Perturbative unitarity of Higgs derivative interactions

We study the perturbative unitarity bound given by dimension six derivative interactions consisting of Higgs doublets. These operators emerge from kinetic terms of composite Higgs models or integrating out heavy particles that interact with Higgs doublets. They lead to new phenomena beyond the Standard Model. One of characteristic contributions by derivative interactions appear in vector boson scattering processes. Longitudinal modes of massive vector bosons can be regarded as Nambu Goldstone bosons eaten by each vector field with the equivalence theorem. Since their effects become larger and larger as the collision energy of vector bosons increases, vector boson scattering processes become important in a high energy region around the TeV scale. On the other hand, in such a high energy region, we have to take the unitarity of amplitudes into account. We have obtained the unitarity condition in terms of the parameter included in the effective Lagrangian for one Higgs doublet models. Applying it to some of models, we have found that contributions of derivative interactions are not so large enough to clearly discriminate them from the Standard Model ones. We also study it in two Higgs doublet models. Because they are too complex to obtain the bound in the general effective Lagrangian, we have calculated it in explicit models. These analyses tell us highly model dependence of the perturbative unitarity bounds.


Introduction
Almost all experimental results are consistent with what the Standard Model (SM) predicts. The recent observations of a new boson [1] have also been predicted by the model as the Higgs boson. Measured properties of the observed particle are still consistent with the SM. However, the model seems to be the effective theory describing physics below the electroweak (EW) scale since it has several theoretical and experimental problems which are maybe explained at the TeV scale.
One of those problems is how to break the EW gauge symmetry. Since the symmetry is broken by hand in the SM, the model does not tell us why and how the EW symmetry is broken. Therefore, we expect the SM should be extended to a model including other sectors responsible for the physics of the electroweak symmetry breaking (EWSB) and phenomena beyond the SM. If the scale is much higher than the EW scale, the observed Higgs mass requires subtle mechanisms or fine tuning. Therefore we expect something new to appear in the TeV region. Eventually, new structure is expected to appear at the TeV scale in order to obtain the EW scale without those artificial constructions.
Many models describing physics beyond the SM have been proposed. The Higgs sectors of these models are extended, and they break the EW symmetry with a certain mechanism. The effects of these new sectors could be first observed as deviations from the SM ones via higher-dimensional operators at a scale lower than their original scales.
When we consider physics beyond the SM with extended Higgs sector, its low energy effective theory probably includes dimension-six derivative interactions as a part of the higher-dimensional operators. These operators have two origins: expansion of kinetic terms if the Higgs doublet is realized as part of a pseudo Nambu Goldstone (NG) field; integrating out heavy new scalar/vector bosons that interact with the Higgs field. The latter case appears even in models including an elementary Higgs field.
If the Higgs boson were removed from the SM, the Higgs sector would be described by the SU (2) L × SU (2) R /SU (2) V nonlinear sigma model. Derivative interactions of NG fields emerge from the kinetic term. These interactions contribute to scattering among longitudinal massive gauge bosons through the equivalence theorem and cross sections of these processes become larger and larger as the energy increases. They finally become so large as to violate perturbative unitarity around 1 TeV [2]. Of course, the recent observation of the Higgs boson showed us the absence of unitarity violation and the validity of the SM description even much above the TeV scale. We confront the similar problem in studying derivative interactions of Higgs doublets.
In this paper, we find the scales where the given perturbative description is available in several models that include derivative interactions of the Higgs doublets.
The rest of this paper is organized as follows. In Sect. 2, we study the unitarity bound given by derivative interactions in one Higgs doublet models (1HDMs). This is extended to the case of the two Higgs doublet models (2HDMs) in Sect. 3. In both of these sections, the unitarity violation scales are explicitly calculated with several models. Finally, our study is concluded in Sect. 4.

Unitarity of derivative interactions in one Higgs doublet models
The perturbative unitarity bounds given by the derivative interaction are discussed on 1HDMs. First, we derive the formula of the unitarity bound and investigate its general properties. Then results are applied to explicit models. The formulae for perturbative unitarity are shown in App. A.

Formulae and general properties of the unitarity bound
The effective Lagrangian of derivative interactions in 1HDM is 3 where f is a scale related to new physics and For the second operator, we replace the covariant derivatives with partial ones because in this paper we consider only longitudinal modes of the gauge bosons. Since the latter term violates the custodial symmetry, our analysis is based on the Lagrangian with c T = 0 4 . Since our focus is entirely on the four point scattering processes given by Eq. (2.1), the vacuum expectation value (VEV) of the Higgs boson plays no role in the following calculation. Therefore, we use where C + /N are a charged/complex-neutral scalar fields. The charged scalar and imaginary part of the neutral scalar are respectively eaten by W ± and Z bosons. Using the above notation, the following amplitudes are obtained 5 : whereŝ andt are the Mandelstam variables and we consider the energy scale where particles can be treated as massless, i.e.,ŝ +t +û = 0.
where we have assumedŝ ≫ m 2 h and have used the result given by Eq. (2.5). As we will see later, the typical unitarity violation scale is a few times larger than the decay constant. Therefore, the above effects are small enough to be neglected. Following Ref. [2], we construct matrices with partial wave amplitudes. The largest eigenvalue of these matrices gives us the strongest bound to the perturbative unitarity. We have found that the zeroth mode gives the strongest bound in 1HDMs, so we focus on this case. With the formulae in App. A, the strongest bound is given by the largest eigenvalue of the following matrix: The perturbative unitarity condition is thereforeŝ Assuming that derivative interactions are purely given by the kinetic term of the nonlinear sigma model, the conservative cut-off scale is expressed in terms of the decay constant, i.e., Λ ∼ 4π f . 6 Using the relation, the unitarity bound is related to the cut-off scale asŝ  [5]. It is therefore necessary to clarify the valid energy scale in the description for each model. We apply the result to cross sections of the scattering of the Higgs boson and longitudinal modes of massive gauge bosons, so called vector boson scattering (VBS) processes, with the equivalence theorem. Since these energy scatterings are dominated by the coefficient, c H , with the custodial symmetry, all of the cross sections are proportional to each other. Here we focus only on the process W + L W − L → hh, and relations with the others are shown in Table. 1. Considering this sort of process, we must remember the importance of the central region 7 which is pointed out in Ref. [6], so that we also show the ratios between the cross section of the Higgs pair production and those of the other processes with the central region cut. The cross section of For this process, Fig. 2.1 shows the region where perturbative unitarity is violated . Assuming that cross sections reach the above bound at √ŝ = 3 TeV, we can obtain the relation: If c H ∼ 1, the effect of the derivative interaction in the process is comparable to the SM background of about √ŝ = 2 TeV, where the cross section is 3 × 10 4 fb without the central region cut; see Ref. [6]. Note that the value of f is typically related to new particle masses. For example, in the little Higgs scenario [7], the top partner mass is given by O( f ). From the viewpoint of the fine tuning, f is required to be below about 1 TeV.
In the Full/Central column, the cross sections of VBS subprocesses with/without the central region cut are shown.

Examples with explicit models
In the rest of this section, we study the unitarity bounds on two models: the minimal composite Higgs model [8] and the littlest Higgs model with T-parity [9]. The latter model has previously been studied in Refs. [10,11]. In Ref. [11], several Little Higgs models are also investigated. 8 Since the normalization of decay constants can be changed, the combination f 2 /c H is meaningful. Here, we follow the normalization given in the original papers. Decay constants have physical meanings through masses of additional massive vector bosons and fermions in each model. The Higgs doublet is embedded in such a way as to preserve the custodial symmetry in both models.

The minimal composite Higgs model
This model is described by the SO(5)/SO(4) nonlinear sigma model including four NG fields [8]. They are identified as the Higgs doublet.
The Lagrangian is where h is the real scalar multiplet of four NG bosons and h is its norm. Expanding these trigonometric functions, we obtain Using Eq. (2.6), the relation between the decay constant and the energy scale of the unitarity violation iŝ (2.14) Assuming that perturbative unitarity is violated at 3 TeV, the decay constant is about 750 GeV. On the other hand, if the decay constant is chosen as 500 GeV, the perturbativity is preserved up to about 2 TeV, where the cross section of W + L W − L → hh is 7 × 10 5 fb. In this case, the cross section of the Higgs boson pair production is one order of magnitude larger than that given by the SM. However, it is challenging to observe this process because the main decay mode is hh → 4b, which is overwhelmed by the QCD background.

The littlest Higgs model with T-parity
Derivative interactions on the littlest Higgs model with T-parity [9] are shown below. Scalar fields are described by the SU (5)/SO(5) nonlinear sigma model which includes 14 NG bosons.
The kinetic term of this model is where Π is the NG field. The Higgs doublet is assigned in the NG field as We omitted the other NG bosons since they don't contribute to the current analysis. Extracting the derivative interaction from the kinetic term, we obtain This result is consistent with previous works [10,11].
If we suppose that f = 750 GeV, the perturbative unitarity is preserved up to about 4 TeV. Hence this model description is valid in higher energy scales while the signals of the derivative interaction are smaller than the previous model.

Unitarity of derivative interactions on two Higgs doublet models
In this section we extend the previous discussion to dimension-six derivative interactions including two Higgs doublets.
The modification is straightforward, and the prescription is also simple. However, the formulae become too complex because of the many degrees of freedom (DOF). Then we cannot obtain the formula of the strongest bound like Eq. (2.6) with the largest eigenvalue of a matrix that consists of partial wave amplitudes. Since the matrix can be diagonalized in individual models, three models are investigated as examples.
In this section, we consider processes whose initial states are electromagnetically neutral. Matrices giving the unitarity bounds for singly or doubly charged initial states are also shown in App. B.

Formulae and general properties of the unitarity bound
The analyses in this section are based on the following effective Lagrangian: where and In the case where we study the custodial symmetric models, the above coefficients are real and follow the relation derived in App. C: The 2HDMs require mixing angles to get mass eigenstates of scalar fields. In this paper, we use the equivalence theorem and focus on only derivative interactions, that is, masses of scalar fields are neglected. In this case, the perturbative unitarity bound is independent of mixing angles. This is also true for models including N Higgs doublets.
The unitarity bound is expressed asŝ where C max is the largest eigenvalue of the matrices given in App. B. 9 . As we will see later, the largest eigenvalue |C max | can be as large as about 10. In this case, the unitarity bound becomes quite strong and leads us to an interesting remark. Consider, for instance, the pair production of a heavy particle whose mass is O( f ) in VBS processes; the energy scale where the pair is produced could be as large as the unitarity violation scale. This means that we couldn't discuss this kind of process by means of these low-energy descriptions.

Examples with explicit models
We study the consequences of the above result with several models including two Higgs doublets. The following three models are studied: the bestest little Higgs model [12]; the UV friendly T-parity little Higgs model [13]; and an inert doublet model. The first and second ones are composite Higgs models and the last one is a toy model including elementary Higgs doublets.

The bestest little Higgs model
The bestest little Higgs model [12] is a little Higgs model which includes two Higgs doublets. We obtain 15 NG fields that parametrize the SO(6) × SO(6)/SO (6) coset. The normalization of the kinetic term is the same as Eq. (2.15), and the NG field is where h 1,2 are real scalar multiplets considered two Higgs doublets and the other NG bosons are eliminated. In this model, Higgs doublets interact with heavy gauge bosons and a singlet scalar. The masses of the heavy gauge bosons depend on the other decay constant that is larger than f in order to avoid the constraints from the electroweak precision measurement (EWPM). Thus the effects coming from the heavy gauge bosons are tiny, and we neglect them. The interaction with a singlet is required to obtain a collective quartic coupling. For simplicity, we introduce the following terms to see the effect: where σ is a neutral singlet scalar 10 . Including this contribution, the coefficients of the derivative interactions are where (3.14) The unitarity bound depends on the value of c σ because the largest eigenvalue is a function of it. For 0 ≤ c σ < 1/8, the bound isŝ For c σ = 0, it is bounded asŝ (3.16) and it becomes weak as c σ increases. For c σ = 1/8, the bound is the weakest: In the region, 1/8 < c σ , the bound isŝ 10 In the original paper [12], m σ = λ 65 + λ 56 f and λ = λ 65 −λ 56 where the right-hand side decreases as c σ increases and the bound becomes the same as the case of c σ = 0 at c σ = 1/7. The unitarity bounds for the cross sections of W + L W − L → hh and W + L W + L → W + L W + L are displayed below. We define the mass eigenstates, h and W ± L , as follows: where N R i is the real part of N i and α and β are mixing angles. Unitarity bounds for these processes are Here the parameters c x and s x are cos x and sin x, and C max = (2 − c σ )/2 for 0 ≤ c σ < 1/8 and C max = (1 + 7c σ )/2 for 1/8 ≤ c σ . If α = β is satisfied, the so-called decoupling limit, we get the relation: The perturbative unitarity bounds of W + L W − L → hh are shown in Fig. 3.1. In order to see the effects of the new parameters, we fix the decay constant to be 750 GeV. The shaded regions in these figures are changed in response to the mixing angles because the cross section depends on the angles. However, the unitarity bound itself depends only on the coefficient, c σ . Hence we can see that the energy scales where each cross section line intersects the unitarity violation regions are independent of the angles, e.g., √ŝ ∼ 1.9 TeV for c σ = 1. For β = 0 and α − β = π/6, the cross sections are independent of the value of c σ ; thus, we have only one line but still the intersecting points are the same.

The UV friendly little Higgs model
The UV friendly T-parity little Higgs model [13] also includes two Higgs doublets as a part of the 14 NG bosons given by the SU (6)/Sp(6) nonlinear sigma model. The normalization of the kinetic term is also the same as Eq. (2.15). This model possesses Z 2 symmetry, so-called T-parity, and one of the Higgs doublets is T-odd. This doublet has no VEV. Since we study only Higgs doublets, the NG field Π can be considered as follows: 11  The decay constant is fixed to be 750 GeV. The mixing angles are set to be (β , α − β ) = (π/6, 0) (upper left), (0, π/6) (upper right), (π/4, π/6) (lower left) and (π/6, π/4) (lower right). The light gray, dark gray and black lines are cross sections for c σ = 1, 0, and 1/8, respectively. The unitarity violation regions depend on the value of c σ , and their brightness corresponds to each line.  These coefficients apparently violate the custodial invariant conditions, Eqs. (3.5) and (3.6). In this model only one of the Higgs doublets has the VEV, so that tree-level contributions to ρ parameter do not appear. With these coefficients, the strongest bound isŝ f 2 π. (3.30) Assuming that perturbative unitarity is violated at 3 TeV, the decay constant, f , is determined as 1.7 TeV. This value looks large from the viewpoint of fine tuning as we have already seen. On the other hand, if the decay constant is about 1 TeV, the unitarity is broken below about 1.7 TeV. The unitarity bounds of W + L W − L → hh and W + (3.32) Note that the cross sections have no mixing angle dependence because only one of the Higgs doublet gets a VEV. These bounds to the cross sections are shown in Fig. 3.2. They correspond to the case c σ = 15/7 for the bestest little Higgs model. The unitarity bound of this model is severe because the largest eigenvalue is much larger than the previous models.

Inert doublet models with odd scalars
We investigate the following Lagrangian consisting of elementary scalar and vector fields: Scalar fields φ 0 , φ a L , and φ a T , are respectively 1 0 , 3 0 , and 3 1 representations of SU (2) L × U (1) Y , and vector fields V 0 , V S , and V a L , are, respectively, 1 0 , 1 1 , and 3 0 representations. We suppose that these new particles and H 2 are odd under an additional Z 2 symmetry, and H 1 and the other SM particles are even under the discrete symmetry. We consider the case that only one of the Higgs doublets has a VEV, such as in the model in Sec. 3.2.2. These choices of couplings and masses for φ a L and φ a T , and V 0 and V S are required to respect SO(4) symmetry. 12 This set up suppresses contributions to the oblique corrections.
After integrating out heavy particles, we obtain the following coefficients of the derivative interactions: where (3.39) Even if additional particles exist, their contributions are included in these four coefficients. We cannot discriminate these multiple contributions from a large contribution of a particle with a large coupling. Using these coefficients, the eigenvalues of Eq. (B.1) are fortunately obtained as the following simple forms: The strongest unitarity condition is given by Eq. (3.7) with the largest eigenvalue in the above. Derivative interactions generated by integrating out T-odd heavy particles must include two H 1 and two H 2 , as in Eqs. (3.35) and (3.37). Furthermore there are no mixing angles in the Higgs doublets because only one of the Higgs doublets has a VEV. They are the reason why cross sections of W + L W − L → hh and W + L W − L → W + L W − L vanish. In this model we have four coefficients Eqs. (3.38) and (3.39) to parametrize the dimension-six differential operators. If we suppose that s 0 = s L = v 0 = v L = 1, the eigenvalue c I 2 becomes the largest: c I 2 = 8. This value gives us a perturbative unitarity condition which is the same as Eq. (3.30).
The unitarity bound, Eq. (3.7), can be interpreted as the perturbativity condition of couplings. For example, if s 0 = s L = v 0 = 0 and v L = 0, we get |C max | = 9v L /2. Then the unitarity bound is In order to preserve unitarity, the unitarity violation scale should be larger than the mass, m vL . As a result, we get the following condition: If the model includes only one 3 0 vector, this requirement is stronger than the naive perturbativity condition g L < 4π. On the other hand, if it includes several 3 0 vectors, the unitarity bound also limits how many there are.

Conclusion
We have studied perturbative unitarity for dimension-six derivative interactions of the Higgs doublets. They are generated by kinetic terms in composite Higgs models, or by integrating out heavy particles that interact with the Higgs doublets. The latter case means that derivative interactions appear even in models consisting of elementary Higgs doublets. We first studied the unitarity bounds in models including only one Higgs doublet. The strongest bounds are expressed by the largest eigenvalue of the matrix given by partial wave amplitudes of VBS processes. We focused on the high-energy region such that derivative interactions could dominate the contributions to the scattering among longitudinal vector bosons. Assuming that the given derivative interactions respect EWPM, only a combination of parameters, c H / f 2 , appears in the analysis. Therefore, the unitarity condition is expressed by the parameter as Eq. (2.6). We have applied it to the cross section of W L W L → hh.
We have calculated the bounds on explicit models: the minimal composite Higgs model, the littlest Higgs model. Their structures of global symmetry are significantly different from each other; SO(5)/SO (4) and SU (5)/SO (5). However, the given bounds are similar; c H = 1 and 1/2. The decay constants f are related to the masses of the top like fermions in composite Higgs models. It is therefore supposed that f / √ c H is larger than about 500 GeV; see e.g., Ref [14], where the perturbative unitarity is violated above the region √ŝ 2 TeV. Even in this case, it is difficult to obtain cross sections large enough to distinguish new physics contributions from the SM ones.
Secondly, similar analyses have been performed in 2HDMs. A simple formula for the unitarity bound could not be obtained in terms of parameters included in the effective Lagrangian (3.1) since the matrix of partial wave amplitudes is too complex to be diagonalized. Hence we have investigated the unitarity bound with explicit models: the bestest little Higgs model; the UV friendly T-parity little Higgs model; and the inert doublet model with heavy Z 2 odd particles. The first and the second ones are literally a kind of little Higgs model and the third one is a toy model including elementary Higgs doublets.
In the first one, derivative interactions are generated not only by the kinetic term but also by the integrating out of a heavy scalar field. The constraints of the former to the unitarity are similar to those given by the 1HDMs discussed in Sec. 2. Including the latter one, the largest eigenvalue depends on the scalar contribution. The unitarity bound can be stronger than the case not including it.
In the second model, the unitarity condition is much more severe compared to the other models mentioned in this paper. This is because the coefficients of the derivative interactions are large in this model. Therefore the unitarity is violated at a low scale compared with the other models. In this kind of model, large coefficients can produce large cross sections, large enough to exceed the SM background. On the other hand, assuming that the masses of additional particles are near the decay constant, they are also near the scale of the unitarity violation. Therefore, contributions of vector resonances probably need to be considered when people investigate, for instance, pair productions of these additional particles with vector boson collisions.
In the last model that includes an elementary Higgs doublet, VBS processes of the SM particles are suppressed as we have shown. In this kind of model, the masses of heavy particles should be smaller than the unitarity violation scale. This condition means that couplings between Higgs doublets and heavy particles are much smaller than the strong coupling, 4π, or the number of these particles is limited.
As a conclusion, we have clarified the importance of studying the unitarity bound when Higgs derivative interactions are investigated because the bound can be significantly lower than the naive cut-off scale.

A Perturbative unitarity
The amplitudes of elastic scattering satisfy the following relation for each partial wave: Eq. (A.1) is the equation of the circle with radius 1/2λ and center (0, 1/2λ ). In the high-energy limit where the masses of produced particles can be neglected, the radius of the circle becomes the maximum. Therefore, the actual amplitudes are in the maximal circle. Finally, partial wave amplitudes at least satisfy M R n ∈ [−1/2, 1/2] and M I n ∈ [0, 1]. If we consider processes involving identical particles in the final state, the bound becomes weaker as M R n ∈ [−1, 1] and M I n ∈ [0, 2]. Considering the unitarity bound for the derivative interactions, in the massless limit, we can express the amplitudes produced by the derivative interactions as whereŝ andt are the Mandelstam variables. The zeroth and the first modes of partial wave amplitudes appear: For type III derivative interactions, the situation is different, that is, certain combinations of SU (2) R violat-  After imposing these conditions to ensure SO(4) symmetry on the derivative interactions, we get their remaining DOF. The result is shown in Table. 2.