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Vacuum polarization of the quantized massive scalar field in the global monopole spacetime I: the field fluctuation

We study the vacuum polarization of a massive scalar field $ϕ$ with arbitrary coupling to gravity in pointlike global monopole spacetime. Using Schwinger-DeWitt proper time formalism, we calculate the vacuum expectation value $<ϕ^{2}>$, when the Compton length of the quantum field is much less than the characteristic radius of the curvature of the background geometry, and we can ignore nonlocal contributions. Explicit analytic expressions are obtained for a general curvature coupling parameter, and specified to the more physical cases of minimal and conformal coupling. Comparing the leading term of $<ϕ^{2}>$, proportional to the coincident limit of the Hadamard-DeWitt coeficcient $a_{2}$, with higher order terms, that include the coincident limits of coefficients up to $a_{5}$, we conclude that the next to next to next to leading approximation need to be used to give a more precise description of vacuum polarization effects in this structures. We also find the trace of the renormalized stress energy tensor for the quantized field in the leading approximation, using the existing relationship between this magnitude, the trace anomaly and the field fluctuation.