How to align and resize uniformly objects in latex table

Question

I am trying to make a table which includes graphics as well.

I am doing

\documentclass[a4paper,12pt]{book}
\usepackage{array}
\begin{document}

    \begin{table}[h!]
         \begin{center}
        \begin{tabular}{ | c | p{4cm} | p{3cm}|  }

         \hline
          Process & Diagrams  &     \sum  $\left| \mathcal{M} \right|^2 / \left ( 4 \pi \alpha_s\right)^2$      \\ \hline\hline
          $qq^\prime \rightarrow qq^\prime$

          &
          \centering
          \vspace{1mm}

       \includegraphics[\width=0.5\textwidth, height=20mm]{./chapter3/figures_susy/qq'_qq'.png}

            & 
        \vspace{1mm}
      $\frac{4}{9} \frac{s^2+u^2}{t^2}$    
          \\ \hline
         \end{tabular}
    \label{tab:scattering_diagrams}
          \end{center}
          \end{table}
\end{document}

but all I get looks like this

What I want is something like this

Answer 1

You had various problems with your code, including a figure that was specified wider than the cell to receive it, a $ placed after \sum rather than before it. The use of the centering environemnt in the midst of the tabular was not helpful. I replaced your image with a black-box placeholder, but this might get you started towards a resolution.

\documentclass{article}
\usepackage{scalerel}
\begin{document}

\begin{table}[h!]
 \begin{center}
\begin{tabular}{ | c | p{6cm} | p{3cm}|  }

 \hline
  Process & Diagrams & $\sum \left| \mathcal{M} \right|^2 / \left (
4 \pi \alpha_s\right)^2$      \\ \hline\hline
  $qq^\prime \rightarrow qq^\prime$

  &
\scalerel*{
\rule{0.5\textwidth}{20mm}%  I REPLACED YOUR INCLUDEGRAPHIC
}{\vphantom{\rule[-10mm]{1ex}{20mm}}}

&
$\frac{4}{9} \frac{s^2+u^2}{t^2}$    
  \\ \hline
 \end{tabular}
\label{tab:scattering_diagrams}
  \end{center}
  \end{table}
\end{document}

Answer 2

The adjustbox package is your friend:

\documentclass[a4paper,12pt]{book}
\usepackage[export]{adjustbox}
\usepackage{graphicx}
\usepackage{array}
\begin{document}

\begin{tabular}{ | c | c | m{3cm}|  }
\hline
Process & Diagrams  & $  \sum  |\mathcal{M}|^2 /( 4 \pi \alpha_s)^2$ \\
\hline
$qq' \rightarrow qq'$
  & \includegraphics[
      width=2cm,   % width
      height=2cm,  % height
      valign=c,    % vertical alignment
      ]{./chapter3/figures_susy/qq'_qq'.png}
  & $\frac{4}{9} \frac{s^2+u^2}{t^2}$ \\
\hline
\end{tabular}
\end{document}

I don't have your picture, so the big black blob.