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By differentiating the dressed quark propagator with respect to a variablebackground field, the linear response of the dressed quark propagator in thepresence of the background field can be obtained. From this general method,using the vector background field as an illustration, we derive a generalformula for the four-quark condensate $<{\tilde 0}|:{\bar q}(0)\gamma_\muq(0){\bar q}(0)\gamma_\mu q(0):|{\tilde 0}>$. This formula contains thecorresponding fully dressed vector vertex and it is shown that factorizationfor $<{\tilde 0}|:{\bar q}(0)\gamma_\mu q(0){\bar q}(0)\gamma_\mu q(0):|{\tilde 0}>$ holds only when the dressed vertex is taken to be the bare one.This property also holds for all other type of four-quark condensate.Comment: Revtex4, 11 pages, no figure

GENERAL FORMULA FOR THE FOUR-QUARK CONDENSATE AND VACUUM FACTORIZATION ASSUMPTION