HelixTrajectory.cc

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#include "FastSimulation/SimplifiedGeometryPropagator/interface/HelixTrajectory.h"
#include "FastSimulation/SimplifiedGeometryPropagator/interface/StraightTrajectory.h"
#include "FastSimulation/SimplifiedGeometryPropagator/interface/BarrelSimplifiedGeometry.h"
#include "FastSimulation/SimplifiedGeometryPropagator/interface/ForwardSimplifiedGeometry.h"
#include "FastSimulation/SimplifiedGeometryPropagator/interface/Particle.h"
#include "FWCore/MessageLogger/interface/MessageLogger.h"
#include "FastSimulation/SimplifiedGeometryPropagator/interface/Constants.h"
#include <cmath>

// helix phi definition
// ranges from 0 to 2PI
// 0 corresponds to the positive x direction
// phi increases counterclockwise

fastsim::HelixTrajectory::HelixTrajectory(const fastsim::Particle& particle, double magneticFieldZ)
    : Trajectory(particle)
      // exact: r = gamma*beta*m_0*c / (q*e*B) = p_T / (q * e * B)
      // momentum in units of GeV/c: r = p_T * 10^9 / (c * q * B)
      // in cmssw units: r = p_T / (c * 10^-4 * q * B)
      ,
      radius_(
          std::abs(momentum_.Pt() / (fastsim::Constants::speedOfLight * 1e-4 * particle.charge() * magneticFieldZ))),
      phi_(std::atan(momentum_.Py() / momentum_.Px()) +
           (momentum_.Px() * particle.charge() < 0 ? 3. * M_PI / 2. : M_PI / 2.))
      // maybe consider (for -pi/2<x<pi/2)
      // cos(atan(x)) = 1 / sqrt(x^2+1)
      // -> cos(atan(x) + pi/2)  = - x / sqrt(x^2+1)
      // -> cos(atan(x) +3*pi/2) = + x / sqrt(x^2+1)
      // sin(atan(x)) = x / sqrt(x^2+1)
      // -> sin(atan(x) + pi/2)  = + 1 / sqrt(x^2+1)
      // -> sin(atan(x) +3*pi/2) = - 1 / sqrt(x^2+1)
      ,
      centerX_(position_.X() -
               radius_ * (momentum_.Py() / momentum_.Px()) /
                   std::sqrt((momentum_.Py() / momentum_.Px()) * (momentum_.Py() / momentum_.Px()) + 1) *
                   (momentum_.Px() * particle.charge() < 0 ? 1. : -1.)),
      centerY_(position_.Y() -
               radius_ * 1 / std::sqrt((momentum_.Py() / momentum_.Px()) * (momentum_.Py() / momentum_.Px()) + 1) *
                   (momentum_.Px() * particle.charge() < 0 ? -1. : 1.))
      //, centerX_(position_.X() - radius_*std::cos(phi_))
      //, centerY_(position_.Y() - radius_*std::sin(phi_))
      ,
      centerR_(std::sqrt(centerX_ * centerX_ + centerY_ * centerY_)),
      minR_(std::abs(centerR_ - radius_)),
      maxR_(centerR_ + radius_)
      // omega = q * e * B / (gamma * m) = q * e *B / (E / c^2) = q * e * B * c^2 / E
      // omega: negative for negative q -> seems to be what we want.
      // energy in units of GeV: omega = q * B * c^2 / (E * 10^9)
      // in cmssw units: omega[1/ns] = q * B * c^2 * 10^-4 / E
      ,
      phiSpeed_(-particle.charge() * magneticFieldZ * fastsim::Constants::speedOfLight *
                fastsim::Constants::speedOfLight * 1e-4 / momentum_.E()) {
  ;
}

bool fastsim::HelixTrajectory::crosses(const BarrelSimplifiedGeometry& layer) const {
  return (minR_ < layer.getRadius() && maxR_ > layer.getRadius());
}

double fastsim::HelixTrajectory::nextCrossingTimeC(const BarrelSimplifiedGeometry& layer, bool onLayer) const {
  if (!crosses(layer))
    return -1;

  // solve the following equation for sin(phi)
  // (x^2 + y^2 = R_L^2)     (1)      the layer
  // x = x_c + R_H*cos(phi)  (2)      the helix in the xy plane
  // y = y_c + R_H*sin(phi)  (3)      the helix in the xy plane
  // with
  // R_L: the radius of the layer
  // x_c,y_c the center of the helix in xy plane
  // R_H, the radius of the helix
  // phi, the phase of the helix
  //
  // substitute (2) and (3) in (1)
  // =>
  //   x_c^2 + 2*x_c*R_H*cos(phi) + R_H^2*cos^2(phi)
  // + y_c^2 + 2*y_c*R_H*sin(phi) + R_H^2*sin^2(phi)
  // = R_L^2
  // =>
  // (x_c^2 + y_c^2 + R_H^2 - R_L^2) + (2*y_c*R_H)*sin(phi) = -(2*x_c*R_H)*cos(phi)
  //
  // rewrite
  //               E                 +       F    *sin(phi) =      G     *cos(phi)
  // =>
  // E^2 + 2*E*F*sin(phi) + F^2*sin^2(phi) = G^2*(1-sin^2(phi))
  // rearrange
  // sin^2(phi)*(F^2 + G^2) + sin(phi)*(2*E*F) + (E^2 - G^2) = 0
  //
  // rewrite
  // sin^2(phi)*     a      + sin(phi)*   b    +      c      = 0
  // => sin(phi) = (-b +/- sqrt(b^2 - 4*ac)) / (2*a)
  // with
  // a = F^2 + G^2
  // b = 2*E*F
  // c = E^2 - G^2

  double E = centerX_ * centerX_ + centerY_ * centerY_ + radius_ * radius_ - layer.getRadius() * layer.getRadius();
  double F = 2 * centerY_ * radius_;
  double G = 2 * centerX_ * radius_;

  double a = F * F + G * G;
  double b = 2 * E * F;
  double c = E * E - G * G;

  double delta = b * b - 4 * a * c;

  // case of no solution
  if (delta < 0) {
    // Should not be reached: Full Propagation does always have a solution "if(crosses(layer)) == -1"
    // Even if particle is outside all layers -> can turn around in magnetic field
    return -1;
  }

  // Uses a numerically more stable procedure:
  // https://people.csail.mit.edu/bkph/articles/Quadratics.pdf
  double sqrtDelta = sqrt(delta);
  double phi1 = 0, phi2 = 0;
  if (b < 0) {
    phi1 = std::asin((2. * c) / (-b + sqrtDelta));
    phi2 = std::asin((-b + sqrtDelta) / (2. * a));
  } else {
    phi1 = std::asin((-b - sqrtDelta) / (2. * a));
    phi2 = std::asin((2. * c) / (-b - sqrtDelta));
  }

  // asin is ambiguous, make sure to have the right solution
  if (std::abs(layer.getRadius() - getRadParticle(phi1)) > 1.0e-2) {
    phi1 = -phi1 + M_PI;
  }
  if (std::abs(layer.getRadius() - getRadParticle(phi2)) > 1.0e-2) {
    phi2 = -phi2 + M_PI;
  }

  // another ambiguity
  if (phi1 < 0) {
    phi1 += 2. * M_PI;
  }
  if (phi2 < 0) {
    phi2 += 2. * M_PI;
  }

  // find the corresponding times when the intersection occurs
  // make sure they are positive
  double t1 = (phi1 - phi_) / phiSpeed_;
  while (t1 < 0) {
    t1 += 2 * M_PI / std::abs(phiSpeed_);
  }
  double t2 = (phi2 - phi_) / phiSpeed_;
  while (t2 < 0) {
    t2 += 2 * M_PI / std::abs(phiSpeed_);
  }

  // Check if propagation successful (numerical reasons): both solutions (phi1, phi2) have to be on the layer (same radius)
  // Can happen due to numerical instabilities of geometrical function (if momentum is almost parallel to x/y axis)
  // Get crossingTimeC from StraightTrajectory as good approximation
  if (std::abs(layer.getRadius() - getRadParticle(phi1)) > 1.0e-2 ||
      std::abs(layer.getRadius() - getRadParticle(phi2)) > 1.0e-2) {
    StraightTrajectory traj(*this);
    return traj.nextCrossingTimeC(layer, onLayer);
  }

  // if the particle is already on the layer, we need to make sure the 2nd solution is picked.
  // happens if particle turns around in the magnetic field instead of hitting the next layer
  if (onLayer) {
    bool particleMovesInwards = momentum_.X() * position_.X() + momentum_.Y() * position_.Y() < 0;

    double posX1 = centerX_ + radius_ * std::cos(phi1);
    double posY1 = centerY_ + radius_ * std::sin(phi1);
    double momX1 = momentum_.X() * std::cos(phi1 - phi_) - momentum_.Y() * std::sin(phi1 - phi_);
    double momY1 = momentum_.X() * std::sin(phi1 - phi_) + momentum_.Y() * std::cos(phi1 - phi_);
    bool particleMovesInwards1 = momX1 * posX1 + momY1 * posY1 < 0;

    double posX2 = centerX_ + radius_ * std::cos(phi2);
    double posY2 = centerY_ + radius_ * std::sin(phi2);
    double momX2 = momentum_.X() * std::cos(phi2 - phi_) - momentum_.Y() * std::sin(phi2 - phi_);
    double momY2 = momentum_.X() * std::sin(phi2 - phi_) + momentum_.Y() * std::cos(phi2 - phi_);
    bool particleMovesInwards2 = momX2 * posX2 + momY2 * posY2 < 0;

    if (particleMovesInwards1 != particleMovesInwards) {
      return t1 * fastsim::Constants::speedOfLight;
    } else if (particleMovesInwards2 != particleMovesInwards) {
      return t2 * fastsim::Constants::speedOfLight;
    }
    // try to catch numerical issues again..
    else {
      return -1;
    }
  }

  return std::min(t1, t2) * fastsim::Constants::speedOfLight;
}

void fastsim::HelixTrajectory::move(double deltaTimeC) {
  double deltaT = deltaTimeC / fastsim::Constants::speedOfLight;
  double deltaPhi = phiSpeed_ * deltaT;
  position_.SetXYZT(centerX_ + radius_ * std::cos(phi_ + deltaPhi),
                    centerY_ + radius_ * std::sin(phi_ + deltaPhi),
                    position_.Z() + momentum_.Z() / momentum_.E() * deltaTimeC,
                    position_.T() + deltaT);
  // Rotation defined by
  // x' = x cos θ - y sin θ
  // y' = x sin θ + y cos θ
  momentum_.SetXYZT(momentum_.X() * std::cos(deltaPhi) - momentum_.Y() * std::sin(deltaPhi),
                    momentum_.X() * std::sin(deltaPhi) + momentum_.Y() * std::cos(deltaPhi),
                    momentum_.Z(),
                    momentum_.E());
}

double fastsim::HelixTrajectory::getRadParticle(double phi) const {
  return sqrt((centerX_ + radius_ * std::cos(phi)) * (centerX_ + radius_ * std::cos(phi)) +
              (centerY_ + radius_ * std::sin(phi)) * (centerY_ + radius_ * std::sin(phi)));
}