Rivet analyses

title: BESIII_2021_I1974025

Measurement of $e^+e^-\to\Lambda^0\bar{\Lambda}^0$ at 3.773 GeV

Experiment: BESIII (BEPC)

Inspire ID: 1974025

Status: VALIDATED NOHEPDATA

Authors: - Peter Richardson

References: - Phys.Rev.D 105 (2022) 1, L011101

Beams: e+ e-

Beam energies: (1.9, 1.9)GeV

Run details: - e+e- to hadrons

Measurement of the angular distribution and polarization for $e^+e^-\to\Lambda^0\bar{\Lambda}^0$ at 3.773 GeV by BESIII.

Source code:BESIII_2021_I1974025.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- > Lambda, Lambdabar class BESIII_2021_I1974025 : public Analysis { public:

/// Constructor
RIVET_DEFAULT_ANALYSIS_CTOR(BESIII_2021_I1974025);


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

/// Book histograms and initialise projections before the run
void init() {
  // Initialise and register projections
  declare(Beam(), "Beams");
  declare(FinalState(), "FS");
  declare(UnstableParticles(), "UFS");
  // histograms
  book(_wsum, "TMP/wsum");
  // for(unsigned int ix=0;ix<6;++ix)
  //    book(_h_F[ix],1,1,1+ix);
  for (unsigned int ix = 0; ix < 6; ++ix) book(_h_F[ix], "TMP/F_" + toString(ix + 1), 20, -1., 1.);
  book(_h_F[5], 1, 1, 6);
  book(_h_mu, 2, 1, 1);
}

void findChildren(const Particle& p, map<long, int>& nRes, int& ncount) {
  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.momentum().p3().unit();
  else
    axis = beams.second.momentum().p3().unit();
  const FinalState& fs = apply<FinalState>(event, "FS");
  // total hadronic and muonic cross sections
  map<long, int> nCount;
  int ntotal(0);
  for (const Particle& p : fs.particles()) {
    nCount[p.pid()] += 1;
    ++ntotal;
  }
  // find the Lambdas
  bool matched = false;
  const FinalState& ufs = apply<UnstableParticles>(event, "UFS");
  Particle Lambda, LamBar;
  for (unsigned int ix = 0; ix < ufs.particles().size(); ++ix) {
    const Particle& p1 = ufs.particles()[ix];
    if (abs(p1.pid()) != 3122) continue;
    // check fs
    bool fs = true;
    for (const Particle& child : p1.children()) {
      if (child.pid() == p1.pid()) {
        fs = false;
        break;
      }
    }
    if (!fs) continue;
    // find the children
    map<long, int> nRes = nCount;
    int ncount = ntotal;
    findChildren(p1, nRes, ncount);
    for (unsigned int iy = ix + 1; iy < ufs.particles().size(); ++iy) {
      matched = false;
      const Particle& p2 = ufs.particles()[iy];
      if (abs(p2.pid()) != 3122) continue;
      // check fs
      bool fs = true;
      for (const Particle& child : p2.children()) {
        if (child.pid() == p2.pid()) {
          fs = false;
          break;
        }
      }
      if (!fs) continue;
      map<long, int> nRes2 = nRes;
      int ncount2 = ncount;
      findChildren(p2, nRes2, ncount2);
      if (ncount2 != 0) continue;
      matched = true;
      for (const auto& val : nRes2) {
        if (val.second != 0) {
          matched = false;
          break;
        }
      }
      if (matched) {
        if (p1.pid() == PID::LAMBDA) {
          Lambda = p1;
          LamBar = p2;
        }
        else {
          Lambda = p2;
          LamBar = p1;
        }
        break;
      }
    }
    if (matched) break;
  }
  // and the children
  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;
  // now for the polarization measurements
  LorentzTransform boost1 = LorentzTransform::mkFrameTransformFromBeta(Lambda.momentum().betaVec());
  Vector3 e1z = Lambda.momentum().p3().unit();
  Vector3 e1y = e1z.cross(axis).unit();
  Vector3 e1x = e1y.cross(e1z).unit();
  Vector3 axis1 = boost1.transform(proton.momentum()).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.momentum().betaVec());
  Vector3 axis2 = boost2.transform(baryon.momentum()).p3().unit();
  double n2x(e1x.dot(axis2)), n2y(e1y.dot(axis2)), n2z(e1z.dot(axis2));
  double cosL = -axis.dot(Lambda.momentum().p3().unit());
  double sinL = sqrt(1. - sqr(cosL));
  double T1 = sqr(sinL) * n1x * n2x + sqr(cosL) * n1z * n2z;
  double T2 = sinL * cosL * (n1x * n2z + n1z * n2x);
  double T3 = sinL * cosL * n1y;
  double T4 = sinL * cosL * n2y;
  double T5 = n1z * n2z - sqr(sinL) * n1y * n2y;
  double mu = n1y - n2y;
  _h_F[0]->fill(cosL, T1);
  _h_F[1]->fill(cosL, T2);
  _h_F[2]->fill(cosL, T3);
  _h_F[3]->fill(cosL, T4);
  _h_F[4]->fill(cosL, T5);
  _h_F[5]->fill(cosL);
  _h_mu->fill(cosL, mu);
  _wsum->fill();
}

pair<double, pair<double, double>> calcAlpha0(Histo1DPtr hist) {
  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(unsigned int imode, Histo1DPtr hist) {
  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() {
  // normalize histograms
  for (unsigned int ix = 0; ix < 6; ++ix) scale(_h_F[ix], 1. / *_wsum);
  scale(_h_mu, 10. / *_wsum);
  // value of aLambda assumed in paper
  double aLambda = 0.754;
  // calculate alpha0
  pair<double, pair<double, double>> alpha0 = calcAlpha0(_h_F[5]);
  Estimate0DPtr _h_alpha0;
  book(_h_alpha0, 3, 1, 1);
  _h_alpha0->set(alpha0.first, make_pair(-alpha0.second.first, alpha0.second.second));
  double s2 = -1. + sqr(alpha0.first);
  double s3 = 3 + alpha0.first;
  double s1 = sqr(s3);
  // alpha- and alpha+ from proton data
  pair<double, double> c_T2_p = calcCoeff(2, _h_F[1]);
  pair<double, double> c_T3_p = calcCoeff(3, _h_F[2]);
  pair<double, double> c_T4_p = calcCoeff(4, _h_F[3]);
  double s4 = sqr(c_T2_p.first);
  double s5 = sqr(c_T3_p.first);
  double s6 = sqr(c_T4_p.first);
  double disc = s1 * s5 * s6 * (-9. * s2 * s4 + 4. * s1 * s5 * s6);
  // now for Delta
  if (disc > 0) {
    double sDelta = (-2. * (3. + alpha0.first) * c_T3_p.first) / (aLambda * sqrt(1 - sqr(alpha0.first)));
    double cDelta = (-3 * (3 + alpha0.first) * c_T2_p.first)
        / (-aLambda * aLambda * sqrt(1 - sqr(alpha0.first)));
    double Delta = asin(sDelta);
    if (cDelta < 0.) Delta = M_PI - Delta;
    double ds_P = (-9 * c_T2_p.first
                   * ((-1 + alpha0.first) * (1 + alpha0.first) * (3 + alpha0.first) * c_T3_p.first
                          * c_T4_p.first * c_T2_p.second
                      + c_T2_p.first * c_T4_p.first
                          * (c_T3_p.first * (alpha0.second.first + 3 * alpha0.first * alpha0.second.first)
                             - (-1 + alpha0.first) * (1 + alpha0.first) * (3 + alpha0.first)
                                 * c_T3_p.second)
                      - (-1 + alpha0.first) * (1 + alpha0.first) * (3 + alpha0.first) * c_T2_p.first
                          * c_T3_p.first * c_T4_p.second)
                   * disc)
        / (pow(1 - pow(alpha0.first, 2), 1.5) * pow(c_T4_p.first, 3)
           * pow(-((disc + 2 * s1 * s5 * s6) / (s2 * s6)), 1.5) * (-9 * s2 * s4 + 4 * s1 * s5 * s6));
    double ds_M = (-9 * c_T2_p.first
                   * ((-1 + alpha0.first) * (1 + alpha0.first) * (3 + alpha0.first) * c_T3_p.first
                          * c_T4_p.first * c_T2_p.second
                      + c_T2_p.first * c_T4_p.first
                          * (c_T3_p.first
                                 * (alpha0.second.second + 3 * alpha0.first * alpha0.second.second)
                             - (-1 + alpha0.first) * (1 + alpha0.first) * (3 + alpha0.first)
                                 * c_T3_p.second)
                      - (-1 + alpha0.first) * (1 + alpha0.first) * (3 + alpha0.first) * c_T2_p.first
                          * c_T3_p.first * c_T4_p.second)
                   * disc)
        / (pow(1 - pow(alpha0.first, 2), 1.5) * pow(c_T4_p.first, 3)
           * pow(-((disc + 2 * s1 * s5 * s6) / (s2 * s6)), 1.5) * (-9 * s2 * s4 + 4 * s1 * s5 * s6));
    ds_P /= sqrt(1. - sqr(sDelta));
    ds_M /= sqrt(1. - sqr(sDelta));
    Estimate0DPtr _h_sin;
    book(_h_sin, 3, 1, 2);
    _h_sin->set(Delta / M_PI * 180., make_pair(ds_M / M_PI * 180., -ds_P / M_PI * 180.));
  }
}

/// @}


/// @name Histograms
/// @{
Histo1DPtr _h_F[6], _h_mu;
CounterPtr _wsum;
/// @}

};

RIVET_DECLARE_PLUGIN(BESIII_2021_I1974025);

} ```