Non-Standard Momentum Dependencies

• Non-Standard Momentum Dependencies are deviations from canonical momentum behaviors, characterized by non-linear, momentum-dependent potentials and symmetry modifications in physical systems.
• They are implemented in models through complex operator expansions and non-associative momentum composition laws, enhancing simulations in heavy-ion collisions and nuclear systems.
• These dependencies are crucial for interpreting experimental data and informing theoretical frameworks in areas ranging from condensed matter to astrophysics.

Non-standard momentum dependencies refer to physical phenomena, model formulations, or theoretical frameworks in which the dependence on momentum (or momentum transfer) deviates significantly from canonical forms dictated by established symmetries, linearity, or minimal effective theory assumptions. Such dependencies arise in a variety of contexts: from non-linear nuclear interactions, generalized kinetic models, deformations of phase space geometry, to momentum-dependent potentials and correlation observables in many-body systems. Their paper is crucial for understanding dynamic equilibration, symmetry breaking, collective flows, and the emergence of non-trivial dynamical behavior across nuclear, condensed matter, high energy, and astrophysical systems.

1. Fundamental Theory and Parameterizations

Non-standard momentum dependencies often manifest through explicit momentum-dependent terms in the effective interaction, transport, or field theory description. These terms encode corrections or entirely new structures beyond simple density or energy dependence, and typically fall into the following types:

• Momentum-dependent mean fields and potentials: For example, in nuclear transport models, the Hamiltonian may be extended as

$$H = \sum_i \frac{p_i^2}{2m_i} + \sum_i \Big[ V_i^{\mathrm{Skyrme}} + V_i^{\mathrm{Yuk}} + V_i^{\mathrm{Coul}} + V_i^{\mathrm{MDI}} \Big],$$

with the momentum-dependent interaction (MDI) parameterized as

$$V_{ij}^{\mathrm{MDI}} = t_4 \ln^2[t_5(\mathbf{p}_i - \mathbf{p}_j)^2 + 1] \delta(\mathbf{r}_i - \mathbf{r}_j)$$

(Sood et al., 2010).

• Momentum and isospin-dependent symmetry potentials: In isospin-asymmetric nuclear matter, the symmetry potential can be augmented to depend both on momentum and isospin, giving rise to effective mass splitting and observables sensitive to high-momentum isovector dynamics (Feng, 2011).
• Non-linear or non-associative momentum composition laws: In deformed or curved phase space/relative locality theories, the composition law for momenta is constructed to be non-linear and (after excluding coordinate artifacts) inherently non-associative, which is a direct signal of intrinsic geometrical curvature in momentum space (Astuti et al., 2015).

2. Physical Realizations in Nuclear and Hadronic Systems

Non-standard momentum dependencies are essential for reproducing and interpreting experimental data in heavy-ion collisions, nuclear matter, and mesonic particle production:

• Transverse flow and energy of vanishing flow (EVF): The balance energy or EVF, at which collective transverse flow disappears in nucleus-nucleus collisions, is strongly sensitive to details of the MDI. In lighter nuclei, MDI enhances the repulsion and lowers EVF; in heavier systems, its effect is opposite (Sood et al., 2010).
• Isospin-sensitive observables: Momentum-dependent symmetry potentials cause a splitting of neutron and proton effective masses, affecting the neutron/proton emission ratios at high transverse momentum and the excitation functions (ratios) of $\pi^-/\pi^+$ and $K^0/K^+$, serving as precision probes of the high-density symmetry energy (Feng, 2011).
• Hyperon and pion production near threshold: Momentum-dependent in-medium potentials derived from chiral mean-field theory (CMF) modify the available phase space for particle production, significantly altering particle yields and spectra, especially for hyperons at low energies (Steinheimer et al., 2 Oct 2024).

3. Mathematical and Computational Frameworks

The formal characterization of non-standard momentum dependencies requires model extensions and new mathematical techniques:

• Hierarchical moment coupling in kinetic theory: A momentum-dependent relaxation time $\tau_R(p) = \bar \tau_R (p \cdot u/T)^\Delta$ in the Boltzmann equation mediates coupling between different moments of the distribution function. The resulting infinite coupled hierarchy (in both energy and angular indices) requires innovative truncation, resummation, or iterative numerical methods to achieve stable and physically meaningful solutions, especially in far-from-equilibrium and low-viscosity regimes (e.g., quark-gluon plasma) (Gangadharan et al., 31 Mar 2025).
• Momentum-dependent operator expansions: In relativistic effective field theory, nucleon-nucleon contact interactions expanded up to fourth-order in soft momentum $p$ yield distinct operators at $O(p^0)$, $O(p^2)$, and $O(p^4)$, with only a subset possessing independent low-energy constants associated with total pair-momentum dependencies. These are critical for matching observables in boosted few-nucleon systems and resolving longstanding discrepancies in three-body scattering (Filandri et al., 2023).
• Nonlocality and non-covariance in fundamental equations: In generalized Dirac equations with operator-valued mass terms (as in ELKO spinor constructions), the Lorentz boost generators inherit non-local, momentum-dependent contributions due to the non-scalar structure of the mass operator. This leads to representations of the Poincaré group that are not manifestly covariant and necessitate momentum-dependent (nonlocal) transformation rules (Nikitin, 2014).

4. Impact on Observable Correlations and Flow

Momentum-dependent correlations are vital for decoding initial-state fluctuations, deformation, and transport coefficients in many-body systems:

• Factorization-breaking and decorrelation observables in heavy-ion collisions: Correlations between flow harmonics in separate kinematic bins (e.g., $$r_n(p_1, p_2) = \langle V_n(p_1) V_n^*(p_2) \rangle/\sqrt{ \langle v_n^2(p_1)\rangle \langle v_n^2(p_2)\rangle }$$) quantify the degree of factorization breaking, directly probing the momentum dependence of collective flow fluctuations and revealing nuclear deformation effects, as shown for U+U, Ru+Ru, and Zr+Zr collisions (Samanta et al., 2023).
• Covariances between mean-$p_T$ and flow harmonics: The momentum-differential Pearson correlation coefficient $\rho([p_T], V_n(q) V_n^*(q))$ captures the interplay between radial flow and harmonic flow strength at a given $p_T$, showing strong sensitivity to initial-state granularity, nuclear deformation, and medium shear viscosity (Samanta et al., 2023).
• Symmetry-induced momentum transfer universality: In inelastic neutron scattering (INS) on high-symmetry spin clusters, the $Q$-dependence of scattering intensity is fully fixed by point-group symmetry, yielding universal functions expressed in spherical Bessel expansions. This structure enables a clean separation between dynamical and geometric effects, dramatically simplifying data analysis for systems such as planar spin rings, cubes, and icosahedra (Tabrizi, 2021).

5. Broader Theoretical and Practical Implications

Non-standard momentum dependencies reshape several conceptual and technical aspects across domains:

• Modification of fundamental conservation laws: In non-Lagrangian dynamical systems where the affine connection of spacetime is defined in terms of external fields (e.g., electromagnetic), the violation of momentum conservation can result from the non-symmetric, field-dependent connection, allowing for internal generation of net thrust without momentum recoil (Arbuzov, 2017).
• Curved and non-associative momentum space: The deformation of momentum space geometry (e.g., with a Lorentz-invariant but non-associative addition law)

$$(p \oplus q)_\mu = p_\mu + q_\mu + \text{(higher order terms)},$$

with associativity breaking directly quantified via curvature tensors, underpins the foundations of relative locality and has direct implications for Planck-scale phenomenology (Astuti et al., 2015).

• Nonstandard Feynman rules and renormalization: Lattice QCD simulations employing interpolating momentum schemes—where momentum injection is not restricted to symmetric points—assist in the disentanglement of genuine physical momentum dependencies from discretization artifacts, essential for robust non-perturbative renormalization and high-accuracy matching to continuum QCD (Garron et al., 2021).

6. Relevance to Precision Physics and Astrophysical Applications

Momentum dependencies that deviate from standard forms often determine the accuracy with which theoretical models can reproduce and predict complex phenomena:

• Constraint from high-precision data: Correct modeling of momentum-dependent potentials and interactions is essential for matching heavy-ion observables (balance energy, particle spectra, flow coefficients), and their interplay with astrophysical observations (e.g., neutron star equation of state) via universal chiral mean-field parameters (Steinheimer et al., 2 Oct 2024).
• Searches for new physics: Modifications of canonical momentum dependencies in QCD and electroweak interactions are central to experimental searches for beyond-standard-model phenomena, including non-pointlike top quark substructure probed via boosted top events and jet substructure at high $p_T$ (Englert et al., 2014), and non-standard dark matter–nucleon interactions explored via momentum-dependent cross sections and astrophysical neutrino fluxes (Peters et al., 2021).

7. Summary

Non-standard momentum dependencies arise from intrinsic physical mechanisms, symmetry constraints, geometrical properties of phase space, or effective theory expansions beyond leading order. Their presence is central to understanding and controlling the dynamical evolution, equilibration, and observable signatures in a wide array of systems, including nuclear transport, high-energy collisions, condensed matter, and beyond-standard-model scenarios. Theoretical frameworks and simulation tools that incorporate and systematically explore such dependencies are indispensable for progress in interpreting experimental results, constraining fundamental parameters, and bridging microscopic dynamics with emergent macroscopic behavior.