π0 DECAY AT CRITICAL TEMPERATURE

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

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 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 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 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.

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 calculate the pion mass from Goldstone modes in the Higgs mechanism related to the neutron decay. The Goldstone pion modes acquire mass by a vacuum misalignment of the Higgs field. The size of the misalignment is controlled by the ratio between the electronic and the nucleonic energy scales. The nucleonic energy scale is involved in the neutron to proton transformation and the electronic scale is involved in the related creation of the electronic state in the course of the electroweak neutron decay. The respective scales influence the mapping of the intrinsic configuration spaces used in our description. The configuration spaces are the Lie groups U(3) for the nucleonic sector and U(2) for the electronic sector. These spaces are both compact and lead to periodic potentials in the Hamiltonians in coordinate space. The periodicity and strengths of these potentials control the vacuum misalignment and lead to a pion mass of 135.2(1.5) MeV with an uncertainty mainly from the fine structure coupling at pionic energies. The pion decay constant 92 MeV results from comparing the fourth-order self-coupling in an effective pion field theory with the corresponding fourth-order term in the Higgs potential. We suggest analogies with the Goldberger-Treiman relation.

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 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.

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 study the influence of the isospin asymmetry on the phase structure of\nstrongly interacting quark matter near the critical point (CP) using a\nGinzburg-Landau approach. The effect is found to be drastic, not only bringing\nabout the shift of the location of the CP, but resulting in a rich phase\nstructure in the vicinity of the CP. In particular, new tricritical and triple\npoints emerge as soon as the isospin density becomes finite. Moreover, we find\nthe CP being washed out from the phase diagram due to the stabilization of a\nhomogeneous charged pion condensate when the isospin chemical potential exceeds\na critical value. We derive a model-independent universal relation between the\ncritical isospin chemical potential and the chiral condensate at the CP. We\nalso study the effect of the $\\mathrm{U}(1)_\\mathrm{A}$ anomaly on the phase\ntransition to the pion condensate in the vicinity of chiral crossover.\n

We study chiral symmetry breaking in the holographic D3-D7 system in a simple model with an arbitrary running coupling. We derive equations for the D7 embedding and show there is a light pion. In particular we present simple integral equations, involving just the running coupling and the quark self energy, for the quark condensate and the pion decay constant. We compare these to the Pagels-Stokar or constituent quark model equivalent. We discuss the implications for walking Technicolour theories. We also perform a similar analysis in the four dimensional field theory whose dual is the non-supersymmetric D3-D5 system and propose that it represents a walking theory in which the quark condensate has dimension 2 + sqrt{3}.

We study chiral symmetry breaking in QCD, using as ingredients in the quark gap equation recent lattice results for the gluon and ghost propagators. The Ansatz employed for the quark‐gluon vertex is purely non‐Abelian, introducing a crucial dependence on the ghost dressing function and the quark‐ghost scattering amplitude. The numerical impact of these quantities is considerable: the need to invoke confinement explicitly is avoided, and the dynamical quark masses generated are of the order of 300 MeV. In addition, the pion decay constant and the quark condensate are computed, and are found to be in good agreement with phenomenology.

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.

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.