amplitudes to the ones due to the usual BPST-instantons. We find that
the small hypercharge coupling suppresses the fractional-instanton
contribution, unless the torus size is taken sub-Planckian, or extra matter is
added above the weak scale. We also discuss these results in light of the
cosmological bounds on the torus size.

##  1 Introduction, summary, and outlook.

It is well established that the Lie algebra of the Standard Model (SM) is
$g_{SM}=su(3)\times su(2)\times u(1)$. The fact that there is freedom to
choose different gauge groups $G_{SM}$ is less broadly appreciated
Aharony:2013hda ; Tong:2017oea . If the gauge group is chosen to be
$G_{SM}^{1}\equiv SU(3)\times SU(2)\times U(1)$, the SM with its known matter
content has a 1-form111The superscript $G^{(q)}$, with $q=1$, is used to
distinguish 1-form symmetries (with group $G$), acting on line operators, from
ordinary 0-form symmetries ($q=0$) acting on local operators. The modern
introduction of higher-form symmetries is found in Gaiotto:2014kfa . The
recent interest in the global aspects of gauge theories and their higher-form
symmetries arose due to the need to resolve puzzles related to the mapping of
Wilson and ’t Hooft line operators under $S$-duality of ${\cal{N}}=4$ super-
Yang-Mills Aharony:2013hda . Further interest is due to the importance of
higher-form symmetries for the recently discovered Gaiotto:2014kfa ;
Gaiotto:2017yup ; Gaiotto:2017tne  generalized ’t Hooft anomalies.
$\mathbb{Z}_{6}^{(1)}$ global symmetry. The global structure of the SM gauge
group can be changed upon gauging its various discrete subgroups, leading to
possible gauge groups $G_{SM}^{N}={SU(3)\times SU(2)\times
U(1)\over\mathbb{{\mathbb{Z}}}_{N}}$, with $N=1,2,3,6$. The global structure
of the SM was discussed in Tong:2017oea , using the language of “genuine line
operators” of Aharony:2013hda . Refs. Garcia-Etxebarria:2018ajm ; Wang:2020iqc
; Davighi:2019rcd ; Wan:2020ynf ; Wan:2019soo ; Wang:2020xyo  studied gauge-
anomaly related aspects of $G_{SM}^{N}$, in particular showing the absence of
global anomalies in the SM with $G_{SM}^{N}$.

There are various ways to exhibit the $\mathbb{Z}_{6}^{(1)}$ 1-form symmetry
of the SM. In this paper, we take an “old-fashioned” approach, initiated by ’t
Hooft tHooft:1979rtg ; tHooft:1981sps  in the study of confinement in Yang-
Mills theory (see also vanBaal:1982ag ; Gonzalez-Arroyo:1997ugn ;
vanBaal:2000zc  and the recent work Unsal:2021cch ; Unsal:2020yeh ). Our main
goal is to begin a discussion of the dynamical effects due to the different
choices of $G_{SM}^{N}$. To this end, we consider the SM in a finite volume,
taking space to be a three-torus222We remind the reader that “cosmic topology”
is a subject with a long history, reviewed in Lachieze-Rey:1995qrb .
${\mathbb{T}}^{3}$. As we review below, the choice of SM gauge group
$G_{SM}^{N}$ is reflected in the boundary conditions on the gauge and matter
fields imposed on the spatial ${\mathbb{T}}^{3}$, in the boundary conditions
in the time direction, and in the way the sum over the sectors with different
boundary conditions is performed in the path integral.

Our main point, using insight due to ’t Hooft tHooft:1981nnx , is to show that
for $G_{SM}^{N}$, with $N\neq 1$, on a finite ${\mathbb{T}}^{3}$ there exist
self-dual instanton solutions of fractional topological charges. They mediate
processes where the baryon number changes by a nonzero amount $\Delta B$, with
integer $\Delta B=\Delta L$ (see Table 35 below). We calculate the exponential
suppression factor for such processes for the various choices of SM gauge
group $G_{SM}^{N}$ and compare it with that for similar processes mediated by
the usual BPST Belavin:1975fg  instantons computed by ’t Hooft tHooft:1976snw
; tHooft:1976rip . Owing to the fact that these fractional instantons are much
less understood333See GarciaPerez:2000aiw ; Gonzalez-Arroyo:2019wpu  for
relevant recent work. than the BPST instantons, we use the constant field
strength self-dual solutions of tHooft:1981nnx ; vanBaal:1984ar . We find that
due to the smallness of the hypercharge coupling, the fractional-instanton
mediated processes are exponentially suppressed compared to the usual BPST
instantons. This suppression can be overcome if the torus size is taken
extremely small (sub-Planckian, with only SM content) or if extra matter is
added to make the hypercharge coupling increase faster above the weak scale.
We also discuss our findings in light of the cosmological bounds on the
${\mathbb{T}}^{3}$ size444See Lachieze-Rey:1995qrb  and, e.g. Levin:1998qq ;
Bond:1997ym ; Aslanyan:2013lsa  for more recent work. and speculate on the
possible cosmological relevance of the global choice $G_{SM}^{N}$.

We leave various interesting aspects for future work. The way the solutions of
tHooft:1981nnx  are modified in the broken phase of the SM has not yet been
studied. It would also be of interest to study the generalization and
extension of the construction of non-constant field strength fractional
instantons of GarciaPerez:2000aiw ; Gonzalez-Arroyo:2019wpu  to the SM case. A
better control of these issues might help clarify the possible role of the
fractional instantons at larger torus sizes.

##  2 The ${\mathbb{Z}}_{6}^{(1)}$ and its gauging via twisted boundary
conditions

In this Section, we explain how the ${\mathbb{Z}}_{6}^{(1)}$ 1-form symmetry
of the SM naturally arises upon formulating the theory on a spatial 3-torus.
This formulation is a convenient starting point to study the fractional
topological charge solutions. We also discuss the gauging of the symmetry and
its relation to the boundary conditions imposed on the gauge fields in the
path integral. (An equivalent description of the gauging of the 1-form
symmetry takes the form of a coupling to a 2-form topological quantum field
theory Kapustin:2014gua  and also requires that the spacetime manifold has
noncontractible 2-cycles.)

To begin, we denote the $SU(3)\times SU(2)\times U(1)$ gauge couplings
$g_{3},g_{2}$, and $g_{1}$, respectively. For use below, note that the
relation between the $U(1)$ coupling and the usual hypercharge coupling
$g_{Y}$ is $g_{1}={g_{Y}\over 6}$, because we take all $U(1)$ charges to be
integer, due to the assumed compactness of the $U(1)$ gauge group. Explicitly,
the charge assignments of the various SM matter fields are standard, except
that the hypercharge is multiplied by $6$:

$\begin{array}[]{c|ccc|cc}\text{field}&SU(3)&SU(2)&U(1)&U(1)_{B}&U(1)_{L}\cr\hline\cr
q_{L}&\square&\square&1&{1\over 3}&0\cr
l_{L}&\bf{1}&\square&-3&0&1\cr\tilde{e}_{R}&\bf{1}&\bf{1}&6&0&-1\cr\tilde{u}_{R}&\overline{\square}&\bf{1}&-4&-{1\over
3}&0\cr\tilde{d}_{R}&\overline{\square}&\bf{1}&2&-{1\over 3}&0\cr
h&\bf{1}&\square&3&0&0\end{array}~{}.$ (1)

All fields above (but the Higgs $h$) are left-handed Weyl spinors and we have
not included a right-handed neutrino as its possible existence will play no
role in our discussion. For use below, in the two r.h. columns we also give
the standard baryon and lepton number assignments. In (1), we only show a
single generation of fermions.

When formulated on a manifold of nontrivial topology, a gauge theory is
defined on coordinate patches covering the manifold, and the gauge fields on
different patches are related by transition functions, which are gauge group
elements. The matter fields on different patches are also related by the
transition functions taken in the relevant representation of the gauge group.
Here, following tHooft:1979rtg ; tHooft:1981sps ; vanBaal:1982ag ; Gonzalez-
Arroyo:1997ugn , we consider the case of a torus, which can be described by
only one patch, but with nontrivial boundary conditions on the fields. If we
were only interested in a Hamiltonian formulation of the theory, we would
consider a spatial three-torus ${\mathbb{T}}^{3}$. In a path integral
formulation, we study the finite temperature partition function of the theory
on a four-torus ${\mathbb{T}}^{4}\simeq{\mathbb{T}}^{3}\times S^{1}$, where
${\mathbb{S}}^{1}$ is interpreted as a thermal circle; if we wish to approach
the zero-temperature limit, we take the size of the circle to infinity.

###  2.1 The ${\mathbb{T}}^{4}$ boundary conditions: transition functions and
gauge transformations

We shall now discuss how the $Z_{6}^{(1)}$ symmetry arises on
${\mathbb{T}}^{4}$. The ${\mathbb{T}}^{4}$ is covered by a single closed
coordinate patch $0\leq x^{\mu}\leq a^{\mu}$, $\mu=1,2,3,4$. This single-patch
construction can be seen as a limit of a multiple patch covering of the
${\mathbb{T}}^{4}$, as is explicitly discussed in vanBaal:1982ag . The gauge
fields on the different sides of the ${\mathbb{T}}^{4}$, e.g. at $x^{\mu}=0$
and $x^{\mu}=a^{\mu}$ are related by transition functions, gauge group
elements $\Omega_{\mu}$, which can be thought of as defined on the
$x^{\mu}=a^{\mu}$ sides of the ${\mathbb{T}}^{4}$; gauge transformations are
also group elements defined on the entire coordinate patch.

To state the boundary conditions, begin by considering an arbitrary $\mu\nu$
2-plane of the torus of sides $a^{\mu}$ and $a^{\nu}$. The coordinates are
identified as $x^{\mu}\equiv x^{\mu}+a^{\mu}$. Thus, the 2-plane is a square
$0\leq x^{\mu}\leq a^{\mu}$, $0\leq x^{\nu}\leq a^{\nu}$ (for fixed values of
the other coordinates). The fields on opposite sides of the square are related
by transition functions, as we now describe. To set our notation, let the
gauge fields of the SM be $A_{(j)}=A_{(j)\;\mu}dx^{\mu}$, with $j=1,2,3$
denoting the $U(1),SU(2)$, and $SU(3)$ gauge fields, respectively. Here
$A_{(j)\;\mu}=A_{(j)\;\mu}^{a}T_{(j)}^{a}$, where $T_{(j)}^{a}$ are the
corresponding group generators in the fundamental representation,
tr$T_{(j)}^{a}T_{(j)}^{b}={1\over 2}\delta^{ab}$, $j=2,3$. For $U(1)$,
$T_{(1)}^{a}$ are replaced by the identity. The gauge field action for the
$j$-th gauge group is

$S_{(j)}={1\over 4g_{(j)}^{2}}\int
d^{4}x\;F_{(j)\;\mu\nu}^{a}F_{(j)}^{\mu\nu\;a}~{},$ (2)

where for $j=1$ there is no sum over $a$. The covariant derivative acting on,
e.g., $q_{L}$ with charges from (1), is then
$D_{\mu}q_{L}=(\partial_{\mu}+iA_{(1)\;\mu}+iA_{(2)\;\mu}^{a}T_{(2)}^{a}+iA_{(3)\;\mu}^{a}T_{(3)}^{a})q_{L}$.
Gauge transformations of the $j$-th group act on the gauge fields as
$A_{(j)}\rightarrow g_{(j)}(A_{(j)}-id)g_{(j)}^{-1}$. On the matter fields,
these are taken in the appropriate representation, e.g. $q_{L}\rightarrow
g_{(j)}q_{L}$, where $g_{(j)}$ is a group element in the fundamental
representation defined in the $0\leq x^{\mu}\leq a^{\mu}$ coordinate patch
($\mu=1,...4$). For $U(1)$, the gauge group element is simply
$g_{(1)}=e^{i\omega_{(1)}}$ when acting on a field of the minimal quantized
unit charge.

The boundary conditions on the gauge and matter fields in the $\mu$-th
direction of the square are then given by

$\displaystyle A_{(j)}(x^{\mu}=a^{\mu})$ $\displaystyle=$
$\displaystyle\Omega_{(j)\;\mu}\left(A_{(j)}(x^{\mu}=0)-id\right)\Omega_{(j)\;\mu}^{-1}~{},~{}j=1,2,3,~{}$
(3) $\displaystyle\psi^{\cal{R}}(x^{\mu}=a^{\mu})$ $\displaystyle=$
$\displaystyle\Omega_{(1)\;\mu}^{\cal{R}}\Omega_{(2)\;\mu}^{\cal{R}}\Omega_{(3)\;\mu}^{\cal{R}}\;\psi^{\cal{R}}(x^{\mu}=0)~{},\text{where}~{}\psi^{\cal{R}}=(q_{L},l_{L},\tilde{e}_{R},\tilde{u}_{R},\tilde{d}_{R},h)~{}.$

Above, we did not explicitly indicate the fact that the fields depend on all
but the $x^{\mu}$ coordinates, i.e. their values are taken on the 3-planes
$x^{\mu}=a^{\mu}$ or $x^{\mu}=0$ in ${\mathbb{T}}^{4}$. The boundary
conditions for the $\nu$-th direction are identical, with $\mu\rightarrow\nu$,
and are given in terms of their own transition functions $\Omega_{(j)\;\nu}$.

L

et us now make some comments regarding the boundary conditions (3):

  1. 1.

The notation in (3) may be somewhat condensed, so let us elaborate. The
functions $\Omega_{(j)\;\mu}$ are the transition functions in the $\mu$-th
direction of the torus for the $j$-th gauge group. They are group elements of
the $j$-th gauge group in the fundamental representation. Note that the
transition functions $\Omega_{(j)\;\mu}$ in the $\mu$-th direction are
independent of $x^{\mu}$ and can be thought to be defined at the
$x^{\mu}=a^{\mu}$ side of the torus. The matter fields at $x^{\mu}=a^{\mu}$
and $x^{\mu}=0$ are related as shown in the second line in (3), where
$\Omega_{(j)\;\mu}^{\cal{R}}$ are the transition functions taken in the
appropriate representation from (1): $\Omega_{(j)\;\mu}^{\cal{R}}$ are the
same group elements $\Omega_{(j)\;\mu}$, but taken in the representation
${\cal{R}}$ of the fermion $\psi^{\cal{R}}$. Explicitly, from eqn. (1), we see
that for the SM and for $j=2,3$ only the fundamental and antifundamental
representations appear, hence $\Omega_{(j)\;\mu}^{\cal{R}}=\Omega_{(j)\;\mu}$
if the field $\psi^{\cal{R}}$ is in the fundamental representation or
$\Omega_{(j)\;\mu}^{\cal{R}}=\Omega_{(j)\;\mu}^{*}$ if it is in the
antifundamental representation. For a $\psi^{\cal{R}}$ field of charge $q_{1}$
under $U(1)$, we have
$\Omega_{(1)\;\mu}^{{\cal{R}}=q_{1}}=(\Omega_{(1)\;\mu})^{q_{1}}$, where
$\Omega_{(1)\;\mu}=e^{i\omega_{(1)\;\mu}}$ is a compact-$U(1)$ group element.
It is clear that with this definition, the matter field and its covariant
derivative obey the boundary conditions (3).

  2. 2.

Consistency of the boundary conditions (3) with gauge transformations of each
gauge group requires that the transition functions $\Omega_{(j)\;\mu}$
themselves transform under gauge transformations:
$\Omega_{(j)\;\mu}\rightarrow
g_{(j)}(x^{\mu}=a^{\mu})\;\Omega_{(j)\;\mu}\;g_{(j)}^{-1}(x^{\mu}=0)$, and
similar for the matter fields, where $\Omega$ and $g$ are to be taken in the
appropriate representation. For example, when studying classical solutions,
one can make use of this freedom to choose a convenient form of the transition
functions. Using this gauge freedom to, e.g., set some transition functions to
unity, amounts to partial gauge fixing (see below).

  3. 3.

Most important for our further discussion is the fact that the transition
functions obey consistency conditions in every $\mu\nu$-plane of the torus.
These are a consequence of the well-known “cocycle conditions” on general
manifolds and we shall also call them by that name. In our single-chart setup
the cocycle conditions follow by requiring that the fields are single valued
on every $\mu\nu$-square. To make the point, consider the matter field
$\psi^{\cal{R}}$. The value of $\psi^{\cal{R}}$ at the
($x^{\mu}=a^{\mu},x^{\nu}=a^{\nu}$) corner of the square can be obtained from
its value at the opposite ($x^{\mu}=x^{\nu}=0$) corner by using the boundary
conditions (3) in two different ways: first go along the $x^{\mu}=0$ side in
the $\nu$-th direction and then along the $x^{\nu}=a^{\nu}$ side in the
$\mu$-th direction, or v.v. Obtaining the same value of the $\psi^{\cal{R}}$
field at the corner then requires that

$\displaystyle\Omega_{(1)\;\mu}^{\cal{R}}\Omega_{(2)\;\mu}^{\cal{R}}\Omega_{(3)\;\mu}^{\cal{R}}(x^{\nu}=a^{\nu})\times\Omega_{(1)\;\nu}^{\cal{R}}\Omega_{(2)\;\nu}^{\cal{R}}\Omega_{(3)\;\nu}^{\cal{R}}(x^{\mu}=0)$
(4) $\displaystyle=$
$\displaystyle\Omega_{(1)\;\nu}^{\cal{R}}\Omega_{(2)\;\nu}^{\cal{R}}\Omega_{(3)\;\nu}^{\cal{R}}(x^{\mu}=a^{\mu})\times\Omega_{(1)\;\mu}^{\cal{R}}\Omega_{(2)\;\mu}^{\cal{R}}\Omega_{(3)\;\mu}^{\cal{R}}(x^{\nu}=0)~{}.$

This should hold for all $\mu,\nu$ planes and all $\psi^{\cal{R}}$ from (1).

  4. 4.

There are consistency conditions similar to (4) for the $\Omega_{(j)\;\mu}$
appearing in the gauge-field boundary conditions. It turns out that there is
some freedom in the gauge field cocycle conditions that we describe below. In
fact, the different $G_{SM}^{N}$ theories are distinguished by the conditions
obeyed by the transition functions for the gauge fields as we now discuss.

###  2.2 $G_{SM}^{1}$ and the ${\mathbb{Z}}_{6}^{(1)}$ global symmetry

Here, we discuss the cocycle conditions on the transitions functions for the
gauge field and introduce the ${\mathbb{Z}}_{6}^{(1)}$ symmetry of the SM.

Using the same logic as for the field $\psi^{\cal{R}}$ to obtain (4), we see
that to obtain a single-valued gauge field, it is sufficient555But not
necessary: running ahead, we immediately notice that because the gauge field
boundary condition in (3) contains both $\Omega$ and $\Omega^{-1}$, one can
allow a phase factor equal to a center element of each gauge group to appear
on the r.h.s. of (5).  to take the gauge field transition functions to obey a
condition identical to (4) for each $j=1,2,3$. We call this cocycle condition
the $G_{SM}^{1}$ condition:

$\displaystyle
G_{SM}^{1}:~{}~{}\Omega_{(j)\;\mu}(x^{\nu}=a^{\nu})\;\Omega_{(j)\;\nu}(x^{\mu}=0)=\Omega_{(j)\;\nu}(x^{\mu}=a^{\mu})\;\Omega_{(j)\;\mu}(x^{\nu}=0)~{},~{}j=1,2,3.$
(5)

The fields summed over in the path integral for $G_{SM}^{1}$ satisfy the
boundary conditions (3) with transition functions obeying the cocycle
conditions (4, 5). Of particular interest to us is the fact that the
topological charge for $G_{SM}^{1}$ is integer tHooft:1979rtg ; tHooft:1981sps
; vanBaal:1982ag , to be discussed below.666It can be shown that, using the
gauge freedom discussed in Section 2.1, one can take the $j=2,3$ transition
functions obeying (5) to be trivial in the spatial ${\mathbb{T}}^{3}$
directions, $\Omega_{(j)\;\mu}=1$ for $\mu=1,2,3$, and only have a nontrivial
$x^{4}$-direction transition function $\Omega_{(j)\;4}\neq 1$. In this gauge,
the transition function in the “time” direction carries all the information
about the integer topological charge.

Now we can explicitly exhibit the 1-form ${\mathbb{Z}}_{6}^{(1)}$ center
symmetry of the SM. This is a ${\mathbb{Z}}_{6}$ global symmetry,
parameterized by two integers, $k_{\mu}^{2}\in{\mathbb{Z}}(\text{mod}\,2)$ and
$k_{\mu}^{3}\in{\mathbb{Z}}(\text{mod}\,3)$, for every spacetime direction
$\mu$. In this formulation, the 1-form center symmetry does not act on any
fields, but acts only on the transition functions as follows

$\displaystyle Z_{6}^{(1)}:~{}\Omega_{(1)\;\mu}$ $\displaystyle\rightarrow$
$\displaystyle e^{i2\pi(-{k_{\mu}^{2}\over 2}-{k_{\mu}^{3}\over
3})}\;\Omega_{(1)\;\mu}~{},~{}k_{\mu}^{2}\in\\{1,2\\},\;k_{\mu}^{3}\in\\{1,2,3\\},\;\mu=1,2,3,4,$
$\displaystyle\Omega_{(2)\;\mu}$ $\displaystyle\rightarrow$ $\displaystyle
e^{i2\pi{k_{\mu}^{2}\over 2}}\;\Omega_{(2)\;\mu}~{},$ (6)
$\displaystyle\Omega_{(3)\;\mu}$ $\displaystyle\rightarrow$ $\displaystyle
e^{i2\pi{k_{\mu}^{3}\over 3}}\;\Omega_{(3)\;\mu}~{}.$

Clearly the action (2.2) is consistent with $\Omega_{(j)\;\mu}$ being (special
unitary, for $j=2,3$) group elements.

That the above ${\mathbb{Z}}_{6}^{(1)}$ is a symmetry follows from the fact
that both the boundary conditions (3) and the gauge and matter fields’ cocycle
conditions (4), (5) are invariant under (2.2). The gauge field boundary
condition, the first line in (3), is clearly invariant since both
$\Omega_{(j)\;\mu}$ and $\Omega_{(j)\;\mu}^{-1}$ appear on the r.h.s. The
cocycle conditions for the gauge field (5) are also obviously invariant, as
$\Omega_{(j)\;\mu}$ appears on both sides of the equations. The matter field
boundary conditions from the second line in (3) are also invariant, for all
$\psi^{\cal{R}}$ from (1), as is easily checked explicitly and follows from
the charge assignments. For example, for $q_{L}$ this follows immediately form
the fact that the phase factors on the r.h.s. of (2.2) simply cancel out. That
the cocycle conditions for the matter fields (4) are also invariant follows
from the invariance of the boundary conditions (3) for $\psi^{\cal{R}}$, as
the products
$\Omega_{(1)\;\mu}^{\cal{R}}\Omega_{(2)\;\mu}^{\cal{R}}\Omega_{(3)\;\mu}^{\cal{R}}$
are invariant under (2.2) for all $\cal{R}$.

We have not yet explained why the symmetry (2.2) is a 1-form symmetry, in the
modern terminology of Gaiotto:2014kfa : this is because the only gauge
invariant operators that (2.2) acts on are Wilson lines winding around the
noncontractible loops of the torus. It is clear that all local gauge invariant
operators are blind to a symmetry like (2.2) that only acts on transition
functions (since a local operator can always be written using fields within a
given coordinate patch). On the other hand, a line operator, such as the
fundamental representation Wilson loop of the $j$-th gauge group, winding once
around the $x^{\mu}$ direction, has the form

$W_{(j)\;\mu}=\text{tr}({\cal{P}}\;e^{\;i\int\limits_{x^{\mu}=0}^{x^{\mu}=a^{\mu}}A_{(j)}}\;\Omega_{(j)\;\mu})~{},$
(7)

where $\cal{P}$ denotes path ordering and the transition function is inserted
in the trace in order to ensure the invariance of $W_{(j)\;\mu}$ under gauge
transformations $g_{(j)}$ on the ${\mathbb{T}}^{4}$ (recall the gauge
transformation law of $\Omega_{(j)\;\mu}$ mentioned above). It is now clear
that under (2.2), we have that

$\displaystyle W_{(3)\;\mu}$ $\displaystyle\rightarrow$ $\displaystyle
e^{i2\pi{k_{\mu}^{3}\over 3}}\;W_{(3)\;\mu}~{},$ (8) $\displaystyle
W_{(2)\;\mu}$ $\displaystyle\rightarrow$ $\displaystyle
e^{i2\pi{k_{\mu}^{2}\over 2}}\;W_{(2)\;\mu}~{},$ (9) $\displaystyle
W_{(1)\;\mu}$ $\displaystyle\rightarrow$ $\displaystyle
e^{i2\pi(-{k_{\mu}^{2}\over 2}-{k_{\mu}^{3}\over 3})}\;W_{(1)\;\mu}~{},$ (10)

showing that ${\mathbb{Z}}_{6}^{(1)}$ acts on the above fundamental Wilson
loops (thus, they are genuine line operators in the SM, as discussed in great
detail in Tong:2017oea ).777We also note that the introduction of the 1-form
symmetry via its action on transition functions is closest to the way the
global ${\mathbb{Z}}_{N}$ center-symmetry is introduced in $SU(N)$ lattice
gauge theory, as described in Greensite:2011zz .

###  2.3 The gauging of ${\mathbb{Z}}_{N}^{(1)}\in{\mathbb{Z}}_{6}^{(1)}$ and
$G_{SM}^{N=2,3,6}$

Having explained the way the global ${\mathbb{Z}}_{6}^{(1)}$ 1-form symmetry
appears in the SM in this language, we can now introduce the different
$G_{SM}^{N}$ theories obtained upon the gauging of the
${\mathbb{Z}}_{N=2,3,6}^{(1)}$ subgroups of ${\mathbb{Z}}_{6}^{(1)}$. We
describe the gauging using the ${\mathbb{T}}^{4}$ formulation of the gauge
theory, as the fractional instantons whose role we want to study are only
explicitly known for ${\mathbb{T}}^{4}$.

When the gauge theory is formulated on ${\mathbb{T}}^{4}$ in the Euclidean
path integral formalism, the integral is performed over all gauge fields and
transition functions obeying the cocycle conditions (4) and (5), modulo gauge
transformations. The cocycle conditions on the gauge field transition
functions in (5) are the ones appropriate to the $G_{SM}^{1}=SU(3)\times
SU(2)\times U(1)$ gauge group. As already mentioned, with these cocycle
conditions, the topological charge is an integer (see (17) and (31) below).

Now we come to the gauging of the 1-form symmetry. Adapting the discussion of
Kapustin:2014gua  to our case, the gauging of subgroups of
${\mathbb{Z}}_{6}^{(1)}$ involves two steps: a modification of the gauge
fields’ cocycle conditions (5) by appropriate center elements and a summation
in the path integral over gauge fields obeying all such modified cocycle
conditions (possibly including ${\mathbb{Z}}_{N}$ phase factors called
“discrete theta angles,” introduced in Aharony:2013hda  and described for
$G_{SM}^{N}$ in Tong:2017oea ).

We begin with modified gauge-field cocycle conditions (5) for the SM case:

$\displaystyle\Omega_{(3)\;\mu}(x^{\nu}=a^{\nu})\;\Omega_{(3)\;\nu}(x^{\mu}=0)$
$\displaystyle=$ $\displaystyle e^{i2\pi{n_{\mu\nu}^{(3)}\over
3}}\;\Omega_{(3)\;\nu}(x^{\mu}=a^{\mu})\;\Omega_{(3)\;\mu}(x^{\nu}=0)~{},$
$\displaystyle\Omega_{(2)\;\mu}(x^{\nu}=a^{\nu})\;\Omega_{(2)\;\nu}(x^{\mu}=0)$
$\displaystyle=$ $\displaystyle e^{i2\pi{n_{\mu\nu}^{(2)}\over
2}}\;\Omega_{(2)\;\nu}(x^{\mu}=a^{\mu})\;\Omega_{(2)\;\mu}(x^{\nu}=0)~{},$
(11)
$\displaystyle\Omega_{(1)\;\mu}(x^{\nu}=a^{\nu})\;\Omega_{(1)\;\nu}(x^{\mu}=0)$
$\displaystyle=$ $\displaystyle e^{i2\pi(-{n_{\mu\nu}^{(3)}\over
3}-{n_{\mu\nu}^{(2)}\over
2})}\;\Omega_{(1)\;\nu}(x^{\mu}=a^{\mu})\;\Omega_{(1)\;\mu}(x^{\nu}=0)~{}.$

The $x$-independent integers appearing in the cocycle conditions for $SU(2)$
transition functions, $n_{\mu\nu}^{(2)}=-n_{\nu\mu}^{(2)}$, are defined (mod
$2$) for every 2-plane, while the ones for $SU(3)$,
$n_{\mu\nu}^{(3)}=-n_{\nu\mu}^{(3)}$, are similarly defined (mod $3$). The
$n_{\mu\nu}^{(2),(3)}$ are known as ’t Hooft twists. In the formalism
introduced in Kapustin:2014gua  (which we shall not elaborate upon) the
integers $n_{\mu\nu}^{(2),(3)}$ correspond to the integrals, taken over the
various 2-planes of the torus, of the appropriate 2-form discrete gauge fields
gauging the 1-form ${\mathbb{Z}}_{N}^{(1)}$ symmetry of the $SU(N)$ theory
(here $N=2,3$). The summation over field configurations obeying (2.3) with
$n_{\mu\nu}^{(2)}=\\{1,2\\}$ and $n_{\mu\nu}^{(3)}=\\{1,2,3\\}$ amounts to
performing the path integral over this topological field.888In a Hilbert space
interpretation (whose details we shall also not need) this is a summation over
discrete magnetic flux sectors and a projection on appropriate discrete
electric flux sectors. These sectors were introduced in tHooft:1979rtg ;
tHooft:1981sps . See also Aharony:2013hda ; Cox:2021vsa  for somewhat related
discussions.

The twists of the gauge field cocycle conditions in (2.3) are consistent with
the cocycle conditions (4) of the SM matter fields
$\psi^{\cal{R}}=(q_{L},l_{L},\tilde{e}_{R},\tilde{u}_{R},\tilde{d}_{R},h)$, as
the phase factors cancel out in the products
$\Omega_{(1)\;\mu}^{\cal{R}}\Omega_{(2)\;\mu}^{\cal{R}}\Omega_{(3)\;\mu}^{\cal{R}}$
for all representations $\cal{R}$ in (1). Once again, this is most obvious for
$q_{L}$, whose cocycle condition (4) is satisfied by the product of the three
lines in (2.3), and is easily verified for the other SM matter fields.

The upshot is that the different $G_{SM}^{N}$ gauge theories are distinguished
by the values of the twists of the cocycle conditions (2.3) obeyed by the
transition functions of the gauge and matter fields summed over in the path
integral:999To avoid confusion, we stress that one can, even in the
$G_{SM}^{1}$ theory, introduce boundary conditions on the ${\mathbb{T}}^{4}$
that correspond to nonzero twists $n_{\mu\nu}^{(j)}$, but without summation
over all allowed twists in the path integral. This corresponds to probing the
$G_{SM}^{1}$ theory by the insertion of nondynamical backgrounds for the
${\mathbb{Z}}_{6}^{(1)}$ global symmetry, needed in studies of generalized ’t
Hooft anomalies involving center symmetry, as shown using this language in
Cox:2021vsa . An unambiguous identification of the different SM gauge groups
with the presence or absence of fractional instanton contributions requires
forbidding such nondynamical insertions. This is natural in the context of
embedding the SM in quantum gravity theories, where global symmetries cannot
appear; for example, in the case of $G_{SM}^{(1)}$, the
${\mathbb{Z}}_{6}^{(1)}$ symmetry is broken by the presence of heavy electric
states whose transition functions do not respect (2.2), disallowing nonzero
twists in $G_{SM}^{(1)}$. Likewise, global ${\mathbb{Z}}_{3}^{(1)}$ twists can
be forbidden in $G_{SM}^{(2)}$ and ${\mathbb{Z}}_{2}^{(1)}$ ones in
$G_{SM}^{(3)}$. It is also worth mentioning that there is a $\mathbb{Z}_{3}$
magnetic symmetry in $G_{SM}^{(3)}$ and $G_{SM}^{(6)}$, which is broken by
heavy magnetic charges in theories of quantum gravity. See Tong:2017oea  and
references therein for a discussion of these issues in the present context.

$\displaystyle G_{SM}^{1}=SU(3)\times SU(2)\times U(1):$ $\displaystyle
n_{\mu\nu}^{(3)}\equiv 0,n_{\mu\nu}^{(2)}\equiv 0,\;\forall\mu,\nu,$
$\displaystyle G_{SM}^{2}={SU(3)\times SU(2)\times
U(1)\over{\mathbb{Z}}_{2}}:$ $\displaystyle
n_{\mu\nu}^{(2)}=\\{1,2\\},\text{with}~{}n_{\mu\nu}^{(3)}\equiv
0\;\forall\mu,\nu,$ $\displaystyle G_{SM}^{3}={SU(3)\times SU(2)\times
U(1)\over{\mathbb{Z}}_{3}}:$ $\displaystyle
n_{\mu\nu}^{(3)}=\\{1,2,3\\},\text{with}~{}n_{\mu\nu}^{(2)}\equiv
0\;\forall\mu,\nu,$ $\displaystyle G_{SM}^{6}={SU(3)\times SU(2)\times
U(1)\over{\mathbb{Z}}_{6}}:$ $\displaystyle
n_{\mu\nu}^{(2)}=\\{1,2\\},n_{\mu\nu}^{(3)}=\\{1,2,3\\}.$ (12)

From our point of view, the most important effect of the gauging of
${\mathbb{Z}}_{6}^{(1)}$ subgroups in the SM is the fact that the topological
charge of gauge fields obeying the modified cocycle conditions (5) can be
fractional. It was first argued by ’t Hooft in tHooft:1979rtg ; tHooft:1981sps
, and rigorously shown in vanBaal:1982ag , that the fractional part of the
topological charge is determined by the twists $n_{\mu\nu}^{(2),(3)}$. In
particular, the topological charge for the $j$-th gauge group is (recalling
$F_{(j)}={1\over 2}F_{(j)\;\mu\nu}dx^{\mu}\wedge dx^{\nu}$ and noting that
there is no trace for $j=1$):

$\displaystyle Q_{(j)}={1\over 8\pi^{2}}\int\;\text{tr}F_{(j)}\wedge
F_{(j)}=\left\\{\begin{array}[]{c}~{}~{}{1\over 32\pi^{2}}\int
d^{4}x\;F_{(j)}^{\mu\nu\;a}\;\tilde{F}_{(j)\;\mu\nu}^{a},~{}j=2,3,{}\cr\cr{1\over
16\pi^{2}}\int
d^{4}x\;F_{(1)}^{\mu\nu}\;\tilde{F}_{(1)\;\mu\nu},~{}j=1,\end{array}\right.~{}$
(16)

where $\tilde{F}_{(j)\;\mu\nu}\equiv{1\over
2}\epsilon_{\mu\nu\lambda\sigma}F_{(j)}^{\lambda\sigma}$. For $j=2$ and $j=3$,
it was shown in tHooft:1979rtg ; tHooft:1981sps ; vanBaal:1982ag  that $SU(j)$
field configurations obeying the cocycle conditions (2.3) have fractional
topological charge

$Q_{(j)}=-{\text{Pf}(n^{(j)})\over
j}+k~{},~{}j=2,3,~{}\text{and}~{}k\in{\mathbb{Z}}~{}.$ (17)

Here, $\text{Pf}(n)\equiv{1\over
8}\epsilon^{\mu\nu\lambda\sigma}n_{\mu\nu}n_{\lambda\sigma}$ is an integer for
an antisymmetric matrix with integer entries. The simplest example is to take
$n_{12}=-n_{21}=n_{34}=-n_{34}=1$ as the only nonzero entries of $n_{\mu\nu}$,
giving $\text{Pf}(n)=n_{12}n_{34}=1$.

We used eqn. (16) to define the quantity $Q_{(1)}$ characterizing the $U(1)$
gauge field configuration, noting that there is no $U(1)$ characteristic class
associated with the ${\mathbb{T}}^{4}$. Instead, the $U(1)$ background is
characterized by the fluxes through noncontractible 2-planes (“first Chern
class”). With nontwisted cocycle conditions for the $U(1)$, i.e. with $U(1)$
transition functions obeying (5), field configurations are characterized by
$2\pi\times$integer fluxes through the various 2-planes, e.g. $\oint
dx^{1}dx^{2}F_{(1)\;12}=2\pi{\mathbb{Z}}$. It is easy to see that our
$Q_{(1)}$ of (17) then takes integer values (see eqn. (31) below). For $U(1)$
field configurations obeying the cocycle conditions from the last line in
(2.3), which are modified by the ${\mathbb{Z}}_{6}$ twists $\bar{n}_{\mu\nu}$
(with ${\bar{n}_{\mu\nu}\over
6}\equiv{-2n_{\mu\nu}^{(3)}-3n_{\mu\nu}^{(2)}\over 6}$), these fluxes through
2-planes are now valued in $2\pi{\mathbb{Z}}\over 6$ and $Q_{(1)}$ is further
reduced, as also seen in (31) below.

The fact that the topological charge is fractional leads one to expect that
the physical effects of such topological fluctuations may be less suppressed
than the ones due to the well-understood integer topological charge BPST
instantons. The gauge field Euclidean action (2) can be rewritten in terms of
the topological charge $Q_{(j)}$ (for $j=2,3$) as

$S_{(j)}=\pm{8\pi^{2}\over g_{(j)}^{2}}Q_{(j)}+{1\over 8g_{(j)}^{2}}\int
d^{4}x(F_{(j)\;\mu\nu}\mp\tilde{F}_{(j)\;\mu\nu})^{2}$ (18)

where the signs are correlated, showing that the action for field
configurations with $Q_{(j)}>0$ are minimized by self-dual fields
$F_{(j)\;\mu\nu}=\tilde{F}_{(j)\;\mu\nu}$ (and anti self-dual ones for
negative topological charge). Thus, the minimal action in each topological
sector $Q_{(j)}$ (for $j=2,3$) is given by the familiar Bogomolny’ bound

$S_{min}={8\pi^{2}\over g_{(j)}^{2}}|Q_{(j)}|={8\pi^{2}\over
g_{(j)}^{2}}|{{\text{Pf}}(n^{(j)})\over j}-k|~{}.$ (19)

For example, with $|Q_{(j)}|={1\over j}$, the action is smaller than that of a
BPST instanton. In particular, solutions whose action scales as
$8\pi^{2}/(g^{2}N)$, as above, with $j\rightarrow N$, give contributions to
the path integral which survive in the large-$N$ limit—which was the initial
motivation of their study in tHooft:1981nnx .

Before we continue, we recall the anomaly equation for the $U(1)_{B}$ baryon
number of (1),

$\partial_{\mu}j^{\mu}_{B}={1\over 32\pi^{2}}\
F_{(2)}^{\mu\nu\;a}\>\tilde{F}_{(2)\;\mu\nu}^{a}-18\;{1\over 16\pi^{2}}\
F_{(1)}^{\mu\nu}\;\tilde{F}_{(1)\;\mu\nu},$ (20)

which implies that the change of baryon number is $\Delta
B=Q_{(2)}-18Q_{(1)}$, for a single generation of SM fields. In the next
Section, we describe the $G_{SM}^{N}$ fractional instantons and study their
effect.

##  3 The fractional instantons of $G_{SM}^{N}$, their actions, and the
$\Delta B\neq 0$ amplitudes

###  3.1 General remarks and concerns

Before we exhibit the fractional instantons for the $G_{SM}^{N}$ theories that
we shall use as our illustrative example, we note that the current knowledge
of such solutions is somewhat limited. The fact that the topological charge is
fractional has been known since tHooft:1979rtg ; tHooft:1981sps ;
vanBaal:1982ag , but there are only a few explicit solutions.

Below, as a first step to investigating the possible role of fractional
instantons, we make two simplifications, both of which should be reconsidered
in future studies. First, we set the vev of the Higgs field to zero assuming a
symmetric phase. Second, we use the constant field strength solutions on
${\mathbb{T}}^{4}$ of tHooft:1981nnx , and, in particular, their abelian
version. These analytic solutions saturate the Bogomolny’ bound (19) and are
stable only for a particular ratio of sides of the ${\mathbb{T}}^{4}$
vanBaal:1984ar . In particular, using these solutions, we are forced to take
the ${\mathbb{T}}^{4}$ sides to obey

${a^{3}a^{4}\over a^{1}a^{2}}=1~{}.$ (21)

This constraint ensures that both the $j=2$ and $j=3$ fractional instantons
are self-dual and obey the Bogomolny’ bound (19). The $SU(3)$ action of these
solutions, however, is only $2/3$ the BPST instanton action (rather than the
minimal101010We note that ’t Hooft tHooft:1981nnx  constructed $SU(N_{c}=3)$
nonabelian constant field strength solutions of the minimal possible action
(19), equal to $1/3$ times the BPST action. However, the relation replacing
(21) and ensuring self-duality then has the ratio $a^{3}a^{4}/(a^{1}a^{2})$
replaced with a quantity $f(N_{c},k,l)$ (with $k+l=N_{c}$) that depends on
$N_{c}$. Hence, self duality is not obeyed for both $N_{c}=2$ and $N_{c}=3$,
prompting us to use the abelian version. The generalization of these solutions
to product-group structures such as the SM has not been studied. $1/3$), while
the $SU(2)$ action is the minimum possible $1/2$ of the BPST action.

Clearly, the the constraint (21) is a source of concern. First, it limits the
ability to independently vary the shape and size of the spatial
${\mathbb{T}}^{3}$ and of the time direction extent $a_{4}$. In particular, it
does not incorporate zero-temperature fractional instantons on
${\mathbb{R}}\times{\mathbb{T}}^{3}$ vanBaal:2000zc . Further, since the
solutions have field strength uniform over the torus, one expects them to be
relevant for possible physical applications only if the size of the torus is
smaller than the causally connected part of the universe. As we discuss below,
this fact, along with the cosmological bounds on the torus size, would then
imply that the SM fractional instantons are mostly of academic interest.

While this may ultimately turn out to be the case, let us make several remarks
pointing towards a possible way out. There exist numerical studies as well as
an analytic approach towards finding fractional instantons on tori not
restricted by (21) or its generalization. More precisely, fractional instanton
solutions can be constructed by relaxing the ratio ${a^{3}a^{4}\over
a^{1}a^{2}}=f(N,k,l)$, with $k+l=N$ (’t Hooft’s generalization of (21))
towards arbitrary values of $a^{\mu}$, using a perturbative approach,
developed in GarciaPerez:2000aiw ; Gonzalez-Arroyo:2019wpu . It is shown there
that the fractional instanton solutions thus obtained have a nontrivial
profile of the field strength over the torus. The instanton size is, however,
still set by the torus dimensions (the equations studied are the scale-free
Yang-Mills equations on a torus with twisted boundary conditions, as in the
top line of (2.3)). It appears, then, that fractional instantons whose size
does not scale with the size of space (time) would require the introduction of
an extra scale. This could be related to the strong scale of QCD or to the
electroweak scale,111111Notice that it is precisely the appearance of an extra
scale, due to the holonomy vev, that reveals the fractional instantons in the
much better understood case of ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$
Lee:1997vp ; Kraan:1998pm .  the Higgs vev that we neglect throughout. It is
thus of interest to study especially (in the present context) the effect of
re-introducing the Higgs vev.

With our concerns spelled out, we now describe the fractional instantons on
${\mathbb{T}}^{4}$.

###  3.2 The fractional instantons in $G_{SM}^{N}$

To describe the fractional instantons explicitly, we introduce $H^{(3)}$ and
$H^{(2)}$, diagonal matrices in the Cartan subalgebra of $SU(3)$ and $SU(2)$:

$\displaystyle H^{(3)}=\text{diag}({1\over 3},{1\over 3},-1+{1\over
3}),~{}~{}H^{(2)}=\text{diag}({1\over 2},-{1\over
2}),~{}\text{tr}H^{(3)}H^{(3)}={2\over 3},~{}\text{tr}H^{(2)}H^{(2)}={1\over
2}.$ (22)

The fractional instantons are the following field configurations, where
$n^{(3),(2)}_{\mu\nu}$ are the twists appearing in (2.3), as we show below:

$\displaystyle A_{(j)}$ $\displaystyle=$
$\displaystyle\sum\limits_{\mu,\nu=1}^{4}{\pi
x^{\nu}dx^{\mu}n_{\mu\nu}^{(j)}\over
a^{\mu}a^{\nu}}H^{(j)},~{}\text{for}~{}j=2,3,$ $\displaystyle A_{(1)}$
$\displaystyle=$ $\displaystyle-\sum\limits_{\mu,\nu=1}^{4}{\pi
x^{\nu}dx^{\mu}\over
a^{\mu}a^{\nu}}\left({3n_{\mu\nu}^{(2)}+2n_{\mu\nu}^{(3)}\over
6}+n^{(1)}_{\mu\nu}\right)~{}.$ (23)

In the last line, we introduced an integer-valued
$n^{(1)}_{\mu\nu}=-n^{(1)}_{\nu\mu}$ twist reflecting the freedom to add
$2\pi{\mathbb{Z}}$ fluxes of the $U(1)$ field. Notice that the gauge fields
(3.2) are not periodic, but instead obey

$\displaystyle A_{(j)}(x^{\lambda}=a^{\lambda})$ $\displaystyle-$
$\displaystyle A_{(j)}(x^{\lambda}=0)=~{}~{}\sum\limits_{\mu}^{4}{\pi
dx^{\mu}n_{\mu\lambda}^{(j)}\over
a^{\mu}}H^{(j)}\equiv-i\Omega_{(j)\;\lambda}d\Omega_{(j)\;\lambda}^{-1},~{}\text{for}~{}j=2,3,$
$\displaystyle A_{(1)}(x^{\lambda}=a^{\lambda})$ $\displaystyle-$
$\displaystyle A_{(2)}(x^{\lambda}=0)=-\sum\limits_{\nu=1}^{4}{\pi
dx^{\mu}\over a^{\mu}}\left({3n_{\mu\lambda}^{(2)}+2n_{\mu\lambda}^{(3)}\over
6}+n^{(1)}_{\mu\nu}\right)\equiv-i\Omega_{(1)\;\lambda}d\Omega_{(1)\;\lambda}^{-1}~{},$

from which we can read off the transition functions

$\displaystyle\Omega_{(j)\;\lambda}=e^{-i\sum\limits_{\mu}{\pi
x^{\mu}n^{(j)}_{\mu\lambda}\over
a^{\mu}}H^{(j)}},~{}\text{for}~{}j=2,3,~{}\text{while}~{}\Omega_{(1)\;\lambda}=e^{i\sum\limits_{\mu}{\pi
x^{\mu}\over a^{\mu}}\left({3n_{\mu\lambda}^{(2)}+2n_{\mu\lambda}^{(3)}\over
6}+n^{(1)}_{\mu\nu}\right)}.$ (25)

We can now ask what cocycle conditions the above transition functions obey. To
this end, we compute the l.h.s. and r.h.s. of (2.3). For $j=3$ and $j=2$, we
have for the l.h.s. of (2.3):

$\Omega_{(j)\;\lambda}(x^{\rho}=a^{\rho})\Omega_{(j)\;\rho}(x^{\lambda}=0)=e^{-i\pi
n^{(j)}_{\rho\lambda}H^{(j)}}e^{-i\sum\limits_{\mu\neq\rho}{\pi
x^{\mu}n^{(j)}_{\mu\lambda}\over
a^{\mu}}H^{(j)}}e^{-i\sum\limits_{\mu\neq\lambda}{\pi
x^{\mu}n^{(j)}_{\mu\rho}\over a^{\mu}}H^{(j)}},$ (26)

while the combination of transition functions (25) appearing on the r.h.s. of
(2.3) is

$\Omega_{(j)\;\rho}(x^{\lambda}=a^{\lambda})\Omega_{(j)\;\lambda}(x^{\rho}=0)=e^{-i\pi
n^{(j)}_{\lambda\rho}H^{(j)}}e^{-i\sum\limits_{\mu\neq\lambda}{\pi
x^{\mu}n^{(j)}_{\mu\rho}\over
a^{\mu}}H^{(j)}}e^{-i\sum\limits_{\mu\neq\rho}{\pi
x^{\mu}n^{(j)}_{\mu\lambda}\over a^{\mu}}H^{(j)}}.$ (27)

Comparing the two expressions (26) and (27), using the antisymmetry of
$n^{(j)}_{\lambda\rho}$, we find that (25) obey the cocycle condition

$\Omega_{(j)\;\lambda}(x^{\rho}=a^{\rho})\Omega_{(j)\;\rho}(x^{\lambda}=0)=e^{i2\pi
n^{(j)}_{\lambda\rho}H^{(j)}}\Omega_{(j)\;\rho}(x^{\lambda}=a^{\lambda})\Omega_{(j)\;\lambda}(x^{\rho}=0)~{}.$
(28)

Now, recalling from (22) that $H^{(3)}=\text{diag}({1\over 3},{1\over
3},-1+{1\over 3})$ and $H^{(2)}=\text{diag}({1\over 2},-{1\over 2})$, we see
that (28) produces precisely (2.3) for $j=2,3$. It is trivial to check that
the remaining $U(1)$ condition in (2.3) also holds; notice that the
$n_{\mu\nu}^{(1)}\in{\mathbb{Z}}$ factor does not enter the cocycle condition.

The upshot of the above discussion is that we have shown that (3.2) are gauge
field backgrounds obeying the appropriate cocycle conditions (2.3) for gauging
the ${\mathbb{Z}}_{6}^{(1)}$ subgroups, as described in (2.3). Having constant
field strength and being abelian, they also solve the gauge field equations of
motion as well as the scalar $h$ equation of motion with vev set to zero, i.e.
assuming the symmetric phase. The stability of (3.2) is a consequence of the
self-duality, which we impose below.

The topological charges (16) of our fractional instantons (3.2) are now easily
computed. In fact, the $Q_{(j)}$ only depend on the transition functions (25)
but for the constant field strength background we can compute them using the
field strength. For $j=2,3$, we have $F_{(j)}=\sum\limits_{\mu\nu}{\pi
dx^{\nu}\wedge dx^{\mu}n_{\mu\nu}^{(j)}\over a^{\mu}a^{\nu}}H^{(j)}$, which
yields, using (22)

$Q_{(2)}={\text{Pf}(n^{(2)})\over 2},~{}Q_{(3)}={2\text{Pf}(n^{(3)})\over
3}~{}.$ (29)

For the $U(1)$, we find instead

$\displaystyle F_{(1)}=-\sum\limits_{\mu\nu}{\pi dx^{\nu}\wedge dx^{\mu}\over
a^{\mu}a^{\nu}}\left({3n_{\mu\nu}^{(2)}+2n_{\mu\nu}^{(3)}\over
6}+n^{(1)}_{\mu\nu}\right)\,,$ (30)

yielding

$Q_{(1)}={1\over 8\pi^{2}}\int F_{(1)}\wedge F_{(1)}={1\over
36}\;\text{Pf}(6n^{(1)}+3n^{(2)}+2n^{(3)})~{}.$ (31)

As promised earlier, the $U(1)$ topological charge $Q_{(1)}$ is integer for
vanishing $n^{(2)}$ and $n^{(3)}$. From (20) we can now immediately compute
the $\Delta B$ due to our fractional instantons

$\Delta B={\text{Pf}(n^{(2)})\over 2}-{1\over
2}\;\text{Pf}(6n^{(1)}+3n^{(2)}+2n^{(3)})~{}.$ (32)

It is easy to see that the change of baryon number (32) is always integer.

The actions of the fractional instantons (3.2) can also be computed. The
smallest actions of the $SU(2)$ and $SU(3)$ fractional instantons are obtained
for solutions on a ${\mathbb{T}}^{4}$ obeying (21). The Bogomolny’ bound (19)
holds after setting $\text{Pf}(n^{(2)})=\text{Pf}(n^{(3)})=1$ by, for example,
taking the only nonzero entries
$n_{12}^{(j)}=n_{34}^{(j)}=-n_{21}^{(j)}=-n_{43}^{(j)}=1$ for $j=2,3$:

$\displaystyle S_{(2)}=\frac{4\pi^{2}n_{12}^{(2)}}{g_{2}^{2}}\,,\quad
S_{(3)}=\frac{16\pi^{2}n_{12}^{(3)}}{3g_{3}^{2}}\,.$ (33)

We note that we have kept $n_{12}^{(j)}$ as a book-keeping device: according
to our discussion near eqn. (2.3), we set $n_{12}^{(2)}=n_{12}^{(3)}=1$ upon
gauging the full ${\mathbb{Z}}_{6}^{(1)}$, we set
$n_{12}^{(2)}=1,n_{12}^{(3)}=0$ when gauging ${\mathbb{Z}}_{2}^{(1)}$, and
$n_{12}^{(2)}=0,n_{12}^{(3)}=1$ when gauging ${\mathbb{Z}}_{3}^{(1)}$. These
values for $n^{(j)}$, $j=2,3$, give rise to the smallest topological charge
for the given gauging. The fractional instantons thus obtained are (anti)
self-dual and are necessary stable, as was shown in vanBaal:1984ar .

The action of the $U(1)$ instantons, on the contrary, needs to be directly
computed starting from the $U(1)$ gauge field background (3.2). Without loss
of generality, we can consider gauge fields in the $1-2$ and $3-4$ planes and
take $n_{12}^{(j)}=-n_{21}^{(j)}=n_{34}^{(j)}=-n_{34}^{(j)}=1$ as the only
nonzero entries of $n_{\mu\nu}^{(j)}$, for $j=2,3$, which gives:

$\displaystyle
S_{(1)}=\frac{2\pi^{2}}{g_{1}^{2}}\left[\left(\frac{n_{12}^{(3)}}{3}+\frac{n_{12}^{(2)}}{2}+n_{12}^{(1)}\right)^{2}+\left(\frac{n_{12}^{(3)}}{3}+\frac{n_{12}^{(2)}}{2}+n_{34}^{(1)}\right)^{2}\right]\,.$
(34)

Notice that $U(1)$ solutions with $n_{12}^{(1)}\neq n_{34}^{(1)}$ are not
(anti) self-dual. Yet, no unstable modes are expected in this case since the
quantum fluctuations about $U(1)$ classical background are always free.

###  3.3 Comparing the $\Delta B\neq 0$ rates

As explained above, different choices of $n_{12}^{(2)}$, $n_{12}^{(3)}$
correspond to gauging different subgroups of $\mathbb{Z}_{6}^{(1)}$ (as per
(2.3)) and yield different values of $\Delta B$; the latter also depends on
the freedom to add integer $U(1)$ fluxes $n_{12}^{(1)}$, $n_{34}^{(1)}$.

Thus, in Table 35, we show, for every gauged $1$-form center symmetry, the
values of $n_{12}^{(1)}$ and $n_{34}^{(1)}$ that yield the smallest $U(1)$
actions and the corresponding baryon number violation for $n_{f}$ SM
generations.121212 Notice that the smallest $U(1)$ action in the
gauged-$\mathbb{Z}^{(1)}_{6}$ case does not result in baryon number violation,
and therefore, we needed to consider the next to the smallest action obtained
by turning on integer-quantized $U(1)$ fluxes denoted by $n^{(1)}$. The way
the scale $\mu_{critical}$ is defined is explained in the text; its value is
given for the case of $n_{f}=3$ generations.

$\begin{tabular}[]{|c|c|c|c|c|c|}\hline\cr Gauged $1$-form
center&$n_{12}^{(1)}$&$n_{34}^{(1)}$&Smallest $U(1)$ action&$\Delta
B$&$\mu_{\scriptsize\mbox{critical}}$ (GeV)\\\
\hline\cr\hline\cr$\mathbb{Z}^{(1)}_{6}$&$-1$&$-1$&$\frac{\pi^{2}}{9g_{1}^{2}}$&$0$&-\\\
\hline\cr$\mathbb{Z}^{(1)}_{6}$&$-1$&$0$&$\frac{13\pi^{2}}{9g_{1}^{2}}$&$3n_{f}$&$6\times
10^{34}$\\\
\hline\cr$\mathbb{Z}^{(1)}_{6}$&$0$&$-1$&$\frac{13\pi^{2}}{9g_{1}^{2}}$&$3n_{f}$&$6\times
10^{34}$\\\
\hline\cr$\mathbb{Z}^{(1)}_{3}$&$0$&$0$&$\frac{4\pi^{2}}{9g_{1}^{2}}$&$-2n_{f}$&$2.7\times
10^{33}$\\\
\hline\cr$\mathbb{Z}^{(1)}_{2}$&$0$&$0$&$\frac{\pi^{2}}{g_{1}^{2}}$&$-4n_{f}$&$1.5\times
10^{24}$\\\ \hline\cr\end{tabular}\,.$ (35)

Our next task is to compare the vacuum to vacuum amplitude of $\Delta B$
mediated by the fractional-charge instantons with the amplitude of the same
quantity due to the weak self-dual BPST instantons. Ideally, one would want to
conduct this study at zero temperature on a background manifold
$\mathbb{M}^{3}\times{\mathbb{R}}$ that admits both types of instantons.
Unfortunately, there is no known background that accommodates analytical
solutions of both types. To make matter worse, even on
$\mathbb{T}^{3}\times{\mathbb{R}}$, as we discussed above, analytical
solutions of fractional instantons are still lacking. With these obstacles,
our best option is to conduct our study on the simple manifold
$\mathbb{T}^{4}$. Here, we argue that our treatment gives the correct order of
magnitude estimate, which should not be greatly altered by the fine details of
the solutions. First, the action of the self-dual fractional instanton, in the
Bogomolny’ bound, depends only on the topological charge and one can proclaim
that deforming $\mathbb{T}^{4}$ into $\mathbb{T}^{3}\times{\mathbb{R}}$ (by
making one of the cycles very large) cannot change the action.131313In the
following, we use the $SU(3)$ solutions with topological charge $\frac{2}{3}$
instead of the solutions with the minimum charge $\frac{1}{3}$ described by ’t
Hooft tHooft:1981nnx , see Footnote 10. We also assume that the $U(1)$
topological charge and action of the constant-field solution (30) does not
change dramatically during this process, i.e., when we make one of the
${\mathbb{T}}^{4}$ cycles very large. Second, although it is true that there
is no solution of self-dual BPST instanton on $\mathbb{T}^{4}$ with zero
twists Braam:1988qk , adding a twist removes the obstruction to the existence
of the solution.141414See vanBaal:2000zc  for a discussion. Simply, if one
starts with a self-dual BPST instanton on $\mathbb{R}^{4}$, then deforming it
to $\mathbb{T}^{4}$ will be obstructed since the latter manifold enjoys a
continuous degeneracy in the moduli space. This degeneracy, however, will be
lifted when we apply the twist. Yet, from a practical point of view, one can
get very close to self-duality even in the absence of twists. In our case the
twists come about naturally, since we need them anyway to obtain the
fractional solutions. This is a win-win situation for both the BPST and
fractional solutions. Again, one can imagine a process where we take one of
the cycles of ${\mathbb{T}}^{4}$ large enough to obtain BPST solutions on
$\mathbb{T}^{3}\times{\mathbb{R}}$. A third important point is that baryon
number violation is governed mainly by anomalies. The latter depend only on
the cohomology classes and not on the specific background manifold, and thus,
we expect the computations on $\mathbb{T}^{4}$ or
$\mathbb{T}^{3}\times\mathbb{R}$ yield the same estimates.

Given all the above caveats and their suggested resolutions, we proceed to our
computations. We begin with $\mathbb{T}^{4}$ of a size much smaller than the
inverse TeV scale and, therefore, we consistently set the Higgs vev to zero
and use the solutions (3.2) and the corresponding action for the fractional
instantons. As we discussed above, finding BPST instanton (approximate)
solutions on $\mathbb{T}^{4}$ should also be possible, with a scale modulus
cutoff by the torus size. We then declare that the actions of the BPST and
fractional instantons will not be greatly modified as we take one of the
$\mathbb{T}^{4}$ cycles large enough, while keeping the $\mathbb{T}^{3}$ size
below the inverse TeV scale, such that we approach the zero-temperature limit
$\mathbb{T}^{3}\times{\mathbb{R}}$. In this limit, the small size of
$\mathbb{T}^{3}$ will set the energy scale of the problem. We also give the SM
fermions periodic boundary conditions along the spatial directions
$\mu=1,2,3$. Effectively, the problem is reduced to quantum mechanics, where
there are two types of nonperturbative solutions that compete with each other
to win the baryon-number-violation race. The action of the solution(s) is the
highest of the barriers between the different vacuum components, and the
solution with the smallest action will more effectively violate the baryon
number. In this regime the schematic form of ’t Hooft vertex in the background
of fractional instantons is given by (we consider the contribution from one
family of quarks and leptons and the field insertions $q_{L}$ etc. are the
fermion zero modes)

$\displaystyle{\cal T}_{\scriptsize\mbox{fraction}}\sim
e^{-\left(S_{(1)}+S_{(2)}+S_{(3)}\right)}(q_{L})^{{\cal
I}_{q_{L}}}(l_{L})^{{\cal I}_{l_{L}}}(\tilde{e}_{R})^{{\cal
I}_{\tilde{e}_{R}}}(\tilde{u}_{R})^{{\cal
I}_{\tilde{u}_{R}}}(\tilde{d}_{R})^{{\cal I}_{\tilde{d}_{R}}}\,,$ (36)

where ${\cal I}_{q_{L}}$, etc. are the Dirac indices:

$\displaystyle{\cal I}_{q_{L}}$ $\displaystyle=$ $\displaystyle
2Q_{(3)}+3Q_{(2)}+6Q_{(1)}\,,\quad{\cal
I}_{l_{L}}=Q_{(2)}+18Q_{(1)}\,,\quad{\cal I}_{\tilde{e}_{R}}=36Q_{(1)}\,,$
$\displaystyle{\cal I}_{\tilde{u}_{R}}$ $\displaystyle=$ $\displaystyle
Q_{(3)}+48Q_{(1)}\,,\quad{\cal I}_{\tilde{d}_{R}}=Q_{(3)}+12Q_{(1)}.$ (37)

The indices are obtained from the index theorem, which gives the number of the
Weyl zero modes in the background of the fractional-charge instanton. The
indices (37), like $\Delta B$ of (32), are always integer, for fermions whose
transition functions are consistent with the cocycle conditions (4, 2.3) (this
can also be explicitly checked). Inserting the vertex into the SM partition
function and applying a $U(1)_{B}$ global transformation gives $\Delta
B={1\over 3}({\cal I}_{q_{L}}-{\cal I}_{\tilde{u}_{R}}-{\cal
I}_{\tilde{d}_{R}})=Q_{(2)}-18Q_{(1)}$, which is exactly the result from the
anomaly equation (20). Let us compare (36) to ’t Hooft vertex from the weak
BPST instantons that give the same $\Delta B$ (per family)

$\displaystyle{\cal T}_{\scriptsize\mbox{BPST}}\sim e^{-|\Delta
B|S_{\scriptsize\mbox{BPST}}}\left(q_{L}^{3}l_{L}\right)^{|\Delta B|}\,.$ (38)

Bearing in mind that the actions are coupling constant dependent, the sum of
the actions $S_{(1)}+S_{(2)}+S_{(3)}$ could become smaller than the action of
the BPST instantons (times $\Delta B$), which, in turn, results in a larger
amplitude for the fractional-instanton-mediated process. Using (33), (34), and
$S_{\scriptsize\mbox{BPST}}=\frac{8\pi^{2}}{g_{2}^{2}}$, the condition
$S_{(1)}+S_{(2)}+S_{(3)}<|\Delta B|S_{\scriptsize\mbox{BPST}}$ can be
rewritten as:

$\displaystyle\frac{16\pi^{2}n_{12}^{(3)}}{3g_{3}^{2}(\mu)}+\frac{2\pi^{2}}{g_{1}^{2}(\mu)}\left[\left(\frac{n_{12}^{(3)}}{3}+\frac{n_{12}^{(2)}}{2}+n_{12}^{(1)}\right)^{2}+\left(\frac{n_{12}^{(3)}}{3}+\frac{n_{12}^{(2)}}{2}+n_{34}^{(1)}\right)^{2}\right]<\frac{8\pi^{2}}{g_{2}^{2}(\mu)}\left(|\Delta
B|-\frac{n_{12}^{(2)}}{2}\right)\,.$ (39)

As above, $n_{12}^{(2)}$ and $n_{12}^{(3)}$ (taken to be either $0$ or $1$)
are used as a book-keeping device to track the gauging of
$\mathbb{Z}_{6}^{(1)}$ or subgroups of it, while $n_{12}^{(1)}$ and
$n_{34}^{(1)}$ are as in Table 35.

As we emphasized in the inequality (39), the coupling constants $g_{1,2,3}$
depend on some energy scale $\mu$. Above the electroweak scale, the Higgs is
in the symmetric phase and the only scale in the problem is the
$\mathbb{T}^{3}$ size, which works as an IR cutoff and sets the energy scale
$\mu$. To determine the size of ${\mathbb{T}}^{3}$ that satisfies the
inequality (39), we run the coupling constants from the weak scale (which is
taken to be the $Z$-boson mass $M_{Z}$) to the scale $\mu$. The running is
governed by:

$\displaystyle\frac{8\pi^{2}}{g_{i}^{2}(\mu)}$ $\displaystyle=$
$\displaystyle\frac{8\pi^{2}}{g_{i}^{2}(M_{Z})}+b_{i}\log\left(\frac{\mu}{M_{Z}}\right)\,,\quad
i=1,2,3$ $\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle-(80n_{f}+6n_{H})\,,\quad
b_{2}=\frac{22}{3}-\frac{4}{3}n_{f}-\frac{n_{H}}{6}\,,\quad
b_{3}=11-\frac{4}{3}n_{f}\,,$ (40)

for $n_{f}$ families of particles and $n_{H}$ Higgs doublets. Notice that
while $b_{2}$ and $b_{3}$ are standard (see, e.g., Martin:1997ns ), the $U(1)$
coupling constant $g_{1}$ as well as its $\beta$-function are related to the
“standard” hypercharge via $g_{1}=g_{Y}/6$. We also use
$g_{3}^{2}(M_{Z})/4\pi\cong 0.118$, $g_{EM}^{2}(M_{Z})/4\pi\cong 1/128$,
$g_{Y}(M_{Z})=g_{EM}(M_{Z})/\cos\theta_{W}(M_{Z})$,
$g_{2}(M_{Z})=g_{EM}(M_{Z})/\sin\theta_{W}(M_{Z})$, and
$\sin^{2}\theta_{W}(M_{Z})\cong 0.23$, where $\theta_{W}(M_{Z})$ is the
Weinberg angle.

Using the SM particle content, setting $n_{f}=3$ and $n_{H}=1$, and making use
of (40) in (39), we solve for the critical energy scale
$\mu_{\scriptsize\mbox{critical}}$ at which the effects of the fractional-
charge instantons surpass the BPST ones. We list the values of
$\mu_{\scriptsize\mbox{critical}}$ in the last column in Table 35. Clearly,
all values of $\mu_{\scriptsize\mbox{critical}}$ are super-Planckian for the
gauging of $\mathbb{Z}_{6}^{(1)}$ or its subgroups. This is attributed to the
fact that the hyper-charge coupling constant is extremely small at the weak
scale. One can push $\mu_{\scriptsize\mbox{critical}}$ into sub-Planckian
values provided that extra matter charged under $U(1)$ participates in the
running.151515It is important to stress that any extra charged matter must
respect the modified cocycle conditions (2.3). We also assume that such new
particles have mass above the TeV scale. As an example, we can assume $\sim
60$ extra scalar particles of mass $\sim 10^{5}$ GeV, which are sterile under
$SU(2)$ and $SU(3)$ but have a charge $6$ under $U(1)$. They push the critical
scale down to $\mu_{\scriptsize\mbox{critical}}\sim 10^{13}$ GeV for
$G_{SM}^{2}$. Even at this energy, the $U(1)$ coupling is still in the
perturbative regime.

Finally, we might be interested in the energy scale at which the fractional
instantons become more important than BPST solutions with minimum topological
charge $Q_{(2)}=1$. In this case we simply set $|\Delta B|=1$ in the
inequality (39) to find that $\mu_{\scriptsize\mbox{critical}}$ are not
dramatically bigger than the values in Table 35.

##  4 Cosmological bounds and discussion

The strongest cosmological constraints on the size of a ${\mathbb{T}}^{3}$
with sides of order $L$ come from the CMB and indicate that
$L>{\cal{O}}(few)\times L_{0}$, where $L_{0}\sim 14$ Gpc is the radius of the
last scattering surface, see e.g. Levin:1998qq ; Bond:1997ym ;
Aslanyan:2013lsa . If compactness of the space is a necessary condition to
make sense of the fractional-charge instantons, which is the conservative
perspective that we have been following throughout our analysis (however, see
the discussion below), then these instantons could have played a role in the
early Universe provided that there is extra charged matter and the size of
${\mathbb{T}}^{3}$ was smaller than or equal to the horizon size when the
Universe was hot and dense.

To this end, we trace back the size of ${\mathbb{T}}^{3}$ before the time of
last scattering, when the Universe was deep in the radiation-dominated era at
temperature $T$ and the instanton effects are expected to be important. We use
the Friedmann equation of a single-component Universe
$3M_{P}^{2}H^{2}(t)=\frac{\pi^{2}}{15}n_{*}T^{4}$ ($M_{P}$ is the reduced
Planck mass, $H(t)$ is the Hubble parameter, and $n_{*}$ is the number of
effective degrees of freedom), along with the fact that the scale factor
behaves as $a\propto 1/\sqrt{H(t)}$. Recalling that the horizon size is
$L_{H}(t)=H^{-1}(t)$, we can use this information to obtain the horizon size,
the physical size $L_{{\mathbb{T}}^{3}}(T)$ of ${\mathbb{T}}^{3}$, and the
ratio $\frac{L_{{\mathbb{T}}^{3}}(T)}{L_{H}(T)}$ at temperature $T$:

$\displaystyle L_{H}(T)\sim\frac{M_{P}}{T^{2}}\,,\quad
L_{{\mathbb{T}}^{3}}(T)={\cal{O}}(few)\frac{M_{P}}{T_{l}T}\,,\quad\frac{L_{{\mathbb{T}}^{3}}(T)}{L_{H}(T)}={\cal{O}}(few)\times\frac{T}{T_{l}}\,,$
(41)

where $T_{l}$ is the CMB temperature at the time of last scattering. Since we
are interested in temperature much higher than $T_{l}$, we immediately see
that the ${\mathbb{T}}^{3}$ was always bigger than the horizon size. One might
immediately run into the conclusion that such fractional solutions should be
dismissed since their size has always been bigger than the horizon. This,
however, is a very conservative view that needs to be carefully examined in a
future study. Moreover, it is conceivable that such solutions can be found at
a scale cut off by the horizon.

Assuming one can make sense of the fractional instantons on scales smaller
than or equal to the horizon size, then things become more interesting at
temperature above the electroweak scale. Now, the electroweak symmetry is
restored and there is no barrier between the pre-vacua of $SU(2)$. In this
case the baryon and lepton number violation happens in units of $3$, one per
family, at a rate set by temperature $\sim T^{4}$. If there is no extra matter
charged under $U(1)$, then our conclusion from the previous section implies
that fractional instantons did not play any role during the thermal history of
the Universe. If, however, there is extra matter that makes the fractional
instantons more important at some critical energy scale
$\mu_{\scriptsize\mbox{critical}}<M_{P}$, then at temperature
$T\gtrsim\mu_{\scriptsize\mbox{critical}}$ the barrier due to the fractional
instantons is removed and the baryon and lepton number violation is expected
to proceed at similar rates $\sim T^{4}$ in units of $\Delta B>3$, as given in
Table 35.

Acknowledgments:

M.A. is supported by STFC through grant ST/T000708/1. EP is supported by a
Discovery Grant from NSERC.

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