Elmar P. Biernat

Confinement, quark mass functions, and spontaneous chiral symmetry breaking in Minkowski space

We formulate the covariant equations for quark-antiquark bound states in Minkowski space in the framework of the Covariant Spectator Theory. The quark propagators are dressed with the same kernel that describes the interaction between different quarks. We show that these equations are charge conjugation invariant, and that in the chiral limit of vanishing bare quark mass, a massless pseudoscalar bound state is produced in a Nambu--Jona-Lasinio (NJL) mechanism, which is associated with the Goldstone boson of spontaneous chiral symmetry breaking. In this introductory paper we test the formalism by using a simplified kernel consisting of a momentum-space

Point-Form Hamiltonian Dynamics and Applications

\delta

Linear confinement in momentum space: singularity-free bound-state equations

-function with a vector Lorentz structure, to which one adds a mixed scalar and vector confining interaction. The scalar part of the confining interaction is not chirally invariant by itself, but decouples from the equations in the chiral limit and therefore allows the NJL mechanism to work. With this model we calculate the quark mass function, and we compare our Minkowski-space results to LQCD data obtained in Euclidean space. In a companion paper we apply this formalism to a calculation of the pion form factor.

Electromagnetic meson form factor from a relativistic coupled-channels approach

The pi-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 pi-pi scattering amplitude allows us to sum the intermediate quark-quark interactions to all orders. The Adler self-consistency zero for pi-pi scattering in the chiral limit emerges as the result for this sum.

Covariant Spectator Theory of heavy–light and heavy mesons and the predictive power of covariant interaction kernels

This short review summarizes recent developments and results in connection with point-form dynamics of relativistic quantum systems. We discuss a Poincaré invariant multichannel formalism which describes particle production and annihilation via vertex interactions that are derived from field theoretical interaction densities. We sketch how this rather general formalism can be used to derive electromagnetic form factors of confined quark–antiquark systems. As a further application it is explained how the chiral constituent quark model leads to hadronic states that can be considered as bare hadrons dressed by meson loops. Within this approach hadron resonances acquire a finite (non-perturbative) decay width. We will also discuss the point-form dynamics of quantum fields. After recalling basic facts of the free-field case we will address some quantum field theoretical problems for which canonical quantization on a space–time hyperboloid could be advantageous.

Charge-conjugation symmetric complete impulse approximation for the pion electromagnetic form factor in the covariant spectator theory

The pion electromagnetic form factor at spacelike momentum transfer is calculated in relativistic impulse approximation using the Covariant Spectator Theory. The same dressed quark mass function and the equation for the pion bound-state vertex function as discussed in the companion paper are used for the calculation, together with a dressed quark current that satisfies the WardTakahashi identity. The results obtained for the pion form factor are in agreement with experimental data, they exhibit the typical monopole behavior at high-momentum transfer, and they satisfy some remarkable scaling relations.

Covariant Spectator Theory of quark-antiquark bound-states: Mass spectra and vertex functions of heavy and heavy-light mesons

Relativistic equations of Bethe-Salpeter type for hadron structure are most conveniently formu- lated in momentum space. The presence of confining interactions causes complications because the corresponding kernels are singular. This occurs not only in the relativistic case but also in the nonrelativistic Schrodinger equation where this problem can be studied more easily. For the linear confining interaction the singularity reduces to one of Cauchy principal value form. Although this singularity is integrable, it still makes accurate numerical solutions difficult. We show that this prin- cipal value singularity can be eliminated by means of a subtraction method. The resulting equation is much easier to solve and yields accurate and stable solutions. To test the methods numerical efficiency, we performed a three-parameter least square fit ofa simple linear-plus-Coulomb potential to the bottomonium spectrum.

Electromagneticρ-meson form factors in point-form relativistic quantum mechanics

Point-form relativistic quantum mechanics is used to derive an expression for the electromagnetic form factor of a pseudoscalar meson for spacelike momentum transfers. The elastic scattering of an electron by a confined quark-antiquark pair is treated as a relativistic two-channel problem for the qqe and qqe{gamma} states. With the approximation that the total velocity of the qqe system is conserved at (electromagnetic) interaction vertices this simplifies to an eigenvalue problem for a Bakamjian-Thomas type mass operator. After elimination of the qqe{gamma} channel the electromagnetic meson current and form factor can be directly read off from the one-photon-exchange optical potential. By choosing the invariant mass of the electron-meson system large enough, cluster separability violations become negligible. An equivalence with the usual front-form expression, resulting from a spectator current in the q{sup +}=0 reference frame, is established. The generalization of this multichannel approach to electroweak form factors for an arbitrary bound few-body system is quite obvious. By an appropriate extension of the Hilbert space this approach is also able to accommodate exchange-current effects.