component ratios for the $h=0$ flows where we can get the exact formulas for them. The (unnormalised) vacuum of the perturbed theory (at finite value of $m$) is $$|v_{0}\rangle=\hat D^{\rm NS}|0\rangle_{\rm NS}$$ that always lies in the NS-sector. From (\[IONS\]) we find \_\^[0]{}=0 , \_\^[0]{} = = = [sign]{}(m) g() where \[gnu\] g()= - . The function $g(\nu)$ asymptoticaly goes to 1 very slowly, as a power function. Here is a plot of $g(\nu)$

\[g\_fig\] ![The component ratio $\Gamma_{\epsilon}^{0}$ for zero magnetic field, $m>0$](gnu4.pdf "fig:")

This function monotonically increases and becomes larger than $0.9$ past $\nu=30$ and larger than $0.99$ past $\nu=313$. These values of the dimensionless mass are very large. This shows that it may be much harder to read off the component ratios from TCSA numerics than say the energy spectrum for which the convergence is typically exponential. The further we have to go into the infrared the greater the TCSA errors can be for a fixed truncation level. We will see that while the perturbative corrections grow the numerics demonstrates that the total truncation error remains bounded.

For $m\to +\infty$ the first excited vector remains in the spectrum. It is $|v_{1}\rangle=\hat D^{\rm R}|0\rangle_{\rm R} $ for which 