Axial-vector Ward-Takahashi Identity Research Articles

Mesons and baryons: Parity partners

We calculate masses of light and heavy mesons as well as baryons of negative parity containing $u,d,s,c$ and $b$ quarks. It is an extension of our previous work where we had studied the positive parity baryons. We adopt a quark-diquark picture of baryons where the diquarks are non-pointlike with a finite spatial extension. The mathematical foundation for this analysis is implemented through a symmetry-preserving Schwinger-Dyson equations treatment of a vector-vector contact interaction, which preserves key features of quantum chromodynamics, such as confinement, chiral symmetry breaking, axial vector Ward-Takahashi identity and low-energy Goldberger-Treiman relations. This treatment simultaneously describes mesons and provides attractive correlations for diquarks in the $\overline{3}$ representation. Employing this model, we compute the spectrum and masses of all spin-1/2 and spin-3/2 baryons of negative parity, supplementing our earlier evaluation of positive parity baryons, containing 1, 2 or 3 heavy quarks. In the process, we calculate masses of a multitude of mesons and corresponding diquarks. Wherever possible, we make comparisons of our results with known experimental observations as well as theoretical predictions of several models and approaches including lattice quantum chromodynamics, finding satisfactory agreement. We also make predictions for heavier states not yet observed in the experiment.

Heavy-light mesons beyond the ladder approximation

The heavy-light mesons are studied within the framework of Dyson-Schwinger equations of QCD. Inspired by the axial-vector Ward-Takahashi identity resulting from the chiral symmetry, we propose a truncation scheme beyond the ladder approximation without introducing any additional parameters. For the pseudoscalar and vector heavy-light mesons, the obtained mass spectrum has the level of relative errors at 5% compared with experimental data and lattice-QCD results. For the leptonic decay constants, our results are comparable with those from experiments and/or lattice QCD. For some channels, the discrepancies are sizable but significantly smaller than those using the equal spacing rule. The truncation scheme proposed in this work is simple and could be improved and applied to study other open flavor hadrons including both mesons and baryons.

From Bethe–Salpeter Wave functions to Generalised Parton Distributions

We review recent works on the modelling of Generalised Parton Distributions within the Dyson-Schwinger formalism. We highlight how covariant computations, using the impulse approximation, allows one to fulfil most of the theoretical constraints of the GPDs. Specific attention is brought to chiral properties and especially the so-called soft pion theorem, and its link with the Axial-Vector Ward-Takahashi identity. The limitation of the impulse approximation are also explained. Beyond impulse approximation computations are reviewed in the forward case. Finally, we stress the advantages of the overlap of lightcone wave functions, and possible ways to construct covariant GPD models within this framework, in a two-body approximation.

Quark model with chiral-symmetry breaking and confinement in the Covariant Spectator Theory

We propose a model for the quark-antiquark interaction in Minkowski space using the Covariant Spectator Theory. We show that with an equal-weighted scalar-pseudoscalar structure for the confining part of our interaction kernel the axial-vector Ward-Takahashi identity is preserved and our model complies with the Adler-zero constraint for pi-pi-scattering imposed by chiral symmetry.

Chiral symmetry andπ-πscattering in the covariant spectator theory

The $\ensuremath{\pi}$-$\ensuremath{\pi}$ scattering amplitude calculated with a model for the quark-antiquark interaction in the framework of the covariant spectator theory (CST) is shown to satisfy the Adler zero constraint imposed by chiral symmetry. The CST formalism is established in Minkowski space and our calculations are performed in momentum space. We prove that the axial-vector Ward-Takahashi identity is satisfied by our model. Then we show that, similar to what happens within the Bethe-Salpeter formalism, application of the axial-vector Ward-Takahashi identity to the CST $\ensuremath{\pi}$-$\ensuremath{\pi}$ scattering amplitude allows us to sum the intermediate quark-quark interactions to all orders. The Adler self-consistency zero for $\ensuremath{\pi}$-$\ensuremath{\pi}$ scattering in the chiral limit emerges as the result for this sum.

Bethe–Salpeter equations: mesons beyond the rainbow-ladder truncation

We investigate masses of light mesons from a coupled system of Dyson–Schwinger (DSE) and Bethe–Salpeter equations (BSE), taking into account dominant non-Abelian, sub-leading Abelian, and dominant pion cloud contributions to the dressed quark-gluon vertex. The axial-vector Ward–Takahashi identity preserving Bethe–Salpeter kernel is constructed and the spectrum of light mesons calculated. Our model goes significantly beyond the rainbow-ladder. We find that sub-leading Abelian corrections are further dynamically suppressed, and that our results supersede early qualitative predictions from simple truncation schemes.

Bethe–Salpeter studies of mesons beyond rainbow-ladder

We investigate the masses of light mesons from a coupled system of Dyson-Schwinger and Bethe-Salpeter equations. The dominant non-Abelian and sub-leading Abelian contributions to the dressed quark-gluon vertex are explicitly taken into account. We also include unquenching effects in the form of hadronic resonance contributions via the back-reaction of pions. We construct the corresponding Bethe--Salpeter kernel that satisfies the axial-vector Ward-Takahashi identity. Our numerical treatment fully includes all momentum dependencies with all equations solved completely in the complex plane. This approach goes well beyond the rainbow-ladder approximation and permits us to investigate the relative impact of different corrections beyond rainbow-ladder on the properties of mesons. We find that sub-leading Abelian corrections are further suppressed dynamically, and that our results supersede early qualitative predictions with significantly simpler truncation schemes.

Probing the Gluon Self-Interaction in Light Mesons

We investigate masses and decay constants of light mesons from a coupled system of Dyson-Schwinger and Bethe-Salpeter equations. We explicitly take into account dominant non-Abelian contributions to the dressed quark-gluon vertex stemming from the gluon self-interaction. We construct the corresponding Bethe-Salpeter kernel that satisfies the axial-vector Ward-Takahashi identity. Our numerical treatment fully includes all momentum dependencies with all equations solved completely in the complex plane. This approach goes well beyond the rainbow-ladder approximation and permits us to investigate the influence of the gluon self-interaction on the properties of mesons. As a first result we find indications of a nonperturbative cancellation of the gluon self-interaction contributions and pion cloud effects in the mass of the rho meson.

2+1flavor lattice QCD toward the physical point

We present the first results of the PACS-CS project which aims to simulate 2+1 flavor lattice QCD on the physical point with the nonperturbatively $O(a)$-improved Wilson quark action and the Iwasaki gauge action. Numerical simulations are carried out at the lattice spacing of $a=0.0907(13)$fm on a $32^3\times 64$ lattice with the use of the DDHMC algorithm to reduce the up-down quark mass. Further algorithmic improvements make possible the simulation whose ud quark mass is as light as the physical value. The resulting PS meson masses range from 702MeV down to 156MeV, which clearly exhibit the presence of chiral logarithms. An analysis of the PS meson sector with SU(3) ChPT reveals that the NLO corrections are large at the physical strange quark mass. In order to estimate the physical ud quark mass, we employ the SU(2) chiral analysis expanding the strange quark contributions analytically around the physical strange quark mass. The SU(2) LECs ${\bar l}_3$ and ${\bar l}_4$ are comparable with the recent estimates by other lattice QCD calculations. We determine the physical point together with the lattice spacing employing $m_\pi$, $m_K$ and $m_\Omega$ as input. The hadron spectrum extrapolated to the physical point shows an agreement with the experimental values at a few % level of statistical errors, albeit there remain possible cutoff effects. We also find that our results of $f_\pi=134.0(4.2)$MeV, $f_K=159.4(3.1)$MeV and $f_K/f_\pi=1.189(20)$ with the perturbative renormalization factors are compatible with the experimental values. For the physical quark masses we obtain $m_{\rm ud}^\msbar=2.527(47)$MeV and $m_{\rm s}^\msbar=72.72(78)$MeV extracted from the axial-vector Ward-Takahashi identity with the perturbative renormalization factors.

Flavor symmetry breaking and meson masses

The axial-vector Ward-Takahashi identity is used to derive mass formulas for neutral pseudoscalar mesons. Flavor symmetry breaking entails nonideal flavor content for these states. Adding that the ${\ensuremath{\eta}}^{'}$ is not a Goldstone mode, exact chiral-limit relations are developed from the identity. They connect the dressed-quark propagator to the topological susceptibility. It is confirmed that in the chiral limit the ${\ensuremath{\eta}}^{'}$ mass is proportional to the matrix element which connects this state to the vacuum via the topological susceptibility. The implications of the mass formulas are illustrated using an elementary dynamical model, which includes an Ansatz for that part of the Bethe-Salpeter kernel related to the non-Abelian anomaly. In addition to the current-quark masses, the model involves two parameters, one of which is a mass-scale. It is employed in an analysis of pseudoscalar- and vector-meson bound-states. While the effects of $\mathit{SU}({N}_{f}=2)$ and $\mathit{SU}({N}_{f}=3)$ flavor symmetry breaking are emphasized, the five-flavor spectra are described. Despite its simplicity, the model is elucidative and phenomenologically efficacious; e.g., it predicts $\ensuremath{\eta}\text{\ensuremath{-}}{\ensuremath{\eta}}^{'}$ mixing angles of $~\ensuremath{-}{15}^{\ifmmode^\circ\else\textdegree\fi{}}$ and ${\ensuremath{\pi}}^{0}\text{\ensuremath{-}}\ensuremath{\eta}$ angles of $~{1}^{\ifmmode^\circ\else\textdegree\fi{}}.$

Transition Temperature of Two-Degenerate-Flavour QCD in theChiral Limit

We present the results for the transition temperature of quantumchromodynamics (QCD) with two degenerate flavours (Nf = 2) ofWilson quarks. On lattice 83×4 with 4 representing thetemporal extent, by using the Ferrenberg–Swendsen reweightingmethod, we determine the critical β = 6/g2 where thetransition occurs, g is the coupling constant. On lattice82×20×4, by using the axial vector Ward–Takahashiidentity, we calculate the current quark mass amq, a is thelattice spacing. Assuming the O(4) scaling, the critical βin the chiral limit is determined. We calculate the ρ mesonmass amρ at zero temperature on lattice 83×20. Byusing the experimental ρ meson mass to set the scale, weobtain 194(1) MeV for the transition temperature in the chiral limit.

Probing unquenching effects in the gluon polarization in light mesons

We introduce an extension to the ladder truncated Bethe-Salpeter equation for mesons and the rainbow truncated quark Schwinger-Dyson equations which includes quark-loop corrections to the gluon propagator. This truncation scheme obeys the axialvector Ward-Takahashi identity relating the quark self-energy and the Bethe-Salpeter kernel. Two different approximations to the Yang-Mills sector are used as input: the first is a sophisticated truncation of the full Yang-Mills Schwinger-Dyson equations, the second is a phenomenologically motivated form. We find that the spectra and decay constants of pseudoscalar and vector mesons are overall described well for either approach. Meson mass results for charge eigenstate vector and pseudoscalar meson masses are compared to lattice data. The effects of unquenching the system are small but not negligible.

Electromagnetic properties of ground-state and excited-state pseudoscalar mesons

The axial-vector Ward-Takahashi identity places constraints on particular properties of every pseudoscalar meson. For example, in the chiral limit all pseudoscalar mesons, except the Goldstone mode, decouple from the axial-vector current. Nevertheless, all neutral pseudoscalar mesons couple to two photons. The strength of the ${\ensuremath{\pi}}_{n}^{0}\ensuremath{\gamma}\ensuremath{\gamma}$ coupling, where $n=0$ denotes the Goldstone mode, is affected by the Abelian anomaly's continuum contribution. The effect is material for $n\ensuremath{\ne}0$. The ${\ensuremath{\gamma}}^{*}{\ensuremath{\pi}}_{n}{\ensuremath{\gamma}}^{*}$ transition form factor, ${\mathcal{T}}_{{\ensuremath{\pi}}_{n}}({Q}^{2})$, is nonzero $\ensuremath{\forall}n$, and ${\mathcal{T}}_{{\ensuremath{\pi}}_{n}}({Q}^{2})\ensuremath{\approx}(4{\ensuremath{\pi}}^{2}/3)({f}_{{\ensuremath{\pi}}_{n}}/{Q}^{2})$ at large ${Q}^{2}$. For all pseudoscalars but the Goldstone mode, this leading contribution vanishes in the chiral limit. In this instance the ultraviolet power-law behavior is $1/{Q}^{4}$ for $n\ensuremath{\ne}0$, and we find numerically ${\mathcal{T}}_{{\ensuremath{\pi}}_{1}}({Q}^{2})\ensuremath{\simeq}(4{\ensuremath{\pi}}^{2}/3)(\ensuremath{-}\ensuremath{\langle}\overline{q}q\ensuremath{\rangle}/{Q}^{4})$. This subleading power-law behavior is always present. In general its coefficient is not simply related to ${f}_{{\ensuremath{\pi}}_{n}}$. The properties of $n\ensuremath{\ne}0$ pseudoscalar mesons are sensitive to the pointwise behavior of the long-range piece of the interaction between light quarks.

Bethe-Salpeter Meson Masses Beyond Ladder Approximation

The effect of quark-gluon vertex dressing on the ground-state masses of the u/d-quark pseudoscalar, vector and axialvector mesons is considered with the Dyson-Schwinger equations. This extends the ladder-rainbow Bethe-Salpeter kernel to two-loops. To render the calculations feasible for this exploratory study, we employ a simple infrared dominant model for the gluon exchange that implements the vertex dressing. The resulting model, involving two distinct representations of the effective gluon exchange kernel, preserves both the axial-vector Ward-Takahashi identity and charge conjugation symmetry. Numerical results confirm that the pseudoscalar meson retains its Goldstone boson character. The vector meson mass, already at a very acceptable value at ladder level, receives only 30 MeV of attraction from this vertex dressing. For the axial-vector states, which are about 300 MeV too low in ladder approximation, the results are mixed: The 1+− state receives 290 MeV of repulsion, but the 1++ state is lowered further by 30 MeV. The exotic channels 0−− and 1−+ are found to have no states below 1.5 GeV in this model.

Pion mass and decay constant

Independent of assumptions about the form of the quark-antiquark scattering kernel we derive the explicit relation between the pion Bethe-Salpeter amplitude, Gamma_pi, and the quark propagator in the chiral limit; Gamma_pi necessarily involves a non-negligible gamma_5 gamma.P term (P is the pion four-momentum). We also obtain exact expressions for the pion decay constant, f_pi, and mass, both of which depend on Gamma_pi; and demonstrate the equivalence between f_pi and the pion Bethe-Salpeter normalisation constant in the chiral limit. We stress the importance of preserving the axial-vector Ward-Takahashi identity in any study of the pion itself, and in any study whose goal is a unified understanding of the properties of the pion and other hadronic bound states.

π- andK-meson Bethe-Salpeter amplitudes

Independent of assumptions about the form of the quark-quark scattering kernel $K$, we derive the explicit relation between the flavor-nonsinglet pseudoscalar-meson Bethe-Salpeter amplitude ${\ensuremath{\Gamma}}_{H}$ and the dressed-quark propagator in the chiral limit. In addition to a term proportional to ${\ensuremath{\gamma}}_{5}$, ${\ensuremath{\Gamma}}_{H}$ necessarily contains qualitatively and quantitatively important terms proportional to ${\ensuremath{\gamma}}_{5}\ensuremath{\gamma}\ensuremath{\cdot}P$ and ${\ensuremath{\gamma}}_{5}\ensuremath{\gamma}\ensuremath{\cdot}\mathrm{kk}\ensuremath{\cdot}P$, where $P$ is the total momentum of the bound state. The axial-vector vertex contains a bound state pole described by ${\ensuremath{\Gamma}}_{H},$ whose residue is the leptonic decay constant for the bound state. The pseudoscalar vertex also contains such a bound state pole and, in the chiral limit, the residue of this pole is related to the vacuum quark condensate. The axial-vector Ward-Takahashi identity relates these pole residues, with the Gell-Mann--Oakes--Renner relation a corollary of this identity. The dominant ultraviolet asymptotic behavior of the scalar functions in the meson Bethe-Salpeter amplitude is fully determined by the behavior of the chiral limit quark mass function, and is characteristic of the QCD renormalization group. The rainbow-ladder Ansatz for $K$, with a simple model for the dressed-quark-quark interaction, is used to illustrate and elucidate these general results. The model preserves the one-loop renormalization group structure of QCD. The numerical studies also provide a means of exploring procedures for solving the Bethe-Salpeter equation without a three-dimensional reduction.

CHIRAL PERTURBATION THEORY AT THE QUARK LEVEL AND NEW NONPERTURBATIVE FORMULAE FOR THE PION DECAY CONSTANT IN QCD

Introducing the most general expression for the corresponding axial-vector vertex, the flavor nonsinglet, chiral axial-vector Ward-Takahashi (WT) identity is investigated in the framework of dynamical chiral symmetry breaking (DCSB). A chiral perturbation theory at the quark level (CHPTq) is proposed in terms of a Taylor series expansions in powers of the external momenta q (momentum of a massless pion) for the direct solution of the above identity at small momentum transfer q (momentum of a massless pion). Correct treatment of initial dynamical singularities at q=0 within the CHPTq approach in accordance with the Ball and Chiu procedure makes it possible to decompose the axial-vector vertex into pole and regular parts in a self-consistent way. The Bethe-Salpeter (BS) bound-state amplitude of a massless pion restored from the identity is shown to coincide with the residue at pole q2=0, which is proportional to the pion decay constant. We find exact solution for the regular piece of the corresponding vertex at zero momentum transfer in terms of the quark propagator dynamical variables alone. This solution automatically satisfies asymptotic freedom (it approaches the point-like vertex at infinity). Applying the proposed CHPTq approach to the matrix element of the axial-vector current determining the pion decay constant, we find “exact” (within the BS bound-state amplitude, restored fom the axial WT identity), nonperturbative expression for the pion decay constant in the current algebra (CA) representation. We show explicitly that the well-known formula of Pagels-Stokar-Cornwall for the pion decay constant is a particular case of the CHPTq approach. We find also new, nonperturbative formulae for the pion decay constant in the Jackiw-Johnson (JJ) representation as well. They now have full physical sense within the CHPTq approach. Renormalization of these expressions as well as their application in technicolor theories with slowly running couplings are briefly discussed. We also propose to distinguish between the scales of DCSB at the quark and hadronic levels (the scale of effective field theory) as well as advocate a simple relation between them based on naive counting arguments.

Chiral-Invariant Regularization and Renormalization of the Axial-Vector Current

The regularization and renormalization procedures for perturbation calculations involving the axial-vector current of the $\ensuremath{\sigma}$ model are considered. It is shown that, even in the absence of electromagnetism, special care must be exercised when introducing fermion regulator fields into the Lagrangian so that the axial-vector Ward-Takahashi identity is not violated in the process. A generalized method of regularization is proposed in which the regulator part of the Lagrangian satisfies chiral invariance and which reduces to the usual one when applied to quantities other than the axial-vector current. By virtue of the chiral-invariant character of this method, a subtraction procedure can be introduced for the axial-vector vertex which eliminates unrenormalizable cutoff-dependent terms. After this subtraction, the regularized axial-vector vertex of the $\ensuremath{\sigma}$ model satisfies the usual Ward-Takahashi identity and is renormalizable by rescaling of the external wave functions.