Massless renormalization-group characteristics with a source ($d\lambda /d\log \mu =\hat{\beta },d\log h/d\log \mu ={\hat{\gamma }}_h$)

For a normalized connected functional obeying $(\mu {\partial }_{\mu }+\hat{\beta }{\partial }_{\lambda }+{\hat{\gamma }}_{h}h{\partial }_h)F=0$, solve the displayed ordinary differential equations and keep $F$ constant along their curves. Where $\hat{\beta }\ne 0$, $\log (\mu /{\mu }_{\ast })={\int }_{{\lambda }_{\ast }}^{\lambda (\mu )}dg/\hat{\beta }(g)$ and $h(\mu )=h_{\ast }\exp {\int }_{{\lambda }_{\ast }}^{\lambda (\mu )}{\hat{\gamma }}_h(g)dg/\hat{\beta }(g)$. For the dimensionless density $f(\lambda ,y)$ of a uniform source, with $y=h/{\mu }^3$, the equation is $[\hat{\beta }{\partial }_{\lambda }+({\hat{\gamma }}_h-3)y{\partial }_y+d]f=0$. Its characteristic invariant is $y\exp [-\int ({\hat{\gamma }}_h-3)dg/\hat{\beta }]$, and its value acquires the factor $\exp [-\int ddg/\hat{\beta }]$. A finite-volume total functional also retains the dimensionless volume argument.