All needed ingredients relevant to one-loop contributions to the decay amplitudes h → Zγ, γγ will be collected in this section. Most generally, the electric charge operator can be written as [29,30]:

where are the generators of the gauge groups SU(2)_{L, R}; B (L) is the baryon (lepton) number defining the U(1)_{B − L} group in the MLRSM. The baryon and lepton numbers of the fermions can be written as in Table 1.

With this information, we can write down the lepton and fermion representations as follows:

where i = 1, 2, 3 is the flavor index.

Gauge boson and fermion masses are originated from the following scalar sector, consisting of a bi-doublet and two triplet scalar fields Δ_{L, R} satisfying

The Higgs components develop vacuum expectation values (VEVs) defined as

where the neutral Higgs components are expanded as follows:

The symmetry-breaking pattern in MLRSM happens in the two following steps: , which corresponds to the reasonable limits that v_R ≫ k₁, k₂ ≫ v_L. Only new gauge bosons will be massive after the first step. The second step is the SM symmetry-breaking generating masses for the SM particles. When the symmetry is broken in step two, only U(1)_Q remains unbroken, where Q is the quantifier. As a result, the photon A_μ has no mass. We stress that the MLRSM contains no more than three scalar multiplets (ϕ, Δ_{L, R}). The physical spectrum and masses of all particles in the model under consideration are summarized as follows.

Physical fermion states and their masses always relate to the Yukawa interactions, which are included in the following Lagrangian parts for leptons and quarks:

Then, the mass terms for leptons and quarks are computed. We will use the results for fermion masses and mixing presented in Refs. [29,30], i.e. all the original and the physical states of fermions are the same. They are identified with the SM ones and will be denoted as e_{aL, R}, u_{aL, R}, and d_{aL, R} in this work. The mass matrices M_ℓ and M_{u, d} for charged leptons and up and down quarks are

where

As we will show below, the matching condition to the SM leads to k = 246 GeV and the Yukawa couplings of quarks defined in the SM can be seen as follows: , , and . Here, we fix α = 0 so that the value of t_β = s_β/c_β can be small, namely t_β ≥ 1.2 [31]. The three above fermion mass matrices are denoted as M_f with f = ℓ, u, d and can be diagonalized by two unitary transformations and as follows: . Here m_{f, i} with i = 1, 2, 3 and f = ℓ, u, d denotes the physical masses of charged leptons and of up and down quarks. The transformation between the flavor basis and the mass basis is . As we will show below, the couplings of the SM-like Higgs boson with charged leptons and quarks are the same as the SM results.

The covariant derivative corresponding to the symmetry of the MLRSM is defined as [29]:

where g_{L, R} and g′ are the SU(2)_{L, R} and U(1)_{B − L} gauge couplings, respectively.

The Lagrangian for scalar kinetic parts is written as

The particular forms of covariant derivatives for the scalar multiplets are

where X = L, R, σ^a is the Pauli matrix corresponding to the SU(2) doublet representation of with a = 1, 2, 3. Therefore, the mass terms of gauge bosons are derived from the VEVs of Higgs components as follows:

where k is defined as in Eq. (9). The mass terms of the neutral and charged gauge bosons read:

where with X = L, R. The mixing angle ξ between two singly charged gauge bosons and is determined by the following formula:

Using the approximation that tan 2ξ ≪ 1⇒tan 2ξ ≈ sin 2ξ ≈ 2sin ξ ≈ 2ξ, and v_L ≪ k₁, k₂ ≪ v_R, the W_L − W_R mixing angle ξ is

The singly charged gauge bosons can be written as functions of the mass basis () as follows:

where c_ξ ≡ cos ξ and s_ξ ≡ sin ξ. The respective charged gauge boson masses are found to be

Identifying that in the SM, we get .

The original neutral gauge basis is expressed in terms of the mass basis (A_μ, Z_1μ, Z_2μ) as follows:

where

and the mixing angles are given by

State synchronization with the SM is as follows: , Z_1μ ≡ Z_μ in the limits , then we also have

The Weinberg angle θ_W is identified from the definition . Then, the neutral gauge boson masses of Z₁, Z₂, and the photon A are given by

In addition, Z₁ ≡ Z and Z₂ ≡ Z′ are, respectively, the SM gauge boson Z found experimentally, and the heavy one appearing in the MLRSM.

The MLRSM scalar potential is written as [29]:

From the minimal conditions of the Higgs potential given in Eq. (24), three parameters , , and are expressed as functions of other independent parameters. Inserting them into the Higgs potential (24), we can determine all Higgs boson masses and physical states. Firstly, the original and the mass base of neutral CP-even Higgs bosons are related to each other as follows:

We note that Eq. (25) does not use the the limit k₁ ≫ k₂ mentioned in Ref. [29], which gives , , s_β ≈ 1, and ϵ₁ ≡ k₁/v_R, ϵ₂ ≡ k₂/k₁. Besides that, from Eq. (25) we get the same result as in Ref. [29] in this limit. In this study, the SM-like Higgs mass is calculated approximately to the order , namely

The SM-like Higgs property appears in Eq. (26) as because when . In this limit, can be identified with the SM-like Higgs boson with mass m_h = 125.38 GeV confirmed experimentally [1]. Then the Higgs self-coupling λ₁ is expressed as follows:

We note that Eq. (26) for SM-like Higgs mass is consistent with Refs. [32–34], implying that the m_h value is still at the electroweak scale even in the case of large t_β. Therefore, a value of t_β ≥ 1.2 is still allowed to get the SM-like Higgs mass consistent with the experiment.

Regarding the SM-like Higgs couplings with charged leptons and fermions, using Eq. (25) for the Yukawa Lagrangian in Eq. (7), we derive easily that

where , and the transformation in Eq. (28) is based on discussion relating to Eq. (8). Therefore, the SM-like Higgs couplings with charged fermions can be identified with the SM results.

Similarly, the original and mass states of the singly charged Higgs bosons have the following relations:

where is massless, corresponding to the Goldstone boson eaten up by , and the remaining squared masses of singly charged Higgs bosons are

Besides that, two components () are also physical states with the following masses:

From the above Higgs potential and the discussion on the masses and mixing of Higgs bosons, all Higgs self-couplings of h giving one-loop contributions to the decays h → γγ, Zγ can be derived analytically. From the general notations in the interacting Lagrangian: , the Feynman rule −iλ_hSS corresponds to the vertex hSS. All nonzero factors λ_hSS are given in Table 2. We note that the vertex factors in Table 2 are derived following the general notation defined in Ref. [35], so that we can use the analytic formulas to compute the partial decay widths of h → γγ, Zγ in the MLRSM mentioned in this work.

The couplings of h with SM fermions can be determined using the Yukawa Lagrangians given in Eq. (7), where the Feynman rule is for each vertex . Because this model does not have exotic charged fermions and the couplings of SM leptons to neutral Higgs/gauge bosons () are defined as in the SM [36–38], we will use the SM results for one-loop fermion contributions to the decay amplitudes of h → Zγ, γγ.

The Higgs–gauge boson couplings giving one-loop contributions to the decays h → Zγ, γγ are derived from the kinetic Lagrangian of the Higgs bosons, namely

where denote charged Higgs bosons in the MLRSM. The Feynman rules for the h couplings to at least one charged gauge boson are shown in Table 3. The momenta appearing in the vertex factors are ∂_μh → −ip_0μh and ∂_μS_{i, j} → −ip_μS_{i, j}, where p₀, p_± are incoming momenta.

The Feynman rules for Z couplings to charged Higgs and gauge bosons as in Eq. (32) are given in Table 4. The couplings are zero in the MLRSM.

The triple gauge couplings of Z and photon to are derived from the kinetic Lagrangian of the non-Abelian gauge bosons

where , and are the SU(2) structure constants. The respective Z couplings to are included in the following part:

Then, the vertex factors corresponding to particular couplings are defined as

where Γ_μνλ(p₀, p₊, p₋) ≡ g_μν(p₀ − p₊)_λ + g_νλ(p₊ − p₋)_μ + g_λμ(p₋ − p₀)_λ, and i, j = 1, 2. The photon always couples to two identical particles as the consequence of the Ward Identity [39], see the second line of Eq. (35). The nonzero factors for triple couplings of Z with charged gauge bosons are collected in Table 5.

To end this section, we emphasize that all couplings determined in this section do not use the assumption k₁ ≫ k₂, equivalently t_β ≫ 1 as used in Refs. [29,30].

In the MLRSM framework, one-loop three-point Feynman diagrams giving contributions to the decay amplitude are shown in Fig. 1, where the unitary gauge is applied to determine the gauge boson contributions. The fermion contributions to the amplitude of the decay h → Zγ coincide with the SM results calculated in Refs. [36,37]. Using the general calculation introduced in Ref. [35], we can write these contributions as follows:

where all form factors are written in terms of the Passarino–Veltman (PV) notations [40].

Similarly, the contribution from the charged Higgs bosons can be given as

The charged gauge boson contributions to the h → Zγ amplitude are

Similarly, the contribution from the charged Higgs and gauge bosons arising from diagrams 3 and 4 in Fig. 1 can be given as

where

Now, the h → Zγ partial decay width is [41,42]:

where the scalar factors are derived as follows [35]:

We note that were omitted in some previous works [5,17,18] because their contributions were expected to be much smaller than the contributions from the SM and are still far from the sensitivity of recent experiments. However, since collider sensitivities have recently been improved and new scales have been established, these contributions are necessary. The branching ratio Br^LR(h → Zγ) in the MLRSM framework is

where is the total decay width of the SM-like Higgs boson h  [41,42]. Although experimental measurements of the SM-like Higgs boson productions and decays are available [43], we focus only on the Higgs production through the gluon fusion process ggF at the LHC, in which the respective signal strengths predicted by the two models SM and MLRSM are equal. Then the signal strength corresponding to the decay mode h → Zγ predicted by the MLRSM is:

where Br^SM(h → Zγ) is the SM branching ratio of the decay h → Zγ. The recent ggF → h → Zγ signal strength is μ_Zγ = 2.4 ± 0.9 at 2.7σ (standard deviation) [2,3].

Similarly, the partial decay width and signal strength of the decay h → γγ can be calculated as [35,42]:

where

and

Here we have used notations to the effect that [6]:

where X₂ = C₁₂ + C₂₂ + C₂ and are PV functions [40] with x = f, s, v implying fermions, charged Higgs, and gauge bosons, respectively. Particular forms given in Eq. (49) are defined precisely in Ref. [6]. In the following section, the numerical results will be evaluated using LoopTools [44].

In this section, we use the following quantities fixed from experiments [45]: m_h = 125.38 GeV, m_W, m_Z, well-known fermion masses, v ≃ 246 GeV, the SU(2)_L gauge coupling g₂ ≃ 0.651, α_em = 1/137, , .

The unknown Higgs self-couplings of the MLRSM are ρ_{1, 2, 3, 4}, α_{1, 2, 3, 4, 5, 6}, λ_{2, 3, 4}. The dependent parameter λ₁ is given by Eq. (27). Some Higgs self-couplings are expressed as functions of the heavy Higgs boson masses, namely

Choosing the masses of , , and as free parameters we get

The other free parameters are λ_2,3,4, , hence the mixing angle and the gauge boson masses will be at the orders of :

We note here that the relations given in Eq. (52) are consistent with the SM because the two couplings hW⁺W⁻ and ZW⁺W⁻ are consistent with the SM predictions.

Apart from the limit g_L = g_R chosen in Eq. (52), in various discussions for the more general case g_R ≠ g_L, which showed that this ratio is allowed in the following range [46,47]:

where the lower bound v_R > 10 TeV.

A recent study showed a lower bound of TeV is still allowed [31], which gives v_R ≥ 17 TeV in this case. On the other hand, the constraint of t_β ≥ 1.2 is allowed, while no lower bounds of charged Higgs masses were given; especially in the limit of the phase, α given in Eq. (6) is zero. Various works discuss the constraints of Higgs masses indirectly [48], or directly from the LHC for doubly charged Higgs bosons [49]. The lower bounds are GeV. Theoretical constraints were discussed in Ref. [33] for Higgs self-couplings satisfying unitarity bounds and vacuum stability criteria, which will be applied in our numerical investigation.

Based on the above discussion for investigating the significant strengths of the two decays h → γγ, Zγ, the values of unknown independent parameters that we choose here will be scanned in the following ranges:

where the Higgs self-couplings satisfy all theoretical constraints discussed in Ref. [33].

To express the differences of prediction between the SM and the MLRSM, we define a quantity Δμ_Zγ as in Ref. [6]:

which is constrained by recent experiments as Δμ_Zγ = 1.4 ± 0.9 [2,3], implying the following 1σ deviation:

The 1σ constraint from h → γγ decay originating from ggF fusion is defined as , leading to the respective 2σ deviation as follows:

The numerical results we discuss in the following will always satisfy this constraint. We have checked numerically that the MLRSM always contains regions of the parameter space where both values of Δμ_Zγ, Δμ_γγ → 0, implying consistency with the SM results. Considering the special case of g_L = g_R, we discuss firstly the dependence of on , which is illustrated in Fig. 2. We just focus on the region satisfying in order to collect interesting points that may support the 1σ range given in Eq. (56). It can be seen that is constrained strictly by , i.e. in the range of 2σ deviation given in Eq. (57). It is noted that negative values of can give larger than the positive ones. The largest values of are still much smaller than the 1σ deviation given by recent experimental data.

We comment here to make the point that the future sensitivities are and , respectively [23]. In the model under consideration, large values of are not allowed with g_L = g_R.

For completeness in the case of g_L = g_R, we discuss the dependence of on t_β and v_R, which are shown in Fig. 3. We can see that depends weakly on v_R, but strongly on t_β. Namely, all values of v_R can give large , while needing small t_β → 1.2.

The correlations between and charged Higgs boson masses are shown in Fig. 4. The results show that all charged Higgs masses do not affect strongly the values of .

Finally, we consider the general case of g_R with allowed values given in Eq. (53). Numerical results for important correlations of with and g_R are depicted in Fig. 5. It can be seen clearly that large corresponds to large g_R, which is consistent with the property that new contributions consist of the factor g_R in the Feynman rules shown in Sect. 3. We emphasize that large g_R is necessary for large that can reach a value of 46%, very close to the recent experimental sensitivity. Furthermore, the expected sensitivity of does not affect large values of that are visible for the incoming experimental sensitivity of 23%.

Finally, we focus on the correlations of with t_β, v_R, and all charged Higgs masses, which are depicted in Fig. 6. It is seen again that large requires small t_β. In contrast, depends weakly on charged Higgs boson masses and v_R as given in Eq. (54).