One-loop corrections to the Higgs self-couplings in the singlet extension

We investigate predictions on the triple Higgs boson couplings with radiative corrections in the model with an additional real singlet scalar field. In this model, the second physical scalar state ($H$) appears in addition to the Higgs boson ($h$) with the mass 125 GeV. The $hhh$ vertex is calculated at the one-loop level, and its possible deviation from the predictions in the standard model is evaluated under various theoretical constraints. The decay rate of $H \to hh$ is also computed at the one-loop level. We also take into account the bound from the precise measurement of the $W$ boson mass, which gives the upper limit on the mixing angle $\alpha$ between two physical Higgs bosons for a given value of the mass of $H$ ($m_H^{}$). We find that the deviation in the $hhh$ coupling from the prediction in the standard model can maximally be about 250\%, 150\% and 75\% for $m_H^{}=300$, 500 and 1000 GeV, respectively, under the requirement that the cutoff scale of the model is higher than 3 TeV. We also discuss deviations from the standard model prediction in double Higgs boson production from the gluon fusion at the LHC using the one-loop corrected Higgs boson vertices.

In such a case, the Higgs sector takes an extended form from the minimal model with an isospin doublet scalar field.
In general, the non-minimal shape of the Higgs sector affects various observables. In particular, it gives deviations in the couplings of the discovered Higgs boson with the mass 125 GeV. Although there is no significant anomaly found in the current LHC data, the deviations might be detected in the future when the data will be more accumulated. Once the deviation is found, we may be able to obtain important information about the physics beyond the SM by fingerprinting the pattern of the deviation in various Higgs observables and the predictions in many new physics models [1].
However, in order to obtain direct information on the Higgs potential, the measurement of the triple Higgs boson coupling is inevitable, which is one of the most important tasks of future collider experiments. From the information of the Higgs potential, we can approach to the physics behind electroweak symmetry breaking. It is known that in extended Higgs sectors physics predicting strongly first order phase transition simultaneously predicts a significant deviation in the triple Higgs boson coupling [29,30]. In Ref. [31], synergy between measurements of gravitational waves and the triple Higgs boson coupling is discussed in probing the first-order electroweak phase transition. Therefore, the measurement of the triple Higgs boson coupling is important not only to test the dynamics of electroweak symmetry breaking but also to investigate physics of the electroweak phase transition and scenarios of electroweak baryogenesis [32][33][34][35].
In this paper, we focus on the Higgs Singlet Model (HSM) whose Higgs sector is composed of an isospin complex doublet field and a real singlet scalar field. The HSM has been drawn much attention in various interests in many papers. For example, works related to the electroweak baryogenesis have been done in Refs. [36][37][38][39][40][41]. Singlet scalar fields have also been studied in the context of the Higgs portal dark matter scenario [42][43][44][45]. The collider phenomenology, especially on the double Higgs boson production process at the LHC gg → hh, has been calculated at the leading order (LO) in Ref. [46] and the next-to-leading order (NLO) in QCD in Ref. [47] . Bounds on the parameter space in the HSM have been comprehensively surveyed by using data at the LHC Run-I in Ref. [49].
In addition to the above studies, there are papers for electroweak radiative corrections to the Higgs boson couplings in the HSM. In Ref. [9], the h couplings with weak bosons and fermions have been calculated at the one-loop level. In Ref. [10], one-loop corrections to the decay rate of the H → hh process with H being a heavier Higgs boson. In this paper, we investigate one-loop corrections to the triple scalar boson couplings hhh and Hhh based on the on-shell renormalization scheme. We apply these one-loop corrected vertices to calculate the decay rate of the H → hh mode and the cross section of the double Higgs boson production via the gluon fusion process gg → hh at the LHC. We find that the one-loop correction to the hhh coupling significantly change the prediction at the tree level to be O(100)% level under the constraint from perturbative unitarity, triviality, vacuum stability and conditions to avoid wrong vacua. Furthermore, the cross section of gg → hh can be more than 20 times larger than the SM prediction due to the resonance effect of

H.
This paper is organized as follows. In Sec. II, we define the Lagrangian of the HSM. In Sec. III, we discuss bounds on the parameter space from theoretical and experimental constraints. In Sec. IV, the renormalization of parameters in the Higgs potential is described based on the on-shell scheme [9]. Numerical analyses for the one-loop corrected hhh coupling, the decay rate of H → hh and the cross section of the double Higgs boson production process via gg → hh are given in Sec. V.
Conclusions are summarized in Sec. VI. In Appendices, we present the analytic expressions for the scalar triple and quartic couplings (Appendix A), the one-loop beta functions for dimensionless coupling constants (Appendix B) and the One Particle Irreducible (1PI) diagram contributions to the hhh and Hhh vertices (Appendix C).

II. THE HIGGS SINGLET MODEL
We define the Lagrangian of the HSM based on the SU (2) L × U (1) Y gauge theory, of which Higgs sector is composed of an isospin complex doublet scalar field Φ and an isospin real singlet scalar field S.
The most scalar potential is given as where the doublet and singlet fields can be parameterized by with G + and G 0 being the Nambu-Goldstone bosons which are absorbed into the longitudinal components of the W + and Z bosons, respectively. The Vacuum Expectation Value (VEV) of the singlet field v S does not contribute to the electroweak symmetry breaking, so that the Fermi constant G F is determined only by the doublet VEV just like the SM: GeV. Moreover, we can show that the shift of the singlet VEV does not change physics [46] as it is proved in the following 1 . If we take the shift S → S + v ′ S , then the potential is rewritten by Therefore, the modification of the potential by the shift v ′ S is absorbed by taking the following reparameterization: Using this shift invariance, we can take v S = 0 without loss of generality, and we set v S = 0 in the following discussion to simplify expressions.
The tadpole terms for h and s are respectively given by From the tadpole condition at the tree level; i.e., T Φ = T S = 0, we can eliminate m 2 Φ and t S . Under this condition, the mass terms in the potential are calculated as where The mass eigenstates of two scalar bosons are defined by introducing the mixing angle α as We identify the mass eigenstate h as the discovered Higgs boson at the LHC with the mass 125 GeV. In this basis, the mass matrix is diagonalized as follows From Eq. (9), the mass eigenvalues and the mixing angle α are expressed by where we introduced the shorthand notation for the trigonometric functions: c θ = cos θ and s θ = sin θ. Using Eqs. (10)- (12), the parameters λ, m 2 S and µ ΦS can be rewritten by From the above discussion, the 7 independent parameters in the potential are expressed by Among them, m h ≃ 125 GeV and v ≃ 246 GeV are known parameters by experiments.
It is important to mention here that the mixing angle α can also be expressed from Eq. (15) as The kinetic Lagrangian L kin and the Yukawa Lagrangian L Y are given by where D µ is the covariant derivative for Φ and σ 2 is the second Pauli matrix. The trilinear interaction terms among the Higgs boson and SM particles are then extracted as Therefore, the scaling factor of the Higgs boson coupling with weak bosons κ V ≡ g HSM hV V /g SM hV V and fermions κ f ≡ g HSM hf f /g SM hf f are universally given at the tree level by From the Higgs potential given in Eq. (1), the scaling factor for the triple Higgs boson coupling hhh is also calculated at the tree level as We see from Eq. (22) that κ h is more sensitive to the mixing angle as compared to κ f and κ V .
We explain how the parameter space can be restricted by taking into account each of four constraints in order.
First, the bound from perturbative unitarity is obtained by requiring that eigenvalues of the S-wave amplitude matrix for the 2 body to 2 body elastic scattering processes are smaller than a given critical value [54]. In our model, all the eigenvalues are calculated at high energies by [51] x For each of eigenvalues, we impose Second, the triviality bound is obtained by requiring that the Landau pole does not appear below a certain energy scale Λ cutoff . Instead of using the scale where the Landau pole appears, we can define the triviality bound as follows where λ i (µ) are the scale dependent dimensionless coupling constants at a scale µ. The scale dependence of λ i is calculated by solving the renormalization group equations (RGEs) for all the dimensionless coupling constants, and the full set of RGEs at the one-loop level are given in Appendix B. Depending on Λ cutoff , we obtain the bound on λ i at the initial scale which is taken to be m Z . In Fig. 1, we show the cutoff scale as a function of λ ΦS (m Z ) for several fixed value of λ S (m Z ) with α = 0. We can see that the cutoff scale immediately becomes low when we take a non-zero value of λ S , because of the large coefficient of the λ 2 S term in the β(λ S ) function given in Eq. (B5). We also see that a larger value of λ ΦS makes the cutoff scale low, e.g., Λ cutoff ≃ 10 3 (3) TeV for λ ΦS = 1 (2) and λ S = 0. For α = 0, the bound becomes stronger than that in the case with α = 0.
Third, the constraint from the vacuum stability is imposed by requiring that the Higgs potential given in Eq. (1) must be bounded from below in any direction with large scalar field values. This requirement can be expressed by where V (4) is the quartic term part of the potential. From Eq. (28), we obtain the following inequalities at a scale µ [40]: If λ ΦS (µ) ≥ 0, the last condition is trivial, while λ ΦS (µ) < 0, that is rewritten by Finally, we explain the bound from wrong vacuum conditions. In the HSM, because of the existence of the scalar trilinear couplings µ S and µ ΦS , non-trivial local extrema can appear in the Higgs potential. Therefore, we have to check whether the true extremum at ( with v ew ≃ 246 GeV corresponds to the minimum of the potential. According to Refs. [38,46], the following five extrema appear where Now, the condition to avoid the wrong vacuum can be expressed by where V nor is the normalized Higgs potential satisfying V nor (v ew , 0) = 0: Before closing this section, we briefly comment on constraints from experimental data. In Ref. [48][49][50], constraints from electroweak precision observables and Higgs boson search data at the LHC have been studied in the HSM with a spontaneously broken discrete Z 2 symmetry. It has been clarified that the constraint from the measurement of the W boson mass gives the strongest upper bound on |s α | in the most of the parameter space. This bound becomes stronger when m H increases, e.g., |s α | 0.3 (0.2) for m H = 300 (800) GeV. We note that this bound can be applied to our model, because it only depends on m H and s α .

IV. RENORMALIZATION
In this section, we calculate the renormalized scalar trilinear verticesΓ hhh andΓ Hhh at the one-loop level based on the on-shell scheme, where for some parameters we apply to the minimal subtraction scheme. The renormalized hV V (V = W, Z) and hff vertices have already been calculated in Ref. [9], so that we focus on the renormalization of the parameters in the Higgs potential. We first shift relevant parameters into the renormalized one and the counter term. We then give a set of renormalization conditions to determine these counter terms. In this paper, the calculations are performed in the 't Hooft-Feynman gauge 2 .
2 It has been pointed out in Ref. [10] that there remain gauge dependences in the mixing parameter α determined by the on-shell scheme. In Refs. [5,7,10,55], a renormalization scheme to remove such a gauge dependence has been proposed. Although in our paper we apply to the usual on-shell renormalization scheme even if there remains the gauge dependence, the ratio of numerical values of physical observables such as the decay rate of the Higgs bosons calculated in the on-shell scheme to the improved scheme without the gauge dependence has been known to be smaller than O(1)% [10]. This means, for example, that if one-loop corrections to a quantity are given to be 1%, the impact on the gauge dependence is less than O(0.01)%.

A. Shift of parameters
The following eight bare parameters in the potential are shifted as In addition, the wave function renormalization for the scalar fields is given by the following way: Using the above counter terms, we can construct the renormalized scalar boson one-and twopoint functions. In the following, we express contributions from 1PI diagrams as Γ 1PI ϕ for the one-point scalar function and Π 1PI ϕϕ ′ for the two-point scalar function. The renormalized one-point function for h and H are given bŷ where The renormalized two-point functions are expressed aŝ

B. Renormalization conditions in the Higgs potential
In the previous subsection, we prepared totally 12 counter terms from Eqs. (42) and (43).
We thus need 12 renormalization conditions to determine them. First, we impose two tadpole conditions at the one-loop level, i.e.,T We then obtain Second, four on-shell conditions for the two-point functions: which determine the following four counter terms and Three counter terms δα, δC hH and δC Hh are determined by imposing the following three conditionŝ by which we obtain From the above discussion, we determine 9 counter terms, but there remain 3 undetermined ones: δλ ΦS , δµ S and δλ S . Among the 3 counter terms, δλ S does not enter the following discussion, which appears in the renormalization of the scalar quartic vertices. For the remaining two counter terms δλ ΦS and δµ S , we apply the minimal subtraction scheme in which they are determined so as to remove the ultra-violet (UV) divergent part of the one-loop correction to the hhh and Hhh vertices. We will further discuss the determination of these counter terms in the next subsection.

C. Renormalized vertices
The renormalized hhh and Hhh vertices are expressed aŝ Γ hhh (p 2 1 , p 2 2 , q 2 ) = 3!λ hhh + δΓ hhh + Γ 1PI hhh (p 2 1 , p 2 2 , q 2 ), Γ Hhh (p 2 1 , p 2 2 , q 2 ) = 2!λ Hhh + δΓ Hhh + Γ 1PI where δΓ Hhh and Γ 1PI Hhh (H = h or H) are the contributions from the counter terms and the 1PI diagrams for the Hhh vertices, respectively. The scalar three point couplings λ hhh and λ Hhh are given in Appendix A. The counter-term contributions are expressed by where δM and δM ′ are the undetermined counter term from the on-shell conditions which are expressed by Applying the minimal subtraction scheme which is discussed in the previous subsection to δM and δM ′ , we obtain where ∆ div expresses the UV divergent part of the loop integral, and N f c is the color factor; i.e., N f c = 3 (1) for f being quarks (leptons).
We note that the counter term of the VEV δv is determined by using the gauge boson two-point functions which have been given in Ref. [9].

V. NUMERICAL RESULTS
In this section, we perform the numerical analysis of some observables, i.e., the deviation in the hhh coupling at the one-loop level from the SM prediction (Sec. VA), the total width and the decay branching ratio of H (Sec. VB) and the double Higgs boson production cross section via the gluon fusion gg → hh at the LHC (Sec. VC) by using the one-loop renormalized hhh and Hhh vertices. In order to constrain the parameter space, we take into account the perturbative unitarity, triviality, vacuum stability and wrong vacuum conditions as we have explained in Sec. III. The triviality and vacuum stability bound depend on the cutoff scale Λ cutoff of the model which is taken to be 3 TeV or 10 TeV in the following analysis. In some plots shown in the following subsections, we also consider the constraint from the electroweak precision test for the W boson mass in Ref. [49] which gives the upper limit on |s α | for a given value of m H .
For the numerical analysis, we have the following five free parameters m H , s α , λ ΦS , λ S , µ S .
As we have seen in Fig. 1, a non-zero value of λ S significantly reduces the cutoff scale because of the RGE evolution of λ S . We thus simply take λ S = 0 throughout this section to have the cutoff scale to be above the multi-TeV scale.

A. One-loop corrected hhh coupling
The scaling factor of the hhh coupling κ h is defined in Eq. (22) at the tree level. Now, we grade up this quantity at the one-loop level as follows: Using this, the deviation in the hhh coupling is expressed by ∆κ h = κ h − 1. if we take s α = 0, the upper limit on m H appears because of the theoretical constraints and the bound from m W , so that we cannot take the decoupling limit in this case. It is also seen that in the region m H 300 GeV, the prediction of ∆κ h does not change so much. When we look at the right panel, we can see that ∆κ h monotonically increases as |s α | becomes large. Because of the bound from m W , we can extract the maximal allowed value of ∆κ h to be about 120%, 70% and 20% for λ ΦS = 1.5, 1.0 and 0.5, respectively.
We note that such a large correction to the hhh coupling happens due to the non-decoupling effect of the H loop when m H mainly comes from the Higgs VEV 3 . In the HSM, this non-decoupling effect appears in the case with λ ΦS v 2 m 2 S or equivalently λ ΦS = O(1) as we can see it in Eq. (7). Similar non-decoupling effects in the hhh coupling have also been found in the THDM as these have been pointed it out in Ref. [3]. In fact, even in the SM the non-decoupling effect in the hhh can be seen on the top loop contribution as its mass purely comes from v, where the magnitude of the correction is proportional to m 4 t , and it can be of order 10% level. It goes without saying that the top loop effect is included in our calculation, but it does not change the value of ∆κ h so much, because it is defined by the deviation from the SM prediction.
Finally, we scan the parameter space to see the possible allowed range of ∆κ h . In order to see the difference between the HSM and the THDM, we also calculate the one-loop corrected hhh coupling based on the previous our work given in Ref. [5]. We take the following scan range for the parameters 300 < m H < 3000 GeV, 0 < |s α | < 0.4, 0 < λ ΦS < 2.5, − 100 < µ S < 100 GeV, for HSM, (67) 300 < m Φ < 3000 GeV, 0.92 < s β−α < 1, 0 < λ ΦΦh /v < 2.5, for THDM, where m Φ = m H (= m A = m H ± ), and λ ΦΦh is defined by vλ ΦΦh = m 2 Φ − M 2 . For the details of the definition of the parameters in the THDM, see, e.g., Ref. [3]. We take into account the current bound on the Higgs boson couplings given at the LHC Run-I experiments. The combined results for the measurements of the scaling factors κ X at the ATLAS and the CMS experiments have been provided in Ref. [58] as follows κ Z = 1.00 +0.11 −0.10 , κ W = 0.91 +0.10 −0.12 , κ τ = 0.90 +0.14 −0. 16 where we pick up the positive allowed values of κ X . We require the predictions of the above scaling factors being inside the 2σ level.
In Fig. 4, we show the scatter plot using the scanned parameter range given in Eqs. in the HSM (THDM). This difference can be explained by the difference of the structure of the sharing of VEVs and the mixing of CP-even Higgs fields at the tree level. In the HSM, the hhh coupling can be decreased due to the non-zero field mixing between h and H, but such reduction can be compensated by the additional contribution of the λ ΦS parameter as it is seen in Eq. (22).
As a result, ∆κ h at the tree level can typically be positive. On the other hand in the THDM, the hhh coupling can be decreased not only by the field mixing but also by the sharing of VEVs such 2 are VEVs of two Higgs doublets), where the latter does not happen in the HSM. This additional reduction by the VEV mixing makes the hhh coupling small as compared to the SM prediction. For reference, we give the tree level expression for κ h in the THDM: From this result, when the second Higgs boson is discovered, and its mass is measured at future collider experiments, we can expect that the big difference in the value of the hhh coupling appears between the HSM and the THDM.

B. Decay of H
Here, we discuss the width of H which is needed to calculate the cross section of the double Higgs boson production as it will be discussed in the next subsection. The total width of H is calculated by where H(x) = 1 (0) for x ≥ 0 (x < 0). In the above formulae, the first term Γ h SM m h SM →m H represents the total width of the SM Higgs boson h SM but the mass m h SM is replaced by m H .
The second term corresponds to the partial width of H → hh decay mode which opens for m H ≥ 2m h = 250 GeV. The analytic expression of the decay rate of H → hh is given by In  We note that the s α dependence of the branching ratio is negligibly small. We can see that the di-Higgs boson channel H → hh with λ ΦS = 2 can be more important than that with λ ΦS = 1, but the branching ratio of this channel becomes small as m H increases.

C. Double Higgs boson production
We now ready to calculate the cross section of the double Higgs boson production via the gluon fusion process: gg → hh at the partial two-loop level, where the meaning of "partial" will be clarified below. other. On the other hand, the latter category has a mixture of two loop momenta, so that we need to evaluate the full two-loop integral. Another important difference between the former and latter contribution is found in the power of the scalar trilinear couplings λ ϕϕ ′ ϕ ′′ . The former (latter) contribution involves a cubic (quadratic) dependence on λ ϕϕ ′ ϕ ′′ . Therefore, when we take a large value of λ ϕϕ ′ ϕ ′′ couplings within the extent allowed by the theoretical constraints, the deviation in the cross section of the double Higgs boson production mainly comes from the diagrams (a)-(e). On the other hand, if we take a small value of λ ϕϕ ′ ϕ ′′ , the contributions from (g) and (h) cannot be neglected. In this subsection, we only take into account the contributions from (a)-(e), and we call this level of the calculation as the partial two-loop level. We note that the diagrams (e) and (f) vanish in our on-shell renormalization scheme explained in Sec. IV when the on-shell Higgs boson h is produced. Consequently, the diagrams (a)-(d) are taken into account in our calculation.
It has been known that QCD corrections largely change the cross section of the gg → hh process.
The NLO calculation in QCD has been evaluated in Ref. [47], and it has been clarified that the amount of the NLO correction is from −30 to +20% level depending on the choice of m H . In this paper, we calculate the cross section at LO in QCD.
In the HSM, the parton level cross section is calculated bŷ where F ∆ is the loop function for the triangle diagram, while F (G ) is that for the box diagram with the same (opposite) helicity of the initial gluons. The analytic formulae for these loop functions are given in Ref. [59]. In Eq. (73), C ∆ is the coefficient of the triangle diagram given as where c ϕ = c α (s α ) for ϕ = h(H) and Γ ϕ is the width of ϕ. The total cross section in the pp collision is calculated by convoluting the di-gluon parton luminosity function L gg : where τ 0 = 4m 2 h /s with s being the collision energy of pp. From now on, we present the numerical results of the double Higgs boson production cross section. For this analysis, we provide four benchmark points (BPs) as shown in Table I, in which we also give the outputs of ∆κ h , Γ H and the ratio of the total gg → hh cross section σ HSM tot /σ SM tot . In Fig. 9, we show the differential hadronic cross section of the gg → hh process in the pp On the other hand, the prediction at the partial two-loop level is smaller than the corresponding SM result mainly due to larger distractive interference effects between the triangle and the box diagram contributions. In the red solid curves, we can observe the small dip at around M hh = 400 GeV (= 2 × m H ) which happens due to the threshold effect ofΓ Hhh . For BP3 and BP4 (lower two panels), we see the significant difference between the results in the SM and in the HSM. In these cases, the peak at around M hh = 400 GeV appears because of the resonance effect of the H propagation, and its shape is quite narrow. This can be explained by the small width of H as compared to the mass of H as we saw in Fig. 5. Thanks to this resonant effect of H, the total cross section significantly increases as compared to the case with m H < 2m h and the SM case, i.e., the ratio of the total cross section becomes 5.32 (20.0) in BP3 (BP4) as it is shown in Table I.

VI. CONCLUSIONS
We have calculated the one-loop correction to the triple scalar boson couplings hhh and Hhh based on the on-shell renormalization scheme in the HSM. We then applied these one-loop corrected couplings to calculate the decay rate of the H → hh mode and the double Higgs boson production process gg → hh at the LHC. It has been clarified that the one-loop correction to the hhh coupling can change its tree level prediction to be the order of 100% under the constraint from the perturbative unitarity, triviality, vacuum stability and conditions to avoid the wrong vacuum.
We have found that the deviation in the hhh coupling from the SM prediction can maximally be about 250%, 150% and 75% for m H = 300, 500 and 1000 GeV, respectively, under the requirement