given in the previous section divided by α $^{′}$(0). In Table I, we present the corresponding results for the W gauge boson and the ρ and K ∗ mesons. Recall that the particles in the loops are restricted to be on-shell. For the W we have included all the leptons and the u , d , s and c quarks, and the numerical values are similar to those obtained in the chiral limit (the largest fermion − to − W mass ratio is only ≈ 0 . 022). As a reference of the magnitude of the modification, we recall several results obtained for contributions of different nature: Ref. [19] finds a correction to the MDM induced by the Higgs of µ = 2 − 0 . 0151, reference [20] finds a correction from quarks, leptons and Higgs loops in the SM of µ = 2 + 0 . 00258, i.e. the width induced corrections is at least two orders of magnitude smaller than other standard contributions. Note that, due to the renormalization condition, the argument that the finite width correction to the vertex is of order Γ /M$_{V}$ ( ≈ 0.03, 0.19 and 0.06 for the W , ρ and K ∗ respectively) does not extend to the electromagnetic multipoles. 

For the ρ meson, pions are the only on-shell particles allowed in the loops. In this case the pion − to − ρ mass ratio ( ≈ 0 . 18) is not as small as the corresponding for the fermion − to − W , and therefore a significant effect from the mass of the particles in the loop is expected. We can compare our results as shown in Table I with those shown in Table II as obtained from several approaches to QCD: Light-front framework with constituent quarks [2–4] and covariant formulations based on the Dyson-Schwinger equations of QCD [1, 5]. In general these last are about the same or one order of magnitude larger than the finite width induced corrections computed in this work. Recently, lattice calculations of the form factors have become available [6] and, in particular, the dependence on the pion mass they are able to reproduce has been exhibited. In Figure 1, we compare our result for the MDM with lattice calculations as a function of the pion mass [6], we also include predictions from the models at the physical mass of the pion. We observe that the pion mass dependence of our results are mostly flat with a slightly tendency to rise for very large masses. Lattice results are also flat with a tendency to increase for low masses. 

The corrections for the K ∗ + meson is dominated by the kaon − to − K ∗ mass ratio ( ≈ 0 . 55) which is very large thus, although the width to mass ratio of the K ∗ is only ≈ 0 . 056, the correction to the multipoles are important. Compared with the predictions listed in Table II, they can be about the same or one order of magnitude smaller. 

As a byproduct, the mean square radius can be computed following [18] as, < R 2 > = ( κ + λ ) /M 2 $_{V}$. For the ρ we obtain a deviation of − 0 . 0012 fm respect to the normal value 

