LauKMatrixPropagator.cc

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// Copyright University of Warwick 2008 - 2013.
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)

// Authors:
// Thomas Latham
// John Back
// Paul Harrison

#include "LauKMatrixPropagator.hh"
#include "LauConstants.hh"
#include "LauTextFileParser.hh"

#include "TMath.h"
#include "TSystem.h"

#include <iostream>
#include <fstream>
#include <cmath>
#include <cstdlib>

using std::cout;
using std::endl;
using std::cerr;

ClassImp(LauKMatrixPropagator)

LauKMatrixPropagator::LauKMatrixPropagator(const TString& name, const TString& paramFile, 
                                           Int_t resPairAmpInt, Int_t nChannels, 
                                           Int_t nPoles, Int_t rowIndex) :
        name_(name),
        paramFileName_(paramFile),
        resPairAmpInt_(resPairAmpInt), 
        index_(rowIndex - 1),
        previousS_(0.0),
        scattSVP_(0.0),
        prodSVP_(0.0),
        nChannels_(nChannels),
        nPoles_(nPoles),
        sAConst_(0.0),
        m2piSq_(4.0*LauConstants::mPiSq),
        m2KSq_( 4.0*LauConstants::mKSq),
        m2EtaSq_(4.0*LauConstants::mEtaSq),
        mEtaEtaPSumSq_((LauConstants::mEta + LauConstants::mEtaPrime)*(LauConstants::mEta + LauConstants::mEtaPrime)),
        mEtaEtaPDiffSq_((LauConstants::mEta - LauConstants::mEtaPrime)*(LauConstants::mEta - LauConstants::mEtaPrime)),
        mKpiSumSq_((LauConstants::mK + LauConstants::mPi)*(LauConstants::mK + LauConstants::mPi)),
        mKpiDiffSq_((LauConstants::mK - LauConstants::mPi)*(LauConstants::mK - LauConstants::mPi)),
        mKEtaPSumSq_((LauConstants::mK + LauConstants::mEtaPrime)*(LauConstants::mK + LauConstants::mEtaPrime)),
        mKEtaPDiffSq_((LauConstants::mK - LauConstants::mEtaPrime)*(LauConstants::mK - LauConstants::mEtaPrime)),
        mK3piDiffSq_((LauConstants::mK - 3.0*LauConstants::mPi)*(LauConstants::mK - 3.0*LauConstants::mPi)),
        k3piFactor_(TMath::Power((1.44 - mK3piDiffSq_)/1.44, -2.5)),
        fourPiFactor1_(16.0*LauConstants::mPiSq),
        fourPiFactor2_(TMath::Sqrt(1.0 - fourPiFactor1_)),
        adlerZeroFactor_(0.0),
        parametersSet_(kFALSE),
        verbose_(kFALSE)
{
        // Constructor

        // Check that the index is OK
        if (index_ < 0 || index_ >= nChannels_) {
                std::cerr << "ERROR in LauKMatrixPropagator constructor. The rowIndex, which is set to "
                          << rowIndex << ", must be between 1 and the number of channels " 
                          << nChannels_ << std::endl;
                gSystem->Exit(EXIT_FAILURE);
        }

        this->setParameters(paramFile);
}

LauKMatrixPropagator::~LauKMatrixPropagator() 
{
        // Destructor
        realProp_.Clear(); 
        negImagProp_.Clear();
        ScattKMatrix_.Clear();
        ReRhoMatrix_.Clear();
        ImRhoMatrix_.Clear();
        ReTMatrix_.Clear();
        ImTMatrix_.Clear();
        IMatrix_.Clear();
        zeroMatrix_.Clear();
}

LauComplex LauKMatrixPropagator::getPropTerm(Int_t channel) const
{
        // Get the (i,j) = (index_, channel) term of the propagator
        // matrix. This allows us not to return the full propagator matrix.
        Double_t realTerm = this->getRealPropTerm(channel);
        Double_t imagTerm = this->getImagPropTerm(channel);

        LauComplex propTerm(realTerm, imagTerm);
        return propTerm;

}

Double_t LauKMatrixPropagator::getRealPropTerm(Int_t channel) const
{
        // Get the real part of the (i,j) = (index_, channel) term of the propagator
        // matrix. This allows us not to return the full propagator matrix.
        if (parametersSet_ == kFALSE) {return 0.0;}

        Double_t propTerm = realProp_[index_][channel];
        return propTerm;

}

Double_t LauKMatrixPropagator::getImagPropTerm(Int_t channel) const
{
        // Get the imaginary part of the (i,j) = (index_, channel) term of the propagator
        // matrix. This allows us not to return the full propagator matrix.
        if (parametersSet_ == kFALSE) {return 0.0;}

        Double_t imagTerm = -1.0*negImagProp_[index_][channel];
        return imagTerm;

}

void LauKMatrixPropagator::updatePropagator(const LauKinematics* kinematics)
{

        // Calculate the K-matrix propagator for the given s value.  
        // The K-matrix amplitude is given by
        // T_i = sum_{ij} (I - iK*rho)^-1 * P_j, where P is the production K-matrix.
        // i = index for the state (e.g. S-wave index = 0).
        // Here, we only find the (I - iK*rho)^-1 matrix part.

        if (!kinematics) {return;}

        // Get the invariant mass squared (s) from the kinematics object. 
        // Use the resPairAmpInt to find which mass-squared combination to use.
        Double_t s(0.0);

        if (resPairAmpInt_ == 1) {
                s = kinematics->getm23Sq();
        } else if (resPairAmpInt_ == 2) {
                s = kinematics->getm13Sq();
        } else if (resPairAmpInt_ == 3) {
                s = kinematics->getm12Sq();
        }

        this->updatePropagator(s);

}

void LauKMatrixPropagator::updatePropagator(Double_t s)
{
        // Calculate the K-matrix propagator for the given s value.  
        // The K-matrix amplitude is given by
        // T_i = sum_{ij} (I - iK*rho)^-1 * P_j, where P is the production K-matrix.
        // i = index for the state (e.g. S-wave index = 0).
        // Here, we only find the (I - iK*rho)^-1 matrix part.

        // Check if we have almost the same s value as before. If so, don't re-calculate
        // the propagator nor any of the pole mass summation terms.
        if (TMath::Abs(s - previousS_) < 1e-6*s) {
                //cout<<"Already got propagator for s = "<<s<<endl;
                return;
        }

        if (parametersSet_ == kFALSE) {
                //cerr<<"ERROR in LauKMatrixPropagator::updatePropagator. Parameters have not been set."<<endl;
                return;
        }

        // Calculate the denominator pole mass terms and Adler zero factor
        this->calcPoleDenomVect(s);
        this->updateAdlerZeroFactor(s);

        // Calculate the scattering K-matrix (real and symmetric)
        this->calcScattKMatrix(s);
        // Calculate the phase space density matrix, which is diagonal, but can be complex 
        // if the quantity s is below various threshold values (analytic continuation).
        this->calcRhoMatrix(s);

        // Calculate K*rho (real and imaginary parts, since rho can be complex)
        TMatrixD K_realRho(ScattKMatrix_);
        K_realRho *= ReRhoMatrix_;
        TMatrixD K_imagRho(ScattKMatrix_);
        K_imagRho *= ImRhoMatrix_;

        // A = I + K*Imag(rho), B = -K*Real(Rho)
        // Calculate C and D matrices such that (A + iB)*(C + iD) = I,
        // ie. C + iD = (I - i K*rho)^-1, separated into real and imaginary parts.
        TMatrixD A(IMatrix_);
        A += K_imagRho;
        TMatrixD B(zeroMatrix_);
        B -= K_realRho;

        TMatrixD invA(TMatrixD::kInverted, A);
        TMatrixD invA_B(invA);
        invA_B *= B;
        TMatrixD B_invA_B(B);
        B_invA_B *= invA_B;

        TMatrixD invC(A);
        invC += B_invA_B;

        // Set the real part of the propagator matrix ("C")
        realProp_ = TMatrixD(TMatrixD::kInverted, invC);

        // Set the (negative) imaginary part of the propagator matrix ("-D")
        TMatrixD BC(B);
        BC *= realProp_;
        negImagProp_ = TMatrixD(invA);
        negImagProp_ *= BC;

        // Also calculate the production SVP term, since this uses Adler-zero parameters
        // defined in the parameter file.
        this->updateProdSVPTerm(s);

        // Finally, keep track of the value of s we just used.
        previousS_ = s;

}

void LauKMatrixPropagator::setParameters(const TString& inputFile)
{

        // Read an input file that specifies the values of the coupling constants g_i for
        // the given number of poles and their (bare) masses. Also provided are the f_{ab} 
        // slow-varying constants. The input file should also provide the Adler zero 
        // constants s_0, s_A and s_A0.

        // Format of input file:
        // Indices (nChannels) of phase space channel types (defined in KMatrixChannels enum)
        // Then the bare pole mass and coupling constants over all channels for each pole:
        // m_{alpha} g_1^{alpha} g_2^{alpha} ... g_n^{alpha}   where n = nChannels
        // Repeat for N poles. Then define the f_{ab} scattering coefficients
        // f_{11} f_{12} ... f_{1n}
        // f_{21} f_{22} ... f_{2n} etc.. where n = nChannels
        // Note that the K-matrix will be symmetrised (by definition), so it is sufficient 
        // to only have the upper half filled with sensible values.
        // Then define the Adler-zero and slowly-varying part (SVP) constants
        // m0^2 s0Scatt s0Prod sA sA0
        // where m0^2 = mass-squared constant for scattering production f_{ab} numerator
        parametersSet_ = kFALSE;

        cout<<"Initialising K-matrix propagator "<<name_<<" parameters from "<<inputFile.Data()<<endl;

        cout<<"nChannels = "<<nChannels_<<", nPoles = "<<nPoles_<<endl;

        LauTextFileParser readFile(inputFile);
        readFile.processFile();

        // Get the first (non-comment) line
        std::vector<std::string> channelTypeLine = readFile.getNextLine();
        Int_t nTypes = static_cast<Int_t>(channelTypeLine.size());
        if (nTypes != nChannels_) {
                cerr<<"Error in LauKMatrixPropagator::setParameters. Input file "<<inputFile
                        <<" defines "<<nTypes<<" channels when "<<nChannels_<<" are expected"<<endl;
                return;
        }

        // Get the list of channel indices to specify what phase space factors should be used
        // e.g. pipi, Kpi, eta eta', 4pi etc..
        Int_t iChannel(0);
        phaseSpaceTypes_.clear();
        for (iChannel = 0; iChannel < nChannels_; iChannel++) {

                Int_t phaseSpaceInt = atoi(channelTypeLine[iChannel].c_str());
                Bool_t checkChannel = this->checkPhaseSpaceType(phaseSpaceInt);

                if (checkChannel == kTRUE) {
                        cout<<"Adding phase space channel index "<<phaseSpaceInt
                                <<" to K-matrix propagator "<<name_<<endl;
                        phaseSpaceTypes_.push_back(phaseSpaceInt);
                } else {
                        cerr<<"Phase space channel index "<<phaseSpaceInt
                                <<" should be between 1 and "<<LauKMatrixPropagator::TotChannels-1<<endl;
                        cerr<<"Stopping initialisation of the K-matrix propagator "<<name_<<endl;
                        cerr<<"Please correct the channel index on the first line of "<<inputFile<<endl;
                        return;
                }

        }

        // Clear vector of pole masses^2 as well as the map of the coupling constant vectors.
        mSqPoles_.clear(); gCouplings_.clear();

        Double_t poleMass(0.0), poleMassSq(0.0), couplingConst(0.0);
        Int_t iPole(0);
        Int_t nPoleNumbers(nChannels_+1);

        for (iPole = 0; iPole < nPoles_; iPole++) {

                // Read the next line of bare pole mass and its coupling constants 
                // over all channels (nChannels_)
                std::vector<std::string> poleLine = readFile.getNextLine();
                Int_t nPoleValues = static_cast<Int_t>(poleLine.size());
                if (nPoleValues != nPoleNumbers) {
                        cerr<<"Error in LauKMatrixPropagator::setParameters. Expecting "
                                <<nPoleNumbers<<" numbers for the bare pole "<<iPole<<" line (mass and "
                                <<nChannels_<<" coupling constants). Found "<<nPoleValues<<" values instead."
                                <<endl;
                        return;
                }

                poleMass = atof(poleLine[0].c_str());
                poleMassSq = poleMass*poleMass;
                LauParameter mPoleParam(poleMassSq);
                mSqPoles_.push_back(mPoleParam);

                std::vector<LauParameter> couplingVector;

                cout<<"Added bare pole mass "<<poleMass<<" GeV for pole number "<<iPole+1<<endl;

                for (iChannel = 0; iChannel < nChannels_; iChannel++) {

                        couplingConst = atof(poleLine[iChannel+1].c_str());
                        LauParameter couplingParam(couplingConst);
                        couplingVector.push_back(couplingParam);

                        cout<<"Added coupling parameter g^"<<iPole+1<<"_"<<iChannel+1<<" = "<<couplingConst<<endl;

                }

                gCouplings_[iPole] = couplingVector;    

        }

        // Scattering (slowly-varying part, or SVP) values

        fScattering_.clear();
        Int_t jChannel(0);

        std::vector<std::string> scatteringLine = readFile.getNextLine();
        Int_t nScatteringValues = static_cast<Int_t>(scatteringLine.size());
        if (nScatteringValues != nChannels_) {
                cerr<<"Error in LauKMatrixPropagator::setParameters. Expecting "
                        <<nChannels_<<" scattering SVP f constants instead of "<<nScatteringValues
                        <<" for the K-matrix row index "<<index_<<endl;
                return;
        }

        std::vector<LauParameter> scatteringVect;
        for (jChannel = 0; jChannel < nChannels_; jChannel++) {

                Double_t fScattValue = atof(scatteringLine[jChannel].c_str());
                LauParameter scatteringParam(fScattValue);
                scatteringVect.push_back(scatteringParam);

                cout<<"Added scattering SVP f("<<index_+1<<","<<jChannel+1<<") = "<<fScattValue<<endl;

        }

        fScattering_[index_] = scatteringVect;

        // Now extract the constants for the "Adler-zero" terms
        std::vector<std::string> constLine = readFile.getNextLine();
        Int_t nConstValues = static_cast<Int_t>(constLine.size());
        if (nConstValues != 5) {
                cerr<<"Error in LauKMatrixPropagator::setParameters. Expecting 5 values for the Adler-zero constants:"
                        <<" m0^2 s0Scatt s0Prod sA sA0"<<endl;
                return;
        }

        Double_t mSq0Value = atof(constLine[0].c_str());
        Double_t s0ScattValue = atof(constLine[1].c_str());
        Double_t s0ProdValue = atof(constLine[2].c_str());
        Double_t sAValue = atof(constLine[3].c_str());
        Double_t sA0Value = atof(constLine[4].c_str());

        cout<<"Adler zero constants:"<<endl;
        cout<<"m0Sq = "<<mSq0Value<<", s0Scattering = "<<s0ScattValue<<", s0Production = "
                <<s0ProdValue<<", sA = "<<sAValue<<" and sA0 = "<<sA0Value<<endl;

        mSq0_ = LauParameter("mSq0", mSq0Value);
        s0Scatt_ = LauParameter("s0Scatt", s0ScattValue);
        s0Prod_ = LauParameter("s0Prod", s0ProdValue);
        sA_ = LauParameter("sA", sAValue);
        sA0_ = LauParameter("sA0", sA0Value);
        sAConst_ = 0.5*sAValue*LauConstants::mPiSq;

        // Identity and null matrices
        IMatrix_.Clear();
        IMatrix_.ResizeTo(nChannels_, nChannels_);
        for (iChannel = 0; iChannel < nChannels_; iChannel++) {
                IMatrix_[iChannel][iChannel] = 1.0;
        }

        zeroMatrix_.Clear();
        zeroMatrix_.ResizeTo(nChannels_, nChannels_);

        // Real K matrix
        ScattKMatrix_.Clear();
        ScattKMatrix_.ResizeTo(nChannels_, nChannels_);

        // Real and imaginary propagator matrices
        realProp_.Clear(); negImagProp_.Clear();
        realProp_.ResizeTo(nChannels_, nChannels_);
        negImagProp_.ResizeTo(nChannels_, nChannels_);

        // Phase space matrices
        ReRhoMatrix_.Clear(); ImRhoMatrix_.Clear();
        ReRhoMatrix_.ResizeTo(nChannels_, nChannels_);
        ImRhoMatrix_.ResizeTo(nChannels_, nChannels_);

        // Square-root phase space matrices
        ReSqrtRhoMatrix_.Clear(); ImSqrtRhoMatrix_.Clear();
        ReSqrtRhoMatrix_.ResizeTo(nChannels_, nChannels_);
        ImSqrtRhoMatrix_.ResizeTo(nChannels_, nChannels_);

        // T matrices
        ReTMatrix_.Clear(); ImTMatrix_.Clear();
        ReTMatrix_.ResizeTo(nChannels_, nChannels_);
        ImTMatrix_.ResizeTo(nChannels_, nChannels_);

        // All required parameters have been set
        parametersSet_ = kTRUE;

        cout<<"Finished initialising K-matrix propagator "<<name_<<endl;

}

void LauKMatrixPropagator::calcScattKMatrix(Double_t s) 
{

        // Calculate the scattering K-matrix for the given value of s.
        // We need to obtain the complete matrix (not just one row/column) 
        // to get the correct inverted (I - i K rho) terms for the propagator.

        if (verbose_) {cout<<"Within calcScattKMatrix for s = "<<s<<endl;}

        // Initialise the K matrix to zero
        ScattKMatrix_.Zero();

        Int_t iChannel(0), jChannel(0), iPole(0);

        // The pole denominator 1/(m^2 - s) terms should already be calculated
        // by the calcPoleDenomVect() function. These same terms are also
        // used for calculating the production K-matrix elements.

        // Calculate the "slowly-varying part" (SVP), e.g. (1 GeV - s0)/(s - s0)
        this->updateScattSVPTerm(s);

        // Now loop over iChannel, jChannel to calculate Kij = Kji.
        for (iChannel = 0; iChannel < nChannels_; iChannel++) {

                std::vector<LauParameter> SVPVect(0);   

                KMatrixParamMap::iterator SVPIter = fScattering_.find(iChannel);
                if (SVPIter != fScattering_.end()) {SVPVect = SVPIter->second;}

                Int_t SVPVectSize = static_cast<Int_t>(SVPVect.size());

                // Scattering matrix is real and symmetric. Start j loop from i.
                for (jChannel = iChannel; jChannel < nChannels_; jChannel++) {

                        Double_t Kij(0.0);

                        // Calculate pole mass summation term
                        for (iPole = 0; iPole < nPoles_; iPole++) {

                                KMatrixParamMap::iterator couplingIter = gCouplings_.find(iPole);

                                std::vector<LauParameter> couplingVect = couplingIter->second;

                                Double_t g_i = couplingVect[iChannel].value();
                                Double_t g_j = couplingVect[jChannel].value();

                                Kij += poleDenomVect_[iPole]*g_i*g_j;
                                if (verbose_) {cout<<"1: Kij for i = "<<iChannel<<", j = "<<jChannel<<" = "<<Kij<<endl;}

                        }

                        if (SVPVectSize != 0 && jChannel < SVPVectSize) {
                                Double_t fij = SVPVect[jChannel].value();
                                Kij += fij*scattSVP_;
                        }

                        Kij *= adlerZeroFactor_;
                        if (verbose_) {cout<<"2: Kij for i = "<<iChannel<<", j = "<<jChannel<<" = "<<Kij<<endl;}

                        // Assign the TMatrix (i,j) element to the variable Kij and Kji (symmetry)
                        ScattKMatrix_(iChannel, jChannel) = Kij;
                        ScattKMatrix_(jChannel, iChannel) = Kij;

                } // j loop

        } // i loop

}

void LauKMatrixPropagator::calcPoleDenomVect(Double_t s)
{
        // Calculate the 1/(m_pole^2 - s) terms for the scattering 
        // and production K-matrix formulae.
        poleDenomVect_.clear();
        Int_t iPole(0);
        for (iPole = 0; iPole < nPoles_; iPole++) {

                Double_t poleTerm = mSqPoles_[iPole] - s;
                Double_t invPoleTerm(0.0);
                if (TMath::Abs(poleTerm) > 1.0e-6) {invPoleTerm = 1.0/poleTerm;}

                poleDenomVect_.push_back(invPoleTerm);

        }

}

Double_t LauKMatrixPropagator::getPoleDenomTerm(Int_t poleIndex) const
{

        if (parametersSet_ == kFALSE) {return 0.0;}

        Double_t poleDenom(0.0);
        poleDenom = poleDenomVect_[poleIndex];
        return poleDenom;

}

Double_t LauKMatrixPropagator::getCouplingConstant(Int_t poleIndex, Int_t channelIndex) const
{

        if (parametersSet_ == kFALSE) {return 0.0;}

        Double_t couplingConst(0.0);
        KMatrixParamMap::const_iterator couplingIter = gCouplings_.find(poleIndex);
        std::vector<LauParameter> couplingVect = couplingIter->second;
        couplingConst = couplingVect[channelIndex].value();

        return couplingConst;

}

Double_t LauKMatrixPropagator::calcSVPTerm(Double_t s, Double_t s0) const
{

        if (parametersSet_ == kFALSE) {return 0.0;}

        // Calculate the "slowly-varying part" (SVP)
        Double_t result(0.0);
        Double_t deltaS = s - s0;
        if (TMath::Abs(deltaS) > 1.0e-6) {
                result = (mSq0_.value() - s0)/deltaS;
        }  

        return result;

}

void LauKMatrixPropagator::updateScattSVPTerm(Double_t s)
{
        // Update the scattering "slowly-varying part" (SVP)
        Double_t s0Scatt = s0Scatt_.value();
        scattSVP_ = this->calcSVPTerm(s, s0Scatt);
}

void LauKMatrixPropagator::updateProdSVPTerm(Double_t s)
{
        // Update the production "slowly-varying part" (SVP)
        Double_t s0Prod = s0Prod_.value();
        prodSVP_ = this->calcSVPTerm(s, s0Prod);
}

void LauKMatrixPropagator::updateAdlerZeroFactor(Double_t s)
{

        // Calculate the multiplicative factor containing various Adler zero
        // constants.
        adlerZeroFactor_ = 0.0;

        Double_t sA0Val = sA0_.value();
        Double_t deltaS = s - sA0Val;
        if (TMath::Abs(deltaS) > 1e-6) {
                adlerZeroFactor_ = (s - sAConst_)*(1.0 - sA0Val)/deltaS;
        }  

}

void LauKMatrixPropagator::calcRhoMatrix(Double_t s)
{

        // Calculate the real and imaginary part of the phase space density
        // diagonal matrix for the given invariant mass squared quantity, s.
        // The matrix can be complex if s is below threshold (so that
        // the amplitude continues analytically).

        // Initialise all entries to zero
        ReRhoMatrix_.Zero(); ImRhoMatrix_.Zero();

        LauComplex rho(0.0, 0.0);
        Int_t phaseSpaceIndex(0);

        for (Int_t iChannel (0); iChannel < nChannels_; ++iChannel) {

                phaseSpaceIndex = phaseSpaceTypes_[iChannel];

                if (phaseSpaceIndex == LauKMatrixPropagator::PiPi) {
                        rho = this->calcPiPiRho(s);
                } else if (phaseSpaceIndex == LauKMatrixPropagator::KK) {
                        rho = this->calcKKRho(s);
                } else if (phaseSpaceIndex == LauKMatrixPropagator::FourPi) {
                        rho = this->calcFourPiRho(s);
                } else if (phaseSpaceIndex == LauKMatrixPropagator::EtaEta) {
                        rho = this->calcEtaEtaRho(s);
                } else if (phaseSpaceIndex == LauKMatrixPropagator::EtaEtaP) {
                        rho = this->calcEtaEtaPRho(s);
                } else if (phaseSpaceIndex == LauKMatrixPropagator::KPi) {
                        rho = this->calcKPiRho(s);
                } else if (phaseSpaceIndex == LauKMatrixPropagator::KEtaP) {
                        rho = this->calcKEtaPRho(s);
                } else if (phaseSpaceIndex == LauKMatrixPropagator::KThreePi) {
                        rho = this->calcKThreePiRho(s);
                }

                if (verbose_) {
                        cout<<"ReRhoMatrix("<<iChannel<<", "<<iChannel<<") = "<<rho.re()<<endl;
                        cout<<"ImRhoMatrix("<<iChannel<<", "<<iChannel<<") = "<<rho.im()<<endl;
                }

                ReRhoMatrix_(iChannel, iChannel) = rho.re();
                ImRhoMatrix_(iChannel, iChannel) = rho.im();    

        }

}

LauComplex LauKMatrixPropagator::calcPiPiRho(Double_t s) const
{
        // Calculate the pipi phase space factor
        LauComplex rho(0.0, 0.0);
        if (TMath::Abs(s) < 1e-10) {return rho;}

        Double_t sqrtTerm = (-m2piSq_/s) + 1.0;
        if (sqrtTerm < 0.0) {
                rho.setImagPart( TMath::Sqrt(-sqrtTerm) );
        } else {
                rho.setRealPart( TMath::Sqrt(sqrtTerm) );
        }

        return rho;
}

LauComplex LauKMatrixPropagator::calcKKRho(Double_t s) const
{
        // Calculate the KK phase space factor
        LauComplex rho(0.0, 0.0);
        if (TMath::Abs(s) < 1e-10) {return rho;}

        Double_t sqrtTerm = (-m2KSq_/s) + 1.0;
        if (sqrtTerm < 0.0) {
                rho.setImagPart( TMath::Sqrt(-sqrtTerm) );
        } else {
                rho.setRealPart( TMath::Sqrt(sqrtTerm) );
        }

        return rho;
}

LauComplex LauKMatrixPropagator::calcFourPiRho(Double_t s) const
{
        // Calculate the 4pi phase space factor. This uses a 6th-order polynomial 
        // parameterisation that approximates the multi-body phase space double integral
        // defined in Eq 4 of the A&S paper hep-ph/0204328. This form agrees with the 
        // BaBar model (another 6th order polynomial from s^4 down to 1/s^2), but avoids the
        // exponential increase at small values of s (~< 0.1) arising from 1/s and 1/s^2.
        // Eq 4 is evaluated for each value of s by assuming incremental steps of 1e-3 GeV^2 
        // for s1 and s2, the invariant energy squared of each of the di-pion states,
        // with the integration limits of s1 = (2*mpi)^2 to (sqrt(s) - 2*mpi)^2 and
        // s2 = (2*mpi)^2 to (sqrt(s) - sqrt(s1))^2. The mass M of the rho is taken to be
        // 0.775 GeV and the energy-dependent width of the 4pi system 
        // Gamma(s) = gamma_0*rho1^3(s), where rho1 = sqrt(1.0 - 4*mpiSq/s) and gamma_0 is 
        // the "width" of the 4pi state at s = 1, which is taken to be 0.3 GeV 
        // (~75% of the total width from PDG estimates of the f0(1370) -> 4pi state).
        // The normalisation term rho_0 is found by ensuring that the phase space integral
        // at s = 1 is equal to sqrt(1.0 - 16*mpiSq/s). Note that the exponent for this 
        // factor in hep-ph/0204328 is wrong; it should be 0.5, i.e. sqrt, not n = 1 to 5.
        // Plotting the value of this double integral as a function of s can then be fitted
        // to a 6th-order polynomial (for s < 1), which is the result used below

        LauComplex rho(0.0, 0.0);
        if (TMath::Abs(s) < 1e-10) {return rho;}

        if (s <= 1.0) {
                Double_t rhoTerm = ((1.07885*s + 0.13655)*s - 0.29744)*s - 0.20840;
                rhoTerm = ((rhoTerm*s + 0.13851)*s - 0.01933)*s + 0.00051;
                // For some values of s (below 2*mpi), this term is a very small 
                // negative number. Check for this and set the rho term to zero.
                if (rhoTerm < 0.0) {rhoTerm = 0.0;}
                rho.setRealPart( rhoTerm );
        } else {
                rho.setRealPart( TMath::Sqrt(1.0 - (fourPiFactor1_/s)) );
        }

        return rho;
}

LauComplex LauKMatrixPropagator::calcEtaEtaRho(Double_t s) const
{
        // Calculate the eta-eta phase space factor
        LauComplex rho(0.0, 0.0);
        if (TMath::Abs(s) < 1e-10) {return rho;}

        Double_t sqrtTerm = (-m2EtaSq_/s) + 1.0;
        if (sqrtTerm < 0.0) {
                rho.setImagPart( TMath::Sqrt(-sqrtTerm) );
        } else {
                rho.setRealPart( TMath::Sqrt(sqrtTerm) );
        }

        return rho;
}

LauComplex LauKMatrixPropagator::calcEtaEtaPRho(Double_t s) const
{
        // Calculate the eta-eta' phase space factor. Note that the
        // mass difference term m_eta - m_eta' is not included,
        // since this corresponds to a "t or u-channel crossing",
        // which means that we cannot simply analytically continue 
        // this part of the phase space factor below threshold, which
        // we can do for s-channel contributions. This is actually an 
        // unsolved problem, e.g. see Guo et al 1409.8652, and 
        // Danilkin et al 1409.7708. Anisovich and Sarantsev in 
        // hep-ph/0204328 "solve" this issue by setting the mass 
        // difference term to unity, which is what we do here...

        LauComplex rho(0.0, 0.0);
        if (TMath::Abs(s) < 1e-10) {return rho;}

        Double_t sqrtTerm = (-mEtaEtaPSumSq_/s) + 1.0;
        if (sqrtTerm < 0.0) {
                rho.setImagPart( TMath::Sqrt(-sqrtTerm) );
        } else {
                rho.setRealPart( TMath::Sqrt(sqrtTerm) );
        }

        return rho;
}


LauComplex LauKMatrixPropagator::calcKPiRho(Double_t s) const
{
        // Calculate the K-pi phase space factor
        LauComplex rho(0.0, 0.0);
        if (TMath::Abs(s) < 1e-10) {return rho;}

        Double_t sqrtTerm1 = (-mKpiSumSq_/s) + 1.0;
        Double_t sqrtTerm2 = (-mKpiDiffSq_/s) + 1.0;
        Double_t sqrtTerm = sqrtTerm1*sqrtTerm2;
        if (sqrtTerm < 0.0) {
                rho.setImagPart( TMath::Sqrt(-sqrtTerm) );
        } else {
                rho.setRealPart( TMath::Sqrt(sqrtTerm) );
        }

        return rho;
}

LauComplex LauKMatrixPropagator::calcKEtaPRho(Double_t s) const
{
        // Calculate the K-eta' phase space factor
        LauComplex rho(0.0, 0.0);
        if (TMath::Abs(s) < 1e-10) {return rho;}

        Double_t sqrtTerm1 = (-mKEtaPSumSq_/s) + 1.0;
        Double_t sqrtTerm2 = (-mKEtaPDiffSq_/s) + 1.0;
        Double_t sqrtTerm = sqrtTerm1*sqrtTerm2;
        if (sqrtTerm < 0.0) {
                rho.setImagPart( TMath::Sqrt(-sqrtTerm) );
        } else {
                rho.setRealPart( TMath::Sqrt(sqrtTerm) );
        }

        return rho;
}

LauComplex LauKMatrixPropagator::calcKThreePiRho(Double_t s) const
{
        // Calculate the Kpipipi + multimeson phase space factor. 
        // Use the simplest definition in hep-ph/9705401 (Eq 14), which is the form 
        // used for the rest of that paper (thankfully, the amplitude does not depend
        // significantly on the form used for the K3pi phase space factor).
        LauComplex rho(0.0, 0.0);
        if (TMath::Abs(s) < 1e-10) {return rho;}

        if (s < 1.44) {

                Double_t powerTerm = (-mK3piDiffSq_/s) + 1.0;
                if (powerTerm < 0.0) {
                        rho.setImagPart( k3piFactor_*TMath::Power(-powerTerm, 2.5) );
                } else {
                        rho.setRealPart( k3piFactor_*TMath::Power(powerTerm, 2.5) );
                }

        } else {
                rho.setRealPart( 1.0 );
        }

        return rho;
}

Bool_t LauKMatrixPropagator::checkPhaseSpaceType(Int_t phaseSpaceInt) const
{
        Bool_t passed(kFALSE);

        if (phaseSpaceInt >= 1 && phaseSpaceInt < LauKMatrixPropagator::TotChannels) {
                passed = kTRUE;
        }

        return passed;
}

LauComplex LauKMatrixPropagator::getTransitionAmp(Double_t s, Int_t channel)
{

        // Get the complex (unitary) transition amplitude T for the given channel
        LauComplex TAmp(0.0, 0.0);

        channel -= 1;
        if (channel < 0 || channel >= nChannels_) {return TAmp;}

        this->getTMatrix(s);
        
        TAmp.setRealPart(ReTMatrix_[index_][channel]);
        TAmp.setImagPart(ImTMatrix_[index_][channel]);

        return TAmp;

}

LauComplex LauKMatrixPropagator::getPhaseSpaceTerm(Double_t s, Int_t channel)
{

        // Get the complex (unitary) transition amplitude T for the given channel
        LauComplex rho(0.0, 0.0);

        channel -= 1;
        if (channel < 0 || channel >= nChannels_) {return rho;}

        // If s has changed from the previous value, recalculate rho
        if (TMath::Abs(s - previousS_) > 1e-6*s) {
                this->calcRhoMatrix(s);
        }

        rho.setRealPart(ReRhoMatrix_[channel][channel]);
        rho.setImagPart(ImRhoMatrix_[channel][channel]);

        return rho;

}

void LauKMatrixPropagator::getTMatrix(const LauKinematics* kinematics) {

        // Find the unitary T matrix, where T = [sqrt(rho)]^{*} T_hat sqrt(rho),
        // and T_hat = (I - i K rho)^-1 * K is the Lorentz-invariant T matrix,
        // which has phase-space factors included (rho). This function is not
        // needed to calculate the K-matrix amplitudes, but allows us
        // to check the variation of T as a function of s (kinematics)

        if (!kinematics) {return;}

        // Get the invariant mass squared (s) from the kinematics object. 
        // Use the resPairAmpInt to find which mass-squared combination to use.
        Double_t s(0.0);

        if (resPairAmpInt_ == 1) {
                s = kinematics->getm23Sq();
        } else if (resPairAmpInt_ == 2) {
                s = kinematics->getm13Sq();
        } else if (resPairAmpInt_ == 3) {
                s = kinematics->getm12Sq();
        }

        this->getTMatrix(s);

}


void LauKMatrixPropagator::getTMatrix(Double_t s)
{

        // Find the unitary transition T matrix, where 
        // T = [sqrt(rho)]^{*} T_hat sqrt(rho), and
        // T_hat = (I - i K rho)^-1 * K is the Lorentz-invariant T matrix,
        // which has phase-space factors included (rho). Note that the first
        // sqrt of the rho matrix is complex conjugated. 

        // This function is not needed to calculate the K-matrix amplitudes, but 
        // allows us to check the variation of T as a function of s (kinematics)

        // Initialse the real and imaginary parts of the T matrix to zero
        ReTMatrix_.Zero(); ImTMatrix_.Zero();

        if (parametersSet_ == kFALSE) {return;}

        // Update K, rho and the propagator (I - i K rho)^-1
        this->updatePropagator(s);
        
        // Find the real and imaginary T_hat matrices
        TMatrixD THatReal = realProp_*ScattKMatrix_;
        TMatrixD THatImag(zeroMatrix_);
        THatImag -= negImagProp_*ScattKMatrix_;

        // Find the square-root of the phase space matrix
        this->getSqrtRhoMatrix();

        // Let sqrt(rho) = A + iB and T_hat = C + iD
        // => T = A(CA-DB) + B(DA+CB) + i[A(DA+CB) + B(DB-CA)]
        TMatrixD CA(THatReal);
        CA *= ReSqrtRhoMatrix_;
        TMatrixD DA(THatImag);
        DA *= ReSqrtRhoMatrix_;
        TMatrixD CB(THatReal);
        CB *= ImSqrtRhoMatrix_;
        TMatrixD DB(THatImag);
        DB *= ImSqrtRhoMatrix_;

        TMatrixD CAmDB(CA);
        CAmDB -= DB;
        TMatrixD DApCB(DA);
        DApCB += CB;
        TMatrixD DBmCA(DB);
        DBmCA -= CA;

        // Find the real and imaginary parts of the transition matrix T
        ReTMatrix_ = ReSqrtRhoMatrix_*CAmDB + ImSqrtRhoMatrix_*DApCB;
        ImTMatrix_ = ReSqrtRhoMatrix_*DApCB + ImSqrtRhoMatrix_*DBmCA;

}

void LauKMatrixPropagator::getSqrtRhoMatrix()
{

        // Find the square root of the (current) phase space matrix so that
        // we can find T = [sqrt(rho)}^{*} T_hat sqrt(rho), where T_hat is the
        // Lorentz-invariant T matrix = (I - i K rho)^-1 * K; note that the first
        // sqrt of rho matrix is complex conjugated

        // If rho = rho_i + i rho_r = a + i b, then sqrt(rho) = c + i d, where
        // c = sqrt(0.5*(r+a)) and d = sqrt(0.5(r-a)), where r = sqrt(a^2 + b^2).
        // Since rho is diagonal, then the square root of rho will also be diagonal,
        // with its real and imaginary matrix elements equal to c and d, respectively

        // Initialise the real and imaginary parts of the square root of 
        // the rho matrix to zero
        ReSqrtRhoMatrix_.Zero(); ImSqrtRhoMatrix_.Zero();

        for (Int_t iChannel (0); iChannel < nChannels_; ++iChannel) {

                Double_t realRho = ReRhoMatrix_[iChannel][iChannel];
                Double_t imagRho = ImRhoMatrix_[iChannel][iChannel];

                Double_t rhoMag = sqrt(realRho*realRho + imagRho*imagRho);
                Double_t rhoSum = rhoMag + realRho;
                Double_t rhoDiff = rhoMag - realRho;

                Double_t reSqrtRho(0.0), imSqrtRho(0.0);
                if (rhoSum > 0.0) {reSqrtRho = sqrt(0.5*rhoSum);}
                if (rhoDiff > 0.0) {imSqrtRho = sqrt(0.5*rhoDiff);}

                ReSqrtRhoMatrix_[iChannel][iChannel] = reSqrtRho;
                ImSqrtRhoMatrix_[iChannel][iChannel] = imSqrtRho;

        }


}