Reference

coordinate_system(::MomType)::CS where {CS <: AbstractCoordinateSystem}

coordinate_names(::AbstractCoordinateSystem)::Tuple{Symbols}

XYZT <: AbstractCoordinateSystem

Cartesian coordinate system for four-vectors. Using this requires the implementation of the following interface functions:

LorentzVectorBase.x(::CustomFourVector)
LorentzVectorBase.y(::CustomFourVector)
LorentzVectorBase.z(::CustomFourVector)
LorentzVectorBase.t(::CustomFourVector)

PxPyPzE <: AbstractCoordinateSystem

Cartesian coordinate system for four-momenta. Using this requires the implementation of the following interface functions:

LorentzVectorBase.px(::CustomFourMomentum)
LorentzVectorBase.py(::CustomFourMomentum)
LorentzVectorBase.pz(::CustomFourMomentum)
LorentzVectorBase.E(::CustomFourMomentum)

PtEtaPhiM <: AbstractCoordinateSystem

Cylindrical coordinate system for four-momenta. Using this requires the implementation of the following interface functions:

pt(::CustomFourMomentum)
eta(::CustomFourMomentum)
phi(::CustomFourMomentum)
mass(::CustomFourMomentum)

x(lv)

Return the x-component (first spatial component) of the Lorentz-vector-like lv.

y(lv)

Return the y-component (second spatial component) of the Lorentz-vector-like lv.

z(lv)

Return the z-component (third spatial component) of the Lorentz-vector-like lv.

t(lv)

Return the time (energy) component of the Lorentz-vector-like lv.

px(lv)

Return the x-component of a given Lorentz-vector-like lv.

Example

For a four-momentum `(px, py, pz, E)`, this function returns the `px` component.

See Also

• LorentzVectorBase.x: The x-component of a Lorentz-vector-like, which is equivalent to px.

py(lv)

Return the y-component of a given Lorentz-vector-like lv.

Example

For a four-momentum (px, py, pz, E), this function returns the py component.

See Also

• LorentzVectorBase.y: The y-component of a Lorentz-vector-like, which is equivalent to py.

pz(lv)

Return the z-component of a given Lorentz-vector-like lv.

Example

For a four-momentum (px, py, pz, E), this function returns the pz component.

See Also

• LorentzVectorBase.z: The z-component of a Lorentz-vector-like, which is equivalent to pz.

E(lv)

Return the energy component (E) of a given Lorentz-vector-like lv.

Example

For a four-momentum (px, py, pz, E), this function returns the E component.

See Also

• LorentzVectorBase.t: The time component of a Lorentz-vector-like, which is equivalent to E.

pt(lv)

Return the transverse momentum ($p_T$) of a given Lorentz-vector-like lv, defined as the Euclidean magnitude of the momentum components in the x-y plane.

Example

For a four-momentum (px, py, pz, E), this function returns sqrt(px^2 + py^2).

Notes

• The transverse components are defined with respect to the z-axis (3-axis), representing the perpendicular momentum to this axis.

See Also

• pt2: For the squared transverse momentum.

pt2(lv)

Return the squared transverse momentum ($p_T^2$) of a given Lorentz-vector-like lv. The transverse momentum is defined with respect to the z-axis (3-axis) and is the sum of the squares of the x- and y-components.

Example

For a four-momentum (px, py, pz, E), this function returns px^2 + py^2.

Notes

• The transverse components are the momentum projections in the x-y plane, perpendicular to the z-axis.

See Also

• pt: For the transverse momentum.

eta(lv)

Return the pseudorapidity ($\eta$) of a given Lorentz-vector-like lv. The pseudorapidity is a commonly used quantity in high-energy physics, particularly in collider experiments, and is defined as:

\[ \eta = -\log(\tan(\theta/ 2))\]

where $\theta$ is the polar angle of the Lorentz-vector-like relative to the z-axis.

Example

For a four-momentum (px, py, pz, E), this function log(tan(theta/2)) where theta is given by the polar_angle.

Notes

• Pseudorapidity is approximately equal to the rapidity $y$ in the ultra-relativistic limit (when the particle's mass is negligible compared to its energy).

See Also

• rapidity: For the rapidity of the Lorentz-vector-like.

phi(lv)

Return the azimuthal angle ($\phi$) of the Lorentz-vector-like lv.

The azimuthal angle is the angle between the x-axis and the projection of the Lorentz-vector-like on the x-y plane.

Example

If ``(x, y, z, t)`` is a four-vector, this is equivalent to ``atan(y / x)``.

See Also

• LorentzVectorBase.sin_phi: for a faster version of the sine of the azimuthal angle.
• LorentzVectorBase.cos_phi: for a faster version of the cosine of the azimuthal angle.

spatial_magnitude(lv)

Return the spatial magnitude of the Lorentz-vector-like lv, i.e., the magnitude of its spatial components.

Example

If the Lorentz-vector-like is a four-vector (x, y, z, t), this function returns sqrt(x^2 + y^2 + z^2).

spatial_magnitude2(lv)

Return the square of the spatial magnitude of the Lorentz-vector-like lv, i.e., the sum of the squares of its spatial components.

Example

If the Lorentz-vector-like is four-vector (x, y, z, t), this function returns x^2 + y^2 + z^2.

mass(lv)

Return the invariant mass of the Lorentz-vector-like lv, computed as the square root of the Minkowski inner product of the four-vector components with itself.

Example

For a four-momentum (px, py, pz, E), this function returns sqrt(E^2 - (px^2 + py^2 + pz^2)).

Notes

• If the squared invariant mass m^2 is negative, this function returns -sqrt(-m^2) to ensure a real result. This can happen in certain unphysical cases where the energy component is smaller than the spatial momentum magnitude.

See Also

• mass2: For the squared invariant mass calculation.

mass2(lv)

Return the squared invariant mass of the Lorentz-vector-like lv, computed as the Minkowski inner product of the four-vector components with itself.

Example

For a four-momentum (px, py, pz, E), this function returns E^2 - (px^2 + py^2 + pz^2).

See Also

• mass: For the invariant mass, i.e., the square root of this value.

boost_beta(lv)

Return the magnitude of the velocity ($\beta$) of a particle as a fraction of the speed of light of the Lorentz-vector-like lv.

Example

For a four-momentum (px, py, pz, E), this function returns sqrt(px^2 + py^2 + pz^2) / E.

Throws

• ArgumentError if the time (energy) component is zero while the spatial components are non-zero, as this results in an undefined velocity.

Notes

• If both the time (energy) and spatial components are zero, the function returns zero ($\beta = 0$), as this represents a stationary object with no motion.

See Also

• boost_gamma: For the relativistic gamma factor, which depends on $\beta$.

boost_gamma(lv)

Return the relativistic gamma factor ($\gamma$) for the Lorentz-vector-like lv.

Example

For a four-momentum (px, py, pz, E) with velocity $\beta$, this function returns $1 / sqrt(1 - \beta^2)$.

See Also

• boost_beta: For the velocity $\beta$, which is used to compute $\gamma$.

mt2(lv)

Return the squared transverse mass ($m_T^2$) of a given Lorentz-vector-like lv, which is the difference between the squared time- (energy-) and the squared z-component.

Example

For a four-momentum (px, py, pz, E), this function returns E^2 - pz^2.

Notes

• The transverse components are defined with respect to the z-axis (3-axis), indicating a projection in the x-y plane.

See Also

• mt: For the transverse mass.

mt(lv)

Return the transverse mass ($m_T$) of a given Lorentz-vector-like lv, calculated as the square root of the squared transverse mass. The transverse mass is often used in high-energy physics to describe the effective mass of a system when only the transverse components are considered.

Example

For a four-momentum (px, py, pz, E), this function returns sqrt(E^2 - pz^2).

Notes

• If the squared transverse mass (mT^2) is negative, the function returns -sqrt(-mT^2) to

handle the imaginary mass situation that can occur in some relativistic systems.

• The transverse components are defined with respect to the z-axis (3-axis), indicating momentum in the x-y plane.

Throws

• May include a warning if the transverse mass is negative, depending on user settings.

See Also

• mt2: For the squared transverse mass.

rapidity(lv)

Return the rapidity for a given Lorentz-vector-like lv.

The rapidity $y$ is defined as:

\[y = \frac{1}{2}\log((E + p_z) / (E - p_z))\]

where E is the energy (time- or 0-component of the Lorentz-vector-like), and pz is the component along the z-axis.

Example

For a four-vector (px, py, pz, E), this function calculates the rapidity as:

y = 0.5 * log((E + pz) / (E - pz))

Notes

• Rapidity is preferred over pseudorapidity when mass effects are significant, as it takes into account the energy and longitudinal momentum of the particle. In contrast, pseudorapidity depends only on the particle's direction and ignores mass.
• The transverse components of the momentum are defined with respect to the 3-axis (beam axis).
• Rapidity is Lorentz-invariant under boosts along the z-axis, making it useful for comparisons between different reference frames in collider experiments.

See Also

• LorentzVectorBase.eta: For the pseudorapidity of the Lorentz-vector-like.

polar_angle(lv)

Return the polar angle ($\theta$) of the Lorentz-vector-like lv.

Example

If the Lorentz-vector-like is a four-vector $(x, y, z, t)$, this is equivalent to $\arccos(z / \sqrt{x^2 + y^2 + z^2})$.

See Also

• LorentzVectorBase.cos_theta: for a faster version of the cosine of the polar angle.

cos_theta(lv)

Return the cosine of the polar angle ($\theta$) of the Lorentz-vector-like lv.

Depending on the coordinate system, this might be an equivalent but faster version of cos(polar_angle(lv)).

Example

If $(x, y, z, t)$ is a four-vector, this is equivalent to $z / \sqrt{x^2 + y^2 + z^2}$.

See Also

• LorentzVectorBase.polar_angle: for the polar angle itself.

cos_phi(lv)

Return the cosine of the azimuthal angle ($\phi$) of the Lorentz-vector-like lv.

Depending on the coordinate system, this might be an equivalent but faster version of cos(phi(lv)).

Example

If $(x, y, z, t)$ is a four-momentum, this is equivalent to $x / \sqrt{x^2 + y^2}$.

See Also

• LorentzVectorBase.phi: for the azimuthal angle itself.
• LorentzVectorBase.sin_phi: for a faster version of the sine of the azimuthal angle.

sin_phi(lv)

Return the sine of the azimuthal angle ($\phi$) of the Lorentz-vector-like lv.

Depending on the coordinate system, this might be an equivalent but faster version of sin(phi(lv)).

Example

If $(x, y, z, t)$ is a four-momentum, this is equivalent to $y / \sqrt{x^2 + y^2}$.

See Also

• LorentzVectorBase.phi: for the azimuthal angle itself.
• LorentzVectorBase.cos_phi: for a faster version of the cosine of the azimuthal angle.

plus_component(lv)

Return the plus component ($p^+$) of the Lorentz-vector-like lv in light-cone coordinates.

This component is defined as (t + z) / 2, where t is the time (or energy) component and z is the third spatial component of the Lorentz-vector-like.

Example

If (x, y, z, t) is a four-vector, this is equivalent to (t + z) / 2.

minus_component(lv)

Return the minus component ($p^-$) of the Lorentz-vector-like lv in light-cone coordinates.

This component is defined as (t - z) / 2, where t is the time (or energy) component and z is the third spatial component of the Lorentz-vector-like.

Example

If (x, y, z, t) is a four-vector, this is equivalent to (t - z) / 2.