Rivet analyses

title: BESIII_2023_I2637702

Transverse $\Lambda$ polarization in $e^+e^-\to\Lambda^0\bar\Lambda^0$ for $\sqrt{s}=3.68\to3.71\,$GeV

Experiment: BESIII (BEPC)

Inspire ID: 2637702

Status: VALIDATED NOHEPDATA

Authors: - Peter Richardson

References: - arXiv: 2303.00271

Beams: e+ e-

Beam energies: (1.8, 1.8); (1.8, 1.8); (1.8, 1.8); (1.8, 1.8); (1.8, 1.8); (1.8, 1.8); (1.9, 1.9)GeV

Run details: - e+ e- -> hadrons

Transverse $\Lambda$ polarization in $e^+e^-\to\Lambda^0\bar\Lambda^0$ for $\sqrt{s}=3.68\to3.71\,$GeV.

Source code:BESIII_2023_I2637702.cc

```c++ // -- C++ --

include "Rivet/Analysis.hh"

include "Rivet/Projections/Beam.hh"

include "Rivet/Projections/FinalState.hh"

include "Rivet/Projections/UnstableParticles.hh"

namespace Rivet {

/// @brief e+ e- > Lambda0 Lambdabar0 class BESIII_2023_I2637702 : public Analysis { public:

/// Constructor
RIVET_DEFAULT_ANALYSIS_CTOR(BESIII_2023_I2637702);


/// @name Analysis methods
/// @{

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

  // Initialise and register projections
  declare(Beam(), "Beams");
  declare(UnstableParticles(Cuts::abspid == 3122), "UFS");
  declare(FinalState(), "FS");

  // Book histograms
  if (!inRange(sqrtS(), 3.68 * GeV, 3.71 * GeV, RangeBoundary::CLOSED, RangeBoundary::CLOSED)) {
    raiseBeamErrorIf(_sqs.empty());
  }
  for (size_t ih = 0; ih < 3; ++ih) {
    book(_e[ih], 1, 1, 1 + ih);
    book(_est[ih], 2, 1, 1 + ih);
  }
  vector<string> edges = _est[0].binning().edges<0>();
  edges.push_back("");
  for (const string& en : edges) {
    book(_h[en + "T2"], "TMP/T2_" + en, 20, -1.0, 1.0);
    book(_h[en + "T3"], "TMP/T3_" + en, 20, -1.0, 1.0);
    book(_h[en + "cThetaL"], "TMP/cThetaL_" + en, 20, -1.0, 1.0);
    if (en != "") {
      const double eval = stod(en);
      if (isCompatibleWithSqrtS(eval)) _sqs = en;
    }
  }
}


void findChildren(const Particle& p, map<long, int>& nRes, int& ncount) const {
  for (const Particle& child : p.children()) {
    if (child.children().empty()) {
      nRes[child.pid()] -= 1;
      --ncount;
    }
    else {
      findChildren(child, nRes, ncount);
    }
  }
}

/// Perform the per-event analysis
void analyze(const Event& event) {
  // get the axis, direction of incoming electron
  const ParticlePair& beams = apply<Beam>(event, "Beams").beams();
  Vector3 axis;
  if (beams.first.pid() > 0)
    axis = beams.first.mom().p3().unit();
  else
    axis = beams.second.mom().p3().unit();
  // types of final state particles
  const FinalState& fs = apply<FinalState>(event, "FS");
  map<long, int> nCount;
  int ntotal(0);
  for (const Particle& p : fs.particles()) {
    nCount[p.pid()] += 1;
    ++ntotal;
  }
  // loop over lambda0 baryons
  const UnstableParticles& ufs = apply<UnstableParticles>(event, "UFS");
  Particle Lambda, LamBar;
  bool matched(false);
  for (const Particle& p : ufs.particles(Cuts::abspid == 3122)) {
    if (p.children().empty()) continue;
    map<long, int> nRes = nCount;
    int ncount = ntotal;
    findChildren(p, nRes, ncount);
    matched = false;
    // check for antiparticle
    for (const Particle& p2 : ufs.particles(Cuts::pid == -p.pid())) {
      if (p2.children().empty()) continue;
      map<long, int> nRes2 = nRes;
      int ncount2 = ncount;
      findChildren(p2, nRes2, ncount2);
      if (ncount2 == 0) {
        matched = true;
        for (const auto& val : nRes2) {
          if (val.second != 0) {
            matched = false;
            break;
          }
        }
        // found baryon and antibaryon
        if (matched) {
          if (p.pid() > 0) {
            Lambda = p;
            LamBar = p2;
          }
          else {
            Lambda = p2;
            LamBar = p;
          }
          break;
        }
      }
    }
    if (matched) break;
  }
  if (!matched) vetoEvent;
  Particle proton;
  matched = false;
  for (const Particle& p : Lambda.children()) {
    if (p.pid() == 2212) {
      matched = true;
      proton = p;
    }
    else if (p.pid() == PID::PHOTON) {
      vetoEvent;
    }
  }
  if (!matched) vetoEvent;
  Particle baryon;
  matched = false;
  for (const Particle& p : LamBar.children()) {
    if (p.pid() == -2212) {
      baryon = p;
      matched = true;
    }
    else if (p.pid() == PID::PHOTON) {
      vetoEvent;
    }
  }
  if (!matched) vetoEvent;
  // boost to the Lambda rest frame
  LorentzTransform boost1 = LorentzTransform::mkFrameTransformFromBeta(Lambda.mom().betaVec());
  Vector3 e1z = Lambda.mom().p3().unit();
  Vector3 e1y = e1z.cross(axis).unit();
  Vector3 e1x = e1y.cross(e1z).unit();
  Vector3 axis1 = boost1.transform(proton.mom()).p3().unit();
  double n1x(e1x.dot(axis1)), n1y(e1y.dot(axis1)), n1z(e1z.dot(axis1));
  // boost to the Lambda bar
  LorentzTransform boost2 = LorentzTransform::mkFrameTransformFromBeta(LamBar.mom().betaVec());
  Vector3 axis2 = boost2.transform(baryon.mom()).p3().unit();
  double n2x(e1x.dot(axis2)), n2z(e1z.dot(axis2));
  double cosL = axis.dot(Lambda.mom().p3().unit());
  double sinL = sqrt(1. - sqr(cosL));
  double T2 = -sinL * cosL * (n1x * n2z + n1z * n2x);
  double T3 = -sinL * cosL * n1y;
  _h[_sqs + "T2"]->fill(cosL, T2);
  _h[_sqs + "T3"]->fill(cosL, T3);
  _h[_sqs + "cThetaL"]->fill(cosL);
  _h["T2"]->fill(cosL, T2);
  _h["T3"]->fill(cosL, T3);
  _h["cThetaL"]->fill(cosL);
}

pair<double, pair<double, double>> calcAlpha0(const Histo1DPtr& hist) const {
  if (hist->numEntries() == 0.) return make_pair(0., make_pair(0., 0.));
  double d = 3. / (pow(hist->xMax(), 3) - pow(hist->xMin(), 3));
  double c = 3. * (hist->xMax() - hist->xMin()) / (pow(hist->xMax(), 3) - pow(hist->xMin(), 3));
  double sum1(0.), sum2(0.), sum3(0.), sum4(0.), sum5(0.);
  for (const auto& bin : hist->bins()) {
    double Oi = bin.sumW();
    if (Oi == 0.) continue;
    double a = d * (bin.xMax() - bin.xMin());
    double b = d / 3. * (pow(bin.xMax(), 3) - pow(bin.xMin(), 3));
    double Ei = bin.errW();
    sum1 += a * Oi / sqr(Ei);
    sum2 += b * Oi / sqr(Ei);
    sum3 += sqr(a) / sqr(Ei);
    sum4 += sqr(b) / sqr(Ei);
    sum5 += a * b / sqr(Ei);
  }
  // calculate alpha
  double alpha = (-c * sum1 + sqr(c) * sum2 + sum3 - c * sum5) / (sum1 - c * sum2 + c * sum4 - sum5);
  // and error
  double cc = -pow((sum3 + sqr(c) * sum4 - 2 * c * sum5), 3);
  double bb = -2 * sqr(sum3 + sqr(c) * sum4 - 2 * c * sum5) * (sum1 - c * sum2 + c * sum4 - sum5);
  double aa = sqr(sum1 - c * sum2 + c * sum4 - sum5)
      * (-sum3 - sqr(c) * sum4 + sqr(sum1 - c * sum2 + c * sum4 - sum5) + 2 * c * sum5);
  double dis = sqr(bb) - 4. * aa * cc;
  if (dis > 0.) {
    dis = sqrt(dis);
    return make_pair(alpha, make_pair(0.5 * (-bb + dis) / aa, -0.5 * (-bb - dis) / aa));
  }
  else {
    return make_pair(alpha, make_pair(0., 0.));
  }
}

pair<double, double> calcCoeff(size_t imode, const Histo1DPtr& hist) const {
  if (hist->numEntries() == 0.) return make_pair(0., 0.);
  double sum1(0.), sum2(0.);
  for (const auto& bin : hist->bins()) {
    double Oi = bin.sumW();
    if (Oi == 0.) continue;
    double ai(0.), bi(0.);
    if (imode == 0) {
      bi = (pow(1. - sqr(bin.xMin()), 1.5) - pow(1. - sqr(bin.xMax()), 1.5)) / 3.;
    }
    else if (imode >= 2 && imode <= 4) {
      bi = (pow(bin.xMin(), 3) * (-5. + 3. * sqr(bin.xMin()))
            + pow(bin.xMax(), 3) * (5. - 3. * sqr(bin.xMax())))
          / 15.;
    }
    else {
      assert(false);
    }
    double Ei = bin.errW();
    sum1 += sqr(bi / Ei);
    sum2 += bi / sqr(Ei) * (Oi - ai);
  }
  return make_pair(sum2 / sum1, sqrt(1. / sum1));
}

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

  const double aLambda = 0.754;
  vector<string> edges = _est[0].binning().edges<0>();
  edges.push_back("");
  scale(_h, crossSectionPerEvent());
  for (const string& en : edges) {

    const double wsum = _h[en + "cThetaL"]->sumW();
    normalize(_h[en + "cThetaL"]);
    scale(_h[en + "T2"], 1.0 / wsum);
    scale(_h[en + "T3"], 1.0 / wsum);

    pair<double, pair<double, double>> alpha0 = calcAlpha0(_h[en + "cThetaL"]);
    pair<double, pair<double, double>> R;
    double tau = sqr(sqrtS() / (2 * 1.115683));
    R.first = sqrt(tau * (1 - alpha0.first) / (1 + alpha0.first));
    R.second.first = R.second.second = R.first / (1. - sqr(alpha0.first));

    pair<double, double> c_T2 = calcCoeff(2, _h[en + "T2"]);
    pair<double, double> c_T3 = calcCoeff(3, _h[en + "T3"]);

    double sDelta = (-2. * (3. + alpha0.first) * c_T3.first) / (-aLambda * sqrt(1 - sqr(alpha0.first)));
    double cDelta = (-3 * (3 + alpha0.first) * c_T2.first)
        / (-sqr(aLambda) * sqrt(1 - sqr(alpha0.first)));

    pair<double, pair<double, double>> Delta;
    Delta.first = asin(sDelta);
    if (cDelta < 0.) Delta.first = M_PI - Delta.first;
    Delta.second.first = (-4
                          * (sqr(c_T3.second) * sqr(1. + alpha0.first) * sqr(1 + alpha0.first)
                                 * sqr(3. + alpha0.first)
                             + sqr(alpha0.second.first) * sqr(c_T3.first) * sqr(1 + 3 * alpha0.first)))
        / (sqr(1. - sqr(alpha0.first))
           * (4 * sqr(c_T3.first) * sqr(3 + alpha0.first) + sqr(aLambda) * (-1 + sqr(alpha0.first))));
    Delta.second.second = (-4
                           * (sqr(c_T3.second) * sqr(1. + alpha0.first) * sqr(1 + alpha0.first)
                                  * sqr(3. + alpha0.first)
                              + sqr(alpha0.second.second) * sqr(c_T3.first) * sqr(1 + 3 * alpha0.first)))
        / (sqr(1. - sqr(alpha0.first))
           * (4 * sqr(c_T3.first) * sqr(3 + alpha0.first) + sqr(aLambda) * (-1 + sqr(alpha0.first))));

    Delta.first *= 180. / M_PI;
    Delta.second.first *= 180. / M_PI;
    Delta.second.second *= 180. / M_PI;

    for (size_t ih = 0; ih < 3; ++ih) {
      double val;
      pair<double, double> err;
      if (ih == 0) {
        val = alpha0.first;
        err = alpha0.second;
      }
      else if (ih == 1) {
        val = Delta.first;
        err = Delta.second;
      }
      else {
        val = R.first;
        err = R.second;
      }
      if (en == "")
        _e[ih]->bin(1).set(val, err);
      else
        _est[ih]->binAt(en).set(val, err);
    }
  }
}

/// @}


/// @name Histograms
/// @{
map<string, Histo1DPtr> _h;
Estimate1DPtr _e[3];
BinnedEstimatePtr<string> _est[3];
string _sqs = "";
/// @}

};

RIVET_DECLARE_PLUGIN(BESIII_2023_I2637702); } ```