# Calculating the quark propagator while dealing with the color, Dirac & flavor matrices all together

$$S^+$$ is a diagonal matrix in color space, and is given as : $$S^+ = S_{rg} ^+ \hat{P_{rg}} + S_b ^+ \hat{P_b}, where, \hat{P_b} = 1_c - \hat{P_{rg}}$$ is the projector on blue quarks, $$\hat{P_{rg}}$$ projects onto the "red" and "green" color components, and, $$S_b ^+ = (\require{cancel}{p_+})^{-1} = \frac{\cancel{p_+}}{p_+ ^2}$$. Now, $$T^+$$ is a component of the Nambu-Gorkov propagator given as $$T^+ = \frac{\triangle ^* \gamma _5 \tau _2 \lambda _2 (\cancel{p_-}\cancel{p_+})}{D(p)} - \frac{\triangle ^* \gamma _5 \tau _2 \lambda _2 |\triangle|^2}{D(p)} \rightarrow T^+ = \bigg[(-) \frac{(\cancel{p_-}\cancel{p_+})\triangle ^* \gamma _5 \tau _2 \lambda _2}{D(p)} -(-) \frac{|\triangle|^2 \triangle ^* \gamma _5 \tau _2 \lambda _2}{D(p)}\bigg]; \\$$ $$\overrightarrow{\tau} and \lambda$$'s are the Pauli and Gell-Mann matrices respectively, $$\triangle ^* and \triangle$$ are scalars, $$\cancel{p_\pm}\equiv\cancel{p} \pm \mu \gamma ^0$$. Question: Are the extra negative signs (shown in round brackets) in the last expression for $$T^+$$ due to the anticommuting nature of $$(\triangle ^* \gamma _5 \tau _2 \lambda _2)$$ with $$(\cancel{p_-}\cancel{p_+})$$, or, beacuse $$(\cancel{p_-}\cancel{p_+})$$ 'lives' in color space while "$$\triangle ^* \gamma _5 \tau _2 \lambda _2$$" do not (but, as per the definitions of $$\cancel{p_\pm}$$ which include the Dirac matrix $$\gamma _0$$, shouldn't it 'be living' in Dirac space)? For further refernce, [please refer to Problem 4 in this paper].1