Chiral logarithms in quenched QCD

We have simulated QCD using 2+1 flavors of domain wall quarks on a $(2.74 {\rm fm})^3$ volume with an inverse lattice scale of $a^{-1} = 1.729(28)$ GeV. The up and down (light) quarks are degenerate in our calculations and we have used four values for the ratio of light quark masses to the strange (heavy) quark mass in our simulations: 0.217, 0.350, 0.617 and 0.884. We have measured pseudoscalar meson masses and decay constants, the kaon bag parameter $B_K$ and vector meson couplings. We have used SU(2) chiral perturbation theory, which assumes only the up and down quark masses are small, and SU(3) chiral perturbation theory to extrapolate to the physical values for the light quark masses. While next-to-leading order formulae from both approaches fit our data for light quarks, we find the higher order corrections for SU(3) very large, making such fits unreliable. We also find that SU(3) does not fit our data when the quark masses are near the physical strange quark mass. Thus, we rely on SU(2) chiral perturbation theory for accurate results. We use the masses of the $\Omega$ baryon, and the $\pi$ and $K$ mesons to set the lattice scale and determine the quark masses. We then find $f_\pi = 124.1(3.6)_{\rm stat}(6.9)_{\rm syst} {\rm MeV}$, $f_K = 149.6(3.6)_{\rm stat}(6.3)_{\rm syst} {\rm MeV}$ and $f_K/f_\pi = 1.205(0.018)_{\rm stat}(0.062)_{\rm syst}$. Using non-perturbative renormalization to relate lattice regularized quark masses to RI-MOM masses, and perturbation theory to relate these to $\bar{\rm MS}$ we find $ m_{ud}^{\bar{\rm MS}}(2 {\rm GeV}) = 3.72(0.16)_{\rm stat}(0.33)_{\rm ren}(0.18)_{\rm syst} {\rm MeV}$ and $m_{s}^{\bar{\rm MS}}(2 {\rm GeV}) = 107.3(4.4)_{\rm stat}(9.7)_{\rm ren}(4.9)_{\rm syst} {\rm MeV}$.

We present results for the isovector nucleon form factors measured on a $96^4$ lattice at almost the physical pion mass with a lattice spacing of 0.085 fm in 2+1 flavor QCD. The configurations are generated with the stout-smeared $O(a)$-improved Wilson quark action and the Iwasaki gauge action at $\beta$=1.82. The pion mass at the simulation point is about 146 MeV. A large spatial volume of $(8.1~{\rm fm})^3$ allows us to investigate the form factors in the small momentum transfer region. We determine the isovector electric radius and magnetic moment from nucleon electric ($G_E$) and magnetic ($G_M$) form factors as well as the axial-vector coupling $g_A$. We also report on the results of the axial-vector ($F_A$), induced pseudoscalar ($F_P$) and pseudoscalar ($G_P$) form factors in order to verify the axial Ward-Takahashi identity in terms of the nucleon matrix elements, which may be called the generalized Goldberger-Treiman relation.

We report on numerical lattice QCD calculations of some of the low moments of the nucleon structure functions. The calculations are carried out with gauge configurations generated by the RBC and UKQCD collaborations with (2+1)-flavors of dynamical domain wall fermions and the Iwasaki gauge action ($\beta = 2.13$). The inverse lattice spacing is $a^{-1} = 1.73$ GeV, and two spatial volumes of ((2.7{\rm fm})^3) and ((1.8 {\rm fm})^3) are used. The up and down quark masses are varied so the pion mass lies between 0.33 and 0.67 GeV while the strange mass is about 12 % heavier than the physical one. The structure function moments we present include fully non-perturbatively renormalized iso-vector quark momentum fraction, (< x >_{u-d}), helicity fraction, (< x >_{\Delta u - \Delta d}), and transversity, (< 1 >_{\delta u - \delta d}), as well as an unrenormalized twist-3 coefficient, (d_1). The ratio of the momentum to helicity fractions, (< x >_{u-d}/< x >_{\Delta u - \Delta d}), does not show dependence on the light quark mass and agrees well with the value obtained from experiment. Their respective absolute values, fully renormalized, show interesting trends toward their respective experimental values at the lightest quark mass. A prediction for the transversity, (0.7 < < 1 >_{\delta u -\delta d} < 1.1), in the (\bar{\rm MS}) scheme at 2 GeV is obtained. The twist-3 coefficient, (d_1), though yet to be renormalized, supports the perturbative Wandzura-Wilczek relation.

We calculate pion mass, pion decay constant, PCAC quark mass and nucleon mass in two flavour lattice QCD with unimproved Wilson fermion and gauge actions. Simulations are performed using DD-HMC algorithm at two lattice spacings and two volumes for several values of the quark mass. The cutoff effects in pion mass and nucleon mass for the explored region of parameter space are found to be negligible. We extract the average value of the up–down quark mass in the MS¯ scheme at 2 GeV, which is in good agreement with the world data. The chiral behaviours of pion mass, pion decay constant and quark condensate are found to be qualitatively consistent with NLO chiral perturbation theory. The extracted values of the pion decay constant and the chiral condensate are in reasonable agreement with the world data.

We calculate the nucleon electric dipole moment (EDM) from the $\ensuremath{\theta}$ term with overlap fermions on three domain wall lattices with different sea pion masses at lattice spacing 0.11 fm. Due to the chiral symmetry conserved by the overlap fermions, we have well-defined topological charge and chiral limit for the EDM. Thus, the chiral extrapolation can be carried out reliably at nonzero lattice spacings. We use three to four different partially quenched valence pion masses for each sea pion mass and find that the EDM dependence on the valence and sea pion masses behaves oppositely, which can be described by partially quenched chiral perturbation theory. With the help of the cluster decomposition error reduction technique, we determine the neutron and proton EDM at the physical pion mass to be ${d}_{n}=\ensuremath{-}0.00148(14)(31)\overline{\ensuremath{\theta}}\text{ }\text{ }\mathrm{e}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{fm}$ and ${d}_{p}=0.0038(11)(8)\overline{\ensuremath{\theta}}\text{ }\text{ }\mathrm{e}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{fm}$. This work is a clear demonstration of the advantages of using chiral fermions in the nucleon EDM calculation and paves the road to future precise studies of the strong $CP$ violation effects.

We consider an inelastic dark matter model, where a fermion is charged under a broken U(1) gauge symmetry, and introduce a tiny Majorana mass term to split the fermion into two states with the light one being a dark matter candidate. If the gauge boson is light, it can mediate both elastic and inelastic dark matter self-interactions in dark halos, leading to observational consequences. Using a numerical technique based on partial wave analysis, we accurately calculate the elastic and inelastic self-scattering cross sections. We assume a thermal freeze-out scenario and fix the gauge coupling constant using the relic density constraint. Then, we focus on six benchmark masses of dark matter, covering a wide range from $10~{\rm MeV}$ to $160~{\rm GeV}$ and map parameter regions where the elastic scattering cross section per unit mass is within $1~{\rm cm^2/g} - 5~{\rm cm^2/g}$, favored to solve small-scale issues of cold dark matter. If the heavy state can decay to the light state and a massless species, the inelastic up-scattering process can cool the halo and lead to core collapse. Taking galaxies with evidence of dark matter density cores, we further derive constraints on the parameter space. For dark matter masses below $10~{\rm GeV}$, the mass splitting must be large enough to forbid up scattering in the dwarf halo for evading the core-collapse constraint; while for higher masses, the up-scattering process can still be allowed. Our results show astrophysical observations can provide powerful tests for dark matter models with large elastic and inelastic self-interactions.

Partially quenched chiral perturbation theory and numerical simulations

We present preliminary results for nucleon form factors including the axial charge and the isovector electric radius in 2+1 flavor QCD at almost the physical pion mass on a $96^4$ lattice with the lattice spacing of 0.085 fm. The configurations are generated with the stout-smeared $O(a)$-improved Wilson quark action and the Iwasaki gauge action at $\beta$=1.82. A large spatial volume of $(8.1 {\rm fm})^3$ allows us to investigate the form factors at small momentum transfer region. We discuss analyses of the momentum dependence of the form factors with the z-expansion.

The pion and proton mass in finite volume

Light quark masses, chiral condensate and quark–gluon condensate in quenched lattice QCD with exact chiral symmetry

We compare the behavior of overlap fermions, which are chirally invariant, and of Wilson twisted mass fermions at full twist in the approach to the chiral limit. Our quenched simulations reveal that with both formulations of lattice fermions pion masses of O(250 MeV) can be reached in practical applications. Our comparison is done at a fixed value of the lattice spacing a=0.123 fm. A number of quantities are measured such as hadron masses, pseudoscalar decay constants and quark masses obtained from Ward identities. We also determine the axial vector renormalization constants in the case of overlap fermions.

All ALEPH measurements of branching ratios of $\tau$ decays involving kaons are summarized including a combination of results obtained with $K^0_S$ and $K^0_L$ detection. The decay dynamics are studied, leading to the determination of contributions from vector $K^*(892)$ and $K^*(1410)$ , and axial-vector $K_1(1270)$ and $K_1(1400)$ resonances. Agreement with isospin symmetry is observed among the different final states. Under the hypothesis of the conserved vector current, the spectral function for the $K\overline{K}\pi$ mode is compared with the corresponding cross section for low energy $e^+e^-$ annihilation, yielding an axial-vector fraction of $(94^{\,+6}_{\,-8})\%$ for this mode. The branching ratio for $\tau$ decay into all strange final states is determined to be $B(\tau^-\to X^-(S=-1)\nu_\tau)=(28.7\pm1.2)\times 10^{-3}$ . The measured mass spectra of the strange $\tau$ decay modes are exploited to derive the $S=-1$ spectral function. A combination of strange and nonstrange spectral functions is used to determine the strange quark mass and nonperturbative contributions to the strange hadronic width. A method is developed to avoid the bad convergence of the spin zero hadronic component, with the result $m_s(M_\tau^2)=(176^{\,+46}_{\,-57})$ MeV/ $c^2$ . The evolution down to 1 GeV gives $m_s(1 {\rm GeV}^2) = (234^{\,+61}_{\,-76}){\rm MeV}/c^2$ .

We present a calculation of the electromagnetic form factor of the pion in N f = 2 + 1 flavor lattice QCD. Calculations are made on the PACS-CS gauge field configurations generated using Iwasaki gauge action and Wilson-clover quark action on a 323 × 64 lattice volume with the lattice spacing estimated as a = 0.0907(13) fm at the physical point. Measurements of the form factor are made using the technique of partially twisted boundary condition to reach small momentum transfer as well as periodic boundary condition with integer momenta. Additional improvements including random wall source techniques and a judicious choice of momenta carried by the incoming and outgoing quarks are employed for error reduction. Analyzing the form factor data for the pion mass at M π ≈ 411 MeV and 296 MeV, we find that the NNLO SU(2) chiral perturbation theory fit yields 〈r 2〉 = 0.441 ± 0.046 fm2 for the pion charge radius at the physical pion mass. Albeit the error is quite large, this is consistent with the experimental value of 0.452 ± 0.011 fm2. Below M π ≈ 300 MeV, we find that statistical fluctuations in the pion two-and three-point functions become too large to extract statistically meaningful averages on a 323 spatial volume. We carry out a sample calculation on a 644 lattice with the quark masses close to the physical point, which suggests that form factor calculations at the physical point become feasible by enlarging lattice sizes to M π L ≈ 4.

We present a lattice calculation of the neutron and proton electric dipole moments (EDM's) with $N_f=2+1$ flavors of domain-wall fermions. The neutron and proton EDM form factors are extracted from three-point functions at the next-to-leading order in the $\theta$ vacuum of QCD. In this computation, we use pion masses 0.33 and 0.42 GeV and 2.7 fm$^3$ lattices with Iwasaki gauge action and a 0.17 GeV pion and 4.6 fm$^3$ lattice with I-DSDR gauge action, all generated by the RBC and UKQCD collaborations. The all-mode-averaging technique enables an efficient and high statistics calculation. Chiral behavior of lattice EDM's is discussed in the context of baryon chiral perturbation theory. In addition, we also show numerical evidence on relationship of three- and two-point correlation function with local topological distribution.

We report our numerical lattice QCD calculations of the isovector nucleon form factors for the vector and axialvector currents: the vector, induced tensor, axialvector, and induced pseudoscalar form factors. The calculation is carried out with the gauge configurations generated with N{sub f} = 2+1 dynamical domain wall fermions and Iwasaki gauge actions at {beta} = 2.13, corresponding to a cutoff a{sup -1} = 1.73 GeV, and a spatial volume of (2.7 fm){sup 3}. The up and down quark masses are varied so the pion mass lies between 0.33 and 0.67 GeV while the strange quark mass is about 12% heavier than the physical one. We calculate the form factors in the range of momentum transfers, 0.2 &amp;lt; q{sup 2} &amp;lt; 0.75 GeV{sup 2}. The vector and induced tensor form factors are well described by the conventional dipole forms and result in significant underestimation of the Dirac and Pauli mean-squared radii and the anomalous magnetic moment compared to the respective experimental values. We show that the axial-vector form factor is significantly affected by the finite spatial volume of the lattice. In particular in the axial charge, g{sub A}/g{sub V}, the finite volume effect scales with a single dimensionless quantity, m{sub {pi}}L, the product of the calculated pion mass and the spatial lattice extent. Our results indicate that for this quantity, m{sub {pi}} L &amp;gt; 6 is required to ensure that finite volume effects are below 1%.