A relativistic quark model for mesons based on numerical solutions of the Bethe-Salpeter equation

ANNALS

OF PHYSICS

82, 407-448

(1974)

A Relativistic Quark Model for Mesons Numerical Solutions of the Bethe-Salpeter

Based on Equation*

ALAN H. GUTH Luboratory

for

Nuclear Science and Department of Physics, Massachusetts Institute Cambridge, Massachusetts 02139 and Joseph ffenry Laboratories Princeton University, Princeton, New Jersey 08540t

of Technology,

Received May 18, 1973

A study is made to determine if the results of the nonrelativistic quark model can be reproduced by a fully relativistic model of deeply bound spin-i quarks. It is found that the relativistic mode1 does not reproduce the nonrelativistic results, even when the quarks have nonrelativistic momenta. However, the model is rather successful in accounting for the known properties of mesons. Numerical solutions to the Bethe-Salpeter equation are obtained for pseudoscalar and vector bound states of equal mass quark-antiquark pairs, with either a scalar, pseudoscalar, or neutral vector exchange interaction. The interaction function corresponds to single particle exchange, with the addition of either one or two regulating terms. It is found that the second regulator allows the internal quark momentum to be nonrelativistic, but that the spinor structure of the wave function remains highly relativistic. Only the scalar interaction can account for the observed spectrum of states. The pseudoscalar interaction yields a vector state of lower mass than the pseudoscalar state, and the vector interaction leads to a vector state which lies approximately one quark mass above the pseudoscalar state. The X quark is taken as slightly heavier than the p and n, and the perturbation treatment of the mass difference leads to a quadratic mass formula. The decay amplitudes for x, K --) SLYare calculated, and it is found, independent of parameters, that f, w fK for either a scalar or vector interaction, in agreement with experiment and in contrast with the nonrelativistic model. The amplitudes for p’, W, c - e’e-, @pL- are also calculated, but in this case the ratios (again parameter independent) are in minor discrepancy with experiment. The question of the additivity of quark amplitudes is examined by calculating (with significant restrictions) the magnetic moments of the vector mesons and the amplitudes for magnetic transitions such as w -+ 2~. The magnetic moments of the vector mesons have the same (trivial) ratios to each other as in the nonrelativistic model, but they are strongly enhanced over the sum of the quark magnetic moments. The amplitude for * Research partially supported by the National 30738X.

+ Present address. 407 Copyright All rights

0 1974 by Academic Press, Inc. of reproduction in any form reserved.

Science Foundation

under Grant No. GP-

408

ALAN

H.

GIJTH

magnetic transitions, however, is related to the quark magnetic moments in approximately the same ratio as in the nonrelativistic model. The model is also used to obtain parameter dependent predictions for the masses and decay amplitudes. These predictions are not experimentally correct, but are generally well within an order of magnitude for a wide range of the parameters. The most significant defect discovered of the model is the presence of ghost states (the daughters of the vector mesons, with J Pc = O+-) with masses of about 2 BeV.

1.

INTR~OUCTI~N

The quark hypothesis, proposed by Gell-Man [l] and Zweig [2], has led to a great deal of success in understanding the properties of hadrons [3]. Most of these successesrely on nonrelativistic calculations, using simple assumptions such as the additivity of quark amplitudes, and simple approximations such as the setting of wave function overlap integrals equal to unity. Inasmuch as quarks have not yet been found, it is unclear why these calculations are so successful. One possibility is that quarks are not physical particles at all, but are rather mathematical entities which occur in some as yet unknown theory which gives approximately nonrelativistic results. The other possibility is that quarks are very massive, but that somehow the deeply bound states have nonrelativistic properties. This paper explores the latter possibility. We have tried to reproduce the success of the nonrelativistic quark model using the fully relativistic formalism of the Bethe-Salpeter (B-S) equation to describe deeply bound states. The study was confined to the low-lying pseudoscalar and vector mesons. Morpurgo [4] has suggested that deeply bound quarks could move nonrelativistitally at the bottom of a broad, deep potential well. The B-S equation involves no potential, but instead uses an interaction function which is expressed in perturbation theory as the sum of all irreducible Feynman diagrams. Since we cannot carry out this sum, we have used instead a phenomenological interaction function which incorporates Morpurgo’s suggestion. Specifically, we have used the one particle exchange propagator modified by the addition of one regulating term:

44) =

1

l

q2 + p2 -

q2 + (12 *

(1.1)

A is not a cutoff mass to be assigned a large value, but is rather a range parameter which is intended to have a small value. The above interaction is found to lead to bound states with average internal quark momenta comparable to the quark mass. We have tried also the doubly regulated interaction function:

44) =

l

q2 + p2 -

1 q2 + A* -

A2 - p2 (42 + (12)” .

(1.2)

RELATIVISTIC

409

QUARK MODEL

This interaction leads to deeply bound states with small quark momenta, but the spinor structure of the B-S wave function remains highly relativistic. We have used interactions which are scalar, pseudoscalar, or neutral vector. Since the interaction represents the sum of irreducible diagrams, this choice of spin and parity does not necessarily correspond to the spin and parity of the gluon which appears in the underlying Lagrangian. A realistic interaction function can be expected to contain terms of these forms and others, but we have just treated these terms one at a time. A rather successful relativistic quark model has recently been developed, in a very different spirit, by Feynman, Kislinger, and Ravndal [5]. These authors lay aside the formalism of quantum field theory with its complexities, substituting a much simpler prescription which allows the calculation of a large number of properties. Sections 2 and 3 are concerned with the reduction and the numerical solution of the B-S equation. Readers who would prefer to first see the results should skip to Section 4.

2. REDUCTION

OF THE BETHE-SALPETER

EQUATION

Numerical solutions to the B-S equation for deeply bound spinless quarks have been investigated by Pagnamenta [6] and by BGhm, Joos, and Krammer [7]. For the case of spin-l/2 quarks, closed form solutions for a restricted class of potentials have been obtained by BGhm, Joos, and Krammer [8]. Numerical solutions have been investigated by Narayanaswamy and Pagnamenta [9, IO], and by Sundaresan and Watson [I I]. The numerical solutions to the spin-4 problem cited above each relied on an expansion of the B-S wave function in terms of scalar and three-vector functions, originally introduced by Gourdin [12]. Here the analysis is done using only Lorentz invariant functions. The method is otherwise similar to that used by Narayanaswamy and Pagnamenta. For simplicity, we will seek solutions to the B-S equation for the bound state of a quark-antiquark pair of equal mass. Such solutions should apply to the nonstrange mesons. The X quark is presumably only slightly heavier than the p and n quarks, so the mass difference will be treated as a first order perturbation. The B-S wave function for the quark-antiquark state can be defined by X,&J, q, a) = (27~)~” 1 d4x e-i*‘o (0 I mPW)

@‘(-@)I

I P, a>,

(2.1)

where 1P, a) is the covariantly normalized bound state ket vector with polarization

410

ALAN

tensor a and four-momentum f&p,

H.

GUTH

P,, = 2p, . The conjugate wave function is defined as

q, a) = (27~)~‘~ 1 d4x eia’r (P, a I R&W)

@)(-x/2))

I Oh

(2.2)

(Other notation conventions are explained in Appendix A.) For a scalar interaction the B-S equation can be written as [13]

x(p,q,a)= _ jg2 [ ir . (4+ PI- ml1 (4+ p)”+ ml2 x

s

-$&

d(q - k) x(p, k, a) [ ‘7, ‘9 ~2p~~1~2],

(2.3)

where g is the coupling constant, and m, and m2 are the masses of the quark and antiquark, respectively, which will be taken for the present to be equal. (In principle the equation should be written with full quark propagators, but we are unable to do this explicitly.) The appropriate equations for the pseudoscalar and neutral vector interactions can be obtained by making the following two substitutions, respectively, on the right side of the equation: X(P, k, a) -

--Y~x(P,

k, a) y5,

(2.4)

X(P, k, a) -

--Y,x(P,

k, a) IL.

(2.5)

(For the case of the neutral vector interaction, the coupling of the vector field to the conserved baryon number current insures that only the 6,” term of the spin-one propagator contributes to the S-matrix. It is therefore reasonable to consider a phenomenological interaction involving only a S,, term.) Now consider the equation in the rest frame of the bound state (p, = (0, zWB/2), where MB is the mass of the bound state). One can perform the Wick rotation [14], analytically continuing k and q into the Euclidean region, where we will denote them by IE and p, respectively. Define p, = -8,

= (0, g&f,),

(2.6)

so that P, is also a real four-vector, provided one stays in the rest frame. The B-S equation for a scalar interaction can then be written as x(P, 4, a) = g2F(p, 4) SUP, 4) s -$$

d(q - rz> x(j,

k a) S+(p, 4),

(2.7)

where 1

WY 4) = (m” + q2 - p2)2 + 4(F * ij)>” ’

(2.8)

RELATIVISTIC

QUARK

411

MODEL

and S*(j.&q)=y-(iijfjg-m.

(2.9)

The symbols p and q refer to the magnitudes of the vectors p and 4. For a pseudoscalar or neutral vector interaction, the substitutions (2.4) and (2.5) still apply. The next step is to expand x(p, q, a) in terms of Lorentz invariant functions. Since there are 16 Dirac matrices there will be at most 16 terms in the expansion. By applying parity invariance, one can show that X(P, 4>4 = Qcl)’ x Y*X(-P,Po

; --9, 40 ; .(9

y4

2

(2.10)

where j and 7p are the spin and parity, respectively, of the bound state, and (2.11) where S is the number of indices with spacelike values. Parity invariance divides the 16 possible terms into two groups of eight, each group with a definite parity. If qp = (- l)‘, then the wave function can be expanded as x(p,

q,

a)

=

ix’l’q +

* a” x’3’q

ix’2’q

* &

* lzy

. q +

*p

f’q

. (?p P q Yc7 II”

+ xt5’Y * a + x(%“Am4iP”%Y~Y5 +

x”‘&up”uu,

+

if%

Lcq YCTPYY

(2.12)

where (2.13) and is a Lorentz invariant function. The factors of 4 and -1 have been chosen to simplify the problem after Wick rotation. Forj = 0, the most general expansion is given by the first four terms of Eq. (2.12), with the factor q . a” omitted. If rip = -(-l)j, then th e expansion can be written as x(p,

q, a)

=

x’l’q +

* (?y5 x@‘q

-

ix’2’q

* 8y

* qy5

* ify -

ix’4’q

* py5 * LZE u”AoPuwn~

+ x’5’%“lL4%P”p,gAYo + x’6’Y * &y,

For j = 0, one uses the same recipe as above. 595/S2/2-S

412

ALAN

H.

GUTH

If the quark and antiquark have equal mass, then x(p, q, a) will also satisfy the charge conjugation symmetry relationship X(P, 4,4

= Ilccx%h

--494

c+,

(2.15)

where yc is the charge conjugation number of the bound state, the superscript T denotes the transposition of the Dirac indices, and (2.16)

C = iy,y, .

When applied to Eq. (2.12), this symmetry implies that x’i’(q2, - P * 4) = (- 1Y 77cx”‘(42, P . 4) (i = 1, 3,4, 5,6, 7),

x’Yq2, - P * 4) = -(- 1Y rlcx”‘(42, P * d (i = 2, 8).

(2.17) (2.18)

When applied to Eq. (2.14), the symmetry implies that Eq. (2.17) holds for and 7, and that Eq. (2.18) holds for the other values. The remaining symmetry is time reversal invariance, which implies that

i = 1,2,4,

X(P, 4,~) = ~~T?J&‘(-P,

PO ; --a qo ; a*) YJ,

(2.19)

where or is an arbitrary phase factor and (2.20)

T = iy,y, .

g can be related to x by analytic continuation

in q,, [15]

R(Pi%%;4 = y4x+cp; q,qo*;a) 74

By combining time reversal invariance with charge conjugation one can show that

x(B, 4,4 = ?1TrlcY5x+(P, 4, a*)

75

(2.21) invariance (CT), (2.22)

provided that p is in its rest frame and q is in the Wick rotated region. This relation can be used to show that all the x ti) defined in Eqs. (2.12) and (2.14) can be taken as real in the Wick rotated region. Equations (2.12) and (2.14) can be summarized by (2.23)

413

RELATIVISTIC QUARK MODEL TABLE

I

Values of kP@ 9 4 1a)

Pseudoscalarbound states

Scalar bound states MU1 = M’2’ = M’S’ = &f(4) =

i y .I y.p i)5IJq”aPV

Vector bound states:

For convenience, the explicit expressions for M(“)( jJ 4, a) used in this paper are listed in Table I. One can also define matrices @ci)( jJ 4, a) which satisfy the relations (2.24) Tr[#j)( j& q, a) W)( j5, q, a)] = aij Explicit expressions for the Mci)( j& 4, a) are listed in Table II. The B-S equation can then be written as p(q2,

p * a = gwp,

q) 7 J- &Mm,

G, E, 4 4

- Q XW2,

P . 0, (2.25)

where ZP(@, q, ]E, a) 3 Tr[i%W(p, q, a) x UF, 4) M’W,

k 4 S+@, a,

(2.26)

414

ALAN

H.

TABLE

GUTH

II

Values of iW’@ 9 q, a)

Scalar bound states

Vector bound states

and the fj are numerical constants, listed in Table III, the values of which depend on whether the interaction is scalar, pseudoscalar, or vector. Equation (2.25) can be reduced to a set of coupled two-dimensional integral equations, but for numerical purposes it is very desirable to have one-dimensional integral equations. Following Narayanaswamy and Pagnamenta [9, lo], this reduction is accomplished by exploiting the approximate O(4) symmetry which becomes exact in the limit when the bound state mass is zero. The functions x(“)(q2, j -4) are expanded in O(4) eigenfunctions: (2.27)

RELATIVISTIC

415

QUARK MODEL

TABLE III Values of h

Pseudoscalar bound states

j

f!S) 2

f!P) 1

1 2 3 4

1 1 1 1

-1

1 1 1 1

-1

1 1 1 1 1 1 1 1

-1

Scalar bound states

Vector bound states

f!"'1 4

1 1

-2 -2 0

-1 -4 1 1

2 2 0

-1 -4 1 1

2 2 0 2

-1 1 1

-2

-1 -1

0 0

where t3* is the angle between q and j& and (2.28)

The Ch?: (cos p) ,are the usual Gegenbauer functions [16]. The U,l(/3) satisfy the orthogonality relation I 0 n sin2 /I d/3 Vi+?) U,“@) = S,,, . The B-S equation then reduces to the infinite set of one-dimensional integral equations

x?(q) = g2 7 $, loa dk JG%(q, k) x?tk), where

(2.29)

coupled (2.30)

416

ALAN

H.

GUTH

The integrals in the above equation can be carried out in closed form. A list of the necessary integrals is given in Table IV. For complete technical details, the reader is referred to the unpublished thesis by the author 17. The explicit expressions for the f&(q, k) which were actually used in the numerical calculations are shown in Tables V, VI, and VII, which refer to pseudoscalar (P) bound states (PC = 0-+), scalar (S) bound states (O+-), and vector (V) bound states (l--), respectively. TABLE IV Necessary Integrals

1.

dQk V,,“Gs,) A(4 - &) = 4(2?r)a’a d,(q, k)C,,‘(cos &), s where (for A(4 - 6) defined by Eq. (1.2)) (r’ - s’)% -- A2 - pL (r’ - .f)” 1 (r - s)r’s’ (r’ + s’)“+” n + 1 [ (r + s)n+2 - (r’ + s’)“+2

AhI, k) = -

r = ((k + 4y -t /w2,

r’ = ((k + q)* + A*)*/*,

s = ((k - qy + py,

s’ = ((k - qy + Ll*p.

(For A@ - &) defined by Eq. (l.l),

1 ’

omit the last term in the brackets.)

where C”’ = c,ycos j3,). For n < 0 , C,” = 0. 3.

s

mc~““Gs,Y(P

- WJ” UC,-, L,d-r &iiv An+acta3 - ___ (n + l)(n + 2) n(n + 2) + n(n + 1) I 42 + [-

nc,’ + 2c:-, 4+aG2 + 4 n(n + 2) (n + l)(n + 2)

2&+d%

+2 [-

(n + l)(n + 2) + 4,

-

4c:-, + PC:-, n(n + 2)

A,-d-a n(n + 1)

6 Ir”

1 +&dY I[ a&% P4

(The term in &fi,, is not necessary.) Table continued

RELATIVISTIC TABLE 4.

&Cd

417

QUARK MODEL IV (continued)

(sin fir (cos fi)” = s: 4 (m2 + qa - pay + 4p2qt cosy

(m” + q2 - p2)2 X(o+b)/2

x [ (1+ ~y”-w where x=

((a+b) /2)-l ma + 1)/a c n!W(a + 1)/2) niO

1

n) x” ’

4p2q2 (m” + q2 - p8)2 .

(The second expression holds only if a and b are even integers, which includes all the necessary cases.) TABLE V k) for Pseudoscalar Bound States

Values of I&(q,

i, n’

130

290

LO

\ i, n

f (ma + q2 + p*)

F

&Adca

391 3

x E&W

290

f

-%04&~

f

(m2 -

q2 + $1

x %oAoks

f

C4&A, +J&(A,, - 4 [~PESSAS

490 F

E4,A,k4

4w T

&A#

1

k’

418

ALAN

H. GUTH

TABLE VI Values of K,$(q, k) for Scalar Bound States

i, n’ \ i, n L1

151 s (m” - q8 - p2)

290 - : mpqE,,A,#

391

4J

- s [4mqE,,A,

x E2,A,ks

+mgE,&‘o

+m2

-$WdW - 4-j

k4

+ qa-P2 P

1

x E&A0 - A,) k’

391 s mEPsA,k8

2 2(ma - qa + pa)

- ;pq&A&’

7

4

x EmA, -@kt, - 4

491 - ; Ea,A,k’

-

0

- $ F

Es,A,k4

1k4 4 ma+q2+P2 7 4 x EmA@

3. NUMERICAL SOLUTIONOF THE BETHE-SALPETER EQUATION

The first step in solving Eq. (2.30) is to truncate the infinite sum over n’. The computer calculations have been carried out using only the lowest nonvanishing term (n = 0 or n = 1) in the expansion of each Lorentz invariant function. With this truncation, it is possible to accurately calculate many properties of the deeply bound states. The inaccuracies introduced by the truncation depend on the specific case, and they are discussed in detail in Appendix B. The next step in the numerical solution is to convert the truncated system of integral equations into a matrix equation. The integral over k was mapped into a Unite region by the change of integration variable k-M *=k

(3.1)

RELATIVISTIC

QUARK

MODEL

\

891

0

2 -+ mEpoAdcs

790

-

0

f W - 4% + p3 EaoAdcs

- ; E,,A&s

690

; w&A,k’

- s mEn&ks

1 ma-qa-pPB 2 4

ws&4P

f

- $ EaoA,,k”

$WdW

5 T Ea,A,k4

- ;P%~X

78

-$ W + (1’ + P”) EdW

0

-

0

60

VII (continued)

58

490

- f

390

GW

2 - ;;s m~%s4&~

5,o

190

i, n

i, n’

TABLE

0

+ Ezo(4 - 4lk*

[ ~PEA

+

m2-q2+pa 2P

f[4 +WMAo -Ad] k4

f

--- Im Em (4, - 4k4 73 P

f wWsA

-

-t E&o - Adlk4

- -!- 1 Ezo (A, - A,)k4 nap

- f~[4W,

891

2

$

F s r

i-5 0

RELATIVISTIC

QUARK

421

MODEL

where A4 is a scaling mass chosen to match the interval over which x(q) is significant. If the resulting integral equation is then converted to a matrix equation by use of Gaussian quadrature [18], one achieves stable solutions provided that the masses ~1and /1 appearing in the interaction function (Eqs. (1.1) or (1.2)) are not too small, compared to m, the quark mass. However, when p and A are both small, the function d,(q, k) (defined in Table IV) becomes sharply peaked in the neighborhood of k w q. For these cases Gaussian quadrature would require more integration points than computer capacity permitted. (The program was normally run with 12 points per invariant function, although it can use up to 24 points per function for the spin zero bound states.) The problem was solved by the use of polynomial interpolating integration with an arbitrary weight function: i M-(xS, s-1’ dx g(x) f(x) - 1=1

(3.2)

where -1

1

1J -1

dx

g(x)

fi 6 WZ==1 m+z

-

xnz).

(3.3)

Regardless of how sharply peaked g(x) is, this approximation is exact if f(x) is a polynomial of degree less than or equal to n - 1, and is a good approximation if f(x) can be well approximated by such a polynomial. Successful results were obtained by setting the function f(x) equal to the expansion function x:!(k) multiplied by any explicit factors of k which occur in K$(q, k). The remaining factors of the integrand were incorporated into g(x). The integral of Eq. (3.3) was performed numerically. The range of integration was divided into three segments, with the center segment just covering the peak, and each segment was then integrated using 64 point Gaussian quadrature. The points of integration for Eq. (3.2) can be chosen somewhat arbitrarily. It proved convenient to use the Gaussian quadrature points. If all other quantities are held tied, the coupling constant g2 becomes the eigenvalue of a matrix equation. This matrix equation could usually be solved by iteration. When the iteration failed to converge, the significantly slower technique of searching for zeros of the characteristic determinant was used. Once the wave functions are calculated, they are normalized by the condition [19]

It is also possible to compute from the B-S equation several perturbation expressions, which serve both as a check on the consistency of the computer calculations

422

ALAN H. GIJTH

and as a tool for limiting the number of computer done. In particular, it can be shown that

calculations

that need to be

and

x Er . (it? - P) + ml x(h S,4[r * GP+ I-9 + ml>. (3.6) If both quark masses are varied together, then the right side of Eq. (3.5) is doubled. For the purpose of consistency checking, one uses the above relations to calculate the partial derivatives

The first expression was checked, by making small variations in m, for pseudoscalar and vector bound states, with each of the three types of interactions. In all six cases there was better than 1% agreement between the directly calculated value of dg2 and the value calculated from the perturbation expression. The second expression above could be checked in only two cases-a pseudoscalar bound state with either a scalar or vector interaction. The agreement was again better than 1%. For vector and scalar bound states, the direct calculation of the second expression is invalidated by the truncation error (see Appendix B), so no comparison is possible. For pseudoscalar bound states with a pseudoscalar interaction, the truncation error prevents the expression from being validly calculated by either means.

4. GENERAL PROPERTIES OF THE SOLUTIONS To discuss the spinor structure of the solutions, it is convenient to express the 4 x 4 matrix x in terms of four 2 x 2 matrices, following the notation of Llewellyn Smith [20]:

For a nonrelativistic bound state, x++ dominates the wave function. For the pseudoscalar bound states, two types of solutions have been found. For scalar and vector interactions, the ground state B-S wave functions were dominated

RELATIVISTIC

QUARK

MODEL

423

by the term x;)(q) y5 . In the limit of p --f 0, these wave functions are invariant under rotations of 4, which means that the states are O(4) singlets. They satisfy x++ w x-- , which Llewellyn Smith calls Model 1. For a pseudoscalar interaction, the wave functions were dominated by the last three terms of the expansion in Lorentz invariant functions. In the O(4) limit, such states can be shown to be degenerate with a state of PC = l++, forming an O(4) quartet which transforms as a four vector ((i, 4) representation). In this case x++ = -x-- , which Llewellyn Smith calls Model 2. Only one type of solution has been found for the vector bound states, corresponding to an O(4) quartet. The wave functions are dominated by the first, third, and fifth terms of the expansion in Lorentz invariant functions, corresponding to Model 2. The only scalar bound states found were the O(4) partners, or “daughters,” of the vector bound states. These solutions are dominated by the first three terms of the expansion in Lorentz invariant functions. Note that in all cases, the spinor structure in the deeply bound limit is highly relativistic. Samples of the momentum space wave functions are shown in Fig. 1. Using the singly regulated interaction function, the wave functions were found to always give internal quark momenta which are comparable to the quark mass. Using the doubly regulated form of the interaction, with small values of TVand II, one finds solutions which are dominated by momenta which are small compared to the quark mass. Unfortunately the computer solutions become unstable for (1 2 0.05 m, so the limit of very small (1 could not be investigated. To understand theoretically the behavior of the wave function as m --+ co with p and (1 fixed, one should look in this limit at the behavior of the kernel of the integral equation, after the range has been mapped into a finite region by Eq. (3.1). For the singly regulated interaction, the kernel is unbounded at the edge of the region of integration corresponding to k ---f co. For the doubly regulated interaction, the kernel is unbounded only at the single point corresponding to q = k + co. One can easily construct interaction functions which lead to kernels which are bounded everywhere in the limit of m + co, such as

t-m3 A(q)= w : A213 - (q2+ p2js*

(4.2)

An interaction of this type, which gives a Fredholm kernel even in the limit m -+ co, would guarantee that the internal quark momenta would remain finite in this limit. Thus, it is clear that there is no problem in constructing phenomenological interactions which give solutions dominated by nonrelativistic momenta. However, the spinor structure remains highly relativistic. An examination of the bound state spectrum shows that only the scalar interaction, either singly or doubly regulated, gives results in accord with reality. It gives

424

ALAN

H.

(0)

GUTH (b),-

90

70

A;50

30

IO 0 I\ (CT

q

O5

1500

x(31000 0

500

0 Ii 0

4

0.5

OOllLu-

q

1. (a) shows the dominant component of the wave function for the pseudoscalar bound state with m = 1.0, p = 0.05, A = 0.10, and MB = 0.01. (b), (c), and (d) show the dominant components of the wave function for the vector bound state, with the same parameters. Note added in proof. All four graphs were calculated using the doubly regulated interaction function. FIG.

a pseudoscalar state as the ground state, with the vector state lying a small fraction of a quark mass above. The vector interaction leads to a vector state which lies above the pseudoscalar state by a mass comparable to the quark mass. The pseudoscalar interaction inverts the order of the pseudoscalar and vector states. Detailed numerical results will be shown in Section 9. One general result of the calculations is the justification of the successful quadratic form of the Gell-Mann-Okubo mass formula for mesons. Equation (3.5) suggests that the perturbation in MB2 caused by the presence of a heavier h quark should be proportional to the extra mass. Actually, the equation will lead to a linear mass formula if the right side is proportional to MB , and to a quadratic mass formula if the right side is independent of MB . For all of the cases calculated, the right side approaches a finite nonzero value as MB --+ 0, leading to a quadratic mass formula for the deeply bound states. (It can also be verified that one obtains a linear mass formula in the nonrelativistic limit.)

RELATIVISTIC

5. DAUGHTERS

425

QUARK MODEL

OF THE VECTOR MESONS AND OTHER GHOSTS

One problem in models of this type is the presence of the unwanted daughters of the vector mesons, with PC = O+-. (In the O(4) limit, these solutions may be obtained from the vector solutions by letting the polarization vector au point in the time direction.) States with this PC cannot be constructed from a quark-antiquark pair in the nonrelativistic quark model. In the relativistic model, the B-S wave function vanishes identically when both quark and antiquark are on their mass shells. There is therefore no corresponding pole in the quark-antiquark S-matrix, but ones does expect a pole in the two quark two antiquark S-matrix, as can be seen in the diagram of Fig. 2.

FIG. 2. This diagram shows how a pole in the off mass shell four point Green’s function can contribute a pole to the two quark two antiquark S-matrix. The dashed line indicates a strong interaction, and the heavy line represents the bound state which leads to the pole in the Green’s function.

The anomalous feature of these states is that they are ghosts. That is, the normalization integral of Eq. (3.4) is calculated to be negative. Thus, these states correspond to poles in the off mass shell Green’s function with residues of the opposite sign from that expected from positive norm intermediate states.

0:

5

t0

m/A FIG. 3. The mass of the daughter of the p meson, with J Pc = O+-. The results are shown for doubly regulated scalar or vector interactions, with p = 42.

426

ALAN

H.

GUTH

The masses of these daughter states have been calculated by using Eq. (3.6) to find aMBz/agz for both the vector bound state and its daughter. Since both these states lie at zero mass for the same value of g2, it follows that the ratio of their masses is approximately the square root of the ratio of these derivatives. Knowing the mass of the p meson, the predicted mass of the daughter has been calculated. The results are shown in Figure 3. The states are found to be very massive, and perhaps they lie outside the range of validity of the O(4) limit calculations which led us to believe that they are ghosts. We have solved for only the O-+, I--, and O+- bound states, so it remains possible that there are other low-lying states in the model. We cannot say if there are other low-lying ghost states [21].

6. r, K+ev,pv

An important success of the relativistic quark model is its ability to describe these decays [20]. One assumes that the hadronic part of the weak current can be written as

J, = %r&

+ Y&$~ ~0s6, + h sin &I,

(6.1)

where p, n, and h are quark indices and 9, is the Cabbibo angle. The matrix elements of this current are conventionally parameterized as (0 1 J,+(x) I r+(P))

= i &f-

cos OcP,,

(0 1J,+(x) 1 K+(P))

= i &fK

sin O,P, .

Given the above equations and the definition it is easy to show that

of the B-S wave function (Eq. (2.1)),

where P = 7r or K. In terms of the expansion functions,

where only the leading terms have been kept. Empirically, if Bc is taken from a fit to hyperon decays (sin 8, = 0.23), one finds that fv M 130 MeV, and fK M 150 MeV. It was pointed out by Van Royen and

RELATIVISTIC

QUARK

MODEL

427

Weisskopf [22] that the approximate equality off* and fK is difficult to reconcile with the nonrelativistic quark model. One must assume that the nonrelativistic wave functions for the 7~and the K obey the following relation at the origin: (6.6)

It is difficult to understand how such a large violation of SU(3) is consistent with the other successes of the symmetry. Llewellyn Smith [20] has already shown that the problem can be resolved in the relativistic model. If the pseudoscalar mesons are O(4) singlets (Model l), then it can be shown that the right side of Eq. (6.5) approaches a constant as MB -+ 0. Thus the value offp does not depend sharply on MB , so fm * fK . If, however, the pseudoscalar mesons are part of an O(4) quartet (Model 2), as was found for the case of a pseudoscalar interaction, then the right side of Eq. (6.5) blows up as l/M, as MB + 0, leading to

Thus, our results show that the ratio of fm to fK is given correctly by the relativistic model with either a scalar or vector interaction, but not with a pseudoscalar interaction. The numerical results are contained in Section 9. 7. p, 644 - e+e-, CL+~The relativistic quark model appears to be unsuccessful in accounting for the ratios of these decay amplitudes, but the discrepancies are not large. One assumes that the hadronic electromagnetic current can be written as j, =

J( ieq$(q)yc~(*) 4=9,n,A

(7.1)

with e, = $e,

e, = e, = -4e.

(7.2)

The matrix elements of the current will be parameterized by

where V = p, w, or 4. g, is related to the frequently used parameter yv by eMv yv=2gv.

(7.4)

428

ALAN

H.

GUTH

The quark content of the mesons is described by the usual mixing angle notation I PO> = u/nK

PF> -

I WI,

1 w) = 1 1) cos 8 - 1 8) sin 8,

(7.5)

1 4) = j 8) cos 8 + 1 1) sin 9, I 8) = (l/~&2

I hX) - I p& -

j nn)) (7.6)

I 1) = U/4~~(I PF) + I nn> + I w.

If the mixing angle is taken from the mass spectrum by assuming perturbations in MB2 (see [3, Chapter S]), one finds 8 = 39”. One might also assume “ideal” mixing, with 8 = arctan (l/d/2) M 35”. One can then show that gv = CVFV 9

(7.7)

where (7.8)

and c, = l/la,

C, = l/G

sin

8,

c, =

--IJ~/~COS

8.

(7.9)

(The wave function x appearing in Eq. (7.8) is to be normalized according to the usual prescription, as the complications due to the superposition of quark states have been absorbed into the constants Cv .) In terms of the expansion functions, g”=

-

(2$fz

Mv s omq34

[4x$‘)(q)+ q2x?(cd1v

where only the leading terms have been kept. All of the vector bound states found belong to O(4) quartets (Model 2) and Llewellyn Smith [20] has shown that for these states the right side of Eq. (7.10) blows up as l/MY as M, + 0. Thus, one would expect that M, gv would be about equal for the three mesons. Empirically, however, it appears that the gv’s are about equal. The compiled average experimental values of gv and Mvgv are listed in Table VIII. They have been compiled from the original data listings in the 1972 Review of Particle Properties [23], using the assumption that measurements of the branching ratios for V + e+e- and V-j p+p- are really measurements of the same quantity, up to a negligible phase space correction. The scale factors S (which indicate inconsistent data) were calculated according to the prescription of the Review of Particle Properties.

RELATIVISTIC

QUARK

TABLE Experimental

429

MODEL

VIII values

of go and M&

24

I9 = 39” &, = 62 & 5 MeV - =60&7MeV 8w & = 74 + 4 MeV M,,&

S=1.8 S = 1.3

= (4.7 *

0.4)

x 104 MeV2

M,,&

= (4.7 f

0.6)

x lo4 MeV”

S = 1.8

M+&

= (7.5 f 0.4)

x lo4 MeV”

S = 1.3

fJ = 35”

&

= 62 i

5 MeV

&J = 66 i 7 MeV &=71 +3MeV Mpg,

S = 1.8 S=

1.3

= (4.7 i

0.4)

x 104 MeV2

M&,,

= (5.2 i

0.6)

x 104 MeV2

S = 1.8

M&

= (7.2 rt 0.4)

x 104 MeV2

S = 1.3

While the model is in discrepancy with the current experimental situation, the discrepancy does not appear to be serious. For either mixing angle, all three values of MV gV lie within 25 % of a central value, and perhaps that is all one should expect from so simple a model. Furthermore, the experiments could be wrong. The inconsistency of the different experiments show that the situation is rather uncertain. The numerical results of these calculations are contained in Section 9.

8. MAGNETIC

MOMENTS

AND

MAGNETIC

DECAYS

OF THE VECTOR

MESONS

One of the most striking successes of the nonrelativistic quark model is the assumption of the additivity of quark magnetic moments. SU(3) symmetry requires each of the three quarks to have a magnetic moment proportional to its charge, so the only undetermined parameter is the constant of proportionality pO. Morpurgo [4] has shown that one can calculate the magnetic moments of the baryon octet, reproducing the successful relations obtained by BCg, Lee, and Pais [25] using the assumptions of SU(6). With p0 determined from the magnetic moment of the proton, the model predicts the magnetic moment of the neutron to

430

ALAN H. GUTH

within 2 %, the magnetic moment of the Z+ safely within the experimental error of 18 % [26], and the magnetic moment of the A within 25x, while the experimental error is 9 % [27]. The rate for the magnetic transition A -+ rrOy has also been calculated [28], and the result is accurate to within 30%, while the experimental error is 13 % [29]. fn this section we will test the assumption of magnetic moment additivity in our relativistic quark model for mesons. To calculate the desired quantities, one needs the matrix element of the electromagnetic current between two bound states. Unlike the somewhat trivial cases discussed in the previous two sections, here one needs the complete formalism developed by Mandelstam [15]. Begin by assuming that the bound state is composed of a quark of charge e, and an antiquark of charge -eB . Superpositions will be considered later. In lowest order, the matrix element is given by

Pi) + %I f z?Apf, q + k 4 GY - (4 + pi> + 4 x&i , q, 4 hy,J,

(8.1)

where k = pi - pf .

t

q-pi \,;I/

(8.2)

2k

q-k-Pf

FIG. 4. Lowest order diagrams for the calculation of the matrix element of the electromagnetic current between two bound states.

RELATIVISTIC

QUARK

MODEL

431

The diagrams corresponding to this expression are shown in Fig. 4. If the quark masses are equal and the B-S wave functions satisfy the charge conjugation relationship of Eq. (2.15), then the second term of Eq. (8.1) is related to the first. The second term can be dropped if the factor of e, in the first term is replaced by e, & e2 , where the + sign holds if the charge conjugation numbers of the initial and final states are opposite, and the -sign holds if they are equal. In contrast to the two previous calculations, this result is not true to all orders in the strong interactions. The inverse “bare” Feynman propagator should be replaced by the inverse of the full propagator. Graphs which correct the electromagnetic vertex are also allowed, so the y,, should be replaced by I’, , the full vertex function. There are also graphs which may be regarded as the polarization of the electromagnetic vertex of one quark by the second, an example of which is shown in Fig. 5. The only excluded diagrams are those which modify the B-S wave functions by either self energy or vertex corrections.

fiG. 5. An example of a graph which may be regarded as the polarization magnetic vertex of one quark by the second quark.

of the electro-

To extract the magnetic moment, note that the general expression for the electromagnetic current of a spin one system can be written

(8.3)

where Q = P, - Pi, P = P, + Pi.

(8.4)

The charge is given by ev = eFl(0)

(8.5)

and the magnetic moment by (8.6)

432

ALAN

H.

GUTH

It is convenient to consider the following choice of kinematic

variables:

Pi =(O, 0, 0, ZMB), ai = (0, 0, 1, 01, Pf = (-t, 0, 0, i(MB2 + ty2>,

(8.7)

af = ((1 + t2/MB2)1/2, 0, 0, -it/MB).

Then

so (8.9)

From Eq. (8.1) pv=

- lim-

t-a 2AifL?t s -(iI;4

Wih

,q +

BQ, 4 ice1 - e2) r2

x xh , 4, 4 GY - (4 - pi> + ml> + higher order terms.

(8.10)

The integral in the above equation can be Wick rotated, but one must remember that the momentum transfer Q is not affected by this distoration of the contour of integration. The argument of R becomes complex, but it can easily be handled by a first order Taylor expansion in Q. Equation (8.10) shows immediately that the magnetic moment of each vector meson should be proportional to its charge e, = e, - e2 . This prediction is in agreement with naive additivity, but the situation is so simple that one would hesitate to extrapolate the result to baryons. For simplicity, we have numerically computed the magnetic moments under the assumption that quarks couple to the electromagnetic field only through an anomalous magnetic moment. Then the r, in Eq. (8.10) can be written as (el - e2) r2 = OLAF- pa21 u2vQy,

(8.11)

where pAl is the anomalous moment of the quark, and -PAZ is the moment of the antiquark. The Q-dependence of z can then be neglected, as Q, is already first order in t. The Wick rotated expression for pV then becomes

x u12x(P,q, ai>SLYf (4 + 2 + ml>,

(8.12)

RELATIVISTIC

QUARK MODEL

433

where a? and ai are taken from Eq. (8.7) with t = 0. These values have been computed and are shown with the data in Section 9. A striking property of Eq. (8.12) is the presence of MB in the denominator. If the B-S wave function belongs to an O(4) quartet, one can easily verify that the integral has contributions which remain finite in the limit of MB + 0, so pV must blow up in this limit. Thus, the hypothesis of magnetic moment additivity does not hold, but instead one expects pV > (p Al - p,& for deeply bound states. One finds the same dependence on MB if one uses pure Dirac coupling, I’, = y3, so this enhancement of the magnetic moment is not a consequence of the anomalous coupling. The enhancement of magnetic moments of deeply bound states is not a new discovery, but has been noticed by previous authors in a variety of specialized and less realistic circumstances [30]. One might try to salvage the idea of quark additivity by saying that bound quarks have “effective” magnetic moments which are of order e/MB , much larger than their “bare” moments pa . This interpretation must be rejected, however, when one calculates the amplitude for the magnetic transition V-t Py and learns that this amplitude is not enhanced in the O(4) limit. Kt 2k,a,

FIG. 6. The kinematic variables describing the decay of a vector meson into a pseudoscalar meson and a photon.

The kinematic variables for the decay V -+ Py are defined in Fig. 6. The S-matrix element is given by Sfi = i &

O’(PJ I j,(O) I UP,, 4)

The current matrix element can be parameterized

The decay rate is then given by

r

V+Py

=

-A-

127r

/32ypt3,

GW4 a*(Pf + K - Pi).

(8.13)

by the constant pVP , defined by

434

ALAN

H.

GUTH

f = Mv2 - M2 2Mv

(8.16)

is the magnitude of the three-momentum of the P (or 7) in the rest frame of the V. In order to calculate /I VP, it is convenient to define fl as the value that ByP would have for the bound state of a quark of charge e and an uncharged antiquark. Then

BVPY= GA

(8.17)

where if one assumes “ideal” mixing for vector mesons and no mixing for pseudoscalar mesons, one has C oono= v3, c Qmo- 1, c*,o = 0, C ptn& = l/3,

(8.18)

Gin = l/45, c,, = 11343, c,, = l/2/3.

Only bLo has been measured, but one can verify that the rates predicted for the other decays using the experimental experimental upper limits [23]. (The with inconsistent results [31]. The order of magnitude of the rate.) p can be calculated by considering

value of flW,,, are consistent with the known reaction + -+ my has also been measured, but measurements at least confirm the expected the following choice of kinematic variables:

Pi = (0, 0, 0, NV), aj = (0, 0, 1, O), P, = (-t,

0, 0, i(Mp2 + t2Y2),

(8.19)

K = 2k = (t, 0, 0, it). Then

Pi) + ml>.

(8.20)

RELATIVISTIC

QUARK

MODEL

435

The above equation is exceedingly difficult for calculations because it does not simplify under a Wick rotation, The momentum transfer k is unaffected, and the argument of zP becomes complex. In order to make the problem tractable, we have again assumed purely anomalous magnetic coupling. We have also limited the calculation to zeroth order in a power series expansion in t, so that one can take )7&, q - k) M zP(pI, q). This latter approximation is reasonable if t is small compared to typical internal quark momenta. (If the physical value of t does not satisfy this criterion, one can imagine a fictitious world where t is arbitrarily small. The calculation then represents a “theoretical experiment” to see if quark magnetic moment additivity holds in this fictitious world.) After these simplifications and a Wick rotation, one has

where p, is evaluated for t = 0. (There is also a term involving ~~~which vanishes due to parity symmetry.) Again the factor of l/M” appears in the expression, but this time the enhancement is cancelled by the vanishing of the integrand in the O(4) limit. To see this cancellation, one inserts for gP the first term in its expansion in Lorentz invariant functions, which is the only term which survives in the p = 0 limit for O(4) singlet states. Similarly one inserts for xv the first, third, and fifth terms, assuming it belongs to an O(4) quartet. One then finds that the trace vanishes exactly in the O(4) limit. (One can show the same result for Dirac coupling, as well.) When the full expansions are used, one finds that fl can be written in the O(4) limit as

Numerical values of p will be presented in Section 9. It is found that the value of fi is generally close to the nonrelativistic value of 2(pA1 + p,&. In order to justify the successful nonrelativistic quark model prediction of the rate for w -+ rrOy in terms of the proton magnetic moment, one would have to calculate the magnetic moment of the three quark bound state in a deeply bound relativistic model. Inasmuch as the relativistic calculation for w --f rrOy nearly reproduces the nonrelativistic result, one would hope to obtain a nonrelativistic answer for the magnetic moment of baryons. If, however, the calculation of baryonic magnetic moments shows the same kind of enhancement as for the vector mesons, then the relativistic quark model would be in serious trouble,

436

ALAN H. GLJTH

9. PRESENTATION OF THE COMPUTER RESULTS In this section we will present the computer results involving the coupling constants, the bound state masses, the decay constantsf, and & , and the magnetic moments pv and decay constants p which emerge from the solutions of the B-S equation. The B-S equation (2.3) depends on four parameters: m (the quark masses), t.~(the inverse range of the attractive interaction), A (the inverse range of the repulsive interaction), and MB (the bound state mass). g2 is treated as an eigenvalue to be found by the solution. The information about the O(4) limit (MB -+ 0) which we seek can be found using any small value of MS , The data was calculated using MB2 = 10-4m2. The computer solution then depends only on the two ratios p/A and A/m. The results will be plotted in terms of the following variables: (1) Coupling go2(P): deeply bound go”(V): deeply bound

Constants:

The value of g2 corresponding to the MB = 0 limit for the most pseudoscalar state. The value of g2 corresponding to the MB = 0 limit for the most vector state.

(2) Masses: m: The value of the p and n quark masses (in BeV) is chosen to scale all the masses so thatf, acquires its empirical value of N 140 MeV. m,: By calculating aMB2/am, using Eq. (3.5), one can choose the mass difference of the X quark so that the quantity MK 2 - MT2 acquires its experimental value. (3) Predicted Ratios: A!,: The derivative aMB2/ag2 for both the pseudoscalar and vector mesons is calculated using Eq. (3.6). The value of g2 is chosen so that Mm2 acquires its empirical value. Then MD2 is determined by using the linear relation between MB2 and g2 which holds in the deeply bound region. Plotted is the ratio of the predicted to the experimental value of M, . Mvgv: The computer program calculates a number for Mvgv , which has the units of mass squared. Once the mass scale has been fixed by f, , this quantity can be translated into MeV2 and plotted as a ratio with its rough experimental average of 6 x lo4 MeV2. M& - A4,,? Once the h quark mass difference has been determined, one can find this quantity by calculating aMB2/amlfor the vector bound state. Plotted is the ratio of the predicted to the experimental value.

RELATIVISTIC

5x104

QUARK

431

MODEL

,

t

4x IO4 90 2w 3x104

9,2(P) 2x104

lX104 v 0

0.25

0.50

0.75

1.0

PiA

FIG. 7. Coupling

constants for a scalar, singly regulated interaction

2m

with A = 0.05 m.

1.0

I 0

1

I

I

0.25

0.50

0.75

0

tL/n

FIG. 8. Masses for a scalar, singly regulated interaction

nP 0

FIG.

0

I 0.25

I 0.50

with A = 0.05 m.

/PV

. I 0.75

1.0

9. Predicted ratios for a scalar, singly regulated interaction

with A = 0.05 m.

438

ALAN

H.

GUTH

pLy: The magnetic moments of the vector mesons are calculated according Al - pA2), which has a finite value in to eq. (8.12). Plotted is the ratio M,&n(,u the MV = 0 limit. 8: The magnetic decay constants are calculated according to Eq. (8.21). They are plotted as a ratio to the nonrelativistic value of 2bA1 + P,Q). /& refers to the calculated value for Mp = 0. ,& refers to the value for Mp = My. Using Eq. (8.22), all other cases can be inferred. IO5

a-

IO4

IO3 4 2

102

1

I IO

0

I 20

1 30

40

m/A FIG.

10.

Coupling

constants for a scalar, singly regulated interaction

2

mX If-

with p = A/2.

,

m

> Q, I co

0

I

I

I

IO

20

30

‘

m/A FIG.

11.

Masses

for a scalar, singly regulated

interaction

with p = A/2.

RELATIVISTIC

QUARK

439

MODEL

3.5 -

0

IO

20

30

40

m/A RG.

12. Predicted ratios for a scalar, singly reguiated interaction

103 0

IO

20

30

with p = A/2.

4( 1

m/A Frc.

13. Coupling constants for a scalar, doubly regulated interaction

with p = A/2.

440

ALAN H. GUTH

Figures 7, 8, and 9 show the effect of varying &‘.l for a fixed A for the case of a scalar, singly regulated interaction. For all the cases studied, it was found that only the coupling constants depend significantly on this ratio. For this reason, the other results (Figs 10-23) are shown only for p = A/2. For the pseudoscalar interaction, it is shown in Appendix B that only the coupling constant calculations can be carried out accurately with the truncations that are used. (This limitation is unimportant because the pseudoscalar interaction gives clearly unphysical results. The vector states lie lower than the pseudoscalar states, and the most deeply bound pseudoscalar state belongs to an O(4) quartet, leading to a very bad prediction of fV,& .)

>/” 5

mX

m

4

03

m

2

1

01

0

I IO

I 20

I 30

40

m/h FIG. 14. Masses for a scalar, doubly regulated interaction

FIG.

15.

Predicted ratios for a scalar, doubly regulated

with p = A/2.

with p = n/2.

IO4

IO3

102. ’ 0

I

I

I

IO

20

30

40

m/A FIG.

16. Coupling constants for a vector, singly regulated interaction with /I = 42.

2-

mA

m

> “I m 7

o------l 0 IO

20

30

40

m/A FIG.

17. Masses for a vector, singly regulated interaction

0

0

IO

20

30

with p = 42.

40

m/A FIG.

18.

Predicted ratios for a vector, singly regulated

interaction

with p = 42.

442

ALAN H. GUTH

2

104a 4 2

IO3

8 4

1

m/A

FlG.

19. Coupling

constants for a vector, doubly regulated interaction with ~1 = A/2,

3-

> m”

2I-

m/A FIG.

20.

Masses for a vector, doubly regulated interaction with p = 42.

RELATIVISTIC

OLI 0

QUARK

”

”

”

5

443

MODEL

”

’

IO

m/A FIG.

21. Predicted ratios for a vector, doubly regulated interaction

with p = A/2.

4 2 102

0

IO

20

30

40

m/h FIG.

22.

Coupling constants for a pseudoscalar, singly regulated interaction with p = 42.

444

FIG.

ALAN

H.

GUTH

23. Coupling constants for a pseudoscalar, doubly regulated interaction with p = A/2.

A rough estimate of the accuracy of the results can be obtained by varying the method of calculation (e.g., the number of integration points and the scaling mass M). It appears that the results are generally accurate to about 1 % or better. However, for m/A 2 20, the results become less stable. These results can probably be trusted to 5 or 10 %. 10.

CONCLUSION

The main goal of the research has been to find out if the fully relativistic formalism of the Bethe-Salpeter equation could produce deeply bound states which are nonetheless nonrelativistic in character. To this question our answer is negative. One can readily construct phenomenological interaction functions which lead to deeply bound solutions with quark momenta which are small compared to the quark mass. However, these states still have spinor structures which are highly relativistic. In particular, we have always found that x++ m fx-- . We have also found that these solutions do not have simple quark additivity properties, as in the nonrelativistic model. In particular, the magnetic moment of a vector meson is much larger than the sum of the quark magnetic moments. A subsidiary goal has been to learn if such a deeply bound relativistic model can successfully account for the properties of the mesons. We have no conclusive answer to the question, but we have discovered some grounds for optimism. It is found that the relativistic model justifies the successful quadratic form of the Gell-Mann-Okubo mass formula, and that it predicts the correct ratio offn,& (in contrast to the nonrelativistic model). Using a very simple interaction function, the predicted meson properties (as shown in Section 9) are generally accurate within a factor of 4. It seems plausible that the meson properties could be matched exactly by using a more complicated interaction function, which would be domi-

RELATIVISTIC

QUARK MODEL

445

nantly but not necessarily entirely scalar. To continue, one should also consider mesons of higher spin. Although the model appears successful in most respects, there remains the problem of ghost states. If the model is to be used simply as a method for making crude predictions, then one can ignore these states. But if the model is to be developed into a full theory, then one must either find a way to eliminate these ghost states or to show that they are somehow acceptable. APPENDIX

A: NOTATION

Coordinates and Momenta: x, = 6, x4) = k %J, 4u = (92 44) = (47 &hl~Lorentz Vector Conjugation: When a Lorentz vector (or vector operator) is denoted with a * (or a t), it is understood that one complex conjugates (or hermitian conjugates) the first three components, but one takes the negative of the conjugate of the fourth component. Wick Rotations: When a momentum is Wick rotated so that all four components are real, then it is indicated by q, It, etc. The symbols q and k are then used to represent the Euclidean lengths of such vectors. We also use the specialized notation ji, = -ip,, where 2p, is the momentum of the bound state. Physical State Normalization: (P’ 1P) = 2P$3(P

- P).

Dirac Matrices (Pauli Notation): u2=

(01 0’11

fJv=

3/j = (ifj

-2)~

(j0 -i 0’) Y4

1 0 o‘z= ( 0-l’ 1

= (:,-49

ULLV = Bi [ru 9%I, 75

Totally Antisymmetric

=

= (“1 -A), h >YJ= wb”.

YlY2Y3y4

Tensor: %Ao

9

with

e1234= 1.

446

ALAN

APPENDIX

B:

THE

H.

GUTH

EFFECT

OF THE

TRUNCATION

In this appendix we will examine the effect of truncating the expansion (Eq. (2.27)) of the Lorentz invariant functions after the first nonvanishing term. The cases will be discussed one by one. For pseudoscalar bound states, solutions were found belonging to O(4) singlets and quartets. The singlet solutions are dominated by the term x;“(q) y6 . For p = 0, the equation for this term decouples completely from all other terms in the expansion. Thus, the truncation allows an exact calculation of the value of g2 corresponding to p = 0. Assuming the dominance of this term, one can see from the Kj$(q, k) that j(p) = U(l),

xi”’ = O(p),

x2’ = 0(l).

(B.1)

(A term Twill be described as O(p”) if the quantity p-“T has a finite limit asp -+ 0.) All three terms give corrections to the wave function of O(p). The three correction terms lead to terms in the equation for x1’(q) which are Q(p2). The next step is to find the magnitude of the leading neglected terms. From Eq. (2.31), it can be seen that if

(E,,(q)

is defined in Table IV) then &(q,

Similarly,

k> K

4E22(q)

-

E2idq)

= O(P3

E2dd

= O(P2) Kz~(q~ M*

(B.2)

if

then K$(q,

k) = U(p2) K;:A,(q, k).

(B.3)

Using these relations, it can be seen that the second nonvanishing term of the expansion of each Lore& invariant function is O(p2) relative to the leading term in that expansion. These neglected terms will lead to errors of O(p2) in the normalization integral, Eq. (3.4). Using the fact that K,$q, k) = U(p2), one sees that the neglected terms would make contributions of O(p4) to the equation for #‘(q). Thus, for a fixed value of g2, p2 is determined up to corrections of U(p4). The quartet solutions for the pseudoscalar bound states, found for pseudoscalar interactions, are dominated by the last three terms of the expansion in Lorentz invariant functions. As before, the truncation allows an exact calculation of the value of g2 corresponding top = 0. This time, however, no other calculations are accurate. The neglected term #(q) would give a contribution of O(p) to the wave

RELATIVISTIC

QUARK MODEL

447

function, and would contribute with O(1) to the normalization integral. Furthermore, the neglected second terms of the expansions of the three dominant Lorentz invariant functions would give corrections to the calculation of p2 of 0(p2), destroying its validity. The only solutions found for the vector bound states are O(4) quartets, dominated by the first, third, and fifth terms of the expansion in Lorentz invariant functions. The neglected terms in the expansion of each Lorentz invariant function are 0(p2) relative to the first term, and the normalization factor is calculated with errors of 0(p2). The value of g2 corresponding to p = 0 has no truncation error, but the direct calculation of p2 is invalid. p2 can however be calculated using the perturbation method of Eq. (3.6). The only scalar bound states found were the O(4) partners, or daughters, of the vector bound states. The solutions are dominated by the first three terms of the expansion in Lorentz invariant functions. The error estimates for the vector bound states apply also to these states.

ACKNOWLEDGMENTS I would like to express my thanks to Professor Francis Low, who both suggested the topic and discussed the work with me as it was progressing. I would also like to acknowledge the computer time made available by the M.I.T. Laboratory for Nuclear Science.

REFERENCES 1. M. GELL-MANN, Phys. Lett. 8 (1964), 214. 2. G. ZWEIG, CERN preprints TH 401, 412, 1964, unpublished. 3. For an excellent review of the quark model, see 3. J. J. KOKKEDEE, “The Quark Model,” W. A. Benjamin, Inc., New York, 1969. 4. G. MORPURGO, Physics 2 (1965), 95. 5. R. P. FEYNMAN, M. KISLINGER, AND F. RAVNDAL, Phys. Rev. D 3 (1971), 2706. 6. A. PAGNAMENTA, Nuovo Cimento 53A (1968), 30. 7. M. B~~HM, H. Jots, AND M. KRAMMER, Nuovo Cimento 7A (1972), 21. 8. M. B&M, H. Joos, AND M. KRAMMER, Nucl. Phys. B51(1973), 397. 9. P. NARAYANA~WAMY AND A. PAGNAMENTA, Nuovo Cimento 53A (1968), 635. 10. P. NARAYANA~WAMY AND A. PAGNAMENTA, Phys. Rev. 172 (1968), 1750. 11. 41. K. SUNDARESANAND P. J. S. WATSON, Ann. Phys. N.Y. 59 (1970), 375. 12. M. GOURDIN, Nuovo Cimento 7 (1958), 338. 13. E. E. SALPETER AND H. A. BETHE, Phys. Rev. 82 (1951), 1232; M. GELL-MANN AND F. E. Low, Phys. Rev. 82 (1951), 350. 14. G. C. WICK, Phys. Rev. 96 (1954), 1124. 15. S. MANDELSTAM, Proc. Roy. Sot. London Ser. A 233 (1955), 248. 16. For example, J. D. TALMAN, “Special Functions: A Group Theoretic Approach,” W. A. Benjamin, Inc., New York, 1968.

448

ALAN

H.

GUTH

17. A. H. GUTH, Ph.D. Thesis, M.I.T., February 1972, unpublished. 18. For example, V. I. KRYLOV, “Approximate Calculation of Integrals,” MacMillan Company, New York, 1962. 19. A normalization condition was first obtained by Mandelstam (Ref. [15]) by calculating the matrix element of a conserved current. Later other authors obtained more general normalization expressions by considering the inhomogeneous B-S equation for the Cpoint Green’s function. See R. E. CUTKOSKY AND M. LEON, P/zJx Rev. B 135 (1964), 1445; N. NAKANISHI, Phys. Rev. B 138 (1965), 1182; D. LURIE, A. J. MACFARLANE, AND Y. TAKAHASHI, Phys. Rev. B 140 (1965), 1091; C. H. LLEWELLYN SMITH, Nuouo Cimento 60A (1969), 348. For our case, both normalization procedures give identical answers. 20. C. H. LLEW~LLYN SMITH, Ann. Physics N.Y. 53 (1969), 521. 21. Ciafaloni, in Ref. 1321, has stated that for scalar interactions the l- ground state is also a ghost, with Jpc = l-f. However, it appears that his conclusion is based on a misapplication of an interesting theorem which he presents in the same paper. The theorem states that for scalar interactions, if the solutions of the Goldstein equation are a subset of the solutions of the B-S equation for MB = 0 and defined Jp, then the greatest eigenvalue (ground state) is given by a solution to the Goldstein equation. This hypothesis does not apply to the lstates, for a wave function of the form x(p = 0, q, a) = f(q”)y . a can be a solution to the Goldstein equation without being a solution to the B-S equation. Thus there is no reason to necessarily expect these l-f ghost solutions to be part of the low-lying spectrum. 22. R. P. VAN ROYEN AND V. F. WEISSKOPF,Nuovo Cimento 50A (1967), 617. 23. Particle Data Group, Phys. Lett. 39B (1972), 1. 24. The p data is from Auslender et al. 69, Lefrancois 71, Hyans et al. 67, Rothwell et al. 69, and Wehmann et aI. 69. The o data is from Binnie et al. 68, Bollini et al. 68, Blakin et al. 71, Chatelus et al. 71, Wehmann et al. 68, Moy 68, Earles et al. 70, and Hayes et al. 71. The $ data is from Binnie et al. 65, Bollini et a/. 68, and Lefrancois 71. Summary of results and complete references in Ref. [23]. 25. M. A. BIG, B. W. LEE, AND A. PAIS, Phys. Rev. Lett. 13 (1964), 514. 26. COOK et al. 66, Kotelchuc et al. 67, Sullivan et al. 67, Combe et al. 68, Mast et al. 68, and Alley et al. 71. Average value and complete references in Ref. [23]. 27. Cool et al. 62, Kernan et al. 63, Anderson et al. 64, Charriere et al. 65, Barkov et al. 71, Dahljense ei al. 71, and Hill et al. 71. Average value and complete references in Ref. [23]. 28. C. BECCHI AND G. MORPURGO, Phys. Rev. B 140 (1965), 687; V. V. ANI~~~~cH et al., Phys. Letf. 16 (1965), 194; W. THIRRING, Phys. Left. 16 (1965), 335; L. D. SOLOVIEV, Phys. Lett. 16 (1965),

345.

The o width has been measured by Armentero et al. 63, D. Miller 65, Abramovic et al. 70, Atherton et aI. 70, Bizzari et al. 71, Coyne et al. 71, and Lefrancois 71. The branching ratio has been measured by Jacquet et al. 69, Balrin et al. 71, and Lefrancois 71. Average value and complete references in Ref. [23]. 30. H. J. LIPKIN AND A. TAVKHELIDZE, Phys. Lett. 17 (1965), 331; 0. W. GREENBERG,Phys. Lett. 19 (1965), 423; P. N. BOGOLXJBOV, Yadernaya Fiz. USSR 5 (1967), 458; M. CIAFALONI AND P. MENOTTI, Nuovo Cimento 46A (1966), 162; M. CIAFALONI, Nuovo Cimento, 51A (1967), 1090. 31. J. LEFRANCOIS,in “1971 International Symposium on Electron and Photon Interactions at High Energies” (N. B. Mistry, Ed.), Cornell Univ. Press, Ithaca, NY, 1972; BA~ILE et al., Phys. Lett. 38B (1972), 117. 32. M. CIAFALONI, Nuovo Cimento 51A (1967), 1090. 29.