Quartic Coupling Unification in the Maximally Symmetric 2HDM

We consider the Maximally Symmetric Two-Higgs Doublet Model (MS-2HDM) in which the so-called Standard Model (SM) alignment can be naturally realised as a consequence of an accidental SO(5) symmetry in the Higgs sector. This symmetry is broken (i) explicitly by renormalization-group (RG) effects and (ii) softly by the bilinear scalar mass term $m^2_{12}$. We find that in the MS-2HDM all quartic couplings can unify at large RG scales $\mu_X \sim 10^{11}\,$-$\,10^{20}$ GeV. In particular, we show that quartic coupling unification can take place in two different conformally invariant points, where all quartic couplings vanish. We perform a vacuum stability analysis of the model in order to ensure that the electro-weak vacuum is sufficiently long-lived. The MS-2HDM is a minimal and very predictive extension of the SM governed by only three additional parameters: the unification scale $\mu_X$, the charged Higgs mass $M_{h^{\pm}}$ (or $m^2_{12}$) and $\tan\beta$, which allow one to determine the entire Higgs sector of the model. In terms of these input parameters, we present illustrative predictions of misalignment for the SM-like Higgs-boson couplings to the $W^\pm$ and $Z$ bosons and, for the first time, to the top and bottom quarks.


I. INTRODUCTION
Despite intense scrutiny, the Standard Model (SM) has proven to be very successful in describing the fundamental interactions of Particle Physics [1][2][3]. The discovery of the Higgs particle [4,5] at the CERN Large Hadron Collider (LHC) was one of the most important achievements towards a minimal ultra-violet (UV) completion of the SM [6,7]. In spite of its great success, the search for new physics beyond the SM still has strong theoretical and experimental motivations, and opens up new possibilities including the study of nonstandard scenarios with extended Higgs sectors. However, latest LHC data dictate that the observed Higgs boson must interact with the electroweak (EW) gauge bosons (Z, W ± ) with coupling strengths that are very close to their SM values [8,9]. This simple fact puts severe limits on possible scalar-sector extensions of the SM.
One class of minimal extensions of the SM is the Two-Higgs Doublet Model (2HDM), where the SM scalar sector is extended by a second Higgs doublet [10,11]. This model can, in principle, account for a SM-like Higgs boson, and contains additionally one charged and two neutral scalars whose observation could be within reach of the LHC [12,13].
In the 2HDM, one may have four different types of Yukawa interactions mediating no Flavor Changing Neutral Currents (FCNCs) at the tree level [14][15][16][17]. More explicitly, in Type-I (Inert-type), all the fermions couple to the first doublet Φ 1 and none to the second doublet Φ 2 . In Type-II (MSSM-Type), the down-type quarks and the charged leptons couple to Φ 1 and the up-type quarks couple to Φ 2 . In Type-III (flipped-model), the down-type quarks couple to Φ 1 and the up-type quarks and the charged leptons couple to Φ 2 . In Type-IV (lepton-specific-model), all quarks couple to Φ 1 and the charged leptons couple to Φ 2 . As mentioned above, in all these different settings of the 2HDM, the couplings of the SM-like Higgs boson to the electroweak gauge bosons (Z, W ± ) must be very close to those predicted by the SM, so as to be in agreement with the current Higgs signals at the LHC. This is only possible within the so-called SM alignment limit of the 2HDM [18][19][20][21][22][23][24][25]. In particular, in the Type-II 2HDM, the couplings of the SM-like Higgs boson to vector bosons lie within 10% of the SM value at 95% CL [26][27][28].
In this paper, we consider the simplest realisation of a Type-II 2HDM, the so-called Maximally Symmetric Two-Higgs Doublet Model (MS-2HDM). In this model, the aforementioned SM alignment can emerge naturally as a consequence of an accidental SO (5) symmetry in the Higgs sector [22,[29][30][31], without resorting to ad-hoc arrangements among the parameters of the theory [21,[32][33][34][35][36]. The SO(5) symmetry is broken explicitly by two sources: (i) by renormalization-group (RG) effects and (ii) softly by the bilinear scalar mass term m 2 12 . A remarkable feature of the MS-2HDM is that all quartic couplings can unify at very large scales µ X ∼ 10 11 -10 20 GeV, for a wide range of tan β values and charged Higgs-boson masses. In particular, we find that quartic coupling unification can happen in two different conformally invariant points, where all quartic couplings vanish. The first conformal point is at relatively low-scale typically of order 10 11 GeV, while the second one is at high scale close to the Planck scale ∼ 10 19 GeV. Most remarkably, we find that the MS-2HDM becomes a very predictive extension of the SM which is governed by only three additional parameters: the quartic coupling unification scale µ X , the charged Higgs mass M h ± (or m 2 12 ) and tan β. These three parameters also suffice to determine the entire Higgsmass spectrum of the model. By means of these input parameters, we are able to obtain definite predictions of misalignment for the SM-like Higgs-boson couplings to the W ± and Z bosons and to the top-and bottom-quarks, which might be testable at future precision high-energy colliders.
The layout of the paper is as follows. After this introductory section, Section II briefly reviews the basic features of the 2HDM and discusses the conditions for achieving exact SM alignment. In Section III, we describe the MS-2HDM, thereby illuminating the origin of natural SM alignment. We also outline the breaking pattern of the SO(5) symmetry, which results from the soft-breaking mass m 2 12 and the RG effects, and discuss its implications for the Higgs-mass spectrum. In Section IV, we analyse in more detail the impact of RG effects up to two loops on all relevant quartic couplings by considering their running from the quartic coupling unification scale µ X to the charged Higgs-boson mass. In particular, we show that the running quartic couplings can be unified at two different conformally-invariant points. In the same section, we present illustrative predictions for the unification of all quartic couplings for typical values of tan β and charged Higgs-boson masses. Section V presents our misalignment predictions for Higgs-boson couplings to gauge bosons and top-and bottomquarks. In Section VI, we analyse the EW vacuum lifetime of the MS-2HDM by considering the bounce solution to the Euclidean equation of motion for the classical potential, with a negative running quartic coupling λ 2 . We estimate the EW vacuum lifetime τ which turns out to be adequately long, being many orders of magnitude larger than the age of the Universe (in the absence of Plank-scale suppressed operators [37,38]). Finally, Section VII contains our conclusions.

II. TYPE-II 2HDM AND SM ALIGNMENT
The Higgs sector of the 2HDM is described by two scalar SU(2) doublets, In terms of these doublets, the most general SU(2) L ⊗U(1) Y -invariant Higgs potential is given by where the mass term m 2 12 and quartic couplings λ 5 , λ 6 and λ 7 are complex parameters. Instead, the remaining mass terms, m 2 11 and m 2 22 , and the quartic couplings λ 1,2,3,4 are real.
Of these 14 theoretical parameters, only 11 are physical, since three can be removed away using a SU(2) reparameterisation of the Higgs doublets Φ 1 and Φ 2 [34].
In the present article, we will restrict our attention to CP conservation and to CPconserving vacua. In the Type-II 2HDM, both scalar doublets Φ 1 and Φ 2 receive nonzero vacuum expectation values (VEVs). Specifically, we have φ 0 is the VEV of the SM Higgs doublet. The minimization conditions resulting from the 2HDM potential in (II.2) give rise to the following relations: where s β ≡ sin β, c β ≡ cos β, t β ≡ tan β = v 2 /v 1 and λ 345 ≡ λ 3 + λ 4 + λ 5 . Following the standard linear expansion of the two scalar doublets Φ j (with j = 1, 2) about their VEVs, we may conveniently re-express them as .

(II.5)
After spontaneous symmetry breaking (SSB), the standard EW gauge fields, the W ± and Z bosons, acquire their masses from the three would-be Goldstone bosons (G ± , G 0 ) [39]. As a consequence, the model has only five physical scalar states: two CP-even scalars (h,H), one CP-odd scalar (a) and two charged bosons (h ± ). The mixings in the CP-odd and charged scalar sectors are individually governed by the same angle β, Correspondingly, the masses of the h ± and a scalars are given by To obtain the masses of the two CP-even scalars, h and H, we need to diagonalise the two-by-two CP-even mass matrix M 2 S , which may explicitly be written down as with λ 34 ≡ λ 3 + λ 4 . The mixing angle α necessary for the diagonalisation of M 2 S may be determined by The SM Higgs field may now be identified by the linear field combination, In this way, one can obtain the SM-normalised couplings of the CP-even h and H scalars to the EW gauge bosons (V = W ± , Z) as follows: with g H SM V V = 1 by definition. In similar manner, we may derive the SM-normalised couplings of the CP-even and CP-odd scalars to up-type and down-type quarks. These couplings are exhibited in Table I.
In these limits, the CP-even H (h) scalar couples to the EW gauge bosons with coupling strength exactly as that of the SM Higgs boson, while h (H) does not couple to them at all [22]. In the above two scenarios, the SM-like Higgs boson is identified with the 125-GeV resonance observed at the LHC [6,7]. In the literature, the neutral Higgs partner (H) in the SM-like h scenario is usually termed the heavy Higgs boson. Instead, in the SM-like H scenario, the partner particle h can only have a mass smaller than ∼ 125 GeV [23]. In this paper, we consider the alignment limit with β = α, which falls in the category of the SM-like H scenario, but the CP-even scalar partner h can be either lighter or heavier than the observed scalar resonance at the LHC. In the alignment limit, the SM-like Higgs boson becomes aligned with one of the neutral eigenstates.
In the so-called Higgs basis [32], the CP-even mass matrix M 2 S given in (II.8) takes on the form The SM alignment limit, cos(β − α) → 1, can be realised in two different ways: The first realisation (i) does not depend on the choice of the non-SM scalar masses, such as M h ± and M a , whereas the second one (ii) is only possible in the well-known decoupling limit [32][33][34]. In the first case, SM alignment is obtained by setting C = 0, which in turn implies the condition [22]: Barring fine-tuning among quartic couplings, (II.14) leads to the following constraints: which are independent of tan β and non-standard scalar masses. In this case, the two CPeven Higgs masses in the alignment limit are given by, In the second realisation (ii) mentioned above, we may simplify matters by expanding In this way, we obtain [22] M 2 Note that at large tan β, the phenomenological properties of the H-boson resemble more and more those of the SM Higgs boson [20,21]. Since we are interested in analysing the deviation of the H-boson couplings from their SM values, we follow an approximate approach inspired by the seesaw mechanism [40]. In particular, we may express all the H-boson couplings in terms of the light-to-heavy scalar-mixing parameter C/ B. Thus, employing (II.9) for the hatted quantities and ignoring A next to C, we may derive the approximate analytic expressions Given the tight experimental limits on the deviation of g HV V from 1, one must have that the light-to-heavy scalar mixing parameter C/ B 1, which justifies our seesaw-inspired approximation. In fact, in the exact SM alignment limit, α → β, the mixing parameter C/ B vanishes identically.
In similar fashion, we may derive approximate analytic expressions for the h-and Hboson couplings to up-and down-type quarks. To leading order in the light-to-heavy scalar mixing C/ B, these are given by In the SM alignment limit, we have g Huu → 1 and g Hdd → 1. Obviously, any deviation of the g Huu and g Hdd couplings from their SM values is controlled by tan β and C/ B.
In the present study, our primary interest lies in natural realisations of SM alignment, for which neither a mass hierarchy M a v, nor a fine-tuning among the quartic couplings will be necessary. To this end, one is therefore compelled to identify possible maximal symmetries of the 2HDM potential that would impose the condition stated in (II.14). In the next section, we will show how SM alignment can be achieved naturally by virtue of an SO(5) symmetry imposed on the theory.

III. THE MAXIMALLY SYMMETRIC 2HDM
A convenient field basis to describe the 2HDM potential will be to make use of an 8dimensional SU(2) L -covariant multiplet representation [41,49], With the help of the multiplet Φ, a 6-dimensional Lorentz vector R A = Φ † Σ A Φ may be constructed, where A = 0, 1, . . . , 5 and Σ A are 8 × 8-dimensional matrices whose precise analytic form is given in [29,41]. In this bilinear field-space formalism, R A transforms under the orthochronous SO(1,5) symmetry group [22,29]. Since the SU(2) L gauge-kinetic terms of the scalar doublets Φ 1,2 must be canonical, the set of SO(1,5) rotations reduces to those of SO (5). So, in the absence of the hypercharge gauge coupling g and fermion Yukawa couplings, the maximal symmetry group of the 2HDM is SO (5). As a consequence, the 2HDM potential has a total of 13 accidental symmetries [41], of which 6 preserve U(1) Y [42][43][44] and 7 are custodially symmetric [29]. Given the isomorphism of the Lie algebras: SO(5) ∼ Sp(4), the maximal symmetry group of the 2HDM in the original [29]. In the Type-II 2HDM, the maximal symmetry group SO (5) is the simplest of the three possible symmetries that can realise natural SM alignment. The other two are in bilinear [original] field space [22]: In what follows, we focus on the simplest realisation of SM alignment, which is called the MS-2HDM.
In the MS-2HDM, the SO(5) symmetry puts severe restrictions on the allowed form of the kinematic parameters of the 2HDM potential in (II.2), Evidently, the MS-2HDM potential obeys naturally the alignment constraints given in (II.15). As a consequence, the scalar potential takes on a very simple form, are soft SO(5)-breaking mass terms, which are introduced here for phenomenological reasons as we will explain below.
After EW symmetry breaking, the following breaking pattern emerges: If ∆V = 0, the CP-even scalar H receives a non-zero squared mass M 2 H = 2λ 2 v 2 , while the other scalars, h, a and h ± , are all massless. These are massless pseudo-Goldstone bosons that have sizeable couplings to the SM gauge bosons. Accordingly, several experimentally excluded decay channels would open, e.g. Z → ha and W ± → h ± h [45]. If the SO(5) symmetry is realised at some high energy scale µ X (much above the EW scale), then due to RG effects the following breaking pattern may take place [22]: Note that the RG effect of the gauge coupling g only lifts the charged Higgs mass M h ± , while the corresponding effect of the Yukawa couplings (particularly that of the top-quark y t ) renders the other CP-even pseudo-Goldstone boson h massive. Instead, the CP-odd scalar a remains massless and can be identified with a Peccei-Quinn (PQ) axion after the SSB of a global U(1) PQ symmetry [46][47][48]. Since weak-scale PQ axions have been ruled out by experiment, we have allowed for the SO(5) symmetry of the MS-2HDM potential in (III.3) to be broken by the soft SO(5)-breaking mass terms in (III.4). Hence, without loss of generality, we may consider that only the soft SO(5)-breaking parameter Re(m 2 12 ) is non-zero. With this minimal addition to the MS-2HDM potential, the scalar-boson masses are given, to a very good approximation, by Hence, all pseudo-Goldstone bosons, h, a and h ± , become massive and almost degenerate in mass. As we will see in the next section, this degeneracy is very stable against RG effects even up to two-loop order.
In our study, we will consider the charged Higgs boson mass M h ± as an input parameter above 500-GeV range, in agreement with B-meson constraints [50]. It will also be our threshold above which all parameters run with 2HDM renormalisation group equations (RGEs). Moreover, we implement the matching conditions with two-loop RG effects of the SM at given M h ± threshold scales. We also employ two-loop 2HDM RGEs to find the running of the gauge, Yukawa and quartic couplings at RG scales larger than M h ± . For reviews on one-, two-and three-loop RGEs in the 2HDM, see [22,[51][52][53][54]. The SM and the 2HDM RGEs have been computed using the public Mathematica package SARAH [55], which has been appropriately adapted for the MS-2HDM.

IV. QUARTIC COUPLING UNIFICATION
As we saw in the previous section, the SO(5) symmetry of the MS-2HDM potential is broken explicitly by RG effects and soft-mass terms. In this section, we will consider a unified theoretical framework in which the SO(5) symmetry is realised at some high-energy scale µ X , where all the conditions in (III.2) are met. Of particular interest is the potential existence of conformally-invariant unification points at which all quartic couplings of the MS-2HDM potential vanish simultaneously.
To address the above issue of quartic coupling unification, we employ two-loop RGEs for the MS-2HDM from the unification scale µ X to the charged Higgs-boson mass M h ± µ X . Below this threshold scale µ thr = M h ± , the SM is a good effective field theory, so we use the two-loop SM RGEs given in [56] to match the relevant MS-2HDM couplings to the corresponding SM quartic coupling λ SM , the Yukawa couplings, the SU(2) L and U(1) Y gauge couplings g 2 and g (with g 1 = 5/3g ). In order to obtain illustrative predictions, we have chosen our threshold scales to be at µ thr = M h ± = 500 GeV, 1 TeV, 10 TeV and 100 TeV.
The theoretical SM parameters are determined in terms of precision observables, such as the Z-boson mass (M Z ), the W ± -boson mass (M W ), the Fermi constant G µ , the strong The values of the two-loop SM couplings at different threshold scales, µ thr = M t , M h ± , are given in Table II.
The matching conditions for the Yukawa couplings at the threshold scale Note that for RG scales µ > µ thr = M h ± , the evolution of the Yukawa couplings h t , h b and h τ is governed by two-loop 2HDM RGEs. As can be seen from Figure 1, the quartic coupling λ 2 , which determines the SM-like Higgs-boson mass M H , decreases at high RG scales due to the running of the top-Yukawa coupling h t and turns negative just above the quartic coupling unification scale µ X ∼ 10 11 GeV, at which all quartic couplings vanish. Thus, for energy scales above the RG scale µ X , we envisage that the MS-2HDM will need to be embedded into another UV-complete theory. Nevertheless, according to our estimates in Section VI, we have checked that the resulting MS-2HDM potential leads to a metastable but sufficiently long-lived EW vacuum, whose lifetime is many orders of magnitude larger than the age of our Universe. In this respect, we regard the usual constraints derived from convexity conditions on 2HDM potentials [58] to be over-restrictive and unnecessary in our theoretical framework.
Of equal importance is a second conformally-invariant unification point µ case, the key quartic coupling λ 2 increases and becomes positive again. Hence, in this second class of settings, any embedding of the MS-2HDM into a UV-complete theory must have to take quantum gravity into account as well.
In Figure 3, we give numerical estimates of the mass spectrum of the MS-2HDM, for tan β = 50 and m 2 12 ≈ 70 2 GeV 2 (M h ± = 500 GeV). We find that all heavy Higgs bosons, h, a and h ± , are approximately degenerate in mass, almost independently of the quartic coupling unification scale µ X . In fact, to leading order, all masses are determined by the soft mass term m 2 12 in (III.7). The curve corresponding to the SM-like Higgs-boson mass M H is mainly tracking λ 2 (µ), which has two roots, one at RG scales around µ  give the corresponding sets of high-scale unification points. From Figure 4, we may also observe the domains in which the λ 2 coupling becomes negative. These are given by the vertical µ-intervals bounded by the lower and the upper curves, for a given choice of M h ± and tan β. Evidently, as the threshold scale µ thr = M h ± increases, the size of the negative λ 2 domain increases and becomes more pronounced for smaller values of tan β. X . Once the values of these three theoretical parameters are given, the entire Higgs sector of the model can be determined. In the next section, we will give typical predictions in terms of these three input parameters.

V. MISALIGNMENT PREDICTIONS FOR HIGGS BOSON COUPLINGS
In this section, we present numerical estimates of the predicted deviations of the SM-like Higgs-boson couplings HV V (with V = W ± , Z), Htt and Hbb, from their respective SM values. As was discussed in Section II, these deviations are controlled by the light-to-heavy scalar mixing parameter C/ B. At the quartic coupling unification scale µ X , the SO(5) symmetry of the MS-2HDM is fully restored and this mixing parameter vanishes. However, as we saw in Section III, RG effects induced by the U(1) Y gauge coupling and the Yukawa couplings of the third generation of fermions break sizeably the SO(5) symmetry, giving rise to a calculable non-zero value for C/ B and so to misalignment predictions for all H-boson couplings to SM particles.
In Figures 5 and 6, we exhibit the dependence of the physical misalignment parameter By analogy, Figures 7 and 8 display misalignment predictions for the H-boson couplings to top-and bottom-quarks, for tan β = 2, 5, 20, 35 and 50 and for lower-and higher-scale quartic coupling unification points, respectively. As before, the deviation of the normalised couplings g Htt and g Hbb from their SM values are larger for low values of tan β, e.g. tan β = 2, and for higher quartic coupling unification points µ (2) X . This effect is more pronounced for g Hbb , as the degree of misalignment might be even larger than 10%. In this case, a comparison between solid and dashed lines demonstrates the goodness of our seesaw-inspired approximation in terms of C/ B.
Finally, we confront our misalignment predictions for the SM-like Higgs boson couplings, g HZZ , g Htt and g Hbb with existing experimental data from ATLAS and CMS, including their statistical and systematic uncertainties [59]. All these results are presented in Table III, for M h ± = 500 GeV. The observed results for g HZZ and g Htt are in excellent agreement with the SM and the MS-2HDM. Instead, the LHC data for g Hbb can be fitted to the SM at the 3σ level. The latter reduces only to 2σ in the MS-2HDM, for tan β = 2 assuming a high-scale quartic coupling unification scenario. Future precision collider experiments will be able to probe such a scenario.  M h ± = 5 0 0 G e V E x a c t A p p r o x t a n β = 5 0 t a n β = 3 5 t a n β = 2 0 t a n β = 5 t a n β = 2 FIG. 5. Numerical estimates of the misalignment parameter |1 − g 2 HV V | pertinent to the HV Vcoupling (with V = W ± , Z) as functions of the RG scale µ, for a low-scale quartic coupling unification scenario, assuming M h ± = 500 GeV and tan β = 2, 5, 20, 35 and 50.
t a n β = 5 0 t a n β = 3 5 t a n β = 2 0 t a n β = 5 t a n β = 2 E x a c t A p p r o x M h ± = 5 0 0 G e V FIG. 6. The same as in Figure 5, but for a high-scale quartic coupling unification scenario.

VI. LIFETIME OF THE METASTABLE ELECTROWEAK VACUUM
According to our discussion in Section IV, we have seen that the MS-2HDM has two conformally-invariant quartic coupling unification points µ E x a c t A p p r o x M h ± = 5 0 0 G e V t a n β = 5 0 t a n β = 3 5 t a n β = 2 0 t a n β = 5 t a n β = 2 t a n β = 5 0 t a n β = 3 5 t a n β = 2 0 t a n β = 5 t a n β = 2  t a n β = 5 0 t a n β = 3 5 t a n β = 2 0 t a n β = 5 t a n β = 2 E x a c t A p p r o x M h ± = 5 0 0 G e V t a n β = 5 0 t a n β = 3 5 t a n β = 2 0 t a n β = 5 t a n β = 2 FIG. 8. The same as in Figure 7, but for a high-scale quartic coupling unification scenario. and the second one at µ (2) X ∼ 10 18 GeV. Between these two RG scales, i.e. for µ (1) X , the running quartic coupling λ 2 (µ) turns negative, which gives rise to a deeper minimum in the effective MS-2HDM potential. This new minimum will then be the true vacuum of the MS-2HDM. In this case, the EW vacuum that we currently live in becomes metastable and is usually called the false vacuum. For the high-scale quartic coupling unification scenario, this vacuum instability might be a problem, unless the lifetime of the EW vacuum is much larger than the age of the Universe.
In a general 2HDM, there are three different types of ground states [10,60,61]: (i) an EW-breaking vacuum that preserves CP and charge (normal minimum); (ii) an EW-breaking vacuum that breaks CP spontaneously, but keeps charge conserved; and (iii) a chargeviolating ground state, where one of the upper components φ + 1,2 of the scalar doublets Φ 1,2 acquires a non-zero VEV. The existence of various minima in the potential may result in tunneling between different vacua. However, it is known since some time [62][63][64] that if the effective 2HDM potential has at least one CP-and charge-preserving local minimum, then any other possible charged-violating minimum cannot be deeper than this and so tunneling into such a local minimum will not be energetically favoured. Hence, our false neutral EW vacuum can only tunnel to another neutral vacuum, and our findings are consistent with this observation in the MS-2HDM.
The probability rate P for quantum tunneling through a barrier is exponentially suppressed and it may be estimated by where ∆S E is the Euclidean action evaluated at the O(4)-symmetric bounce solution [66]. The action ∆S E interpolates between the new phase at high field values and the EW phase. As we will see below, ∆S E is the Euclidean action of the corresponding bounce minus the action of the false vacuum configuration [65]. To determine the lifetime of the false vacuum, we have to look for the so-called bounce solutions that satisfy the equations of motion, where the index i = 1, 2, . . . 5 labels all scalar fields φ i = (h, H, a, h + , h − ) and φ ≡ {φ i }. Note that the fields φ i depend only on the Euclidean radial coordinate r = (t 2 E + x 2 ) 1/2 , as a consequence of the O(4)-symmetry of the problem. The boundary conditions of the equations of motion are where φ fv i are the values of the fields φ i at the false vacuum. More explicitly, the action ∆S E,i along a φ i -direction is given by [66] Here, φ b ≡ {φ b i } denotes collectively the bounce solutions satisfying the boundary conditions stated in (VI.3). For our MS-2HDM scenario, we have five second-order differential equations to determine the bounce configurations, where r = 0 is the center of the bounce which asymptotically approaches the false vacuum φ fv ≡ {φ fv i } as r → ∞. As the second deeper minimum of the potential occurs at field values φ i v, where typically φ i > ∼ 10 10 GeV, it will be a good approximation if we only keep its quartic terms. Therefore, the EW vacuum lifetime is computed by considering only the scale-invariant part of the effective potential: where the running quartic coupling λ 2 is negative. This analytic approximation leads to the following Euclidean action for the Fubini instanton [67]: where i = 1, 2. Knowing ∆S E,i , an approximate formula for the EW vacuum lifetime τ in units of the age of the Universe (T U ∼ 13.7×10 9 years) for the individual quantum-tunnelings from the EW vacuum to a deeper minimum in the φ i -direction (φ 1 = h and φ 2 = H) may be derived, (VI.7) Figure 9 gives the profile of the MS-2HDM potential along the H-(black curves) and h-(red curves) field directions. By considering the running quartic couplings, the instability shown in Figure 9 occurs when λ 2 crosses zero and becomes negative after the first unification point µ (1) X . Note that the scalar potential along the field direction of the SM-like Higgs boson H has a higher maximum and a deeper minimum compared to those found along the h direction.  We find that the EW vacuum lifetime τ is many orders of magnitude larger than the age of the Universe, and even much larger than the SM EW vacuum lifetime τ SM ∼ 10 640 T U in the flat space (in the absence of Plank-scale suppressed operators [37,38]). Therefore, we safely conclude that our Universe is adequately stable and so our high-scale quartic coupling unification scenario is phenomenologically viable.

VII. CONCLUSIONS
We have considered one of the simplest realisations of a Type-II 2HDM, the so-called Maximally Symmetric Two-Higgs Doublet Model (MS-2HDM). The scalar potential of this model is determined by a single quartic coupling and a single mass parameter. This minimal form of the potential can be reinforced by an accidental SO(5) symmetry in the bilinear field space, which is isomorphic to Sp(4)/Z 2 in the field basis Φ given in (III.1). The MS-2HDM can naturally realise the so-called SM alignment limit, in which all SM-like Higgs boson couplings to W ± and Z bosons and to all fermions are identical to their SM strength independently of tan β and the mass of the charged Higgs boson.
The SO(5) symmetry of the MS-2HDM is broken explicitly by RG effects due to the U(1) Y gauge coupling and equally sizeably by the Yukawa couplings of the third generation of quarks and charged leptons. For phenomenological reasons, we have also added a soft SO(5)-breaking bilinear mass parameter m 2 12 to the scalar potential, which lifts the masses of all pseudo-Goldstone bosons h ± , h and a above 500-GeV range in agreement with Bmeson constraints. To evaluate the RG running of the quartic couplings and the relevant SM couplings, we have employed two-loop 2HDM RGEs from the unification scale µ X up to charged Higgs-boson mass M h ± . At the RG scale M h ± , we have implemented matching conditions between the MS-2HDM and the SM parameters.
Improving upon an earlier study [22], we have now explicitly demonstrated that in the MS-2HDM all quartic couplings can unify at much larger RG scales µ X , where µ X lies between µ (1) X ∼ 10 11 GeV and µ (2) X ∼ 10 20 GeV. In particular, we have shown that quartic coupling unification can take place in two different conformally invariant points, at which all quartic couplings vanish. This property is unique for this model and can happen at different threshold scales M h ± = 500 GeV, 1 TeV, 10 TeV and 100 TeV. More precisely, the low-scale (high-scale) unification point arises when λ 2 crosses zero and becomes negative (positive) at the RG scale µ X ). The region between these two scales corresponds to negative values of λ 2 . For this reason, we have performed a vacuum stability analysis of the model in order to ensure that the EW vacuum is sufficiently long-lived. In this respect, we have estimated the EW vacuum lifetime τ which was found to be reassuringly long, i.e. much larger than that of the EW vacuum lifetime in the SM, τ SM ∼ 10 640 T U , assuming the absence of harmful Planck-scale suppressed operators.
It is important to reiterate here that the MS-2HDM is a very predictive extension of the SM, as it is governed by only three additional parameters: (i) the charged Higgs-boson mass M h ± (or m 2 12 ), (ii) the ratio of VEVs tan β, and (iii) the conformally-invariant quartic coupling unification scale µ X which can only take two discrete values: µ (1) X and µ (2) X . Given these three parameters, the entire Higgs sector of the model can be determined. In this context, we have presented illustrative predictions of misalignment for the SM-like Higgsboson couplings to the W ± and Z bosons and, for the first time, to the top-and bottomquarks. The predicted deviations to Hbb-coupling is of order 10% and may be observable to future precision collider experiments.
Evidently, our novel theoretical framework can be straightforwardly extended to multi-HDMs with n scalar-doublets based on the maximal symmetry group Sp(2n)/Z 2 ⊗ SU(2) L . We plan to report progress on this issue in an upcoming publication.