Small Current Quark Mass Research Articles

Effective potential of composite operators in the first order region of the QCD phase transition

We propose a method to determine the effective potential of QCD from the gap equation by introducing the homotopy method between the solutions of the equation of motion. Via this method, the effective potential beyond the bare vertex approximation is obtained, which then generalizes the Cornwall-Jackiw-Tomboulis effective potential for the bilocal composite operators. Moreover, the extended effective potential is set to be a function of self-energy instead of the propagator, which is the key point for the potential to be bounded from below. We then investigate the extended effective potential in the cases of phase transition of the QCD vacuum with a small current quark mass, the first order phase transition of QCD at finite temperature, and high baryon chemical potential. In the former case, the effective potential shows as an inflection point at the critical mass where the multiple solutions of the Dyson-Schwinger equation (DSE) vanishes, which is consistent with that obtained by solving the DSE directly. For the latter case, the in-medium properties, such as the latent heat and the difference of trace anomaly, of QCD is obtained. Published by the American Physical Society 2024

Bound states in gauge theories as the Poincaré group representations

The bound-state generating functional is constructed in gauge theories. This construction is based on the Dirac Hamiltonian approach to gauge theories, the Poincaré group classification of fields and their nonlocal bound states, and the Markov-Yukawa constraint of irreducibility. The generating functional contains additional anomalous creations of pseudoscalar bound states: para-positronium in QED and mesons inQCDin the two-gamma processes of the type of γ + γ → π 0 +para-positronium. The functional allows us to establish physically clear and transparent relations between the perturbativeQCD to its nonperturbative low-energy model by means of normal ordering and the quark and gluon condensates. In the limit of small current quark masses, the Gell-Mann-Oakes-Renner relation is derived from the Schwinger-Dyson and Bethe-Salpeter equations. The constituent quark masses can be calculated from a self-consistent nonlinear equation.

Bound States in Gauge Theories as the Poincaré Group Representations

The bound state generating functional is constructed in gauge theories. This construction is based on the Dirac Hamiltonian approach to gauge theories, the Poincar\'e group classification of fields and their nonlocal bound states, and the Markov-Yukawa constraint of irreducibility. The generating functional contains additional anomalous creations of pseudoscalar bound states: para-positronium in QED and mesons in QCD in the two gamma processes of the type of \gamma + \gamma = \pi_0+para-positronium. The functional allows us to establish physically clear and transparent relations between the perturbative QCD to its nonperturbative low energy model by means of normal ordering and the quark and gluon condensates. In the limit of small current quark masses, the Gell-Mann-Oakes-Renner relation is derived from the Schwinger-Dyson (SD) and Bethe-Salpeter (BS) equations. The constituent quark masses can be calculated from a self-consistent non-linear equation.

Mass spectrum in supersymmetric QCD and problems with the seiberg duality. Equal quark masses

The dynamical scenario is considered for N=1 SQCD, with N_c colors and N_c<N_F<3N_c flavors with small but nonzero current quark masses m_Q\neq 0, in which quarks form the diquark-condensate phase. This means that colorless chiral quark pairs condense coherently in the vacuum, <{\bar Q}Q>\neq 0, while quarks alone don't condense, <Q>=<\bar Q>=0, so that the color is confined. Such condensation of quarks results in formation of dynamical constituent masses \mu_C \gg m_Q of quarks and appearance of light "pions" (similarly to QCD). The mass spectrum of SQCD in this phase is described and comparison with the Seiberg dual description is performed. It is shown that the direct and dual theories are different (except, possibly, for the perturbative strictly superconformal regime).

Volume and quark mass dependence of the chiral phase transition

We investigate chiral symmetry restoration in finite spatial volume and at finite temperature by calculating the dependence of the chiral phase transition temperature on the size of the spatial volume and the current-quark mass for the quark-meson model, using the proper-time Renormalization Group approach. We find that the critical temperature is weakly dependent on the size of the spatial volume for large current-quark masses, but depends strongly on it for small current-quark masses. In addition, for small volumes we observe a dependence on the choice of quark boundary conditions.

Volume dependence of the pion mass in the quark-meson model

We consider the quark-meson-model in a finite three-dimensional volume using the Schwinger proper-time renormalization group. We derive and solve the flow equations for finite volume in local potential approximation. In order to break chiral symmetry in the finite volume, we introduce a small current quark mass. The corresponding effective meson potential breaks chiral O(4) symmetry explicitly, depending on $\ensuremath{\sigma}$ and $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\pi}}$ fields separately. We calculate the volume dependence of the pion mass and of the pion decay constant with the renormalization group flow equations and compare with recent results from chiral perturbation theory in a finite volume.

Effect of a Small Current Quark Mass on Dressed Gluon and Quark Propagator**The project supported in part by National Natural Science Foundation of China under Grant Nos. 10175033 and 10135030 and the Research Fund for the Doctoral Program of Higher Education under Grant No. 20030284009

Based on the Dyson–Schwinger approach, a method for obtaining the small current quark mass effect on the dressed gluon and quark propagator is developed. A comparison with the results of the previous approach is given.

Effect of a Small Current Quark Mass on Bag Constant**The project supported in part by National Natural Science Foundation of China under Grant Nos. 10175033 and 10135030 and the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20030284009

A method for obtaining the small current quark mass effect on the dressed quark propagator within the Dyson-Schwinger approach is developed. From this the small current quark mass dependence of the bag constant is evaluated. It is found that the bag constant decreases with the increasing current quark mass and the contribution of the current quark mass cannot be dropped.

Explicit and Dynamical Chiral Symmetry Breaking in anEffective Quark–Quark Interaction Model

A method for obtaining the small current quark mass effect on thedressed quark propagator from an effective quark–quark interaction modelis developed. Within this approach both the explicit and dynamical chiralsymmetry breakings are analysed. A comparison with the previous resultsis given.

Small Current Quark Mass Effects on Dressed-Quark Propagator in an Effective Quark-Quark Interaction Model**The project supported in part by National Natural Science Foundation of China under Grant Nos. 19975062, 10175033, and 10135030

A method for obtaining the small current quark mass dependence of the dressed quark propagator from an effective quark-quark interaction model is developed. Within this approach the small current quark mass effects on dressed-quark propagator have been studied. A comparison with previous results is given.

BARYON MAGNETIC MOMENTS AND SPIN DEPENDENT QUARK FORCES

The J=3/2 Δ, J=1/2 nucleon mass difference shows that quark energies can be spin dependent. It is natural to expect that quark wave functions also depend on spin. In the octet, such spin dependent forces lead to different wave functions for quarks with spin parallel or antiparallel to the nucleon spin. A two component Dirac equation wave function is used for the quarks assuming small current quark masses for the u and d quarks. Then, the neutron/proton magnetic moment ratio, the nucleon axial charge, and the spin content of the nucleon can all be simultaneously fit assuming isospin invariance between the u and d quarks, but allowing for spin dependent forces. The breakdown of the Coleman–Glashow sum rule for octet magnetic moments follows naturally in this Dirac approach as the bound quark energy also effects the magnetic moment. Empirically the bound quark energy increases with the number of strange quarks in the system. Allowing the strange quark wave function similar spin dependence predicts the magnetic moments of the octet, in close agreement with experiment. Differences between the octet and decuplet magnetic moments are also explained immediately with spin dependent wave functions.

Parton distribution functions from large Nc QCD

In the Bjorken limit, the transverse momenta of partons can be treated perturbatively. The non-perturbative part of the problem of determining Deep Inelastic structure functions becomes then that of solving two dimensional QCD (2DQCD). We show an isomorphism of 2DQCD with a bilocal field theory of hadrons (two dimensional Quantum Hadron Dynamics, 2DQHD). Baryons are topological solitons of this theory. The classical limit of 2DQHD is the large N c limit of 2DQCD; by solving its equations of motion we can determine the structure function of the baryon. We compare the answer with measurements in neutrino scattering and get good agreement; also we find that the anti-quark content of the baryon is small in the limit of large N c and small current quark mass.

Monopole Excitation of the Nucleon in a Relativistic Three-Quark Model

The densities and form factors of the proton andthe Roper monopole excitedstate resonance are calculatedusing a relativistic three-quark model. Small currentquark masses are used with the three-body Dirac equation solved in hypercentralapproximation. A QCD-based three-body potential,proportional to a minimum string length between thethree quarks, is used for confinement. The calculatedelectric form factor for the proton reproduces closelya dipole fit to the data. The proton density is morecompact than is the Roper resonance density. The centraldensity of the proton is about five times that for the Roper resonance. The hyperradial nodein the Roper resonance composite three-quark wavefunction shows up as a node in the transition densitybetween the proton and the Roper resonance. This node also causes the calculated transition formfactor to be larger than either the proton or Roperresonance form factors, all evaluated at the same valueof momentum transfer. The Roper resonance form factor is smaller than the proton form factor, asexpected, indicative of the Roper resonance being a morediffuse system than the proton.

Variational quark mass expansion and the order parameters of chiral symmetry breaking

We investigate in some detail a "variational mass" expansion approach, generalized from a similar construction developed in the Gross-Neveu model, to evaluate the basic order parameters of the dynamical breaking of the $SU(2)_L \times SU(2)_R$ and $SU(3)_L \times SU(3)_R$ chiral symmetries in QCD. The method starts with a reorganization of the ordinary perturbation theory with the addition of an arbitrary quark mass $m$. The new perturbative series can be summed to all orders thanks to renormalization group properties, with specific boundary conditions, and advocated analytic continuation in $m$ properties. In the approximation where the explicit breakdown of the chiral symmetries due to small current quark masses is neglected, we derive ansatzes for the dynamical contribution to the "constituent" masses $M_q$ of the $u,d,s$ quarks; the pion decay constant $F_\pi$; and the quark condensate $< \bar q q > $ in terms of the basic QCD scale $\Lambda_{\bar{MS}} $. Those ansatzes are then optimized, in a sense to be specified, and also explicit symmetry breaking mass terms can be consistently introduced in the framework. The obtained values of $F_\pi$ and $M_q$ are roughly in agreement with what is expected from other non-perturbative methods. In contrast we obtain quite a small value of $|< \bar q q >|$ within our approach. The possible interpretation of the latter results is briefly discussed.

Rest Frame Properties of the Proton

The proton is modeled as three quarks of smallcurrent quark mass. The threebody Dirac equation issolved with spin-independent central diagonal linearconfining potentials with an attractive Coulombic term in a relativistic threequark model.Hyperspherical coordinates are used, and the bound stateis found analytically. After integrating over thehyperangles, the Hamiltonian is an 8 by 8 matrix ofcoupled first-order differential equations in onevariable, the hyperradius. These are analytically solvedin hypercentral approximation. For the(1/2+)3 ground-state configurationin the nonrelativistic large-quark-mass limit, there are no nodes in the wave function.However, in the extreme relativistic limit of smallcurrent quark masses of a few MeV, the expectation valueof the number of nodes is about 1.30 when the potential parameters are chosen to reproducethe proton rms charge radius. The quarks are assumed topossess a Pauli anomalous magnetic moment, like that ofthe electron and muon of (α/2π)(e/m). Assuming all three quarks have equal mass, one can fitthe rest energy, magnetic moment, rms charge radius, andaxial charge of the proton with this relativisticthree-body Dirac equation model. The solution found shows the necessity of including all componentsof the composite three-quark wave function, as the uppercomponent contributes only 0.585 to the norm.

Confinement and the composite “pionic” goldstone mode via the NJL mechanism in a bag-like potential model

We start from an effective lagrangian for massless u- and d-quarks containing a chiral-symmetric effective interaction with a position-dependent coupling strength. We study the appearance of the pseudoscalar isovector Goldstone mode, which is generated by the spontaneous breakdown of chiral symmetry due to the selfconsistent confining potential. The introduction of small current quark masses (~8 MeV) leads to a finite energy eigenvalue of ~140 MeV. The “wave function” of the mode is determined up to a normalization by the single-quark propagator and depends strongly on the cutoff parameter. We compare it with that of a single q- q “bag” state.