[PWGCF] JCorran/Core: Bug Fix to C0-scaling term (#18366)

Author: nmallick19

PWGCF/JCorran/Core/FlowJSPCAnalysis.cxx

@@ -121,15 +121,21 @@ void FlowJSPCAnalysis::calculateCorrelators(const int fCentBin)
     fillHistograms(fCentBin, j, correlationNum, correlationDenom, weightCorrelationNum, weightCorrelationDenom);
   }
 
-  // N_m = Re(Q_{0,1}); weight M_m. 3SPC → fN3, 4SPC → fN4.
+  // Effective multiplicity N_m (arXiv:2606.10258 Eq. IV.2 / manuscript eq:Neff):
+  // 1/N_m^{m-1} = sum_i M_m(N_i) N_i^{-(m-1)} / sum_i M_m(N_i),
+  // with M_m = N(N-1)...(N-m+1) = fCorrelDenoms[m-1] and N = Re(Q_{0,1}).
+  // TProfile stores <N^{-(m-1)}>_comb; invert offline as N_m = content^{-1/(m-1)}.
+  // 3SPC → fN3 (m=3); 4SPC → fN4 (m=4).
   if (mHistRegistry && qvecs) {
     const double nSel = qvecs->QvectorQC[0][1].Re();
     const float centX = static_cast<float>(fCentBin) + 0.5f;
     if (nSel > 0.0 && std::isfinite(nSel)) {
       if (mWhichSPC == 0 && fCorrelDenoms[2] > 0.0) {
-        mHistRegistry->fill(HIST("fN3"), centX, nSel, fCorrelDenoms[2]);
+        const double invN3Pow = 1.0 / (nSel * nSel); // N^{-(3-1)}
+        mHistRegistry->fill(HIST("fN3"), centX, invN3Pow, fCorrelDenoms[2]);
       } else if (mWhichSPC == 1 && fCorrelDenoms[3] > 0.0) {
-        mHistRegistry->fill(HIST("fN4"), centX, nSel, fCorrelDenoms[3]);
+        const double invN4Pow = 1.0 / (nSel * nSel * nSel); // N^{-(4-1)}
+        mHistRegistry->fill(HIST("fN4"), centX, invN4Pow, fCorrelDenoms[3]);
       }
     }
   }

PWGCF/JCorran/Core/FlowJSPCAnalysis.h

@@ -79,9 +79,10 @@ class FlowJSPCAnalysis
       return;
     }
     mHistRegistry->add("FullCentrality", "FullCentrality", o2::framework::HistType::kTH1D, {{100, 0., 100.}}, true);
-    // Effective N_m per centrality class, weighted by M_m (arXiv:2606.10258 c0).
-    mHistRegistry->add("fN3", "Effective N_{3};centrality class;N_{3}", {o2::framework::HistType::kTProfile, {{9, 0., 9.}}}, true);
-    mHistRegistry->add("fN4", "Effective N_{4};centrality class;N_{4}", {o2::framework::HistType::kTProfile, {{9, 0., 9.}}}, true);
+    // Stores <N^{-(m-1)}>_comb = 1/N_m^{m-1} (arXiv:2606.10258 Eq. IV.2).
+    // Invert offline: N_m = content^{-1/(m-1)}. Weight is M_m.
+    mHistRegistry->add("fN3", "1/N_{3}^{2};centrality class;1/N_{3}^{2}", {o2::framework::HistType::kTProfile, {{9, 0., 9.}}}, true);
+    mHistRegistry->add("fN4", "1/N_{4}^{3};centrality class;1/N_{4}^{3}", {o2::framework::HistType::kTProfile, {{9, 0., 9.}}}, true);
     mHistRegistry->add("Centrality_0/fResults", "Numerators and denominators", {o2::framework::HistType::kTProfile, {{24, 0., 24.}}}, true);
     mHistRegistry->add("Centrality_0/fCovResults", "Covariance N*D", {o2::framework::HistType::kTProfile, {{48, 0., 48.}}}, true);
     mHistRegistry->add("Centrality_0/phiBefore", "Phi before", {o2::framework::HistType::kTH1D, {{100, 0., o2::constants::math::TwoPI}}}, true);