The recent theoretical progress in uncovering the phase diagram of strongly coupled theories, has lead us to identify a number of models which can dynamically break the electroweak gauge symmetry and pass the precision tests.

The simplest of these models are:

- Minimal Walking Technicolor (MWT) – which include a new strong sector based on the SU(2) gauge group with two Dirac flavors in the adjoint representation;

- Next to Minimal Walking Technicolor (NMWT) – similar to MWT but based on the SU(3) gauge theory with two flavors in the two-index symmetric representation.

To close the gap between the theoretical community and the experimentalists working at particle colliders, we provide a public implementation of our models, to investigate their phenomenology.

We have implemented MWT and NMWT using the FeynRules package. We also provide the MadGraph and CalcHEP interfaces.

- Matti Järvinen ( mjarvine @ physics.uoc.gr )
- Tuomas Hapola ( hapola @ cp3.sdu.dk )
- Eugenio Del Nobile ( delnobile @ cp3.sdu.dk )
- Claudio Pica ( pica @ cp3.sdu.dk )

Jeppe R. Andersen, Oleg Antipin and Marco Nardecchia helped in the validation.

An earlier implementation of this model on LanHEP was written by A. Belyaev, M. Frandsen, R. Foadi, and M. Järvinen.

Also see the HEPMDB database.

Our implementation makes use of the effective low-energy model containing scalars, pseudoscalars, vector mesons and other fields predicted by the models. The implemented model is the simplest one, which contains only the lightest composite states, which are expected to be the most important ones for collider phenomenology. These are the composite Higgs and the vector and axial spin-one resonances. For these states the effective theories of MWT and NMWT coincide.

The most relevant references for this model implementation are:

- Phys. Rev. D 71, 051901 (2005) http://arxiv.org/abs/hep-ph/0405209 – F. Sannino and K. Tuominen,
*Orientifold Theory Dynamics and Symmetry Breaking*. This article introduces MWT and NMWT. Note that the original name was*Techniorientifold*. - Phys. Rev. D 76, 055005 ( 2007) http://arxiv.org/abs/0706.1696 – R. Foadi, M.T. Frandsen, T. A. Ryttov, F. Sannino,
*Minimal Walking Technicolor: Set Up for Collider Physics*. This article derives the effective theory for MWT. - Phys. Rev. D 79, 035006 (2009) http://arxiv.org/abs/0809.0793 – A. Belyaev, R. Foadi, M.T. Frandsen, M. Jarvinen, A. Pukhov, F. Sannino,
*Technicolor Walks at the LHC*. This article presents the Lagrangian used in this implementation, and analyses LHC phenomenology by using the earlier LanHEP implementation.

See also:

- Phys. Lett. B597:89-93,2004 http://arxiv.org/abs/hep-ph/0406200 – Deog Ki Hong, Stephen D.H. Hsu, F. Sannino,
*Composite Higgs from higher representations*. - Phys. Rev. D72:055001, 2005 http://arxiv.org/abs/hep-ph/0505059 – D.D. Dietrich, F. Sannino, K. Tuominen
*Light composite Higgs from higher representations versus electroweak precision measurements: Predictions for CERN LHC*. - Phys. Rev. D 75, 085018 (2007) http://arxiv.org/abs/hep-ph/0611341 – D. D. Dietrich and F. Sannino,
*Conformal window of SU(N) gauge theories with fermions in higher dimensional representations*. Note that the original name was*Walking in the SU(N)*. - For the construction at the Lagrangian level of the terms involving the space-time epsilon tensor – representing the correct generalization of the Wess-Zumino-Witten topological term – involving massive spin one fields see Acta Phys. Polon. B40:3533-3743, 2009; http://arxiv.org/abs/0911.0931 – F. Sannino,
*Conformal Dynamics for TeV Physics and Cosmology*.

- MWT_101030.fr: the main FeynRules file (last version: 30/10/2010)
- MWT_101030.nb: an example notebook to use the FeynRules file.

- CalcHEP: MWT_ch.zip
- MadGraph: MWT_mg.zip
- MWT_Calculator.zip: the calculator needed in MadGraph in order to generate the param_card.dat for a different set of model parameters.

FeynArts and Sherpa interfaces have not been tested.

The model file is loaded as usual. The implementation supports only unitary gauge. The standard model section has only Cabibbo mixing, and the electron and the muon, as well as the up, down and strange quarks, are taken to be massless.

The calculator is needed by the MadGraph implementation in order to change the parameters of the model. The directory is provided by a README file with the instructions on the usage.

The model file implements a (linear) effective theory for the spin-zero and spin-one sectors in technicolor, with the minimal SU(2)_{L} x SU(2)_{R} -> SU(2)_{V} chiral symmetry breaking pattern. The strong technicolor interactions are linked to the electroweak sector as stipulated by the electroweak gauge transformations of the techniquarks. The modifications to the effective theory due to the electroweak interactions are mostly small. The composite scalar sector contains the composite Higgs boson and a triplet of massless technipions, which are eaten by the heavy gauge boson Z and W. The Higgs is expected to be relatively light (mass less than 500 GeV). We also have vector and axial spin-one triplets, which mix with each other and with the electroweak gauge bosons.

In addition to the standard model fermions, we thus have the following new particles:

- Composite Higgs scalar H
- Neutral heavy vector R
_{1}^{0} - Charged heavy vectors R
_{1}^{+}, R_{1}^{-} - Neutral heavy vector R
_{2}^{0} - Charged heavy vectors R
_{2}^{+}, R_{2}^{-}

The numbering convention for the heavy spin-one states is such that R_{1} is always the lightest one. When the mass scale is below 1 TeV, R_{1} (R_{2}) has larger component of the axial (vector) spin-one composite state than of the vector (axial) state. When masses are increased to about 2 TeV, the situation is reversed.

Using the effective theory introduces several new coupling constants. These can be constrained by linking to the underlying gauge theory via the Weinberg sum rules and the definition of the electroweak S parameter. After taking into account the Weinberg sum rules, the free parameters can be expressed in terms of:

- MA: The spin-one mass scale. More precisely, the mass of the axial spin-one state in the limit where the electroweak interactions are turned off. Allowed range is from about 500 GeV to about 3 TeV, depending on the values of other parameters.
- gt: The effective strength of technicolor interactions. Parametrizes the corrections of the electroweak interactions to the technicolor sector, which are typically O(g/gt), with g being the weak coupling constant. In particular, the mixing of the composite spin-one states with the electroweak gauge bosons, and therefore also the coupling of the composite spin-one states to the standard model fermions, is O(g/gt). Allowed values range from about 1 to about 10.
- S: The (contribution of the lowest spin-one states to the) S parameter. Recommended values come from naive estimates of the S parameter (calculation of techniquark loops), which gives S=0.15 for MWT and S=0.3 for NMWT.
- MH: The mass of the composite Higgs boson.
- rs: Parametrizes the couplings of the Higgs to the composite spin-one states. Expected to be of order 1.

The implementation of the following processes through the FeynRules interface was cross-checked with the already existing implementation in LanHEP (see references):

- pp>jj at 14 TeV
- pp>mu+mu- at 14 TeV

Among others, also the processes

- e+e->mu+mu- at 14 TeV
- uu~>hZ at 14 TeV

were cross-checked between the MadGraph and CalcHEP implementations.

Standard model processes like

- gg>gg
- ug>ug
- ud>us

were checked by comparing to the standard model implementation both in CalcHEP and in MadGraph.

Furthermore, the matrix elements generated for

- l+l->r1,r2>l+l-
- uu~>r1,r2>l+l-
- uu~>r1>dd~
- l+l->r1>uu~g
- l+vl>W->W-h and l-vl>W+>W+h

were checked by hand for a few phase space points.

Values of constrained parameters (e.g. rotation matrices) in CalcHEP, some randomly chosen Lagrangian terms, as well as all the partial widths for the decay of any of the composite states to any two particles, were checked against the earlier LanHEP implementation.

- HiggsEffective.fr is an extension introducing the effective coupling HGG (Higgs + two gluons) arising at loop level.
- MWT_HEC_Calculator.zip: the MWT parameters calculator for MadGraph extended so to include the Higgs effective coupling to two gluons.

HiggsEffective.fr has to be loaded together with the FeynRules file for the MWT model, i.e. using the command `LoadModel[HiggsEffective.fr, MWT.fr]` where MWT.fr indicates the name of the MWT model file.

The value of the effective coupling is a function of the other parameters of the model, that has been calculated in full generality at the one loop order. For practical reasons though the effective coupling is declared as an external parameter. Anyway its correct value doesn’t need to be introduced in the FeynRules file but it will be computed by the modified calculator for MadGraph. Therefore this extension can be currently used only with MadGraph.

Note that the effective coupling is independent of the new dynamics because the Higgs-fermions couplings are not modified in the MWT implementation with respect to the SM.