Rivet analyses

title: BESIII_2023_I2683607

$e^+e^-\to\Sigma^+\bar{\Sigma}^-$ form factors

Experiment: BESIII (BEPC)

Inspire ID: 2683607

Status: VALIDATED NOHEPDATA

Authors: - Peter Richardson

References: - Phys.Rev.Lett. 132 (2024) 8, 081904 - arXiv: 2307.15894

Beams: e+ e-

Beam energies: (1.2, 1.2); (1.3, 1.3); (1.4, 1.4)GeV

Run details: - e+ e- -> hadrons

Form factors for $e^+e^-\to\Sigma^+\bar{\Sigma}^-$ at three energies measured by the BESIII collaboration.

Source code:BESIII_2023_I2683607.cc

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

include "Rivet/Analysis.hh"

include "Rivet/Projections/Beam.hh"

include "Rivet/Projections/UnstableParticles.hh"

namespace Rivet {

/// @brief e+ e- > Sigma+ Sigmbar- class BESIII_2023_I2683607 : public Analysis { public:

/// Constructor
RIVET_DEFAULT_ANALYSIS_CTOR(BESIII_2023_I2683607);


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

/// Book histograms and initialise projections before the run
void init() {
  // Initialise and register projections
  declare(Beam(), "Beams");
  declare(UnstableParticles(Cuts::abspid == 3222), "UFS");
  declare(FinalState(), "FS");

  // Book histograms
  book(_h_T2, "TMP/T2", 20, -1., 1.);
  book(_h_T3, "TMP/T3", 20, -1., 1.);
  book(_h_cThetaL, "TMP/cThetaL", 20, -1., 1.);
  book(_wsum, "TMP/wsum");

  for (const string& en : refData<YODA::BinnedEstimate<string>>(1, 1, 1).xEdges()) {
    const double eval = stod(en) * GeV;
    if (isCompatibleWithSqrtS(eval)) {
      _sqs = en;
      break;
    }
  }
  raiseBeamErrorIf(_sqs.empty());
}

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 sigma baryons
  const UnstableParticles& ufs = apply<UnstableParticles>(event, "UFS");
  Particle Sigma, SigBar;
  bool matched(false);
  for (const Particle& p : ufs.particles(Cuts::abspid == 3222)) {
    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) {
            Sigma = p;
            SigBar = p2;
          }
          else {
            Sigma = p2;
            SigBar = p;
          }
          break;
        }
      }
    }
    if (matched) break;
  }
  if (!matched) vetoEvent;
  Particle proton;
  matched = false;
  for (const Particle& p : Sigma.children()) {
    if (p.pid() == 2212) {
      matched = true;
      proton = p;
    }
    else if (p.pid() != PID::PI0) {
      vetoEvent;
    }
  }
  if (!matched) vetoEvent;
  Particle baryon;
  matched = false;
  for (const Particle& p : SigBar.children()) {
    if (p.pid() == -2212) {
      baryon = p;
      matched = true;
    }
    else if (p.pid() != PID::PI0) {
      vetoEvent;
    }
  }
  if (!matched) vetoEvent;
  // boost to the Sigma rest frame
  LorentzTransform boost1 = LorentzTransform::mkFrameTransformFromBeta(Sigma.mom().betaVec());
  Vector3 e1z = Sigma.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 Sigma bar
  LorentzTransform boost2 = LorentzTransform::mkFrameTransformFromBeta(SigBar.mom().betaVec());
  Vector3 axis2 = boost2.transform(baryon.mom()).p3().unit();
  double n2x(e1x.dot(axis2)), n2z(e1z.dot(axis2));
  double cosL = axis.dot(Sigma.mom().p3().unit());
  double sinL = sqrt(1. - sqr(cosL));
  double T2 = -sinL * cosL * (n1x * n2z + n1z * n2x);
  double T3 = -sinL * cosL * n1y;
  _h_T2->fill(cosL, T2);
  _h_T3->fill(cosL, T3);
  _wsum->fill();
  _h_cThetaL->fill(cosL);
}

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() {
  const double aLambda = -0.980;
  normalize(_h_cThetaL);
  scale(_h_T2, 1. / *_wsum);
  scale(_h_T3, 1. / *_wsum);
  pair<double, pair<double, double>> alpha0 = calcAlpha0(_h_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_T2);
  pair<double, double> c_T3 = calcCoeff(3, _h_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 iy = 0; iy < 3; ++iy) {
    double val;
    pair<double, double> err;
    if (iy == 0) {
      val = alpha0.first;
      err = alpha0.second;
    }
    else if (iy == 1) {
      val = Delta.first;
      err = Delta.second;
    }
    else {
      val = R.first;
      err = R.second;
    }
    BinnedEstimatePtr<string> tmp2;
    book(tmp2, 1, 1, 1 + iy);
    tmp2->binAt(_sqs).set(val, err);
    if (iy == 1 && _sqs == "2.3960"s) {
      tmp2->binAt(_sqs + "B").set(val, err);
    }
  }
}

/// @}


/// @name Histograms
/// @{
Histo1DPtr _h_T2, _h_T3;
Histo1DPtr _h_cThetaL;
CounterPtr _wsum;
string _sqs = "";
/// @}

};

RIVET_DECLARE_PLUGIN(BESIII_2023_I2683607); } ```