Line Operators in the Standard Model

There is an ambiguity in the gauge group of the Standard Model. The group is $G = SU(3) \times SU(2) \times U(1)/\Gamma$, where $\Gamma$ is a subgroup of ${\bf Z}_6$ which cannot be determined by current experiments. We describe how the electric, magnetic and dyonic line operators of the theory depend on the choice of $\Gamma$. We also explain how the periodicity of the theta angles, associated to each factor of $G$, differ.


Introduction
At the parochial distance scales accessible by experiment, the world in which we live is governed by the Standard Model. The gauge sector is one of the most beautiful constructs in theoretical physics, involving an intricate interplay between chiral fermions to ensure the cancellation of anomalies. Indeed, the matter content in one generation forms what is arguably the simplest non-Abelian four-dimensional chiral gauge theory.
Despite the fact that the Standard Model is built around the idea of gauge symmetry, there is a little-advertised ambiguity in the choice of gauge group. We learn in kindergarten that we should takẽ But this is not quite accurate. Experimental considerations tell us only that the gauge group is where Γ is a discrete group. As we review below, the matter content of the Standard Model is invariant under a suitably chosen Z 6 subgroup ofG. For this reason, it is sometimes stated that one should take the gauge group to include the quotient Γ = Z 6 . action configurations carrying topological charge. This means that, unlike their non-Abelian counterparts, the electric spectrum and correlation functions do not depend on θ Y . Nonetheless, Abelian θ-angles can change the physics in more subtle ways. This happens, for example, in the magnetic sector through the Witten effect [2,3]. It also happens when θ-angles vary in space or, relatedly, in the presence of boundaries. Indeed, much of the rich and beautiful phenomenology of topological insulators can be understood as a domain wall between θ em = 0 and θ em = π [4,5]. We will see below that the periodicity of θ Y is determined by Γ and that, after electroweak symmetry breaking, this affects the periodicity of the electromagnetic theta angle, which is a particular combination of θ 2 and θ Y .
The paper is organised as follows. Section 2 is a review of the results of [1], specifically how the line operators and associated theta angles differ for SU(N) and SU(N)/Z N . Section 3 contains all the main results of the paper, including how the line operators and θ-angles in the Standard Model depend on the choice of discrete quotient Γ. Particular attention is paid to the values of θ for which the gauge sector is invariant under CP or, equivalently, under time reversal. We also describe the fate of the line operators under electroweak symmetry breaking, since this determines the allowed electromagnetic charges of particles.
The distinctions between the different gauge groups G =G/Γ described here are rather formal in nature. We end in Section 4 with some speculations on how these distinctions may manifest themselves physically.

A Review of Line Operators
We start by reviewing the properties of line operators, described in [1], that we will later need.
Wilson lines are operators which describe the insertion of an infinitely massive, electrically charged particle sitting at the origin of space [6]. They are labelled by representations R of the gauge group G, and given by Wilson lines exist for all representations R, regardless of the dynamical matter content of the theory. This means, in particular, that different gauge groups G will have a different spectrum of Wilson lines, even if they share a common Lie algebra g. We denote the weight lattice of g as Λ w . Then the representations of the universal cover of G -which we denote asG -are labelled by points in the lattice Λ w /W where W is the Weyl group. Representations of G =G/Γ are labelled by an appropriate sublattice Λ w /W . 't Hooft lines are operators which describe the insertion of an infinitely massive, magnetically charged particle sitting at the origin of space [7]. They are best thought of as defect operators, in which the gauge fields have suitable boundary conditions imposed on the worldline of the magnetic source [8]. For a gauge group G, 't Hooft lines are labelled by some sublattice of Λ mw /W where Λ mw is the magnetic weight lattice [9]; it is the weight lattice of the dual Lie algebra g ⋆ , or the dual of the root lattice of g.
Given a choice of gauge group G =G/Γ, we would like to determine the allowed spectrum of line operators. This problem was solved in [1] for connected groups. (Here we consider only line operators which enjoy an independent existence, as opposed to those which survive only on the boundary of a surface operator.) Certain line operators exist regardless of the choice of the discrete quotient Γ. For the Wilson lines, these correspond to the root lattice Λ r ⊂ Λ w . They include, for example, the Wilson line in the adjoint representation. Different choices of Γ then determine which of the remaining Wilson lines are allowed. These can be characterised by their transformation under Λ w /Λ r = Z(G), the centre ofG.
A similar story plays out for the 't Hooft lines: for every choice of G =G/Γ there is a 't Hooft line corresponding to the co-root lattice Λ cr ⊂ Λ mw or, equivalently the root lattice of g ⋆ . The remaining 't Hooft lines are characterised by Λ mw /Λ cr = Z(G) and their admissibility depends on the choice of Γ.
The upshot of this is that line operators decompose into classes, labelled by the Furthermore, the line operators form a group; the operator product gives rise to an addition law while the reversal of orientation provides an inverse. These properties are enough to ensure that if one member of the class labelled by (z e , z m ) is present in the theory the the entire class is present.
This is the Dirac quantisation condition for non-Abelian lines. The important role played by the centre of the gauge group in Dirac quantisation was first pointed out by Corrigan and Olive [10]. For G = SU(N)/Z N (but not for all other gauge groups [1]), other solutions to (2.1) are generated by the theta angle. This arises through the Witten effect [2] -which holds for line operators as well as dynamical particles [8,11] -and ensures that, in the presence of θ = 0, purely magnetic line operators turn into dyonic line operators. In particular, as θ → θ + 2π, each line operator picks up an electric charge given by

How Does the Dynamics Differ?
In four-dimensional Minkowski spacetime, R 3,1 , the difference between Yang-Mills with gauge group SU(N) and SU(N)/Z N is rather formal. In particular, any local observer is blind to the distinction. Nonetheless, the different line operators mean that there are subtle differences between the two theories. This appears, for example, after confinement. In either theory, the confining phase can be viewed as arising through the condensation of magnetic monopoles with charge λ m ∈ Λ mr . For SU(N) Yang-Mills, these are the minimally charged monopoles. However, for SU(N)/Z N , these are not the minimum charge. This means that (at θ = 0) this theory exhibits topological order, with an emergent magnetic discrete Z N gauge symmetry in the infra-red [1].
The difference between the two theories takes on a more meaningful role when the theory is compactified on a space with non-trivial topology, since now the Wilson line for the Z N gauge symmetry can get an expectation value, resulting in different physics in the lower dimension. This is perhaps clearest in the N = 1 super Yang-Mills, where one has more control over the dynamics. When compactified on a circle, or on a higher dimensional torus, the Witten indices for SU(N) and SU(N)/Z N differ [12,13].

Line Operators in the Standard Model
In this section, we extend the analysis of [1] to the non-connected gauge group The quotient group Γ lies in the centres of SU(2) and SU (3), combined with a suitable U(1)Ỹ rotation. The quotient Γ = Z 6 is generated by (2)) obeys η 2 = 1 and ω ∈ Z(SU(3)) obeys ω 3 = 1 and q is the U(1)Ỹ charge. The quotient Γ = Z 3 is generated by ξ 2 and the quotient Γ = Z 2 is generated by ξ 3 .
Line operators are now labelled by three electric charges and three magnetic charges, one pair for each factor of the gauge group. As reviewed in Section 2, for non-Abelian gauge groups the line operators fall into classes, labelled by z e 2 , z m 2 = 0, 1 for SU(2) and z e 3 , z m 3 = 0, 1, 2 for SU (3). We also require the additional labels (q, g) to describe the electric and magnetic charge under U(1)Ỹ . We chose conventions 2 such that q ∈ Z and, in the absence of any discrete quotient, g ∈ Z as well. However, as we will see, the presence of a discrete quotient Γ = 1 means that g can take fractional values.
The Dirac quantisation condition is simplest to state between purely electric and purely magnetic lines: it is We deal with each choice of Γ = 1, Z 2 , Z 3 and Z 6 in turn. We start by describing the spectrum of line operators when θ = 0 for each factor of the gauge group; we will subsequently see how the spectrum changes with θ.   The quantization condition condition (3.1) still requires z m 3 = 0 mod 3. However, now magnetic lines exist with z m 2 = 1 provided they are accompanied by Abelian magnetic charge g = 1 2 . We can add to these Abelian line operators with (q, g) = (2, 0) and (q, g) = (0, 1). The resulting spectrum of line operators is that of U(2) × SU(3) and is shown in Figure 3. Γ = Z 3 : Wilson lines must now be invariant under ξ 2 . This mean that electric lines must have q = z e 3 mod 3. Each of these can have any z e 2 = 0, 1.
The quantization condition condition (3.1) now allows lines with SU(3) magnetic charge z m 3 = 0, 1, 2, as long as they are accompanied by Abelian magnetic charge g = z m 3 /3. No SU(2) magnetic charge is allowed: z m 2 = 0 mod 2. We can add to these Abelian line operators with (q, g) = (3, 0) and (q, g) = (0, 1). The resulting spectrum of line operators is that of SU(2) × U(3) and is shown in We could add to this the right-handed neutrino ν R which is a gauge singlet. The Higgs also obeys the relationship between electric charges, sitting in the representation (2, 1) 3 ⇒ (z e 2 , z e 3 ) q = (1, 0) 3 . The fact that all Standard Model fields satisfy q = 3z e 2 − 2z e 3 mod 6 is, of course, equivalent to the statement made in the introduction that the Standard Model gauge group is consistent with U(1) × SU(2) × SU(3)/Z 6 .
The quotient Γ = Z 6 allows for the richest spectrum of magnetic line operators. Now purely magnetic operators exist for any choice of (z m 2 , z m 3 ) provided they are accompanied by an Abelian magnetic charge 6g = 3z m 2 + 2z m 3 mod 6. For example, a basis of magnetic operators is (z m 2 , z m 3 ) g = (1, 0) 1/2 and (0, 1) 1/3 . We can add to these Abelian line operators with (q, g) = (6, 0) and (0, 1). The resulting spectrum is shown in Figure  5.

Theta Angles
We can now ask how the spectrum of line operators changes as we vary the θ-angles. A priori, there are three such angles, one for each factor of the gauge group. We call these θỸ , θ 2 and θ 3 . Within the framework discussed in this paper it makes sense to ask how the line operators vary under each of these.
Before we proceed, it is worth reviewing the role of theta angles in the Standard Model. The most discussed is the QCD theta angle, θ 3 . Bounds on strong CP violation restrict θ 3 10 −10 . (The value θ 3 = π also preserves CP but the meson spectrum derived from the chiral Lagrangian differs from the observed values [14,15].) It is usually stated that the weak theta angle, θ 2 , can be rotated away in the Standard Model . This follows from the existence of an anomalous global symmetry B +L → Z N f where N f = 3 is the number of generations. We will revisit this below.
Finally, there is very little, if any, discussion of the theta angle for hypercharge θỸ . This changes neither the spectrum nor correlation functions of local operators. Nonetheless, it can play a role in the presence of magnetic monopoles or boundaries of space. Correspondingly, it also changes the spectrum of line operators.
Here we start by ignoring the effects of global anomalies and focus on the spectrum of line operators and the Witten effect. As reviewed in Section 2, for simple gauge groups, a quotient by the centre has the effect of extending the range of θ. In the present context, the quotient Γ = Z p extends the range of the Abelian theta angle only 4 . We have This is simplest to see for the case of Z 2 and Z 3 where the gauge group is G = U(2) × SU(3) and G = SU(2) × U(3) respectively. Here We denote the U(1) gauge field asã, the SU(N) gauge field as a and their corresponding field strengths asf and f . The theta terms for the U(1) × SU(N) theory are To describe the U(N) theory, we introduce the canonically normalised gauge field b = a +ã1 N with corresponding field strength g. The theta terms can then be written as Similarly, for the Z 6 quotient, one can check that the spectrum of line operators is invariant under the identification (3.2), withθ ∈ [0, 72π).

The Effect of Global Anomalies
The chiral nature of the Standard Model means that the theta angle θ 2 can be rotated away. Here we review this argument. In fact, as we will see, a more careful statement is that a linear combination of θ 2 andθ can be removed.
In general, theta angles can be rotated away if the theory admits a continuous global symmetry which suffers a mixed anomaly with the gauge symmetry. This arises most naturally in the presence of a massless chiral fermion. (For example, a massless up quark provides an elegant solution to the strong CP problem, albeit one that appears not to be favoured by Nature). However, even with non-vanishing Yukawa couplings, so that all fermions have a mass, the Standard Model still admits two global symmetries: these are lepton and baryon number: where, for once, we have bowed to tradition and employed the non-integer normalisation of the baryon current. Both L and B suffer mixed anomalies with both SU(2) and U(1)Ỹ . They are L SU(2) 2 = B SU(2) 2 = −1 and LỸ 2 = BỸ 2 = +18 We recover the well known fact that the combination B − L is non-anomalous. Meanwhile, under a transformation of L, parameterised by α L , we have The linear combinationθ + 18θ 2 is physical and cannot be rotated away. We will see the interpretation of this shortly.

Electroweak Symmetry Breaking
Let's now see what becomes of our line operators after electroweak symmetry breaking. The Higgs field H lies in the (2, 1) 3 representation and condenses, breaking U(1)Ỹ × SU(2) → U(1) em of electromagnetism. We will denote the electric charges of U(1) em as Q and the magnetic charges as G. We choose to normalise electric charges such that the electron has Q = −1.
The allowed electric and magnetic charges under U(1) em depend on our choice of discrete quotient Γ. The electric charge under U(1) em is given by with q the U(1)Ỹ charge and with the normalisation λ 2 2 ∈ Z so that, for example, λ e 2 = ±1 corresponds to the fundamental representation of SU (2). (Written in terms of Y =Ỹ /6, this takes the more familiar form Q = Y + λ e 2 .) Meanwhile, after condensation of the Higgs, most 't Hooft and dyonic line operators with U(1)Ỹ or SU(2) magnetic charge exhibit an area law. Those that remain deconfined obey the condition The resulting magnetic charge under U(1) em is given by In this normalisation, the Dirac quantisation condition for pure electromagnetism reads QG ∈ Z. We can now describe the spectrum of electric and magnetic U(1) em charges for each quotient. Note that for Γ = Z 3 and Γ = Z 6 , the spectrum is not consistent with the naive, electromagnetic Dirac quantisation QG ∈ Z. This is simply the statement that a minimum Dirac monopole is inconsistent with the fractional charge of the quarks. The resolution to this was given long ago [10]: the magnetic monopole must also carry colour magnetic charge, and this provides an extra contribution to the Dirac quantisation condition, rendering the spectrum consistent. (This fact is also emphasised in [17] in the context of GUT monopoles.) Indeed, it is simple to check from Figures 4 and 5 that the relevant 't Hooft lines do indeed carry SU(3) magnetic charge. This is tantamount to the fact that, in these cases, the low-energy gauge group is actually U(3) rather than U(1) × SU(3).
We denote the electromagnetic field strength as F , again normalised such that the electron carries charge −1. After symmetry breaking, the electromagnetic theta term is We see that θ em = (θỸ + 18θ 2 )/4, precisely the combination that cannot be removed by a chiral rotation. This, of course, is no coincidence: it follows from 't Hooft anomaly matching and the fact that the Higgs multiplet (2, 1) 3 can give mass to all chiral fermions, leaving behind the vector-like theory of QED.

Summary
The global structure of the Standard Model gauge group depends on the choice of quotient Γ. The differences in these theories described above are rather formal in nature. Just because a line operator exists in a theory does not mean that it is available to experimenters. It is clearly interesting to better understand the physical implications of the different choices of Γ to see if they may reveal themselves in some way in our world.
Some minor, and fairly cheap, respite can be found in the conjecture that, in any theory with gravity, the full set of charges carried by line operators are also carried by dynamical objects [18]. The best arguments for this come from black hole physics [19].
In the absence of gravity, one can always decouple fields by taking their mass to infinity, leaving behind only the non-dynamical line operators. In a theory with gravity, this is not possible: the backreaction of the line operator will eventually form a black hole, which carries the appropriate electric and magnetic charges.
Conversely, the global structure of the gauge group determines the fluxes that are allowed through cycles in a non-trivial spacetime. A black hole provides such a spacetime, with an S 2 horizon, and the fluxes which can thread this are determined by the choice of Γ.
The further requirement that elementary particles should not form black holes [20] suggests that there are new magnetic (and possibly electric) particles to be found whose charges depend on Γ. Obviously, if a neutral quark is discovered, transforming in the (1, 3) 0 representation of G, then we must take Γ = Z 2 or 1. In contrast, as we saw above, the discovery of a magnetic monopole, consistent with the minimum Dirac quantisation with respect to the electron, but not with respect to the quark, would mean that Γ = Z 6 . Of course, these predictions are rather toothless: the particles have a mass which is constrained only by the Planck scale are unlikely to be abundant; one must hope that Nature is kind [21].
One can ask if there are more subtle ways to distinguish between the theories. As explained in [1], and reviewed in Section 2, after confinement the Yang-Mills theory with gauge group SU(N)/Z N exhibits topological order, with an emergent Z N magnetic gauge symmetry. This gives rise to the possibility of more interesting physics arising in spacetimes with non-trivial topologies or boundaries. For the Standard Model, there exists a magnetic Z 3 symmetry arising when Γ = Z 3 or Z 6 but the states which carry charge under this also carry magnetic charge under under U(1) em . It would be interesting to see if this has any implications for physics in the presence of non-trivial topology or, more interestingly, in dynamical spacetime. This may allow us to answer the basic question: what is the gauge group of the Standard Model?