KAHIBARO
21.2. Particle Kinematics
Table of Contents
Transverse momentum
In collider and fixed target experiments it is natural to split a particle momentum into components parallel and perpendicular to the beam direction. By convention the beam defines the $z$ axis, so the transverse plane is the $x$–$y$ plane.
The transverse momentum, written $p_{\mathrm{T}}$ or $p_T$, is the size of the momentum component in this plane. If a particle has three–momentum components $(p_x, p_y, p_z)$, then
$$
p_T = \sqrt{p_x^2 + p_y^2}.
$$
Key relation for transverse momentum:
$$
p_T = \sqrt{p_x^2 + p_y^2}.
$$
This definition has two important features in accelerator physics. First, in a symmetric collider the initial state has very small net transverse momentum, which means the sum of $p_T$ of all final state particles should be close to zero. Large imbalances are linked to missing transverse momentum and often indicate particles that escape detection, such as neutrinos. Second, many detector systems have uniform acceptance and resolution in azimuth around the beam but a limited range in polar angle, so $p_T$ is usually measured and quoted more reliably than $p_z$.
In ROOT you often encounter transverse momentum when working with four–vectors. For example, a TLorentzVector called p provides p.Pt() to return $p_T$. If you store track or jet information in TTrees, it is common to keep separate branches for px, py, and pz. You can then calculate $p_T$ either in your event loop, for example with sqrt(pxpx + pypy), or by defining a new column with RDataFrame and using it directly in histograms and plots.
Pseudorapidity
The direction of a particle relative to the beam is often described not by the polar angle $\theta$ but by pseudorapidity, written $\eta$. For a particle that makes a polar angle $\theta$ with respect to the $z$ axis, pseudorapidity is defined as
$$
\eta = -\ln\left(\tan\frac{\theta}{2}\right).
$$
Definition of pseudorapidity:
$$
\eta = -\ln\left(\tan\frac{\theta}{2}\right).
$$
This quantity grows large and positive when $\theta$ is small and the particle travels close to the positive beam direction, and large and negative when the particle is close to the negative beam direction. Around the transverse plane at $\theta = 90^\circ$ the pseudorapidity is zero. For ultrarelativistic particles where the mass is negligible compared to the momentum, $\eta$ closely approximates the true rapidity used in relativistic kinematics.
Pseudorapidity has practical advantages in detector design and analysis. Coverage of detectors is often specified in terms of $|\eta|$ ranges, for example a calorimeter might measure particles up to $|\eta| < 2.5$. Intervals in $\eta$ correspond, for highly energetic particles, to regions that transform simply under Lorentz boosts along the beam axis. That feature makes distributions in $\eta$ convenient to compare between different collision energies and reference frames.
ROOT uses pseudorapidity in many particle physics tools. For a TLorentzVector you can obtain $\eta$ directly through p.Eta(), provided the vector encodes the full energy and momentum. When you analyze TTrees, it is common to start from transverse momentum $p_T$, pseudorapidity $\eta$, and azimuthal angle $\phi$ instead of the Cartesian components. If you need to convert from $\theta$ to $\eta` when preparing data, you can implement the formula once as a function or as a column defined with RDataFrame and then reuse it throughout your analysis.
Azimuthal angle
The azimuthal angle, written $\phi$, specifies the orientation of a particle momentum in the transverse plane around the beam axis. In a right–handed coordinate system, the beam is along the $z$ axis, and $\phi$ is measured in the $x$–$y$ plane from the positive $x$ axis toward the positive $y$ axis. For a momentum with components $(p_x, p_y)$, the azimuthal angle satisfies
$$
\phi = \arctan\left(\frac{p_y}{p_x}\right),
$$
with the understanding that one uses the two–argument arctangent to place $\phi$ in the correct quadrant.
Transverse momentum and azimuthal angle from components:
\[
p_T = \sqrt{p_x^2 + p_y^2}, \quad
\phi = \operatorname{atan2}(p_y, p_x).
\]
The range of $\phi$ is usually taken as $[-\pi, \pi)$ or $[0, 2\pi)$. Because of the cylindrical symmetry of many collider setups, distributions of $\phi$ are often expected to be uniform if there is no preferred direction. Deviations from uniformity can indicate detector effects or interesting physics such as azimuthal correlations between particles.
In ROOT, TLorentzVector provides p.Phi() to return the azimuthal angle associated with the vector, while detector objects like jets or tracks stored in TTrees often come with a phi branch directly. When comparing directions of two particles, analysts often look at the difference in azimuth,
$$
\Delta\phi = \phi_1 - \phi_2,
$$
adjusted to lie within a standard range such as $[-\pi, \pi)$. That quantity is central when studying angular separations or constructing distances in the $\eta$–$\phi$ plane, which are heavily used for jet algorithms and isolation criteria in modern particle physics analyses.
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