Field Roles and Material Dependence Justification

1. EME Field Components and Their Roles

The EME Lagrangian (Section 3.1) includes two distinct field components: the scalar field $\phi$ and the vector field $A_\mu^{EME}$.

1.1. Justification for Dual Fields

The EME theory requires both fields to resolve the "like-charges-repel" paradox.

• The Scalar Field $\phi$ couples to the mass-induced scalar charge ($\rho_{eff} \propto \rho_{mass}$) and mediates the universal attractive force (gravity).
• The Vector Field $A_\mu^{EME}$ couples to the standard electric charge ($J^\mu$) and mediates the standard repulsive electrostatic force. The observed net force is the sum of these two interactions, which is why the EME theory can explain both attraction (gravity) and repulsion (buoyancy, electrostatic repulsion) from a unified electromagnetic foundation.

1.2. Justification for Scalar Coupling to the Trace $T$

The scalar field $\phi$ couples to the trace of the energy-momentum tensor $T = T^\mu_\mu$ via the term $\kappa \phi T$.

• Why the Trace? The trace $T$ is the only Lorentz-invariant scalar quantity that can be constructed from the energy-momentum tensor. In the non-relativistic limit, $T \approx -\rho c^2$, making the coupling $\kappa \phi T$ proportional to the mass density $\rho$. This is the necessary condition for the scalar field to act as the source of the mass-proportional force.
• Contrast with EM: The standard electromagnetic field $A_\mu$ couples to the 4-current $J^\mu$. The EME theory maintains this standard coupling for $A_\mu^{EME}$ but introduces the novel $\phi T$ coupling for the scalar field, which is essential for replacing gravity with a scalar interaction.

2. Material Dependence Through $\delta(Z,A)$

The material-dependent factor $\delta(Z,A)$ has a derived symmetry structure and a partially derived normalisation. The differential contribution of the neutron-proton mass difference to the mass-induced QVP fixes the $Z/A$ dependence, whilst the overall loop normalisation is encoded in $C_{\text{QFT}}$ and anchored by hadronic matching plus lattice-QCD input. The EFT-level form is:

Where $C_{\text{QFT}}$ is a dimensionless constant resulting from the loop integral, benchmarked at $C_{\text{QFT}} \approx 0.03$. Using the known mass difference, the coefficient for the $Z/A$ term is calculated to be:

The $Z/A$ dependence is a direct consequence of underlying nuclear physics and the EME QVP mechanism. The form of the material dependence is therefore predictive inside the EFT, whilst the explicit loop normalisation remains part of the finite-density/UV-completion programme rather than a loose empirical fit.

2.1. Mechanism for $Z/A$ Dependence

The effective charge $\rho_{eff}$ is generated by the mass of the constituent particles (protons, neutrons, electrons).

• Neutron-Proton Asymmetry: The mass of a neutron ($m_n$) is slightly greater than the mass of a proton ($m_p$). This mass difference, coupled with the different electromagnetic structures of the neutron (which has a complex charge distribution despite being neutral overall), leads to a differential contribution to the mass-induced QVP.
• Reference Point: Silicon ($Z/A \approx 0.5$) is the reference point because it represents a nucleus where the number of protons and neutrons is approximately equal, leading to a cancellation of the first-order QVP asymmetry terms.
• Origin of $2.36 \times 10^{-7}$: This factor arises from the neutron-proton mass splitting multiplied by the benchmark value of $C_{\text{QFT}}$. It is not inserted ad hoc, but the full loop-level derivation of the normalisation is still beyond the scope of the present EFT formulation and remains part of the subsequent UV-completion / finite-density paper. The explicit EFT formula is:

Where $C_{\text{QFT}}$ is a dimensionless constant resulting from the loop integral, benchmarked at $C_{\text{QFT}} \approx 0.03$. Using the known mass difference, the coefficient for the $Z/A$ term is calculated to be: