TRICERATOPS: QFT Computation & Exoplanet Vetting

• The Mathematica TRICERATOPS framework provides analytic evaluation of triple-K integrals, automated ε-expansion, and solutions for conformal Ward identities in QFT.
• The Python TRICERATOPS toolkit employs Bayesian inference and extensive scenario modeling to estimate false positive probabilities for TESS exoplanet candidates.
• Both implementations feature modular design and automated calculation pipelines, offering actionable insights for QFT research and exoplanet candidate vetting.

TRICERATOPS is the designation for two independent scientific software packages in distinct domains: (1) a Mathematica framework for analytic computation of triple-K integrals and conformal correlators in quantum field theory, and (2) a Python-based Bayesian toolkit used for statistical vetting and validation of exoplanet candidates identified by the Transiting Exoplanet Survey Satellite (TESS). Although both share the TRICERATOPS moniker, there is no direct relationship between these packages; their functional scopes, user bases, and technical architectures are entirely separate.

1. Analytic Evaluation of Triple-K Integrals and Conformal Correlators

The Mathematica/Wolfram-Language package known as TRICERATOPS (also TripleK) is primarily tailored for manipulation and analytic evaluation of triple-K integrals, construction of conformal 2- and 3-point functions in momentum space, and symbolic/numeric assessment of multi-loop massless Feynman integrals with generalized propagators $1/p^{2\nu}$ (Bzowski, 2020). The central computational object is the triple-K integral,

$$I_{\alpha\{\beta_1,\beta_2,\beta_3\}}(p_1,p_2,p_3) = \int_0^\infty dx\, x^\alpha \prod_{j=1}^3 K_{\beta_j}(p_j x),$$

where $K_\nu(z)$ is the modified Bessel function of the second kind. The package extends these definitions to n-point multiple-K integrals for broader tensor structures.

Key algorithmic advances include implementation of the reduction framework of Bzowski–McFadden–Skenderis (JHEP 02 (2016) 068), analytic continuation, dimensional regularization $\epsilon$-expansion, systematic extraction of divergences (via KDivergence), and automated series simplification and symbolic algebra for scalar and tensor correlators. Support for primary, secondary, and transverse conformal Ward identities (CWIs), as well as manipulation of tensorial decompositions, is embedded within the package modules TripleK.wl and Konformal.wl.

Module functionalities include:

• Analytic/numeric triple-K evaluation (KEvaluate, KCanEvaluateQ)
• Series expansion and simplification (KExpand, KSimplify)
• Automated regularization and $\epsilon$-pole extraction
• Loop integral representation and conversion to K-integrals (LoopToK, LoopEvaluate)
• Conformal Ward identity algebra and solution construction

The system is distributed with five example notebooks, providing detailed calculations for renormalization in free field theories (scalar, Weyl fermion), chiral anomaly correlators, explicit CWI derivations and checks, and computation of central charges/Euler anomalies.

2. Bayesian Validation of TESS Exoplanet Candidates

TRICERATOPS in the context of exoplanet science constitutes a Python-based, fully automated Bayesian engine for rapid vetting and statistical validation of TESS Objects of Interest (TOIs) (Giacalone et al., 2020). Its core function is precise estimation of the False Positive Probability (FPP)—the probability that a signal is not from a transiting planet on the target star—and the Nearby False Positive Probability (NFPP)—the probability the photometric event originates from a transit/eclipse on a resolved companion star.

The package supports a comprehensive set of astrophysical scenarios, incorporating statistical models for planets or eclipsing binaries on the target, unresolved bound or background stars, and all Gaia-resolved neighbors. Bayesian inference is anchored in scenario priors (with broken power-law parametrizations for orbital period and physical parameters) and marginal likelihoods computed via $N=10^6$ prior draws, integrating light-curve model likelihoods, Gaussian noise assumptions, and secondary-eclipse constraints. Scenario probabilities $\mathcal{P}_j$ define the FPP and NFPP statistics:

$$\mathrm{FPP} = 1 - (\mathcal{P}_{\mathrm{TP}} + \mathcal{P}_{\mathrm{PTP}} + \mathcal{P}_{\mathrm{DTP}}),$$

$$\mathrm{NFPP} = \sum_{s\in\{\mathrm{NTP,\, NEB,\, NEBx2P}\}} \mathcal{P}_s$$

Classification thresholds are empirically calibrated on 68 previously confirmed or rejected TOIs:

• Validated planet: FPP < 0.015 and NFPP < $10^{-3}$
• Likely planet: FPP < 0.5 and NFPP < $10^{-3}$
• Likely nearby false positive: NFPP > $10^{-1}$

The signal-to-noise ratio (SNR) is defined as $$\mathrm{SNR} = \delta_{\mathrm{obs}}/(\sigma_{\mathrm{CDPP}} \sqrt{N_{\mathrm{tra}}})$$, with reliable FP discrimination above SNR > 15 for 2-min cadence TESS data.

Modeling of contaminating stars utilizes TIC queries within 210″, PSF dilution calculation via 2D Gaussian profiles, flux-ratio based depth thresholds ($\delta_s = \delta_{\mathrm{obs}}/X_s$), and multi-scenario light-curve synthesis (using batman and TRILEGAL). High-resolution imaging constraints can be incorporated to down-weight unresolved-companion FPs.

3. Multi-Color Transit Photometry and TRICERATOPS+

Recent developments detailed in "Validation of TESS Planet Candidates with Multi-Color Transit Photometry and TRICERATOPS+" (Barrientos et al., 4 Aug 2025) describe the upgraded TRICERATOPS+ pipeline, extending Bayesian validation by integrating ground-based transit photometry across multiple bandpasses. This architecture utilizes joint likelihood evaluation:

$$\ln L_{\mathrm{total}} = \ln L_{\mathrm{TESS}} + \sum_k \ln L_k$$

where $\ln L_k$ is the log-likelihood from forward-modeling each ground-based photometric dataset in filter $k$, matching observed depth/shape against band-specific synthetic curves (incorporating limb-darkening and chromatic dilution).

Key technical advances in TRICERATOPS+ include:

• Multi-band limb-darkening coefficients via ExoTiC-LD
• Bandpass-specific dilution modeling using TRILEGAL stellar populations
• Augmented handling of resolved Gaia companions and user-supplied flux ratios
• Modular computation of per-band likelihoods to quantify the impact of additional photometric follow-up

Validation decision boundaries are now set as:

• VP (Validated Planet): FPP < 1.5% and NFPP < 0.1%
• PP (Possible Planet): $1.5\%\le \mathrm{FPP}\le 50\%$ and NFPP < 0.1%
• FP (False Positive): FPP > 70% or NFPP > 10%

Case studies demonstrate enhanced discrimination power—events with ambiguous TESS-only SNR or shape can be resolved via chromatic transit analysis, with FPP dropping by 1–5 orders of magnitude upon inclusion of multi-color data.

4. Software Architecture, Installation, and Usage paradigms

The exoplanet-focused TRICERATOPS is implemented in Python ≥3.6, with critical dependencies including astroquery, batman, numpy/scipy/pandas, astropy, and TRILEGAL. The code base is modular, with scenario definition, prior generation, likelihood evaluation, Bayesian summarization, and I/O separated into distinct submodules.

A typical installation is achieved via

pip install triceratops

or via Git clone/build. Minimal API usage entails:

from triceratops import Triceratops tc = Triceratops(tic=123456789, sectors=[4], aperture_mask=[...], lightcurve='2min') tc.run() print(tc.result.fpp, tc.result.nfpp, tc.result.classification)

Command-line and JSON outputs are supported.

TRICERATOPS+ (multi-color pipeline) is available at https://github.com/JGB276/TRICERATOPS-plus and on PyPI, with installation via

pip install triceratops-plus

Example usage incorporates loading TESS and ground-based light curves, contrast curves, and stellar parameters for initialization and validation. Documentation is provided at https://triceratops-plus.readthedocs.io/.

5. Limitations, Caveats, and Prospective Extensions

Both TRICERATOPS implementations are subject to domain-specific constraints:

• TRICERATOPS (TripleK/Mathematica):
  • Limited by CPU/RAM; large integrals and high-derivative correlators may incur slow runtimes
  • Bugs possible due to single-author development
  • Proposed extensions: FeynCalc/Package-X interface, support for quadruple-K and 4-point correlators, and robust numeric integration for arbitrary parameters
• TRICERATOPS/TRICERATOPS+:
  • Assumes circular orbits; eccentricity is not modeled
  • Fixed transit periods; uncertainties or TTVs are not natively accommodated
  • Universal Gaussian PSF approximation introduces minor systematics
  • FPPs derived using power-law priors for $R_p$ and $P_{\mathrm{orb}}$; studies requiring unbiased occurrence rates should use/switch to uniform priors (in development)
  • Low SNR events ($\lesssim 15$) can yield unreliable FPP; NFPP preferred for FP rejection
  • Instrumental FPs (cosmic rays, wheels, scattered light) are not modeled; use in conjunction with DAVE, Robovetter, or manual vetting
  • Multi-planet system priors not implemented (roadmap: higher prior odds for multiplanet TOIs)
  • Multi-color validation may both decrease and increase FPP, depending on chromaticity revealed in follow-up

A plausible implication is that future TRICERATOPS versions will incorporate additional astrophysical scenarios, improved treatment of orbital eccentricity, dynamic period inference, and enhanced instrumental systematics modeling.

6. Scientific Impact and Application Strategies

The Mathematica-based TRICERATOPS/TripleK package provides a critical infrastructure for analytic calculations in conformal field theories, tensor correlator algebra, and symbolic momentum-space evaluations, directly supporting quantum field theory and holography research. Its reduction and regularization capabilities streamline analytic calculations previously performed manually or via ad hoc scripts.

The Python-based TRICERATOPS and TRICERATOPS+ frameworks constitute the only TESS vetting and validation tools modeling transits from resolved nearby contaminant stars and quantifying both FPP and NFPP. These capabilities substantially refine the statistical validation pipeline for exoplanet candidates, guide allocation of high-resolution imaging and ground-based follow-up, and synergize with complementary vetters (e.g., VESPA, DAVE). Best-practice recommendations advocate validation of “likely planets” near threshold values via AO/speckle imaging, prioritization of high-NFPP candidates for targeted time series, and usage in conjunction with centroid offset diagnostics.

TRICERATOPS+ demonstrates that multi-color photometry measurably improves false-positive discrimination, with validation examples showing orders-of-magnitude decreases in FPP upon joint TESS + J-band + optical transit analysis. This suggests that ground-based follow-up campaigns targeting ambiguous candidates (e.g., V-shaped or shallow events, low SNR) should integrate parallel TRICERATOPS+ validation to maximize scientific yield and resource allocation efficiency.