How to make appear right expectation symbol on second line of a splitted equation into align environment

Question

Here below the rendering of Latex code snippet : I can't get to make display the right expectation symbol ( \right> ) on the second line of a splitted equations (on both eq(3) and eq(4))

\noindent To calculate the variance we assume ${a_{\ell m, p h}^2}$ is constant in order to not have a random denominator:
\begin{align} 
\operatorname{Var}\left(\hat{o}_{\ell}\right) &=\dfrac{1}{(2 \ell+1)^2C_{\ell, p h}^2 f_{\text{sky}}} \left<\left(\sum_{m=-\ell}^{\ell} (|b_{\ell m}|^2 - B_\ell) +  2Re(a_{\ell m}^*b_{\ell m}) \right)^2 \right> \nonumber \\
&=\alpha \sum_{m,m' =-\ell}^{\ell} 
\left<\left(b_{\ell m}b_{\ell m}^* - B_\ell + a_{\ell m}^*b_{\ell m}+a_{\ell m}b_{\ell m}^* \right)\nonumber \\
&\quad\quad\left(b_{\ell m'}b_{\ell m'}^* - B_\ell +  a_{\ell m'}^*b_{\ell m'}+a_{\ell m'}b_{\ell m'}^*\right)\right>
\end{align}
\text{with $=\alpha= \dfrac{1}{(2 \ell+1)^2C_{\ell, p h}^2 f_{\text{sky}}}$}
The terms like $B_\ell \left< a\dots b \dots\right>$ or $\left<bb\dots a\dots b \dots\right>$ where $a_{\ell m}$  appears only once are 0 since a is centered and independent of the rest. After simplification and rearranging of the terms we obtain : 
\begin{align}
\operatorname{Var}\left(\hat{o}_{\ell}\right)&=\alpha \sum_{m,m' =-\ell}^{\ell} 
\left<b_{\ell m}b_{\ell m}^*b_{\ell m'}b_{\ell m'}^*\right> - B_\ell^2 +\left<a_{\ell m}^*b_{\ell m}a_{\ell m'}^*b_{\ell m'}\nonumber \\
&+a_{\ell m}b_{\ell m}^*a_{\ell m'}^*b_{\ell m'}+a_{\ell m}^*b_{\ell m}a_{\ell m'}b_{\ell m'}^*+a_{\ell m}b_{\ell m}^*a_{\ell m'}b_{\ell m'}^*\right> \nonumber \\
\end{align}

Here the capture of rendering :

How to make appear these missing \right> expectation symbols ?

Answer 1

Your code is a good reminder of the downsides of overusing \left and \right. I'd use explicit sizing instructions, especially \bigl, \bigr, \biggl and \biggr. Indeed, the code shown below doesn't contain any instances of \left and \right.

Separately, I'd use judiciously-placed ^{} particles ("empty exponents") to make sure all the subscript terms are placed at a uniform depth. Oh, and please don't use \dfrac for inline-math expressions.

\documentclass{article} % or some other suitable document class
\usepackage[T1]{fontenc}
\usepackage{mathtools} % load 'amsmath' automatically
\DeclareMathOperator{\Var}{Var} % variance operator
\usepackage{microtype}
\begin{document}
\noindent 
To calculate the variance we assume $a^2_{\ell m, p h}$ 
is constant in order to not have a random denominator:
\begin{align} 
\Var(\hat{o}{\ell}) 
&=\frac{1}{(2 \ell+1)^2C{\ell, p h}^2 f^{}{\mathrm{sky}}} 
  \biggl<\biggl(,\sum{m=-\ell}^{\ell} 
    (|b^{}{\ell m}|^2 - B^{}{\ell}) 
    +2\Re(a^{\ell m}b^{}{\ell m}) \biggr)^{!2}, 
  \biggr> \nonumber \
&=\alpha \sum_{\mathclap{m,m' =-\ell}}^{\ell} 
\bigl\langle (b^{}_{\ell m}b^{\ell m} - B^{}{\ell} 
    + a^{\ell m}  b^{}{\ell m}
    + a^{}_{\ell m} b^{\ell m} ) \nonumber \
&\qquad\qquad\times(b^{}{\ell m'}b^{\ell m'} - B^{}{\ell} 
  +a^{\ell m'}b^{}{\ell m'} 
  +a^{}{\ell m'}b^*{\ell m'}) \bigr\rangle
\end{align}
with $\alpha= 1/\bigl[(2 \ell+1)^2C_{\ell, p h}^2 f^{}{\mathrm{sky}}\bigr]$.
The terms like $B^{}{\ell} \langle a\dots b \dots\rangle$ 
or $\langle bb\dots a\dots b \dots\rangle$ 
in which $a^{}{\ell m}$ appears only once are $0$ 
since $a$ is centered and independent of the rest. After 
simplification and rearranging of the terms we obtain: 
\begin{align}
\Var(\hat{o}{\ell})
&=\alpha \sum_{\mathclap{m,m' =-\ell}}^{\ell} 
\langle b^{}{\ell m}b^*{\ell m} b^{}{\ell m'}b^*{\ell m'}\rangle

B^2_{\ell} +\bigl\langle 
    a^{\ell m} b^{}{\ell m}a^{\ell m'} b^{}{\ell m'} \nonumber \

&\qquad+a^{}{\ell m}b^*{\ell m} a^{\ell m'} b^{}{\ell m'} 
       +a^{\ell m} b^{}{\ell m}a^{}{\ell m'}b^*{\ell m'} 
       +a^{}{\ell m}b^*{\ell m} a^{}{\ell m'}b^*{\ell m'}\bigr\rangle

\end{align}
\end{document}