Partially Quenched Nucleon-Nucleon Interactions

S.R. Beane and M.J. Savage

In previous work, we showed that the long-distance part of the potential between nucleons in partially-quenched QCD is very different to that in QCD due to the contribution from hairpin interactions [15]. Such interactions give rise to an exponentially falling potential as opposed to the Yukawa behavior that one finds in QCD. The coefficient of this exponential component depends upon the difference between sea and valence quark masses. In our recent work we explore the scattering amplitudes directly in PQ$\chi$PT. We find that calculations are straightforward with BBSvK power-counting, and we were able to produce amplitudes in the ${}^1\kern -.14em S_0$ channel and potentials that are to be used in the Schrodinger equation in the ${}^3\kern-.14em S_1-{}^3\kern-.14em D_1$ coupled-channels. For instance the scattering volumes in the P-waves are

$$a (^1 P_1) = { g_A^2 M_N\over 4\pi f^2 m_\pi^2} +\ {g_0^2 M_N\over 12\pi f^2 m_\pi^2} {m_{SS}^2-m_\pi^2\over m_\pi^2}$$

$$a (^3 P_0) = -{ g_A^2 M_N\over 4\pi f^2 m_\pi^2} +\ {g_0^2 M_N\over 4\pi f^2 m_\pi^2} {m_{SS}^2-m_\pi^2\over m_\pi^2}$$

$$a (^3 P_1 ) = +{ g_A^2 M_N\over 6\pi f^2 m_\pi^2} -\ {g_0^2 M_N\over 6\pi f^2 m_\pi^2} {m_{SS}^2-m_\pi^2\over m_\pi^2}$$

$$a (^3 P_2 ) = 0 ,$$ (1)

where $m_{SS}$ is the mass of a meson containing two sea quarks and $g_0$ is the singlet axial coupling constant. The scattering length in the ${}^1\kern -.14em S_0$ channel in the partially-quenched EFT is found to be

$${1\over a^{({}^1\kern-.14em S_0)}} = \gamma - {M_N\over 4\pi} (\mu-\gamma)^2 D_2^{({}^1\kern-.14e... ...\gamma)^2 D_{2B}^{({}^1\kern-.14em S_0)} (\mu)\ \left(m_{SS}^2-m_\pi^2\right)$$

$$+\ {g_A^2 M_N\over 8\pi f^2}\left[\ m_\pi^2\log\left({\mu\over m_\pi}\right) +\ (m_\pi-\gamma)^2 - (\mu-\gamma)^2 \right]$$

$$+\ {g_0^2 M_N\over 8\pi f^2}\ \left( m_{SS}^2-m_\pi^2 \righ... ...u\over m_\pi}\right) +\ {1\over 2} - {\gamma \over m_\pi} \right] ,$$ (2)

where $\gamma$ is a constant that enters at LO in the expansion and is a $\mu$-independent linear combination of $C_0^{({}^1\kern-.14em S_0)}$ and $\mu$ that must be determined from data. The hope is that the RG scale dependent constants $D_2^{({}^1\kern-.14em S_0)} (\mu)$ and $D_{2B}^{({}^1\kern-.14em S_0)} (\mu)$ will be determined from lattice calculations.