Casimir scaling in glueballs in SU($N$) and Sp($2N$) gauge theories: hints from constituent approaches

We show that the lattice glueball masses $M_G$ versus $N$ in SU($N$) and Sp($2N$) Yang-Mills theories scale as $\frac{M_G}{\sqrtσ}\sim \sqrt{\frac{C_2(adj)}{C_2(f)}}$, with $σ$ the fundamental string tension and $C_2(adj)$ and $C_2(f)$ the quadratic Casimir of the gauge algebra in the adjoint and fundamental representations. This scaling behaviour is followed by the great majority of available lattice glueball states, and may set constraints on $SU(3)$ models by imposing a specific behaviour at $N\neq 3$. The observed scaling is compatible with two assumptions: (1) The glueball masses are proportional to the square root of the adjoint string tension, $M_G\sim \sqrtσ_{adj}$; (2) The string tension follows the Casimir scaling, i.e. $σ_{adj}=\frac{C_2(adj)}{C_2(f)}σ$. In a constituent gluon picture, our results suggest a low-lying glueball spectrum made of two transverse constituent gluons bound by an adjoint string, completed by three transverse constituent gluons bound by a Y-junction of adjoint strings rather than a $Δ-$shaped junction of fundamental strings.