Simple Example
==============

This example is supposed to serve as a quick start to the ConstructTerms
module of FlavorPy.

We determine the symmetry invariant terms of the flavor model introduced
in “Are neutrino masses modular forms?” by F. Feruglio
(https://arxiv.org/pdf/1706.08749) as so-called Example 1 in section
3.12. The chiral supermultiplets and their transformation properties of
this model are

.. image:: simpleexample_constructterms_table.png

where :math:`k_L=1`, :math:`k_u=0`, :math:`k_\varphi=-3`,
and :math:`k_{E_i}+k_L+k_d+k_\varphi=0`. Table taken from
https://arxiv.org/pdf/1706.08749.

Import
------

After installing FlavorPy with 
:code:`pip install flavorpy`, 
import the modelfitting module.

.. code:: ipython3

    # Import the constructterms module of FlavorPy
    import flavorpy.constructterms as ct

Define Groups
-------------

Let us start by defining the non-Abelian group :math:`A_4`.

.. code:: ipython3

    # Representations of A4
    A4_reps = ['1', '1p', '1pp', '3']
    
    # Tensor products of A4_reps as a matrix, i.e. A_tensor_procuts[i,j] = A4_reps[i] x A4_reps[j]
    A4_tensor_products = [[['1'], ['1p'], ['1pp'], ['3']],
                          [['1p'], ['1pp'], ['1'], ['3']],
                          [['1pp'], ['1'], ['1p'], ['3']],
                          [['3'], ['3'], ['3'], ['1', '1p', '1pp', '3', '3']]]
    
    # Construct the group
    A4 = ct.NonAbelianGroup('A4', representations=A4_reps, tensor_products=A4_tensor_products)

Next, we also need the modular weight. Since modular weights multiply in
the same way as charges of a U(1) group, we can define the modular
weight as a U(1) group

.. code:: ipython3

    mod_weight = ct.U1Group('mod weight')

It often also helps to define the gauge :math:`U(1)_Y` for the
hypercharge

.. code:: ipython3

    U1y = ct.U1Group('U1y')

Define Fields
-------------

Next, we can define the fields of the flavor model.

The paper does not explicitly specifies the modular weight of
:math:`E_1^c`, :math:`E_2^c`, :math:`E_3^c`, and :math:`H_d`, it only
says that :math:`k_{E_i}+k_L+k_d+k_\phi=0`. Hence we choose them
accordingly.

.. code:: ipython3

    ke = 20
    kd = - ke + 1 - 3
    
    E1 = ct.Field('E1', charges={A4:['1'], mod_weight:ke, U1y:1})
    E2 = ct.Field('E2', charges={A4:['1pp'], mod_weight:ke, U1y:1})
    E3 = ct.Field('E3', charges={A4:['1p'], mod_weight:ke, U1y:1})
    L = ct.Field('L', charges={A4:['3'], mod_weight:-1, U1y:-1/2})
    Hd = ct.Field('Hd', charges={A4:['1'], mod_weight:kd, U1y:-1/2})
    Hu = ct.Field('Hu', charges={A4:['1'], mod_weight:0, U1y:+1/2})
    PhiT = ct.Field('PhiT', charges={A4:['3'], mod_weight:3, U1y:0})
    Y = ct.Field('Y', charges={A4:['3'], mod_weight:2, U1y:0})
    
    all_fields = [E1, E2, E3, L, Hd, Hu, PhiT, Y]

We also need to define the superpotential as a “Field”

.. code:: ipython3

    W = ct.Field('W', charges={A4:['1'], mod_weight:0, U1y:0})

Determine symmetry invariant terms in the superpotential
--------------------------------------------------------

Then we can determine the symmetry invariant terms up to a specific
order of the superpotential with

.. code:: ipython3

    allowed_terms = ct.list_allowed_terms(all_fields, W, order=5)
    allowed_terms




.. parsed-literal::

    [L L Hu Hu Y, E1 L Hd PhiT, E2 L Hd PhiT, E3 L Hd PhiT]



Hence, these four terms are the only symmetry invariant terms in the
superpotential up to order 5. Note that this is exactly the same result
as in the paper, see eqs. (35) and (37)