Vector4.hh

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#ifndef RIVET_MATH_VECTOR4
#define RIVET_MATH_VECTOR4

#include "Rivet/Math/MathHeader.hh"
#include "Rivet/Math/MathUtils.hh"
#include "Rivet/Math/VectorN.hh"
#include "Rivet/Math/Vector3.hh"

namespace Rivet {


  class FourVector;
  class FourMomentum;
  class LorentzTransform;
  typedef FourVector Vector4;
  FourVector transform(const LorentzTransform& lt, const FourVector& v4);


  /// @brief Specialisation of VectorN to a general (non-momentum) Lorentz 4-vector.
  class FourVector : public Vector<4> {
    friend FourVector multiply(const double a, const FourVector& v);
    friend FourVector multiply(const FourVector& v, const double a);
    friend FourVector add(const FourVector& a, const FourVector& b);
    friend FourVector transform(const LorentzTransform& lt, const FourVector& v4);

  public:

    FourVector() : Vector<4>() { }

    template<typename V4>
    FourVector(const V4& other) {
      this->setT(other.t());
      this->setX(other.x());
      this->setY(other.y());
      this->setZ(other.z());
    }

    FourVector(const Vector<4>& other)
    : Vector<4>(other) { }

    FourVector(const double t, const double x, const double y, const double z) {
      this->setT(t);
      this->setX(x);
      this->setY(y);
      this->setZ(z);
    }

    virtual ~FourVector() { }

  public:

    double t() const { return get(0); }
    double x() const { return get(1); }
    double y() const { return get(2); }
    double z() const { return get(3); }
    FourVector& setT(const double t) { set(0, t); return *this; }
    FourVector& setX(const double x) { set(1, x); return *this; }
    FourVector& setY(const double y) { set(2, y); return *this; }
    FourVector& setZ(const double z) { set(3, z); return *this; }

    double invariant() const {
      // Done this way for numerical precision
      return (t() + z())*(t() - z()) - x()*x() - y()*y();
    }

    /// Angle between this vector and another
    double angle(const FourVector& v) const {
      return vector3().angle( v.vector3() );
    }

    /// Angle between this vector and another (3-vector)
    double angle(const Vector3& v3) const {
      return vector3().angle(v3);
    }

    /// @brief Square of the projection of the 3-vector on to the \f$ x-y \f$ plane
    /// This is a more efficient function than @c polarRadius, as it avoids the square root.
    /// Use it if you only need the squared value, or e.g. an ordering by magnitude.
    double polarRadius2() const {
      return vector3().polarRadius2();
    }

    /// Synonym for polarRadius2
    double perp2() const {
      return vector3().perp2();
    }

    /// Synonym for polarRadius2
    double rho2() const {
      return vector3().rho2();
    }

    /// Projection of 3-vector on to the \f$ x-y \f$ plane
    double polarRadius() const {
      return vector3().polarRadius();
    }

    /// Synonym for polarRadius
    double perp() const {
      return vector3().perp();
    }

    /// Synonym for polarRadius
    double rho() const {
      return vector3().rho();
    }

    /// Angle subtended by the 3-vector's projection in x-y and the x-axis.
    double azimuthalAngle(const PhiMapping mapping = ZERO_2PI) const {
      return vector3().azimuthalAngle(mapping);
    }

    /// Synonym for azimuthalAngle.
    double phi(const PhiMapping mapping = ZERO_2PI) const {
      return vector3().phi(mapping);
    }

    /// Angle subtended by the 3-vector and the z-axis.
    double polarAngle() const {
      return vector3().polarAngle();
    }

    /// Synonym for polarAngle.
    double theta() const {
      return vector3().theta();
    }

    /// Pseudorapidity (defined purely by the 3-vector components)
    double pseudorapidity() const {
      return vector3().pseudorapidity();
    }

    /// Synonym for pseudorapidity.
    double eta() const {
      return vector3().eta();
    }

    /// Get the spatial part of the 4-vector as a 3-vector.
    Vector3 vector3() const {
      return Vector3(get(1), get(2), get(3));
    }


  public:

    /// Contract two 4-vectors, with metric signature (+ - - -).
    double contract(const FourVector& v) const {
      const double result = t()*v.t() - x()*v.x() - y()*v.y() - z()*v.z();
      return result;
    }

    /// Contract two 4-vectors, with metric signature (+ - - -).
    double dot(const FourVector& v) const {
      return contract(v);
    }

    /// Contract two 4-vectors, with metric signature (+ - - -).
    double operator*(const FourVector& v) const {
      return contract(v);
    }

    /// Multiply by a scalar.
    FourVector& operator*=(double a) {
      _vec = multiply(a, *this)._vec;
      return *this;
    }

    /// Divide by a scalar.
    FourVector& operator/=(double a) {
      _vec = multiply(1.0/a, *this)._vec;
      return *this;
    }

    /// Add to this 4-vector.
    FourVector& operator+=(const FourVector& v) {
      _vec = add(*this, v)._vec;
      return *this;
    }

    /// Subtract from this 4-vector. NB time as well as space components are subtracted.
    FourVector& operator-=(const FourVector& v) {
      _vec = add(*this, -v)._vec;
      return *this;
    }

    /// Multiply all components (space and time) by -1.
    FourVector operator-() const {
      FourVector result;
      result._vec = -_vec;
      return result;
    }

  };


  /// Contract two 4-vectors, with metric signature (+ - - -).
  inline double contract(const FourVector& a, const FourVector& b) {
    return a.contract(b);
  }

  /// Contract two 4-vectors, with metric signature (+ - - -).
  inline double dot(const FourVector& a, const FourVector& b) {
    return contract(a, b);
  }

  inline FourVector multiply(const double a, const FourVector& v) {
    FourVector result;
    result._vec = a * v._vec;
    return result;
  }

  inline FourVector multiply(const FourVector& v, const double a) {
    return multiply(a, v);
  }

  inline FourVector operator*(const double a, const FourVector& v) {
    return multiply(a, v);
  }

  inline FourVector operator*(const FourVector& v, const double a) {
    return multiply(a, v);
  }

  inline FourVector operator/(const FourVector& v, const double a) {
    return multiply(1.0/a, v);
  }

  inline FourVector add(const FourVector& a, const FourVector& b) {
    FourVector result;
    result._vec = a._vec + b._vec;
    return result;
  }

  inline FourVector operator+(const FourVector& a, const FourVector& b) {
    return add(a, b);
  }

  inline FourVector operator-(const FourVector& a, const FourVector& b) {
    return add(a, -b);
  }

  /// Calculate the Lorentz self-invariant of a 4-vector.
  /// \f$ v_\mu v^\mu = g_{\mu\nu} x^\mu x^\nu \f$.
  inline double invariant(const FourVector& lv) {
    return lv.invariant();
  }

  /// Angle (in radians) between spatial parts of two Lorentz vectors.
  inline double angle(const FourVector& a, const FourVector& b) {
    return a.angle(b);
  }

  /// Angle (in radians) between spatial parts of two Lorentz vectors.
  inline double angle(const Vector3& a, const FourVector& b) {
    return angle( a, b.vector3() );
  }

  /// Angle (in radians) between spatial parts of two Lorentz vectors.
  inline double angle(const FourVector& a, const Vector3& b) {
    return a.angle(b);
  }

  /// Calculate transverse length sq. \f$ \rho^2 \f$ of a Lorentz vector.
  inline double polarRadius2(const FourVector& v) {
    return v.polarRadius2();
  }
  /// Synonym for polarRadius2.
  inline double perp2(const FourVector& v) {
    return v.perp2();
  }
  /// Synonym for polarRadius2.
  inline double rho2(const FourVector& v) {
    return v.rho2();
  }

  /// Calculate transverse length \f$ \rho \f$ of a Lorentz vector.
  inline double polarRadius(const FourVector& v) {
    return v.polarRadius();
  }
  /// Synonym for polarRadius.
  inline double perp(const FourVector& v) {
    return v.perp();
  }
  /// Synonym for polarRadius.
  inline double rho(const FourVector& v) {
    return v.rho();
  }

  /// Calculate azimuthal angle of a Lorentz vector.
  inline double azimuthalAngle(const FourVector& v, const PhiMapping mapping = ZERO_2PI) {
    return v.azimuthalAngle(mapping);
  }
  /// Synonym for azimuthalAngle.
  inline double phi(const FourVector& v, const PhiMapping mapping = ZERO_2PI) {
    return v.phi(mapping);
  }


  /// Calculate polar angle of a Lorentz vector.
  inline double polarAngle(const FourVector& v) {
    return v.polarAngle();
  }
  /// Synonym for polarAngle.
  inline double theta(const FourVector& v) {
    return v.theta();
  }

  /// Calculate pseudorapidity of a Lorentz vector.
  inline double pseudorapidity(const FourVector& v) {
    return v.pseudorapidity();
  }
  /// Synonym for pseudorapidity.
  inline double eta(const FourVector& v) {
    return v.eta();
  }



  ////////////////////////////////////////////////



  /// Specialized version of the FourVector with momentum/energy functionality.
  class FourMomentum : public FourVector {
    friend FourMomentum multiply(const double a, const FourMomentum& v);
    friend FourMomentum multiply(const FourMomentum& v, const double a);
    friend FourMomentum add(const FourMomentum& a, const FourMomentum& b);
    friend FourMomentum transform(const LorentzTransform& lt, const FourMomentum& v4);

  public:
    FourMomentum() { }

    template<typename V4>
    FourMomentum(const V4& other) {
      this->setE(other.t());
      this->setPx(other.x());
      this->setPy(other.y());
      this->setPz(other.z());
    }

    FourMomentum(const Vector<4>& other)
      : FourVector(other) { }

    FourMomentum(const double E, const double px, const double py, const double pz) {
      this->setE(E);
      this->setPx(px);
      this->setPy(py);
      this->setPz(pz);
    }

    ~FourMomentum() {}

  public:
    /// Get energy \f$ E \f$ (time component of momentum).
    double E() const { return t(); }

    /// Get 3-momentum part, \f$ p \f$.
    Vector3 p() const { return vector3(); }

    /// Get x-component of momentum \f$ p_x \f$.
    double px() const { return x(); }

    /// Get y-component of momentum \f$ p_y \f$.
    double py() const { return y(); }

    /// Get z-component of momentum \f$ p_z \f$.
    double pz() const { return z(); }

    /// Set energy \f$ E \f$ (time component of momentum).
    FourMomentum& setE(double E)   { setT(E); return *this; }

    /// Set x-component of momentum \f$ p_x \f$.
    FourMomentum& setPx(double px) { setX(px); return *this; }

    /// Set y-component of momentum \f$ p_y \f$.
    FourMomentum& setPy(double py) { setY(py); return *this; }

    /// Set z-component of momentum \f$ p_z \f$.
    FourMomentum& setPz(double pz) { setZ(pz); return *this; }

    /// @brief Get the mass \f$ m = \sqrt{E^2 - p^2} \f$ (the Lorentz self-invariant).
    ///
    /// For spacelike momenta, the mass will be -sqrt(|mass2|).
    double mass() const {
      // assert(Rivet::isZero(mass2()) || mass2() > 0);
      // if (Rivet::isZero(mass2())) {
      //   return 0.0;
      // } else {
      //   return sqrt(mass2());
      // }
      return sign(mass2()) * sqrt(fabs(mass2()));
    }

    /// Get the squared mass \f$ m^2 = E^2 - p^2 \f$ (the Lorentz self-invariant).
    double mass2() const {
      return invariant();
    }

    /// Calculate the rapidity.
    double rapidity() const {
      return 0.5 * std::log( (E() + pz()) / (E() - pz()) );
    }

    /// Calculate the squared transverse momentum \f$ p_T^2 \f$.
    double pT2() const {
      return vector3().polarRadius2();
    }

    /// Calculate the transverse momentum \f$ p_T \f$.
    double pT() const {
      return sqrt(pT2());
    }

    /// Calculate the transverse energy \f$ E_T^2 = E^2 \sin^2{\theta} \f$.
    double Et2() const {
      return Et() * Et();
    }

    /// Calculate the transverse energy \f$ E_T = E \sin{\theta} \f$.
    double Et() const {
      return E() * sin(polarAngle());
    }

    /// Calculate the boost vector (in units of \f$ \beta \f$).
    Vector3 boostVector() const {
      // const Vector3 p3 = vector3();
      // const double m2 = mass2();
      // if (Rivet::isZero(m2)) return p3.unit();
      // else {
      //   // Could also do this via beta = tanh(rapidity), but that's
      //   // probably more messy from a numerical hygiene point of view.
      //   const double p2 = p3.mod2();
      //   const double beta = sqrt( p2 / (m2 + p2) );
      //   return beta * p3.unit();
      // }
      /// @todo Be careful about c=1 convention...
      return Vector3(px()/E(), py()/E(), pz()/E());
    }

    /// Struct for sorting by increasing energy
    struct byEAscending {
      bool operator()(const FourMomentum& left, const FourMomentum& right) const{
        double pt2left = left.E();
        double pt2right = right.E();
        return pt2left < pt2right;
      }

      bool operator()(const FourMomentum* left, const FourMomentum* right) const{
        return (*this)(left, right);
      }
    };

    /// Struct for sorting by decreasing energy
    struct byEDescending {
      bool operator()(const FourMomentum& left, const FourMomentum& right) const{
        return byEAscending()(right, left);
      }

      bool operator()(const FourMomentum* left, const FourVector* right) const{
        return (*this)(left, right);
      }
    };


    /// Multiply by a scalar
    FourMomentum& operator*=(double a) {
      _vec = multiply(a, *this)._vec;
      return *this;
    }

    /// Divide by a scalar
    FourMomentum& operator/=(double a) {
      _vec = multiply(1.0/a, *this)._vec;
      return *this;
    }

    /// Add to this 4-vector. NB time as well as space components are added.
    FourMomentum& operator+=(const FourMomentum& v) {
      _vec = add(*this, v)._vec;
      return *this;
    }

    /// Subtract from this 4-vector. NB time as well as space components are subtracted.
    FourMomentum& operator-=(const FourMomentum& v) {
      _vec = add(*this, -v)._vec;
      return *this;
    }

    /// Multiply all components (time and space) by -1.
    FourMomentum operator-() const {
      FourMomentum result;
      result._vec = -_vec;
      return result;
    }


  };


  inline FourMomentum multiply(const double a, const FourMomentum& v) {
    FourMomentum result;
    result._vec = a * v._vec;
    return result;
  }

  inline FourMomentum multiply(const FourMomentum& v, const double a) {
    return multiply(a, v);
  }

  inline FourMomentum operator*(const double a, const FourMomentum& v) {
    return multiply(a, v);
  }

  inline FourMomentum operator*(const FourMomentum& v, const double a) {
    return multiply(a, v);
  }

  inline FourMomentum operator/(const FourMomentum& v, const double a) {
    return multiply(1.0/a, v);
  }

  inline FourMomentum add(const FourMomentum& a, const FourMomentum& b) {
    FourMomentum result;
    result._vec = a._vec + b._vec;
    return result;
  }

  inline FourMomentum operator+(const FourMomentum& a, const FourMomentum& b) {
    return add(a, b);
  }

  inline FourMomentum operator-(const FourMomentum& a, const FourMomentum& b) {
    return add(a, -b);
  }



  /// Get the mass \f$ m = \sqrt{E^2 - p^2} \f$ (the Lorentz self-invariant) of a momentum 4-vector.
  inline double mass(const FourMomentum& v) {
    return v.mass();
  }

  /// Get the squared mass \f$ m^2 = E^2 - p^2 \f$ (the Lorentz self-invariant) of a momentum 4-vector.
  inline double mass2(const FourMomentum& v) {
    return v.mass2();
  }

  /// Calculate the rapidity of a momentum 4-vector.
  inline double rapidity(const FourMomentum& v) {
    return v.rapidity();
  }

  /// Calculate the squared transverse momentum \f$ p_T^2 \f$ of a momentum 4-vector.
  inline double pT2(const FourMomentum& v) {
    return v.pT2();
  }

  /// Calculate the transverse momentum \f$ p_T \f$ of a momentum 4-vector.
  inline double pT(const FourMomentum& v) {
    return v.pT();
  }

  /// Calculate the transverse energy squared, \f$ E_T^2 = E^2 \sin^2{\theta} \f$ of a momentum 4-vector.
  inline double Et2(const FourMomentum& v) {
    return v.Et2();}

  /// Calculate the transverse energy \f$ E_T = E \sin{\theta} \f$ of a momentum 4-vector.
  inline double Et(const FourMomentum& v) {
    return v.Et();
  }

  /// Calculate the velocity boost vector of a momentum 4-vector.
  inline Vector3 boostVector(const FourMomentum& v) {
    return v.boostVector();
  }


  //////////////////////////////////////////////////////


  /// @name \f$ \Delta R \f$ calculations from 4-vectors
  //@{

  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between two four-vectors.
  /// There is a scheme ambiguity for momentum-type four vectors as to whether
  /// the pseudorapidity (a purely geometric concept) or the rapidity (a
  /// relativistic energy-momentum quantity) is to be used: this can be chosen
  /// via the optional scheme parameter. Use of this scheme option is
  /// discouraged in this case since @c RAPIDITY is only a valid option for
  /// vectors whose type is really the FourMomentum derived class.
  inline double deltaR(const FourVector& a, const FourVector& b,
                       RapScheme scheme = PSEUDORAPIDITY) {
    switch (scheme) {
    case PSEUDORAPIDITY :
      return deltaR(a.vector3(), b.vector3());
    case RAPIDITY:
      {
        const FourMomentum* ma = dynamic_cast<const FourMomentum*>(&a);
        const FourMomentum* mb = dynamic_cast<const FourMomentum*>(&b);
        if (!ma || !mb) {
          string err = "deltaR with scheme RAPIDITY can only be called with FourMomentum objects, not FourVectors";
          throw std::runtime_error(err);
        }
        return deltaR(*ma, *mb, scheme);
      }
    default:
      throw std::runtime_error("The specified deltaR scheme is not yet implemented");
    }
  }


  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between two four-vectors.
  /// There is a scheme ambiguity for momentum-type four vectors
  /// as to whether the pseudorapidity (a purely geometric concept) or the
  /// rapidity (a relativistic energy-momentum quantity) is to be used: this can
  /// be chosen via the optional scheme parameter.
  inline double deltaR(const FourVector& v,
                       double eta2, double phi2,
                       RapScheme scheme = PSEUDORAPIDITY) {
    switch (scheme) {
    case PSEUDORAPIDITY :
      return deltaR(v.vector3(), eta2, phi2);
    case RAPIDITY:
      {
        const FourMomentum* mv = dynamic_cast<const FourMomentum*>(&v);
        if (!mv) {
          string err = "deltaR with scheme RAPIDITY can only be called with FourMomentum objects, not FourVectors";
          throw std::runtime_error(err);
        }
        return deltaR(*mv, eta2, phi2, scheme);
      }
    default:
      throw std::runtime_error("The specified deltaR scheme is not yet implemented");
    }
  }


  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between two four-vectors.
  /// There is a scheme ambiguity for momentum-type four vectors
  /// as to whether the pseudorapidity (a purely geometric concept) or the
  /// rapidity (a relativistic energy-momentum quantity) is to be used: this can
  /// be chosen via the optional scheme parameter.
  inline double deltaR(double eta1, double phi1,
                       const FourVector& v,
                       RapScheme scheme = PSEUDORAPIDITY) {
    switch (scheme) {
    case PSEUDORAPIDITY :
      return deltaR(eta1, phi1, v.vector3());
    case RAPIDITY:
      {
        const FourMomentum* mv = dynamic_cast<const FourMomentum*>(&v);
        if (!mv) {
          string err = "deltaR with scheme RAPIDITY can only be called with FourMomentum objects, not FourVectors";
          throw std::runtime_error(err);
        }
        return deltaR(eta1, phi1, *mv, scheme);
      }
    default:
      throw std::runtime_error("The specified deltaR scheme is not yet implemented");
    }
  }


  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between two four-vectors.
  /// There is a scheme ambiguity for momentum-type four vectors
  /// as to whether the pseudorapidity (a purely geometric concept) or the
  /// rapidity (a relativistic energy-momentum quantity) is to be used: this can
  /// be chosen via the optional scheme parameter.
  inline double deltaR(const FourMomentum& a, const FourMomentum& b,
                       RapScheme scheme = PSEUDORAPIDITY) {
    switch (scheme) {
    case PSEUDORAPIDITY:
      return deltaR(a.vector3(), b.vector3());
    case RAPIDITY:
      return deltaR(a.rapidity(), a.azimuthalAngle(), b.rapidity(), b.azimuthalAngle());
    default:
      throw std::runtime_error("The specified deltaR scheme is not yet implemented");
    }
  }

  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between two four-vectors.
  /// There is a scheme ambiguity for momentum-type four vectors
  /// as to whether the pseudorapidity (a purely geometric concept) or the
  /// rapidity (a relativistic energy-momentum quantity) is to be used: this can
  /// be chosen via the optional scheme parameter.
  inline double deltaR(const FourMomentum& v,
                       double eta2, double phi2,
                       RapScheme scheme = PSEUDORAPIDITY) {
    switch (scheme) {
    case PSEUDORAPIDITY:
      return deltaR(v.vector3(), eta2, phi2);
    case RAPIDITY:
      return deltaR(v.rapidity(), v.azimuthalAngle(), eta2, phi2);
    default:
      throw std::runtime_error("The specified deltaR scheme is not yet implemented");
    }
  }


  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between two four-vectors.
  /// There is a scheme ambiguity for momentum-type four vectors
  /// as to whether the pseudorapidity (a purely geometric concept) or the
  /// rapidity (a relativistic energy-momentum quantity) is to be used: this can
  /// be chosen via the optional scheme parameter.
  inline double deltaR(double eta1, double phi1,
                       const FourMomentum& v,
                       RapScheme scheme = PSEUDORAPIDITY) {
    switch (scheme) {
    case PSEUDORAPIDITY:
      return deltaR(eta1, phi1, v.vector3());
    case RAPIDITY:
      return deltaR(eta1, phi1, v.rapidity(), v.azimuthalAngle());
    default:
      throw std::runtime_error("The specified deltaR scheme is not yet implemented");
    }
  }

  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between two four-vectors.
  /// There is a scheme ambiguity for momentum-type four vectors
  /// as to whether the pseudorapidity (a purely geometric concept) or the
  /// rapidity (a relativistic energy-momentum quantity) is to be used: this can
  /// be chosen via the optional scheme parameter.
  inline double deltaR(const FourMomentum& a, const FourVector& b,
                       RapScheme scheme = PSEUDORAPIDITY) {
    switch (scheme) {
    case PSEUDORAPIDITY:
      return deltaR(a.vector3(), b.vector3());
    case RAPIDITY:
      return deltaR(a.rapidity(), a.azimuthalAngle(), FourMomentum(b).rapidity(), b.azimuthalAngle());
    default:
      throw std::runtime_error("The specified deltaR scheme is not yet implemented");
    }
  }

  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between two four-vectors.
  /// There is a scheme ambiguity for momentum-type four vectors
  /// as to whether the pseudorapidity (a purely geometric concept) or the
  /// rapidity (a relativistic energy-momentum quantity) is to be used: this can
  /// be chosen via the optional scheme parameter.
  inline double deltaR(const FourVector& a, const FourMomentum& b,
                       RapScheme scheme = PSEUDORAPIDITY) {
    switch (scheme) {
    case PSEUDORAPIDITY:
      return deltaR(a.vector3(), b.vector3());
    case RAPIDITY:
      return deltaR(FourMomentum(a).rapidity(), a.azimuthalAngle(), b.rapidity(), b.azimuthalAngle());
    default:
      throw std::runtime_error("The specified deltaR scheme is not yet implemented");
    }
  }

  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between a
  /// three-vector and a four-vector.
  inline double deltaR(const FourMomentum& a, const Vector3& b) {
    return deltaR(a.vector3(), b);
  }

  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between a
  /// three-vector and a four-vector.
  inline double deltaR(const Vector3& a, const FourMomentum& b) {
    return deltaR(a, b.vector3());
  }

  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between a
  /// three-vector and a four-vector.
  inline double deltaR(const FourVector& a, const Vector3& b) {
    return deltaR(a.vector3(), b);
  }

  /// @brief Calculate the 2D rapidity-azimuthal ("eta-phi") distance between a
  /// three-vector and a four-vector.
  inline double deltaR(const Vector3& a, const FourVector& b) {
    return deltaR(a, b.vector3());
  }

  //@}

  /// @name \f$ \Delta phi \f$ calculations from 4-vectors
  //@{

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const FourMomentum& a, const FourMomentum& b) {
    return deltaPhi(a.vector3(), b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const FourMomentum& v, double phi2) {
    return deltaPhi(v.vector3(), phi2);
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(double phi1, const FourMomentum& v) {
    return deltaPhi(phi1, v.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const FourVector& a, const FourVector& b) {
    return deltaPhi(a.vector3(), b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const FourVector& v, double phi2) {
    return deltaPhi(v.vector3(), phi2);
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(double phi1, const FourVector& v) {
    return deltaPhi(phi1, v.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const FourVector& a, const FourMomentum& b) {
    return deltaPhi(a.vector3(), b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const FourMomentum& a, const FourVector& b) {
    return deltaPhi(a.vector3(), b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const FourVector& a, const Vector3& b) {
    return deltaPhi(a.vector3(), b);
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const Vector3& a, const FourVector& b) {
    return deltaPhi(a, b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const FourMomentum& a, const Vector3& b) {
    return deltaPhi(a.vector3(), b);
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaPhi(const Vector3& a, const FourMomentum& b) {
    return deltaPhi(a, b.vector3());
  }

  //@}

  /// @name \f$ |\Delta eta| \f$ calculations from 4-vectors
  //@{

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const FourMomentum& a, const FourMomentum& b) {
    return deltaEta(a.vector3(), b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const FourMomentum& v, double eta2) {
    return deltaEta(v.vector3(), eta2);
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(double eta1, const FourMomentum& v) {
    return deltaEta(eta1, v.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const FourVector& a, const FourVector& b) {
    return deltaEta(a.vector3(), b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const FourVector& v, double eta2) {
    return deltaEta(v.vector3(), eta2);
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(double eta1, const FourVector& v) {
    return deltaEta(eta1, v.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const FourVector& a, const FourMomentum& b) {
    return deltaEta(a.vector3(), b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const FourMomentum& a, const FourVector& b) {
    return deltaEta(a.vector3(), b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const FourVector& a, const Vector3& b) {
    return deltaEta(a.vector3(), b);
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const Vector3& a, const FourVector& b) {
    return deltaEta(a, b.vector3());
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const FourMomentum& a, const Vector3& b) {
    return deltaEta(a.vector3(), b);
  }

  /// Calculate the difference in azimuthal angle between two spatial vectors.
  inline double deltaEta(const Vector3& a, const FourMomentum& b) {
    return deltaEta(a, b.vector3());
  }

  //@}

  //////////////////////////////////////////////////////


  /// @name 4-vector string representations
  //@{

  /// Render a 4-vector as a string.
  inline std::string toString(const FourVector& lv) {
    ostringstream out;
    out << "("  << (fabs(lv.t()) < 1E-30 ? 0.0 : lv.t())
        << "; " << (fabs(lv.x()) < 1E-30 ? 0.0 : lv.x())
        << ", " << (fabs(lv.y()) < 1E-30 ? 0.0 : lv.y())
        << ", " << (fabs(lv.z()) < 1E-30 ? 0.0 : lv.z())
        << ")";
    return out.str();
  }

  /// Write a 4-vector to an ostream.
  inline std::ostream& operator<<(std::ostream& out, const FourVector& lv) {
    out << toString(lv);
    return out;
  }

  //@}


}

#endif