Sixth-order time-convolutionless master equation and beyond: Late-time resummations, two types of divergences, and the limits of validity

Perturbative master equations are essential for modeling open quantum systems but often exhibit late-time divergences when environmental correlations decay algebraically. In this work, we analyze the time-convolutionless (TCL) master equation expanded to order ${\mathrm{TCL}}_{2n}$ and demonstrate that, while van Kampen's cumulants suppress early-time secular growth, they ultimately diverge at long times. To overcome this, we introduce a resummation technique based on the Hadamard trick, which incorporates time integrals directly into the bath's spectral density via elementwise multiplication. This approach establishes a maximum expansion order, ${n}_{\text{max}}=\ensuremath{\lceil}s+1\ensuremath{\rceil}$, and defines a precision limit of the asymptotic states of $O({\ensuremath{\lambda}}^{2\ensuremath{\lceil}s\ensuremath{\rceil}})$, where $s$ is the power-law exponent and $\ensuremath{\lambda}$ is the weak-coupling constant. The resummed master equation features renormalized Bohr frequencies that capture decoherence and spectral overlap effects. In the unbiased spin-boson model, this results in an inflation of the generator at a temperature-independent rate equal to the decoherence rate ${\ensuremath{\nu}}_{2}$ and a finite validity time ${t}_{L}\ensuremath{\approx}(s+1)/{\ensuremath{\nu}}_{2}$. For exponentially decaying correlations, the method recovers a proper Markovian limit below a critical coupling threshold.