STTAHK CONFINEMENT AMP THE WASK MODEL The first successful scheme of the nuclear shell structure to describe nucle• J. Kuti ar spins, magnetic moments, and various Central Research Institute for Physics, ß -decay spectra is that where the nucle• Budapest, Hungary ar energy levels are treated as due to filling-up of individual particle levels
1. THE STATIC QUARK MODEL for nucléons in a spherical box. It is assumed that the strong interaction of each nucleón with all the other nucléons The purpose of these lectures is to in the nucleus can be approximated as a discuss recent work on the quark model and roughly constant interaction potential its applications to hadron spectroscopy extending over the nuclear volume such and some high-energy phenomena. that the assemblage of nucléons forms a "self-consistent" box. It has become widely accepted that hadrons are composite objects with frac• Similarly, we shall assume that the tionally charged quark constituents. Hadrcn average field of force which confines the spectroscopy may be explained then in terms quarks may be represented by a scalar of the excitations of valence quarks inside square-well potential of radius R as composite hadrons. Perhaps even more strik• shown in Fig.1.1. The radius R is taken ingly, under powerful electron and neutrino to be 1 i'ermi, in accordance with our microscopes, the elementary quark degrees knowledge about the typical size of a of freedom (quark-partons) have been re• hadron. solved in deep inelastic lepton-nucleon scattering. V(r)
Nevertheless, quarks as ordinary quark 1 Mq elementary particles have never teen iso• mq lated frcm composite hadrons. This negative 1 * experimental result motivates the idea of R quark confinement and accordingly, quarks Fig.1.1. Scalar square-well are assumed tc be permanently bound inside potential to confine quarks strongly interacting particles. within a spherical box of radius R .
The final microscopic theory for de• 11 scribing this strange situation in hadron The agent which supports this self - -consistent" square-well potential Is not physics is not known yet. The path we shall specified yet. It may be due to some effect follow here is a recent attempt tc approach of the other constituents, or related to the problem of quark confinement from the the structure of the vacuum as a physical phenomenological side. medium. We shall return to this problem in the third lecture. 1.1. THE SINGLE PARTICLE MODEL With a non-relativistic description Jn a first approximation we shall of quarks inside the spherical box we would picture a strongly interacting particle as run immediately into trouble. In the non- a small domain of space which is occupied -relativistic quark model the nucleon's by quark and gluon ortits. It will be magnetic moment is determined by the free assumed, therefore, that there is an aver• quark magnetic moment jt = q/2mq where q age field of force which confines the is the quark charge and m^ its mass. In quark and gluon constituents to the inte• the standard, non-relativistic quark model rior of hadrcns. First, we shall describe the mass of up quarks and down quarks is the quark orbits in the spirit of the sin• taken as mq s- 300 MeV in order to fit the gle particle model from nuclear physics. proton's magnetic moment.
- 79 - However, the formula yu. = q/2m Is in the static quark model, where the word valid only for a relativistic free quark. "static" refers to the rigid spherical po• For the hour.d Dirac quark the result is tential4'' . In the forthcoming lectures we correct only in the non-relativistic limit shall see that the static potential may be
when mq » p , where p is the momentum replaced by some ful]y relativistic and of the bound quark. If we solve the l-S) Schrödinger equation for a particle of dynamical confinement mechanism mass niq in the square-well potential of
Fig. 1.1 in the Mq •*•
so that the momentum p is of the order Each eigenmode is characterized by some of MJC/H . Thus, p ~6C0 MeV and hence eigenfrequency £ , the square of the total the formula fl = q/2m cannot be used for angular momentum J , and the projection iHq ~ 300 MeV . Consequently, this standard Jz of the total angular momentum along, result of the non-relativistic quark model say, the z-axis. has no derivation for a physical proton. The angular momentum operator J is It would be valid only for a gigantic pro• the sum of the orbital angular momentum L, ton with a radius of several ferais. —> ->
Therefore, we shall describe the sta• L * f xf , tionary states of the quark orbits by the -» relativistic Dirac equation and the spin operator ~ J£ , [f/] , ft + fi + fl Vit) (1.1) so that we may write -. , ft where tt- and fl are the standard Dirac matrices : -» The components of J satisfy the usual " • U oj , I3' I. -ij commutation relations [i\ J1] - ;JJ . and 6 with k=l,2,3 stands for the 2x2 Pauli matrices. The momentum p* is • represented in Eq.(l.l) by -it V in coor• dinate representation. We shall set ti=c=l Besides, the square of the total angular here and later on, unless they are kept momentum commutes with each component, for pedagogical purposes. Our metric is
A 11 (+ ) with nÄp *= g^VV = P*- P* • [j2 , Jk] = 0 , k = 1,2,3 .
The scalar potential V is that of Fig. 1.1 in the Mq -»• <*> limit. The components of L and ¿ £ satisfy Thus, the mass of the quark is mq inside similar commutation relations. the spherical box (hadron.) , and it is m + M •* o» outside. The potential V(r) It is easy to show that the Hamilton implements our initial assumption that operator quarks with finite energy cannot exist outside hadrons, while they are freely H ' ¿ ? + /3 + fl]/<*) moving particles in the interior points. Hence the terminology : quark confinement commutes with J and J , - 80 - lar momentum operator are denoted by , [H , J2J = 0 , /i. * * í , where f*. is the projection of [H , Jj = o . the spin on the z-axis. Since the square ?2 of the spin operator Therefore, the operators H , J have common eigenfur.ctions ¿ a ? / 4 is a c-number, we have one eigenvalue equa• tion to determine u^* : where S , J(3+l), and M are the common Let us denote the spin variable of the eigenvalues of the corresponding operators. eigenfunction Uy«. by <¡t whose two values The four-component spinor wave func• are chosen as oC=± j . Since tion HJJM may be represented as T£iH (I.2; we find where %fn and #£JM are two-component spinors. Substituting (1.2) into the Dirac The eigenfunctions S^JM of the oper• ía •* i ;* equation we find ator J * L *j o are given by a.3) -> where n « f is a unit normal vector along The two-component spinors are eigenfunc- T , and the Cl*^-* are Clebsch-Gordan 2 coefficients. For a given J there are two tions of the operators J and Jz : independent ways of constructing the eigen• function from two different values of the orbital angular momentum : M = vn Eq.íl.3) may be solved in spherical polar •A coordinates rt Q *t by separation of the variables. The function VfjM defined in the space of the angular variables Ô , f and First, we shall find the angular de• the spinor variable J. is expanded in pendence of the two-component spinors Eq. (1.4) in terms of the orthonormal spinors and XfjM . This can be dene by Uyx where the quantities applying the addition rule of two angular momenta in quantum mechanics. The eigen- il functions of the orbital angular momentum are well-known from non-relativistic are the contravariant spinor components. quantum mechanics as the spherical har• The quantity Jl^/*?.) with transforms monics as a spinor V - Vi«, V£,M-1 The eigenfunctions of the spin angu• - 81 - Jljj^ are known as the spinor spherical harmonics. The Clebsch-Gordan coefficients are given in Table 1.1. Table 1.1. The Clebsch- + r) + -Gordan coefficients for •37- — V (e-^%-vc-rj) ¡ The spinor spherical harmonics SLJF^(N) determine the angular dependence of the spinors "ffejM and /¿JM .If ^£JM is expressed in terms of Jl}tM , then /i?M is given by Jlj^ , where g+f'.jj. Tnis -j^r ~ f 6<-yy + <£ - « j - Foy » o. (i.6~b) follows from the fact that /¿JM is pro• portional to ¿^f* ^TJM in Eq.íl.3J. Under spatial rotations 5*-p* behaves as • S-n , The function G(r) can be expressed and from Eq.(1.6a) as < ¿fio IC F where 0 if r<< where Jo * nnflJflJf is the element of solid M» ¿f r > K angle around the unit vector n . Both and XÎÏM are arbitrary functions of the radial variable r which was The solutions of Eq.il.8) can be omitted in the separation of the angular expressed in terms of Bessel functions. dependence. The general solution of the differential equation The complete four-component spinor wave function H*£j{M is given in polar coordinates by (1.5) YX) * Ct j< J„ i/3x; + Cjfx J.„ i/3x; IF) iW*; where , and . fir) and where 5„C*> is the Eessel function. Thus, g(v) are the radial wave functions. By we may write for the general solution of substituting (1.5) into Eq.il.3) we find Eq.il.8 ; after elementary calculations a coupled system of first order differential equa• F«v - '« (f 3^ . eA¡? lv_k w f (1. 9) tions for the radial functions : - 82 - are where the constant c is the same as in where c4 and cx arbitrary coeffi• cients and (1.11) . The asymptotic forms (1.11) and (.1.12) are valid for any value of r out• side the square-well potential when •*•«>. TP = It follows from Eqs.(l.ll) and (.1.12) that fC-c) - - G<*) (1.13) The eigenvalue spectrum can be calculated from the requirement of continuity for the if r>R and Mq -*• oo . The relation (1.13) solutions at r = R . remains valid if we approach the boundary r = R from outside, and it must be re• quired also for the interior solution on 1.3. BOUNDARY CONDITION AND EIGENVALUES the boundary by the condition of continuity. We shall calculate the eigenvalue The spectrum of a confined quark in spectrum of a confined quark inside the the Mq -*oo limit can be calculated then spherical potential wall in the M -» «o from the solution of the free Dirac equa - limit. Consider first the outside solution. tion where the boundary condition Since £ is some fixed eigenvalue, p iM^ if r >R and -> oo . Using the well- (I.13a) -known relation is required for H^JÍM in Eq.<1.5) at }W OX) r = R . The wave functions ifejgKi vanish outside the spherical wall. we may write Í1.9) as The boundary condition (1.13) can be written as for r > R , and Mq -> *> . (1.14) are the standard ones. Eq.(l.l4) is of gen• eral validity for arbitrary quark states inside the cavity because the stationary require a particular choice for the ratio states "VÍJÍM subject to (1.14) form a cj/c¿ in order to cancel the exponen• complete system inside the sphere (rfR) . tially growing terms in the exterior part The boundary condition (1.14^ guar• of the bound state solution as given by antees that there is no charge or other Eq.(1.10). Thus, the radial wave function quantum number lost through the potential F(r) exponentially decreases outside the wall at r = R . Indeed, the local charge spherical square-well potential : and current densities in the interior are cot) ~ c eM,Y" , tt.n) i^cx) -- "fit) i") , where c is some arbitrary constant. where we omitted the indices £ ï t M of the The function G(r) is calculable spinor wave function. If the quantum number asymptotically by substituting (1.11) into associated with the current is not to be Eq.(1.7) , lost through the surface of the sphere, then it is necessary that G«) a. -c « ' (Lia; - 83 - subject to the boundary condition £1.14). k F\ I NF I K 3 (x) . - o Prom the condition of regularity of the radial wave functions when r •» o it if r = R .It follows from the boundary follows that condition (1.14) that + 0 %>0 j f-t r Therefore, for both values of OC we may ¿ 3-k . fin? I = -ÏI - t . write Therefore, G(r) is calculated for K>0 from the YT î f <*> = o , 3¿ <•*) = <*) - ~ J, <•*) are consequences of (1.14) . which yields The stress tensor for the quark wave function which describes the momentum and energy flow inside the hadron is If % A" I*) = ~ L*> (•*) T il gives and The momentum and energy flow through the Therefore, for both values of "K we find potential wall is given by evaluating „zn^-pk* on the surface : a. i0 The constant c in fir; and g(r) is de• termined from the normalization condition The boundary condition of Eq.(1.14) was used in the derivation of ïïq.(l.lb). We 4 . have found before that 'F'J'm O on the sur• face, and hence its derivative points The eigenvalues ¿n for a given J and X along the normal, are determined from the boundary condition Él.14) which gives a transcendent equation for £n : Therefore, we find that where Pq is interpreted as the quark PARITY pressure exerted on the surface. The wave functions ^EITIA are eigen- let us calculate now the eigenstates functions of the Hamilton operator H , of a quark inside the sphere of radius R . the square of the total angular momentum •*2 The wave function +{}eM L , though the two-component spinors j£jgM - 84 - The result (1.21) shows that the intrinsic parity "¿^ of the quark is multiplied by a state-dependent factor X * C-*)* <1.22) are eigenfunctions of with eigen• values l(t*') and «'(«'+<) , which may be called the parity of the quark respectively ; state. The two states for a given J differ here l*}t{ and i'-Jïj . in their parity 1t according to Eq.(1.22). These remarks follow from the simple observation that the Hamilton op• 1.4. THE NUCLEON WAVE FUNCTION erator H does not commute with The nucleón is represented in our static quark model by a three-quark wave We conclude, therefore, that for a rel• function where each quark occupies the ativistic quark there is no meaning in lowest eigenmode of the spherical cavity separating the angular momentum into or• whose radius is R .We shall discuss now bital part and spin part. This separation some of the eigenfunctions and the solu• is meaningful only in the non-relativistic tions of the eigenvalue equations^ limit when X^JÍ'M is verv small in compar• ison with YtjiM • Then the square of the For massless quarks, with m^= o in orbital angular momentum, L = I (l*t) , is the Dirac equation, if J = 1/2 , either a good quantum number. In the relativistic case the two tic**-,) values of I label two different quantum J4rr e ' (1.23) states which are distinguished by their parity quantum number under spatial re• flections : or + < r —> - r . (1.20) (*) ¿V (1.24) The four-component spinor r\('f(i) trans• forms under the parity transformation (1.20) as where the normalization constant is where the intrinsic parity of the quark is independent of its quantum me- The spherical Bessel functions Cf ) are chanical state, and fya - and U^. is a two-component Pauli spinor. The formulae CI.23) and Í1.24) can be trivi• ally derived on the basis of the discussion in Subsections 1.2 and 1.3. Prom The linear boundary condition (1.14) yields an eigenvalue condition for the mode frequencies XTI x » and (- - 85 - tributes a term £Onvfl) to the energy of the where Y-n,* is related to the energy £rxtc nucleón. We shall return to the construc• of the eigenmode by tion of the nucleón wave function after C^ ' 1? introducing the quark field operator of the spherical cavity. One notes that Eq.(1.25)is a special case of (1.19) for mq= o , and 3=1/2 . By convention we choose positive (negative) n sequentially to label the positive (negative) roots of Eq.(1.25). The first few solutions to Eq.(1.25) are •K. = -1 2.04 ; x, 5.40 VI 2,-1 c 3.81 ; s 7.00 K = +1 l.l = 2.1 The eigenvalue condition (1.25) implies the relation which gives immediately the negative en• ergy eigenvalues. The negative energy Pig. 1.2. Eigenvalue x(iriqR) states are interpreted in the spirit of of the lowest quark mode with the Dirac equation as positive energy mass in a spherical cavity of radius R . antiquark solutions. The quark field operator q(r,t) can Eigenstates with higher total angular be defined in terms of the complete system momentum may be constructed in analogous of cavity eigenstates manner. Instead of this construction we shall give here the J = 1/2 states for massive quarks. The lowest quark state for J = 1/2 and can be written as where there is an infinite sum over integer values of n for each J , 1 , M ; n la• (1.26) bels the radial excitations of quarks for given angular momentum quantum numbers. *^WM stands for the spinor wave func• where the normalization constant is tion of the eigenmodes. Like in positron theory, we redefine ' 00 £(€-yn^) the quark annihilation operators ajijj(n) The eigenfrequency (or energy) n > o *J1M fa) where X» tf("ljty obeys the eigenvalue equation + ¿ n) n < o ajlM^ = JlM<-- (1.19) for J = 1/2 and : Thus, we may write the quark field oper• (1.27) ator as The smallest positive root of (1.27,) , x(mqR) , is shown in Pig.1.2. •n>o (1.28) Each occupied mode of mass m^ con• - 86 - where kj1MCn> annihilates a quark and and the lowest root x-L_1 = 2.04 of (1.25) djjjjCti,) creates an antiquark in the corre• are chosen. You may have noticed that I sponding eigenmode. The antiquark wave alternatingly substitute the value of I or functions ¿'«M ^) wH-h positive energy K. for the second label of the wave func• 8,.' =i„i for n >o are defined by tions and creation operators. with.all other anticommutators zero. The vacuum, or empty cavity, is defined as a state I o > such that - d4(t;u4c+;^aj - u4a;vt;d3 and a similar construction is valid for As the first interesting exercise in our neutron state. the static quark model with relativistic quarks we calculate now the static prop• One notes that the wave function (1.30) erties of the nucleón '»^. I trust that you is symmetric in the quark variables if the are familiar with the elements of SU(3) two up quarks and the down quark occupy the and SU(6) , though some group properties same lowest orbit in the nucleón. This is will be discussed in the second lecture. forbidden for two identical up quarks be• Here we need only a minimal knowledge to cause of the Permi statistics for half- treat the static nucleón with relativistic -integer particles. In the next lecture quark orbits. we shall introduce three colors for each quark which solves the problem then. How• Por simplicity, we take massless up ever, the results of the following discus• and down quarks. With R = 1 fermi the sion are not affected by this old paradox kinetic energy of the quark model. You may think about the nucleón simply as built of a red, yellow, and blue quark and hence the wave function of the confined three-quark state in the with the lowest mode occupied can be made lowest mode is about 1220 MeV not very far antisymmetric. from the average mass of 1180 MeV for the N(938) - A (1236) system. 1.5 STATIC PARAMETERS OP THE MJCLEON The quark state We calculate now the magnetic moment Clol (-«'o) l°> (1.29) and charge radius of the proton and neutron Í x and the axial-vector charge £ß -decay cou• is described by the wave function (1.23) pling constant) of the proton. The magnetic if moment operator is defined by jML A i rxf?«î f 3i) - 87 - where the quark operator q has an index origin of the magnetic moment in our pic• i = 1,2 for up quarks and down quarks, ture is completely different from that of respectively. A summation is understood the non-relativistic quark model. If it were over the omitted index i in Eq. (1.31.). not confined, the massless Dirac field would The charge matrix possess no magnetic moment (in Eq.(1.32) -» 00 , asR->oo), Confinement sets a Q = scale through the radius R of the spher• ical square-well potential of Fig.1.1 and in the internal Cu d) space is familiar a magnetic moment arises from the cross from the elements. terms between the upper and lower compo• nents of the wave function (1.23,) . Given the quark field operator (l.28) and the state vector (1.30,) it is straight- The mean square of the charge radius forward to calculate the value of /*• in for a quark eigenmode is defined by (l.3l) for the nucleón states. For the proton we find «*> - J clV «r*V;qf cf) j 0-33) = (1.32) r-9* Mu XtfàOiift'i) where Q is the quark charge and is the wave function from Eq.(1.23). The for• where R was fixed at the beginning to be mula (I.33) can be easily evaluated and 1 fermi. we obtain The result (1.32) is understood with• out fancy field theory. Let us calculate the magnetic moment of a single quark of For the proton Eq.(1.34) gives charge Q in the eigenmode of Eq. (1.23) <**>* = 0.73 fer*u- ; by using (1.31) as a quantum mechanical formula in terms of the wave function q(r) for the neutron of(1.23) . The result is '/V r\ « n *,/( (*,,.,-<) . The proton's charge radius is measured to be The rest is SU(6), since the spin-isospin h IJ MÙtfMlrM) (1.35) while the experimental value is gp = 2.79. where the matrix The agreement is not too bad for a first 4 rough guess. The neutron's magnetic moment f °1 can be calculated analogously and L 0 -* J acts in (u , d) space. The coupling constant gA contributes to the weak decay of the neutron, is obtained which is a famous result of H -* V + e + v£ the non-relativistic quark model. It is interesting to note that the which is interpreted in the quark model as - 88 - the quark decay- function (1.26). The result is a + e + v¿ 1? (2.1) inside the nucleón. The matrix element of the axial charge where A = mqR and x = x(mqR) is the operator in Eq.(1.35) can be evaluated for smallest positive root of Eq. (1.27) as shown the proton state vector (I.30) in a in Pig.1.2. JL stands for 1? X(*Ji) with 4 4 straightforward manner. Por massless quarks ^2 = J +x . The formula (2.1) goes over with the wave function (1.23) we obtain into the previous result for m^ = o . The axial-vector charge (^fl ) ft) -f ¿'t ^*CT)€ In the non-relativistic quark model (2.2) connects either non-strange quarks i=j, SA - 3 S=0 or one strange and one non-strange The result for relativistic quarks differs quark i=s, j=u, or d ; S=l . In Eq.(2.2) because the lower components in Eq.(l.23) ^¡It) denotes the quark wave function are important in their contribution to g^ of the i~th species in the lowest cavity eigenmode. We obtain a fairly complicated and have opposite spin orientation from expression for the axial vector coupling the upper components. constant : We have seen here that the static quark model with confined relativistic (Q \ ¿XjXj Q:-A¿ (2.3) quarks incorporates many of the successful features of the non-relativistic quark p model, and where it is different improve• The mean charge radius 4r > of Eq.(1.33) ments are made, as, for example, in the for a massive quark in the lowest cavity value of g^ eigenmode is i i d.[ix*c<-o+u +ia-33-1Mn+a-j ««- 3) QUARK EXCITATIONS 2. HADRON SPECTROSCOPY In order to display the rich spectrum of confined quarks inside the potential wall, we shall discuss now the Hamilton 2.1. RADIAL NUCLEON EXCITATIONS operator of the quantized field theory of Before I get to the discussion of quarks inside a sphere of radius R . The orbital excitations of confined quarks, Hamilton operator is defined by there is a technical point to add to the topics of the first lecture. The static U - f* • (2.5) parameters of the nucleón were calculated Í there for massless quarks. In later appli• cations some mass is certainly required in normal ordered form where the field for the strange quark so that we have to operator of the i^h quark species has been given in Eq.(1.28) ; i = 1,2,3 stands for calculate the single particle matrix ele- U) up, down, and strange quarks, respectively. ments of massive quarks for completeness ' , A summation is understood over the repeated The contribution of a single quark indices of the different quark types in of mass rriq in the lowest cavity eigen- Eq(2.5). The mass matrix m¿_j in (2.5) is mode to the magnetic moment is calculated diagonal : from ( 1.31) with the help of the wave - 89 - reasonable. 0 o As we shall see, three-quark baryons 0 O «*»3 are to be color singlets and are therefore to be constructed of quarks in totally Here m^ and m.2 are equal in the ab• symmetric spin-isospin-spatial states. Sev• sence of electromagnetic interaction. The eral spatial states (quantum modes) are mass of the strange quark mj, or ms in available in order of increasing energy : a different notation is always larger than mu=m]_ . lSi/2 with xx-i =2.04 , iPl/2 with x11 = 3.84 , The Hamilton operator can be written in terms of creation and annihilation 2Si/2 with *2-l 5.4 operators as etc. It is elementary group theory to find the totally symmetric ways to distribute up and down quarks with spin = 1/2 among + WHERE ^MJ^JJCIM these orbitals^*. Strange baryons are not considered yet. The resulting spectrum is given in Fig.2.1. Here,as you can see from is the occupation number operator of a the above list of the eigenvalues of differ• given eigenmode. Eq.(2.7) implies that ent orbitale,we take massless up and there is an infinite spectrum of quark down quarks, for simplicity. modes and an infinite variety of different occupations inside the sphere of radius R . N(i-) _ 1PÍ _A(H J.N(R2N(|-- We shall see a little later that the 2200 -1VVV physical hadron states must be colorless in the presence of quark-gluon coupling. Z000- Therefore, only those occupations of quark 1W and antiquark orbitals will be allowed in 2Ntt.).N(R '«ÉM.4H the potential well which have zero tri- 1*00 - ality quantum number, such as qqq or qq , 1 etc. 1C00 - •«tt-)N(t-)- •1*4 \\ Another practical restriction on our first investigation of the baryon spectrum VtOO - in the spherical potential well is that we do not consider quark orbits with total is angular momentum J>l/2 . The reason is 1200 L N(H that a quark with JI 3/2 exerts a non- Fig.2.1. Low-lying three-quark spherical pressure on the surface and it nonstrange baryon states with 3 £3/2 in the static quark model. is expected that the corresponding state Nucléons C1=1/2) are in the left column, A'a ¿1=3/2) in the becomes deformed from the spherical sym• right. metry when we go beyond the static poten• tial approximation. The lowest states, (lSi/2)"^, are to to be identified with the NÉ938) and The lowest quark eigenmodes with A(l236). One notes from Fig.2.1 that the J = 1/2 and their radial excitations with symmetry scheme of the static quark model the same angular momentum correspond to with relativistic quark orbits is not a spherically symmetric pressure where the the SU(6) of the non-relativistic quark static potential picture seems to be more model. For example, the low-lying negative - 90 - 3* S parity multiplet does not include a fined to the interior of the sphere under state since each quark is in a J = 1/2 these conditions for the vacuum against the state. Baryons with three quark modes oc• vector gluon field. cupied and Ji 5/2 are states in which the surface is probably not dominantly The repulsion of the gluon field from spherical. There may arise a non-spherical the outside phase of the vacuum can be pressure on the surface even for baryon understood as an exercise from electro• states with J ¿3/2 in Fig.2.1 if P-states statics. Consider a point charge Q em• are excited and there is some interference bedded in a semi-infinite dielectric between different quark orbitals. a distance d away from a plane interface which separates the first medium from an• We expect the energies in Fig. 2.1 other semi-infinite dielectric £t . The to be shifted and the degeneracies removed surface is taken as the plane z=o , as when the quark-gluon interaction is incor• shown in Fig.2.3. porated in the model. PHYSICAL VACUUM 2.2 CONFINED GLUONS Quarks are not the only hadron con• stituents in our picture. They are coupled MAGE to massless vector particles, or gluons, CHARGE which are the mediators of important inter• actions between quarks. Gluons must be confined inside hadrons which have been represented so far as a spherical square- -well potential with infinite walls for Fig.2.3. The solid lines with arrows show the electric flux quarks. lines of a charge Q embedded in a semi-infinite dielectric^. Gluons as vector particles are de• scribed similarly to the photon by the We have to find the solution to the Maxwell equations. We shall assume now equations of electrostatics that inside the sphere of radius R there 47T is a vacuum phase (or hadron phase) with (2.a; dielectric constant 6=1 and magnetic o fl permeability = 1 . However, outside the and potential well the physical vacuum acts o for gluons as a strange medium with £ = o 7*f everywhere. The boundary conditions at z=o and f*. = eo (see Fig.2.2). are given by (2.9) £-1 1 £-0 A"1 The solution to the problem is given •r by the method of image charges. The elec- trie field E is derivable from a poten• tial <)> which is given for z > o by the formula . Fig.2.2. The vacuum in two phases against vector gluons 2> is characterized by some 1'T4U* *J, °. (2.10) dielectric constant & and magnetic permeability . The image charge Q* in (2.10) is located at the symmetrical position with respect We wish to show that gluons become con• to Q as shown in Fig.2.3. For z - 91 - potential is equivalent to that of a they always drag along. charge Q1' at the position of the actual Let us turn now to the description of charge Q confined gluons inside a sphere of radius •R R . Here we shall follow a different pro• cedure as compared with the description of The boundary conditions (2.9) determine the quark orbitals in the first lecture. the image charges Q ' and Q1' : Instead of starting out with quantum me• in!* a , chanical gluon orbitals and constructing (2.11) the quantum field from these states, we 2£* a shall study first the classical gluon field. The semiinfinite slab of dielectric It is characterized by the gluon elec- corresponds in our terminology to trie field E and gluon magnetic field B the "hadron phase" of the vacuum inside which satisfy the Maxwell equations C¿*f**i) the potential wall, if £, » I is chosen. V 2 = o The semiinfinite slab of dielectric €z (.2.12) may be associated with the "outside phase" when vector D vanishes in the left half-space Gluon field confinement is implemented by and it becomes tangential to the surface the boundary conditions in the "hadron phase" (see Pig.2.3) . This picture follows from Eq.fë.ll). n • E n x 3 = o (2.13) There is a similar situation in the at r = R , in accordance with our two- presence of magnetic field. Since the -phase picture of the vacuum. The first permeability ju of the "outside phase" is two equations of (2.12) are satisfied auto• infinite, the magnetic field H becomes matically if the field strengths are ex• perpendicular to the surface in the pressed in terms of the gluon four-potential "hadron phase". This is understandable if A^ff) as we note that the houndary condition on the -> -> —> —> 3X Vf 3 = Vx fl magnetic field H at the surface of a E - very high-permeability material is the There is some freedom in our choice of the same as for the electric field at the sur• four-potential A^. as expressed by the face of a conductor. gauge transformation -> The magnetic field H cannot pene• A 31 trate into the "outside phase", similarly The second pair of equations in (2.12) to the induction vector D in electro• are equivalent to the wave equation! statics. Consequently, no wave propagation is supported in the "outside phase" ! • A = o , a? = o , n = — - V Indeed, the reflection coefficient if the Lorentz condition R on the surface between the two phases, ~* —» VA 1Ï for electromagnetic waves of the gluon 3t is imposed on A field propagating in the "hadron phase" , is given by The gluon field energy is given by the well-known expression When £4-»0 there is total reflection on H. if JV(*'.*}, (2.14) the surface. The gluon field becomes con• fined to the "hadron phase" . Charged point and the angular momentum of the field is particles are also confined to the "hadron phase" because of the gluon gauge field •r * x (£*3) (2.15) - 92 The components of the stress-energy tensor between the charge Q and the surface is are repulsive implicating a positive electric pressure on the surface. The situation is opposite in the mag- (2.16) netostatic case where the electric charge Q is replaced by a fictive magnetic mo• Too - - ¿ C ( nopole g . Due to the highly polarizable where the expressions magnetic property of the outside phase, a magnetic surface charge polarization appears, which has the opposite sign of g . There• describe the energy and momentum densities fore, an attractive force arises between of the gluon field. the monopole g and the- surface explaining the negative sign of the magnetic pressure It follows from the boundary condi• in (2.17) . tions (2.13) that there is no energy flow across the surface : CAVITY EIGEOTODES KO R - O . There are two different types of spherical waves which satisfy the Maxwell The momentum flow through the surface is equations (2.12) inside a cavity of radius given by evaluating fi^T'1* with the help R . The transverse magnetic (TM) , or elec• of Eq.(2.l6) and the boundary conditions tric, fields are given in multipole form by (2.13) : 'CI <*; • < Jl [ £ u The negative sign of the magnetic field pressure in (2.17) requires some where Oi = k . The radial function gj(ta) explanation. The repulsive force exerted is defined by by the gluon electric field on the walls of the spherical potential well can be (2.20) understood from the simple model of the dielectric slab of Fig.2.3. There is a and J^(x) is the Bessel function. The polarization surface charge density on the vector spherical harmonics in (2.19a,b) plane at z=o between the two semiinfinite follow the standard definitions : slabs given by ( HM^' Y** <*>, (2.21a; o„ i = —: (2.18) V,M^V°< IX £ C£,+€I) 3 H " 4 density ^j»ol . When £4"0 and £a-»o , the polarization charge in (2.18) is of the <•%<*>)*••>h same sign as Q . Therefore, the force - 93 and while the square of the total angular mo• mentum J in(2.15,)is J(J+1) as expected. There is an infinite sequence of modes Since the vector spherical harmonics for given angular momentum values with YjjU in (2.21a) is transverse to r , increasing numbers of radial nodes. The we find that the magnetic field 3 is parity in TM mode is transverse in the electric, or transverse magnetic, multipole eigenmode : V = ( --))* • The discussion is similar for the -* transverse electric TE , or magnetic, The electric field E in the TM multipole modes. The fields in multipole form are eigenmode is not transverse, however. Indeed, ¿f^^rity in (2.19a) can be written as where the first term is longitudinal along the radial vector r while the second term is transverse to v . Eqs. (2.19a,b) change to the form ¿2.26a,b) Since the boundary condition (2.13) under the substitution E -> -iB, B -> iE , for B requires it to be longitudinal, which corresponds to the invariance of the and according to (2.22) B is transverse Maxwell equations (2.12) with respect to in the TM mode, we find the eigenvalue the transformations B -* E , E -* -B . equation from (2.19b) , -+OT) + The electric field (fjt) is trans• îj - O (2.24) + The frequency of the lowest TM mode in 3 fa+4&V %.t CkH)'0 (2.21) units of the inverse of the cavity radius is whose lowest eigenfrequency in units of the T inverse cavity radius is xQ(TM) = 4.49 , for J = 1~ (2.25) P + xo(TE) = 2.74 , for J = 1 . (2.28) where P is the parity of the eigenmode. It describes the state-dependent change The parity of the TE eigenmodes is ?• <-')>+*. of sign of the field strengths under spa• tial reflections. The normalization in Eq.(2.19a,b) is QUANTIZATION chosen in such a way that the electro• Similarly to the field strengths, the magnetic field energy (2.14) is ¿O y gluon four-potential A/A. can be expanded into spherical eigenmodes. In radiation - 94 - gauge ( AQ = 0^ the expansion is gluon orbitals inside the potential well. The spectroscopic implications^ of + A Table 2.1. The quantum numbers The Hamilton operator can be calcu• of the four quarks are given here : isospin, third component lated from (2.14) using the expansion of of isospin, charge, and charm. Eq.(2.29) : J. C2.3i; quark «3Ml species I Y Q c Similarly, the third component of the 1 Î 1 I 0 3 angular momentum J is calculated from u 1 Í 3 Eq.(2.15) to be 1 4 T 0 d Í 3 "5 _ I / O O 0 s 3 " 3 T I Introducing the occupation number operator 0 •f c O 1 3 /t we may write Eqs.(2.31) and (2.32) as The quark eigenmode inside the spher• ical cavity is labelled then with a new index «C = 1,2,3,4 with reference to the quark species which may occupy it. The suffix «(• is known as the flavor index of 3* - H *H (AW*! +l) , the quark. Antiquarks carry the same quantum where the eigenvalues of the operator A&JÏMJ numbers as quarks, but with opposite signs, are positive integers and zero. The zero of course. eigenvalue corresponds to the zero-point The same mass m^ = ma is assigned to fluctuations of the gluon field. the non-strange and charmless up and down In the expansion ¿2.29) for the vec• quarks. In what follows here we shall assume tor potential the summation over "X in• mu = o , for simplicity. The charmed quark cludes the TE and TM modes. We are not is supposed to be heavy with a mass between concerned here with the problem of longi• 1 and 2 GeV , while the mass of the strange tudinal and scalar gluon modes which quark is about 3oo MeV . The mass parameter appear only in intermediate states in is defined here as the value of m in the q perturbation theory, and not as physical free Dirac equation (l.l) . - 95 - We shall assume that each flavor where occurs in three different colors with a suffix i = 1,2,3 . The twelve different U - ZXf íl i Ü i 1 k'* J quark field operators are a straightfor• ward generalization of the expansion in The standard Gell-Mann matrices ClK Eq.(1.28) : act in the internal color space and « n.HM formation U, . The commutation relations of the %n matrices are well-known : Tahle 2.2 is a short summary of the twelve different quark fields : Here f.-v are the structure constants of Table 2.2 color SHO) • As a reminder( we give here up down strange charmed two A matrices, red %* K Q O A o o O -i O O 4 o yellow % %3 \* o o O o o -i blue %3i %z ?33 The color generators of SUO) c are defined by Each quark with given flavor, say the up quark, comes in three colors with the same particle properties except the color quan• tum numbers. The third component of the color isospin is The field theoretical Hamiltonian r = r, is defined with the help of the quark -3 2 fields as while the color hypercharge is defined by .= Í VU'' i-'M"*^* C2.34, The eigenvalues of the color generators F3 and Fg unambiguously characterize the where a summation is understood for the colored quark states of the triplet repre• same indices. A normal ordering is under• sentation in Table 2.3 . stood in the Hamilton operator (2.34) to eliminate the zero-point energy of the Table 2.3 empty cavity (vacuum) . Therefore, H 1 ( 3tn m 1 < 0 ~i. 0 V2 i* ¡oí Z "Ä Ï The Hamiltonian of Eq.(2.34) is in• variant under the transformations of the à 1 2 1 1 Z iP8 color group of SU(3) • Quarks are trans• ÏÏ Tf {3 ~Í3 ~(3 (3 formed as members of the triplet represen• tation of color SUG) : The eigenvalues of Ij and Y for colored quark states are given by the diagonal ele• ments of the matrices i 3 and J, g , respectively. The antiquarks are charac- - 96 - terized by complementary colors. The vector gluon fields are designated by Ay* where the first index i refers The flavor and color properties of to color. Gluons belong to an octet rep• gluons were not specified yet. Gluons are resentation of the color gauge group SU(3), assumed to be flavor singlets under the and the eight colors are labelled by CA) , flavor group of SU and they belong i = 1,2,...,8 . These vector particles are to the octet representation of color SU(3), flavor singlets under the hadronic symmetry There are eight colored gluon fields group SU(4) . A^ír.t; , i = 1,2 8 The action W inside the sphere of radius R is invariant under the color labelled by the color index i gauge group SU<3) , and we shall write for it 2.4. QUANTUM CHROMODYKÂMICS IN A CAVITY Many theorists believe today in quan• tum chromodynamics (q.c.d.) as the micro• (2.35) scopic dynamics of quarks and gluons -3 ii*' A «y.} providing a key understanding of hadron structure. This field theoretical model is where the flavor and color indices of the very elegant and simple in its formulation. quark fields are not written out explicitly. It became popular among many theorists for Terms without some indices are understood the following reasons : to be summed over those omitted indices (1) it explains scaling and the parton here, and later on. picture in deep inelastic lepton- nucleon scattering in terms of asymp• The non-Abelian field strength tensor totic freedom ; in Eq.(2.35) is given by (2) it allows for a spatially symmetric ground state of the nucleón in the terminology of the static quark model; (3) it gives a total cross section of The structure constants of the color gauge electron-positron annihilation into group SU 13) are denoted by f. , and i jk hadrons which is three times larger g is the small, fundamental quark-gluon than without color, in approximate coupling constant in the action W . The agreement with the data ; eight Gell-Mann matrices A; act in the (4) there is a hope that only color sin• internal color space of quarks. glet states are observable in the theory with a possible explanation Apart from the mass term with the di• for quark confinement. agonal mass matrix The last point is only a conjecture and m = there are some doubts about its validity. In our phenomenological approach quark and gluon confinement are provided as the the action W in (2.35) is invariant under initial assumption of the model. the hadronic symmetry group SU (4). We shall designate the quark fields The boundary condition for the quark by where the first index i=l,2,3 fields is given on the surface of the refers to the triplet representation of sphere by color SU(3) . Quarks belong to the lowest representation of the flavor group SU(4) (2.36) as the approximate hadronic symmetry and the second index << refers to this group. The boundary condition for the gauge fields is - 97 - magnetic fields, Y\ • 11 - O (2.42a) vx* - ( o , -n ) , (2.37) «n x 3; » o where n = — is the unit normal to the (2.42b) surface. so that the normal components of the color -* The field equations can be derived electric fields E. , and the tangential from the action principle + ;fl r 0 ¿-i ^R^)n Î^ V-I V .(2.39j field strength tensor Fiju.n , respectively. In Eq.(2.38) we have introduced the gauge It follows from Eq.(2.42a) that there is no color electric flux through the sur• covariant derivative face of the hadron (potential wall) . As a consequence of Gauss's theorem, the total color charges for i = 1,2,...,8 There are eight conserved color cur- x must vanish in the model for an extended ; < ) in the theory, and they hadron with closed boundary : are given by Consequently, only color singlet states are allowed inside the potential wall ! The two terms in Eq.(2.40) are the con• tributions to the color currents from quarks and gluons, respectively. 2.5. THE BARYON AND MESON SPECTRUM In our perturbative calculations the • We shall calculate now the spectrum confined gluon fields will be treated in of light baryons and mesons in the lowest the zeroth order of the quark-gluon cou• order of the quark-gluon coupling. Consider pling constant g . In this approximation, a hadron with static, spherical boundary ignoring the self coupling of the gauge (R = 1 fermi) whose interior is populated fields, the field equations (2.38) and with quark orbitals in color singlet state. (2.39) become identical with those of eight The quark content of the lowest baryon independent Maxwell fields coupled to color states is qqq , while for mesons qq . The charged matter. hadron states are classified by the rep• The eight color generators of the resentations of the flavor group which we model are determined from the color cur^ take as SUO) for baryons and mesons with• rents as out charmed quarks .0 J »* (2.41) In lowest order of «*c = the gluon exchange graphs are shown in Pig.2.4. They are constants of motion and measure Since the quarks remain in the lowest eigen• the color charges of the hadron with col• mode in Pig.2.4a , the color current at the ored quark and gluon constituents. vertices is time-independent. Consequently, The boundary condition (2.37) for the only the static part of the gluon propagator gauge fields may be written in a more contributes in Pig.2.4a. The gluon propagator familiar form in terms of the electric and is defined as in ordinary field theory : - 98 where \o> is the empty cavity ( vacuum) where j"/^ is the color current of the k^*1 in free field theory . The gluon fields quark orbital : are confined, subject to the boundary con• ditions (2.42a,b) . (it) + (*)-* Clt) (k) ¡i Here ¡^K.^) is the scalar color magnetization density of a quark of mass m^ in the lowest eigenstate. The integral n (a) (b) /*• ( WK i ) " I Pig.2.4- Gluon interaction gives the color magnetic moment (2.1) of diagrams for a baryon in the quark orbital. lowest order of < (b) gluon self-energy. k to determine B.'i ' : To lowest order in <¿e the non- Abelian gluon self-coupling does not con• tribute and the gluons act as eight inde• pendent Abelian fields without self-inter• action. It is like having eight "photons" where /*C*iK,r) is the integral of er') as the mediators of quark interactions to a radius r , and inside the hadron. c à*' .tkt r at' ~ In the self-energy diagrams of •r r> Pig.2.4b the intermediate quark may be in Since the color magnetic field B,-(k) any cavity mode. After renormalization, is radial in Eq.(2.45) it satisfies the the diagram gives a Lamb shift type con• boundary condition (2.42b) automatically. tribution to the splitting of hadron energy levels. We do not calculate this The expression (2.45) for B(- may be self-energy diagram here with some hope substituted into the color magnetostatic that its contribution does not change the interaction energy (2.43/" character of the spectrum significantly. f AF„'-^ ZL 3;t?)ic (?) , (2.46) STATIC QUARK-GLUON INTERACTION ENERGY JSfkete »»< k>i , ' ' We shall calculate now the contri• where B.^ is the color magnetic field bution of Pig.2.4a to hadron energies. generated by the ktn quark of the hadron. The color magnetostatic interaction energy After straightforward calculation of (2.46) may be written as we find AEH - - { fí \ B, C?) 3. (?); (2.43) (2.47; where the magnetic fields Bi are deter• mined from the quark current distributions •R3 by Maxwell's equations and the boundary where conditions (2.42b) . Therefore, we write (2.44a) l 1 -2X*xe si*xK iiV xi + j xkH (*xK) + i*¿Sit**t) V • 3; = O (2.44b) - 99 - K« is ne with y^x^-sinXjj.cos xk ; xk is the root of Here Í V * color charge density of Eq.(1.27) for a given mkR and a quark of mass mK in the lowest eigenmode with o The color and spin dependence of We obtain the color electric field by the Eq.(2.47) can be simplified by the follow• application of Gauss's law ing observations. For a color singlet meson (k, (2.52) Squaring íl; + 3t- and using to a radius r (2.48) Eq. (2.52) can be used in the expression we find (2.49) for the color electrostatic interaction The trick is similar for baryons energy. The sum over the color index i can be performed using Eqs.(2.48), (2.49) , k and (2.50), with the following result whence (2.53) (2.50) The final expression for the magnetic Here \ = 1 for a baryon, 2 for a meson. interaction energy is The function f (x^x-^) is given by the integral H (2.51) 1 o * K*HkK, *a/«; = k>Z£ i Mk¿-(¿^ ?e; where Here % = 1 for a baryon, 2 for a meson. l ¿[^-(WJfKj/XK ]--m [s;njfKcoíXK-(ííM xK)/cKJ We turn now to the calculation of the gluon electrostatic energy. We shall ap• with x(rnR) given by Eq.(1.27), and proximate in Fig.2.4a the static gluon propagator with the free one in the electro• (2.54) static case. This probably overestimate somewhat the color electrostatic inter• HADRON MASSES action energy. We are ready to evaluate the hadron The color electric field of a quark masses now by writing down the expression satisfies the equations of electrostatics, for the total energy of the system inside the square-well potential of Fig.1.1. The quarks contribute their rest and -.Oik) kinetic energies to the hadron's mass : where ¿¡¿ «y is a sigle quark s color charge density llt) Here L -, and ïf are the respectivr e u, d. s numbers of the nonstrange and strange - 100 - quarks, and the eigenfrequency is defined With m = m. = o , and ni = 3oo MeV we in Eq.(2.54) . U Q S find fu4 = 0.3 , fu4 * fu.S - fss • The color magnetic interaction energy can he written in the form r E/w, Muj +aufHui-»-a«Mff (2.56) Table 2.4. Parameters which specify the gluon magneto• by evaluating „ -» static energy of light hadrons in each state. In Eq.(2.56/) Mud is the a a Hadron ud ss color magnetic interaction between two us nonstrange quarks, is that between V -3 0 0 a nonstrange and strange quark and M 0 0 S s A -3 is the interaction energy between two Z 1 -4 0 ~^ strange quarks. The values of , 0 -4 1 A 0 0 Mus , and Msg are shown in Pig.2.5. 3 1 2 0 r—« 0 2 1 if 0 0 3 2 0 0 K* 0 2 0 Ol 2 0 0 4 0 0 2 or -6 0 0 K 0 -6 0 The mass of a hadron of radius R is then given by E¿ Pig.2.5. Magnetic gluon exchange M - f ^ + £TM + (2.58) energy of two quarks as a func• tion of mR . The solid line gives the interaction energy between where the individual terms are given hy equal-mass quarks (Mud or WiSB) , Eqs.(2.55)-(2.57) . The various mass split• the dashed line is the interaction energy between a massless quark tings follow a simple pattern. If <¿c were and a quark of mass m (Mus) • zero, baryons would be heavier than mesons since baryons have three quarks and mesons The coefficients a , , a „ , a have two. The A and the proton would be ucl us ss degenerate as would be the § and JC . in (2.56) are state-dependent and tabu• lated in Table 2.4. When we turn on the color magnetic The color electrostatic energy (2.53) interaction the A and the proton are split a f will be written in the form since ud=3 or the A and -3 for the proton. The $ and W remain degenerate, 2 57 but they are split from the ÍÍ . The ratio £ £ = ¿uJ • lud. + Li |uj *• Ist fíf j í- ) of the magnetic interaction 1 : -3 for 10 where f -, , f, , and f have analogous the (J; ) and 3f follows from the fact ud ' us ' ss ° meaning as the magnetic terms above. The that the quarks are in a triplet state 10 (6Vft"0 in the ^J, ) and are in a singlet coefficients bud , bug , and bsg are 6~, • 6"x)"-3 ÎC . also state-dependent. Por the proton, for state ( in the example The binding from the color electro• V = 3 ' bus = bss = 0 • static interaction energy is larger for - 101 - mesons which further splits them from model. We face the same difficulty, of baryons, though this splitting is not large. course. If the ^ and ^' are treated on the same footing as the ^> and to then The SU (3)symmetry breaking is intro• one state will be duced by the mass splitting of the strange and nonstrange quarks. If the only effect 1 = SS of this SU (3) breaking were in the quark and the other mass-kinetic term E^ , the £ and A 7> . I-Í-UÜ+dl; would remain degenerate. However, the presence of the strange quark's mass modi• in which case the ^' will be degenerate fies the wave function of the strange with the 3T (as the u> is with the f) quark and therefore causes a secondary Experimentally the is very massive SU<3) breaking through the gluon magneto- (958 MeV). static interaction. This splits the £ and A in the right direction. For mu=o There is, however, an annihilation this can be seen from Fig.2.5 where M process, shown in Fig.2.7, which will Lud is larger than M If they were equal, raise the and lower the *l without us there were no splitting. significant changes in the other states. Since the gluons are invariant under flavor The resulting spectrum of light SU (3), The diagrams in Pig.2.7 couple only baryons and mesons is shown in Fig.2.6 to the SU<3) -singlet components of the states 7 and ' . Because the gluons are vectors, the diagrams vanish for the vector meson states. They act only on the */ and y' . The annihilation diagrams are diagonal in the SU<3) octet and singlet states which correspond to the quark states ^=R (UÜ 4-cU -ZSS ) t J= (UU + CLI + SS) There is no coupling to the octet state and we shall assume phenomenologically a large singlet matrix element which will mix the states ss and uü+d3 . Applying the standard procedure to describe the mixing, we introduce the mass matrix r + Í — M = » s n «3 ft J, . BARYONS ME50N5 where a is a number proportional to c The octet energy E0 and the singlet ener- gy E,j are calculable from Eq.^.58). Fig.2.6. The hadron masses are shown for m =m.=o , m =3oo MeV , THE "2 -"l' PROBLEM It is well-known that the 7 and 7' Fig.2.7. Lowest order gluon anni• have special problems in the ordinary quark hilation diagrams for spin-zero flavor SUC3) singlet mesons. - 102 - We can no.w diagonalize M. an compute 3. THE QUARK BAG MODEL and it^i . The parameter a is deter• mined to fit M^i = 958 MeV.. The mass is calculated then. This is shown in 3.1. INTRODUCTION Pig.2.6. Motivated by recent field theoretical It remains an open question whether investigations, we shall assume that the the explicit calculation of the diagrams physical vacuum is characterized by some in Pig.2.7 would confirm the phenomenolog- microscopic structure which in "normal ical analysis above. This brings up the phase", outside hadrons, cannot support the most disturbing point about the spectrum propagation of quark and gluon fields. The in Fig. 2.6 : the large value of <¿c which vacuum acts like a strange medium against was required to account for the fairly hadronic constituent fields, though Lorentz large mass splittings due to spin inter• invariance will be maintained. actions Now, by concentration of energy, a small MAGNETIC MOMENTS domain of a different phase may be created in the "medium" of the physical vacuum. It We shall briefly discuss the predic• is like boiling the vacuum and creating small tions for the magnetic moments of baryons. bubbles with a characteristic size of 1 fermi. The calculation is rather straightforward, Inside the bubble (hadron phase_) quark and since the magnetic moment of a single gluon fields can propagate in the ordinary quark orbital was already given in Eq^2.lj . manner. The procedure is the same what we fol• lowed for the proton and neutron in the We. picture the hadron then as a small first lecture. The results are given in domain in the new.phase with quark and gluon Table 2.5. constituents. This is the. bag. The boundary surface of the bag between the two phases Table 2.5. Baryon magnetic moments is impermeable against the vector gluon in units of f^p . The predictions of the static quark model with rel• fields, therefore they cannot penetrate into ativistic auarks are for mu=md=o", the normal phase of the vacuum. The imper• and ms=2R0*MeV. The SU Q) predic• tions' depend on the P/D ratio for meability of the surface is expressed in the the magnetic moment operator. The form of boundary conditions for the gluon figures in parentheses correspond to a conventional choice. fields? The gluon electric fields E. i=l,2, ...,8 in an octet of eight colors are tan• MAGNETIC MOMENT Lj^jf-r) gential whereas the gluon magnetic fields B^ are normal to the surface in the in• Hadron Experiment Quark mod SU O) stantaneous rest frame of the surface ele• 2 ment. Consequently, there is no gluon field W -0.685 " 3 energy flux through the surface of the hadron A -0.24 -0.26 domain in space. 0.93 0.97 1 (D The dynamics of the quark and gluon r 0. 31 -h <*> fields inside the bag is governed locally by the field equations of quantum chromo- -0.53 -0.36 -1-c (-1) z~ dynamics (q.c.d.) . Gluons are confined -0.56 o (-f ) inside the hadron phase and quarks become also confined because of the gluon gauge -0.69Í0.27 -0.23 -1-c (-1) fields they always drag along. - 103 - the physical vacuum in gauge theories, though We shall consider the quark hag model as a step forward from the static quark it did not escape our attention that the above discussed physical picture may be re• model of the previous lectures. In the ía; forthcoming applications we have to discuss lated to instantons in q.c.d. , or to some those properties of the bag model which other vacuum phenomena (merons ?) . distinguish it from a static square-well t* ) potential. Apart from the pleasing feature According to a recent suggestion ' , of the fully relativistic formulation, the physical vacuum in quantum chromody- perhaps the most important new element in namics may exist in two different phases. the bag model is that hadrons are de• Both phases are characterized by pseudo- scribed as deformable droplets whose particles with one-half topological charge shapes are determined dynamically. merons The name meron comes from the Greek root fxêfoC meaning part or frac• Now we come to the important question tion. In the presence of separated quarks why the bag as a small droplet of hadron the meron gas in plasma phase which con• phase is stable against the internal fines quarks may find it favorable to be pressure of the quark and gluon constit• in a dielectric phase at a cost of some uents. We assume that to create a vacuum energy per unit volume and unit surface. bubble in hadron phase takes an amount of The quarks would be almost free particles energy B per unit volume, and an amount inside this region, thus producing an effec• g of energy per unit surface. The con• tive "bag" with fairly sharp boundary (see stant B may be associated with the vac• Pig.3.1) . uum pressure exerted on the bubble : the energy required to increase the volume of the bubble by an amount The pressure exerted by the gluon fields on the boundary of a hadron is balanced then by volume energy B per unit volume and surface energy 6* per Pig.3.1. Separated quarks are unit surface. The quark bag model is the connected by color flux lines invention of the ingenious MIT group 2~*) in hadron phase which may be associated with some dielectric who have introduced volume tension to phase of merons. The outside stabilize hadrons. Later the surface en• "normal" phase is a quark con• fining plasma phase for merons. ergy (f per unit surface was introduced by a group of us upon dynamical and phys• ical considerations 3.2. LIQUID DROP MODEL AND SINGLE PARTICLE The boundary of the bag is trans• SPECTRA parent against leptons and the mediators Our step from the static square-well of electromagnetic and weak interactions. potential to the bag model is somewhat These particles (or fields) may propagate analogous to an interesting example from in both phases of the vacuum in the normal the history of nuclear physics. manner. The two phase picture of the vac• uum in the bag model is a strong inter• A heavy nucleus is best described in action phenomenon. telegraphic style as a bag of nucléons. In a more sophisticated manner, the close We do not attempt to derive B or packing of the nucléons in the nucleus and 6" from some microscopic structure of the existence of a relatively sharp nuclear - 104 - boundary have led to the comparison of the centrifugal pressure exerted by the particles nucleus with a liquid drop. The empirical on the walls of the nucleus may lead to a binding energies can be interpreted then considerable deformation. The quadrupole as a sum of surface energy, volume energy, moments associated with those deformations and electrostatic energy of the nuclear are in accordance with observations. droplet. The treatment of the nucleus as a deformable droplet is quite successful An instructive scheme to demonstrate in the theory of nuclear fission. this idea is that where the nuclear energy levels are treated as due to a filling-up According to the liquid drop model, of individual particle levels for nucléons the fundamental modes of nuclear excit• in a spherical box with infinite walls. ations correspond to collective types of It is assumed here that the strong inter• motion, such as surface oscillations and action of each nucleón with all other elastic vibrations. nucléons can be approximated in the nucleus as a roughly constant interaction potential New progress in the theory of nuclear over the nuclear volume so that the nucléons spectra was obtained through the develop• form a "self-consistent" box (or bag) . ment of the so-called single particle model This model assumes that nuclear stationary This bag is deformable, and there are states, like electron configurations in dynamical degrees of freedom associated atoms, can be approximately described in with the surface deformations. For odd A terms of the motion of the individual par• nuclei great success is obtained by asso• ticles in an average field of force. ciating the spin and magnetic moment of the nucleus with the odd valence nucleón alone The single particle model explains the outside the core. The valence nucleón is stability of certain nuclei, those which coupled to the core through the collective possess closed shells of protons and variables of the droplet. This coupling neutrons. The model is also successful in arises from the boundary condition for the accounting for the spins of nuclear ground valence nucleon's wave function at the sur• states and nuclear magnetic moments. face of the nucleus. The liquid drop model and the single The description of odd A nuclei in particle model represent opposite ap• terms of the dynamical variables of the va• proaches to the problem of nuclear struc• lence nucléons outside the core, together ture. Each refers to essential aspects of with the collective variables of the nucle• nuclear structure, and it is expected that ar droplet may be called the bag model of a synthesis is necessary for a detailed the nucleus, in our terminology. The model H) description of nuclear properties. is quite successful in describing the single particle excitation spectra of odd A nu• The necessity of this synthesis is clei. clearly indicated by the observed behav• ior of nuclear quadrupole moments. Though The analogy with the quark bag model the quadrupole moments of nuclear ground is almost self-explanatory. The valence states give definite evidence of the shell quarks of hadrons are similar to the va• structure, for many nuclei, the magnitude lence nucléons of nuclei, and the dynamical of the quadrupole moments is too large in variables of the bag (hadron droplet) are comparison with the predictions of the analogous to the collective variables of single particle model. This suggest that the nuclear droplet. The volume energy of the eauilibrium shape of those nuclei the nucleus is like the volume energy BV deviates from the spherical symmetry. in the quark bag model. In both models, the surface energy is an essential part of A simple explanation arises if one the droplet's dynamics. Similarly to the considers the motion of the individual nuclear deformations in high angular momen- particles in a deformatle nucleus. The - 105 - tum states, we expect cigar-like bag shapes out by the surface of the bag in Minkowski for high angular momentum excitations of space-time. The strength of surface tension hadrons. is set by the constant ff with dimension energy/area, or length . The last inte• If the quark bag model turns out to gral in Eq.O.l) is proportional to the be a successful phenomenological device, four-dimensional volume swept out by the and that is a big if, then it becomes im• bag as a whole in Minkowski space-time. portant to understand whether the collec• Por a given instant of time it is propor• tive variables of the bag are associated tional to the three-dimensional volume of with the internal microscopic dynamics of the bag. The constant B has the dimen• the constituents, or they are related to sion of energy/volume, or length-^ . This the microscopic structure of the physical term may be interpreted as vacuum pressure vacuum. against the bubble. The action W in (3.1) is Lorentz 3.3. ACTION PRINCIPLE. BAGGED Q.C.D. invariant, since it is defined in a geo• metrical manner. Following the considerations of the introduction to this lecture, we want to Prom the variation of the quark and find the action W for the bag which is gluon fields in the interior points of the pictured as a bubble in hadron phase , bag we can derive the field equations (2.38) embedded in the field non-supporting phys• and (2.39) as a consequence of the action ical vacuum in a Lorentz invariant manner. principle The relativistic action W for the (3.2b) bag is written as so that the normal components of the color electric fields E^ vanish on the boundary of the bag. In Eqs.(3.2a,b) n is a unit normal vector to the surface of the bag SURF and nQ is the surface velocity along n , Therefore, the tangential components of the The first part on the right-hand side of color magnetic fields B^ vanish on the Eq. Í3.1) is recognized as the action of surface of the bag in the instantaneous q.c.d. (see Eq.(2.35)) as restricted to the rest frame of the surface element. interior points of the bag. Hence the ter• minology : bagged quantum chromodynamics. It follows from Eq.í3.2a) that there is no color electric flux through the sur• The second integral in (3.1) describes face of the bag. Consequently, the eight 2 the surface part of the action where 4 5 color generators , i=l,2,...,8 , must is the area of the surface element, and vanish in the bag model for an extended VL is the velocity of the surface ele• hadron with closed boundary (Gauss's theo• ment along the normal vector of the sur• rem ) face in a given point. In other words, the second integral in (3.1) is the three- dimensional area of the hypersurface swept 4« - 106 - /"V The color currents ii <•*) are defined in Eq. (3.5) has a transparent physical Eq.(2.40) . interpretation. It describes the balance of The covariant generalization of the forces in pquilibrium. The first term on static boundary conditions (2.36) for the the right-hand side of (3.5) describes the quark fields is straightforward : surface tension of the boundary between the two phases of the vacuum like for an air bubble in liquid phase. The second term B acts like vacuum pressure against the where tf * C*\»tn) was defined before. bubble. Variation of the surface variables The left-hand side of (3.5) is the in (3.1) leads to the equation of motion gluon field pressure on the surface of the for the boundary of the extended hadron : bag. It is balanced out in equilibrium by surface tension and vacuum pressure. - i F- F- 26 % + 2 (3.3; We did not include the quark pressure on the left-hand side of Eq.(3.3) . It can where % is the mean curvature of the be shown that the quark pressure is present surface in Minkowski space. in (3.3) in an indirect way when the quark- gluon coupling constant is not zero. Quarks The mean curvature X. is best de• are coupled to the gauge fields which exert scribed in curvilinear coordinates î M . a pressure on the surface whose «C-»o limit The surface of the bag is given by J1 = is equivalent to using the quark boundary The metric of the JH system is condition (2.36) and the quark pressure a 3X% ¿X* "R^ defined in Subsection 1.3. «/*>» * ¿3* jjw We picture the classical bag equations where X denote the rectilinear and in the following manner : orthogonal coordinates. With these defini• tions we may write for 1¿ field equations gauge field of q.c.d. boundary (3.4; conditions ««4. CZ-31)) where is the Jacobian of the coordinate trans• formation. One can see by inspection of Eq.(3.4) that K depends on the velocity and acceleration of the corresponding sur• face point. bubble dynamics Eq.(3.3) governs the dynamics of the = ¿CK +3 bubble under surface tension with outward gluon field pressure and inward vacuum pressure. For a static surface there is great simplification in (3.4) and we obtain The bag as a dynamical system requires a rather complicated description in terms of several dynamical variables, even if only a few quark and gluon constituents are present and the collective variables where l/fy and 1/R* are the principal of the bag's surface are represented by curvatures of the static surface in a given a minimal number of dynamical degrees of point. freedom. A great simplification arises if - 107 - inside the dynamical system of the bag we may identify a slowly moving subsystem whose motion is instantaneously followed by the rest. This separation of the system into two parts is the working hypothesis of the adiabatic bag dynamics. gives the stationary states *ßi (H,r) for fixed values of the coordinates R of the heavy particles. The index n stands for 3.4. NUCLEON in ADIABATIC APPROXIMATION the quantum numbers of the stationary states ; the energies and the wave In order to understand the basic idea v functions Bn<"R/'**) depend upon R as of the adiabatic approximation method, let upon a set of parameters. us consider a molecule which consists of a given number of electrons with mass m „ Assuming that the solutions of (3.6) and of atomic nuclei with mass M . The are known, the eigenfunctions of the com• Hamiltonian can be written in the form plete Schrödinger equation where can be written as is the operator for the kinetic energy of 11 the electrons (light particles) , and "\> i*,')- Zk™)^' ,*) (3.8) K 4-J * ÏH *í since the functions Hn t*,*) form a com• is the kinetic energy of the nuclei (heavy plete system for given R . particles) . The electron coordinates with respect to the center of mass are denoted Substituting (3.8) into (3.7) , after by r , and R stands for the relative coor• multiplication by "f,£ ¿ -R,*) and integrating dinates of the nuclei. V(r,R) is the poten• over the coordinates r , we find the fol• tial energy of the interaction. lowing system of equations for k^C*) : BORN-OPPENHEIMER APPROXIMATION In molecular physics due to the large ratio M/m of nuclear mass to electron Here the operator is defined "by mass the nuclear periods are much longer than the electronic periods. It is then a good approximation to regard the nuclei as (3.10) fixed calculating the electronic motion. In the second step the nuclear motion can _ pff^tt.VK*^*,«-) . be. calculated under the assumption that the The system of differential equations electrons have their steady motion for each ¿3.9) is exact. In adiaratio approximation instantaneous arrangament of the nuclei. the right-hand side of (3.9) is set to zero This is the Born-Oppenheimer approximation, and the system of differential equations for or adiabatic approach. 4>^CÄ) decouples into independent equa• tions Mathematically it is based upon the hypothesis that the operator l^n for the f + £~,«>"W° <*)• E° 4° kinetic energy of the heavy particles can be treated as a small perturbation. Thus, for each stationary state -m of the light in the zeroth order approximation the Schrödinger equation particles. One notes from C3.ll) that the - 108 - colored gauge fields may be neglected in motion of the heavy particles is governed zeroth approximation. Their effects can be by the potential energy £nCH) , which is included perturbatively. the energy of the light particles (elec• trons) for fixed position of the heavy par• We shall apply the adiabatic approx• ticles ( nuclei ) . imation scheme for the description of the bag's shape in quantum mechanics with light Thus, in adiabatic approximation, the quarks inside. The collective (surface) wave function ^CH,*) reduces to the simple variables of the bag are regarded as the product slowly moving part of the system. Thus, we ^wii = ^"ßiv C*i+) (3.12) shall conjecture the following correspon• dence with molecular physics : so that for each stationary state m of the electrons there are several states of nuclei - surface variables motion of the heavy nuclei which are dis• tinguished by the quantum numbers i> electrons - light quarks Using perturbation theory it can be As a first step, the bag equations shown that the condition for the appli• have to be solved for a fixed static sur^ cability of the adiabatic approximation face, so that the energy levels of the reduces to the inequality confined quarks become analogous to the energy levels £«(*) of the Schrödinger \ - 109 - in spherical approximation. is represented by i* Introducing the canonical momentum t .I A t - -* jf« - ^ which is conjugate to the surface The Hamiltonian in Eq.(3.l6) acts upon %>(f) variable £ , AT as a nonlocal integral operator and the 1* > stationary Schrödinger equation 3 the Hamiltonian of the surface in spher• i ) e [lT'+KI*S"jt+ ^3f -i'£ determines the quantized bag in spherical approximation. The potential energy ¿CF) is taken from (3.18) . The wave function The quantum mechanical commutation rela• K^S) vanishes at F'O and "thus %x is tion Hermitian in the j> E EO) interval. The mathematical definition of the is valid between the canonically conjugate square root operator variables j* and ^ In adiabatic approximation the wave requires some elaboration. First, we choose function of the surface and three quarks a complete orthonormal set of wave func• may be written as a product (see (3.12)) , tions from the solution of the eigenvalue equation + V £ [- |! V]t^' ^V (3.20) where the spinor wave function Af «•) is the eigenfunction of the free Dirac equa• for the square of the operator, where tion inside a spherical surface of radius § under the constraint of the boundary \JC$) - HX'e'-J'I . condition (1.14) at r = f . In Eq. Í3.17) The boundary condition *ßt diagonal in the above defined basis ( (3.IB) for the surface Hamiltonian (3.16) if the it can be written as a nonlocal integral three massless quarks of the nucleón occupy operator the lowest eigenmode inside the sphere of radius £ . The adiabatic quark wave func• If+/¿*vy \1¡> -> I-tyKw^, tions in (3.17) are taken from (.1.23) • In coordinate representation the sur• face of the bag is described by the wave The integral equation 0.22) can be function <|> tj>; , and the kinetic operator solved by numerical methods. The first few - 110 - eigenvalues of Eq. (3.22) are given by As we can see from Fig.3.3 , the sur• face of the bag is blurred quantum mechani• cally, though the probability distribution is sharply peaked around the classical ^ « SM 3.23 radius of the static bag. This observation serves as a basis for the static bag approx for B = 0 . The corresponding wave func• imation where the kinetic energy of the tions are shown in Fig.3.2. The energy surface is neglected and a sharp boundary levels appear as the radial excitations of is assumed approximately. the surface ( compare with vibrations of From the minimum of the static energy nuclei in molecules) . with pure volume tension, * we can determine the mass of the nucleón in the static bag approximation as it was done first by the MIT group three years 3) ago. The parameter B is chosen in (3.24) to fit the average mass of the NC938) - A (1236) system: i CtrSp (3 x4).<) = -MÍO HeV whence B*^ = 120 MeV . This value of the vacuum pressure sets the hadronic scale and Fig.3.2. The wave functions are the static radius of the nucleón from the shown for the radial excitations of the surface when the three minimum of (3.24) is found to be 1.4 fermi massless quarks of the nucleón with a mean charge square radius of occupy the lowest orbital in our adiabatic approximation. P Fig.3.3 shows the probability distri• bution for finding the bag in gro-and state The radii of the different hadrons are with a radius between ^ and f +• 4. CHARMONIUM. STRING. GIUONIUM 4.1. CHARMONIUM The charmonium bound state may be treated in adiabatic approximation as a quark molecule whose charmed quarks corre• spond to the slowly moving heavy particles of the system. In close analogy with the Fig.3.3 hydrogen molecule we may conjecture the correspondence s gies of the bag, is protons - heavy charmed quarks, electrons - light quarks, massless Therefore, the charmonium bound state is gluons, collective bag described by the Hamiltonian variables, Coulomb 1Î* potential - instantaneous cc-inter- H - *¿ +¿ £4.1) action. where the kinetic energy of the heavy The adiabatic approximation is de• charmed quarks is added to the rest masses, fined first in terms of the simplified and the potential energy V(r) . problem where the dynamical degrees of freedom are the non-Abelian gauge fields In a further approximation we shall fly-. , the ordinary, light quark fields ignore the transverse gluon and light quark fiK."^ » "the collective variables of the degrees of freedom together with the kinetic bag and a pair of static sources of the quantum energy of the surface which is treated classically. gauge fields at positions r-^ and rg The static sources consist of a pair of color spins represented by the ¿ 5¿ matri• THE STATIC BAG ces of color 3U<3) with the interaction The static non-Abelian gluon field between the static quark sources may be re• placed by an effective Abelian field fyt i ables include the gauge fields, light 3 CMrf-l £ * owl 35 o quarks, color spins, and the collective ¿iv 3 variables of the bag are given by ' (4.2) «u £ 3 1«' «"-v-J inside the static bag with the effective where r^ and r2 are parameters in the quark-gluon coupling constant g . The equa• Hamiltonian, similarly to the variables R tion of motion for the bag's surface is a in the Schrödinger equation £3.6) . static relation here between the gluon electric field pressure and surface tension In adiatatic approximation at any plus vacuum pressure : instant of time when the heavy quarks are 1 located at r^ and ?2 , the other dy• i "* *• • i \ -3 namical degrees of freedom are described (4.3) by the state \j¿(<,j?i)> , that is the bag with the light variables can instantaneous• where 1^ and 1/Rg are the principal ly readjust itself to the slowly moving curvatures in two orthogonal directions at sources of the charmed quark-ar.tiquark a given point of the static surface. The pair and remains in ground state. electric field E is subject to the bound• ary condition In this molecular approximation at -n £ every instant of time the energy stored (4.4) in the gauge fields and light quarks, where n is a normal vector to the static together with the surface and volume ener• surface. - 112 - In order to be able to calculate the is an exact solution to the static bag equations if the flux of the homogeneous potential energy V*(r) in IA.1) , we have —» to solve the static bag equations (4.2) - color electric field E longitudinal in• - (4.4) . Eq.(4.2) together with the bound• side the tube is determined by the color ary condition may be solved for any bag charge g . The tube is of cylindrical shape by computer. Eq.£4.3) determines geometry (see Fig.4.2) . then the static equilibrium shape of the bag with fixed sources inside. In the numerical calculations the strength of the quark-gluon coupling was set at the value Fig.4.2 Ï_£_ -= o.Z (4.5) 4JT while Indeed, if the longitudinal color elec• ZZO Me/ tric field E forms a homogeneous vortex 6" * (4.6) tube of radius R with cylindrical surface, the Maxwell equations are satisfied inside was chosen for surface tension. For sim• —* plicity, B = 0 was taken. The numbers the tube trivially. Since E is tangential (4.5) and (4.6) satisfy several require• at the surface, Eq.(4.4) is also trivial. ments in charmonium physics which will be It follows from (4.3) that discussed a little later. i (4.8) fi The shape of the bag from the comput• is valid on the cylindrical surface. From er calculation is shown in Fig.4.1 for Gauss's law, different values of the distance between the static sources'.3'' We estimate the error j = \t\n*r in the calculation to be within ten per• the radius R of the vortex tube can be cent. expressed using (4.8) . We get then Eq.(4.7) The static energy stored in the color electric field on a length r , is proportional to the length with the proportionality factor (4.9) which is the field energy per unit length. 1 2 3 A The surface energy per unit length is Fig.4.1. The first quadrant of the static bag for different '/s values of the distance between (4.10) the static sources. The arrow indicates the radius of an exact vortex tube between the quarks at infinite separation. A similar vortex tube develops between two point-like color charges at large sepa• ration. Only at the ends of the cigar-like COLOR ELECTRIC VORTEX TUBE bag are the field E and the shape of the bag modified in comparison with the exact, We shall prove now that an infinite infinite vortex tube solution of Fig.4.2. color electric vortex tube with radius R, The potential energy V(r) of the cc- (4.7) -pair is shown for the numerical solution v .Z7r* ys -* M which follows from Eqs. (4.9) and (4.10) is well approximated by the numerical solution. The slope of surface energy is twice of the slope for the gluon field energy in Pig.4.3. The radius of the ideal vortex tube can be calculated from Eq.(4.7) with the previously fixed values of g /kit and G : This value of R is shown in Pig.4.1 indicating the rapid convergence of the bag shape to the exact vortex tube. The vortex energy \ - ys * V Pig.4^3. The potential energy of per unit length may be identified for long the cc-pair as a function of the hadrons with a string-like tension between separation r . The straight line is the surface energy. the quark and antiquark. Por high angular momentum states of the elongated hadron, the slope of the Regge trajectory is re• The potential energy V(r) is well lated to % asymptotically as ^ approximated by o( It is remarkable that % = 1 GeV/fermi from at small distances where a dominant the charmonium fit is very close to the -2 Coulomb interaction energy is expected universal Regge slope at'to) =0.9 GeV in between the color charges. At large dis• the string model. tances V(r) is proportional to the dis• tance r , CHARMONIUM SPECTRUM The potential energy of Pig.4.3 can be used in the Schrödinger equation and the proportionality factor A is given by the exact vortex tube solution, ~j£ V* * BÍ (4.11) to calculate the charmonium spectrum. Here r stands for the relative three-vector With the parameters of Eqs.(4.5) and of the cc-pair and m is the reduced mass (4.6) the numerical value of A is of the charmed quark. An additive constant is not determined in the potential V(rJ 1 GeV/fermi , well-tailored for charm- from Pig.4.3 which is well approximated onium calculations. The color electric by vortex tube solution with linearly rising a* i potential energy sets in already at rather - -± - + \r . (4.14) 114 - tu with a classical radius R = 0.5 fermi for Some authors7 have used a similar the vortex tube. This is not a very large potential to V(r) of Eq.i4.14) . We are energy on the scale of charmonium physics, close to their results in our calculation'3'' so that the notion of an effective potential of the charmonium spectrum if V Cr) , and the adiabatic picture in general, •m s i Pi Kogut and Susskind have considered S.4.4) . Mass(G.v) 17 the effects of the light quarks. As the length of the vortex tube between the color charges and the field energy increases, it becomes energetically favorable at a certain 15 point to lower the energy by creating ordi• nary qq-pairs inside the tube-like bag : the long range force can be screened by 3.3 producing a color neutralizing light quark near each heavy quark. Therefore, as r -» the energy of a cc-pair surrounded by screening cloud becomes the sum of the 3.1 masses of two charmed mesons. The bag under• goes fission and disintegrates into a pair of D and D mesons (see Pig.4.5) . 1 1+- r- 2** , o* t»9 spft Pig.4.4. The charmonium spectrum in the quark bag model in adiab• atic approximation. The spin-dependence of the cc inter• action energy remains an unsolved problem in the quark bag model. The force between Pig.4.5 a color charge and the bag surface is repulsive classically. A color magnetic moment, however, exerts an attracive force on the surface which may explain the 4.2. STRING-LIKE EXCITATIONS OP HADRONS rather strong spin splitting of the ^ and - 115 - and baryon trajectories. field is consistent with the linear bound• ary conditions in Eqs.(3.2a,b) . If from (4.16) and B± from (4.17) are substituted into Eq.44.15) , the DIQUARK cross section A can be expressed as a function of the orbital velocity v : (4.18) where we have used j MESON BARYON Pig.4.6. Elongated bag rotating »-< about the center of mass with an angular frequency ¿o . for a color triplet state. The cross section A in (4.18) shows the expected Lorentz contraction. Vie shall construct now the solution The total mass M of the rotating which yields maximal angular momentum for bag consists of three terms, fixed mass. Accordingly, let us consider a bag with elongated shape rotating about (4.19) the center of mass with an angular fre• quency CO . It is assumed that the end where E_ _q is the quark energy, E^ the points move with light velocity. gluon field energy, and B V is the volume A given point inside the bag, at a energy. Surface tension is set to zero here, distance x" from the axis of rotation, for simplicity. moves with a velocity The length of the stretched bag ( L ) X x is determined , for a given value of the V = wx - — total angular momentum J , by the condi• where L is the length of the cigar-like tion bag. In this calculation the bag surface â m _ will be given by balancing the gluon field TT ' - (4.20) pressure against B , the confining vac• uum pressure : This is the condition that the angular momentum be a maximum for fixed M .. The different pieces of the total mass The color electric field E; is M in (4.19) are the following. The volume determined by Gauss's law, energy is IBV = 3-l\¿V ñcvj = blÍJ JL ¡AV[¡^ * ft - J Ï (4.16) where A is the cross section of the bag and the energy of the colored flux lines at a point where the mean field strength is associated with the color charge 3 ï ^' is E^ . The color magnetic field associated with the rotation of the color electric field is given by Since we assume that the valence quark wave functions are localized near the ends, the £ - 7* ÎL (4.17) total quark energy E^ is -> at a point moving with velocity v . This (4.21) - 116 for a convective quark momentum p at the ends. The total angular momentum J = Jq+J.^ Is the sum of the quark and gluon field where <^l0> reduces to (4.23) with C * | angular momenta. The angular momentum of in color triplet representation. the color fields is We shall discuas now some approxima• tions of the above calculations. The formu• la (4.21) for the total quark energy as• sumes that the energy ~ i/fK associated whereas the total quark angular momentum with the confinement of quarks may be ig• is given by nored in comparison to the total energy of the system. In this approximation the quark energy With the calculated ingredients we E = 2p is zero for the following reason. may express the total mass M as a func• 1 tion of L , Por a given angular momentum J the quark energy E^ can be expressed with the help £4.22) of the relation J = pi + Jf as for fixed total angular momentum J . Prom Eqs. (4.20) and (4.22) we find an where L is given by asymptotically linear trajectory 4 3 - oi'o; M ¡ (4.26) with the slope The quark energy Eq in (4.25? is zero for the value of L from (4.26) . Therefore, ignoring corrections of the order of 1//Ä , the total energy and The parameters B and °£c have been angular momentum of the elongated bag is determined from the hadron spectrum to be associated with the color flux lines. The about B^*= 140 MeV and o(c«¿ . With quarks at the ends serve to terminate the these values in (4.23) we find color flux, otherwise they do not affect the dynamics. d'(o) = 0.93 (W)'* This picture of the leading Regge in very good agreement with the experimen• trajectory is similar to the dual string tally determined slope which is about model. However, the bag dynamics is associ• ¿O) = 0.9 CeV~2 . **f ated with the color flux lines and the geometrical variables of the string merely The slope ^<») in (4.23) has been serve to parametrize the motion of these calculated for a state in which the color• fields. Further, since the cross section A ed objects at the two ends belong to the in (4.18) is independent of J , the elon• 3 or 3 representation of color SU(3). gated bag as a string is "fat", with the If the expectation value of the Casimir transverse dimension of ordinary hadron operator j ground states. 1*1 The calculation presented here is is C in a given representation, the expected to be valid for an asymptotic formula for the slope may be written as trajectory with large angular momenta. The value of J should be of the order of five - 117 - spherical pressure on the surface of the or so. In that case gluonic hadron. The energies of gluon states with non-zero angular momentum are less reliable in spherical approximation. and the string energy dominates over quark energies at the ends. The hope is that the asymptotic calculation remains sensible Table 4.1. Lowest-lying gluonium even at lower values of the angular mo• states with zero angular momentum. The last two states with J=l and mentum. J = 2 are not well described in spherical approximation. They are included for later reference. 4.3. GLUONIUM occupied modes J mass MeV One of the most spectacular predic• TE TE 0++ 1096 tions of the quark bag model is the exis• + tence of quarkless gluonic hadrons*K They TE TM 0~ 1446 are constructed in close analogy with the TE TE TE 0+" 1644 quark orbitals of ordinary hadrons. Gluons TM TM 0++ 1796 populate then the eigenmodes of a spheric• al bag of radius 1? TE TE TM 1 1994 TE TE 2++ 1096 To lowest order in °¿e the gluon self-coupling may be ignored, and the colored vector gluon fields behave as Abelian vector fields. They satisfy The high angular momentum states of Maxwell's equations inside the bag subject glueballs are elongated string-like objects to the boundary conditions (3.2a,b) . on straight line Regge trajectories. This The gluon orbitals were carefully description is motivated by the discussion discussed in the second lecture where TE before, in Subsection 4.2. and TM eigenmodes were introduced with One notes that the slope of the Regge definite angular momentum quantum numbers. trajectories associated with "fat" gluon When these gluon eigenmodes become popu• strings is given by lated with valence gluons they must form overall color singlets. Color singlet states of two gluons i -t may be constructed with - 118 - The spherical cavity state 2 in spherical gluon pressure on the walls. 2 TE mode whose mass was given in Table 4.1 ++ If we accept the trajectory structure at 1096 MeV may become the 2 particle on the pomeron trajectory when the hadron for all-glue bags where the pomeron has deformation from the non-spherical pressure intercept one and its daughters are integer is included in the calculations. spaced below, the mass of the vector glue- ball should be around wo, a I.3M with the i One finds motivation for the existence slope di'f~tC*)*0.eM~. of all-glue hadron states outside the quark UJ bag model, too. Freund and Nambu predicted Recently Robson has suggested that the this new form of hadronic matter in the vector glueball might already have been string model They identified the pomeron found in a DESY-Prascati experiment at and its daughters with closed quarkless 1.11 GeV. He argues that the main decay strings whose slope is one-half of the properties of the recent DESY-Frascati ordinary slope^ c^^n * I «C1. resonance are certainly consistent with a vector glueball if the basic assumption of The leading pomeron trajectory and Freund and Nambu is accepted : the only some daughters of the closed string are poles needed in the Zweig violation cal• shown in Fig.4.7 where the intercept of culations are the (V , oo , ^ t «uU "ty , as the pomeron is put to one arbitrarily. shown in Pig.4.8 for some sequential pole model diagrams. GLUEBALL PHENOMENOLOGY In the experiment the reaction One way to produce quarkless states o V * f> in any of the above mentioned models is to annihilate quarks with antiquarks. Some was studied by the interference between the authorshave explored the possibility Compton and Bethe-Heitler processes and a that when such annihilations occur, as in resonance-like structure of mass 1110 MeV Zweig-rule violating meson decays, the was found. The observed width was consis• intermediate quarkless resonances (glue- tent with the experimental resolution of balls) dominate the dynamics. Unfortunate• 30 MeV and it was parametrized as a reso• ly, the discussions remain on the level of nance of width 20 MeV, and a gross form of phenomenology. 3.12. ' if I ' U Ceif* Nevertheless, we hope that a first dt |t-o ' ) qualitative insight into the problem may a factor of twenty smaller than for co - be useful and stimulating for an experi• - production. mental search, and it may lead to the We shall follow Robson's argument to construction of a genuine theory of all- get some insight into this kind of phenom• -glue hadron states. enology even if this resonance-like struc• Let us consider first the vector ture disappears for some reason in the future. glueball 0v with quantum numbers J=l . In the string model 0y is the lowest If one assumes the SU(4) broken cou• physical state on the first daughter of plings of the 0v to the known vector the pomeron trajectory. Its mass may be mesons, estimated from Pig.4.7 where the parallel and equidistant trajectories give a mass to rj (4.28) -mo, « l.kGtV , if ci'f.„ lo)* is taken. The lowest lying 1 state has a and i*fl„ = l.ll GeV , fo„ =20 MeV are mass of about 2 GeV in the static model accepted for input, then of Table A.l which may be modified in the future due to deformations by the non- r(it ' i.c lut - 119 - is obtained from the sequential pole as the second daughter are expected in the model of Pig.4.8, only a factor of two mass region of 1-2 GeV. Their phenomenology above experiment. This is taken by Robson is very interesting and may be found else• as an encouragement for ignoring the where . contribution of higher mass vector glue- balls in the sequential pole model. Their inclusion would not be an appealing pos• sibility anyway. 5. EXOTIC STATES. THE POMERON 5.1. MULTIQUARK HADRONS The quark bag model may shed some light on the old problem of exotics. In this model the mass of a hadron increases roughly in proportion to the total number of quarks inside the bag. This is due to the fact that the quark kinetic energy dominates the to• tal energy of the bag. This would implicate that multiquark hadrons are heavier then ordinary ones. The crucial question is, of course, whether these multiquark states bind to form relatively stable hadrons. (a) (b) Pig.4.8. Sequential pole model It has been known for a long time with 0V intermediate state, (a.) that the Coulomb-like color electric forces phenomenological diagram ; Cb) string picture. saturate inside hadrons supporting rather strong evidence against low-lying exotic states in quantum chromodynamics. It means Using the mass 1.11 GeV and Eq.(4.28) that there are no strong forces between as input, the width may also be predic• color singlet mesons and baryons which ted from the sequential pole model with might be related to confinement structure. the following results With this argument a low-lying qqqq state, say, would be rather like a loosely bound S = 0.032 GeV'4 , molecule with color singlet qq "atoms" , if such a molecule may exist at all. K+K) - O.liel/, ( S.Stf.Setf uf) f Cl»-* la) Recently Jaffe has called attention to the presence of color magnetic forces, r (0V->K*K)- 0.84 MeV * r¿u„-> K.&.)l however. Two color singlet hadrons sitting T (o„-.ee) = if ef . close to each other are not an eigenstate of the magnetic gluon exchange operator of Eq.¿2.47) . They can exchange a gluon -> f The dominant decay mode is 0V 3T becoming color octets still forming an over• 0 evolving into a JT^JC'IT final state, all color singlet state. This force may mix and the decay into two kaons is sup• and split multiquark states. pressed. Since the spin splittings among qqq The tensor glueball 0T as the baryons and qq mesons are a significant lowest physical state on the pomeron fraction of their masses, it may happen trajectory, and the scalar glueball 0 that a multiquark state qqqq or qqqqq may - 120 - lose so rauch energy in color-spin inter• A more detailed discussion on the action that it becomes bound, relative to phenomenological aspects of the multiquark the decay into normal qq or qqq hadrons. states in the bag model is given elsewherei0y! We turn now to the interesting problem of On the basis of group theoretical high angular momentum excitations of qqqq calculations Jaffe argues that the ground states. states of qqqq and qqqqq are not exotic, they are nonets. This is good news, since they may be misclassified as conventional 5.2. BARYONIUM qq or qqq states. The weight diagrams and quark content of these multiplets is The quark bag model accomodates a fam• shown in Fig.5.1. ily of mesons which are probably dominant- ly coupled to baryon-antibaryon channels. Experimental results support the existence • usdd dsuu • of these peculiar mesons which are known as states of baryonium . The baryonium is pictured in the quark (uG-dd)ss bag model as a fat string-like spinning uds object with a diquark and anti-diquark pair udud ^(uü*dd)ss at the two ends. The dominant decay mode of the spinning string is to create a qq pair in the color electric field of the elongated bag so that the quark joins the südd» -1 J sduu diquark and the antiquark joins the anti- Q2^2 NONET diquark breaking the string into a baryon- MESONS -antibaryon pair. We have seen in Subsection 4.2 that ,uudss uddss 0 the slopes of Regge trajectories associ• ated with rotating string-like objects depend on the color separation at the two ends. The slope formula (4.24) depends on uds(uû-dd) the value of the Casimir operator. We may ddsuû have color charges in triplet, sextet, or udsss uds(uû +dd) octet representation at the ends of the V2 spinning bag. The triplet representation applies udssü • -IA • UDSSD" for the large angular momentum excitations d'à NONET of ordinary mesons and baryons. Octet sep• BARYONS aration is characteristic of the excitations of glueballs into spinning objects at the ends. In baryonium the diquark (qq) may be in color triplet or color sextet representation. Fig.5.1. The quark content of qqqq and qqqqq nonets. The slopes of the corresponding trajectories are given in Table 5.1 : Table 5.1 The exotic states turn out to be v heavier in Jaffe s calculation, far above triplet sextet octet the threshold for decaying into (qq)(qq) to/3 3 c =Í(¿A/ or (qqq)(qq) states, so that they must be (»1 broad if resonant at all. slope OC'ÍO; ff''« - 121 - Therefore, we expect baryonium tra• -Further search for baryonium spates remains jectories with ordinary slope <*'<°) when one of the most exciting projects in hadron the diquark is in color triplet represen• spectroscopy. tation, and with a slope $^d'<°) in sextet representation of the diquark. The latter states are very difficult to excite from 5.3. QUARK BAG STAR ordinary hadrons, since this 6-E string can only be produced in higher order There is a general belief that pulsars requering an extra gluon exchange in com• are neutron stars compressed to densities parison with ordinary processes. greater than the density of atomic nuclei. We may guess that when the density of mat• If, however, such a state is produced ter is further increased and becomes so somehow, it remains quite stable since its large that the mean distance between quarks decay is also of higher order in the quark- in different baryons is much less than -gluon coupling. Consequently, the quark 1 fermi, the description of matter as nu• bag model predicts the existence of ex• cléons interacting via potentials does not tremely narrow spinning states with large remain valid. A new description of matter angular momentum which predominantly decay composed of quarks may become relevant then. into baryon-antibaryon channels. Here the The existence of quark matter on a mac• expression "extremely narrow" means rela• roscopic scale would be rather spectacular tively narrow, in comparison with ordinary evidence for multiquark states. resonances. It is conjectured that at sufficient• Recently Chew suggested that two ly high densities matter will behave as a natural parity trajectories, with isospin relativistic gas of free quarks with degeneracy as well as exchange degeneracy are able to account for most of the avail• able evidence concerning meson states that where p is the pressure and ? is the communicate preferentially with baryon- density of matter. We shall bring now some -antibaryon channels. The two baryonium qualitative arguments that the vacuum trajectories are shown in Pig.5.2. They pressure (or surface tension) in the quark have ordinary slopes <¿'t°) corresponding bag model may be large enough to compress to diquarks in color triplet representation. the neutron phase of pulsars into quark S T U phase(see Pig.5.3) . (a) (b) Pig.5.2. The two leading natu• Pig.5.3. Phase transition of ral parity baryonium trajecto• neutron matter into quark ries proposed by G.P.Chew. phase under large pressure, (a) Neutron matter ; Cb) quark matter. - 122 - The normal density of nuclear matter When the quark masses may be neglected 18 in comparison with the fermi energy of -nucleón. f\o * 0.16 quarks, one can write for the energy den• sity j> at zero temperature or III + 3 S . A n (5.2) ° cm* on dimensional grounds. Here A is a con• In highly compressed nuclear matter ex• stant proportional to 110 , and n is the perts estimate the density to be about baryon number density which is conserved i - £ , even when baryons do not exist anymore cm* ' individually. and the corresponding pressure in neutron stars may be about The constant A was calculated by Chaplin and Nauenberg to second order in the quark-gluon coupling constant g , Now we recall that a fit(or an esti• mate, more precisely) for the lowest baryon and meson states suggested the value B*« 120 MeV . This value of the where lit is the number of quark flavors vacuum pressure in c.g.s. units is contributing to the energy density f . From Eqs.i.5-1) - ^5.3) we can express the chemical potential (Gibbs energy per so that the possibility for the formation unit baryon number) as a function of the of quark bag stars is there. pressure p for zero temperature tt) Following Chaplin and Nauenberg we A \3A . „J/h shall apply Gibbs'criterion for a phase transition to calculate the density where From the condition /^¿ary«t * /*- Pandharipande-Smith 2.7 determines then the transition point. Bethe-Johnson I 6.5 First, we calculate the chemical Bethe-Johnson II 13 potential in the quark phase of matter. The ground state of quark matter inside a gigantic bag (quark star) under vacuum pressure from the outside is a fermi gas In the calculations the values B *=• lAfO d 3..X with all color states occupied for each and c * were used from the best MIT eigenmode up to the fermi level. fit to the low-lying hadron spectrum. - 123 - Prom Table 5-2 it turns out that the sufficiently weak coupling, so that the phase transition occurs at baryon den• problem of rising cross sections and related sities which are 10-50 times the baryon logarithmic effects may be viewed as higher energy density in normal nuclei. Now, it order corrections which are ignored here. is argued that the maximum allowed energy The constancy of the total cross sec• density for siable quark stars as derived tions is usually associated with the fol• from general relativity is lowing experimental observations : 1. Elastic cross sections are also y„„x - 8.3 3! , Jt ' - r3 . (5.4) approximately constant over the same ener• gy range, though they are much smaller than The value of in (5.4) seems the corresponding total cross sections. The to be smaller than in Table 5.2 by a fac• elastic amplitudes mainly appear as the tor of three to five. Therefore, the diffraction due to multi-particle production existence of stable quark bag stars or processes. The elastic processes themselves quark matter at the center of neutron are only secondarily reflected in the elas• stars remains under que-stion mark. tic amplitudes. Since we are in a very sensitive en• 2. The real parts of forward scattering ergy region both in baryon phase and quark amplitudes are small compared to imaginary phase, the results are far from being parts. The real part associated via dis• conclusive. We seem to be on the border persion relations with a constant 6¿o¿ is line of being capable of determining the zero, so that its presence is a measure of possible existence of this incredible the non-constancy of 6¿,¿ . object. There is certainly more to come in 3. There is a diffractive peak which is the theoretical development of the quark approximately Gaussian in the momentum model and nuclear matter calculations at transfer. extreme densities before the question can 4. There is an approximate factorization be settled satisfactorily. of diffractive amplitudes and total cross sections 5. Approximate factorization and Peynman scaling, (or limiting fragmentation) are 5.4. HIGH ENERGY SCATTERING PROCESSES observed in inclusive processes. In the We shall study in the quark bag model fragmentation region of b the inclusive the qualitative description of hadronic cross section for a fragment c is final states associated with high energy processes. It will be shown that color b. . d£f - fiedle *c) Gab ¿'le separation above .1 fermi is the governing mechanism to generate the final state in demonstrating the independenc from the high energy collisions. This mechanism may projectile a . is the momentum trans• explain the great similarity of final fer from b to c and X = 4- MZA is Peynman's states in hadron-hadron and lepton-hadron scaling variable with M the missing mass, reactions, as well as in e+-e~ annihila• and s the center of mass energy squared. tion. 6. An universal plateau is observed in the central region in rapidity space. This LOW'S MODEL OP THE POMERON plateau seems to be independent of the ini• tial state of the reaction in which it is A remarkable feature of total hadron created. cross sections is that they are approxi• mately constant over a broad energy range, Low s model of the pomeron seems to between 10 GeV and 300 GeV laboratory account for all the above mentioned obser• energy. Low presents an attractive and vations, together with the constancy of simple model of the "bare" pomeron with éío¿ . The only exception is factorization which is accidental in the model. - 124 - BAGS IN INTERACTION requires the exchange of two colored gluons which is a higher order process in <¿c Following Low we describe the colli• sion process of two hadrons qualitatively as the specific interaction of the hadron COLOR OCTET bags with color gluon exchange. Consider two bags approaching each other in the • VORTEX cm. system with a definite impact param• eter b as shown in Fig.5.4. We assume the same radius and mass for the two bags. COLOR OCTET t Fig.5.5. Bags after fusion and color gluon exchange form a b 2R stretched vortex tube with the 2R two ends racing away with light \ velocity, approximately. I One notes that a similar color electric 5 Fig. .4. Two bags colliding vortex develops in deep inelastic lepton- with impact parameter b . Dots symbolize the valence - nucleón scattering when a colored quark quarks of incoming nucléons. inside the nucleón receives enormous momen• tum transfer by, say, a very energetic electron. As a result, the kicked out quark In a classical picture with sharp carrying color charge is running away from bag boundaries the bags will pass each the rest of the nucleón approximately with other without interaction for b > 2R light velocity. Since the diquark left They cannot interact, since the colored behind must be in color triplet state, the vector gluon fields are confined to the bag stretches and a color electric vortex interior of the bags. In the quantum the• appears between the quark in escape and the ory the surface of the bag becomes fuzzy, rest of the nucleón. but a well-defined meaning is maintained for the impact parameter b In electron-positron annihilation at very large energies when a qq-pair is cre• When b <. 2R the two bags will touch ated inside a "hadronic domain" of the in some point, and evolve for intermediate physical vacuum, the pair is running away times as a fused single bag in highly in back-to-back configuration with a rel• excited state. The most probable inter• ative velocity 2v~2c . Again, a color action which may arise is the exchange of electric vortex tube develops with color a soft colored gluon between the two parts triplet charges at the two ends. of the intermediate bag running away with a relative velocity 2v~2c . The two We have seen that three different color singlet parts become color octets reactions with different initial states due to the exchange of the soft gluon. have led to very similar intermediate states Since the color flux lines are confined, ("stretched color electric vortex tubes ) the intermediate bag stretches and a color before decaying into the final state. This electric vortex tube develops between the observation may serve as án explanation for two color octet parts (see Fig.5.5 ) . the experimental fact that the three reac• The overlap of the initial config• tions have similar multi-particle distri• uration with the stretched intermediate butions in the hadronic final states. bag is negligible so that the lowest order real part of the elastic scattering ampli• FRAGMENTATION IN THE BAG MODEL tude fCb) vanishes. Elastic scattering For the sake of definiteness, we shall - 125 - describe first the decay of the elongated tudinally excited objects are similar to intermediate bag of the e+- e- annihilation the elongated intermediate bag, so that process. they split again. After n splittings we get 2n similar objects. The decay con• Prom Gauss's theorem the effective tinues until the longitudinal excitation color electric field strength inside the of the new objects becomes comparable with vortex tube is their transverse excitation. These are not stable hadrons yet, though they decay into 3 **JT (5.5) the final state as ordinary resonances (fire balls). where R stands for the radius, and g is the fundamental quark-gluon coupling. Consider the above process in the The factor {ATS in (5.5) comes from rapidity variable. Initially, the rapidity the triplet representation of color at the of the baek-to-back qq-pair is -Inl-f and two ends. in If , respectively : Since the bag surface is classical in this discussion , it makes little dif• _L_ ference whether we apply surface tension or volume energy for providing confine• The pair creation energy somewhat decreases ment. We shall use volume energy. The the momentum of the quark and antiquark, energy of a vortex tube of length L is from p to p-m . The rapidity, on the other hand, remains almost the same : ¿ * i n'r-i + 3HlirL where the second term comes from the vol• ume energy. Prom the minimum of £ for In other words, if the quark and antiquark a given L , we find the radius of the longitudinally excited bag are suf• ficiently separated in rapidity space (àyiî), then pair creation does not change their rapidity. and After splitting the bag, we find the following rapidity diagram : When the vortex tube is long enough,* a new qq-pair may be created inside. Pair creation may occur through tunneling. Por the balance of energy it is neces• This simple picture already yields the sary that important properties of fragmentation. Indeed, the multiplicity of hadrons from the elongated bag is the same as the where m is the minimum quark energy sum of hadron multiplicities of two newly required by momentum uncertainty due to created bags. The extension of bags in the transverse dimension R : rapidity space is also additive, A3 = A^+A-J* I - IM so that the multiplicity is proportional Since the flux lines become abrupted to the length of the rapidity interval between the newly created q and q , the bag may break there producing two color• •Yl «N. o-n-íí • 2 In. 2- f less objects. The newly born and longi- - 126 - or In pp collision at least two qq pairs must be created to neutralize the color octet content of the fragmenting parts (see The above picture is valid if the Fig.5.6). The central plateau develops in mass of the splitting bags is sufficiently- large, The existence of the universal plateau and the fragmentation regions is Fig.5.6 a natural consequence of the model. Since the color electric field is invariant the above described manner, though its under flavor S1K3), the flavor of the height may depend on the different pair created pair in the middle of the expand• creation in the middle, and the different ing bag does not depend on the type of strength of the color electric field gener• the originally incoming particles. There• ated by the color octet charges at the ends fore, the rapidity space becomes populated of the elongated bag. uniformly by fireballs of 2-3 GeV, and only the ones at the end of the rapidity If the length of the initial system distribution remember the specific proper• before breakup is L0 in the cm. system ties of the original quarks. in pp collision, then the corresponding time for which the combined system holds Fireballs from the interior of the together is ^ a¿o since V*C*i . In the rapidity distribution decay into the final lab system, the corresponding event happens particles of the universal plateau, while at a time after collision the two fireballs at the end of the rapid• ity distribution populate the fragmentation i - T' region of the target and projectile. It follows then that the width of a fragmen• and distance from the collision point tation region is the same as the extension 1/. f. of a fireball's decay products in rapidity space, . where -is the velocity of the trans• At higher center of mass energy the formation from the cm. system to the only change is that the distance between laboratory system, the two fragmentation regions becomes larger, so that the distribution depends 4 only on iJm At 300 GeV, for example, THE POMEROIÍ so that the combined state holds together for a long time before breakup. This may The mechanism of multi-particle + give a natural explanation for particle production is the same in e - e~ collision production in nuclei. as in, say, proton-proton scattering. The fragmentation regions are, of course, The exchange of a soft gluon in pp different. collision leads automatically to a constant cross section which is mainly inelastic - 127 - due to color separation. Low estimates REFERENCES Inside the bag the quark fields are 15. J.Kogut and L.Susskind, Phys.Rev.Letters 34 (1975) 767. coupled to the non-Abelian gluon fields 16. K.Johnson and C.B.Thorn, Phvs.Rev. D13 with a coupling constant which is assumed (1976) 1934. to be small. So apart from a region close 17. P.G.O.Freund and Y.Nambu, Phys.Rev.Letters to the boundary the quarks are moving 34 (1975) 1645. relatively freely and it is natural to 18. J.P.Willemsen, Phys.Rev. D13 (1976) 1327. assume that a highly virtual photon will 19. D.Robson, Phys.Letters 66B (1977) 267. see a free field short distance structure. 20. R.L.Jaffe, Phys.Rev. D15 (1977) 267; D15 (1977) 281. The parton picture follows then. 21. G.F.Chew, LBL-5391 report (1976; . 22. G.Chapline and Nauenberg, Univ. of Cal. Santa Cruz report Q.97G) ; B.D.Keister and L.S.Kisslinger, Phys. Letters 64B (1976,' 117 and refs. therein. 23. P.E.Low, Phys.Rev. D12 (1975) 163. - 128 -