Gauge Theory of Pairing and Spin Fluctuations Near the Quantum Critical Point

We have solved the spin Fermion (periodic Kondo) model for the superconductor transition temperature $T_c $ and for the electron energy gap function $\phi $ as $T\to T_c $. We find for realistic parameters$W$, the electron band width, $N_{B} \left( \omega \right)$, the Boson density of states and $J_q ,$ the Kondo exhange interaction, that $T_c =1.14{\kern 1pt}\,\omega _s \;\,e-\frac{(1+\lambda _Z )}{\lambda _\phi }$ where $\lambda _Z $ is the normal state renormalization constant and $\lambda _\phi $ is the pairing interaction strength. We find $T_c $ is exponentially higher for $\ell =1$ (p-wave), $S=1$ (spin triplet) pairing than for $s$- wave pairing $S=0$. We note $\lambda _Z =0$ for $p$-wave pairing due to the odd parity of the relevant. For realistic parameters the solution of Eliashberg's equation for $T_c $ predicts $T_c \tilde {-}\;5\times 10^5\,^0K$ with $H_{c2} \sim 10^8T$ and$j_c \sim 10^8\,Amps/cm^2$. When $T_c $ and $\phi $ are simultaneously maximized, with respect to $N_B \left( \omega \right)$ and $J_q $ considerably high $T_c ,H_{c2} ,j_c $ values are predicted, namely $T_c $ of order $5\times 10^8\,^0K \quad H_{c2} \sim 10^{13}T$ and $j_c \sim 10^{13}\,Amps/cm^2$There values are predicted to exist in systems such as the Heusler alloys, e.g. for $Au_2 \left( {Mn_{2-x} \;A\ell _x } \right)$ for $x\tilde {-}0.1-0.5.$