mt2w_bisect.cc

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/***********************************************************************/
/*                                                                     */
/*              Finding MT2W                                           */
/*              Reference:  arXiv:1203.4813 [hep-ph]                   */
/*              Authors: Yang Bai, Hsin-Chia Cheng,                    */
/*                       Jason Gallicchio, Jiayin Gu                   */
/*              Based on MT2 by: Hsin-Chia Cheng, Zhenyu Han           */
/*              May 8, 2012, v1.00a                                    */
/*                                                                     */
/***********************************************************************/

/*******************************************************************************
  Usage: 

  1. Define an object of type "mt2w":
     
     mt2w_bisect::mt2w mt2w_event;
 
  2. Set momenta:
 
     mt2w_event.set_momenta(pl,pb1,pb2,pmiss);
 
     where array pl[0..3], pb1[0..3], pb2[0..3] contains (E,px,py,pz), pmiss[0..2] contains (0,px,py) 
     for the visible particles and the missing momentum. pmiss[0] is not used. 
     All quantities are given in double.    
     
     (Or use non-pointer method to do the same.)

  3. Use mt2w::get_mt2w() to obtain the value of mt2w:

     double mt2w_value = mt2w_event.get_mt2w();       
          
*******************************************************************************/

#include <iostream>
#include <cmath>
#include "PhysicsTools/Heppy/interface/mt2w_bisect.h"

using namespace std;

namespace heppy {

  namespace mt2w_bisect {

    /*The user can change the desired precision below, the larger one of the following two definitions is used. Relative precision less than 0.00001 is not guaranteed to be achievable--use with caution*/

    const float mt2w::RELATIVE_PRECISION = 0.00001;
    const float mt2w::ABSOLUTE_PRECISION = 0.0;
    const float mt2w::MIN_MASS = 0.1;
    const float mt2w::ZERO_MASS = 0.0;
    const float mt2w::SCANSTEP = 0.1;

    mt2w::mt2w(double upper_bound, double error_value, double scan_step) {
      solved = false;
      momenta_set = false;
      mt2w_b = 0.;                      // The result field.  Start it off at zero.
      this->upper_bound = upper_bound;  // the upper bound of search for MT2W, default value is 500 GeV
      this->error_value =
          error_value;  // if we couldn't find any compatible region below the upper_bound, output mt2w = error_value;
      this->scan_step = scan_step;  // if we need to scan to find the compatible region, this is the step of the scan
    }

    double mt2w::get_mt2w() {
      if (!momenta_set) {
        cout << " Please set momenta first!" << endl;
        return error_value;
      }

      if (!solved)
        mt2w_bisect();
      return mt2w_b;
    }

    void mt2w::set_momenta(double *pl, double *pb1, double *pb2, double *pmiss) {
      // Pass in pointers to 4-vectors {E, px, py, px} of doubles.
      // and pmiss must have [1] and [2] components for x and y.  The [0] component is ignored.
      set_momenta(
          pl[0], pl[1], pl[2], pl[3], pb1[0], pb1[1], pb1[2], pb1[3], pb2[0], pb2[1], pb2[2], pb2[3], pmiss[1], pmiss[2]);
    }

    void mt2w::set_momenta(double El,
                           double plx,
                           double ply,
                           double plz,
                           double Eb1,
                           double pb1x,
                           double pb1y,
                           double pb1z,
                           double Eb2,
                           double pb2x,
                           double pb2y,
                           double pb2z,
                           double pmissx,
                           double pmissy) {
      solved = false;  //reset solved tag when momenta are changed.
      momenta_set = true;

      double msqtemp;  //used for saving the mass squared temporarily

      //l is the visible lepton

      this->El = El;
      this->plx = plx;
      this->ply = ply;
      this->plz = plz;

      Elsq = El * El;

      msqtemp = El * El - plx * plx - ply * ply - plz * plz;
      if (msqtemp > 0.0) {
        mlsq = msqtemp;
      } else {
        mlsq = 0.0;
      }                 //mass squared can not be negative
      ml = sqrt(mlsq);  // all the input masses are calculated from sqrt(p^2)

      //b1 is the bottom on the same side as the visible lepton

      this->Eb1 = Eb1;
      this->pb1x = pb1x;
      this->pb1y = pb1y;
      this->pb1z = pb1z;

      Eb1sq = Eb1 * Eb1;

      msqtemp = Eb1 * Eb1 - pb1x * pb1x - pb1y * pb1y - pb1z * pb1z;
      if (msqtemp > 0.0) {
        mb1sq = msqtemp;
      } else {
        mb1sq = 0.0;
      }                   //mass squared can not be negative
      mb1 = sqrt(mb1sq);  // all the input masses are calculated from sqrt(p^2)

      //b2 is the other bottom (paired with the invisible W)

      this->Eb2 = Eb2;
      this->pb2x = pb2x;
      this->pb2y = pb2y;
      this->pb2z = pb2z;

      Eb2sq = Eb2 * Eb2;

      msqtemp = Eb2 * Eb2 - pb2x * pb2x - pb2y * pb2y - pb2z * pb2z;
      if (msqtemp > 0.0) {
        mb2sq = msqtemp;
      } else {
        mb2sq = 0.0;
      }                   //mass squared can not be negative
      mb2 = sqrt(mb2sq);  // all the input masses are calculated from sqrt(p^2)

      //missing pt

      this->pmissx = pmissx;
      this->pmissy = pmissy;

      //set the values of masses

      mv = 0.0;   //mass of neutrino
      mw = 80.4;  //mass of W-boson

      //precision?

      if (ABSOLUTE_PRECISION > 100. * RELATIVE_PRECISION)
        precision = ABSOLUTE_PRECISION;
      else
        precision = 100. * RELATIVE_PRECISION;
    }

    void mt2w::mt2w_bisect() {
      solved = true;
      cout.precision(11);

      // In normal running, mtop_high WILL be compatible, and mtop_low will NOT.
      double mtop_high = upper_bound;  //set the upper bound of the search region
      double mtop_low;                 //the lower bound of the search region is best chosen as m_W + m_b

      if (mb1 >= mb2) {
        mtop_low = mw + mb1;
      } else {
        mtop_low = mw + mb2;
      }

      // The following if and while deal with the case where there might be a compatable region
      // between mtop_low and 500 GeV, but it doesn't extend all the way up to 500.
      //

      // If our starting high guess is not compatible, start the high guess from the low guess...
      if (teco(mtop_high) == 0) {
        mtop_high = mtop_low;
      }

      // .. and scan up until a compatible high bound is found.
      //We can also raise the lower bound since we scaned over a region that is not compatible
      while (teco(mtop_high) == 0 && mtop_high < upper_bound + 2. * scan_step) {
        mtop_low = mtop_high;
        mtop_high = mtop_high + scan_step;
      }

      // if we can not find a compatible region under the upper bound, output the error value
      if (mtop_high > upper_bound) {
        mt2w_b = error_value;
        return;
      }

      // Once we have an compatible mtop_high, we can find mt2w using bisection method
      while (mtop_high - mtop_low > precision) {
        double mtop_mid, teco_mid;
        //bisect
        mtop_mid = (mtop_high + mtop_low) / 2.;
        teco_mid = teco(mtop_mid);

        if (teco_mid == 0) {
          mtop_low = mtop_mid;
        } else {
          mtop_high = mtop_mid;
        }
      }
      mt2w_b = mtop_high;  //output the value of mt2w
      return;
    }

    // for a given event, teco ( mtop ) gives 1 if trial top mass mtop is compatible, 0 if mtop is not.

    int mt2w::teco(double mtop) {
      //first test if mtop is larger than mb+mw

      if (mtop < mb1 + mw || mtop < mb2 + mw) {
        return 0;
      }

      //define delta for convenience, note the definition is different from the one in mathematica code by 2*E^2_{b2}

      double ETb2sq = Eb2sq - pb2z * pb2z;  //transverse energy of b2
      double delta = (mtop * mtop - mw * mw - mb2sq) / (2. * ETb2sq);

      //del1 and del2 are \Delta'_1 and \Delta'_2 in the notes eq. 10,11

      double del1 = mw * mw - mv * mv - mlsq;
      double del2 = mtop * mtop - mw * mw - mb1sq - 2 * (El * Eb1 - plx * pb1x - ply * pb1y - plz * pb1z);

      // aa bb cc are A B C in the notes eq.15

      double aa = (El * pb1x - Eb1 * plx) / (Eb1 * plz - El * pb1z);
      double bb = (El * pb1y - Eb1 * ply) / (Eb1 * plz - El * pb1z);
      double cc = (El * del2 - Eb1 * del1) / (2. * Eb1 * plz - 2. * El * pb1z);

      //calculate coefficients for the two quadratic equations (ellipses), which are
      //
      //  a1 x^2 + 2 b1 x y + c1 y^2 + 2 d1 x + 2 e1 y + f1 = 0 ,  from the 2 steps decay chain (with visible lepton)
      //
      //  a2 x^2 + 2 b2 x y + c2 y^2 + 2 d2 x + 2 e2 y + f2 <= 0 , from the 1 stop decay chain (with W missing)
      //
      //  where x and y are px and py of the neutrino on the visible lepton chain

      a1 = Eb1sq * (1. + aa * aa) - (pb1x + pb1z * aa) * (pb1x + pb1z * aa);
      b1 = Eb1sq * aa * bb - (pb1x + pb1z * aa) * (pb1y + pb1z * bb);
      c1 = Eb1sq * (1. + bb * bb) - (pb1y + pb1z * bb) * (pb1y + pb1z * bb);
      d1 = Eb1sq * aa * cc - (pb1x + pb1z * aa) * (pb1z * cc + del2 / 2.0);
      e1 = Eb1sq * bb * cc - (pb1y + pb1z * bb) * (pb1z * cc + del2 / 2.0);
      f1 = Eb1sq * (mv * mv + cc * cc) - (pb1z * cc + del2 / 2.0) * (pb1z * cc + del2 / 2.0);

      //  First check if ellipse 1 is real (don't need to do this for ellipse 2, ellipse 2 is always real for mtop > mw+mb)

      double det1 = (a1 * (c1 * f1 - e1 * e1) - b1 * (b1 * f1 - d1 * e1) + d1 * (b1 * e1 - c1 * d1)) / (a1 + c1);

      if (det1 > 0.0) {
        return 0;
      }

      //coefficients of the ellptical region

      a2 = 1 - pb2x * pb2x / (ETb2sq);
      b2 = -pb2x * pb2y / (ETb2sq);
      c2 = 1 - pb2y * pb2y / (ETb2sq);

      // d2o e2o f2o are coefficients in the p2x p2y plane (p2 is the momentum of the missing W-boson)
      // it is convenient to calculate them first and transfer the ellipse to the p1x p1y plane
      d2o = -delta * pb2x;
      e2o = -delta * pb2y;
      f2o = mw * mw - delta * delta * ETb2sq;

      d2 = -d2o - a2 * pmissx - b2 * pmissy;
      e2 = -e2o - c2 * pmissy - b2 * pmissx;
      f2 = a2 * pmissx * pmissx + 2 * b2 * pmissx * pmissy + c2 * pmissy * pmissy + 2 * d2o * pmissx +
           2 * e2o * pmissy + f2o;

      //find a point in ellipse 1 and see if it's within the ellipse 2, define h0 for convenience
      double x0, h0, y0, r0;
      x0 = (c1 * d1 - b1 * e1) / (b1 * b1 - a1 * c1);
      h0 = (b1 * x0 + e1) * (b1 * x0 + e1) - c1 * (a1 * x0 * x0 + 2 * d1 * x0 + f1);
      if (h0 < 0.0) {
        return 0;
      }  // if h0 < 0, y0 is not real and ellipse 1 is not real, this is a redundant check.
      y0 = (-b1 * x0 - e1 + sqrt(h0)) / c1;
      r0 = a2 * x0 * x0 + 2 * b2 * x0 * y0 + c2 * y0 * y0 + 2 * d2 * x0 + 2 * e2 * y0 + f2;
      if (r0 < 0.0) {
        return 1;
      }  // if the point is within the 2nd ellipse, mtop is compatible

      //obtain the coefficients for the 4th order equation
      //devided by Eb1^n to make the variable dimensionless
      long double A4, A3, A2, A1, A0;

      A4 = -4 * a2 * b1 * b2 * c1 + 4 * a1 * b2 * b2 * c1 + a2 * a2 * c1 * c1 + 4 * a2 * b1 * b1 * c2 -
           4 * a1 * b1 * b2 * c2 - 2 * a1 * a2 * c1 * c2 + a1 * a1 * c2 * c2;

      A3 = (-4 * a2 * b2 * c1 * d1 + 8 * a2 * b1 * c2 * d1 - 4 * a1 * b2 * c2 * d1 - 4 * a2 * b1 * c1 * d2 +
            8 * a1 * b2 * c1 * d2 - 4 * a1 * b1 * c2 * d2 - 8 * a2 * b1 * b2 * e1 + 8 * a1 * b2 * b2 * e1 +
            4 * a2 * a2 * c1 * e1 - 4 * a1 * a2 * c2 * e1 + 8 * a2 * b1 * b1 * e2 - 8 * a1 * b1 * b2 * e2 -
            4 * a1 * a2 * c1 * e2 + 4 * a1 * a1 * c2 * e2) /
           Eb1;

      A2 = (4 * a2 * c2 * d1 * d1 - 4 * a2 * c1 * d1 * d2 - 4 * a1 * c2 * d1 * d2 + 4 * a1 * c1 * d2 * d2 -
            8 * a2 * b2 * d1 * e1 - 8 * a2 * b1 * d2 * e1 + 16 * a1 * b2 * d2 * e1 + 4 * a2 * a2 * e1 * e1 +
            16 * a2 * b1 * d1 * e2 - 8 * a1 * b2 * d1 * e2 - 8 * a1 * b1 * d2 * e2 - 8 * a1 * a2 * e1 * e2 +
            4 * a1 * a1 * e2 * e2 - 4 * a2 * b1 * b2 * f1 + 4 * a1 * b2 * b2 * f1 + 2 * a2 * a2 * c1 * f1 -
            2 * a1 * a2 * c2 * f1 + 4 * a2 * b1 * b1 * f2 - 4 * a1 * b1 * b2 * f2 - 2 * a1 * a2 * c1 * f2 +
            2 * a1 * a1 * c2 * f2) /
           Eb1sq;

      A1 = (-8 * a2 * d1 * d2 * e1 + 8 * a1 * d2 * d2 * e1 + 8 * a2 * d1 * d1 * e2 - 8 * a1 * d1 * d2 * e2 -
            4 * a2 * b2 * d1 * f1 - 4 * a2 * b1 * d2 * f1 + 8 * a1 * b2 * d2 * f1 + 4 * a2 * a2 * e1 * f1 -
            4 * a1 * a2 * e2 * f1 + 8 * a2 * b1 * d1 * f2 - 4 * a1 * b2 * d1 * f2 - 4 * a1 * b1 * d2 * f2 -
            4 * a1 * a2 * e1 * f2 + 4 * a1 * a1 * e2 * f2) /
           (Eb1sq * Eb1);

      A0 = (-4 * a2 * d1 * d2 * f1 + 4 * a1 * d2 * d2 * f1 + a2 * a2 * f1 * f1 + 4 * a2 * d1 * d1 * f2 -
            4 * a1 * d1 * d2 * f2 - 2 * a1 * a2 * f1 * f2 + a1 * a1 * f2 * f2) /
           (Eb1sq * Eb1sq);

      long double A3sq;
      /*
    long  double A0sq, A1sq, A2sq, A3sq, A4sq;
    A0sq = A0*A0;
    A1sq = A1*A1;
    A2sq = A2*A2;
    A4sq = A4*A4;
    */
      A3sq = A3 * A3;

      long double B3, B2, B1, B0;
      B3 = 4 * A4;
      B2 = 3 * A3;
      B1 = 2 * A2;
      B0 = A1;

      long double C2, C1, C0;
      C2 = -(A2 / 2 - 3 * A3sq / (16 * A4));
      C1 = -(3 * A1 / 4. - A2 * A3 / (8 * A4));
      C0 = -A0 + A1 * A3 / (16 * A4);

      long double D1, D0;
      D1 = -B1 - (B3 * C1 * C1 / C2 - B3 * C0 - B2 * C1) / C2;
      D0 = -B0 - B3 * C0 * C1 / (C2 * C2) + B2 * C0 / C2;

      long double E0;
      E0 = -C0 - C2 * D0 * D0 / (D1 * D1) + C1 * D0 / D1;

      long double t1, t2, t3, t4, t5;
      //find the coefficients for the leading term in the Sturm sequence
      t1 = A4;
      t2 = A4;
      t3 = C2;
      t4 = D1;
      t5 = E0;

      //The number of solutions depends on diffence of number of sign changes for x->Inf and x->-Inf
      int nsol;
      nsol = signchange_n(t1, t2, t3, t4, t5) - signchange_p(t1, t2, t3, t4, t5);

      //Cannot have negative number of solutions, must be roundoff effect
      if (nsol < 0)
        nsol = 0;

      int out;
      if (nsol == 0) {
        out = 0;
      }  //output 0 if there is no solution, 1 if there is solution
      else {
        out = 1;
      }

      return out;
    }

    inline int mt2w::signchange_n(long double t1, long double t2, long double t3, long double t4, long double t5) {
      int nsc;
      nsc = 0;
      if (t1 * t2 > 0)
        nsc++;
      if (t2 * t3 > 0)
        nsc++;
      if (t3 * t4 > 0)
        nsc++;
      if (t4 * t5 > 0)
        nsc++;
      return nsc;
    }
    inline int mt2w::signchange_p(long double t1, long double t2, long double t3, long double t4, long double t5) {
      int nsc;
      nsc = 0;
      if (t1 * t2 < 0)
        nsc++;
      if (t2 * t3 < 0)
        nsc++;
      if (t3 * t4 < 0)
        nsc++;
      if (t4 * t5 < 0)
        nsc++;
      return nsc;
    }

  }  //end namespace mt2w_bisect

}  // namespace heppy