Chern Simons condensate from misaligned axions

We obtain the nonequilibrium condensate of the Chern Simons density induced by a misaligned homogeneous coherent axion field in linear response. The Chern-Simons dynamical susceptibility is simply related to the axion self-energy, a result that is valid to leading order in the axion coupling but to all orders in the couplings of the gauge fields to other fields within or beyond the standard model except the axion. The induced Chern-Simons density requires renormalization which is achieved by vacuum subtraction. For ultralight axions of mass $m_a$ coupled to electromagnetic fields with coupling $g$, the renormalized high temperature contribution postrecombination is $⟨\overrightarrow{E}·\overrightarrow{B}⟩(t)=-\frac{g{\pi }^2{T}^4}{15}\overline{a}(t)+\frac{g{m}_a^{2}T}{16\pi }\dot{\overline{a}}(t)$ with $\overline{a}(t)$ the dynamical homogeneous axion condensate. We conjecture that emergent axionlike quasiparticle excitations in condensed matter systems may be harnessed to probe cosmological axions and the Chern-Simons condensate. Furthermore, it is argued that a misaligned axion can also induce a non-Abelian Chern-Simons condensate of similar qualitative form, and can also “seed” chiral symmetry breaking and induce a neutral pion condensate after the QCD phase transition.

Figure 1

$⟨\overrightarrow{E}·\overrightarrow{B}⟩(x)$ in linear response (3.3) for the case of free photons. The wavy lines correspond to the retarded correlation function (3.4), the heavy solid line to the misaligned axion condensate $\overline{a}(x^{\prime })$.

Figure 2

Mixing between the emergent axion field $\mathrm{\Theta }(\overrightarrow{x},t)$ and the cosmological axion field $a(\overrightarrow{x},t)$ via the Chern-Simons correlation function. The wavy lines correspond to the correlation function (3.4), the heavy solid line to the axion field $a(x^{\prime })$, and the double solid line to the emergent axion quasiparticle field $\mathrm{\Theta }(\overrightarrow{x},t)$.

Figure 3

Replacing the axion field by the misaligned axion condensate $\overline{a}(\overrightarrow{x},t)$ the photon loop with $\overline{a}$ as the external leg is identified with the Chern-Simons condensate $⟨\overrightarrow{E}·\overrightarrow{B}⟩$ (see Fig. 1), which acts as an external source $h(\overrightarrow{x},t)=⟨\overrightarrow{E}·\overrightarrow{B}⟩$ linearly coupled to the quasiparticle axion field $\mathrm{\Theta }(\overrightarrow{x},t)$.