content_image/345257.jpg content_image/345258.jpg content_image/345259.jpg content_image/345260.jpg
## C. Chiral limit 

The chiral limit correction to the vertex is known to be proportional to the tree level, in both the Fermion and boson loops corrections [11]. Therefore, in this limit we can write the modification to the form factors in a generic form as follows: 

## III. NUMERICAL RESULTS 

<img src='content_image/345257.jpg'>

The correction to the vertex seems to induce a modification to the the electric charge. However, since the Ward identity is fulfilled, the modification to the vertex is followed by a modification to the propagator, which produces an exact cancelation of the correction to the electric charge. Let us illustrate this point in more detail, for the sake of clarity we consider the expression in the chiral limit: The modified propagator can be set as [11, 15]: 

<img src='content_image/345258.jpg'>

where ¯ M$_{V}$ , ¯ Γ ≡ M$_{V}$ , Γ / √ 1 + γ $^{2}$, ( γ ≡ Γ /M$_{V}$ ), i.e., it can be seen as a renormalization of the vector field by the inclusion of the finite width 

<img src='content_image/345259.jpg'>

where Z 1 / 2 = 1 / (1 + iγ ) 1 / $^{2}$. Then, the electromagnetic current becomes 

<img src='content_image/345260.jpg'>

Therefore, gauge invariance requires that Zα $^{′}$( k $^{2}$) = 1, which is indeed the case, and the electric charge does not receive any correction. Note that the modification to the vertex given by equation 21 implies that none of the multipoles receive corrections in the chiral limit. An analysis for unstable spin-1/2 particles has also been performed in ref. [16], pointing out to complex renormalization factors as a requirement for properly defined physical quantities. Further considerations on the renormalizability of the wave function can be seen in ref. [17]. The proper values of the modifications to the form factors are then found by the expressions 

