- #1

spaghetti3451

- 1,344

- 33

##\displaystyle{\mu \frac{\partial g_{3}}{\partial\mu}(\mu) = - \frac{23}{3} \frac{g_{3}^{3}(\mu)}{16\pi^{2}},}##

with

##\alpha_{3}(\mu = M_{Z}) = 0.118.##

We can use separation of variables to solve the beta function equation:

##\displaystyle{\int \frac{dg_{3}}{g_{3}^{3}} = - \frac{23}{48\pi^{2}} \int \frac{d\mu}{\mu}}##

##\displaystyle{\frac{1}{g_{3}^{2}} = \frac{23}{24\pi^{2}}\ln\left(\frac{\mu}{\Lambda_{\text{QCD}}}\right)}##

##\displaystyle{\frac{1}{\alpha_{3}} = \frac{23}{6\pi}\ln\left(\frac{\mu}{\Lambda_{\text{QCD}}}\right).}##

Using the physical condition, we then find that

##\displaystyle{\Lambda_{\text{QCD}} = \left(91\ \text{GeV}\right) e^{-6\pi/2.714}.}##

--------------------------------------------------------------------------------------------------------------------------------------

What is the physical significance of ##\Lambda_{\text{QCD}}## as obtained by solving for the the strong coupling ##g_3##?