Extending the Kinetic Mass to Higher Orders in $1/m_Q$

Currently, the kinetic mass is defined in terms of the pole mass and operators at order $1/m_Q^2$, which are known to N$^3$LO accuracy in $\alpha_s$. At the same time, the Heavy Quark Expansion (HQE) for inclusive semileptonic decays is known up to and including terms of order $1/m_Q^5$. Therefore, it is desirable to extend the definition of the kinetic mass to higher orders in $1/m_Q$. The original kinetic mass is based on the hadron-mass formula in Heavy Quark Effective Theory (HQET). However, the HQE is formulated in terms of matrix elements defined in full QCD to avoid the appearance of non-local matrix elements. To avoid this, we develop a definition of the kinetic mass rooted in full QCD. Starting from the hadron-mass formula derived from the energy-momentum tensor of full QCD, we define a relation between a general mass and the pole mass. Using a simple cut-off scheme, we compute a generalized kinetic mass at one loop to all powers of $1/m_Q$, which reproduces the well-known results for the kinetic mass up to $1/m_Q^2$. Our approach opens the road to a consistent use of the kinetic mass at higher-orders in the heavy quark expansion.