Universal content of the proper time flow in scalar and Yang-Mills theories

We investigate the perturbative structure of the proper time renormalization group flow in scalar and Yang-Mills theories. Although the proper time flow does not belong to the class of exact functional renormalization group equations, we show that it correctly reproduces the universal coefficients of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>β</a:mi> </a:math> functions at one and two loops. For the O(N) scalar theory, we derive the one- and two-loop contributions to the running quartic coupling and also confirm the expected anomalous dimension. For the SU(N) Yang-Mills theory, using the background field method, we compute the gauge coupling renormalization recovering the correct two-loop <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>β</c:mi> </c:math> function without generating any gauge-symmetry–violating terms. These results highlight that, despite its limitations for reconstructing the full effective action, the proper time flow retains the essential universal content of renormalization, accounting for its reliability in diverse applications ranging from statistical models to quantum gravity.