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Simple Model for the QCD Vacuum A1110? 31045(3 NBSIR 83-2759 Simple Model for the QCD Vacuum U S. DEPARTMENT OF COMMERCE National Bureau of Standards Center for Radiation Research Washington, DC 20234 Centre d'Etudes de Bruyeres-le-Chatel 92542 Montrouge CEDEX, France July 1983 U S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANOAROS - - - i rx. NBSIR 83-2759 SIMPLE MODEL FOR THE QCD VACUUM m3 C 5- Michael Danos U S DEPARTMENT OF COMMERCE National Bureau of Standards Center for Radiation Research Washington, DC 20234 Daniel Gogny and Daniel Irakane Centre d'Etudes de Bruyeres-le-Chatel 92542 Montrouge CEDEX, France July 1983 U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director SIMPLE MODEL FOR THE QCD VACUUM Michael Danos, Nastional Bureau of Standards, Washington, D.C. 20234, USA and Daniel Gogny and Daniel Irakane Centre d' Etudes de Bruyeres-le-Chatel , 92542 Montrouge CEDEX, France Abstract B.v treating the high-momentum gluon and the quark sector as an in principle calculable effective Lagrangian we obtain a non-perturbati ve vacuum state for OCD as an infrdred gluon condensate. This vacuum is removed from the perturbative vacuum by an energy gap and supports a Meissner-Ochsenfeld effect. It is unstable below a minimum size and it also suggests the existence of a universal hadroni zation time. This vacuum thus exhibits all the properties required for color confinement. I. Introduction By now it is widely believed that the confinement in QCD, in analogy with superconductivity, results from the existence of a physical vacuum which is removed from the remainder of the spectrum by an energy density gap and which exhibits a Meissner-Ochsenfeld effect. 1 More particularly, it is believed that these characteristics of the physical vacuum result from the infrared properties of QCD. With this in mind we will construct in this paper a simple model for QCD which concentrates on the low-energy part and parametrizes the high-energy part of the theory, and we will show that this simple model has a non-perturbati ve solution for the vacuum which indeed possesses the desired characteristics. Even though this model is extremely simple we believe that it is sufficiently accurate to reproduce the essential qualitative features of the QCD vacuum. Furthermore, the accuracy of the model can be improved; in principle the model could be enlarqed in a strai qht-forward manner to yield approximations to the actual solution of the QCD vacuum problem, and hence, to QCD. The most extensive recent treatments of the confinment problem are based on the evaluation of Wilson loops. 2 ’ 3 This method avoids the explicit intro- duction of the structure of the vacuum. The properties of the vacuum itself are discussed in several papers which show that it can have the character of a superconductor, i.e., that a superconductive vacuum is compatible with QCD. 4 Our paper supplements these references in that it provides an explicit construction of the vacuum which demonstrates that a superconductive QCD vacuum not only is possible but actually exists. The present paper is organized as follows. In section II we define the model and show how it follows from the QCD Lagrangian. We perform the analysis in the Hamiltonian formalism in the SchrO'dinger picture, discussed in detail in references 5-7. The drawback of not being manifestly covariant is outweighed by the fact that this treatment lends itself immediately to achieving non-perturbati ve solutions. In this section we show how the property of anti -screeni ng inherent in QCD leads to a aap, and we discuss the conditions for the existence of the gap. Even though an exhaustive description of the Mei ssner-Ochsenfeld effect requires a correct treatment of the vacuum-baq interface we show in section III that our present simple model already points toward the existence of that effect. In section IV we discuss the effect of our neglect of the center-of-mass motion and indicate that this leads to an error on the conservative side, i.e., that our treatment yields a lower limit on the gap size. We sketch the manner in which the physical vacuum condenses from the perturbative vacuum in section V, using as an example the decay of a tt° into two photons. We find indications for the existence of a universal time constant for this process, or, equivalently, for a universal hadroni zation time, similar for all hadrons. Finally, we give a summary of our results. 2 II. The Gap In order for the results of the model theory to have any bearing on QCD, the model should contain as many as possible of the essential ingredients of QCD. We now discuss our reasoning. Firstly, we share the belief that the gauge sector by itself. d a F F 1 yv yv ( ) a r abc F 3 A 3 A gf 2 yv y v v y ( ) is incapable of yielding a gap since the Lagranqian (1) contains no scale. Namely, if one should obtain from (1) a gap, say, A = 300, then this result would mean simultaneously A = 300 eV and A = 300 GeV, which clearly is impossible. In fact, the Fermion sector of QCD does contain scale factors in the form of the quark mass terms. (The origin of the quark masses is still rather mysterious; however this need not concern us here.) In a covariant treatment the Fermion sector would intrude into the Boson sector via modifications of the Boson propagators and the vertex functions. We now shall describe how we approximate this effect in our treatment. As explained in references 5-7, one may derive in the SchrO'dinqer picture of the fields a field-theoretic SchrO'dinqer equation from the f i el d- theoric Lagrangian: = H | S> E | S> (3) where |S> is the state vector describing an eigenstate of the system. We break the Hilbert space into two parts and rewrite eq (3) as 3 . - E(q) 4 <) (J* \) (£) ( ) from which one may eliminate G to obtain XF + Y Y+F = H F + H F = EF eTz 0 eff (5) The form of the operators X,Y,Z depends on the division of the Hilbert space, and in QCD also on the choice of the qauge. We shall take the space F to consist of that part of the Hilbert space which contains only low momentum transverse gluons, while G contains the rest of the Hilbert space, i.e., quarks, longitudinal gluons and high momentum transverse qluons. The precise specification of F will beqiven below. Supposedly the confinement problem is an infrared, i.e., low momentum phenomenon. Hence it should be possible to divide the the Hilbert space such that all high-momentum phenomena are taken care of in the space G. In other words, one may hope that an approximate separation, asymptotic freedom- infrared slavery, is possible in the sense that the "renormalized" parameters calculated solely in the part G of the Hilbert space are reasonably indepen- dent of the size of the part F, for "sufficiently small" F, while at the same time F is "sufficiently large" as far as the description of the infrared properties is concerned. As a consequence of this separation only the part F of the Hilbert space must be treated non-perturbati vely; the part G could be treated by perturbation methods. These considerations form the basis of our model Before discussing the specifics of our model we specify our choice of the 5-7 representation of the transversal vector field operators. To wit, we expand the (Schrffdi nger picture) field operators in terms of the complete. 4 « ) orthonormal, discretized set of vector spherical functions of good angular momentum, J, and parity, specified by the index k = magnetic or electric, i.e., the analogue of the multipole expansion of electrodynamics, augmented by the color index a: A = z(a (r) + h.c. 6 J<vma ^Jicvma ( ) The creation-annihilation operators fulfill • [a ( 7 ) jKvma* J' k' v'm' a' ] jj‘ «' vv' mm' aa' The discretization has been achieved by forming wave packets in the momentum | with a set of orthonormal weight functions specified by the index v. This k | 2 way the functions have a well defined finite mean square momentum <k >, 4 2 2 which depends on v, and limited spread <k > - <k > . At the same time the functions have a limited size in position space, i.e., both <r 2> and <r 4 > - <r 2 > 2 have well-defined, finite values. We describe this size by the 1 2 parameter i - <r > computed for v = 1. At this point we must specify the model. We shall test whether employing a non-perturbative solution suffices for developing an energy gap for the vacuum. To that end we make the simplest possible choice for F, and we consider the particular coherent pairing state constructed with one of the components of eq (6) only. -0/2 (a Jkv Jkv ) F 0 > 8 ' |0> | ( ) 5 where the dot indicates formation of a singlet in both Minkowski and color spaces and, of course, we take v = 1. We have omitted in eq (8) the time dependence; it will have to be added to achieve the complete Schrd’di nger picture state vector. This state contains an unspecified number of pairs. It connot be achieved in a perturbative treatment as it is connected with the perturbative vacuum |0> only by an infinite number of applications of the Hamiltonian. The parameter 0 is a variational parameter which will be used to find a (local) minimum of the Hamiltonian. For technical reasons, similar to those in superconductivity it is advantageous to replace eq (8) by -0/2 B.
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