Glueball scattering cross section in lattice $SU(2)$ Yang-Mills theory

We calculate the scattering cross section between two $0^{++}$ glueballs in $SU(2)$ Yang-Mills theory on a lattice at $\beta =2.1$, 2.2, 2.3, 2.4, and 2.5 using the indirect (HAL QCD) method. We employ the cluster-decomposition error reduction technique and use all space-time symmetries to improve the signal. In the use of the HAL QCD method, the centrifugal force was subtracted to remove the systematic effect due to the nonzero angular momenta of lattice discretization. From the extracted interglueball potential, we determine the low energy glueball effective theory by matching with the one-glueball exchange process. We then calculate the scattering phase shift and derive the relation between the interglueball cross section and the scale parameter $\mathrm{\Lambda }$ as ${\sigma }_{\varphi \varphi }=(2–51){\mathrm{\Lambda }}^{-2}$ ($\text{stat}+\text{sys}$). From the observational constraints of galactic collisions, we obtain the lower bound of the scale parameter as $\mathrm{\Lambda }>60\mathrm{MeV}$. We also discuss the naturalness of the Yang-Mills theory as the theory explaining dark matter.

Figure 1

Glueball effective mass calculated with the standard glueball operator (3) (green points) and with APE smearing (blue points) at $\beta =2.5$. The result of the previous work [312] (red line, with the uncertainty band) is also shown for comparison.

Figure 2

NBS amplitudes for the 1-body wall source calculations on the lattice with $\beta =2.1$, 2.2, 2.3, 2.4, and 2.5.

Figure 3

NBS amplitudes for the 1-, 2-, and 3-body wall source calculations on ${16}^3\times 24$ lattice with $\beta =2.4$.

Figure 4

Comparison of the calculations of the NBS amplitude using the 1-body wall source with the volumes ${16}^3\times 24$ and ${32}^3\times 24$ ($\beta =2.5$). The NBS amplitude was calculated with 1045000 and 126000 configurations for the ${16}^3\times 24$ and ${32}^3\times 24$ lattices, respectively.

Figure 5

CDERT applied to the glueball two-point function calculated with 100 configurations and the smeared operator on ${16}^3\times 24$ lattice with $\beta =2.4$. We see that the correlator saturates at the cutoff $\rho =7$ (lattice unit), and further increase of the cutoff enlarges the statistical error bar.

Figure 6

Glueball NBS amplitude with a 1-body source ($\mathcal{J}=\tilde{\varphi }$) obtained by applying the CDERT with the cutoff $\rho =7$ (lattice unit) on ${16}^3\times 24$ lattice at $\beta =2.4$. We compare it with the wall source calculation to visualize the improvement of the signal.

Figure 7

Effective mass plot of the glueball NBS amplitude in the momentum space, with a 1-body source ($\mathcal{J}=\tilde{\varphi }$) on ${16}^3\times 24$ lattice with $\beta =2.5$. We see that the energy of the system saturates at the single glueball mass.

Figure 8

Comparison of the interglueball potentials extracted using the HAL QCD method with and without the subtraction of centrifugal force (${14}^3\times 16$ lattice, $\beta =2.3$). The potential becomes attractive after the subtraction in the region around $r\sim 0.3$.

Figure 9

Interglueball potential calculated on the lattice in $SU(2)$ YMT at $\beta =2.1$, 2.2, 2.3, 2.4, and 2.5. The fits with the Yukawa (14) and 2-Gaussian (15) fitting forms are also displayed. The colored band denotes the uncertainty (statistical and systematic). For each $\beta$, we do not displayed data points at $r=0$, 1, and $r\ge 4$ (lattice unit) because we did not use them in the fit.

Figure 10

One-glueball exchange process in the t channel generated by the glueball trilinear interaction, with the exchanged momentum $k$.

Figure 11

Feynman rule for the glueball trilinear interaction.