One-Loop Radiative Correction to the Triple Higgs Coupling in the Higgs Singlet Model

Though the 125 GeV Higgs boson is consistent with the standard model (SM) prediction until now, the triple coupling can deviate from the SM value in the physics beyond the SM (BSM). In this paper, the radiative correction to the triple Higgs coupling is calculated in the minimal extension of the SM by adding a real gauge singlet scalar. In this model there are two scalars $h$ and $H$ and both of them are mixed states of the doublet and singlet. Provided that the mixing angle is set to be zero, $h$ is the pure left-over of the doublet and its behavior is the same as that of the SM except the triple $h$ couping. In this SM limit case, the effect of the singlet $H$ will decouple from the fermions and gauge bosons, and firstly shown up in the triple $h$ coupling. Our numerical results show that the deviation is sizable. For $\lambda_{\Phi{S}}=1$ (see text for the parameter definition), the deviation $\delta_{hhh}^{(1)}$ can be $40\%$. For $\lambda_{\Phi{S}}=1.5$, the $\delta_{hhh}^{(1)}$ can reach $140\%$. The sizable radiative correction is mainly caused by three reasons: the magnitude of the coupling $\lambda_{\Phi{S}}$, light mass of the additional scalar and the threshold enhancement. In the optimal case, the triple $h$ coupling is very sensitive to the BSM physics, and this model can be tested at future high luminosity hadron colliders and electron-positron colliders.


I. INTRODUCTION
The standard model (SM) has been extensively tested, especially the deviations for the gauge sector are strongly constrained by the electro-weak precision measurements from the Large Electron-Positron Collider (LEP) [1], Tevatron and the large hadron collider (LHC).
However, the Yukawa sector and the scalar sector are two regimes which are still not well probed. Since the discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012 [2,3], the most important task is to measure the properties of the scalar accurately. The measurements will help us understand the nature of the electro-weak symmetry breaking mechanism (EWSB) [4][5][6][7]. If there exists new physics beyond the SM (BSM), it is believed that it is related with the Higgs couplings more or less. The Higgs boson is a door to the unknown new world.
Current measurements of the Higgs couplings with gauge bosons tend to be the SM values.
At the same time Higgs couplings with the third generation fermions are inferred from the Higgs production processes at the LHC, which are also consistent with those in the SM. Usually for the model construction, the Higgs couplings with fermions and gauge bosons will have the SM limit at the electro-weak scale. However the triple Higgs coupling can deviate from the SM value largely in this limit. Such feature of the triple Higgs coupling has been studied extensively in the two Higgs doublet model (THDM) [8,9], inert Higgs doublet model (IHDM) [10], Higgs triplet model (HTM) [11] and models with an additional heavy neutrino [12].
Searching for BSM physics is one of the most important goals of high energy physics.
The most direct way is to increase the energy of the colliders and see whether there are new heavy resonances, while it is always hard or even impossible to construct the very high energy colliders because of the limitations from the expanses, technologies and so on.
However there are other methods to achieve this goal. The new heavy particles will leave footprints at the electro-weak scale through loop effects. We may have indirect signals for the BSM through some physical quantities which are sensitive to the heavy particles.
The minimal extension of the SM in the scalar sector is to add a real gauge singlet. The Higgs singlet model (HSM) has been studied exhaustively in a lot of papers. For example, Ref. [13] studied a model which includes a Z 2 symmetry spontaneously breaking real Higgs singlet and the author considered the theoretical and phenomenological constraints of this model. Ref. [14] explored the resonant di-Higgs production in the 14TeV hadron collider with an additional intermediate, heavy mass Higgs boson. Ref. [15] considered two scenarios: there was (no) mixing between the SM Higgs and the singlet. Then, they analyzed the constraints from electro-weak precision observables, LHC Higgs phenomenology and dark matter phenomenology. Ref. [16,17] emphasized the heavy-to-light Higgs boson decay at the NLO. Ref. [18] focused on the one-loop radiative corrections in the HSM and they performed the numerical calculations for the hZZ, hW W , hff , hγγ, hγZ, hgg couplings, but not for triple h coupling, which is the main topic in this paper.
In the following, we will make a careful analysis of the triple h coupling up to one-loop level in this model in the SM limit. There will be an universal deviation from the SM predictions for the hV V , hff couplings arising from the wave-function renormalization constant δZ h .
The numerical results show that the universal correction is small. For the triple h coupling, there are still hHH, hhHH couplings (see Appendix A) in this limit. When the mass of the additional scalar is [90,150] GeV and the coupling λ ΦS is order one, the radiative correction to the triple h coupling can be 40% or even larger in the vicinity of double Higgs production.
It may be measured at future hadron colliders and electron-positron colliders. This paper is organized as follows. In Sec. II, we give a detailed description of the model including the theoretical constraints on the parameter space and the radiative correction to the triple h & hZZ couplings. In Sec. III, we present the numerical results. Sec. IV is devoted to the conclusions and discussions. Feynman rules, related Feynman diagrams and calculational details are collected in the Appendix.

II. MODEL
We introduce a real additional gauge singlet S with hyper-charge Y = 0 besides the SM Higgs doublet Φ. Then, we can write the scalar potential V (Φ, S) as Evidently, the singlet doesn't have any Yukawa interactions or gauge interactions with the SM fields. The scalar fields Φ, S in the unitary gauge can be parametrized as Without loss of generality, we can set v S to be zero by shifting the S, namely the redefinition of the field. After EWSB, the two tadpoles are Mass terms of the scalar fields are After diagonalizing the mass matrix, we get the following expressions In the above expressions, we use m H instead of m S to avoid the confusion with the parameter in the Lagrangian. s α , c α , s 2α are the simplified notations for sinα, cosα, sin2α. From now on, we will choose the parameters m 2 h , m 2 H , α, λ ΦS , λ S , µ S , v as the inputs. According to the definitions of m 2 h , m 2 H , tan2α in Eq. 2 and Eq. 3 , λ Φ , m 2 S , µ ΦS can be expressed by the new inputs as A. Constraints on the parameter space In the SM limit (α → 0), H will decouple from the fermions and gauge bosons because of the scaling factor s α . Thus, it will evade all the present experimental constraints. But the SM Higgs will still couple with H by the interacting vertices hHH, hhHH. And this makes great influence on the triple h coupling which is discussed later. All the analyses below will be carried out under the SM limit assumption, namely α = 0. When Φ, S is very large, the scalar potential will become V (Φ, S) = λ Φ (Φ † Φ) 2 + λ ΦS Φ † ΦS 2 + λ S S 4 . It must be bounded from below, so we have Another constraint we should consider is the so-called perturbative unitarity [19]. S-wave amplitude a 0 should satisfy the relation |Re(a 0 )| < 1 2 , where a 0 is given by a 0 = 1 16πs 0 −s dtM (t). Here, s, t are the Mandelstam variables as usual, and M is the scattering amplitude. According to the Goldstone equivalence theorem, massive vector boson is dominated by the longitudinal polarization at high energy. So we need only to consider the two-to-two scattering processes with initial and final states: W + L W − L , Z L Z L , Z L h, Z L H, hh, HH, hH. Similar analyses have been discussed in many papers [20,21]. This is a 7 × 7 matrix, but it will be reduced into a 3 × 3 matrix in the SM limit. A subtlety one may caution is an extra 1 √ 2 for the same initial and final states, which is often ignored in many papers. After some trivial calculations, we have a 3 × 3 matrix Three eigenvalues of the matrix are a 1, , a 3 = − λ ΦS 8π . Then, we have the constraints from perturbative unitarity In the SM limit, λ Φ = m 2 h 2v 2 . Together with the bounded constraints in Eq. 5, we get the following parameter space in Fig. 1. The interesting feature is that there are no constraints on m H , µ ΦS . We will calculate the deviation of the triple h coupling from the SM value originated from one-loop radiative correction in the SM limit. During the calculations, we adopt the conventions from Ref. [22]. There is no doubt that the loop particles must be the additional scalar H. To gauge the deviation from the SM value, we define δ In the following, we will present the numerical results for δ (1) hhh for the chosen model parameters.
Similarly, The deviation of the hZZ coupling from the SM value originated from one-loop radiative correction in the SM limit is defined as The analytical expressions can be found in Appendix C. The related Feynman rules and calculations in the SM limit are given in the Appendix.

III. NUMERICAL RESULTS
In this section, we will do some numerical evaluations of δ (1) hhh for different model parameters. We set m h = 125GeV, v = 246GeV as in the SM. The deviation of δ (1) hhh is mainly determined by λ ΦS , m H , p 2 , where one of the Higggs boson with momentum p is off-shell. The dominant contribution is from the triangle diagram which is proportional to λ 3 ΦS . We choose the allowed value of λ ΦS = 1, 1.5, respectively, and study the dependence on m H , p 2 using LoopTools [23]. Behaviours of δ (1) hhh are shown in Fig. 2, Fig. 3. It is easy to see that the correction to the triple h coupling is sensitive to λ ΦS , m H , p 2 . If the coupling λ ΦS can reach order one, the deviation can be very large for m H ∈ [90, 150] GeV in the vicinity of double Higgs production ( p 2 ≈ 250 GeV). For λ ΦS = 1, the δ gg → h * → hh production channel at future hadron colliders [24][25][26]. The model may also be probed through e + e − → Z * → Zh * → Zhh and e + e − → ν eνe W + * W − * → ν eνe h * → ν eνe hh production channels at future electron-positron colliders [27][28][29]. At a low energy electronpositron colliders with 240GeV or so and high luminosity, δ (1) hhh can also be detected indirectly [30][31][32].
Additionally, we consider the comparison between the triple h and hZZ coupling. Numerical results are shown in Fig. 4: We can find that the δ (1) hZZ is very small, compared to the triple h coupling. Owing to the high precision measurement of hZZ coupling, it can be complementary to the triple h coupling.

IV. CONCLUSIONS
The radiative correction to the triple h coupling is calculated in the minimal extension of the SM by adding a real gauge singlet scalar. In this model there are two scalars h and H and both of them are mixed states of the doublet and singlet. Provided that the mixing angle is set to be zero, h is the pure left-over of the doublet and its behavior is the same as that of the SM at the tree level. However the radiative corrections from the singlet H can alter h-related couplings which deviate from the SM values. Our numerical results show that the deviation δ (1) hhh is sizable. For λ ΦS = 1, the δ (1) hhh can be 40%. For λ ΦS = 1.5, the δ (1) hhh can reach 140%. The sizable radiative correction is mainly caused by three reasons: the magnitude of the coupling λ ΦS , light mass of the additional scalar and the threshold enhancement. The radiative correction for the hZZ coupling can be a complementarity to the triple h coupling because of the high precision measurement. In the optimal case, the triple h coupling is very sensitive to BSM physics, and this model can be tested at future high luminosity hadron colliders and electron-positron colliders.