Standard Model + Wprime + Zprime

This model implements the SM at NLO QCD supplemented by interaction terms with a \(\mathrm{W}'\) and \(\mathrm{Z}'\) based on [JKS12], [BJK+16] (see also [Sul02], [DS12]):

\[\begin{split}\mathcal{L}_{\mathrm{W}'\mathrm{Z}'} =& k_{\mathrm{W},\pm} \frac{e}{\sqrt{2}s_\mathrm{w}} W_\mu' \left(C_\mathrm{W,\pm}^l \bar \nu^i \gamma^\mu w^\pm l^j + C_\mathrm{W,\pm}^q \bar u^i \gamma^\mu w^\pm d^j\right) +\\ &k_{\mathrm{Z},\pm} \frac{e}{\sqrt{2}s_\mathrm{w}c_\mathrm{w}} Z_\mu' \left(C_\mathrm{Z,\pm}^l \bar l^i \gamma^\mu w^\pm l^j + C_\mathrm{Z,\pm}^u \bar u^i \gamma^\mu w^\pm u^j + C_\mathrm{Z,\pm}^d \bar d^i \gamma^\mu w^\pm d^j \right) +\\ &\mathrm{h.c.},\end{split}\]

where \(C_\mathrm{W,\pm}^l\), \(C_\mathrm{W,\pm}^q\), \(C_\mathrm{Z,\pm}^l\), \(C_\mathrm{Z,\pm}^u\), \(C_\mathrm{Z,\pm}^d\) are CKM-like coupling matrices.

Parameter conventions

The SMWZP doesn’t come with a dedicated interface, thus, parameters can be set only with the generic function set_parameter_rcl(). In addition to the SM parameters, the SMWZP has the following new parameters

Parameter Recola identifier default value
\(k_{\mathrm{W},+}\), \(k_{\mathrm{W},-}\) 'kWR', 'kWL' 0.3, 1.1
\(k_{\mathrm{Z},+}\), \(k_{\mathrm{Z},-}\) 'kZR', 'kZL' 0.1, 1.3
\(C_{W,-}^l\) 'CWlL1x1', 'CWlL1x2', .. \(I_3\)
\(C_{W,+}^l\) 'CWlR1x1', 'CWlR1x2', .. \(I_3\)
\(C_{W,-}^q\) 'CWqL1x1', 'CWqL1x2', .. \(I_3\)
\(C_{W,+}^q\) 'CWqR1x1', 'CWqR1x2', .. \(I_3\)
\(C_{Z,-}^l\) 'CZlL1x1', 'CZlL1x2', .. \(I_3\)
\(C_{Z,+}^l\) 'CZlR1x1', 'CZlR1x2', .. \(I_3\)
\(C_{Z,-}^d\) 'CZdL1x1', 'CZdL1x2', .. \(I_3\)
\(C_{Z,+}^d\) 'CZdR1x1', 'CZdR1x2', .. \(I_3\)
\(C_{Z,-}^d\) 'CZdL1x1', 'CZdL1x2', .. \(I_3\)

The original CKM matrix is also implemented, but only for the first two generations and parametrized by the cabbibo angle:

Parameter Recola identifier default value
\(\theta_{\mathrm{c}}\) 'cabi'

Field conventions

Fields Recola identifier
\(\mathrm{W}'^+\), \(\mathrm{W}'^-\) 'Wp+', 'Wp-'
\(\mathrm{Z}'\) 'Zp'

Power counting

The model has been implemented with a power counting (see SM power counting) for the couplings \(k_{\mathrm{W},+}\), \(k_{\mathrm{W},-}\), \(k_{\mathrm{Z},+}\), and \(k_{\mathrm{Z},-}\) which allows to select individual contributions. See the examples below on how to use that feature.

Snippet code using the SMWZP

from pyrecola import *


# Change masses of W',Z'
set_parameter_rcl("MWp", 1500.)
set_parameter_rcl("MZp", 3500.)

# enable to draw off-shell currents
# set_draw_level_branches_rcl(1)

define_process_rcl(1, 'u d~ -> t b~', 'NLO')
unselect_power_LoopAmpl_rcl(1, 'QCD', 0)


p1 = [500., 0., 0.,  500.]
p2 = [500., 0., 0., -500.]

# generate a sample PSP using RAMBO
p = set_outgoing_momenta_rcl(1, [p1, p2])

# compute tree squared and tree one-loop interference
compute_process_rcl(1, p, 'NLO')

# get all different contributions (pow=[n,m,o] == gs^n e^m k^o)
# pure SM
A1_0 = get_squared_amplitude_rcl(1, 'NLO', pow=[2, 4, 0])
# SM interference with W'Z'
A1_1 = get_squared_amplitude_rcl(1, 'NLO', pow=[2, 4, 2])
# pure W'Z'
A1_2 = get_squared_amplitude_rcl(1, 'NLO', pow=[2, 4, 4])

use recola

implicit none
integer, parameter :: dp = kind (23d0)
real(dp) :: p(0:3,1:4), A2(2)

call set_print_level_squared_amplitude_rcl(2)

! enable to draw off-shell currents
! call set_draw_level_branches_rcl(1)

call define_process_rcl(1, 'u d~ -> t b~', 'NLO')
call unselect_power_LoopAmpl_rcl(1, 'QCD', 0)

call generate_processes_rcl

p(:,1) = [500d0, 0d0, 0d0,  500d0]
p(:,2) = [500d0, 0d0, 0d0, -500d0]
! generate a sample PSP using RAMBO
call set_outgoing_momenta_rcl(1, p(:,1:2), p)

! compute tree squared and tree one-loop interference
call compute_process_rcl(1, p, 'NLO', A2)
#include "recola.hpp"
#include <iostream>

int main(int argc, char *argv[])


Recola::set_parameter_rcl("MWp", 1500.);
Recola::set_parameter_rcl("MZp", 3500.);

// enable to draw off-shell currents
// Recola::set_draw_level_branches_rcl(1);

Recola::define_process_rcl(1, "u d~ -> t b~", "NLO");

Recola::unselect_power_LoopAmpl_rcl(1, "QCD", 0);

// generate it

// generate a sample PSP using RAMBO
double pin[2][4] =
{{500., 0., 0., 500.},
 {500., 0., 0., -500.}};
double p[4][4];
Recola::set_outgoing_momenta_rcl(1, pin, p);

// compute tree squared and tree one-loop interference
double A2[2];
Recola::compute_process_rcl(1, p, "NLO", A2);

double A1_0,A1_1,A1_2;
int pow[3] = {2, 4, 0};
Recola::get_squared_amplitude_rcl(1, pow, "NLO", A1_0);
pow[2] = 2;
Recola::get_squared_amplitude_rcl(1, pow, "NLO", A1_1);
pow[2] = 4;
Recola::get_squared_amplitude_rcl(1, pow, "NLO", A1_2);
std::cout << "A1_0: " << A1_0 << std::endl;
std::cout << "A1_1: " << A1_1 << std::endl;
std::cout << "A1_2: " << A1_2 << std::endl;

return 0;

Releases Standard Model + Wprime + Zprime


[BJK+16]Roberto Bonciani, Tomás Jezo, Michael Klasen, Florian Lyonnet, and Ingo Schienbein. Electroweak top-quark pair production at the LHC with $Z’$ bosons to NLO QCD in POWHEG. JHEP, 02:141, 2016. arXiv:1511.08185, doi:10.1007/JHEP02(2016)141.
[DS12]Daniel Duffty and Zack Sullivan. Model independent reach for W-prime bosons at the LHC. Phys. Rev., D86:075018, 2012. arXiv:1208.4858, doi:10.1103/PhysRevD.86.075018.
[JKS12]Tomas Jezo, Michael Klasen, and Ingo Schienbein. LHC phenomenology of general SU(2)xSU(2)xU(1) models. Phys. Rev., D86:035005, 2012. arXiv:1203.5314, doi:10.1103/PhysRevD.86.035005.
[Sul02]Zack Sullivan. Fully Differential $W^\prime $ Production and Decay at Next-to-Leading Order in QCD. Phys. Rev., D66:075011, 2002. arXiv:hep-ph/0207290, doi:10.1103/PhysRevD.66.075011.