.. _smatgc_warsaw:

Standard Model + ATGC (Warsaw basis)
------------------------------------

This model is a reparametrisation of the :ref:`Standard Model + ATGC <smatgc>` model file, which parametrises the
anomalous triple gauge boson couplings of mass dimension six in terms of the Standard Model effective field theory
in the Warsaw basis :cite:`Haisch:2025jqr`:

.. math:: \mathcal{L}^{\mathrm{eff}.} =
      \mathcal{L}^{\mathrm{SM}}
      +\sum_i \frac{c_6^i}{\Lambda^2} \mathcal{O}_{6, \, \mathrm{Warsaw}}^i
      +\sum_i \frac{c_8^i}{\Lambda^4} \mathcal{O}_8^i
      ,

The fields correspond to the ones in the :ref:`SM <fieldconventions>`.

Parameters and power-counting
*****************************

Besides the :ref:`usual power-counting <powercounting>` Wilson coefficients carry
the power :m:`\mathrm{LAM}` (similar to :m:`\mathrm{QED}`, :m:`\mathrm{QCD}`)
corresponding to :m:`\Lambda^{-2}`.  For the dimension 6 operators the following
parameters can be set

.. list-table::
   :widths: 4 8 12 4
   :header-rows: 1
   :align: center

   * - Parameter
     - |recola| identifier
     - Operator
     - Order

   * - :m:`c_{W}/\Lambda^2`
     - ``'CWD6'``
     - :m:`\epsilon_{ijk} \hspace{0.25mm} W^{i, \nu}_{\mu} \hspace{0.25mm} W^{j, \lambda}_{\nu} \hspace{0.25mm}W^{k, \mu}_{\lambda}`
     - :m:`\mathrm{LAM}^1`

   * - :m:`c_{HB}/\Lambda^2`
     - ``'CHBD6'``
     - :m:`H^{\dagger} H \hspace{0.25mm} B_{\mu \nu} B^{\mu \nu}`
     - :m:`\mathrm{LAM}^1`

   * - :m:`c_{HW}/\Lambda^2`
     - ``'CHWD6'``
     - :m:`H^{\dagger} H \hspace{0.25mm} W_{\mu \nu}^i W^{i, \mu \nu}`
     - :m:`\mathrm{LAM}^1`

   * - :m:`c_{HW\!B}/\Lambda^2`
     - ``'CHWBD6'``
     - :m:`H^{\dagger} \sigma^{i} H \hspace{0.25mm} W^{i}_{\mu\nu} B^{\mu\nu}`
     - :m:`\mathrm{LAM}^1`


   * - :m:`c_{\tilde{W}}/\Lambda^2`
     - ``'CWtildeD6'``
     - :m:`\epsilon_{ijk} \hspace{0.25mm} W^{i, \nu}_{\mu} \hspace{0.25mm} W^{j, \lambda}_{\nu} \hspace{0.25mm} \widetilde{W}^{k, \mu}_{\lambda}`
     - :m:`\mathrm{LAM}^1`

   * - :m:`c_{H\tilde{B}}/\Lambda^2`
     - ``'CHBtildeD6'``
     - :m:`H^{\dagger} H \hspace{0.25mm} B_{\mu \nu} \widetilde{B}^{\mu \nu}`
     - :m:`\mathrm{LAM}^1`

   * - :m:`c_{H\tilde{W}}/\Lambda^2`
     - ``'CHWtildeD6'``
     - :m:`H^{\dagger} H \hspace{0.25mm} W_{\mu \nu}^{i} \widetilde{W}^{i, \mu \nu}`
     - :m:`\mathrm{LAM}^1`

   * - :m:`c_{H\tilde{W}\!B}/\Lambda^2`
     - ``'CHWtildeBD6'``
     - :m:`H^{\dagger} \sigma^{i} H \hspace{0.25mm} \widetilde{W}^{i}_{\mu\nu} B^{\mu\nu}`
     - :m:`\mathrm{LAM}^1`

The dimension eight contributions for the neutral sector are identical to the ones in
the :ref:`Standard Model + ATGC <smatgc>` model file.

.. list-table::
   :widths: 4 8 12 4
   :header-rows: 1
   :align: center

   * - Parameter
     - |recola| identifier
     - Operator
     - Order

   * - :m:`c_{BW}/\Lambda^4`
     - ``'CBWL4'``
     - :m:`-\mathrm{i}
       \Phi^\dagger
       B_{\mu\nu}
       \frac{\tau_i}{2}
       W^{\mu\rho\;i}
       \left\{D_\rho, D^\nu\right\}\Phi + \mathrm{h.c.}`
     - :m:`\mathrm{LAM}^2`

   * - :m:`c_{WW}/\Lambda^4`
     - ``'CWWL4'``
     - :m:`\mathrm{i}
       \Phi^\dagger
       \frac{\tau_i}{2}
       \frac{\tau_j}{2}
       W^{i}_{\mu\nu}
       W^{\mu\rho\;j}
       \left\{D_\rho, D^\nu\right\}\Phi + \mathrm{h.c.}`
     - :m:`\mathrm{LAM}^2`

   * - :m:`c_{BB}/\Lambda^4`
     - ``'CBBL4'``
     - :m:`\mathrm{i}
       \Phi^\dagger
       B_{\mu\nu}
       B^{\mu\rho}
       \left\{D_\rho, D^\nu\right\}\Phi + \mathrm{h.c.}`
     - :m:`\mathrm{LAM}^2`

   * - :m:`c_{\tilde BW}/\Lambda^4`
     - ``'CBtWL4'``
     - :m:`-\mathrm{i}
       \Phi^\dagger
       \tilde B_{\mu\nu}
       \frac{\tau_i}{2}
       W^{\mu\rho\;i}
       \left\{D_\rho, D^\nu\right\}\Phi + \mathrm{h.c.}`
     - :m:`\mathrm{LAM}^2`


The user has to make sure that no corrections other than pure QCD ones are
selected. An example for diboson production is given below.


Snippet code
************


.. tabs::

  .. group-tab:: WZ-Fortran90

    .. code-block:: fortran

      program atgc_warsaw_example
        use recola
        implicit none
        integer, parameter :: dp = kind(23d0)
        real (dp)          :: p(0:3,1:6), pp(0:3,1:4), pdw(0:3,1:3), pdz(0:3,1:3), A2(0:2)
        real (dp)          :: MW, MZ, GW, GZ
        character(len=100) :: modelname
        complex (dp) :: cone
        integer :: i
        parameter ( cone=(1d0,0d0) )

        call set_output_file_rcl('*')

        MW = 8.0349970922628273E+01
        GW = 2.0842988587989093E+00
        MZ = 9.1153480619182758E+01
        GZ = 2.4942663787728243E+00

        call set_output_file_rcl('*')
        call set_complex_mass_scheme_rcl

        call set_mu_ir_rcl(0.5d0*(MW+MZ))
        call set_mu_uv_rcl(0.5d0*(MW+MZ))
        call set_mu_ms_rcl(0.5d0*(MW+MZ))

        call set_light_fermions_rcl(1d-3)
        call set_alphas_rcl(0.118d+00, MZ, 5)

        call set_parameter_rcl("MW", cone*MW)
        call set_parameter_rcl("WW", cone*GW)
        call set_parameter_rcl("MZ", cone*MZ)
        call set_parameter_rcl("WZ", cone*GZ)

        call set_resonant_particle_rcl('W+')
        call set_resonant_particle_rcl('Z')

        call set_parameter_rcl('CWD6',        cone*1.0d-6)
        call set_parameter_rcl('CHBD6',       cone*0.0d-6)
        call set_parameter_rcl('CHWD6',       cone*0.0d-6)
        call set_parameter_rcl('CHWBD6',      cone*0.0d-6)
        call set_parameter_rcl('CWtildeD6',   cone*0.0d-6)
        call set_parameter_rcl('CHBtildeD6',  cone*0.0d-6)
        call set_parameter_rcl('CHWtildeD6',  cone*0.0d-6)
        call set_parameter_rcl('CHWtildeBD6', cone*0.0d-6)

        call set_parameter_rcl('CBWL4',   cone*0.0d0)
        call set_parameter_rcl('CWWL4',   cone*0.0d0)
        call set_parameter_rcl('CBBL4',   cone*0.0d0)
        call set_parameter_rcl('CBtWL4',  cone*0.0d0)


        ! Define process and set coupling orders
        call define_process_rcl(1,'u d~ -> W+(mu+ nu_mu) Z(e+ e-)','NLO')

        call select_power_BornAmpl_rcl(1,'QCD',0)
        call select_power_LoopAmpl_rcl(1,'QCD',2)
        call select_power_BornAmpl_rcl(1,'QED',4)
        call select_power_LoopAmpl_rcl(1,'QED',4)

        ! Generate process
        call generate_processes_rcl

        ! Example phase-space point for process 1
        p(:, 1) = [227.39345312954683d0,  0.0000000000000000d0,  0.0000000000000000d0,  227.39345312954683d0]
        p(:, 2) = [227.39345312954683d0,  0.0000000000000000d0,  0.0000000000000000d0, -227.39345312954683d0]
        p(:, 3) = [42.091253388702761d0,  7.0682379749920559d0,  22.640966407137846d0,  34.772119059840406d0]
        p(:, 4) = [183.26515868246659d0, -29.942025082444019d0, -56.364534205011779d0,  171.79241195327580d0]
        p(:, 5) = [19.329309891962328d0,  12.844761211993173d0, -14.265740621156976d0, -2.2633989564945494d0]
        p(:, 6) = [210.10118429596204d0,  10.029025895458791d0,  47.989308419030905d0, -204.30113205662164d0]

        pp(:, 1) = p(:, 1)
        pp(:, 2) = p(:, 2)
        pp(:, 3) = p(:, 3) + p(:, 4)
        pp(:, 4) = p(:, 5) + p(:, 6)

        write(*,*) "sqrt(pW^2): ", sqrt(pp(0,3)**2 - sum(pp(1:3,3)**2))
        write(*,*) "sqrt(pZ^2): ", sqrt(pp(0,4)**2 - sum(pp(1:3,4)**2))

        pdw(:, 1) = pp(:, 3)
        pdw(:, 2) =  p(:, 3)
        pdw(:, 3) =  p(:, 4)

        pdz(:, 1) = pp(:, 4)
        pdz(:, 2) =  p(:, 5)
        pdz(:, 3) =  p(:, 6)

        ! Compute process and print its results
        call compute_process_rcl(1,p,'NLO')
        call writeResults(1)

        do i=0,2
          call get_squared_amplitude_rcl(pr, [2, 8, i], 'NLO', A2(i))
          write(*,'(A,X,I2,X,A,I1,A,E15.8)') 'A2 for process', pr, '(lam = -', i, ') = ', A2(i)
        end do

        call reset_recola_rcl
      end program atgc_warsaw_example

Releases
^^^^^^^^

* :sm_atgc_warsaw:`2.4.0`

.. rubric :: References

.. bibliography:: ../references.bib
   :filter: docname in docnames