## Solving and Plottings RGEs from [email protected]¶

In [1]:
#Firts thing first we have to load pylab in order to be able to plot the results
%pylab inline
import numpy as np

Populating the interactive namespace from numpy and matplotlib


### 1. Solving the RGEs¶

In [2]:
#We asssume that we have a BetaFunction.py file as well as a SolveRGEs.py file both produced by [email protected] during a previous run

In [3]:
#Make the RGEclass available
import sys
sys.path.append('./Source/Output/')
import RGEclass as rge
#The BetaFunction.py must be loaded (assumed in the current working directory, if not add the path to sys.path like above)
from BetaFunction import beta_function_SM as bSM


In order to create an object of the rgecls class and then solve the rges one needs to know how many equations are there and in which order they have be written in the BetaFunction.py file. First solution is to look directly into this file. Second solution is to look at the BetaFunction_SolveRGEs.py file where everything has been declared automatically and to copy the information here

In [4]:
#Now the first thing is to create an RGE object of the RGEclass for or result
OurRGEs = rge.RGE(bSM,32,labels=['betaY_u[0,0]', 'betaY_u[0,1]', 'betaY_u[0,2]', 'betaY_u[1,0]', 'betaY_u[1,1]', 'betaY_u[1,2]', 'betaY_u[2,0]', 'betaY_u[2,1]', 'betaY_u[2,2]', 'betaY_e[0,0]', 'betaY_e[0,1]', 'betaY_e[0,2]', 'betaY_e[1,0]', 'betaY_e[1,1]', 'betaY_e[1,2]', 'betaY_e[2,0]', 'betaY_e[2,1]', 'betaY_e[2,2]', 'betaY_d[0,0]', 'betaY_d[0,1]', 'betaY_d[0,2]', 'betaY_d[1,0]', 'betaY_d[1,1]', 'betaY_d[1,2]', 'betaY_d[2,0]', 'betaY_d[2,1]', 'betaY_d[2,2]', 'bg_SU2L', 'bg1', 'bg_SU3c', 'bLambda_1', 'bmu_1'])
#note that the labels have to correspond to the ordering of the beta functions in BetaFunction.py


Now, we set all the initial values and pass them to the RGE object using the Y0 attribute of the instance. Note that there are pre-defined values for the SM initial values at the Z mass inside the RGEclass object. To have an exaustive list one can print the instance itself

In [5]:
#list of all pre-defined values
OurRGEs

Out[5]:
Instance of the RGE class. Physical parameters predefined calculated at the Z boson mass :
Pi 3.14159265359
alpha 0.00775735008921
alphaS 0.1184
sinthetasq 0.2316
ZbosonMass 91.1876
Gf 1.16637e-05
Higgs vev 246.22
Top pole mass 173.3
Up matrix [0.0024, 1.27, 165.3308]
Down matrix [0.00475, 0.104, 4.19]
Lepton matrix [0.000511, 0.10566, 1.777]
.
In [6]:
#Set the initial values, here we use the SM values predefine in the RGE class
lbd0 = 0.07
mu0 = -lbd0*OurRGEs.vev**2
#Initial all the values to zero and modify the one we want to be different
Y0 = [0]*32
#Up diagonal Yukawas
Y0[0],Y0[4],Y0[8]=OurRGEs.yU[0],OurRGEs.yU[1],OurRGEs.yU[2]
#Down diagonal Yukawas
Y0[9],Y0[13],Y0[17]=OurRGEs.yD[0],OurRGEs.yD[1],OurRGEs.yD[2]
#Gauge Couplings
Y0[-4],Y0[-5],Y0[-3]=OurRGEs.g10,OurRGEs.g20,OurRGEs.g30
#lambda and mu1
Y0[-2],Y0[-1] = lbd0,mu0
OurRGEs.Y0 = Y0

In [7]:
#Set initial and final values for the scale directly e.g. Mz to 10**16 (same as initial values)
tmin,tmax,tstep = np.log(OurRGEs.Mz),19,0.5
#By default the assumptions is set to 'two-loop': True,'diag':True which means that the two loop equations are used if provided and Yukawas are diagonal
#Note that the user is free to modify the betaFunction.py file to introduce new assumptions and then add them to the dictionnary.
OurRGEs.solve_rges(tmin,tmax,tstep,assumptions={'two-loop': False,'diag':True})

In [8]:
#Now we can acces all the results via rge.Sol
OurRGEs.Sol['bLambda_1'][:10]#ten first points of lambda(t)

Out[8]:
array([ 0.04357418,  0.01944982, -0.00264243, -0.02292316, -0.04157585,
-0.05875525, -0.07459352, -0.08920489, -0.1026891 , -0.11513409])
In [9]:
#There is also an interpolated object accessible by,Can be called in any point
OurRGEs.Solint['bLambda_1'],OurRGEs.Solint['bLambda_1'](6)

Out[9]:
(<scipy.interpolate.interpolate.interp1d at 0x10544ac90>,
array(-0.07219024007593562))

### 2. Plotting the results¶

In [10]:
#Let's first plot the quartic coupling. To do so we use the plot function of the pylab library
plot(OurRGEs.Sol['t'],OurRGEs.Solint['bLambda_1'](OurRGEs.Sol['t']),label=r'$\Lambda_1$')
grid(True)

In [11]:
#Let's now look at the gauge couplings
plot(OurRGEs.Sol['t'],OurRGEs.Sol['bg1']**-2*np.sqrt(3/5.)**2,'k',label=r'$\alpha_1^{-1}$')
plot(OurRGEs.Sol['t'],OurRGEs.Sol['bg_SU2L']**-2,'b',label=r'$\alpha_2^{-1}$')
plot(OurRGEs.Sol['t'],OurRGEs.Sol['bg_SU3c']**-2,'r',label=r'$\alpha_3^{-1}$')
grid(True)
#titles
xlabel('$t=\log_{10}(Q\ \mathrm{GeV})$')
ylabel('Running in the SM')
title('Running of the gauge couplings in the SM obtained from [email protected]')
#number of column of location of the legend, collects all the label entries above
legend(loc=4,ncol=3)

Out[11]:
<matplotlib.legend.Legend at 0x10574b5d0>
In [12]:
#To illustrate the difference between one loop and two loop we can create a new object with the same RGEs but soolve it for two loop.
#Note that we could use the same instance we have already created but we won't be able to compare the results then.
rge2L = rge.RGE(bSM,32,labels=['betaY_u[0,0]', 'betaY_u[0,1]', 'betaY_u[0,2]', 'betaY_u[1,0]', 'betaY_u[1,1]', 'betaY_u[1,2]', 'betaY_u[2,0]', 'betaY_u[2,1]', 'betaY_u[2,2]', 'betaY_e[0,0]', 'betaY_e[0,1]', 'betaY_e[0,2]', 'betaY_e[1,0]', 'betaY_e[1,1]', 'betaY_e[1,2]', 'betaY_e[2,0]', 'betaY_e[2,1]', 'betaY_e[2,2]', 'betaY_d[0,0]', 'betaY_d[0,1]', 'betaY_d[0,2]', 'betaY_d[1,0]', 'betaY_d[1,1]', 'betaY_d[1,2]', 'betaY_d[2,0]', 'betaY_d[2,1]', 'betaY_d[2,2]', 'bg_SU2L', 'bg1', 'bg_SU3c', 'bLambda_1', 'bmu_1'])

In [13]:
rge2L.Y0 = Y0

In [14]:
rge2L.solve_rges(tmin,tmax,tstep,assumptions={'two-loop': True,'diag':True})

In [15]:
#Now we can compare the top yukawa at one and two loop
plot(rge2L.Sol['t'],rge2L.Solint['betaY_u[2,2]'](rge2L.Sol['t']),label=r'Two loop $Y_t$')
plot(OurRGEs.Sol['t'],OurRGEs.Solint['betaY_u[2,2]'](OurRGEs.Sol['t']),label=r'One loop $Y_t$')
grid(True)
xlabel('$t=\log_{10}(Q\ \mathrm{GeV})$')
ylabel(r'$Y_t$')
title('Comparison 1L vs 2L of $Y_t$')
#number of column of location of the legend, collects all the label entries above
legend(loc=4,ncol=3)

Out[15]:
<matplotlib.legend.Legend at 0x1056f23d0>