Soft-gluon production due to a gluon loop in a constant chromoelectric background field

We obtain an exact result for the soft-gluon production and its ${p}_{T}$ distribution due to a gluon loop in a constant chromoelectric background field ${E}^{a}$ with arbitrary color. Unlike Schwinger's result for ${e}^{+}{e}^{\ensuremath{-}}$ pair production in QED which depends only on one gauge invariant quantity, the electric field $E$, we find that the ${p}_{T}$ distribution of the gluons depend on two gauge invariant quantities, ${E}^{a}{E}^{a}$ and $[{d}_{abc}{E}^{a}{E}^{b}{E}^{c}{]}^{2}$.

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