Green's functions of Nambu-Goldstone modes and Higgs modes in superconductors

We examine fundamental properties of Green's functions of Nambu-Goldstone and Higgs modes in superconductors with multiple order parameters. Nambu-Goldstone and Higgs modes are determined once the symmetry of the system and that of the order parameters are specified. Multiple Nambu-Goldstone modes and Higgs modes exist when we have multiple order parameters. The Nambu-Goldstone Green function $D(\omega,{\bf q})$ has the form $1/(gN(0))^2\cdot (2\Delta)^2/(\omega^2-c_s^2{\bf q}^2)$ with the coupling constant $g$ and $c_s=v_F/\sqrt{3}$ for small $\omega$ and ${\bf q}$, with a pole at $\omega=0$ and ${\bf q}=0$ indicating the existence of a massless mode. It is shown, based on the Ward-Takahashi identity, that the massless mode remains massless in the presence of intraband scattering due to nonmagnetic and magnetic impurities. The pole of $D(\omega,{\bf q})$, however, disappears as $\omega$ increases as large as $2\Delta$: $\omega\sim 2\Delta$. The Green function $H(\omega,{\bf q})$ of the Higgs mode is given by $H(\omega,{\bf q})\propto (2\Delta)^2/((2\Delta)^2-\frac{1}{3}\omega^2+\frac{1}{3}c_s^2{\bf q}^2)$ for small $\omega$ and ${\bf q}$. $H(\omega,{\bf q})$ is proportional to $1/(gN(0))^2\cdot \Delta/\sqrt{ (2\Delta)^2+c_s^2{\bf q}^2-\omega^2}$ for $\omega\sim 2\Delta$ and $\omega < \omega({\bf q})$. This behavior is similar to that of the $\sigma$-particle Green function in the Gross-Neveu model. That is, the Higgs Green function $H(\omega, {\bf q})$ has the same singularity as the Green function of the $\sigma$ boson of the Gross-Neveu model. The constant part of the action for the Higgs modes is important since it determines the coherence length of a superconductor. There is the case that it has a large eigenvalue, indicating that the large upper critical field $H_{c2}$ may be realized in a superconductor with multiple order parameters.