If the Weak Were Strong and the Strong Were Weak

We explore the phase structure of the Standard Model as the relative strengths of the SU(2) weak force and SU(3) strong force are varied. With a single generation of fermions, the structure of chiral symmetry breaking suggests that there is no phase transition as we interpolate between the SU(3)-confining phase and the SU(2)-confining phase. Remarkably, the massless left-handed neutrino, familiar in our world, morphs smoothly into a massless right-handed down-quark. With multiple generations, a similar metamorphosis occurs, but now proceeding via a phase transition. In the second half of the paper we introduce a two-parameter extension of the Standard Model, a chiral gauge theory with gauge group U(1) x Sp(r) x SU(N). We again explore the phase structure of the theory as the relative strengths of the Sp(r) and SU(N) gauge couplings vary.

There is also a third, counterfactual scale in the Standard Model, which doesn't play any role in our world. This is the infra-red scale at which the weak force would become strong if other effects didn't first intervene. It is a rather academic exercise to specify this scale but if we were to ignore electroweak symmetry breaking and run the SU (2) beta function down, assuming a single massless generation, it is given by 1 The purpose of this paper is to understand the phase structure and possible quantum phase transitions of the theory as the three scales, v, Λ strong and Λ weak , vary.
The question of what happens if the Higgs mechanism is turned off, and the strong force dominates, has been well studied. This situation occurs in the regime, v ≪ Λ weak ≪ Λ strong (1.1) It was pointed out long ago that the chiral condensate of QCD transforms under electroweak symmetry. This means that the pions act as a substitute for the Higgs boson, giving a mass of order f π to the W-and Z-bosons, an observation that motivated the subsequent development of technicolor models [1,2]. The phenomenology of this regime was described in [3] and, in more detail, by Quigg and Shrock [4].
In this paper, we are interested in what happens as we vary the couplings, to interpolate from (1.1) to the regime v ≪ Λ strong ≪ Λ weak (1. 2) The pattern of chiral symmetry breaking in this regime was mentioned briefly in [4] and explored further in [5] and will be reviewed in some detail below. First the weak force with SU (2) gauge group confines, with a particular pattern of chiral symmetry breaking. This condensate breaks the strong gauge group, SU (3) → SU (2), which itself subsequently confines, breaking chiral symmetry yet further. The resulting physics shares some similarities with the early work of Abbott and Farhi [6][7][8], exploring the possibility that the SU (2) weak force is actually confining, rather than spontaneously broken.
Our goal is to understand the spectrum of massless fermions and Goldstone bosons of the Standard Model, and a closely related chiral gauge theory, in the two regimes (1.1) and (1.2). Our primary motivation for undertaking this work is simple: we thought it was a fun question. More generally, this paper sits within a larger programme aimed at understanding the dynamics of chiral gauge theories. Early work on this topic is summarised in the review article [9]. Since then, a number of articles have studied the dynamics and phase structure of large classes of chiral gauge theories [10][11][12][13][14][15][16][17][18][19]. A number of proposals have been made for lattice regularisations of chiral gauge theories [20][21][22][23][24].

Summary of Results
As we vary the coupling constants of the theory, to interpolate from regime (1.1) to regime (1.2), the physics depends strongly on the number of generations, which we denote as N f . Perhaps the biggest surprise arises when we have just a single generation, N f = 1. In this case, the massless fermion spectrum is comprised of a single, left-handed neutrino in the regime (1.1), a fact that is familiar from our world 2 . In contrast, in regime (1.2) the massless fermion spectrum contains a single colour component of the right-handed down quark. Furthermore, the unbroken symmetries are identical in the two regimes. In particular, the massless down quark that emerges when the weak force dominates is neutral under electromagnetism, a fact that can be understood by noting that the U (1) Q electromagnetic subgroup twists within the SU (2) × SU (3) gauge group as we vary the ratio Λ strong /Λ weak . These facts suggest that the two regimes sit in the same phase, and the neutrino morphs smoothly into the down quark 3 .
With N f ≥ 2 generations, there is a similar story: in regime (1.1), one finds massless lefthanded leptons, while in regime (1.2) there are massless, neutral right-handed quarks. This time, however, the symmetry breaking structure in the two regimes differs, ensuring that there is a phase transition between the two. The exact structure of the symmetry breaking depends on the fields and couplings that are present, and we consider a number of variations of the Standard Model, both with and without hypercharge and Yukawa couplings.
In Section 3, we introduce a novel chiral gauge theory, based on the gauge group When coupled to specific fermion and scalar fields, this can be thought of as a two parameter extension of the Standard Model. We again explore the phase structure as the relative couplings of the two non-Abelian gauge groups are varied and find a pattern analogous to that of the Standard Model.
Finally, we include two extended Appendices which describe a number of features of vacuum alignment, the dynamical process that determines the vacuum structure of theories with chiral symmetry breaking and multiple gauge groups [25,26]. This, it turns out, is important in order to understand the structure of chiral symmetry breaking in regime (1.2).

Note Added
As we started to write this paper, we became aware of the [31] by Berger, Long and Turner which asks essentially the same questions, motivated by early universe baryogenesis. Our results largely agree where there is overlap.

Variations on the Standard Model
We start by discussing a simple chiral gauge theory with gauge group G = SU (2) × SU (3) We stick to convention and refer to SU (2) as the weak force and SU (3) as the strong force. However, we will be interested both in situations where these names are appropriate, and also in situations where the weak is strong and the strong is weak. We will encounter a large number of different groups below, both gauge and global; where there is a possibility of confusion, we will refer to the gauge groups as SU (2) weak and SU (3) strong .
We couple four Weyl fermions to G: these are the left-handed quarks q L , left-handed leptons l L , and right-handed quarks q R = (d R , u R ). At this stage, we include neither the right-handed electron, nor right-handed neutrino since both are singlets under G. These will be introduced in Sections 2.2 and 3 respectively.
We start by considering just a single generation of fermions; we then turn to multiple generations in Section 2.1. The non-anomalous global symmetry is where here, and elsewhere, we ignore discrete factors. Here U (1) V is the familiar B − L symmetry of the Standard Model. The transformations of the various fermions under the gauge and global symmetries are summarised as Both baryon and lepton number are anomalous, meaning that these quantum numbers are not individually conserved in the quantum theory. We will see a dramatic illustration of this fact as we adiabatically vary the coupling constants.
There are 't Hooft anomalies for both U (1) 3 V and SU (2) R [32]. (The latter is a Z 2 anomaly [33].) This ensures that either these symmetries are spontaneously broken in the infra-red, or there are massless fermions. We will find that the anomalies are saturated in the infrared by massless fermions, albeit with different microscopic representatives playing the role in different regimes.
Both gauge group factors are asymptotically free. This means that the gauge couplings become large in the infra-red where, left to the their own devices, they result in two dynamically generated scales. We refer to the scales for SU (2) and SU (3) as Λ weak and Λ strong respectively. We have little understanding of the dynamics when Λ strong ≈ Λ weak . However, in the two limits Λ strong ≫ Λ weak and Λ weak ≫ Λ strong , some simple intuition about confinement and chiral symmetry breaking is enough to understand what happens. We will then try to match the two regimes.
The limit where the strong force dominates is well studied [1][2][3][4]. The strong dynamics results in a quark condensate which takes the form If we ignore the weak force, then this is a condensate in two-flavour QCD. The SU (2) L × SU (2) R flavour symmetry is spontaneously broken to the diagonal subgroup SU (2) diag , resulting in three Goldstone bosons.
Turning on the weak force, we identify SU (2) L with the SU (2) weak ⊂ G gauge group. The condensate acts as a Higgs field, completely breaking the SU (2) gauge group. All three of the would-be Goldstone bosons are eaten, giving mass to the W-bosons. This mass is of order f π , the pion decay constant.
A global symmetry survives, formed from a diagonal combination SU (2) diag ⊂ SU (2) weak × SU (2) R , and the infra-red global symmetry takes the same form as the ultra-violet symmetry (2.1), The quark bound states are all massive and form representations of F strong . Meanwhile, the leptons l L remain massless, transforming as These massless leptons saturate the 't Hooft anomalies of the global symmetry.
When the weak force dominates, we expect a condensate of left-handed fermions to form.
If we ignore the strong force, then the SU (2) gauge theory enjoys an SU (4) global symmetry, under which ψ transforms in the 4. The condensate breaks 4 this to Sp(2), resulting in dim SU (4) − dim Sp(2) = 15 − 10 = 5 Goldstone bosons. Now we turn the strong force back on, and see the effect of the condensate (2.3). This was discussed previously in [5]. It turns out that the choice of J mn does not affect the physics in this case. (This statement no longer holds when we discuss multiple generations; we will discuss this in more detail in Section 2.1 and in much more detail in the Appendices.) For any choice of J mn , the condensate (2.3) includes a quark-bilinear of the form where a, b, c = 1, 2, 3 are SU (3) strong indices and we've now suppressed the SU (2) weak index structure which remains as in (2.3). For any choice of σ c , the condensate acts as Higgs field for the strong force, breaking All 5 of the would-be Goldstone bosons described above are eaten by the now-massive gluons.
Without loss of generality, we choose σ c = (0, 0, 1), so that the condensate (2.4) involves only the q a with a = 1, 2 coloured quarks. We denote the remaining quark asq L = q L 3 . It forms a condensate with the lepton where the anti-symmetry of (2.3) is assured because the condensate is symmetrised over both spinor and weak indices, leaving the Grassmann nature of the fermions to do its job.
Both condensates (2.4) and (2.5) would appear to break the U (1) V symmetry of the original theory; they have charges +2 and −2 respectively. However, it is straightforward to find a global U (1) symmetry that survives by locking U (1) V with a suitable SU (3) strong gauge transformation. If we denote the generator of U (1) V as Q V , then the generator of the surviving global symmetry is defined as At this point, the left-handed quarks q L and leptons l L have become gapped. We're left just with the right-handed quarks q R , which now transform under the unbroken SU (2) strong gauge group. Under the combined symmetry breaking 4 We use the convention Sp(1) ≡ SU (2).
the right-handed quarks decompose as Now we let SU (2) strong flow to the infra-red where it too confines. The (2, 2) 0 quarks above will form a condensate Hereσ c specifies the direction in SU (3) colour space determined by the weak condensate (2.4). Importantly this new condensate breaks neither SU (2) R nor U (1)V . We're left with the infra-red symmetry which, once again, is unchanged in form from the ultra-violet (2.1), There is a single massless fermion transforming under F weak ; this is the right-handed quark q R that transforms in the (1, 2) +3 representation in the decomposition (2.7), or Once again this saturates the 't Hooft anomalies. Note, in particular, that the U (1) charge +3 (as opposed to −3 seen when Λ strong ≫ Λ weak ) is compensated by the fact that we have a massless right-handed fermion, rather than left-handed. We see that the massless lepton l L in the regime Λ strong ≫ Λ weak has transmuted into a massless right-handed quark in the regime Λ strong ≪ Λ weak . This provides a striking example of the lack of individual baryon and lepton number conservation in the theory. However, this transmutation occurs without violating the B − L symmetry, a feat which is made possible by the twisting (2.6) which means that the infra-red gauge-invariant down quarkq R carries a different B − L quantum number from its gauge-dependent microscopic parent.

No Phase Transition?
For a single generation, the global symmetry group of the theory remains unbroken both when Λ strong ≫ Λ weak and when Λ weak ≫ Λ strong . While it is true that the UV symmetry group is locked with different gauge symmetries in each case, there is no gauge invariant way to distinguish them. This suggests that there is no phase transition as we vary the ratio Λ strong /Λ weak , and the massless lepton transforms smoothly into the massless quark.
This picture resonates with an old story. Recall that QCD with two flavours is one of the few cases where the 't Hooft anomalies can be saturated by massless baryons [32]. There is a complementary way of viewing this [34][35][36], in which a q L q L condensate forms, Higgses SU (3) strong → SU (2) strong , and leaving behind a massless quark. The fact that there is no phase transition between the Higgs and confining phases means that the massless baryon can be viewed as a continuously connected to this massless quark. Ultimately, the physics described in the above paragraph is thought not to occur for QCD. However, it does occur in the regime Λ weak ≫ Λ strong , where the q L q L quark condensate (2.4) is encouraged by the SU (2) weak force rather than SU (3) strong . This suggests that as we head into the regime Λ strong ≈ Λ weak , it may be appropriate to better think of the massless quarkq R as a massless baryon. Indeed, the baryon B ∼ q L · q L · q R has the same quantum numbers as the massless fermion. This means that, starting from the regime Λ strong ≫ Λ weak , the massless lepton can mix with the baryon, and ultimately emerge in the other regime Λ weak ≫ Λ strong as a massless quark.

Multiple Generations
We now repeat the analysis of the previous section, but with N f generations of fermions. The gauge group remains G = SU (2) × SU (3), but the global symmetry group is now (again omitting discrete factors) The quantum numbers of the fermions are There are now numerous 't Hooft anomalies for F . This time we will see that some of these symmetries are broken, with the 't Hooft anomalies in the surviving symmetries saturated by massless fermions.
Both SU (2) weak and SU (3) strong remain asymptotically free for N f ≤ 5. (This bound comes from the weak force; the strong force remains asymptotically free up to N f = 8 generations.) For suitably large N f , the individual gauge theories sit in a conformal window while, for suitably low N f , they undergo chiral symmetry breaking. The lower end of the conformal window is not well understood, but it is thought that it sits around 8 Dirac fermions for SU (3) [37] and around 6 Dirac fermions for SU (2) [38][39][40].
We analyse the theory in the regime in which both gauge groups undergo chiral symmetry breaking. This means that our analysis is restricted to N f = 2 and, possibly, N f = 3 which is a marginal case for SU (2) weak . Λ strong ≫ Λ weak Once again, the limit where the strong force dominates is well understood. The usual QCD condensate forms, with a moduli space parameterised by Σ ij . If we ignore the weak force, then this condensate breaks the SU (2N f ) L × SU (2N f ) R → SU (2N f ) diag flavour symmetry in the usual fashion, resulting in 4N 2 f − 1 Goldstone bosons. We now turn on the SU (2) weak gauge coupling. Often in such situations, different points on the moduli space give rise to different symmetry breaking patterns and one must work harder to determine which of the original possible vacua becomes the true vacuum [25,26]. We will see a number of examples of this shortly. However, in the present situation this issue does not arise. Instead, each point in the moduli space breaks the SU (2) weak gauge symmetry completely.
The condensate (2.9) breaks the global symmetry group (2.8) to This results in a moduli space of Goldstone modes, There are dim M strong = 4(N 2 f − 1) Goldstone bosons. Note that this is three fewer than the counting before we turned on SU (2) weak ; these three were sacrificed on the altar of the Higgs mechanism.
As in the previous section, the leptons remain massless. They transform under the surviving symmetry group F strong as It is simple to check that these massless leptons saturate the 't Hooft anomalies of the surviving global symmetry F strong .
When the weak force dominates, we again expect a condensate of left-handed fermions to form. We write the collection of SU (2) weak doublets as ψ mi , with m = 1, 2, 3, 4 labelling the quarks and leptons as in (2.2), and i = 1, . . . , N f , the flavour index. The condensate takes the general form If we ignore the strong force, then the SU (2) weak gauge theory has an SU (4N f ) global symmetry which is broken by the condensate to Sp(2N f ), resulting in an intermediate moduli space The question that we want to answer is: what becomes of these modes when we turn on SU (3) strong ?
This time there is a slightly involved calculation to do. Different choices ofJ mi,nj give different symmetry breaking patterns for SU (3) strong and a different mass spectrum for the resulting gauge bosons. For example, if the condensate forms in a flavour-diagonal fashion, withJ mi,nj = J mn δ ij , then it breaks the strong gauge group to which is the same symmetry breaking pattern that we saw in the N f = 1 case. Such a condensate also breaks the global symmetry (2.8) tõ where Alternatively, a condensateJ mi,nj which is off-diagonal in the flavour basis will break the SU (3) strong gauge group completely and further breakF [5]. (We provide a number of specific examples in Appendix B.2.) The question that we must ask is: what is the preferred choice of breaking?
The tools to answer this question were introduced long ago by Peskin [25] and Preskill [26]. They showed how introducing a second gauge group induces a potential on the moduli space M 0 . The true ground state of the system is determined by the minimum of this potential. We review this mechanism in some detail in Appendix A. Furthermore in Appendix B.1 we show that the flavour-diagonal condensate, with symmetry breaking (2.13) and (2.14) is a local, stable minimum of the potential. Although we have been unable to prove, in generality, that there are not other local minima, we argue that generically one expects all other condensates to exhibit tachyonic modes and, in Appendix B.2, we show this explicitly for a number of putative vacua with different symmetry breaking patterns. The upshot is that the flavourdiagonal symmetry breaking pattern (2.13) and (2.14) appears to be dynamically preferred.
With the global symmetry F defined in (2.8) broken toF in (2.14), the moduli space of ground states arising from the weak dynamics, is weak is greater than the 5 Goldstone bosons eaten by the Higgs mechanism (2.13). This reflects the existence of a potential on M 0 induced by gauge symmetry SU (3) strong . We compute the masses of the resulting pseudo-Goldstone bosons in Appendix B.1.
We're still left with the dynamics of the unbroken SU (2) strong gauge symmetry to contend with. Under the residual symmetry SU and Goldstone bosons parameterise the space As in the previous section, the massless fermion is now identified withq R , corresponding to the (1, 2N f ) +3 representation in (2.16). This transformation properties of this fermion are Again, the fermionq R saturates the surviving 't Hooft anomalies.
In contrast to the case with N f = 1, the symmetry breaking pattern (2.10) and (2.17) differs in the two regimes, meaning that there is certainly a quantum phase transition as we vary the relative strengths of Λ strong and Λ weak . It is natural to ask the order of this phase transition.
Sadly, the symmetry breaking structure gives little guidance. Note that neither of the surviving symmetry groups, F strong and F weak , is a subgroup of the other, reflecting the fact that the order parameters associated to the two phases are different. Most phase transitions in Nature that exhibit this property are first order; indeed, this "sub-group criterion" is sometimes stated to be a clear indication of a first order phase transition. However, there is nothing that guarantees that this has to be the case. The two phases could be reached by two second order phase transitions, with an intermediate phase in between. This intermediate phase must have a global symmetry group that contains both F weak and F strong as subgroups, for example the UV global symmetry F .
It is also possible that the transition proceeds through a single, continuous phase transition. In Landau theory, this requires tuning to a multi-critical point. However, more exotic phase transitions, in which a gauge symmetry emerges and no fine tuning is needed, are also possible [41]. Needless to say, it would be interesting to better understand the nature of the transition.

Adding Hypercharge and Yukawa Couplings
We now extend our study by including U (1) Y hypercharge and Yukawa couplings. The gauge group is To ensure cancellation of anomalies, we must now also include a right-handed electron e R in our theory. We further include a single Higgs field, φ. We will omit the right-handed neutrino for now, but revisit this in Section 3.
We include N f generations of fermions, coupled to the Higgs through the Yukawa couplings Here the flavour index i = 1, . . . , N f is summed over so that, in contrast to the Standard Model, there is an independent SU (N f ) flavour symmetry for quarks and leptons, as well as the B − L symmetry that we denote as The representations of the fields under G and F are shown in the table.
Because we haven't included a right-handed neutrino, the SU (N f ) l × U (1) V symmetries have various 't Hooft anomalies, all of which arise from the leptons. The contribution to the 't Hooft anomalies from quarks vanish.
We are interested in this theory in the regime v ≪ Λ weak , Λ strong where the Higgs expectation value, v, is much smaller than all other scales so that the dynamics is dominated by the gauge interactions. We now repeat the analysis of previous sections. As before, we assume that N f is sufficiently small so that both gauge groups undergo chiral symmetry breaking; N f = 2 appears to surely be safe; N f = 3 is unclear.
When the strong force dominates, a condensate (2.9) forms as before. In terms of the up and down quarks, this reads where the 1, 2 labels on q L are SU (2) weak indices. As in the Standard Model, this condensate breaks This, of course, is the usual symmetry breaking pattern of electroweak down to electromagnetism.
As the theory no longer has a chiral symmetry, the full global symmetry (2.20) survives in the infra-red Because the full symmetry group survives, there are no Goldstone bosons. There are, however, light pion modes. These are the usual massless Goldstone bosons arising from the chiral symmetry breaking of QCD, which get a mass through the Yukawa couplings. (Even in the absence of a Higgs vev, the pions get a mass through mixing with φ.) Some aspects of these pions, as well as the associated baryons, were discussed in [4].
The leptons l L and e R remain unaffected by the gauge dynamics. They are distinguished by their charges under U (1) Q ; the left-handed lepton splits into e L with charge Q = −1 and ν L with charge Q = 0. Meanwhile, the right-handed electron e R has charge Q = −1.
The electron pair gets a mass through the Yukawa coupling, while the left-handed neutrino remains massless, transforming as This saturates the 't Hooft anomalies of F .
When the weak force dominates, the condensate (2.12) forms. When we subsequently turn on both SU (3) strong and U (1) Y gauge groups, we must again determine the correct vacuum.
One might be tempted to think that since U (1) Y is free in the infra-red, it does not affect the vacuum state described in the previous section. This, it turns out, is correct but it takes a calculation to show it. Indeed, in [26], Preskill gave examples of chiral symmetry breaking where a subsequent gauging of a U (1) subgroup changes the vacuum structure when the U (1) coupling constant becomes sufficiently strong. In Appendix B.3 we show that this doesn't happen in the present case.
The upshot of this argument is that the condensate (2.12) that minimises the potential remains unchanged by U (1) Y . The quarks once again condense in a flavour-diagonal basis, as in (2.4), to with a, b, c = 1, 2, 3 colour indices and i, j = 1, . . . N f flavour indices. The remaining condensate pairs theq L i = σ a q L ai quark with the leptons as in (2.5) The condensate breaks the global symmetry F in (2.20) to The two condensates break the remaining gauge group to We have seen the breaking to SU (2) strong previously. To see that a U (1)Q survives, note that both of the condensates (2.21) and (2.22) carry U (1) Y charge +2. If we pick σ c = (0, 0, 1) in the condensate (2.21), then we can construct the unbroken gauge generator The surviving SU (2) strong gauge symmetry is coupled to the right-handed quarks. Under the breaking The fermions that transform as doublets under SU (2) strong condense and become gapped as the gauge group becomes strong. The resulting condensate does not further break F weak from (2.23). This means that, in contrast to the regime Λ strong ≫ Λ weak , there is now a moduli space of Goldstone We're left with three gapless Weyl fermions, which were singlets under SU (2) strong . Two of these, arising from u R and e R , carry equal and opposite U (1)Q × U (1)V charge. Although these are not coupled directly through the Yukawa coupling (2.19), there is nothing to prohibit this pair becoming gapped as they interact with the scalar field. This leavesd R , the neutral component of the down quark, as the surviving massless fermion. It transforms as It is noticeable that in the UV theory, the quarks did not appear to play any role in the computation of 't Hooft anomalies. Yet, by the time we flow to the infra-red, the sole remaining fermion is a quark and saturates the surviving 't Hooft anomalies of F weak .
Note that in both Λ strong ≫ Λ weak and Λ weak ≫ Λ strong regimes, there is a surviving U (1) gauge symmetry that we may identify with electromagnetism, and a surviving U (1) global symmetry that we may identify with B − L. These symmetries are twisted with different gauge symmetries in the two regimes, but this does not impede us from identifying them. This conclusion differs from [4] were it is claimed that both electromagnetic and B − L symmetries are broken in the Λ weak ≫ Λ strong regime.
The addition of hypercharge and Yukawa couplings does not change the conclusions of our earlier sections. If we have N f = 1 generation of fermions, then it seems plausible that the transition between the two regimes proceeds without a phase transition. Meanwhile, for N f ≥ 2, a phase transition must occur.
However, in contrast to the situation in Section 2.1, there is a fairly simple symmetry breaking pattern between the two regimes, with SU (N f ) q ×SU (N f ) l , which survives when the strong force dominates, breaking to SO(N f ) diag when the weak force dominates, suggesting that a mean-field description of the phase transition in terms of Landau theory may be appropriate.

A Novel Chiral Gauge Theory
In this section, we extend our analysis to a chiral gauge theory with gauge group Anomaly considerations, to be described below, mean that we must take N odd. For the simplest values of r = 1 and N = 3 this gauge group coincides with that of the Standard Model.
The chiral fermion content is a natural extension of that of the Standard Model: we take left-handed fermions q L and l L to transform under Sp(r), while the right-handed fermions are singlets under Sp(r). One key difference is that we must take r copies of each of the right-handed fermions, including r copies of the right-handed neutrinos ν R . We introduce an index α = 1, . . . , r to distinguish these fields. For later convenience, we also introduce r distinct Higgs fields φ α at this time too. The full set of fermions and scalars and their transformations is given by with α = 1, . . . , r. It is straightforward to show that, with these charge assignments, all gauge anomalies vanish. The Z 2 anomaly of Sp(r) is vanishing only for N odd. Notice that the first right-handed neutrino is decoupled from the gauge fields, as in the Standard Model, but the other r − 1 carry U (1) Y charge.
The U (1) Y charge assignments also allow us to construct Yukawa interactions. For a single generation, we have Here (q † L · φ † ) denotes the Sp(r) singlet that one can construct from these two fields. (This is analogous to the SU (2) singlet constructed using ǫ ab .) The generalisation of the Standard Model with SU (3) gauge group replaced by SU (N ) is fairly well explored. (See, for example, [42].) The generalisation with SU (2) replaced by Sp(r), with the anomaly-free charge assignments shown in the table above, appears to be novel. We note the possibility that such a theory with gauge group U (1) × Sp(r) × SU (3) may describe our world, with the additional Higgs fields φ α , α = 2, . . . , r, breaking Sp(r) to SU (2) at some high scale. Moreover, this two parameter extension of the Standard Model may lend itself to a large r, large N expansion; we leave this possibility to future work. (A large N expansion of certain chiral gauge theories was previously proposed in [43,44].) We will adopt the convention of the Standard Model and refer to the Sp(r) gauge group as weak and the SU (N ) gauge group as strong. As in the previous section, we will be interested in the phase diagram of the theory, with the two asymptotic regimes in which one of the gauge groups dominates over the other. We will study different variants of this problem, both with and without hypercharge interactions and Yukawa couplings.

Beta Functions
We will discuss the chiral theory coupled to N f generations of fermions and focus on situations where both gauge groups are asymptotically free . The SU (N ) gauge group is coupled to 2rN f Dirac fermions, each in the fundamental representation, and is asymptotically free provided 11N > 4rN f Meanwhile, the Sp(r) factor is coupled to N f (N + 1) Weyl fermions, each in the pseudo-real fundamental representation. If we ignore the Higgs fields for now, Sp(r) is asymptotically free provided 11(r + 1) > (N + 1)N f For N f ≥ 6, at least one of the gauge groups is infra-red free. In contrast, for any N f ≤ 5, there are always choices of N and r for which both gauge groups become strongly coupled in the infra-red. This conclusion persists in the presence of Higgs fields.
As before, our analysis will rely on chiral symmetry breaking in the regime where one or the other gauge group becomes strongly coupled. This takes place for suitably low N f , below the conformal window. The lower-edge of the conformal window is not well established. The SU (N ) gauge factor has 2rN f Dirac fermions in the fundamental representation, and undergoes chiral symmetry breaking for 2rN f < C ⋆ N for some C ⋆ which is expected to sit somewhere around 3 to 4. Meanwhile the Sp(r) gauge factor has N f (N + 1) Weyl fermions in the pseudo-real fundamental representation, and is expected to undergo chiral symmetry breaking when whereĈ ⋆ is around 6 to 8. (See, for example, [45].) For N f ≤ 2 there are an infinite number of choices of N and r for which chiral symmetry breaking occurs, while for N f = 4 it seems likely there are none. The situation for N f = 3 is, in all cases, more murky.

Sp(r) × SU (N )
We start by neglecting the U (1) Y factor and focussing only on the gauge group Because the right-handed electrons e R and neutrinos ν R are singlets under the non-Abelian part of the gauge group, we may ignore them for the purpose of this discussion. We will also discard the Higgs field for now, focussing only on the fermions. As in the case of the Standard Model, here we will find the richest symmetry breaking patterns, unconstrained by hypercharge assignments and Yukawa couplings. We will then reintroduce both of these in Section 3.2 With N f generations of fermions, the global, non-anomalous, symmetry group is Under the gauge and global symmetry groups, the fermions transform as , the two right-handed quarks now undistinguished by hypercharge. There are numerous 't Hooft anomalies between the various subgroups of F .
When the SU (N ) strong force dominates, the usual quark condensate (2.9) forms and the quarks become gapped, leaving behind a number Goldstone bosons.
If we ignore the Sp(r) weak force, the theory has a SU (2rN f ) L × SU (2rN f ) R × U (1) V global symmetry, broken by the condensate to SU (2rN f ) diag × U (1) V . Gauging Sp(r) weak , means that the global symmetry F in (3.2) breaks to Here Sp(r) diag ⊂ Sp(r) weak × SU (2rN f ) diag arises from a simultaneous gauge transformation and surviving SU (2rN f ) diag rotation. This ensures that the Sp(r) gauge symmetry is fully broken, with only this "weak-flavour-locked" global symmetry surviving. The remaining The Goldstone bosons therefore parameterise the moduli space There are dim M strong = 4r 2 N 2 f − 2r 2 − r − 1 of them. With the Sp(r) weak gauge group fully Higgsed, the left-handed leptons remain massless. They transform under F strong as with ψ mi · ψ nj a Sp(r) weak singlet. (The gauge group indices are contracted using the Sp(r) invariant anti-symmetric tensor) andJ mi,nj an anti-symmetric matrix.
Once again, we must determine the choice ofJ mi,nj that minimizes the potential induced by gauging the SU (N ) strong group. This is a fairly involved calculation and is presented in Appendix B.4, where we show that the flavour-diagonal condensate is again a (local) minimum of the potential, with no tachyonic modes. This means that the dynamically preferred vacuum condensate breaks the gauge group to generalising the earlier result (2.13). At the same time, the global symmetry F is broken tõ where SO(N f ) ⊂ SU (N f ) L ′ × SU (N f ) L and U (1)V ⊂ U (1) V × SU (N ) strong is defined in analogy with (2.6).
We're still left with the right-handed quarks q R , which are now coupled to the surviving Sp( 1 2 (N + 1)) gauge group. Under the symmetry breaking SU (N ) strong × U (1) V → Sp((N − 1)/2) strong × U (1)V these right-handed quarks decompose as As Sp( 1 2 (N − 1) becomes strong and confines, those quarks transforming in the N − 1 representation condense. This further breaks the flavour symmetry group to The final result is that we have a moduli space of vacua, For a single generation of fermions, N f = 1, we again see that the infra-red global symmetry in the two regimes coincides: both are Sp(r) × U (1). This now differs from the UV symmetry (3.2), meaning that there are Goldstone bosons even in this case. Nonetheless, it appears plausible that there is no phase transition for N f = 1 as we vary the gauge coupling constants. As in Section 2, the left-handed lepton in one regime transmutes into a right-handed quark in the other.
For N f ≥ 2, the symmetry breaking patterns on either side differ and there must be a phase transition as we vary between them. Once again, neither of the symmetries F strong and F weak , defined in (3.5) and (3.7), are subgroups of the other.

Adding Hypercharge and Yukawa Couplings
Finally we discuss the theory introduced at the beginning of this section, replete with U (1) Y coupling and flavour-diagonal Yukawa interactions (3.1). The gauge symmetry is and the global symmetry is where the matter fields transform as This time, the presence of the right-handed neutrinos ensure that the global symmetries F suffer no 't Hooft anomalies. The arguments of the previous section allow us to quickly determine the symmetry breaking pattern in the two regimes.
When the strong force dominates, the full UV symmetry (3.8) survives. There are no Goldstone bosons. Although the leptons remain massless after the gauge interactions become strong, they interact with the Higgs fields and, indirectly, with the mesons and there is nothing to prevent them gaining a mass, suppressed by the Yukawa coupling. For generic values of the Yukawa couplings, we therefore expect the fermions to be gapped.
When the weak force dominates, the condensate (2.12) forms. A computation of the correct vacuum alignment can be found in Appendix B.4, which shows that the ground state preserves the global symmetry The condensate also breaks the SU (N ) strong gauge group as in (3.4). As the surviving subgroup of SU (N ) strong confines, the resulting condensate does not further break the global symmetry F weak . Once again, no symmetry principle ensures massless fermions and, generically, none are expected to survive. Instead, the gapless modes are supplied by the Goldstone bosons which parameterise The story is, by now, familiar. There is no evidence of a phase transition in the symmetry breaking pattern when N f = 1. Such a phase transition must occur for N f ≥ 2 although now the symmetry breaking pattern suggests that such a phase transition can be captured

A Appendix: Vacuum Alignment
When a gauge theory spontaneously breaks chiral symmetry, the resulting Goldstone bosons parameterise a moduli space of vacua M 0 . If this theory is subsequently coupled to a second gauge group, which becomes strong at a lower scale, then much of the the vacuum moduli space M 0 is lifted. The preferred ground state is chosen dynamically in a process known as vacuum alignment.
The physics of vacuum alignment was explained in two beautiful papers by Peskin [25] and Preskill [26]. In this appendix, we review the results of these papers. In Appendix B, we then apply these results to understand the ground states in situations of interest in Sections 2 and 3.
We consider a general gauge theory with gauge group as with the convention that the gauge group G 1 will always run to strong coupling before G 2 , meaning that the dynamically generated scales are ordered as We couple our gauge theory to fermions. The full theory will have a global symmetry group that we denote as F . However, if we first turn off the second gauge group G 2 by setting its coupling to zero, the global symmetry group of remaining theory, with only G 1 , will be larger: we denote this global symmetry group as K.
We are interested in situation where the confinement of G 1 and subsequent condensation of fermion bilinears breaks this global symmetry to a smaller subgroup The are three basic symmetry breaking patterns, suggested by the maximally attractive channel hypothesis, that can arise with a single gauge group G 1 [25]. Only two of them will be needed in the bulk of the paper, but we list all three for completeness: • If there are n massless Dirac fermions in a complex representation of G 1 , we have the global symmetry K = SU (n) L × SU (n) R , with the two factors acting on left-and right-handed Weyl fermions which we denote as ψ L and ψ R . The condensate takes the general form . . , n and U L/R ∈ SU (N ) L/R . The subgroup H = SU (n) diag leaves the condensate untouched, meaning that we have the familiar QCD-like breaking pattern This form of condensate arises in the bulk of the paper when the SU (N ) gauge group, with N ≥ 3, first becomes strong.
• If there are 2n Weyl fermions in a pseudo-real representation of G 1 then we have a global symmetry K = SU (2n). The condensate forms through the invariant anti-symmetric tensor, ǫ ab . It takes the general form Here U ∈ SU (2n) and J is the 2n × 2n anti-symmetric matrix given in block form above. The resulting symmetry breaking pattern is This form of the condensate arises in the bulk of the paper when the gauge group Sp(r) or SU (2) becomes strong.
• If there are 2n Weyl fermions in a real representation of G 1 , then the global symmetry is again K = SU (2n). This time the condensate forms through the invariant symmetry tensor of the reperesentation, d ab . It takes the general form Again, U ∈ SU (2n) and D is a 2n × 2n symmetric matrix given in block form above. Now, the symmetry breaking pattern is This form of the condensate does not play a direct role in this paper although, as we show in Section 2.1, we have a similar symmetry breaking pattern when first Sp(r) and subsequently SU (N ) becomes strong.
Each of the symmetry breaking patterns described above results in a vacuum moduli space Each point of M 0 corresponds to a different orientation of H ⊂ K.

A.1 A Potential Over the Moduli Space
We now turn on the coupling for the second gauge group G 2 ⊂ K. The global symmetry of the theory is reduced to F . Correspondingly, the symmetry breaking pattern K → H is reduced. Different orientations of H in K descend to different symmetry breaking patterns, each of the form The question we need to address is: what symmetry breaking pattern is preferred? This is the question of vacuum alignment.
As explained in [25,26], the choice of vacuum is determined dynamically. To see why this is the case note that, after gauging G 2 , there are three different fates for the would-be Goldstone modes in M 0 . Some will be charged under G 2 ; these act as Higgs bosons, breaking G 2 to the smaller groupG ⊂ G 2 and are eaten by the Higgs mechanism. Other scalars in M 0 are not eaten, but are no longer protected by symmetry constraints; they will gain a mass, as we explain more fully below, and are referred to as pseudo-Goldstone bosons. Finally, some scalars remain exactly massless; these are Goldstone modes of the full theory, whose moduli space includes the factor M ⊂ F/F Note that this need not be the full moduli space because when the surviving gauge group G becomes strong, it too may break some chiral symmetry, resulting in further Goldstone bosons.
We now describe how the potential is generated over M 0 , following [25,26]. The minimum determines the locus of ground states in M 0 and, correspondingly, the surviving symmetries G andF in (A.1). The potential is generated by the coupling The current is given by where G α are the generators of G 2 .
At one-loop, exchange of the W-bosons gives rise to a potential on the moduli space. A point Ω ∈ M 0 corresponds to a putative vacuum state |Ω . The energy of this state is given by This correlation function, and those below, are time-ordered. Here g 2 is the gauge coupling associated to the gauge group G 2 and the gauge propagator ∆ µν (x) is defined in the usual way by It will prove to be useful to change perspective, somewhat analogous to the shift from an active to passive viewpoint. To this end, we fix a reference vacuum state |0 . A general ground state |Ω is given by the unitary action with U ∈ K/H. (Strictly speaking, U is a unitary representation of K acting on the Hilbert space.) We now parameterise the point in M 0 by U ∈ K/H. We define the rotated currents, In this notation, the potential (A.2) becomes In the vacuum |0 , there is a particular embedding of the unbroken subgroup H ⊂ K. We introduce the following notation for the generators of the Lie algebra of K and its subalgebras 5 • Let T m , m = 1, . . . , dim K, denote the generators of K • Let H a , a = 1, . . . , dim H, denote the generators of H ⊂ K.
Any generator, T of K, can be decomposed into components projected along the two subalgebras H and X = K/H. We write the projection along H as T H and the projection along X as T X , so we have To avoid an explosion of notation, we denote the Lie group and Lie algebra by the same letter.
with T a H = Tr(T H a ) the projection along H and T i X = Tr(T X i ) the projection along X. Implementing a decomposition of this kind for the current (A.3), we have with J a Hµ = iψγ µ H a ψ the currents that lie in the unbroken H ⊂ K and J i Xµ = iψγ µ X i ψ the currents that lie in the broken K/H. Substituting this decomposition into the potential (A.4), we have three terms: J 2 H , J 2 X and J H J X . The cross-term J H J X vanishes. The other two terms also simplify. In particular, using the fact that K/H is a symmetric space, we have where J X denotes any choice of normalised generator, e.g. J X = J 1 X ; the exact choice doesn't matter precisely because it's a symmetric space. We use a similar convention for J H . The potential (A.4) can then be written as We can further simplify this using Both terms in the potential (A.5) can then be written in terms of Tr (U † G α U ) 2 X , giving where V 0 is independent of U . Here we've introduced f π , the characteristic energy scale associated to chiral symmetry breaking, defined in the usual manner as The mass scale M 2 in (A.6) is given in terms of broken and unbroken current by Importantly, M 2 can be shown to be positive [25,26]. We postpone the derivation of this result to Appendix A.2 below.
The expression for the potential (A.6) has a particularly elegant interpretation: the group theoretic factor is simply the sum of the G 2 gauge boson masses, We see that, with M 2 > 0, the minimum of the potential V (U ) occurs when the gauge group is broken the least, in the sense that the sum of the gauge boson masses is smallest.
In practice, life is simplest if we are able to pick the reference state |0 to be a local minimum. For this to be the case, the generators G α of G 2 ⊂ K must obey a number of properties. To see this, we parameterise the vacua in the neighbourhood of |0 by To leading order in ρ, the potential (A.6) then reads For ρ = 0 to be an extremum of V , we need where the second equality follows from the fact that, for K/H a symmetric space, [H, H] ∼ iH and [H, X] ∼ iX, and [X, X] ∼ iT . The third equality is of course the cyclic property of trace. We learn that the reference vacuum |0 is a stationary point of V provided that [G α H , G α X ] = 0 for each α (no sum) (A.8) Next we must ensure that |0 is a local minimum, as opposed to a maximum or saddle point. For this, we must compute the Hessian of V . In a mass-diagonal basis for the broken generators X i , one can show that the mass eigenstates are given by This combination will show up regularly in what follows; we denote it as We see that we have a local minimum only if m 2 X > 0 for each of the pseudo-Goldstone bosons X. In contrast, if there is any direction with m 2 X < 0 then there is a tachyonic mode which destabilises the would-be vacuum.
In fact, life is not quite as simple as we have described. We will encounter a number of situations in which the leading order result (A.9) gives m 2 X = 0 for some pseudo-Goldstone boson X, even though there is no symmetry protecting the mass. In this case, we must work harder and look to the second-order terms.
Since the right-hand side is positive, we learn that M 2 X > M 2 H . This is sufficient to guarantee positivity of M 2 which can be related to the meson masses as The machinery of spectral functions is also needed to get an expression for the secondorder correction to the masses of the pseudo-Goldstone bosons. The essence of the idea is simple: the masses M 2 are computed using the gapless gauge boson propagator ∆ µν (x) in (A.7). However, the condensate partially breaks the gauge group G 2 , giving some of the gauge bosons a mass. For these gauge bosons, the propagator should be replaced by the massive propagator.
Here for simplicity, we assume that each of the massive G 2 gauge bosons has the same mass, which we denote as µ 2 . (In the examples of Appendix B, this is too naive and there will be gauge bosons with different masses. This adds an extra complication, but here we deal with just the simplest case.) The mass µ 2 changes the propagator of the gauge boson and, correspondingly, shifts the mass M 2 to M 2 µ , which we will compute shortly. Note that the mass µ 2 will be proportional to g 2 2 , meaning that M 2 µ differs from M 2 only at order g 4 2 . With this correction, the mass of the pseudo-Goldstone bosons (A.10) becomes Our interest lies in those pseudo-Goldstone bosons whose mass m 2 X vanishes at leading order. For these, it must be the case that unbroken C X (G α ) = broken C X (G α ), so we can write Note, however, that for an unbroken generator G α we have, by definition, G α X = 0 (since G α X is the projection onto the broken part). Using the definition of C X (G α ) in (A.9) and (A.10), we see that This is the key to showing that the second order correction to the mass terms is positive. (It is also the step that needs revisiting when the broken gauge bosons have different masses.) Invoking the spectral representation, the mass (A.12) can be written as The fact that this is positive definite follows once again from the observation that M 2 X > M 2 H . This ensures that those pseudo-Goldstone bosons that remain massless at leading order receive a positive mass at the next order. Note also that the gauge boson mass is of order µ 2 ∼ g 2 2 f 2 π , ensuring that this mass m 2 X is indeed of order g 4 2 as expected. As we stressed above, this calculation assumed that the massive G 2 gauge bosons have a common mass µ 2 . This allowed us to write the second-order correction to the massless pseudo-Goldstone bosons as (A.12). Below we will meet situations in which this step needs revisiting, and the positivity of the mass correction is no longer so straightforward. Nonetheless, we will see that the positivity remains.

B Examples
We now apply the results of Appendix A to the models considered in the bulk of the paper.

B.1 Vacuum Alignment for SU (2) × SU (3)
We start by applying the ideas above to the chiral gauge theory with gauge group G = SU (2)× SU (3), coupled to N f generations of fermions. For now, we include neither hypercharge nor Yukawa interactions. This is the theory described in Section 2.1.
When Λ strong ≫ Λ weak , so the strong force dominates, the original chiral symmetry breaking gives rise to a moduli space In this case, there is no calculation to do: each point in M 0 breaks the SU (2) gauge group completely. As described in the main text, the true moduli space of the theory is M strong defined in (2.11). We have dim M 0 − dim M strong = 3, with this difference accounted for by the Higgs mechanism which means that three pions are eaten when SU (2) is broken. This simple counting means that there are no pseudo-Goldstone bosons in this case and no potential over M 0 is generated.
The regime Λ weak ≫ Λ strong is more involved. When the SU (2) gauge group becomes strong, the resulting condensate (2.12) allows for a number of different symmetry breaking patterns. These include SU (3) strong → SU (2) strong , and SU (3) strong → ∅. We show here that the former symmetry breaking pattern is a (local) minimum of the potential. In Appendix B.2 we show that putative vacua in which SU (3) strong is completely broken have a tachyon and are unstable.
We denote the fermions as Li , l Li with m = 1, 2, 3, 4 and i = 1, . . . N f (Note that the colour components q 3 and q 2 are exchanged compared to the main text. This doesn't change the conclusions, but makes some of the generators below a little simpler.) If we ignore the SU (3) strong gauge fields, we have a moduli space of vacua given by We now turn on the SU (3) strong gauge fields. We will show that the flavour diagonal ground state is a minimum of the resulting potential.
It is trivial to show that this vacuum is an extremum of the potential, with the generators obeying (A.8); this follows from the flavour-diagonal nature and the fact that there is no vacuum alignment problem for a single generation. It remains to show that the masses (A.9) of the pseudo-Goldstone bosons are non-tachyonic. For this, we will need explicit expressions for the generators of G 2 = SU (3) strong and X ∈ SU (4N f )/Sp(2N f ).
First the gauge generators. Since the vacuum (B.1) breaks SU (3) strong −→ SU (2) strong , it makes sense to classify the generators in terms of their representation under SU (2) weak . They decompose as 8 −→ 3 ⊕ 2(2) ⊕ 1. The triplet are the generators of the unbroken SU (2) strong where, here and below, the generators are normalised as Tr G α G β = δ αβ . The two pairs of generators transforming in the doublet are Finally the singlet is given by (2.15), the expected number of exact Goldstone modes. This leaves us with the fate of the pair of (2, S) 0 representations unaccounted for. These are not exact Goldstone bosons, so we expect that the vanishing of the mass is an artefact of working to leading order in perturbation theory; we must look to second order to see if the resulting mass-squared is positive or negative.

B.2 Unstable Vacua
We have shown that the flavour-diagonal condensate (B.1) is a local minimum of the potential. We have not, however, shown that it is global minimum.
This, it appears, is a challenging problem. The moduli space is large, and there may be many saddle points. However, the simple observation that the potential (A.6) is proportional to the sum of the W-boson masses means that those ground states which break the gauge symmetry the least are favoured. For this reason, it seems likely that the flavour-diagonal vacuum (B.1) is, in fact, the true ground state of the system.
In this appendix, we give some calculations to back up this intuition. We have not been able to find other local minima of the potential. Instead, we will show that a number of obvious candidates for ground states have tachyonic modes and so are unstable. We work with N f = 2 and give two examples of putative ground states, each with different symmetry breaking patterns, which turn out to be saddle points.
For the first example, consider a condensate of the form where a = 1, 2, 3 is the SU (3) strong colour index, and the 1,2 labels on the quarks and leptons refer to flavour. With such a condensate, the gauge groups breaks as If we choose to order the 8 Weyl spinors as ψ = q 1 1 L , q 2 1 L , q 3 1 L , l 1 L |q 1 2 L , q 2 2 L , q 3 2 L , l 2 L T , then the SU (3) strong generators act as with λ α are the usual 3 × 3 Gell-Mann matrices. It is straightforward to show that the broken and unbroken generators obey (A.8), so this condensate is at least a saddle point of the potential. However, one finds that this condensate has a higher energy than (B.1). More importantly, there are also tachyonic modes. One of this is associated to the would-be Goldstone bosons transforming in the octet of SU (3) strong The mass matrix for these 8 generators can be found using (A.9); it is diagonal, and given by diag (0, 1, 0, 0, 1, 0, 1, 0) The overall minus sign means that this vacuum is unstable.
As a second example, we consider the condensate that arises in the single flavour case, but now with two flavours which align differently within the SU (3) strong gauge group. For the first generation we take, which picks out the a = 3 colour direction. Meanwhile, for the second generation we take which picks out the a = 2 colour direction. The combination of both condensates breaks the SU (3) strong gauge group completely.
To compute the masses, it is simplest to note that the condensate is related to (B.1) by a permutation of the ψ components. There are two ways to proceed; we could fix the action of the gauge generators G α on ψ, in which case the permutation acts as conjugation on the broken generators X defined in Appendix B.1. Alternatively, we could fix the action of the unbroken generators, in which case the permutation acts by conjugation of G. In either case, a simple calculation shows that the condensates (B.6) and (B.7) are indeed a saddle point of the potential, but with energy higher than both the local minimum (B.1) and the unstable vacuum (B.5).
It is a little more involved to demonstrate that the condensates (B.6) and (B.7) are unstable and we refrain from giving all the details. . Because the gauge group is broken completely, many of the broken generators X given in Appendix B.1 now mix. Diagonalising the resulting mass matrix, one finds that there are massive modes, massless modes and, crucially, two tachyonic modes. This vacuum is unstable.

B.3 Adding Hypercharge
We now repeat the calculation of Appendix B.1 in the presence of a U (1) Y hypercharge interaction. This corresponds to an additional gauge generator which, in the notation of Appendix B.1, takes the form The unfamiliar normalisation factor ensures that this generator obeys Tr G 2 Y = 1. There is a simple generalisation of the mass formula (A.9) in which the different gauge generators are summed over, weighted with their gauge couplings. We denote the gauge coupling associated to U (1) Y as g Y . Then the masses of the pseudo-Goldstone bosons (B. The (1, S) 0 and (1, S ⊕ A) 0 states remain as exact Goldstone bosons as previously. There is, however, a correction to the second-order mass of the (2, S) 0 states due to hypercharge. To see this, we need the usual mixing of the generators G 1 and G Y which yield the Z-boson and the photon. The mass-matrix for the corresponding gauge fields is Y which has eigenvalues and µ 2 γ = 0 Correspondingly, the generators mix and take the form Armed with these results, we can now revisit the calculation of Appendix B.1. At leading order, only the mass of (1, A) is modified by the hypercharge to While the mass of (2, S) is modified by replacing the contribution from G 1 in (B.4) by the contribution from the Z-boson, which is The group theoretic factor remains negative and the mass remains positive.
The upshot of these short calculations is that hypercharge does not destabilise the vacuum. Indeed, it is clear from the calculations above why this is: both the chosen vacuum, and the hypercharge, are flavour-diagonal. The analysis of [25,26] shows that the flavour-diagonal vacuum is likely to be destabilised only by the introduction of U (1) gauge symmetry under which different generations carry different charges.

B.4 Vacuum Alignment for Sp(r) × SU (N )
The analysis of vacuum alignment for Sp(r) × SU (N ) follows that of Section B.1; only the group theory is a little more involved.
Before we turn on the SU (N ) strong gauge symmetry, the chiral condensate induces the K → H symmetry breaking pattern expected of a pseudo-real representation, SU ((N + 1)N f ) −→ Sp((N + 1)N f /2) We now gauge SU (N ) strong . We postulate that vacuum is again formed by a flavour-diagonal condensate, under which the gauge group is broken to SU (N ) strong −→ Sp(ν) strong with ν = N − 1 2 More generally, as explained in Section 3, the surviving global symmetry H is broken to As before, it will prove useful to decompose all generators into representations of this unbroken group. The calculation is very similar to that in Appendix B.1 apart from one complication arising from the presence of the traceless antisymmetric tensor representation of Sp(ν) strong which is absent when ν = 1.
The SU (N ) strong generators decompose as where we use the hat to distinguish these Sp(ν) strong representations from similar SO(N f ) representations that we will meet below. HereŜ is the symmetric tensor representation with dimensions ν(2ν + 1),Â is the traceless antisymmetric tensor representation with dimensions (ν − 1)(2ν + 1), andF is the fundamental representation 2ν. The remaining representations are pseudo-Goldstone bosons. At leading order, we can compute their masses using the formula (A.9). The full calculation is a little involved, and here we quote only the final result: Meanwhile, at leading order, two of the pseudo-Goldstone bosons remain massless, m 2 (Â,S) = m 2 (F ,S) = 0 We write these as m 2 A and m 2 F for short. As in previous examples, to see whether these massless modes will destabilise the vacuum we need to look at higher order.
The broken gauge generators sit in theÂ,F and singlet representations in (B.9). The masses of the corresponding gauge bosons are given by We now use the formula (B.4); each of the X = (Â, S) and X = (F, S) pseudo-Goldstone modes get mass at second order, given by Once again, each of the representations contributes a positive contribution to the mass. Note that there is no A representation when N = 3, and these formulae hold only for N ≥ 5. Meanwhile, for the X = (F , S) state, we have The fact that each of these is negative ensures that the masses of the pseudo-Goldstone bosons are positive and the vacuum is stable. One can further show that the vacuum is not destabilised by the addition of hypercharge.