Sea quark contributions to the electric polarizability of hadrons

We present a lattice QCD calculation of the polarizability of the neutron and other neutral hadrons that includes the effects of the background field on the sea quarks. This is done by perturbatively reweighting the charges of the sea quarks to couple them to the background field. The main challenge in such a calculation is stochastic estimation of the weight factors, and we discuss the difficulties in this estimation. Here we use an extremely aggressive dilution scheme to reduce the stochastic noise to a manageable level. The pion mass in our calculation is 300 MeV and the lattice size is 3 fm. For the neutron, we find that ${\alpha }_E=2.70(55)\times 10^{-4}{\mathrm{fm}}^3$, which is the most precise lattice QCD determination of the polarizability to date that includes sea effects.

Figure 1

Dependence of $(\log \frac{G_{\pi }(t,\eta )}{G_{\pi }(t,0)})/(t{\eta }^2)$ for the neutral pion on a ${24}^3\times 48$ lattice, as a function of $\eta$ for different correlator times. This quantity is roughly equivalent to the shift in the effective mass at the chosen correlator time divided by ${\eta }^2$, and should be constant in the range where $\eta$ creates a purely quadratic effect. As this was only computed on a single configuration there are no error bars; we emphasize that these data are only useful for determining the perturbative regime of $\eta$.

Figure 2

Dependence of the mean and standard deviation of the reweighting factor as a function of $\eta$ for the ${24}^3\times 48$ ensemble used in this study. The estimator is computed using Eq. (8). The solid lines include the quadratic effects in $\eta$, whereas the dotted lines include only the linear term. Vertical lines indicate the $\eta$ value we used.

Figure 3

Exact values for $[w(\eta )-1]/\eta$ on a $4^4$ lattice, compared with the value predicted by the stochastic estimator for $w^{\prime }$ and its error band. The black circles are derived from the exact values of the determinant ratio at different $\eta$; the solid red line shows the central value of the stochastic estimator, and the dashed red lines the stochastic error bands.

Figure 4

Mapping of a representative set of the off-diagonal elements of $M^{\prime }{M}_0^{-1}$ (left) and $M^{\prime \prime }{M}_0^{-1}-(M^{\prime }{M}_0^{-1}{)}^2$ (right). The average of the square of the 12 diagonal elements, the ones that define the signal, is shown as a horizontal line.

Figure 5

Uncertainty estimates for different dilution schemes for the first-order term (left) and second-order term (right) as a function of the estimator cost, given by the number of partitions in each dilution scheme. The solid (dashed) line indicates the expected uncertainty for the estimates based on repeated use of the undiluted (spin-color-diluted) estimator. The horizontal line in the left panel corresponds to the standard deviation of the gauge fluctuations, as estimated in the next section. The point in the bottom right labeled by a black arrow is the BCHC-$6^4$ dilution scheme actually used in the computation.

Figure 6

Distribution of the values of the stochastic estimators on the full 300-configuration ensemble. The two leftmost panels are the stochastic estimators for $w_q^{\prime }$ and ${\tilde{w}}_q^{\prime \prime }$ using BCHC dilution. The right panel is the estimator for $w_q^{\prime 2}$ using a combination of BCHC dilution and hopping parameter expansion estimators, as described in the text. Note that for the center panel we shifted the distribution by subtracting its mean.

Figure 7

Effective mass (above) and effective mass shift (below) for the neutron. For the effective mass, the mass shift is sufficiently small that the plot symbols with and without the electric field are indistinguishable. In both cases, the value of the mass or mass shift extracted from the full correlated fit is shown as a colored error band; its horizontal extent indicates the chosen fitting range of 9–21 for the neutron. For the effective mass shift, the data at larger values of $t$ lie outside the plot range and are not shown.