Scale hierarchies, symmetry breaking and particle spectra in SU(3)-family extended SUSY trinification

A unification of left-right $\rm{SU}(3)_\rm{L}\times \rm{SU}(3)_\rm{R}$, colour $\rm{SU}(3)_\rm{C}$ and family $\rm{SU}(3)_\rm{F}$ symmetries in a maximal rank-8 subgroup of ${\rm{E}}_8$ is proposed as a landmark for future explorations beyond the Standard Model (SM). We discuss the implications of this scheme in a supersymmetric (SUSY) model based on the trinification gauge $\left[\rm{SU}(3)\right]^3$ and global $\rm{SU}(3)_\rm{F}$ family symmetries. Among the key properties of this model are the unification of SM Higgs and lepton sectors, a common Yukawa coupling for chiral fermions, the absence of the $\mu$-problem, gauge couplings unification and proton stability to all orders in perturbation theory. The minimal field content consistent with a SM-like effective theory at low energies is composed of one $\mathrm{E}_6$ $27$-plet per generation as well as three gauge and one family $\rm{SU}(3)$ octets inspired by the fundamental sector of ${\rm{E}}_8$. The details of the corresponding (SUSY and gauge) symmetry breaking scheme, multi-scale gauge couplings' evolution, and resulting effective low-energy scenarios are discussed.


I. INTRODUCTION
Finding successful candidate theories unifying the strong and electroweak interactions, leading to a detailed understanding of the SM origin, with all its parameters, hierarchies, symmetries and particle content remain a big challenge for the theoretical physics community. Some of the most popular SM extensions are based on supersymmetric (SUSY) GUTs where the SM gauge interactions are unified under symmetry groups such as SU (5) and SO (10) [1][2][3][4][5][6][7] as well as E 6 1 and E 7 [11]. A particularly appealing scenario proposed by Glashow in 1984 [12] is based upon the rank-6 trinification symmetry [SU (3) There have been many phenomenological and theoret- * eliel@thep.lu.se † aapmorais@ua.pt ‡ astrid.ordell@thep.lu.se § Roman.Pasechnik@thep.lu.se ¶ jonas.wessen@thep.lu.se 1 The E 6 -based models are typically motivated by heterotic string theories where massless sectors consistent with the chiral structure of the SM are naturally described by an E 8 × E 8 gauge theory. For more details we refer the reader to Refs. [8][9][10] ical studies of T-GUTs, in both SUSY and non-SUSY formulations, motivated by their unique features (see e.g. Refs. ). For example, due to the fact that quarks and leptons belong to different gauge representations in T-GUT scenarios, the baryon number is naturally conserved by the gauge sector [15], only allowing for proton decay via Yukawa and scalar interactions, if at all present. As was shown for a particular T-GUT realisation in Ref. [26], the proton decay rates were consistent with experimental limits in the case of low-scale SUSY, or completely unobservable in the case of split SUSY. Many T-GUTs can also accommodate any quark and lepton masses and mixing angles [15,30] whereas neutrino masses are generated by a see-saw mechanism [23] of radiative [26] or inverse [28] type.
Despite a notable progress in exploring gauge coupling unification, neutrino masses, Dark Matter candidates, TeV-scale Higgs partners, collider and other phenomenological implications of GUTs, there are several yet unresolved problems. A general challenge in GUT model building (and particularly so in T-GUT), has to do with the existence of an appropriate minimum of the potential with a spontaneously broken GUT symmetry leading to the SM gauge group. Namely, the lower the number of free parameters at the GUT scale, the more difficult it is to find a realizable GUT scenario with a SM-like EFT limit at low energies.
Another problem in the case of SUSY T-GUT model building is the longstanding issue of avoiding GUT scale masses for the would-be SM leptons. To circumvent this, the usual solution is to add several 27-plets of E 6 with scalar components responsible for SSB of gauge trinification [15, 18, 20, 21, 25, 26, 28-31, 33, 37], or to simply add higher dimensional operators [20,21,25,28,38]. These approaches typically require a significant fine-tuning in high-scale parameter space (especially, in the Yukawa sector) [26]. Otherwise, they exhibit phenomenological issues with proton stability [15,21,26] and with a large amount of unobserved light states [12,20,30,31,34,38]. Despite continuous progress, the SM-like EFTs originating from T-GUTs still remain underdeveloped in comparison to other GUT models such as SU (5), SO (10) or even E 6 (see e.g. Ref. [32] and references therein).
In this paper, we explore in detail the SUSY T-GUT model proposed in [39] with a global SU(3) F family symmetry inspired by the embedding of E 6 × SU(3) into E 8 .
We will refer to this model as the SUSY Higgs-Unified Trinification (SHUT) model (for alternative ways of extending the SM by means of an SU(3) F symmetry see e.g. Refs. [40][41][42][43]). As we will see, the SHUT model offers solutions to some of the problems faced by previous T-GUTs. As the light Higgs and lepton sectors are unified, the model can be embedded into a single E 8 representation. Furthermore, the embedding suggests the introduction of adjoint scalars and a family SU(3) F , where the former protects a sufficient amount of fermionic states from acquiring masses before EWSB to be in agreement with the SM. The interplay of the family SU(3) F also provides a unification of the high-scale Yukawa sector into a single coupling. This is in contrast to well-known SO (10) and Pati-Salam models where the Yukawa unification is constrained to the third family only (see e.g. Refs. [44][45][46][47][48][49][50][51][52][53][54][55][56]).
The Yukawa and gauge couplings unification in the SHUT model largely reduces its parameter space, making a complete analysis of its low-energy EFT scenarios technically feasible. The model also has a particular feature in that no further spontaneous breaking of the symmetry towards the SM gauge group is provided by the SUSY conserving part of the model, and that the energy scales at which the symmetry is further broken are instead associated with the soft SUSY-breaking operators. As such, both the electro-weak scale and the scales of intermediate symmetry breaking are naturally suppressed relative to the GUT scale.
In Sect. II we briefly discuss the key features of the SHUT model and its SSB scheme, and in Sect. III the high-scale SHUT model is introduced in its minimal setup in detail.
In particular, we discuss its features and the details on how it solves the longstanding problems of previous T-GUT realizations and how the GUT scale SSB in this model leads to a Left-Right (LR) symmetric SUSY theory. In Sect. IV we discuss the inclusion of soft SUSYbreaking interactions and how they lead to a breaking of the remaining gauge symmetries down to the SM gauge group, and in Sect V we present a short overview of the low-energy limits of the SHUT model. Finally, Sect. VI contains an analysis of RG evolution of gauge couplings at one loop and extraction of characteristic values of the GUT and soft scales, before concluding in Sect. VII.

A short note on notation
In this article we adopt the following notations: • Supermultiplets are always written in bold (e.g. ∆).
As usual, the scalar components of chiral supermultiplets and fermionic components of vector supermultiplets carry a tilde (e.g. ∆), except for the Higgs-Higgsino sector where the tilde serves to identify the fermion SU(2) L × SU(2) R bi-doublets (e.g. H).
• Fundamental representations carry superscript indices while anti-fundamental representations carry subscript indices.
• If a field transforms both under gauge and global symmetry groups, the index corresponding to the global one is placed within the parenthesis around the field, while the indices corresponding to the gauge symmetries are placed outside.

II. LEFT-RIGHT-COLOR-FAMILY UNIFICATION
In Glashow's formulation of the trinified [SU(3) L × SU(3) R ×SU(3) C ] Z 3 ⊂ E 6 (LRC-symmetric) gauge theory [12], three families of the fermion fields from the SM are arranged over three 27-plet copies of the E 6 group, namely, while the Higgs fields responsible for a high-scale SSB are typically introduced via e.g. an additional 27-plet. Here and below, the left, right, and color SU(3) indices are denoted by l, r, and x, respectively, while the fermion families are labelled by an index i = 1, 2, 3.
The SHUT model first presented in Ref. [39], in contrast to the Glashow's trinification, introduces the global family symmetry SU(3) F which acts in the generation-space. In this case, the light Higgs and lepton superfields, as The symmetry breaking scheme in the SHUT model studied in this work. The symmetry groups in red correspond to the accidental symmetries of the high-scale theory. The global accidental U(1) W and, consequently, its low-energy counterparts U(1) S ,T discussed below are considered to be softly broken at low-energy scales and thus are shown as crossed-out symmetry groups.
The leptonic tri-triplet superfield L i l r that unifies the SM left-and right-handed leptons and SM Higgs doublets can be conveniently represented as In addition, the SHUT model also incorporates the adjoint (namely, SU(3) L,R,C,F octet) superfields ∆ L,R,C,F . The first SSB step in the SHUT model SU(3) L,R,F → SU(2) L,R,F × U(1) L,R,F is triggered at the GUT scale by the SUSY-preserving vacuum expectation values (VEVs) in the scalar components of the corresponding octet su-perfields while all the subsequent low-scale SSB steps are triggered by VEVs in the leptonic tri-triplet L i l r through the soft SUSY-breaking operators.
Along this work, we will be focused on the symmetry breaking scheme shown in Fig. 1. There it can be seen that an accidental global U(1) B × U(1) W symmetry (which is marked in red and will be discussed in detail in the next section) appears in the high-scale theory. As we will see, although alternative breaking schemes are possible, this is the one leading to the low energy SM-like scenarios we find most interesting. As we shall see in Sec. V, dimension-3 operators that softly break U(1) W , and consequently its low-energy descendants (that will be denoted below as U(1) S ,T ), are needed for a phenomenologically viable low-scale fermion spectrum. Such interactions do not have a perturbative origin from the high-scale theory and are added to the effective theory that emerges once the heavy degrees of freedom of the SHUT model are integrated out.

III. SUPERSYMMETRIC TRINIFICATION WITH GLOBAL SU(3)F
This section contains a review of the SHUT model before and after the T-GUT symmetry is broken spontaneously by adjoint field VEVs. We here present the symmetries, particle content and interactions of the model at both stages, in addition to showing how it addresses the shortcomings of previous T-GUTs.

A. Tri-triplet sector
In the following, we consider the SHUT model -a SUSY T-GUT theory based on the trinification gauge group with an accompanying global SU(3) F family symmetry, i.e.
Here and below, curly brackets indicate global (nongauge) symmetries. The minimal chiral superfield content (shown in Tab. I) that can accommodate the SM (Higgs and fermion) fields, is comprised of three tritriplet representations of G 333{3} which we label as L, Q L and Q R respectively (for their explicit relation to the SM field content up to a possible mixing, see Eqs. (1) and (2)). The Z symmetry enforces the gauge couplings of the SU(3) L,R,C groups to unify, i.e. g L = g R = g C ≡ g U . As mentioned previously, all fields in Tab. I can be contained in a (27,3) representation of E 6 ×SU(3) F . In turn, the group E 6 × SU(3) F is a maximal subgroup of E 8 , where the (27, 3) fits neatly into the 248 irrep of E 8 whose branching rule is given by Note, for clarity, that we are only considering representations of the subgroup [SU(3)] 4 of E 8 , which are chiral rather than vector-like, in agreement with the chiral fermion content of the SM. In this work, we treat SU(3) F as a global symmetry. While considerably simpler, the trinification model with global SU(3) F can be viewed as the principal part of the fully gauged version in the limit of a vanishingly small family-gauge coupling g F g U . In that case, Goldstone bosons would become the longitudinal d.o.f of massive SU(3) F gauge bosons instead of remaining as massless scalars. Such a restricted model can thus be a first step towards the fully gauged E 8 -inspired version.
Considering only renormalizable interactions, the symmetry group G 333{3} allows for just a single term in the superpotential with the tri-triplet superfields, where λ 27 can be taken to be real without any loss of generality, as any phase can be absorbed with a field redefinition. As the light Higgs and lepton sectors are fully contained in the single tri-triplet L, this construction provides an exact unification of Yukawa interactions of the fundamental superchiral sector and the corresponding scalar quartic couplings to a common origin, λ 27 .
The superpotential in Eq. (6) has an accidental U(1) W × U(1) B symmetry as we can perform independent phase rotations on two of the tri-triplets as long as we do a compensating phase rotation on the third. We can arrange the charges of the tri-triplets under U(1) W × U(1) B as shown in Tab. II, such that U(1) B is identified as the symmetry responsible for baryon number conservation. With this, we have proton stability to all orders in perturbation theory.
The model with the superpotential in Eq. (6) also exhibits an accidental symmetry under LR-parity P. This accompanied by Here, α is the spinor index on the Grassman valued superspace coordinate θ. Note that s and t in Eq. (7) label both SU(3) L,R indices as such representations are swapped under LR-parity. At the Lagrangian level, the LR-parity transformation rules become which can be verified by expanding out the components of the superfields in Eq. (7). In this model, LRparity exists already at the SU(3) level, unlike common SU(2) L × SU(2) R LR-symmetric realisations. Note also that there exist the corresponding accidental Right-Colour and Colour-Left parity symmetries due to the Z (LRC) 3 permutation symmetry imposed in the SHUT model.
As mentioned in the introduction, one of the main drawbacks of a SUSY T-GUT (as well as any SUSY GUT with very few free parameters) is the difficulty for spontaneous breaking of high-scale symmetries. For example, while the non-SUSY T-GUT in Ref. [36] has no problem with SSB down to a LR-symmetric theory, when including SUSY the additional relations between potential and gauge couplings make it so that there is no minimum of the potential allowing for that breaking. Moreover, even when relaxing the family symmetry, any VEV in e.g. L i induces mass terms that mix the L i fermions with the gauginos λ a L,R through D-term interactions of the type This is a common problem in the previous T-GUT realizations as the number of light fields would not be enough to accommodate the particle content of the SM at low energies. While it is possible to get around this issue by adding extra Higgs multiplets to the theory and making them responsible for the SSB, this significantly increases the amount of light exotic fields that might be present at low energies but are unobserved. Such theories typically contain a very large number of free parameters and a fair amount of fine tuning which significantly reduces their predictive power.
In the SHUT model, this issue is instead solved by the inclusion of adjoint SU(3) L,R,C,F chiral supermultiplets, ∆ L,R,C,F . By triggering the first SSB, while preserving SUSY, VEVs in scalar components of ∆ L,R,F do not lead to heavy would-be SM lepton fields. In addition, the scalar and fermion components of ∆ L,R,C are all automatically heavy after the breaking and thus do not remain in the low-energy theory.

B. SU(3) adjoint superfields
The addition of gauge adjoint superfields is the main feature preventing SM-like leptons from getting a GUT scale mass. As was briefly mentioned above, the gauge and family SU(3) adjoints are motivated by the (78, 1) and (1,8) representations of E 6 × SU(3) F (which can be inspired by the branching rule of the 248-rep in its embedding into E 8 as shown in Eq. (5)). Indeed, the 78-rep, in turn, branches as 3 . We include three gauge-adjoint chiral superfields ∆ L,R,C corresponding to (8, 1, 1), (1,8,1) and (1,1,8) in Eq. (11), respectively, as well as the family SU(3) F adjoint, ∆ F (all listed in Table III). The transformation rule for the Z (LRC) 3 symmetry in G 333{3} of Eq. (3) is now accompanied by the cyclic permutation of {∆ L , ∆ C , ∆ R } fields.
In order to keep the minimal setup, in this work we will not consider the fields that correspond to 3, 3, 3 and 3, 3, 3 from Eq. (11). In practice, they can be made very heavy and only couple to the tri-triplets via gauge interactions. By introducing the adjoint chiral superfields, we have to add the following terms to the superpotential in Eq. (6).
Here, d abc = 2Tr[{T a , T b }T c ] are the totally symmetric SU(3) coefficients.
Note that bilinear terms are only present for the adjoint superfields and not for the fundamental ones, as they are forbidden by the T-GUT symmetry. Since the VEVs of the adjoint scalars set the first scale where the T-GUT symmetry is spontaneously broken, while all subsequent breaking steps occur at scales given by the soft parameters. In other words, the model is free of the so-called µ-problem.
We can pick the phase of ∆ L,R,C,F to make µ 78 and µ 1 real, which makes λ 78 and λ 1 complex, in general. Notice that the superpotential provides no renormalisable interaction terms between the adjoint superfields and the tri-triplets. The accidental U(1) W × U(1) B symmetry of the tri-triplet sector is not affected by ∆ L,R,C,F as we can take these fields simply to not transform under this symmetry. The gauge interactions are parity-invariant with the following definitions for the transformation rules, or, equivalently, ∆ a L,R,C,F P → ∆ * a R,L,C,F at the superfield level. However, LR-parity is not generally respected by the F-term interactions unless λ 78 and λ 1 are real. In what follows, we assume a real λ 78 , whereas the accidental LR-parity can be explicitly broken by the soft SUSYbreaking sector of the theory, at or below the GUT scale. Now, for illustration, let us discuss briefly the first symmetry breaking step which determines the GUT scale in the SHUT model (see Fig. 1). Eq. (12) leads to a scalar potential containing several SUSY-preserving minima with VEVs that can be rotated to the eighth component of∆ 8 L,R,F . In particular, there is an SU(3) C and LR-parity preserving minimum with for the gauge-adjoints, and for the family-adjoint, setting the GUT scale v ∼ v F . The vacuum structure ∆ 8 L,R,F = 0 leads to the spontaneous breaking SU(3) L,R,F → SU(2) L,R,F × U(1) L,R,F (see Appendix A for the corresponding generators and U(1) charges), resulting in the unbroken group LR-parity also remains unbroken since v L = v * R , which is true as long as λ 78 is taken to be real.
By making the shift and substituting µ 78 = λ78 v 2 √ 6 , µ 1 = λ1 vF 2 √ 6 in the superpotential, we obtain The quadratic terms in the superpotential vanish for ∆ 4,5,6,7 L,R,F , since d aa8 = −1/(2 √ 3) for a = 4, 5, 6, 7, meaning that these fields receive no F-term contribution to their masses (contrary to the other components of ∆ L,R and ∆ F which receive GUT scale masses m 2 ∆ ∼ λ 2 78 v 2 and λ 2 1 v 2 F , respectively). While the global Goldstone bosons Re[∆ 4,5,6,7 F ] are present in the physical spectrum, the gauge ones become the longitudinal polarisation states of the heavy gauge bosons related to the breaking G 333 → G 32211 .
The presence of massless scalar degrees of freedom can only be avoided in the extended model with the gauged family symmetry. It is clear, however, that even in the case of an approximately global SU(3) F with g F g U there are no massless Goldstones in the spectrum (provided that the accidental symmetries are softly broken at low energies) but a set of relatively light family gauge bosons very weakly interacting with the rest of the spectrum.
By performing the shifts in Eq. (17) in the D-terms, we obtain In this section we describe the details of the supersymmetric theory left after the adjoint fields acquire VEVs. As shown in the previous section, all components of the gauge adjoint chiral superfields receive masses of the order of the GUT scale (O(v)) in the vacuum given by Eq. (17). This means that to study the low-energy predictions of the theory, we need to integrate out ∆ L,R,C , as well as components 1, 2, 3 and 8 of ∆ F .
For the gauge sector of the SHUT model, ∆ L,R naturally triggers a SU(3) L,R → SU(2) L,R × U(1) L,R breaking also for the tri-triplets (whose interactions with∆ L,R are mediated via V a L,R gauge bosons). For the global SU(3) F sector, there is no coupling of∆ F to the tri-triplets and, thus, the SU(3) F symmetry remains intact (or approximate in the case of g F g U ) in the tri-triplet sector, resulting in G 32211{3} rather than G 32211{21} . Integrating out ∆ L,R,C , and components 1, 2, 3 and 8 of ∆ F , therefore leaves us with a supersymmetric theory based on the symmetry group G 32211{3} , with a chiral superfield content given by ∆ 4−7 F and by the branching of L, Q L and Q R .
Writing the trinification tri-triplets in terms of G 32211{3} representations one gets, where the vertical and horizontal lines denote the separation of the original tri-triplets into SU(2)-doublets and singlets after the first SSB step. We will refer to the lepton and quark SU(2) L,R doublets as E L,R and q L,R . With this, we find that the most general superpotential consistent with G 32211{3} is (21) Note, in this effective SUSY LR theory one could naively add a mass term like ε ijμ H i F G i F (that is symmetric under SU(2) F × U(1) F but not under full SU(3) F ) between the massless components of the family-adjoint superfield, H i F , and the massless superfield G i F containing the Goldstone bosons. Such an effective µ-term is matched to zero at tree level at the GUT scale. Due to SUSY nonrenormalisation theorems [57], in the exact SUSY limit this term cannot be regenerated radiatively at low energies soμ is identically zero and was not included in the superpotential given by Eq. (21). So, the resulting superpotential contains only fundamental superfields coming from L, Q L and Q R and is indeed invariant under In the GUT scale theory, a complex λ 78 would be the only source of LR-parity violation. In the low energy theory this should lead to y 3 = y * 4 . Otherwise, y 3 = y * 4 and after the matching is performed we can always make any y 1,2,3,4 real by field redefinitions. The same argument applies for the equality of the corresponding LR gauge couplings for SU(2) L,R × U(1) L,R symmetries.
Since we now have an effective LR-symmetric SUSY model with a U(1) L,R symmetry, there is a possibility of having gauge kinetic mixing. The U(1) L,R D-term contribution to the Lagrangian is given by where the terms proportional to κ are the Fayet-Iliopoulos terms, while the D-terms and the expressions for X L,R are shown in Appendix D 3 b.
The values of the parameters {y 1,2,3,4 , g C , g L, R , g L, R , χ, κ} in the LR-symmetric SUSY theory are determined by the values of the parameters {λ 27 , λ 78 , g U , v} in the high-scale trinification theory at the GUT scale boundary through a matching procedure 2 . Regarding the RG evolution of the couplings, we note that the only dimensionful parameter in the effective theory is the Fayet-Iliopoulos parameter κ. This means that β κ ∝ κ so that if κ = 0 at the matching scale (which is true, at least, at tree level), then κ will remain zero throughout the RG flow yielding no spontaneous SUSY-breaking. Thus, we stick to the concept of soft SUSY-breaking in what follows.

IV. SOFTLY BROKEN SUSY AT THE GUT SCALE
In this section we describe the details of adding soft SUSY-breaking terms before the SHUT symmetry is broken spontaneously by adjoint field VEVs. One of the most important results is treated in Sec. IV B, where it is shown that the symmetry breakings below the GUT scale are triggered solely by the soft SUSY-breaking sector. This in turn allows for a strong hierarchy between the GUT scale and the scale of the following VEVs.

A. The soft SUSY-breaking Lagrangian
The soft SUSY-breaking scalar potential terms respecting the imposed G 333{3} symmetry, are bilinear and trilinear interactions given by for the gauge-adjoints and pure tri-triplet terms, and 2 Before adding soft SUSY-breaking interactions, ∆ F is completely decoupled from the fundamental sector when taking SU(3) F to be global, meaning that λ 1 and v F do not enter in the matching conditions.
for the family adjoint. All parameters here are assumed to be real for simplicity. We note that although trilinear terms with the gauge singlets (such as∆ * F∆F∆F above) are not in general soft, due to the family symmetry and the fact that a d aab = 0, the dangerous tadpole diagrams do indeed cancel and do not lead to quadratic divergences.
The terms in Eq. (23) and (24), which account for the most general soft SUSY-breaking scalar potential consistent with G 333{3} and real parameters, also respect the accidental U(1) W × U(1) B symmetry of the original SUSY theory. However, accidental LR-parity is, in general, softly-broken as long as A G = AḠ, and this breaking can then be transmitted to the other sectors of the effective theory radiatively (e.g. via RG evolution and radiative corrections at the matching scale).
The only dimensionful parameters entering in the treelevel tri-triplet masses come from soft SUSY-breaking parameters, such that the corresponding scalar fields receive masses of the order of the soft SUSY-breaking scale. The full expressions are given in Appendix B, from which we notice that positive squared masses requires For more details, see Sect. B 1 a.
Note that the A F -term in the soft sector introduces small SU(3) F violating (but SU(2) F × U(1) F preserving) effects on the interactions in the effective theory once ∆ F = 0.
Consider, for example, effective quartic interactions between components ofL that come from two A F tri-linear vertices connected by an internal∆ 1,2,3 The possible fermion soft SUSY-breaking terms are the Majorana mass terms for the gauginos and the Dirac mass terms between the gauginos and the fermion components of ∆ L,R,C , namely, From the transformation rules in Eqs. (7) and (13) it follows that LR-parity is not respected by L fermion soft unless M 0 = 0.

B. Vacuum in the presence of soft SUSY-breaking terms
Here we show how the scalar potential changes in the presence of soft SUSY-breaking interactions. In particular, how soft SUSY-breaking terms trigger a VEV in L 3 3 3 ≡ φ 3 of the same order as the soft SUSY-breaking scale.
2 v ϕ being the VEVs present, our potential evaluated in the vacuum is given by As all other fields (that do not acquire VEVs) only enter in bi-linear combinations, it suffices to consider the above terms to solve the conditions for vanishing first derivatives of the scalar potential. We retain the notation v = 2 √ 6µ 78 /λ 78 for the VEVs of∆ 8 L,R in the absence of soft terms. Assuming that the soft terms are much smaller than the GUT scale, i.e. m soft v, we can approximately solve the extremum conditions for v L,R,ϕ by Taylor expanding them to the leading order in soft terms. Doing so we find where in the top equation we see that the φ 3 VEV is of the order of the soft SUSY-breaking scale. In other words, the φ 3 VEV cannot be triggered unless soft terms are introduced. As is described in Sec. IV A, the soft trilinear couplings A G,Ḡ , A 78 and C 78 need to be m 2 27 /v for having positive squared masses.
Adding the soft terms shifts the values of the VEVs v L,R described in Sec. III B by a relative amount behaving as Furthermore, we note that the presence of v ϕ slightly affects the equality of as long as A G = AḠ. The relative difference between v L,R , therefore, behaves as That is, although the VEVs of∆ L,R are shifted by the soft terms, the effect is very small, if not negligible, for m soft v.
With a non-zero v ϕ ∼ m soft v, the symmetry is further broken as where U(1) L+R consists of simultaneous U(1) L,R phase rotations by the same phase. U(1) S and U(1) S are also simultaneous U(1) L,R phase rotations, but with opposite phase, which is compensated by an appropriate U(1) F and U(1) W transformation, respectively. All generators are presented in Appendix A.
In the limit of vanishingly small A F → 0 in Eq. (24) The inclusion of soft SUSY-breaking interactions results in non-zero masses for the fundamental scalars contained in the L, Q L and Q R superfields as well as for the gauginos. By construction, the soft SUSY-breaking parameters are small in comparison to the GUT scale, i.e. m soft v, which means that the heavy states in the SUSY theory discussed in Sect. III will remain heavy and only those that were massless will receive contributions whose size is relevant for the low-energy EFT.
The masses of the fundamental scalars are purely generated in the soft SUSY-breaking sector. Furthermore, for a vacuum where only adjoint scalars acquire VEVs as in Eq. (17), there is no mixing among the components of the fundamental scalars corresponding to the physical eigenstates at the first breaking stage shown in Fig. 1.
The Higgs-slepton masses (no summation over the indices is implied) read while the corresponding squark masses are given by In Tab. VII of Appendix B we show the masses for each fundamental scalar component in the LR-parity symmetric limit corresponding to A G = AḠ, for simplicity.
Moreover, the H F mass is given by The exact expressions for scalar fields' squared masses can be found in Tab. VIII of Appendix B.
The massless superpartners of the gauge bosons associated with the unbroken symmetries also acquire soft-scale masses. In particular, they mix with the chiral adjoint fermions via Dirac-terms whose strength, M 0 in Eq. (26), is also of the order m soft . Typically, for minimal Diracgaugino models, the ad-hoc introduction of adjoint chiral superfields has the undesirable side effect of spoiling the gauge couplings' unification. However, in the model studied in Refs. [58,59], this problem is resolved by evoking trinification as the natural embedding for the required adjoint chiral scalars needed to form Dirac mass terms with gauginos. With this point in mind, we want to note that the SHUT model, with softly broken SUSY at the GUT scale, is on its own a Dirac-gaugino model and a possible high-scale framework for such a class of models.
The mass matrix for the adjoint fermions in the basis We denote the resulting mass eigenstates as The same effect is observed for the gluinosg a whose masses, in the limit M 0 ∼ M 0 v ∼ µ 78 , are equal to M 0 , for the light states, and µ 78 , for the heavy states. There is also an SU(2) F -doublet fermion H F that acquires a mass of the order of soft SUSY-breaking scale m soft . Note that H F as well as its superpartner H F receive Dterm contributions if SU(3) F is gauged. Finally, the chiral fundamental fermions are massless at this stage.

V. PARTICLE MASSES AT LOWER SCALES -A QUALITATIVE ANALYSIS
In this section we give a short overview of the low-energy limits of the SHUT model, i.e. the spectrum after φ 3 , φ 2 and ν 1 R acquires VEVs. In particular, we investigate whether the SM-extended symmetry, G SM × U(1) T × U(1) T as represented at the bottom of Fig. 1, leaves enough freedom to realise the SM particle spectrum.

A. Colour-neutral fermions
Once the SU(2) R ×SU(2) F symmetries are broken, the tridoublets H f l r and the bi-doublets h l r are split into three distinct generations of SU(2) L doublets. We will then rename them as where i = 1, 2, and where their scalar counterparts follow the same notation but without tildes. From this we can build mass terms for the charged lepton and charged Higgsinos as Let us start by classifying all possible EW Higgs doublet and complex-singlet bosons, whose VEVs may have a role in the SM-like fermion mass spectrum. There are three types of Higgs doublets distinguished in terms of their U(1) Y × U(1) T charges and one possibility for complex singlets (and their complex conjugates). In particular, we can have Note that the doublets in each line can mix, in particular, in the last line the two complex singlets emerge from the mixing φ * 1 , ν 2 R , ν 3 R → S 1 , S 2 , G s induced by the third breaking step in Fig. 1, with G s being a complex Goldstone boson 3 .
According to the quantum numbers shown in Tab. VI of Appendix A, the matrix M C has the structure where the symbols denote the type of VEV contributing to the entry. In this case, the rank of the matrix M C is at most three, which means that while we may be able to identify the correct patterns for the masses of the charged leptons in the SM, there will be massless charged Higgsinos remaining in the spectrum after EWSB, which is unacceptable. The mass terms are forbidden by the U(1) T symmetry, which remains unbroken after EWSB independent on the number of Higgs doublets involved.
In order to get a particle content consistent with the SM, one needs to break the U(1) W symmetry, thus avoiding the remnant U(1) T symmetry. The most general U(1) W violating terms after ∆ 8 L,R,F (obeying all other symmetries) are with A ijk v. The charged lepton mass matrix now reads where labels entries related to the ν 1 R VEV and can thus be well above the EW scale. We now have a mass matrix of rank-6 which means that no charged leptons and Higgsinos are left massless after EWSB. Note that before the EW symmetry is broken there are three massless lepton doublets, as the matrix in (41) with only -type entires has rank 3, in accordance with the SM. Furthermore, due to large -type entries, the structure of M C allows for three exotic lepton eigenstates heavier than the EW scale. Similarly, in the neutrino sector, no massless states remain after EWSB.
We see from the structure of Eq. (41) that, while the maximal amount of light SU(2) L Higgs doublets is nine, the minimal low-scale model needs at least two Higgs doublets, one of the -type and one of the •-type, for the rank of the matrix to remain at 6. Note also that the low-scale remnant of the family symmetry, U(1) T , is non-universal in the space of fermion generations. As such, the various generations of Higgs bosons couple differently to different families of the SM-like fermions, offering a starting point for a mechanism explaining the mass and mixing hierarchies among the charged leptons. In addition, with the only tree-level interaction among fundamental multiplets arising from the high scale term L i Q j L Q k R ijk , the masses for all leptons must be generated at loop-level, providing a possible explanation for the lightness of the charged leptons observed in nature. Whether or not the exact pattern of charged SM-like leptons masses can be obtained, remains to be seen when the RG evolution of all parameters has been carried out.

B. Quark sector
In the absence of the accidental U(1) T symmetry, the low-energy limit of the SHUT model also offers good candidates for SM quarks without exotic massless states after EWSB. To see this we first note that once φ 3 develops a VEV at the second SSB stage shown in Fig. 1, two generations of D-quarks mix and acquire mass terms of the form Then, at the third breaking stage, the ν 1 R and φ 2 VEVs trigger a mixing between the R-type quarks D i R and d i 1 0 0 0 0 0 0 a 1 a 2 0 0 0 0 a 3 a 4 0 0 0 0 0 0 a 5 a 6 a 7 0 0 0 a 8 a 9 a 10 0 0 0 a 11 a 12 a 13 where the parameters a 1 through a 13 are not all independent as the matrix is unitary. At the classical level, and with no hierarchy between φ 3 , φ 2 and ν 1 R , the parameters are given by while the corresponding expressions after RG evolution are functions of φ 3 , φ 2 , ν 1 R and Yukawa couplings, and are too extensive to be presented here.
Defining the components of the SU(2) L quark doublets as Q 1,2 L ≡ u 1,2 L , d 1,2 L T and q L ≡ u 3 L , d 3 L T , we can construct the Lagrangian for the SM-like quarks as With the different possibilities found for the Higgs sector, the most generic structure for M u and M d matrices obey the following patterns: In order for all quarks to gain a mass after EWSB, the matrices in Eq. (44) must be of rank-3. As such, the low-scale limit of the SHUT model requires, at least, two Higgs doublets, where both •-type and -type ones are present. In contrast to charged leptons, for which the contributions arise solely from effective Yukawa couplings, in Eq. (44) there are allowed tree-level bilinears for the SM-like quarks.
Next, let us consider the possible flavour structure in the low-scale limit. At the classical level, we have Cabbibo where v 1,2,d and y d , y where the Cabbibo mixing is always between a massless and a massive state at tree-level, i.e. it is valid to interpret the mixing as being between the first two generations of quarks.
With this, we have the following relations among the quark masses Note again that a realistic spectrum needs to be generated from quantum effects. The classical limit of the theory, where results in a Cabibbo angle satisfying tan θ C = v1 v2 and the quark masses being given by i.e. the lowest order contributions to the particle spectrum imply a degeneracy of charm and top quark masses, while strange and bottom quark masses squared are related with a factor 3. Higher order corrections are then of crucial importance and should be responsible for the emergence of a realistic spectrum.

VI. ESTIMATING THE SCALES OF THE THEORY
In this section we estimate the symmetry breaking scales of the model, i.e. the GUT scale ∆ 8 L,R,F ∼ v, and the intermediate scales φ 3 , φ 2 and ν 1 R , by forcing the unified gauge coupling at the GUT scale to evolve such that it reproduces the measured values of the SU(3) C × SU(2) L ×U(1) Y gauge couplings at the EW scale. This is done through a matching and running procedure, where the gauge couplings are matched at tree-level accuracy and evolved with one-loop RGEs, as a first step before matching at one-loop in future work. At each breaking scale, fermions obtaining a mass from the associated VEV are integrated out, giving rise to four intermediate energy ranges of RG evolution with different β-functions. We will refer to these regions as The symmetry alone does not dictate the structure of the scalar mass spectrum, and we will therefore have to make assumptions about what scalars are to be integrated out at each matching scale. However, by studying the extreme cases we will show that the soft SUSY-breaking scale (which we associate with the scale of the largest tri-triplet VEV, φ 3 ) is bounded from below by roughly 10 11 GeV, independently of the scalar content.
With the β-functions and matching conditions presented in Appendix C, we may set up a system of equations with three known values, the SM couplings at the Z-mass scale, and five unknown quantities, α −1 g (v), ln( φ 3 /v), ln(m Z / φ 2 ), ln( φ 2 / φ 3 ) and ln( ν 1 R / φ 2 ): with the following known parameters at the m Z scale (∼ 91.2 GeV) [60] As we have more than three unknowns, the scales cannot be solved for uniquely, but are functions of ln( φ 2 / φ 3 ) and ln( ν 1 R / φ 2 ). If we take, for example, the scenario of having no hierarchies between these three scales, we end up with the following values m soft ∼ 8.8 · 10 10 GeV, v ∼ 4.9 · 10 17 GeV, where hence the unified gauge coupling satisfies the perturbativity constraint, the GUT scale is below M Planck and the soft scale is well separated from both the GUT scale and the EW scale. Note that while the hierarchy between the GUT scale and the soft SUSY-breaking scale is stable with respect to radiative corrections, the hierarchy between the EW scale and the soft SUSY-breaking scale needs to be finely tuned.
Let us investigate whether the introduction of a hierarchy between φ 3 , φ 2 and ν 1 R can lower the soft scale φ 3 . By solving for ν 1 R in Eq. (50) and inserting all known values, we have the equation The b-values will vary depending on the scalar field content with the extreme values presented in Appendix C. To minimise the argument of the exponential (and thereby minimising the value of ν When ranging over various hierarchies using the b-values in (56), we see that the scale of ν 1 R decreases as the hierarchy between φ 2 and φ 3 increases. The soft scale φ 3 , on the other hand, is minimised when it is equal to ν 1 R , i.e. when there are no hierarchies, as shown in Fig. 2 (left), by which we conclude that Eq. (53) is in fact the optimal scenario in the sense that it provides the strongest hierarchy between the GUT scale and the soft SUSY-breaking scale. In Fig. 2 (right) we show the evolution of the gauge couplings for this scenario.
It is important to mention that these scales are obtained from gauge couplings evolved to one-loop accuracy but matched at tree-level, where one-loop matching conditions could introduce significant corrections, due to the many fields involved, as indicated in Ref. [61]. As the resulting scales could be sensitive to potentially significant threshold corrections, we are careful not to draw any strong conclusions at this point.
Furthermore, there is a possibility for lowering the soft scale by relaxing the Z 3 symmetry at the GUT scale, with gauge unification instead happening at the E 6 level. In fact, as was demonstrated in [62], a non-universal gauge coupling at the GUT breaking scale may arise from corrections to the gauge kinetic terms induced by dimension 5 operators, emerging due to higher dimensional E 6 representations. This would also open up the possibility for the emergence of new gauge bosons at, or at least close to, the TeV scale. We leave the question about a significance of such effects and its phenomenological implications for a further study.

VII. SUMMARY
Here, we would like to summarise the basic features of the LRCF-symmetric SHUT theory considered in this paper: • In contrast to previous GUT scale formulations based on gauge trinification, all three fermion generations are unified into a single (27, 3)-plet of SU(3) F × E 6 , and no copies of any fundamental E 6 reps are required for its consistent breaking down to the gauge symmetry of the SM. The considered SU(3) F × E 6 symmetry can be embedded into E 8 , motivating the addition of (1, 8) and (78, 1) multiplets corresponding to four SU(3)-octet reps. The gauge couplings are enforced to unify by means of a cyclic permutation symmetry Z 3 acting on the trinification subgroup of the LRCF-symmetry in the same way as in the Glashow's formulation.
• The chiral-adjoint sector ∆ a F = (1, 8) and ∆ a L,R,C ⊂ (78, 1) is necessary for a consistent breaking of the LRCF-symmetry down to the SM gauge symmetry in the softly-broken SUSY formulation of the theory while none of the adjoint fields remain at the EW scale. In our model, the fields developing VEVs at lower energies (the tri-triplets) happen to have the mass terms of O(m soft ), while the fields whose VEVs spontaneously break the high-scale SHUT LRCF-symmetry (the adjoints) have their GUT scale mass term in the superpoten-tial. Hence, our model does not exhibit an analogue of the µ-problem in the MSSM.
• With the first symmetry breaking being triggered at the GUT scale by VEVs in the adjoint (octet) scalars, mass terms in the fundamential (L, Q L , Q R tri-triplet) sector are forbidden. This means that the SM-like quarks and leptons remain massless until EWSB.
• In the SHUT model, all possible tree-level masses for fermions come from a single term in the superpotential, L i Q j L Q k R ijk . As we have seen, only two generations of would-be SM quarks get such contributions to their masses. As such, the model offers a starting point for a mechanism explaining the mass hierarchies of the SM, where, for example, the charged leptons are all light as they have no allowed tree-level masses and instead attain their masses radiatively (i.e. via loop-induced threshold corrections). Also, with three Higgs doublets at low energies, the model has Cabbibo quark mixing at tree-level, while radiatively generated (and RG evolved) Yukawa interactions open the possibility of reproducing the complete structure.
• The symmetry breaking scales below the GUT scale (including the EW scale) are fully determined by the dynamics of the soft SUSY-breaking interactions and are thus naturally protected from the GUT scale radiative corrections. A particularly rel-evant multi-stage symmetry breaking scheme in the SHUT theory down to the SM-like gauge effective theory has been shown in Fig. 1.
• The LRCF-symmetric theory contains an accidential U(1) B baryon symmetry, by which the proton remains stable to all orders in perturbation theory. Other accidental U(1) W and LR-parity symmetries can be (softly) broken in the low-energy EFT ensuring there being no massless charged leptons below the EWSB scale, and allowing the breaking of SU(2) R and SU(2) L symmetries at different energy scales, respectively.
• The smallest possible hierarchy between the EW scale and the soft scale, and the largest possible hierarchy between the soft scale and the GUT scale, occurs as the VEVs of φ 3 , φ 2 and ν 1 R are all put at the same scale. For this scenario, the soft scale ends up at ∼ 9 · 10 10 GeV and the GUT scale at ∼ 5·10 17 GeV. However, these numbers do not take into account potentially large one-loop threshold corrections.
Given the above properties, the SHUT model offers interesting new possibilities for deriving the structure and parameters of the SM from the GUT scale physics. This is a good motivation for investigations of this model, its multi-scale symmetry breaking patterns, loop-level matching and RG flow. Among the first natural steps would be to uncover some of the features of the simplest SM-like low-energy EFTs in a symmetry-based study without invoking the full-fledged radiative analysis of the SHUT theory. The EFT scenarios studied in this work pave the ground for further phenomenological studies of trinification based GUTs and move beyond the most common issues of such theories in the past. In this appendix we provide a summary of the SSB scheme from the high-scale GUT symmetry down to that of the SM.

Breaking path and generators
The breaking path from the GUT symmetry down to a LR-symmetric effective theory reads where global symmetries (including the accidental ones) are indicated by {· · · }. The generators of the U(1) groups after the GUT SSB are whereas after the φ 3 VEV we have with normalization factors conveniently chosen to provide integer charges for leptons and scalar bosons.
Note that, according to the discussion in Sect. IV A the LR-parity can be explicitly broken in the soft SUSYbreaking sector and is therefore absent in the effective theory.
We may also place a VEV in φ 2 and ν 1 R . In such a case the breaking scheme takes the form where the generators of U(1) Y , U(1) T and U(1) T read

Quantum numbers
In this section we present the representations and charges of the light states after each breaking step. We consider as light states all fields that are decoupled from the GUT scale after the first SSB step.
In what follows, the Higgs bi-doublets are referred to as H 1,2,3 , the singlet Higgs-lepton fields denoted as φ 1,2,3 and the lepton doublets as E 1,2,3 L,R , while the quark multiplets split up into Q 1,2,3 L,R and D 1,2,3 L,R , where Q are the 3 × 2 blocks and D the 3 × 1 blocks. The superscript 1, 2, 3 is the generation number. Whenever convenient we will adopt a simplifying notation according to where f is a family index running over the first two generations with X representing any of such SU(2) F doublets.
The minimisation conditions are then used in the Hessian matrix whose eigenvalues corresponding to the fundamental and adjoint scalar sectors are shown in Tabs. VII and VIII, respectively. Note that, for simplicity, we use the LR-symmetric case with AḠ = A G .
The branching rule for a fundamental representation of SU(3) A , A = L, R, F when it is broken down to SU(2) A × U(1) A reads where, up to an overall normalization factor, the subscripts represent the U(1) A charge. Therefore, after the SSB, the eigenstates shown in Tab. VII form representations of the G 32211{21} symmetry given in Eq. (16) and transform as singlets, doublets, bi-doublets and tridoublets under the SU(2) L,R,F symmetries, as schematically represented by the blocks in Eq. (19) 4 . The LRparity discussed in Sect. III A yields identical masses for the SU(2) L and SU(2) R eigenstates at the trinification SSB scale.
The adjoint scalars ∆ a A=L,R,F are complex octets whose branching rule is given by where the complex octet is a reducible representation while its real and imaginary parts are the irreducible representations. As such, we end up with two real triplets, two real singlets and two complex doublets and their complex conjugates after the SSB. Each broken symmetry provides four Goldstone degrees of freedom out of  VI: Field content and quantum numbers after the ν 1 R and φ 2 VEVs as in Eq. (A4). The charge for U(1) T is to be rescaled with a factor −1/6, and the charge for U(1) T with a factor −1/ √ 3.
which eight correspond to breaking of the local symmetries whereas four of them -to the global ones. While the triplet mass eigenstates, 3 0 , can be written as the two real singlets 1 0 read Finally, there are two complex doublets from the real part of ∆ a L,R,F , transforming as 2 −1 and 2 1 and two complex doublets from the imaginary part of R , e L , e 21 , H 12 , H L , u , respectively, where the subscript −1 stands for the doublet with negative T 8 eigenvalue.

a. Scalar mass spectrum
It is possible to write the minimisation conditions in a convenient way by recasting the scalar masses. In particular, the fundamental scalar masses can be collectively written as where c i 1,2 are constants with index i running over all fundamental scalar eigenstates. For simplicity, the soft SUSY-breaking parameters and the family breaking VEV can be redefined in terms of a dimensionless parameter times a common scale v as follows where, in the limit of low-scale SUSY-breaking, α 27 , σ G , σ F 1 and β ∼ O (1) such that both gauge and family SSBs occur simultaneously at the GUT scale. Eq. (B9) allows one to rewrite the scalar masses in terms of the common scale v such that ω ϕi 1. As the expression for the fundamental scalar masses contains three independent parameters, we may characterize the entire spectrum by the following three definitions where the dimensionless parameters ξ, δ and κ can span the entire spectrum by laying in the interval of 0 to 1, as the common mass scale is chosen to be the largest scale in the model, i.e. the GUT scale v. With this, we can recast the scalar mass terms in the resulting EFT as (B12) Using Eq. (B12) the general set of conditions necessary to set the positivity of the fundamental scalar mass spectrum reads Following the same procedure, we may redefine the parameters of the adjoint sector in terms of the GUT SSB scale v as follows Substituting Eqs. (B14) in Tab. VIII and, similarly to Eq. (B10), choosing where now ωφ i =H F ∼ O(1) since only H F does not contain large F-and D-term contributions. Solving the system of equations w.r.t σ 1 , τ 1 , α 1 , σ 78 , τ 78 , α 78 we obtain The scalar field components of the gauge and family adjoint sectors are treated separately. Noting that ρ F 1, the general stability condition for the masses of the family sector read Finally, the positivity conditions for the gauge sector are where we have defined λ 2 78 ≡ y > 0 and g 2 U ≡ z > 0. When conditions (B13), (B17) and (B18) are simultaneously satisfied, the tree-level vacuum of the SHUT model is stable.

Fermion Masses
The masses of the fermions that originate from the gaugeadjoint sector are somewhat more complicated. For the sake of simplicity, we use a shortened notation and show the exact expressions for the fermion masses squared in Tab. IX.

# of Weyl spinors (mass) 2
Fermionic components  In particular, we parametrize the octet masses by X 8 C , Y 8 C and Z 8 C , where the number in the superscript denotes the representation under the symmetry labeled in the subscript. The explicit form of such parameters reads The singlet and triplet fermion masses depend on the X 1,3 L,R , Y 1,3 L,R and Z 1,3 L,R parameters which are given by For the new doublet fermions, the mass eigenstates are written in terms of X 2 L,R , Y 2 L,R and Z 2 L,R which read Note that the doublets H A , which are the left-handed Weyl fermions defined to transform as 2 1 , form mass terms of the form m H A H A with H A being also the lefthanded Weyl fermions transforming as 2 −1 .

Gauge boson masses
The gauge bosons of the SU(3) C group remain massless and are identified with the SM gluons whereas the massive gauge bosons are generated upon the SSB of the SU(3) L,R symmetries. The covariant derivative of the GUT symmetry reads where G µa L are the gauge fields of the SU(3) L symmetry which cyclically transform into G µa R and G µa C by means of Z 3 -permutations. Considering the gauge-breaking VEVs ∆ c L,R = δ c 8 v, the relevant kinetic terms that couple the vector and scalar fields evaluated in the vacuum of the theory are given by Therefore, there are eight massive gauge bosons in the model which transform as complex 2 1 representations of SU(2) L,R × U(1) L,R whose charge eigenstates read with mass m 2 G = 3 4 g 2 U v 2 . In addition to the unbroken colour sector, the remaining gauge bosons are also massless at the SHUT SSB scale. In general, the one-loop β-function for a gauge coupling is given by [63] β where κ = 1/2 for Weyl fermions, C 2 (G) = N is the Casimir index, S 2 (F ) is the Dynkin index for a fermion and S 2 (S) is the Dynkin index for a complex scalar. The one-loop β-function for the gauge coupling of a U(1) theory reads where again κ is equal to 1/2 for Weyl fermions, and where Q f and Q s are, respectively, the charges for all fermions and scalars in the theory.
Rewriting the gauge couplings in terms of the inverse of the structure constants, α −1 = 4π/g 2 , the solutions of (C1) and (C2) reads where the b i -coefficients are dependent on the number of particles and respective charges of a given EFT. Below, we specify such information for each of the four regions and provide the corresponding results for the one-loop β-functions.

Region I
As discussed in Sec. IV C, all components of the fundamental scalars and fermions remain in the spectrum after the breaking of the T-GUT symmetry. In this region, the fermion sector also contains two adjoint triplets, T L,R , two adjoint singlets, S L,R and one adjoint octet in color g a . Here adjoint triplets/doublets/singlets refers to triplet/doublet/singlet representations coming from an SU ( Here g C is the gauge coupling for SU(3) C and g L,R is the gauge coupling for SU(2) L × SU(2) R .
For the U(1) L × U(1) R coupling, g L,R , the β-function is calculated using the charges in Tab. IV of Appendix A. With this we obtain b I gL,R = 9. (C5)

Region II
In region II, the adjoint scalars are integrated out, in addition to D L,R in the second and third generation, which are the only fermions able to form a Dirac mass at this stage. When it comes to the fundamental scalars, there are no clear hierarchies in the spectrum, so here we will instead present the possible extreme values.
As apparent from Eq. With this, the b-values lie in the following intervals where hence the upper bound corresponds to the maximal field content and the lower bound to the minimal field content.

Region III
In region III, the fermion spectrum remains the same, while for the scalar sector we once again investigate the extreme values. The maximal field content is still to keep all fundamental scalars, while for the minimal field content we may now remove E 2 L , as SU(2) F is broken and only E 1 L is involved in the breaking scheme down to the SM.
With this, all b-values are identical to those in region II, apart from the lower bound of b gL+R where again the upper bound corresponds to the maximal field content and the lower bound to the minimal field content.

Region IV
In region IV, the minimal field content corresponds to integrating out all scalars apart from three Higgs doublets, e.g. H 1,2 u and H 2 d and the field responsible for breaking the U(1) T symmetry, e.g. φ 1 . A minimum of two Higgs doublets are required to remain in order for all SM particles to gain a mass, while a third is needed for getting the appropriate Cabbibo mixing at tree level, as discussed in Sec. V B.
where g L is the gauge coupling for SU(2) L .
For U(1) Y , the charges in Tab. VI of Appendix A, results in where g Y is the gauge coupling for U(1) Y .

Matching conditions
The gauge couplings unification condition at the GUT scale reads with the charges in Tab. IV of Appendix A.
At the soft scale, the gauge coupling matching conditions are obtained by finding the gauge boson mass eigenstates after the VEVs φ 2 , φ 3 and ν 1 R , respectively, by expanding our old basis in terms of the new one, e.g.
at the φ 3 scale, and at the ν 1 R scale, while the matching at the φ 2 scale is trivial, . Finally, at the Z-boson mass scale, the matching conditions between the electromagnetic coupling, the hypercharge coupling and the SU(2) L coupling are already well-known where θ W is the weak mixing angle, sin 2 (θ W ) ∼ 0.2312 [60]. 5 Here, G 3 R is the gauge boson corresponding to the third generator of SU(2) R , B L,R the gauge bosons for U(1) L,R and B L+R the gauge boson for U(1) L+R .
Appendix D: Lagrangian of the LR-symmetric effective theory The field content of the EFT is derived from the mass spectrum after the T-GUT symmetry breaking. As a general rule, the light fields, i.e. those with a mass scale much smaller than the GUT scale v, are kept in the EFT spectrum whereas those with masses of the same order of magnitude as v are integrated out.
The light field components and their group transformations under the LR-symmetry obtained after v and v F VEVs (see Eq. (A1)) are shown in Tab. IV, where we use the notation given in Eq. (A6).
1. The scalar potential of the LR-symmetric effective model The scalar potential of the effective LR-symmetric theory generated after the T-GUT breaking can be summarized by where V 2 , V 3 and V 4 denote the quadratic, cubic and quartic scalar self-interactions, respectively. For simplicity, we will suppress colour indices in V LR and, for all those terms that can be written from LR-parity transformations on the fields, we will show them within square brackets as P LR [· · · ]. Note that here we use this notation for both the cases of invariance or not under LR-parity. For instance, while for the LR-parity symmetric case we should preserve the couplings, for the LR-parity broken case we should also read m →m, A →Ā, λ →λ whenever LR-parity transformation is applied.
We start by writing the scalar mass terms, whereas the trilinear interactions are expressed as Due to a large number of possible contractions of four scalar fields in the effective LR-symmetric model, we will employ a condensed notation to express the scalar quartic self-interactions. We describe below the five possible types of terms.
For the first type, which we denote "sc1", we consider terms with one reoccurring index, where we define the reoccurring index as an index possessed by all the four fields. For such a combination there are three possible contractions, out of which two of them are linearly independent. In particular, we have where colour indices are suppressed in the condensed form.
For terms with two reoccurring indices, denoted as "sc2", no matter if they are SU(2) indices or SU(3) indices 6 , there are four linearly independent contractions that read The third type involves terms with two reoccurring indices (either SU(2) or SU(3) indices) but identical fields. We denote this case as "sc3" and observe that there are only two linearly independent terms of the form Note that the case with three reoccurring indices and different fields does not exist and the only case with one reoccurring index and identical fields is the one involving the gauge singlet φ f .
Finally, the fifth type ("sc5") involves terms without reoccurring indices or terms with one reoccurring index but four identical fields such as Note that, for ease of notation, we assume that combinatorial factors were absorbed by various λ i and λ i − j .
We will then consider five different scenarios organized according to the type of index contractions as described in detail in Eqs. (D3), (D4), (D5), (D6) and (D7): The first contribution reads 6 The two types coincide since for SU(2) the three combinations reduce down to two, using that ε ij ε kl = δ k i δ l j − δ l i δ k j , while for SU(3) there are only two possible contractions to begin with, and no Levi-Civita tensor to impose a reduction.
with V gen sc1 corresponding to the interactions generated only after the matching procedure, i.e. not directly obtained by expansion of the Lagrangian of the original theory, and given by The effective quartic interactions with two reoccurring indices are given by The third contribution, which accounts for identical multiplets and two reoccurring indices, has the form V sc3 = λ 107 − 108 h * r l h l r h * r l h l r + P LR λ 101 − 102 q * l L q L l q * l L q L l while the forth scenario, where identical fields with three reoccurring indices are considered, reads Finally, for those terms that contain only one independent type of contraction we have Here, the terms generated after the breaking are

The fermion sector of the LR-symmetric EFT
The part of the Lagrangian of the effective LR-symmetric theory that involves purely quadratic fermion interac-tions as well as the Yukawa terms reads For the mass terms we have while for the Yukawa ones we write for convenience, where the first three terms, which involve only the fields from the fundamental representations of the trinification group, denote three, two and one SU(2) contractions, respectively, whereas the last ones describe the Yukawa interactions of the singlet S, triplet T and octet g a fermions.
The terms with three SU(2) contractions are given by L 3c = ε f f P LR y 1 Q f r R h l r Q f L l + P LR y 2 q r R H f l r Q f L l +y 3 Q f r R h l r Q f L l + y 4 q r R H f l r Q f L l + c.c. , those with two SU(2) contractions are written as and for those with one SU(2) contraction we have The part of the Lagrangian involving the singlets S L,R reads L S = P LR y 19 Q * l L f S L Q f L l + y 20 q * l L S L q L l + y 21 Finally, the Yukawa interactions involving gluinos are given by L g = P LR y 41 Q * l L f T a g a Q f L l + y 41 q * l L T a g a q L l +y 43 D * L f T a g a D f L + y 44 B * L T a g a B L + c.c. .

(D18)
3. The gauge sector of the LR-symmetric EFT In this section, we consider interactions involving the gauge bosons of the effective SHUT-LR model. For ease of reading, we separate those into the gauge-scalar (gs), gauge-fermion (gf) and pure-gauge (pg) interaction types,

a. Covariant derivatives and field strengths
The covariant derivatives of the LR-symmetric effective model can be written in a compact matrix form as follows where summation is assumed over each pair of repeated indices, Y A is the U(1) A hypercharge and 1 A and 1 adj A are the identity matrices with the same dimensions of the fundamental and adjoint representations, respectively. The field strength tensors of the U(1) A , SU(2) A and SU(3) C gauge symmetries are given by The U(1) L,R D-terms of the LR-symmetric theory read χ (X R − κ) + X L + κ , with f = 1, 2, 3.