The value of Wilson loop defines static potential, compare the plots with fixed length , see how the value of Wilson loop is varying with smearing. The Wilson loop must be same for fixed length(as Wilson Loop is observable). You will find that the smearing steps decrease lattice spacing, and calculate the new lattice spacing after each smearing step.

The value of \(\beta\) (inverse gauge coupling) depends upon the action used for the calculations.

We know that the inverse temperature \(\beta\) is related to lattice spacing and temperature in the following way:

\[ \beta = a_{\tau}.N_{\tau} = \frac{1}{T} \\ \]

we have one more beta (gauge coupling (\(\beta=2N_{c}/g^{2}\))), we identify that with absence of any units or dimensions.

Our aim is to calculate the \(\beta\) (gauge coupling) of HISQ from corresponding \(\beta\) values of P4 gauge action.
We will do that in following manner:

1. Find lattice spacing \(a_{\tau}\) for corresponding \(\beta\)(gauge coupling). Which can be found by using the function a_div_r1_2014(beta_input) from ms.py of AnalysisToolBox.

   	a/r1=a_div_r1_2014(beta_input)
   	# we know that r1=0.3157 fm, so we can find lattice spacing a
   	a=0.3157 fm * a/r1
   	T=(1/(a*n_t)*197.7) #MeV, Here the factor 197.7 can be find at [[conversion of units]].
   

2. Use above relation to find the corresponding temperature(T) = \(\frac{1}{a_{\tau}.N_{\tau}}\)
3. Once we found the temperature, we can find lattice spacing for any \(N_{\tau}\).
4. from the lattice spacing we have to guess the inverse gauge coupling for HISQ and find the closest value of a. Which is found at scales_hisq.py. Or find the inverse function of a=f(\(\beta\)), i.e. \(\beta=f^{-1}(a)\).
5. Note that lattice spacing depends on action and the inverse gauge coupling. Once we change the action (P4 or hisq), the whole analysis has to be done again.

\(\beta_{P4}\)	\(a_{p4}\) in fm	T(MeV)	\(\beta_{HISQ}\)
3.53	0.12933716702207762	191.07036723466828	5.88
3.57	0.11663534132654599	211.87831851764366	5.975
3.585	0.11249240092955154	219.68150555766184	6.01
3.76	0.07841014722354198	315.16966712926006	6.34
3.82	0.07033094944455437	351.37446878180106	6.435
3.92	0.0592236517369858	417.27416792448065	6.595
4.00	0.05193808031732161	475.8069579972184	6.71
4.08	0.04573994551368463	540.2826724532594	6.83