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For a boson like the W , the scheme consider that such loops are produced by fermions, while for vector mesons, like the ρ , bosons are the natural particles in the loop. 

In general, the CP conserving electromagnetic vertex can be decomposed into the follow- ing Lorentz structure 

<img src='content_image/345426.jpg'>

where the electromagnetic form factors can be identified as: |Q| ≡ α ( k $^{2}$) is the electric charge form factor ( in e units), | ⃗ µ | = β ( k $^{2}$) ≡ 1 + κ + λ is the magnetic dipole moment form factor (in e/ 2 M$_{V}$ units) and the electric quadrupole form factor is | X$_{E}$ | = κ − γ ( k $^{2}$) M 2 V ≡ κ − λ (in e/M 2 V units). The parameters κ and λ are of common use in the literature to refer to the electromagnetic multipoles [12, 13]. The static electromagnetic properties of a particle are defined for the case when the particle is on-shell and in the limit of ⃗ k → 0. At tree level, for example, the standard model predicts for the W to have α (0) = 1, β (0) = 2 and γ (0) = 0 ( κ = 1 and λ = 0), corresponding to |Q| = 1, | ⃗ µ | = 2 and | X$_{E}$ | = 1. Deviations from these values are generically called anomalous and are produced by the inclusion of higher order contributions [14]. In the present case such contributions are exclusively those required to maintain the electromagnetic gauge invariance, upon the introduction of the finite decay width. 

## A. W boson form factors 

Let us identify the modification to the W boson form factors introduced by the correction to the electromagnetic vertex. For that purpouse we consider the explicit expression of the vertex obtained in ref. [10], where the mass of the emitting particles in the loop ( m ) and its weak partner ( m $^{′}$) have been considered. The Lorentz structure, transversality and on- shell condition of the boson along with the proper limit for k → 0 leads to the following expressions for the form factors, defined in equation (8): 

## • Electric charge 

where Σ 2 ≡ m $^{2}$+ m $^{$^{′}$2}$, Δ 2 ≡ m $^{2}$− m $^{′}$2 and Q$_{i}$ is the electric charge of the radiating particle in the loop, Γ$_{i}$ /M$_{W}$ ≡ g 2 $_{i}$λ 3 / $^{2}$/ 48 π is the partial decay width for the modes corresponding 

<img src='content_image/345427.jpg'>

