Comment on "Nucleon-nucleon scattering lengths in QCD sum rules"

In a recent Letter [1], Kondo and Morimatsu present a QCD sum rule calculation of nucleon-nucleon scattering lengths. They also relate the empirical scattering lengths to the nucleon mass shift δM in nuclear matter to cast doubt on the " linear density approximation. " In this Comment, we point out flaws in both parts of their analysis. It is useful to start with the (nonrelativistic) optical potential for a nucleon in nuclear matter in the lowest order of the multiple scattering expansion [2] V (q) ∼ k F τ (q, k) d 3 k , (1) where τ (q, k) is the effective forward-scattering amplitude for an incoming nucleon with momentum q and a nucleon in nuclear matter with momentum k [2]. If |q| ≫ k F , one can approximate V (q) by neglecting the momenta of nucleons in the Fermi sea and replacing τ (q, k) by the amplitude in free space t(q, k): V (q) ∼ t(q, 0) ρ , which is the usual impulse approximation. However, in Ref. [1], q = 0 is taken after the impulse approximation. Applying this approximation with |q| ≪ k F drastically overestimates 1