Lattice calculation of coordinate-space vector and axial-vector current correlators in QCD

We study the vector and axial-vector current correlators in perturbative and nonperturbative regimes of QCD. The correlators in Euclidean coordinate space are calculated on the lattice using the Möbius domain-wall fermion formulation at three lattice spacings covering 0.044–0.080 fm. The dynamical quark effects of $2+1$ light flavors are included. The sum $V+A$ and the difference $V-A$ of the vector ($V$) and axial-vector ($A$) current correlators calculated on the lattice after extrapolating to the physical point agree with those converted from the ALEPH experimental data of hadronic $\tau$ decays. The level of the agreement in the $V+A$ channel is about $1.3\sigma$ or smaller in the region of $|x|\ge 0.4\mathrm{fm}$, while that in the $V-A$ channel is about $1.8\sigma$ at $|x|=0.74\mathrm{fm}$ and smaller at other distances. We also extract the chiral condensate from the short-distance correlators on the lattice using the partially conserved axial current relation. Its result extrapolated to the chiral and continuum limit is compatible with other estimates at low energies.

Figure 1

Spectral functions of the vector (upper) and axial-vector (lower) channels measured by the ALEPH Collaboration (circles) [6] as functions of $s$. The prediction of the parton model (dotted line), perturbation theory (solid line) and the fit result (dashed curve and band) using the fit function (8) are also shown.

Figure 2

Vector and axial-vector correlators reproduced using the dispersion relation (7). The spectral functions measured by the ALEPH Collaboration are used for $s\le 2.7{\mathrm{GeV}}^2$, while those in $s>2.7{\mathrm{GeV}}^2$ are calculated perturbatively with (solid band) and without (hatched band) the duality-violating term. The prediction of the massless perturbation theory is also shown (dotted line).

Figure 3

Decomposition of the vector correlators into contributions from the spectral function in different regions of $s$.

Figure 4

Same as Fig. 3 but for the axial-vector channel.

Figure 5

$R_{V+A}$ calculated on the ensembles with $\beta =4.35$, $a{m}_s=0.0180$ and three input light quark masses: $a{m}_q=0.0042$ (diamonds), 0.080 (squares) and 0.0120 (circles). The prediction of massless perturbation theory (dashed curve) and the results from experiment calculated with the duality-violating term in (8) (band) are also shown.

Figure 6

Same as Fig. 5 but calculated on ensembles with different lattice cutoffs. Pion masses are $M_{\pi }\simeq 300\mathrm{MeV}$.

Figure 7

Lattice result for $R_{V+A}$ after the chiral and continuum extrapolations. Data in each bin are extrapolated assuming (18). The bin size is larger in the short-distance region $|x|\lesssim 0.4\mathrm{fm}$ (blue crosses) than others (red squares) as there are fewer lattice points.

Figure 8

Fit parameter $d_i$ obtained from the extrapolation of $R_{V+A}(x)$ to the physical point. The result for each bin is plotted as a function of $|x|=x_i$.

Figure 9

${\chi }^2$ divided by the degrees of freedom for the extrapolation of $R_{V+A}(x)$ to the physical point. The result for each bin is plotted as a function of $|x|=x_i$.

Figure 10

Significance of the difference $D_{V+A}(x)$ between the lattice calculation and experiment plotted as a function of $|x|$.

Figure 11

$R_{V-A}$ calculated on the ensembles with $\beta =4.35$, $a{m}_s=0.0180$ and three input light quark masses: $a{m}_q=0.0042$ (diamonds), 0.080 (squares) and 0.0120 (circles). The results from experiment converted including the duality-violating term in (8) (band) are also shown.

Figure 12

Same as Fig. 11 but calculated on ensembles with different lattice cutoffs. Pion masses are $M_{\pi }\simeq 300\mathrm{MeV}$.

Figure 13

Lattice result for $R_{V-A}$ after the chiral and continuum extrapolations. Data in each bin are extrapolated assuming (18). The bin size is larger in the short-distance region $|x|\lesssim 0.4\mathrm{fm}$ (blue crosses) than others (red squares) as there are fewer lattice points. The experimental result (band) and the predictions of the OPE including up to dimension-4 (dotted curve) and dimension-6 (dashed curve) operators are also plotted.

Figure 14

Same as Fig. 8 but for the $V-A$ channel.

Figure 15

Same as Fig. 9 but for the $V-A$ channel.

Figure 16

Significance of the difference $D_{V-A}(x)$ between the lattice calculation and experiment plotted as a function of $|x|$.

Figure 17

${\mathrm{\Sigma }}_{m_q}^{V/A}$ as functions of $(x/a{)}^2$ calculated on the ensemble with $\beta =4.35$ and $(a{m}_q,a{m}_s)=(0.0042,0.0180)$.

Figure 18

$({\mathrm{\Sigma }}_{m_q}^{\overline{\mathrm{MS}}}(2\mathrm{GeV}){)}^{1/3}$ calculated on the ensembles with $\beta =4.35$, $a{m}_s=0.0180$ and at three input light quark masses: $a{m}_q=0.0042$ (diamonds), 0.080 (squares) and 0.0120 (circles). The gray band shows the FLAG average $({\mathrm{\Sigma }}^{\overline{\mathrm{MS}}}(2\mathrm{GeV}){)}^{1/3}=274(3)\mathrm{MeV}$.

Figure 19

Same as Fig. 18 but calculated on ensembles with different lattice cutoffs and pion masses are $M_{\pi }\simeq 300\mathrm{MeV}$.