Renormalization of the Higgs sector in the triplet model

We study radiative corrections to the mass spectrum and the triple Higgs boson coupling in the model with an additional Y=1 triplet field. In this model, the vacuum expectation value for the triplet field is strongly constrained from the electroweak precision data, under which characteristic mass spectrum appear at the tree level; i.e., $m_{H^{++}}^2-m_{H^+}^2\simeq m_{H^+}^2-m_A^2$ and $m_A^2\simeq m_H^2$, where the CP-even ($H$), the CP-odd ($A$) and the doubly-charged ($H^{\pm\pm}$) as well as the singly-charged ($H^\pm$) Higgs bosons are the triplet-like. We evaluate how the tree-level formulae are modified at the one-loop level. The $hhh$ coupling for the standard model-like Higgs boson ($h$) is also calculated at the one-loop level. One-loop corrections to these quantities can be large enough for identification of the model by future precision data at the LHC or the International Linear Collider.

extensions of the Higgs sector in order to explain these phenomena. First, by imposing an unbroken discrete symmetry to the Higgs sector, we can obtain dark matter candidates, such as the inert doublet model [7]. Second, the type-II seesaw model [8], where a Higgs triplet field is added to the SM, can generate neutrino masses at the tree level. Radiative seesaw models [9][10][11][12][13] can also explain neutrino masses at loop levels, where additional scalar bosons, e.g., charged scalar bosons, are running in the loop. Finally, the scenario based on the electroweak baryogenesis can explain the baryon asymmetry of the Universe by an additional CP-phase and nondecoupling effects in extended Higgs sectors [14,15] .
Discoveries of additional scalar bosons such as charged Higgs bosons will be direct evidence of extended Higgs sectors. In addition, as indirect signatures, detecting the deviation of the Higgs boson couplings with gauge bosons hV V , those with quarks and leptons hf f and Higgs selfcoupling constant hhh, where h is the SM-like Higgs boson, from those predicted values in the SM can also be evidence of extended Higgs sectors. In particular, once h is discovered at the LHC in near future, precision measurements of these coupling constants at the LHC and at the International Linear Collider (ILC) turns to be very important. Therefore, in order to distinguish various extended Higgs sectors, it is necessary to prepare precise calculations for the Higgs boson interactions including radiative corrections.
In this Letter, we construct the renormalization scheme for the one-loop calculation of the observables in the Higgs sector such as mass spectrum and coupling constants in the Higgs triplet model (HTM), where the hypercharge Y = 1 Higgs triplet field is added to the SM. The HTM is motivated to generate tiny neutrino masses by the type-II seesaw mechanism. This model is unique because the electroweak rho parameter deviates from unity at the tree level. In such a model, unlike the SM, four independently measured parameters such as α em , G F , m Z and sin 2 θ W are necessary to describe the electroweak observables [16]. In the model with the Y = 0 Higgs triplet field, the rho parameter and the W boson mass were calculated at the one-loop level in the on-shell scheme in Ref. [16,17]. Recently, in the HTM, one-loop corrections to the rho parameter as well as the W boson mass have been studied [18], and it is clarified that relatively large mass differences among the triplet-like Higgs bosons are favored by the precision data at the LEP/SLC [3,19]. However, radiative corrections to the Higgs sector with renormalization have not been studied in the HTM so far.
In the HTM, there are seven physics scalar states; i.e., the doubly-charged H ±± , the singlycharged H ± , a CP-odd A as well as two CP-even (h and H) scalar states. When the vacuum expectation value (VEV) of the triplet field v ∆ is much smaller than that of the doublet field v φ as required by the electroweak precision data, h behaves as the SM-like Higgs boson while H ±± , H ± , H and A are the triplet-like ones. Phenomenology for the HTM has been studied in Refs. [20? -24].
When the triplet-like Higgs bosons are degenerated in mass, H ++ mainly decays into the same sign dilepton or the diboson depending on the magnitude of v ∆ . On the other hand, in the case with the mass difference among the triplet-like fields, cascade decays of the triplet-like Higgs bosons can be dominant [22][23][24]. For example, when H ++ (H or A) is the heaviest of all the triplet-like Higgs bosons, the main decay mode of H ++ (H or A) can be H ++ → H + W +( * ) → HW +( * ) W +( * ) or ). In any case, measuring the masses of the triplet-like Higgs bosons is important to test the HTM.
For v 2 ∆ /v 2 φ ≪ 1, characteristic relationships among the masses of the triplet-like Higgs bosons are predicted at the tree level [24]: where m H ++ , m H + , m H and m A are the masses of H ±± , H ± , H and A, respectively. These formulae are useful to distinguish the model from the others which also contain charged scalar bosons when the masses of the triplet-like Higgs bosons are measured with sufficient accuracy. In particular, in order to compare the mass spectrum with future precision measurements, it is very important to evaluate the radiative correction to the above tree-level formulae. We first define A ) − 1 in order to investigate how the mass formulae in Eq. (1) are modified by the quantum effect. We find that the magnitude for ∆R can be as large as 10% when we take parameter sets favored by the electroweak precision data: [18].
The theoretical prediction on the triple Higgs boson coupling hhh is also quite important because the Higgs potential is reconstructed by measuring the mass of h and the coupling of hhh. In the case of the two Higgs doublet model, radiative corrections to the coupling of hhh have been studied by using the effective potential method in Ref. [25] as well as the diagrammatical method in the on-shell scheme in Ref. [26], in which the renormalized hhh coupling can deviate from the SM prediction by about 100% due to the nondecoupling property of the additional scalar bosons without contradiction of the perturbative unitarity [27]. We here calculate the renormalized hhh coupling in the HTM, and find that the deviation of this coupling from the SM prediction can also be of

II. MODEL
The HTM is one of the minimal extension of the SM, where the Higgs triplet field ∆ with Y = 1 is added to the SM. The most general Higgs potential under the SU (2) L × U (1) Y gauge invariance is given by The Higgs fields are parameterized as where v φ and v ∆ are the VEVs of the doublet field and the triplet field, respectively, which satisfy When the triplet field ∆ carries lepton number of two units (L = −2), the µ term breaks the lepton number, so that it is the source of Majorana masses of neutrinos.
The tree-level contributions to the tadpoles for the component fields ϕ 0 and δ 0 are given by By using the stationary conditions T φ = T ∆ = 0, we obtain where The mass of the doubly-charged scalar bosons ∆ ±± (≡ H ±± ) is given as The mass matrices can be expressed for the singly-charged scalar states in the basis of (w + , ∆ + ) and the CP-odd scalar states in the basis of (z 0 , ξ 0 ) as The mass eigenstates of the scalar fields are obtained by diagonalizing these mass matrices by the mixing angle β and β ′ as where G ± and G 0 are the Nambu-Goldstone bosons which are absorbed by the longitudinal component of the weak gauge bosons, and the mixing angles are given by We then obtain the masses of the singly-charged scalar bosons and the CP-odd scalar boson as For the CP-even scalar states, the mass matrix is given in the basis of (ϕ 0 , δ 0 ) by By the diagonalization of this matrix with the mixing angle α as we obtain the masses of the CP-even Higgs bosons and the mixing angle: where (M 2 CP−even ) ij , (i, j = 1, 2) are elements of the mass matrix in Eq. (15). From Eqs. (7), (8), (9), (13), (14), (17), (18) and (19), eight parameters: m, M , µ and λ 1 -λ 5 in the Higgs potential can be translated into five mass parameters: m H ++ , m H + , m H , m A and m h , one mixing angle α and two VEVs: v φ and v ∆ . In the next section, we take v as one of the input parameters instead Masses of neutrinos are generated through the Yukawa interaction: where h ij is a 3 × 3 symmetric complex matrix, L i L is the i-th generation of the left-handed lepton doublet. After the electroweak symmetry breaking, the neutrino masses are obtained as It can be seen that when the lepton number violating parameter µ is taken to be of eV which is required by the data. In such a case, the HTM can be tested at TeV-scale collider experiments. The smallness of µ is just the assumption in the minimal version of the HTM.
However, if the parameter µ is forbidden at the tree level but is generated at higher orders of perturbation, such a tiny value of µ may be explained in a natural way [28].
In the HTM, the electroweak rho parameter ρ deviates from unity at the tree level: The experimental value of the rho parameter ρ exp is quite close to unity: ρ exp = 1.0008 +0.0017 −0.0007 [29], so that the value of v ∆ is bounded from above: The relation in Eq. (23) can also be described as The mass formulae Eqs. (23) and (24) can be understood by considering global symmetries in the Higgs potential in Eq. (3), which is invariant under the SU (2) L × U (1) Y gauge symmetry. The potential has accidental global symmetries in some specific cases as discussed below. First, let us consider the case where the µ term is absent in the potential. The potential then respects the global U (1) symmetry which conserves the lepton number. Unless the lepton number is spontaneously broken the triplet field does not carry the VEV; i.e., v ∆ = 0. The terms of λ 2 and λ 3 then do not contribute to the four mass parameters m H ++ , m H + , m A and m H , so that they are described by the λ 4 and λ 5 couplings. Therefore, two of the four are independent and the rest two are determined via Eqs. (23) and (24). Next, we consider the case where both the µ term and the λ 5 term are zero.
In this case, an additional global SU (2) symmetry (we call this symmetry as SU (2)

III. RENORMALIZATION OF THE HIGGS SECTOR
In this section, we define the scheme for the one-loop calculations by setting the renormalization conditions for the parameters of the HTM in the similar way to the case in the two Higgs doublet model constructed in Ref. [26].
First of all, while at the tree level the tadpoles T h and T H are set to be zero, at the one-loop level they are nonzero and used to eliminate the one-point functions of h and H. We here prepare the tadpole counter-terms δT h and δT H by shifting the tadpoles as where i represents H ++ , H + , A, H and h. In addition to these counter-terms, those for mixing angles as well as the wave function renormalization factors are defined by In order to determine these 22 counter-terms δT h , δT H , δv, δv ∆ , δm 2 i , δZ i , δZ G 0 ,δZ G + , δα, δβ, δβ ′ , δC GH , δC GA , and δC hH , we give the renormalization conditions below.
We first discuss δv and δv ∆ . At the tree level, the VEVs v and v ∆ are determined together with the gauge coupling constants g and g ′ for SU(2) L and U(1) Y by the four electroweak observables whereŝ 2 W is defined as the ratio of the coefficients of the vector part and the axial vector part in the Zēe vertex as in Refs. [16,18] We note that in the HTM, because the rho parameter deviates from unity at the tree level, the relation of sin 2 θ W = 1 − m 2 W /m 2 Z does not hold. Thus, we have to introduceŝ 2 W as the independent input parameter to describe the electroweak parameters. Consequently, the counter-terms δv and δv ∆ are determined as where δ V B and δ ′ V are the vertex and box diagram corrections to the muon decay process and the vertex diagram corrections to the Zēe vertex, respectively, whose explicit formulae are given in Refs. [16,18], and Π AB T (p 2 ), for (AB = W W , ZZ, γγ, γZ) are the transverse component of the gauge boson two-point functions. The counter-terms δβ and δβ ′ are related to δv and δv ∆ as The tadpole counter-terms δT h and δT H are determined so as to eliminate the one-point function for h (Γ h ) and H (Γ H ) at the one-loop level: The counter-terms for doubly-charged Higgs bosons (δm 2 H ++ and δZ H ++ ) are determined by the on-shell conditions as where Γ XY (p 2 ) represents the two-point function for XY . The four counter-terms relevant to the singly-charged scalar states (δm 2 H + , δZ H + , δZ G + and δC GH ) are determined by Similarly, the four counter-terms relevant to the CP-odd scalar states (δm 2 A , δZ A , δZ G 0 and δC GA ) are determined by Finally, the six counter-terms relevant to the CP-even scalar states (δm 2 h , δm 2 H , δα, δZ h , δZ H and δC hH ) are determined by the following on-shell conditions Therefore, all the 22 counter-terms are determined completely by the above conditions, and oneloop calculations for the other observables are now predictable.
We note that Γ G + G − (0) = 0 and Γ G 0 G 0 (0) = 0 are automatically satisfied as a consequence of the Nambu-Goldstone theorem, when we impose the renormalization conditions of Γ G + H − (0) = 0 and Γ GA (0) = 0 in Eqs. (40) and (43), respectively. We also note that because of the Ward-Takahashi identities the conditions in Eqs. (40) and (43) are respectively equivalent to the conditions on the mixings between W ± and H ± and between Z and A: where Γ W H (m 2 W ) and Γ ZA (m 2 Z ) are defined from the W ± H ∓ and ZA two-point functions as

IV. RESULTS
We here evaluate observables at the one-loop level by using the renormalization method defined in the previous section. First, we calculate the deviation from the tree-level formula among the masses of the triplet-like Higgs bosons in Eq. (25) which holds approximately in the limit of v 2 ∆ /v 2 ≪ 1. Then, we calculate the one-loop contribution of the triplet-like Higgs bosons to the triple Higgs boson coupling hhh.
At the tree level, R can be expressed in terms of the quartic coupling constants as Under the situation of v 2 ∆ /v 2 ≪ 1, we have In the limit of v 2 ∆ /v 2 → 0, R = 1, so that the squared mass of A is no more an independent parameter, which is determined by (m 2 A ) tree = 2m 2 H + − m 2 H ++ . The one-loop corrected quantity of R can be written as where ∆R is the one-loop correction to R in the limit of v ∆ /v → 0, which is evaluated as The squared renormalized pole mass of A, (m 2 A ) pole , is not the input parameter in this limit. Therefore, we cannot adopt the on-shell renormalization condition for A given in Eq. (42) in the present case. Instead, (m 2 A ) pole is predicted in this case by where Π 1PI XY are the 1PI diagrams for the two-point functions of XY , and T 1PI h,H are the 1PI diagrams for the one-point functions of h and H. Consequently, we obtain We note that ∆R is a function of three input parameters, e.g., m H ++ , ∆m (= m H ++ − m H + ) and m h , because this quantity is evaluated in the limit of v ∆ /v → 0. We here comment on the contribution of the term which is proportional to v 2 ∆ /v 2 to R. As we mentioned before, the upper bound of v ∆ is about 8 GeV by the electroweak data, which means v 2 ∆ /v 2 10 −3 . Therefore, this contribution to R can be neglected, and main contribution to R should come from ∆R. In the following numerical analysis, we take m h = 125 GeV and α = 0.
In Fig. 1 (left), ∆R is shown as a function of m H ++ for several values of ∆m (< 0). From the electroweak precision data, ∆m = −O(100) GeV is favored for m H ++ = O(100) GeV [18]. It is found that negative values of ∆R are predicted in the case with non-zero values of ∆m. The magnitude of ∆R is 0.06 to 0.13 when we take 100 GeV< m H ++ < 300 GeV and ∆m = −100 GeV.
In Fig. 1 (right), ∆R is shown as a function of the pole mass of A for several values of ∆m (> 0) although these cases are not necessarily favored by the electroweak precision data [18]. The mass of the SM-like Higgs boson h is again taken to be 125 GeV. Positive values of ∆R are predicted.
The magnitude of ∆R is 0.11 to 0.17 for 100 GeV< m A < 300 GeV and ∆m = +100 GeV.
We here give a comment regarding to the current experimental bound for the mass of the doubly-charged Higgs boson at the LHC. When v ∆ 10 −4 GeV and ∆m ≤ 0, the doubly-charged Higgs boson can mainly decay into the same sign dilepton. In such a case, the mass of the doublycharged Higgs boson is bounded from below about 300 GeV and about 400 GeV by the results from the ATLAS collaboration and the CMS collaboration [30], respectively. Although we perform the numerical analysis in the limit of v ∆ /v → 0, results are not so changed even if we take v ∆ > 10 −4 GeV, because of the tiny contribution of O(v 2 ∆ /v 2 ). In the case of v ∆ > 10 −4 GeV and ∆m ≤ 0, the main decay mode of H ±± can be the same sign diboson, so that the current bound for m H ++ cannot be applied.
Next, we discuss the one-loop contributions of the triplet-like Higgs bosons to the triple Higgs boson coupling hhh for the SM-like Higgs boson h. We calculate the hhh vertex function at the one-loop level both in the diagrammatic method in the on-shell scheme defined in the previous section and in the effective potential method. The leading contribution to the deviation in the hhh coupling constant λ hhh from the SM prediction is obtained for Neglecting the terms proportional to v 2 ∆ /v 2 , ∆λ hhh /λ We have confirmed the correctness of this result in both the methods of calculation; i.e., the diagrammatical method in the on-shell scheme and the effective potential method.
We note that the one-loop contribution of the triplet-like Higgs bosons are quartic power-like in mass, so that it does not decouple in the large mass limit. The deviation from the SM prediction can be very large when m 2 h ≪ m 2 ∆ even under the constraint from perturbative unitarity [31], where m ∆ represents the mass of the heaviest of all the triplet-like Higgs bosons. It is known that the similar effect can also appear in the calculation of the one-loop corrected hhh coupling in the two Higgs doublet model when the masses of the scalar bosons in the loop are given mainly by the VEV [25,26].
We here show the numerical results for ∆λ hhh /λ SM(loop) hhh for m h = 125 GeV and α = 0. In Fig. 2 (left), the deviation ∆λ hhh /λ SM(loop) hhh is shown as a function of m H ++ for several values of ∆m (< 0), where H ++ is the lightest of all the triplet-like Higgs bosons. The solid lines represent the result by using the diagrammatical method in the on-shell scheme, while the dashed ones represent that by using the effective potential method. It can be seen that the results by using the two ways are almost the same. The small deviation comes from the effect from the wave function renormalization. The deviation from the SM prediction can be significant for large m H ++ and ∆m values due to the quartic power dependence of these parameters (see Eq. (56)). In the case of ∆m = 0, the magnitude of the deviation to be 50% for m H ++ = 300 GeV. The larger deviation is obtained for the larger value of |∆m|. For example when we take ∆m = −60 GeV (−100 GeV) and m H ++ = 300 GeV, it amounts to about 100% (150%). In Fig. 2 ( be significant. When we take ∆m = +60 GeV (+100 GeV) and m A = 300 GeV, it amounts to about 100% (180%). We note that these large deviations satisfy the tree-level unitarity bounds.
For instance, the case with m A = 300 GeV and ∆m = +100 GeV implies m H + = 413 GeV and m H ++ = 513 GeV. We have confirmed that this parameter set satisfies the unitarity bounds [31].
Finally, at the ILC, the triple Higgs boson coupling would be expected to be measured with several times 10 % accuracy, so that the deviation in the hhh coupling in the HTM suggested by the electroweak precision data could be detectable.

V. CONCLUSIONS
We have investigated the one-loop renormalization for the Higgs potential in the HTM by using the on-shell scheme. We have studied how the characteristic relation among the masses of the triplet-like Higgs bosons (R ≃ 1) is changed at the one-loop level. We have found that R can be modified by the radiative corrections around 10% for v 2 ∆ /v 2 ≪ 1. We have also calculated the renormalized triple Higgs boson coupling in the HTM and its deviation from the SM prediction.
It has been found that for the parameter sets suggested by the electroweak precision data the magnitude of the deviation can be as large as O(100)% due to the quartic power-like dependence of the masses of the triplet-like Higgs bosons. Therefore, by measuring the mass spectrum of the triplet-like Higgs bosons and also the triple Higgs boson coupling accurately, the HTM can be tested at future collider experiments. Detailed analysis is shown elsewhere [32].