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field theory the width is naturally included in the imaginary part of the self-energy of the particles and, the Ward identity is respected at all orders in perturbation theory. These facts are exploited in those schemes by including the resummation of the fermion/boson loops in the propagator and the corrections in the electromagnetic vertices. Then, the imaginary part of the fermion/boson loops introduces the tree level width in the gauge boson propagator and, the gauge invariance is not violated since the fermion/boson loops obey the Ward identity order by order. 

At tree level, the propagator for a vector boson of mass M$_{V}$ can be set as 

<img src='content_image/369794.jpg'>

where T $^{µν}$( q ) ≡ g µν − q $^{µ}$q$^{ν}$/q 2 and L $^{µν}$( q ) ≡ q $^{µ}$q$^{ν}$/q $^{2}$, are the transversal and longitudinal projectors, respectively. The vertex for the process V ( q 1 ) → V ( q 2 ) γ ( k ) is defined from the electromagnetic current 

<img src='content_image/369795.jpg'>

At tree level, the vertex can be set as that given by the standard model for the W boson: 

<img src='content_image/369796.jpg'>

These expressions satisfy the Ward identity 

Upon the inclusion of the finite width of the boson, by considering the loop contributions, the propagator is modified in a generic form as: 

<img src='content_image/369797.jpg'>

<img src='content_image/369798.jpg'>

where Im Π $^{T}$( q $^{2}$) and Im Π $^{L}$( q $^{2}$), are the transverse and longitudinal part of the absortive contribution of the self-energy induced by the particles in the loop. Similarly, the vertex becomes 

<img src='content_image/369799.jpg'>

where Γ µνλ 1 contains the loop corrections. The Ward identity relates the loop contributions by requiring to satisfy 

<img src='content_image/369800.jpg'>

