= {{{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:2401.17365 [ [#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:2401.17365 [ [#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:2401.17365 [ [#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, 40%)]]

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 ==


==== ZPolar ====
* The {{{SM_Loop_ZPolar}}} UFO contains the SM Lagrangian and includes QCD counter terms up to NLO in QCD. (This means it can be used for both tree-level computations and loop-level computations up to one loop in QCD.) Importantly, the SM Z boson has been replaced with the longitudinal state {{{Z0}}}, the transverse state {{{ZT}}}, the auxiliary state {{{ZA}}}, and the unpolarized state {{{ZX}}}. The state {{{ZX}}} is essentially the original Z boson. 
 - The PIDs are: 23 for {{{ZX}}}, 230 for {{{Z0}}}, 231 for {{{ZT}}}, and 232 for {{{ZA}}}
 - The W boson is '''NOT''' polarized in this UFO
 - ~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/VPolarization/SM_Loop_ZPolar.tgz ./
 - ~/Path $ tar -zxvf SM_Loop_ZPolar.tgz

==== WPolar ====
* The {{{SM_Loop_WPolar}}} UFO is like {{{SM_Loop_ZPolar}}} but for the W boson
 - The PIDs are: +24 for {{{WX^+}}}, +240 for {{{W0^+}}}, +241 for {{{WT^+}}}, and +242 for {{{WA^+}}}
 - For the {{{W^-}}} fields, use {{{-PID}}}
 - The Z boson is '''NOT''' polarized in this UFO
 - ~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/VPolarization/SM_Loop_WPolar.tgz ./
 - ~/Path $ tar -zxvf SM_Loop_WPolar.tgz


==== VPolar ====
* The {{{SM_Loop_VPolar}}} UFO is like {{{SM_Loop_WPolar}}} and {{{SM_Loop_ZPolar}}} but for both the W and the Z
 - ~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/VPolarization/SM_Loop_VPolar.tgz ./
 - ~/Path $ tar -zxvf SM_Loop_VPolar.tgz

==== LO UFOs ====
* The {{{SM_VPolar_XLO}}}, {{{SM_WPolar_XLO}}}, and {{{SM_ZPolar_XLO}}} UFOs are like the three UFOs above but '''DO NOT CONTAIN''' QCD counter terms
 - ~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/VPolarization/SM_VPolar_XLO.tgz ./
 - ~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/VPolarization/SM_WPolar_XLO.tgz ./
 - ~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/VPolarization/SM_ZPolar_XLO.tgz ./

==== Regenerating NLO UFOs ====
Computing the counter terms ({{{WriteCT[]}}} in NLOCT) is incredibly memory intensive. {{{WPolar}}} and {{{ZPolar}}} each requires 10+ GB of RAM. {{{VPolar}}} requires 20ish GB of RAM. The output of NLOCT for {{{VPolar}}}  ({{{2023_0201_SM_Loop_VPolar_FA.nlo}}}) is provided for convenience (see attachments).

==== Usage in MG5aMC@NLO ====
For usage in MadGraph5aMC@NLO, see the appendix of M. Javurkova, et al, arXiv:2401.17365 [ [#Javurkova 1] ].



== 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:2401.17365

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