double Analytic_Mt2_2220_Calculator::mt2_2220_Sq(const TwoVector& visA,
const TwoVector& visB,
const TwoVector& ptmiss){{
// WARNING -- FOR TESTING PURPOSES I AM HIDING THE INPUTS!
/* For testing, the following example matches 20100622a-mt2 series mathematica notebooks */
/* const TwoVector ptmiss(1.0,0.0);
const TwoVector visA(TwoVector(cos(1.0),sin(1.0))*1.2);
const TwoVector visB(TwoVector(cos(2.1),sin(2.1))*0.8);
*/
m_info = Info();
const double a = visA.pt();
const double b = visB.pt();
const double magPtmiss = ptmiss.pt();
const TwoVector movedP(-(ptmiss.py()),+(ptmiss.px()));
const TwoVector movedB(-(visB.py()),+(visB.px()));
const double adotp = visA.dot(ptmiss);
const double bdotp = visB.dot(ptmiss);
const double ahdotbh = visA.dot(visB)/sqrt((visA.ptsq())*(visB.ptsq()));
const double eap = visA.dot(movedP);
const double ebp = visB.dot(movedP);
const double eahbh = visA.dot(movedB)/sqrt((visA.ptsq())*(movedB.ptsq()));
// For the special case of ma=0, mb=0, chi=0, MT2 is *zero* whenever the ptmiss vector is "between" visA and visB, since this always admits a solution in which p and q are parallel to a and b (respectively) making MT_a = MT_b = 0. So find coeffs of ptmiss in the a,b basis, and see if they are both positive. If they are, return zero:
if (eap/eahbh >= 0 && -ebp/eahbh >=0) {
return 0;
}
if (m_debugMode) {
std::cout << "example=({"
<< " a -> " << a
<< ", b -> " << b
<< ", p -> " << magPtmiss
<< ", thetaap -> ArcTan[" << adotp << "," << eap << "]"
<< ", thetabp -> ArcTan[" << bdotp << "," << ebp << "]"
<< "})/.correct" << std::endl;
std::cout << "adotp eap " << adotp << " " << eap << std::endl;
std::cout << "bdotp ebp " << bdotp << " " << ebp << std::endl;
std::cout << "ahdotbh eahbh " << ahdotbh << " " << eahbh << std::endl;
}
const double deltadotp = adotp - bdotp;
const double sigmadotp = adotp + bdotp;
const double esigmap = eap + ebp;
const double edeltap = eap - ebp;
const double Kss = - deltadotp * eahbh;
const double Kcc = - esigmap * ahdotbh;
const double Ks = sigmadotp;
const double Kc = esigmap;
const double Kcs = -sigmadotp * ahdotbh - edeltap * eahbh;
const double K1 = 0;
m_info.Kss = Kss;
m_info.Kcc = Kcc;
m_info.Ks = Ks;
m_info.Kc = Kc;
m_info.Kcs = Kcs;
m_info.K1 = K1;
if (m_debugMode) {
std::cout << "{Nss, Ncc, Ns, Nc, Ncs, N1} " << Kss << " " << Kcc << " " << Ks << " " << Kc << " " << Kcs << " " << K1 << std::endl;
}
// For a polynomial where Nss is the coeff of sin^2 and Nsc is the coefficent of the sin cos term, etc, then if you substitute sin=sqrt(1-cos^2) and then rearrange for a polynomial exclusively in cos, you get a polynomial with the following coeffs ... see 20100622a-mt2-masslessinvisiblesandmasslesschi-lookingfornicetriganswers-dumpdotprods-1.nb
/* For Cos coeffs
const double coeffCos0 = (K1 - Ks + Kss)*(K1 + Ks + Kss);
const double coeffCos1 = 2.*(-Kcs* Ks + Kc*(K1 + Kss));
const double coeffCos2 = square(Kc) - square(Kcs) + square(Ks) + 2.*(Kcc - Kss)*(K1 + Kss);
const double coeffCos3 = 2.*(Kcs*Ks + Kc*(Kcc - Kss));
const double coeffCos4 = square(Kcs) + square(Kcc - Kss);
*/
// For Sine Coeffs
const double coeffSin0 = (K1 - Kc + Kcc)*(K1 + Kc + Kcc);
const double coeffSin1 = 2.*(-Kcs* Kc + Ks*(K1 + Kcc));
const double coeffSin2 = square(Ks) - square(Kcs) + square(Kc) + 2.*(Kss - Kcc)*(K1 + Kcc);
const double coeffSin3 = 2.*(Kcs*Kc + Ks*(Kss - Kcc));
const double coeffSin4 = square(Kcs) + square(Kss - Kcc);
// NB we could get the same for coeffs of sin by interchanging s with c (and notint that Ksc should be re-written as Kcs).
// now find the roots of this quartic by the method of Farrari.
const double AA=coeffSin4;
const double BB=coeffSin3;
const double CC=coeffSin2;
const double DD=coeffSin1;
const double EE=coeffSin0;
m_info.A = AA;
m_info.B = BB;
m_info.C = CC;
m_info.D = DD;
m_info.E = EE;
std::vector<double> goodSinValues;
if (m_rootFindingMethod == FERRARI) {
std::vector<double> sinValues(4);
// FIND ROOTS OURSELVES
if (m_debugMode) {
std::cout << "{AA,BB,CC,DD,EE} "
<< AA/magPtmiss/magPtmiss << " "
<< BB/magPtmiss/magPtmiss << " "
<< CC/magPtmiss/magPtmiss << " "
<< DD/magPtmiss/magPtmiss << " "
<< EE/magPtmiss/magPtmiss << " "
<< std::endl;
}
const double aa = -(3./8.)*square(BB/AA) + CC/AA;
const double bb = +(1./8.)*cube(BB/AA) - BB*CC/(2.* square(AA)) + DD/AA;
const double gg =
- (3./256.)*square(square(BB/AA))
+ (1./16.)*CC*square(BB)/cube(AA)
- BB*DD/(4. *square(AA)) + EE/AA;
if (m_debugMode) {
std::cout << "{aa,bb,gg} " << aa << " " << bb << " " << gg << std::endl;
}
const double PP = -square(aa)/12. - gg;
const double QQ = -cube(aa)/108. + aa*gg/3. - square(bb)/8.;
const double BITINROOT = square(QQ)/4. + cube(PP)/27.;
// BITINROOT might be negative, and it is OK if it is, because things should be real again by the time we contstruct yy. So temprarily drift into complex numbers:
if (m_debugMode) {
std::cout << "PP QQ BITINROOT " << PP <<" " << QQ << " " << BITINROOT << std::endl;
}
/* THIS IS FOR REAL ONLY
const double RR = -QQ/2. + sqrt(BITINROOT);
if (m_debugMode) {
std::cout << "PP QQ BITINROOT RR " << PP << " " << QQ << " " << BITINROOT << " " << RR << std::endl;
}
const double UU = pow(RR,1./3.);
const double yy = -(5./6.)*aa + UU - PP/(3.*UU);
*/
const std::complex<double> complexBITINROOT(BITINROOT,0);
const std::complex<double> complexRR = sqrt(complexBITINROOT)-QQ/2.;
const std::complex<double> complexUU = pow(complexRR,1./3.);
// Note the special line should have a separate case to deal with UU=0.
std::complex<double> complexyy = + complexUU - PP/(complexUU*3.) -(5./6.)*aa ;
if (m_debugMode) {
std::cout << "complexBITINROOT " << complexBITINROOT << std::endl;
std::cout << "complexRR " << complexRR << std::endl;
std::cout << "complexUU " << complexUU << std::endl;
std::cout << "complexyy " << complexyy << std::endl;
}
double yy = complexyy.real();
double tmp1 = aa + 2.*yy;
if (tmp1<0) {
if (m_debugMode) {
std::cout << "WARNING: Debug message from Analytic_Mt2_2220_Calculator ... send this infor to Lester. About to throw exception." << std::endl;
std::cout << "example=({"
<< " a -> " << a
<< ", b -> " << b
<< ", p -> " << magPtmiss
<< ", thetaap -> ArcTan[" << adotp << "," << eap << "]"
<< ", thetabp -> ArcTan[" << bdotp << "," << ebp << "]"
<< "})/.correct" << std::endl;
std::cout << "WARNING: Oh dear ... a thing that should not be negative is negative and has val " << tmp1 << " ! Attempting to fix." << std::endl;
std::cout << "WARNING: A this point PP = " << PP << " complexUU = " << complexUU << " QQ = " << QQ << " aa = " << aa << " complexyy = " << complexyy << " tmp1 = " << tmp1 << std::endl;
}
// This is most probably a sign that UU was getting very close to zero, and the subtraction UU-PP/(3*UU) became unstable. In the limit of small UU, PP/(3UU) goes to the cube root of QQ (and QQ should be positive) .. so let's try that again:
complexyy = + complexUU - pow(std::complex<double>(QQ,0),1./3.) -(5./6.)*aa ;
yy = complexyy.real();
tmp1 = aa + 2.*yy;
// is tmp1 STILL less than zero?
if (tmp1<0) {
if (m_debugMode) {
std::cout << "WARNING: Oh dear, even after attempt to fix things, tmp1 is still negative. Now complexyy = " << complexyy << " tmp1 = " << tmp1 << std::endl;
}
// let us hope this is rounding, so that we may proceed without crashing.
tmp1=0;
} else {
std::cout << "WARNING: appears to have been fixed" << std::endl;
}
}
const double WW = sqrt(tmp1);
if (m_debugMode) {
std::cout << "yy WW " << " " << yy << " " << WW << std::endl;
}
{
unsigned int rootcount = 0;
unsigned int goodrootcount = 0;
for (int sign1 = -1; sign1<=+1; sign1 += 2) {
for (int sign2 = -1; sign2<=+1; sign2 += 2) {
const double tmp2 = -(3.*aa + 2.* yy + sign1 * 2.*bb/WW);
const double root =
- BB/(4.* AA)
+ (sign1 * WW
-
sign2 *sqrt(tmp2)
)/2. ;
sinValues[rootcount] = root;
++rootcount;
if (m_debugMode) {
std::cout << "A ROOT = " << root << std::endl;
}
if (tmp2<0) {
if (m_debugMode) {
std::cout << "ROOT " << rootcount << " is complex." << std::endl;
}
}else {
if (m_debugMode) {
std::cout << "ROOT " << rootcount << " is real." << std::endl;
}
goodSinValues.push_back(root);
++goodrootcount;
}
}
}
}
// by this stage, goodSinValues should have a set of good sin vales, roots,
// END OF finding quartic roots by OURSELVES
} else {
// get someone else to calculate the quartic roots:
std::complex<double> r1;
std::complex<double> r2;
std::complex<double> r3;
std::complex<double> r4;
Mt2::quartic::r8poly4_root(AA,BB,CC,DD,EE,r1,r2,r3,r4);
m_info.r1 = r1;
m_info.r2 = r2;
m_info.r3 = r3;
m_info.r4 = r4;
// Now how do we separate the real roots from the "almost" real roots from the complex roots?
insertIfOK(r1,goodSinValues);
insertIfOK(r2,goodSinValues);
insertIfOK(r3,goodSinValues);
insertIfOK(r4,goodSinValues);
}
std::set<double> answers;
for (int i=0; i<(int)goodSinValues.size(); ++i) {
if (m_debugMode) {
std::cout << "GOOD ROOT " << i+1 << " = " << goodSinValues[i] << std::endl;
}
// this valuie for sin(theta) could correspond EITHER to
// cos(theta) = +sqrt(1-sin^2(theta))
// OR to
// cos(theta) = -sqrt(1-sin^2(theta))
// since when we squared our multinomial in {ss, cc, sc, c, s} to get rid of the sqrt() signs, we lost the sign in front of them,
const double s = goodSinValues[i];
const double modc = sqrt(1.-s*s); /* modulus of cosine */
// In the case that the positive square root is needed, then the following two quantities should be identical, otherwise they should differ in sign:
const double LHS = K1 + Kcc + s *(Ks + (-Kcc + Kss)* s);
const double RHS = -(Kc + Kcs* s) * modc;
if (m_debugMode) {
std::cout << "LHS RHS " << LHS << " " << RHS << std::endl;
}
const bool usePositiveSquareRoot = (fabs(LHS-RHS)<fabs(LHS+RHS));
if (m_debugMode) {
if (usePositiveSquareRoot) {
std::cout << "NEED positive SQUARE ROOT" << std::endl;
} else {
std::cout << "NEED negative SQUARE ROOT" << std::endl;
}
}
const double sinTheta = s;
const double cosTheta = (usePositiveSquareRoot?+modc:-modc);
if (m_debugMode) {
std::cout << "cosTheta = " << cosTheta << " sinTheta = " << sinTheta << std::endl;
}
// Now the balanced condition implies that 2 a_mu p^mu = 2 b_mu q^mu hence
//const double p_over_q =
// (b*(1 - cosTheta * ahdotbh - sinTheta * eahbh)) /
// (a*(1 - cosTheta * ahdotbh + sinTheta * eahbh)) ;
const double p_over_q = -
(( cosTheta * eap + sinTheta * adotp) * b )/
(( cosTheta * ebp + sinTheta * bdotp) * a );
if (m_debugMode) {
std::cout << "a b = " << a << " " << b << std::endl;
std::cout << "p over q = " << p_over_q << std::endl;
}
// The following quantity should be zero as it (times q) is the cpt of p+q in the direc perp to ptmiss:
const double shouldBeZero = p_over_q*(cosTheta * ebp + sinTheta * bdotp)/b + (cosTheta*eap + sinTheta*adotp)/a;
if (m_debugMode) {
std::cout << "should be zero : " << shouldBeZero << std::endl;
}
// The following quantity should NOT be zero as it (times q) is the square of the cpt of p+q in the direc of ptmiss:
const double thing = p_over_q*(cosTheta * bdotp - sinTheta * ebp)/b + (cosTheta*adotp - sinTheta*eap)/a;
if (m_debugMode) {
std::cout << "should NOT be zero : " << thing << std::endl;
}
// thing*q = magPtmiss^2 ... hence ... q = magPtmiss^2 /thing;
const double q= square(magPtmiss) /thing;
const double p = p_over_q * q;
if (m_debugMode) {
std::cout << "p and q are " << p << " " << q << std::endl;
}
if (p>=0 && q>=0) {
const double MT2SQ1 = 2.*a*p*(1 - cosTheta * ahdotbh - sinTheta * eahbh);
const double MT2SQ2 = 2.*b*q*(1 - cosTheta * ahdotbh + sinTheta * eahbh);
if (m_debugMode) {
std::cout << "MT2SQ 1 and 2 are " << MT2SQ1 << " " << MT2SQ2 << std::endl;
}
// MT2SQ1 and MT2SQ2 should be equal, but due to rounding errors they may not be ... so lets return the mean:
answers.insert( (MT2SQ1+MT2SQ2)*0.5);
}
}
if (answers.empty()) {
// If we got here, then there was no real root with p and q >0 ... so something went wrong:
if (m_debugMode) {
std::cout << "Warning ... no physical root found in Analytic_Mt2_2220_Calculator!" << std::endl;
}
throw no_mt2_value_found; // no answer found
} else {
// answers should be sorted, and the first one should be the smallest, so
return *(answers.begin());
}
}} // end of mt2_2220_Sq method
double Basic_MCT2_332_Calculator::mct2_332_Sq(const LorentzTransverseVector& visA,
const LorentzTransverseVector& visB,
const TwoVector& ptmiss,
const double mEachInvisible){
mCT2Fcn theFCN(ptmiss.px(),
ptmiss.py(),
mEachInvisible,
visA.px(),
visA.py(),
visA.mass(),
visB.px(),
visB.py(),
visB.mass());
const double massScale = (
ptmiss.pt() +
mEachInvisible +
visA.pt() +
visA.mass() +
visB.pt() +
visB.mass()
)/6.0;
// DANG! Try to get rid of Minuit output:
//std::ofstream DANG_log("/dev/null");
//std::streambuf * DANG_save = std::cerr.rdbuf();
//if (DANG_log.is_open()) {
// std::cerr.rdbuf(DANG_log.rdbuf());
// }
double guessx = 0.5*(ptmiss.px());
double guessy = 0.5*(ptmiss.py());
MnUserParameters upar;
upar.Add("etx", guessx, 0.02*massScale);
upar.Add("ety", guessy, 0.02*massScale);
const int highQuality=2;
// Usually migrad produces the best minumum.
// But when the minimum is in a fold, migrad can fail badly.
// On the fold, simplex does well. We therefore do both separately
// and record the answer of the one that did best.
// Further to the above notes, it now seems that by choosing the massScale sensibly, and by making the "tolerance" (the second argument to the call that extracts the FunctionMinimum below) much smaller, the simplex algorithm seems to work so well that we don't need the migrad algorithm. This is good news, as the migrad algorithm produces lots of error output that we can't get rid of.
MnSimplex simplex(theFCN, upar, highQuality);
FunctionMinimum minS = simplex(0,massScale*0.000001);
//const double etxAtMin = minS.UserState().Value("etx");
//const double etyAtMin = minS.UserState().Value("ety");
//MnMigrad migrad(theFCN, upar, highQuality);
//FunctionMinimum minM = migrad(0,massScale*0.000001);
//const double best = fmin(minS.Fval(), minM.Fval());
const double best = minS.Fval();
// DANG! Undoing our attempt to get rid of Minuit output:
//if (DANG_log.is_open()) {
// std::cerr.rdbuf(DANG_save);
//}
return best;
}