2
$\begingroup$

I'm sure many of us are familiar with the following plot showing the running of the inverse of the fine-structure constants of the SM.

running of SM couplings (I got the picture from google)

At one-loop, the expressions for the fine-structure constants of the hypercharge, weak, and color gauge groups are,

$$\frac{1}{\alpha_1 (\mu)} \approx 59 - \frac{41}{20 \pi}\log[\frac{\mu}{M_Z}]$$ $$\frac{1}{\alpha_2 (\mu)} \approx 30 + \frac{19}{12 \pi}\log[\frac{\mu}{M_Z}],$$ $$\frac{1}{\alpha_3 (\mu)} \approx 8.5 + \frac{7}{2 \pi}\log[\frac{\mu}{M_Z}],$$

where the numerical values of $\alpha$'s in the first terms in RHS are at the mass of the Z boson $M_Z$. Now, if I plot these (from $\mu=100$ GeV to $\mu=2\times 10^16$ GeV) using Mathematica, I get this:

enter image description here

The slopes of the $log$'s in the previous equations do not produce the shape appearing in the first plot.

What am I missing here? How is the first plot, which we usually see in the literature, produced?

Update: What I was missing is skills with Mathematica :p, Now I get the correct plot: enter image description here

$\endgroup$
7
  • 2
    $\begingroup$ Your plot doesn't pass a sniff test: The equation for the running of $\alpha_3$ is a line with a slope of roughly 1 and a $y$-intercept of 8.5. It should rise to about 25 by $\mu = 10^{16} M_Z$. Perhaps you're taking log twice? $\endgroup$
    – user1504
    Aug 6, 2013 at 18:08
  • $\begingroup$ Yes you are right, I took it twice like any stupid would do.. I suspected I did something wrong in my plotting. Now I get the desired plot when I use the LogLogPlot function. Previously I used Plot, with the interval being from Log[MZ] to Log[10^16] so the log was taken twice. $\endgroup$
    – stupidity
    Aug 6, 2013 at 18:31
  • 1
    $\begingroup$ Or even better is the LogLinearPlot function which plots log scale on the x-axis only. $\endgroup$
    – stupidity
    Aug 6, 2013 at 18:47
  • 2
    $\begingroup$ You have to take care of the difference between $log_{10}$ units and $ln$ units. Your formulae are expressed in $ln$ units, but the graph uses $log_{10}$ units. Why $\alpha_3$ is not a straight line ? See test plot $\endgroup$
    – Trimok
    Aug 6, 2013 at 18:54
  • $\begingroup$ This question appears to be off-topic because it is about using a software tool. $\endgroup$ Aug 6, 2013 at 19:22

0