Stability of $\varphi^4$-vector model: four-loop $\varepsilon$ expansion study

The stability of $O(n)$-symmetric fixed point regarding the presence of vector-field term ($\sim h p_{\alpha}p_{\beta}$) in the $\varphi^4$ field theory is analyzed. For this purpose, the four-loop renormalization group expansions in $d=4-2\varepsilon$ within Minimal Subtraction (MS) scheme are obtained. This frequently neglected term in the action requires a detailed and accurate study on the issue of existing of new fixed points and their stability, that can lead to the possible change of the corresponding universality class. We found that within lower order of perturbation theory the only $O(n)$-symmetric fixed point $(g_{\text{H}},h=0)$ exists but the corresponding positive value of stability exponent $\omega_h$ is tiny. This led us to analyze this constant in higher orders of perturbation theory by calculating the 4-loop contributions to the $\varepsilon$ expansion for $\omega_h$, that should be enough to infer positivity or negativity of this exponent. The value turned out to be undoubtedly positive, although still small even in higher loops: $0.0156(3)$. These results cause that the corresponding vector term should be neglected in the action when analyzing the critical behaviour of $O(n)$-symmetric model. At the same time, the small value of the $\omega_h$ shows that the corresponding corrections to the critical scaling are significant in a wide range.