Underbrace/ Overbrace

Question

How do I modify \underbrace such that there is more vertical space between the formula and the beginning of the braces?

\documentclass[a4paper,13pt]{scrartcl}
\usepackage[utf8]{inputenc}

\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amstext}

\begin{document}

\begin{align*}
\liminf _{M \ni m \rightarrow \infty} \rho_{m} \geq 1
    &-\frac{2}{\mu \varepsilon} C_{H} \overbrace{\limsup _{M \ni m \rightarrow \infty} \Delta_{m}}^{=0} \\
    &-\frac{4}{\mu \delta} \underbrace{\limsup _{M \ni m \rightarrow \infty} \sup _{d \in B_{\Delta m}(0)} \left[ \frac{f(x^{m}+d)-f(x^{m})-\phi(x^{m}, \Delta_{m} ; d)}{\Delta_{m}} \right]}_{\leq 0} \geq 1 \, .
\end{align*}

\end{document}

Here is a picture of the result

Answer 1

In case you want to retain manual control over the vertical distance between the first and second arguments of \overbrace, I suggest you change \Delta to \Delta^{\mathstrut}, where \mathstrut is defined in the LaTeX kernel (and in PlainTeX) as \vphantom{(}.

I would also like to suggest that you insert \textstyle directives at the start of the second arguments of \overbrace and \underbrace in order to give the associated material just a bit more visual prominence. And, to keep the arguments of \limsup and \sup from getting too close, I'd insert \, ("thinspace") between \limsup_{M\ni m \rightarrow \infty} and `\sup_{d\in B_{\Delta m}(0)}

\documentclass[a4paper,13pt]{scrartcl}
%%\usepackage[utf8]{inputenc} % that's the default nowawadays

\usepackage{mathtools}
%%\usepackage{amsmath}  % <-- is loaded automatically by 'mathtools'
\usepackage{amsthm}
%%\usepackage{amsfonts} % <-- is loaded automatically by 'amssymb'
\usepackage{amssymb}
%%\usepackage{amstext}  % <-- is loaded automatically by 'amsmath'

\begin{document}

\begin{align*}
\liminf _{M \ni m \rightarrow \infty} \rho_{m} \geq 1
    &-\frac{2}{\mu \varepsilon} C_{H} 
     \overbrace{\limsup_{\mathclap{M \ni m \rightarrow\infty}} 
     \Delta_{m}^{\mathstrut}}^{\textstyle =0} \\
    &-\frac{4}{\mu \delta} 
     \underbrace{
     \limsup_{M\ni m \rightarrow \infty} \, 
     \sup_{d\in B_{\Delta m}(0)} 
     \biggl[ 
     \frac{f(x^{m}+d)-f(x^{m})-\phi(x^{m}, \Delta_{m} ; d)}{\Delta_{m}} 
     \biggr]}_{\textstyle \leq 0}
     \geq 1 \, .
\end{align*}

\end{document}