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Rivet analyses reference

CMS_2019_I1744604

$t$-channel single top-quark differential cross sections and charge ratios at 13 TeV
Experiment: CMS (LHC)
Inspire ID: 1744604
Status: VALIDATED REENTRANT
Authors:
  • Matthias Komm
References:
  • Eur.Phys.J. C80 (2020) 370, 2020
  • DOI:10.1140/epjc/s10052-020-7858-1
  • arXiv: 1907.08330
Beams: p+ p+
Beam energies: (6500.0, 6500.0) GeV
Run details:
  • t-channel single top quark and antiquark events at $\sqrt{s} = 13$ TeV; charged lepton (including those from intermediate $\tau$ lepton decays) + two jets selected at particle level

Abstract: A measurement is presented of differential cross sections for $t$-channel single top quark and antiquark production in proton-proton collisions at a centre-of-mass energy of 13 TeV by the CMS experiment at the LHC. From a data set corresponding to an integrated luminosity of 35.9 $\textrm{fb}^{-1}$, events containing one muon or electron and two or three jets are analysed. The cross section is measured as a function of the top quark transverse momentum ($p_\textrm{T}$), rapidity, and polarisation angle, the charged lepton $p_\textrm{T}$ and rapidity, and the $p_\textrm{T}$ of the W boson from the top quark decay. In addition, the charge ratio is measured differentially as a function of the top quark, charged lepton, and W boson kinematic observables. The results are found to be in agreement with standard model predictions using various next-to-leading-order event generators and sets of parton distribution functions. Additionally, the spin asymmetry, sensitive to the top quark polarisation, is determined from the differential distribution of the polarisation angle at parton level to be $0.440 \pm 0.070$, in agreement with the standard model prediction. Rivet: This analysis is to be run separately on t-channel single top quark and top antiquark simulation and the resulting outputs have to be merged for caculating the top quark and antiquark cross section sums and ratios. The particle level selection requires: exactly one dressed lepton (electron or muon; including those from intermediate $\tau$ lepton decay) with $p_{T} > 26$ GeV and $|\eta| < 2.4$; exactly two jets with $p_{T} > 40$ GeV and $|\eta| < 4.7$ that are separated from the dressed lepton by $\Delta R>0.4$. A neutrino candidate reconstructed from the summed transverse momenta of all prompt neutrinos using a W boson mass constraint ($m_\textrm{W}=80.4$ GeV). The four momentum of the top quark is found by summing the dressed lepton, neutrino, and the jet momenta, where the latter is chosen such that a top quark mass closest to 172.5 GeV is obtained. The other jet is taken to be the one that recoils against the W boson in production and used for calculating the top quark polarization angle.

Source code: CMS_2019_I1744604.cc
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// -*- C++ -*-
#include "Rivet/Analysis.hh"
#include "Rivet/Projections/FinalState.hh"
#include "Rivet/Projections/FastJets.hh"
#include "Rivet/Projections/ChargedLeptons.hh"
#include "Rivet/Projections/DressedLeptons.hh"
#include "Rivet/Projections/MissingMomentum.hh"
#include "Rivet/Projections/PromptFinalState.hh"
#include "Rivet/Projections/VetoedFinalState.hh"
#include "Rivet/Projections/PartonicTops.hh"

namespace Rivet {


  /// Differential cross sections and charge ratios for 13 TeV t-channel single top-quark production
  class CMS_2019_I1744604 : public Analysis {
  public:

    /// Constructor
    DEFAULT_RIVET_ANALYSIS_CTOR(CMS_2019_I1744604);


    /// @brief Book histograms and initialise projections before the run
    void init() override {

      // final state of all stable particles
      Cut particle_cut = (Cuts::abseta < 5.0) and (Cuts::pT > 0.*MeV);
      FinalState fs(particle_cut);

      // select charged leptons
      ChargedLeptons charged_leptons(fs);
      // select final state photons for dressed lepton clustering
      IdentifiedFinalState photons(fs);
      photons.acceptIdPair(PID::PHOTON);

      // select final state prompt charged leptons
      PromptFinalState prompt_leptons(charged_leptons);
      prompt_leptons.acceptMuonDecays(true);
      prompt_leptons.acceptTauDecays(true);
      // select final state prompt photons
      PromptFinalState prompt_photons(photons);
      prompt_photons.acceptMuonDecays(true);
      prompt_photons.acceptTauDecays(true);

      // Dressed leptons from selected prompt charged leptons and photons
      Cut lepton_cut   = (Cuts::abseta < 2.4) and (Cuts::pT > 26.*GeV);
      DressedLeptons dressed_leptons(
        prompt_photons, prompt_leptons, 0.1,
        lepton_cut, true
      );
      declare(dressed_leptons, "DressedLeptons");

      // Jets
      VetoedFinalState fsForJets(fs);
      fsForJets.addVetoOnThisFinalState(dressed_leptons);
      declare(
        // excludes all neutrinos by default
        FastJets(fsForJets, FastJets::ANTIKT, 0.4),
        "Jets"
      );

      // Neutrinos
      IdentifiedFinalState neutrinos(fs);
      neutrinos.acceptNeutrinos();
      PromptFinalState prompt_neutrinos(neutrinos);
      prompt_neutrinos.acceptMuonDecays(true);
      prompt_neutrinos.acceptTauDecays(true);
      declare(prompt_neutrinos, "Neutrinos");

      // Partonic top for differentiating between t and tbar events
      declare(PartonicTops(),"TopQuarks");


      // book top quark pt histograms
      book(_hist_abs_top_pt, "d13-x01-y01"); // absolute cross section
      book(_hist_norm_top_pt, "d37-x01-y01");  // normalized cross section
      book(_hist_ratio_top_pt, "d59-x01-y01"); // charge ratio
      // temporary histograms of absolute top quark and antiquark cross sections
      book(_hist_t_top_pt, "t_top_pt", refData(13, 1, 1));
      book(_hist_tbar_top_pt, "tbar_top_pt", refData(13, 1, 1));


      // book top quark rapidity histograms
      book(_hist_abs_top_y, "d15-x01-y01"); // absolute cross section
      book(_hist_norm_top_y, "d39-x01-y01"); // normalized cross section
      book(_hist_ratio_top_y, "d61-x01-y01"); // charge ratio
      // temporary histograms of absolute top quark and antiquark cross sections
      book(_hist_t_top_y, "t_top_y", refData(15, 1, 1));
      book(_hist_tbar_top_y, "tbar_top_y", refData(15, 1, 1));


      // book charged lepton pt histograms
      book(_hist_abs_lepton_pt,"d17-x01-y01"); // absolute cross section
      book(_hist_norm_lepton_pt,"d41-x01-y01"); // normalized cross section
      book(_hist_ratio_lepton_pt,"d63-x01-y01");  // charge ratio
      // temporary histograms of absolute top quark and antiquark cross sections
      book(_hist_t_lepton_pt,"t_lepton_pt",refData(17, 1, 1));
      book(_hist_tbar_lepton_pt,"tbar_lepton_pt",refData(17, 1, 1));


      // book charged lepton rapidity histograms
      book(_hist_abs_lepton_y, "d19-x01-y01"); // absolute cross section
      book(_hist_norm_lepton_y, "d43-x01-y01"); // normalized cross section
      book(_hist_ratio_lepton_y, "d65-x01-y01"); // charge ratio
      // temporary histograms of absolute top quark and antiquark cross sections
      book(_hist_t_lepton_y, "t_lepton_y", refData(19, 1, 1));
      book(_hist_tbar_lepton_y, "tbar_lepton_y", refData(19, 1, 1));


      // book W boson pt histograms
      book(_hist_abs_w_pt, "d21-x01-y01"); // absolute cross section
      book(_hist_norm_w_pt, "d45-x01-y01"); // normalized cross section
      book(_hist_ratio_w_pt, "d67-x01-y01"); // charge ratio
      // temporary histograms of absolute top quark and antiquark cross sections
      book(_hist_t_w_pt, "t_w_pt", refData(21, 1, 1));
      book(_hist_tbar_w_pt, "tbar_w_pt", refData(21, 1, 1));


      // book top quark polarization angle histograms
      book(_hist_abs_top_cos, "d23-x01-y01"); // absolute cross section
      book(_hist_norm_top_cos, "d47-x01-y01"); // normalized cross section
      // temporary histograms of absolute top quark and antiquark cross sections
      book(_hist_t_top_cos, "t_top_cos", refData(23, 1, 1));
      book(_hist_tbar_top_cos, "tbar_top_cos", refData(23, 1, 1));

    }


    /// @brief Perform the per-event analysis
    void analyze(const Event& event) override {
      vector<Particle> topQuarks = applyProjection<PartonicTops>(
        event,
        "TopQuarks"
      ).tops();


      // skip events with no partonic top quark
      if (topQuarks.size() != 1) {
        return;
      }

      vector<DressedLepton> dressedLeptons = applyProjection<DressedLeptons>(
        event,
        "DressedLeptons"
      ).dressedLeptons();

      // only analyze events with one dressed lepton (muon or electron)
      if (dressedLeptons.size()!=1) {
        return;
      }

      Cut jet_cut((Cuts::abseta < 4.7) and (Cuts::pT > 40.*GeV));
      vector<Jet> jets = applyProjection<FastJets>(
        event,
        "Jets"
      ).jets(jet_cut);

      // ignore jets that overlap with dressed leptons within dR<0.4
      std::vector<Jet> cleanedJets;
      DeltaRLess dRFct(dressedLeptons[0], 0.4);
      for (const auto& jet: jets) {
        if (not dRFct(jet)) {
          cleanedJets.push_back(jet);
        }
      }

      // select events with exactly two jets
      if (cleanedJets.size() != 2) {
        return;
      }

      vector<Particle> neutrinos = applyProjection<PromptFinalState>(
        event,
        "Neutrinos"
      ).particles();

      // construct missing transverse momentum by summing over all prompt neutrinos
      FourMomentum met(0, 0, 0, 0);
      for (const auto& neutrino: neutrinos) {
        met += neutrino.momentum();
      }

      /* find unknown pz component of missing transverse momentum by imposing
         a W boson mass constraint */
      std::pair<FourMomentum,FourMomentum> nuMomentum = NuMomentum(
        dressedLeptons[0].px(), dressedLeptons[0].py(), dressedLeptons[0].pz(),
        dressedLeptons[0].E(), met.px(), met.py()
      );

      // define the W boson momentum as the sum of the dressed lepton + neutrino
      FourMomentum wboson = nuMomentum.first + dressedLeptons[0].momentum();

      /** construct the pseudo top quark momentum by summing the W boson and
       *  the jet that yields a top quark mass closest to TOPMASS
       */
      FourMomentum topQuark(0, 0, 0, 0);
      int bjetIndex = -1;
      for (size_t i = 0; i < cleanedJets.size(); ++i) {
        const auto& jet = cleanedJets[i];
        FourMomentum topCandidate = jet.momentum() + wboson;
        if (fabs(topQuark.mass() - TOPMASS) > fabs(topCandidate.mass() - TOPMASS)) {
          bjetIndex = i;
          topQuark = topCandidate;
        }
      }

      if (bjetIndex < 0) {
        return;
      }

      // define the jet used to construct the pseudo top quark as the b jet
      Jet bjet = cleanedJets[bjetIndex];

      // define the other jet as the spectator jet
      Jet lightjet = cleanedJets[(bjetIndex + 1) % 2];

      // calculate the cosine of the polarization angle that is defined as the
      //   angle between the charged lepton and the spectator jet in the top
      //   quark rest frame
      LorentzTransform boostToTopFrame = LorentzTransform::mkFrameTransform(topQuark);
      Vector3 ljetInTopFrame = boostToTopFrame.transform(lightjet.momentum()).vector3().unit();
      Vector3 leptonInTopFrame = boostToTopFrame.transform(dressedLeptons[0].momentum()).vector3().unit();
      double polarizationAngle = ljetInTopFrame.dot(leptonInTopFrame);

      // fill the histograms depending on the partonic top quark charge
      if (topQuarks[0].charge() > 0) {
        _hist_t_top_pt->fill(topQuark.pt() / GeV);
        _hist_t_top_y->fill(topQuark.absrapidity());
        _hist_t_lepton_pt->fill(dressedLeptons[0].pt() / GeV);
        _hist_t_lepton_y->fill(dressedLeptons[0].absrapidity());
        _hist_t_w_pt->fill(wboson.pt() / GeV);
        _hist_t_top_cos->fill(polarizationAngle);

      } else {
        _hist_tbar_top_pt->fill(topQuark.pt() / GeV);
        _hist_tbar_top_y->fill(topQuark.absrapidity());
        _hist_tbar_lepton_pt->fill(dressedLeptons[0].pt() / GeV);
        _hist_tbar_lepton_y->fill(dressedLeptons[0].absrapidity());
        _hist_tbar_w_pt->fill(wboson.pt() / GeV);
        _hist_tbar_top_cos->fill(polarizationAngle);
      }
    }


    /// @brief Normalise histograms etc., after the run
    void finalize() override {

      // multiply by 0.5 to average electron/muon decay channels
      scale(_hist_t_top_pt, 0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_t_top_y, 0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_t_lepton_pt, 0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_t_lepton_y, 0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_t_w_pt,0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_t_top_cos, 0.5 * crossSection() / picobarn / sumOfWeights());

      scale(_hist_tbar_top_pt, 0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_tbar_top_y, 0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_tbar_lepton_pt,0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_tbar_lepton_y, 0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_tbar_w_pt, 0.5 * crossSection() / picobarn / sumOfWeights());
      scale(_hist_tbar_top_cos, 0.5 * crossSection() /picobarn / sumOfWeights());

      // populate absolute, normalized, and ratio histograms once top quark and
      // antiquark histograms have been populated
      if (_hist_t_top_pt->numEntries() > 0 and _hist_tbar_top_pt->numEntries() > 0) {
        fillAbsHist(_hist_abs_top_pt, _hist_t_top_pt, _hist_tbar_top_pt);
        fillNormHist(_hist_norm_top_pt, _hist_t_top_pt, _hist_tbar_top_pt);
        divide(_hist_t_top_pt, _hist_abs_top_pt, _hist_ratio_top_pt);

        fillAbsHist(_hist_abs_top_y, _hist_t_top_y, _hist_tbar_top_y);
        fillNormHist(_hist_norm_top_y, _hist_t_top_y, _hist_tbar_top_y);
        divide(_hist_t_top_y, _hist_abs_top_y, _hist_ratio_top_y);

        fillAbsHist(_hist_abs_lepton_pt, _hist_t_lepton_pt, _hist_tbar_lepton_pt);
        fillNormHist(_hist_norm_lepton_pt, _hist_t_lepton_pt, _hist_tbar_lepton_pt);
        divide(_hist_t_lepton_pt, _hist_abs_lepton_pt, _hist_ratio_lepton_pt);

        fillAbsHist(_hist_abs_lepton_y, _hist_t_lepton_y, _hist_tbar_lepton_y);
        fillNormHist(_hist_norm_lepton_y, _hist_t_lepton_y, _hist_tbar_lepton_y);
        divide(_hist_t_lepton_y, _hist_abs_lepton_y, _hist_ratio_lepton_y);

        fillAbsHist(_hist_abs_w_pt, _hist_t_w_pt, _hist_tbar_w_pt);
        fillNormHist(_hist_norm_w_pt, _hist_t_w_pt, _hist_tbar_w_pt);
        divide(_hist_t_w_pt, _hist_abs_w_pt, _hist_ratio_w_pt);

        fillAbsHist(_hist_abs_top_cos, _hist_t_top_cos, _hist_tbar_top_cos);
        fillNormHist(_hist_norm_top_cos, _hist_t_top_cos, _hist_tbar_top_cos);
      }
    }

    //@}


  private:

    // for reconstruction only
    const double WMASS = 80.399;
    const double TOPMASS = 172.5;

    // Top quark pt histograms and ratio
    Histo1DPtr _hist_abs_top_pt;
    Histo1DPtr _hist_norm_top_pt;
    Scatter2DPtr _hist_ratio_top_pt;
    Histo1DPtr _hist_t_top_pt;
    Histo1DPtr _hist_tbar_top_pt;

    // Top quark rapidity histograms and ratio
    Histo1DPtr _hist_abs_top_y;
    Histo1DPtr _hist_norm_top_y;
    Scatter2DPtr _hist_ratio_top_y;
    Histo1DPtr _hist_t_top_y;
    Histo1DPtr _hist_tbar_top_y;

    // Charged lepton pt histograms and ratio
    Histo1DPtr _hist_abs_lepton_pt;
    Histo1DPtr _hist_norm_lepton_pt;
    Scatter2DPtr _hist_ratio_lepton_pt;
    Histo1DPtr _hist_t_lepton_pt;
    Histo1DPtr _hist_tbar_lepton_pt;

    // Charged lepton rapidity histograms and ratio
    Histo1DPtr _hist_abs_lepton_y;
    Histo1DPtr _hist_norm_lepton_y;
    Scatter2DPtr _hist_ratio_lepton_y;
    Histo1DPtr _hist_t_lepton_y;
    Histo1DPtr _hist_tbar_lepton_y;

    // W boson pt histograms and ratio
    Histo1DPtr _hist_abs_w_pt;
    Histo1DPtr _hist_norm_w_pt;
    Scatter2DPtr _hist_ratio_w_pt;
    Histo1DPtr _hist_t_w_pt;
    Histo1DPtr _hist_tbar_w_pt;

    // Top quark polarization angle histograms
    Histo1DPtr _hist_abs_top_cos;
    Histo1DPtr _hist_norm_top_cos;
    Histo1DPtr _hist_t_top_cos;
    Histo1DPtr _hist_tbar_top_cos;


    /// @brief helper function to fill absolute cross section histograms
    void fillAbsHist(
      Histo1DPtr& hist_abs, const Histo1DPtr& hist_t,
      const Histo1DPtr& hist_tbar
    ) {
      (*hist_abs) += (*hist_t);
      (*hist_abs) += (*hist_tbar);
    }

    /// @brief helper function to fill normalized cross section histograms
    void fillNormHist(
      Histo1DPtr& hist_norm, const Histo1DPtr& hist_t,
      const Histo1DPtr& hist_tbar
    ) {
      (*hist_norm) += (*hist_t);
      (*hist_norm) += (*hist_tbar);
      hist_norm->normalize();
    }


    /** @brief helper function to solve for the unknown neutrino pz momentum
     *  using a W boson mass constraint
     */
    std::pair<FourMomentum,FourMomentum> NuMomentum(
      double pxlep, double pylep, double pzlep,
      double elep, double metpx, double metpy
    ) {

      FourMomentum result(0, 0, 0, 0);
      FourMomentum result2(0, 0, 0, 0);

      double misET2 = (metpx * metpx + metpy * metpy);
      double mu = (WMASS * WMASS) / 2 + metpx * pxlep + metpy * pylep;
      double a  = (mu * pzlep) / (elep * elep - pzlep * pzlep);
      double a2 = std::pow(a, 2);

      double b  = (std::pow(elep, 2.) * (misET2) - std::pow(mu, 2.))
                  / (std::pow(elep, 2) - std::pow(pzlep, 2));

      double pz1(0), pz2(0), pznu(0), pznu2(0);

      FourMomentum p4W_rec;
      FourMomentum p4b_rec;
      FourMomentum p4Top_rec;
      FourMomentum p4lep_rec;

      p4lep_rec.setXYZE(pxlep, pylep, pzlep, elep);

      FourMomentum p40_rec(0, 0, 0, 0);

      // there are two real solutions
      if (a2 - b > 0 ) {
        double root = sqrt(a2 - b);
        pz1 = a + root;
        pz2 = a - root;

        pznu = pz1;
        pznu2 = pz2;

        // first solution is the one with the smallest |pz|
        if (fabs(pz1) > fabs(pz2)) {
          pznu = pz2;
          pznu2 = pz1;
        }

        double Enu = sqrt(misET2 + pznu * pznu);
        double Enu2 = sqrt(misET2 + pznu2 * pznu2);

        result.setXYZE(metpx, metpy, pznu, Enu);
        result2.setXYZE(metpx, metpy, pznu2, Enu2);

      } else {

        // there are only complex solutions; set pz=0 and vary px/py such
        // that mT=mW while keeping px^2+py^2 close to the original pT^2
        double ptlep = sqrt(pxlep * pxlep + pylep * pylep);

        double EquationA = 1;
        double EquationB = -3 * pylep * WMASS / (ptlep);

        double EquationC = WMASS * WMASS * (2 * pylep * pylep) / (ptlep * ptlep)
                           + WMASS * WMASS
                           - 4 * pxlep * pxlep * pxlep * metpx / (ptlep * ptlep)
                           - 4 * pxlep * pxlep * pylep * metpy / (ptlep * ptlep);

        double EquationD = 4 * pxlep * pxlep * WMASS * metpy / (ptlep)
                           - pylep * WMASS * WMASS * WMASS / ptlep;

        vector<double> solutions = EquationSolve(EquationA, EquationB, EquationC, EquationD);

        vector<double> solutions2 = EquationSolve(EquationA, -EquationB, EquationC, -EquationD);

        double deltaMin = 14000 * 14000;
        double zeroValue = -WMASS * WMASS / (4 * pxlep);
        double minPx = 0;
        double minPy = 0;

        for ( size_t i = 0; i < solutions.size(); ++i) {
          if (solutions[i] < 0) continue;
          double p_x = (solutions[i] * solutions[i] - WMASS * WMASS) / (4 * pxlep);
          double p_y = (WMASS * WMASS * pylep
                        + 2 * pxlep * pylep * p_x
                        - WMASS * ptlep * solutions[i]
                        ) / (2 * pxlep * pxlep);
          double Delta2 = (p_x - metpx) * (p_x - metpx) + (p_y - metpy) * (p_y - metpy);

          if (Delta2 < deltaMin && Delta2 > 0) {
            deltaMin = Delta2;
            minPx = p_x;
            minPy = p_y;
          }

        }

        for ( size_t i = 0; i < solutions2.size(); ++i) {
          if (solutions2[i] < 0) continue;
          double p_x = (solutions2[i] * solutions2[i] - WMASS * WMASS) / (4 * pxlep);
          double p_y = (WMASS * WMASS * pylep
                        + 2 * pxlep * pylep * p_x
                        + WMASS * ptlep * solutions2[i]
                       ) / (2 * pxlep * pxlep);
          double Delta2 = (p_x - metpx) * (p_x - metpx) + (p_y - metpy) * (p_y - metpy);
          if (Delta2 < deltaMin && Delta2 > 0) {
            deltaMin = Delta2;
            minPx = p_x;
            minPy = p_y;
          }
        }

        double pyZeroValue = (WMASS * WMASS * pxlep + 2 * pxlep * pylep * zeroValue);
        double delta2ZeroValue = (zeroValue - metpx) * (zeroValue - metpx)
                                 + (pyZeroValue - metpy) * (pyZeroValue - metpy);

        if (deltaMin == 14000 * 14000) {
          return std::make_pair(result, result2);
        }

        if (delta2ZeroValue < deltaMin) {
          deltaMin = delta2ZeroValue;
          minPx = zeroValue;
          minPy = pyZeroValue;
        }


        double mu_Minimum = (WMASS * WMASS) / 2 + minPx * pxlep + minPy * pylep;
        double a_Minimum  = (mu_Minimum * pzlep) /
                            (elep * elep - pzlep * pzlep);
        pznu = a_Minimum;

        double Enu = sqrt(minPx * minPx + minPy * minPy + pznu * pznu);
        result.setXYZE(minPx, minPy, pznu , Enu);
      }
      return std::make_pair(result, result2);
    }


    /// @brief helper function find root of the cubic equation a*x^3 + b*x^2 + c*x + d = 0
    std::vector<double> EquationSolve(
      double a, double b,
      double c, double d
    ) {
      std::vector<double> result;

      std::complex<double> x1;
      std::complex<double> x2;
      std::complex<double> x3;

      double q = (3 * a * c - b * b) / (9 * a * a);
      double r = (9 * a * b * c - 27 * a * a * d - 2 * b * b * b
                 ) / (54 * a * a * a);
      double Delta = q * q * q + r * r;

      std::complex<double> s;
      std::complex<double> t;

      double rho = 0;
      double theta = 0;

      if (Delta <= 0) {
        rho = sqrt(-(q * q * q));

        theta = acos(r / rho);

        s = std::polar<double>(sqrt(-q), theta / 3.0);
        t = std::polar<double>(sqrt(-q), -theta / 3.0);
      }

      if (Delta > 0) {
        s = std::complex<double>(cbrt(r + sqrt(Delta)), 0);
        t = std::complex<double>(cbrt(r - sqrt(Delta)), 0);
      }

      std::complex<double> i(0, 1.0);


      x1 = s + t + std::complex<double>(-b / (3.0 * a), 0);

      x2 = (s + t) * std::complex<double>(-0.5, 0)
           - std::complex<double>(b / (3.0 * a), 0)
           + (s - t) * i * std::complex<double>(sqrt(3) / 2.0, 0);

      x3 = (s + t) * std::complex<double>(-0.5, 0)
           - std::complex<double>(b / (3.0 * a), 0)
           - (s - t) * i * std::complex<double>(sqrt(3) / 2.0, 0);

      if (fabs(x1.imag()) < 0.0001) result.push_back(x1.real());
      if (fabs(x2.imag()) < 0.0001) result.push_back(x2.real());
      if (fabs(x3.imag()) < 0.0001) result.push_back(x3.real());

      return result;
    }

  };


  DECLARE_RIVET_PLUGIN(CMS_2019_I1744604);

}