S₂-Only Background Model

• The S₂-Only Background Model is an approach that relies exclusively on the S₂ signal to characterize background events in experiments like dark matter detection and medical image segmentation.
• It employs dedicated frameworks, including conditional normalizing flows and control region strategies, to accurately model various backgrounds such as cathode events, delayed electrons, and solar neutrino recoils.
• Demonstrated efficacy through enhanced sensitivity and precision, the model achieves competitive exclusion limits in particle physics and improved Dice scores in imaging applications.

An S₂-only background model refers to a framework in which the primary observable or computational objective is restricted to the “S₂” signal or segmentation outcome, with background characterization, modeling, or inference performed exclusively through this domain. This concept finds rigorous implementation in weakly interacting particle detection (notably liquid-xenon TPC dark matter searches) as well as in semi-supervised medical image segmentation. In both contexts, the S₂-only background model facilitates highly sensitive analysis by isolating background-like events or regions, supporting both statistical discrimination and auxiliary supervision.

1. S₂-Only Background in Dark Matter Direct Detection

Within the XENONnT experiment, the S₂-only analysis employs the corrected S2 area (cS2)—the total charge (photoelectrons, PE) released by ionization events—as the sole selection observable, exploiting 80 PE ≤ cS2 ≤ 500 PE (∼3–16 electrons) to access low-energy backgrounds and dark matter signals. The finalized model characterizes four background components (Aprile et al., 16 Jan 2026):

• Cathode events (B₁): β–γ decays on cathode wires, large S2 width.
• Delayed electrons (B₂): Photoionization-induced afterpulses, described by conditional normalizing flows.
• Accidental electrons (B₃): Random single-electron pileup, temporally isolated.
• Solar ⁸B CEνNS (B₄): Irreducible nuclear recoil events from solar neutrinos.

The expectation for bin $j$ in cS2 is

$$\mu_j = \sum_{i=1}^4 N_i \int_{\text{bin }j} f_i(\mathrm{cS2})\,d(\mathrm{cS2})$$

where $f_i$ is the normalized PDF of each component, and $N_i$ is its rate in the science ROI with Gaussian uncertainty.

2. Explicit Construction and Validation of Background Components

Each background class is constructed and normalized with dedicated control regions and data-driven or simulation-based templates:

• Cathode background: Extracted via cathode-dominated sidebands and waveform simulation, normalized by sideband subtraction and scale-propagation.
• Delayed electron background: Modeled with conditional normalizing flow to generate a joint emission distribution; normalized from sideband-selected DE events, AE leakage corrected.
• Accidental electron background: Pure AE sample defined by temporal isolation and ML-classified PMT pattern; cS2/width shape taken directly, normalization from low-cS2 sidebands after DE correction.
• CEνNS background: Nuclear recoil spectrum integrated over solar neutrino flux, charge yield from calibration, template normalized with flux and yield uncertainty.

Validation is performed with calibration datasets (²²⁰Rn, ²²²Rn), blinded science region tests, and sideband/nuisance propagation for systematic uncertainties (Aprile et al., 16 Jan 2026).

Background Component	Normalization (SR0/SR1/SR2)	Origin/Selection
Cathode (B₁)	480±70 / 660±70 / 1210±90	S2 width/BDT
Delayed-e (B₂)	1.3±0.5 / 0.34±0.07 / 17.2±2.3	DE sidebands
Accidental-e (B₃)	97±17 / 108±8 / –	AE isolation/ML
CEνNS (B₄)	21±5 / 29±7 / 32.3±8	Solar ν spectrum

3. Statistical Inference and Goodness-of-Fit

Statistical inference proceeds by extended binned likelihood fits in cS2 across science runs. The full likelihood is:

$$\mathcal{L}(\sigma,\boldsymbol\theta) = \prod_{i}\prod_{j} \mathrm{Poisson}(n_{ij}^{\mathrm{obs}}\mid \mu_{ij}(\sigma,\boldsymbol{\theta})) \prod_m G(\theta_m\mid\theta_m^0,\sigma_{\theta_m})$$

$\sigma$ is the DM cross-section, $\boldsymbol{\theta}$ includes per-background normalizations and nuisance parameters.

Upper limits are set via profile-likelihood Asimov (PCL) procedure. No excess observed; a 90% C.L. exclusion of $\sigma_{\rm SI}<6.0\times10^{-45}~\mathrm{cm}^2$ at $m_\chi=5~\mathrm{GeV}/c^2$ is quoted, pushing sensitivity to the CEνNS “neutrino floor” (Aprile et al., 16 Jan 2026).

4. S₂-Only Background Branch in Semi-Supervised Segmentation

In medical image segmentation, a "S₂-only background" branch refers to training the background decoder $D_{\rm bg}$ (with shared encoder $E$) in isolation, to predict the background mask $y_{\rm bg}=1-y_{\rm fg}$ with high confidence (Cao et al., 22 May 2025).

• Network: Background branch uses a VNet/U-Net encoder-decoder, outputs $p_b(x)\in(0,1)$ as background probability for each voxel.
• Objective: Voxel-wise binary cross-entropy on labeled samples plus cross-view bidirectional consistency with foreground decoder:

$$L_{S₂-\rm bg} = L_{\rm bg}^1 + \lambda_c (L_{b\to f} + L_{f\to b})$$

where $L_{b\to f}$ and $L_{f\to b}$ penalize deviations from complementary outputs $(p_b(x),1-p_f(x))$ over all voxels.

Pseudo-labeling and mutual alignment ensure the background output serves as authoritative context for the foreground segmentation. Training alternates minibatches of labeled and unlabeled images, optimizing only $D_{\rm bg}$ and $E$.

5. Theoretical and Empirical Impact of S₂-Only Background Modeling

Bidirectional consistency regularization enforces that confident background predictions force confident foreground predictions, theoretically lowering segmentation entropy and boosting model certainty. If $|q-½|>|μ-½|$ ($q=p_b$, $μ=p_f$), the gradient drives $μ$ further from $½$, reducing uncertainty [Appendix B, Theorem 2 in (Cao et al., 22 May 2025)].

Empirically, the isolated S₂-only background model achieves Dice scores on par with the corresponding foreground-only branch, e.g. on LA:

• S₂-only bg: 88.61%–90.06%
• FG-only: 88.45%–90.03%

and on Pancreas:

• S₂-only bg: 83.65%–84.57%
• FG-only: 82.03%–83.03%

When reintegrated in full CVBM, joint DSC exceeds prior state-of-the-art (Cao et al., 22 May 2025).

Dataset	DSC S₂-only bg	DSC fg-only
4/76-LA	88.61%	88.45%
8/72-LA	90.06%	90.03%
6/56-Pancreas	83.65%	82.03%
12/50-Pancreas	84.57%	83.03%

6. S₂-Only Background Model in Astrospectroscopy: S-Stars

In Galactic-center astrometry, S₂-only background models have been constructed for orbit fitting and redshift analysis under non-standard central mass hypotheses. A key example is the Ruffini–Argüelles–Rueda (RAR) core-halo fermionic dark-matter model (Becerra-Vergara et al., 2020). Here, "background" refers to the proposed quantum core (of $m c^2=56$ keV fermions and $M_c=3.5\times10^6~M_\odot$), as opposed to a pointlike black hole.

• Density Profile: Generated by solving the relativistic Fermi gas distribution and Tolman–Oppenheimer–Volkoff structure equations.
• Geodesic and Redshift Modeling: S2’s motion and photon redshift computed from numerical integration of test-particle equations in the RAR spacetime, with boundary conditions from sky-projected astrometry.
• Fit Quality: The RAR model yields slightly improved reduced chi-squares for S2’s orbital data ($\langle\bar{\chi}^2\rangle_{\rm RAR}=3.0725$ vs BH $3.3586$), with comparable redshift fits ($\bar\chi^2_{z,\rm RAR}\approx1.28$, $\bar\chi^2_{z,\rm BH}\approx 1.04$). No statistically significant deviation appears for S2 alone.

The RAR core background avoids the need for a singularity and allows for further discrimination through future pericenter precession measurements and multi-messenger signatures (Becerra-Vergara et al., 2020).

7. Relevance, Generalization, and Future Directions

S₂-only background modeling isolates the background contribution in otherwise signal-dominated analyses, enabling increased sensitivity and robustness in both physical experiments and computational inference tasks. In particle astrophysics, this allows approaching irreducible limits set by neutrino backgrounds; in medical imaging, it elevates confidence and generalization through auxiliary supervision and cross-view consistency.

A plausible implication is that such model isolation can serve as both as a foundation for likelihood-based inference (as in XENONnT) and for regularization and calibration in learning-based segmentation (as in CVBM and similar frameworks). Further research may refine nuisance treatment, calibration transfer, and theoretical justification across domains.