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The Z, Z' → γγγ decays in the minimal 331 model

A complete calculation at one-loop level for the Z, Z' → γγγ decays is presented in the context of the minimal 331 model, which predicts the existence of a new Z' neutral gauge boson, Y++, Y+ charged gauge bosons, new exotic quarks and charged scalars. Bose symmetry is exploited to write a compact and manifest SUC[(3)-invariant vertex function for the γγγV0 coupling. Previous results on the Z → γγγ are reproduced (Br(Z → γγγ) ~ 10−10). It is found that this decay is insensitive to the effects of new heavy particles. This contrasts with the Z' → γγγ decay where the associated branching ratio is of the order of 10−6.


Introduction
In this work, we will study some properties of the Z ′ gauge boson predicted by the minimal 331 model [1]. Particularly, we are interested in studying the rare decay of this Z ′ gauge boson into three photons. This type of decay is naturally suppressed in renormalizable theories, as they first arise at the one-loop level. The analogous decay in the Standard Model (SM) Z → γγγ is very suppressed, with a branching ratio of the order of 10 −10 GeV [2].

The minimal 331 model
The minimal 331 model is based on the SU C (3) ⊗ SU L (3) ⊗ U X (1) gauge group. The SU C (3) group represents strong interactions and the SU L (3) ⊗ U X (1) group is the simplest extension of the electroweak gauge group. The minimal 331 model predicts new exotic quarks, new gauge bosons and scalars, among them a new neutral gauge boson Z ′ is expected at the TeV scale [1]. This model offers a possible solution to the family replication problem: anomalies cancel out when all families are summed over, which suggests that the family number must be 3. The lepton spectrum of the model is accommodated as antitriplets of SU L (3): Two quark generations are arranged as triplets and the other one as an antitriplet: The right-handed quarks are given as follows: The D, S and T particles are new exotic quarks with charges −4/3, −4/3 and +5/3, respectively. Henceforth, we denote exotic quarks with Q. The scalar sector of the minimal 331 model is comprised of three triplets and one sextet of SU L (3): The spontaneous symmetry breaking occurs in the following way:

masses to the new exotic quarks D , S , T and to the new heavy gauge bosons
The amplitude of the γγγV 0 vertex can be written according to the spin of particles circulating in the loops (see Figure 1). This is possible since we calculate the gauge particles contribution using covariant R ξ gauges [3], which render finite and gauge invariant contributions where M 1/2 , M 1 and M 0 are, respectively, the spinorial, vectorial and scalar amplitudes.
Once the loop integrals are computed, the amplitudes can be expressed in terms of gauge structures and their associated form factors as follows: Bose statistics imposes that the amplitudes must be symmetric under the interchanges of pairs of photons: (p 1 , µ 1 ) ↔ (p 2 , µ 2 ) ↔ (p 3 , µ 3 ). Gauge invariance is satisfied: p iµ i M µ 1 µ 2 µ 3 µ 4 γγγV 0 = 0. There are 18 Lorentz structures: The rest of structures are obtained from these ones by Bose symmetry. F V 0 i are finite scalar form factors in terms of Passarino-Veltman functions [3].

The Z → γγγ and Z ′ → γγγ decays
By using the formula for calculating the decay width [3,4], we can discuss the impact of new physics on the Z → γγγ decay. To perform this, we consider the scenario: m Q = 500 GeV and m Y = m H =250 GeV. The relative importance of each type of contribution is shown in Table 1. The Particle Data Group reports that Γ(Z → γγγ) < 10 −5 GeV [4]. However, the SM prediction states that Γ(Z → γγγ) = 1.35 × 10 −9 GeV with an associated branching ratio of 5.41 × 10 −10 . In contrast, the minimal 331 model prediction is Γ(Z → γγγ) = 1.31 × 10 −9 GeV with a respective branching ratio of 5.26 × 10 −10 [3]. As referred to the minimal 331 model, it imposes the theoretical restriction m Y m Z ′ < 0.26 [5], whereas lower bounds on m Y and m Z ′ obtained from experimental data restrict this ratio to be 0.19 < m Y m Z ′ [5]. The most promising scenario occurs when m Q = 500 GeV, m Y m Z ′ = 0.19, m Y =275.5 GeV and m H = 250 GeV. The total contribution to Br(Z ′ → γγγ) can be appreciated in Fig. 2. From this figure, we can see that the branching ratio for the Z ′ → γγγ is mainly of the order of 10 −6 . For further details of calculation and analysis see Reference [3].

Final remarks
The minimal 331 model predicts new physic at energy relatively near to the Fermi scale and gives a possible solution to the fermion family replication problem. The Z → γγγ decay is insensitive to the presence of the new physics provided by the minimal 331 model. We corroborate that