BSM-Induced Contributions in Observables

• BSM-induced contributions are deviations in measured observables that signal the presence of new physics beyond the Standard Model.
• They are extracted through a systematic workflow involving model construction, event simulation, fiducial projection, and rigorous statistical testing.
• These contributions influence experimental designs across collider, flavor, neutrino, and cosmological studies by refining searches for BSM signals.

A Beyond the Standard Model (BSM)-induced contribution refers to any deviation, enhancement, or modification to measurable observables (cross sections, decay rates, spectra, distributions, etc.) arising explicitly from new physics beyond the Standard Model. BSM-induced contributions are an essential focal point of modern phenomenology, providing both the theoretical foundation for interpreting null results and the workflow for extracting constraints or evidence of new physics across collider, flavor, neutrino, astrophysical, and cosmological probes.

1. Formal Definition: BSM-Induced Excess in Observables

For any observable, the measured or predicted quantity can be decomposed as

$$X^{\text{BSM+SM}} = X^{\text{SM}} + \Delta X^{\text{BSM}}$$

where $\Delta X^{\text{BSM}}$ encapsulates the distinctive effect of the BSM sector, including new amplitudes, particles, or operator insertions that are absent in $X^{\text{SM}}$. In experimental analyses, the BSM-induced contribution is typically isolated by first predicting the SM background—including all detector corrections and modeling uncertainties—then computing or measuring the total rate, and subtracting to obtain the BSM-induced excess. In binned distributions, this yields per-bin contributions:

$$\Delta\sigma_i^{\text{BSM}} = \sigma_{\text{fid}}^{\text{BSM+SM}}(dX_i) - \sigma_{\text{fid}}^{\text{SM}}(dX_i)$$

where $i$ indexes bins in the relevant observable (momentum, mass, etc.), and $\sigma_{\text{fid}}$ denotes the unfolded, fiducial-region cross section with all experimental cuts identically applied to both prediction and measurement (Butterworth, 2019).

2. Calculation and Extraction Workflow

BSM-induced contributions are extracted through a sequence of model implementation, event generation, fiducial projection, and statistical testing:

• Model Construction: BSM scenarios are encoded in UFO models via tools such as FeynRules, specifying Lagrangians with new particles, couplings, and operators.
• Event Generation and Simulation: Fully showered and hadronized events are generated (e.g., with Herwig), including all relevant production and decay channels, and possible interference with SM amplitudes (Butterworth, 2019).
• Fiducial Analysis: Detector- and analysis-level observables are constructed with identical kinematic and object selection criteria as experimental analyses, typically using analysis frameworks like Rivet.
• BSM-Induced Excess: BSM contributions are determined by rerunning the observable calculation with and without new physics, holding all other settings fixed.
• Statistical Comparison: The BSM-induced excess in each bin, $\Delta\sigma_i^{\text{BSM}}$, is subjected to hypothesis testing (e.g., $\chi^2$ or likelihood ratios) comparing SM+BSM predictions to data, to extract exclusion regions in BSM parameter space.

This workflow is central to the Contur package and similar methodologies that leverage unfolded, particle-level measurements for robust, model-independent BSM limits (Butterworth, 2019).

3. Theoretical Decomposition: EFT, Interference, and Hierarchy

In effective field theory (EFT), BSM-induced contributions arise as higher-dimensional operators suppressed by the new-physics scale $\Lambda$,

$$\mathcal{L}_{\text{EFT}} = \mathcal{L}_{\text{SM}} + \sum_i \frac{C_i}{\Lambda^2} \mathcal{O}_i^{(6)} + \sum_j \frac{C'_j}{\Lambda^4} \mathcal{O}_j^{(8)} + \cdots$$

BSM-induced effects enter observables via SM–BSM interference and pure BSM squared terms, e.g.,

$$\sigma \propto |\mathcal{M}_{\text{SM}}|^2 + 2\operatorname{Re}(\mathcal{M}_{\text{SM}}\mathcal{M}_{\text{BSM}}^*) + |\mathcal{M}_{\text{BSM}}|^2$$

Helicity selection rules can nullify certain interference contributions: for many high-energy processes involving transversely-polarized vector bosons, the leading dimension-6 operator insertions ($\mathcal{O}(1/\Lambda^2)$) fail to interfere with the SM, making the dominant BSM-induced contribution arise from dimension-8 operators or squared dim-6 terms ($\mathcal{O}(1/\Lambda^4)$) (Azatov et al., 2016). Precise characterization of BSM-induced effects therefore necessitates careful operator enumeration and interference analysis.

4. Representative Examples Across Frontiers

High-Energy Colliders

At LHC, BSM-induced contributions manifest as deviations in fiducial cross-section shapes due to new mediators (e.g., $Z'$), dark sector particles, or effective operators. For instance, in a gauged $B{-}L$ model: increasing $g'$ at fixed $M_{Z'}$ amplifies $\Delta\sigma_{\text{BSM}}$ quadratically, while exceeding decay thresholds ($M_{Z'}<2M_{\text{DM}}$) shifts the BSM-induced contribution between visible (jets, leptons) and invisible ($E_T^{\text{miss}}$) final states (Butterworth, 2019).

Precision Flavor Physics

In flavor observables, BSM-induced contributions enter effective weak Hamiltonians as new or modified Wilson coefficients,

$$\mathcal{H}_{\text{eff}} = - \frac{4G_F}{\sqrt{2}} V_{tb}V_{ts}^* \sum_i [C_i O_i + C_i' O_i']$$

Modifications to branching ratios and CP-violating observables—e.g., $\mathcal{B}(B_{s,d} \to \mu^+\mu^-)$, $S_{\psi\phi}$—arise from nonzero BSM-induced $C_S$, $C_P$, and new CP phases. Distinct “fingerprints” in the pattern of deviations enable discrimination among BSM scenarios via their induced contributions (Buras et al., 2012).

Neutrino-Electron Scattering

In dedicated neutrino experiments or DM detectors, BSM-induced contributions alter recoil spectra through non-standard interactions, effective four-fermion (SPVAT) operators, or light mediator exchange. The BSM-induced shift in the differential cross section,

$$\frac{d\sigma}{dT} = \frac{d\sigma_{\text{SM}}}{dT} + \Delta \left( \frac{d\sigma}{dT} \right)_{\text{BSM}}$$

provides sensitivity to Wilson coefficients (e.g., $\epsilon_{ee}^{L,R}$, $g_{Z'}$, $\delta C_a$) well below existing bounds, with distinctive spectral distortions at recoil endpoints (Link et al., 2019).

Forward Physics and Cosmology

In far-forward LHC experiments, BSM-induced yields are computed via QCD-driven production cross sections with BSM couplings inserted into matrix elements (e.g., dark photon $\epsilon$), and signals are separated from overwhelming backgrounds using topological and kinematic cuts on BSM-induced event characteristics (Garzelli, 2022). In primordial gravitational wave backgrounds, BSM-induced contributions from hidden sectors or altered expansion histories directly reshape the predicted $\Omega_{\text{GW}}(f)$, shifting amplitude and peak frequency according to the number of relativistic degrees of freedom and nonstandard equations of state (Muia et al., 2023).

5. Statistical Treatment and Parameter Constraints

BSM-induced contributions shape likelihood construction:

$$\chi^2(p) = \sum_i \frac{[\Delta\sigma_i^{\text{BSM}}(p)]^2}{[\delta\sigma_i^{\text{exp}}]^2}$$

with $p$ denoting BSM parameters. Exclusion at a given CL corresponds to $\chi^2(p) > \chi^2_{1;\,0.95} = 3.84$ (for one parameter), or via $p$-value comparison to the appropriate $\chi^2$ distribution (Butterworth, 2019). Systematic uncertainties, correlations, and bin selection strategies further modulate sensitivity to BSM-induced effects.

6. Model-Dependent Structure of BSM-Induced Terms

The dependence of $\Delta X^{\text{BSM}}$ on model parameters is highly process-specific:

• Gauge Boson Mediators: Cross sections scale as $g'^2$ for gauge couplings, and mass thresholds reorder decay topologies (Butterworth, 2019).
• Effective Operators: For dimension-$d$ operators, BSM-induced terms scale as $(E/\Lambda)^{d-4}$. When interference vanishes, pure BSM-squared or higher-dimension terms govern the leading deviations (Azatov et al., 2016).
• Flavor Transitions: BSM-induced contributions enter as shifts or new terms in all relevant $C_i$. Their chiral and flavor structure controls the observable impact across $B$, $K$, and $D$ sectors, as demonstrated by tools such as FlavorKit (Porod et al., 2014).
• Neutrino and Light Mediator Models: Sensitivity to BSM-induced couplings ($g_{Z'}$, $\epsilon$) can extend many orders of magnitude below previous exclusions by leveraging non-SM recoil shapes and statistical power (Link et al., 2019).

7. Implications for Experimental Design and Global Fits

The centrality of BSM-induced contributions in modern data analyses mandates:

• Maintaining strict correspondence between theory and experiment in analysis design to robustly extract per-bin $\Delta X^{\text{BSM}}$.
• Accounting for possible breakdown of EFT truncation, especially when helicity selection rules nullify dim-6 interference, requiring explicit inclusion of dim-8 terms (Azatov et al., 2016).
• Leveraging global and multi-observable fits, since many BSM-induced deformations in one channel are correlated or even predicted from shifts in “primary” observables elsewhere (Gupta et al., 2014).
• Incorporating experimental systematics, detector effects, and bin-by-bin uncertainty modeling to avoid overstating exclusions from small BSM-induced contributions (Butterworth, 2019).

BSM-induced contributions are the organizing principle in interpreting precision particle physics, guiding both the extraction of robust limits and the search for signals across all energy, flavor, and cosmic frontiers.