Physics:Rapidity Mass Matrix

The Rapidity Mass Matrix (RMM) is a method of transforming a list of particles for particle physics experiments into a 2D sparse Matrix for Machine learning algorithms.

The RMMs encapsulate information on single- and two-particle densities of identified particles and jets can lead to a systematic approach for defining input variables ("feature space") for various artificial neural networks (ANNs) used in particle physics, independent of event signatures. By construction, the RMMs are expected to be sensitive to a wide range of popular event signatures of the Physics:Standard Model, and thus can be used for various searches of new signatures . The diagonal elements of RMM represent transverse momenta of all objects, the upper-right elements are invariant masses of each two-particle combination, while the lower-left cells reflect rapidity differences.Event signatures with missing transverse energies and Lorentz factors are also included.

Definition of RMM

The RMM is a physics-motivated $mxm$ array that encodes pairwise correlations among all reconstructed objects in a particle collision event. In the RMM representation, each row and column corresponds to a predefined "slot" reserved for a specific object class.

For five types of objects ($T=5$), the ordering typically used is: $\text{MET}$, jets, $b$-jets, muons, electrons, and photons.

For each type, $N$ object slots are reserved, typically with $N=10$. The matrix dimension is then

$$m=1+TxN=51$$

yielding a $51x51$ matrix with $2601$ elements per event. This configuration is commonly referred to as T5N10.

This data structure enables the visualization of structurally complex kinematic information for many different classes of reconstructed objects.

Definition of RMM

Let $R_{ij}$ denote the element in row $i$ and column $j$ of the event RMM. The indices are arranged into contiguous blocks as follows:

$$ \begin{aligned} \text{MET:} &\quad i=1,\; j=1\\ \text{Jets:} &\quad i \text{ or } j ∈ [2,\;N+1]\\ b\text{-jets:} &\quad i \text{ or } j ∈ [N+2,\;2N+1]\\ \text{Muons:} &\quad i \text{ or } j ∈ [2N+2,\;3N+1]\\ \text{Electrons:} &\quad i \text{ or } j ∈ [3N+2,\;4N+1]\\ \text{Photons:} &\quad i \text{ or } j ∈ [4N+2,\;5N+1] \end{aligned} $$

In this construction, missing transverse energy (MET) is scaled by the centre-of-mass energy $\sqrt{s}$. The first row and first column are defined by

$$ R_{i1}\;(i>1) = \frac{m_T(t_i)}{√{s}}, \qquad R_{1j}\;(j>1) = h_L(t_j). $$

Here $m_T(t_i)$ is the transverse mass of object $t_i$, and $h_L$ represents the longitudinal Lorentz factor in the beam direction.

The remaining elements are chosen to be invariant under Lorentz boosts along the longitudinal axis:

$$ R_{ij} = \begin{cases} \frac{E_T(t_i)}{√{s}}, & i>1,\ i=j=L, \\[4pt] δ E_T(t_i) \equiv \frac{E_T(t_{i-1}) - E_T(t_i)}{E_T(t_{i-1}) + E_T(t_i)}, & i>2,\ i=j \ne L, \\[6pt] \frac{m(t_i,t_j)}{√{s}}, & i>j, \\[6pt] h(t_i,t_j), & i<j. \end{cases} $$

The symbol $L$ denotes a set of five diagonal elements, one for each object class when $T=5$, given by

$$ L ∈ [2,\;N+2,\;2N+2,\;3N+2,\;4N+2]. $$

These positions are reserved for the scaled transverse energies. The remaining diagonal entries store transverse momentum imbalance variables $\delta E_T(t_i)$.

The quantities $m(t_i,t_j)$ are two-body invariant masses, while $h(t_i,t_j)$ are rapidity differences between pairs of objects.

Thus, the diagonal of the matrix stores transverse energies and transverse momentum imbalances, the lower triangle stores pairwise invariant masses, and the upper triangle stores rapidity differences. Azimuthal angles between objects are not explicitly included, since they can be inferred from the RMM variables.

Properties and use in machine learning

The block-structured form of the RMM provides an interpretable geometric map of an event within a single matrix. However, the full $51x51$ RMM contains about 2,600 entries, many of which are redundant for downstream machine-learning applications.

In some studies, the number of reserved slots for electromagnetic objects, such as leptons and photons, was reduced from 10 to 5 in order to decrease the number of empty cells, since energies available at the Large Hadron Collider are generally insufficient to populate all such slots. Although this reduces the total number of variables from 2,601 to 1,287, many entries still remain zero for Standard Model events and for many commonly studied beyond-the-Standard-Model signals.