Rivet Analyses Reference

BESIII_2020_I1791570

Analysis of $J/\psi$, $\psi(2S)$ decays to $\Sigma^+\bar\Sigma^-$
Experiment: BESIII (BEPC)
Inspire ID: 1791570
Status: VALIDATED
Authors:
  • Peter Richardson
References:
  • Phys.Rev.Lett. 125 (2020) 5, 052004
Beams: e- e+
Beam energies: (1.6, 1.6); (1.8, 1.8) GeV
Run details:
  • e+e- > J/psi, psi 2s. Beam energy must be specified as analysis option "ENERGY" when rivet-merging samples.

Analysis of the angular distribution of the baryons, and decay products, produced in $e^+e^-\to J/\psi, \psi(2S) \to \Sigma^+\bar\Sigma^-$. Gives information about the decay and is useful for testing correlations in hadron decays. N.B. The moment data is not corrected for efficiency/acceptance and should therefore only be used qualatively. Beam energy must be specified as analysis option "ENERGY" when rivet-merging samples.

Source code: BESIII_2020_I1791570.cc
  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
// -*- C++ -*-
#include "Rivet/Analysis.hh"
#include "Rivet/Projections/Beam.hh"
#include "Rivet/Projections/FinalState.hh"
#include "Rivet/Projections/UnstableParticles.hh"

namespace Rivet {


  /// @brief J/Psi, psi(2S) -> Sigma+ Sigmabar-
  class BESIII_2020_I1791570 : public Analysis {
  public:

    /// Constructor
    RIVET_DEFAULT_ANALYSIS_CTOR(BESIII_2020_I1791570);


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

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

      // Initialise and register projections
      declare(Beam(), "Beams");
      declare(UnstableParticles(), "UFS");
      declare(FinalState(), "FS");
      
      // Book histograms
      book(_h_T1, "/TMP/T1",20,-1.,1.);
      book(_h_T2, "/TMP/T2",20,-1.,1.);
      book(_h_T3, "/TMP/T3",20,-1.,1.);
      book(_h_T4, "/TMP/T4",20,-1.,1.);
      book(_h_T5, "/TMP/T5",20,-1.,1.);
      
      book(_h_cThetaL,"/TMP/cThetaL",20,-1.,1.);
      if(isCompatibleWithSqrtS(3.1,1e-2))
	book(_h_mu,1,1,1);
      else if(isCompatibleWithSqrtS(3.686,1e-2))
	book(_h_mu,1,1,2);
      else
	throw Error("Unexpected sqrtS ! Only 3.1 and 3.686 GeV atr supported");
      book(_wsum,"/TMP/wsum");
    }

    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();
      // 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(auto const & val : nRes2) {
      	      if(val.second!=0) {
      		matched = false;
      		break;
      	      }
      	    }
      	    // fond 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;
      // find proton
      Particle proton;
      matched = false;
      for (const Particle & p : Sigma.children()) {
	if(p.pid()==2212) {
	  matched=true;
	  proton=p;
	}
	else if(p.pid()!=111) {
	  matched = false;
	  break;
	}
      }
      if(!matched) vetoEvent;
      // find antiproton
      Particle pbar;
      matched = false;
      for (const Particle & p : SigBar.children()) {
	if(p.pid()==-2212) {
	  matched=true;
	  pbar=p;
	}
	else if(p.pid()!=111) {
	  matched = false;
	  break;
	}
      }
      if(!matched) vetoEvent;
      // boost to the Sigma rest frame
      LorentzTransform boost1 = LorentzTransform::mkFrameTransformFromBeta(Sigma.momentum().betaVec());
      Vector3 e1z = Sigma.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 Sigma bar
      LorentzTransform boost2 = LorentzTransform::mkFrameTransformFromBeta(SigBar.momentum().betaVec());
      Vector3 axis2 = boost2.transform(pbar.momentum()).p3().unit();
      double n2x(e1x.dot(axis2)),n2y(e1y.dot(axis2)),n2z(e1z.dot(axis2));
      double cosL = axis.dot(Sigma.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_T1->fill(cosL,T1);
      _h_T2->fill(cosL,T2);
      _h_T3->fill(cosL,T3);
      _h_T4->fill(cosL,T4);
      _h_T5->fill(cosL,T5);
      _h_mu->fill(cosL,mu);
      _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 (auto bin : hist->bins() ) {
       	double Oi = bin.area();
	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.areaErr();
	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 (auto bin : hist->bins() ) {
	double Oi = bin.area();
	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.areaErr();
	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(_h_cThetaL);
      scale(_h_T1, 1./ *_wsum);
      scale(_h_T2, 1./ *_wsum);
      scale(_h_T3, 1./ *_wsum);
      scale(_h_T4, 1./ *_wsum);
      scale(_h_T5, 1./ *_wsum);
      scale(_h_mu, 2./ *_wsum);
      // histos for J/psi or psi(2s)
      int ih = isCompatibleWithSqrtS(3.1,1e-2) ? 1 : 2;
      // calculate alpha0
      pair<double,pair<double,double> > alpha0 = calcAlpha0(_h_cThetaL);
      Scatter2DPtr _h_alpha0;
      book(_h_alpha0,4,1,ih);
      _h_alpha0->addPoint(0.5, alpha0.first, make_pair(0.5,0.5),
			  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 = calcCoeff(2,_h_T2);
      pair<double,double> c_T3 = calcCoeff(3,_h_T3);
      pair<double,double> c_T4 = calcCoeff(4,_h_T4);
      double s4 = sqr(c_T2.first);
      double s5 = sqr(c_T3.first);
      double s6 = sqr(c_T4.first);
      double disc = s1*s5*s6*(-9.*s2*s4 + 4.*s1*s5*s6);
      if(disc>=0.) {
	disc = sqrt(disc);
	double aM = -sqrt(-1./s2/s6*(2.*s1*s5*s6+disc));
	double aP = c_T4.first/c_T3.first*aM;
	double aM_P = (2*(alpha0.first*c_T4.first*alpha0.second.first + c_T4.second*s2)*(disc + 2*s1*s5*s6)
		       - c_T4.first*s2*(4*s3*c_T3.first*c_T4.first*(c_T3.first*c_T4.first*alpha0.second.first +s3*c_T4.first*c_T3.second +s3*c_T3.first*c_T4.second) +
				(disc*(- 9*s2*s3*c_T2.first*c_T3.first*c_T4.first* c_T2.second
				       + 9*((1 -  alpha0.first*(3 + 2*alpha0.first))* c_T3.first*c_T4.first*alpha0.second.first -  s2*s3*c_T4.first*c_T3.second
					     - s2*s3*c_T3.first*c_T4.second)* s4
				       + 8*(c_T3.first*c_T4.first*alpha0.second.first +  s3*c_T4.first*c_T3.second +  s3*c_T3.first*c_T4.second)* s1*s5*s6))
				/(4*pow(3 + alpha0.first,3)*pow(c_T3.first,3)*pow(c_T4.first,3) -9*s2*s3*c_T3.first*c_T4.first*s4)))/
	  (2.*pow(c_T4.first,3)*pow(s2,2)*sqrt(-((disc + 2*s1*s5*s6)/(s2*s6))));
	double aM_M = (2*(alpha0.first*c_T4.first*alpha0.second.second + c_T4.second*s2)*(disc + 2*s1*s5*s6)
		       - c_T4.first*s2*(4*s3*c_T3.first*c_T4.first*(c_T3.first*c_T4.first*alpha0.second.second +s3*c_T4.first*c_T3.second +s3*c_T3.first*c_T4.second) +
				(disc*(- 9*s2*s3*c_T2.first*c_T3.first*c_T4.first* c_T2.second
				       + 9*((1 -  alpha0.first*(3 + 2*alpha0.first))* c_T3.first*c_T4.first*alpha0.second.second -  s2*s3*c_T4.first*c_T3.second
					     - s2*s3*c_T3.first*c_T4.second)* s4
				       + 8*(c_T3.first*c_T4.first*alpha0.second.second +  s3*c_T4.first*c_T3.second +  s3*c_T3.first*c_T4.second)* s1*s5*s6))
				/(4*pow(3 + alpha0.first,3)*pow(c_T3.first,3)*pow(c_T4.first,3) -9*s2*s3*c_T3.first*c_T4.first*s4)))/
	  (2.*pow(c_T4.first,3)*pow(s2,2)*sqrt(-((disc + 2*s1*s5*s6)/(s2*s6))));
	double aP_M = (c_T4.first*sqrt(-((disc + 2*s1*s5*s6)/   (s2*s6)))*
		       (-2*c_T3.second -  (2*alpha0.first*c_T3.first*alpha0.second.first)/s2 +  (c_T3.first*(4*s3*c_T3.first*c_T4.first*(c_T3.first*c_T4.first*alpha0.second.first +  s3*c_T4.first*c_T3.second +  s3*c_T3.first*c_T4.second)
							    + (disc*(-9*s2*s3*c_T2.first*c_T3.first*c_T4.first* c_T2.second
								      +  9*((1 -  alpha0.first*(3 + 2*alpha0.first))* c_T3.first*c_T4.first*alpha0.second.first -  s2*s3*c_T4.first*c_T3.second
									     -  s2*s3*c_T3.first*c_T4.second)* s4 +
								      8*(c_T3.first*c_T4.first*alpha0.second.first +  s3*c_T4.first*c_T3.second +  s3*c_T3.first*c_T4.second)* s1*s5*s6))/
							    (4* pow(3 + alpha0.first,3)* pow(c_T3.first,3)* pow(c_T4.first,3) -  9*s2*s3*c_T3.first*c_T4.first*s4)))/
			(disc + 2*s1*s5*s6)))/(2.*pow(c_T3.first,2));
	double aP_P = (c_T4.first*sqrt(-((disc + 2*s1*s5*s6)/   (s2*s6)))*
		       (-2*c_T3.second -  (2*alpha0.first*c_T3.first*alpha0.second.second)/s2 +  (c_T3.first*(4*s3*c_T3.first*c_T4.first*(c_T3.first*c_T4.first*alpha0.second.second +  s3*c_T4.first*c_T3.second +  s3*c_T3.first*c_T4.second)
							    + (disc*(-9*s2*s3*c_T2.first*c_T3.first*c_T4.first* c_T2.second
								      +  9*((1 -  alpha0.first*(3 + 2*alpha0.first))* c_T3.first*c_T4.first*alpha0.second.second -  s2*s3*c_T4.first*c_T3.second
									     -  s2*s3*c_T3.first*c_T4.second)* s4 +
								      8*(c_T3.first*c_T4.first*alpha0.second.second +  s3*c_T4.first*c_T3.second +  s3*c_T3.first*c_T4.second)* s1*s5*s6))/
							    (4* pow(3 + alpha0.first,3)* pow(c_T3.first,3)* pow(c_T4.first,3) -  9*s2*s3*c_T3.first*c_T4.first*s4)))/
			(disc + 2*s1*s5*s6)))/(2.*pow(c_T3.first,2));
	Scatter2DPtr _h_alphaM;
	book(_h_alphaM,2,1,1);
	_h_alphaM->addPoint(0.5, aM, make_pair(0.5,0.5),
			    make_pair(-aM_M , -aM_P ) );
	Scatter2DPtr _h_alphaP;
	book(_h_alphaP,2,1,2);
	_h_alphaP->addPoint(0.5, aP, make_pair(0.5,0.5),
			    make_pair(-aP_M , -aP_P  ) );
	Scatter2DPtr _h_alphabar;
	book(_h_alphabar,2,1,3);
	_h_alphabar->addPoint(0.5, 0.5*(aM-aP), make_pair(0.5,0.5),
			      make_pair(0.5*sqrt(sqr(aM_M)+sqr(aP_P)) ,
					0.5*sqrt(sqr(aM_P)+sqr(aP_M))) );
	// now for Delta
	double sDelta = (-2.*(3. + alpha0.first)*c_T3.first)/(aM*sqrt(1 - sqr(alpha0.first)));
	double cDelta = (-3*(3 + alpha0.first)*c_T2.first)/(aM*aP*sqrt(1 - sqr(alpha0.first)));

	double Delta = asin(sDelta);
	if(cDelta<0.) Delta = M_PI-Delta;
	double ds_P = (-9*c_T2.first*((-1 + alpha0.first)*(1 + alpha0.first)*  (3 + alpha0.first)*c_T3.first*c_T4.first*c_T2.second +  c_T2.first*c_T4.first*(c_T3.first*(alpha0.second.first + 3*alpha0.first*alpha0.second.first) -(-1 + alpha0.first)*(1 + alpha0.first)*(3 + alpha0.first)*c_T3.second)
			      -  (-1 + alpha0.first)*(1 + alpha0.first)*  (3 + alpha0.first)*c_T2.first*c_T3.first*c_T4.second)*disc)/
	  (pow(1 - pow(alpha0.first,2),1.5)*pow(c_T4.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.first*((-1 + alpha0.first)*(1 + alpha0.first)*  (3 + alpha0.first)*c_T3.first*c_T4.first*c_T2.second +  c_T2.first*c_T4.first*(c_T3.first*(alpha0.second.second + 3*alpha0.first*alpha0.second.second) -(-1 + alpha0.first)*(1 + alpha0.first)*(3 + alpha0.first)*c_T3.second)
			      -  (-1 + alpha0.first)*(1 + alpha0.first)*  (3 + alpha0.first)*c_T2.first*c_T3.first*c_T4.second)*disc)/
	  (pow(1 - pow(alpha0.first,2),1.5)*pow(c_T4.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));
	Scatter2DPtr _h_sin;
	book(_h_sin,3,1,ih);
	_h_sin->addPoint(0.5, Delta/M_PI*180., make_pair(0.5,0.5), make_pair( -ds_P/M_PI*180., -ds_M/M_PI*180. ) );
      }
    }

    ///@}


    /// @name Histograms
    ///@{
    Histo1DPtr _h_T1,_h_T2,_h_T3,_h_T4,_h_T5;
    Histo1DPtr _h_cThetaL;
    Histo1DPtr _h_mu;
    CounterPtr _wsum;
    ///@}


  };


  RIVET_DECLARE_PLUGIN(BESIII_2020_I1791570);

}