Higgs Triplet Extension of the Standard Model

This Higgs Triplet Extension of the Standard Model (HT) is a simple extended Higgs sector with one additional real vevless SU(2) Higgs triplet field \(\Sigma\) with hypercharge \(Y_\Sigma=0\). Our conventions follow mainly [FPPRMW09]. See also [CDJ08].

In the vevless case the most general renormalizable scalar potential reads

(1)\[V_{\mathrm{HT}} = -m_\Phi^2 \Phi^\dagger \Phi -m_\Sigma^2 F +\frac{\lambda}{4} \left(\Phi^\dagger \Phi\right)^2 +\frac{b_4}{4} F^2 +\frac{a_2}{2} \Phi^\dagger \Phi F,\]

with \(\Phi\) being the SM Higgs doublet and \(F\) being defined as

\[F = 2 \mathrm{Tr} \left[ \Sigma\cdot\Sigma\right].\]

Parameter conventions

Parameters of the model can be set with the generic function set_parameter_rcl(). All parameters in the HT are defined to be real and we choose the following set of physical parameters:

Basis

HT potential

Gauge part

before SSB

\(m_\Phi\), \(m_\Sigma\), \(\lambda\), \(a_2\), \(b_4\)

\(g\), \(g^\prime\)

Recola2 input

\(M_{\mathrm{H}_1}\), \(M_{\mathrm{H}^\pm}\), \(a_2\), \(b_4\), \(M_\mathrm{W}\)

\(\alpha_\mathrm{em}\), \(M_\mathrm{Z}\)

The default values are

Parameter

Recola identifier

default value

\(M_{\mathrm{H}_1}\)

'MH1'

125

\(M_{\mathrm{H}^\pm}\)

'MHP'

500

\(a_{2}\)

'a2'

0.1

\(b_{4}\)

'b4'

0.2

The fields extend the ones in the SM by

Fields

Recola identifier

\(H_\mathrm{1}\)

'H1'

\(H_\mathrm{2}\)

'H2'

\(H^+\)

'H+'

\(H^-\)

'H-'

where \(H_\mathrm{1}\) is the lighter Higgs-boson which typically takes the role of the SM one.

Power counting and renormalization

The model has been implemented with a power counting (see SM power counting) that assumes the couplings \(a_2\), \(b_2\) to scale as \(\mathrm{QED}^2\). The renormalization has been performed in the complete on-shell scheme with the possibility to switch between standard EW renormalization schemes. The new Higgs-potential parameters are renormalized MSbar. Concerning the renormalization see also [BH98]. The mass correction to the second neutral Higgs-boson can be printed by increasing the print level set_print_level_parameters_rcl().

Snippet code using the HT

from pyrecola import *

set_output_file_rcl('*')
set_print_level_squared_amplitude_rcl(2)

# Change HT parameters
set_parameter_rcl("MH1", 125.)
set_parameter_rcl("MHP", 350.)
set_parameter_rcl("a2", 0.3)
set_parameter_rcl("b4", -0.7)

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

define_process_rcl(1, 'e+ e- -> H+ H-', 'NLO')

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)
A0 = get_squared_amplitude_rcl(1, 'LO', pow=[0, 4])
# born-virtual interference (EW corrections)
A1 = get_squared_amplitude_rcl(1, 'NLO', pow=[0, 6])

print("A0:", A0)
print("A1:", A1)

reset_recola_rcl()
  program main

  use recola

  implicit none
  integer, parameter :: dp = kind (23d0)
  real(dp) :: p(0:3,1:4), A0, A1

  call set_parameter_rcl("MH1", complex(125d0,0d0))
  call set_parameter_rcl("MHP", complex(350,0d0))
  call set_parameter_rcl("a2", complex(0.3d0,0d0))
  call set_parameter_rcl("b4", complex(-0.7d0,0d0))

  call set_output_file_rcl('*')
  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, 'e+ e- -> H+ H-', 'NLO')

  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')

  call get_squared_amplitude_rcl(1,[0,4], 'LO' , A0)
  call get_squared_amplitude_rcl(1,[0,6], 'NLO', A1)
  write(*,*) "A0:", A0
  write(*,*) "A1:", A1

end program main
#include "recola.hpp"
#include <iostream>

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

Recola::set_output_file_rcl("*");
Recola::set_print_level_squared_amplitude_rcl(2);

Recola::set_parameter_rcl("MH1", 125.);
Recola::set_parameter_rcl("MHP", 350.);
Recola::set_parameter_rcl("a2", 0.3);
Recola::set_parameter_rcl("b4", -0.7);

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

Recola::define_process_rcl(1, "e+ e- -> H+ H-", "NLO");

// 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 A0,A1;
int pow[2] = {0, 4};
Recola::get_squared_amplitude_rcl(1, pow, "LO", A0);
pow[1] = 6;
Recola::get_squared_amplitude_rcl(1, pow, "NLO", A1);
std::cout << "A0: " << A0 << std::endl;
std::cout << "A1: " << A1 << std::endl;

return 0;
}

Releases

UFO model files

References

BH98

T. Blank and W. Hollik. Precision observables in SU(2) x U(1) models with an additional Higgs triplet. Nucl. Phys. B, 514:113–134, 1998. arXiv:hep-ph/9703392, doi:10.1016/S0550-3213(97)00785-2.

CDJ08

Mu-Chun Chen, Sally Dawson, and C.B. Jackson. Higgs Triplets, Decoupling, and Precision Measurements. Phys. Rev. D, 78:093001, 2008. arXiv:0809.4185, doi:10.1103/PhysRevD.78.093001.

FPPRMW09

Pavel Fileviez Perez, Hiren H. Patel, Michael.J. Ramsey-Musolf, and Kai Wang. Triplet Scalars and Dark Matter at the LHC. Phys. Rev. D, 79:055024, 2009. arXiv:0811.3957, doi:10.1103/PhysRevD.79.055024.