Track-Then-Sum in HEP & Sensor Fusion

• Track-then-sum is a framework that first constructs fine-grained track-level objects and then aggregates them through summation or convolution to form global observables.
• It employs techniques such as track functions and DGLAP-type evolution to manage collinear divergences and nonperturbative effects in high-energy physics measurements.
• The method also underpins multi-sensor fusion by combining linear and circular state estimates using weighted averaging, enhancing precision in cluttered and dynamic environments.

Track-then-sum refers to a class of methodologies in which one first forms track-level objects (typically, reconstructed trajectories in high energy physics, or state estimates in tracking and sensor fusion), and subsequently combines these through summation or convolution to construct observables or global state estimates. This paradigm systematically partitions the inferential or measurement problem: tracking is performed at the finest relevant level and the results inform downstream, aggregate observables. Track-then-sum has been developed for both jet-based LHC observables with sensitivity to charged particles (Chang et al., 2013) and high-level multi-sensor circular fusion (Kohnert et al., 2022).

1. Field-Theoretic Track-Then-Sum in High-Energy Physics

The archetypal high-energy track-then-sum framework is the computation of track-based observables at the LHC, where only charged particles (“tracks”) are considered to suppress pile-up contamination. However, such observables are not infrared safe in perturbative QCD, necessitating a systematic treatment of hadronization effects.

The key ingredient is the track function $T_i(x,\mu)$, defined for parton species $i$ as the probability density that a parton of light-cone momentum $k^-$ hadronizes into charged hadrons carrying a total fraction $x = p_C^-/k^-$. The operator definition (bare, quark case) is: $$T_{q}(x) = \int d y^{+} d^{2} y_{\perp} \, e^{i k^{-} y^{+}/2} \frac{1}{2N_c} \sum_{C,N} \delta \left( x - \frac{p_C^-}{k^-} \right) \text{Tr} \left[ \frac{\gamma^-}{2} \langle 0 | \psi( y^{+}, 0, y_{\perp} ) W[y,0] | C N \rangle \langle C N | \bar{\psi}(0) | 0 \rangle \right]$$ enforcing $\int_0^1 dx\,T_i(x, \mu) = 1$. Analogous definitions exist for gluons.

Track functions absorb the full set of collinear infrared divergences associated with restricting to charged hadrons, precisely in analogy with PDFs and FFs in the factorization of collinear QCD.

2. Renormalization Group Evolution and Matching

Track functions $T_i(x, \mu)$ obey a DGLAP-type evolution equation: $$\mu \frac{d T_i(x,\mu)}{d\mu} = \frac{1}{2} \sum_{j,k} \int_0^1 dz \int_0^1 dx_1 \int_0^1 dx_2 \frac{\alpha_s(\mu)}{\pi} P_{i \to jk}(z) T_j(x_1,\mu) T_k(x_2,\mu) \delta\left[ x - z x_1 - (1-z)x_2 \right]$$ where $P_{i \to jk}(z)$ is the timelike Altarelli–Parisi splitting kernel. At NLO, the RG mixes pairs of track functions reflecting the independence of charged-hadron fragmentation following a partonic splitting.

To compute track-based observables, the partonic cross-section is matched to track functions. For a general observable measured only on charged hadrons, the cross-section is: $$d\sigma_\text{track} = \sum_N \int d\Phi_N \left[ \frac{d\overline{\sigma}_N}{d\Phi_N} \right] \prod_{i=1}^N \int_0^1 dx_i\, T_i(x_i,\mu) \delta\left[ \bar{e} - \hat{e}\left( \{x_i p_i\} \right) \right]$$ where $i$0 is the IR-finite matching coefficient, obtained by subtracting all collinear divergences into the $i$1. This convolution structure enables systematic inclusion of hadronization effects in any jet-based observable restricted to charged particles (Chang et al., 2013).

3. NLO Example: Total Charged-Particle Energy Fraction in $i$2 Hadrons

For $i$3 at NLO, the track-then-sum calculation for the total charged-particle energy fraction $i$4 proceeds as: $i$5 At LO, the matching coefficient is the Born cross section; at NLO, collinear divergences are subtracted by the track functions. The resultant charged-fraction spectrum displays robust perturbative convergence and matches detailed parton-shower simulations, validating the track-then-sum formalism for collider observables (Chang et al., 2013).

4. Application: Track Mass and Dimensionless Ratios at the LHC

In LHC analyses, such as $i$6jet, the track mass observable is constructed as: $i$7 where $i$8 are sampled from $i$9. The full track-mass spectrum is reconstructed by reclustering these rescaled four-momenta.

To reduce hadronization-induced smearing, one uses dimensionless ratios such as $k^-$0. Because fragmentation fluctuations enter both numerator and denominator, such ratios display significant cancellation of nonperturbative effects, making track-based substructure observables viable for high-precision LHC phenomenology (Chang et al., 2013).

5. Track-Then-Sum in High-Level Track Fusion for Circular Quantities

Track-then-sum is also foundational in high-level multi-sensor fusion, as in the circular track fusion scheme of (Kohnert et al., 2022). Here, each sensor independently tracks state vectors $k^-$1 and produces local track estimates with state covariance.

Pipeline:

1. Spatial and Temporal Alignment: Each sensor's tracks are buffered and time-aligned to the fusion epoch using constant-velocity prediction.
2. Track History: A rolling buffer of $k^-$2 fused-epoch track sets protects against clutter and track-crossings.
3. Data Association: Global Nearest Neighbour (GNN) matching associates sensor tracks to system-level tracks, using a Mahalanobis-type metric augmented by likelihood terms.
4. State Fusion (Track-Then-Sum):
   • Linear Components: Classic Kalman/information fusion:

   $k^-$3

• Circular Components: Weighted mean fusion. For $k^-$4 independent sensors supplying circular means $k^-$5 and concentration/variance $k^-$6:

  $k^-$7

  $k^-$8

  with $k^-$9 (VM) or $x = p_C^-/k^-$0 (WN). The fused VM concentration is $x = p_C^-/k^-$1, and the fused WN variance is $x = p_C^-/k^-$2 (Kohnert et al., 2022).

Fused track identifiers are reconciled to preserve target continuity. This architecture is robust to latency, out-of-sequence updates, and multi-sensor crossing ambiguities.

6. Validation and Limitations

Monte Carlo simulation demonstrates that the WN fusion formula reproduces sample variance, while the VM fusion rule slightly overestimates concentration (VM not closed under convolution). As the dispersion in one sensor becomes large, the fusion variance converges to the more precise sensor, a key property unattainable in earlier circular-variance based fusion mechanisms (Kohnert et al., 2022).

Limitations include:

• Growth of weights ($x = p_C^-/k^-$3, $x = p_C^-/k^-$4) in large sensor networks, requiring normalization.
• GNN association complexity and the necessity to avoid double-counting when fusing tracks previously fused.
• Validity limited to unimodal, symmetric uncertainties (VM/WN), with necessity for mixture or particle-based methods in highly ambiguous domains.
• For $x = p_C^-/k^-$5 or $x = p_C^-/k^-$6 (uniform regime), weighted-average fusion becomes meaningless.

7. Impact and Outlook

Track-then-sum methods offer systematic, theoretically grounded procedures for incorporating fundamentally nonperturbative or irreducible uncertainties—be it in hadron collider observables sensitive to fragmentation, or in high-level track fusion across heterogeneous sensor suites. Their core structure—tracking at the finest level followed by consistent summation or convolution—parallels factorization principles in field theory and optimal data combination in estimation theory. In collider physics, this framework enables precision calculations using only charged-particle observables, facilitating pile-up suppression and opening new avenues for jet substructure analyses. In sensor fusion, track-then-sum allows seamless integration of both linear and circular states, with demonstrably improved estimation performance in real-world multi-sensor deployments.

The mathematical parallels across these domains suggest potential generalizations and further methodological cross-fertilization. Constraints associated with multi-modality, extreme dispersion, or incremental fusion highlight directions for ongoing research (Chang et al., 2013, Kohnert et al., 2022).