= {{{VPolar}}}: The Standard Model at NLO in QCD with helicity-polarized W and Z bosons =

=== Contact Author ===
Richard Ruiz
* Institute of Nuclear Physics Polish Academy of Science (IFJ PAN)
* rruiz AT ifj.edu.pl

In collaboration with: 
M. Javurkova,  R.C.L. de Sá, and J. Sandesara arXiv:xxx.yy [ [#Javurkova 1] ]
D. Buarque Franzosi, O.  Mattelaer, and Sujay Shil arXiv:1912.01725 [ [#BuarqueFranzosi 2] ]




==== Usage resources ====

* For instructions and examples on using the VPolar UFO libraries, see M. Javurkova, et al, arXiv:xxx.yy [ [#Javurkova 1] ]
* For additional background, see also D. Buarque Franzosi, et al, arXiv:1912.01725 [ [#BuarqueFranzosi 2] ]
* See '''Validation''' section below for additional information

* '''Special note:''' this UFO was developed using MG5aMC and calls the '''1L''', '''1T''', and '''1A''' propagators defined in ALOHA (see '''aloha_object.py''' and ''create_aloha.py''). These may be defined differently in other generators. If they are not defined in your favorite generators, they must be added to the ''propagators.py'' file in the {{{VPolar}}} UFO. The file '''particles.py''' must then be updated to reflect the propagator names. R. Ruiz is happy to assist with this.

==== Citation requests ====

* If using the UFO, please cite  , see M. Javurkova, et al, arXiv:xxx.yy [ [#Javurkova 1] ]



== Model Description -- helicity polarization as a Feynman rule ==

The broad idea of the ''helicity polarization as a Feynman rule'' is to treat the helicity-truncated propagator (see [ [#BuarqueFranzosi 2] ] for details) as the Feynman rule for a particle that sits in a definite helicity polarization.
The helicity-truncated propagator is given by
{{{
#!latex
\begin{align}
\Pi_{\mu\nu}^{V\lambda}(q) = \frac{-i\varepsilon_\mu(q,\lambda)\ \varepsilon^*_\nu(q,\lambda)}{q^2-M_V^2 +iM_V\Gamma_V}
\end{align}

and is related to the full propagator by

\begin{align}
    &\Pi_{\mu\nu}^V (q) =
    \frac{-i\left(g_{\mu\nu} - q_\mu q_\nu / M_V^2\right)}{q^2-M_V^2 +iM_V\Gamma_V}\
    \\
    &=\sum_{\lambda\in\{0,\pm1,A\}} 
    \eta_\lambda\ \left(
    \frac{-i\varepsilon_\mu(q,\lambda)\ \varepsilon^*_\nu(q,\lambda)}{q^2-M_V^2 +iM_V\Gamma_V} \right)\ .
\end{align}
}}}

Here, {{{$\eta_\lambda=+1$}}}, unless {{{$\lambda=0$}}} and {{{$V_{\lambda}$}}} is in the t-channel; in that case {{{$\eta_\lambda=-1$}}}.


By making the graphical identification 

[[Image(mgPolar_FeynmanRule.png, 50%)]]

then one can interpret the full propagator in Eq. 3 as the sum of propagators (or interfering graphs) for a collection of particles {{{$V_\lambda$}}}, where each {{{$V_\lambda$}}} has its own propagator.

The {{{VPolar}}} UFOs implement this idea for the W and Z bosons.

== UFO Description and Usage ==






== Studies that have used the above model files ==
* Please email to update this space.

== References ==

[=#Javurkova] [1] M. Javurkova, R. Ruiz, R. Coelho Lopes de Sa, and J. Sandesara, ''Polarized ZZ pairs in gluon fusion and vector boson fusion at the LHC,'' arXiv:xx.yyyy

[=#BuarqueFranzosi] [2] D. Buarque Franzosi, O. Mattelaer, R. Ruiz, S. Shil, ''Automated Predictions from Polarized Matrix Elements,'' '''JHEP''' 2020, 82 (2020) arXiv:1912.01725