Running AutoTherm#

The code can be run entirely within a jupyter notebook or through shell commands and python scripts. Here we discuss extensively the latter option. An example notebook is also provided. To run it, you need to install jupyter within the virtual environment wherein AutoTherm has been installed, or vice-versa.

The analytical, Mathematica-based part#

We start by activating the virtual environment, if not already active

cd $AutothermDir
source .venv/bin/activate

Here $AutothermDir should be replaced by its actual value, as discussed in the installation section.

We then move to the $ModelDir directory where the model file and config file have been created in the previous section and we launch the python executable handling the creation of configuration files and the execution of the Wolfram language scripts. This is done through

cd $ModelDir
autotherm-run axionSM.cfg

Barring any bugs with the model file or with the code, this should go through all steps. The final lines of the output should then read

Analytical pipeline completed
The matrix elements squared read:
{(1, -1, -1): '(24*ct^2*ht^2*s)/fPQ^2 + ((5*c1^2*g1^6 + 9*c2^2*g2^6 + 24*c3^2*g3^6)*(t^2 + u^2))/(32*fPQ^2*pi^4*s)', (-1, -1, 1): '-1/32*(768*ct^2*ht^2*pi^4*t^2 + 5*c1^2*g1^6*(t^2 + 2*t*u + 2*u^2) + 9*c2^2*g2^6*(t^2 + 2*t*u + 2*u^2) + 24*c3^2*g3^6*(t^2 + 2*t*u + 2*u^2))/(fPQ^2*pi^4*t)', (-1, 1, -1): '-1/32*(768*ct^2*ht^2*pi^4*u^2 + 5*c1^2*g1^6*(2*t^2 + 2*t*u + u^2) + 9*c2^2*g2^6*(2*t^2 + 2*t*u + u^2) + 24*c3^2*g3^6*(2*t^2 + 2*t*u + u^2))/(fPQ^2*pi^4*u)', (1, 1, 1): '((c1^2*g1^6 + 15*c2^2*g2^6 + 48*c3^2*g3^6)*(t^2 + t*u + u^2)^2)/(64*fPQ^2*pi^4*s*t*u)'}
The couplings are:
{'gauge': ('g1', 'g2', 'g3'), 'noneq': ('fPQ',), 'others': ('c1', 'c2', 'c3', 'ct', 'ht')}
The thermal masses are:
('(11*g1^2)/6', '(11*g2^2)/6', '2*g3^2')

These are the main results of this step: the matrix elements squared and the thermal masses that are needed. They have been written in the axionSM_result.json file within $ModelDir. On an M3 MacBook Pro with Python 3.12 and Mathematica 14 this takes about 20 seconds.

The numerical part#

The final step is to perform the phase-space integration over these matrix elements. This is carried out by the numerical module. To this end, we can create this simple python script within $ModelDir, which we call get_rate.py.

#start by importing the analytical and numerical modules
from analytical.controller import *
from numerical.manipulate import *


#this function calls the analytical pipeline. To avoid re-running all the steps,
#set the conf, rules and proc flags in axionSM.cfg to no
axion_analytical=analytical_pipeline("axionSM.cfg")

#we now create an instance of the NumRate class using the output of the
#analytical pipeline. 1 is the degeneracy of the axion (no spin or any other quantum number)
axion_rate=NumRate(*axion_analytical,1)

#now all methods of the class are available. For illustration purposes we output
#the full rate at 10 values of k/T for an axion-gluon coupling only. We rescale
#fpq by setting its value to 1 and we use a strong coupling g_3 corresponding to
#alpha_s=0.06
k=numpy.linspace(1,10,10)
#the tuple for the couplings is the union of the gauge and other couplings
#from the output of the analytical pipeline we have
# {'gauge': ('g1', 'g2', 'g3'), 'noneq': ('fPQ',), 'others': ('c1', 'c2', 'c3', 'ct', 'ht')}
rate = axion_rate.rate(k,1.,(0.,0.,numpy.sqrt(4.*numpy.pi*0.06),0,0,1.,0,0),0)
print(rate)

Before executing this file, it is recommended to set the conf, rules and proc flags in axionSM.cfg to no, so as to avoid re-running the Wolfram language scripts. The script is executed with

python get_rate.py

Barring bugs, it should output

Assuming SM thermal bath
Configuration check completed successfully.
Skipping configuration.
Skipping generation of Feynman rules.
Skipping generation of all processes.
Analytical pipeline completed
/Users/jacopo/Nextcloud/AUTOTHERM/autotherm/MyModels/axion/get_rate.py:12: AutothermWarning: The ratio between the gauge boson mass 11*g1**2/6 and the coefficient of the IR divergence -11*c1**2*g1**6/(384*pi**4) depends on the coupling constants entering the mass.
This is most likely not problematic, continuing with evaluation.
axion_rate=NumRate(*axion_analytical,1)
/Users/jacopo/Nextcloud/AUTOTHERM/autotherm/MyModels/axion/get_rate.py:12: AutothermWarning: The ratio between the gauge boson mass 11*g2**2/6 and the coefficient of the IR divergence -11*c2**2*g2**6/(128*pi**4) depends on the coupling constants entering the mass.
This is most likely not problematic, continuing with evaluation.
axion_rate=NumRate(*axion_analytical,1)
/Users/jacopo/Nextcloud/AUTOTHERM/autotherm/MyModels/axion/get_rate.py:12: AutothermWarning: The ratio between the gauge boson mass 2*g3**2 and the coefficient of the IR divergence -c3**2*g3**6/(4*pi**4) depends on the coupling constants entering the mass.
This is most likely not problematic, continuing with evaluation.
axion_rate=NumRate(*axion_analytical,1)
[array([ 1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9., 10.]), array([6.24468720482499e-6, 2.48806377023693e-5, 3.35216329472938e-5,
    3.89866431064388e-5, 4.30820774272845e-5, 4.64391827454089e-5,
    4.93197028763330e-5, 5.18530080251627e-5, 5.41151703842284e-5,
    5.61573931901464e-5], dtype=object), array([1.32458357276370e-5, 2.68517915080078e-5, 3.44196852859558e-5,
    3.94963238927934e-5, 4.34096375330550e-5, 4.66671741757908e-5,
    4.94874378831309e-5, 5.19815453260401e-5, 5.42167932436927e-5,
    5.62397439824078e-5], dtype=object), array([2.13516715648551e-5, 3.50716746267359e-5, 4.25672158519041e-5,
    4.74800571298319e-5, 5.12012593583113e-5, 5.42733060334197e-5,
    5.69285281412678e-5, 5.92808393892183e-5, 6.13957319484079e-5,
    6.33165674566682e-5], dtype=object)]

The warnings about proportionality are harmless. The numerical integration in the three schemes has been successfully carried out. On an M3 MacBook Pro with Python 3.12 this takes about 3 seconds.

Running AutoTherm#

The analytical, Mathematica-based part#

The numerical part#

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