Dispersion relations and higher symmetries

ANNALS

OF

PHYSICS:

39,

381-434

Dispersion

(1966)

Relations

and

Higher

Symmetries*

S. FUBINI~ Institute

of Theoretical

Department

Physics, Department of Physics, Stanford University, Stanford, California and of Physics, University of California, Los Angeles, CaliforniaS

G. SEGR~ Lawrence

Radiation

Laboratory,

University

of

California,

Berkeley,

California

AND

J. D. WALECKA§ Institute

of Theoretical

Physics, Department of Physics, Stanford, California

Stanford

University,

A dispersion approach is employed to obtain a covariant formulation of higher symmetry groups. This is done by studying the commutation relations of the 144 generators of compact U(12) in the framework of a quark model. The generators (or generalized charges) are classified in a covariant manner and the problem of the convergence of the sum rules obtained from the commutators is analyzed. It is found that only some of the charges may be viewed as approximate constants of the motion, that is to say their off-diagonal matrix elements have a gentle high energy behavior and the corresponding dispersion sum rules converge. The above mentioned “good” charges coincide with those of the collinear SU(6) group. In this way we have a covariant derivation of SU(6)W; inserting in the sum rules the bayon fk+ octet and ah+ decnplet, we reproduce some of the consequences of SU(6)W symmetry. The matrix elements of our “good” charges can be related to scattering amplitudes of pseudoscalar and vector particles and thereby we obtain an estimate of the limits of validity of the group symmetry. * Supported in part by the U.S. Air Force through Air Force Office of Scientific Contract AF 49(638)-1389 and by the U. S. Atomic Energy Commission. t Permanent address: Instituto di Fisica dell’Universit& Torino, Italy. $ Supported in part through a grant from the National Science Foundation. 5 A. P. Sloan Foundation Fellow. 381

Research

382

FUBINI,

SEGRh,

I.

AND

WALECKA

INTRODUCTION

The use of group theoretical techniques has proven to be a very valuable tool in the study of elementary particle physics, in both the classification of particles and the relating of transition amplitudes. Spurred on by the success of SU(3) (I), attempts have been made to find higher symmetry groups which include internal and dynamical variables in a nontrivial way, that is, other than as the direct product of two symmetry groups. The most famous of these schemes, SU(6) (2), which includes the internal symmetry group SU(3) and the group of spin rotations SU(2), has met with considerable success in correlating experimental data. The group of spin rotations is a symmetry of states at rest, whose orbital angular momentum is therefore zero; one is faced however by apparently insurmountable problems in trying to embed this zero-momentum symmetry SU(6) in a fully relativistic structure (3). To say this in another way, the only possible invariance group of physical states containing both the Poincare group and an internal symmetry group is a direct product of the two. One is immediately led to ask then whether it is possible to obtain, at least in a given approximation, the so called “good results of XU( 6)” without postulating SU( 6) as a symmetry of the physical states. Gell-Mann (4)) over the course of the past few years, has repeatedly emphasized what may be the solution, namely that the currents, whose integrals over all space form the generators of the group, can be taken to obey equal-time commutation rules. By assuming the commutation relations to be saturated by some set of physical states, we may reproduce the results previously obtained by group theory. It is clear in fact that if we take the commutation rules of an algebra of currents between states belonging to what would be an irreducible representation of the group and then assume the commutator to be saturated by the states which would form the same representation of the group to which the initial and final state belong, we will by consistency reproduce the same results as saying directly that the states formed a representation of the group. The so called “group theoretical result” has been obtained however without assuming the group to be an even approximate symmetry. This point of view has been stressed by R. Dashen and M. Gell-Mann and B. W. Lee (5). A Lorentz-covariant dispersive approach to the matrix elements of commutation relations of SU( 3) and chiral XU( 3) X SU( 3) has recently been proposed (6). Poles in the dispersion relation lead to the previously discussed “group theoretical result” and the higher singularities correspond to the symmetry breaking corrections. This technique has been successfully applied to several cases of physical interest (7). In this paper we shall study the commutation relations of higher symmetries (SU( 6) and U( 12) in particular) by this method. Consider the 144 Hermitian charges

Qi, = 1 $(x)x”r,+(z)

dx

a = 0, 1, . . . 8, /J, = 1, 2, . . e 16

(1.1)

DISPERSION

RELATIONS

AND

SYMMETRIES

383

where X” are the nine SU(3) matrices and rlr are the 16 Dirac matrices between the fields 4 and 1c,which we take to be free quark fields. We obtain the commutation relations by explicitly assuming a quark model’ though the matrix elements of the charges have of course a well-defined meaning independent of how we have derived their commutation relations. Out of the 144 charges of ( 1.1) whose commutation relations close to form the compact U( 12) group, 72 are, up to a U( 3) rotation, charges that appear in the weak and electromagnetic interactions and therefore have directly measurable matrix elements. They form the ohiral U( 6) X U( 6) subgroup of U( 12) with generators (8).

Q,* = j- $h”yfi# dx (1.2)

The other 72 do not have directly measurable matrix elements. Using the method of ref. 6, we shall attempt to write dispersion relations for the commutators of the U( 12) charges. These relations could be used either as sum rules into which we would insert observable physical quantities or, limiting ourselves to a given set of particles and resonances, as a means for obtaining selfconsistent relations among this set, which we then call a “supermultiplet,” reproducing results very much like those obtained if the “supermultiplet” formed an irreducible representation of the group. One of the problems we must face immediately in this dispersive treatment is that of subtractions, for we are calculating the value of the amplitude at a fixed symmetry point. If a subtraction is needed, our relations become completely useless. As we shall see later, there are very strong reasons for believing that less than half of the 144 charges give rise to commutation relations which can be expressed as a dispersion relation not requiring subt,ractions. Unfortunately those that do, the so called “good charges” do not form a subgroup of U(12); the largest group which can be formed from these charges is Dhe chiral U(3) X U(3) whose generators are

(1.3)

It therefore

seems impossible

to reconcile in a general manner a purely group

1 Recently Dashen and Frautschi directly by means of a strong interaction this interesting possibility.

have tried bootstrap,

to

obtain these commutation but we shall have nothing

relations to say about

384

FUBINI,

SEGRk,

AND

WALECKA

theoretical description of higher symmetries with the unsubtracted dispersion relation approach. The situation can be viewed as follows: in U( 3) X U( 3) the symmetry is broken by the mass term in the Hamiltonian, which becomes negligible at high momenta whereas the higher symmetries are broken by the kinetic energy term in the Hamiltonian which does not vanish at large momenta and prevents our dispersion relations from converging. Some of our sum rules are convergent however and it seems useful to investigate them in detail. As an example, we shall reproduce some of the good results of SU( 6) W by keeping in the matrix element of commutators between octet $h+ baryon states only octet $+jf and decuplet 45+ states. In Section II, we shall summarize, for the sake of completeness, the main features of the dispersive approach. Section III contains a covariant classification and discussion of the charges, currents and divergences appearing in U( 12). We shall there separate the “good” from the “bad” charges on the basis of which lead to unsubtracted dispersion relations and discuss generally the subtraction problem. Section IV is an analysis of the convergent dispersion relations as sum rules and as means for obtaining self-consistent relations between particles by approximating with a “supermultiplet” the intermediate states appearing in the commutation relation. In section V we shall use the decuplet-dominance model and compare the predictions so obtained with those of dynamical groups and with experiment. Several appendices are included to explain the details of the calculations. II. GENERAL

DISCUSSION

In order to make the paper self-contained we will summarize in this section the main features of the dispersion approach. Although there are now general and more satisfactory derivations of the dispersion sum rules (9), we shall follow here the simple intuitive approach of Fubini, Furlan and Rossetti (6). Let us start by writing the charge asthe flux of the currentj, through the threedimensional surface d3x ’

Q = -jM,,t

da

(2.1)

Now using Gauss’ theorem we can rewrite Eq. (2.1) in terms of a four-dimensional volume integral eWe useB metric such that v,, = (v, ivo) and a-b = a.b - U&O. Our y matrices are Hermitian and satisfy 7r~v + Y”Y,, =26,. . The Dirac equation is (ir. P + m)u(P) = 0 and we set c = h = 1. Note y6 = -yly2y3y4.

DISPERSION

Q = +J

RELATIONS

N4ebd

385

AND SYMIMETRIES

d4x + Q(-

cc> (2.2)

= - $ D(xM--2,rlJ d4x+ Q(Oc 1 where D(x) E s!g

P

Since the time integration appearing in matrix elements of this expression are to be defined by an adiabatic limiting procedure, the contribution at f co connects only states of equal energy so we can write Q 2 1 D(x)e(x,,,?,> d4x G -j-

0(x)0( -x,r],)

d*x

(2.3)

the symbol A meaning that the equation is valid only when taken between states of different energy.3 In this way the equal time commutator between a charge and a current can be simply written (2.4) where

Fp(pJ = -1 (p~j[D(z),j,(0)llpz)ei*‘“~(--x,rl~)d4x

(2.5)

The commutator between two charges can then be easily obtained from Eq. (3.5) by letting pl -+ 712. This procedure has however the disadvantage of requiring the use of a limiting procedure applied to a nonforward amplitude. The most direct method is to use Eq. (2.2) twice in the commutator (pl I[&“, &#]I p2) which leads to (2.6) where P(q)

= I/‘-“=

/

e-+8(x0) d4x(P 1W(x),

D”(O>lI P)

(2.7)

The vector r], is the timelike vector indicating the normal to the spacelike surface &. We may, without lossof generality, simply take q,, = P&/m. Equation 8 If equal energy states appear, one can use suitable limiting fied in ref. 9.

procedures

which are justi-

386

FUBINI,

SEGRh,

AND

WALECKA

(2.7) forms the basis for obtaining dispersion relations for the invariants of 5( CJ,). The absorptive parts of these dispersion relations are obtained from the usual formula

Abss”%d

=;

(2d4c

{(PID”I~)(~ID~IP)~(~‘(P~

-P

- q) (2.8)

- ip, Do In>{n 1D” 1P)t4’(P,

- P + q)} ,j,,/T

Our sums rules contain matrix elements of the form (P 1D ( n). In the case of currents like $r,\c/, or &ysy,,# these matrix elements can be directly related to amplitudes appearing in weak or electromagnetic interaction. It is however possible, as pointed out in refs. 6 and 7, to relate those matrix elements to strong interaction amplitudes using a procedure analogous to that leading to the Goldberger Treiman relation. An alternative, though lesselegant, way of obtaining dispersion relations is to use completenessdirectly in the commutator of charges.

(P I[Q”, &‘I I P> = F {(P I Qa I n>b I Q’ I P> - 0’ I QB I 4(n I Q” I P>I Since Q” and Q” contain a three-dimensional integral the difference between initial, intermediate, and final three-momenta will be zero. Therefore the fourdimensional momentum transfer between initial and intermediate state will only have a nonvanishing time component E, - E, = 48, + p” - dn? + p2 where 4% is the mass of the intermediate state. This causes the transition matrix element to depend on a virtual four-momentum p, - p which becomes more timelike asthe intermediate state massincreases.This is unpleasant because singularities in $ appearing for q2in the timelike region may tend to emphasize large S, contributions to our dispersion integrals. The simplest way of avoiding this situation is to consider the special frame in which p -+ 00 and therefore E, - E, m (8, - nz2)/2p + 0. This means that our matrix elements are on “the massshell” in the sensethat all four components of the momentum transfer are zero. The sum rules obtained in this way coincide completely with those obtained by meansof the covariant method since the matrix elements are put into the mass shell, and the noncovariant factors compensate each otherq4This equivalence is actually more general since it can be proven at least in simple casesthat covariant dispersion relations can be obtained from the Low equation by meansof completenessused in the p + 00 frame. Let us now say a few words about the most troublesome problem, that of subtractions. We do not believe there is any completely rigorous argument for determining whether a dispersion relation requires subtractions. In our case we can *We

wish

to thank

Dr.

R. Norton

for very

illuminating

discussions

on this

point.

DISPERSIOK

RELATIONS

AND

387

SYMMETRIES

however find some clear criteria for deciding this fundamental point. First of all, if it is possible to express the sum rule in terms of observable amplitudes, high energy theorems like the Pomeranchuk one can be used to estimate the convergence of the integrals. For example in the Adler-Weisberger sum rule for GA/Gv the presence of the mt - u- difference makes us confident of the convergence of the sum rule. Another more restrictive test is obtained by the use of completeness in the p 4 5~ limit where the pole term comes entirely from the one-particle contribution in the completeness sum. In the free field model, where we know the answer, we have no cuts and the complete absence of renormalization. Therefore, if there are no subtra&ions in our dispersion relations, the single-particle term must reproduce the other side of the commutator identity.5 If this is not the case, we are led to believe that subtraction constants are needed. The general application of this method will be discussed in the next section. Wa wish however to show, by a simple example, how the criterion works. Consider the two rommutators

T(-$ czx] = - [ ST12)+ czx 03) [ j ih5 d+‘# clx, [ iL-15 The r.h.s. is the same and the “charges” In the limit p + cc

on the 1.h.s. differ by a y4 factor.

where

We see that in the sum rule (A) the one-particle term leads to the correct answer. This is not the case for sum rule (B) in which subtraction constants are required. III.

CLASSIFICATION

OF

CHARGES

AND

CURRENTS

this section we wish to consider in more detail the transformation properties under the Lorenta group of the generators of U( 12). Our primary concern will be In

5 Of course using completeness, the commutation relations for the free field model are saturated using one-particle and one-particle-plus-pair states corresponding to the two different time orderings of a unique Feynman graph. In the p --t 00 limit the one particle contribution give rise to the pole term whereas the pair term goes into a subtraction term.

388

FUBINI,

SEGRik,

AND

WALECKA

the possibility of deriving dispersion relations from the commutators of charges, laying special emphasis on the question of subtractions. To simplify the discussion we shall drop, for the moment, all SU( 3) indices. The sixteen Hermitian operators $I’,,# can be classified, according to their transformation properties under the Lore&z group as a scalar (S), vector (V) , axial vector (A), tensor (T), or pseudoscalar (I’). These five types of currents can in turn be expressed in two equivalent forms as is shown in Table I. We have written the current, operators in two forms corresponding to the covariant tensors and their duals as we will need both forms for our subsequent discussion. The tensors can be related to their duals by using the completely antisymmetrical tensor ~cllrzp3p4as, e.g., for the tensor current

(3.1) The charges in turn are defined as the space integral of the currents $I’#. We list, the sixteen charges in Table II. In order to apply the procedure of Section II for deriving dispersion relation from the equal time commutators of the charges, we have to explicitly exhibit their Lorentz character. We recall from the last, section that the electromagnetic charge Q” may be envisaged as the flux of the vector current J, through the three-dimensional surface element du = dx

Qv = -$

JNv~M cla

(3.2)

q being the normal to the surface. All other charges can be cast into a covariant form by analogy to Eq. (3.2). Since we have to construct charges from integrals of spacelike as well as timelike components of currents, Eq. (3.1) tells us that we have to consider the fluxes of both dual forms of the currents given in Table I. In Table III we list the covariant form of the charges with their nonexplicitly covariant form (corresponding to IJ&along the time axis) given in parenthesis. We note that the covariant charges are completely antisymmetric in their indices and have no components parallel to q,, . For example Q:1y2yshas only one nonzero component, QFZ3, for qrr along the time axis.

DISPERSION

RELATIONS

AND

TABLE THE

QUARK

TABLE COVARIANT

FORMS

OF THE CHARGES

AND

389

SYMMETRIES

II CHARGES

III THEIR

NONCOVARIANT

EQUIVALENTS

By using Gauss’ theorem, these integrals of fluxes over surfaces may be converted into volume integrals of the four-divergences as discussed in Section II. Thus, for example,

Q"lllP2=

- s J

(3.3)

where

D;,,(x)

= dJ’;;’

(xl

(3.4)

8

We have to deal with the following divergences

D-Sa/.tz,ia; Dv, %,

; kT,

fitiT;

D&a , fi”;

Dk.,m

.

The divergences are antisymmetric tensors whose divergences, becauseof (3.4)) are identically zero, e.g.,

(3.5) This reduces the number of effective components to sixteen. Using the dispersion method and Gauss’ theorem, as shown above, we may then directly express charge commutation relations in terms of dispersion relations involving divergences of currents, as shown in Section II.

390

FUBINI,

SEGRk,

AND

WALECKA

To make use of these dispersion relations we must give an interpretation to matrix elements of the divergence D taken between two physical states a and 5; (a 1D 1 b). Let us therefore discuss the possibility of relating this matrix element to a strong interaction amplitude as is done with the Goldberger-Treiman relations. Consider the analytic properties of (a 1D ) b) as a function of 4’ = (p, - pb)‘. We can write

where the states which contribute to the spectral function Rd( 1’) must have the same transformation properties and quantum numbers as D. We might therefore hope to be successful by keeping only the lowest mass mesonic resonances having the same angular momentum, parity, and SU( 3) quantum numbers as D. This approach has proven fruitful in the study of electromagnetic form factors (vector mesons) and in the Goldberger-Treiman relation (pseudoscalar mesons) for the decay rate of the charged n meson. We shall therefore write

where Ir) represents the one-particle state. We are now in a position to give a classification of the divergences or charges. Since the sum rules involving a commutator of the charge with a current are taken in the limit qp + 0 (see Eq. (2.5) ), we must ask for the behavior of the matrix elements (r ) D ) 0) in this limit. There are two kinds of tensor indices available through the state (r): The indices of the spin polarization vector which are necessarily symmetrized among each other and qr. Therefore if there is more than one vector index on the divergence, then, since the divergences are antisymmetric in their indices we must necessarily use one or more powers of q,, in writing down the invariant form of this matrix element. Thus we see immediately that there is a difference between Dv, aA, DlrT, BPT and the other divergences. These transform respectively like a scalar, pseudoscalar, vector, and axial vector particle so that one can introduce for \r) the appropriate meson state6 and the matrix element (r 1 D ) 0) can tend to a finite limit as q# + 0. The situation is rather different in the case of the other divergences, where, even if we can introduce a one-particle state as in ( 3.6)) we find that its contribution vanishes as qe + 0. For example D& receives a contribution from a vector particle, but by invariance b I D:‘,ll, IO> = (~Pl(u)qPz 6 Assuming it exists; the presence being confirmed experimentBally.

of octets

of scalar

d~h,) and axial

(3.6) vector

mesons

is far from

DISPERSION

RELATIONS

AND

SYMMETRIES

391

where E*(U) is the polarization vector of the vector particle in the state U. This vanishes in the limit Q + 0. In the approximation of keeping the pole contribution to the matrix element of (a 1D / b) we then see that only the commutation relations of Qv, o*, Q,‘, o,,;’ will lead to dispersion relations in which we may use polology to relate the absorptive part to the absorptive part of scattering by a meson which can in turn be related to a total cross section; the two terms in the commutator then combine to give a difference of cross sections corresponding to scattering of particles with opposite spins or opposite charges (see Appendixes 1, 2). By then using Pomeranchuk-like arguments on the asymptotic behavior of differences of cross sections, we will be able to infer convergence of the dispersion relations corresponding to these commutation relations. We therefore distinguish the above eight charges by calling them “good charges” as opposed to the eight “bad charges.” The convergence of the sum rules arising from the latter is rather suspicious. The hope t#hat the commutators involving the “good” charges lead to convergent sum rules is based on the gentleness of the divergence related to those charges. The gentleness of the divergence of the tensor currents is related to vector mesons in the same way as the divergence of the axial vector current is related t.o pseudoscalar mesons. To examine the convergence question in another manner as discussed in the previous section, let us consider the p + r, limit of our commutation relations

(P I[&(..., , Qc-,I1P> and check whether the one-particle intermediate state gives us the correct answer in the free field limit. We are therefore led to examine the matrix elements of the currents between, because of the space integration, states of equal momentum. The matrix elements of J equal

p) = ~(p)$V,TM’)U(p) (P I J (S,V,T,*,P)( there being no renormalization.

(3.7)

As ?, -+ ~4’

u(P) +55(*f,) fo)-+g(-:.,1+ where v is the unit vector V = P/P0 . Introducing the notation u, = (v, i) 7 In this limit, we revert to the spinor normalization particles.

u+u = 1 as in the case of massless

392

FUBINI,

SEGRh,

AND

WALECKA

we may write

(3.9)

= 0

By5u

We see that the scalar and pseudoscalar one-particle matrix elements are zero in this limit, the vector matrix elements (convection current) are proportional to momentum, the axial vector (chirality current) to 6-v times momentum, and the tensor current is proportional to the spin components orthogonal to the direction of motion. We see here, comparing, e.g., au and tiy*u, the role the y4 matrix plays in the p + 00 limit, a role analogous to that of y5 in the p + 0 limit. Let us now see for which charges the commutation relations are satisfied keeping only a one-particle intermediate state. It is immediately clear that this is not true for J $# dx and J $y& dx. It is also not true for J $r# dx and J &,y# dx as one can see by considering the nonvanishing commutator

[ja(;)~i*dx, jj.(:>xtdx]

where i and j are spatial components

and i # j and using (3.9). On the other hand

prothe charges J $74 ,16 # dx and J $y4y*g z5 # dx meet our requirement 0 0 vided that i.v = 0.’ This condition, together with the requirement that c is spacelike, is met by imposing the two conditions E.P = E’Q = 0. The vector spurion must therefore have the same spin properties as a real transverse (massless) photon.g We now note that our classification of charges by the consistency of oneparticle intermediate states for p + 00 is in agreement with the classification based on polology. 8 Since we must also the anticommutation 9 We thank Professor points.

that

add

internal symmetry rules are satisfied. M. Gell-Mann for

operators (Aa) it is necessary to also verify This is actually the case. very useful discussions and advice on these

DISPERSION

RELATIONS

AND

393

SYMMETRIES

In conclusion we are led to believe that the commutation relations for the eight “good charges”, Qv, aA, Q,‘, ofiT (transverse components of Q,’ and Qti;’ only) can be related to physical scattering amplitudes by way of an unsubtracted dispersion relation. The commutators of the other eight charges lead to dispersion relations probably requiring subtractions; the commutation relations of these charges may themselves require additional terms. If we now include explicitly the nine SU( 3) A matrices we can say that out of the 1-M operators generating the algebra of U( 12) only 5-l are likely to lead to unsubtracted dispersion relations and these are”

(3.10)

We wish to emphasize that the charges coming from the spatial components of JPv and J,* are bad charges while those coming from the time components are classified as good. Let us now discuss the equal time commutation rules of these operators (3.10). They can readily be obtained by means of the first quantization forms, using the identity fr,h”,

rvxa] = %{r,

) r”) [A”, xal + $5 w,

A@][I-, ) r,1

(3.11)

We shall see that, the two terms in the right hand side of Eq. ( 3.11) will play a rather different role; it is therefore convenient to distinguish them by considering separately the terms symmetric and antisymmetric in p and v (we shall refer to these as r-even and r-odd). Since the anticommutator of two operators is a quantity which depends on the representation with which one works, we see that the r-even and r-odd parts of our commutator have a different degree of generality and test different aspects of the underlying theory. The r-even part depends on the fact that we have used a spin ?,a representation for our underlying fields while the r-odd part, since it depends on (A”, A’) tests the fact that we have assigned our basic fields to the fundamental (triplet) representation of XU( 3). This means that for the y-matrix 10 From

here

on we shall

use $+(s)

= &a since

{k(z),

$p+(~‘))~-t’

= S,pYs)(x

- x’).

394

FUBINI,

SEGR&,

AND

WALECKA

part of Eq. (3.11) we use the usual relations between Dirac matrices whereas for the X’S we use (see Appendix 2)

The commutation and anti-commutation use only of good charges can be symbolically

relations for the y-matrices written as

{V, V) = v

{T, V) = T

{A, A) = V

(T, Tf = V

making

(3.12)

(V, A] = A [T, T] = A (3.13)

[T, A] = T

all others being zero.‘l The meaning of these equations is that if one takes a component of a good charge of the type indicated on the left hand side of these equations, one finds Some component of the type indicated on the right hand side. For example, let us look at two examples of the last of the set of Eqs. (3.12). We have xir2, fiir1,

721

=

1

(3.14a)

Y5Y21

=

--Y473

(3.14b)

Whereas the left hand sides of (3.12) are obtained from good components of the tensor current, the right hand sides give rise to good and bad components respectively of the vector current. This discussion shows us that the good charges in (3.10) are not closed under the commutation rules and therefore do not generate a subgroup. The largest subgroup they contain is U( 3) X U(3) generated by J #+(z)X*#(x) dx and J J;‘( x)y5Xa$( x) dx. Therefore on a completely general basis, we will not be able to derive from our covariant dispersion sum rules results corresponding to the assumption of the full symmetry groups U( 12) or even SU( 6). We note, however, that even if from a strict second quantization viewpoint we have no higher symmetry group at all, our sum rules still have a group theoretical character since the good and the bad charges arising from J,v and J,” are interconnected by Lorentz invariance. In the particular case of matrix elements between single-particle states of the same four momentum (i.e., between particles having the same mass) there is a unique number for each matrix element of I1 This

is true,

of course,

only

for

the components

giving

rise to good

charges.

DISPERSION

RELATIONS

AND

395

SYMMETRIES

J,” and JpA, and Eqs. (3.12) and (3.13) give a closed set of equations for those numbers. We thus recover a group theoretical structure which is of course dependent on the particularly simple kinematical situation. The preceding discussion becomes particularly clear in the p + CC limit (see Eq. (3.9) ) where the matrix element of J,,v is just the four-velocity which involves no Pauli matrices, the matrix element of J,* is proportional to the spin component in the direction of motion and the matrix elements of J,‘y to the spin components orthogonal to that direction. It is clear that one makes the identification,

Equations (3.12) and (3.13) become the usual algebraic relations for the Pauli matrices. Therefore, our dispersion approach gives a justification of the so-called collinear group SU( 6) W (IO), whose validity might be hard to understand on the basis of pure symmetry arguments. IV.

THE

DISPERSION

SUM

RULES

A. In this section we shall derive the dispersion sum rules corresponding to the commutators discussed previously. For simplicity we shall classify those sum rules in the exact SlJ( 3) limit. Let us start with the well known relations between &*a and &@.

1. The [da,

According

rsh8] Sum Rules

to our general discussion

55”58= &qe

+‘e(zo)

we need the invariant d4x(PS(p)

j [Dsa(x),

amplitude

l&‘(O)]

( PS(u))

(4.2)

Using SU( 3), S$ can be expanded in terms of three invariants Plo, l’~, I’,,, , P,,, as shown in Appendices 2 and 6. The labels correspond to the SU(3) representation in the crossed channel (see Appendix 2). The dispersion relations for these amplitudes arelZ

(4.3a) I2 The

sum

rules

for 10 and n

are the negative

of each

other.

396

FUBINI,

0= o

=

SEGRk,

WALECKA

_ T SWIn-l P8a$‘) dv’ “0

-2fd $$

AND

_

F

lw

Im

Pl;I;y’)

(4.3b) (4.3c)

dv’

"0

with VZ

-q.P

and

vo = f$[(m + m,)’ - m2]

and where f and d are the weak axial vector coupling constantsi

As extensively discussed in the literature (11) Im P,,, can be experimentally estimated in terms of scattering cross sections for the pseudoscalar meson octet on the nucleon.14 Using the SU( 3) crossing matrix, or the 6-p symbols, Im P,,,, can be related to the imaginary part of a pure SU( 3) amplitude in the direct channel. This is done in Appendix 2. 2. The [yxXa, ypXa] Sum Rules Let us discuss the sum rules generated by the commutator

We will need the invariant

function

la The relation of d and f to the weak interaction coupling constants and Cabibbo (16) is discussed at the end of Section V. I4 One linear combination of Eqs. (4.3a, b, c) is the Adler-Weisberger two are discussed in ref. 11.

of Gell-Mann relation,

(1)

the other

DISPERSION

Carrying invariant

RELATIONS

AND

out the symmetrization procedure form of this amplitude as &f(

v) $)

of Appendix

= iq P) (e (*)-PA;;8)(v)

+ [P,

5, we can write

~“‘]A~ZQY>}U(P)

in a! + 0 and A 2 is symmetric.

where A1 is antisymmetric

397

SYMMETRIES

(4.8)

The sum rules coming from the r-even part of this amplitude to Eqs. (4.3). We find

= - (F”+ 0”) - 2” I‘ w Im ‘IT

Q = -2FD

7T

s “0

Ai::

are closely analogous dv

VI2

YO

* Im Ali:(v’)

-2”

(4.7)

We take (see Section III)

E-P = E.Q = q* = 0

-12

the

dv’

(4.9a) (4.9b)

V!=

(4.9c) The sum rules coming from the r-odd

F and D are the two coupling constants

= [(;

;

;)

D + (;

part of the amplitude

(4.7) are

defined by

f

;)

F] a(P)

;[r,,r.lu(P)

(4’11)

and a is simply given by (4.12) The sum rules we have just obtained might at first look rather formal since at the present time no known interaction takes advantage of the tensor coupling

398

FUBINI,

SEGRk,

AND

WALECKA

$[r,, , yy]Xa#. However we shall show in the next section that using a mass extrapolation analogous to the one used in deriving the Goldberger-Treiman formula, the quantities appearing on the right hand side of Eqs. (4.9) and (4.10) can be related to physical transition matrix elements involving vector mesons. More specifically F and D are connected to the magnetic coupling of vector mesons to nucleons and Im A (I) is related to the difference of total cross sections for charge conjugated mesons on nucleons whereas Im A(” is related to the difference of spin-up and spin-down vector meson cross sections. Therefore the convergence of Eqs. (4.9) like that of Eq. (4.3) relies on the Pomeranchuk theorem while the convergence of Eqs. (4.10) is based on the spin independence of high energy cross sections. B. THE DECUPLET DOMINANCE MODEL Let us now examine the implications and interrelations of the sum rules we have derived. As in many other dispersion approaches this can be done by means of models in which one uses some simplifying assumptions as to the states which give the leading contribution to the sum rules. The most natural model comes from assuming the sum rules to be saturated by the SU( 3) decuplet of baryon resonances and baryon octet. This is for two reasons: first of all, the theoretical analysis of low energy pion nucleon scattering and photo-production assuming the dominance of the 3$-3$ resonance has been quite successful. Secondly, the SU(6)W group puts the nucleon octet and the J = $54’ decuplet in the same [56] representation. We shall see that, in the decuplet dominance model, our sum rules show a remarkable degree of self-consistency reproducing some of the results of XU( 6). This is of course a consequence of the arguments given at the end of Section III. We want to point out that we shall not make any a priori assumption about the spin of the decuplet state since it will follow from the comparison of the r-even and I’-odd sum rules. Let us first consider all F symmetric sum rules. Keeping only the decuplet contribution they assume the form [-

(cl2 +p>

I

= -1‘2

- 2fd - a10 = 0

~~+;cY,,

(4.13)

= 0

- (0’ + F”) = -12

j -2FD

- plo = 0

(4.14)

DISPERSION

RELATIONS

AND

399

SYMMETRIES

where (~~0and PI0 are the contribution to the dispersion integral of the decuplet state. We note that the coefficients of (~1~and /IlO in (4.13) and (4.14) are pure SU(3) crossing coefficients and (~10and @lomay have contributions from any spin state. From these equations we find”

(4.15)

PlO

=

-

16~5 3

__

The results of Eq. (4.15) are well known SU(6) results concerning the weak axial vector coupling (12). The interpretation of Eq. (4.16) in terms of observable quantities will be discussed in the next section. The parts of Eqs. (4.15) and (4.16) concerning the nucleon octet can be derived from the r-even relations in the decuplet dominance model without ever mentioning the spin of the decuplet state. Let us now consider the I’-odd sum rules in the decuplet dominance model. We can write

-~(;D2-F2)-;&lo=~ FD--I -$(D2

-~F~)+-$=~~~=

2 YIO =

(1

(4.17)

if

j/id /

+(D2+F2)

+?!$o=

m4sa

3

(4.18)

15In Eqs. (4.15) and (4.16) F and D are determined up to a common sign as aref and cl. Equations (4.15) were derived independently by Bergia and Lannoy and by Gerstein (lb).

400 If we introduce

FUBINI,

SEGRi$

AND

WALECKA

in Eq. (4.17) the values off and cl given by (4.15) we find that F = +g (4.19)

D=2fl 710 = %1/5 and from Eq. (4.18) a=1

(4.20)

We see from Eqs. (4.19) and (4.16) that the values of F and D obtained from the r-even and r-odd sum rules coincide. This remarkable degree of consistency is a consequence of the existence of the collinear SU( 6) group as discussed in the previous section.‘” Equation (4.19) leads also to a determination of the value of ylo . The ratio ~~o//.?Io between the spin-flip and nonspin-flip parts of the amplitude depends on the spin of the decuplet state. Using an isobaric model for a J = $$* particle one obtains for this ratio a value (see Appendix 5) 010 -= YlO

-2

(1

42)

(1

+

(4.21)

xl

where M2 - m2 2M2

x=

In the limit of equal masses (x = 0) this coincides with the prediction based on the decuplet dominant model so, to obtain consistency in this limit, we require the decuplet to have spin 31. Another interesting point concerns the relation f=F (4.22)

d=D

which can be called “spherical symmetry” since (f, d) and (P, D) represent the components of spin parallel and orthogonol to the direction of motion respectively. The validity of these equations can be understood more simply by introducing the operators EKQKT

=

*+e+

dx

s

(4.23) &” = 1 ++y5+ dx 16We thank Professor S. Berman for advice on this point.

DISPERSION

RELATIONS

AND

401

SYMMETRIES

As these charges are SU( 3) scalars, they can only connect our initial (octet) and final (octet) states to intermediate octet states. Let us now saturate the commutation relations involving these operators under the assumption (weaker than decuplet dominance) that no octet other than the nucleon gives a sizable contribution. From the commutation relations between the operators (4.23), one finds that the one-particle matrix elements are not renormalized (i.e., a = 1). If we now consider the commutator between the operators (4.23) and oAa = $+(x)-yhX”$(x) dx we immediately obtain f = F and d = D. Therefore we conclude that the “spherical symmetry” leading to Eq. (4.22) is still valid even in circumstances where the larger group SU( 6) W is broken. V. RELATION

WITH

SCATTERING

OF

VECTOR

MESONS

The sum rules (4.3)) (4.9), (4.10) discussed in the previous section are an exact consequence of the colmmutation relations between scalar and vector “charges.” We have now to provide a physical interpretation of the quantities appearing in those sum rules. For the scalar and pseudoscalar charges, related to vector and pseudovector currents, this interpretation can readily be given in terms of weak or electromagnetic interaction. However even in that case it has proven very useful to use analytic continuation in the mass of the pseudoscalar octet, in order to introduce observable scattering amplitudes into the absorptive part of the dispersion relations. On the other hand the vector charges Q,’ seem to have no connection with the weak interactions (they do not show any significant tensor coupling) and so the only possible interpretation is through analytic continuation in mass, leading to scattering amplitudes for vector mesons. The relation between pseudoscalar divergences and pseudoscalar particles can be symbolically written following Gell-Mann and Levy (IS) DAB = const. I$’ where 4’ is the field of the pseudoscalar symbolically be expressed as 0,‘” where c$,,’ is the field of the vector Let us consider this situation in of the matrix element (al DTB In) in this matrix element is dominated meson pole. We can write

octet. Likewise = const. &,’

(5.1) the vector divergences

can

(5.2)

meson. more detail studying the analytic dependence the variable q2 = (pL1- p,)’ and assuming that by the contribution coming from the vector

402

FUBINI,

SEGRti,

AND

WALECKA

where TB(n) is the scattering amplitude of a vector particle on the nucleon into the state 1n) and a SV is the strength of coupling the vector meson to the vacuum through the tensor current

In this way using Eqs. (5.3) and (5.4)) the absorptive amplitudes appearing in the sum rules (4.9) and (4.10), A”~2)(~)Pr6 may be expressed in terms of equivalent absorptive amplitudes for vector meson scattering V”,‘)( v)rya )see Appendix 3). Im Abet

= f$

Im P)(v),,,

Im A(‘)(Y)~~~

= -f$ Im V’2)(~)p76

(r-even)

(5.5)

(r-odd)

(5.6)

The r-even amplitude in Eq. (5.5) can be related (see Eqs. (Al.21) and (A2.23) ) to the difference of cross sections of charge conjugate vector mesons on nucleons. The r-odd amplitude of Eq. (5.6) is related to the difference between spin-up and spin-down cross section by Eq. (A1.22). For the one nucleon intermediate state we will need matrix elements of the form Q,(P + q( $[y.q, r& [P). This vertex has the same structure as the nucleon electromagnetic vertex. However, since we will want in general matrix elements between states of slightly different mass and since (5.7) is an operator

identity,

we must write (5.3)

= Gz(P + q)&i[q2Y, - r.qqJ + fih.Q, r/Ju1(P) We now require that this amplitude (using c-q = 0) E*

(P + qI$[r.q,

has no poles as q2 -+ 0 and finally obtain

!?I#(P) = CAP + Qw7iq2~+ fih.Q, dl~lu3 q2-.bG2(P + q)lf2h.q, dl~l(~)

(5.9)

The electric part of this matrix element must vanish as q2 -+ 0 exactly as with electromagnetic monopole transitions. If we now keep just the vector mesons in the q” channel we arrive at the formula

DISPERSION

RELATIONS

AND

SYMMETRIES

403

If we interpret the nucleon electromagnetic form factors by means of vector meson polology then &$’ are proportional to the magnetic moments of the nucleon octet

PD= g$rtlN = my2 gvrg* ELF where gV-/ is the coupling constant of the vector mesons to the vacuum through the electromagnetic current (see Appendix 3). At this point we wish to emphasize that our extrapolation from the vector particle pole to a2 = 0 relies heavily on the particular choice of invariants which we have made. Our previous choice was essentially dictated by the criterion of simplicity and led to the appearance of the anomolous magnetic coupling constant in Eqs. (5.11), (5.12). However, it is quite conceivable that the first invariant might be of the form Q2W + cr>P -

(2r.P + rd

%I

(5.13)

corresponding to a no-subtraction philosophy in the Sachs form factors rather than the Dirac form factors. This choice would have led to the appearances of the total instead of the anomolous magnetic moments in Eqs. (5.11), (5.12). Work is still in progress to understand in a more satisfactory way this point. It may be that the difference in the choice of the anomolous or total moments is a measure of the uncertainty in the extrapolation procedure. Within the framework of polology in q2, we are therefore able to give a physical interpretation of our coupling constants D and F in terms of the magnetic moments of the nucleon. We have (5.14a) (5.14b) Since the quantity to getI

asv/gvr

is unknown, we can take the ratio of these equations

= 1.18 = 2.61 17 Note

(total moments) (anomolous moments)

(5*15(

404

FUBINI,

SEGRh,

AND

WALECKA

Our decuplet dominance model gave us a value (decuplet dominance) This is just the result of the static SU( 6) theory where the agreement with the values using the total magnetic moments is well known (14). If the only octet which enters into our theory is that of the nucleon, then as we have seen in the last section we also have the prediction

The experimental value of the ratio cl/j can be deduced from weak interactions. If one writes the SU( 3) structure of the weak axial vector matrix element in the more familiar form GJ(1

- a>$$? + a dijk]

(1) (see also Cabibbo

where the jiik and diik are the SU( 3) matrices of Gell-Mann (16) ) then one easily finds the relation

The decuplet dominance

model thus gives”

a = s = 0.60

(decuplet dominance)

which again is just the static SU( 6) result. The experimental tity is (16)

value of this quan-

%Zp = 0.67 f 0.03

(Brene et al)

= 0.63

(Willis

et al.)

Thus taking the value aeXpFZ ?+$one finds a value

4

y- 2x4 = 1.49 exP We may also give an interpretation to the quantity 010if we assume that we can u The decuplet dominance

model alao gives the well-known

-%=;[+&)+2(&)]=;=1.67 to be compared with the experimental

value -G.JGv

= 1.18.

SU(6) result

DISPERSION

RELATIONS

do polology in q2 on the amplitude case we find

AND

405

SYMMETRIES

for photoexcitation

of the nucleon.

In this

when C1,2(~) are the amplitudes for forward Compton scattering discussed in Appendix 1. If we keep just the J = 35’ state in an isobar model then we find (Appendix 4)

pJ(g>’ =g---->’ = -gl

Jz,2)

(5.17)

where CB = 0.298 is the experimental value of the rNN* (17). Thus we can writeI

coupling

constant

given by Mathews

( gT>k = $(1 -l2,2)(R) = 16.2 Our decuplet dominance

solution gives the value

= 12

(decuplet

this is the static SU( 6) prediction; l/[l and the experimental

(5.18)

(experiment)

dominance

using the mass splitting

M = m, pn e - 2) correction

- x/2] = 1.12

value pn = - 1.91 we get

= 12.4

(r-even

relations-decuplet

dominance)

In this section we have finally shown how the sum rules derived in Section III can be given a physical interpretation by using polology in q2 and, if desired, limiting ourselves to a decuplet dominance model. We only considered the commutator of two tensor charges, but it is clear that all commutators of “good charges” can be treated in a similar manner. Among these other commutators, the one involving an axial vector charge and a tensor charge is probably the most interesting as the sum rule, by polology, involves an integral of pseudoscalar meson-nucleon to vector meson-nucleon scattering which in principle is a measurable yuantity. We shall present in Appendix 8 some of the details of the calcula19As already pointed out, the decuplet dominance model is only internally consistent for vanishing 2. We here make a first attempt to include the mass difference correctionby using the I’-symmetric relations only.

406 tion; F, d, large. inner

FUBINI,

SEGRi%,

AND

WALECKA

in the decuplet dominance model the sum and f already obtained, though in practice It is a general feature that in the decuplet contradictions between the commutation

rules admit the solutions for D, the mass corrections are rather dominance model there are no relations of “good charges.“”

VI. CONCLUSION

The purpose of this paper is an attempt to understand higher symmetries from the point of view of a covariant dispersion analysis of the equal-time commutators derived from a quark model. The basic idea of the dispersion approach is to generate the group theoretical results from the contribution of the single particle states belonging to a “supermultiplet” and to evaluate the corrections in terms of the off -diagonal one-to-many-particle-state matrix elements. Therefore in the convergence of the sum rules we have an “experimental” measurement of the validity of the group. Our original aim was to derive in a covariant manner the results of the compact symmetry group U( 12) based on the commutation relations between charges of the form &i* = / ~+(x)J?iX”~(x)

dx

where I’i is one of the 16 Dirac matrices. We found, however, that most of the dispersion relations which could be derived in this way very likely need subtractions. In other words, the group is so badly broken that the off-diagonal matrix elements do not damp down fast enough at high energy to give convergent integrals. Using, however, the relations which we do believe converge, we still maintain someof the underlying group theoretical structure: for if we take matrix elements of the commutation relations between single-particle states we find relations corresponding to the collinear group SU( 6)W. The internal consistency following from XU( 6)W is obtained in the limit of particles of the supermultiplet (e.g., the nucleon octet and isobar decuplet) having the same mass. These results and their comparison with experiment are discussed at the end of Section V. The over-all agreement is reasonable but far from perfect. One of the advantages of the dispersion approach is the possibility of discussing in a meaningful way the correction terms. The simplest effects to look at are those of the massdifferences of the particles within a multiplet. This effect introduces a factor 1+x ___ - 1.4 x = M;,,-2) 1 - x/2 ( 10In the decuplet dominance charges give no new information.

model, the commutation

rules between

the dual tensor

DISPERSION

RELATIONS

AND

407

SYMMETRIES

into the ratio of the isobar contribution to the r-odd and I-even relations spoiling our consistency relations which hold in the equal-mass limit. Finally, a possible way of proceeding is to transform our sum rules by means of a suitable extrapolation procedure, into relations involving forward Compton scattering. This is done in Appendix 7. It is interesting in principle to compare these relations with experiment since they come from the r-antisymmetric part of the commutators and therefore are a test of the SU(3) quark commutation relations. If one tries to test these relations keeping only the nucleon and the 3-3 resonance, one gets the correct sign and order of magnitude but discrepancies which range up to ~50% . This may be due to our use of the pole approximation in the vector meson channel. On the other hand, one of the relations we obtain exhibits an interesting analogy with a sum rule involving magnetic moments and the charge radius derived on a more rigorous basis by several people (19). This relation also shows a discrepancy of the same order of magnitude if one keeps just the nucleon and 3-3 resonance. This discrepancy appears to be somewhat too large to be explained in terms of higher resonances and may require other kinds of effects like peripheral contributions to photoexcitation. If this turns out to be the case, it would put a limit on the possibility of explaining quantitatively properties of elementary particles from pure considerations of symmetry. APPENDIX

1. “INVARIANTOLOGY”

In this appendix we discuss the kinematic structure of the amplitudes we are interested in and their relations to cross sections. We shall discuss the dependence on the internal symmetry variables [SU( 3)] in the next section.

1. Fortcad

Compton Scattering

We start with a discussion of forward Compton scattering from a nucleon since this amplitude is the one most closely related in kinematic structure to the one we are studying. The S-matrix for Compton scattering can be written as d4’(P2

m2

+ K2 - PI - K1) 4~1

Tfi

(Al.l)

w2E1E2

where Tfi is the Lorentz invariant transition matrix (D is our normalization volume). A count of the linearly independent helicity amplitudes which contribute to forward scattering shows there are only two invariants in this case. Thus we can write

2’;; = G(P)[c(~).~%l(v) + [&‘), e”‘]c2z(P)~uA(p)

(A1.2)

where we have defined v = -P.K

(A1.3)

408

FUBINI,

and I, e(2) are the initial can take to satisfy

SEGRh,

AND

WALECKA

and final (transverse)

polarization

vectors which we (A1.4)

c.K = e-P = 0

Our spinors are normalized to GU = 1. The differential and c.m. systems are given in terms of T/i by

cross sections in the lab

(A1.5) (Al.@ where W = Ki+ E is the total energy in the c.m. system. From unitarity have the optical theorem = 4zw Im Tii = g ~7 7r

Imfi(OLm.

we

(A1.7)

or in the lab = -&

Irnfi(O)Isb where i denotes the initial

helicities.

Tii = rl~+[e(2) .E%(

where 7 are now two-component polarization vectors [& = 81

Im Tii = 2

~7’

(A1.8)

Thus in the lab we have (A1.9)

v) + 2id. ( zc2) A k1))C2( V)]TJX

Pauli spinors. If we now use the correct circular

e:l’ = T-&

[e, f ie,]

x = fl (A1.lO)

e:2) = exm+ we find the relations T, = C,(Y) - 2Cz(v)

(Al.ll)

Ta = C,(Y)

(A1.12)

where the subscripts p and a indicate using the optical theorem we have

+ Z,(v)

parallel and antiparallel

helicities.

Thus

Im Cl(v) = +

[a?’ + fJTOtl

(A1.13)

Im C2(v) = p

[ay

(A1.14)

- f7fot]

DISPERSlON

RELATIONS

AND

409

SYMMETRIES

9. Fomard Vector-Meson Nucleon Scattering

We again write the S-matrix s,i = 6fi - Q$

is’4’(P2 + q.2- PI

-

41)

4~1

1722 wzEI Ez

Tfi

(A1.15)

and find that the differential cross sections are given by

(A1.17) A count of linearly independent helicity amplitudes shows there are four invariants for forward vector-meson nucleon scattering. Actually we shall seethat we will only need the amplitude for scattering of vector-mesons with helicities X, = f 1 and therefore our amplitude is reduced to two invariants as in Compton scattering Tii = GA(P)[e @)*PV1( v) + [P,

tP]Vz( Y)]Uh(P)

Xv = ~1

(A1.18)

just as before we have the optical theorem Imfi(OL.

= -4+W

Im T;i = 2 Q?

(A1.19)

or in the lab (A1.20) Exactly repeating our previous considerations (note (qV/m)

= qL)

(A1.21) (A1.22) 3. Spurion-Nucleon Amplitude We want to analyze the kinematic structure of the amplitude

(A1.23) (or and ,Bdenote the internal quantum numbers.) Our previous discussiontells us that we have to deal with masslessparticles and

410

FUBINI,

SEGRk,

AND

WALECKA

because of colinearity we have to use space components of E orthogonal direction of motion, that is, we take q2 = 0

to the (A1.24)

q-e = t*P = 0

(Al.25)

so that the kinematics are exactly the same as in Compton scattering. Therefore from our previous considerations we can immediately write the invariant form (2) as (1) = ?&(P)[e (2). ,(‘)A;a( v) + [$c2),&“]A;‘( Y)]u~( P) (Al.26) CA 5x&3$ We will also need the absorbtive Abs &&f)

part of this expression which we can write

= -a( p)[e(2). e(1)Im A?(V)

+ [fc2), e(l)1Im Az”‘(v)]u( P) - P - q)

(A1.27)

- (PIqw IPn>(Pn Ia”(o)Ip)~‘4’m - P+ q>l APPENDIX

2. SOME

SU(3) PROPERTIES

In this section we summarize some useful (well-known) SU( 3) relations and give a discussion of the perhaps less familiar properties of the recoupling or 6-p coefficients of SU( 3). We need these recoupling identities to cast both sides of our dispersion relation sum rules into the same form. 1. Clebsch-Gordon Coeficients We follow the notation of deSwart (20). We denote the irreducible representations of SU( 3) by F(V) where k denotes the dimension of the representation and v are the quantum numbers which label the members of a representation. We take v = [Y, I, 131 where Y = S + B. Consider the direct product state vectors 1~Iv~zv2). The C-G coefficients allow us to construct the states which transform according to an irreducible representation of SU( 3)

We have included the index y since a representation of dimension p may occur several times in the reduction of the direct product ~1 @ ~2 . The C-G coefficients generate a unitary transformation, that is, (A2.2) (A2.3)

DISPERSION

They also have the symmetry

RELATIONS

AND

properties

(deswart)

411

SYMMETRIES

(A2.4)

where 8 = I, + $$Y, N is the dimension of the representation, of y’ to y can be inferred from the tables of C-G coefficients.

and the relation (20). (A2.6)

The phase factors .$1.2,3can be obtained from deSwart or by simply looking up the C-G coefficients on both sides of these relations. The C-G coefficients also allow us to reduce the direct product of two irreducible tensor operators into its irreducible components by (A2.7) The Wigner-Eckart theorem allows us to extract elements of an irreducible tensor operator

the v dependence of the matrix

Putting these results together, we can extract the v dependence of the matrix elements of the products of operators which appear in our dispersion relations in the following manner

2. Recoupling or 6-p CoeBcients We see that in performing sums over intermediate relations we need quantities like

states in our dispersion

(A2.10) while on the other side of the commutators,

when we take the matrix

elements

412

FUBINI,

SEGRh,

of the charges, we have quantities

AND

WALECKA

like (A2.11)

where the representations are coupled in a different order. The relation between these two coupling schemes is given by the recoupling or 6-p coefficients. Consider the states formed by the direct product of three-state vectors [ EI~v~~Y~~vJ. We can now form an irreducible basis in two ways. Consider the states

where we first couple ~1 and k2 to get p12, and then couple to cl3 to get p8 . Using properties (i) and (ii) of the C-G coefficients one easily sees ((~11*2)1-1:2,&3&’

1 (I.L#ZhlZ,ti3&9~)

=

(A2.13)

6,a’6v”‘6,,1888’6~12r’12

Therefore we have a new complete orthonormal basis and the transformation is unitary. We could also have coupled ~1 to the result of ~2 and p3 coupled to give ~23 . This new basis is also complete and orthonormal. The unitary transformation coefficients between these two distinct bases 1 (~~2)~12&3~d

=

#z6

(d~~3)~23&

1 (~@2h2,~31.LB)

(A2.15)

[ /-+c(2EL3h23,&

are independent of v as can be seen by applying the raising and lowering operators to both sides of this equation and are defined to be the 6-p coefficients (al)

Taking matrix elements of the defining onality relation

equation

above and using the orthog-

(/&~2v2;3v3

we find

\

G42.18)

DISPERSION

RELATIONS

TABLE 6-p

AND

413

SYMMETRIES

Al

COEFFICIENTS

m-f

88

81 82

1

34

81

-l/d8 0

82

0 -1/1/S

10

-a

i6

v%

27

2/25/S

414

FUBINI,

SEGRti, AND WALECKA

TABLE A2 PEASE FACTOR El

El

1

1

which is the desired relation. Table A2.21

-1

-1

-1

1

We give a table of the relevant 6-p coefficients in

3. Charge Exchange Amplitudes

Consider the physical process (A2.19)

S(P) + 8(a) --j 8(a) + S(P)

The scattering amplitude

for this process transforms T aS- (8(p) 1P[s”‘P]

Since we adopt the usual convention

as

18(a))

(A2.20)

- V)

(A2.21)

(deswart)

o(jLY)+ = (-1>“&*

(Note: ii = I, + $$Y is just the charge of the field) we see that for the SU(3) part of our amplitude we have

We can therefore relate the scattering amplitude for a given S-channel process to the SU(3) amplitudes. If we look at just the charge exchange amplitudes we find 21 The 6-p coefficients can be related to the crossing matrices of deSwart by (20)

as well as the relation of p’, 6’ to p, 8 can easily be obtained by looking up the appropriate C-G coefficients in different orders using (iv), (v), and (vi).

DISPERSION

RELATIONS

AND

; [T@P)

- T@“P>I - 6d5 -![TIO - Tiiil + &

; tTb+p>

-

T(7)]

= - $&

415

SYMMETRIES

[TM - Tiol + $&

Tsz, - f T,,,

(A2.23)

Ts,, + ; Tszz

Exactly the same relations hold for the scattering of the vector-meson octet. We therefore conclude (i) The absorptive parts of the odd XU( 3) amplitudes in our dispersion relations (the I’-symmetric amplitudes) can be related by polology to the difference of cross sections for ( Kf, K-), ( aoK”), and (r+, r-) scattering, or the corresponding vector-meson scattering, from a proton. (Since the 10 and a amplitudes are always just the negatives of each other, we can just work with their difference.) (ii) By t a-ihn g 1inear combinations of our sum rules for the I’-symmetric amplitudes we can get three sum rules involving the diflerence of physical cross sections. 4. Relation to the XV(S)

Amplitudes

in the Direct Channel

Suppose we keep a given amplitude cr, in the direct or S-channel. in this channel we would write the amplitude as

Coupling

(A2.24) Our amplitudes

The relation We have

in the crossed or t channel are defined by

between these two amplitudes

Im A,,, These coefficients

=

t1(f%)

is evidently

given by a 6-p symbol.

dx-x

are all given in Table Ad.3 We can write

(A2.26) out these equations

416

FUBINI,

SEGRA,

AND

WALECKA

19 U -1-e

lo, U -1-a

+

I 2 U cD11(3

s

U -Id+

03 U I*

-

I

+

+

+

2

U Ic;I

+

i -100

I

DISPERSION

RELATIONS

AND

417

SYMMETRIES

For the nucleon contribution a811 = (~8~~so that the 10 and ti amplitudes just the negatives of each other. APPENDIX

1. Single Nucleon Matrix Consider

3. POLOLOGY

IN

are

q”

Elements

the vertex function

We know from Section V that this vertex has the kinematic

structure

QJviI= C(P + a)lh(q”rr - r.aqd + fzMkY4, rJb(o~

(A3.2)

We want to disperse fi( q’) in the variable q2 and in the absorptive part we will keep only the vector-meson resonances. (We do not have to consider fi(q’) since E. q = 0, q2 + 0.) We shall do this by simply making an isobar model and carrying out the calculation of the absorptive part in perturbation theory. (See Fig. 1.) We have

-f$q(f ; $A’:,+(I ; ;)P:N]}u(P) The first term is the coupling of the vector-meson to the vacuum through the current [email protected], r,]x”$ with strength aso . Now keeping just the contribution to the invariant [y.q, e] and evaluating the numerator as q2 + -rn*! we can write

FIG.

1.

“Spurion “-nucleon

vertex

418

FUBINI,

SEGRk,

(note a#,, + (n,d~v2)lq2=--my2 6Yq2)

= Q2I’m,

AND

WALECKA

= 0) [c

,” ;)&+

(z

,” ;)g]

(A3.4)

Since we showed in Appendix 2 that (A3.5) we can conclude

(A3.6)

We could carry out an exactly equivalent calculation for the electromagnetic vertex of the nucleon. The electromagnetic current is assumed to have the form JI”

= z2 $-f,( XCrnO)i- 73l XCooo))#and we have

Again, as illustrated

in Fig. 2, we have for the absorptive

part

DISPERSION

RELATIONS

AND

SYMMETRIES

8(p)

_p+q

FIG.

2. Photon-nucleon

419

vertex

We assume that we can only write an unsubtracted dispersion relation for the magnetic moment. Therefore, exactly as before, (A3.9) or

(A3.10)

Combining with our results for D and F we have

(A3.11)

Theorefore D/F = pD/pF and the tensor couplings have the same ratios as the nuclear magnetic moments. 2. Absorptive Part of the Dispemiml Integral We will keep just the vector-meson poles in q? and qt in the absorptive part of our dispersion relations as indicated in Fig. 3. Referring to the last appendix we write

420

FUBINI,

SEGR&

AND

WALECKA

8(B) 7c; ‘9

FIG.

8(a) +e

3. Polology in q2in the absorptive amplitude

(2) a!9 (1) Abs Q $7 $7 (912.g22,v)

.(s,,+~)(T~(qle+ my’))i$s - 6rlv ” T;!(v)vN( + 5)

(

. (7d(q2* + mv*n

-

q226,,

q2pq221mvz

&W

1

e3.12) P

where T$( Y) v-N is the vector meson-nucleon T matrix defined in the last section. In this expression we are to evaluate the numerators on the vector meson poles. We also have qr abq,, T,, = 0 so that all the terms in q disappear and going to q* = 0 we find Abs &F;,!(Y)

$) = Abs Gd” $1 T;;(~) “--NE$)

and we pick out just the transverse forms in Appendix 1 we conclude

parts of TvmN. Comparing

Im AL~(Y),,~~ = $$ Im VI,~(Y)~~~

(A3.13) with

our general

(A3.14)

We could carry out an exactly similar analysis to relate the absorptive part of Compton scattering to vector-meson-nucleon scattering. The relation is evidently (note e” = 4?ra! where (Y = xs7) (A3.15)

Im CI,~(Y)~~~ = or combining

results 2

Im A1.d~)~~~ = &a

Im

CI.P(Y)~~~

(A3.16)

DISPERSION

RELATIONS

AND

4’11

SYMMETRIES

3. Isobar Contribution Consider just the contribution of the isobar. We know from the work of Gourdin and Salin (I’?‘) that one amplitude completely dominates the electromagnetic excitation of the nucleon isobar. ‘*

(P + q, proton* ( Jiy) 1P, proton} (A3.17) - r.qhMP>

where wx( P) is a Rarita-Schwinger wave function (see next Appendix). We shall therefore keep the same amplitude in our dispersion relations as we shall show that these processes are related by polology in q2. That is, we write

Using our polology in q2 on this amplitude

just as before we can conclude

(A3.19)

or finally (A3.20) Mathews

gives ( 17) 63 = 0.298 APPENDIX

4. ISOBAR

(experiment)

CALCULATIONS

We describe the y$+ nucleon isobar by a Rarita Schwinger wave function wh(P) satisfying the subsidiary conditions = 0

(A4.1)

Pxwx(P) = 0

(A4.2)

Yxwx(q

422

FUBINI,

SEGRk,

AND

WALECKA

and the Dirac equation (A4.3)

(i7.P + M)wx( P) = 0 For the intermediate

state sums we use (from the subsidiary

,,igB>o

= (M - ir*P) 2M

w(P)&(P)

-[

hp + &

(YAP.0 - PXYP) + A2

conditions)

PAP, - f YXYp

In evaluating the contribution of the isobar to the absorptive persion relations we need to evaluate 2%

Spins.

1

part of our dis-

- &.P)~X(P + q)Ox(P + Q)Y5 c W’hdq~~(~)

(A4.5)

E>O

* ( q&(l) - py.q)u(P)S(v where VR = (M2 - m2)/2 and easily be reduced to 2VR2 _ = m u(P)

v

= -P-q.

(A4.4)

-

VEl)

Using q2 = c.q = c. P = 0 this can

1 ec2).ec1)- 3 P(~)E(~)- 3+2

u(P>S(v

- vR)

(A4.6)

The last term in the parenthesis gives us a correction term which depends on the mass difference M - m. We have for the correction factor J? antisymmetric

1 + M;G2m2 lAPPENDIX

Consider

M2 - m” 4M2 5. SUM

r symmetric RULES

FOR

TENSOR

relations

(A4.7)

relations

(A4.S)

CHARGES

the tensor charges (A5.1)

These charges satisfy

the commutation

rules

(A5.2)

We use the usual spherical component notations

of Appendix

2. We now take the

DISPERSION

RELATIONS

AND

423

SYMMETRIES

matrix elements of this relation between nucleon states of momentum IP). We observe the following (i) Since s #+( .c) A”+( X) rlx is just one of the generators of XU( 3) we know its matrix elements (ii) We write #+[rx , r,lx’+

= %&r4

, YX , r,l~V

= %~4Xr”~Y~Y”~Y~

(A5.3)

and the matrix element of this quantity gives us just the weak interaction axial vector coupling constants since only one kinematic invariant, ~(p)y~y,u(p) remains in this case. (iii) The quantity #‘[ri , r,]# = j$~&ysy~J/ has only one kinematic invariant for the same reasons. We use a to denote the strength of this matrix element. Thus we can write the right hand side of this matrix element

For the left hand side we use the dispersion define F:;(Y)

= dv

relation as discussed in the text. We

1 ~Z~xe-~~‘%(z,,)(P 1[&J,“,(z),

a,J~,(O)]

1P)

(A5.5)

where G(s)

= ; &dh,

dAa$(X)

(A5.6)

and we take q2 = 0. In order to insure that our dispersion relations have the symmetry of the commutator built in it is more convenient to first write

[TX”, T,Bl = Si([TA”l, T,pl - [q!, TX”11 and then define

(A5.7)

424

FUBINI,

SEGRb,

AND

WALECKA

The first term in parenthesis is symmetric in CII= p and antisymmetric while the second has the opposite properties. From Appendix 1 we know

t:2)SXq08( vp = G(P)(E (2).e(1)A:B(v) +

[e('), e"']A;'(v))u(P)

in X = p

(A5.9)

where we take E*q = e+P = 0. As discussed in the text, our sum rules are now obtained from

iPoD -a t:2%;V1) p p Iv=0 = r.h.s. av and extracting sum rules

the SU( 3) dependence of 5 x”,” as shown in Appendix

-m&(O);

= -44%~

-mAdO);,,

= o

= ad

-A(O);,,

= -12

-m&(O);,, = dgf --mAz(O):, = 0

-m&(O):o

--mA2(O)i,,

= [(;

2 we find the

(A5.11)

= 0

-mA1(0)&, = 0

with the prime denoting a derivative

j/‘y(p

(A5.10)

with respect to

V.

Using

+ @h$$h.q, d+‘R(I)) ;

18,)D +(;

;

fp]ii(P+

&.q,slu(P)

(A5*12)

for the nucleon matrix element as discussed in Section V and the recoupling identity of Appendix 2 we find for the contribution of the nucleon pole term

We have given the nucleon a slightly Thus

different mass m2 in the intermediate state.

vo = (7~12~ - m12)/2 This mass difference drops out of our final result as it must. Therefore we finally find the sum rules

o = -2FD -12

_ ;

[*ImA(;!$l PO

= -(F2

+ 02) _ T

o = SD2

~[“ImAC.$hdv’

dv’

Jc, Im A(pl

(A5.14)

dv

vo 0

=

-32

o

=

-&(iD2

_

$l

JW

-

vo =

$$[(?n

;2)

+

mJ2

ImA($i-,

-

fh’

y~ImA,,;~2~~~

-

m’l

If we now approximate the integral by simply taking cussed in Appendix 4 we find

0)

the decuplet

isobar as dis-

\

- 42/2 3 a = -1

(D2 + F’) 4s +q+&l+x)(,~)

@ll)

( D2 - ;F’)

+; /

+ $(&y(l

G312)

(2m

5,f

+x)

= FD + @8 (-&(1+x)(-;) o=

--j,,/~(?~~-

F~)

(-&)b

r-even

(A5.15)

426

FUBINI,

(821) (822)

0= -12

SEGRh,

AND

WALECKA

?g(*+-;)(-;)

-2FD= -(D2

+ F’)

(A5.16)

x is a correction factor coming from the mass difference M - m. For the invariant of Appendix 4 we have (see Appendix 5) x= APPENDIX

M2-dg21 2Mz

6. SUM RULES

(A5.17)

’

FOR AXIAL

VECTOR

CHARGES

1. Invariantology Consider the scattering of a pseudoscalar meson by a nucleon. Just as before we can write the S-matrix as (2*)4i s,i = 6,i - --$-a

(4)

(Pz+qz-PI-d

(A6.1)

GT,i

and the crosssection in the c.m. system (A6.2) The optical theorem is

Irnfi(())

= I?$

= 2 OTot

(A6.3)

In our subsequent analysis we need the invariant form of s;! = /y

1 e+3(~~)

d4x(Pj

[DJx),

D!(O)] ( p)

(A6.4)

There is only one invariant amplitude for this processand we write

5;; = ia( P)P”+y Y)U(P)

(A6.5)

6’. Commutation Relations and Sum Rules We deal with the charges (A6.6) These charges satisfy the commutation relations

DISPERSION

RELATIONS

AND

427

SYMMETRIES

(A6.7) Taking matrix P we find

elements of this expression

E+ (PWlL&501,&!I Dispersing

between

nucleon states of momentum

I Pi)

the left hand side we find our sum rules from iPo $2 a; Sb( v) Jvzo = r.h.s.

(A6.9)

This leads to the relations -w&(O)

= 0

-T&j(O)

= 0

-mP&l(o)

= 0

-w&(O)

= -12

(A6.10)

Note that only the odd SU(3) amplitudes enter in this case since we only have T-symmetric relations. Inserting the nucleon pole term with weak axial vector coupling constants defined by

we find the sum rules

p I7 even

(A6. 12)

428

FUBINI,

SEGRk,

AND

WALECKA

3. Decuplet Dominance

If we include the isobar through a coupling

+4Iihm5 X’lc, Imt)) m (lO(rl)P

then we easily find just as before

00)

(A6.13)

(821) (822)

-12 APPENDIX

= -(f

+ d2)

7. SUM RULES

INVOLVING

COMPTON

SCATTERING

In order to arrive at a set of sum rules involving only completely measurable quantities we make the following two assumptions: (i) In the absorptive part of Compton scattering, just as in that of the amplitude involving the divergence of the tensor currents, we shall do polology in the 4’ channel and keep just the vector mesons. (ii) The electromagnetic current operator is of the form .JF’ = (i/2)&,. (X(“‘) + ( l/&)X’ooo’)J/. We shall assume that we can experimentally separate the (010) “isovector” from the (000) “isoscalar” contributions to the imaginary part of the forward Compton scattering amplitude. This can easily be done in any semiphenomenological fit to the data involving isobars and peripheral graphs. Writing out the Compton amplitudes in terms of our SU( 3) amplitudes in the crossed channel (see Appendix 2) we have

Ipp= c”10~010 Inn= - -& Cl - & c,,, c 010~010 1 + 21/s % ; p*om

Ipp = f cooo*woIILn.= - -&Cl

1 + 30,/G c27

(A7.1)

+ $j CSl,

3 - 21/E C81, + lo& 1

(A7.2) cm

DISPERSION

RELATIONS

AND

429

SYYMETRIES

+ po~~o]pp = - -A1/3[c”@J.OlO + (po.ynn (A7.3) --&1546 Since these amplitudes all involve symmetric SU( 3) amplitudes only the I’-oddinvariants. Let us introduce the notation 3 d rv=z~+q~=

f

f

-

c2,

we will deal with

GA 73

(A7.4)

1 d 2fl

rs=z---

F 3 __D tv = 5 d15 -+2-\/3 __F 1”=243-2fi

(A7.5)

1D

P (A7.6)

corresponding Appendix 3,

to the isoscalar and isovector

amplitudes.

D -=-

F z-z-t.V

ts

P

PF

PS

P’

Then by our polology of

(A7.7)

and (A7.8) Now by taking the linear combinations (A7.1, 2, 3) of our sum rules or, more directly, by simply taking matrix elements between proton states we find 2a -

3rs = (3tS)Z - (%J

2 a + 2 = (p)’ 3

_ (CJ

rv = (CJtstv) - (??)

. g!g

J-r Irn yv’)

dv

(A7.9)

. L!&

l;

dv’

(A7.10)

. +&

Im cy,y l;

Irn [‘:)(‘$-

cib2)b’)1 ,jv’

(A7.11)

430

FUBINI,

SEGR%,

AND

WALECKA

where all the absorptive parts are for the spin-flip part of Compton scattering on the proton and neutron and are related to the difference of the spin parallel and anti-parallel cross sections by

Im C”‘(y)= + + [ar - opt]

(A7.12)

We therefore have three equations in two unknowns a and ts or tv[ts/tv = $/~v]. If we keep only AT = 1 transitions into intermediate states then the integrals in (A7.9) and (A7.11) are zero and eliminating the unknowns we arrive at a sum rule involving only experimental quantities 3ryf!c+4-

(pv)L

We can get an approximate model and we find2’

-~Jc:lmcf$‘)d’(~T=

evaluation

l]

(A7.13)

of the right hand side by using the isobar

C3Yl + x>

(A7.14)

= -6.45 Evaluating

the left hand side [we use o+

S N which gives rg/rv E s]

1.h.s. = -4.26

[total moments]

= -3.50

[anomalous moments].

(A7.15)

These relations are interesting to test in detail since they come from that part of the commutation relation involving {X”, X’j and thus are a test of the quark model. It is possible that these sum rules are more general than the derivation through the vector mesons would lead one to believe. More rigorous relations similar to the above but involving the F-even part of the commutators have recently been derived by Adler and others (21) . The experimental test of those relations inserting only the nucleon and the 3-3 resonance leads to a discrepancy which is of the same order as that of (A7.14) and (A7.15). This is an indication that higher energy contributions, probably not of the resonance type, might play an important role in sum rules involving Compton scattering.23 22The decuplet dominance solution gives the value r.h.s. = 1.h.s. = -4. Hearn and Drell(22) have shown that a direct experimental evaluation of the right hand side of (A7.13) gives a number much closer to those of (A7.15). 23We wish to thank G. Patsakos and A. Hearn for many useful discussions on this appendix .

DISPERSION

APPENDIX

RELATIONS

AND

8. SUM RULES FOR AXIAL CHARGE COMMUTATOR

431

SYMXETRIES

VECTOR-TENSOR

1. Just as in Appendices 1, 2, 6 we define the S-matrix and the invariant form

-iq3&J

d4x(P /[D;(x),

D:(O)]

1P)EX

(A8.1)

There is only one invariant in the forward direction:

GL = iq P)y5eR@(v)u( p>

(AS.2)

2. Cornrwtation Relations The charges QSaand TX”, defined in (A6.6) and (A5.1) respectively, obey the commutation relation

Dispersing the left hand side, as in Appendices 5, 6 we find four relations corresponding to the 27, SIX, S12and 1 SU( 3) representations in the cross channel. Only even representations occur because we have I‘ antisymmetric relations. Using the definitions (4.4)) (4.11) and setting (for X # 4)

where D and F equal D and F as SU( 3) octets of the tensor currents have one specified coupling to baryons. The four relations then are

432

FUBINI,

SEGR&,

AND

WALECKA

(A8.5)

(27)

where R can be related by polology to pseudoscalar meson-nucleon to vector meson-nucleon inelastic scattering. Being inelastic, the amplitude presumably tends to zero fast enough for our sum rule integrals to converge. We may once again assume that our sum rules are saturated by the nucleon octet and the 35+ decuplet. In the limit of equal masses the four relations (A8.5) reduce to -- 4v53

Go = - -&

(Ff + Dd)

-&d-;F~)+~(&--aal(~) (A8.6)

These equations are formally identical to (4.17), (4.18) and admit the previously determined solutions for F, D, f, d, G31, and ~~31. In addition we determine the relative sign of D to d, G31to (~~1which are both positive, with our definitions of current matrix elements. Of the other commutation relations [V, V], [V, A], and [A, A] are well known and V, T merely leads to SU( 3) sum rules among magnetic moments. This exhausts the commutation relations between good charges listed in formulas (3.12) and (3.13). ACKNOWLEDGMENTS We wish to thank Professor M. Gell-Mann for very illuminating discussions and advice. One of us (S. F.) wishes to thank the Physics Departments of Stanford University and the University of California, Los Angeles for their hospitality. The last part of this paper was carried out while he was at the Institute for Advanced Study, Princeton, New Jersey. He acknowledges the kind hospitality of Professor R. Oppenheimer and his interest in this work.

DISPERSION

RELATIONS

AND

433

SYMMETRIES

Another one of us (G. S.) wishes to thank Professor L. Van Hove and the Theoretical Division of CERN for support during the initial stages of this work. RECEIVED:

February 7,1966 REFERENCES

1. M. GELL-MANN, 2. F. GURSEY AND

Phys. Rev. L. RADICATI,

126,1067 (1962); Y. NE’EMAN, Nucl. Phys. 26,222 (1961). Phys. Rev. Letters 13, 173 (1964); B. SAKITA, Phys. Rev.

136, B1756 (1964). 3. W. D.

MCGLINN,

Phys.

(1965); S. COLEMAN,

Rev. Letters Phys. Rev. 133,

12, 467 (1964); L. MICHEL, B1262 (1965).

Phys.

Rev.

137, B405

GELL-MANN, Phys. 1,63 (1964). W. LEE, Phys. Rev. Letters 14, 676 (1965); R. DASHEN AND M. GELL-MANN, Phys. Letters 17, 142 (1965); Phys. Letters 17, 145 (1965). 6. S. FUBINI AND G. FURLAN, Phys. 1, 229 (1965); S. ADLER, Phys. Rev. Letters 14, 1051 (1965); W. WEISBERGER, Phys. Rev. Letters 14, 1047 (1965); S. FUBINI, G. FURLAN, AND C. ROSSETTI, Nuovo Cimento 40, 1171 (1965). 7. S. ADLER, ref.6; W. WEISBEROER, ref. 6; S. FUBINI AND G. FURLAN, ref. 6; G. FURLAN, F. LANNOY, C. ROSSETTI, AND G. SEGR~, Nuovo Cimento 38, 1747 (1965); Nuovo Cimento 40, 497 (1965). 8. R. FEYNMAN, M. GELL-MANN, AND G. ZWEIG, Phys. Rev. Letters 13, 678 (1964); K. BARDAHCI, J. M. CORNWALL, P. G. 0. FREUND, AND B. W. LEE, Phys. Rev. Letters 13, 698 (1964); Phys. Rev. Letters 14, 48 (1965); A. SALAM, R. DELBOURGO, AND J. STRATHEDEE, Proc. Roy. Sot. (London) A284, 146 (1965). 9. V. ALESSANDRINI, M. A. B. BEG, AND L. S. BROWN, “Remarks on the Saturation of Equal Time Commutat,ors and Physical Sum Rules,” Univ. of California preprint; W. WEISBERQER, “Unsubtracted Dispersion Relations and the Renormalization of the Weak Axial Vector Coupling Constants,” SLAC preprint; S. FUBINI, “Equal Time Commutators and Dispersion Relations,” Institute For Advanced Study preprint; S. OKUBO, “Algebra of Currents and a Low Energy Theorem,” University of Rochester preprint. 10. K. J. BARNES, P. CARRUTHERS, AND F. VON HIPPEL, Phys. Rev. Letters 14, 82 (1965); K. J. BARNES, Phys. Rev. Letters 14,796 (1965); H. J. LIPKIN AND S. MESHKOV, Phys. Rev. Letters 14, 670 (1965); R. DASHEN AND M. GELL-MANN, ref. 6. 11. D. AMATI, C. BOUCHIAT, AND J. NUYTS, Phys. Letters 19, 59 (1965); L. K. PANDIT AND J. SCHECTER, Phys. Letters 19, 56 (1965); C. A. LEVINSON AND I. MUZINICH, Phys. Rev. Letters 16, 715 (1965); W. WEISBERGER, ref. 9. 18. S. BERGIA AND F. LANNOY, CERN preprint; I. GERSTEIN, “SU(6) Results Derived from SU(3) X SU(3) Algebra,” University of Pennsylvania preprint. IS. M. GELL-MANN AND M. LBvY, Nuovo Cimento 16, 560 (1960); Y. NAMBU, Phys. Rev. Letters 4, 380 (1960); J. BERNSTEIN, S. FUBINI, M. GELL-MANN, AND W. THIRRING, Nuovo Cimento 17,757 (1960); K. C. CHOU, Zh. Eksperim. i Teoret. Fiz. 12,492 (1961). 14. M. A. B. BEG, B. W. LEE, AND A. PAIS, Phys. Rev. Letters 13, 514 (1964); B. SAHITA, Phys. Rev. Letters 13, 643 (1964). 16. N. CABIBBO, Phys. Rev. Letters 10, 531 (1963). 16. N. BRENE, B. HELLESEN, AND M. Roos, Phys. Letters 11, 344 (1964); W. WILLIS, et al., Phys. Rev. Letters 13, 231 (1964). 17. M. GOURDIN AND P. SALIN, Nuovo Cimento 27, 309 (1963); J. MATHEWS, Phys. Rev. 137, B444 (1965). 4. M. 5. B.

434

FUBINI,

SEGRk,

AND

WALECKA

18. S. FUBINI, G. FURLAN, AND C. ROSSETTI, “Nucleon Magnetic Moments ‘and Photoproduction Sum Rules,” University of Trieste preprint. 19. See, e.g., S. ADLER “Tests of Local Commutation Relations in High Energy Neutrino Reactions,” CERN preprint TH 598, Eq. (47b); N. CABIBBO AND L. RADICATI, Phys. Letters 19,697 (1966); J. BJORKEN, private communication. 20. J. J. DESWART, Rev. Mod. Phys. 36, 916 (1963); Nuovo Cimento 31, 420 (1964); P. McNAMEE AND F. CHILTON, Rev. Mod. Phys. 36,1005 (1964). 21. G. MURTAZA, preprints, Imperial College, London; M. NIETO, Phys. Rev. 140, B434 (1965). 2%‘. S. D. DRELL AND A. C. HEARN, Phys. Rev. Letters 16,908 (1966).