Estimating the unquenched strange quark mass from the lattice axial Ward identity

We present a determination of the strange quark mass for two flavors ($n_f=2$) of light dynamical quarks using the axial Ward identity. The calculations are performed on the lattice using $O(a)$ improved Wilson fermions and include a fully nonperturbative determination of the renormalization constant. In the continuum limit we find $m_s^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=111(6)(4)(6)\mathrm{MeV}$, using the force scale $r_0=0.467\mathrm{fm}$, where the first error is statistical, the second and third are systematic due to the fit and scale uncertainties, respectively. Results are also presented for the light quark mass and the chiral condensate. The corresponding results are also given for $r_0=0.5\mathrm{fm}$.

Figure 1

One-, two-, three- and four-loop results for $[\Delta Z_m^{\overline{\mathrm{MS}}}(\mu ){]}^{-1}$ in units of ${\Lambda }^{\overline{\mathrm{MS}}}$.

Figure 2

$[\Delta Z_m^{\mathrm{R}{\mathrm{I}}^{\prime }-\mathrm{MOM}}({\mu }_p){]}^{-1}$ versus ${\mu }_p/{\Lambda }^{\overline{\mathrm{MS}}}$.

Figure 3

$A_P$ and $B_P$ for $Z_P^{\mathrm{R}{\mathrm{I}}^{\prime }-\mathrm{MOM}}$ for $\beta =5.20$ and $\beta =5.40$.

Figure 4

$Z_{\tilde{m}}^{\mathrm{RGI}}$ for $\beta =5.20$ (filled circles), $\beta =5.25$ (filled squares), $\beta =5.29$ (filled upper triangles), $\beta =5.40$ (filled lower triangles) together with fits as in Eq. (50).

Figure 5

$Z_{\tilde{m}}^{\mathrm{RGI}}$ versus $\beta$. The black circles are the results from Table 3, while the open squares are the TRG-PT results from Table 4. Furthermore, the open diamonds and triangles are the NP results from [12], using the two different results for the axial renormalization constant [56]. (The empty triangle results have been slightly displaced for clarity.)

Figure 6

$R_{\tilde{m}}^X$ versus $(a/r_0{)}_c^2$ for $X=\mathrm{TRB}-\mathrm{PT}$ (open squares), $X=\mathrm{ALPHA}-Z_A$ (open diamonds) and $X=\mathrm{ALPHA}-Z_A^{\mathrm{CON}}$ (open triangles).

Figure 7

The ratio $r_0{m}_q^{\mathrm{RGI}}/(r_0{m}_{ps}{)}^2$ against $(r_0{m}_{ps}{)}^2$ for $\beta =5.20$. The fit is described in the text. Results for equal sea and valence quark masses are denoted by an open symbol; partially quenched results by filled symbols. The labeled dashed and dashed-dotted lines are also explained in the text.

Figure 8

The ratio $r_0{m}_q^{\mathrm{RGI}}/(r_0{m}_{ps}{)}^2$ against $(r_0{m}_{ps}{)}^2$ for $\beta =5.25$. Notation as for Fig. 7.

Figure 9

The ratio $r_0{m}_q^{\mathrm{RGI}}/(r_0{m}_{ps}{)}^2$ against $(r_0{m}_{ps}{)}^2$ for $\beta =5.29$. Notation as for Fig. 7.

Figure 10

The ratio $r_0{m}_q^{\mathrm{RGI}}/(r_0{m}_{ps}{)}^2$ against $(r_0{m}_{ps}{)}^2$ for $\beta =5.40$. Notation as for Fig. 7.

Figure 11

$c_a^{\mathrm{RGI}}$ (filled circles), $c_b^{\mathrm{RGI}}$ (filled upper triangles) $c_c^{\mathrm{RGI}}$ (filled lower triangles) and $c_d^{\mathrm{RGI}}$ (filled diamonds) versus $(a/r_0{)}_c^2$. The $(r_0/a{)}_c$ i.e. the chirally extrapolated values for $r_0/a$ are used. The open symbols represent the values of $c_a^{\mathrm{RGI}}$ and $c_i^{\mathrm{RGI}}$ $(i=b,c,d)$ in the continuum limit.

Figure 12

$m_s^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$ versus $(a/r_0{)}_c^2$ (filled circles), together with a linear continuum extrapolation (empty circle). The scale used is $r_0=0.5\mathrm{fm}$. For comparison we also show the results from using the VWI, open squares [11].

Figure 13

Results for $m_s^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$ versus $a^2$ using the AWI (upper plot) and VWI (lower plot) methods. The results are presented with the collaborations preferred units and scales. Circles (together with a linear continuum extrapolation) are from this work and [11]; diamonds from [12]; squares from [13]; up triangles from [3]; down triangles from [4]; left triangles from [6]; right triangles from [8]. NPR denotes nonperturbative renormalization, while TB-PT denotes TI boosted perturbation theory. [6, 8] are for $n_f=2+1$ flavors; the other results are all for $n_f=2$ flavors.