Is quantum criticality relevant to upper critical field ($H_{c2}$) scaling in the heavy-fermion compound CeRhIn$_5$?

We report disparate forms of upper critical field ($H_{c2}$) scaling below and above a critical pressure $P_{c1}$, where long- range antiferromagnetic order (AFM) abruptly disappears in the pressure-tuned heavy-fermion superconductor CeRhIn$_5$. For $P > P_{c1}$, $h_{c2} = H_{c2}/H_{c2}(T=0$~K) shows normal two- fluid type scaling with $t = T/T_c$, i.e., $h_{c2} = (1-t^2)(1- at^2)$, where $T_c$ is superconducting transition temperature at zero field and a is a fitting parameter. In this regime, the Maki parameter $\alpha$, a measure of Pauli limiting effects, is large, suggesting that an inhomogeneous superconducting state (FFLO) may be observable. For $P < P_{c1}$, in stark contrast, $h_{c2}$ linearly increases with decreasing $t$ down to $t = 0.1 $ and the specific heat discontinuity $\Delta C/C_N$ at $T_c$ is much smaller than the BCS value (=1.43). These observations manifest the interplay between superconductivity and antiferromagnetic fluctuations that exist in the vicinity of $P_ {c1}$.