Positive-semidefinite testing of the complete next-to-leading Hamiltonian

Develop a test that determines positive semidefiniteness of the complete next-to-leading Hamiltonian, beyond the partial JIMWLK-kernel tests discussed in the literature, in order to assess whether a Langevin formulation is possible.

Background

The paper discusses related work testing the JIMWLK kernel as a quadratic form in transverse separations. Positive semidefiniteness is identified as the condition needed for a Langevin representation, and some prescriptions satisfy it while the fixed-order kernel and conventional running-coupling prescriptions violate it.

The cited analysis does not test the entire next-to-leading Hamiltonian. The paper explicitly records the unresolved methodological problem of determining how the full Hamiltonian should be tested, rather than only its examined components or prescriptions.