pSPSS: Phenomenological symmetry protected seesaw

The motivation and implementation of the pSPSS is discussed in

19. Antusch, J. Hajer, and J. Rosskopp. ‘Simulating heavy neutrino-antineutrino oscillations at colliders’ (2022).

Model description

The pSPSS describes the interactions of a pseudo-Dirac pair of two Majorana degrees of freedom

$N_1$

and

$N_2$

generically appearing in low-scale seesaw models protected by a lepton-number like symmetry.
In the lepton number conserving (LNC) limit the interactions of the symmetry protected seesaw (SPSS) with the Standard Model are

$\mathcal L_\text{SPSS}^L = \overline N_i \, i\!\! \not\! \partial N_i -  y_{\alpha1} \widetilde H^\dagger \bar \ell_\alpha N_1^c - \overline N_1 m_M^{} N_2^c + \text{H.c.}$

The additional lepton number violating (LNV) interactions must be small in order to generate light neutrino masses.
Additionally, they introduce a small mass splitting

$\Delta m$

between the two heavy neutrino mass eigen states

$m_{4/5}^{} = m_M^{} + \frac12 m_M^{} |\theta|^2 \mp \frac12 \Delta m$

where the contribution with the active sterile mixing parameter

$\theta = m_D / m_M$

with

$m_D = y_1 v$

and

$v\simeq174 \text{ GeV}$

appears already in the LNC case.
The smallness of the LNV interactions ensures unobservable collider effects, with the exception of heavy neutrino-antineutrino oscillations since these are a macroscopic interference effect

$P^{\text{LNC}/\text{LNV}}_\text{osc}(\tau) = (1 \pm \cos\left(\Delta m \tau \right) \exp(-\lambda))/2$

where

$\lambda$

captures the damping of the oscillations due to decoherence.
Therefore, the details of the seesaw model besides the generated mass splitting can be neglected when simulating pseudo-Dirac heavy neutrinos as long as the neutrino-antineutrino oscillations are taken into account.

FeynRules implementation

The FeynRules model file contains in addition to the Standard Model parameter as free parameter the heavy neutrino

• mixing parameter

  $\theta_\alpha$
  

  theta1, theta2, theta3
• Majorana mass

  $m_M$
  

  Mmaj
• Mass splitting

  $\Delta M$
  

  deltaM
• Damping parameter

  $\lambda$
  

  damping

MadGraph patch

In order to generate events with heavy neutrino-antineutrino oscillations it is necessary to patch the ./bin/internal/common_run_interface.py file in MadGraph using

for event in lhe:
    leptonnumber = 0
    write_event = True
    for particle in event:
        if particle.status == 1:
            if particle.pid in [11, 13, 15]:
                leptonnumber += 1
            elif particle.pid in [-11, -13, -15]:
                leptonnumber -= 1
    for particle in event:
        id = particle.pid
        width = param_card['decay'].get((abs(id),)).value
        mass splitting = param_card.get_value('FRBLOCK', 1)
        if width:
            if id in [8000011, 8000012]:
                tau0 = random.expovariate(width / cst)
                if 0.5 * (1 + math.cos(mass splitting * tau0 / cst)) >= random.random():
                    write_event = (leptonnumber == 0)
                else:
                    write_event = (leptonnumber != 0)
                vtim = tau0 * c
            else:
                vtim = c * random.expovariate(width / cst)
            if vtim > threshold:
                particle.vtim = vtim
    # write this modify event
    if write_event:
        output.write(str(event))
output.write('</LesHouchesEvents>\n')
output.close()