Implications of Higgs Discovery for the Strong CP Problem and Unification

A $Z_2$ symmetry that extends the weak interaction, $SU(2)_L \rightarrow SU(2)_L \times SU(2)'$, and the Higgs sector, $H(2) \rightarrow H(2,1) + H'(1,2)$, yields a Standard Model quartic coupling that vanishes at scale $v' =~\gg~$. Near $v'$, theories either have a"prime"sector, or possess"Left-Right"(LR) symmetry with $SU(2)' = SU(2)_R$. If the $Z_2$ symmetry incorporates spacetime parity, these theories can solve the strong CP problem. The LR theories have all quark and lepton masses arising from operators of dimension 5 or more, requiring Froggatt-Nielsen structures. Two-loop contributions to $\bar{\theta}$ are estimated and typically lead to a neutron electric dipole moment of order $10^{-27}$e cm that can be observed in future experiments. Minimal models, with gauge group $SU(3) \times SU(2)_L \times SU(2)_L \times U(1)_{B-L}$, have precise gauge coupling unification for $v' = 10^{10\pm1}$ GeV, successfully correlating gauge unification with the observed Higgs mass of $125$ GeV. With $SU(3) \times U(1)_{B-L}$ embedded in $SU(4)$, the central value of the unification scale is reduced from $10^{16-17}$ GeV to below $10^{16}$ GeV, improving the likelihood of proton decay discovery. Unified theories based on $SO(10) \times CP$ are constructed that have $H+H'$ in a ${\bf 16}$ or ${\bf 144}$ and generate higher-dimensional flavor operators, while maintaining perturbative gauge couplings.


JHEP10(2018)130 1 Introduction
In moving towards a UV completion of the Standard Model (SM), the vast majority of work in recent decades has assumed new physics at around the TeV scale. However, the Large Hadron Collider has discovered a highly perturbative Higgs boson with mass 125 GeV, but no clear evidence for physics beyond the SM. These results suggest that an unconventional framework of particle physics should be taken seriously, with the SM the correct effective theory to very high energy scales. Remarkably, the observed value of the Higgs mass results in the SM Higgs quartic coupling vanishing.
at a scale µ c (10 9 − 3 × 10 12 ) GeV [1] (see [2][3][4][5][6][7][8][9][10] for earlier works), or even higher if the top mass is below its measured value by more than 2σ. In this framework, we take the scale for new physics to be µ c . The weak scale is highly fine-tuned; this might result from environmental requirements [11,12], and should not prevent an exploration of this new picture.
Several ideas for the new physics that lead to the vanishing of the quartic have been proposed. It could be that a continuous global symmetry of an extended Higgs potential is spontaneously broken at µ c , leading to the Higgs boson becoming a Pseudo-Nambu-Goldstone Boson (PNGB), with potential arising from loop corrections [13]. Other possibilities include supersymmetry at µ c with tan β = 1 [14][15][16], and anthropic arguments associated with vacuum instability [17].
In this paper we introduce a new mechanism for physics at µ c that leads to (1.1). We assume that the entire theory at µ c is invariant under an exact Z 2 symmetry that interchanges the SM Higgs doublet H with its Z 2 partner H . This Z 2 is spontaneously broken by the condensation H = v at scale µ c . H must be neutral under SU(2) L because v v, the electroweak scale. Z 2 symmetry requires that H is a doublet under a partner SU(2) gauge symmetry. We study the most general Z 2 -invariant potential for (H , H). In the required limit of v v , the potential possesses an accidental SU(4) symmetry so that the SM Higgs boson is a PNGB at scale v ∼ µ c , leading to (1.1) before including the usual SM radiative corrections. Indeed, we find that this mechanism is highly constrained and requires that SU(2) L × SU(2) symmetry breaking is essentially unique and pristine in its simplicity: H(2, 1) + H (1,2). This simple gauge structure for the Higgs has several key implications.
We assume the SM SU(2) L -doublet fermions are singlets under SU (2 ), so that the theory contains q(2, 1) + (2, 1) + q (1, 2) + (1,2). Two classes of theories then emerge. In the first class (q , ) are identified as the SM SU(2) L -singlet fermions, so that SU(2) contains the right-handed W and we call it SU(2) R . In the second class (q , ) do not have the correct color and hypercharge to be identified as SM states and they form part of a sector that acquires mass from v , which we call the prime sector.
The Z 2 symmetry at v solves the strong CP problem [18] if it is extended to include spacetime parity and if it does not replicate the QCD gauge group. With parity included, at scale v we call the discrete symmetry P LR in models with SU(2) L × SU(2) R , and P in JHEP10(2018)130 models with a prime sector. The strong CP problem is solved in both classes of theories: P LR forces the quark mass matrices to be Hermitian, while P forces the strong CP phase from colored triplets in the prime sector to cancel that from quarks in the SM sector. Quantum corrections may generate a small but non-zero strong CP phase, which we study.
Parity solutions to the strong CP problem have a long history, starting in 1978 when the possibility of Hermitan quark mass matrices in left-right (LR) symmetric models was stressed [19,20]. However, a viable solution in conventional LR models, where the SM Higgs doublet is incorporated into φ(2, 2) under SU(2) L × SU(2) R , is problematic [21]. Phases in the Higgs potential lead to a phase in the vacuum, reintroducing the strong CP problem, unless additional symmetry, such supersymmetry [22][23][24] or CP symmetry [25], is added. Nevertheless, in LR theories, with the non-standard embedding of the SM Higgs into H L (2, 1) + H R (1, 2), a simple solution to the strong CP problem was discovered by Babu and Mohapatra in 1989 [26]. However, it was based on an approximate parity that was softly broken to obtain a large symmetry breaking scale for SU(2) R . Our starting point is different. We find that (1.1) results only if Z 2 is exact and only if the SM Higgs is embedded as H(2, 1) + H (1, 2), and we follow the implications of this understanding of the Higgs boson mass.
The gauge charges of SM fermions can be simply understood from grand unification, but gauge coupling unification in the SM lacks precision and leads to unacceptable proton decay, unless very large threshold effects are invoked. Our new understanding of the Higgs mass implies that new physics lies near µ c where λ SM vanishes. Could this new physics lead to precise unification? Running of gauge couplings is substantially altered because there is an additional SU(2) gauge group but, to solve strong CP, the color gauge group is not replicated. In the class of theories involving a prime sector at v , the new heavy states couple to the color gauge group but not to SU(2) L , so for v ∼ 10 9−13 GeV, the SM couplings g 2 and g 3 do not meet by the Planck scale. However, in the minimal LR symmetric model with gauge group SU(3) c × SU(2) L × SU(2) R × U(1)B−L (3221), gauge coupling unification occurs with very high precision for v ∼ 10 10 GeV, right in the region that yields the observed Higgs mass. Furthermore, the unification scale is of order 10 16−17 GeV. If the gauge group above v is SU(4) × SU(2) L × SU(2) R (422), v is increased and the unification scale decreases so that planned proton decay searches become more powerful probes of the theory.
We find that the LR symmetric models may be unified very neatly into SO(10), with matter in 16s [27,28]. For λ SM (µ c ) = 0, the Higgs must transform as H(2, 1) + H (1, 2) under SU(2) L × SU(2) R and therefore as a 16 rather than a 10 of SO(10), greatly affecting the structure of the theory. SO(10) contains a generator C LR that is charge conjugation together with SU(2) L ↔ SU(2) R , so P LR = CP * C LR . Hence the unified theory must have a symmetry CP × SO(10) broken by the condensation of a field that is both C LR and CP odd, such as a CP odd 45 In the SO(10) theory, the discrete symmetry that ultimately leads to strong CP conservation is CP; P is not defined.

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Gauge coupling unification and the SO(10) embedding only work for the LR theories, not for the theories with prime sectors. In any case, the LR theories have minimal field content and appear more elegant. However, with the Higgs in H(2, 1)+H (1, 2) they do not have Yukawa couplings. Thus the flavor problem cannot be postponed, and we construct Froggatt-Nielsen type theories [29]. Flavor is particularly pressing for the heavy quarks and leptons where the corresponding heavy vector fermions cannot be far above v .
In section 2 we demonstrate that a Z 2 with H(2, 1) ↔ H (1, 2) leads to (1.1). In section 3 we explore minimal models at scale v of both LR and prime sector classes, showing explicitly the possibilities for constructing flavor operators. In section 4 we show that, by including parity in the definition of the Z 2 symmetry at the scale v , the strong CP problem can be solved in all these models, provided the color group is Z 2 -invariant. In section 5 we explore gauge coupling unification in both 3221 and 422 schemes, and draw conclusions for proton decay. In section 6 we construct a variety of SO(10) models, including operators that lead to quark (u, d), charged lepton (e) and neutrino (ν) masses. We show how d/e mass splittings arise, and how ν masses become decoupled from u masses. We find that precision gauge coupling unification is possible even in the presence of the required Froggatt-Nielsen states.

Vanishing Higgs quartic from a Z symmetry
In this section we show that the near vanishing of the SM Higgs quartic coupling at a high energy scale v can be explained by a Z 2 symmetry spontaneously broken at v .

Vanishing quartic coupling
We introduce a Z 2 symmetry under which the Higgs field H and its partner H are exchanged. The renormalizable potential of H and H is given by We assume that the mass scale m is much larger than the electroweak scale, m 2 > 0 and λ = O(1). To obtain a small electroweak scale, λ may not be large and negative. Then H acquires a large vacuum expectation value (vev) of H = v , with v 2 = m 2 /λ, and the Z 2 symmetry is spontaneously broken. After integrating out H at tree-level, the Low Energy potential in the effective theory for H is In order to obtain the hierarchy H = v v , it is necessary that |λ | 1. After this fine-tuning, the quartic coupling of the Higgs H, λ SM , also vanishes. The structure of the vacuum is dominated by quantum corrections, which are investigated in appendix B. We find that λ is small and negative and the vacuum with v v is stable. The vanishing quartic can be understood by an accidental SU(4) symmetry under which (H, H ) is a fundamental representation. For λ = 0, the potential in eq. (2.1) is  (GeV) 3 × 10 12 1 × 10 11 2 × 10 10 3 × 10 9 8 × 10 8 manifestly SU(4) symmetric. After H obtains a vacuum expectation value, the Standard Model Higgs is understood as a Nambu-Goldstone boson with a vanishing potential. Below the scale v , quantum corrections from SM particles renormalize the quartic coupling, and it becomes positive. From the perspective of running from low to high energies, the scale at which the SM Higgs quartic coupling vanishes, µ c , is to be identified with v as in (1.1) v µ c . The value of µ c depends on uncertainties in the top quark mass m top , and is listed in table 1, following the calculation of ref. [1]. As shown in appendix B, quantum corrections give a non-zero value of λ SM (v ) ∼ −y 4 t /(16π 2 ) ∼ −10 −3 , where y t is the top Yukawa coupling, slightly changing the prediction for v ; but the effect is smaller than the uncertainty from the top quark mass.
Although the scale v is much smaller than the Planck scale and the typical unification scale, the theory is no more fine-tuned than the Standard Model because of the Z 2 symmetry. The required fine-tuning is of the order of where the first factor in the left hand side is the fine-tuning to obtain the scale m much smaller than the cut off scale Λ, and the second one is the fine-tuning to obtain λ (v ) which produces the small electroweak scale from m. The total tuning is the same order as in the Standard Model, v 2 /Λ 2 , and may be explained by environment requirements [11,12].

The symmetry breaking sector at scale v
Since H obtains a large vacuum expectation value, it cannot have the same Standard Model gauge charges as H. We must introduce an additional SU(2) gauge symmetry, under which H is charged. The Z 2 symmetry exchanges the two SU(2) gauge symmetry groups: SU(2) L ↔ SU(2) , for example   There are several options for the action of Z 2 on SU(3) c and U(1)Y . The theory with partners of both SU(3) c and U(1)Y is nothing but the mirror world (see [30,31] for reviews). The gauge group is ( , and all of the SM particles have their mirror counterparts. However, it is not necessary to introduce partners of SU(3) c and U(1)Y . Indeed, in the next section, we show that in minimal cases, where SU(3) c × U(1)Y is not replicated, the strong CP problem [18] may be solved by including space-time parity in the Z 2 symmetry.

Minimal models at scale v
In the previous section, we showed that the essentially unique Higgs structure for breaking SU(2) L × SU(2) is remarkably minimal, H(2, 1) + H (1, 2), implying that in the theory well below scale v the SM Higgs is H. Since the SM Higgs carries non-zero hypercharge and is an SU(2) singlet, the electroweak group must be extended beyond SU(2) L × SU(2) . Hence, the gauge group with fewest generators that yields our understanding of the SM quartic is SU(3) c × SU(2) L × SU(2) × U(1). We choose the normalization of the U(1) generator so that it is conventional hypercharge on SU(2) singlets, so that without loss of generality we have H(1, 2, 1, −1/2) and H (1, 1, 2, ±1/2). The vacuum expectation value of H breaks SU(2) R × U(1) → U(1)Y . 1 We assume that the SU(2) L doublet quarks and leptons of the SM are SU(2) singlets, so they must transform as q(3, 2, 1, 1/6) and (1, 2, 1, −1/2). The Z 2 symmetry requires that q, , H have partners transforming as (1, 2) under (SU(2) L , SU(2) ). There are four possible SU(3) c × U(1) charge assignments for (q , , H ), as listed in table 2, where the fermions are left-handed Weyl spinors. We designate these cases as A(−, −), B(+, −), C(−, +), and D(+, +), where the signs indicates whether (SU(3) c , U(1)) charges are conjugated. We find that, with minimal field content consistent with gauge anomaly freedom, the right-handed SM quarks and leptons transform as (1, 2) under (SU(2) L , SU(2) ) in model A but as (1,1) in models B, C and D. In the first sub-section we study Model A and identify the Z 2 partner of SU(2) L as SU(2) R . In the second sub-section we study models B, C and D.
Model A is free of gauge anomalies with (q, q , , ). While there are no gauge-invariant Yukawa couplings, there are interactions between fermions and scalars at dimension 5  Table 3. Possible X particles for generating Yukawa couplings in Model A. We also show the embedding of X into the Pati-Salam group SU(4) × SU(2) L × SU(2) R as well as the SO (10) group.
Hereỹ u,d,e are dimensionless flavor matrices, with flavor indices suppressed, while M u,d,e are mass scales. On breaking SU(2) R , the theory below scale v is the SM (with right-handed neutrinos to be discussed). q and contain the SU(2) L -singlet SM quarks and leptons and U(1) is identified as (B − L)/2. SM Yukawa couplings arise from (3.1) and are given bỹ The dimension 5 operators of (3.1) can be generated by the exchange of heavy states. Since the top Yukawa coupling is near unity, at least some of these states must be close to v , and we take these to be fermions, X (andX when Dirac), as extra scalars near v require further fine-tuning. The possible gauge charges of X for each Yukawa coupling are listed in table 3. Anticipating the next section, we also show the possible embedding of these fermions into SO(10) representations with a dimension 210 or smaller. The couplings break baryon number if a colored Higgs is involved. The resultant proton decay is suppressed by Yukawa couplings and is negligible compared to that by heavy gauge boson exchanges. In the f = u, d, e sectors, if these heavy fermions have mass matrices M X f and Yukawa couplings x f to q/ and x f to q / , then the resulting 6 × 6 mass matrices are where M X f , x f and x f are 3 × 3 matrices. We show the result for X(1, 1); for X(2, 2) the same result applies except with v ↔ v . After integrating out the heavy fermions, we obtain the dimension-5 operator in eq. (3.1) with The effect of Z 2 on these couplings will be discussed later in this section.

JHEP10(2018)130
The following dimension-5 operators give masses to both left and right-handed neutrinos, ) they obtain Dirac masses with the right-handed neutrinos in .
With just (q, q , , ), Models B, C and D contain gauge anomalies. In Models C and D, q and do not have the right charges to be identified with SM SU(2) L -singlet quarks or leptons, and there are no Yukawa-like interactions for electrically-charged fermions at any dimension. For these theories, the minimal additions for anomaly freedom are SU(2) Lsinglet fermions of the SM,ū,d andē, and their Z 2 partnersū ,d andē with gauge charges shown in table 4. The following Yukawa couplings are allowed for Model C, where generation indices are suppressed. In Model D, the allowed Yukawa couplings are After H obtains a large vacuum expectation value, the partner fermions obtain a large mass and decouple. The theory has a SM sector, similar to the SM but at scale v with weak interactions from SU(2) . The effective theory below the SM sector is just the SM (with right-handed neutrinos to be discussed). In both sectors, the U(1) gauge symmetry is now hypercharge; but in theory C the two sectors have opposite color.
In Model B, has the right charge to be identified with the SM SU(2) L -singlet lepton, so singlet fieldsē andē are not added. The electron Yukawa couplings is obtained from the third term in eq. (3.1). q , on the other hand, cannot be identified with SM SU(2) L -singlet quarks, and hence we addū,d,ū andd as shown in table 4, and the up and down Yukawa couplings are as in (3.5):

JHEP10(2018)130
In this hybrid theory, the gauge U(1) can be interpreted as (B − L)/2 on leptons and hypercharge on quarks. The SM sector contains only quarks, and they have opposite hypercharges to the SM quarks.
In Models B and C the 6 × 6 mass matrices for the quarks of the two sectors are Neutrino masses are generated by the operators of (3.4), as in Model A. Models B, C and D contain heavy exotic matter. The exotic leptons (present only in Models C and D) mix with the SM leptons via (3.5) and are unstable. The exotic quarks of Model D mix with the SM quarks via (3.6) and are unstable. However, in Models B and C, exotic quarks cannot mix with SM quarks, so that the lightest exotic quark, u , is stable.
H and H are embedded into (4, 2, 1) and (4, 1, 2), respectively. The vev of H breaks the SU(4) × SU(2) L × SU(2) symmetry to the SM gauge group. The stability of this vacuum, as well as possible quantum corrections to the SM Higgs quartic coupling, is discussed in appendix C.
The embedding of (q, q , , , H, H ) is given in table 5. Again, Model D contains gauge anomalies and (q , ) do not have the right charges to be identified with SM SU(2) L -singlet quarks or leptons: further SU(2) L -singlet fermions must be added. Their SU(4) ×SU(2) L × SU(2) charges are given by embedding the particles in table 4 into SU(4)×SU(2) L ×SU(2) . Here we focus on Model A where q are the SM SU(2) L -singlet quarks, and we identify SU(2) with SU(2) R .

Z 2 symmetry and the strong CP problem
If the action of the Z 2 symmetry is the simple exchange of fermions, ψ ↔ ψ , the Yukawa couplings are required to satisfyỹ ij =ỹ ji for eqs. (3.1), (3.10), or y ij = y ij as well as λ ij = λ ij for eqs. (3.5), (3.6). If, on the other hand, the Z 2 symmetry involves the space-time parity transformation, ψ(t, x) → iσ 2 ψ * (t, −x), the requirements areỹ ij =ỹ * ji (i.e. Hermitian) or y ij = y * ij and λ ij = λ * ij . Here σ 2 acts on the spinor index. The contribution to the strong CP phase from the quark Yukawa couplings is proportional to the phases of detỹ, dety × dety * , det y * λ * λ y , (4.1) for Models A, B and C, and D, which vanish. Space-time parity also forces θ QCD = 0, so thatθ = 0 at tree-level.
We summarize this solution of the strong CP problem by P : which applies whether the gauge group is 3221 or 422. P does not forbid phases inỹ or y, and the CKM phase is obtained as usual from the physical phase ofỹ or y. For cases where SU(2) is identified as SU(2) R , we call the parity transformation P LR , since it exchanges SU(2) L ↔ SU(2) R . In Model A, before the heavy Dirac X fermions are integrated out, the 6 × 6 color triplet mass matrices take the form of (3.2). At this level it is easy to see that the strong CP problem is solved because For models B and C the 6 × 6 quark mass matrices take the form of (3.8). Since y u,d = y * u,d , the contribution toθ from the SM sector cancels that from the SM sector.
Model A, with gauge group 3221 andX having the same charge asū,d andē, was proposed in ref. [26] as a solution to the strong CP problem based on parity. There parity was assumed to be softly broken to obtain the hierarchy between the vacuum expectation values of H and the electroweak scale. As we have shown in the previous section, soft breaking is not required. Parity symmetry can be spontaneously broken by the condensation of H , thereby explaining the vanishing of the SM quartic at scale v .
Since parity is spontaneously broken, the strong CP phase may be generated by higherdimensional operators. The following operator is composed only of bosonic fields, and is JHEP10(2018)130 not controlled by any symmetry at scale v , 2 where M * is a cut-off scale and G is the field strength of SU(3) c . Condensation of H yields the strong CP phase For a cut-off scale of the Planck mass, satisfying the experimental constraint θ < 10 −10 [33][34][35] requires v < 10 13 GeV. Quantum corrections also gives a non-zero strong CP phase. In Models B and C they are essentially the same as in the SM model, and are negligibly small [36]. The quantum correction in Model D is not known and will be investigated elsewhere. In Model A, after integrating out heavy states XX, any radiative corrections to the dimension 5 operators do not induceθ; but contributions arise from parity-invariant operators of dimension 7 After H obtains a vev, this leads to non-Hermitian Yukawa couplings. As shown in appendix D, non-Hermitian contributions to the flavor matrices c u and c d arise first at 2-loop level, and require flavor mixing. 3 The typical correction is of the form where g is a gauge coupling constant at scale v and C is a numerical constant which depends on the theory of flavor. The constant C is at most O(10 3 ) and such large values result when there is no hierarchy among the various X masses. However, with a Frogatt-Nielsen structure the quark mass hierarchy naturally follows from an X mass hierarchy, and this gives small values for C, with the leading contribution typically Note that |V cb | 2 of (4.6) arises from the product of the 23 mixing in the up and down sectors, θ u 23 θ d 23 , so that ∆θ is further suppressed if one is much smaller than the other. 2 The operator may be controlled by a symmetry realized at a high energy scale. For example, in the SO(10) model discussed in the next section, the symmetry at high scales is SO(10) × CP and CP forbids the operator in eq. (4.3). After SO(10) × CP symmetry breaking, the operator is generated with a further suppression factor of vG/M * . Supersymmetry can also suppress the operator. 3 Ref. [26] shows that the one-loop correction to θ is absent. It also claims that the two-loop corrections are suppressed by v/v , which we could not confirm.

JHEP10(2018)130 5 Gauge coupling unification
We investigate the running of gauge couplings in the LR theories to determine whether a more precise unification is possible than in the SM, which requires large threshold corrections to avoid exclusion from proton decay. We anticipate section 6 where we show that these theories can be successfully embedded into SO (10). We treat separately the cases where the gauge group above v is SU The gauge couplings evolve from IR to UV as follows. Between the electroweak scale and the scale v , SU(3) c × SU(2) L × U(1)Y couplings (g 3 , g 2 , g 1 ) evolve as in the SM, d dlnµ where the U(1)Y coupling g 1 is suitably normalized for unification, g 2 1 = 5/3g 2 . At the scale v , the U(1) coupling g B−L , suitably normalized for unification, is obtained from the relation 1 The SU(2) R coupling is the same as the SU(2) L coupling.
where ∆b i denotes the contribution from heavy X states. We assume that the X states form nearly degenerate SO(10) multiplets, so that ∆b i do not affect relative running of the gauge couplings. If there are many large X multiplets that are light, the unified gauge coupling grows so that unified threshold corrections and two loop running effects are expected to give contributions to ∆(M G ), defined below in (5.5), in excess of 10. However, with X in 10 and 45 dimensional representations this is easy to avoid by taking where generation indices a, b run over multiplets X a lighter than M G .
To quantify the quality of the unification, we define Interestingly, around this central value precise gauge coupling unification is achieved. Note however that the heavy states from each SO(10) multiplet (e.g. the gauge bosons, the SO(10) symmetry breaking field, and XX) are not expected to be degenerate and will typically have mass ratios of O(1). This is expected to generate a threshold correction to ∆ of ∼ 10. Similar contributions may arise from 2-loop running. Thus the remarkable agreement between gauge coupling unification and the observed value of the Higgs boson mass, allows for v anywhere in the range of (10 9 − 3 × 10 12 ) GeV at the 2σ level. A more precise determination results if the uncertainties on the top quark mass are reduced. Figure 2 shows contours of ∆(M G ) in the (v , M G ) plane, and the constraint on M G from the proton decay [37]. The parameter point which minimizes ∆ has a large M G , and thus cannot be probed by near future searches for nucleon decay. However, the above mentioned threshold corrections to ∆ of ∼ 10 implies that there is an interesting region of parameter space with lower M G that will be probed by near future searches.
We quantify the quality of the unification in the following way. For a given v , we define We then match the gauge couplings at v , g 4 (v ) = g 3 (v ), and evolve them towards the unification scale M G , where we define .

SO(10) unification
In this section we embed Model A into SO(10) grand unified theories. In Model A, q and have the opposite SU(3) c × U(1) charges to q and . Thus in this section we denote them asq and¯ .

Matter unification
The embedding of q, , H and their parity partners into SO(10) multiplets is shown in table 6. We also show the embedding into the SU(4) × SU(2) L × SU(2) R subgroup. Quarks and leptons are unified into the 16 of SO(10); the U(1) gauge symmetry is nothing but B − L. The Higgs doublets H and H are also unified into a 16 or 144; they cannot live in a 10, since this contains a (1, 2, 2). The embedding of the heavy states XX is shown in  Table 6. SO(10) embedding of quarks, leptons and Higgses.

SO(10) symmetry breaking and Z 2 symmetry
In the previous section we consider two types of Z 2 symmetries. One is the simple exchange of fields, ( , q) → (¯ ,q ). We call this Z 2 transformation C LR , as it involves charge conjugation and the exchange of SU(2) L with SU(2) R . The other Z 2 involves space-time parity, with (t, x) → (t, −x) and ( , q) → iσ 2 (¯ ,q ) * , which we call P LR . C LR is present in all SO(10) theories, as it is a subgroup of the SO(10) gauge group [38,39], and is sometimes called D parity in the literature [40,41]. On the other hand, as in the SM, space-time parity is not a symmetry of SO(10) theories because the fermions are chiral. Nevertheless, if the SO(10) theory is CP invariant it is also P LR invariant, because For some patterns of the SO(10) symmetry breaking, C LR (P LR ) remains unbroken; the vanishing Higgs quartic coupling is then explained by SO(10) gauge symmetry (and CP ). Figure 4 shows the required symmetry breaking pattern. A scalar 54 (≡ φ 54 ) and 210 (≡ φ 210 ) give the symmetry breaking patterns SO(10) respectively.
Having P LR unbroken below the unified scale is particularly interesting, as it solves the strong CP problem. This requires the symmetry above the unification scale to be SO(10)×CP . P LR remains unbroken if the SO(10) symmetry breaking field vev is odd under both C LR and CP . The smallest representation that leaves SU(2) L × SU(2) R unbroken via a vev odd under C LR is 45 (≡ φ 45 ), [40,41], giving the symmetry breaking pattern 4 The next such smallest representation is 210 (≡ φ 210 ) with

Fermion masses
We briefly comment on how realistic SM Yukawa couplings can be obtained, leaving the detailed analysis to future work.

Down-type quarks and charged leptons
As we discuss in the previous section, Yukawa couplings arise from the exchange of heavy states XX. We introduce three 10 (≡ X 10 ) 5 with the following interactions, where O G denotes possible insertions of SO(10) symmetry breaking fields. After integrating out X 10 , the dimension 5 operators of eq. (3.1) for down-type quarks and charged leptons are obtained. Without O G , down-type quarks and charged leptons of the same generation have identical Higgs couplings at M G , in contradiction with the observed spectrum. The observed masses can result, for example, by non-degeneracies in X 10 from SO(10) symmetry breaking.

Up-type quarks and neutrinos
For the up Yukawa couplings we introduce 54 and/or 45 (≡ X 54 , X 45 ). We first discuss the case with 54. The following interaction leads to the up Yukawa couplings, After H and H condense, this interaction gives a mass only to a linear combination of ν and ν , which is dominantly ν . The SM left-handed neutrino masses are not related to the SM up-type quark masses. Next we consider the case with 45. The up Yukawa couplings are obtained from In contrast to the case with 54, the first term is non-zero even if the Higgs is embedded into 16 and without insertion of the SO(10) symmetry breaking fields. The interaction, however, in general generates left-handed neutrino masses via the exchange of the SU(2) L triplet in X 45 . Without any SO(10) symmetry breaking from O G , the left-handed neutrino masses at tree-level are m u,c,t ×v/v , which are too large and hierarchical. This can be avoided by 1) generating the up Yukawa couplings predominantly JHEP10(2018)130 from 54 or 2) picking up SO(10) symmetry breaking effects from O G . For the latter option, for example, consider the dimension-5 operator where m, n, k = 1 · · · 10 are the vector indices of SO (10). Because of the missing vev structure of φ 45 , this operator does not give a Yukawa coupling to the SU(2) L triplet, and hence leaves the SM neutrinos massless while giving up-type quark masses. The particular SO(10) structure in eq. (6.10) can be obtained by introducing a pair of 45s, Y 45 andȲ 45 , and the interaction with (ψ 16 X 45 )φ * H suppressed by some parity.

CKM phase
The strong CP problem is solved by imposing SO(10) × CP symmetry. An SO(10) symmetry breaking field, odd under both CP and the C LR parity contained in SO (10), leaves the product, P LR , unbroken. To obtain the CKM phase, the up or down Yukawa couplings must feel this symmetry breaking, since otherwise they are CP symmetric. One simple possibility is to obtain M X d,e in eq. (6.6) via L = (M ij + iλ ij φ 45 ) X 10,i X 10,j + h.c., (6.12) where M ij and λ ij are symmetric and anti-symmetry, respectively. CP requires that M ij and λ ij are real. After φ 45 obtains its vev, the mass matrix of X is Hermitian, and can explain the CKM phase without introducing the strong CP phase.

A top-down view
We began this paper by introducing a Z 2 symmetry and showing that the SM Higgs quartic vanishes at the scale v where Z 2 is spontaneously broken, and we proceeded to explore the consequences. We argued that there must be a new SU(2) gauge group and that the Higgs sector for SU(2) × SU(2) must be essentially minimal: H(2, 1) + H (1, 2). We constructed the set of minimal theories at v : those based on gauge group 3221 with a prime sector and those based on 3221 or 422 with a left-right symmetry. We demonstrated that if the Z 2 included spacetime parity these theories could all solve the strong CP problem, and then we proceeded to explore the possibility of unification. Here we take an opposite view and start from a particular grand unified theory and show how the SM appears. Consider a grand unified theory based on symmetry group SO(10) × CP and broken by the vev of 45 − to 3221 × P LR . Light matter is contained in three 16, ψ i , and the Higgs surviving below the unified scale, (H, H ), lies in a 16, φ H , rather than the usual 10. Since the Yukawa interaction ψ i ψ j φ H is not gauge invariant, quark and lepton masses must arise JHEP10(2018)130 from interactions with other fields X. On integrating out X, flavor appears in higherdimensional operators which contain SO(10) breaking, avoiding certain unacceptable u : d : e mass ratios. For example, d/e masses can arise from X 10 , and u/ν masses from X 45 or X 54 . In the latter case, care must be taken to ensure a decoupling of u and ν masses.
The intermediate symmetry 3221 × P LR is broken by H = v to the SM. Taking v ∼ 10 10 GeV, leads to both precision gauge coupling unification and λ SM (v ) ∼ 0; there is a successful prediction that correlates the observed gauge couplings with the Higgs and top quark masses (five low-energy observables are given in terms of four parameters: g G , M G , y t , v ). The H vev converts the higher dimensional flavor operators to SM Yukawa couplings that are Hermitian, including 1-loop radiative corrections, solving the strong CP problem.

Final comments
We have not explored cosmological consequences of the ideas and theories discussed in this paper. The maximum temperature that the SM sector is reheated to after inflation must be less than v to avoid domain walls from spontaneous Z 2 breaking. This is still consistent with leptogenesis from ν R decay [47], although the constraints become more severe from lower v , and ν R are present in all the LR models. Finally, unlike the axion solution of the strong CP problem [48][49][50][51], the simplest theories discussed here do not include an exotic dark matter candidate.
There are several key measurements for our scenario. A more precise determination of the top quark mass would better constrain µ c and therefore the Z 2 -breaking scale v . In LR theories, a significant fraction of the parameter space with precision gauge coupling unification has proton decay at a rate observable in planned experiments [52], especially if the intermediate symmetry is 422. Finally, LR models have 2-loop contributions toθ that are likely within 1 or 2 orders of magnitude of the current experimental bound from the neutron electric dipole moment and may be observable by future experiments [53][54][55]. Discovery of such a dipole moment would point to these theories over axion theories, and would provide an important new constraint in constructing theories of flavor.

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which has dimensions and λ /λ = x which is dimensionless. Hence, the low energy potential for H must take the form At the SU(4) invariant point x = 0 both f and g must vanish. Hence the fine-tune on the quadratic term for v v necessarily forces λ SM 1. (From explicit computation, we find that at tree level f (x) = x and g(x) = x(1 + x/2), corresponding to (2.2)). Now consider additional scalar fields, χ, that acquire vevs at scale v , so that the Z 2 invariant potential of the full theory V (H, H , χ) has additional dimensionless couplings that break the SU(4) invariance. The low energy effective theory for H now arises from more than one SU(4) breaking parameters, x i , and takes the form of (A.1), with f (x) → f (x i ) and g(x) → g(x i ). The fine-tune f (x i ) 1, necessary for v v , restricts x i to lie on a surface in the parameter space. Although the surface passes through the origin, where SU(4) is restored, a generic point on the surface has large SU(4) breaking and hence an order unity value for g(x i ). To understand the near vanishing of λ SM at v , the scalar potential of the full theory must not have any SU(4) breaking couplings beyond λ of (2.1).
There are two possible such additions to the scalar sector that contribute to SU(2) symmetry breaking at scale v • Multiple copies of doublets H a (2, 1) ↔ H a (1, 2), leading to SU(4) violating via giving multiple contributions to f (x i ) and g(x i ).
Additional scalars at scale v are possible if they don't acquire a vev. We do not consider them as they introduce extra fine-tuning to the theory. Our mechanism for understanding why λ SM (v ) is very small requires that the low energy field that breaks SU(2) L arises predominantly from H(2, 1). For example, consider a bi-doublet with φ(2,2) ↔ φ † (2,2). This leads to the SU(4) breaking interaction with λ φ real. The H vev now leads to mass mixing between H and φ, so that an additional SU(4) breaking parameters appear in H/φ mass terms. Fine-tuning one combination to be light no longer guarantees that g(x) vanishes. Requiring λ SM (v ) < ∼ 10 −2 implies that the light SM Higgs contains no more than 10% of φ. It is possible that additional scalars are in the theory at scale v, but they cannot substantially take part in electroweak symmetry breaking, and we do not add them as they imply further fine-tuning.

JHEP10(2018)130 B Quantum corrections and the stability of the vacuum
In this appendix we investigate the potential of H and H , including one-loop corrections, and show the stability of the vacuum. We only include the dominant correction, namely that from the top Yukawa coupling.
We first discuss models B, C and D. For simplicity, in model D we take the limit λ u 1. The one-loop corrected potential is given by where M is an arbitrary scale. A change of M can be absorbed by a change of λ. We take M to be the vev of H , which is given by After integrating out H , the potential of H is, to the leading order in c and λ , given by To obtain the electroweak scale much smaller than v , requires the fine tune λ To One can easily verify that α = 0 is a minimum of the potential. We next discuss model A. The squared diagonalized masses of the top quark and the X state for a generic value of H and H are The one-loop corrected potential is given by We take the scale M to be m + at H = 0, m +,0 = M 2 X + x 2 v 2 . The vev of H is given by

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After integrating out H , the potential of H is given by where we have replaced M X using the relation The electroweak fine-tuning requires λ + c The α-dependent part of the potential is where r ≡ y 4 t /(2x 4 ). One can verify that α = 0 is a minimum of the potential.

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Let us first investigate the conditions for the mass of h to vanish and for vacuum stability. The potential of h and h is At the tree level, the vev of h is h = m/ √ λ v . The condition for the h mass to vanish is y − g = λ. For this value of y, the mass-squareds of colored particles are given by u is the Nambu-Goldstone boson eaten by the colored gauge boson. The stability of the vacuum requires that g, k > 0. Next we discuss quantum correction to the SM Higgs quartic coupling. The couplings of h and h with the colored Higgses explicitly break the SU(4) symmetry on (H, H ) (SU(2) on (h, h ) after eliminating the (4, 1) components), and hence these couplings give quantum corrections to the SM Higgs quartic coupling at scale v . Since we are interested in the one-loop correction, we may use the tree-level relation y = λ + g in the coupling of h and h with the colored Higgses. For this value, the coupling between h, h and u,ū is of the form which is SU(2) invariant. Thus, quantum corrections from u,ū loops do not generate a SM Higgs quartic coupling at the one-loop level. The masses of d andd, which mix with each other for generic values of h and h , are given by The one-loop corrected effective potential of h and h is given by Here we take the renormalization scale to be m. We minimize this potential with respect to h , obtain the condition that the mass of h vanishes at the minimum, and find that the SM Higgs quartic coupling at v is  Here we choose X to transform as (1,1) under (2 L , 2 R ). The alternative of X(2,2) gives Yukawa interactions qXH + q XH, but the estimation is essentially the same. Ref. [26] shows that there is no one-loop quantum correction to the strong CP phase. As we will see two-loop corrections do not vanish and may give an observable neutron electric dipole moment in future experiments. We do not consider theories having X states with both charges. We can assign the spurious symmetry shown in table 7. The parity requires that x ij = x * ij and M a are real. To make use of the spurious symmetry, we keep x as an independent coupling and M a as a complex number until we evaluate the reality of the quantum correction.
After integrating out XX, the following higher-dimensional operators are generated, so that the quark Yukawa matrices are no longer Hermitian. The correction to the strong CP phase is given by where the Yukawa matrix and its inverse are given by  Table 8. Maximal possible corrections to the strong CP phase from (i, j, k) terms in eq. (D.11).
With hierarchical M a , the correction is much smaller.
The contribution to ∆θ from c u(2,2) ij depends on the structure of x u and x d , but we can derive a rough upper bound on the correction. Using the relation y ij = v k x ia x * ja /M a , we obtain |x ia x † aj | < ∼ |y ij |M a /v , giving In table 8, we list the contribution to ∆θ for each combination (i, j, k) in eq. (D.11) which exceeds O(10 −9 ); the maximal correction to the strong CP phase is 10 −6 .
The maximal value is achieved by assuming that the relevant tree-level Yukawa couplings, y of (D.4), are generated by the exchange of heavy states with masses around v . A more natural assumption is that the quark mass hierarchies reflect hierarchies in the masses

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