One-loop electron-muon QED threshold matching and running

Use natural units $c=\mathrm{\hslash }=1$ and Minkowski signature $\eta =\mathrm{diag}(-,+,+,+)$. In four spacetime dimensions assign mass dimensions $[x^{\mu }]=-1$, $[{\mathrm{\partial }}_{\mu }]=1$, $[A_{\mu }]=1$, $[F_{\mu \nu }]=2$, $[{\psi }_f]=3\mathrm{/}2$, $[e_{\mathrm{D}\mathrm{S}}]=[{\alpha }_{\mathrm{D}\mathrm{S}}]=0$, and $[m_e]=[M_{\mu }]=[\mu ]=[Q]=1$. For $x\in {\mathbb{R}}^{1,3}$, regard $A_{\mu }(x)$, ${\psi }_e(x)$, and ${\psi }_{\mu }(x)$ as formal perturbative fields, with no classical-solution regularity or nonperturbative operator domain assumed. The potential $A=A_{\mu }d{x}^{\mu }$ is a real $U(1)$ one-form and $F=dA$ has components $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$. The fields ${\psi }_e,{\psi }_{\mu }$ are four-component complex Dirac spinors; use Clifford matrices satisfying $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=-2{\eta }^{\mu \nu }$ and the Dirac adjoint $\bar{\psi }={\psi }^{†}{\gamma }^0$. At each fixed renormalization scale $\mu$ in an active interval, let $e_{\mathrm{D}\mathrm{S}}(\mu )>0$ and use $D_{\nu }={\mathrm{\partial }}_{\nu }+i{e}_{\mathrm{D}\mathrm{S}}(\mu )A_{\nu }$ for fields of charge $-e_{\mathrm{D}\mathrm{S}}(\mu )$, with gauge kinetic term $-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$. For a real dimensionless gauge parameter $\chi (x)$, the convention is $A\mapsto A+d\chi$ and ${\psi }_f\mapsto e^{-i{e}_{\mathrm{D}\mathrm{S}}(\mu )\chi }{\psi }_f$. Define the dimensionless running coupling by ${\alpha }_{\mathrm{D}\mathrm{S}}(\mu )=e_{\mathrm{D}\mathrm{S}}(\mu {)}^2\mathrm{/}(4\pi )$. The decoupling-subtraction (DS) convention uses minimal subtraction within each interval of active charged fields, removes a charged field from the low-energy theory below its threshold, and matches the coupling continuously at that threshold. Set ${\alpha }_{\mathrm{D}\mathrm{S}}(m_e)={\alpha }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}>0$ as the low-energy boundary condition. The source-reported one-loop DS flow has two active Dirac fields for $\mu >M_{\mu }$, one for $m_e<\mu <M_{\mu }$, and none for $\mu <m_e$. For $q>0$, define $R_e(q)=\frac{1}{{\alpha }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}-\frac{1}{3\pi }\log \text{\hspace{0.17em}⁣}\left(\frac{q^2}{m_e^2}\right)$ and $R_{\mu }(q)=\frac{1}{{\alpha }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}-\frac{1}{3\pi }\log \text{\hspace{0.17em}⁣}\left(\frac{M_{\mu }^2}{m_e^2}\right)-\frac{2}{3\pi }\log \text{\hspace{0.17em}⁣}\left(\frac{q^2}{M_{\mu }^2}\right)$. For positive evaluation scale $Q$ of mass dimension one, require $R_e(q)>0$ for every $q\in [m_e,Q]$ when $m_e<Q<M_{\mu }$; for $Q>M_{\mu }$, require $R_e(q)>0$ for every $q\in [m_e,M_{\mu }]$ and $R_{\mu }(q)>0$ for every $q\in [M_{\mu },Q]$. The corresponding one-loop solutions are: $\frac{1}{{\alpha }_{\mathrm{D}\mathrm{S}}(Q)}=\frac{1}{{\alpha }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}-\frac{1}{3\pi }\log \text{\hspace{0.17em}⁣}\left(\frac{Q^2}{m_e^2}\right)(m_e<Q<M_{\mu }),$ and $\frac{1}{{\alpha }_{\mathrm{D}\mathrm{S}}(Q)}=\frac{1}{{\alpha }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}-\frac{1}{3\pi }\log \text{\hspace{0.17em}⁣}\left(\frac{M_{\mu }^2}{m_e^2}\right)-\frac{2}{3\pi }\log \text{\hspace{0.17em}⁣}\left(\frac{Q^2}{M_{\mu }^2}\right)(Q>M_{\mu })\mathrm{.}$ The two expressions agree continuously at $Q=M_{\mu }$. These are solutions of the supplied one-loop renormalization-group equations; they are not measured values, exact all-orders predictions, or numerical error estimates.

Finite renormalization schemes and Wilson coefficient vectors

Finite-order EFT threshold matching map

Burgess, full EFT review

Manohar, full EFT lecture notes