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2

First note that $$\frac{\partial}{\partial m} \frac{i}{\not p - m} = \frac{i}{(\not p-m)^2}$$ Now the exact answer for finite $q$ for that diagram is $$\mathcal{A} = \left( \frac{im}{v} \right) \frac{i}{\not p-m} \frac{i}{\not p + \not q - m} G(q)$$ where $G(q)$ is the propagator for whatever ...

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I am not sure whether the standard model has a Landau pole at $10^{34}$ GeV but there are two obstacles to providing a definite answer to the question: (1) perturbation theory is no longer valid when the coupling constants get large, and (2) $10^{34}$ GeV is well beyond the Planck scale, so that ignoring the effects of (quantum) gravity is not valid. If ...

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Every phase transition has an order parameter: something that vanishes above the transition temperature and is finite below. In superconductors, the order parameter is a complex quantity related to the superconducting gap: $\Delta = |\Delta| e^{i \phi}$. In BCS theory, there is a self-consistent equation for the gap: $\Delta_k = -\sum_q V_{kq} ... 1 We have to change the renormalization scale to a typical scale of the process we are considering because otherwise perturbation theory becomes rather useless, since process far away from the renormalization scale get large higher order corrections due to the "large log"$\ln(-p^2/\mu^2)$, where$p$is the process scale and$\mu\$ the renormalization scale. ...

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