I have an exercise where I did not manage to understand the questions. Basically, I have this Lagrangian

\begin{equation} \mathcal{L}=\frac{1}{2}(\partial \pi)^2-\frac{1}{\Lambda^3}(\partial \pi)^2\square \pi +\alpha \pi \delta^3(\vec{x}) \end{equation}

The questions are:

  1. Discuss at what distance from the source the non-linearities of the $\pi$ field become important in the spherically symmetric solution.
  2. Compute the field $E(r)$ generated by the point like source in the spherically symmetric soultion, where $\vec{E}=\vec{\nabla} \pi=\hat{r}E(r)$

What does question $1$ mean? I need to solve the equation of motion in order to find the spherically symmetric solution and then maybe take some limit for $r\rightarrow 0$, how can I solve the equation of motion?

Question $2$ maybe is not difficult if I manage to understand question $1$, so I would like to ask something else: when I have a Lagrangian of a particle coupled to a gauge field, how can I "define" an electromagnetism? I mean, how I find the electric and magnetic field generated by the particle inside an external field, in this case represented by a point-like source?

In both cases I will be happy if just someone can tell me some books or notes where I can find the answers to both my questions.

  1. For this question you don't need to obtain the e.o.m. It suffices to look at the Lagrangian and check when the second term is of the same order as the third term, i.e., you want to find the scale at which

$$ \frac{ (\partial \pi)^2 \Box \pi }{\Lambda^3} \sim \alpha\, \pi\, \delta(\vec{x})$$

  1. Here you are being asked to solve the e.o.m, in the case where $\pi =\pi(r)$. Indeed, using spherical coordinates for the metric, once you plug $ r E(r)= \partial_r \pi$ into the e.o.m and solve it you will end up obtaining a fairly simple expression for $E$.

In case you continue stuck try checking this paper: https://arxiv.org/abs/0811.2197


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