Decays of Standard Model like Higgs boson h → γγ, Zγ in a minimal left-right symmetric model

Two decay channels h → γγ, Zγ of the Standard Model-like Higgs in a left-right symmetry model are investigated under recent experimental data. We will show there exist one-loop contributions that affect the h → Zγ amplitude


INTRODUCTION
The standard model-like (SM-like) Higgs decay h → Zγ is one of the most important channels being searched at the experimental center [1].Meanwhile, the experimental evidence of this loop-induced decay relating to the effective coupling hZγ has been reported by ATLAS and CMS recently [2? ], in agreement with the SM prediction within 1.9 standard deviation.Experimental data shows that the effective coupling hγγ derived from h → γγ decay rates is constrained very strictly [4].In contrast, the effective coupling hZγ in many models beyond SM (BSM) might differ considerably from the SM prediction, because the Z couplings to new particles are less strict than those of the photon.Hence, studying the effective hZγ couplings will be an indirect channel to determine the properties of new particles.Controlled by the strict experimental constraint of the decay h → γγ, constraints of the SM-like Higgs decay h → Zγ affected by new fermions and charged scalars were studied in several BSMs such as 3-3-1 models [5,6], only Higgs extended SM versions [7][8][9][10], U (1) gauge extensions from SM [11,12], supersymmetric models [13][14][15], chiral extension of the SM [16], . . . .Previous studies h → Zγ in left-right symmetric models ignored one-loop contributions relating to the diagrams consisting of both virtual Higgs and gauge particles in the loops [17,18], where the h-Higgs-gauge boson couplings were assumed to be suppressed.
One interesting extension of the beyond the SM models is an extension of the lepton sector.Namely, the minimal left-right version (MLRSM) is constructed based on the parity symmetry SU (2) L ⊗ SU (2) R ⊗ U (1) B−L [26][27][28], which contains Higgs fields included in two SU (2) L triplets denoted as ∆ L,R and a bi-doublet field Φ playing the SM Higgs role.
Therefore, the MLRSM allows us to solve the parity problem of the SM as well as the neutrino oscillation data through the seesaw mechanism.Besides, it contains extended particles which may result in interesting consequences for rare decays such as Higgs boson All needed ingredients relevant to one-loop contributions to the decay amplitudes h → Zγ, γγ will be collected in this section.In most general, the electric charge operator can be written as [29,30] where T L,R i 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 number of the fermions can be written in the table I.

TABLE I: The baryon and the lepton numbers of the fermions in MLRSM
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 ∆ L,R satisfying The Higgs components develop vacuum expectation values (VEV) defined as where the neutral Higgs components are expanded as follows The symmetry breaking pattern in MLRSM happens in two following steps: − −−−−− → U (1) Q , which corresponds to the reasonable limits that v R ≫ k 1 , k 2 ≫ 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 to 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, ∆ L,R ).The physical spectrum and masses of all particles in the model under consideration are summarized as follows.

B. Fermions
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 c β f e + s β f e → y e , s β f q + c β f q → y u and k 1 f q + k 2 f q → y d .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 can be diagonalized by two unitary transformations V L f and V R f as follows: ).Here m f,i with i = 1, 2, 3 and f = ℓ, u, d denotes the physical masses of charged leptons, up and down quarks.The transformations between the flavor basis f . 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.

C. Gauge bosons
The covariant derivative corresponding to the symmetry of the MLRSM is defined as [29] D 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 to the scalar multiplets are where X = L, R, σ a is the Pauli matrix corresponding to the SU (2) doublet representation of T a L,R with a = 1, 2, 3. Therefore, the mass terms of gauge bosons are derived from the vev of Higgs components as follows where k is defined in Eq. ( 9).The mass terms of the neutral and charged gauge bosons read: where Xµ with X = L, R. The mixing angle ξ between two singly charged gauge bosons W ± L and W ± R is determined by the following formula Using the approximation that tan 2ξ ≪ 1 ⇒ tan 2ξ ≈ sin 2ξ ≈ 2 sin ξ ≈ 2ξ, and The singly charged gauge bosons W ± L,R can be written as functions of the mass basis where c ξ ≡ cos ξ and s ξ ≡ sin ξ.The respective charged gauge boson masses are found to be Identifying the

The original neutral gauge basis (W 3
Lµ , W 3 Rµ , B µ ) are expressed in terms of the mass basis (A µ , Z 1µ , Z 2µ ) as follows where and the mixing angles t R , t z 2 , t z 3 are given by State synchronization with the SM as follows: W 3 Lµ ≡ W 3 µ , Z 1µ ≡ Z µ in the limits c z 3 → 1, then we also have The Weinberg angle θ W is identified from the definition cos θ Then, the neutral gauge boson masses of Z 1 , Z 2 , and the photon A are given by , and m 2 A = 0.
In addition, Z 1 ≡ Z, and Z 2 ≡ Z ′ are respectively SM gauge boson Z found experimentally, and the heavy one appearing in the MLRSM.

D. Higgs bosons
The MLRSM scalar potential is written as [29] V From the minimal conditions of the Higgs potential given in Eq. ( 24), three parameters Φ , and µ 2 ∆ 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 not that Eq. ( 25) do not use the the limit k 1 ≫ k 2 mentioned in Ref. [29], which gives 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 The SM-like Higgs property appears in Eq. ( 26) as In this limit, h 0 ≡ h can be identified with the SM-like Higgs boson with mass m h = 125.38GeV confirmed experimentally [1].Then the Higgs selfcoupling λ 1 is expressed as follows We note that Eq. ( 26) for SM-like Higgs mass is consistent with Ref. [32][33][34], implying that the m h value is still at the electroweak scale even in the case of large t β .Therefore, the 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 Yukawa Lagrangian in Eq. ( 7), we derive easily that where k = 246 = g/( √ 2m W ), 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 G ± W 2 is massless, corresponding to the Goldstone boson eaten up by W ± 2 , and the remaining squared masses of singly charged Higgs bosons are Besides that, two components (δ ++ L , δ ++ R ) are also physical states with the following masses

III. COUPLINGS AND ANALYTIC FORMULAS INVOLVED WITH LOOP-INDUCED HIGGS DECAYS A. Couplings
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: + h.c.) + . . ., the Feynman rule −iλ hSS corresponds to the vertex hSS.All non-zero factors λ hSS are given in Table II.We note that Vertex Coupling: the vertex factors in Table II are derived following the general notation defined in Ref. [35], so that we can use the analytic formulas to compute the partial decay widths 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 Eqs.(7), where the Feynman rule is Because this model does not have exotic charged fermions and the couplings of SM leptons to neutral Higgs/gauge bosons (g h f f , g Z f f ) are defined as in the SM [36][37][38], we will use the SM results for one-loop fermion contributions to the decay amplitudes h → Zγ, γγ.
The Higgs-gauge boson couplings giving one-loop contribution to the decays h → Zγ, γγ are derived from the kinetic Lagrangian of the Higgs bosons, namely where S i , S j = H ± 1,2 , H ±± 1,2 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 III.The momenta Vertex Coupling appearing in the vertex factors are ∂ µ h → −ip 0µ h and ∂ µ S i,j → −ip µ S i,j , where p 0 , p ± are incoming momenta.
The Feynman rules for Z couplings to charged Higgs and gauge bosons in Eq. ( 32) are given in Table IV.The couplings The triple gauge couplings of Z and photon to W ± 1,2 are derived from the kinetic Lagrangian of the non-Abelian gauge bosons where , and ϵ abc (a, b, c = 1, 2, 3) are the SU (2) structure constants.The respective Z couplings to W ± 1,2 are included in the following part: Then, the vertex factors corresponding to particular couplings are defined as where The photon always couples to two identical particles as the consequence of the Ward Identity [39], see the second line of Eq. ( 35).The non-zero factors for triple couplings of Z with charged gauge bosons are collected in Table V.
To end this section, we emphasize that all couplings determined in this section do not use the assumption k 1 ≫ k 2 , equivalently t β ≫ 1 used in Ref. [29,30].
to determine the gauge boson contributions.The fermion contributions to amplitude of the decay h → Zγ coincide with the SM results calculated in Ref. [36,37].Using the general calculation introduced in Ref. [35], we can write these contributions as follows where all form factors F SM 21,f i are written in terms of the Passrino-Veltman (PV) notations [40].
Similarly, the contribution from the charged Higgs bosons can be given as The charged gauge boson contributions W ± 1,2 to the h → Zγ amplitude are Similarly, the contribution from charged Higgs and gauge boson arising from two 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 F LR 21 is derived as follows [35] We note that F LR 21,V S were omitted in some previous works [5,17,18] because it was expected to be much smaller than the contributions from the SM and are still far from the sensitivity of the recent experiments.However, since collider sensitivities have recently been improved and new scales have been established, these contributions are necessary.The branching ratio where Γ LR h is the total decay width of the SM-like Higgs boson h [41,42].Although there are available experimental measurements of the SM-like Higgs boson productions and decays [43], we focus only on the Higgs production through the gluon fusion process ggF at LHC, in which the respective signal strength predicted by two models SM and MLSM 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 singal strength is µ Zγ = 2.4 ± 0.9 at 2.7σ (standard deviation) [2? ].
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 the notations that [6] where x ) 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].

NUMERICAL DISCUSSIONS A. Setup parameters
In this section, there are 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 g 2 ≃ 0.651, α em = 1/137, e = √ 4πα em , s 2 W = 0.231.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 λ 1 is given by Eq. ( 27).Some Higgs self-couplings are expressed as functions of the heavy Higgs boson masses, namely Choosing the mass of m H + 1 , m H + 2 , and m H ++ 1 as free parameters we get The other free parameters are λ 2,3,4 , R ) > 0, the mixing angle and the gauge bosons 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), various discussions for the more general case g R ̸ = g L , which showed that this ratio is allowed in the follows range [46,47]: where the lower bound v R > 10 TeV.
A recent study showed a lower bound of m W R > 5.5 TeV is still allowed [31], which gives v R ≥ 17 TeV in this case.On the other hand, the constraints 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 of works discuss the constraints of Higgs masses indirectly [48], or directly from LHC for doubly charged Higgs bosons [49].The lower bounds are m H ±± ≥ 1080 GeV.Theoretical constraints was 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 we choose here will be scanned in the following ranges: where the Higgs self-couplings satisfy all theoretical constraints discussed on Ref. [33].

B. Results and discussions
To express the differences of the prediction between the SM and the MLRSM, we define a quantity ∆µ Zγ as in Ref. [6] ∆µ which is constrained by recent experiments ∆µ Zγ = 1.4 ± 0.9 [2? ], implying the following 1σ deviation: The 1σ constraint from h → γγ decay originating from ggF fusion is defined as ∆µ LR γγ ≡ (µ LR γγ − 1) × 100%, 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 that both values of ∆µ Zγ , ∆µ γγ → 0, implying the consistency with the SM results.Considering the special case of g L = g R , we discuss firstly on the dependence of ∆µ LR Zγ on ∆µ LR γγ , which is illustrated in Fig. 2. We just focus in the region satisfying |∆µ LR Zγ | ≥ 5% in order to collect interesting points that may support the 1σ range given in Eq. ( 56) It can be seen that ∆µ LR Zγ is constrained strictly by ∆µ LR γγ , i.e., ∆µ LR Zγ ≤ 18% in the range of 2σ deviation given in Eq. (57).It is noted that negative values of ∆µ LR γγ < 0 can give large ∆µ LR Zγ than the positive ones.Largest values of ∆µ LR Zγ is still much smaller than the 1σ deviation given by recent experimental data.
For completeness the case of g L = g R , we discuss on the dependence of ∆µ LR Zγ on t β and v R , which are shown in Fig. 3.We can see that ∆µ LR Zγ depends weakly on v R , but strongly on t β .Namely, all values v R can give large ∆µ LR Zγ , while needs small t β → 1.

The correlations between ∆µ LR
Zγ and charged Higgs boson masses are shown in Fig. 4. The results show that all charged Higgs masses do not affect strongly on values of ∆µ LR Zγ .Finally, we consider the general case of g R with allowed values given in Eq. (53).Numerical results for important correlations between ∆µ LR Zγ with ∆µ LR γγ and g R are depicted in Fig.

It can be seen clearly that large ∆µ LR
Zγ corresponds to large g R , which is consistent with the property that new contributions consisting of factor g R in the Feynman rules shown in      section III.We emphasize that lagre g R is necessary for large ∆µ LR Zγ that can reach value of 46%, very close to the recent experimental sensitivity.Furthermore, the expected sensitivity of ∆µ LR Zγ = 4% does not affect large values of ∆µ LR Zγ that are visible for the incoming experimental sensitivity of 23%.
Finally, we focus on the correlations between ∆µ LR Zγ verus t β , v R , and all charged Higgs masses, which are depicted in Fig. 6.It is seen again that large ∆µ LR Zγ requires small t β .In contrast, ∆µ LR Zγ depends weakly on charged Higgs boson masses and v R given in Eq. (54).

V. CONCLUSIONS
We have studied all one-loop contributions to the SM-like Higgs decays h → γγ, Zγ in the MLRSM framework.Interesting properties of the new gauge and Higgs bosons were explored.Namely, the SM-like Higgs couplings were identified with the SM prediction   and experimental data.All masses, physical states of gauge and Higgs bosons and their were presented clearly so that all couplings relate to one-loop contributions the decay amplitudes h → γγ, Zγ are derived analytically.From this, decays h → γγ, Zγ in MLRSM have been discussed using the relevant recent experimental results.The one-loop contributions from the diagrams containing both gauge and Higgs mediation were included in the decay amplitude h → Zγ.These contributions were ignored in previous studies, although they may enhance the h → Zγ amplitude, but do not affect the h → γγ one, leading to the possibility that large ∆µ Zγ may be allowed under the strict experimental constraint of ∆µ γγ .We have shown that the mentioned h decay rates depend weakly on t β , the SU (2) R vacuum scale v R .The 2σ deviation of µ γγ results in a rather strict constraint |∆µ Zγ | ≤ 46%.On the other hand, the large values of ∆µ Zγ > 23% can appear under the very strict constraint of |∆µ γγ | ≤ 4% corresponding to the future experimental sensitivities, provided the two requirements that enough small t β and large g R are necessary.Therefore, the future experimental searches of the two decays mentioned in this work will be important FIG.1: One-loop three-point Feynman diagrams contributing to the decay h → Zγ in the unitary gauge, where f i,j are the SM fermions,s i,j = H ± 1,2 , H ±± 1,2 , V i,j = W ± 1 , W ± 2 .

TABLE II :
Feynman rules for the SM-like Higgs boson couplings with charged Higgs bosons

TABLE III :
Feynman rules for couplings of the SM-like Higgs boson to charged Higgs and gauge bosons.

TABLE IV :
Feynman rules of couplings of Z to charged Higgs and gauge bosons.Notations p + and p − are incoming momenta.
FIG. 3: Correlations between ∆µ LRZγ with different values of t β and v R with g R = g L .