Two-Loop Turbulent Helical Magnetohydrodynamics: Large-Scale Dynamo and Energy Spectrum

We present a two-loop field-theoretic analysis of incompressible helical magnetohydrodynamics (MHD) in fully developed stationary turbulence. A key feature of helical MHD is the appearance of an infrared-unstable ``mass-like'' term in the loop diagrams of the magnetic response function. Physically, this term corresponds to the relevant perturbation of the Joule damping, proportional to $\boldsymbol{\nabla} \times \boldsymbol{b}$ ($\boldsymbol{b} =$ magnetic field). Its presence destabilizes the trivial ground state $\langle \boldsymbol{b} \rangle = 0$ and forces us to look for a mechanism for stabilizing the system. We show that such stabilization can be achieved in two ways: (i) by introducing into induction equation an external mass-like parameter that precisely cancels these dangerous loop corrections (kinematic regime), or (ii) via spontaneous breaking of the rotational symmetry, leading to a new ground state with nonzero large-scale magnetic field (turbulent dynamo regime). For the latter case, we study the two-loop correction to the spontaneously generated magnetic field and demonstrate that Goldstone-like corrections to Alfv\'en modes along with some other anisotropic structures arise. Our results also confirm that the emergent mean magnetic field leads to a steeper slope of the magnetic energy spectrum, $-11/3 + 2\gamma_{b\star}$ (with $\gamma_{b\star} = -0.1039 - 0.4202\rho^2$, for $|\rho| \leqslant 1$ as the degree of helicity), compared to the Kolmogorov velocity spectrum of $-11/3$, thereby breaking equipartition.