Maximally Symmetric Three Higgs Doublet Model

We consider the general Three-Higgs Doublet Model (3HDM) and identify all limits that lead to exact SM alignment. After discussing the underlying symmetries that can naturally enforce such an alignment, we focus on the most economic setting, called here the Maximally Symmetric Three-Higgs Doublet Model (MS-3HDM). The potential of the MS-3HDM obeys an $\mathrm{Sp(6)}$ symmetry, softly broken by bilinear masses and explicitly by hypercharge and Yukawa couplings through renormalisation-group effects, whilst the theory allows for quartic coupling unification up to the Planck scale. Besides the two ratios of vacuum expectation values, $\tan\beta_{1,2}$, the MS-3HDM is predominantly governed by only three input parameters: the masses of the two charged Higgs bosons, $M_{h_{1,2}^{\pm}}$, and their mixing angle $\sigma$. Most remarkably, with these input parameters, we obtain definite predictions for the entire scalar mass spectrum of the theory, as well as for the SM-like Higgs-boson couplings to the gauge bosons and fermions. The predicted deviations of these couplings from their SM values might be probed at future precision high-energy colliders. The new phenomenological aspects of the MS-3HDM with respect to the earlier studied MS-2HDM are discussed.


I. INTRODUCTION
The expedition for physics Beyond the Standard Model (BSM), such as the exploration of non-standard scenarios with an extended Higgs sector, has strong theoretical and experimental motivations. The data collected from the CERN Large Hadron Collider (LHC) impose constraints on the coupling strengths of the Higgs boson, primarily on its interaction with the electroweak (EW) gauge bosons, namely the W ± and Z bosons. These LHC data show that the coupling strengths of the observed 125-GeV scalar particle must be very close to those predicted by the Standard Model (SM) [1][2][3]. This simple fact severely restricts the form of possible scalar-sector extensions of the SM [4].
The simplest nHDM is the Two-Higgs Doublet Model (2HDM). In the 2HDM, two SMlike Higgs scenarios accommodating a scalar resonance of mass ∼ 125 GeV as observed at the LHC can be realised. In these scenarios, the interaction strength of the observed scalar particle to the W ± and Z bosons can approach its SM prediction. In such a SM-aligned 2HDM, the other CP-even states do not couple to the EW gauge bosons at the tree level. In a general 2HDM without the imposition of any symmetry, exact SM alignment limit can be achieved in two different ways: (i) either by postulating for all new-physics mass scales to be sufficiently large, or (ii) by resorting to a fine-tuning among the parameters of the model [21][22][23][24][25][26][27]. This generic feature persists even within more extended frameworks, such as the Three-Higgs Doublet (3HDM) [28][29][30][31][32][33][34][35].
In this paper we analyse the general 3HDM, for which we find three distinct SM-like Higgs scenarios that possess an exact SM alignment limit. We then discuss the underlying symmetries that can naturally enforce such an alignment without the need to decouple all new-physics mass scales or to resort to ad-hoc arrangements among the parameters [13,15,19,36]. In this context, earlier studies [19,30] have shown that the potential of nHDMs contains a large number of SU(2) L -preserving accidental symmetries as subgroups of the symplectic group Sp(2n). This maximal symmetry group plays an instrumental role in classifying accidental symmetries that may occur in the scalar potentials of nHDMs and nHDM-Effective Field Theories with higher-order operators [15,19,30,37]. Thus far, this classification has been done for: (i) the 2HDM [36], (ii) the 2HDM Effective Field Theory for higher-order operators up to dimension-6 and dimension-8 [37], and (iii) the 3HDM [28,30]. Interestingly enough, there are three continuous symmetries [15] which, when imposed on the nHDM scalar potential, lead to SM alignment. In the present study, we will consider the most economic class of such scenarios, i.e. the Maximally Symmetric nHDM (MS-nHDM). In particular, we will focus on the MS-3HDM and show how one can have successful quartic coupling unification up to the Planck scale.
The present study of the MS-3HDM extends a previous work on the MS-2HDM [13,[38][39][40][41]. Like in the MS-2HDM, the Sp(6) symmetry of the MS-3HDM potential gets violated by two sources: (i) softly by bilinear scalar mass terms m 2 ij (with i, j = 1, 2, 3), and (ii) explicitly by renormalization-group (RG) effects involving the hypercharge and Yukawa couplings. The MS-3HDM is a very predictive scenario, as it only depends on fewer theoretical parameters than those in the general 3HDM. In addition to the two ratios tan β 1,2 of vacuum expectation values (VEVs) involving the three Higgs doublets, the model is mainly governed by only three input parameters: the masses of the two charged Higgs bosons, M h ± 1,2 , and their mixing angle σ. Most notably, with these input parameters, we obtain sharp predictions for the entire scalar mass spectrum of the theory, including the interactions of all Higgs particles to the SM fields and all scalar self-interactions.
In analogy to the findings in the MS-2HDM [40], we show how all quartic couplings in the MS-3HDM can unify at high-energy scales µ X and vanish identically at two distinct conformal points, called here µ (1,2) X , for which µ (1) X 10 13 GeV and µ (2) X 10 21 GeV. Assuming that all quartic couplings unify at µ (1,2) X , we present definite predictions for the full scalar mass spectrum, as well as for the SM-like Higgs-boson interactions with the W ± and Z bosons, the top-and bottom-quarks, and tau-leptons. The deviations found for all these couplings from their SM values might be testable at future high-energy e + e − colliders.
The layout of this paper is as follows. In Section II, we discuss the basic features of the general 3HDM and identify all possible SM alignment limits that can take place in this model. For definiteness, we focus on the canonical SM-like Higgs scenario in the Type-V 3HDM and derive the conditions for achieving exact SM alignment. In the same section, we present analytic expressions that describe the misalignment predictions for the SM-like Higgs boson couplings to the EW gauge bosons and SM fermions. In Section III, we define the 3HDM in the bilinear scalar-field formalism. Given that Sp(6) is the maximal symmetry of the 3HDM potential, we review the complete set of continuous maximal symmetries for the SM alignment limit that may take place in the 3HDM potential. Subsequently, we concentrate on the most minimal setting, the MS-3HDM, and clarify the origin of natural SM alignment in this model. In Section IV, we describe the Higgs-mass spectrum of the model and determine the breaking pattern of the Sp(6) symmetry due to RG effects and the soft-breaking mass terms m 2 ij . In Section V, we evaluate two-loop RG effects on all relevant couplings in the 3HDM. Specifically, we show results obtained by a RG evolution of these couplings from the quartic coupling unification points µ (1,2) X down to the threshold scale µ thr = M h ± 1 . In this analysis, we consider illustrative benchmark scenarios for the VEV ratios, tan β 1 and tan β 2 , and the kinematic parameters of the charged Higgs sector. We also present misalignment predictions for Higgs-boson couplings to the W ± and Z bosons, τ -leptons, and t-and b-quarks. Section VI summarises our results and discusses the new phenomenological aspects of the MS-3HDM with respect to the MS-2HDM. Finally, technical details, as well as the one-and two-loop RG equations pertinent to the 3HDM, are presented in Appendices A, B and C.

Yukawa Types
The five independent types of Yukawa interaction for nHDMs, with n ≥ 3 scalar doublets.
In the 2HDM, only the top four interaction types can be realised.

II. THE GENERAL 3HDM
The general 3HDM augments the SM Higgs-sector with three Higgs doublets, denoted here as where the scalar doublets Φ i (with i = 1, 2, 3) carry the same U(1) Y -hypercharge quantum number: Y Φ = 1/2. The general 3HDM potential invariant under SU(2) L ⊗ U(1) Y may be succinctly expressed as follows [47]: with λ ijkl = λ klij . Thus, the SU(2) L ⊗ U(1) Y invariant 3HDM potential contains 9 real bilinear mass terms, m 2 ij , along with 45 real quartic couplings. Moreover, the scalar potential V may realise up to 19 accidental symmetries, and this number can further increase by another 21 custodial symmetries for which only the SU(2) L group is preserved [30]. The explicit form of the 3HDM potential is given in Appendix A.
Moreover, extending the Higgs-sector of the SM with additional Higgs doublets leads generically to Flavor Changing Neutral Currents (FCNCs) at tree level, which are severely constrained by experiment. One simple way to suppress such tree-level FCNCs will be to enforce that fermions belonging to a specific family do not couple simultaneously to two or more different Higgs doublets in the Yukawa sector [42]. In the 3HDM, this condition allows for five independent types of Yukawa interactions as exhibited in Table I. More explicitly, in Type-I all fermions couple to the first doublet Φ 1 , and none to the other two doublets Φ 2,3 . In Type-II, the down-type quarks, d i L,R , and the charged leptons, e i L,R , couple to Φ 1 , and the up-type quarks, u i L,R , couple to Φ 2 , where the superscript i = 1, 2, 3 labels the three generations of fermions. In Type-III, the down-type quarks couple to Φ 1 and the up-type quarks and the charged leptons couple to Φ 2 . In Type-IV, all quarks couple to Φ 1 and the charged leptons to Φ 2 . In Type-V, the down-type quarks couple to Φ 1 , the up-type quarks to Φ 2 , and the charged leptons to Φ 3 .
In this paper, we consider the Type-V realisation of the 3HDM, where each of the Higgs doublets Φ 1,2,3 couples to one fermionic family only. Specifically, the Yukawa Lagrangian of the Type-V 3HDM reads is the hyperchargeconjugate of Φ 2 , and σ 2 is the second Pauli matrix. To endow all SM fermions with a mass at tree level, all three scalar doublets must receive a non-zero VEV, i.e. Φ 0 Here, we will restrict our attention to CP conservation and to CP-conserving vacua.
Performing the usual linear expansion of the scalar doublets Φ j (with j = 1, 2, 3) about their VEVs, we may re-express them as . (II.4) Taking into account minimization conditions on the CP-conserving 3HDM potential in (II.2) gives rise to the following relation: (II.5) where i, j, k = 1, 2, 3 and i = j = k. In (II.5), the VEVs v 1,2,3 of the three scalar doublets are given by Note that the VEVs may equivalently be determined by the two ratios, tan β 1 = v 2 /v 1 and tan β 2 = v 3 / v 2 1 + v 2 2 , given the constraint that v ≡ v 2 1 + v 2 2 + v 2 3 is the VEV of the SM Higgs doublet.
For later convenience, we now define the three-dimensional rotational matrices about the individual axes z, y and x as follows: as well as the two-dimensional rotational matrix Our first step is to transform all scalar fields from a weak basis with generic choice of vacua to the so-called Higgs basis [21,23], where only one Higgs doublet acquires the SM VEV v. This can be achieved by virtue of a common orthogonal transformation that involves the two mixing angles β 1 and β 2 , i.e.
with O β ≡ R 13 (β 2 )R 12 (β 1 ). After spontaneous symmetry breaking (SSB), the Z and W ± gauge bosons become massive after absorbing in the unitary gauge the three would-be Goldstone bosons G 0 and G ± , respectively [43]. Consequently, the model has nine scalar mass eigenstates: (i) three CP-even scalars (H, h 1,2 ), (ii) two CP-odd scalars (a 1,2 ), and (iii) four charged scalars (h ± 1,2 ). Our next step is therefore to determine the composition of the above scalar mass eigenstates in terms of their respective weak fields in the Higgs basis. This can be done by the orthogonal transformations In the Higgs basis, spanned by {η 1,2 } and {η ± 1,2 }, the CP-odd and charged scalar mass matrices reduce to the 2 × 2 matrices given by Finally, the masses for the three CP-even scalars, H, h 1 and h 2 can be evaluated by diagonalising the squared mass matrix M 2 S , expressed in the general weak basis {φ 1,2,3 }, by employing the orthogonal matrix O defined in (II.11). In this general weak basis, M 2 S reads where the analytic form of all its entries is given in Appendix B 3. Therefore, the diagonal mass-basis matrix M 2 S for the physical CP-even scalars takes on the form with the convention of mass ordering: In order to determine all possible limits of SM alignment, it proves convenient to perform a diagonalisation of the CP-even mass matrix M 2 S in two steps. In the first step, we use two angles, e.g. α 1,2 , to project out the mass eigenvalue of the SM-like Higgs state. For instance, if H is identified with the observed SM-like Higgs boson, then M 2 S may be block-diagonalised as follows: with O α = R 13 (α 2 )R 12 (α 1 ). Here, the mixing angles α 1,2 are given in terms of the massmatrix elements in (II.16) by where C 3 /C 2 = tan α 1 . The second step of diagonalisation consists in bringing the matrix M 2 S in a fully diagonal form. This can be done by using the orthogonal transformation R 23 (α), where the squared masses of the two heavy CP-even scalars, h 1,2 , are Thus, the mixing angle α can be evaluated in terms of the above mass parameters as (II. 23) In this CP-conserving 3HDM, the SM-normalised couplings of the SM-like Higgs boson to the EW gauge bosons (V = Z, W ± ) are calculated to be: g HV V = cos α 2 cos β 2 cos(β 1 − α 1 ) + sin α 2 sin β 2 , (II. 24) g h 1 V V = cos α cos β 2 sin(β 1 − α 1 ) + sin α cos α 2 sin β 2 − cos β 2 sin α 2 cos(β 1 − α 1 ) , (II.25) which obey the sum rule: g 2 HV V + g 2 h 1 V V + g 2 h 2 V V = 1. Evidently, there are three possible scenarios for which the 125-GeV resonance can identified with the SM-like Higgs boson. The first one is the so-called canonical SM-like Higgs scenario, where M H ≈ 125 GeV with coupling strength g HV V = 1, but g h 1,2 V V = 0. In this case, there are two possible arrangements for the mixing angles: The second possibility is that the h 1 boson represents the observed scalar resonance at 125 GeV, with g h 1 V V = 1. In this case, the H scalar is lighter than the h 1 , whereas the h 2 is heavier. For this SM-like h 1 scenario, we have four possibilities to arrange the mixing angles: (i) β 1 = α 1 , β 2 = α 2 ± π/2, α = ±π/2 , The last possibility is the option that h 2 represents the observed SM-like Higgs boson, with g h 2 V V = 1. Here, the H and h 1 scalars are lighter than the SM-like h 2 boson, with vanishing couplings to the EW gauge bosons. This scenario may be achieved for the following four choices of mixing angles: (i) β 1 = α 1 , β 2 = α 2 + π/2, α = 0, (ii) β 1 = α 1 + π/2, β 2 = α + π/2, α 2 = 0, (iii) β 1 = α 1 ± π/2, β 2 = 0, α = ∓π/2, (iv) β 2 = π/2, α 2 = α = 0 .
(II. 29) In this paper, we consider the canonical SM-like H scenario, in which SM alignment is obtained when β 1 = α 1 and β 2 = α 2 , according to the option (i) in (II.27).
In the Higgs basis {H 1,2,3 }, the 3 × 3 mass matrix of the CP-even scalars is given by where we employed the shorthand notation: c x ≡ cos x and s x ≡ sin x. In the SM alignment limit β 1 = α 1 and β 2 = α 2 under consideration, the mass parameter A becomes equal to M 2 H , while the parameters C 1 and C 2 vanish. Thus, taking the limit C 1,2 → 0, the following relationships between the quartic couplings may be derived: while the remaining quartic couplings are zero. These conditions may be obtained independently of any values for the mixing angles β 1 and β 2 . Moreover, for a small but calculable value of misalignment, the deviations of β 1 − α 1 and β 2 − α 2 from zero can be parameterised as follows: where relations analogous to (II. 19) and (II.20) have been used for the hat quantities occurring in the CP-even mass matrix of (II.30).
Having determined the mixing angles in terms of the matrix elements of M 2 S , we may now calculate the reduced H-boson couplings to the EW gauge bosons in a power expansion of C 1,2 / B 1,2 . To order C 2 1,2 / B 2 1,2 , these are given by the following approximate analytic expressions: Likewise, the SM-normalised couplings of the CP-even scalars to up-type, down-type quarks and charged leptons are To order C 2 1,2 / B 2 1,2 , the SM-normalised couplings of the H boson to fermions are dictated by the following approximate analytic formulae: We note that the analytic expressions of the reduced H-boson couplings in (II.35) and in (II.41)-(II.43) go to the SM value 1, when the exact SM alignment limit C 1,2 → 0 is considered, or when the new-physics mass scales B 1,2 ∼ M 2 h 1,2 are taken to infinity. Since we are interested in the former possibility which in turn implies a richer collider phenomenology, we will study scenarios in which SM alignment is accomplished by virtue of symmetries.
In the next section, we will discuss restrictions on the model parameters of the 3HDM that emanate from SM alignment, and identify possible maximal symmetries that can be imposed on the 3HDM potential to fulfill these restrictions.

III. SYMMETRIES FOR SM ALIGNMENT IN 3HDM
After outlining the basic features of the 3HDM in the previous section, we now review the key symmetries for SM alignment for this class of models [15]. To this end, we introduce the 12-dimensional SU(2) L -covariant Φ-multiplet [30,52], are the U(1) Y hypercharge-conjugates of Φ i and σ 1,2,3 are the Pauli matrices. Moreover, the Φ-multiplet satisfies the Majorana-type property [52], is the charge conjugation operator and 1 3 is the 3 × 3 identity matrix. With the help of the Φ-multiplet, one may now define the bilinear field vector [30], The Σ A matrices have 12 × 12 elements and can be expressed in terms of double tensor products as follows: where t a S and t b A stand for the symmetric and the anti-symmetric generators of the SU(3) group, respectively. Requiring that the canonical SU(2) L gauge-kinetic terms of the scalar doublets Φ 1 , Φ 2 and Φ 3 remain invariant, then the maximal symmetry of the 3HDM is the symplectic group Sp(6) for vanishing hypercharge gauge coupling g and fermion Yukawa couplings. In an earlier study [30], the full classification of all 40 accidental symmetries for the 3HDM has been presented. This classification is obtained by working out all distinct subgroups of the maximal symmetry group G Φ 3HDM = [Sp(6)/Z 2 ] ⊗ SU(2) L in a bilinear fieldspace formalism. From these 40 symmetries, only a few possess the desirable property of natural SM alignment.
In the 3HDM, the constraints required for SM alignment can be realised naturally by virtue of three symmetries [15,19]: (i) the maximal symmetry group Sp(6), (ii) SU(3) HF , and (iii) SO(3) HF × CP symmetries. Here, the abbreviation HF indicates Higgs Family symmetries that only involve the elements of Φ = (Φ 1 , Φ 2 , Φ 3 ) T and not their complex conjugates. The construction of the potential invariant under these symmetries is facilitated by an earlier developed technique based on prime invariants [15,19,30].
The MS-3HDM potential can be constructed by means of the Sp(6)-invariant expression, In particular, it takes on the following minimal form: and all other potential parameters are set to zero.
The SU(3) HF -invariant 3HDM potential may be expressed as a function of S, given in (III.5), and the SU( with a = 1, 2, 3 and Φ = Φ 1 , Φ 2 , Φ 3 T . Hence, the model parameters of the SU(3) HFinvariant 3HDM potential satisfy the following relations: In this case, the following relationships among the potential parameters may be derived: and all other parameters not listed above vanish.
In summary, if a 3HDM potential is invariant under one of the three symmetries: Sp(6), SU(3) HF and SO(3) HF ×CP, then its parameters will satisfy the alignment conditions stated in (II.32), for any value of the mixing angles β 1 and β 2 . To render such constrained 3HDM potentials phenomenologically viable, we may have to include arbitrary soft symmetrybreaking bilinear masses, m 2 ij (with i, j = 1, 2, 3). However, such soft symmetry breakings do not spoil SM alignment [15,19]. Obviously, the MS-3HDM is the most economic setting that realises naturally such an alignment from the general class of 3HDMs. In particular, we find that in the MS-3HDM, the mixing angles that diagonalise the heavy sectors of the CP-even, CP-odd and charged-scalar mass matrices are all equal, i.e. (III.11) As will see in the next sections, this is a distinct and unique feature of the MS-3HDM under study with respect to other aligned 3HDMs.

IV. BREAKING PATTERN OF THE TYPE-V MS-3HDM
In this section, we will discuss the breaking pattern of the Type-V MS-3HDM, where each Higgs doublet Φ 1,2,3 couples to one type of fermions only, as given in Table I. In particular, we notice that after SSB and in the Born approximation, the MS-3HDM predicts one CP-even scalar H with non-zero squared mass M 2 H = 2λ 22 v 2 , while all other scalars, h 1,2 , a 1,2 and h ± 1,2 , are massless pseudo-Goldstone bosons with sizeable gauge and Yukawa interactions. Hence, as we will see below, we need to consider two dominant sources that violate the Sp(6) symmetry of the theory: (i) the RG effects of the gauge and Yukawa couplings on the potential parameters, and (ii) soft symmetry-breaking bilinear masses, so as to make all the pseudo-Goldstone fields sufficiently heavy in agreement with current LHC data and other low-energy experiments.
To start with, let us first consider the effects of RG evolution, within a perturbative framework of the MS-3HDM. Given that physical observables, such as S-matrix elements, e.g. S(µ, g(µ), m(µ)), are invariant under changes of the RG scale µ, one derives the wellknown fundamental relation: In the above, the generic coupling constant g and mass parameter m 2 satisfy typical RG equations: where β(g) and γ m 2 (g) are the beta function and the anomalous dimension, respectively. As explicitly indicated, β(g) and γ m 2 (g) depend only on g in the Minimal-Subtraction (MS) scheme. In Appendix C we present the complete set of one-and two-loop beta functions of all gauge, Yukawa and quartic couplings, for the general 3HDM, including beta functions for potential mass parameters and the VEVs of the scalar doublets.
Although the aforementioned RG effects generate masses for the CP-even and charged pseudo-Goldstone bosons, h 1,2 and h ± 1,2 , the corresponding CP-odd pseudo-Goldstone bosons, a 1,2 , remain massless. The latter two states play the role of axions thanks to the presence of two Peccei-Quinn (PQ) symmetries U(1) PQ ⊗ U(1) PQ [54][55][56]. However, these visible EW-scale axions have strong couplings to gauge and SM fermions, and as such, they are ruled out experimentally. For this reason, we admit a second source for violating the Sp(6) symmetry in the MS-3HDM, through the introduction of soft Sp(6)-breaking bilinear mass terms m 2 ij . As a consequence, the complete MS-3HDM potential is given by After EW symmetry breaking, the diagonal mass terms m 2 ii may be eliminated in favour of the VEVs v 1,2,3 and the off-diagonal terms m 2 i =j (with i, j = 1, 2, 3), according to (II.5). Hence, to a very good approximation, the scalar masses are found to be: Notice that with the introduction of soft symmetry-breaking bilinears, all pseudo-Goldstone bosons, h 1,2 , a 1,2 and h ± 1,2 , receive appreciable masses at the tree level. To sum up, we consider an Sp(6) symmetry imposed on the MS-3HDM potential which is exact (up to soft symmetry-breaking masses) at some high-energy scale µ X , at which quartic coupling unification occurs. Then, the following breaking pattern will generally take place for the Type-V MS-3HDM: In our analysis detailed in the next section, we take charged scalar masses M h ± 1,2 > ∼ 500 GeV as input parameters, while maintaining agreement with B-meson constraints [59]. For the benchmark scenarios that we will be studying, we will have M h ± 1 ∼ M h ± 2 , such that we can ignore any RG effects between the two charged scalar masses, and so perform a matching of the MS-3HDM to the SM at a single threshold.

V. QUARTIC COUPLING UNIFICATION
In this section we will study quartic coupling unification in the Type-V 3HDM. In analogy to a previous analysis for the MS-2HDM [40], we find that in the MS-3HDM all quartic couplings can unify to a single value λ at very high-energy scales µ X . As the highest scale of unification, we take the values at which λ(µ X ) = 0. Like in the MS-2HDM, we will see that there are two such conformally invariant unification points in the MS-3HDM which we distinguish them as µ In our analysis we employ two-loop Renormalization Group Equations (RGEs) to evaluate the running of all relevant MS-3HDM parameters from the unification point µ X to the threshold mass scale Below µ thr , we assume that the SM is a good effective field theory, and as such, we use the two-loop SM RGEs given in [57] to match the relevant MS-3HDM couplings to the corresponding SM quartic coupling λ SM , the Yukawa couplings, and the SU(2) L and U(1) Y gauge couplings: g 2 and g . Unless stated explicitly otherwise, we use the following baseline model for our analysis, where all input parameters are given at the RG scale µ = M h ± 1 : In addition, the values of the two-loop SM couplings at different threshold scales, µ thr = m t , M h ± 1 , are determined in a fashion similar to [40]. In Figure 1, we display the RG evolution of all quartic couplings for the benchmark model of (V.1), with a low charged Higgs mass M h ± 1 = 500 GeV. We observe that the quartic coupling λ 22 , which determines the SM-like Higgs-boson mass M H , decreases at high RG scales, due to the running of the top-Yukawa coupling y t . The coupling λ 22 turns negative just above the quartic coupling unification scale µ (1) X ∼ 10 13 GeV, at which all quartic couplings vanish. Below µ (1) X , the MS-3HDM quartic couplings exhibit different RG runnings, and especially the couplings λ ijij (with i = j and i, j = 1, 2, 3) take on non-zero values. Also, for energy scales above µ (1) X , we understand that the MS-3HDM has to be embedded into a higher-scale UV-complete theory. Nevertheless, according to the results from [40], such a potential leads to a metastable but sufficiently long-lived EW vacuum, whose lifetime is many orders of magnitude longer than the age of our Universe. For this reason, we do not apply the usual, over-restrictive constraints derived from positivity conditions on the scalar potential that would imply an absolutely stable EW vacuum.
In addition to the conformal unification point µ (1) X , there is in general a second and higher conformal point µ (2) X ∼ 10 21 GeV in the MS-3HDM. This is exemplified in Figure 2. This higher conformal point occurs, when the quartic coupling λ 22 increases at high RG scales and crosses zero for the second time. This happens when the running of the top-Yukawa coupling overtakes that of the gauge coupling. Obviously, in such a theoretical setting, any embedding of the MS-3HDM into a candidate UV-complete theory must primarily include UV aspects of quantum gravity.  the MS-3HDM becomes degenerate at the unification point µ X , clustering about the two different charged Higgs masses, M h ± 1 = 500 GeV and M h ± 2 = 525 GeV, with a mixing angle σ [rad] = 0.012 for the h ± 1 -h ± 2 system. As shown in Figure 3, RG effects will break these mass degeneracies from a few MeV, for M h 1 − M a 1 and M h 2 − M a 2 , up to about 30 GeV, for M h ± 2 − M a 2 . Figures 4 and 5 show all conformally-invariant quartic coupling unification points in the (tan β 1 , log 10 µ) and (tan β 2 , log 10 µ) planes, by considering different values of threshold scales µ thr , i.e. for µ thr = M h ± 1 = 500 GeV, 1 TeV and 10 TeV. In both figures, the lower curves (dashed curves) correspond to sets of low-scale quartic coupling unification points, while the upper curves (solid curves) give the corresponding sets of high-scale unification points. From Figures 4 and 5, we may also observe the domains in which the λ 22 coupling becomes negative. Evidently, as the threshold scale µ thr = M h ± 1 increases, the size of the negative λ 22 domain increases. This becomes more pronounced for larger values of tan β 1 and smaller values of tan β 2 . So, one may obtain lower and upper bounds on tan β 1 and tan β 2 for this unified theoretical framework at different threshold scales µ thr . For example, if µ thr = 500 GeV, we may deduce the bounds: 40 tan β 1 55 and 0.013 tan β 2 0.022.
We have already seen how at the conformal points, µ (1) X and µ (2) X , all quartic couplings vanish simultaneously, leading to an exact SM alignment. Nevertheless, for lower RG scales, the Sp(6) symmetry is broken, giving rise to calculable non-zero misalignment predictions. As was discussed in Section II, this misalignment can be derived using our analytic expressions given in (II.35). In Figure 6, we present our numerical estimates of the predicted deviations for the SM-like Higgs-boson coupling HXX (with X = W ± , Z, t, b, τ ) from its respective SM value. Specifically, Figure 6 exhibits the dependence of the misalignment parameter |1 − g 2 HXX | (with g H SM XX = 1) as functions of the RG scale µ, for both the low-and high-scale quartic coupling unification scenarios. Evidently, the deviation of the normalised coupling g HXX from its SM value gets larger for the higher-scale unification scenario. Moreover, the degree of misalignment for g Hbb and g Hτ τ approach values larger than 10%. The ATLAS and CMS data [58] for g Hbb = 0.49 +0.26 −0.19 and g Hbb = 0.57 +0.16 −0.16 , which can be fitted to the SM at the 3σ level, reduces only to 2σ in the MS-3HDM in the high-scale unification M h ± 1 = 5 0 0 G e V t a n β 1 = 5 0 t a n β 2 = 0 . 0 1 8 Numerical estimates of the misalignment parameter |1 − g 2 HXX | pertinent to the HXXcoupling (with X = V, t, b, τ and V = W ± , Z) as functions of the RG scale µ, for the low-scale and the high-scale quartic coupling unification scenarios, considering M h ± scenario. We observe that the normalised couplings, g HV V and g Htt , approach their SM values g H SM V V = g H SM tt = 1 at the two quartic coupling unification points, µ (1) X and µ (2) X .

VI. CONCLUSIONS
We have analysed the basic low-energy structure of the general 3HDM. We have found that this model can realise three distinct SM-like Higgs scenarios of SM alignment. Our study was focused on the canonical SM-like Higgs scenario of the Type-V, for which conditions on the model parameters for achieving exact SM alignment were derived. Interestingly enough, there are three continuous symmetries which, when imposed on the nHDM scalar potential, are sufficient to ensure SM alignment. These are: (i) Sp(6), (ii) SU(3) HF , and (iii) SO(3) HF . Amongst these symmetries, the most economic setting is the Maximally Symmetric Three-Higgs Doublet Model (MS-3HDM), whose potential obeys an Sp(6) symmetry. The Sp(6) symmetry is softly broken by bilinear masses m 2 ij (with i, j = 1, 2, 3), as well as explicitly by hypercharge and Yukawa couplings through RG effects, whilst the theory allows for quartic coupling unification up to the Planck scale.
The MS-3HDM is a remarkably predictive scenario, as it only depends on a few theoretical parameters when compared to the large number of independent parameters that are required in the general 3HDM. In fact, besides the ratios of the Higgs-doublet VEVs, tan β 1,2 , the model is mainly governed by only three input parameters: the masses of the two charged Higgs bosons, M h ± 1,2 , and their mixing angle σ. Most notably, with the help of these input parameters, we have obtained misalignment predictions for the entire scalar mass spectrum of the theory, including the interactions of all Higgs particles to the SM fields.
We have presented the one-and two-loop RG equations pertinent to the general 3HDM. We have used these to evaluate the two-loop RG effects on all relevant couplings in the MS-3HDM. In particular, we have shown that all quartic couplings in the MS-3HDM can unify at high-energy scales µ X and vanish simultaneously at two distinct conformal points, that are denoted by µ (1,2) X with µ (1) X 10 13 GeV and µ (2) X 10 21 GeV. These limits have been obtained by considering the RG evolution of the quartic couplings from the unification points µ (1,2) X down to the threshold scale µ thr = M h ± 1 = 500 GeV, 1 TeV and 10 TeV. For our analysis, we considered a typical benchmark scenario for the VEV ratios, tan β 1,2 , and the kinematic parameters of the charged Higgs sector. We have obtained misalignment predictions for the Higgs-boson couplings to the W ± , the Z bosons, and t-quarks. These turned out to be very close to their SM value, g HZZ = g Htt = g Hτ τ = 1, and so they are in excellent agreement with the current LHC observations. On the other hand, the LHC data measuring the strength of the Hbb-coupling, g exp Hbb , differs from its SM value by 3σ, but this deviation reduces significantly to less than 2σ in the MS-3HDM.
The present results for the MS-3HDM, along with an earlier study of the MS-2HDM [40], demonstrate the high predictive power of maximally symmetric settings in nHDMs. Such settings not only can naturally provide the experimentally favoured SM alignment, but also allow us to obtain sharp predictions for the entire scalar mass spectrum of the theory, including the interactions of all Higgs particles to the SM fields and all scalar self-interactions. This fact opens up new interesting theoretical vistas that merit detailed exploration in the near future. We therefore plan to return to dedicated investigations of the Higgs selfinteractions in this framework that may be probed via multi-Higgs and charged Higgs production events at the LHC and future high-energy colliders.

Appendix A: The 3HDM Potential
The most general 3HDM potential invariant under SU(2) L ⊗ U(1) Y may be explicitly expressed in terms of three Higgs doublets Φ i (i = 1, 2, 3) as follows: where λ ii ≡ λ iiii /2. Furthermore, assuming a CP-conserving 3HDM potential the following three minimisation conditions can be obtained: As discussed in Section II, there are nine physical Higgs states in 3HDM: (i) three CPeven scalars (H, h 1,2 ), (ii) two CP-odd scalars (a 1,2 ), and (iii) four charged scalars (h ± 1,2 ). The details for deriving the masses of these particles are given in Subsections B 1, B 2 and B 3, respectively.

Appendix C: Renormalization Group Equations for 3HDM
In this section, we present the complete set of one-and two-loop beta functions of all gauge, Yukawa and quartic couplings, for the Type-V 3HDM, including beta functions for potential mass parameters and the VEVs of the scalar doublets. These are presented in Subsections C 1, C 2, C 3, C 4 and C 5. To this end, we use the following conventions [60]: where β (1) and β (2) refer to the one-and two-loop RGEs, respectively.

One-and Two-Loop RGEs of Gauge Couplings
The one-and two-loop RGEs of the gauge couplings take on the following forms: Tr y e y † e . (C.5)

One-and Two-Loop RGEs of Yukawa Couplings
The one-and two-loop RGEs of the Yukawa couplings can be given as follows: Tr y u y † u y u y † u y d − 12λ 11 y d y † d y d − 2λ 1122 y u y † u y d + 2λ 1221 y u y † u y d + 6λ 2 11 y d + λ 2 1122 y d + λ 1122 λ 1221 y d + λ 2 1221 y d + λ 2 1133 y d Tr y u y † u y u y † u y u −