Callan-Symanzik equation

The Callan-Symanzik equation , also Gell-Mann-Low equation , 't Hooft-Weinberg equation or Georgi-Politzer equation , after Curtis Callan , Kurt Symanzik , Murray Gell-Mann , Francis Low , Gerardus' t Hooft , Steven Weinberg , Howard Georgi, and David Politzer , is an equation in quantum field theory . It describes how the renormalized Green functions of the theory behave depending on the energy scale. It is therefore a renormalization group equation.

Green's function is the vacuum expectation value of the time-ordered product of all fields (particles) occurring in the theory. Assuming there are two types of particles, the electron and the photon , then Green's function for a system of photons and electrons is: ${\displaystyle \psi }$ ${\displaystyle A}$ ${\displaystyle G_{k,l}}$ ${\displaystyle k}$ ${\displaystyle l}$

${\displaystyle G_{k,l}=\left\langle \Omega \left|T\left(A_{\mu _{1}}\dots A_{\mu _{k}}\psi _{1}\dots \psi _{l}\right)\right|\Omega \right\rangle }$

with the time order operator and the vacuum state . In general, the renormalized Green function is dependent on all momentum of the particles, the renormalized coupling constants and their renormalized masses as well as a renormalization parameter . The Callan-Symanzik equation is: ${\displaystyle T}$ ${\displaystyle |\Omega \rangle }$ ${\displaystyle p}$ ${\displaystyle e_{R}}$ ${\displaystyle m_{R}}$ ${\displaystyle \mu }$

${\displaystyle \left(\mu {\frac {\partial }{\partial \mu }}+{\frac {k}{2}}\gamma _{3}+{\frac {l}{2}}\gamma _{2}+\beta {\frac {\partial }{\partial e_{R}}}+\gamma _{m}m_{R}{\frac {\partial }{\partial m_{R}}}\right)G_{k,l}=0}$

In this equation the abbreviations were

• ${\displaystyle \gamma _{3}={\frac {\mu }{Z_{3}}}{\frac {\mathrm {d} Z_{3}}{\mathrm {d} \mu }}}$ with the renormalization factor for the photon ${\displaystyle Z_{3}}$
• ${\displaystyle \gamma _{2}={\frac {\mu }{Z_{2}}}{\frac {\mathrm {d} Z_{2}}{\mathrm {d} \mu }}}$ with the renormalization factor for the electron ${\displaystyle Z_{2}}$
• ${\displaystyle \gamma _{m}={\frac {\mu }{m_{R}}}{\frac {\partial m_{R}}{\partial \mu }}}$
• ${\displaystyle \beta =\mu {\frac {\partial e_{R}}{\partial \mu }}}$

used. The function is also called Symanzik's beta function and shows the course of the coupling constants with the observed scale . ${\displaystyle \beta }$ ${\displaystyle \mu }$

Individual evidence

1. ^ ^{A b c} Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 433-434 (English).