Nonfactorizable Position in Physics

• Nonfactorizable position is defined as the failure of independent decomposition of probabilities or amplitudes due to intrinsic dynamical and informational correlations.
• In quantum field theory and QCD, nonfactorizable corrections from soft-gluon exchanges and charm-quark loops significantly influence cross section calculations and decay rates.
• The concept is pivotal in diverse domains such as quantum foundations, flavor physics, and cosmology, challenging conventional hidden-variable and clustering analyses.

A nonfactorizable position comprises theoretical contexts, experimental observations, and interpretive stances where the usual “factorization” of amplitudes, probabilities, or correlations—typically into products of independent subsystem quantities—fails. In quantum foundations, high-energy physics, and statistical mechanics, such phenomena arise when hidden-variable descriptions, QCD amplitudes, electroweak vertex corrections, or multidimensional gravitational geometries preclude a simple decomposition, reflecting intrinsic dynamical or informational interconnectedness. This notion has become central in the analysis of Bell inequalities, precision cross section calculations, flavor physics, and cosmological clustering.

1. Foundations: Factorability versus Nonfactorizable Structure

Factorability refers to the property that joint probabilities for outcomes of distinct measurements can be written as products of probabilities for each outcome, conditioned on a complete set of hidden variables. For a set of measurements $\{M_i\}$ and a hidden variable $\lambda$, factorability is expressed as:

$$P(M_1 = m_1, M_2 = m_2 | \lambda) = P(M_1 = m_1 | \lambda) \cdot P(M_2 = m_2 | \lambda)$$

As discussed in "Revisiting factorability and indeterminism" (Rodriguez, 2011), factorability on $\lambda$ is only guaranteed when the hidden variable renders the outcomes deterministic (i.e., $P(M = m_k | \lambda)\in\{0,1\}$ for all $m_k$). In cases where indeterminism remains, only by extending to a larger hidden-variable space (e.g., $\gamma = \lambda \oplus \mu$ with $\mu$ an additional independent variable) can one hope to restore factorability. If such an extension is not accessible or known, the system exhibits a nonfactorizable position: observed correlations cannot be decomposed into independent subsystem probabilities.

2. Nonfactorizable Mechanisms in Quantum Field Theory and QCD

Nonfactorizable QCD effects are prominent in higher-order perturbation theory, where soft-gluon exchanges connect distinct quark lines in processes such as Vector Boson Fusion (VBF) Higgs production. Here, nonfactorizable corrections arise from diagrams that do not admit separation into corrections localized to a single scattering line; instead, Glauber-phase correlations link different scattering sectors (Liu et al., 2019, Gates, 2023). Mathematically, at NNLO, the nonfactorizable correction to the VBF differential cross section takes the form

$$d\sigma^{(\text{NNLO})}_{\text{nf}} = \frac{N_c^2-1}{4 N_c^2} \, \alpha_s^2\,\chi_{\text{nf}}\ d\sigma^{(\text{LO})}$$

where $\chi_{\text{nf}}=[\chi^{(1)}]^2 - \chi^{(2)}$ is an explicit functional of transverse momentum integrals (see detailed expressions in (Gates, 2023)). Infrared divergences cancel in this combination, but nonfactorizable contributions can reach percent-level corrections in relevant kinematic distributions despite color suppression, driven by incomplete cancellation of the Glauber phase and $\pi^2$ enhancement.

Similarly, in exclusive pion-induced Drell-Yan at J-PARC, soft nonfactorizable mechanisms—arising in the absence of hard gluon exchange—lead to amplitudes not expressible as convolutions of standard distribution amplitudes and GPDs (Tanaka, 2017). These mechanisms dominate in certain kinematics, exceeding the factorizable contributions and providing sensitive probes of nonperturbative nucleon structure.

3. Flavor Physics: Nonfactorizable Corrections and Isospin/Flavor Symmetries

In meson and baryon decays, nonfactorizable effects arise from contributions that cannot be captured by naive factorization (spectator diagrams and color-singlet currents). They typically reflect the influence of color-octet currents, W-exchange diagrams in baryons, soft gluon rescattering, and final-state interactions. For two-body decays of doubly charmed baryons, nonfactorizable amplitudes are computed via current algebra (S-wave) and pole models (P-wave), entering coherently and interfering with factorizable amplitudes (Cheng et al., 2020). The interplay determines whether branching ratios are enhanced or suppressed—even leading to destructive interference and substantial deviations from naive predictions.

SU(2) isospin and SU(3) flavor symmetry analyses can systematize nonfactorizable contributions in weak meson decays. The isospin decomposition of B decays yields reduced amplitudes $A_{1/2}^{\prime\prime}$ and $A_{3/2}^{\prime\prime}$, whose ratio exhibits universality ($a\approx 0.22$) across multiple channels (Kaur et al., 2021). The SU(3) framework generalizes this, allowing the full amplitude to be written as a sum of factorizable and nonfactorizable components parameterized by a small number of reduced matrix elements (labeled $E$, $I$, $X$), facilitating comprehensive predictions (Kaur et al., 27 Feb 2025).

4. Nonfactorizable Corrections in Semileptonic and FCNC Decays

Electroweak processes and rare decays can acquire nonfactorizable QED (photon exchange) corrections between quark and lepton lines, notably in $\overline{B}\to D^{(*)}\ell^-\bar\nu_\ell$ and $\overline{B}_s\to D_s^{(*)}\ell\bar\nu_\ell$ (Yang et al., 8 Oct 2024, Yang et al., 14 Feb 2025). These corrections cannot be absorbed into hadronic or leptonic form factors and introduce vertex corrections that depend intricately on kinematics and lepton flavor. The amplitude is modified as:

$$\mathcal{A} = \mathcal{A}_0 (1 + \alpha_{\mathrm{em}} \eta), \quad \eta = \eta_b + \eta_c$$

where $\eta$ encodes mass- and process-dependent loop contributions. As a result, branching ratios are systematically enhanced for lighter lepton species, decreasing lepton flavor universality (LFU) ratios $R(D^{(*)})$. The nonfactorizable QED contributions are more pronounced for electrons than muons and negligible for taus, resulting in $R(D^{(*)})_e < R(D^{(*)})_\mu$ within the Standard Model (Yang et al., 8 Oct 2024, Yang et al., 14 Feb 2025).

Nonfactorizable charm-quark loops in exclusive FCNC B decays and rare radiative decays (e.g., $B_s\to \gamma l^+l^-$) require explicit treatment of the B-meson three-particle distribution amplitude, often in double collinear light-cone configurations (Kozachuk et al., 2018, Melikhov, 10 Aug 2025, Belov et al., 1 Apr 2024). Unlike semileptonic decays (collinear configuration), nonfactorizable FCNC amplitudes couple soft QCD physics from both sides of the process, necessitating two-variable fit functions for form factors $H_{A,V}^{\text{NF}}(k'^2,k^2)$ and reaction-dependent corrections to Wilson coefficients, notably $\Delta^{\text{NF}}C_7(q^2)/C_7>0$ (Belov et al., 1 Apr 2024). The sign opposition between factorizable and nonfactorizable charm contributions has significant phenomenological impact.

5. Statistical Mechanics and Nonfactorizable Geometries

Nonfactorizable background geometries (e.g., warped extra-dimensional models) modify gravitational potentials and, consequently, the thermodynamics of clustering in cosmological systems (Khanday et al., 2022). Here, the canonical partition function and entropy are altered via corrections to Newton’s law, reflected in a modified clustering parameter:

$$b^* = \frac{Y(\beta_1 + \beta_2)}{1 + Y(\beta_1 + \beta_2)}$$

where $\beta_2$ captures the nonfactorizable geometric correction. The statistical properties—internal energy, specific heat, and distribution function—retain their standard structural forms but with b* controlling the details of phase transition and time scale of clustering. Larger b* signifies stronger correlations and faster clustering, with direct observable consequences in galaxy distributions.

6. Quantum Foundations and the Measurement Independence Debate

Philosophically, the nonfactorizable position refers in Bell-type analyses to the stance of retaining measurement independence (i.e., settings are statistically uncorrelated with hidden variables), and associating violations of Bell inequalities with failure of outcome or parameter independence (Kitajima, 17 Sep 2025). This maintains locality of settings but rejects factorability of probabilities:

$$P(a, b | A, B, \lambda) \neq P(a|A,\lambda)P(b|B,\lambda)$$

unless outcome/parameter independence holds. The article argues, via de Regt’s contextual theory, Kuhn’s theory choice criteria, and Lakatos’s methodology, that the nonfactorizable position is more intelligible, consistent, and progressive than superdeterminism (which denies measurement independence). Intermediate stances, quantifying partial measurement dependence with mutual information $C_{\text{MD}}$, offer a spectrum bridging the extremes—demonstrating that even minimal correlations ($C_{\text{MD}}\ll 1$) suffice to impact Bell-type violations and quantum randomness certification.

7. Implications, Applications, and Future Directions

Nonfactorizable positions manifest in diverse domains:

• In collider physics, NNLO cross section corrections (e.g., Higgs VBF) cannot be neglected in precision analyses.
• In flavor physics, nonfactorizable amplitudes are critical for reconciling theoretical predictions with branching ratios and CP asymmetries.
• In quantum foundations, nonfactorizability informs debates on the interpretation of quantum nonlocality, contextuality, and randomness.
• In cosmology, nonfactorizable geometries influence clustering and the phase structure of gravitating systems.

The failure of factorization typically signals deeper entanglement of dynamics, correlations, or information than permitted by naive independent-subsystem reasoning. Precise modeling requires the correct identification of the relevant configuration (collinear, double collinear, or geometrical), inclusion of all necessary nonperturbative effects, and careful symmetry analysis. As experimental techniques sharpen and theoretical frameworks evolve, nonfactorizable phenomena continue to serve as crucial touchstones for assessing the completeness and validity of prevailing physical models.