On pion mass and decay constant from theory

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

We solve the coupled system of Dyson-Schwinger and Bethe-Salpeter equations for the quark propagator and the pion Bethe-Salpeter amplitude on a finite volume. To this end we use a truncation scheme that includes pion cloud effects in the quark propagator and light mesons. We study volume effects in the quark condensate, the pion mass and the pion decay constant and compare to corresponding results in other approaches. In general we find large effects for volumes below V=(1.8 fm)^4.

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.

The rainbow truncation of the quark Dyson-Schwinger equation is combined with the ladder Bethe-Salpeter equation for the meson bound state amplitudes and the dressed quark-W vertex in a manifestly covariant calculation of the K_{l3} transition form factors and decay width in impulse approximation. With model gluon parameters previously fixed by the chiral condensate, the pion mass and decay constant, and the kaon mass, our results for the K_{l3} form factors and the kaon semileptonic decay width are in good agreement with the experimental data.

We present a lattice calculation of the electromagnetic form factor of the pion obtained using the tree-level Symanzik improved gauge action with two flavors of dynamical twisted Wilson quarks. The simulated pion masses range approximately from 260 to 580 MeV, and the lattice box sizes are chosen in order to guarantee that ${M}_{\ensuremath{\pi}}L\ensuremath{\gtrsim}4$. Accurate results for the form factor are obtained using all-to-all quark propagators evaluated by a stochastic procedure. The momentum dependence of the pion form factor is investigated up to values of the squared four-momentum transfer ${Q}^{2}\ensuremath{\simeq}0.8\text{ }\text{ }{\mathrm{GeV}}^{2}$ and, thanks to the use of twisted boundary conditions, down to ${Q}^{2}\ensuremath{\simeq}0.05\text{ }\text{ }{\mathrm{GeV}}^{2}$. Volume and discretization effects on the form factor appear to be within the statistical errors. Our results for the pion mass, decay constant and form factor are analyzed using (continuum) chiral perturbation theory at next-to-next-to-leading order. The extrapolated value of the pion charge radius is $⟨{r}^{2}{⟩}^{\mathrm{phys}}=0.456\ifmmode\pm\else\textpm\fi{}{0.030}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{0.024}_{\mathrm{syst}}$ in nice agreement with the experimental result. The extrapolated values of the pion form factor agree very well with the experimental data up to ${Q}^{2}\ensuremath{\simeq}0.8\text{ }\text{ }{\mathrm{GeV}}^{2}$ within uncertainties which become competitive with the experimental errors for ${Q}^{2}\ensuremath{\gtrsim}0.3\text{ }\text{ }{\mathrm{GeV}}^{2}$. The relevant low-energy constants appearing in the chiral expansion of the pion form factor are extracted from our lattice data, which come essentially from a single lattice spacing, adding the experimental value of the pion scalar radius in the fitting procedure. Our findings are in nice agreement with the available results of chiral perturbation theory analyses of $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering data as well as with other analyses of our collaboration.

The charge form factor and weak decay constant of the pion as well as the pion-quark coupling constant in symmetric nuclear matter are explored in the framework of the Nambu--Jona-Lasinio model, where the pion is described as a bound state of dressed quark-antiquark pair obtained by the Bethe-Salpeter equation. For the in-medium current quark properties, we adopt the quark-meson coupling model, which describes successfully many hadron properties in a nuclear medium. The pion decay constant and the pion-quark coupling constant are found to decrease with increasing density as well as the magnitude of the light quark condensate. But the pion mass is found to be insensitive to density up to $1.25$ times the normal nuclear density. The pion charge form factor in the space-like region is also explored and is found to have a similar $Q^2$ dependence as the form factor in vacuum showing $1/Q^2$-behavior in large $Q^2$ region, where $Q^2$ is the negative of the four-momentum transfer squared. The modifications of the charge radius of the charged pion in nuclear matter are then estimated and the root-mean-square radius at the normal nuclear density is predicted to be larger than that in vacuum by about 20\%.

We present lattice QCD results for masses and magnetic polarizabilities of light and strange pseudoscalar mesons, chiral condensates, decay constants of neutral pion, and neutral kaon in the presence of background magnetic fields with $eB$ ranging up to around $3.35\text{ }\text{ }{\mathrm{GeV}}^{2}$ ($\ensuremath{\sim}70{M}_{\ensuremath{\pi}}^{2}$) in the vacuum. The computations were carried out in ($2+1$)-flavor QCD mostly on ${32}^{3}\ifmmode\times\else\texttimes\fi{}96$ lattices using the highly improved staggered quark action with ${M}_{\ensuremath{\pi}}\ensuremath{\approx}220\text{ }\text{ }\mathrm{MeV}$ at zero temperature. We find that the masses of neutral pseudoscalar mesons monotonously decrease as the magnetic field strength grows and then saturate at a nonzero value, while there exists a nonmonotonous behavior of charged pion and kaon masses in the magnetic field. We observe a $qB$ scaling of the up and down quark flavor components of neutral pion mass, neutral pion decay constant as well as the quark chiral condensates at $0.05\ensuremath{\lesssim}eB\ensuremath{\lesssim}3.35\text{ }\text{ }{\mathrm{GeV}}^{2}$. We show that the correction to the Gell-Mann-Oakes-Renner relation involving the neutral pion is less than 6% and the correction for the relation involving neutral kaon is less than 30% at $eB\ensuremath{\lesssim}3.35\text{ }\text{ }{\mathrm{GeV}}^{2}$. We also derive the Ward-Takahashi identities for QCD in the magnetic field in the continuum formulation including the relation between integrated neutral pseudoscalar meson correlators and chiral condensates.

The quenched chiral logs are examined on a $16^3 \times 28$ lattice with Iwasaki gauge action and overlap fermions. The pion decay constant $f_{\pi}$ is used to set the lattice spacing, $a = 0.200(3) {\rm fm}$. With pion mass as low as $\sim 180 {\rm MeV}$, we see the quenched chiral logs clearly in $m_{\pi}^2/m$ and $f_P$, the pseudoscalar decay constant. We analyze the data to determine how low the pion mass needs to be in order for the quenched one-loop chiral perturbation theory ($\chi$PT) to apply. With the constrained curve-fitting method, we are able to extract the quenched chiral log parameter $\delta$ together with other low-energy parameters. Only for $m_{\pi} \leq 300 {\rm MeV}$ do we obtain a consistent and stable fit with a constant $\delta$ which we determine to be 0.24(3)(4) (at the chiral scale $\Lambda_{\chi}=0.8 {\rm GeV}$). By comparing to the $12^3 \times 28$ lattice, we estimate the finite volume effect to be about 2.7% for the smallest pion mass. We also fitted the pion mass to the form for the re-summed cactus diagrams and found that its applicable region is extended farther than the range for the one-loop formula, perhaps up to $m_{\pi} \sim 500-600$ MeV. The scale independent $\delta$ is determined to be 0.20(3) in this case. We study the quenched non-analytic terms in the nucleon mass and find that the coefficient $C_{1/2}$ in the nucleon mass is consistent with the prediction of one-loop $\chi$PT\@. We also obtain the low energy constant $L_5$ from $f_{\pi}$. We conclude from this study that it is imperative to cover only the range of data with the pion mass less than $\sim 300 {\rm MeV}$ in order to examine the chiral behavior of the hadron masses and decay constants in quenched QCD and match them with quenched one-loop $\chi$PT\@.

We suggest the gauge groups SU(3), SU(2) and U(1) to share a common origin in U(3). We take the Lie group U(3) to serve as an intrinsic configuration space for baryons. A spontaneous symmetry break in the baryonic state selects a U(2) subgroup for the Higgs mechanism. The Higgs field enters the symmetry break to relate the strong and electroweak energy scales by exchange of one quantum of action between the two sectors. This shapes the Higgs potential to fourth order. Recently intrinsic quantum mechanics has given a suggestion for the Cabibbo angle from theory (EPL124-2018) and a prediction for the Higgs couplings to gauge bosons (EPL125-2019). Previously it has given the nucleon mass and the parton distribution functions for u and d quarks in the proton (EPL102-2013). It has given a quite accurate equation for the Higgs mass in closed form (IJMPA30-2015) and an N and Delta spectrum essentially without missing resonances (arXiv:1109.4732). The intrinsic space is to be distinguished from an interior space. The intrinsic space is non-spatial, i.e. no gravity in intrinsic space. The configuration variable is like a generalized spin variable excited from laboratory space by kinematic generators: momentum, spin and Laplace-Runge-Lenz operators. The baryon dynamics resides in a Hamiltonian on U(3) and projects to laboratory space by the momentum form of the wavefunction. The momentum form generates conjugate quark and gluon fields. Local gauge invariance in laboratory space follows from unitarity of the configuration variable and left invariance of the coordinate fields on the intrinsic space. Future work should aim to invoke leptons in the second and third generations and quarks in the third.

We study the difficulties associated with detecting chiral singularities predicted by chiral perturbation theory (ChPT) in lattice QCD. We focus on the physics of the remnant O(2) chiral symmetry of staggered fermions in the strong coupling limit using the recently discovered directed path algorithm. Since it is easier to look for power-like singularities as compared to logarithmic ones, our calculations are performed at a fixed finite temperature in the chirally broken phase. We show that the behavior of the chiral condensate, the pion mass and the pion decay constant, for small masses, are all consistent with the predictions of ChPT. However, the values of the quark masses that we need to demonstrate this are much smaller than those being used in dynamical QCD simulations. We also need to use higher order terms in the chiral expansion to fit our data.

We present a systematic study of the effectiveness of light quark mass reweighting. This method allows a single lattice QCD ensemble, generated with a specific value of the dynamical light quark mass, to be used to determine results for other, nearby light dynamical quark masses. We study two gauge field ensembles generated with 2+1 flavors of dynamical domain wall fermions with light quark masses m_l=0.02 (m_\pi=620 MeV) and m_l=0.01 (m_\pi=420 MeV). We reweight each ensemble to determine results which could be computed directly from the other and check the consistency of the reweighted results with the direct results. The large difference between the 0.02 and 0.01 light quark masses suggests that this is an aggressive application of reweighting as can be seen from fluctuations in the magnitude of the reweighting factor by four orders of magnitude. Never-the-less, a comparison of the reweighed topological charge, average plaquette, residual mass, pion mass, pion decay constant, and scalar correlator between these two ensembles shows agreement well described by the statistical errors. The issues of the effective number of configurations and finite sample size bias are discussed. An examination of the topological charge distribution implies that it is more favorable to reweight from heavier mass to lighter quark mass.

The influence of temperature on the pion decay constant and the two-photon decay of π0 is considered near the chiral symmetry restoration point. We analyze how the mass and decay width relate to each other and find that the π0 decay width in hot matter is suppressed, provided that the pion mass vanishes at the critical point; it diverges if pion mass remains finite while quark condensate is dissolved at T c .

We determine the topological susceptibility of the gauge configurations generated by lattice simulations using 2 flavors of optimal domain-wall fermion on the 16 3 32 lattice with length 16 in the fifth dimension, at the inverse lattice spacing a -1 1.9 GeV. Using the adaptive thick-restart Lanczos algorithm, we project the low-lying eigenmodes of the overlap Dirac operator, and obtain the topological charge Q t of each gauge configuration in eight ensembles with pion masses in the range 230 -580 MeV. From our result of Q t , we compute the topological susceptibility and the second normalized cumulant. Our result of the topological susceptibility agrees with the sea-quark mass dependence predicted by the next-to-leading order chiral perturbation theory, and provides a determination of both the chiral condensate and the pion decay constant.

We investigate the behaviour of the pion decay constant and the pion mass in two-flavour lattice QCD, with the physical and chiral points as ultimate goal. Measurements come from the ensembles generated by the CLS initiative using the O(a)-improved Wilson formulation, with lattice spacing down to about 0.05 fermi and pion masses as low as 190 MeV. The applicability of SU(2) chiral perturbation theory is investigated, and various functional forms, and their range of validity, are compared. The physical scale is set through the kaon decay constant, whose measurement is enabled by inserting a third, heavier valence strange quark.

Update on pion weak decay constants in nuclear matter