Axion-Higgs interplay in the two Higgs-doublet model

We study the Dine-Fischler-Srednicki (DFS) model in the light of the recent Higgs LHC results and electroweak precision data. The DFS model is a natural extension of the two-Higgs doublet model endowed with a Peccei-Quinn symmetry and leading to a physically acceptable axion. For generic couplings, the model reproduces the minimal Standard Model showing only tiny deviations (extreme decoupling scenario) whereas all additional degrees of freedom (with the exception of the axion) are very heavy. Recently, new corners of this model have been highlighted where it may exhibit enlarged global symmetries making the corresponding models technically natural (naturalness scenario). In some cases an additional Higgs could be present at the weak scale. In this case, the new light $0^+$ state would be accompanied by relatively light charged and neutral pseudoscalar Higgses. We will use the oblique corrections, particularly $\Delta\rho$, to constrain the mass spectrum in this case. As a final result, we also work out the non-linear parametrization of the DFS model in the generic case where all scalars except the lightest Higgs and the axion have masses at or beyond the TeV scale.


I. INTRODUCTION
An invisible axion [1][2][3] constitutes to this date a very firm candidate to provide all or part of the dark matter component of the cosmological budget. There are several extensions of the Minimal Standard Model (MSM) providing a particle with the characteristics and couplings of the axion [4,5]. In our view a particularly interesting possibility is the model suggested by Dine, Fischler and Srednicki (DFS) more than 30 years ago that consists in a fairly simple extension of the popular two Higgs-doublet model (2HDM). As a matter of fact it could probably be argued that a good motivation to consider the 2HDM is that it allows for the inclusion of a (nearly) invisible axion [6][7][8]. Of course there are other reasons why the 2HDM should be considered as a possible extension of the MSM. Apart from purely aestethic reasons, it is easy to reconcile such models with the existing phenomenology without unnatural fine tuning (at least at tree level). They may give rise to a rich phenomenology, including possibly (but not necessarily) flavour changing neutral currents at some level, custodial symmetry breaking terms or even new sources of CP violation [9,10].
Following the discovery of a Higgs-like particle with m h ∼ 126 GeV there have been a number of works considering the implications of such a finding on the 2HDM, together with the constrains arising from the lack of detection of new scalars and from the electroweak precision observables [11]. Depending on the way that the two doublets couple to fermions, they are classified as type I, II or III (see e.g. [9] for details), with different implications on the flavour sector. Consideration of all the different types of 2HDM plus all the rich phenomenology that can be encoded in the Higgs potential leads to a wide variety of possibilities with different experimental implications, even after applying all the known phenomenological low-energy constraints.
Requiring a Peccei-Quinn (PQ) symmetry leading to an axion does however severely restrict the possibilities, and this is in our view an important asset of the DFS model. This turns out to be particularly the case when one includes all the recent experimental information concerning the 126 GeV scalar state and its couplings. Exploring this model, taking into account all these constraints is the motivation for the present work.
The structure of this paper is as follows. In section II we discuss the possible global symmetries of the DFS model, namely U (1) P Q (always present), and SU (2) L × SU (2) R (the SU (2) R subgroup may or may not be present). Symmetries are best discussed by using a matrix formalism that we review and extend. Section III is devoted to the determination of the spectrum of the theory. We present up to four generic cases that range from the extreme decoupling, where the model -apart from the presence of the axion-is indistinguishable from the MSM at low energies, to the one where there are extra light Higgses below the TeV scale. This last case necessarily requires some couplings in the potential to be very small; a possibility that is nevertheless natural in a technical sense and therefore should be contemplated as a theoretical hypothesis. We also discuss the situation where custodial symmetry is exact or approximately valid in this model. However, even if light scalars can exist in some corners of the parameter space, the presence of a substantial gap between the Higgs at 126 GeV and the rest of scalar states, with masses in the multi-TeV region or even beyond, is a rather generic characteristic of DFS models (therefore this hierarchy could be claimed to be an indirect consequence of the existence of a light invisible axion). In section IV we discuss the resulting non-linear effective theory emerging in this generic situation. Next in section V we analyze the impact of the model on the (light) Higgs effective couplings to gauge bosons. Finally in section VI the restrictions that the electroweak precision parameters, particularly ∆ρ, impose on the model are discussed. These restrictions, for reasons that will become clear in the subsequent sections, are relevant only in the case where all or part of the additional spectrum of scalars is light.

II. MODEL AND SYMMETRIES
The DFS model contains two Higgs doublets and one complex scalar singlet, namely Moreover, we define the usual electroweak vacuum expectation value v = 246 GeV as v 2 = (v 2 1 + v 2 2 )/2 and tan β = v 2 /v 1 . The implementation of the PQ symmetry is only possible for type II models, where the Yukawa terms are withφ i = iτ 2 φ * i . The PQ transformation acts on the scalars as and on the fermions as For the Yukawa terms to be PQ-invariant we need Let us now turn to the potential involving the two doublets and the new complex singlet.
The most general potential compatible with PQ symmetry is The c term imposes the condition −X 1 +X 2 +2X φ = 0. If we impose that the PQ current does not couple to the Goldstone boson that is eaten by the Z, we also get X 1 cos 2 β + X 2 sin 2 β = 0. If furthermore we choose 1 X φ = −1/2 the PQ charges of the doublets are X 1 = − sin 2 β, X 2 = cos 2 β.
Global symmetries are not very evident in the way fields are introduced above. To remedy this let us define the matrices [12] and Defining also the constant matrix W = (V 2 1 + V 2 2 )I/2 + (V 2 1 − V 2 2 )τ 3 /2, we can write the potential as There is arbitrariness in this choice. This election conforms to the conventions existing in the literature.

Parameter
Custodial limit λ 1 , λ 2 , λ 4 λ 1 = λ 2 = λ and λ 4 = 2λ A SU (2) L × SU (2) R global transformation acts on our fields as We now we are in a better position to discuss the global symmetries of the potential. The behavior of the different parameters under SU (2) R is shown in Table I. See also [13].
Finally, let us establish the action of the PQ symmetry previously discussed in this parametrization. Under the PQ transformation: with Using the values for X 1,2 given in Eq. (7) We have two doublets and a singlet, so a total of 4 + 4 + 2 = 10 spin-zero particles.
Three particles are eaten by the W ± and Z and 7 scalars fields are left on the spectrum; two charged Higgs, two 0 − states and three neutral 0 + states. Our field definitions will be worked out in full detail in section IV. Here we want only to illustrate the spectrum. For the charged Higgs mass we have 2 The quantity v φ is proportional to the axion decay constant. Its value is known to be very large (at least 10 7 GeV and probably substantially larger if astrophysical constraints are taken into account, see [14] for several experimental and cosmological bounds). It does definitely make sense to organize the calculations as an expansion in v/v φ .
In the 0 − sector there are two degrees of freedom that mix with each other with a mass matrix which has a vanishing eigenvalue. The eigenstate with zero mass is the axion and A 0 is the pseudoscalar Higgs with mass Eq. (16) implies c ≥ 0. For c = 0, the mass matrix in the 0 − sector has a second zero, i.e.
in practice the A 0 field behaves as another axion.
In the 0 + sector, there are three neutral particles that mix with each other. With h i we denote the corresponding 0 + mass eigenstates. The mass matrix is given in Appendix B.
In the limit of large v φ , the mass matrix in the 0 + sector can be easily diagonalized [7] and presents one eigenvalue nominally of order v 2 and two of order v 2 φ . Up to O(v 2 /v 2 φ ), these masses are LHC.
It is worth it to stress that there are several situations where the above formulae are non-applicable, since the nominal expansion in powers of v/v φ may fail. This may be the case where the coupling constants a, b, c connecting the singlet to the usual 2HDM are very small, of order say v/v φ or v 2 /v 2 φ . One should also pay attention to the case λ φ → 0 (we have termed this latter case as the 'quasi-free singlet limit'). Leaving this last case aside, we have found that the above expressions for m h i apply in the following situations: Case 1: The couplings a, b and c are generically of O(1), If c λ i v 2 /v 2 φ the 0 − state is lighter than the lightest 0 + Higgs and this case is therefore already phenomenologically unacceptable. The only other case that deserves a separate discussion is Case 4: Same as in case 3 but c ∼ λ i v 2 /v 2 φ In this case, the masses in the 0 + sector read, up to O(v 2 /v 2 φ ), as where Recall that here we assume c to be of O(v 2 /v 2 φ ). Note that In the 'quasi-free singlet' limit, when λ φ → 0 or more generically λ φ a, b, c it is impossible to sustain the hierarchy v v φ , so again this case is phenomenologically uninteresting (see Appendix C for details).
We note that once we set tan β to a fixed value, the lightest Higgs to 126 GeV and v φ to some large value compatible with the experimental bounds, the mass spectrum in Eq. (15), (16) and (17)- (19) grossly depends on the parameters: c, λ 4 and λ φ , the latter only affecting the third 0 + state that is anyway very heavy and definitely out of reach of LHC experiments; therefore the spectrum depends on only two parameters. If case 4 is applicable, the situation is slightly different and an additional combination of parameters dictates the mass of the second (lightish) 0 + state. This can be seen in the sum rule of Eq. (22) after requiring that m h 1 = 126 GeV. Actually this sum rule is also obeyed in cases 1, 2 and 3, but the r.h.s is dominated then by the contribution from parameter c alone.

A. Heavy and light states
Here we want to discuss the spectrum of the theory according to the different scenarios that we have alluded to in the previous discussion. Let us remember that it is always possible to identify one of the Higgses to the scalar boson found at LHC, namely h 1 .
Case 1: all Higgses except h 1 acquire a mass of order v φ . This includes the charged and 0 − scalars too. We term this situation 'extreme decoupling'. The only light states are h 1 , the gauge sector and the massless axion. This is the original DFS scenario [4] Case 2. This situation is similar to case 1 but now the typical scale of masses of This range is beyond the LHC reach but it could perhaps be explored with an accelerator in the 100 TeV region, a possibility being currently debated. Again the only light particles are h 1 , the axion and the gauge sector. This possibility is natural in a technical sense as discussed in [7] as an approximate extra symmetry would protect the hierarchy.
Cases 3 and 4 are phenomenologically more interesting. Here we can at last have new states at the weak scale. In the 0 + sector, h 3 is definitely very heavy but m 2 h 1 and m 2 h 2 are proportional to v 2 once we assume that c ∼ v 2 /v 2 φ . Depending on the relative size of λ i and cv 2 φ /v 2 one would have to use Eq. (17) or (20). Because in case 3 one assumes that cv 2 φ /v 2 is much larger than λ i , h 1 would still be the lightest Higgs and m h 2 could easily be in the TeV region. When examining case 4 it would be convenient to use the sum rule (22).
We note that in case 4 the hierarchy between the different couplings is quite marked: typically to be realized one needs c ∼ 10 −10 λ i , where λ i is a generic coupling of the potential. The smallness of this number results in the presence of light states at the weak scale. For a discussion on the 'naturalness' of this possibility see [7].

B. Custodially symmetric potential
While in the usual one doublet model, if we neglect the Yukawa couplings and set the U Y (1) interactions to zero, custodial symmetry is automatic, the latter is somewhat unnatural in 2HDM as one can write a fairly general potential. These terms are generically not invariant under global transformations SU (2) L × SU (2) R and therefore in the general case after the breaking there is no custodial symmetry to speak of. Let us consider now the case where a global symmetry SU (2) L × SU (2) R is nevertheless present as there are probably good reasons to consider this limit. We may refer somewhat improperly to this situation as to being 'custodially symmetric' although after the breaking custodial symmetry proper may or may not be present. If SU (2) L × SU (2) R is to be a symmetry, the parameters of the potential have to be set according to the custodial relations in Table I. Now, there are two possibilities to spontaneously break SU (2) L × SU (2) R and to give mass to the gauge bosons.
If the VEVs of the two Higgs fields are different (tan β = 1), the custodial symmetry is spontaneously broken to U (1). In this case, one can use the minimization equations of Appendix A to eliminate V , V φ and c of Eq. (10). c turns out to be of order (v/v φ ) 2 . In this case there are two extra Goldstone bosons: the charged Higgs is massless Furthermore, the A 0 is light: This situation is clearly phenomenologically excluded.
In this case, the VEVs of the Higgs doublets are equal, so tan β = 1. The masses are These three states are parametrically heavy, but they may be light in cases 3 and 4.
The rest of the 0 + mass matrix is 2 × 2 and has eigenvalues (up to second order in v/v φ ) It is interesting to explore in this relatively simple case what sort of masses can be obtained by varying the values of the couplings in the potential (λ, λ 3 and c). We are basically interested in the possibility of obtaining a lightish spectrum (case 4 previously discussed) and accordingly we assume that the natural scale of c is ∼ v 2 /v 2 φ . We have to require the stability of the potential discussed in Appendix D as well as m h 1 = 126 GeV.
The allowed region is shown in Fig. 1. Since we are in a custodially symmetric case there are no further restrictions to be obtained from ∆ρ.

IV. NON-LINEAR EFFECTIVE LAGRANGIAN
We have seen in the previous section that the spectrum of scalars resulting from the potential of the DFS model is generically heavy. It is somewhat difficult to have all the scalar masses at the weak scale, although the additional scalars can be made to have masses in weak scale region in case 4. The only exceptions are the three Goldstone bosons, the h 1 Higgs and the axion. It is therefore somehow natural to use a non-linear realization to describe the low energy sector formed by gauge bosons (and their associated Goldstone bosons), the lightest Higgs 0 + state h 1 , and the axion. Deriving this effective action is one of the motivations of this work.
To construct the effective action we have to identify the proper variables and in order to do so we will follow the technique described in [12]. In that paper the case of a generic 2HDM where all scalar fields were very massive was considered. Now we have to modify the method to allow for a light state (the h 1 ) and also to include the axion degree of freedom.
We decompose the matrix-valued Φ 12 field introduced in Section II in the following form potential. Note that c is assumed to be of order v 2 /v 2 φ and cv 2 φ /v 2 has to be ∼ λ i (case 4 discussed in the text).
U is a 2 × 2 matrix that contains the three Goldstone bosons associated to the breaking of SU (2) L (or more precisely of SU (2) L × U (1) Y to U (1) em ). We denote by G i these Goldstone Note that the matrices I and J of Eq. (9) entering the DFS potential are actually independent of U. This is immediate to see in the case of I while for J one has to use the property τ 2 U * = Uτ 2 valid for SU (2) matrices. The effective potential then does depend only on the degrees of freedom contained in M 12 whereas the Goldstone bosons drop from the potential, since, under a global SU (2) L × SU (2) R rotation, Φ 12 and U transform as Obviously the same applies to the locally gauged subgroup.
Let us now discuss the potential and M 12 further. Inspection of the potential shows that because of the term proportional to c the phase of the singlet field φ does not drop automatically from the potential and thus it cannot be immediately identified with the axion.
In other words, the phase of the φ field mixes with the usual 0 − scalar from the 2HDM. To deal with this let us find a suitable phase both in M 12 and in φ that drops from the effective potential -this will single out the massless state to be identified with the axion.
We write M 12 = M 12 U a , where U a is a unitary matrix containing the axion. An immediate choice is to take the generator of U a to be the identity, which obviously can remove the phase of the singlet in the term in the effective potential proportional to c while leaving the other terms manifestly invariant. This does not exhaust all freedom however as we can include in the exponent of U a a term proportional to τ 3 . It can be seen immediately that this would again drop from all the terms in the effective potential, including the one proportional to c when taking into account that φ is a singlet under the action of τ 3 that of course is nothing but the hypercharge generator. We will use the remaining freedom just discussed to properly normalize the axion and A 0 fields in the kinetic terms to which we now turn.
The gauge invariant kinetic term will be where the covariant derivative is defined by By defining U a = exp 2ia φ X/ v 2 φ + v 2 s 2 2β with X in Eq. (14), all terms in the kinetic term are diagonal and exhibit the canonical normalization. Moreover the field a φ disappears from the potential. Note that the phase redefinition implied in U a exactly coincides with the realization of the PQ symmetry on Φ 12 in Eq. (12) as is to be expected (this identifies uniquely the axion degree of freedom).
Finally, the non-linear parametrization of Φ 12 reads as with and in terms of the fields in Eq. (1). The singlet field φ is non-linearly parametrized as With the parametrizations above the kinetic term is diagonal in terms of the fields of M 12 and ρ. Moreover, the potential is independent of the axion and Goldstone bosons. All the fields appearing in Eqs. (33) and (35) The rotation matrix R as well as the corresponding mass matrix are given in Appendix B.
H and S are the so called interaction eigenstates. In particular, H couples to the gauge fields in the same way that the usual MSM Higgs does.

A. Integrating out the heavy Higgs fields
In this section we want to integrate out the heavy scalars in Φ 12 of Eq. (32) in order to build a low-energy effective theory at the TeV scale with an axion and a light Higgs.
As a first step, let us imagine that all the states in Φ 12 are heavy; upon their integration we will recover the Effective Chiral Lagrangian [15] where the O i is a set of local gauge invariant operators [16], and the symbol D µ represents the covariant derivative defined in (31). The corresponding effective couplings a i collect the low energy information (up to energies E 4πv) pertaining to the heavy states integrated out. In the unitarity gauge, the term D µ U † D µ U would generate the gauge boson masses.
If a light Higgs (h = h 1 ) and axion are present, they have to be included explicitly as dynamical states [17], and the corresponding effective Lagrangian will be (gauge terms are omitted in the present discussion) formally amounting to a redefinition of the 'right' gauge field and Here h is the lightest 0 + mass eigenstate, with mass 126 GeV but couplings in principle different from the ones of a MSM Higgs. The terms in L ren are required for renormalizability [18] at the one-loop level and play no role in the discussion.
The couplings a i are now functions of h/v, a i (h/v), which are assumed to have a regular expansion and contribute to different effective vertices. Their constant parts a i (0) are related to the electroweak precision parameters ('oblique corrections').
Let us see how the previous Lagrangian (38) can be derived. First, we integrate out from Φ 12 = UM 12 U a all heavy degrees of freedom, such as H ± and A 0 , whereas we retain H and S because they contain a h 1 component, namely where H and S stand respectively for R H1 h 1 and R S1 h 1 .
When the derivatives of the kinetic term of Eq. (30) act on M 12 , we get the contribution ∂ µ h∂ µ h in Eq. (38). Since the unitarity matrices, U and U a drop from the potential of Eq. (10) only V (h) remains. 3 Note that the axion kinetic term is not well normalized in this expression yet. Extra contributions to the axion kinetic term also come from the term in the first line of Eq. (38). Only once we include these extra contributions, the axion kinetic term gets well normalized. See also discussion below.
kinetic term of Eq. (30) the contribution Here we used that M 12 M † 12 has a piece proportional to the identity matrix and another proportional to τ 3 that cannot contribute to the coupling with the gauge bosons since TrD µ U † D µ Uτ 3 vanishes identically. The linear contribution in S is of this type thus decoupling from the gauge sector and as a result only terms linear in H survive. Using that [U a , M 12 ] = [U a , τ 3 ] = 0, the matrix U a cancels out in all traces and the only remains of the axion in the low energy action is the modification D µ → D µ . The resulting effective action is invariant under global transformations U → LUR † but now R is an SU (2) matrix only if custodial symmetry is preserved (i.e. tan β = 1). Otherwise the right global symmetry group is reduced to the U (1) subgroup. It commutes with U (1) P Q .
We then reproduce (38) with g 1 = 1. However, this is true for the field H on the l.h.s. of Eq. (43), not h = h 1 and this will reflect in a reduction in the value of the g i when one considers the coupling to the lightest Higgs only.
A coupling among the S field, the axion and the neutral Goldstone or the neutral gauge boson survives in Eq. (43). This will be discussed in Section V. As for the axion kinetic term, it is reconstructed with the proper normalization from the first term in (30)  See e.g. [15,16] The first one is universal, its coefficient being fixed by the W mass.
In the above expression W µν and B µν are the field strength tensors associated to the SU (2) L and U (1) Y gauge fields, respectively. In this paper we shall only consider the self-energy, or oblique, corrections, which are dominant in the 2HDM model just as they are in the MSM.
The oblique corrections are often parametrized in terms of the parameters ε 1 , ε 2 and ε 3 introduced in [19]. In an effective theory such as the one described by the Lagrangian (44) and (45) ε 1 ,ε 2 and ε 3 receive one loop (universal) contributions from the leading O(p 2 ) term v 2 Tr(D µ UD µ U † ) and tree level contributions from the a i (0). Thus where the dots symbolize the one-loop O(p 2 ) contributions. The latter are totally independent of the specific symmetry breaking sector. See e.g. [12] for more details.
A systematic integration of the heavy degrees of freedom, including the lightest Higgs as external legs, would provide the dependence of the low-energy coefficient functions on h/v, i.e. the form of the functions a i (h/v). However this is of no interest to us here.

V. HIGGS AND AXION EFFECTIVE COUPLINGS
The coupling of h 1 can be worked out from the one of H, which is exactly as in the MSM, where R H1 = 1 − (v/v φ ) 2 A 2 13 /2 and g SM 1 ≡ 1. With the expression of A 13 given in Appendix B, bosons are very small, experimentally indistinguishable from the MSM case. In any case the correction is negative and g 1 < g SM 1 . Case 4 falls in a different category. Let us remember that this case corresponds to the situation where c ∼ λ i v 2 /v 2 φ . Then the corresponding rotation matrix is effectively 2 × 2, with an angle θ that is given in Appendix B. Then In the custodial limit, λ 1 = λ 2 and tan β = 1, this angle vanishes exactly and g 1 = g SM 1 . Otherwise this angle could have any value. Note however that when c λ i v 2 /v 2 φ then θ → 0 and the value g 1 g SM 1 is recovered. This is expected as when c grows case 4 moves into case 3. Experimentally, from the LHC results we know [20] that g 1 = [0.67, 1.25] at 95% CL.
Let us now discuss the Higgs-photon-photon coupling in this type of models. Let us first consider the contribution from gauge and scalar fields in the loop. The diagrams contributing to the coupling between the lightest scalar state h 1 and photons are exactly the same ones as in a generic 2HDM, via a loop of gauge bosons and one of charged Higgses. In the DFS case the only change with respect to a generic 2HDM could be a modification in the h 1 W W (or Higgs-Goldstone bosons coupling) or in the h 1 H + H + tree-level couplings. The former has already been discussed while the triple coupling of the lightest Higgs to two charged Higgses gets modified in the DFS model to Note that the first line involves only constants that are already present in a generic 2HDM, while the second one does involve the couplings a, b and c characteristic of the DFS model.
The corresponding entries of the rotation matrix in the 0 + sector can be found in Appendix B. In cases 1, 2 and 3 the relevant entries are R H1 ∼ 1, R S1 ∼ v 2 /v 2 φ and R ρ1 ∼ v/v φ , respectively. Therefore the second term in the first line is always negligible but the piece in the second one can give a sizeable contribution if c is of O(1) (case 1). This case could therefore be excluded or confirmed from a precise determination of this coupling. In cases 2 and 3 this effective coupling aligns itself with a generic 2HDM but with large (typically ∼ 100 TeV) or moderately large (few TeV) charged Higgs masses. Case 4 is slightly different again. In this case R H1 = cos θ and R S1 = sin θ but R ρ1 = 0.
The situation is again similar to a generic 2HDM, now with masses that can be made relatively light, but with a mixing angle that because of the presence of the c terms may differ slightly from the 2HDM. For a review of current experimental fits in 2HDM the interested reader can see [11].
In this section we will also list the tree-level couplings of the axion to the light fields, thus completing the derivation of the effective low energy theory. The tree-level couplings are very few actually as the axion does not appear in the potential, and they are necessarily derivative in the bosonic part. From the kinetic term we get From the Yukawa terms (2) we get The loop-induced couplings between the axion and gauge bosons (such as the anomalyinduced coupling a φF F , of extreme importance for direct axion detection [14]) will not be discussed here as they are amply reported in the literature.

VI. CONSTRAINTS FROM ELECTROWEAK PARAMETERS
For the purposes of setting bounds on the masses of the new scalars in the 2HDM, ε 1 = ∆ρ is the most effective one. For this reason we will postpone the analysis of ε 2 and ε 3 to a future publication.
ε 1 can be computed by [19] with the gauge boson vacuum polarization functions defined as We need to compute loops of the type of Fig. 2. These diagrams produce three kinds of terms. The terms proportional to two powers of the external momentum, q µ q ν , do not enter in Π V V (q 2 ). The terms proportional to just one power vanish upon integration. Only the terms proportional to k µ k ν survive and contribute.
Although it is an unessential approximation, to keep formulae relatively simple we will compute ε 1 in the approximation g = 0. The term proportional to (g ) 2 is actually the largest contribution in the MSM (leaving aside the breaking due to the Yukawa couplings) but it is only logarithmically dependent on the masses of any putative scalar state and it can be safely omitted for our purposes [12]. The underlying reason is that in the 2HDM custodial symmetry is 'optional' in the scalar sector and it is natural to investigate powerlike contributions that would provide the strongest constraints. We obtain, in terms of the mass eigenstates and the rotation matrix of Eq. (36), where f (a, b) = ab/(b−a) log b/a and f (a, a) = a. Setting v φ → ∞ and keeping Higgs masses fixed, we formally recover the ∆ρ expression in the 2HDM (see the Appendix in [12]), namely Now, in the limit v φ → ∞ and m H ± → m A 0 (cases 1, 2 or 3 previously discussed) the ∆ρ above will go to zero as v/v φ at least and the experimental bound is fulfilled automatically.
However, we are particularly interested in case 4 that allows for a light spectrum of new scalar states. We will study this in two steps. First we assume a 'quasi-custodial' setting whereby we assume that custodial symmetry is broken only via the coupling λ 4B = λ 4 − 2λ being non-zero. Imposing vacuum stability and the experimental bound of (ε 1 − ε SM 1 )/α = It is also interesting to show (in this same 'quasi-custodial' limit) the range of masses allowed by the present constraints on ∆T , without any reference to the parameters in the potential. This is shown for two reference values of m A 0 in Fig. 4. Note the severe constraints due to the requirement of vacuum stability.
Finally let us turn to the consideration of the general case 4. We now completely give up custodial symmetry and hence the three masses m A 0 , m H ± and m h 2 are unrelated, except for the eventual lack of stability of the potential. In this case, the rotation R can be different form the identity which was the case in the 'quasi-custodial' scenario above. In particular, R S2 = cos θ from Appendix B and the angle θ is not vanishing. However, experimentally cos θ is known to be very close to one (see section V). If we assume that cos θ is exactly equal to one, we get the exclusion/acceptance regions shown in Fig. 5. Finally, Fig. 6 depicts the analogous plot for cos θ = 0.95 that is still allowed by existing constraints. We wee that the allowed range of masses are much more severely restricted in this case.

VII. CONCLUSIONS
With the LHC experiments gathering more data, the exploration of the symmetry breaking sector of the Standard Model will gain renewed impetus. Likewise, it is important to search for dark matter candidates as this is a degree of freedom certainly missing in the minimal Standard Model. An invisible axion is an interesting candidate for dark matter; however trying to look for direct evidence of its existence at the LHC is hopeless as it is extremely weakly coupled. Therefore we have to resort to less direct ways to explore this sector by formulating consistent models that include the axion and deriving consequences that could be experimentally tested.
In this work we have explored such consequences in the DFS model, an extension of the popular 2HDM. A necessary characteristic of models with an invisible axion is the presence of the Peccei-Quinn symmetry. This restricts the form of the effective potential. We have taken into account the recent data on the Higgs mass and several effective couplings, and included the constraints from electroweak precision parameters.
Four possible scenarios have been considered. In virtually all points of parameter space of the DFS model we do not really expect to see any relevant modifications with respect to the minimal Standard Model predictions. The new scalars have masses of order v φ or √ vv φ in two of the cases discussed. The latter could perhaps be reachable with a 100 TeV circular collider although this is not totally guaranteed. In a third case, it would be possible to get scalars in the multi-TeV region, making this case testable in the future at the LHC.
Finally, we have identified a fourth situation where a relatively light spectrum emerges. The last two cases correspond to a situation where the coupling between the singlet and the two doublets is of order v 2 /v 2 φ ; i.e. very small (10 −10 or less) and in order to get a relatively light spectrum in addition one has to require some couplings to be commensurate (but not necessarily fine-tuned).
The fact that some specific couplings are required to be very small may seem odd, but as it has been argued elsewhere it is technically natural, as the couplings in question do break The results on the scalar spectrum are derived here at tree level only and are of course subject to large radiative corrections. However one should note two ingredients that should ameliorate the hierarchy problem. The first observation is that the mass of the 0 − scalar is directly proportional to c; it is exactly zero if the additional symmetries discussed in [7] hold. It is therefore somehow protected. On the other hand custodial symmetry relates different masses, helping to maintain other relations. Some hierarchy problem should still remain but of a magnitude very similar to the one already present in the minimal Standard Model. Work on this point is in progress.
We have imposed on the model known constraints such as the fulfilment of the bounds on the ρ-parameter. These bounds turn out to be automatically fulfilled in most of parameter space and become only relevant when the spectrum is light (case 4). This is particularly relevant as custodial symmetry is by no means automatic in the 2HDM. Somehow the introduction of the axion and the related Peccei-Quinn symmetry makes possible custodially violating consequences naturally small. We have also considered the experimental bounds on the Higgs-gauge bosons and Higgs-two photons couplings.
In conclusion, DFS models containing an invisible axion are natural and, in spite of the large scale that appears in the model to make the axion nearly invisible, there is the possibility that they lead to an spectrum that can be tested at the LHC. This spectrum is severely constrained, making it easier to prove or disprove such possibility in the near future. On the other hand it is perhaps more likely that the new states predicted by the model lie beyond the LHC range. In this situation the model hides itself by making indirect contributions to most observables quite small. This is diagonalized with a rotation (B2) We write the rotation matrix as To compute Π ZZ entering Eq. (53), we need diagrams like Fig. 2 with V = Z. The X, Y pairs are