Running coupling in SU(2) gauge theory with two adjoint fermions

We study SU(2) gauge theory with two Dirac fermions in the adjoint representation of the gauge group on the lattice. Using clover improved Wilson fermion action with hypercubic truncated stout smearing we perform simulations at larger coupling than before. We measure the evolution of the coupling constant using the step scaling method with the Schrödinger functional and study the remaining discretization effects. At weak coupling we observe significant discretization effects, which make it difficult to obtain a fully controlled continuum limit. Nevertheless, the data remains consistent with the existence of a fixed point in the interval $2.2\lesssim g^{*2}\lesssim 3$. We also measure the anomalous dimension and find that its value at the fixed point is ${\gamma }^{*}\simeq 0.2\pm 0.03$.

Figure 1

The measured values of the Schrödinger functional coupling $g^2(g_0^2,L/a)$ against $L/a$ at different ${\beta }_L$’s.

Figure 2

$g^2(g_0^2,L/a)$ against $L/a$ for ${\beta }_L=6$ and 8, compared against the running of the coupling in two-loop perturbation theory (the dashed lines).

Figure 3

The scaled lattice step scaling function $\mathrm{\Sigma }(g^2,2,L/a)/g^2=g^2(g_0^2,2L/a)/g^2(g_0^2,L/a)$ calculated directly from the data in Table 2. For comparison, the black dashed line gives the continuum two-loop perturbative result.

Figure 4

The scaled step scaling function $\sigma (u,2)/u$, $u=g^2$, using only the largest volume pairs ($L/a=10$ and 20) (the red hashed band) and with continuum extrapolation (the green shaded band). The black dashed curve shows the universal two-loop perturbative result.

Figure 5

The continuum step scaling function resulting from extrapolations of the type in Eq. (20). The shaded bands show the result when the fit is constrained by the universal $\beta$-function coefficients, where the wider band includes terms up to $u^5$, the narrower band up to $u^4$. The red hashed band shows the unconstrained result. For details, see the text.

Figure 6

The estimate for the mass anomalous dimension $\overline{\gamma }(u)$. (Left panel) Examples of the continuum extrapolation. (Right panel) The green shaded band shows the continuum limit of $\overline{\gamma }$, and the red hashed band shows the result using only the largest lattice size $L/a=12$. The black dashed line gives the one-loop perturbative result.