Minimal Higgsless Model or 3-Site Model

Implementation Author

• Neil Christensen
  • Michigan State University
  • neil@…

This implementation is based on an earlier implementation done in collaboration with Alexander Belyaev which can be found ​here.

References

• ​Phys.Rev.D74:075011,2006: This is the first description of the 3-Site Model.
• ​Phys.Rev.D78:031701,2008 : This is a study of the LHC collider phenomenology of the 3-Site Model that uses the earlier !LanHEP version that this implementation was based on.
• ​arXiv:0906.2472 : This is the 2nd FeynRules paper where this implementation was published. It contains a shortened version of the notes below.

Model Files

• 3-Site-v1.3.tgz: 3-Site Model model-files.

Model Implementation

Details about the Minimal Higgsless Model can be found in the reference above. This implementation goes beyond that reference in two ways. It uses exact formulas for all the internal parameters and wavefunctions and the Goldstone bosons and ghosts are worked out in complete detail as is necessary for Feynman gauge. Full details about this implementation can be found in:

• 3-Site-Notes.pdf​: Notes giving the details of the model implementation.
• 3-Site-TeX.pdf​: Automatic !LaTeX output from this model.

Instructions

The 3-Site Model is implemented in both Feynman and unitary gauge. A switch ' FeynmanGauge ' has been created. To switch between the two simply set FeynmanGauge True/False= inside your Mathematica notebook after loading the model but before doing any calculations with it (finding vertices for example.) FeynmanGauge is set to False by default.

Examples

We provide a basic notebook giving examples of how to run the interfaces on this model:

• Example-v1.0.nb.tgz

Interfaces

This model implementation is known to work with the following interfaces:

• CalcHEP/CompHEP Interface
• MadGraph Interface
• Sherpa Interface: This model works with a beta version of Sherpa that will be released later this year.
• FeynArts Interface: A handful of calculations have been done but a thorough validation has not been done.

Validation

Over 200 2→2 processes were run in a variety of ways. First, each process was compared between the original !LanHEP implementation and the current FeynRules implementation. Second, each process was run across multiple monte-carlos including !CalcHEP, !CompHEP, MadGraph and Sherpa. Third, each process was run in two different gauges, namely Feynman gauge (in !CalcHEP and !CompHEP) and in unitary gauge (in !CalcHEP, MadGraph and Sherpa). The cross section was computed for each process and compared to one another. Agreement to better than 1% was found for all processes. The parameters for these calculations were taken as in the model files above. The energies and cuts for these calculations were:

<table align="center" border="0"><thead><tr><td align="left" valign="middle" style="width: 200px"> Particles involved </td><td align="center" valign="middle" style="width: 100px"><strong>&radic;s <br /></strong></td><td align="center" valign="middle" style="width: 100px"><strong>p_T> <br /></strong></td></tr></thead><tbody><tr><td align="left" valign="middle">Only SM<br /></td><td align="center" valign="middle">600GeV </td><td align="center" valign="middle">20GeV <br /></td></tr><tr><td align="left" valign="middle">W',Z' <br /></td><td align="center" valign="middle">1200GeV <br /></td><td align="center" valign="middle">200GeV <br /></td></tr><tr><td align="left" valign="middle"> Heavy Fermion Partners </td><td align="center" valign="middle">10TeV <br /></td><td align="center" valign="middle">2TeV <br /></td></tr></tbody></table>

where "Particles" refers to what particles are involved in the process. The results of these validations can be seen in the following images: <table align="center" border="0"><tbody><tr><td align="center" valign="middle" style="width: 200px">Strong Processes​ <br /></td><td align="center" valign="middle" style="width: 200px">ff&rarr;AW Processes​ <br /></td><td align="center" valign="middle" style="width: 200px">

ll&rarr;ll Processes​ </td></tr><tr><td align="center" valign="middle"> ff&rarr;AA Processes​<br /></td><td align="center" valign="middle"> ff&rarr;ZW Processes​<br /></td><td align="center" valign="middle"> ll&rarr;qq Processes​ </td></tr><tr><td align="center" valign="middle"> ff&rarr;AZ Processes​<br /></td><td align="center" valign="middle"> ff&rarr;WW Processes​<br /></td><td align="center" valign="middle"> ln&rarr;ln Processes​</td></tr><tr><td align="center" valign="middle"> ff&rarr;ZZ Processes​<br /></td><td align="center" valign="middle">ChargedVV&rarr;VVProcesses​</td><td align="center" valign="middle">

ln&rarr;qq Processes​ </td></tr><tr><td> </td><td align="center" valign="middle">

Neutral VV&rarr;VV Processes​ </td><td> </td></tr></tbody></table>

Each of these processes was also run at a single phase space point of the squared amplitude. In this test, only !CalcHEP in Feynman and unitary gauge and MadGraph were used. It is planned to include Sherpa at a later date. The energies were chosen as in the cross section comparison. The angle was chosen to be 73.3 degrees. Agreement to better than 0.1% was found in all cases. Here are images of the results: <table align="center" border="0">

<tbody><tr><td align="center" valign="middle" style="width: 200px">Strong Processes​ <br /></td><td align="center" valign="middle" style="width: 200px">ff&rarr;AW Processes​ <br /></td><td align="center" valign="middle" style="width: 200px">

ll&rarr;ll Processes​ </td></tr><tr><td align="center" valign="middle"> ff&rarr;AA Processes​<br /></td><td align="center" valign="middle"> ff&rarr;ZW Processes​<br /></td><td align="center" valign="middle"> ll&rarr;qq Processes​ </td></tr><tr><td align="center" valign="middle"> ff&rarr;AZ Processes​<br /></td><td align="center" valign="middle">

ff&rarr;WW Processes​ </td><td align="center" valign="middle"> ln&rarr;ln Processes​</td></tr><tr><td align="center" valign="middle">ff&rarr;ZZ Processes​<br /></td><td align="center" valign="middle">ChargedVV&rarr;VVProcesses​</td><td align="center" valign="middle">

ln&rarr;qq Processes​ </td></tr><tr><td> </td><td align="center" valign="middle">

Neutral VV&rarr;VV Processes​ </td><td> </td></tr></tbody>

</table>

These tests were performed with the following versions of the software: <table align="center" border="0"><thead><tr><td> Software </td><td> Version </td></tr></thead><tbody><tr><td>!Mathematica</td><td>7.0.0 <br /></td></tr><tr><td>FeynRules <br /></td><td>1.4.0 <br /></td></tr><tr><td>!CalcHEP <br /></td><td>2.5.3 <br /></td></tr><tr><td>!CompHEP</td><td>4.4.104 <br /></td></tr><tr><td>MadGraph </td><td>4.4.21 <br /></td></tr><tr><td>!Sherpa <br /></td><td>Private development version <br /></td></tr></tbody></table>