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]):

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

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.

Note

The $\mathrm{h.c.}$ is applied to the full new physics part of the lagrangian. Goldstone bosons associated to $\mathrm{W}'$ and $\mathrm{Z}'$ have been added by hand.

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 SM CKM matrix is also implemented giving rise to modified charged currents.

Parameter	Recola identifier	default value
$\mathrm{CKM}$	`'CKM1x1'`, `'CKM1x1'`, ..	$I_3$

Z and Z’ vertices can be matched by setting their masses equal and furthermore adjusting the diagonal couplings as follows:

\[\begin{split}C_{Z,-}^d &= \frac{-1 + 2 c_\mathrm{w}^2}{6}, \quad C_{Z,+}^d = \frac{s_\mathrm{w}^2}{3}\\ C_{Z,-}^u &= \frac{-1 + 4 c_\mathrm{w}^2}{6}, \quad C_{Z,+}^u = -\frac{2 s_\mathrm{w}^2}{3}\end{split}\]

Note

In older versions $\le 2.2.3$ the SM CKM was implemented only for the first two generations and parametrized by the Cabibbo angle:

Parameter	Recola identifier	default value
$\theta_{\mathrm{c}}$	`'cabi'`

Note that setting a non-diagonal SM CKM matrix does not interfere with the Z’, W’ coupling, i.e. the couplings of the fermions with Z’,W’ are independent of the SM CKM matrix.

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 *

set_print_level_squared_amplitude_rcl(2)

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


# set CKM to include Cabibbo mixing angle
tc = 13.02/180.*pi
cb = cos(tc)
sb = sin(tc)
set_parameter_rcl("CKM1x1", cb)
set_parameter_rcl("CKM1x2", sb)
set_parameter_rcl("CKM2x1", -sb)
set_parameter_rcl("CKM2x2", cb)

# 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)

generate_processes_rcl()

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])

reset_recola_rcl()

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_print_level_squared_amplitude_rcl(2);

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
Recola::generate_processes_rcl();

// 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¶

UFO model files¶

SMWZP-UFO

References

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.

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'`, ..	\(I_3\)

Standard Model + Wprime + Zprime¶

Parameter conventions¶

Field conventions¶

Power counting¶

Snippet code using the SMWZP¶

Releases Standard Model + Wprime + Zprime¶

UFO model files¶

Navigation

Related Topics

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