= `pSPSS`: Phenomenological symmetry protected seesaw =

The motivation and implementation of the `pSPSS` is discussed in 

S. 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 [[latex($N_1$)]] and [[latex($N_2$)]] generically appearing in low-scale seesaw models protected by a lepton-number like symmetry.[[BR]]
In the lepton number conserving (LNC) limit the interactions of the symmetry protected seesaw (SPSS) with the Standard Model are

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

The additional lepton number violating (LNV) interactions must be small in order to generate light neutrino masses.[[BR]]
Additionally, they introduce a small mass splitting [[latex($\Delta m$)]] between the two heavy neutrino mass eigen states

[[latex($m_{4/5}^{} = m_M^{} + \frac12 m_M^{} |\theta|^2 \mp \frac12 \Delta m$)]]

where the contribution with the active sterile mixing parameter [[latex($\theta = m_D / m_M$)]] with [[latex($m_D = y_1 v$)]] and [[latex($v\simeq174 \text{ GeV}$)]] appears already in the LNC case.[[BR]]
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 

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

where [[latex($\lambda$)]] captures the damping of the oscillations due to decoherence.[[BR]]
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 [[latex($\theta_\alpha$)]] `theta1`, `theta2`, `theta3`
* Majorana mass [[latex($m_M$)]] `Mmaj`
* Mass splitting [[latex($\Delta M$)]] `deltaM`
* Damping parameter [[latex($\lambda$)]] `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()
}}}