Chiral Abelian gauge theories with few fermions

We construct chiral theories with the smallest number $n_\chi$ of Weyl fermions that form an anomaly-free set under various Abelian gauge groups. For the $U(1)$ group, where $n_\chi = 5$, we show that the general solution to the anomaly equations is a set of charges given by cubic polynomials in three integer parameters. For the $U(1) \times U(1)$ gauge group we find $n_\chi = 6$, and derive the general solution to the anomaly equations, in terms of 6 parameters. For $U(1) \times U(1) \times U(1)$ we show that $n_\chi = 8$, and present some families of solutions. These chiral gauge theories have potential applications to dark matter models, right-handed neutrino interactions, and other extensions of the Standard Model. As an example, we present a simple dark sector with a natural mass hierarchy between three dark matter components.


Introduction
Gauge theories with chiral fermions have remarkable properties. The fermion masses can be generated only if the chiral gauge group is spontaneously broken. This provides an upper limit on the fermion masses, given approximately by the vacuum expectation value that breaks the gauge symmetry. By contrast, vectorlike fermions (i.e., fermions whose masses are gauge invariant) are generically expected to be as heavy as the largest mass scale present in the theory.
The gauge charges of chiral fermions are strongly constrained by the requirement that the gauge symmetry is anomaly-free, i.e., it is preserved at the quantum level [1]. The only potential loophole would be the cancellation of the anomalies by an axion which interacts with the gauge bosons via a dimension-5 coupling. Nevertheless, if the scale of gauge symmetry breaking is much smaller than the Planck scale, as is the case in experimentallyaccessible models, then the dimension-5 coupling should arise from integrating out some chiral fermions of mass comparable to the gauge boson masses [2]. Thus, the charges of all fermions should still form an anomaly-free set. In other words, the fermion charges must be solutions to certain anomaly equations. Generically, these are cubic equations because the gauge anomalies in four spacetime dimensions arise from triangle diagrams with the gauge bosons on the external lines and the fermions running in the loop.
In Abelian gauge theories there are no interactions involving only gauge bosons, so the gauge couplings grow with the energy scale. Thus, these theories are well-behaved at high energies only if they can be embedded in some non-Abelian structures. A necessary condition for that embedding is that the Abelian charges are commensurate, which implies that they can be taken as integers upon an appropriate redefinition of the gauge couplings.
A discussion of some additional arguments leading to the same conclusion can be found in [3]. Thus, the cubic anomaly equations must be solved with integer variables. This is a vexing mathematical challenge given that it is impossible to construct an universal algorithm for solving such Diophantine equations [4], and that general solutions are not known in most nontrivial cases [5,6].
For the simplest Abelian gauge group, namely a single U (1), the most general solution to the anomaly equations has been found in [7]. That result shows that the most general U (1) charges of n Weyl fermions are given by certain quartic polynomials in n − 2 integer parameters.
No such general solution is known for larger Abelian gauge symmetries, which are products of U (1) groups. The numerical search for solutions to the anomaly equations is useful only when the range of fermion charges is very small [8]. To see that, note that in the case of m U (1)'s with n fermions carrying integer charges ranging from −z to +z, the search needs to cover a number of configurations of the order of (2z + 1) m n . For z = O(10), m = 2 and n ≥ 6, that number is already above 10 15 .
A method useful in the case of arbitrary gauge groups, including product groups, is the construction of an anomaly-free chiral set starting from any anomalous set of fermions and adding some new fermions of certain charges [9]. Another method is to start from some non-Abelian gauge group with a known anomaly-free representation and analyze its U (1) subgroups [10]. It is not yet clear whether algebraic geometry methods [11] may lead to general solutions for multiple U (1) groups.
In this paper we analyze the Abelian gauge theories with the smallest number n χ of chiral fermions. For a single U (1), it is known that n χ = 5 [12,13], so that the minimal chiral theory includes five Weyl fermions with charges depending on three integer parameters [7]. We show that the general solution to the anomaly equations simplifies in this case, being given by cubic polynomials in the three parameters. This parametrization allows us to prove that the set of solutions is infinite, and that the relative signs of all five charges are fixed.
For the U (1) × U (1) gauge theories we find that the smallest number of chiral fermions is n χ = 6, and we construct the general solution to the anomaly equations for this chiral theory, in terms of six integer parameters. For the U (1) × U (1) × U (1) gauge group, we show that chiral theories exist only for n ≥ 8 fermions, and we identify some anomaly-free chiral sets of charges which depend on three integer parameters.
The chiral gauge theories constructed here have potential applications to various model building efforts related to gauge extensions of the Standard Model [14,15], flavor structures [16], right-handed neutrino interactions [12][13][14][17][18][19], dark matter sectors [19,20], and other phenomena. Chiral models have great advantages over models with vectorlike fermions. The chiral fermions are ensured to be much lighter than the Planck scale, and furthermore large mass hierarchies between fermion flavors naturally arise in chiral models. As an example, we show that even one of the simplest U (1) × U (1) chiral models obtained here is a viable dark sector with complex phenomenological implications, which includes three dark matter components with hierarchical masses.
In Section 2 we display the anomaly equations for arbitrary Abelian gauge theories, summarize the most general solution for a single U (1), and define some key concepts (primitive solutions, canonical order for chiral sets of charges, etc.) used in later Sections. The U (1) gauge theory with 5 fermions is analyzed in Section 3. The general solution to the U (1) × U (1) gauge theory with 6 fermions is derived in Section 4. The smallest number of chiral fermions charged under three U (1) groups is derived in Section 5, along with some families of solutions. Our conclusions are collected in Section 6.

Anomaly equations for Abelian gauge groups
We study Abelian gauge theories with a number m of U (1) groups whose charges are carried by n Weyl fermions. The charges of the ith fermion, for any 1 ≤ i ≤ n, under the all the Weyl fermions are taken to have the same chirality (e.g., left-handed). The set of charges of the n fermions under the U (1) j gauge group is labelled by As the charges within the { z j } set are commensurate, we can choose all z ji charges to be integers by rescaling the g j gauge coupling of U (1) j , for 1 ≤ j ≤ m.
The fermion charges must satisfy several anomaly equations. For each U (1) group there is an [U (1)] 3 anomaly equation, due to triangle diagrams with three external gauge bosons [1], and a gravitational-U (1) anomaly equation, due to diagrams with one external gauge boson and two external gravitons [21]. The first type of equation is cubic in charges, while the second one is linear in charges: Eliminating z jn from Eq. (2.3), we can rewrite Eq. (2.2) in the form "sum of the cubes equals the cube of the sum", with n − 1 integer variables: There are also anomaly equations involving each pair of U (1) groups, due to triangle diagrams with two external gauge bosons of one kind and a single external gauge boson of the other type.
Finally, there are U (1) j U (1) j U (1) j anomaly equations (these are new only when no two indices are equal, which requires m ≥ 3), due to diagrams with three different external gauge bosons: Altogether, the number of anomaly equations is where 2m is due to the U (1) and [U (1)] 3 anomalies, 2 m 2 is the number of U (1) j [U (1) j ] 2 mixed anomalies, and m 3 is due to the U (1) j U (1) j U (1) j anomalies. For each U (1) j gauge group, the most general solution to the [U (1) j ] 3 anomaly equation (2.2) and the gravitational-U (1) j anomaly equation (2.3), depends on n − 2 integer parameters. That general U (1) j solution was found in [7], and can be written as the "merger" of two vectorlike sets of charges. The merger operation 1 , labelled by ⊕, acts on two sets of charges of equal length, similar to those of Eq. (2.1), and is defined by For even n, the general U (1) solution (here we drop the j index to simplify the notation) is the merger of two vectorlike sets of charges, and takes the form where 1 , ..., n/2−1 and k 1 , ..., k n/2−1 are integer parameters. For odd n, the general U (1) solution can be written as with the n−2 integer parameters labelled in this case by k 1 , ... , k (n−1)/2 and 1 , ... , (n−3)/2 . Notice that the above forms of the general U (1) solution imply that charges are given by polynomials in the integer parameters, which collectively are quartic. We use the phrase "general solution" not as a reminder that it applies to any number of fermions, but rather to emphasize that any chiral set of charges that satisfy the two U (1) anomaly equations for fixed n corresponds to a particular choice of the integer parameters. fermions. Thus, it is sufficient to consider chiral sets of fermions, which means that the following condition must be satisfied: where the above inequality does not preclude z ji = −z ji for some or most (but not all) values of j in the 1 ≤ j ≤ m interval. That is, two Weyl fermions may form a vectorlike pair with respect to some of the U (1) j subgroups and still be chiral with respect to the whole U (1) m gauge group.
As all anomaly equations (2.2)-(2.7) are homogeneous in the charges of each U (1) group, for any chiral set of integer charges which satisfies them there is an infinite set of solutions where all the charges under any U (1) group are multiplied by an arbitrary integer. It is thus sufficient to consider sets of charges which are coprime with respect to each U (1), i.e., the greatest common divisor of the n charges under each U (1) is 1. We will refer to such a set as coprime.
Thus a solution to the anomaly equations is nontrivial if it is chiral and coprime. For large enough n the solutions include composite sets, which are formed of two or more subsets that independently satisfy the anomaly equations. Extending the terminology of [7] to multiple U (1)'s, we define "primitive" solutions as anomaly-free chiral sets which are coprime but not composite. identical to the canonical form defined in [7]. To be more explicit, let us label the set of This set is in canonical form if and only if (2.14) Note that even for m ≥ 2 primitive solutions in the canonical form belong to families of m! sets related by interchanges of the U (1) groups. That additional trivial multiplicity can also be eliminated by imposing the prescription that z 11 has the largest absolute value of all z ji charges, with additional prescriptions in the cases where there are two or more such charges. However, for the purpose of this article it is not necessary to impose these prescriptions.
In the case of a single U (1) gauge symmetry, we simplify the notation for fermion charges introduced in Section 2 by dropping the index that labels the gauge group: z 1i ≡ z i , for 1 ≤ j ≤ n. The anomaly equation of the type "sum of the cubes equals the cube of the sum", Eq. (2.4) for n = 5 (and m = 1, i.e., a single U (1) group) is while the fifth charge is given by The chiral conditions (2.12) become Adapting Eq. (2.14) to the case of m = 1, n = 5, a chiral set {z 1 , ..., z 5 } is in the canonical form (as defined in [7]) if it satisfies Next we derive some generic properties of the anomaly-free chiral sets of 5 Weyl fermions. Later in this Section we will present and analyze the general solution to the anomaly equations for a single U (1) with n = 5.

Properties of the 5-charge chiral sets
We first point out that all chiral solutions to Eqs.
Using the canonical ordering (2.14) and imposing the chiral conditions (3.3) we find when the charges are ordered in canonical form. The uniqueness of the above signature is a special property of the n = 5 case. For n = 6 there are two possible signatures [7], and for larger n the number of signatures grows quickly.
We next prove that no two charges are equal in the chiral set for n = 5. Due to the signature established above, it is sufficient to show that z 2 = z 3 and z 4 = z 5 . If z 2 = z 3 were true, then Eqs. (3.5) and (3.2) would lead to the following quadratic equation in z 4 : The discriminant ∆ of this equation can be written as the difference of two perfect squares: A necessary condition for Eq. (3.7) to have an integer solution is that ∆ is a perfect square. Thus, the three integers 2z 2 (z 1 + z 2 ), ∆ and z 2 1 − 2z 2 2 must form a Pythagorean triple, which implies (see, e.g., pages 245-247 of [5]) that there must exist two integers m and m such that z 2 (z 1 + z 2 ) = m m and z 2 1 − 2z 2 2 = m 2 + m 2 . These equations give z 1 (z 1 + 2z 2 ) = (m + m ) 2 > 0, which is in contradiction with the inequalities z 1 > 0 and z 1 + 2z 2 = −z 4 − z 5 < 0. Thus, we have shown that z 2 < z 3 .
Our proof that z 4 = z 5 is more elaborate. If z 4 = z 5 were true, first we follow the same steps as above with z 2 ↔ z 4 and z 1 ↔ z 3 , which leads to z 3 (z 3 + 2z 4 ) > 0, implying −z 3 > 2z 4 . Second, Eqs. (3.5) and (3.2) with z 4 = z 5 imply that the following quadratic equation in r = −z 1 /z 3 > 1 would need to be satisfied: where the parameter introduced here is 0 < a = −2z 4 /z 3 < 1. It is straightforward to show that Eq. (3.9) has no solutions with r > 1, and consequently the assumption z 4 = z 5 is false. This completes the proof that the chiral set does not include two charges which are equal. Combining this result with the signature (3.6), we find that the charges of an anomaly-free chiral set in the canonical form satisfy a remarkable constraint: We now turn to the question of whether the number of primitive solutions to Eqs. (3.1) and (3.2) is infinite. To that end we construct a 1-parameter family of solutions: where k ∈ Z and k ≥ 1. Notice that this chiral set is already in the canonical form for k ≥ 3. We still need to find for which values of k this set is coprime. Let us concentrate on the first and third charges of this set, namely (2k + 1) 2 and −2(k + 1) 2 . If these have a common divisor z ≥ 1, then 2k + 1 ≡ 0 (mod z ) and k + 1 ≡ 0 (mod z ), which implies 1 ≡ 0 (mod z ) so that z = 1. It follows that the first and third charges are always coprime, so that the whole set is coprime for any k ≥ 1. Thus, the chiral set (3.11) represents a countably infinite family of primitive solutions.
It is worth noting that the set (3.11) for k 1 has two charges of order ±4k 2 , two charges of order ±2k 2 , and one of order 3k. Therefore, in the large k limit this chiral set is a small (but essential) perturbation away from a vectorlike set.

General solution to the 5-charge anomaly equations
The most general nontrivial solution to the U (1) anomaly equations (2.2) and (2.3) for an odd number of Weyl fermions is given by the merger shown in Eq. (2.11), as proved in [7]. Only the chiral solutions are nontrivial, as all vectorlike ones are obtained from either set used in (2.11) and depend on (n − 1)/2 parameters.
In the case of the smallest number of fermions, n χ = 5, the anomaly equations take the form (3.1) and (3.2), and the general solution becomes Given that there are 5 charges and two equations, the general solution depends on 3 integer parameters labelled in this case by k 1 , k 2 and .
As explained in Section 2, the general U (1) solution gives the n charges as quartic polynomials in the integer parameters. We point out that in the n = 5 case a simplification occurs: the 5 charges arising from the merger (3.12) are proportional to k 1 = 0, so dividing all charges by k 1 we find that the solution is cubic in the integer parameters. Explicitly, our 3-parameter solution to the U (1) anomaly equations for n = 5 charges is Notice that each charge is a homogeneous cubic polynomial in the three integer parameters, but as functions of individual parameters the charges are quadratic or in some cases linear. It is straightforward, albeit tedious, to check that this is an anomaly-free chiral set.
The necessary and sufficient condition for any 5-charge set to be chiral, i.e., to satisfy Eq. (2.12), is that none of the charges is 0. Thus, the above set is chiral if and only if all of the following conditions are satisfied: Any choice of the three integers k 1 , k 2 , corresponds to a 5-charge solution, which can also be obtained up to a reordering of charges, or an overall sign flip, by a few other choices of k 1 , k 2 , . The simplest transformations of this type are as well as Consequently, no primitive solutions are removed when k 1 , k 2 , are restricted to A more complicated transformation that keeps the set invariant is This implies, in particular, that any solution corresponding to parameters of the type (k 1 , k 2 + k 1 , −k 2 + k 1 ) is equivalent to the solution arising from (k 1 , k 2 , −k 2 ). There is also a more particular case of invariance, which seems related but does not arise from the transformation (3.18): For certain values of k 1 , k 2 , the set (3.13) represents a primitive solution (not necessarily in the canonical form), while for other values it is a primitive solution rescaled by a common integer factor. To reduce the latter category, it is useful to impose that the three integers (k 1 , k 2 , ) are coprime (not necessarily pairwise coprime). A particular transformation that changes both the ordering of the charges within the set, and the greatest common divisor (GCD) of the set, while leaving invariant the primitive solution is the following: In Table 1 we display the primitive solutions in canonical form with z 1 ≤ 32. For each of the primitive solution we indicate a choice of (k 1 , k 2 , ) that generates that solution. In many cases, retrieving the canonical form requires a reordering of charges within the set The proof that all chiral solutions for arbitrary n can be generated by the merger operations of Eqs. (2.10) and (2.11) was given in [7] for an arbitrary number of fermions.
It is instructive to adapt that proof of generality to the case of n = 5. Consider an arbitrary chiral set of 5 charges, { q }, which is a solution to the anomaly equations (3.1) and (3.2). We need to show that there exists three integers (k 1 , k 2 , ) such that the set { z } of 5 charges given in (3.13) is the same as { q } up to an overall rescaling by an integer.
To that end, we identify (k 1 , k 2 , ) in terms of the { q } components as follows: Replacing these expressions in the set (3.13) we obtain each of the five z i quantities as 6th-degree polynomials in the q 1 , q 2 , q 3 and q 4 charges of the { q } set (after using the linear equation to eliminate q 5 ). The next step is to eliminate the higher powers of one of the charges, for instance z 4 , using the cubic equation (3.1) written in terms of q i 's, The result of this lengthy computation is that the { z } set is proportional to { q }: The final step is to notice that Eq. (3.22) implies Primitive solution The relation between (k 1 , k 2 , ) and the { q } set shown in Eq. (3.21) indicates that there is a more restrictive range for the integer parameters which still spans the general solution. When { q } is in the canonical form, analogous to (3.10), the parameters satisfy Consequently, this restriction is sufficient for the set (3.13), with appropriate reorderings and an overall integer rescaling, to generate all the anomaly-free chiral solutions. This restriction is also sufficient for the chirality conditions (3.14) to be satisfied, and it implies Thus, imposing (3.25) and (3.26) brings the set (3.13) almost to the canonical form: the only remaining conditions are z 4 > z 5 , which sets an upper limit on /k 1 as a function of k 1 /k 2 , and |z 2 | > |z 3 | which sets a lower limit on /k 1 for a range of k 1 /k 2 . The restriction (3.25), however, has a drawback: to generate various solutions with small charges one needs large values of (k 1 , k 2 , ), and thus the set { z } needs to be divided by a large GCD( z ). By contrast, the range (3.17) works with lower absolute values of (k 1 , k 2 , ) (see Table 1) at the cost of having multiple coprime choices for (k 1 , k 2 , ) associated with a single solution.

General solution for U (1) × U (1) with 6 fermions
We now turn to the smallest set of fermions charged under a U (1) 1 × U (1) 2 gauge group that is chiral and anomaly free. Using the notation introduced in Section 2, the charges of n Weyl fermions under the two U (1) groups are { z 1 } and { z 2 }, respectively. We impose that z 1 is not parallel to z 2 -otherwise there is a field redefinition which removes couplings to one of the gauge bosons. If the whole set of  cannot transform a vectorlike solution into a chiral one, or vice-versa. Thus, the minimum number of fermions such that U (1) × U (1) is chiral and anomaly-free is n χ ≥ 6.
As we will show, by explicit construction, this minimum number of fermions is n χ = 6. In This is useful provided z 11 +z 12 = 0, which is always satisfied when { z 1 } is chiral. Since no We start by analyzing the first charge pairing (4.4). The mixed anomaly equations (2.5) and (2.6) now take the form In order to solve these equations, we consider a few cases depending on the relations between the { z 1 } charges. Although a reordering of charges in { z 1 } does not alter the physics, and can avoid some of the special cases discussed below, for simplicity we hold the ordering of charges fixed in our analysis. The case where z 11 = z 12 is discussed later in this Section. It is convenient to use the short-hand notation An identity valid for any anomaly-free set of 6 charges is ζ 12 + ζ 34 + ζ 56 = −(z 11 + z 12 )(z 13 + z 14 )(z 15 + z 16 ) . Note that here v + 5 = v − 5 , but later on this equality will not hold as we will use different Consider now the special case ζ 12 + ζ 34 = 0, which combined with the identity (4.10) implies (z 15 − z 16 ) 2 = − (z 11 + z 12 ) (z 13 + z 14 ). A solution for Eq. (4.11) is then v 5 = 0, which implies v 1 = −z 2 13 + z 2 14 and v 3 = z 2 11 − z 2 12 . This solution turns out to be the same as one of the solutions shown in Eq. (4.12) extended to the case of ζ 12 + ζ 34 = 0 (the other solution becomes trivial, v 1 = v 3 = v 5 = 0). For v 5 = 0, the quadratic equation Extracting v 1 from Eq. (4.7), and removing an overall factor of (z 11 + z 12 ) (z 13 + z 14 ), we find the following solution for v 1 , v 3 , v 5 , which we label with an upper index 0: This solution cannot be obtained from the ones given in (4.12), even after allowing an arbitrary change in the overall normalization.
where κ is a parameter that allows for changes in the normalizations of v ±,0 displayed in Eqs. (4.12) and (4.13). By rescaling v ±,0 and z 1 their coefficients in (4.14) can always be made integer. Furthermore, from Eq. (4.2) any integer choice for the coefficients will lead to a solution for z ±,0 2 . Thus, the most general sets of U (1) 2 charges which satisfy all anomaly equations are with κ and λ integers and the most general solutions to the system of four anomaly equations (for two mixed U (1) 1 × U (1) 2 anomalies, and two U (1) 2 anomalies) are the following sets: where the ±, 0 upper indices are correlated, i.e., there is a solution with index +, another solution with index −, and a special solution with index 0 that applies only for a set z 1 that satisfies ζ 12 +ζ 34 = 0, in which case only one of v ± is nontrivial. Without loss of generality, given the freedom of redefining the U (1) 2 gauge coupling, we take the parameters κ and λ as integers. One should keep in mind that in order to obtain a coprime set the six integers that represent the U (1) 2 charges shown above may need to be divided by their greatest common divisor. Note that the [ z 1 , z 2 ] sets are typically chiral under each of the U (1)'s; we refer to such sets as doubly-chiral under U (1) 1 × U (1) 2 .
We now turn to the second charge pairing in the set { v }, which is given in Eq. (4.5). (4.17) Thus, the general solution is defined by 6 integer parameters, 4 integers necessary to specify { z 1 }, κ, and λ, and a discrete choice from six solutions, two for each choice of charge pairing in { v }. We will refer to these discrete choices as "branches".

Parametrization of the U (1) × U (1) solutions
We now construct the parametrization of the chiral-vectorlike and doubly-chiral solutions for 6 fermions charged under U (1) 1 × U (1) 2 . The general set of charges { z 1 }, which is a solution to the U (1) 1 anomaly equations, depends on 4 integer parameters, k 1,2 , 1,2 , as follows from Eq. (2.10) and the detailed discussion for 6-charge sets given in [7]. Since the z 1j are quartic polynomials in the k 1,2 , 1,2 , one would expect the z ± 2 charges to be collectively 12 th order in the k 1,2 and 1,2 parameters but various common polynomial factors arise allowing simplification. The simplification is maximal when the last three charges of the solution generated from the merger (2.10) are reordered. More precisely, which gives the six U (1) 1 charges as the following quartic polynomials: Inserting this parametrization in Eq. (4.12) we obtain the components of the vectorlike set { v + } after removing the greatest common divisor (which is an 8th degree polynomial in the integer parameters): } are parametrized as in Eq. (4.20). Altogether, this solution depends on 6 integer parameters: k 1 , k 2 , 1 , 2 , κ, λ.
Instead of using the { v + } vectorlike set, one could construct a different solution with the { v − } set, whose components are similarly given by In the case where ζ 12 + ζ 34 = 0, the [ z 1 , z 0 2 ] solution from Eq.

Example of a chiral dark sector
The U (1) 1 × U (1) 2 chiral sets obtained here may have interesting applications in a variety of hidden-sector models, related to right-handed neutrinos, dark matter, and other phenomena. To give an example, we take the second set of (4.22), fix λ = 0 and κ = 1, and introduce a scalar φ carrying charges (−2, 0) under The field content of this model is shown in Table 2. The gauge charges allow the Weyl fermions ψ i , i = 1, ..., 6, to have the following Lagrangian interactions with the scalar: The first term is renormalizable, and its dimensionless coefficient, the Yukawa coupling y 45 > 0, is presumably of order one. The last two terms are effective interactions generated at the scale M * where the two U (1) groups are embedded in some non-Abelian structure.
We will not provide an explicit renormalizable origin of these two terms, but they may be generated at tree level, or at loop level, by various heavy particles which are integrated out. The two dimensionless coefficients, c 16 , c 23 > 0, may thus be of order one or much smaller than that.
A scalar potential with a minimum at φ > 0 leads to three Dirac masses. The 4-component fermion formed of ψ 2 and ψ 3 has a mass m 23 = c 23 φ 4 /M 3 * . Similarly, the Dirac fermion formed of ψ 1 and ψ 6 has a mass m 16 = c 16 φ 3 /M 2 * , while the remaining mass is m 45 = y 45 φ . Parametrically, we expect large hierarchies between these masses: These 3 Dirac fermions are stable due to a global U (1) 3 symmetry. Thus, the model described here is a good candidate for a dark sector, with three dark matter components.
The gauge boson associated with U (1) 2 is a massless dark photon, while the one associated with U (1) 1 is a massive "dark Z" boson. Only the lightest and the heaviest fermions interact with the dark photon, while all fermions interact with the dark Z.
The first pair of { z 2 } cannot be vectorlike (because we seek the cases where the [ z 1 , z 2 ] set is chiral). Thus, the first two charges in { z 2 } can be taken to be j 1 and j 2 .  Table 3: All doubly-vectorlike solutions to the U (1) 1 × U (1) 2 anomaly equations for 6 chiral fermions. The [ z 1 , z 2 ] solution shown in the first column depends only on the j 1 , j 2 , j 3 integer parameters, with the z 2 set of U (1) 2 charges written in terms of j 1 , j 2 , j 3 , which are the quadratic polynomials (4.25) in j 1 , j 2 , j 3 . The other 7 columns represent all remaining solutions for z 2 , with j 1 , j 2 , j 3 for each of them obtained from (4.25) by flipping the sign for one of the j parameters (specified in the second-to-last row), or by interchanging j 1 and j 2 , or in three cases by performing both transformations.
For the first branch of this type, the anomaly equations determine j 1 , j 2 , j 3 in terms of j 1 , j 2 , j 3 as follows: For each of the other 7 arrangements of the second set { z 2 }, the expressions for j 1 , j 2 , j 3 in terms of j 1 , j 2 , j 3 are obtained from the above equations by flipping the sign of one of the j 1 , j 2 , j 3 parameters, or by the interchange j 1 ↔ j 2 . In three of these cases, as indicated in Table 3, both the sign flip and the interchange need to be performed on Eqs. (4.25) in order to obtain the correct j 1 , j 2 , j 3 dependence on j 1 , j 2 , j 3 .
Likewise, if j 2 = j 3 , then −j 1 = j 2 = j 3 , leading to a vectorlike set. It is thus sufficient to impose j 1 > j 2 > j 3 ≥ 0 . (4.26) Out of the 8 possible solutions for z 2 shown in Table 3, only 3 are linearly independent.
Nevertheless, when the masses of the two U (1) 1 × U (1) 2 gauge bosons are different, each of those 8 solutions generically leads to different observables, as the couplings between the gauge bosons and fermions are determined by the charges. As a numerical example, j 1 = 3, j 2 = 2, j 3 = 1 gives the following 8 solutions: where we show [ z 1 , z 2 ] for the first solution, and only { z 2 } for the other 7 solutions (as z 1 remains unchanged).
When j 3 = 0 only 4 of the 8 solutions are distinct (for example, the third and seventh solutions shown in Table 3 become identical). In the particular case j 1 = 2, j 2 = 1, j 3 = 0, the four solutions are Although we focus on theories with the smallest chiral sets, it is instructive to comment on larger sets of fermions charged under U (1)×U (1). For a general doubly-vectorlike solution [ z 1 , z 2 ] with n = 2p fermions, where p ∈ Z, the number b(p) of possible arrangements of two sets of vectorlike charges such that the solution is chiral satisfies a recurrence relation: This implies that the number of branches b(p) for doubly-vectorlike solutions grows quickly with the number n = 2p of fermions, e.g., b(4) = 60 and b(5) = 544. The sequence of integers generated by the recurrence relation (4.29) arises in various combinatorics problems [23]. For p 1, we find the following approximate behavior: To summarize the main result of this Section, we have identified the general solution for the charges of 6 fermions which form an anomaly-free chiral set under the U (1) × U (1) gauge group. The general doubly-vectorlike solution depends only on the three integer parameters j 1 , j 2 , j 3 , and on a discrete parameter that labels up to 8 branches shown in Table 3. The general doubly-chiral solution depends on six integer parameters, and has 6 branches given in (4.17) and (4.17). Any chiral-vectorlike solution can be obtained from the doubly-chiral solution by setting λ = 0.

Three or more U (1) gauge groups
We now turn to a theory with n Weyl fermions charged under a U (1) m gauge group where m ≥ 3. The number of anomaly equations, N eq , is given in Eq. (2.8). In the case m = 3, i.e., when the gauge group is Next we will make a sequence of invertible transformations, as in Eq. (4.2), that will place two zero charges under each of the three U (1)'s for different fermions. This will allow us to show, using the anomaly equations, that the theory with 7 fermions is vectorlike.
Since the linear transformation (4.2) cannot transform chiral solutions into vectorlike ones, this will prove that any chiral theory with m = 3 requires more than 7 fermions.
First, use some nonzero elements of { z 2 } and { z 3 } to form a linear combination of charges that leads to two zeros in { z 1 }, and then sort the order of the fermions to get Now, using the U (1) 1 U (1) 2 U (1) 3 anomaly equation, we find that at least one of z 17 , z 27 and z 37 is zero. Without loss of generality, we take z 37 = 0, implying that { z 3 } has exactly 4 nonzero entries. As a result, { z 3 } includes two vectorlike pairs, so that there are only two possibilities for the anomaly-free solutions with 7 fermions:

Solutions for U (1) 3 anomaly equations with 8 fermions
We now show that chiral solutions for the U (1) 1 × U (1) 2 × U (1) 3 anomaly equations exist when there are 8 Weyl fermions. We will not attempt to derive the general solution of this type. Instead, we explicitly construct solutions which are vectorlike with respect to each U (1) group taken separately, but are chiral with respect to the whole gauge group.
Extending the terminology of Section 4.4, these are triply-vectorlike solutions.
For the case of U (1) 1 × U (1) 2 we showed how to take any 6-fermion solution to the  Table 3. The solution has the structure, where j 1 , j 2 , j 3 are the quadratic polynomials (4.25) in the j 1 , j 2 , j 3 parameters. This completes the proof that the smallest number of Weyl fermions charged under U (1) 3 that leads to chiral anomaly-free theories is n χ = 8.
Note that these solutions contain fermions which are vectorlike with respect to two of the U (1)'s but chiral with respect to all three. These triply-vectorlike solutions can be made chiral under each U (1) by taking linear combinations of charges, as in Eq.(4.2). As a numeric example consider the solution in Eq. (5.3) with j 1 = 1, j 2 = 2, j 3 = 3, put into canonical form, for the second, these two triply-vectorlike solutions will transform to ones which are either composite or fully chiral under each U (1),

More U (1) gauge groups
Consider the case of U (1) m gauge theories with m ≥ 4. As before, n χ denotes the smallest number of fermions such that the theory is chiral and anomaly free, so the total number of charges is n χ m. Since the number of equations that these charges need to satisfy, N eq given in (2.8), grows as m 3 , one could expect that n χ will grow quadratically with m. It turns out that is not true: n χ grows at most linearly with m.
To see that, notice that we can use the As an example, we construct a composite solution for m = 5 starting from a 6-fermion chiral-vectorlike solution to U (1) × U (1) and an 8-fermion triply-vectorlike solution to U (1) × U (1) × U (1). In particular we use the chiral-vectorlike solution generated by Eqs. (4.19) and (4.20) with (k 1 , k 2 , 1 , 2 ) = (2, 0, 1, −1) and the first triply-vectorlike solution of Eq. (5.3) with (j 1 , j 2 , j 3 ) = (1, 2, 3). By taking linear combinations of charges, the block diagonal structure is removed and we obtain the following set in canonical form: Although no longer immediately apparent, this solution is still composite and consists of two subsets of six and eight fermions which are separately anomaly free.

Conclusions
The gauge charges of chiral fermions cannot be freely chosen. The requirement of anomaly cancellation, dictated by a set of cubic Diophantine equations, places strong and complicated constraints on both the number and charges of the fermions that can be present in consistent theories. In this article we have analyzed the solutions to the anomaly equations for various Abelian gauge theories with the smallest number of Weyl fermions necessary to form anomaly-free chiral sets.
For a single U (1) gauge group the anomaly equations take the form "the sum of the cubes equals the cube of the sum", and the most general solution was derived in [7] for an arbitrary number of fermions. We have shown here that for the smallest chiral theory, which contains 5 Weyl fermions, the general solution has different properties than theories with more fermions: the charges are given by cubic polynomials in 3 integer parameters [see Eq. (3.13)], the relative signs of the 5 charges are unique, and no two charges can be equal. We also proved that there is a countably infinite number of such solutions.
For the U (1) × U (1) gauge theories, in addition to the anomalies for each U (1), there are two mixed anomaly equations. We showed that at least 6 chiral fermions are necessary to satisfy all anomaly equations, and we constructed the general U (1) × U (1) solution for the 6-fermion set. Remarkably, any anomaly-free chiral set of 6 charges under a single  Table 3 and Eq. (4.25). Finding the general solution for U (1)×U (1) with n ≥ 7 fermions is a challenging problem since the number of branches grows quickly with n, see Eq. (4.29).
As the number m of U (1) gauge groups increases, the number of anomaly equations that must be satisfied grows as m 3 . For the U (1) × U (1) × U (1) gauge theory, which has 13 anomaly equations, we proved that the minimum number of chiral fermions is n χ = 8, and we constructed a couple of 3-parameter chiral solutions, see Eq. The results and methods presented here have broad potential application, such as dark matter models, extensions of the Standard Model with Z bosons, or neutrino model building. As an illustration, we presented a chiral dark matter model with U (1) × U (1) gauge group and three dark fermions of hierarchical masses.