University of Orléans**, F-45067 Orléans, France

We consider the classical radiative solution to the QCD (quantum chromodynamics) dynamical equations and find trivial solutions similar to the electrodynamic situation when one considers singlet (colorless) radiation sources. We transpose the findings to quantum fields and identify radiation from hadron (meson) sources with gluon packages of 3 (2) gluons. Invoking the singlet character of the QCD matter and radiation field, we deduce that the blank (singlet) gluon packages are the elementary building blocks of QCD radiation, just as blank hadrons and mesons are the building blocks of baryon matter. Colorlessness of the gluon packages suggests a possibly simpler quasi-abelian behaviour, as far as linear superposition is involved, of the QCD singlet radiation field.

PACS: 12.38.-t, 12.38.Aw

keywords: quantum chromodynamics, gluon radiation

The dynamic behaviour of the strong interaction, giving rise to phenomena such as quark confinement and asymptotic freedom, still eludes understanding, despite intense efforts during the past forty years. Considering the difficulty of the task, it is essential to take advantage of any simplification and symmetry the actual physical world has to offer. In particular, we should avoid first solving the bigger problem of general self-interacting Yang-Mills theories if we find that reality corresponds to a, perhaps much simpler, subclass of solutions. It is the object of this work to exploit the singlet structure of matter, hadrons and mesons, to give a preliminary to the derivation of the structure of the quantum chromodynamic radiation field. Quantumchromodynamics (QCD) is an SU(3) gauge theory acting between nucleon constituents (quarks(Gell-Mann) or partons(Feynman)). Each parton carries one color charge out of three, the QCD radiation field (gluons) acting between quarks transforms one color into another; gluons thus carry a gluon charge and interact among themselves. The QCD energy operator governing the dynamical evolution is given by

(1)

The T ^a (a=1,..,8) are 3x3 matrices which obey the SU(3) defining property

(2)

The C^abc are the structure constants, odd under cyclic permutation of indices, whose non-vanishing members are given for SU(3) by [1]

C¹²³=1, C¹⁴⁷=1/2, C¹⁵⁶=1/2, C²⁴⁶=1/2, C²⁵⁷=1/2, C³⁴⁵=1/2, C³⁶⁷=-1/2, C⁴⁵⁸=Ö (3/2), C⁶⁷⁸=Ö (3/2)

The gluon electric, magnetic and vector potential fields have the equal-time commutation relations

(3)

(4)

(5)

with k,l = x,y,z; e is the completely antisymmetric tensor; are Kronecker-delta and Dirac deltafunction, respectively. All other commutators vanish. The are Dirac spinors, the Dirac matrices.

From the energy operator and the commutation relations, we obtain the QCD analog of Maxwell's equations:

(6)

where and

Contrary to electrodynamics, the general solution of the radiation QCD equations (j=0) is unknown, both for the classical and the quantum case. There do, however, exist subclasses of simple solutions for the free classical case. Examples are or . These solutions reproduce the non-interacting case and have plane wave solutions as well known from electrodynamics.

If we want to learn something about the gluon radiation field, we have to look at the structure of the sources. In fact, the sources determine radiation, not the inverse. Think of quantum electrodynamics (QED) where matter is capable of producing photons, but photons are only able to produce matter-antimatter pairs.

Sources for QCD are organized in color singlets. Formally, that means that they are eigenstates of the eight conserved color charge operators with eigenvalue zero.

Singlets with particles only (no antiparticles) contain three quarks whose wavefunction is of the form g(1)h(2)k(3) , where g,h,k corresponds to parton creations, 1,2,3 to color charge. In order to be a color singlet, one has to form superpositions of this state with color-interchanged states. The superposition operation is not trivial. Remember that in QED, states like ((electron)+(proton)) make no sense; only product states of particles of different species are allowed (Fock space). Here, it is just the contrary, pure product states of partons are unphysical.

Let us investigate the currents (sources) generated by quark threesomes in the classical case. We then treat the operators as c-numbers. We find that the (m), m=1,2,3 are all equal and give the currents for a = 1,4,6 and zero otherwise (for the Gell-Mann representation of the ). They are just proportional to the matrix elements

.

If we assume that there is only radiation from the sources considered, the QCD Maxwell equations give

(7)

for a = 1,4,6 and zero otherwise.

Hence, a parton threesome generates a gluon threesome with identical vectorpotential for each gluon ().

Three-quark hadrons are not the only color singlet that can occur in nature. We also have mesons, composed of particle-antiparticle combinations. Their wavefunctions are of the form , suitably symmetrized by superpositions. In order to find the currents corresponding to mesons, we have to consider terms like in the interaction energy operator. The only non-zero diagonal elements occur for a =3, 8. A purely classical treatment for the operator is not adequate here, for we have antiparticles involved (in fact, setting all s to equal c-number functions would give zero), so we have to take this cum grano salis.

The first step in transposing the findings from our classical treatment to a quantum field theoretical one is determining the energy of the plane wave modes obtained from equ. (7). For hadron (meson) radiation, attributing an energy to each mode A(k) gives () from equ. (1). We identify this with 3 (2) gluon radiation.

Next, we have insisted all along on the singlet character of matter. This is true for both the classical and quantum situation. The singlet space of physical states is a subspace of the whole possible vector space. Time evolution maps this space into itself, and this includes all virtual states as well. We never leave, neither virtual nor real, the physical singlet space. The radiation generated from this matter will then also have singlet character, i.e. it will be colorless. The classical treatment suggests that this singlet radiation consists of 3 (2) gluon packages from hadron (meson) sources. In a sense, the gluon threesome is the pendant to the parton threesome in the nucleon, the twosome is the gluon equivalent to the quark-antiquark in the meson. We thus recover a structure similar to that of matter in the QCD radiation field. Just as color singlets are the elementary building blocks of our baryon world, these blank gluon combinations are then the true elementary building blocks of the QCD radiation field. Individual gluons make as little physical sense as individual quarks. Furthermore, their singlet nature suggests that they should not interact among themselves but should rather create a behaviour similar to the known U(1) behaviour in QED, in particular as far as linear superposition is concerned, a perspective that might make it treatable one day. This point, very speculative, evidently needs much further elaboration.

To summarize, we have considered the classical radiative solution to the QCD dynamical equations and have found that one obtains trivial solutions similar to the electrodynamic situation when one considers singlet (colorless) radiation sources. We transposed the findings to quantum fields and identified radiation from hadron (meson) sources with gluon packages of 3 (2) gluons. Invoking the singlet character of the QCD matter and radiation field, we deduced that the blank (singlet) gluon packages are the elementary building blocks of QCD radiation, just as blank hadrons and mesons are the building blocks of baryon matter. Colorlessness of the gluon packages suggests a possibly simpler quasi-abelian behaviour, as far as linear superposition is involved, of the QCD singlet radiation field.

*address for correspondence: 1 Impasse du Crucifix, F-45000 Orléans, France

email: reinhold.brueckner@free.fr

**independent association

[1] Ta-Pei Cheng, Ling-Fong Li ;Gauge theory of elementary particle physics; Oxford University press, Oxford (1989)