Strongly Coupled Quantum Forces

Abstract

Quantum forces are long-range interactions originating from vacuum fluctuations of mediator fields. Such forces inevitably arise between ordinary matter particles whenever they couple to light mediator species. Conventional computations of quantum forces rely on evaluating one-loop Feynman diagrams of the relevant scattering processes. In this work, we introduce a novel framework to compute quantum forces. Instead of relying on perturbative scattering amplitudes, we directly evaluate the quantum fluctuations of the mediator field by solving its quantized equation of motion with appropriate boundary conditions. This approach remains valid beyond the Born approximation and thus applies to regimes of strong coupling between the mediator and matter fields. In the weak-coupling limit, our results reproduce the known expressions from the Feynman diagram approach. In the strong-coupling regime, the result is modified by a factor that can suppress or enhance the effect. In contrast to classical forces, quantum forces intrinsically violate the superposition principle. Our approach may therefore offer a useful tool for probing non-perturbative effects in the infrared regime.

Figures (4)

• Figure 1: Intuitive pictures of how classical (left panel) and quantum (right panel) forces are generated. Classical forces such as the Coulomb force are generated by a static field $\phi$ (dashed lines) induced by an object (the blue blob labeled "$\chi$ cluster") via a coupling of the form $\phi\overline{\chi}\chi$. With a spherically symmetric profile for the $\chi$ cluster, the $\phi$ field decreases as $1/r$ outside the sphere, exerting a force $\propto 1/r^2$ on the test particle (smaller blue blob). Quantum forces are generated by quantum fluctuations of the $\phi$ field. Given the interaction $\phi^{2}\overline{\chi}\chi$ indicated on the right panel, the $\chi$ cluster emits and absorbs $\phi$ simultaneously. So it cannot generate a static nonzero distribution of $\phi$ (i.e. $\langle\phi\rangle=0$). However, quantum fluctuations still cause non-vanishing $\langle\phi^2\rangle$ which decreases as $1/r^{3}$ in this model. The Gauss surface (green dashed) is added to demonstrate that the flux of the corresponding "electric field" through the surface is nonzero for the classical case but vanishes in the quantum case.
• Figure 2: The field fluctuation (left panel) as a function of $|m_{\rm M}|R$, computed from Eq. (\ref{['eq:phisqnonlinear']}). For comparison, the two individual terms appearing in Eq. (\ref{['eq:phisqnonlinear']}) are shown in the right panel. For convenience, all quantities are normalized to unity in the weak-coupling limit ($z\ll 1$). Numerical results are evaluated at $r = 10R$.
• Figure 3: Comparison between the scalar-mediator form factor $F_\phi(z)$ [defined in Eq. (\ref{['eq:formfactorscalardef']})] and the fermionic-mediator form factor $F_\psi(z)$ [defined in Eq. (\ref{['eq:FormFactorFermion']})] in the regime where $z\equiv m_{\rm M}R > 0$. Both form factors approach $1$ in the weak-coupling limit $z \ll 1$, while in the strong-coupling limit $z \gg 1$ they scale as $3/z^{2}$ and $3/z$, respectively.
• Figure 4: The contour of integration in the complex $k_{\rm out}$-plane. The path along the real axis is denoted by $C_0$. The contour is deformed to wrap around the branch cut starting at $i m_\phi$, with vertical segments $C_1$ and $C_2$.