PhysRevD.9.3471 1974 New extended model of hadrons MIT bag model

CONSTRUCTION OF DUAL AMPLITUDES ~%'ork supported in part by Scientific and Technical Research Council of Turkey. D. D. Goon, Phys. Lett. 29B, 669 (1969);M. Baker and D. D. Coon, Phys. Rev. D 2, 2349 (1970). 2M. Baker and D. D. Coon (unpublished). 3G. Veneziano, Nuovo Cimento 57A, 190 (1968). 4D. D. Coon, U. Sukhatme, and J, Tran Thanh Van„Phys. Lett. 45B, 287 (1973). M. Arik and D. D. Coon, Phys. Lett. 48B, 141 (1974). 8J. F. Gunion, S. J. Brodsky, and R. Blankenbecler, Phys. Lett. 39B, 649 (1972); P. V. Landshoff and J. C. Polklnghorne, iMd. 44B, 293 (1973};S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153 (1973). 7R. Blankenbec1er, S. J. Brodsky, J. F. Gunion, and R. Savit, Phys. Rev. D 8, 4117 (1973). SE. Cremmer and J. Nuyts, Nucl. Phys. B26, 151 (1971). J. D. Bjorken and J. Kogut;, Phys. Rev. D 8, 1341 (1973). PHYSICAL REVIEW D VOLUME 9, NUMBER 12 New extended model of hadrons* 1 5 JUNE 1974 A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. %'eisskopf Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 25 March 1974) We propose that a strongly interacting particle is a finite region of space to which fields are confined. The confinement is accomplished in a Lorentz-invariant way by endowing the finite region with a constant energy per unit volume, B. We call this finite region a "bag." The contained fields may be either fermions or bosons and may have any spin; they may or may not be coupled to one another. Equations of motion and boundary conditions are obtained from a variational principle. The confining region has no dynamical freedom but constrains the fields inside: There are no excitations of the coordinates determining the confining region. The model possesses many desirable features of hadron dynamics: (i) a parton interpretation and presumably Bjorken scaling; the confined fields are free or weakly inter- acting except close to the boundary; (ii) infinitely rising Begge trajectories as a consequence of the bag's finite extent; (iii) the Hagedorn degeneracy or limiting temperature; (iv) all physical hadrons are singlets under hadronic gauge s~~etries. For example, in a theory of fractionally charged, "colored" quarks interacting with colored, massless gauge vector gluons, if both quark and gluon fields are confined to the bag, only color-singlet solutions exist. In addition to establishing these general properties, we present complete classical and quantum solutions for free scalars and also for free fermions inside a bag of one space and one time dimension. Both systems have linear mass-squared spectra. We demonstrate Poincare invariance at the classical level in any dimension and at the quantum level for the above-mentioned explicit solutions in two dimensions. We discuss the behavior of specific solutions in one and three space dimensions. We also discuss in detail the problem of fermlon boundary conditions, which follow only indirectly from the variational principle. I. INTRODUCTION In this paper we shall propose a new model for the structure of hadrons. It is a model which will be formulated in exact, quantitative language. However, it is conceptually simple and, conse- quently, we shall see immediately that it possess- es many features which are in accord with the present understanding of hadron structure. %'e assume that a region of space which is capa- ble of containing hadronic fields has a constant, positive potential energy, B, per unit volume. B will be the only parameter of the theory, at least at the start. B will be of the order 1 GeV/(fm)', and the chaxacteristic linear dimension of a had- ron will be scaled by (I/B)'~'. For short, we will call a region of space which contains hadron fields a "bag." Because the action associated with B is propor- tional to the volume of the space-time hypertube swept out by it, the model is relativistically in- variant. As an example, the simplest such sys- tem is described classically by the action R where the spatial region of integration extends over a closed, finite part of space (the bag). In (1.1), p is the prototype of a hadromc fieM, that is, the field for partons or hadron constituents. To obtain the equations of motion and associated boundary conditions, we require 5' to be stationary CHODOS, JAF FE, JOHNSON, THORN, AND %'EISSKOPF with respect to variations 5p of p which are arbi- trary inside of and on the surface of the bag. We also require 5' to be stationary with respect to independent variations of the position of the surface of the bag [6R(n, f)]. R will depend on two parameters a„a2, which vary over the surface, and time. This feature is, of course, also es- sential for relativistic invariance. Since there are no kinetic terms in (1.1) which involve the bag surface, the equations which result from this re- quirement will be equations of constraint which (implicitly) define the geometrical variables [R(n, t)] to be functions of the field degrees of freedom in the bag. Thus, this model of an ex- tended relativistic object is distinct from earlier ones" in that in those models the geometrical variables were also, in part, dynamical. The field equations and boundary conditions which re- sult from the variational principle are p =0 inside the bag, BR n p+n van=0 on the surface8 f &p' —&(Vp)'=8 on the surface, where ~ is the normal to the surface at any point. In Sec. III we discuss these in detail. We emphasize that (1.1) is just a prototype of our model. The hadron constituent fields which are confined in the bag can carry any spin or quantum number. In this paper we generally shall assume that the fields confined in a bag are "mass- less, " that is, we shall take the free Lagrangian for the fields to be the part which consists of the derivative terms. Thus, the only dimensional parameter will be B. In this class of models the fields contained in a bag need have no interaction terms in the Lagran- gian. Therefore, our model is capable of realiz- ing in a covariant context the free-parton sub- structure for hadrons. Indeed, this feature was instrumental in suggesting the model. It is intui- tively clear that a short-distance probe (q' » MB) when scattering from a constituent quantum in the bag will scatter from a free pointlike particle far from the walls, and hence Bjorken scaling of the scattering amplitude should result. Up to this point we have only a model of a single hadron, albeit, in all of its possible states. In order to have a theory of strong interactions we must provide for a local coupling among hadrons. We may visualize this interaction to be one which allows a bag to fission or different bags to co- alesce. This may be described classically in a local, causal way, if we couple two bags to form Q=I3+g Y+ 3C, (1.3) where I, and I'belong to the ordinary SU(3) and C belongs to the colored SU(3). The quarks have ord1nary Fermi stat1stles. We construct the mod- el so C =0 for all physical hadrons. In this way all physical states will belong to zero-triality ordinary SU(3) representations. To achieve this we allow the current qy" Cq to be coupled to a massless Abelian gauge field (gluon) with a small coupling constant. This will weakly break color symmetry and give small radiative corrections to the parton structure. W'e assume that the gluon field is hadronic, that is, it is confined in the bag with the colored quark fields. This will prevent the appearance of hadron states with C a 0. In Sec. VI we show that solutions of the dynamical equations exist only if C =—0. This is consistent with our fission interaction, for if a bag with C=0 a single bag (or allow a single bag to become two) at points on their respective surfaces. We then integrate over the point. To obtain the quantum amplitude we would then (for example) further "sum over histories. " Clearly, the constant B must be universal among bags for this interaction to make sense. It is needless to say that the cal- culation of this amplitude would be a formidable task. %'e make no apologies for this. The test of our mode1 will result from the class of predic- tions of quantitative results which can be obtained from it. In particular, asymptotic calculations of many sorts can be made in a fairly simple way as we shall show in detail in this paper. Because the bag model describes an extended hadron, it shares with other extended models such as the string a leading Regge trajectory which is infinitely rising. Furthermore, the bag yields an asymptotic density of states of the form p-e ~~0 (see Sec. II) in common with the string model (and other extended models with different internal di- mensions). Although we have suggested that the fields in a bag should be free in first approximation, at the next level we shall propose that they be coupled weakly. %e shall argue that such a weak coupling can account for the observed quantum numbers of the hadrons. The coupling should be weak enough so that the parton currents will show only small deviation from sealing by means of radiative cor- rections. We propose that the hadronie fieMs con- tained in the bag are "colored" quarks and gluons. ' The simplest model, which will have the correct quantum numbers for physical hadrons and approx- imately conserved color symmetry, is the Han- Nambu model with the integer-charge rule for the colored quarks, NE% EXTENDED MODEL. OF HADHONS begins to fission (Fig. 1) by means of our assumed hadronic interaction into two bags, mith C and -C, the bags will be connected with total flux lines C since the flux is confined in the bag. If me imagine, classically, that the tmo bags are con- nected by a neck of area A and length R, the gluon field energy in the neck will be proportional to (C/A)'AR = (C'R/A). This neck energy diverges if the bags try to recede from each other (R -~) or if the bags fission (A-O), that iS, the flux lines cannot be broken by the strong interaction. The gluon coupling constant need not be large to achieve this. Thus, the massless gluon field confined in the bag alloms for an intuitively simple, classiea1 may of understanding mhy C = 0 for all physical hadrons. ' An alternative scheme with color an exact sym- metry (and, therefore, with many fewer states than in the Han-Nambu model) can be based on a massless non-Abelian colored- gluon field confined in the bag. Here, the masslessness is also nec- essary to ensure renormalizability. ' In this mod- el color symmetry mould be exact, and a trivial generalization of the above argument to the non- Abelian case shows that it is hidden. However, in order that the non-Abelian gauge theory be con- sistent mith electromagnetic interactions, it would be necessary to replace (1.3) by Q=I, +-,'F. That is, in the theory with exact color symmetry the quarks mould have fractional charge. The remainder of the paper is organized as fol- lows. In Sec. II me examine the highly excited states of the bag from a semiclassical point of view. We are able to calculate very simply horn energy is shared between the bag and the fields in- side. %'e derive the level density, and discuss the large-quantum-number behavior of the Regge trajectories. In Sec. III me treat in detail both the classical and the quantum meehanies of a one- syace, one-time dimensional bag containing scalar fields. In Sec. IV we offer some illustrative solu- tions of the equations of motion, in both one and three space dimensions. %e turn in See. V to a discussion of the proper boundary conditions for Fermi fields, and we solve the quantum-mechani- cal fermion bag in two dimensions, analogous to the scalar solution of Sec. ID. The problem of interactions within the bag is briefly treated in Sec. VI, where we derive the boundary conditions for a system of colored quarks interact- ing with non-Abelian gauge fields, and we prove from these boundary conditions that colored had- rons cannot exist. In Sec. VII we outline the prob- lems and challenges which lie ahead. IMAM)mijs = ~ FIG. 1. A color-singlet bag attempting to fission into two bags which are not color singlets. The flux lines of the colored gluon field are shown explicitly. II. SEMICLASSICAL DESCRIPTION OF A HADRON AT HIGH EXCITATION Because of 'the slmpllclty of Gill' model 1't 18 pos- sible to draw a number of general, semiquantitative conclusions regarding the properties of the hadron. Some of these will be in the form of "virial theo- rems, "which relate the time averages of dynami- cal quantities. These are rigorous on the classi- cal level and probably remain so in the quantum theory, at least in the semiclassical limit when interpreted as relations between expectation val- ues. Many of our results are based upon a statistical treatment of the model at high excitation. In this "thermodynamic" limit we approximate the bag by a gas of free, massless particles —the quanta of the p field which we shall call "partons" —enclosed in a region 8 and subject to an external pressure B. The extent to which this approximation is valid will be discussed below. For the moment me note that relations derived as time averages from virial theorems are reproduced as ensemble averages in our thermodynamics. Our conclusions may be summarized as follows (the derivations will follow): (a) The fieid in the bag behaves on the average like a perfect relativistic gas; that is, the trace of the energy-momentum tensor associated with the field, when averaged over space and time, is zero: CHODOS, JA F F E, JOHNSON, THORN, AND WE ISSKOP F (b) The time-averaged volume of a bag is pro- portional to its energy: E =4B(V) . (c) The ground state and lowest excited states of the bag contain a few partons of average mo- mentum of order B'/ enclosed inavolume of order B ' ' [B.hast he di mensi on(le ngt h) 'with 5 =c =1, andenergiesare expressed as reciprocal lengths. ] (d) In the thermodynamic limit the bag has a fixed temperature, To, independent of its energy. T, is of order B ' '. This is equivalent to the fol- lowing statements: (d,) The average kinetic energy of the partons is of order To independent of the bag's energy E pro- vided the latter is larger than T,: E»T,. (d,) The asymptotic level density g(E) of the sys- tem is an exponential function of E: g-e '" (d, ) The number, N, of partons plus antipartons present in the hadron is proportional to its energy: N~E/T(). (e) If the classical dynamics is such that there is a maximum angular momentum of the hadron at a given total energy E, that maximum must be =km-'"E'", where k is a dimensionless constant determined by the detailed dynamics. If the classical limit (5 -0) exists, quantum corrections to this formula would be down by powers of E. If there is no clas- sical leading trajectory, a plausibility argument suggests that the leading trajectory might be (for large E) Z.„=I 'B -'"E' (e =I) . (f) The most likely angular momentum for large E is given by g ~ (B-1/4E)5/6 Several of these results are familiar phenomeno- logieal attributes of hadrons. In particular if B'/' is of the order of 3 GeV, (c) is familar from quark models and (d) is characteristic of statistical models. As we shall see, points (d) summarize the thermodynamics of a radiation fieM confined under constant pressure. Point (b) is as yet untested since no experimen- tal indication of the size of highly excited hadrons is available. According to (e) the highest Regge trajectory rises proportional to (M')"/' for large M, which, for n=2 is the usually assumed linear trajectory and for g =+, it is still compatible with present experimental evidence. (f) indicates that the angular momentum of the most frequent states increases with a smaller power of M, namely, 3f'/'. The 7'eneziano model comes to a similar result with J ~M.' To begin deriving these results we turn to sim- ple thermodynamic and statistical arguments. Points (a) and (b) which are virial theorems wiII be discussed subsequently. According to our mod- el the hadron is described by a field (or fields) and confined to a volume V. The boundary condi- tions on the fields ensure that they vanish outside V. Since we assume the fields confined in the bag to be quasifree and massless, it is natural to ap- proximate their properties by those of a confined relativistic gas of massless particles. The quan- tum excitations of the fields, the "partons, "cor- respond to the particles of the gas. As the basis of (1.1) the total energy of the gas is given by where E, is the internal energy of the gas ("radia- tion energy") and B is the constant defined in (1.1). The gas interacts at the boundary. Because the boundary conditions are nonlinear, this interac- tion allows an exchange of energy between the radiation (E„) and the bag (BV), and also allows for the transformation of parton energy into new partons (if bosons) or parton pairs (if fermions). This is demonstrated explicitly. in Sec. III for classical solutions in two dimensions. Thus the bag's surface serves as a means of establishing a thermal equilibrium in the gas. Since the field interacts at the boundary, we would expect the re- lativistic gas approximation to be valid only when the wavelength of the partons is much shorter than Vl/3 For the lowest excitation states we do not apply thermodynamics. For these the number of partons is lowland their wavelengths will be of the order V' '. Hence, E„=NV ' ' where N is a small in- teger. Minimization of (2.1) gives immediately V='B 3/ and, as a consequence, the estimates quoted in point (c). These results follow from di- mensional analysis (8 = c = 1) provided B is the only important dimensional parameter. In particu- lar, the zero-point energy of the fields in the bag has been left out. In Sec. III we show that (at least for the case of one spatial dimension) this zero- point energy is not fixed by the theory and decou- plea from the dynamics. Returning to the relativistic-gas approximation (and of necessity to states of relatively high ex- citation), we note that the term BV in (2.1) may be interpreted as the energy associated with an ex- ternal pressure I3. The system corresponds there- NEW EXTENDED MODEL OF HADRONS 3475 fore to a bubble of ideal, ' relativistic gas within an ideal liquid under constant pressure, B. Equi- librium will obtain only when the radiation pres- sure of the gas balances the pressure exerted by the liquid. In a gas of particles of negligible rest mass the pressure is p = &E„/V where E„ is the energy of the gas. Equilibrium then requires where = E/Te+ Se, 4 1/4S(E)=—— E +S3 3B (2.5) (2.6) p= eiE„/V =B, (2.2} E =E„+BV=4BV. (2.3} The assumptions made here are compatible with the virial theorems discussed at the end of this section, which state that and d xa —0 time average B&V&time evevele ~ Since the trace of the energy-momentum tensor associated with the field is proportional to (ep)' the first is equivalent to p = eE,/V and the second is just the time-average analog of our equilibrium (ensemble average) result (2.3). Continuing our approximation we estimate the entroyy of the bag by calculating that of a free massless gas enclosed in a container of volume V and with total energy E„=3BV. There are no de- grees of freedom associated with the walls of the bag (the bag coordinates are determined from the motion of the field), so there will be no added con- tribution from the walls. It is true that the bound- ary conditions place constraints on the fields, but these should have negligible effect at high excita- tions (i.e., many short-wavelength partons). Also if the confined field theory were interacting our approximations would be valid only for small val- ues of the coupling constants. We compute the entropy of a free massless gas in thermal equi- librium using the second law of thermodynamics dS= " +—dVT 'T and the familiar black-body law E„=AT V, (2.4) dS dE„3B (l+ e) where n =(gv'