= Massless renormalization-group characteristics with a source
{title2=$d\lambda/d\log\mu=\widehat\beta,\quad d\log h/d\log\mu=\widehat\gamma_h$}
For a normalized connected functional obeying $(\mu\partial_\mu+\widehat\beta\partial_\lambda+\widehat\gamma_hh\partial_h)F=0$, solve the displayed ordinary differential equations and keep $F$ constant along their curves. Where $\widehat\beta\ne0$, $\log(\mu/\mu_*)=\int_{\lambda_*}^{\lambda(\mu)}dg/\widehat\beta(g)$ and $h(\mu)=h_*\exp\int_{\lambda_*}^{\lambda(\mu)}\widehat\gamma_h(g)dg/\widehat\beta(g)$. For the dimensionless density $f(\lambda,y)$ of a uniform source, with $y=h/\mu^3$, the equation is $[\widehat\beta\partial_\lambda+(\widehat\gamma_h-3)y\partial_y+d]f=0$. Its characteristic invariant is $y\exp[-\int(\widehat\gamma_h-3)dg/\widehat\beta]$, and its value acquires the factor $\exp[-\int d\,dg/\widehat\beta]$. A finite-volume total functional also retains the dimensionless volume argument.
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