Power counting in peripheral partial waves: The singlet channels

We analyze the power counting of the peripheral singlet partial waves in nucleon-nucleon scattering. In agreement with conventional wisdom, we find that pion exchanges are perturbative in the peripheral singlets. We quantify from the effective field theory perspective the well-known suppression induced by the centrifugal barrier in the pion-exchange interactions. By exploring perturbation theory up to fourth order, we find that the one-pion-exchange potential in these channels is demoted from leading to subleading order by a given power of the expansion parameter that grows with the orbital angular momentum. We discuss the implications of these demotions for few-body calculations: though higher partial waves have been known for a long time to be irrelevant in these calculations (and are hence ignored), here we explain how to systematize the procedure in a way that is compatible with the effective field theory expansion.

Figure 1

Cutoff dependence of the nonperturbative phase shifts for the OPE potential in the singlet peripheral waves ${}^{1}P_1, {}^{1}F_3$, and ${}^{1}H_5$. We show the isoscalar partial waves because for them the OPE potential is three times stronger than in the isovector partial waves. Hence the cutoff dependence is also expected to be more evident in the isoscalar channels. We show the $r_c=0.3–2.0\mathrm{fm}$ cutoff range for the c.m. momenta $k_{\mathrm{c}.\mathrm{m}.}=75$, 150, and $300\mathrm{MeV}$.

Figure 2

Convergence of the perturbative expansion of the phase shifts for the OPE potential in the singlet peripheral partial waves ${}^{1}P_1, {}^{1}D_2, {}^{1}F_3, {}^{1}G_4, {}^{1}H_5$, and ${}^{1}I_6$. We compare the full, nonperturbative OPE phase shifts—the black solid lines in the figures—with the perturbative ones computed at increasing orders, where T stands for tree level (i.e., the Born approximation) and 1L, 2L, and 3L stand for the one-, two- and three-loop calculations (i.e., second-, third-, and fourth-order perturbation theory). The calculations have been performed with a finite cutoff of $r_c=0.3\mathrm{fm}$.

Figure 3

Inverse of the ratio of the the diagonal potential $V_l$ in momentum space between the $l$ wave and the $p$ wave. (See the main text for a detailed explanation of the ratio.)

Figure 4

Ratio between the actual value of ${\mathrm{\Lambda }}_{NN}$ appearing in the OPE potential and the critical ${\mathrm{\Lambda }}_{NN}^{*}$ generating a bound state at threshold for the $l$ wave. The ratio is calculated for isoscalar and isovector channels. In the isoscalar case the ratio is negative as the OPE potential is actually repulsive in the isoscalar channels ${}^{1}P_1, {}^{1}F_3, {}^{1}H_5$, and so on.

Figure 5

Inverse ratios between the $n\mathrm{th}$ order over the $(n-1)\mathrm{th}$ order perturbation theory for OPE and the singlet partial waves with $l\le 6$. (See the main text for a detailed explanation of the ratios.) The theoretical value to which the ratios should converge is shown by the black solid line.