A Direct Construction of the Nuclear Effective Interaction

Traditionally the nuclear physics effective interactions problem is attacked in two steps, the encoding of phase-shift information in a rather singular ``realistic'' NN interaction $V_{NN}$, followed by a reduction of $H=T+V_{NN}$ to the included or P-space $H^{eff}$ by integrating out numerically the effects of $H$ in $Q=1-P$. Here we show that $H^{eff}$ can be determined directly in P, eliminating the need for any knowledge of $V_{NN}$ in Q. The method exploits the Haxton-Luu form of the Bloch-Horowitz equation, in which long and short-range contributions to $H^{eff}$ are separated. This decomposition allows one to build into an effective theory the correct infrared behavior, which for continuum states is governed by the energy-dependent phase shifts $\delta(E)$. The effects of $V_{NN}$ in Q can then be absorbed into a small number of nearly energy-independent low-energy constants (LECs), the coefficients of short-range operators. I show that the experimental knowledge of $\delta(E)$ that traditionally is encoded in $V_{NN}$ can instead be used directly in P to determine the LECs. The method reduces the task of finding a precise $H^{eff}$ to that of solving a self-consistent eigenvalue problem in P.