The three- and four-Higgs couplings in the general two-Higgs-doublet model

We apply the unitarity bounds and the bounded-from-below (BFB) bounds to the most general scalar potential of the two-Higgs-doublet model (2HDM). We do this in the Higgs basis, i.e. in the basis for the scalar doublets where only one doublet has vacuum expectation value. In this way we obtain bounds on the scalar masses and couplings that are valid for all 2HDMs. We compare those bounds to the analogous bounds that we have obtained for other simple extensions of the Standard Model (SM), namely the 2HDM extended by one scalar singlet and the extension of the SM through two scalar singlets.


Introduction
In order to unveil the detailed mechanism of electroweak symmetry breaking it is crucial to measure the self-couplings of the boson with mass 125 GeV discovered in 2012 at the LHC [1]. In this paper we call that boson h 1 . The Standard Model (SM) predicts h 1 to be a scalar and predicts its cubic and quartic couplings g 3 and g 4 , which we define through to be g SM 3 ≈ 32 GeV and g SM 4 ≈ 0.032, respectively. However, in Nature the scalar sector may be more complicated than in the SM and then g 3 and g 4 might have very different values. In this paper we survey the allowed values of g 3 and g 4 in three extensions of the SM: • The SM plus two real, neutral scalar singlets and with a reflection symmetry on each of those singlets. Let SM2S denote this model, which we treat in section 2.
• The two-Higgs-doublet model (2HDM), which is the focus object of section 3.
• The 2HDM with the addition of one real, neutral scalar singlet and with a reflection symmetry of that singlet. This model, which we dubb the 2HDM1S, is dealt with in section 4.
Our ingredients for bounding g 3 and g 4 in each of these models are: • The bounded-from-below (BFB) and the unitarity conditions on the quartic part of the scalar potential of each model. We apply those conditions directly in the basis for the scalar doublets where only one of them has vacuum expectation value (VEV).
• The experimental bound on the oblique parameter T [2].
• The (approximate) bound cos ϑ > 0.9 on the h 1 component cos ϑ of the scalar doublet with nonzero VEV.
Other authors before us [3]- [7] have used the BFB and unitarity constraints in order to bound the scalar masses and couplings of the 2HDM. However, they have done it in the context of a constrained version of the model, viz. the 2HDM with a reflection symmetry acting on one of the scalar doublets, leading to λ 6 = λ 7 = 0 in the scalar potential of equation (33). In this paper we deal on the fully general 2HDM. We enforce the BFB and unitarity constraints in the so-called Higgs basis, i.e. the basis where only one of the doublets has VEV. Since that basis exists for every 2HDM, we thus obtain results that apply to every 2HDM. At present there are only indirect, very rough bounds on g 3 . Using the Standard Model Effective Theory developed in ref. [8] and experimental data [9], ref. [10] has found that −8.4 < g 3 g SM unitarity of h 1 h 1 → h 1 h 1 scattering has been used [14] to obtain g 3 g SM 3 6.5 and g 4 g SM 4 65. In an analysis of a specific three-Higgs-doublet model, ref. [15] has found that in that model −1.3 < g 3 g SM 3 < 20.0 and 1.05 < g 4 g SM 4 < 1.6. The measurement of g 3 should be possible at future colliders. Reference [16] concluded that one may be able to measure g 3 provided −0.72 < g 3 g SM 3 < 7.05. Unfortunately, measuring g 4 is probably more challenging [17].

g 3 and g 4 in the SM
The Standard Model has only one scalar doublet φ 1 . We write it where v is the VEV, which is real and positive, and G + and G 0 are (unphysical) Goldstone bosons. In the SM H coincides with the observed scalar h 1 . The scalar potential is The minimization condition of V is µ 1 = −λ 1 v 2 . Therefore, in the unitary gauge where G ± and G 0 do not exist, The second term in the right-hand side of equation (4) indicates that the squared mass M 1 of the observed scalar is given by M 1 = 2λ 1 v 2 . Therefore, Using the approximate experimental values one gathers from equation (5a) that It should be noted that the sign of g 3 implicitly depends on the sign of h 1 . We fix that sign by noting that the covariant derivative of φ 1 gives rise to a term Thus, the coupling W + µ W µ− h 1 , viz. g 2 v √ 2 , is positive.

The Standard Model plus two singlets
We consider the Standard Model with the addition of two real SU(2) × U(1)-invariant scalar fields S 1 and S 2 . We assume two symmetries S 1 → −S 1 and S 2 → −S 2 . We call this model the SM2S. 1 The scalar potential is

Unitarity condidions
We derive the unitarity conditions on the parameters of V 4 . 2 We follow closely the method of ref. [20]. We write where a and b are complex fields. Then, There are seven two-particle scattering channels (Q is the electric charge, T 3 is the third component of weak isospin): 1 The SM2S has already been mentioned in the literature as a model for Dark Matter, see ref. [18]. 2 Strictly speaking, the unitarity conditions derived and utilized in this paper are the ones valid in the limit of infinite Mandelstam parameter s. For finite s one must take into account the trilinear vertices that are induced from the quartic vertices when one substitutes one of the fields by its VEV. The unitarity conditions then become s-dependent and may be either more or less restrictive than the conditions in the limit of infinite s. See ref. [19].
In order to derive the unitarity conditions one must write the scattering matrices for pairs of one incoming state and one outgoing state with the same Q and T 3 . Let the incoming state be xy and let the outgoing state by zw, where x, y, z, and w may be either a, a * , b, b * , S 1 , or S 2 . The corresponding entry in the scattering matrix is the coefficient of xyz * w * in V 4 , with the following additions: For each n identical operators in xyz * w * there is an additional factor n! in the entry. If x = y there is additional factor 2 −1/2 in the entry.
If z = w there is additional factor 2 −1/2 in the entry.
One finds in this way that the scattering matrices for the channels 1, 2, 3, and 4 are The scattering matrices for the channels 5 and 6 are The scattering matrix for channel 7 is The matrix (14) is similar to the matrix The unitarity conditions are the following: the eigenvalues of all the scattering matrices should be smaller, in modulus, than 4π. Thus, in our case, and the eigenvalues of should have moduli smaller than 4π.

Bounded-from-below conditions
One may write where X = φ † φ, Y = S 2 1 , and Z = S 2 2 are positive definite quantities independent of each other. Therefore, the BFB conditions are [21]

Procedure
Let the VEV of S 1 be w 1 and let the VEV of S 2 be w 2 . Then, the vacuum stability conditions are Using equation (2) with G + = 0 and G 0 = 0, i.e. in the unitary gauge, together with S 1 = w 1 + σ 1 and S 2 = w 2 + σ 2 , one obtains where One diagonalizes the real symmetric matrix M as where R is a 3 × 3 orthogonal matrix that may be parameterized as Here, c j = cos ϑ j and s j = sin ϑ j for j = 1, 2, 3. One has where the h j are the physical scalars, i.e. the eigenstates of mass; the scalar h j has squared mass M j . We assume that h 1 is the already-observed scalar. The interactions of the scalars with W + W − are given by equation (8b), i.e.
We define the sign of the field h 1 to be such that the coupling of h 1 to W + W − has the same sign as in the Standard Model. Thus, we choose −π/2 < ϑ 1 < π/2. According to equation (21), and The oblique parameter T is given by [22] T = T singlets = 3s 2 where In our numerical work we use as input the nine quantities v, w 1 , w 2 , M 1 , M 2 , M 3 , ϑ 1 , ϑ 2 , and ϑ 3 , which are equivalent to the nine parameters of the scalar potential µ 1 , m 2 1 , m 2 2 , λ 1 , ψ 1 , ψ 2 , ψ 3 , ξ 1 , and ξ 2 . We input equations (6) and choose arbitrary values for M 2 > 0 and M 3 > 0 such that M 2 ≤ M 3 (this represents no lack of generality, it is just the naming convention for h 2 and h 3 ). The VEVs w 1 and w 2 are chosen positive; this corresponds to the freedom of choice of the signs of S 1 and S 2 . The angle ϑ 1 is in either the first or the fourth quadrant, with so that the h 1 W + W − coupling is within 10% of its Standard Model value. The angle ϑ 2 is in the first quadrant; this corresponds to a choice of the signs of the fields h 2 and h 3 . The angle ϑ 3 may be in any quadrant. We firstly compute T according to equation (29) and check that it is inside its experimentally allowed domain [2] −0.04 < T < 0.20. We then compute We validate the input if the inequalities (16) and (19) hold and if the moduli of all three eigenvalues of the matrix (17) are smaller than 4π.

Results
A remarkable result of our numerical work is that there is an upper bound on the mass √ M 2 ; even if the VEVs w 1 and w 2 are allowed to be as high as 100 TeV-and, correspondingly, the mass √ M 3 also grows to a value of that order-the mass √ M 2 remains much smaller. In figure 1 we depict the upper bound on √ M 2 as a function of c 1 ; when c 1 → 1 the upper bound disappears, i.e. it tends to infinity.   In figure 2 we display the predictions for g 3 and g 4 . In order to produce that figure we have randomly generated √ M 2 and √ M 3 in the range 0 to 3 TeV and the VEVs w 1 and w 2 in the range 0 to 1.1 TeV. (Actually g 4 may become slightly larger, viz. g 4 0.5, if w 1 and w 2 are allowed to reach 10 TeV.) One sees that g 3 is always below its SM value and is always positive, while g 4 is almost always above its SM value. If the masses of the new scalars are higher, then g 3 takes values closer to the SM value. An important point is that g 3 remains of the same order of magnitude ∼ 30 GeV as in the SM, but g 4 may easily be 15 times larger than in the SM.
In the left panel of figure 3 one sees that g 3 is correlated with cos ϑ 1 : when cos ϑ 1 → 1 the coupling g 3 necessarily approaches its SM value. This behaviour is because of equation (27d) and c 1 > 0.9, which implies |s 1 | ≪ c 1 . On the other hand, g 4 is not correlated with cos ϑ 1 , as one sees in the right panel of figure 3.

The two-Higgs-doublet model
We next consider the model with two scalar gauge-SU(2) doublets φ 1 and φ 2 having the same weak hypercharge. This is usually known as 2HDM. The scalar potential is given by equation (9a), where where µ 1,2 and λ 1,2,3,4 are real. The ten (real) coefficients in V 4 may be grouped as [23] Under a (unitary) change of basis of the scalar doublets, η 00 is invariant while where O is an SO(3) matrix. Only quantities and procedures that are invariant under the transformation (35) are meaningful.

Unitarity conditions
We write Then, The relevant scattering channels are [20]: 1. The channel Q = 2, T 3 = 1, with three states aa, cc, and ac.
Channel 5 produces the scattering matrix A similarity transformation transforms the matrix (38) into the direct sum of two 4 × 4 matrices (39b) Here, is invariant under a change of basis of the doublets. It is obvious that the eigenvalues of the matrices (39) are invariant under such a change too. Channel (4) produces the scattering matrix which may readily be shown to be similar to M 1 . Channel (3) produces the scattering matrix     The matrix (42) is similar to where Channels (1) and (2) (40), should have moduli smaller than 4π. These conditions were first derived in ref. [24]. We emphasize that they are, as they should, invariant under a change of basis of the two doublets.  The case λ 1 = λ 2 = λ 3 = λ 4 = λ 5 = 0 is not realistic because it produces a potential unbounded from below. Still, one may compute the unitarity conditions in that case and one obtains

Bounded-from-below conditions
Necessary and sufficient conditions for the scalar potential of the 2HDM to be BFB were first derived in ref. [23]. Ivanov [25] and Silva [26] later produced other, equivalent conditions to the same effect. We have implemented numerically both the conditions of ref. [23] and those of ref. [26]. We have found that the Ivanov-Silva algorithm runs several times faster than the one of ref. [23]. We have also checked that all the points produced by either algorithm were validated by the other one. The points in our scatter plots were produced by using the algorithm of ref. [26]. That algorithm runs as follows. One constructs the 4 × 4 matrix and one computes its four eigenvalues. Then the potential is BFB if all the following conditions apply: • All four eigenvalues are real.
• All four eigenvalues are different from each other.

Procedure
We consider the most general 2HDM and purport to find out its ranges for g 3 and g 4 . We use the Higgs basis for the scalar doublets; in that basis only φ 0 1 has VEV and therefore φ 1 has the expression (2), while In equation (60), σ 1 and σ 2 are real fields and C + is the physical charged scalar of the 2HDM. We emphasize that using the Higgs basis represents no lack of generality, because both the unitarity and the BFB conditions are the same in any basis. Since only φ 1 has VEV, the vacuum stability conditions are µ 1 = −λ 1 v 2 and µ 3 = −λ 6 v 2 [30]. The coupling µ 2 in equation (33a) is unrelated to the parameters of V 4 ; one may trade it for the charged-Higgs squared mass M C = µ 2 + λ 3 v 2 . The mass terms of H, σ 1 , and σ 2 are given by line (21b), with [30] The matrix M is diagonalized through equations (23)- (25). The three invariants of M are We input parameters λ 1,2,··· ,7 that satisfy both the unitarity conditions and the BFB conditions of subsections 3.1 and 3.2, respectively. We also use the values of M 1 and v in equations (6). The two equations are quadratic in M C . By affirming the fact that both quadratic equations (63) must hold for the same value of M C , one is able to compute both M C and cos 2 ϑ 1 . We thus get to know the full matrix M, hence its eigenvalues M 2 and M 3 and its diagonalizing matrix R.
We require cos ϑ 1 > 0.9. We also compute the oblique parameter where T singlets is given by equation (29). We require −0.04 < T < 0.20. The four-Higgs vertex is given by The three-Higgs vertex is given by We also want to consider the h 1 C + C − vertex, which may be relevant in the discovery of the charged scalar. That vertex is given by where, in the 2HDM,

Results
As we know from subsections 3.1 and 3.2, in general λ 1 can take any value in between 0 and 4π/3. Once the constraint cos ϑ 1 > 0.9 is imposed, however, λ 1 can be no larger than ∼ 1; this is illustrated in figure 7. The closer cos ϑ 1 is to 1, the closer λ 1 must be to its SM value M 1 /(2v 2 ) = 0.258; note that λ 1 is almost always larger than its SM value when cos ϑ 1 > 0.9; the minimum value that we have obtained for λ 1 is 0.217. If cos ϑ 1 0.99, then the masses of the new scalar particles of the 2HDM, namely √ M C , √ M 2 , and √ M 3 can be no larger than ∼ 700 GeV; if cos ϑ 1 0.95, they can be no larger than ∼ 550 GeV. When cos ϑ 1 becomes close to 1, the masses of the new scalar particles all grow in tandem to reach O(TeV); this is illustrated in figure 8. Moreover, when c 1 → 1 the masses of all three new scalars become almost identical, as seen in figure 9. While √ M C − √ M 2 may be as large as 400 GeV if M 2 is close to zero (the minimum value that we have obtained for √ M 2 was 12 GeV), that mass difference becomes smaller than 100 GeV if both masses are larger than 1 TeV. (Note that M C may be either larger or smaller than M 2 ; also remember that by convention M 2 is always smaller than M 3 , but they may be smaller than M 1 .) We now come to the predictions for g 3 and g 4 in the 2HDM, which are depicted in figure 10.
One sees that g 3 and g 4 are broadly correlated with each other; g 3 may be up to three times larger than in the SM and g 4 may be up to six times larger than in the SM. An interesting feature is that g 3 may be zero or even negative, i.e. it may have sign opposite to the one in the SM. (We recall that the sign of g 3 is measured relative to the sign of c 1 ; we arrange that c 1 is always positive.) On the other hand, g 4 is always positive because     of the boundedness from below of the potential. Notice that, while g 3 has a much larger range in the 2HDM than in the SM2S, for g 4 the opposite thing happens-it may be as high as 0.45 in the SM2S, but no more than one half of that value in the 2HDM.
In figure 11 we depict the coupling g 1CC of the 125 GeV neutral scalar to a pair of charged scalars in the 2HDM. One sees that that coupling may be as large as 1,500 GeV (with either sign) when the mass of the charged scalars is close to 500 GeV.

The two-Higgs-doublet model plus one singlet
We consider in this section the two-Higgs-doublet model with the addition of one real SU(2) × U(1)-invariant scalar field S. We assume a symmetry S → −S. As a shorthand, we shall dub this model the 2HDM1S. The quartic part of the scalar potential is

Bounded-from-below conditions
Deriving necessary and sufficient BFB conditions for even a rather simple potential like the one in equation (69) is a notoriously difficult problem [31]. If V 4 were negative for some possible values of S 2 , φ † 1 φ 1 , φ † 2 φ 2 , and φ † 1 φ 2 , then V 4 would tend to −∞ upon multiplication of those four values by an ever-larger positive constant. Therefore, we want V 4 to be positive for all possible values of S 2 , φ † 1 φ 1 , φ † 2 φ 2 , and φ † 1 φ 2 . In order to guarantee this, we proceed in the following fashion.
Necessary condition 1: When S 2 = 0, equation (69) reduces to its first two lines, i.e. to the quartic potential of the 2HDM. Therefore, one must require the fulfilment of the conditions of subsection 3.2, viz. the four conditions in between equations (48) and (50).
Necessary condition 2: When φ † 1 φ 2 = 0, Since φ † 1 φ 1 , φ † 2 φ 2 , and S 2 are positive definite quantites, we must require [21] ψ > 0, (71a) After enforcing these two necessary conditions, we know that V 4 > 0 either when only the first two lines of the potential (69) exist or when only the third line exists. If we guarantee that the fourth line is always positive too, then we will be sure that V 4 is always positive. We therefore have the following 4 Sufficient condition: If, besides the two necessary conditions, then V 4 is BFB.
We have numerically found the absolute minimum of V 4 for any set of parameters of the potential (69) that satisfies the two necessary conditions but does not meet the sufficient condition (72). We have done this by using S 2 = 1 together with equations (51) and by minimizing V 4 in the domain r 2 > 0, 0 ≤ θ ≤ π/2, 0 ≤ h ≤ 1, and 0 ≤ α < 2π. If the minimum of V 4 is positive, then the set of input parameters is good, else the set of input parameters is bad and one must discard it.

Unitarity conditions
There are the same five scattering channels as in the 2HDM, cf. subsection 3.1; but the channel Q = T 3 = 0 has an additional scattering state S 2 . Additionally, there are two extra scattering channels: • The channel Q = 1, T 3 = 1/2 with the two states aS and cS.
• The channel Q = 0, T 3 = −1/2 with the two states bS and dS.
Both these channels produce a scattering matrix Channels 1 and 2 of subsection 3.1 again produce the scattering matrix (44). Channel (3) produces that matrix together with the additional eigenvalue I of equation (40). Channel (2) produces the scattering matrix (39a). Finally, channel 5 has the additional scattering state S 2 and therefore, instead of producing both the matrix M 1 of equation (39a) and the matrix M 2 of equation (39b), it produces M 1 together with Thus, the unitarity conditions for the 2HDM1S are the following: both |I| and the moduli of all the eigenvalues of the 2×2 matrix M 4 , of the 3×3 matrix M 3 , of the 4×4 matrix M 1 , and of the 5 × 5 matrix M ′ 2 must be smaller than 4π.

Procedure
Just as in the previous section, we utilize the Higgs basis for the two doublets, i.e. equations (2) and (60). We also write S = w + σ, where w is the VEV of the scalar S and σ is a field. The mass terms of the scalars are with cf. equation (61). One diagonalizes M as where R is a 4 × 4 orthogonal matrix. The squared mass M 1 is given by equation (6a). Without loss of generality, M 2 < M 3 < M 4 . Just as in the previous sections, we require The expression for the oblique parameter T is [22] and we demand −0.04 < T < 0.20. We input random values for the 15 real parameters M C , λ 1,2,3,4 , |λ 5,6,7 |, ψ, ξ 1,2 , |ξ 3 |, arg (λ * 5 λ 6 λ 7 ), arg (λ * 6 λ 7 ), and arg (λ * 6 ξ 3 ). We moreover input M 1 and v 2 given in equations (6). Then, 1. We require the input parameters to satisfy the BFB conditions of subsection 4.1this may imply a numerical minimization of V 4 to check that V 4 > 0.
2. We require the input parameters to satisfy the unitarity conditions written after equation (74).
3. We compute the VEV w from the condition that M 1 should be an eigenvalue of the matrix M.
4. We compute the full matrix M, its eigenvalues M 2,3,4 , and its diagonalizing matrix R; we choose the overall sign of R such that R 11 ≡ c 1 > 0.
5. We impose both the condition (78) and the condition that the oblique parameter T is within its experimental bounds.
6. We compute the couplings Figure 12: The differences between the masses of the two lightest non-SM neutral scalars and the mass of the charged scalar versus the mass of the charged scalar in the 2HDM1S. Green points have all the scalars with mass larger than 500 GeV; magenta points have all the scalars with mass larger than 1 TeV.

Results
In figure 12 we have plotted the differences among the masses of the scalars against the mass of the charged scalar. One sees that √ M C and √ M 3 cannot be more than ∼ 300 GeV from each other, but √ M 2 may be as much as 2 TeV smaller than both of them. In figure 13 we present a scatter plot of the mass of the lightest non-SM neutral scalar against c 1 . One sees that, contrary to what happens in the 2HDM (cf. figure 8), √ M 2 Figure 13: The mass of the lightest non-SM neutral scalar versus R 11 in the 2HDM1S. Green points have all the scalars with mass larger than 500 GeV; magenta points have all the scalars with mass larger than 1 TeV. may reach 1 TeV even when c 1 is as low as 0.9. We depict in figure 14 the three-and four-Higgs couplings g 3 and g 4 . The ranges of the couplings are markedly different from the analogous ranges in the previous two models. In particular, g 3 in the 2HDM1S may be some eight times larger than in the 2HDM, and thirty times larger than in the SM; while g 3 ∈ ]−30 GeV, 120 GeV[ in the 2HDM, g 3 ∈ ]−250 GeV, 1 TeV[ in the 2HDM1S. Also, g 4 in the 2HDM1S may be twice as large as in the 2HDM (0.48 instead of 0.24). In the 2HDM1S there is no clear correlation between g 3 and g 4 .
In figure 15 we have plotted the h 1 C + C − coupling g 1CC . Once again, that coupling in the 2HDM1S may be three times larger than in the 2HDM; very large values of g 1CC occur even for c 1 very close to 1.

Conclusions
In this paper we have emphasized that both the bounded-from-below (BFB) conditions and the unitarity conditions for the two-Higgs-doublet model (2HDM) are invariant under a change of the basis used for the two doublets. Therefore, one may implement those conditions directly in the Higgs basis, viz. the basis where only one doublet has vacuum expectation value. This procedure allows one to extract bounds on the masses and couplings of the scalar particles of the most general 2HDM, disregarding any symmetry that a particular 2HDM may possess.
We have also utilized this procedure for two other models, namely the Standard Model with the addition of two real singlets (SM2S) and the 2HDM with the addition of one real singlet (2HDM1S), in both cases with reflection symmetries acting on each of the singlets. We have discovered that there are large variations among the couplings in these   three very simple extensions of the SM. We have focussed on the three couplings g 3 (h 1 ) 3 , g 4 (h 1 ) 4 , and g 1CC h 1 C + C − , where h 1 is the observed neutral scalar with mass 125 GeV and C ± are the charged scalars of the 2HDM. We have found, for instance, that: • 0.6 < g 3 g SM 3 ≤ 1 and 0.6 ≤ g 4 g SM 4 < 15 is the SM2S. Thus, if one adds singlets to the SM, then g 3 becomes smaller but does not change order of magnitude, while g 4 may increase by one order of magnitude.
• −0.5 < g 3 g SM 3 < 3 and 0 < g 4 g SM 4 < 6 is the 2HDM. In this case g 3 may have opposite sign and g 4 may be smaller than in the SM. Also, g 1CC may be as high as 1.5 TeV (positive or negative) in the 2HDM.
• −6.5 < g 3 g SM 3 < 30 and 0 < g 4 g SM 4 < 15 is the 2HDM1S. This model distinguishes itself by the possibility of very large-either positive or negative-values of g 3 , and also large values of g 4 (uncorrelated with the ones of g 3 ). Once again, g 1CC may be of order TeV in the 2HDM1S.
A comparison of the predictions of the three models for g 3 and g 4 is depicted in figure 16.
We emphasize that our method may be used to obtain bounds and/or correlations among other parameters and/or observables of these models. Unfortunately, it may be difficult to generalize our work to more complicated models, both because they may contain too many parameters and because it is very difficult to derive full BFB conditions for even rather simple models.