Flatland: abelian extensions of the Standard Model with semi-simple completions

We parametrise the space of all possible flavour non-universal $\mathfrak{u}(1)_X$ extensions of the Standard Model that embed inside anomaly-free semi-simple gauge theories, including up to three right-handed neutrinos. More generally, we parametrise all abelian extensions (i.e.) by any number of $\mathfrak{u}(1)$'s) of the SM with such semi-simple completions. The resulting space of abelian extensions is a collection of planes of dimensions $\leq 6$. Numerically, we find that roughly $2.5\%$ of anomaly-free $\mathfrak{u}(1)_X$ extensions of the SM with a maximum charge ratio of $\pm 10$ can be embedded in such semi-simple gauge theories. Any vector-like anomaly-free abelian extension embeds (at least) inside $\mathfrak{g} = \mathfrak{su}(12)\oplus \mathfrak{su}(2)_L\oplus \mathfrak{su}(2)_R$. We also provide a simple computer program that tests whether a given $\mathfrak{u}(1)_{X^1}\oplus \mathfrak{u}(1)_{X^2}\oplus \dots$ charge assignment has a semi-simple completion and, if it does, outputs a set of maximal gauge algebras in which the $\mathfrak{sm}\oplus\mathfrak{u}(1)_{X^1}\oplus \mathfrak{u}(1)_{X^2}\oplus \dots$ model may be embedded. We hope this is a useful tool in pointing the way from $\mathfrak{sm} \oplus\mathfrak{u}(1)_{X^1}\oplus \mathfrak{u}(1)_{X^2}\oplus \dots$ models, which have many phenomenological uses, to their unified gauge completions in the ultraviolet.


Introduction
There are many phenomenological reasons to entertain an extension of the Standard Model (SM) gauge algebra sm := su(3) ⊕ su(2) L ⊕ u(1) Y by u(1) X summands which, after being spontaneously broken, would give rise to neutral Z gauge bosons. For example, if weakly coupled to the SM fields, a Z boson could mediate interactions with a dark sector. If light (m X 4 GeV [1]) and suitably coupled to leptons, a Z can mediate a 1-loop contribution to the anomalous magnetic moment of the muon that resolves an estimated 4.2σ discrepancy [2,3] between data and the SM prediction. If heavier (m X 2 TeV, say) and equipped with quark flavour violating and lepton flavour universality violating interactions, a Z can resolve a collection of measurements [4][5][6][7][8][9][10][11][12][13][14] (the 'b → sµµ anomalies') in semi-leptonic B-meson decays that are in tension with the SM. Finally, the u(1) X gauge symmetry itself can be put to good phenomenological use, for example in explaining the flavour puzzleà la Froggatt and Nielsen [15].
To make life (or at least model-building) simpler, it is a good idea to ensure the u(1) X extension of the SM gauge algebra is free of perturbative gauge anomalies. 1 If this is not the case, the UV completion of the model must feature extra chiral fermions that restore anomaly cancellation. If the extra fermions are chiral under sm, then it will be difficult to give them big enough masses to have eluded discovery at the LHC. On the other hand, if the extra fermions are SM singlets, then their masses can comfortably reside at the heavy scale of u(1) X breaking, so there is no tension here with collider bounds. 2 Thus, the simplest strategies for building a consistent sm⊕u(1) X model are to ensure gauge anomaly cancellation amongst the SM fermions on their own, or allowing some number N of SM singlets. In the cases N ≤ 3, a.k.a. the 'SM+3ν R ', the complete space of flavour non-universal anomaly-free u(1) X extensions of the SM has been numerically scanned in Ref. [19], and even parametrised analytically (for the N = 3 case only) in Ref. [20].
Despite their many uses, extending the SM by an anomaly-free u(1) X gauge symmetry goes against the popular idea of unification, whereby one asks for fewer forces not more, or at least fewer fields, at high energies. There is not a huge variety of unified gauge theories in which one can embed the SM on its own, given in particular its intricate pattern of hypercharge quantum numbers, and this is clearly made more difficult by seeking to embed a second set of u(1) X charges also. To illustrate the point, if there were a single generation of SM fermions, there is no additional u(1) X whatsoever that could be embedded in the su(5) GUT [21], while there is an unique option that embeds inside so (10) [22,23], namely X = B − L. Of course, there is not one generation but three in Nature, and this opens up a much wider arena of flavour non-universal gauge models, to which we return our attention shortly.
Extending the SM by u(1) X also exacerbates the issue of the Landau pole in the SM; now there are two U(1) gauge symmetries, each of which becomes more strongly coupled at higher energy scales. Even though we got lucky with hypercharge, for which the Landau pole is at a very high scale and so can perhaps be waved away by appealing to quantum gravity effects, this is not necessarily the case for a phenomenologically-motivated u(1) X . It was recently shown [24] that Z models for the b → sµµ anomalies suffer from sub-Planckian Landau poles, that can be as low as 100 TeV for realistic models. This means that Z models for the b → sµµ anomalies require new physics of some kind to tame this running -in Ref. [24] extra scalars and fermions were included to soften the running of the u(1) X gauge coupling. Arguably, the resulting theories are starting to look even more complicated at higher energies, taking us further from the ideals of unification.
The other option for curing a low scale Landau pole is to try to reverse the direction of its running through self-interacting gluon contributions; in other words, to ameliorate the Landau pole by embedding the u(1) X inside a semi-simple gauge algebra, and taking seriously the idea of unification. But, given any old anomaly-free u(1) X extension of the SM, it is usually not obvious whether the sm ⊕ u(1) X theory can be embedded in a semi-simple gauge algebra, nor is it so obvious what strategy to employ to find this algebra.
If one restricts the matter content of the UV model, however, the question at least becomes sharply posed, because the number of possible Lie algebras g in which sm ⊕ u(1) X embeds becomes finite. Recently, Ref. [25] classified all possible semi-simple gauge extensions of sm that do not require extra fermions beyond those of the SM+3ν R , and that are free of perturbative gauge anomalies. 3 The result was a finite set, call it S, of 340 inequivalent Lie algebras g. The richness of this list is in large part a result of there being three families of SM fermions. This allows for options in which gauge and flavour symmetries are unified that have been little studied until now; for example, g = su(4) ⊕ sp(6) L ⊕ sp(6) R in which electroweak and flavour symmetries are unified (a model that was explored in detail in [26]), or g = su(12) ⊕ su(2) L ⊕ su(2) R in which colour and flavour are unified.
Equipped with the list S, one can find all possible anomaly-free sm ⊕ u(1) X subgroups, in fact all possible subgroups of the more general form sm ⊕ u(1) X 1 ⊕ u(1) X 2 ⊕ . . . , 4 contained in any of the algebras g ∈ S. In this paper, we have two related goals: 1. To parametrise the space of all anomaly-free abelian extensions of the SM, by any number of u(1)'s, that embed in anomaly-free semi-simple algebras g ∈ S.
2. To provide a simple way of testing whether a given anomaly-free abelian extension of the SM sits in one of the anomaly-free semi-simple algebras g ∈ S.
After achieving the first goal, the second goal follows swiftly. To achieve goal 1., our method is straightforward: for a sufficiently large subset of algebras g in S, we first compute the centraliser C g (sm) of sm in g, then find the Cartan subalgebra h Cg(sm) thereof, from which we can extract the most general sets of charges that can be embedded in g.
Unlike the space of anomaly-free u(1) X extensions of sm, which is some complicated set of rational points cut out by the intersection of a quadratic and a cubic equation, the space of anomaly-free abelian extensions with semi-simple completions is a 'nice' linear space, being a union of planes. For want of a better name, at times we refer to this space as 'Flatland' [29]. 5 To give an explicit example of one of these component planes, the set of u(1) X 1 ⊕u(1) X 2 ⊕ . . . charge assignments that embeds in the flavour non-universal algebra g 2 = so(10) 1 ⊕ so(10) 2 ⊕ so(10) 3 is the plane where BL is an abbreviation for B − L. To give another simple example, for the colour-flavour unification group g 6 := su(12) ⊕ su(2) ⊕ su (2), the corresponding plane is P 6 = Span(Y, BL, B 12 , B 23 , L 12 , L 23 ), where B 12 is an abbreviation for B 1 − B 2 . Equivalently, the plane P 6 is, up to hypercharge, the space of traceless combinations of B 1 , B 2 , B 3 , L 1 , L 2 , and L 3 , which coincides precisely with the space of anomaly-free abelian extensions with vector-like charges (see Section 4.1). The fact that such vector-like u(1) X models embed in the colour-flavour unification algebra su(12)⊕su(2)⊕su(2) has already been put to phenomenological use in Ref. [30], which reveals a novel connection between flavour non-universality and the stability of the proton.
We find that a total of 8 such planes, like P 2 and P 6 just described, are needed to parametrise Flatland, the space of all possible sm ⊕ u(1) X 1 ⊕ u(1) X 2 ⊕ . . . theories with semi-simple completions, taking care to ensure there is a possible gauge group that is free of not just local anomalies, but also global anomalies. 6 For some of these planes, we show how to derive predicates (i.e. true-or-false statements) that test whether given sets X n of charges lie in that plane. Such a predicate must take into account the freedom to permute family indices of each species of SM fermion, and some planes in Flatland are difficult to characterise by a simple permutation-invariant predicate. Nonetheless, by cycling through family permutations on a computer, one can straightforwardly determine whether a given abelian extension by u(1) X 1 ⊕ u(1) X 2 ⊕ . . . sits in any of these 8 planes, and thence identify its possible semi-simple completions. 7 We write a computer program that performs these tasks, and share the code with the arXiv submission of this article. We hope that this provides a useful tool when model-building with (multiple) u(1) extensions of the SM, by pointing the way towards the possible unified gauge models in the UV. We apply our results to many anomaly-free u(1) X models from the literature. As mentioned, any vector-like anomaly-free u(1) X , such as B − L, B 3 − L 2 , B − 3L µ , or L µ − L τ , necessarily embeds (at least) in the colour-flavour unification algebra g 6 . It is therefore for the chiral u(1) X models that the situation is more interesting. We give a summary of results in Table 4. To give two more examples, we find that the 'DY 3 model' of [31,32], which is the unique anomaly-free u(1) X model in which a single family of quarks and two families of leptons are charged, does not sit in any semi-simple g. Nor do any of the 21 chiral 'muoquark models' from [1,33] have semi-simple completions.
Finally, we ask and answer the quantitative question of 'how many' anomaly-free sm ⊕ u(1) X models can be embedded in semi-simple gauge theories. By scanning through the 'anomaly-free atlas' of Ref. [19], we find that roughly 2.5% of solutions in the anomaly-free atlas with a maximum charge of 10 have semi-simple completions. This fraction falls roughly exponentially with the maximum charge (see Fig. 2).
The rest of the paper is structured as follows. In Section 2 we introduce the notion of an abelian extension of the SM, setting out the definitions and formalism we shall need in the paper. In Section 3 we describe our method for finding which abelian extensions have semi-simple completions, in detail: this Section can be skipped without loss of continuity, if one is only interested in our results. In Section 4 we find the collection of planes parametrising all possible abelian extensions that have semi-simple completions. We apply these results to a selection of explicit u(1) X models from the literature, and to all solutions in the 'anomalyfree atlas' of [19]. Finally, in Section 5 we explain how to use the short computer program Test your own charges.nb, included with the arXiv submission of this article, with which the user can test whether an input sm ⊕ u(1) X 1 ⊕ u(1) X 2 ⊕ . . . theory has a semi-simple completion (and if so, find out in which of the 8 planes of Flatland it sits). We also itemise a selection of other computer programs that are included with the article.

Abelian extensions from semi-simple algebras
Let sm := su(3) ⊕ su(2) L ⊕ u(1) Y denote the gauge algebra of the SM, and let γ : sm → su (48) denote an embedding that defines the representation of the 48 Weyl fermions of the SM+3ν R . To set our conventions, the SM Weyl fermions are in the following representations of sm, where j ∈ {1, 2, 3} is the family index: , and ν c j ∼ (1, 1) 0 . All these fields are left-handed Weyl fermions. Ref. [25] recently catalogued all possible semi-simple gauge extensions of the SM+3ν R ; roughly, this means finding all (complex) semi-simple Lie algebras g such that there is a pair of embeddings α : sm → g and β : g → su(48) that are compatible with γ, in the sense that the diagram g sm su (48) β γ α (2.1) commutes, and such that g is moreover free of local gauge anomalies. Two triples are said to be equivalent (g, α, β) ∼ (g , α , β ) if there is an inner automorphism j of su (48) and an . A grand total of 340 such inequivalent (g, α, β) were found. Let us call this set of extensions S (for 'semi-simple'). Of these algebras, 24 are maximal, meaning there is no inequivalent (g , α , β ) such that there are embeddings j : g → g and i : su(48) → su (48) with j • α = α and β • j = i • β. These 24 maximal algebras, as taken from [25], are reproduced in the first 24 rows of Table 1. We return to the issue of maximal algebras in the next Subsection.
Let sm S := su(3) ⊕ su(2) L be the semi-simple part of the SM gauge algebra. An abelian extension of the SM is a representation where a is abelian, and where there is an embedding such that • ι is the usual SM fermionic representation γ. It is preferable to include u(1) Y in the abelian part a because of the freedom to do linear field redefinitions on the abelian gauge fields (at the expense of introducing kinetic mixing). Including u(1) Y in the abelian part thus allows us to properly account for all possible extensions. We restrict our attention to those extensions where for some choice of Cartan subalgebras, since these are the ones which act on fermions via a phase. 8 The same information contained in the embedding : sm → su (48) can be expressed, less formally (but perhaps more familiarly), by a particular 5) for N specific (independent) sets of charge assignments X n ∈ Q 18 (where a point in Q 18 records the charges of the 18 fermions of SM+3ν R ).
In this work, we will find all possible abelian extensions of the SM+3ν R that can be embedded into at least one of the anomaly-free semi-simple gauge models in S. Namely, we find abelian extensions such that there exists a (g, α, β) and an embedding such that τ | sm = α and the diagram g sm S ⊕ a su (48)  Clearly, if an abelian extension embeds into any of the 340 algebras in S then it must also embed into a maximal one. Hence we can restrict our attention to maximal (g, α, β).
Since a commutes with sm = sm S ⊕ u(1) Y , τ (a) must be a subalgebra of the centraliser C g (sm) of sm in g, which is defined to be (2.8) More formally, we denote the algebra itself C g (sm) and its embedding into g as ρ : C g (sm) → g. Using standard theorems in Lie algebra theory (see e.g. [34]), it is easy to show that C g (sm) is a reductive Lie algebra, where recall that a reductive Lie algebra is one that can be written as a direct sum of simple factors and u(1) factors. Given a chosen Cartan subalgebra h Cg(sm) of C g (sm) we have a diagram This condition would exclude embeddings analogous to u(1) → su(2) : X → J±, where J± = σ1 ± iσ2 are the usual ladder operators. The unpleasantry of such an embedding is that J± is not diagonalisable (i.e. is not a semi-simple element), and it will not act reducibly on the SM fermions.
where τ C is induced by the embedding of h Cg(sm) into g and C can be defined as the unique embedding such that this diagram commutes. The important property of Diagram (2.9) is that it is maximal with respect to the Diagrams in (2.7). By 'maximal' we mean that, for any valid : sm S ⊕ a → su(48) satisfying (2.7), there is an embedding t : sm S ⊕ a → sm S ⊕ h Cg such that there exists a commutative diagram where all morphisms are embeddings and i 1 and i 2 are inner automorphisms. In particular, commutativity of the 'left face' implies the maps τ and τ C •t agree up to an inner automorphism i 1 on g; colloquially, we would say that τ 'factors through' sm S ⊕ h Cg . Note that the choice of the specific h Cg(sm) does not matter, since all choices are related by inner automorphism.
The map β : g → su (48) can then be used to determine how h Cg(sm) acts on the fermions, via the weight system. This can be translated into a set of u(1) X charges, which span a plane P g in the rational 'charge space' Q 18 . The question of whether some abelian extension : sm S ⊕ a → su(48) sits in any (g, α, β) then reduces to the question of asking whether the charges associated with a sit inside the (maximal) planes P g (up to family permutations that correspond to relabeling).
Our procedure can thus be summarised as follows: for each of the maximal algebras (g, α, β) in S, we 1. find the centraliser C g (sm), 2. find a Cartan subalgebra h Cg(sm) , 3. extract the weights associated to the map β, 4. thence find a basis of independent charge vectors, thence the plane P g that they span. This procedure, which is detailed in the next Section, can already be made fairly 'algorithmic'. It can, in fact, be carried out for all 340 algebras in S.
Lie algebra g of a gauge group G, this condition could be imposed without ambiguity. But of course, it is not enough to ensure that a gauge theory is free of local (perturbative) anomalies, because there may be more subtle global anomalies that are not associated with infinitesimal gauge transformations, which can arise non-perturbatively. The canonical example is the anomaly associated with a single Weyl fermion in the doublet representation of SU (2), in 4d [35]. By definition, such global anomalies cannot be computed knowing only the Lie algebra g, but depend on the global structure of the group G itself. Some of the gauge algebras listed in Ref. [25] do not correspond to any anomaly-free gauge theory, irrespective of the freedom to consider gauge groups that differ globally. For example, consider the maximal algebra listed as number 14 in Table 1, g = su(5) ⊕2 ⊕so(10)⊕su (2). All fermion fields are singlets under the su(2) factor except for a doublet of right-handed neutrinos transforming in the (1, 1, 1, 2) representation. The associated gauge group G necessarily features an SU (2) factor, which has a global anomaly [35].
In a similar way, all the algebras in Table 1 that are highlighted in red and with an asterisk necessarily suffer from global anomalies. Thus, we should remove these from the list of possible g if we are to consider UV gauge groups that are properly anomaly-free, without any extra fermion content. Of course, after doing this, the resulting cut-down list of algebras is no longer guaranteed to be the complete list of maximal anomaly-free algebras; there may be valid (i.e. not necessarily anomalous) algebras in S which embed into one of those maximal algebras we have crossed off, but which do not embed into any algebra remaining in the maximal list. To account for this, one must include three further algebras, listed 25-27 in Table 1, that become maximal when considering all possible UV gauge models which are free of both local and global anomalies.
In fact, we will present results in two cases: firstly, ignoring the issue of global anomalies (for which we consider the g labelled 1-24 in Table 1), and secondly, taking into account the issue of global anomalies in the manner just described (which amounts to crossing off all the 'starred' algebras, coloured red in Table 1, and instead including the algebras 25-27 coloured green.) The result shall be two parametrisations of abelian extensions of the SM that fit in either list of semi-simple completions. In both cases, the parametrisation takes the form of a union of planes, which are different. Clearly, the second set is a subset of the first.

Method
We now proceed to give the details of the procedure outlined in Section 2, for each of the maximal algebras in Table 1.
Before we start, we note that the information contained in Ref. [25] for each element of S is not exactly a triple (g, α, β). Instead, it outputs a list of triples consisting of a semi-simple g, a so-called 'projection matrix' Λα, and a 48-dimensional representation of g, up to some notion of equivalence. The fact that this set is equivalent to S is proven in Appendix B of Ref. [36].
Since projection matrices feature heavily in this Section, we think it useful to give a selfcontained definition, as follows. Let us start by reviewing the weight space of a reductive Lie algebra r : The weights then live in the dual of this space, h * r = h * s ⊕ (u(1) N ) * . Given any semi-simple Lie algebra s, a suitable basis of h * s is the 'fundamental weights'. Throughout this paper we always write weights in this basis. A basis of (u(1) N ) * can be formed by taking the dual of a basis X n of u(1) N .
Given an embedding f : r → g of the reductive Lie algebra r into a semi-simple Lie algebra g, we can form the pull-back Λf : (Note that in the literature Λf is often restricted to certain subsets of h * g ). We will call the operator Λf the projection matrix of f . When written in the bases just described, we denote it [Λf ].
For the present work, we need not just the projection matrices Λα, but the embeddings α : sm → g themselves for each of the maximal algebras g i in Table 1. The first step of our method is thus to take each projection Λα i associated to each g i , and from it deduce a fully fledged embedding α i : sm → g i .

3.1
Step 1: Finding the embeddings of sm in g To start, it is helpful to notice that the problem 'factorises' in a convenient way. Because each g i is semi-simple, one can decompose it uniquely into a direct sum of simple ideals. These decompositions take the form: where each b a is a simple ideal for which Im(α i ) ∩ b a = ∅, and where hor i is the largest (semisimple) ideal for which Im(α i ) ∩ hor i = ∅, and I i is a g i -specific list that we define shortly. For each b a , there is a non-zero embedding α a : sm → b a which can be extracted (as we will do in this Subsection) from a projection matrix Λα a . From Ref. [25], one can see that the 27 maximal gauge algebras in Table 1 are built out of a small set of 12 recurring building blocks (b a , Λα a ), a ∈ {1, . . . , 12}. 9 The 'building blocks' (b a , Λα a ), along with other important data that we describe below, are listed in Table 2. The list I i appearing in Eq. (3.1), which may feature repeats, specifies which of these building blocks appear in g i . For example, • for g 2 we have I 2 = {2, 2, 2} and hor 2 = 0, • for g 6 we have I 6 = {12, 7, 3} and hor 6 = 0, • for g 27 we have I 27 = {1, 1} and hor 27 = su(2) ⊕2 .
For each of the projection matrices Λα a , we can find associated embeddings α a in the following way. First, let us define the simple roots of sm: let h Y be the generator of u(1) Y , let κ be the simple root of su(2), and let {τ 1 , τ 2 } be the simple roots of su(3). Let ∆(b a ) := {λ r } be the simple roots of b a . For each Lie algebra we will work with the Chevalley basis. Roughly, given any semi-simple Lie algebra g we choose a basis {h λr } for its Cartan subalgebra that is 10 su(4) ps  Table 1 can be constructed. For each building block b a , we record the projection matrix [ Λα a ] for the map from SM into b a , as well as its centraliser C a , and the projection matrix [ Λρ a ] for the embedding of the centraliser in b a . dual to the fundamental weights, which when adjoined with a set of vectors {e λ }, where λ is in the root system Φ(g) of g, forms a full basis of g. For the details of this construction, and examples of Chevalley bases, are given in Appendix A.
A projection matrix Λα a tells us explicitly how its associated embedding α a acts on the Cartan subalgebra of sm. Namely, we have assuming summation on the index r. The first two rows of the projection matrix tell us how su (3) is embedded, the third row tells us how su(2) L is embedded, and the last row tells us how hypercharge is embedded. To find out how the embedding α a acts on the non-Cartan basis elements, we define, for each τ ∈ Φ(su(3)) and for ±κ (the roots of su(2)), the following sets (see e.g. [37]) For us, the intersection of any two such Γ τ or Γ ±κ is always the empty set. The image of α a is then given by Conditions are put on f by ensuring that α a preserves commutators. Any such f satisfying these conditions provides a valid choice of α a . Different valid choices are related and will lead to the same final result (a consequence of the theorems proven in Appendix B of Ref. [36]).

Example: so(6) R
We now give a couple of explicit examples to illustrate how the embeddings α a for the various building blocks b a listed in Table 2 can be inferred from the projection matrices Λα a . For our first example, which is especially straightforward, consider the building block b 6 = so(6) R . As can be read from Table 2, the projection matrix for so(6) R is given by (3.10) From this, it is easy to read off α 6 since it acts trivially on su(2) L and su (3), whilst mapping the hypercharge generator to For a second, slightly more involved, example, consider the building block b 8 = sp(4) L , that appears in models with electroweak flavour unification [26] in two generations. The projection matrix for the embedding α 8 : sm → sp(4) L is From this we see that only su(2) L embeds non-trivially into sp(4) L , and we can ignore the other summands in sm. Eq. (3.12) tells us that From the fact that α 8 must be an embedding, the map f must satisfy one of two sets of conditions. Either We choose the values This gives us the valid embedding of su(2) L in sp(4) L : Using similar methods, one can find valid embeddings α a for all 12 of the building blocks b a listed in Table 2.

Assembling the building blocks (I)
From there, it is easy to construct the embeddings α i : sm → g i for each of the 27 maximal algebras g i in Table 1. In particular, the corresponding projection matrix is given by where I i = {a 1 , a 2 , a 3 , . . . , a n }.  We do not need α 27 in what follows explicitly so will not report it here.

3.2
Step 2: Computing the centralisers of sm in g Given an embedding α i : sm → g i , the next step is to compute the centraliser of sm in g i . Fixing sm, for each maximal gauge algebra g i the centraliser is itself a pair (C i , ρ i ) consisting of an algebra C i := C g i (sm), as defined in Eq. (2.8), and another embedding ρ i : C i → g i . The factorisation of g i into the building blocks {b a }, as in (3.1), implies there is a similar factorisation of the centraliser (C i , ρ i ) where (C a , ρ a ) is the centraliser of α a . Clearly, the centraliser of hor i is isomorphic to itself. Thus, the embedding ρ hor i can be chosen to be the identity. Given our knowledge of all the embeddings {α a }, the task of finding each 'building block centraliser' C a becomes reasonably straightforward. It simply amounts to finding all elements of b a which commute with the image of α a . Regarding the embedding ρ a , all the information of interest to us is captured by the projection matrix [ Λρ a ]. The centralisers C a and associated [ Λρ a ] for each of our building blocks b a are displayed in Table 2. In the next two Subsections we show how these are calculated, continuing the explicit examples of b 6 = so(6) R and b 8 = sp(4) L . We continue to use the Chevalley basis for each Lie algebra.

Example (contd.): so(6) R
A general element of the Lie algebra b 6 = so(6) R , which is 15-dimensional, is expanded in the Chevalley basis as follows, so(6) R u = c 1 h λ 1 + c 2 h λ 2 + c 3 h λ 3 + c 4 e λ 123 + c 5 e λ 23 + c 6 e λ 12 + c 7 e λ 2 + c 8 e λ 3 + c 9 e λ 1 + c 10 e −λ 123 + c 11 e −λ 23 + c 12 e −λ 12 + c 13 e −λ 2 + c 14 e −λ 3 + c 15 e −λ 1 , where each coefficient c i ∈ C, and e.g. λ 123 := λ 1 + λ 2 + λ 3 . We want to find all such u that satisfy the condition The most general u which satisfies this condition is given by The space of such u defines a Lie subalgebra of so(6) R , the centraliser of u(1) Y , which can be shown to be isomorphic to C 6 = su(3) ⊕ u(1). The su (3) here is a family symmetry that acts by complex rotations on the family index of each right-handed species of field, i.e. on d i , u i , e i , and ν i . (But note that this information, namely how C 6 acts on the fermions, cannot yet be deduced; doing so requires the steps detailed in Section 3.3.) The u(1) summand in C 6 is of course just u(1) Y . The embedding ρ 6 : C 6 → so(6) R is given by Here we want to enforce the condition that i.e. that v commutes with the image of su(2) L in sp(4) L . The most general v which satisfies all these conditions is simply Thus, we have that the centraliser is a 1-parameter subalgebra of sp(4) L , C 8 = u(1). Letting h X denote its generator, we have which tells us that Physically, this u(1) acts as a rotation in family-space on (two families of) the q and lefthanded SU (2) L doublets.

Assembling the building blocks (II)
Once we have computed Λρ a for each building block b a (as per the two examples above), one can then build the projection operator associated to each g i . For the horizontal factor, the projection operator ρ hor i is simply the identity map. In terms of matrices (in our chosen basis), where again I i = {a 1 , a 2 , a 3 , . . . , a n }, and where 1 rank(hor i ) denotes the rank(hor i ) × rank(hor i ) identity matrix. Note that the projection matrix here is block diagonal, which should be contrasted with the 'horizontal concatenation' used in the corresponding Eq. (3.22) that assembles the projection matrix for the embedding. 10 10 Lest there is any confusion, the reason for the difference is that the projections [Λαa] for each embedding αa : sm → ba has a row index that always runs over the four Cartan generators of sm (while the column index runs over the Cartan of ba), and so the number of rows is unchanged upon concatenating the building blocks. Here, on the other hand, each ρa is a map from a different summand Ca of the centraliser Ci, to the corresponding ba; thus, to assemble the building blocks, the projection is here a direct sum.

Example (contd. II):
Recall that for g 27 we have I 27 = {1, 1}, and hor 27 = su(2) ⊕2 . Using the results for the building blocks (Table 2), the centraliser is Then from Eq. (3.40) and Table 2 Table 1; then in Section 3.2 we showed how, given these embeddings, to compute the centralisers (C i , ρ i ) of the sm in each g i . The remaining step, which is the subject of this Subsection, is to parametrise the plane of charges associated with a Cartan h C i . To do this, we also need the information encoded in the maps β i : g i → su (48), defined in Eq. (2.1), which specify how the SM fermions are embedded in each g i . In [25], the maps β i : g i → su (48) are not given explicitly. Rather, the program in Ref. [25] outputs the representation of g i that each β i corresponds to. From this information, we can find the weight system of the representation, which we denote Φ(β i ). Since the weights tell us the eigenvalues of all fermion components in the representation under the Cartan subalgebra h g i , the weight system contains the information we need to extract the charges of each fermion under h C i .
To do this, let us first define an indexing set of SM fermion fields, Then for each F j ∈ Q we can assign a choice of weight w F j ∈ Φ(β i ) such that no two w F j 's are the same, and There is of course freedom to permute the family labels here, for example one could assign a given weight to w q 1 or w q 2 . For each w ψ there are exactly three possible choices in Φ(β i ), so each possible choice appears exactly once. We define a vector space Q 18 with a basis {b ψ } ψ∈Q . This vector space can be interpreted as our 'space of charges'. The sm ⊕ u(1) X anomaly cancellation conditions (ACCs), for example, define a complicated surface in this space, whilst here we will construct planes within it. There is a linear map χ from h Q g i , which we define to be the rational space spanned by {h λ } λ∈∆(g i ) , to Q 18 , defined by our choice of w ψ . Namely, letting h ∈ h Q g i , the map The physics interpretation of this equation is that the 'component' w ψ (h) in the b ψ direction is the 'charge' of the fermion ψ with respect to any Cartan generator h ∈ h Q g i (where, at the moment, this includes the generators corresponding to the embedding of sm).
The embedding of the centraliser ρ i : C i → g i , then defines a subplane of Q 18 , which is given by Each point in P i corresponds to the charges of all 18 fermions under a particular element in the Cartan of C i . In this way, we construct the planes P i that parametrise the fermion charges of any abelian extension : sm S ⊕ a → su(48) that embeds in g i , for each of the maximal g i listed in Table 1. We emphasise that the generator of u(1) Y (that will remain unbroken) is included in this plane.
To be more explicit, it is easy to show that the plane P i is spanned by the d i := rank(h C i ) vectors where [Λρ i ] k * denotes the kth row of the projection matrix [Λρ i ] that is constructed as in Eq. (3.40) using the results in Table 2. One can interpret the vectors S i,k ∈ Q 18 , for each value of k, as a basis for the possible u X 1 ⊕ u(1) X 2 ⊕ . . . charges which can be embedded into the g i model. The dimension of P i is d i .  Table 3: Charges of the SM+3ν R fermions under a set of u(1) X symmetries that play a central role in this paper. Namely, every possible sm ⊕ u(1) X gauge model that embeds inside a semi-simple g can be expressed as a linear combination of a particular subset of these rows (that we determine), up to family permutations within each species. Here 'B' and 'L' stand for baryon and lepton as usual, 'Y ' stands for (global) hypercharge, 'S' stands for sinistral (meaning left-handed), 'D' stands for dextral (meaning right-handed), 'F ' stands for five, 'T ' for ten, and 'N ' for neutrino.

Example (contd. III):
We now illustrate this construction by continuing with the explicit example of g 27 . The representation corresponding to β 27 , which can be read off from , where each weight has 10 columns corresponding to the 10 fundamental weights of su(5) ⊕2 ⊕ su(2) ⊕2 , using the same ordering as in Subsections 3.1.4 and 3.2.4. We have assigned the weights (3.52) such that w F 1 < w F 2 < w F 3 lexicographically. 11 Since

Step 4: Testing if an abelian extension sits in a semi-simple algebra
In this Subsection we explain how to test whether a given abelian extension : sm S ⊕ a → su(48) sits in the semi-simple extension g i . Associated with is a set of u(1) charges which we denote {X n }, with X n ∈ Q 18 . The abelian extension sits in g i , in the sense of (2.7), if the plane spanned by {X n } sits in P i , up to family permutations.
For each plane P i , whose dimension we label d i , we choose a basis {R i,j } and a set of d i 'dual' vectors {n i,j }, such that R i,j · n i,j = δ j,j . (3.54) This then allows us to define a projection P i : 3 /S i and aσ ∈ S ×6 3 such that for each X n P i (σ(X n )) =σ(X n ). (3.56) Note that the LHS is independent of the choice of representative in the class [σ]. The existence ofσ can be checked via lexicographic ordering. However, one has to scan through all possible [σ] ∈ S ×6 3 /S i . Thus, for computational efficiency, we wish to find a choice of basis {n i,j } such that the stabiliser group S i is as large as possible. We do this by choosing a subset Q i ⊆ Q of cardinality d i and defining our {n i,j } = {bψ}ψ ∈Q i for which there exists an associated basis {R i,ψ }ψ ∈Q i . The elements of Q i are chosen so that Q i contains as few different species as possible. In practice, we find both Q i and R i,ψ via row-reduction in matrices with permuted species. One way to interpret the list Q i is as the minimum number of charges one needs to specify a point in P i .
As a representative of this class, let σ act trivially on all other species (except b q i and b ν c i ), and send (b ν c . Namely, we choose the representative , P 27 (σ(T 12 )) =σ(T 12 ), P 27 (σ(N 12 )) =σ(N 12 ). (3.62) Thus, up to equivalence under family permutations, the plane Span(Y, Y 3 , T 12 , N 12 ) lies within P 27 . Moreover, since the dimension of both planes is 4 they must coincide. Hence, we may write As we soon see, all the planes in Flatland can be similarly represented as a span of some of the vectors appearing in Table 3.

Examples of permutation invariant diagnostics
For certain planes in our list a short predicate (a true-or-false statement) can be contrived specifying if a given X sits in a plane up to permutation. Two pertinent examples where this is the case are the planes P 4 and P 6 , which recall correspond to the gauge algebras g 4 = su(4) ⊕ sp(6) ⊕2 and g 6 = su(12) ⊕ su(2) ⊕2 , the predicates for which we study here. The planes P 4 and P 6 can be written in a similar fashion to Eq. where the sets of charges appearing in this equation are given in Table 3.
We require a little more notation to continue, as follows. Given a set of charges X satisfying the ACCs, we let X F for each F ∈ {q, u c , d c , , e c , ν c } denote a diagonal 3 × 3 matrix with diagonal entries equal to X F j , j ∈ {1, 2, 3}. The values of TrX F , TrX 2 F , and det X F determine these charges uniquely up to permutation. We can moreover trade each X F for a traceless matrixX F , by the addition of some amount of BL and Y . Specifically, we defineX for some a ∈ Z. Since TrX = 0 and TrX q = 0, these conditions are equivalent to detX = detX q = 0 , (3.70) A similar argument can be applied for the fields d c , e c , ν c and u c , which gives us the predicate Likewise, for the plane P 6 we get These predicates could be used to test whether a set of charges sits in P 4 or P 6 very efficiently, without having to cycle through permutations in the manner described in Subsection 3.4.

Results
Having described our method in detail in the previous two Sections, we here summarise our results. We obtain the planes of X-charge assignments which parametrise all abelian extensions of the SM that embed in each of the 27 maximal algebras listed in Table 1. The planes are, however, not independent from one another. We thus identify planes that are equal, as follows: where the plane in parentheses at the end of each line indicates how we denote the whole equivalence class. All other planes sit in their own equivalence class. Moreover, some (equivalence classes of) planes sit entirely within others. These relationships between planes are summarised in Fig. 1, and can be used to deduce a minimal set of planes that need to be considered (i.e. such that everything embeds in at least one of these planes, but no two of these planes embed in one another). To give one example, the plane P 4 , that parametrises abelian extensions that embed in the electroweak flavour unification algebra g 4 = su(4) ⊕ sp(6) L ⊕ sp(6) R , itself embeds in the plane P 7 associated to g 7 = su(4) ⊕ sp(4) L ⊕ sp(4) R ⊕ so (10). This relationship is represented in Fig. 1 by the (directed) arrow connecting nodes labelled '4' and '7'.
The following planes (4.2-4.5) need to be considered, whether or not we are accounting for global gauge anomalies:   Figure 1: The interrelationships between the different (equivalence classes of) planes. An arrow from i to j indicates that P i is contained entirely within P j (up to family permutations). If the dot associated with i is coloured red then every g k with plane P k equivalent to P i has global anomalies. If the dot is coloured green then every g k with plane P k equivalent to P i is only maximal once those g i 's with global anomalies are removed (i.e. k = 25, 26 or 27).
Recall from Section 2.1 that the list of maximal gauge algebras changes depending on whether we account for global anomaly cancellation or not. If we ignore global anomalies, the following planes (4.6-4.7) should be considered, in addition to (4.2-4.5), (4.6) If, however, we restrict to gauge algebras for which there are possible gauge groups that are free of local and global anomalies, then the algebras g 14 and g 22 (amongst others) are struck out, and other g i ∈ S become maximal as described in Section 2.1. In this case, accounting for global anomalies, the following planes (4.8-4.11) need also be considered, in place of (4.6-4.7):  If a given anomaly-free set of u(1) X 1 ⊕ u(1) X 2 ⊕ . . . charge assignments sits in one of these planes, then this abelian extension of the SM embeds inside the corresponding g i in that equivalence class. If a X-charge assignment sits in none of these planes, then it has no semi-simple completion. We emphasise that these planes P i parametrise the abelian SM extensions : sm S ⊕ a → su (48) which sit in semi-simple extensions of the SM, as introduced carefully in Section 2. We refer to the collection of planes, that covers all cases of abelian extensions with semi-simple completions, as 'Flatland'. As explained, we distinguish two cases, in which we consider completions which are free of only local anomalies (F ), and completions which are free of both local and global anomalies (F), which are: F := (P 2 ∪ P 6 ∪ P 7 ∪ P 16 ) ∪ (P 8 ∪ P 15 ∪ P 26 ∪ P 27 ) . (4.13) As explained in Section 2, the abelian part a necessarily includes the generator of hypercharge that will remain unbroken (as can indeed be seen from the formulae (4.2-4.11) for the 8 planes), thanks to our requiring the embedding ι in (2.3). If we interpret such a gauge model sm S ⊕ a as an extension of the SM, the idea is that sm S ⊕ a breaks to sm. For each plane P i , the maximum number of independent Z gauge bosons that will be produced is thus (4.14) Since the dimension of the planes is ≤ 6, we conclude that the largest number of independent Z bosons we can produce from a semi-simple extension is 5.

Vector-like solutions and the plane P 6
An important subclass of anomaly-free sm ⊕ u(1) X 1 ⊕ u(1) X 2 ⊕ . . . theories is given by those where each X n charge assignment is vector-like, which means that left-and right-handed fields are charged equally. 12 From (4.3), it is easy to see that every set of charges in X ∈ P 6 is vector-like, up-to the addition of hypercharge, where recall that the gauge algebra g 6 = su(12) ⊕ su(2) ⊕ su(2). As we show shortly, an inverted statement is also true, namely 13 Every anomaly-free vector-like extension sm ⊕ u(1) X 1 ⊕ u(1) X 2 ⊕ . . . sits in g 6 . These results are not that surprising; the algebra g 6 fully unifies 3-family flavour symmetry with the quark-lepton unified su(4) colour symmetry of Pati-Salam [38], and colour is vector-like. This should be contrasted, for example, with the corresponding plane P 4 for the  13 The fact that a class of such anomaly-free vector-like sm ⊕ u(1)X models embeds in su(12) ⊕ su(2) ⊕ su (2) was appreciated in Ref. [30].
electroweak flavour unification algebra g 4 ; in that case, where flavour is unified with chiral electroweak symmetries, we get a plane P 4 containing many chiral 'directions', namely S 12 , D 12 , and D 23 . To see that our inverted statement (4.15) is true, first note that a general vector-like X can be parametrised as a linear combination of family-specific baryon and lepton numbers for coefficients {β i , λ i } ∈ Q 6 . The fact that X is vector-like means that the cubic anomaly, the gravitational anomaly, and the mixed anomaly with su(3) all vanish automatically. Substituting (4.16) into the sm ⊕ u(1) X anomaly cancellation equations, one finds that the mixed anomaly involving one hypercharge boson and two u(1) X bosons vanishes also, even though hypercharge is chiral. This leaves only a pair of mixed anomalies between u(1) X and su(2)⊕u(1) Y that are linear in the X charges, and both these anomaly coefficients are proportional to 3 i=1 (β i + λ i ), which must therefore vanish. Thus, the most general anomaly-free vector-like X corresponds to gauging a linear combination of baryon and lepton numbers, with coefficients that sum to zero. This tells us that a generic such X must sit in the plane P 6 .
Turning to a generic anomaly-free vector-like extension sm ⊕ u(1) X 1 ⊕ u(1) X 2 ⊕ . . . , our above results tells us that, since each X n is vector-like and sm ⊕ u(1) X n is anomaly free, each X n must lie in P 6 (we note that P 6 is invariant under family permutations). Thus, the span {Y, X n } forms a sub-plane of P 6 and is automatically anomaly free. This leads us directly to the result in (4.15). But also the further statement, that any extension of the sm by any number of vector-like charges is guaranteed to be anomaly free, provided each sm ⊕ u(1) X n is anomaly free (without having to check any of the extra mixed anomalies explicitly).

Testing a selection of known models
Given any particular anomaly-free abelian extension of the SM, of which examples abound in the literature (especially featuring a single u(1) X extension), using our method we can easily ascertain whether the model embeds in a semi-simple gauge algebra g ∈ S. We can moreover identify which equivalence classes of planes it embeds in, from the list (4.2-4.11). We collect results for a selection of models from the literature in Table 4. The tests of all these models are included in the computer program Test you own charges.nb that is included with this arXiv submission, and that we describe how to use in Section 5.
We introduce the notation U i 1 ,i 2 ,...in to indicate the set of solutions in the planes P i 1 , P i 2 , . . . P in but not in any others in the list (4.2-4.11). In particular, U ∅ is the set of solutions that do not embed in any of the planes, corresponding to sm ⊕ u(1) X extensions that do not have any semi-simple gauge completions.
To draw out some notable results, we see that all vector-like extensions, including the rank-3 u(1) Le−Lµ ⊕ u(1) Le−Lτ ⊕ u(1) Lµ−Lτ extension, embed at least in P 6 in accordance with Subsection 4.1. This includes various models that have been used to explain the b → sµµ anomalies, such as models based on B 3 − L 2 [39][40][41] (which embeds in many more planes besides P 6 ), on B − 3L 2 [42], and its class of generalisations from Ref. [30] that also feature exact proton stability. The Third Family Hypercharge (Y 3 ) Model [43] and the X = Y 3 − 3(B − L) 3 variant [44], despite being chiral, embed in many different planes. Other chiral models, such as the DY 3 model of [31], the chiral muoquark models of [1,33], and the sm⊕u(1) ⊕2 'anti-GUT inspired' neutrino mass model of [28], 14 do not sit in any of our planes, and so none of these chiral models have a semi-simple completion.

Comparison with the anomaly-free atlas
In this section, we investigate 'what fraction' of anomaly-free u(1) X extensions of the SM+3ν R can be embedded in any anomaly-free semi-simple extension of the SM+3ν R . For a set of charges X ∈ Q 18 (that we label Q j , U j , D j , L j , E j and N j ) to be anomaly-free it must satisfy the ACCs, which are a set of Diophantine equations: Due to the non-linear quadratic and cubic equations, the total space of anomaly-free X charges in Q 18 is complicated. An analytic parametrisation of this space is given in Ref. [20]. This followed a numerical scan in Ref. [19] of points up to some maximum height of Q max = 10.

Model
Refs  Table 4: Selection of anomaly-free U (1) X extensions of the SM, and the planes in which they embed. The notation in the far column U i1,...in indicates that the given charge assignment embeds in the planes P i1 , P i2 , . . . P in but not in any others in the list (4.2-4.11). Embedding in a plane P i means that the sm⊕u(1) X extension embeds in a semi-simple gauge theory with gauge algebra g i , corresponding to the numbered list in Table 1 (as well as other algebras for which the planes are equivalent, as explained in the main text). In addition to the particular solution labelledL µ−τ , we tested all the other 20 'chiral muoquark' models from [1,33], finding that all are in U ∅ , meaning that none of these sm ⊕ u(1) X models have semi-simple completions. In the bottom three rows, we also include examples of multi-Z extensions, specifically two u(1) ⊕2 extensions and one u(1) ⊕3 extension, as a proof of principle.
Here Q max is defined as the maximum absolute value of a set of charges X, once it has been rescaled to contain co-prime integers. (Note that two sets of charges that differ by family permutations or by a constant rescaling should be considered 'the same'.) Our planes define linear subspaces of the overall space of anomaly-free X charges. We can perform the task of determining how many solutions up to a given Q max lie in each plane. There are two ways to do this. Firstly we can scan through the lists of solutions provided in Ref. [19] and test, which lie in our planes. Secondly, we can generate these solutions explicitly by scanning through permissible coefficients of the vectors {R i,j }. As a check, we perform both analyses. By using the U i 1 ,i 2 ,...,in notation introduced in the previous Subsection, we can give complete information, short of providing the solutions themselves, about how many (inequivalent) solutions sit in each of the 10 planes up to Q max = 10. The results are: where we have written in red those planes which should only be considered when we are ignoring the effects of global anomalies, and we have written in green those planes which are only 'maximal' when one is taking account of global anomalies. There are exactly two solutions in all of our planes, which unsurprisingly, correspond to hypercharge and the zerosolution.
In Figure 2 we plot the fraction of all anomaly-free sets of charges that sit in any semisimple extension, as a function of Q max . We summarise the same data in a table below the plot. The red points are for the list of semi-simple algebras including those that suffer from global anomalies. The green points are for the list of semi-simple algebras when care is taken to exclude those which necessarily suffer from global anomalies (see Section 2.1). Since there are more planes the former can sit in, the red points are always higher than the green points. The total fraction of anomaly-free solutions that have semi-simple completions is, for Q max = 10, around 2.5% in both cases.
5 Testing your own u(1) X 1 ⊕ u(1) X 2 ⊕ . . . extension of the SM As part of the ancillary directory to the arXiv submission of this paper, we have included several computer programs. These are described below:

Test your own charges.nb
The Mathematica program Test your own charges.nb in our ancillary directory contains a function testCharges that takes as an input a single set of charges X, or multiple sets of charges {X n }.
For a general single set of charges X, with all Weyl fermions of the SM+3ν R taken to be left-handed, the function is called as follows The fraction of anomaly-free u(1) X extensions of the SM, including three right-handed neutrinos, that embed in any semi-simple gauge model. This is plotted as a function of Q max , the maximum absolute charge in the solution X after clearing common divisors. The red points consider embedding each sm ⊕ u(1) X in any g that is free of local anomalies, while the green points restrict to g for which there is a corresponding gauge group that is further free of global anomalies. We see that, out to Q max = 10, to which the anomaly-free atlas was charted in [19], roughly 2.5 % of solutions are anomaly-free.
Anomaly free Vector-like P 2

Other programs
We include several other programs in the ancillary directory to our arXiv submission. These are • Test your own charges.cpp: similar to Test your own charges.nb, this is a C++ program which of the planes in Eqs. (4.2-4.11) a single set of charges X sits. From this it can be deduced weather it sits in any semi-simple extension absent of local anomalies or absent of local and global anomalies.
• Generate up to Qmax.cpp: this is a C++ program which given a maximum charge Q max generates, and counts all charges in each of our 10 planes in Eqs. (4.2-4.11) up to this Q max . This is related to our discussion in Section 4.3.
• Atlas scan.cpp: this is a C++ program that scans through the Anomaly Free Atlas of Ref. [19], and checks how many sit in each plane. Again, this is related to our discussion in Section 4.3.
• Plane checks.h: this is an auxiliary script for the C++ programs above.
• Flatland.nb: this is a Mathematica script in which much of the workings out in this paper have been explicitly written.
• Flatland aux.m: this is a Mathematica script which is auxiliary to Flatland.nb. It contains functions related to the Chevalley basis, described in Appendix A.

Summary
In this paper, we introduced the notion of a general abelian extension of the Standard Model (including three right-handed neutrinos), and found all such abelian extensions that embed in anomaly-free semi-simple gauge theories, allowing for generic flavour non-universality. Such abelian extensions are widely used in model building, providing a gauge theory framework for extensions of the SM by heavy neutral Z gauge bosons. This space of abelian extensions is a linear space, which we call 'Flatland', composed of a set of planes whose dimensions do not exceed six. Consequently, starting from any such semi-simple gauge theory and breaking down to the SM, one can produce at most five neutral Z bosons with linearly independent fermion charges (the offset with the dimensionality of Flatland is because hypercharge, which remains unbroken, is always included in the abelian part). Many different semi-simple gauge models can give this maximal number of independent Z bosons; examples include su(12) ⊕ su(2) ⊕ su(2) and so(10) ⊕3 . We explicitly parametrise the planes that compose Flatland analytically. We prove various results along the way: for example, we show that every anomaly-free vector-like abelian extension of the SM necessarily embeds in an su(12) ⊕ su(2) ⊕ su(2) gauge model.
We provide a short computer program, described in Section 5, to enable users to test whether an abelian extension of their choosing (for example, their favourite anomaly-free Z model) fits inside any semi-simple gauge model. The code moreover lists a set of maximal gauge algebras (if there are any) in which the user's model embeds. We applied this code to a selection of models from the literature, including all anomaly-free flavour non-universal sm ⊕ u(1) X extensions with maximum charge 10. We hope that this work provides a useful bridge for model builders wanting to connect Z models, which have many uses, with unified gauge models that might describe physics in the UV.