$SU(3)^p$ Quiver Theories with ${\cal N} = 0$ for $p=8$ and $9$

We close a gap in previous studies of nonsupersymmetric ${\cal N}=0$ quiver gauge theories from a phenomenological point of view aimed at acquiring specific proposals for models beyond the Standard Model (BSM). Because $SU(3)$ is the gauge group of QCD we fix $N=3$ and vary only the $Z_p$ abelian orbifold. The values $1 \leq p \leq 7$ have been previously fully discussed as well as one special case, discovered by happenstance, of $p=12$. The values $p = 8$ and $p=9$ are discussed comprehensively in the present paper including the electroweak mixing angle, gauge coupling unification, spontaneous symmetry breakdown to the standard model, and the occurrence of three quark-lepton families. Two promising quiver node identifications are discovered for $p=8$ and three for $p=9$. All of these merit further study as BSM candidates.


Introduction
A possible approach to generate new models beyond the Standard Model (BSM) is to use nonsupersymmetric gauge theories derived from the most highly supersymmetric N = 4 gauge theories. Such N = 0 theories can be systematically constructed from N = 4 ones by using suitable abelian Z p orbifolding [1,2]. These constructions are encoded by quiver diagrams [3,4], in which the i th node represents the U (N ) i gauge symmetry and oriented arrows from the i th to the j th node represent fermions in the bifundamental (N i , N j ) representation of the two gauge groups at nodes i and j. Scalars are usually denoted by dashed lines connecting two nodes, in a related representation of the gauge group (N,N ) + (N , N ), if the parameters a i of the quiver theory, that we will define below, are all nonzero. If any of the parameters a i is zero then such scalars can be in singlet or in adjoint representations of the gauge group, but it has been shown that in such a case chiral fermions are not allowed by the theory, and as such they are of no physical interest. Explicit examples with Z p orbifolding have been considered in the past for several p values. For example, in [5] it has been discussed a Z 7 model which contains all the states of the Standard Model (SM) and in [6] a Z 12 model allowing grand unification at a scale of 4 TeV. The result of this construction is a gauge theory with a gauge structure of the form SU (3) p which contains a colour gauge symmetry SU (3) n C C , a weak SU (3) n W W symmetry and a SU (3) n H H of hypercolour, with n H + n W + n C = p. This symmetry is characterised by a single coupling g above the scale of grand unification (GUT) µ GU T , where the p factors are all independent copies of SU (3), with a Z p symmetry which renders the p nodes of the quiver diagram identical.
The issue whether such classically scale invariant theory may be conformal invariant at quantum level, with a vanishing β function beyond one loop, has been matter of debate in the past, and conclusive arguments in this context are still missing [7,8] Recent discussions of classically conformally gauge filed theories include [9][10][11][12]. The structure of the theory below µ GU T (µ < µ GU T ) is of the form SU  (3) factors, at this scale, is the surviving diagonal subgroup of the the colour, weak and hypercolour symmetries, with couplings which are renormalized and reduced by the same multiplicites n i (g → g i = g/ √ n i ).
The SU (3) 3 symmetry of the diagonals is indeed a trinification [13], but with gauge couplings which are different in size and that can be unified at a far smaller scale compared to the typical 10 15 − 10 16 GeV GUT scale. In ordinary trinification, the 3 couplings meet at a specific (usually very large) scale, after a large logarithmic running, which is not necessary in this case, with the result that the GUT scale can be as low as 4 TeV. Above such scale, as we have already mentioned, the quiver theory is probably characterised by a quasi conformal behaviour, since the one-loop beta function vanishes, while its vanishing at two loops is not guaranteed. The appearance of double trace operators, due to the breaking of supersymmetry of the mother theory, with their non-vanishing beta-functions, has been brought up as an argument against its quantum conformal behaviour. In these theories the hierarchy is significantly ameliorated since the one loop quadratic divergences, which emerge in the Higgs sector of the SM, are absent. This is due to a precise cancellation between bosonic and fermionic contributions in the scalar 2-point function, a property which is inherited by the quiver theory from the N = 4 mother theory.
In the models that we study below these features are all present and render them quite interesting from the phenomenological viewpoint. In the absence of any supersymmetric signal at the LHC, it is therefore tempting to reconsider such models in some generality, building on previous analysis and extending their classification, since they provide an alternative view to unification based on ordinary GUT's. This in an energy range which can probed at the LHC or at least at the next generation of colliders. The goal of our work is to present some additional quiver theories which are consistent with the particle content of the SM and which have not been noticed before. In the sequence of Z p models that we consider, as we shall see, the first with chiral fermions is Z 4 but the Z 7 and Z 12 examples also fall into the class we shall investigate.

General features of quiver theories
We consider the compactification of the type-IIB superstring on the orbifold AdS 5 × S 5 /Γ where Γ is an abelian group Γ = Z p of order p with elements exp (2πiA/p), 0 ≤ A ≤ (p − 1). The resultant quiver gauge theory has N residual supersymmetries with N = 2, 1, 0 depending on the details of the embedding of Γ in the SU (4) group which is the isotropy of the S 5 . This embedding is specified by the four integers which are manifestly non-supersymmetric because no fermions are in adjoint representations of the gauge group. Scalars appear in representations in which the six integers (a i , −a i ) characterize the transformation of the antisymmetric secondrank tensor representation of SU (4). The a i are given by It is possible for one or more of the a i to vanish, in which case the corresponding scalar representation in the summation in Eq.(3) is to be interpreted as an adjoint representation of one particular U (N ) j . One may therefore have zero, two, four or all six of the scalar representations, in Eq.(3), in such adjoints.
Note that there is one model with all scalars in adjoints for each even value of p (see Model Nos 1,3,12). For general even p the embedding is A m = ( p 2 , p 2 , p 2 , p 2 ). This series is the complete list of N = 0 abelian quivers with all scalars in adjoints.
To be of more phenomenolgical interest the model should contain chiral fermions. This requires that the embedding be complex: A m ≡ −A m (mod p). It has been shown that for the presence of chiral fermions all scalars must be in bifundamentals.  Table 1: List of all abelian chiral quiver models for p ≤ 7.
The proof of this assertion follows by assuming the contrary, that there is at least one adjoint arising from, say, a 1 = 0. Therefore A 3 = −A 2 (mod p). But then it follows from Eq.(1) that A 1 = −A 4 (mod p). The fundamental representation of SU (4) is thus real and fermions are non-chiral.
The converse also holds: If all a i = 0 then there are chiral fermions. This follows since by assumption 4 . Therefore reality of the fundamental representation would require A 1 ≡ −A 1 hence, since A 1 = 0, p is even and A 1 ≡ p 2 ; but then the other A m cannot combine to give only vector-like fermions. It follows that in an N = 0 quiver gauge theory, chiral fermions are possible if and only if all scalars are in bifundamental representations.
For the lowest few orders of the group Γ, the members of the infinite class of N = 0 abelian quiver gauge theories are tabulated below.
We show in Table 1 the list of quiver models for p ≤ 7, the first is at p = 4. In this paper we shall discuss the cases p = 8 and 9. We stop at p = 9 because we can already satisfy all of the requisite constraints from three generations, electroweak mixing and gauge coupling unification. More mundanely this keeps the number of generators of the gauge group not above 72 which is smaller than E 6 . In [14] it was shown that the condition necessary for the presence of chiral fermions, that all the scalars must be in bifundamentals, coincides with the condition necessary for the cancellation of one-loop quadratic divergences. This is encouraging since, if these two conditions had been contradictory, the quiver approach would be seriously compromised. The coincidence supports the idea that quiver gauge field theories are a promising and potentially fruitful future direction for BSM physics.

2.1
Quivers with p > 7 For p ≥ 7, we shall keep only the chiral solutions because non-chiral examples are of no phenomenological interest. We continue to number the retained models sequentially. Let n p be the number of inequivalent chiral quiver theories for fixed p then our search, checked by a computer  program, yields the following results: n 2 = n 3 = 0, n 4 = 1, n 5 = 2, n 6 = 3, n 7 = 6 all agreeing with the 1999 result [5] and summarized above. Note that for these first 12 chiral models only the p = 7 models numbered 7B, 7D, and 7E can have their p modes labelled such that they contain the three chiral families of the SM. For p = 8, we find n 8 = 9 with the inequivalent solutions given in  For p = 9, we find n 9 = 13 with the inequivalent solutions given in Table 3 (p=9).

Model building with quiver theories
So far we have used only mathematics to arrive at potentially interesting chiral theories with gauge group SU (3) p where 4 ≤ p ≤ 9. More experimental data would be very welcome to guide us beyond the SM but for the present we have to do without. The physics of the situation enters when we attempt to assign the p nodes to colour (C), weak (W) and hypercharge (H) preparatory to spontaneous symmetry breaking to the SM. The labels C, W , and H are for convenience with book-keeping only. More physics constraints arise from three families, the electroweak mixing, gauge coupling unification and the requirement of a scalar sector sufficient to permit spontaneous symmetry breaking to the SM. As mentioned in the introduction, we shall use the notation for the general gauge structure of a quiver theory. The general understanding will be that the n C , n W , n H sectors will undergo spontaneously symmetry breaking to the corresponding diagonal subgroups in which, by virtue of the choices of the diagonal subgroups, the gauge couplings of the C, W, and H sectors are related to the original common quiver gauge coupling by If we define α i ≡ g 2 i /(4π) then we have from Eq.
which will play a role in gauge coupling unification. Notice that the other two independent ratios involving α 1 are not necessary given the fact that the normalization of the U(1) generator is arbitrary. This will only occur if the U (1) is embedded in a non abelian gauge symmetry, which is not the case here, since U (1) Y emerges both from SU (3) W and SU (3) H after the lumping of the original symmetry to diagonal. The electroweak mixing angle Θ W depends on g W and on g Y where Y is the weak hypercharge according to From the PDG tables [15], the values of We shall use the RG equations for I = C, W, Y where the RG β-functions are, at one-loop order [16] b with N f am = 5 2 for M ≤ M t = 173.2 GeV and N f am = 3 for M > M t . Using these relations, we can determine that has the value R(µ) = 3, 2 for the µ values µ ≃ 800 GeV, µ ≃ 200 TeV, respectively, and that has the value sin 2 Θ(µ) = 1 4 for µ ≃ 3.8 TeV. In general, for a large value of p, one could explore various possibilities for R(µ), linked to the ratio (8), which would fix appropriately the unification scale µ GU T .

Model Building for p = 8
There are n 8 = 9 possibilities for the A m and a i listed in Table 2 (p=8) which we may label (8A) through (8I) and analyse them in turn. We will be labelling the nodes in a quiver clockwise as nodes on a hectagon, according to their C, W or H nature and represent them in a sequence, with the edges represented by hyphens. For p = 8 (8A) A m = (1115), a i = (222). With one color (C) node and two weak (W) node, the node assignments allowed, when we require that there are three families and sufficient scalars to break SU For this case, we may try either

C -H -W -H -H -W -H -H or C -H -H -H -W -W -H -H.
In both assignments, however, the breaking of the H's fails.

Model Building for p = 9
There are n 9 = 13 possibilities for the A m and a i listed in Table 2 (p=9) which we may label (9A) through (9M) and analyse them in turn.

Discussion
For p = 8, only the cases (8B) and (8F) allow the spontaneous symmetry breaking to the threefamily standard model, and these both have unification possible between the gauge couplings, provided that the energy scale µ is chosen correctly.
In (8B), the C and W embeddings require the matching condition In (8F), on the other hand, the SM embedding requires the different condition Using an RGE running of the couplings α i (µ) up from the Z mass gives the energy scales µ = M GU T ≃ 200 TeV and µ = M GU T ≃ 800 GeV corresponding to Eqs. (15) and (16) respectively. For p = 9, we have identified (9F), (9G) and (9L) as the only consistent node identifications. The first and third require the unification implied by Eq.(15) while the second needs Eq.(16) for unification. These p = 8 and p = 9 quivers merit further study, including whether there is the possibility of a conformal window for at least a part of the extensive energy range between M GU T and M P lanck .

Conclusions
The objective of this work has been to present some additional examples of quivers which are compatible with the spectrum of the Standard Model. At the same time they involve scalars which can have VEVs to break the products of the SU(3)'s to the diagonal subgroups, and as such they merit further analysis. The surviving models are (8B), 8F), (9F), (9G) and (9L).
They have a type of grand unification which is quite different than the way it was envisioned long ago [17,18], where a single group contained the Standard Model group and that there was a desert between the weak scale and the GUT scale. The predictions of that approach were connected to proton decay and neutrino masses. In this approach, by contrast, there is no need for assumption of a desert extending over 10 or more orders of magnitude in energy. In these models the unification takes place at 800 GeV or 200 TeV which are scales within the foreseeable realm of accelerators in existence or of the next generation. They predict a wealth of new particles, including gauge bosons and further quarks and leptons. We eagerly await more data from the LHC to identify which BSM is chosen by Nature.