Rm Fm Research Articles

Derivation of transition density from the observed 4He(e, e′)4He(02+) form factor raising the α-particle monopole puzzle

Abstract Recently, the monopole transition form factor of the electron-scattering excitation of the $0^+_2$ state (Ex = 20.21 MeV) of the 4He nucleus was observed over a broad momentum transfer range ($0.5 \le q^2 \le 5.0 \, {\rm fm}^{-2}$) with dramatically improved precision compared with older sets of data; modern nuclear forces, including those derived from the chiral effective field theory, failed to reproduce the form factor, which is called the α-particle monopole puzzle. To resolve this puzzle by improving the study of the spatial structure of the $0^+_2$ state, we derive in this letter a possible $0^+_1\!\rightarrow \! 0^+_2$ transition density ρtr(r) for r ≳ 1 fm from the observed form factor. The shape of the transition density is significantly different from that obtained theoretically in the literature.

Equation of state for strange quark matter: Linking the Nambu–Jona-Lasinio model to perturbative QCD

Neutron star constraints and {\it ab initio} pQCD evaluations require the EoS representing cold quark matter to be stiff at intermediate baryonic densities and soft at high-$n_B$. Here, I suggest that the three flavor NJL model with a density dependent repulsive coupling, $G_V(\mu)$, can generate an EoS which interpolates between these two regimes. Such an interpolation requires repulsion to start decreasing with the chemical potential just after chiral transition takes place. The conjecture behind this mechanism is that repulsion should be necessary only as long as the quark condensates, which dress the effective masses, have non-vanishing values. This assumption guarantees that an initially hard EoS suffers a conspicuous change of slope at ${\cal E} \simeq 0.7 \,{\rm GeV fm^{-3}}$ converging to the pQCD results at higher energy densities. Then, the speed of sound naturally reaches a non-conformal maximum at $n_B = 3.23 \, n_0 = 0.52 \, {\rm fm}^{-3}$ while the trace anomaly remains positive for all densities, in agreement with recent investigations. These non-trivial results {\it cannot} be simultaneously obtained when $G_V$ vanishes or has a fixed value. Therefore, the simple model proposed here is able to link the (non-perturbative) region of intermediate densities to the region where pQCD becomes reliable.

Nucleon isovector couplings in Nf=2+1 lattice QCD at the physical point

We present results for the scalar and tensor isovector-couplings ($g_S$ and $g_T$) of the nucleon measured at the physical point ($M_{\pi}=135$ MeV) with a single lattice spacing of $0.085\ \mathrm{fm}$ in 2+1 flavor QCD. Our calculations are carried out with two ensembles of gauge configurations generated by the PACS Collaboration with nonperturbatively ${\cal O}(a)$ improved Wilson quark action and Iwasaki gauge action on $(10.9\ {\rm fm})^4$ and $(5.5\ {\rm fm})^4$ lattices, where the finite-size effect on the nucleon mass was not shown at the level of the statistical precision less than 0.5%. We compute the nucleon three-point correlation functions in the vector, axial, scalar, and tensor channels. We confirm that our previous result of the nucleon axial coupling on the large spatial volume of $(10.9\ {\rm fm})^4$ has no finite-size effect at the level of the statistical precision of 1.9%. For the renormalization, we first renormalize $g_S$ and $g_T$ nonperturbatively using the RI/SMOM$_{(\gamma_\mu)}$ scheme, a variant of Rome-Southampton RI/MOM scheme with reduced systematic errors, as the intermediate scheme. We evaluate our final results at the renormalization scale of 2 GeV in the $\overline{\rm MS}$ scheme through matching procedure between the RI/SMOM$_{(\gamma_\mu)}$ and $\overline{\rm MS}$ schemes with the help of perturbation theory, and then obtain $g_S=0.927(71)_{\rm stat}(22)_{\rm syst}$ and $g_T=1.036(6)_{\rm stat}(20)_{\rm syst}$.

Baryonic form factors of the pion and kaon in a chiral quark model

The baryonic form factor of charged pions is studied in detail in the Nambu--Jona-Lasinio model with constituent quarks, where the spontaneously broken chiral symmetry is the key dynamical ingredient guaranteeing the would-be Goldstone boson nature of the pseudoscalar mesons octet. In general, this form factor arises when the isospin symmetry is broken, which is the case if the $u$ and $d$ quark masses are split, as in the real world, or if the electromagnetic effects were taken into account. We obtain estimates for this basic property of the pion resulting from the quark mass splitting for a range of model parameters, and importantly, for different pion masses, going up to the values used in lattice QCD. We find very stable model results, with the mean square radius of $\pi^+$ in the range $(0.05-0.07~{\rm fm})^2$. From charge conjugation, the baryonic form factor of $\pi^+$ and $\pi^-$ are equal and opposite. We also obtain the transverse-coordinate, relativistically invariant baryonic density of the charged pion. In $\pi^+$, the inner region carries a negative, and the outside -- a positive baryon number density, both cancelling to zero, as obviously the pion carries no net baryon charge. We also carry out an analogous analysis for the kaon, where the effect is much larger due to the sizable $s$ and $u/d$ quark mass splitting. We discuss the prospects of lattice QCD measurements of the baryonic form factors of charged pions and kaons.

Lattice calculation of the pion mass difference Mπ+−Mπ0 at order O(αem)

We present a lattice calculation of the charged/neutral pion mass difference $M_{\pi^{+}}-M_{\pi^{0}}$ at order $\mathcal{O}(\alpha_{em})$ using the gauge configurations produced by the Extended Twisted Mass Collaboration with $N_{f}=2+1+1$ dynamical quark flavours at three values of the lattice spacing ($a \simeq 0.062, 0.082, 0.089~{\rm fm}$) and pion masses in the range $M_{\pi} \simeq 250-500~{\rm MeV}$. We employ the RM123 method and expand the path-integral around the isospin symmetric point at leading order in the electromagnetic coupling $\alpha_{em}$. Making use of the recently proposed RTM scheme, we evaluate the full $\mathcal{O}(\alpha_{em})$ contribution, with the inclusion of the disconnected diagram. At the physical point, after performing the continuum and infinite volume extrapolation, we obtain the value $M_{\pi^{+}}-M_{\pi^{0}}= 4.622~(95)~{\rm MeV}$ which is in good agreement with the experimental result $[ M_{\pi^{+}} - M_{\pi^{0}} ]^{exp.} = 4.5936(5)~{\rm MeV}$.

Pion form factor and charge radius from lattice QCD at the physical point

We present our results on the electromagnetic form factor of pion over a wide range of $Q^2$ using lattice QCD simulations with Wilson-clover valence quarks and HISQ sea quarks. We study the form factor at the physical point with a lattice spacing $a=0.076$ fm. To study the lattice spacing and quark mass effects, we also present results for 300 MeV pion at two different lattice spacings $a=0.04$ and 0.06 fm. The lattice calculations at the physical quark mass appear to agree with the experimental results. Through fits to the form factor, we estimate the charge radius of pion for physical pion mass to be $\langle r_{\pi}^2 \rangle=0.42(2)~{\rm fm}^2$.

Chiral condensates for neutron stars in hadron-quark crossover: From a parity doublet nucleon model to a Nambu–Jona-Lasinio quark model

We study the chiral condensates in neutron star matter from nuclear to quark matter domain. We describe nuclear matter with a parity doublet model (PDM), quark matter with the Nambu--Jona-Lasino (NJL) model, and a matter at the intermediate density by interpolating nuclear and quark matter equations of state. The model parameters are constrained by nuclear physics and neutron star observations. Various condensates in the interpolated domain are estimated from the chemical potential dependence of the condensates at the boundaries of the interpolation. The use of the PDM with substantial chiral invariant mass ($m_0 \gtrsim 500$ MeV, which is favored by the neutron star observations) predicts the mild chiral restoration, and the significant chiral condensate remains to baryon density $n_B \sim 2-3n_0$ ($n_0\simeq 0.16\,{\rm fm}^{-3}$: nuclear saturation density), smoothly approaching the NJL predictions for the color-flavor-locked phase at $n_B \gtrsim 5n_0$. The same method is applied to estimate diquark condensates, number densities of up-, down- and strange-quarks, and the lepton fraction. In our descriptions the chiral restoration in the interpolated domain proceeds with two conceptually distinct chiral restoration effects; the first is associated with the positive scalar density in a nucleon, relevant in dilute regime, and the other primarily arises from the modification of the quark Dirac sea, which is triggered by the growth of the quark Fermi sea. We discuss several qualitative conjectures to interpolate the microphysics in nuclear and quark matter.

Constraints on the phase transition and nuclear symmetry parameters from PSR J0740+6620 and multimessenger data of other neutron stars

Recently, the radius of neutron star (NS) PSR J0740+6620 was measured by Neutron Star Interior Composition Explorer (NICER) and an updated measurement of neutron skin thickness of ${}^{208}$Pb ($R_{\rm skin}^{208}$) was reported by the PREX-II experiment. These new measurements can help us better understand the unknown equation of state (EOS) of dense matter. In this work, we adopt a hybrid parameterization method, which incorporates the nuclear empirical parameterization and some widely used phenomenological parameterizations, to analyze the results of nuclear experiments and astrophysical observations. With the joint Bayesian analysis of GW170817, PSR J0030+0451, and PSR J0740+6620, the parameters that characterize the ultradense matter EOS are constrained. We find that the slope parameter $L$ is approximately constrained to $70_{-18}^{+21}$ MeV, which predicts $R_{\rm skin}^{208}=0.204^{+0.030}_{-0.026}\,{\rm fm}$ by using the universal relation between $R_{\rm skin}^{208}$ and $L$. The bulk properties of canonical $1.4\,M_\odot$ NS (e.g., $R_{1.4}$ and $\Lambda_{1.4}$) as well as the pressure ($P_{2\rho_{\rm sat}}$) at two times the nuclear saturation density are well constrained by the data; i.e., $R_{1.4}$, $\Lambda_{1.4}$, and $P_{2\rho_{\rm sat}}$ are approximately constrained to $12.3\pm0.7$ km, $330_{-100}^{+140}$, and $4.1_{-1.2}^{+1.5}\times10^{34}\,{\rm dyn\,cm^{-2}}$, respectively. Besides, we find that the Bayes evidences of the hybrid star and normal NS assumptions are comparable, which indicates that current observation data are compatible with quarkyonic matter existing in the core of massive star. Finally, in the case of normal NS assumption, we obtain a constraint for the maximum mass of nonrotating NS $M_{\rm TOV}=2.30^{+0.30}_{-0.18}$ $M_\odot$. All of the uncertainties reported above are for 68.3% credible levels.

Phenomenological QCD equations of state for neutron star dynamics: Nuclear-2SC continuity and evolving effective couplings

We delineate the quark-hadron continuity by constructing QCD equations of state for neutron star dynamics, covering the wide range of charge chemical potential ($\mu_Q$) and temperatures ($T$). Based on the nuclear-2SC continuity scenario, we match equations of state for nuclear and two-flavor color-superconducting (2SC) quark matter, where the matching baryon density is $n_B\simeq 1.5n_0$ ($n_0\simeq 0.16\, {\rm fm^{-3}}$: nuclear saturation density). The effective vector and diquark couplings in a quark matter model evolve as functions of $n_B, \mu_Q, T$, whose low density values are constrained by the nuclear matter properties and neutron star radii, with the high density behavior by the two-solar mass ($2M_\odot$) constraint. With couplings dependent on $n_B$, we examined how smooth the nuclear-2SC continuity can be, and found problems in matching nuclear and 2SC entropies at low temperatures. To proceed with the continuity scenario, we enforce smooth matching by making the couplings ($n_B, \mu_Q, T$)-dependent. In effect, this adds phenomenological contributions which we call "X". After the phenomenological matching, we take the rest as our predictions. The 2SC and color-flavor-locked (CFL) phases computed with these evolving couplings are called 2SCX and CFLX. The CFLX appears around $n_B\simeq 2$-$4n_0$ and, in contrast to the conventional CFL, has non-negligible dependence on $(\mu_Q,T)$. To examine the astrophysical consequences of our modeling, we add charged leptons and neutrinos, and study the composition of matter for lepton fractions relevant for protoneutron stars and neutron star mergers. The abundance of neutrinos and thermal effects reduce the strangeness fraction and stiffen equations of state. For a neutrino trapped neutron star at $T\simeq 30 $ MeV with a lepton fraction $Y_L\simeq 0.05$, the mass is larger than its cold static counterpart by $\sim 0.1M_\odot$.

Accurate Determination of the Neutron Skin Thickness of ^{208}Pb through Parity-Violation in Electron Scattering.

We report a precision measurement of the parity-violating asymmetry A_{PV} in the elastic scattering of longitudinally polarized electrons from ^{208}Pb. We measure A_{PV}=550±16(stat)±8(syst) parts per billion, leading to an extraction of the neutral weak form factor F_{W}(Q^{2}=0.00616 GeV^{2})=0.368±0.013. Combined with our previous measurement, the extracted neutron skin thickness is R_{n}-R_{p}=0.283±0.071 fm. The result also yields the first significant direct measurement of the interior weak density of ^{208}Pb: ρ_{W}^{0}=-0.0796±0.0036(exp)±0.0013(theo) fm^{-3} leading to the interior baryon density ρ_{b}^{0}=0.1480±0.0036(exp)±0.0013(theo) fm^{-3}. The measurement accurately constrains the density dependence of the symmetry energy of nuclear matter near saturation density, with implications for the size and composition of neutron stars.

QCD color superconductivity in compact stars: Color-flavor locked quark star candidate for the gravitational-wave signal GW190814

At sufficiently high densities and low temperatures matter is expected to behave as a degenerate Fermi gas of quarks forming Cooper pairs, namely a color superconductor, as was originally suggested by Alford, Rajagopal and Wilczek [Nuclear Physics B 537, 443 (1999)]. The ground state is a superfluid, an electromagnetic insulator that breaks chiral symmetry, called the color-flavor locked phase. If such a phase occurs in the cores of compact stars, the maximum mass may exceed that of hadronic matter. The gravitational-wave signal GW190814 involves a compact object with mass $2.6{\rm M}_\odot$, within the so-called low mass gap. Since it is too heavy to be a neutron star and too light to be a black hole, its nature has not been identified with certainty yet. Here, we show not only that a color-flavor locked quark star with this mass is viable, but also we calculate the range of the model-parameters, namely the superconducting gap $\Delta$ and the bag constant $B$, that satisfies the strict LIGO constraints on the equation of state. We find that a color-flavor locked quark star with mass $2.6{\rm M}_\odot$ satisfies the observational constraints on the equation of state if $\Delta \geq 200{\rm MeV}$ and $B\geq 83{ \rm MeV}/{\rm fm^3}$ for a strange quark mass $m_s=95~{\rm MeV}/c^2$, and attains a radius $(12.7-13.6) {\rm km}$ and central density $(7.5-9.8) 10^{14}{\rm g}/{\rm cm}^3$.

Neutron electric dipole moment using lattice QCD simulations at the physical point

We extract the neutron electric dipole moment $\vert \vec{d}_N\vert$ within the lattice QCD formalism. We analyse one ensemble of $N_f=2+1+1$ twisted mass clover-improved fermions with lattice spacing of $a \simeq 0.08 \ {\rm fm}$ and physical values of the quark masses corresponding to a pion mass $m_{\pi} \simeq 139 \ {\rm MeV}$. The neutron electric dipole moment is extracted by computing the $CP$-odd electromagnetic form factor $F_3(Q^2 \to 0)$ through small $\theta$-expansion of the action. This approach requires the calculation of the topological charge for which we employ a fermionic definition by means of spectral projectors while we also provide a comparison with the gluonic definition accompanied by the gradient flow. We show that using the topological charge from spectral projectors leads to absolute errors that are more than two times smaller than those provided when the field theoretic definition is employed. We find a value of $\vert \vec{d}_N\vert = 0.0009(24) \ \theta \ e \cdot {\rm fm}$ when using the fermionic definition, which is statistically consistent with zero.

Charge radii of the nucleon from its flavor dependent Dirac form factors

We have determined the proton and the neutron charge radii from a global analysis of the proton and the neutron elastic form factors, after first performing a flavor decomposition of these form factors under charge symmetry in the light cone frame formulation. We then extracted the transverse mean-square radii of the flavor dependent quark distributions. In turn, these are related in a model-independent way to the proton and neutron charge radii but allow us to take into account motion effects of the recoiling nucleon for data at finite but high momentum transfer. In the proton case we find $\langle r_p \rangle = 0.852 \pm0.002_{\rm (stat.)} \pm0.009_{\rm (syst.)}~({\rm fm})$, consistent with the proton charge radius obtained from muonic hydrogen spectroscopy \cite{pohl:2010,antog2013}. The current method improves on the precision of the $\langle r_p \rangle$ extraction based on the form factor measurements. Furthermore, we find no discrepancy in the $\langle r_p \rangle$ determination among the different electron scattering measurements, all of which, utilizing the current method of extraction, result in a value that is consistent with the smallest $\langle r_p \rangle$ extraction from the electron scattering measurements \cite{Xiong:2019umf}. Concerning the neutron case, past results relied solely on the neutron-electron scattering length measurements, which suffer from an underestimation of underlying systematic uncertainties inherent to the extraction technique. Utilizing the present method we have performed the first extraction of the neutron charge radius based on nucleon form factor data, and we find $\langle r_n^2 \rangle = -0.122 \pm0.004_{\rm (stat.)} \pm0.010_{\rm (syst.)}~({\rm fm}^2)$.

Deconfinement critical point of lattice QCD with Nf=2 Wilson fermions

The ${\rm SU}(3)$ pure gauge theory exhibits a first-order thermal deconfinement transition due to spontaneous breaking of its global $Z_3$ center symmetry. When heavy dynamical quarks are added, this symmetry is broken explicitly and the transition weakens with decreasing quark mass until it disappears at a critical point. We compute the critical hopping parameter and the associated pion mass for lattice QCD with $N_f=2$ degenerate standard Wilson fermions on $N_\tau\in\{6,8,10\}$ lattices, corresponding to lattice spacings $a=0.12\, {\rm fm}$, $a=0.09\, {\rm fm}$, $a=0.07\, {\rm fm}$, respectively. Significant cut-off effects are observed, with the first-order region growing as the lattice gets finer. While current lattices are still too coarse for a continuum extrapolation, we estimate $m_\pi^c\approx 4 {\rm GeV}$ with a remaining systematic error of $\sim 20\%$. Our results allow to assess the accuracy of the LO and NLO hopping expanded fermion determinant used in the literature for various purposes. We also provide a detailed investigation of the statistics required for this type of calculation, which is useful for similar investigations of the chiral transition.

Confirming the existence of the strong CP problem in lattice QCD with the gradient flow

We calculate the electric dipole moment of the nucleon induced by the QCD theta term. We use the gradient flow to define the topological charge and use $N_f = 2+1$ flavors of dynamical quarks corresponding to pion masses of $700$, $570$, and $410$ MeV, and perform an extrapolation to the physical point based on chiral perturbation theory. We perform calculations at $3$ different lattice spacings in the range of $0.07~{\rm fm} < a < 0.11$ fm at a single value of the pion mass, to enable control on discretization effects. We also investigate finite size effects using $2$ different volumes. A novel technique is applied to improve the signal-to-noise ratio in the form factor calculations. The very mild discretization effects observed suggest a continuum-like behavior of the nucleon EDM towards the chiral limit. Under this assumption our results read $d_{n}=-0.00152(71)\ \bar\theta\ e~\text{fm}$ and $d_{p}=0.0011(10)\ \bar\theta\ e~\text{fm}$. Assuming the theta term is the only source of CP violation, the experimental bound on the neutron electric dipole moment limits $\left|\bar\theta\right| < 1.98\times 10^{-10}$ ($90\%$ CL). A first attempt at calculating the nucleon Schiff moment in the continuum resulted in $S_{p} = 0.50(59)\times 10^{-4}\ \bar\theta\ e~\text{fm}^3$ and $S_{n} = -0.10(43)\times 10^{-4}\ \bar\theta\ e~\text{fm}^3$.

Chiral pasta: Mixed phases at the chiral phase transition

Interiors of neutron stars are ultra-dense and may contain a core of deconfined quark matter. Such a core connects to the outer layers smoothly or through a sharp microscopic interface or through an intermediate macroscopic layer of inhomogeneous mixed phases, which is globally neutral but locally contains electrically charged domains. Here I employ a nucleon-meson model under neutron star conditions that shows a first-order chiral phase transition at large densities. In the vicinity of this chiral transition I calculate the free energies of various mixed phases - different 'pasta structures' - in the Wigner-Seitz approximation. Crucially, chirally broken nuclear matter and the approximately chirally symmetric phase (loosely interpreted as quark matter) are treated on the same footing. This allows me to compute the interface profiles of bubbles, rods, and slabs fully consistently, taking into account electromagnetic screening effects and without needing the surface tension as an input. I find that the full numerical results tend to disfavor mixed phases compared to a simple step-like approximation used frequently in the literature and that the predominantly favored pasta structure consists of slabs with a surface tension $\Sigma \simeq 6\, {\rm MeV}/{\rm fm}^2$.

Determination of the QCD coupling from the static energy and the free energy

We present two determinations of the strong coupling \(\alpha_s\). The first one is from the static energy at three-loop accuracy, and may be considered an update of earlier determinations by some of us. The new analysis includes new lattice data at smaller lattice spacings, and reaches distances as short as \(0.0237\,{\rm fm}\). We present a comprehensive and detailed estimate of the error sources that contribute to the uncertainty of the final result, $\alpha_s(M_Z)=0.11660^{+0.00110}_{-0.00056}$. The second determination is based on lattice data for the singlet free energy at finite temperature up to distances as small as \(0.0081\,{\rm fm}\), from which we obtain $\alpha_s(M_Z)=0.11638^{+0.0009 5}_{-0.00087}$.

Deuteron charge radius from the Lamb-shift measurement in muonic deuterium

The deuteron charge radius is calculated from the measurement of the Lamb shift in muonic deuterium, taking into account the electron vacuum polarization correction to the nuclear-structure effects. This correction is unexpectedly large and gives a mean-square charge-radii difference $ r_d^2-r_p^2 = 3.81747(346)\, {\rm fm}^2 $, which is now consistent with that obtained from the ordinary H-D isotope shift in the 1$S$-2$S$ transition. This suggests that the long-standing discrepancy in the proton charge radius obtained from electronic and muonic systems is due to an underestimated uncertainty in ordinary hydrogen spectroscopy.

Surface tension of dense matter at the chiral phase transition

If a first-order phase transition separates nuclear and quark matter at large baryon density, an interface between these two phases has a nonzero surface tension. We calculate this surface tension within a nucleon-meson model for domain walls and bubbles. Various methods and approximations are discussed and compared, including a numerical evaluation of the spatial profile of the interface. We also compute the surface tension at the other first-order phase transitions of the model: the nuclear liquid-gas transition and, in the parameter regime where it exists, the direct transition from the vacuum to the (approximately) chirally symmetric phase. Identifying the chirally symmetric phase with quark matter - our model does not contain explicit quark degrees of freedom - we find maximal surface tensions of the vacuum-quark transition $\Sigma_{\rm VQ}\sim 15 \, {\rm MeV}/{\rm fm}^2$, relevant for the surface of quark stars, and of the nuclear-quark transition $\Sigma_{\rm NQ}\sim 10 \, {\rm MeV}/{\rm fm}^2$, relevant for hybrid stars and for quark matter nucleation in supernovae and neutron star mergers.

Calculation of K→ππ decay amplitudes with an improved Wilson fermion action in a nonzero momentum frame in lattice QCD

We present our result for the $K\to\pi\pi$ decay amplitudes for both the $\Delta I=1/2$ and $3/2$ processes with the improved Wilson fermion action. In order to realize the physical kinematics, where the pions in the final state have finite momenta, we consider the decay process $K({\bf p}) \to \pi({\bf p}) + \pi({\bf 0})$ in the nonzero momentum frame with momentum ${\bf p}=(0,0,2\pi/L)$ on the lattice. Our calculations are carried out with $N_f=2+1$ gauge configurations generated with the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson fermion action at $a=0.091\,{\rm fm}$ ($1/a=2.176\,{\rm GeV}$), $m_\pi=260\,{\rm MeV}$, and $m_K=570\,{\rm MeV}$ on a $48^3\times 64$ ($La=4.4\,{\rm fm}$) lattice. For these parameters the energy of the $K$ meson is set at that of two-pion in the final state. We obtain ${\rm Re}A_2 = 2.431(19) \times10^{ -8}\,{\rm GeV}$, ${\rm Re}A_0 = 51(28) \times10^{ -8}\,{\rm GeV}$, and $\epsilon'/\epsilon = 1.9(5.7) \times10^{-3}$ for a matching scale $q^* =1/a$ where the errors are statistical. The dependence on the matching scale $q^*$ of these values is weak. The systematic error arising from the renormalization factors is expected to be around $1.3\%$ for ${\rm Re}A_2$ and $11 \%$ for ${\rm Re}A_0$. Prospects toward calculations with the physical quark mass are discussed.