HNLs: README.txt

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README
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The files effective_HeavyN_Dirac_v103.fr and effective_HeavyN_Majorana_v103.fr implement in FeynRules the Lagrangian interactions of an extra heavy neutrino N4 or "heavy neutral lepton" (HNL) for the cases in which N4 is a Dirac or Majorana particle, respectively. Here we summarize the main features of the implemented models, and point out their main limitations as well as some caveats. See the companion paper arXiv: 2007.03701 for details.  

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1. Mesons included:
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Both files contain a "Mesons" switch, which is set to True by default:
   Mesons = True;
so that the quarks are removed from the Lagrangian and are instead replaced by the relevant pseudoscalar and vector mesons in the effective theory at low energies. In particular, the model files include the following particle definitions: 
  - charged pseudoscalar mesons: pions, kaons, D, Ds, B, Bc; 
  - neutral pseudoscalar mesons: pi0, K0, eta, eta', D0, B0, B0s; 
  - charged vector mesons: rho, K^*, B^*;
  - neutral vector mesons: rho0, phi and omega. 
These meson interactions are only complete in their interactions with the HNL and other SM leptons. Purely hadronic processes are not implemented. The models are only intended to describe the low-energy, tree level effective interactions of the HNL. 

These effective operators involve on-shell mesons, neutrinos and charged leptons, which appear in external legs only. Thus, the diagrams in which these particles propagate in internal legs may be generated by MadGraph, but should be disregarded. This can be easily done by imposing the process of interest to be computed at leading order in QED, since all effective vertices with mesons are higher order and the presence of an internal leg would at least involve order 4 in QED. Note that processes involving off-shell charged leptons or neutrinos may still be computed if those particles do not enter any effective vertex. 

If the "Mesons" switch is set to False instead, the Lagrangian will rather include the SM quarks so as to describe the HNL phenomenology at higher energies, suitable for colliders. In this case, no meson interactions will be included in the Lagrangian.

Notice that the HNL decays with 3 or more mesons discussed in Section 5.3 of the companion paper cannot be implemented in the model file with the "Mesons" switch set to True. However, these decay widths can be estimated from the corresponding decays to quarks with the "Mesons" switch set to False and applying the correction in Eqs. (5.14) and (5.15) of the paper.  

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2. Free parameters:
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The HNL phenomenology is controlled by two sets of parameters:
  - the HNL mass, which is set by the variable MN4 in the model files. By default, MN4 = 1 GeV. 
  - the set of Yukawa couplings Y_a of the HNL with the SM electron, muon and tau lepton doublets. These are encoded in the parameters theta_a = Y_a vev/MN4, where vev is the Higgs vacuum expectation value. At leading order, they correspond to the HNL mixing with the electron, muon and tau flavours respectively (see eqs. (B.1) and (B.2) in the companion paper for details). These three parameters are defined in terms of their modulus and argument (between 0 and 360 degrees). By default, the three moduli are set to 10^-5, and the three arguments to 0.  

Of course, the mixings and mass of the heavy neutrino may be changed to explore other regions of the parameter space. Please note that, in order to modify the mixings, one should specify the values of the modtheta_i and argtheta_i parameters, and not the theta_i themselves (which will be automatically set accounting to the introduced modulus and argument). 

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3. Neutrino mass eigenstates:
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Both models are written in terms of the physical neutrino mass eigenstates in the Lagrangian (see eqs. (B.1) and (B.2) in the companion paper for details), defined as n = (v1, v2, v3, N4). To clearly distinguish these states from the flavour eigenstates, we gave v1, v2, v3 and N4 new PDG particles codes: 12000,14000,16000 and 18000, respectively.

In this implementation, v1, v2 and v3 are exactly massless in the Dirac case, while v3 has a negligible mass in the Majorana scenario. The mass of N4 corresponds to MN4 to leading order in theta_a. Notice in particular that: 
  - v1, v2, v3 do *not* correspond to the SM neutrino flavour eigenstates (except for the Dirac case in the limit of small theta_a). 
  - v1, v2, v3 do *not* correspond to the neutrino oscillation mass eigenstates either (since most of them are massless at this level of approximation). 
Thus, the correct way of treating them is to add incoherently over v1, v2 and v3 when there are light neutrinos in the final state of any process. If the models are exported to MadGraph5, a convenient way of doing this is by defining the multiparticle v = v1 v2 v3 (and v~ = v1~ v2~ v3~ in the Dirac case). 

An example bash script is provided: "Example_Script.sh". The script shows how to load the Dirac neutrino model and generate 1e6 events for the decay N -> v pi0. It also shows how to change the heavy neutrino parameters at running time, as well as how to define the v multiparticle as outlined above.

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4. Semileptonic meson decays:
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Semileptonic meson decays include form factors that depend on the squared momentum transfer between the parent and daughter mesons, q^2. By default, these are included in the model files evaluated at an average <q^2>, which depends on the HNL mass MN4 (see Appendix D in the companion paper). We find that this is a relatively good approximation, and allows the user to use the model out-of-the-box, without having to modify anything in it. This approximation can be extrapolated to very small values of MN4, and thus can also be used to simulate semileptonic meson decays to the SM massless neutrinos.

However, we also provide a script (written in Python3) that can be used to modify the relevant UFO files so as to correctly implement the full q^2 dependence of the form factors. In order for this to work, the user should first export the model to UFO format, and then run the Python3 script as:
  python3 generate_form_factors_v103.py PATH_TO_UFO_MODEL 
where PATH_TO_UFO_MODEL is the absolute path to the folder containing the UFO model files. We have checked that this feature is compatible with MadGraph5 (v2.6.5 and above); unfortunately, it is not compatible with the MadWidth module. 

Finally, notice that the form factors implemented correspond to those for semileptonic *meson* decays. Therefore, these models should *not* be used to simulate semileptonic decays of charged leptons or the HNL: although the user may see that the corresponding effective vertices exist in the model, the form factor would be incorrect in this case!

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5. How to export the model to UFO format:
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Notice that, in the default Mathematica file distributed with FeynRules, the command used as an example to export a model to UFO format is:
    WriteUFO[LGauge,LHiggs,LFermions,LYukawa,LGhost];
When using our models, this command should be changed to 
    WriteUFO[LagHeavyN];
in order to include all new terms in the Lagrangian. In this case, the "Mesons" switch (see point 1 above) will select the interactions with quarks or mesons as indicated by the user. 

Alternatively, you may use
    WriteUFO[LGauge,LHiggs,LFermions,LYukawa,LGhost,LHadr]; 
to include interactions with mesons, or
    WriteUFO[LGauge,LHiggs,LFermions,LYukawa,LGhost,LQuarks];
to include interactions with quarks (high-energy interactions).

We strongly discourage including both interactions with mesons and quarks simultaneously, as this may lead to double-counting and other inconsistencies.

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6. Other issues and clarifications:
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  - We have checked that these model files work in MadGraph5 v2.6.5 and above. 
  - When generating events, both the renormalization and factorization scales should be set to FIXED.
  - The parameter r defined in the Majorana model file corresponds to the rho parameter in Eq. (B.2) in the companion paper.