quantum field theory - Dimension analysis in Derrick theorem

The following image is taken from p. 85 in the textbook Topological Solitons by N. Manton and P.M. Sutcliffe:

What I don't understand from the above statement:

• why $e(\mu)$ has minimum for $d=2,3$, whereas when $d=4$, $e(\mu)$ is scale independent and stationary points and vacuum solutions are possible?


• How $e(\mu)$ is a continuous function bounded by zero?

Answer

Generalized versions of Derrick's No-Go Theorem compare spatially scaled, non-trivial, time-independent, finite-energy, classical, field-configurations to exclude the existence of static solitons. (Since we are only considering time-independent field-configurations, there is no kinetic energy $T=0$. Therefore the stationary action principle $\delta S=0$ amounts to minimize the (potential) energy $T+V$. Here $S:=\int \! dt ~L$, and $L:=T-V$. See also this Phys.SE post.)

Let $d$ be the number of spatial dimensions. The (potential) energy of the $\mu$-scaled configuration is

$$\tag{4.22} e(\mu)~=~\sum_{n\in\{0,2,4\}} \mu^{n-d} E_n\geq 0,$$

where we assume that the scale parameter

$$\tag{A}\mu~\in~]0,\infty[$$

is strictly positive, and that the energies

$$\tag{B} E_n~\geq ~0, \qquad n\in\{0,2,4\}$$

are non-negative. From this it already follows that the ($\mu$-scaled potential) energy $e:]0,\infty[\to [0,\infty[$ is a non-negative and continuous function, and in particular that it is bounded from below, cf. some of OP's subquestions (v1).

1. 
   

   Case $E_0,E_4>0$ and $d\leq 3$: Then $$\tag{C} e(0)~:=~\lim_{\mu\to 0^{+}}e(\mu)~=~\infty~=~\lim_{\mu\to \infty}e(\mu)~=:~e(\infty),$$ so that there must exist an interior$^1$ (relative) minimum, and therefore Derrick's No-Go conclusion does not apply.

   
   


2. 
   

   Case $d\geq 4$: The function $e$ is monotonically weakly decreasing. [The word weakly means here that it could be (locally) constant.] The only possibility to have an interior minimum (and therefore evade Derrick's No-Go conclusion) is if $E_0=0=E_2$ and if moreover either (i) $d=4$ or (ii) $E_4=0$. The former case corresponds to pure 4+1 gauge theory, which indeed has non-trivial static soliton solutions with $E_4>0$. The latter case corresponds to vacuum solutions $e\equiv 0$. The function $e$ is in both cases a constant function, i.e. independent of the scale $\mu$.

   
   


3. 
   

   Case $E_0=0=E_2$: The function $e$ is monotonic. The only possibility to have an interior minimum (and therefore evade Derrick's No-Go conclusion) is, if (i) $d=4$, or if (ii) $E_4=0$, i.e. we are back in the previous case (2).

   
   
   

References:

1. 
   

   N. Manton and P.M. Sutcliffe, Topological Solitons, 2004, Section 4.2.

   
   


2. 
   

   S. Coleman, Aspects of symmetry, 1985. Note that Sidney calls solitons for lumps.

   
   



3. 
   

   R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory, 1987.

   
   

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$^1$ An interior minimum point $\mu$ means that $\mu$ is different from the boundary $0$ and $\infty$.