Collider Templates for New Physics

• Collider templates are defined as functional forms representing expected kinematics from specific physical processes, facilitating signal detection.
• They are constructed via model-inspired jet substructure and bispectrum analyses to effectively discriminate signal from background.
• Applications include precision tagging of boosted heavy-particle decays and extracting gluon-jet fractions as well as primordial non-Gaussian features.

Collider templates are model-inspired or model-independent functional forms used as analysis tools for identifying signals of new physics in experimental data from collider experiments and cosmological surveys. Their core purpose is to encode the energy and angular distributions expected from specific physical processes, such as boosted heavy-particle decays or heavy particle exchange during inflation, enabling optimal discrimination between signal and background or among different theoretical scenarios. Collider template methodologies span high-energy particle colliders—where event-level jet substructure information is exploited—and cosmological collider approaches, which seek non-Gaussian primordial correlators as relic probes of inflationary heavy fields.

1. Definitions and Conceptual Foundation

Collider templates are statistical or deterministic models that represent specific particle-level or field-level kinematic patterns expected under a given physics hypothesis. At hadron colliders, a template typically encodes the idealized energy flow or constituent momentum patterns resulting from the decay of a boosted heavy particle, encapsulated as sets of N-body four-vectors constrained by the parent-candidate kinematics (Backović et al., 2012, Almeida et al., 2011). In cosmological contexts, collider templates correspond to analytic or semi-analytic forms of the bispectrum or higher-order correlators, designed to capture the non-analytic, oscillatory, and angular features induced by the exchange of massive particles during the inflationary epoch (Meerburg et al., 2016, Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025, Sohn et al., 2024).

Templates are central to likelihood-based, overlap-based, or fit-based analyses, supporting consistent quantification of how well observed events (e.g., jets, bispectra) match hypothetical signal or background models.

2. Collider Templates in Jet Substructure Analyses

In high-energy collider experiments, such as those conducted at the LHC, collider templates are implemented as sets of fixed-order partonic configurations, constructed for identifying decays of boosted heavy objects (e.g., Higgs boson, top quark) embedded within jets (Backović et al., 2012).

The canonical template workflow involves:

• Template Generation: For each signal decay hypothesis (e.g., Higgs→bb̄ or top→bqq̄), multi-body partonic configurations $\{p_1, ..., p_N\}$ satisfying energy-momentum conservation and additional constraints (mass shells, invariant mass windows) are generated and discretized over the relevant degrees of freedom (rapidity, azimuth, substructure kinematics).
• Overlap Computation: Observed jets are clustered (typically with anti-$k_T$, $R = 1.0$), and the overlap between the measured jet constituent distribution and each template is quantified using a kernelized functional form. Two standard kernel choices are a flat "cone" $F(\Delta R) = 1$ for $\Delta R < R_a$, else 0, and a Gaussian $F(\Delta R) = \exp[-\Delta R^2/(2 \omega_a^2)]$.
• Peak Overlap and Tagging: The maximum overlap $Ov_N(j) = \max_f O v_N(j,f)$, with $O v_N(j,f)$ defined as

$$O v_N(j,f) = \exp\left[-\sum_{a=1}^N \frac{1}{2\sigma_a^2} \left( \epsilon\, p_{T,a}^{(f)} - \sum_{i \in j} p_{T,i}\,F(\Delta R_{i,a}) \right)^2 \right]$$

is used for tagging: jets with $Ov_N$ above a threshold are classified as signal-like, and the "peak template" gives the most probable partonic directions within the jet (Backović et al., 2012).

Typical template construction parameters include discretization in rapidity ($n_\eta$), azimuth ($n_\phi$), and optionally, transverse momenta ($n_{p_T}$), with constraints applied to ensure physical viability (e.g., minimal $p_T$, minimal angular separation).

Performance metrics indicate that, for example, with $Ov_2 > 0.8$, $\gtrsim 60\%$ signal (Higgs) tagging efficiency is attainable versus $\sim 90\%$ rejection of QCD jets. For three-prong top tags, $Ov_3$ analyses achieve $\sim 50\%$ efficiency at $90\%$ QCD rejection (Backović et al., 2012).

3. Collider Templates in Jet Flavor Fraction Extraction

Collider templates are also employed for precision measurement of signal fractions in admixtures, notably to extract the gluon-jet fraction $f_g$ in collider datasets (Shulha, 2023). Here, templates take the form of normalized probability distributions $T_{q}(x)$, $T_{g}(x)$ over a chosen jet macro-parameter $x$, constructed from MC samples with explicit quark/gluon character.

Given an observed data distribution $D_{\mathrm{obs}}(x)$, a linear mixture model

$$D_{\mathrm{obs}}(x) = f_q T_q(x) + f_g T_g(x), \quad (f_q + f_g = 1)$$

enables $f_g$ extraction by binned $\chi^2$ or unbinned extended-likelihood fits. Template-shape uncertainties (from mismatches between MC and true templates) are propagated via linear error formulas. Data-driven control samples with different known $f_g$ allow construction of corrective scale factors $C_q(x), C_g(x)$ to align MC templates with empirical distributions, supporting robust estimation of both $f_g$ and its systematic uncertainty (Shulha, 2023).

4. Collider Templates in Cosmological Bispectrum Analyses

Cosmological collider templates are analytic or semi-analytic bispectrum (three-point function) shapes that encode the leading squeezed-limit non-Gaussianity features from heavy field exchange during inflation (Meerburg et al., 2016, Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025, Sohn et al., 2024).

The prototypical cosmological collider signal is characterized by non-analytic, oscillatory scaling in the squeezed limit:

$$S_{\text{squeezed}}(k_L, k_S) \propto \left( \frac{k_L}{k_S} \right)^{1/2 \pm i \mu} P_s(\cos\Theta),$$

with $\mu = \sqrt{m^2/H^2 - 9/4}$ (real for $m > 3H/2$), $P_s$ a Legendre polynomial encoding spin $s$, and $\Theta$ the angle between long and short modes (Meerburg et al., 2016, Suman et al., 21 Nov 2025, Sohn et al., 2024).

Templates are constructed for different interaction scenarios:

• Scalar exchange (type I/II): $\dot\phi^2 \sigma$ or $(\partial_i \phi)^2 \sigma$ cubic couplings, yielding shapes with different squeezed-limit angular and oscillatory structure (Suman et al., 26 Dec 2025).
• Spin-1 and spin-2 exchange: Involve corresponding coupling terms and result in bispectrum templates that include characteristic angular modulation.

Parametric forms, e.g.,

$$S^{\rm I}_{\mathrm{col}}(k_1,k_2,k_3) = \frac{k_1 k_2}{(k_1+k_2)^2} \left( \frac{k_3}{k_1+k_2} \right)^{1/2} \cos\left[\mu \ln\frac{k_3}{2 c_s (k_1+k_2)} + \delta\right] + 2\,\text{perms.}$$

are used, with mass $m$ mapped to $\mu$, and $c_s$ parameterizing sound speed differences.

The efficacy of cosmological collider template analysis is critically determined by the suppression of overlap with standard large-scale structure bispectrum shapes (e.g., equilateral, orthogonal) to isolate truly new-physics-induced features.

5. Orthogonalization and Statistical Implementation

Because collider templates can possess substantial overlap with standard model bispectrum shapes, a key methodology is orthogonalization: adjusting template coefficients so that the collider component is statistically uncorrelated with equilateral and orthogonal (single-field) shapes under an inner product in primordial shape-space (Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025). The inner product is defined as

$$\langle S^{(1)}, S^{(2)} \rangle = \iiint_{\mathcal{V}_k} \frac{S^{(1)}(k_1, k_2, k_3)\, S^{(2)}(k_1, k_2, k_3)}{k_1 + k_2 + k_3}\, d\mathcal{V}_k,$$

and orthogonalization proceeds by solving linear equations so that overlap cosines vanish with respect to the unwanted shapes.

For numerical data analysis, the Modal Bispectrum Estimator (Modal pipeline) and the CMB Bispectrum Estimator (CMB-BEST) project the primordial shapes through transfer functions onto observable CMB multipole space using separable mode expansions (Suman et al., 26 Dec 2025, Sohn et al., 2024). Observed bispectrum coefficients are estimated, and the optimal amplitude $f_{\mathrm{NL}}$ for each template is computed, with the look-elsewhere effect accounted for by trial-factor-adjusted significance scans over template parameter grids (e.g., $(\mu, c_s)$).

6. Performance, Current Constraints, and Outlook

Template tagger approaches in jet physics deliver high signal-to-background discrimination, with demonstrated robustness to pileup and diffuse contamination. In Higgs and top tagging, peak-overlap taggers maintain high efficiency and QCD rejection with manageable CPU cost, scaling linearly with the number of templates (Backović et al., 2012, Almeida et al., 2011). For quark/gluon fraction measurement, template-based fits systematically account for both statistical and template-systematic uncertainties, with the dominant model-dependence from MC shape mismodeling mitigated by empirical template corrections (Shulha, 2023).

In cosmological collider analyses, orthogonalized templates have enabled the first unbiased searches for primordial non-Gaussianity signals in Planck CMB data. Recent results find no statistically robust detection yet, with best-fit $f_{\mathrm{NL}}$ excursions of $2.35\sigma$ (spin-0, type II) after look-elsewhere correction in the Modal pipeline; similar results are obtained with CMB-BEST (Suman et al., 26 Dec 2025, Suman et al., 21 Nov 2025, Sohn et al., 2024). Projections for next-generation CMB and LSS surveys forecast $\sim$factor $\times 2-5$ improvements in $\sigma_{f_{\mathrm{NL}}}$. For 21-cm interferometric arrays targeting the cosmic dark ages, the sensitivity to collider template signatures (with cosmic-variance limit and moderate baselines) allows percent-level mass determination for $m \sim H$ fields, provided $f_{\mathrm{NL}}$ exceeds minimal gravitational-coupling thresholds (Meerburg et al., 2016).

Collider template methodologies are extensible to higher-point functions (trispectra), multiple spins, variable sound speeds, and broader classes of physics signals, supporting both experimental discovery and sharp model exclusion as data sensitivity improves.

7. Software, Limitations, and Extensibility

Software packages like TemplateTagger provide fast, extensible C++ implementations of template-overlap methods for collider jet analyses, supporting user customization to any decay model and kernel (Backović et al., 2012). Modal and CMB-BEST pipelines support arbitrary analytic or numerical bispectrum templates in CMB and LSS applications (Suman et al., 26 Dec 2025, Sohn et al., 2024). Limitations include the reliance on fixed-order partonic configurations for collider jets, potentially missing effects from broad resonances or non-trivial hadronization, and computational scaling with the size of the template catalog. In cosmological analyses, reducing overlap with secondary shapes and robustly modeling signal covariance remain active sources of systematic challenge.

Extensibility is ensured by template-generation architectures (e.g., new decay models, hybrid kernels via subclassing), and by the generality of analytical template construction for cosmological collider signatures. Cross-pipeline agreement has been explicitly verified for representative benchmark templates at the $1\sigma$ level (Suman et al., 26 Dec 2025).

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References:

• (Backović et al., 2012) TemplateTagger v1.0.0: A Template Matching Tool for Jet Substructure
• (Almeida et al., 2011) Three-particle templates for boosted Higgs
• (Shulha, 2023) Model uncertainty in measuring the gluon jet fraction at the hadron collider
• (Meerburg et al., 2016) Prospects for Cosmological Collider Physics
• (Suman et al., 26 Dec 2025) Searching for Cosmological Collider in the Planck CMB Data II: collider templates and Modal analysis
• (Suman et al., 21 Nov 2025) How Significant are Cosmological Collider Signals in the Planck Data?
• (Sohn et al., 2024) Searching for Cosmological Collider in the Planck CMB Data