Color-singlet State Research Articles

Phenomenological equation of state for quark matter

A model for quark matter, similar to the SLAC bag model, is used as a phenomenological basis in view of astrophysical illustrations. This model, although crude, is expected to be sufficient for this type of application where orders of magnitude only are required at this stage. This model consists of colored quarks interacting via scalar gluons (responsible for the confinement) and colored vector gluons. Although color-singlet states are assumed to be the states physically realized in nature, this assumption is not essential and can easily be relaxed. A covariant statistical-mechanical formalism based on the use of techniques similar to those of plasma physics is given. A Bogolubov-Born-Green-Kirkwood-Yvon hierarchy is obtained for the relevant statistical quantities (i.e., covariant Wigner functions, moments of the gluon fields, etc.). It is truncated at the third order (neglect of three-body correlations). The thermodynamical quantities are then evaluated and the resulting equation of state is applied to neutron stars.

Feynman rules of quantum chromodynamics inside a hadron

We start from quantum chromodynamics in a finite volume of linear size $L$ and examine its color-dielectric constant ${\ensuremath{\kappa}}_{L}$, especially the limit ${\ensuremath{\kappa}}_{\ensuremath{\infty}}$ as $L\ensuremath{\rightarrow}\ensuremath{\infty}$. By choosing as our standard ${\ensuremath{\kappa}}_{L}=1$ when $L=\mathrm{some}\mathrm{hadron}\mathrm{size} R$, we conclude that ${\ensuremath{\kappa}}_{\ensuremath{\infty}}$ must be 1; furthermore, from the fact that a free quark has not been observed we can estimate an upper bound: ${\ensuremath{\kappa}}_{\ensuremath{\infty}}1.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}\ensuremath{\alpha}$ where $\ensuremath{\alpha}$ is the fine-structure constant of QCD inside the hadron. A permanent quark confinement corresponds to the limit ${\ensuremath{\kappa}}_{\ensuremath{\infty}}=0$. The hadrons are viewed as small domain structures (with color-dielectric constant = 1) immersed in a perfect, or nearly perfect, color-dia-electric medium, which is the vacuum. The Feynman rules of QCD inside the hadron are derived; they are found to depend on the color-dielectric constant ${\ensuremath{\kappa}}_{\ensuremath{\infty}}$ of the vacuum that lies outside. We show that, when ${\ensuremath{\kappa}}_{\ensuremath{\infty}}\ensuremath{\rightarrow}0$, the mass of any color-nonsinglet state becomes $\ensuremath{\infty}$, but for color-singlet states their masses and scattering amplitudes remain finite. These new Feynman rules also depend on the hadron size $R$. Only at high energy and large four-momentum transfer can such $R$ dependence be neglected and, for color-singlet states, these new rules be reduced to the usual ones.

Confinement of three quarks by infrared singularities

An infrared-divergent interaction previously shown to lead to confinement in the quark-antiquark bound-state problem is used in the three-quark problem. Once again the divergent quark self-energy is canceled by the binding energy, leading to a finite Bethe-Salpeter bound-state equation, only in a color-singlet state. The key to the method is the decomposition of the bound-state amplitude into a sum of three terms, each of which depends on only one energy variable; analysis of the amplitude's pole structure then becomes possible. Confinement is demonstrated and ground-state energy is estimated variationally.

Effect of instantons on the heavy-quark potential

A systematic large-mass expansion of the quark-antiquark effective Hamiltonian due to instantons is developed. The $O({m}^{\ensuremath{-}2})$ spin-spin and spin-orbit contributions are evaluated in the dilute-gas approximation. These can be expressed in terms of differential operators acting on the spin-independent potential. The three-quark effective Hamiltonian (in a color-singlet state) is shown to be a sum of quark-antiquark Hamiltonians. The structure and numerical value of the effective Hamiltonian is discussed.

Gluonic long-range forces between hadrons

Long-range forces between hadrons due to the exchange of two or more gluons in the color singlet state are derived. The calculation iss facilitated by assuming a simplified effective interaction and using the method of dimensional regularization. In addition to the naive QCD, the dipole form for the gluon propagator is studied.

Effects of confinement on analyticity in two-dimensional quantum chromodynamics

We study the analytic properties of scattering amplitudes in two-dimensional quantum chromodynamics. The long-range confining force leads to nonanalytic behavior for individual Feynman diagrams and amplitudes involving colored external states calculated in the light-cone gauge. Nevertheless physical amplitudes involving only color-singlet external states have the normal analyticity properties. We explicitly demonstrate analyticity for the meson inelastic form factor and the elastic scattering amplitude.

Classical algebraic chromodynamics

I develop an extension of the usual equations of $\mathrm{SU}(n)$ chromodynamics which permits the consistent introduction of classical, noncommuting quark source charges. The extension involves adding a singlet gluon, giving a $\mathrm{U}(n)$-based theory with outer product ${P}^{a}(u, v)=(\frac{1}{2})({d}^{\mathrm{abc}}+i{f}^{\mathrm{abc}})({u}^{b}{v}^{c}\ensuremath{-}{v}^{b}{u}^{c})$ which obeys the Jacobi identity, inner product $S(u, v)=(\frac{1}{2})({u}^{a}{v}^{a}+{v}^{a}{u}^{a})$, and with the ${n}^{2}$ gluon fields elevated to algebraic fields over the quark color charge ${C}^{*}$ algebra. I show that provided the color charge algebra satisfies the condition $S(P(u, v), w)=S(u, P(v, w))$ for all elements $u$, $v$, $w$ of the algebra, all the standard derivations of Lagrangian chromodynamics continue to hold in the algebraic chromodynamics case. I analyze in detail the color charge algebra in the two-particle ($qq, q\overline{q}, \stackrel{-}{\mathrm{qq}}$) case and show that the above consistency condition is satisfied for the following unique (and, interstingly, asymmetric) choice of quark and antiquark charges: ${Q}_{q}^{a}={\ensuremath{\xi}}^{a}$ ${Q}_{\overline{q}}^{a}={\overline{\ensuremath{\xi}}}^{a}+{\ensuremath{\delta}}^{a0}{(\frac{n}{2})}^{\frac{3}{2}}1$, with ${\ensuremath{\xi}}^{a}{\ensuremath{\xi}}^{b}=(\frac{1}{2})({d}^{\mathrm{abc}}+i{f}^{\mathrm{abc}}){\ensuremath{\xi}}^{c}$, ${\overline{\ensuremath{\xi}}}^{a}{\overline{\ensuremath{\xi}}}^{b}=\ensuremath{-}(\frac{1}{2})({d}^{\mathrm{abc}}\ensuremath{-}i{f}^{\mathrm{abc}}){\overline{\ensuremath{\xi}}}^{c}$. The algebraic structure of the two-particle $\mathrm{U}(n)$ force problem, when expressed on an appropriately diagonalized basis, leads for all $n$ to a classical dynamics problem involving an ordinary SU(2) Yang-Mills field with uniquely specified classical source charges which are nonparallel in the color-singlet state. An explicit calculation shows that local algebraic $\mathrm{U}(n)$ gauge transformations lead only to a rigid global rotation of axes in the overlying classical SU(2) problem, which implies that the relative orientations of the classical source charges have physical significance. (For an application to the static $q\overline{q}$ force problem, see my later paper). I conclude with a series of conjectures about the extension of the algebraic results to the general $N$-particle case, and about the extension of the classical theory developed here to a full field theory.

S-wave Colour Singlet States of Multiquark Configurations

Simple methods for determining and systematically labelling the S-wave colour singlet states of multiquark configurations are developed and results are given for the quark configurations QN (N = 0 (mod 3)) and for the quark-antiquark configurations Q2Q2, Q4Q and Q3Q3. The masses of the colour singlet states associated with the Q6 and Q9 configurations are calculated using the MIT bag model and a number of conclusions concerning these mass spectra are drawn.

Infrared behavior in non-Abelian theories for forward processes

We show, by means of a simple example, that the Bloch-Nordsieck program cannot be applied to non-Abelian theories in an identical manner to quantum electrodynamics. The simple example is single-gluon bremsstrahlung in quark-quark scattering to lowest order in perturbation theory, the cross section for which is found to be quadratically divergent, i.e., approx. 1/lambda/sup 2/, where lambda is the fictitious gluon mass. Radiation from internal gluon lines is found to contribute to this divergence. Application to the scattering of color-singlet states is discussed. (AIP)

Masses of colored vector mesons

Mass formulas for colored vector mesons are derived under the assumption of one common mixing angle forSU(3)′, independent of the color quantum numbers, and correspondingly one common mixing angle (ϑ″) forSU(3)″. This a priori plausible assumption turns out to have strong implications and thus might be too restrictive. We allow for a non-trivial spatial overlap integralρ between color-singlet and color-octet states. Various cases are treated and physical possibilities are identified. The best agreement is obtained for ϑ″ = 0. There are two models of this type with and without a part of the symmetry breaking Hamiltonian density transforming as (Y, Y) underSU(3)′⊗SU(3)″. Models with ϑ″ ≠ 0 are also possible. They cannot have a (Y, Y) and predict 3.34 GeV as the mass of theψ′ (3.7). This error of 10% may however be used to reject this possibility. Masses of colored vector mesons are predicted in the various models. If a (Y, Y) is present, ideal mixing inSU (3)′ is implied by the general formalism of the model.

Algebraic approach to the quark problem

An algebraic framework is constructed for the description of color singlet states composed of colored quarks. Quark fields are elements of a local octonionic module. Invariant actions are constructed for the quark fields. The actions are symmetric under local color transformations belonging to the exceptional Lie groupG2. The theory is quantized in a generalized path integral formalism. It is found that only color singlet states propagate, whereas quarks are «confined».

Color Gluons and the Decay of theψ(3700)intoψ(3100)

The decay of $\ensuremath{\psi}(3700)$ into $\ensuremath{\psi}(3100)$ plus hadrons constitutes a violation of Zweig's rule. On the assumption that the de-excitation proceeds through the emission of Yang-Mills gluons in a color-singlet state, I predict a width of 190 \ifmmode\pm\else\textpm\fi{} 40 keV for this process. All parameters entering the calculation are taken from previous fits to the decay widths of the $\ensuremath{\psi}(3100)$.