Standard Model + ATGC (Warsaw basis)¶
This model is a reparametrisation of the Standard Model + ATGC 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 [HLP+25]:
The fields correspond to the ones in the SM.
Parameters and power-counting¶
Besides the usual power-counting Wilson coefficients carry the power \(\mathrm{LAM}\) (similar to \(\mathrm{QED}\), \(\mathrm{QCD}\)) corresponding to \(\Lambda^{-2}\). For the dimension 6 operators the following parameters can be set
Parameter |
Recola identifier |
Operator |
Order |
|---|---|---|---|
\(c_{W}/\Lambda^2\) |
|
\(\epsilon_{ijk} \hspace{0.25mm} W^{i, \nu}_{\mu} \hspace{0.25mm} W^{j, \lambda}_{\nu} \hspace{0.25mm}W^{k, \mu}_{\lambda}\) |
\(\mathrm{LAM}^1\) |
\(c_{HB}/\Lambda^2\) |
|
\(H^{\dagger} H \hspace{0.25mm} B_{\mu \nu} B^{\mu \nu}\) |
\(\mathrm{LAM}^1\) |
\(c_{HW}/\Lambda^2\) |
|
\(H^{\dagger} H \hspace{0.25mm} W_{\mu \nu}^i W^{i, \mu \nu}\) |
\(\mathrm{LAM}^1\) |
\(c_{HW\!B}/\Lambda^2\) |
|
\(H^{\dagger} \sigma^{i} H \hspace{0.25mm} W^{i}_{\mu\nu} B^{\mu\nu}\) |
\(\mathrm{LAM}^1\) |
\(c_{\tilde{W}}/\Lambda^2\) |
|
\(\epsilon_{ijk} \hspace{0.25mm} W^{i, \nu}_{\mu} \hspace{0.25mm} W^{j, \lambda}_{\nu} \hspace{0.25mm} \widetilde{W}^{k, \mu}_{\lambda}\) |
\(\mathrm{LAM}^1\) |
\(c_{H\tilde{B}}/\Lambda^2\) |
|
\(H^{\dagger} H \hspace{0.25mm} B_{\mu \nu} \widetilde{B}^{\mu \nu}\) |
\(\mathrm{LAM}^1\) |
\(c_{H\tilde{W}}/\Lambda^2\) |
|
\(H^{\dagger} H \hspace{0.25mm} W_{\mu \nu}^{i} \widetilde{W}^{i, \mu \nu}\) |
\(\mathrm{LAM}^1\) |
\(c_{H\tilde{W}\!B}/\Lambda^2\) |
|
\(H^{\dagger} \sigma^{i} H \hspace{0.25mm} \widetilde{W}^{i}_{\mu\nu} B^{\mu\nu}\) |
\(\mathrm{LAM}^1\) |
The dimension eight contributions for the neutral sector are identical to the ones in the Standard Model + ATGC model file.
Parameter |
Recola identifier |
Operator |
Order |
|---|---|---|---|
\(c_{BW}/\Lambda^4\) |
|
\(-\mathrm{i} \Phi^\dagger B_{\mu\nu} \frac{\tau_i}{2} W^{\mu\rho\;i} \left\{D_\rho, D^\nu\right\}\Phi + \mathrm{h.c.}\) |
\(\mathrm{LAM}^2\) |
\(c_{WW}/\Lambda^4\) |
|
\(\mathrm{i} \Phi^\dagger \frac{\tau_i}{2} \frac{\tau_j}{2} W^{i}_{\mu\nu} W^{\mu\rho\;j} \left\{D_\rho, D^\nu\right\}\Phi + \mathrm{h.c.}\) |
\(\mathrm{LAM}^2\) |
\(c_{BB}/\Lambda^4\) |
|
\(\mathrm{i} \Phi^\dagger B_{\mu\nu} B^{\mu\rho} \left\{D_\rho, D^\nu\right\}\Phi + \mathrm{h.c.}\) |
\(\mathrm{LAM}^2\) |
\(c_{\tilde BW}/\Lambda^4\) |
|
\(-\mathrm{i} \Phi^\dagger \tilde B_{\mu\nu} \frac{\tau_i}{2} W^{\mu\rho\;i} \left\{D_\rho, D^\nu\right\}\Phi + \mathrm{h.c.}\) |
\(\mathrm{LAM}^2\) |
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¶
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¶
References
- HLP+25
Ulrich Haisch, Jakob Linder, Giovanni Pelliccioli, Emanuele Re, and Giulia Zanderighi. Polarized-boson pairs at NLO in the SMEFT. JHEP, 11:080, 2025. arXiv:2507.21768, doi:10.1007/JHEP11(2025)080.
