Heavy Quark $\bar{q}q$ Matrix Elements in the Nucleon from Perturbative QCD
Andrei Kryjevski
TL;DR
The paper tackles the problem of determining the nucleon scalar matrix elements $\mathcal{M}_q = \langle N| m_q \bar q q|N\rangle$, which are relevant for dark matter interactions and dense-matter physics. It develops a six-flavor QCD framework and computes the heavy-quark contents $\mathcal{M}_c$, $\mathcal{M}_b$, and $\mathcal{M}_t$ to ${\cal O}(\alpha^3)$ using threshold matching and perturbative running, with results proportional to $M(1-x_{uds})$ and small positive radiative corrections; the leading piece decreases from $t$ to $c$ as the number of light quarks drops. The dominant uncertainty comes from the poorly known strange content entering $\Delta_s M$, and a related analysis suggests a potentially large $\mathcal{M}_s$ for the physical strange mass. The methods combine a heavy-quark expansion with the Feynman-Hellman theorem to relate derivatives of the nucleon mass to scalar matrix elements, and the results have implications for dark matter scattering cross sections and kaon-condensation phenomena in dense matter.
Abstract
The scalar heavy quark content of the nucleon, ${\cal M}_q = \bra{N} m_q \bar q q \ket{N},$ is relevant for computing the interaction of dark matter candidates with ordinary matter, while ${\cal M}_s$ is important for predicting the properties of dense matter. We compute ${\cal M}_q$ in perturbative QCD to ${\cal{O}}{(α^3)}.$ As one goes from ${\cal M}_t$ to ${\cal M}_c$ the leading order contribution decreases as the number of light quarks is dropping, while the radiative corrections grow and are all positive. The leading source of uncertainty in the calculation is due to the poorly known value of ${\cal M}_s.$ A related calculation suggests that a large value for ${\cal M}_s$ may be reasonable.