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We investigate the effect of a small, gauge-invariant mass of the gluon on the anomalous chromomagnetic moment of quarks (ACM) by perturbative calculations at one loop level. The mass of the gluon is taken to have been generated via a topological mass generation mechanism, in which the gluon acquires a mass through its interaction with an antisymmetric tensor field $B_{mu u}$. For a small gluon mass $(<10$ MeV), we calculate the ACM at momentum transfer $q^2=-M_Z^2$. We compare those with the ACM calculated for the gluon mass arising from a Proca mass term. We find that the ACM of up, down, strange and charm quarks vary significantly with the gluon mass, while the ACM of top and bottom quarks show negligible gluon mass dependence. The mechanism of gluon mass generation is most important for the strange quarks ACM, but not so much for the other quarks. We also show the results at $q^2=-m_t^2$. We find that the dependence on gluon mass at $q^2=-m_t^2$ is much less than at $q^2=-M_Z^2$ for all quarks.
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