Relativistic quark models based on the Bethe-Salpeter equation

ANNALS

71, 443-465 (1972)

OF PHYSICS:

Relativistic Quark Models Based on the Bethe-Salpeter Equation* M. K. Department

of Physics,

SUNDARESAN

Carleton

P. J. S. Department

of Theoretical

Physics,

University,

Ottawa,

Canada

WATSON’ University

of Oxford,

Oxford,

England

Received March 29. 1971

We consider a model of spinor quarks interacting via simple harmonic interactions in the BetheSalpeter equation. By discussion of a number of explicit solutions, we demonstrate that a large amount of nonrelativistic intuition is wrong for this model of tightly bound systems. We describe a few not implausible quark models and their properties: one of these has a reasonably physical mass spectrum and, in addition to satisfying some theoretical prejudices, gives rise to rather natural explanations for the split AS and some of the success of calculations with the nonrelativistic quark model.

1.

INTRODUCTION

In a previous paper [l] we have discusseda model of deeply bound quarks interacting via a simple harmonic force, which is described by the Bethe-Salpeter equation. In this paper we have extended the model to include a wider class of couplings. We feel that it is of interest for a number of reasons:firstly becauseof the lack of a satisfactory relativistic quark model of any kind which would allow us to calculate properties of mesons(such as their weak couplings, decay widths and electromagnetic mass splittings) from a comparatively unambiguous model. Secondly it has been shown [2] that a model for the Veneziano amplitude may be obtained by considering quarks interacting via a superimposition of harmonic oscillator forces; however, this requires spinlessquarks and it is so far impossible * Research partly supported by the National Research Council of Canada, and by the Science Research Council. + Present address: Department of Physics, University of Guelph, Guelph, Ontario, Canada.

443 Copyright All rights

0 1972 by Academic Press, Inc. of reproduction in any form reserved.

444

SUNDARESAN

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to extend it to spinor quarks. Thus this calculation might be considered as some sort of lowest-order approximation to the Veneziano model. In particular the crucial requirement of linearly rising trajectories is already contained in our model. Finally there are few solutions of the spinor Bethe-Salpeter equation1 which make any statements about the form of the wave functions in the deep-binding case, and so we feel that the solutions have some intrinsic interest. Section 2 discusses the model and method of solution, along with some of the anomalies of the Bethe-Salpeter equation, Sections 3 and 4 describe some numerical results and Section 5 discusses the progress towards the formulation of a relativistic quark model. Section 6 summarizes the problems and successes of the present calculations.

2. THE MODEL

The model and its method of solution have been described in [l] in some detail. We have made essentially two modifications; the first allows us to consider a rather wider class of problems and the second improves the numerical methods. The starting point is again the spinor Bethe-Salpeter equation (d + P - w x(4, PM - F - M) = c MYh + ~“0,2) air&q,

i

P) ri )

(2.1)

where the quarks have total momentum 24 and relative momentum 9: for a bound state P2 = p2 where p is the bound-state mass in units of the quark mass. The ai are essentially coupling constants, ill is the oscillator-range parameter and X the depth. Further, r, = 1, r, = y6,

rv=

9yu,

r, = -iyuy5,

r, = h.

(2.2)

It is easy to show that if we write

x(q,P) = c rjxj = (X-+ x++) x-- x+i (which is equivalent to [l, Eq. (3.2)]}, then C C a,I’Jixir, i i

= C bjrixj , j

(2.4)

1 The deeply bound Bethe-Salpeter equation has been solved for a separable potential by C. Llewyllyn Smith [3]. Various authors have solved the equation for zero bound-state mass and/or massless exchange (see Ref. [4] for a complete list), but these solutions suffer from a number of problems, and anyway refer only to Yukawa-type forces.

RELATIVISTIC

QUARK

445

MODELS

where b=Aa, s

-1

B

GT

UC

FV

1 1 1 1 -1 -1

1 (2.5)

This modifies [ 1, Eq. (3.5)] to

where the matrix M is given in Tables I and II of [I] in which the lower sign must always be taken. This allows us to consider any parity-conserving superimposition of couplings. It should perhaps be pointed out that we are solving a somewhat peculiar eigenvalue problem; schematically we can write Eq. (2.6) as AijXj = XbiXi ,

where some of the quantities bi may be zero, and obviously a conventional value routine will not work. We must therefore write (l/h) Xi = (A-l)ji biXi )

(2.7)

eigen-

(2.8)

find the eigenvalues, and invert them. Numerically we invert the matrix A, and multiply it by the diagonal (and possibly singular)2 matrix b. Then to find the corresponding eigenvectors we must return to the numerical equivalent of Eq. (2.7), which we treat as a conventional set of algebraic equations for a given X. A comment about our nomenclature is in order here. It is sometimes convenient to consider X expressed in terms of the invariant functions S, T, _V, _V,_F, G, B, C introduced in [I, Eq. (3.2)] and sometimes in terms of the form given above [Eq. (2.3)], the explicit correspondence between the two different sets of invariant functions is given below in Eq. (2.12). We talk of a B-type bound state, or eigenvalue, by which we mean that the 2 This may lead to rows and columns of A-% which are identically zero. We eliminate these by rearranging the matrix and calculate the remaining eigenvalues conventionally.

446 corresponding wavefunction there are other components,

SUNDARESAN

AND

WATSON

is predominantly of the form y&P, q2). In general so a (B + @)-type state implies that

where B and C have the same numerical magnitude, and the mass t.~is small for a tightly bound state. For this form, it is clear that x++ ‘v x-- and it corresponds to Llewyllyn Smith’s [3] type I state. We are inclined to regard this as the natural form for the wavefunction of a deeply bound state because it resolves the Van RoyenWeisskoff 15J paradox and appears to give a basis for some of the apparently ad hoc prescriptions of the nonrelativistic quark model, as further discussed in Section 6. In this framework, a nonrelativistic wavefunction would have the spinor structure ys(B + 7°C) where B GX C again and other amplitudes are small. Note that there is no p multiplying the C in this case, so that x++ > (x+-, x-+) > x--. This is Llewyllyn Smith’s type III state; we refer to this type of state as N-R. In addition to these two there are various other possibilities: Llewyllyn Smith’s type II states for which x+-k N -x-and others small, and the queer states for which x+- and x-3. are dominant. These last we regard as being somewhat unphysical from the point of view of the quark model. We have attempted to improve the numerical methods by allowing the asymptotic behaviour to vary. We assume that the wavefunction goes asymptotically as x N e-B@.

(2.9)

For the B-type solutions, at p2 = 0, fi = l/(Ma); however, there is obviously no need for the same asymptotic behaviour specified by this value of j3 to apply to any other solution, and so the problem arises as to what asymptotic behaviour should be ascribed to, for example, an S-type solution. There are various possible solutions to this quandary, of which we have investigated two. Firstly we have attempted to use a traditional variational method. We have varied p and chosen the values of h and /3 at dAfd/l = 0. We have no certainty that this method will work, since we are not dealing with a conventional Hermitian eigenvalue problem. Nevertheless the procedure works in almost all cases; the h vs. /I plots have well-defined minima and these do not change significantly under an increase in the matrix size. The only exception is F--type eigenvalues which, when the r summation is increased beyond r = 0 and 1 (notation of [l]), have no minima, and it in fact appears that x N -l/p as p + 0. It is easy to find model calculations which show a similar effect. However, we may note that the problem arises for very small /3, which implies that the wavefunction is still large for large values of q2; in other words the motion is highly relativistic. This gives rise to very large off-diagonal elements in the matrix, and renders the calculation unstable.

RELATIVISTIC

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MODELS

447

It has been pointed out to us [6] that a possibly similar failure of the variational method can occur in potential theory if the potential changes sign, as with a tensor force. It is also perhaps relevant that we have solved the equation varying h against p, whereas we really require to fix X and to vary ,LLagainst p. We have therefore effectively ignored the problem by using a stable matrix size, but it should be borne in mind that all the results quoted for P-type eigenvalues may be wrong. However the stability of calculations for other types of states suggests that they are more reliable. Secondly we have tackled the problem head on by solving the asymptotic equation first; in other words we rewrite the equations keeping the terms that are dominant at infinity [i.e., O(q2) rather than O(M)]. For a given interaction, we solve first these asymptotic equations to find an “asymptotic eigenvalue” ,& which is the same as the j.3in Eq. (2.9). There will be an eigenvector with a particular dominant term corresponding to this ,8. We then go back and solve the full equations for the given interaction with this value of /3,and select the solutionwith the same dominant term in the eigenvector. These methods agree for B-type solutions, but unfortunately they disagree for practically all others. In particular the latter criterion gives no S-type eigenvalues at all, and it is worth pointing out why. If we assume G+ and T are small, so [A + a2D(q)]T = [A + a2D(q)] G+ = 0, then the S equation takes the form (for p2 = 0, J = 0) [A + GD(q)]

s = @If2 - q2 +

;2M-;2

) s.

(2.10)

Now this looks rather like a Schrodinger equation with a potential well surrounded by a repulsive barrier. Thus one has solutions which look like a resonance inside the well, not unlike resonances produced by a centrifugal barrier in the Schrodinger equation. However the asymptotic behaviour in this case is given by dD(q)S

= -q?!$

(2.1 l),

which is not capable of giving rise to resonant-like structure. Thus the solution of Eq. (2.11) does not have a great deal of relevance to the solutions of Eq. (2.10). In addition there is no real reason to expect that if a certain term was dominant near q2 = 0, then the same term should be also dominant asymptotically. We may summarize the above discussion by saying that, apart from B- and GO-type solutions, which seem to be stable enough to merit some confidence, the equations are so complicated that some of the other solutions may not have the interpretation that we have placed on them. We have also improved the stability of the eigenvalue routine which now allows us to obtain solutions for the scalar interaction.

448

SUNDARESAN

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Before passing on to a fairly detailed discussion of some particular interactions it is perhaps useful to give a heuristic discussion of the sorts of solutions expected. The correspondence between Eq. (2.3) in terms of [l, Eq. (3.2)] is XS-fS XP

-

xo-

B T

x05

--+

c

xoi

-

vi

(2.12)

(ignoring irrelevant factors). It turns out that we can find a rather convenient semiempirical rule to predict what type of bound states a given interaction will produce. We may find the “pseudo couplings” bi from Eq. (2.5); then the rule states that for small mass (say, p - 0.05), Ai II l/hi . (2.13) As an example, if the coupling is purely pseudoscalar (up = 1, aL = a, = uA = aT = 0) then from Eq. (2.3) b, = b, = br = 1, b, = bA = -1 and we may expect S-, B-, _V- and F-type eigenvalues with h N 1. This is in fact what we find; the lowest O-i- state is (B + &)-type and so on. (The C-, G-, T-, g-type eigenvalues with X N - 1 are of course unphysical.) This rule appears to be necessary rather than sufficient because occasionally as pz varies two eigenvalues may come together and become complex. We note two further empirical rules which help to obtain a feeling for the solutions. Firstly the X - p2 plots are approximately linear for small p2, and the linearity and the stability are particularly improved if those couplings which are not important are made large and negative; thus the pseudoscalar (P”) interaction (bs = bp = bT = 1, by = b, = - 1) gives rise to much straighter h - $ plots than the P2 + S2 coupling (b, = bp = br = 1, by = b, = 0). Secondly, the wavefunction appears to take up an N-R form whenever possible. Thus if we choose b, N bA N 1, brest = - 1 the .empirical rule would imply that h, N h, c1 1 (for O-+). This occurs at p2 = 0, but for p2 # 0 these eigenvalues mix strongly so the lower eigenvalue is sharply depressed and the eigenvector has B = C almost exactly (we might expect this from degenerate perturbation theory). The curiosity here is that this appears to occur only for those states which we wouId expect to arise in an N-R calculation. Thus we find strong B-C mixing and strong F-G mixing (given the correct couplings), but if b, N b, N 1 where one might expect strong S-T mixing, there is no S-type state at all anywhere between b, = 0.7 and bs = 1.5 (with b, = 1). We note for future reference that the P2 interaction does not allow mixing of this type to occur.

RELATIVISTIC

QUARK

MODELS

449

These rules allow us to reduce drastically the number of couplings that can be used in a quark model. Thus the purely vector case a, = -1, arest = 0 would have (at p2 - 0) X, N 1, h, N X, N 2 and hence by fixing h _N 1 to give a low mass 0-+ bound state, we would find, e.g., the I-- bound state would be roughly as massive as the quarks, and would be bound in the G- sector. Numerical calculations confirm this. Next there is the phenomenon that certain eigenvalues may annihilate each other for certain values of $; what in fact happens is that they collide and become complex. It seems reasonable that this in turn implies that the corresponding Regge trajectories become complex (we cannot directly verify this, unfortunately), which one might suspect reflected some aspect of the numerical methods used. However a similar effect has been noted with scalar Bethe-Salpeter models by Cutkosky and Deo [7] (who showed that trajectory functions can “turn back” which is probably equivalent). A further unexpected result found from our numerical work is that the BetheSalpeter equation is capable of producing a bound state with an apparently repulsive interaction. As we have shown, a y5y6 interaction gives a (B + &)-type solution; starting from the same interaction with the opposite sign, we find a (C + @)-type solution. Again nonrelativistic intuition is at fault; a rough analogy may be found with exchange forces which are attractive in one partial wave and repulsive in the next. A related point is the possibility of a vastly richer spectrum of particles that arise from a spinor Bethe-Salpeter equation. For example, even with the scalar interaction (which one might expect to be the simplest) we find two O-f solutions at pL2N 0, one B-type and the second (U+ + C)-type. Two final points relate to massless particles. Firstly some eigenvalues are present for p2 = 0 which disappear for pL2> 0 (because they become complex). It is remotely possible that this has some physical significance. Secondly it is widely believed that daughter states are separated by dJ = 1,2,... from their parent trajectory at p2 = 0. For certain sectors this is true, but for others not. This knotty point is discussed in more detail in the appendix.

3.

RESULTS

Since in this problem there are essentially five free parameters, it is quite clearly impossible to give an exhaustive account. However, since the results are so essentially relativistic in that the mass spectrum depends vitally upon the interaction (as opposed to the N-R case where, say, a scalar and a vector exchange will give a rather similar basic potential) it seems interesting to describe a number of cases. This is done by giving the coefficients of the pL2vs. X plot in the tables below.

450

SUNDARESAN AND WATSON

For a given Jpc state we make an approximate expansion3 P2 = co + C,@ - 1) + C,(X - 1)2.

(3.1)

For most purposes it is sufficient to consider A = I, (i.e., p2 = C,), which gives a rough feel for the massspectrum obtained. The linearity of the plot is shown by the smallness of C, . For some of the obviously pathological cases, we have considered only the O-f, l--, 2++ states, but for a few of the more interesting ones we have looked at most of the low Jpc combinations. A few casesare numerically unstable, by which we mean either that we have been unable to find any solutions at all or that there are solutions present at p2 = 0 which disappear (become complex) for p2 > 0. In a few casesthe program has failed to find an intelligent fit to Eq. (3.1), but as this invariably happens when the eigenvalue is very large, implying a very large bound-state mass,we have not bothered to find a better fit. It is convenient to work with the “pseudo couplings” bi rather than the ai , because this gives more of a feel for the results. We have always normalized these so that the largest bi is 1. We discuss first the interactions involving only one type of coupling and then those invoIving simple mixtures. “Plots” refer to plots of p2 as a function of h [Eq. (3.1)]. (Note that in [I] we plot X as a function of p2.) Since our ideas are colored by the quark model, we have used the particle names to describe what are, strictly speaking, simply certain Jpc combinations.

4. COMMENTS ON TABLES Table I. The wavefunctions are of a mixed, but principally N-R type. The trajectories and plots are shallow and curved and the equations have a general tendency towards instability. Table II. Characteristically very stable solutions, and straight plots and trajectories. Solutions are mainly L-S type I (see Footnote 4). This is the most reasonable starting point for the relativistic quark model (seeSection 5). Table III.

Rather similar to P2 (which in itself is extremely peculiar), but gives

rise to L-S type II states. Note that A, lies slightly lower than B.

Let us now note the nonrelativistic limit of the +P2 interactions where we have p2 = 1. This is of rather academic interest, but it is a rather strong test of the model that the wavefunctions should take up N-R form when the massbecomes large. As always this model is surprising; the P2 interaction leadsto almost perfectly N-R wavefunctions with very much the sort of properties we would guess, but 8 In fact the computer fits Eq. (3.1) to the lowest positive eigenvalue at pL2= 0.0, 0.01 and 0.02. 4LS is an abbreviation for Llewyllyn Smith.

RELATIVISTIC

QUARK

4.51

MODELS

unfortunately the -P2 also leads to a stable wavefunction. We have discussed in greater detail the P2 interaction; this is firstly because it is numerically very stable and secondly because it provides the most appealing starting point for a quark model. TabZe IV. Normal parity states lie much lower particle is 6 and the 7r is very heavy.

than abnormal,

the lightest

Table V. Reasonable mass spectrum except that there is no O-+ at all (the 3-is anomalous because two eigenvalues have collided). Table VI. Normal parity states have very straight plots and trajectories, with the 6 as lightest (this is natural because all are S-type solutions), while the abnormal parities are all lighter, and have more steeply curving plots with (l/3) the slope. Table VII. No abnormal trajectories straight.

parity states at all (obviously!).

Normal

plots and

Table VIII. A very reasonable mass spectrum except for the lack of a r. An interesting feature is the very close degeneracy of the 6, A,, and A, . The B lies rather higher than the A, . Table IX. Rather similar to T2, which is surprising in quite different sectors. Table because different unstable

as the particles are bound

X. Taken with Tables VI and IX this shows the peculiarities of the model, here the coupling looks rather similar to the T2, but the results are totally in that the normal parity states are P-type, the solutions tend to be and the plots are sharply curved.

Table XI. V+ sector.

Abnormal

parity

states are too high, and the rr is bound in the

TabZe XII. The spectrum here is quite good, notably the B is heavier than the Al, although the 6 is heavier than the A2. Here it is curious that the different types have different slopes on the h - p2 plot: e.g., at p2 = 0 both the 1++ and l+- are B-type, but the PO-type solutions have a sufficiently steep slope to lie below the B-type at p2 = 0.02. Also the F+ slopes are steeper than the F-, in a way that is reminiscent of Dalitz’s proposal for the A2 splitting [8]. Table XIII. These and the next couplings are not simple mixtures, but plausible guesses for quark models. This one is a reasonable starting point, although the rr is too heavy. Note that B and A, are degenerate but of different types; in fact at p2 = 0 the B is pure B-type (i.e., it is a r recurrence) while the A, is pure Go-type. Contrary to the empirical rule, there are no S-type states at all.

452

SUNDARESAN

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TABLE S2 (Pure

I

Scalar

Interaction)

bs = 1, bp = 1, bv = 1, bA = 1, br = 1 JPC

Type

O-+(n)

B+C

l--co)

G-

c, (x 103)

2++(A)

G-+F-

o++(s) l++(4)

F+ + G+ Go + V-

2.28 1.69 4.23 3.45 4.21

G- + F-

8.11

+ F-

l+-(B) 3--(G)

o++ 1++ 1+3--

-0.0597 -0.0561 -0.0921 -0.0738 -0.0838

0.216 0.271 0.320 0.271 0.221

-0.13

0.414

II

P2 Interaction bs = 1, bp = 1, bv = -1, bA = -1,

l-2++

G

unstable

TABLE

0-t

Cl

B+& F- + V” + pGF-+ V”+pG-

4.04 1.28 4.45 4.08 3.82 4.06 6.77

S + G+

V- + F” v+ + u

F- + V” + pG-

TABLE

br = 1

-1.01 -1.00 -1.00 -1.01 -1.02 -1.01 -1.00

0.0089 0.0092 0.0044 0.0093 0.0014 0.0092 0.0039

III

P2 Interaction bp = -1, bv = 1, bA = 1, bT = -1 -

bs = -1,

c, ( x 102)

Type

o-+

c + v+ + PB

l-2++ o++ 1++ 1+3--

G- + T+pFG-+

T+pF-

G+ + T vu-

G-+

+ c + c

T+pF-

4.07 2.89 4.99 3.56 4.07 4.09 7.08

Cl

ca

-1.01 -1.01 -1.00 -1.01 -1.01 -1.02 -1.00

0.0091 0.0086 0.0059 0.0122 0.0093 0.0163 0.0049

RELATIVISTIC

453

QUARK MODELS

TABLE

IV

V21nteraction bs = 1, bp = -1, .5V = -4, bA = 4, br = 0

Type O-f

u+ + c

l--

S + G+

2++ O’f

S + G+ S + G+

lff 1+-

u- + u+ u- + u+

3--

S + G+

C,(xlOZ) very heavy 1.56 10.0

-1.00 -1.00 -1.01

5.24 46.0 21.6 12.7

TABLE

-0.333 -0.553

-1.00

0.0083 0.0069 0.0088 -0.0851 -0.290 -0.0054

V

A2 Interaction bs = 1, bp = -1, bV = &, bA = -8, /Jr = 0

Type l--

S + G+

2++ Off

S + G+ S + Gf G” + F” Go + F” S+G+

1+3--

Cl

C-2

none

O-f

1++

c, ( x 102)

0.227 0.486 0.124 0.224 0.233 1.26

TABLE

-1.57 -2.94 -1.18 -0.248 -0.253 -7.17

1.14 3.48 0.365 0.038 0.039 9.67

VI

PInteraction bs = 1, bp = 1, bV = 0, bA = 0, br = -fr c, ( x 102) O-f l--

B + PC S + G+

10.2

2++ o++ I++ 1+3--

S + G+ S + G+ B+pC B+pC S + G+

13.6 8.2 2.78 2.04 17.0

1.34

-0.345 -0.995 -0.986 -1.00 -0.360 -0.349 -0.978

0.155 0.0058 0.0029 0.0014 0.156 0.153 0.0001

454

SUNDARESAN

TABLE A2 + bs = 1, bp =

Type 0-f

l-2++ o++ lff lf3--

none S + S + S + none none S +

WATSON

AND

-1,

VII

V2 Interaction by = 0, bA = 0, br = 0

C,(xlP)

Cl

G+ G+ G+

1.04 1.40 0.71

-1.016 -1.018 -1.014

0.0133 0.0148 0.0120

G+

1.78

-1.020

0.0160

TABLE

VIII

A2 - V2 Interaction bs = 0, bp = 0, bv = 1, bA =

Type 0-+ l-2++ o++ 1++ 1+3--

G- + G- + T+G+ Go + Go + G- +

c, (x 1oy

pF” pF” G+

TABLE

B+PC S + S + S + B+PC B+rC S +

G+ G+ G+

G+

bT = 0

G

-0.374 -0.366 -0.473 -0.348 -0.358 -0.359

-0.0063 -0.0067 -3.12 0.153 0.156 -0.0063

IX

A* Interaction

bs = 1, bp = 1, bv = O,bA

Type

-1,

Cl

none 1.17 2.02 2.01 2.03 2.77 2.82

T T

V2 -

0-+ l-2++ o++ 1+-c 1+3--

G

c, ( x 102) 1.37 10.3 14.0 8.34 2.81 2.07 17.7

= 0, br = 0

G -0.348 -1.016 -1.018 - 1.026 -0.364 -0.353 -1.021

G 0.155 0.0137 0.0147 0.0247 0.155 0.153 0.0163

RELATIVISTIC

QUARK TABLE

455

MODELS

X

P2 + S2 Interaction b~=l,bp=l,by=0,b‘.j==0,b7.=l

Type FF-

c, (x 10:)

B + PC + v” + CLC+ V" + $I-

Cl

1.72 1.18 2.70

TABLE

C-2

-0.392 -0.484 -0.433

-0.961 -0.639 -1.145

XI

P2 + V2 Interaction bs = 1, bp = 0, by = -0.75, bA = 0.25, br = 1

Type 0-C l-2++

co (x

v+ + c F- + S F-+F+

103) 11.4 7.3 4.8

TABLE

Cl

C2

-1.07 -1.22 -0.843

0.088 1.89 -0.23

XII

Ps - PInteraction bs = 0, bp = 1, bv = 0.5, bA = -0.75,

Type

c, ( x 102)

O-f

B + PC

l-2++ o++ 1++ 1+3--

FFF+ F” F” F-

0.323 0.348 0.872 0.959 0.826 1.25 1.30

+ + + + + +

GGG+ Go Go G-

TABLE

bT. =

1

Cl -0.977 -0.196 -0.187 -0.173 -0.202 -0.181 -0.191

-0.204 0.098 0.129 0.222 -0.188 0.207 0.137

XIII

bs = 1, bp = 1, by = 1, bA = 0, br = 0

Type

o--e l-2++ o++ 1++ 1+3--

B+pC G- +

G+

C-

G’

+

T+G+ Go + pF” B+rC G- + G+

c, (x

102)

1.36 1.23 1.11 2.37 2.06 2.06 2.99

-0.348 -0.379 -0.373 -0.684 -0.353 -0.353 -0.366

0.155 -0.058 -0.086 -0.0775 0.153 0.153 -0.088

456

SUNDARESAN

AND

TABLE

WATSON

XIV

bs = ---I, bp = 1, bv = 1, bA = -1, Type 0-+ l-2++ o++ 1+* 1+3--

C,(

v+c N.R. N.R. N.R. N.R. N.R. N.R.

x 102)

Cl

1.92 0.0142 0.0557 0.0425 0.0557 0.0121 0.0121

Type O-f

co (x

N.R.

O’i

bA = 1, bT = 1

G

102)

0.218 2.91 5.82 6.06 0.596 0.499 8.24

N.R. N.R.

lf’ lf3--

F- + V”

0.0353 0.325 0.319 0.318 0.315 0.314 0.314

XV

bp = 1, bv = -1,

F- + V” F- + V” F+ + pG+

l-2++

G

-1.04 -0.0136 -0.0277 -0.0239 -0.0766 -0.0395 -0.0395

TABLE

bs = -1,

bT = 1

-0.059 -1.00 -1.01 -1.01 -0.0935 -0.0899 -1.00

TABLE

0.215 0.0068 0.0039 0.0092 0.244 0.156 0.0032

XVI

bs = 1, bp = 1, by = 0, bA = 1, bT = 0

o-+ l-2++

Type

c, ( x 102)

B+rC U0 + FUo + F-

0.043 9.7

TABLE

O-f

-0.0234 -1.017 unstable

3.11 0.015

XVII l--

bs

br

a

ha

b

0.99 0.995 0.99 0.99 0.995 0.99

0.96 0.96 0.97 0.96 0.96 0.97

0.01 0.01 0.01 0.001 0.001 0.001

1.04 1.04 1.04 1.004 1.004 1.004

1.0540 1.0540 1.0439 1.0442 1.0442 1.0335

2++ As 1.1068 1.1018 1.1068 1.0158 1.0108 1.0158

b 1.0877 1.0865 1.0756 1.0463 1.0463 1.0356

As 1.0950 1.0834 1.0885 1.0178 1.0127 1.0178

o++ As 1.0505 1.0454 1.0505 1.0141 1.0090 1.0141

RELATIVISTIC

QUARK

457

MODELS

Table XIV.

This gives rise to a mixture of nonrelativistic and relativistic states. Thus the 7~is B-type and relatively steep, while the other states have N-R wavefunctions and have very shallow plots. In fact mixed in with the N-R states there are other relativistic states with much steeper plots. Not only do the wavefunctions have N-R form, but the mass splitting is characteristically N-R, and the F+G+ states and the F-G- states (i.e., the 3L + I., and the 3L - lJ) lie much closer than they do in general. Table XV. Table XVI.

Abnormal

parity states too low and have very shallow plots.

No good for numerical reasons.

Table XVII. To give some feeling for the effect of varying the parameters on the solutions, we show the results for a basically pseudoscalar interaction with small variations of bs , br and (Y. This table has a different interpretation; the columns show the eigenvalues of various types corresponding to p2 = 0. The slopes may be taken as 1 for all the plots; in fact it varies between 1.01 and 0.985. As in Table II above, the second-order term is completely negligible. It turns out that we can write Ai - 1 = (l/hi) - 1 + C + AU,

where A and C have the following values (approximately): Type O-f

l-2++

B FS F-

S

A

c

4

0.0

1.1 10

0.0 0.0

3.5 9

0.01 -0.005

There appears to be no rationale behind these numbers; moreover it seems most odd that all the plots should have the same slope and that this should be so nearly -1. There appear to be some effects of “eigenvalue repulsion” when h,- = h, , but a@art from this, the eigenvalue of one type is almost unaffected by a change in the b6 of the other. Of the interactions discussed above, we see that the interactions in Tables VI, IX, XII-XV appear ri priori attractive alternatives for a quark model, but all of these suffer from some flaws on closer examination. We rule out (1) the interaction in Table VI because 6 is inevitably lighter than p, as the states are of the same type. (2) the interaction in Table IX because it gives an A, inevitably heavier than B (see the appendix).

4.58

SUNDARESAN AND WATSON

(3) the interaction in Table XII becauseit is rather similar to P2. (4) the interaction in Table XIII becauseit cannot give rise to a split A, in a very attractive fashion since S-type eigenvalues never appear (in violation of the empirical rule). (5) the interaction in Table XIV because it makes the normal parity particles N-R type. (6) the interaction in Table XV because it makes the abnormal parity particles N-R type. Vector and axial-vector currents seem to have a fundamental importance in physics, and one might like to consider interactions of the form

However our V2 and A2 (and combinations) interactions seemto lead to unsatisfactory results. Table VIII, which involves the most reasonableof theseinteractions has wavefunctions of an unacceptable type in addition to the lack of a r.

5. RELATIVISTIC

QUARK

MODELS

The discussion above confirms the widely held view that the Bethe-Salpeter equation has too many solutions. Nevertheless we have attempted by a judicious choice of the interaction to construct a relativistic quark model whose mass spectrum is broadly in agreement with the observed mass spectrum and whose wavefunctions agree with our prejudices about the type of solutions. Although there is difficulty in identifying the states of the model with the observed states, the identification we have produced seemsthe most reasonable from the point of nonrelativistic models, and it seemsthe extra states are not physically significant. The computational problem is fairly large; starting from a given interaction we must first construct the matrix M for a given Jpc and find the eigenvalues and eigenvectors. We calculate X as a function of p2, and repeat the whole processfor several Jpc values with the sameinteraction. Then the program selectsa value of h to make (M,,,/M,) physically correct (the choice of making this massratio correct is purely for convenience) and calculates the other massesby interpolation. Unfortunately we have five parameters (bL , b, , b, , b, , bT and cy.which have one relation between them) and hence must use a minimization procedure to vary the couplings until a physical massspectrum is obtained. The phrase “a physical mass spectrum” begs a number of questions. These calculations apply only to equal-mass quarks, and hence are restricted to Z = 0 and Z = 1 mesons. Z = 0 mesonsunfortunately mix, and this is difficult to take into account, so we consider the charged mesonsonly. There is still the trouble of

RELATIVISTIC

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MODELS

459

what physical states are there and we have assumed that the 7~,p, 6, A, , B, AzL , A,, and g (= R, ?) are all physical. As “the problem of the split A, peak is obviously the central problem in mesonic physics today” [8] it would be fainthearted not to use the freedom inherent in this model to explain it. The g is assumed to be 3-- in line with a number of experiments. The minimization is far from simple because the dependence of the physical masses on the couplings is hideously nonlinear; in addition to the complicated but presumably analytic dependence of the eigenvalues on the couplings, we are interested principally in the lowest eigenvalue and this may completely change its type for a small change of the quantities bi . Example 17 of the previous section (for the 2’+) shows an explicit occurrence of this. Thus the program may attempt to interpolate between an S-type eigenvalue in one region of the b-space and an F--type in a nearby region. We then use a fitting program (VAOSA) including a number of checks. Clearly we cannot hope to start from any interaction and arrive at a comprehensible solution, so we have to choose our starting points using the empirical rules above. Even then there are too many parameters for us to carry out an exhaustive study of the possible solutions,5 and we must thus include our prejudices as to what sort of wavefunctions we want. Firstly we feel that the O-+(r) must be of the (B + PC)type which Llewellyn Smith [3] has shown resolves the Van Royen-Weisskopf paradox. Llewellyn Smith and subsequently Dias de Deus [9] have given fairly strong arguments that the p wavefunction must be (F- + PG-)-type. The first constraint implies that b, > 0, and we may choose b, = 1; then the second implies that bT ‘v 1. Then it would be of great interest to produce a split A, ; this can most easily be done by making b, N I so that the F- and the S states are almost superimposed. To avoid a plethora of states, and furthermore to avoid N-R wavefunctions we then require b, ‘v bV < 1, and for simplicity and stability we may choose b, = b, = - I. Thus our starting point is almost exactly the P2 interaction. We show the resultant mass spectrum and some wavefunctions in Tables XVIII and XIX. It is difficult to obtain a mass spectrum which is everywhere correct; the minimization program will happily provide two almost degenerate 2++ states,6 but then the 6 is more or less degenerate with the p, while the A, and the B tend to be in the wrong order. If the A, is V--type, then choosing b, sufficiently large to make the A, lighter than the B in turn drags the p below the 7r. In other words the experimental x - B trajectory is too flat for this model. To obtain these we have used a 16 x 16 matrix (r = 0, n = Jand r = 1, n = Jstates) with /3 = l/(Mcu). The underlined particles are fitted by the program, but the r and AzL are correct 5 Of course our limiting to 5 parameters is artificial. We could easily imagine interactions of the form p,#jz , or the relativistic generalization of k’(c) = P could be V(r) = r2S(P * r) rather than V(r) = P. B Exact degeneracy leads to numerical instability.

460

SUNDARESAN

AND

TABLE Particle

Jpc

XVIII”

Mass Exp. mass

o++ to+-1 (&

0.617 0.680 0.139

n

WATSON

Type

0.960

S,G+ S, G+

0.139

B, C

Particle

2nd mass

Type

1.44 1.27 1.28 Complex 1.08 1.40 1.04 1.38

F+, G+ S, G+ V+, B

Complex

l-W’) l’1++

0.879 1.02 0.965

P

B Al A 2L

2++

(2f-1 2-3-(3-t)

F-, V” s, s

1.22 1.05 1.29

1.01 1.29 1.39 1.35 1.38 1.61 1.68

F,(1540)? *“(&IO)? g

2-t

0.770

B, C V-, F” F-, GS, G+ B, VV-, B F-, GS, G+

1.66

1.43 1.67 1.39 1.68 1.71 1.90

‘hi

p(1710)?

Complex

3+-

R(1750)? S(1930)?

3++ 4++ (4f-) 4-+ 4--

1.67 1.87 1.93

Complex 1.94 1.95 Complex Complex

V-, F” F-, V” S, G+

Complex v-, c

1.51

S, G+ F-, V” V-, B B, C S, G+ V”, FV-, B B, VS, G+ F-, GV-, F” S, G-

& C

a The parameters that give this solution are: bL = 0.99, b,y = 1.00, by = -1.01,

b,4 = -1.01,

h = 0.04,

bT = 0.97,

a = 0.0124.

These parameters imply a quark mass of 42 GeV. TABLE D Go:

S 00

Tf 0.135 0.987

BOO GO

P

Fill G-0 KTI u:, VA

XIX

0.905 0.135 0.142

0.104 -0.351

0.9997 0.0235

B

(p = 0.0236)

0.387 0.353 0.308

0.982 0.166

A 2L

Al

Fh

2:

FOl 61 V&? VZO G

0.893 0.201 0.142 0.124 -0.336

(/I = 0.164) ‘hi

G,+, S GYl

0.160 0.965 -0.128

RELATIVISTIC

QUARK

MODELS

461

by definition. Some speculative assignments are given for states which are not well established. In the table “2nd mass” refers to the first excited states of a given .F. This means either a radial excitation or a queer state. We give next the wavefunctions for some of the more interesting particles. The 6 and QTcontain 8 components and the rest 16, but we have only written down the largest components. The numbers have been arbitrarily normalized to 1; the overall normalization constant will be provided by the correct normalization of the wavefunction. There are clearly a number of states for which there is no physical evidence even bearing in mind a potential inaccuracy of 300400 MeV in the masses. We may speculate that these states are unobservably wide, or unobservably narrow. As an example of the problems of interpretation which may arise, consider the 0++ states. It is a very serious problem (not only for the quark model) to explain why the S(963) should have a width of about 8 MeV while the ~(750) has a width of about 300 MeV, when it is assumed that both are in the same octet. The evidence that the 6 is 0++ comes from various experiments which observe a ~‘7)enhancement, but this is invariably higher (~985 MeV) and wider (-40 MeV) than the 6 as seen in missing massexperiments. Thus it seemsplausible that the latter particle is the true O++,and so the widths are no longer so incompatible, while the 6(963) seenin missingmassexperiments is a more exotic beast altogether, such as the 0+- required by our model. This would require a 4~ decay mode, which would inevitably be small. This model, in common with a number of others, has a state which could be called p’. However, becausethe wavefunction is dominantly S-type, the coupling to nn will be very strong and to the electromagnetic current very weak. Hence such a state would be difficult to observe. It also seemslikely that the odd daughter states in general will have very small widths; for this reason we have bracketed them in Table XVIII.

6.

CONCLUSIONS

We have described some results obtained from the spinor Bethe-Salpeter equation, with particular reference to the formulation of a relativistic quark model. However some of our results have somewhat wider implications. We summarize first the more general conclusions for this type of deeply bound system by contrasting them with some widely held prejudices which originate from the nonrelativistic domain. (1) It is usually stated that the quark model does not allow exotic states of the second kind (e.g., l-f). We seethat we do have statesof this kind in the present 595/7+-I

1

462

SUNDARESAN

AND

WATSON

model, in other words, exotic states of the second kind are possible even in the qq model (relativistic). (2) It is usually stated that for any given J pc the wavefunctions can differ only by the radial quantum number (with the exception of J = L - I and J = L + 1 states). It is clear in our present model that this is not true, we have the extra quantum number n, and in addition the spinor structure can be quite different for different states of the same Jpc. (3) It is found in our work that there need not be a close connection between normal and abnormal parities for the samespin. (4) It is usually assumed that there is no strong dependence of the mass spectrum on the type of coupling assumed. We seein our model that this is far from true, different assumed couplings giving quite different mass spectra and eigenvectors. (5) It is usually stated that trajectories are “naturally” straight in mass (rather than masssquared). In fact, the solutions we have found give trajectories, which are straight in masssquared, to a good approximation. (6) One encounters the usual statement that daughter states are separated by one unit of angular momentum from their parent trajectory at mass squared equal to zero. We find that this is true only in certain sectors. (7) It is usually stated that Regge trajectories are real below threshold. Here we have examples of trajectories colliding and thus inducing complex values of $. The Bethe-Salpeter equation, as used above, leadsto only a partially satisfactory relativistic quark model. However it is probably true to say that it is the only rigorously correct relativistic two-body equation available for application to deeply bound systems. Since the quark model is essentially a relativistic problem, it is clearly necessary to construct some sort of covariant wavefunctions to make any progress, and we feel that the quark model described in the preceding section has at least some interesting features. Two interesting conclusions relate to the nonrelativistic quark model; the successof the baryon spectroscopy and electromagnetic properties is in sharp contrast to that of the mesons. It is possible that the interaction of 3 quarks may be of the type to give rise to an almost perfectly N-R wavefunction. Secondly the strong decays N* + NT are often calculated [lo] by assuming that the interaction is of the form a . b, which in turn is simply the nonrelativistic reduction of Uy5u. Now our calculations suggeststrongly that

Xn(4)- Y@(q) + WmN and hence a not unreasonable approximation would be x,, N By, which is of course just that used above. This is an interesting example of the subtlety of the relativistic model, becausethis approximation would be totally unable to explain the Van Royen-Weisskopf paradox [5].

RELATIVISTIC

QUARK

MODELS

463

We have found a plausible mechanism for the A, splitting, which seems to require p’ at a somewhat lower mass. The main deficiency of the model is the comparative lightness of the B-meson. This could probably be overcome fairly simply by allowing the o( to vary a little for the different couplings. This could flatten out our r - B trajectory with little effect on the remaining particles. One of the more disturbing features of the present discussion is that the success of SU(6) and/or the naive quark model appears to be a mere coincidence. It would be perfectly easy for nature to choose an interaction which would give rise only to normal parity particles; in this case it is impossible to believe that the naive quark model would ever have been proposed! The P2 interaction appears almost unique; not only is the mass spectrum very reasonable, but it has remarkable numerical stability. The matrix is nearly symmetric, while the plots are very straight and all have the same slope. A most important question is how general these results are. The first point is the use of the oscillator potential, as opposed to the usual ladder approximation. The reason that we choose the oscillator is of course that we can then “solve” the equations! The ladder approximation, however, (which is supposed by many to be more “realistic”) not only ignores an infinitude of diagrams and presumably an infinitude of exchanged particles but, more important, does not seem to give linear trajectories. It is very easy to find models which give potentials which are at least soft-centered.’ Even assuming that the interaction is correct, we have had to make the approximation of ignoring other channels; thus the p mass will be shifted because it is coupled to the ~ZTchannel, etc. Dispersion theoretic arguments [12] suggest that the shift will satisfy Re 6111~= A Im 6m2, where A is roughly of the order of I, and hence the mo2 may differ by as much as 0.3 (GeV)2 from its position as calculated here. It would be useful to carry out a similar calculation in Bethe-Salpeter language. The most important content of the present wavefunctions is the possibility of calculating properties of mesonsin a covariant manner. As yet we have not done so, but it is worth noting that Dias de Deus [9] and Llewyllyn Smith [3] have shown that the wavefunctions rather similar to the type we find satisfy such diverse requirements as vector-meson dominance, P.C.A.C., a good value for the w -+ r’y coupling and corrections to the baryon magnetic moments which are in agreement with experiment. 7 The most trivial is three scalar particles with their couplings and masses arranged to cancel the first three terms in their expansion at r = 0. More plausibly, a straight-line Regge trajectory with a suitable residue anda suitablenonlinearLagrangian[ll] both givesoft-centered potentials.

464

SUNDARESAN AND WATSON

APPENDIX:

DAUGHTER

STATES

Daughters provide one of the more awkward problems here, probably because our equations are not crossing symmetric. In the scalar equation, it is easy to see that the so-called abnormal solutions of the Bethe-Salpeter equation lie integer units of angular momentum below the parent at p2 = 0 [l, Eqs. (2.13)-(2.22)]. From these it is simple to see that at pL2= 0 the equation is independent of J, and so there is a set of degenerate daughter states corresponding to the different values of n. It is unfortunate that we can solve the equation for any p2 < 0 and any h but only integral J. Hence to study daughters we must take an integral spin particle at p2 = 0 and see whether the daughters have the same eigenvalue. It is hard to believe that the daughter structure will change fundamentally for nonintegral J. In the spinor problem the situation is vastly more confused. At p2 = 0, the B and G, vectors are completely decoupled, and so the above argument holds. However, there is a coupling between (e.g.) the S and F- vectors. Hence the S- and F--type eigenvalues mix, and so do their daughters, but in an n- and J-dependent way, and hence there is no need for the equal spacing to hold except as a dynamical accident. If we then make the (almost obvious) equation of daughters with abnormal states the statements in Section 2 follow. The interpretation of odd daughters in particular is very difficult; as is well known they have negative norm in the scalar case, but the wavefunctions vanish8 when the external particles are on the mass shell [3,4]. To see this, consider the kinematic conditions P2 = 11.2,

PI2 = M2,

P22 = M2.

(B.1)

From this it follows that (P + q)2 = (P - q)2 or P * q = 0, which gives [I, (2.6) and (2.7)] cos # = 0. Now &l(O) = 0 if 12- L’is odd, so the wavefunctions vanish identically. This implies that they are not poles of the two-body S-matrix, and we may thus feel that they are quite unphysical. However, it is then somewhat embarrassing that triplet J = L states may be (in a manner of speaking) daughters of the singlet J + 1 = L + 1 states. To see this, consider B-type states at p2 = 0 (when they decouple). Then

If J = 2, then n = 2, 4, 6 ,... [i.e., (J - it ) even] states have even-time parity (under t -+ -t) while y1= 3, 5, 7 [i.e., (J - n) odd] have odd-time parity (in fact, of course, these two states cannot simultaneously be massless). Since they both have 8 The identical problem arises in the relativistic quark model of Feynman et al. [13], and there appears to be no satisfactory solution.

RELATIVISTIC

QUARK MODELS

465

positive spatial parity, the charge conjugation must change; hence the first is 2-f and the second 2--. This is an unconventional meaning for daughters, but by continuing the 2-- trajectory we can see that there would be (at p2 = 0) a J = 1, n = 2(1++) state degenerate with the J = 2, n = 2, (2-+) state. Hence we would expect the l++ wavefunction to vanish on mass shell. This will not happen if the 1++ is bound in the F”, Go, U+ or U- sectors, because in these sectors the even-time parity states vanish on mass shell. This is understandable nonrelativistically speaking because we expect J = L triplet states to have F” = Go, rest small, while J = L singlet states will have B ‘v C, rest small. Thus for the P2 interaction it is a considerable relief that the 1+r is bound largely in the P sector. However, if we assume that we may disregard those parts of the wavefunction which vanish on mass shell, then we are in the even more peculiar situation of having particles some part of whose wavefunction may be disregarded. This seems to lead to conceptual difficulties. We have not investigated whether the odd daughters from our spinor equation have normalization problems. It is a reasonable speculation that the equal spacing of the daughters might be restored in a truly crossing symmetric theory.

ACKNOWLEDGMENTS We would like to thank Professor R. H. Dalitz for his interest in this work and a number of useful conversations. Computing facilities at the Rutherford Laboratory are gratefully acknowledged. One of us (P.J.S.W.) would like to thank Professor Sir Rudolf Peierls for his hospitality at Oxford. REFERENCEB I. 2.

3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13.

M. K. SUNDARE~AN AND P. J. S. WATSON, Ann. Phys. 59 (1970), 375. L. SIJSSKIND, Phys. Rev. Lett. 23 (1969), 545. C. LLEWYLLYN SMITH, Ann. Phys. 53 (1969), 521. N. NAKANISHI, Prog. Theor. Phys. Supp. No. 43 (1969). R. P. VAN ROVEN AND V. F. WEISSKOPF, Nuovo Cimento A 50 (1967), 617. R. H. DALITZ, private communication. R. E. CUTKOSKY AND B. DEO, Phys. Rev. Lett. 19 (1967), 1256. R. H. DALITZ, contributed talk to the International Conference on Symmetries and Quark Models, Wayne State University, Detroit, 1969. J. DIAS DE DEUS, “PCAC and Vector meson dominance in the Relativistic Quark Model,” preprint, University College, London. D. FAIMAN AND A. W. HENDRY, Phys. Rev. 180 (1969), 1609. P. K. MITTER, private communication. E. J. SQIJIRES AND P. J. S. WATSON, Ann. Phys. 41 (1967), 409. R. P. FEYNMANN, M. IOSLINGER, AND F. RAVNDAL, Phys. Rev. D 3 (1971), 2706.