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I am reading this paper by E. Weinberg - Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups.

I am having a problem with a calculation. I don't have much experience with such calculations, and have never come across the gamma matrices mentioned here. So, it would be great if someone could help. M is an position independent quantity.

The author claims that the divergence of the following quantity $$J_i=\frac{1}{2}tr<x| \gamma_5 \gamma_i(\gamma \cdot D) \frac{1}{-(\gamma.D)^2+M^2} |x>$$ The divergence of $J_i$ is the expression given below $$tr <x| \gamma_5 \frac{M^2}{-(\gamma.D)^2+M^2}|x>$$

tr is the trace over group and spinor indices.

$\gamma$ are the gamma matrices forming the Clifford algebra. $D_4=\Phi$ where $\Phi^{ac}=f_{abc}\phi^b$, $\phi$ is the scalar higgs field. and $D$ is the usual covariant derivative in Yang-Mills theory given by $(D_{\mu}\phi)^a=\partial_{\mu}\phi^a + f_{abc}A_{\mu}^b\phi^c$

Attempt at a solution

$$\partial_iJ_i=\frac{1}{2}tr<x| \gamma_5 \gamma_i \partial_i(\gamma \cdot D) \frac{1}{-(\gamma.D)^2+M^2}|x>-\frac{1}{2}tr<x| \gamma_5 \gamma_i(\gamma \cdot D) \frac{-2(\gamma.D)\partial_i(\gamma.D)}{(-(\gamma.D)^2+M^2)^2}|x>$$ Now in the second expression I can combine the two $\gamma \cdot D$. After taking $\partial_i(\gamma \cdot D)$ common from both the terms, you can add and subtract $M^2$ from the second term. Divergence=$$\partial_iJ_i=\frac{1}{2}tr<x| \gamma_5 \gamma_i \partial_i(\gamma \cdot D) \frac{1}{-(\gamma.D)^2+M^2}|x>-\frac{1}{2}tr<x| \gamma_5 \gamma_i[(\gamma \cdot D)^2 + M^2 - M^2] \frac{-2\partial_i(\gamma.D)}{(-(\gamma.D)^2+M^2)^2}|x>$$ The first part of the second term would reduced the power of the denominator by one, and then combine with the first term, and $M^2$ in the numerator would get stranded with the inverse squared term.

Divergence = $$\frac{1}{2}tr \ \gamma_5 \gamma_i \partial_i(\gamma \cdot D) <x|- \frac{1}{-(\gamma.D)^2+M^2} + \frac{2M^2}{(-(\gamma.D)^2+M^2)^2}|x>$$

I don't know how to proceed after this. The problem is the denominator is squared in the second term, where is should have power 1. Help will be very much appreciated.

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