Radiative corrections to the Higgs boson couplings in the model with an additional real singlet scalar field

We calculate renormalized Higgs boson couplings with gauge bosons and fermions at the one-loop level in the model with an additional isospin singlet real scalar field. These coupling constants can deviate from the predictions in the standard model due to tree-level mixing effects and one-loop contributions of the extra neutral scalar boson. We investigate how they can be significant under the theoretical constraints from perturbative unitarity and vacuum stability and also the condition of avoiding the wrong vacuum. Furthermore, comparing with the predictions in the Type I two Higgs doublet model, we numerically demonstrate how the singlet extension model can be distinguished and identified by using precision measurements of the Higgs boson couplings at future collider experiments.


I. INTRODUCTION
Although a Higgs boson was discovered by the LHC experiments in 2012 [1,2], the structure of the Higgs sector and the physics behind the Higgs sector remain unknown. Deep understanding for the Higgs sector is a key to explore new physics beyond the standard model (SM).
The minimal Higgs sector of the SM satisfies the current LHC data [3,4], while most of nonminimal Higgs sectors do so as well. As there is no theoretical reason to choose the minimal form of the Higgs sector like in the SM, there are many possibilities for extended Higgs sectors which contain additional scalar isospin multiplets. In principle, there are infinite kinds of extended Higgs sectors. However, particular importance exists in the second simplest Higgs sectors, where only one isospin multiplet is added to the SM Higgs sector, such as a model with an additional singlet, doublet or triplet scalar field. There are many new physics models which predict one of these extended Higgs sectors such as the B − L extended SM with the B − L symmetry breaking [5] which contains an additional singlet scalar field, the minimal supersymmetric SM [6,7] whose Higgs sector has the two Higgs doublets, and the model for the Type II seesaw mechanism [8] which can generate Majorana neutrino masses by introducing a complex triplet Higgs field, and so on. These second simplest Higgs sectors can also be regarded as low-energy effective theories of more complicated Higgs sectors.
How can we test extended Higgs sectors by experiments? Obviously direct discovery of the second Higgs boson is manifest evidence for extended Higgs sectors. By detailed measurements of such a particle, we can determine the structure of the Higgs sector. On the other hand, we can also test extended Higgs sectors by precisely measuring low energy observables such as those in flavor physics [9], electroweak precision observables [10], etc. As additional observables we can consider a set of coupling constants of the discovered Higgs boson. In general, a pattern of deviations in these observables strongly depends on the effects of extra Higgs bosons and other new physics particles, so that we may be able to fingerprint extended Higgs sectors and new physics models if we can detect a special pattern of the deviations at future experiments [11][12][13][14][15][16][17].
After the Higgs boson discovery, coupling constants of the discovered Higgs boson with SM particles became new observables to be measured as precisely as possible at current and future colliders. Currently the measured accuracies for the Higgs boson couplings are typically of the order of 10 % [3,4]. They will be improved drastically to the order of 1 % or even better at future lepton colliders, such as the International Linear Collider (ILC) [11,18], the Compact LInear Collider (CLIC) [19] and Future e + e − Circular Collider (FCCee). Therefore, these future electron-positron colliders are idealistic tools for fingerprinting Higgs sector and new physics models via precise measurements of the Higgs boson couplings. In order to compare theory predictions with such precision measurements, calculations with higher order corrections are clearly necessary.
In this paper, we calculate one-loop corrections to the Higgs boson couplings with gauge bosons and with fermions in the model with a real isospin singlet scalar field (HSM) [34][35][36][37][38][39]. The renormalized couplings can deviate from the SM predictions due to the mixing effect and the one-loop contributions of the extra neutral scalar boson. The one-loop contributions are calculated in the on-shell scheme. We numerically investigate how they can be significant under the theoretical constraints from perturbative unitarity and vacuum stability and also the conditions of avoiding the wrong vacuum. Furthermore, we compare the results with the predictions at the one-loop level in the THDM with Type I Yukawa interactions [15,16]. We study how the HSM can be distinguished from these models and identified by using precision measurements of the Higgs boson couplings at future collider experiments. This paper is organized as follows. In Sec. II, we define the HSM, and briefly review the tree level properties to fix notation. In Sec. III, we present our calculational scheme for one-loop corrections to the Higgs boson couplings in the HSM. Sec. IV is devoted to showing the numerical results of the renormalized scaling factors of the Higgs boson couplings. In Sec. V, we show deviations in the hZZ, hbb and hγγ couplings in the HSM together with those in the THDM with the Type I Yukawa interaction, and see how we can discriminate these models by using future precision measurements of these couplings. Conclusions are given in Sec. VI. Various formulae are collected in Appendix.

II. MODEL
The scalar sector of the HSM is composed of a complex isospin doublet field Φ with hypercharge Y = 1/2 and a real singlet field S with Y = 0. The most general Higgs potential is given by where all parameters are real. The Higgs fields Φ and S can be parametrized, where v and v S are vacuum expectation values (VEVs) of Φ and S, respectively. The fields G + and G 0 are Nambu-Goldstone bosons to be absorbed in longitudinally polarized weak gauge bosons.
Notice that v is determined by the Fermi constant G F by v = 1/( does not affect electroweak symmetry breaking. As it has been pointed it out in Refs. [38,40], the potential in Eq. (1) is invariant under the transformation of v S → v S by redefining all the potential parameters associated with S.
At the tree level, tadpoles are given by By imposing the stationary condition T φ = 0 and T s = 0, m 2 Φ and t S are related to the other parameters as After the electroweak symmetry breaking, mass terms of the scalar fields can be expressed as where with We diagonalize the mass matrix by introducing the mixing angle α, and express the scalar fields by mass eigenstates H and h, where the mass eigenstates H and h are related to the original fields s and φ by The masses of H and h are given by where h identified to be the discovered Higgs boson with m h 125 GeV. The mixing angle α can be written in terms of the parameters in the potential as We note that the SM limit is realized by taking M 2 to be infinity. In the following discussion, we use s α and c α to express sin α and cos α, respectively.
By using physical parameters m 2 h , m 2 H and α, the three parameters in the potential, λ, m 2 S , and µ ΦS , can be expressed as There are eight real parameters in the Higgs potential m 2 Φ , λ, µ ΦS , λ ΦS , t S , m 2 S , µ S and λ S , which are replaced by v, m 2 h , m 2 H , α, v S , λ ΦS , λ S and µ S . The kinetic terms for the scalar fields are given by where We obtain interaction terms between weak gauge fields and scalar fields as where m W and m Z are the masses of W and Z bosons, respectively. Although the Yukawa interaction is the same form as that in the SM, Yukawa couplings are modified from the SM predictions by the field mixing, where m f represents the mass of a fermion f .
We define the scaling factors as ratios of the Higgs boson couplings in the HSM from those in the SM,  (22) and (1) as We take into account several theoretical constraints in the HSM; i.e., perturbative unitarity [41], vacuum stability [36] and the condition to avoid a wrong vacuum [38,40]. We give the explanation of these theoretical constraints in Appendix A.
In addition to these theoretical constraints, the parameter space in the HSM is constrained by using experimental data. In Refs. [42,43], the one-loop corrections to m W has been calculated in the HSM with a discrete Z 2 symmetry. The limits on s α and m H have been derived by comparing the prediction of m W and its measured value at the LEP experiment, namely, |s α | 0.3 (0.2) with m H = 300 (800) GeV is excluded at the 2σ level. Although the electroweak S, T and U parameters have also been calculated in Ref. [43], constraints from those parameters are weaker than those from m W . We show the formulae of one-loop corrected m W and also the S, T and U parameters in Appendix B.

III. RENORMALIZATION
In this section, we define the renormalization scheme of the HSM in order to calculate the one-loop corrected Higgs boson couplings. We describe how to determine each counter term in the gauge sector, the Yukawa sector and the Higgs sector. We employ the same renormalization procedure as those given in Refs. [15,16] for the gauge sector and the Yukawa sector, because the parameters in these sectors are exactly the same as those in the SM.

A. Renormalization in the gauge sector
The gauge sector is described by three independent parameters as in the SM. When we choose m W , m Z and α em as the input parameters, all the other parameters such as v and weak mixing angle sin θ W (s W ) are given in terms of these three input parameters as These parameters and weak gauge fields; namely W ± µ , Z µ and A µ , are shifted as follows Renormalized two point functions of gauge fields, W + W − , ZZ, γγ and the γZ mixing, are given by using the above counter terms and the 1PI diagrams denoted by Π 1PI XY aŝ where The explicit expressions of 1PI diagrams for gauge boson two point functions are given in Appendix. D 2.
Imposing following five renormalization conditions as [44] ReΠ five independent counter terms δm 2 W , δm 2 Z , δα em , δZ γ and δZ γZ are determined as, Because of the relations Eqs. (26) and (38), other counter terms can be expressed by using above counter terms, Because we have obtained explicit forms of counter terms in the gauge sector, we can calculate the one-loop level predictions for electroweak observables such as electroweak precision parameter ∆r and renormalized W boson mass m reno W . Their formulae are given in Appendix B.

B. Renormalization in the fermion sector
We here discuss renormalization in the fermion sector. The Lagrangian of the fermion sector is given by where Ψ L (Φ R ) is a left (right) handed fermion field. They are shifted into a renormalized parameter and renormalized fields, and counter terms, Two point functions of fermion fields are composed of following two parts, Each part is expressed, where We determine the counter terms by following conditions, Then we obtain each counter term,

C. Renormalization in the Higgs sector
There are eight following parameters in the Higgs potential, As described in Sec. II, four of them can be rewritten in terms of the physical parameters m 2 h , m 2 H , α and v by using Eqs. (5), (17), (18), (19). Remained parameters are λ ΦS , v S , µ S , λ S , where t S is replaces by v S as described in Eq. (4).
First, we shift the bare parameters into renormalized parameters, Two physical scalar fields are shifted to the renormalized fields and the wave function renormal- We also shift the tadpoles as where T H and T h are related with T φ and T s as Renormalized one and two point functions at the one-loop level are given bŷ where analytic expressions of 1PI diagram parts are given in the Appendix D.
We note that there are 14 independent counter terms in the Higgs sector. By imposing following nine renormalized on-shell conditions, we determine following nine counter terms, As shown in Sec. III A, δv can be determined by the renormalization in the gauge sector. We note that forms of δλ ΦS , δv S , δλ S and δµ S cannot be determine above conditions. These do not appear in the one-loop calculation of the hV V and hff vertices. When one-loop corrections to the triple scalar couplings such as the hhh coupling, these counter terms have to be determined by additional renormalization conditions as discussed in Ref. [16,20] in the context of the THDM. The study of one-loop corrections to the triple Higgs boson coupling in the HSM is discussed elsewhere [45].

D. Renormalized Higgs couplings
In this subsection, we give formulae of the renormalized Higgs boson couplings. They are composed of the tree level part, the counter term part and the 1PI diagram part. The renormalized hV V and hff couplings are expressed aŝ Each renormalized form factor is given by, where the counter terms are expressed as where δm 2 W , δZ V and δv are given in Sec. III A and δm f and δZ V f are given in Sec. III B. Tree level scaling factors κ i V and κ j f are given by

A. Renormalized scaling factors
In this section, we show numerical results for the renormalized Higgs boson couplings, i.e., hV V and hff . We also calculate the leading order results of the decay rate of the h to γγ process. Our numerical program is written as a FORTRAN program, and the package; LoopTools [46] is used for the one-loop integrations.
Our numerical results are shown in terms of the scaling factors. Deviations in the one-loop corrected scaling factors for hV V and hff couplings are defined as whereΓ 1 hV V,SM andΓ S hf f,SM are the one-loop corrected hV V and hff couplings in the SM. The formulae for the one-loop decay rates h → γγ, h → Zγ and h → gg are given in Appendix E. We numerically evaluate deviations in the scaling factor of the hγγ , hγZ and hgg effective coupling defined as In our numerical evaluation, we use the following values for the input parameters [47]: where ∆α em is defined as 1− α em α em (m Z ) withα em (m Z ) being the fine structure constant at the scale of m Z . Furthermore, we set the momenta (p 2 1 , p 2 2 , q 2 ) to be (m 2 and (m 2 f , m 2 f , m 2 h ) for ∆κ V and ∆κ f , respectively. As we mentioned in Sec. II, we can take the value of v S freely without changing physics. We fix v S to be 0 in the following numerical analyses. (105) The most right hand side of Eq. (105) comes from the H loop contributions of δZ h . The structures of these one-loop contributions are the same as those in the THDMs as described in Ref. [16].  Fig. 3 (left), we learn that ∆κ Z is zero in the large mass limit for H. For a nonzero negative value of ∆κ Z there is an upper bound on m H . The upper bound evaluated at the one-loop level is almost the same as that at the tree level for each value of negative ∆κ Z . If by future precision measurements ∆κ Z is determined as ∆κ Z = −2.0 ± 0.5%, the upper bound on m H is obtained to be about 4 TeV. In Fig. 3 (right), the tree level results are on the curve described by ∼ −s 2 α /2 for small |s α |. At the one-loop level the magnitude of the deviation from the tree level prediction is typically about 1%.
For smaller values of |s α |, ∆κ Z is smaller than the tree level prediction, while for larger |s α | the one-loop corrected value ∆κ Z can be larger than the tree level prediction but the sign of ∆κ Z is   Fig. 5 and Fig. 6.
For the definition of the parameters in the THDM, see; e.g., Ref. [16] HSM THDM the precision measurement of the hV V coupling and the universal coupling of hff as long as the deviations in κ V is detected. One of the notable features of the predictions in the exotic extended Higgs sectors such as the GM model and the model with the septet field is the prediction that the scaling factor κ V can be greater than unity [8,12,51,52], while both THDMs and the HSM always In order to compare the theory calculations with precision measurements at future lepton colliders such as the ILC, where most of the Higgs couplings are expected to be measured with high accuracies at the typically O(1) % level or even better [53], the above tree level analyses in Ref. [48] must be improved by using the predictions with radiative corrections. In Refs. [15,16], the oneloop corrected scaling factors in the four types of THDMs have been calculated in the on-shell scheme, and the above tree level discussions in Ref. [48] have been repeated but at the one-loop level. Even in the case including one-loop corrections, it is useful to discriminate types of Yukawa interactions by using the pattern of deviations among the hff couplings. It is also demonstrated in Ref. [16] that information of inner parameters can be considerably extracted by combination of the precision measurements on the Higgs boson couplings when a deviation in κ V is large enough to be detected.
We here show the one-loop corrected scaling factors of hZZ and hbb couplings in the HSM in comparison with those in the Type I THDM. The expected 1σ uncertainties for these scaling factors at the LHC with the center-of-mass energy ( √ s) to be 14 TeV and the integrated luminosity (L) to be 3000 fb −1 (HL-LHC) and also the ILC with the combination of the run with √ s = 250 GeV with L = 250 fb −1 and that with √ s = 500 GeV with L = 500 fb −1 (ILC500) are given by [53] [σ(κ Z ), σ(κ b ), σ(κ γ )] = [2%, 4%, 2%], HL-LHC, For the predictions at the one-loop level in the THDM, we fully use the formulae and the numerical program developed in Ref. [16]. Finally, we discuss how we can discriminate the HSM and the Type I THDM by using theoretical predictions of ∆κ Z , ∆κ b and ∆κ γ with radiative corrections and Higgs boson coupling measurements at the HL-LHC and the ILC500. We find that if κ V will be measured to be deviated In this section, we summarize three theoretical constraints; i.e., perturbative unitarity, vacuum stability and the condition to avoid the wrong vacuum.

Perturbative unitarity
The constraints from the perturbative unitarity in the HSM had discussed in Ref. [41]. Under the perturbative unitarity bound, the matrix of the S-wave amplitude for the two-body to two-body scattering of scalar fields has to be satisfied in following conditions, In the HSM, there are seven neutral scattering processes. 2 Digonalizing the matrix of the neutral scattering processes, we obtain following independent eigenvalues, Because we take the constraint with ξ = 1 2 , specific bounds of Eq. (A1) are 3λ + 6λ S ± (3λ − 6λ S ) 2 + 4λ 2 ΦS < 8π, λ < 4π, λ ΦS < 4π. (A5)

Vacuum stability
As conditions of vacuum stability, we require the value of the potential to be positive at large Φ and S. Because terms of the quartic interactions are dominant in the potential with large values of the fields, must be satisfied. In order to satisfy A6, following bounds for λ parameters are imposed, where the third bound is applied when λ ΦS is negative.

To avoid the wrong vacuum
We are free to choose the value of v S . We take to be (v, v S ) = (v EW , 0), because the singlet field does not contribute to electroweak symmetry breaking. However, even (v EW , 0) is the extrema, there is a possibility that there are lower extremes at other points. According to Refs [38,40], five kinds of other extrema. If one or more than one extrema given in Eq. (24) and (B1) Ref. [38] become deeper than V (v EW , 0), then such a vacuum should be regarded as a wrong vacuum. In the analyses of this paper, we use the condition to avoid the wrong vacuum given in Ref. [38].

Appendix B: One-loop level corrected electroweak observables
We here list the renormalized electroweak parameter ∆r and renormalized W boson mass m reno W . They can be expressed as [44] where δ V B is the box and the vertex diagram contributions to the muon decay process, which is given by [44] δ V B = α em 4πs 2 Moreover, we also can calculate electroweak S, T and U parameters as where g Z = g/c W .

Appendix C: Tree level Higgs boson couplings
First, we give feynman rules of trilinear vertices and quartic vertices obtained from the Higgs kinetic term. There are two kinds of trilinear vertices and one kind of quartic vertices; i.e., Scalar-Gauge-Gauge, Scalar-Scalar-Gauge and Scalar-Scalar-Gauge-Gauge type. Their couplings are expressed as The coefficients of trilinear vertices g φV 1 V 2 and g µ φ 1 φ 2 V , and those of quartic vertices g φ 1 φ 2 V 1 V 2 are listed in Tab. II and in Tab. III, where p µ 1 (p µ 2 ) indicates incoming momentum of φ 1 (φ 2 ).
We give feynman rules of the scalar trilinear and the quartic vertices. When we express those couplings as In this section, we give one-loop fermion, vector boson and scalar boson contributions to the one, two and three point functions by using Passarino-Veltman functions [61] whose notation is same as those defined in Ref. [62]. We calculate 1PI diagrams in the 't Hooft-Feynman gauge so

One point functions
The 1PI tadpole contributions are calculated by where D = 4 − 2 and N f c indicates the color number of each particle.

Two point functions
The 1PI diagram contributions to the scalar boson two point functions are expressed as The fermion loop contributions to the gauge boson two point functions are calculated as   − g 3 m W c 2 W c α C 24 (c ± , c 0 , c ± ) − e 2 gm W s α C 24 (c ± , c γ , c ± ) − g 3 m W c α C 24 (c 0 , c ± , c 0 ) and C 1223 = C 12 + C 23 .