Nonrelativistic effective field theory for axions

Axions can be described by a relativistic field theory with a real scalar field ϕ whose self-interaction potential is a periodic function of $\varphi$. Low-energy axions, such as those produced in the early Universe by the vacuum misalignment mechanism, can be described more simply by a nonrelativistic effective field theory with a complex scalar field $\psi$ whose effective potential is a function of ${\psi }^{*}\psi$. We determine the coefficients in the expansion of the effective potential to fifth order in ${\psi }^{*}\psi$ by matching low-energy axion scattering amplitudes. In order to describe a Bose-Einstein condensate of axions that is too dense to truncate the expansion of the effective potential in powers of ${\psi }^{*}\psi$, we develop a sequence of systematically improvable approximations to the effective potential that resum terms of all orders in ${\psi }^{*}\psi$.

Figure 1

Relativistic axion potentials $\mathcal{V}$ as functions of $\varphi$: instanton potential (dotted curve) and chiral potential for $z=0.48$ (thick solid curve) and for $z=0.45$ and 0.51 (higher and lower thin solid curves).

Figure 2

Polynomial truncations of the potentials $\mathcal{V}$ as functions of $\varphi$: instanton potential (left panel) and chiral potential with $z=0.48$ (right panel). The truncations of the potential after the second, third, fourth, and fifth powers of ${\varphi }^2$ are shown as successively thicker solid lines. The vertical dotted line in the right panel marks the radius of convergence of the chiral potential.

Figure 3

The tree-level diagrams for low-energy $3\to 3$ scattering in the relativistic axion theory. The first two diagrams are also diagrams in axion EFT. In the last diagram, the thicker line indicates a virtual axion whose invariant mass is approximately $3m$.

Figure 4

The tree-level diagrams for low-energy $4\to 4$ scattering in the relativistic axion theory. The first six diagrams are also diagrams in axion EFT. In the last five diagrams, the thicker lines indicate virtual axions whose invariant mass is approximately $3m$.

Figure 5

Polynomial truncations of the effective potentials ${\mathcal{V}}_{\mathrm{eff}}$ as functions of $|\psi |$: instanton effective potential (left panel) and chiral effective potential with $z=0.48$ (right panel). The truncations of the potential after the second, third, fourth, and fifth powers of ${\psi }^{*}\psi$ are shown as successively thicker solid lines. The vertical dotted line in the right panel marks the radius of convergence of the chiral nonrelativistic reduction potential.

Figure 6

Nonrelativistic reduction potentials ${\mathcal{V}}_{\mathrm{eff}}^{(0)}$ with $\frac{1}{2}{m}_a{\psi }^{*}\psi$ subtracted as functions of $|\psi |$: instanton potential (dotted curve) and chiral potentials for $z=0.48$ (thicker solid curve) and for $z=0.45$ and 0.51 (thinner solid curves). The horizontal lines are the asymptotic values of the subtracted potentials at large $|\psi |$.

Figure 7

The tree-level diagrams for $n\to n$ scattering in the relativistic axion theory with no virtual axion lines (left diagram) and with one virtual axion line (right diagram). The corresponding diagrams in axion EFT are all those with no virtual axion lines and those with one virtual axion line for which $|p-q|=1$.

Figure 8

First improved effective potentials ${\mathcal{V}}_{\mathrm{eff}}^{(1)}$ with $0.374m_a{\psi }^{*}\psi$ subtracted as functions of $|\psi |$: instanton potential (dotted curve) and chiral potential with $z=0.48$ (solid curve). The horizontal lines are the asymptotic values of the corresponding subtracted nonrelativistic reduction potentials in Fig. 6.