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My query is with Weinberg Vol3 equation just above 26.7.22 Weinberg follows Majorana Superfield formalism. Where, covariant derivative is defined as,

$$D_{R\alpha}=-\epsilon_{\alpha \beta}\frac{\partial}{\partial \theta_{R\beta}}-(\gamma^{\mu}\theta_L)_\alpha\frac{\partial}{\partial x^{\mu}},\tag{26.3.25}$$

and

$$D_{L\alpha}=+\epsilon_{\alpha \beta}\frac{\partial}{\partial \theta_{L\beta}}-(\gamma^{\mu}\theta_R)_\alpha\frac{\partial}{\partial x^{\mu}}.\tag{26.3.26}$$

Now my question is the following action of derivative,

$$\gamma^{\mu}D_L((\bar{D}_R\Phi^*)\gamma_{\mu}(D_L\Phi))=4(\gamma^{\nu}\partial_{\nu}\Phi^*)D_L\Phi+2D_R\Phi^*(\bar{D}_LD_L)\Phi.\tag{26.7.22}$$

I was able to get the first term , but I am unable to get the coefficient for the second term. The trouble is that whenever $\gamma$ passes through one of the covariant derivatives it picks up non-trivial factors. I dont see how the coefficient of 2 is coming in the second term. Any help would be greatly appreciated.

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