Infinite multiplets and the deep inelastic lepton pair annihilation processes

ANNALS

OF PHYSICS:

64,

474-509 (1971)

Infinite Multiplets and the Deep Inelastic Lepton Pair Annihilation Processes* I.

MARKO

PAVKOVI~

Department of Physics & Astronomy, Tel-Aviv University, Ramat Aviv, Israel Received

May

14, 1970

The lepton pair annihilation channel is investigated in terms of infinite multiplets. We consider only those experiments which record the single hadron p in the final state and have the content and the energy momentum distribution of the rest of the produced hadron matter X unknown. In the described circumstances the complete information about the process can be condensed into the two invariant structure functions ml(q2, V) and m2(qa, Y), provided that the process involves the exchange of only one virtual photon between the lepton and the hadron systems. All our results are derived from the hypothesis that the hadron matter X is obtained as a result of a decay of a resonance with high mass value. The resonances are described by the infinite component wave equations. Particular attention is devoted to the kinematical region q2 --f co, w = 1 - Ma/q2 fixed, where 2/p is the center-of-mass energy of the lepton pair and M is the overall mass of X. The conditions are stated for the existence of Bjorken limits,

Two specific models based on the Nambu-Fronsdal and the Abers-Grodsky-Norton wave equations, respectively, are investigated, and it is found that the first wave equation predicts vanishing of the quantities RX andFe , while in the second wave equation, i”, and p2 diverge. These opposite extremes entitle us to speculate about the feasibility of an infinite multiplet model which would allow the ml and vm2 functions to remain finite on the Bjorken limit.

1.

INTRODUCTION

In the previous paper,l henceforth to be quoted as I, a model for the deep inelastic electron nucleon scattering was investigated, based exclusively on an infinite number of isobaric resonances. The total multitude of resonances with unlimited mass and spin was condensed into a single infinite component field operator whose space-time properties have been determined through an infinite * Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force., under AFOSR grant number EOOAR-68-0010, through the European Office of Aerospace Research.

474

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ANNIHILATION

475

component wave equation. The two specific models that were considered, based on two different infinite component wave equations, gave the following result. The first model, based on the Nambu-Fronsdal wave equation, predicted a vanishing Bjorken limit for the invariant functions WI and v W, , while in the second model the same limiting quantities diverged. Such a situation, though not satisfactory, allowed us to speculate about the possibility of constructing an infinite component wave equation which would have the important property of predicting the finite Bjorken limit for the quantities W, and VW, . However, such an equation, which would, in addition, have to yield form factors, mass spectrum, and degeneraties in approximate match with physical reality, may not be immediately available. In the meantime, we are undertaking the less ambitious task of investigating, in terms of computationally accessible but physically rudimentary infinite multiplets, the lepton pair annihilation channel as well. The two models that we employ are admittedly too crude to give reliable numerical predictions. However, they do not lack the essential qualitative features of the hadron phenomenology, such as the rapid decrease of form factors (unfortunately, not sufficiently rapid to fit the available proton form factor data), the observed similarity in the t-functional dependence of elastic and excitation form factors, the rising Regge trajectories, etc. It is instructive to consider more closely the last of the listed aspects, i.e., the question of mass spectrum. It is a generally held belief that the Regge trajectories display the linear mass2-spin relationship MJ2 - J. Recently, however, it has been speculated that not all resonances participate on an equal basis in the scattering processes.” Those resonances which participate in larger measure happen to lie on parabolic M, N .I trajectories. But characteristically enough, these trajectories are among the predictions of many infinite component wave equations, including the two which are considered in the present article. This is one more reason to regard the two selected wave equations as a qualitatively good first approximation. When attempting to go, within the framework of an infinite component field theoretical model, from one channel of the process into another, one is confronted with the two typical peculiarities of the infinite multiplets: The absence of CPT symmetry in some models, and the lack of crossing symmetry in others.3 The first problem is not serious and can be solved by artificially doubling the particle space to include antiparticles4 as well. The second problem is more serious but may not be irreparable. Certainly, at the moment, there is no evidence that crossing symmetry may not hold in nature. It is therefore imperative to explore all avenues toward constructing a crossing symmetric infinite multiplet. But this task may require more time and effort. Our ambitions here are modest. We will calculate the process in the crossed channel, openly admitting that the scattering amplitudes in two channels are not related by the operation of crossing. The immediate consequence, on the technical level, of the lack of crossing

476

PAVKOVIC

symmetry is doubled computational effort. In contrast to the local finite component relativistic quantum field theory, the infinite multiplets are not in possession of powerful general theorems that would guarantee the same computational result in two different crossed channels. Thus, for each individual model, one is forced to calculate the relevant S-matrix elements in each channel separately. In the investigation of the properties of the isobaric resonance model in the deep inelastic electron-nucleon scattering, the main effort was directed towards verifying scale invariance in the sense of Bjorken, i.e., answering the question if and under which conditions the invariant functions Wl(q2, V) and vWz(q2, V) become nontrivial functions of a single variable w = 1 - M2/q2 when q2 and M are allowed to go simultaneously to infinity. The conditions for the existence of Bjorken limits for the annihilation channel structure functions Wl(q2, V) and vF2(q2, V) are stated in Section (7). In Sections (5)--(8), we consider two particular models based on the Nambu-Fronsdal infinite component wave equation and the Abers-GrodskyNorton wave equation, respectively.5 The first model predicts vanishing Bjorken limit for the quantities V1 and up2 , while in the second model, V1 and uW2 diverge on the same limit. Speculations are presented concerning the significance of constructing an infinite resonance model which would allow the W1 and vp2 functions to remain finite on the Bjorken limit.

2.

KINEMATICS

Our first task is to establish the kinematic process I- +1++p

region for the lepton pair annihilation +x.

Here p stands for the outgoing antiproton and X denotes all other hadron matter that is being produced in the reaction. The conservation of energy momentum requires that k+k’=p+P,

(2.1)

where the four vectors k and k’ correspond to the colliding lepton and antilepton, respectively, p is the four momentum of the antiproton and P denotes the overall four momentum of the rest of the created hadrons. The invariant quantities M2 = P2, q2 = (k + kr)2,

1

v = ~PCL

q=k+k’

p2 = MN2

LEPTON

are not independent,

PAIR

but are related in a fundamental q2 - 2M,v

477

ANNIHILATION

kinematical

equation

+ MN2 = W.

(2.4

The Eq. (2.2) follows directly from (2.1). In the annihilation channel, the variable q2 has the physical interpretation squared mass of total hadron matter produced and its physical range is

of the

4MN2 < q2 < co.

The variable v is the total energy of the lepton pair, measured in the rest frame of the outgoing antinucleon. v is positive definite, and, moreover

This lower bound can be easily obtained by considering the invariant quantity v in the center-of-mass Lorentz frame of the I-l-:- pair. In this frame q = 0, q. := d\/s’, and mj$&pq = yq

1

poqo

In the limiting case when X refers only to an outgoing antinucleon M == MN , the Eq. (2.2) collapses into

with the mass

0 = q2 - 2M,v.

For larger values of M, 0 < q2 - 2M,v.

(2.3)

Most of these kinematical considerations can receive their compact graphical expression through the use of the (v, q2/2MN) diagram depicted in Fig. 1. When the total center of mass energy of the system grows larger, the total mass of the produced hadron matter X can either remain constant, or can increase as well. In this latter case, i.e., when MN~ < q2 and MN2 < Ma,

478

PAVKOVI6 q2/2MN

- v

FIG. 1. A (v, q2/2MN)

graph of the kinematical region of the lepton pair annihilation

process.

we can again talk about the double Bjorken limit, provided that the quantity

=l--2-M~2 q2

remains finite. The range of w in the physical region of the annihilation follows immediately from (2.2) and (2.3). It is

channel

O
3. THE INVARIANT STRUCTUREFUNCTIONSWI AND m2 The lepton pair annihilation process is, like the inelastic electron-nucleon scattering, completely characterized by the two invariant structure functions7*8 WA@, 4 and W2(q2, 4

x [2~(qz,v)+~(l-+)

2

‘) sin2e .

‘~2(4’,

2M

N

1

(3.1)

This statement is valid under the assumption that only one hadron is being detected in the final state of the process and that the nature and the energy momentum distribution of the rest of the produced hadron matter is left unknown. This is the

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ANNIHILATION

origin of the summation sign, referring to these undetected final states, in the upcoming formula (3.2). 6’ is the angle between the momentum of the outgoing hadron which is being detected and the collision axis of the Z-Z+ pair in the center-of-mass coordinate frame. 01is the fine structure coupling constant, N = l/ 137. It should also be noted that the formula (3.1) is valid provided that only one virtual photon leaves the volume of I-I+ annihilation. The structure functions W1 and WZ are defined as the invariant objects in v”“(q”,

v) = 47r2 ; -&-

(0 I j,(O) I p, p,Xp,

pn I j,(O) i 0) * (271)’ a4(9 - p ---- p,)

N

1p, p,j is a final state.It includ es an outgoing selected hadron (antinucleon) with momentum p, and the rest of the hadron matter with the overall four momentum pn . j,(O) is the total hadronic electromagnetic current taken at the origin of the Minkowski space. The symbol C, in the expression (3.2) also includes the averaging over the helicity states of the antinucleon p. Inverting the defining Eq. (3.2) for the invariant functions r;i;i, and WZ, one obtains

and F2C9”?

v> = *

[; (W1dq2, v) + m2$Aq2,v)) - $ W&q2, v)].

(3.3)

In the resonance model that we are considering, the states p and pn which form the final state are actually members of an infinite multiplet which comprises particle states with arbitrary high fractional spin values. The normalization for these oneparticle states, which is consistent with (3.2) is given by (P’J’JZ’

I PJJ*) = &P’ - PI &&J,

1 3 5 J = 2 , z , z ,...,

For reasons of simplicity,

J, = -J,

-J

3

+ l,..., J.

the other quantum numbers such as isospin and hyper-

480

PAVKOVIb

charge, which are normally necessary for the complete specification of the single particle states, have been suppressed. The current entering the expression for V,,y is locally conserved. The ensuing continuity equation, restricted here to the oneparticle subspace of the Hilbert space of all physical states, reads (P’ -PI”


(3.4)

or, in the crossed channel (hadron pair creation) (P’ + PI“ (P’J’J,‘,

pJJz I h(O) I 0) = 0

(3.5)

The Eqs. (3.4) and (3.5) are supposed to be valid for any p’ = (z/M;?; + P’~ 1p’), J + p p > an d any two ordered pairs (J’, J,‘) and (J, J& p=(-I We can now return to the expressions (3.2) and (3.3) for V1 and FK2 structure functions. The Bjorken conjecture,s applied here in the lepton pair annihilation channel, is contained in the statement that in the asymptotic region of larger q2 and v, the limits lim

q%m

Vl(q2, v) = F,(w)

w fixed

and lim

*Lu

viV2(q2, v) = F,(w)

w fixed

exist and are generally different from zero. Our aim is to state the conditions which must be satisfied within the framework of an infinite resonance model in order that the Bjorken limit q2 -+ co, w fixed give rise to the nontrivial functions F1 and F2 . But first we want to explain in more detail the simple physical picture behind the use of infinite multiplets in the annihilation channel.

4. INFINITE MULTIPLETS IN THE ANNIHILATION

CHANNEL

The use of infinite component fields in the description of lepton pair annihilation processes offers a simple picture of the reaction. Let us denote by p an antinucleon with mass MN that is being detected and let X stand for the rest of the produced hadron matter I- + If + p + x.

(4.1)

Then, in the spirit of infinite multiplets, one can visualize (4.1) as a process in three stages. In the fist stage, the lepton-antilepton pair is annihilated and con-

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ANNIHILATION

verted into the highly virtual photon of mass dq2. In the second stage, the virtual photon splits into two parts-the antinucleon p and the resonance of spin J and mass MJ carrying the same quantum numbers as X does. Visualized from the center-of-mass Lorentz frame, antinucleon p and the created higher spin resonance leave the volume of reaction with three momenta of the same magnitude but of the opposite direction. The numerical value of the

length of the three momentum matical relation

is implicitly

contained in the fundamental

MJ2 = MN2 - 2MNv

kine-

~- q2.

Within the framework of the infinite multiplets, all resonances have a strictly infinite lifetime. In reality, they are generally short-lived objects which promptly decay into a multitude of more stable hadrons. It is the resonance decay which completes the third stage of the process (Fig. 2).

Resonance

with

spin

J

FIG. 2. A diagram showing schematically a resonance creation and its subsequent decay. Note that the infinite component wave equation does not provide the means to calculate the last step of the process-namely, the lifetime of the resonance and the number and nature of decay products.

For the classification group of resonances in their rest frame, we consider in the following sections highly idealized SU(3) x g and SU(3) x [d x g], respectively, where SU(3) stands for the usual group of isospin and hypercharge quantum numbers, g represents an infinite dimensional unitary irreducible representation of the group SL(2, c) and d denotes the usual space of four component Dirac spinors. The mass spectra of the resonances are described in two different models by two different infinite component wave equations. The same wave equations specify the locally conserved electromagnetic currents and determine the form factors in both scattering and annihilation channels.

482

PAVKOVIC 5. MODEL I: ABERS-GRODSKY-NORTON

The infinite component

WAVE EQUATION

wave equation

(5.1) m, , m, real parameters, was first investigated by Abers, Grodsky, and Norton.lO Later, it was found that the associated conserved vector current

appears among the factorized, relativistic quark model representations of the local current algebra at infinite momentum. 11-r3 In Eq. (5.1), yU are the familiar 4 x 4 Dirac matrices, while a,, and Luv are the generators of the SL(2, c) group, acting in the nonunitary space of Dirac spinors d and in the infinite dimensional unitary representation space g of the direct product d x g, respectively.14 The classification group for the infinite multiplet in its rest frame is d x g. Though other possibilities are certainly open, we will restrict our attention to the two Majorana representations. Thus, either g = (0, $) or g = (3 , 0) in the Gelfand notation.15 In the first case, g = (0, $), the internal “orbital” angular momenta run through the values 1 = 0, 1, 2 )...) while in the second case, g = (+ , 0), the analogous sequence reads 1

=

+

) 8

) Q

)...

.

The spin of the resonance is the sum of two parts J = L + $a.

(5.2)

The first term in (5.2) corresponds to the “orbital” angular momentum, while the second adend in (5.2) stands for the spin 4 of the quark. Combining orbital angular momentum with the quark spin, one obtains either fractional overall particle spin J = 4 , 3 , $ ,..., or the integral spins J = 0, 1, 2 ,... .

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ANNIHILATION

The former case is suitable for the description of baryons, while the latter is suitable for the description of mesons. The mass spectrum associated to the model contains four brancheslO MJ* = m,(J + 4) + drn12(J + i-y + Mo2, - MJ’ = m,(J + f) & v’m12(J + $)” + M,“?, MO2 = mo2 - 2m,m, + Qm,2.

(5.3)

The first two branches correspond to the “particles”, while the second two branches describe the “antiparticles”. The equality of absolute values of particle and antiparticle masses can be formulated in terms of CPT invariance. However, we will not discuss here the transformation properties of various objects that appear in (5.1) under the action of discrete symmetries C, P, and T. The first branch MJ+ of the particle mass spectrum is ascending and physical, while the MJ- branch leads to an unphysical accumulation point zero. For J + cc, lim,,, MJ- = 0. We will therefore eliminate the branch MJ- and the corresponding antiparticle branch -MJ- from further considerations and deal exclusively with the masses MJ+ = MJ, MJ = m,(J + 4) + dm,B(J

+ @m

(5.4)

1 3 5

J = z , z , z ,... .

Besides the described real mass spectrum, the wave equation (5.1) also possesses the continuous multitude of the imaginary mass solutions p2 < 0, as well as zero mass solutions p2 = 0.12 We do not consider these additional solutions of the wave equation in the applications that are discussed in the present article. It should be mentioned, however, that those less familiar solutions are needed for completeness and cannot be ignored in the construction of local field operators. Since the object of the present article is the analysis of creation and annihilation processes, it is convenient to emphasize the field theoretical aspect of the formalism. The procedure of constructing infinite component fields involves, as is customary when one deals with free fields, the creation and the annihilation operators of the single particle / a, u+ / and antiparticle / b, b+ / states {aA,( MP’), @,W), {Mp’), {w(P’),

h(P):

= 0,

%‘(P>> = &,qP’

- Ph

hi(p)) = 0,

(5.5)

h+(p))

= %,&P’

- P),

b,(p))

= {a,,(~‘),

h+(p))

= 0.

484

PAVKOVI

6

The correct quantization scheme is Fermi-Dirac,lO as indicated by the curvy brackets. h stands for the ordered set of quantum numbers that are needed for the complete specification of a single particle state. In the case of real mass, h includes mass, spin, helicity, and parity. In this case, the multiple index h is purely discrete. Notice that the concept of spin is intimately related to the reduction of the classification group SL(2, c) with respect to its compact subgroup SU(2). If we want the field operators to be local, we must include the imaginary and zero mass states as well. This is the reason why we have introduced the index h which is a more general concept than ordinary spin and helicity. For example, in the case of imaginary mass a “spin” piece of h is continuous. This time, the “spin” corresponds to the eigenvalues of the invariant operator which is associated to the noncompact subgroup SU(1, 1) of the SL(2, c) group.12 The physical significance of this new quantum number has yet to be clarified. In the case of the continuous h, the symbols a,<, on the rhs of the Eqs. (5.5) should be replaced by the 8(x’ - h) functions. Also, when appropriate, the sum over h will be understood as an integral. The field operators are defined by the expressions

where the amplitudes

and 1

XdP) = mA(p) e--ipYb

I cb)

I 0)

with I hP) = h+(P) I (07

%(P) I (0 = 0,

I hP> = b+(P) I o>,

b,(P) I 0) = 0

are the infinite component analogs of the ordinary Dirac spinors. They satisfy the wave equations (YP - d)

#(P> = 0

and

(5.7) (YP + =Jo X(P) = 0,

respectively.

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485

ANNIHILATION

Thus, apart from a constant of proportionality, (5.8)

X(P) = Y&J)Without loosing much generality, we can now assume that the normalization satisfy N,(P) = R*(P). This relationship

enables us to define the unitary charge conjugation

factors

operator C,

C&(P) C-l = BAG C%‘(P> c-’

= h’(P),

c I 0) = IO)

and verify that C induces the following transformation

on the field operators

and q+‘(x)

y& c-’ = - 1Clp(x,%,$I.

Here, the 4 x 4 “charge conjugation” product d x g. The equalities q+ = w-1,

(5.9)

matrix 92 acts in the piece d of the tensor GfT = -$$

and vy,G?-1 = --yoT are already familiar from the formalism of the ordinary Dirac equation. The justification for the interpretation of C as a charge conjugation operator comes from Cj,(x) C-l = -jJx). where the current operator .Lw is the local Hermitian satisfies the continuity

= : P(x) YoY,W):

(5.10)

vector object which, by virtue of the wave equation (5.1), equation au j,(x) = 0.

The normal ordering assures that (0 I “L(x) I 0) = 0.

486

PAVKOVIC

The current matrix elements in the scattering channel are given by
I hP) = G4P’)

N,(P) YGP’, YmMP),

while the analogous matrix elements in the lepton (creation of the (X’p’, jp) pair), read Q’P’Y AP I.im

I 0) = KTP’) = mP’1

K(P) h*(P)

channel

4’CP’) WtLXn(P) $&P’)

In the case of real mass h = (J, J,) and the normalization equal to1

It is understood that, in the applications, of the mass spectrum.

pair annihilation

(5.11a)

YOYeY5#A(P).

(Kllb)

factors are numerically

MJ will always refer to the physical branch

6. CURRENT MATRIX ELEMENTS AND THE STRUCTURE FUNCTIONS WI AND T2 The amplitudes #.,.,= and #aQ , corresponding to the resonance and the detected antinucleon, respectively, can now be expanded, as was explained in detail in I, in terms of the eigenvectors of the commuting set yO , J2, L2, and J, :

In fact, the problem employs only two l’s, I = J f 4, since that is the way quark spin-& couples with the internal orbital angular momentum. The numerical values of the coefficients in the expansion

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ANNIHILATION

have been determined in 1. They are matrix elements of the matrix S, which diagonalizes the mass operator --1

m, - inI

MJ+ - MJ-

S=

-’

2

‘1

m. - ml M,+ - MJ-

21

Tj t-

Note that we again omit the symbol which would serve to distinguish the physical reference from the unphysical branch of the mass spectrum. Thus, z,!J~~, is actually But before expressing current matrix elements in terms of c$$$ amplitudes, a, . we rewrite the current matrix elements in a form which makes use of the boost operators. This form turns out to be convenient for computational purposes,

where &(p,

, p) is defined as in I,

e(PJ >PI = B+(PJ) YuYu~(P~

t =e T=

F. YoYue?

= @

e+itQe-icQ,

Ci=

=

YOY

and I&, #t0 are rest frame spinors. Now, in the center-of-mass Lorentz frame which is utilized in the description of lepton pair annihilation processes p = (MN cash 5 j 0, 0, MN sinh .$), pJ = (MJ cash .$, I 0, 0, MJ sinh &)

and MN sinh .$ + MJ sinh fJ = 0.

This alignment of linear momenta of resonance and outgoing along the z-axis simplifies the computation of FFyS quantities E(P,

, P>y5 = eixo” [cash -$ + 01~sinh -$I

where x = C-J- 5 ’ 595/64/2-12

antinucleon

state

yOyU [cash $ + 01~sinh $1 y5,

488

PAVKOVIC

We are now forced to introduce another hyperbolic angle I& # = 5, + 6 An explicit evaluation of sM(pJ, p)y5 for p = 1,2,3 gives 4E;y5 = u, cash -$ + icx, sinh s, szy5 = uz cash $- - icx, sinh $-, 9t,y, = u3 cash T* + y6 sinh T,dJ a=

a (

0'1

a=

Expressed in terms of rjJtJ$ amplitudes,

(0

Q 1'

y5 = (1

').

the current matrix elements read

IC’:J,~y5#fo = [ S,, I2cash $- &‘,,eiyQ~q&~

+ iS,*,S,, sinh -$- ~#~$~e”~~~cu,~~~~ + 1S,, I2 cash x &)+ JJ+ 2

tJse

‘“QE~~.)~~~

+ iS.$S,, sinh $- $~;~~Jze’xQ~a2~~~;

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ANNIHILATION

The matrix elements of 9&, do not enter in the expressions for WI and VZ , and besides, as a consequence of the current conservation, they are zero in the center-ofmass Lorentz frame. For computational purposes, it is now convenient to descend from the level of 4 x 4 matrices to the level of u matrices. We introduce the new 2-component (in the Dirac space) spinors u and Y:

and write the current matrix elements as

j- is*11 S12 sinh X ujJ..+ JzeixQ%& &lo 2

+ i&Y,*, sinh x2 vtJJf +

1 S12 I2

cash

x

2

iXQ, IJZe

V:J+!J

~2UuIo

eixQz

z

TV @7;

= ( sll I2 cash x u~J.+J,eixQZcyi & 2 - iS$12 sinh X us,-. aJ eixQ%,c Ilo 2 2 _

- iS,*,Sl, sinh -& vtJJ++JZe

iXQZ

+ / S,, I2 cash x v+JJ++Jae

ixQ,

2

2

~1UAOo

a,~jlo;

1sll 1’ cash -II, u~J-~J,eiXQ”uguiOo 2

i S&S,, sinh T# u:J&gJ,eiYQzc&lo + ~,*,~ll

sinh

* 7

V!J+~J,eixQ’ugOo

-i- 1s12 I2 cash $ t$J+ $JzeixVio&lo . (6.1)

The numerical evaluation of the current matrix elements follows much the same pattern as in the e - N scattering channel. Matrix elements which correspond to

490

PAVKOVIC

the first two terms (j.~ = I, 2) in (6.1) have already been calculated in I. The matrix elements for 9&, are given in the Appendix. The knowledge of current matrix elements enables us now to construct the invariant structure functions IV1 and IV2 for any values of v and q2 inside the physical region of the process. Our primary concern is the Bjorken limit.

7. wl,w2

AND THE BJORKEN LIMIT

Expressed in terms of the current matrix elements (6.1), the quantities FU, read 1 0=-l/2

Je=-J

* I &(P)l”

5=1/2,3/2.5/2

f d”p,, a4(q - p - pn) ,...

I WPn)12 I &Jz(PJ w~X~o(P)12~

where the antinucleon by assumption is placed at the bottom . Writing X(P)antinucleon= x&p)

of the spin ladder,

Go - PO- Pno) = 2p,o@(qo- PO- Pno) wqo - Po12 - P2no) = 2/&2

- w) 0 (@-

J?f@ -pno) 8(q2- 2M,v + iwig - i&2) d/42

or = $

~cQo-Po--Pno)=~~~

w(2 - w) 0 (@

- yg

- pno) S(w - WA

N

N

where w, = w.J(q%)= 1 -

and performing

0=-l/2

the integration

JL=-J

* $g

2lbf,V

w=---$-4

over the phase space of the resonances, one obtains

5=1/2.3/2,...

42 - w) 0 (dip N

ikfJ2 - MN2 cl2 ’

- f$

- PJO) %w - WA.

(7.1)

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ANNIHILATION

By taking into account (5.8), one is led to an equivalent form

*&

lV(2 - w) 0 (\/gi

- y+

- PJ,) qw - WJ).

(7.2)

N

Note that this expression for VU, is still valid for any pair of physical values of q2 and v, not necessarily in the “deep” kinematical region yet. Let us now consider the Bjorken limit q3 - co, w fixed. A short examination of (3.3) reveals that the quantities IT1 and VZ will possess the nontrivial Bjorken limit only if the objects fii(w) =

lim ; q3,m w fixed

’ / #:J,k

F

ii

0=-l/2

J,=-J

-

P)

cw

I &(P)l”

~O%?%~&>~’

&F-

I N,(q - P)l” w(2 - IV),

i== 1,2

(7.3a)

N

and f,,(w) =

lim qb,

f

.i

1f e--1/2

IO fixed

- 1 &J,b

-

d

i

(27~)~/ m,(,)12 I N,(q - p)i2

J.=-J

%I~3%$&>12

w(2 - w)

&-

(7.3b)

N

exist and are different from zero. As in I, in the definition of jifii and f33, it is understood that the spin J, wherever it appear, has to be replaced by the physical branch of the trajectory J = U(M), where the mass M is, in turn, related to the quantities q2 an dw through the mass shell equation hf2 = MN2 + q2(1 - w). As in the scattering channel, the nontrivial functions fifii and f33 by themselves will not assure the scale invariance of the quantities16 F,(w) =

lim qbm

w fixed

Vll

+

2

v22

=

Fll

+.f22

2

c

J=1/2.3/2....

@

(&+pJ,,)

&w--w,),

492

PAVKOVIb

F,(w) =

zzz

2 vu + ~ q2”4 v2 [ 2

lim $a& --

~MN

W

fn

+

- $ wsti]

K2

.f22

c

2

@ (dF

- 3

- PJO) &w - WJ)

J-1/2.3/2,...

(7.4) in the strict Bjorken sense. The reason is as follows: Theoretically, the quantities F1 and P, , plotted against the variable w will be represented by a forest of 6 functions. In the experimental data one expects to find the resonances as an infinite sequence of more or less pronounced peaks located at the points wJ

=

1 _

MJ2

-

MN2

,

cl2

and whose height will vary as prescribed by the scale invariant functions j;, , = depend upon the value of q2 and move f 22 2 and fs3 . Now, the points rightwards toward w = 1 when q2 increases. Thus, by knowing the value of q2, one can, at least in principle, identify and follow the path of the individual resonance. This is the mechanism which spoils the scale invariance in the strict Bjorken sense. Nevertheless, one can introduce the concept of scale invariance in an average sense by defining wJ

wJ(q2)

(F,(w)),= fil ;fe2 and

(F,(w)),= - *

[fl1; f22-&I.

Experimentally, one expects the data points to oscillate around the smooth curves which represent (F,),, and (F2), , plotted against w. One can obtain scale invariance in this generalized sense, provided that the quantities JJw), i = 1,2, and fa’,, are nontrivial functions of w. For very large q2, w fixed, the graph of vV2 plotted against w will be characterized by an enormous number of resonances, all piled up on the left side and very close to w = 1. (F,(w)), will appear as an average of these resonances (See Fig. 3.). Clearly, in these circumstances, it is not the properties of the individual resonances, but their statistical averages that matter. Finally, we remark that the applicability of the formulas derived in this section

LEPTON

PAIR

493

ANNIHILATION

p w,=l-

MJ= - A4pJ2 q2 ’

I 2

6(w - WJ) + J 77 (w - WJY +

C’ e i 2 1

FIG. 3. A graph of the function vmZ plotted against the scaling variable w as one would expect in the case of an infinite-multiplet model with finite Bjorken limits for the structure functions PI and VIVA. The idealized zero-width resonances, which are described by the a-functions are replaced on the graph by more realistic resonances of finite widths described by the Breit-Wigner functions. The value of ,, need not be of only one sign. On the other hand, both W,(q2,V) and ~Jq”. V) are positive definite.

is not limited to the AGN wave equation. lo With the appropriate changes of the normalization factors and the current matrix elements, the formulas can be utilized in any model based on an infinite component wave equation. In particular, to derive the final result, we use them in the second model that is considered in Section 8. WI, m2 and the Bjorken Limit-Continuation All the computational details are now at hand. To reach the final result, one need only know how the hyperbolic angles x and $ are related to the invariant quantities v and q2.

494

PAVKOVI

6

In the center-of-mass Lorentz frame cash x = cosh(& - 6) = cash .& cash 5 - sinh & sinh .$ -

PJP MIMN

=

q2 - MJ2 - MN2 2kfJMN ’

q2 = (PJ + P)“.

(7.5)

This gives 1

2

cosh2 x

~&MN

= 1 + cash x = q2 - (MJ - MN)2 ’

2

At this point we will make a short digression and see how and why the crossing symmetry is spoiled in the case of infinite multiplets. Consider, for example, the matrix element

(00 1IF” 100) =

l

,

(7.6)

cash x

2

which, in case one employs Majorana representations as the Hilbert space framework for the single particle states, is intimately related to the elastic form factors. In the scattering channel cash x =

1

MJ2 + MN~ - q2

2ibfJM~

’

~MJMN

cosh2 L

= (MJ +

MN)~

- q2

2

and the matrix element (7.6) is equal to / MJ = MN /:
scattering channel.

On the other hand, in the crossed channel, cash x =

42 - MJ2 - MN2

2ikf,M,

1 ’

cosh2 -$

~&MN

= qa - (MJ - MN)’

LEPTON

PAIR

495

ANNIHILATION

and the matrix element (7.6) gives entirely different functional dependence upon 9”: (0 0 1.ziXQ”\OO) 23,

crossed channel. h2

This situation is typical. The relevant matrix elements are analytic in the angle variable x but not in the invariant momentum transfer q”, since the relationship between x and q2 is different in the two different channels. Though the problem of crossing symmetry is far from solved by this illustrative example, one thing is already clear-namely, that the loss of crossing symmetry in the framework of infinite multiplets is intimately related to the use of unitary representations of X(2, c) in the description of single particle states. It was the unitarity of the relativistic boost operators e@~Q,e-f5Qz

_

@Qz,

x=&-5

that introduced the hyperbolic angle x. In the case of Dirac relativistic boosts, for example, which are not unitary, similar equation leads to a different hyperbolic angle CJ 5 e2 nseT ‘3 = ,$ %) * = 6 + 5, cash # = cash EJ cash f + sinh fJ sinh f = AN

(POPJO - P”>

=-------v+MN

2M~-

MJ

V2 MJ

‘F>

because

and p” = MN2 (5

- 1).

(7.7)

Note that in the special case of equal masses for the bra and ket states *=o and the question of crossing symmetry is left to be resolved by detailed dynamical considerations. In contrast to the situation with finite dimensional nonunitary representations of SL(2, c), within the framework of infinite multiplets, where the

496

PAVKOVIk

kinematical and dynamical questions are much more intertwined, the issue of crossing symmetry is already urgent on the level of relativistic boost operators. We return now to the problem of T1 and m2 functions and their Bjorken limits. In the infinite resonance model for the lepton pair annihilation process, calculations are performed on the sequence of mass shells MJ2 = MN= - 2MNv + q2,

J = + , $ , Q ,...,

where it is assumed that the antinucleon is detected at the end of the reaction. For large values of q2 and v such that w = 2MNv/q2 is kept fixed, the masses M, grow according to the formula MJ2 = MN2 - 2M,v

+ q2

= MN2 + q2(1 - w). On the other hand, the Model I predicts the mass spectrum (physical branch) MJ = ml(J + 4) + dm12(J + ii)>”+ Mo2, Inverting

J = +, j,... .

(7.8)

(7.8), we obtain the Regge trajectory J = a(M) =--

:+

M2 - MO2 2m,M

(7.9)

or

As explained previously, wherever J appears in the calculations, be substituted for by the continuous trajectory (7.9). On the Bjorken limit we find 1 cosh2 x 2

~MNMJ

= q2 -

(MJ

-4M,(+l)&,, qbx co fixed

tanh $- - &cc w fixed

1,

-

MN)~

it will have to

LEPTON

[tanh+]”

PAIR

= [l -

491

ANNIHILATION

cos;,x]J 2

co&2

.k

2

=

MJ

+ ~MJ

x

tanhTF*

MN

L&

+

v 2M,

(1

-

M’)

-“Jy

qLz0 1

w fixed

With all the technical details at hand, the computation vVz is straightforward. The net result is lim

Wl(q2, v) = d,g,(w)

C

qhm

w fixed

of the quantities

J=1/2.3/2,5/2,...

and lim vFV2(q2, v) - - ?l!!f!t qbcv 11’

IU fixed

lim wI(q2, v), q2-m

w fixed

where

- 2 & = (1 - w) e wJ = w,(qz) = 1 _ MJ2 --2 MN2 ,

($ - 1) , J=’

2 5

2 ’ 2 ’ 2 ‘... *

VI and

498

PAVKOVIC 8. MODEL II-NAMBU-FRONSDAL

WAVE EQUATION

In this model, the Hilbert space of one-particle spin states at rest is the Majorana representation g = (9 , 0). The spin content of g is J = & , # , Q ,...

a series which is suitable, in a most rudimentary manner, for describing an idealized infinite sequence of baryon excited states. The infinite component wave equation (0 + 2iar,a~ + b2) #(x) = 0,

a, b real parameters

(8.1)

was first considered by Nambu and Fronsdal. I7 Later, it was found that the associated locally conserved current satisfies the Gell-Mann-Dashen current algebra at infinite momentum.lsJ9 Although it is possible to consider the infinite dimensional matrices I’, , E.L= 0, 1,2, 3 acting in the selected infinite-dimensional unitary representation spaces of noncompact groups of a larger order than SL(2, c),17 for reasons of computational simplicity we will restrict ourselves to the already described Majorana representation. For this case, the matrix elements of r, are well known.20 The equation (8.1) does not have positive and negative frequences placed symmetrically. To remedy the situation and make space for antiparticles, we artificially double the space of particle states by introducing the 2 x 2 matrix TV,

(0 + 2iaT,r,ap

+ b2) #(x) = 0.

(8.2)

Now, besides the two particle brancheP MJ* = a(J + 4) & da2(J + 4)s + b2,

the mass spectrum also comprises the symmetrically placed antiparticle branches --M,*. Of the two particle branches, &+ is ascending and physical, while MJhas an unphysical accumulation point zero, lim,, MJ- = 0. This unphysical feature is sufficient to eliminate MJ- from a role in the applications that are considered here. Henceforth, under the mass spectrum, we will always mean its physical branch, MJ E MJ+. Besides the real mass spectrum, the wave equation (8.2) also possesses spacelike, p2 < 0, and lightlike, p2 = 0, solutions. Unless b = 0, there are no null solutions p = 0. Although there is nothing in the kinematics of lepton pair annihilation processes which would preclude the appearance of states with p2 < 0 and p2 = 0, the question of whether these states are part of physical reality is not appropriate in the present article. Thus, we mention the

LEPTON

PAIR

499

ANNIHILATION

p2 < 0 and p2 = 0 states only in connection with completeness. We stress that it is the property of completeness which plays the crutial role in the construction of the local field operators. The quantized form of the model, specified by the Eq. (8.2), is obtained by defining the field operators

with

d3pCn+= Mx)= SPo,”

NA(P>UA(P)G(P)

#2(4= jp,,o”3P;eiasfly z&Jb,+(p) sh+w

= I, >“d3P?

0

einL NA*(p)

KS’(P)

a,,+(p)

(8.4)

~2+(X)= j, ,. d3pC~ @px ~~A*(PI ~5+(p)h(p) 0

In this second model, the particle creation and annihilation operators a, a+ (particles) and b, b+ (antiparticles), form the cannonical commutation instead of anticommutation relations. The nonvanishing commutators are

h(P’>, Q+(P)1= hxP’

- P)

and

(8.5)

Rdp’), h+(p)1= &A%P’ - P>. In the case of real mass, the multivalued index h includes mass, spin, helicity, and parity. In this case, h is discrete. In the case of imaginary and zero mass, a “spin” piece of X is continuous. As in the preceeding model, in the case of continuous h, the symbols on the rhs of (8.5) should be understood as the 6(X’ - X) functions. Also, the signs of summation in (8.4) should be replaced with an integral when appropriate. The amplitudes

and

500

PAVKOVI

6

with I AP) = 4(P) I o>,

6(P) I 0) = 0,

I A,> = G(P) I o>,

b,(P) I 0) = 0

are the infinite component analogs of the familiar p-dependent Dirac spinors. They satisfy the same infinite component wave equation: (p” - 2arp - b2) u(p) = 0 (8.6)

and

(p2 - 2arp - b2) u(p) = 0.

Finally, NA(p) and flA(p) are the normalization we will assume that N,(P)

factors. Without

= R?(P).

losing generality, (8.7)

In this case, one can simply establish the relationship between particles and antiparticles by first introducing the 2 x 2 matrix V defined by21 cp z.zzg-1, Vr,F

w = -97, = -5-3 ) a = 1,2,

@%3*;(P) = ~~(P)9 h(P)

= U(P),

#2(P)

and then also the unitary charge conjugation identical, to the matrix %‘) defined by W(p)

=

4.d

operator C (which is related, but not

C-l = MO9

Caf;(p) C-l = b:(p).

and c I 0) = I 0). The relationship between the normalization factors (8.7) now enables us to shift the operation of charge conjugation from the particle level to the level of field operators

LEPTON

PAIR

501

ANNIHILATION

The physical meaning of the charge conjugation on the locally conserved current operators

operator C is reflected in its action

j,(x) = : #'(x)(iG - u7ar)u #(x):, (0 I.jJx) j OI> = 0.

Cj,(x) C-l = -jJx),

In the scattering channel, the current matrix elements are

while in the lepton pair annihilation sponding matrix elements read

channel (hadron creation channel), the corre-

O'P'? & i.im IO> = Ni+G) mJ) Gm'

- P + 24,

%(I+

In order to avoid serious misunderstanding, it is necessary at this point to reconsider the delicate problem of parity and its relationship to the concept of an antiparticle. Experience informs us that in the case of fractional spins the particles and the corresponding antiparticles carry parities of opposite signs. Now, a single Majorana representation yields the parity sequence of either (-l)J-l/” or -( -1)J-1/2 type, depending upon the overall parity assignment of the tower. It is the first sequence which is suitable for description of the tower in which the lowest state l/2+ is reserved for the nucleon. In order to assure that the antinucleon and, moreover, any other antiparticle in the associated tower is equiped with the required opposite parity, namely -(-1)J-1/2, one is forced to employ another Majorana representation, g = (- 4 , 0), characterized by different overall parity assignment.22 Thus, in conclusion, PUJ’(P) P-l = (- l)J-3 aJ+(-p), PbJ+(p)

P-l

=

-(-I)“-*

b,+(--p),

where P+ = P,

P2 = I

and also PIO)

= IO).

The procedure of calculating the current matrix elements in the lepton pair annihilation channel follows the same pattern as in model I. Note that the particles and the corresponding antiparticles satisfy the same Eqs. (8.6) and therefore employ the same boost operator. The proton is described by the spinor

502 while the antiproton

PAVKOVIC

state is represented by the spinor wave function

It is convenient to introduce the non-Hermitian

current variables

For jk , we obtain:

&,(PJ)(PJ - p + 2ar)4

uadp) = (JJ, / eibQz(2d’)* = 2a c x

(It is the antiproton equalities,l

=

55

-

(JJ, 1eixQz I J’Jz’XJ’J,’

which is detected in the final state.) Exploiting

we obtain

(J - NJ+

B)(J+ 8) * 3!

I r, I W,

5.

and

and23

eeitQz 1 Qo)

1

the familiar

LEPTON

Also, the z-component

PAIR

503

ANNIHILATION

of the current gives

Note that

Now, in the process of going into the kinematical limit q2 + co, w = 2MNv/q2 fixed, we foIlow the same procedures that were outlined in the treatment of Model T. In particular, we will replace J + & , wherever it appears, with M,/2u, since pz MJ = 2u(J + g>,

The explicit expressions for the normalization

w fixed

595/6412-13

etc.

factors are

504

PAVKOVIt

and I N,(P,)12 = (jy&

=

M,+yM,-

(%13) 1 _ 1MNv

1 ’ l/s

MI+ ’ MJf - MJ-

q2

w fixed

Again, as in Model I, we are able to assemble the exact expressions for v-(q2, v) and V2(q2, v) valid for all values of the variables q2 and v. However, the present model is too crude to give useful numerical predictions in any kinematical region. Our primary interest is in the behavior of WI and V2 functions on the Bjorken limit.

c

0 (* - $y - p,,)--& qw- w,)

+1/2.3/2,5/2....

w-fixed

a, = 3MN3 4a

1 da-F’

M,2 - MN2

w, = w,(q2) = 1 -

q2

and lim

g&c

8 (@-

Wl(q2, v) = dJ;(w)

- $y

- p,,) &

qw - w,).

&l/2.3/2.6/2....

wfixed

fl(w)

=

w

($

-

q3

e-F

(it

-

‘).

Thus, the Bjorken limit does not exist in the sense that for large values of q2, w fixed, the contribution of resonances decreases as l/M,. This is the opposite extreme from the Model I, which predicted diverging rand vV2 . Obviously, the existence of the nontrivial Bjorken limit is a strongly model-dependent feature.

LEPTON PAIR ANNIHILATION

505

CONCLUSIONS

For not too large values of the total center of mass energy of the colliding leptonantilepton pair we can expect the final hadronic state of the reaction to be dominated by high mass and high spin resonances. This belief is supported by the experimental data on the similar process of proton-antiproton annihilation, which shows an abundance of resonances for all center of mass energies that have been investigated thus far.24 It is clear that the infinite component fields are a suitable tool for the description of resonances, whether they appear in the e - N scattering or in the lepton pair annihilation channel. It is less clear at the moment from which modei, based on an infinite component wave equation, one can expect best quantitative results. Though satisfactory in the qualitative sense, the two models that were considered in the present article are presumably too crude to give reliable numerical predictions. This applies to both mass spectra and form factors. However, it was not our intention here to achieve a good qualitative description of the resonance region. Instead, we wanted to learn something about the behavior of the available infinite resonance models in the kinematical region MN2 Q q2, MN2 < 2MNv, w = 2MNv/q2 fixed. This region will hopefully become accessible in the near future, with the construction of powerful l-Z+ colliding beams machines. It is also intimately related, through the operation of crossing, to the similar kinematical region now under investigation in the deep inelastic electron-nucleon scattering. The two models that were considered in the present article failed to fulfill our expectations-namely, they failed to give the nontrivial Bjorken limits for the structure functions r1 and w2 . The first model, based on the AGN wave equation gave a diverging answer, while the second model, based on the NF wave equation, predicted the vanishing of V1 and vF2 on the Bjorken limit. Nevertheless, the mere existence of the two opposite extremes suggests the possibility that the construction of an infinite resonance model with nontrivial Bjorken limits for the functions m1 and vW2 is at least theoretically feasible. Of course, the ultimate answer to the state of resonances in the deep lepton-pair annihilation processes will have to come from experiment itself. Theoretically, strong support for resonances, even in the new kinematical region q2 - co, w = 2MNv/q2 kept fixed, comes from the existence of presumably infinitely rising Regge trajectories. Indeed, at the moment, the evidence about the existence of maximal stable mass in the hadron world is very meager.25 It is only natural to assume that the objects which lie on infinitely rising Regge trajectories will always show up in the experiments, wherever this is allowed by kinematics and the selection rules. However, here we are touching the delicate and extremely important issue of whether experimentally each individual resonance is indeed accessible or whether one can best hope in certain experiments to observe an

506

PAVKOVIi:

accumulated effect of many resonances, not their individual features. Experimentally, the rate with which the new resonances are being discovered does not show any sign of slowing down. 26On the other hand, theoretically, the very rich hadron spectrum seems to be an inescapable consequence of duality.27 In these circumstances, one is justified in questioning whether the quantum numbers of individual resonances are really what matters. It is not hard to imagine the situation, which can realistically be expected to occur in the higher mass region, where the only relevant experimental information which one can hope to extract will be the average properties of the clusters of resonances. Besides this more fundamental issue of the occurance of too many resonances on the missing mass scale, we also have a completely kinematical effect of piling up of resonances on the w-scale. For large values of q2, the distances between individual resonances on the w-scale will become extremely small. In the case of the linearly rising trajectories MJ2 - J, the distance dw, between neighboring resonances will be equal to

while for the trajectories M, - J, we will have, similarly,

AwJ M -MJ q2 Rd (1 -V)&. What one can expect to observe on the graph vr2 plotted versus w, then, are not the resonances themselves, but rather, an average scale invariant function (~2(4>,v

(Fig. 3).

The question of whether one can distinguish tightly packed resonances from the diffraction on elementary constituents2s by other experimental means is now of crucial importance. APPENDIX .

t uJJ-+Jse

iXQ,

vtlo = c (J - ; 0 - u”, Qo’ 1J a - u” + o’)*(l 0’0” . x:~~+,-(J

u - u”, &a” 1$0)

- $ u - u” 1 eiXQs 1 1 u - a”)

= ; (J - * u - u’, Qu’ 1Ju) *( 1 u - u’, &a’ 1&a) . (J - 4 u -

U'

1 eixQ'

I

1 a - a’)

LEPTON

t

iXQ

vJJ+sJ;e

zu$Oo

=

5

(J

+

PAIR

$0,

. x~~~Q~(J

$0

+

j Ju)*(o

3

0

507

ANNIHILATION

0,

/ eixQ”

j

+J

1 $crJ

0 0)

or

t vJJ+$Jze

2xQ”u3v&,= uge (J + 4 u - u”, $a’ ) J CT- a” + o’)*(l . x:,,*u~x,,(J

+ i& u - u” j efXQzj 1 u - a”)

= 5 (J + Q u - a’, Qu’ / Ju)*( . (-

I)*-“‘(

J +

+ 0 -

0’

1 u - u’, $a’ 1 &T)

1 e”Qz / 1 u -

or

iXQ. USA-

a J,e

“U3U,ou

= 5 (J - 4 0, *a’ 1J a)*(0

0, ;u j +u)

. x~~*u~x~~
$Jze ixQ”u3uto**

=

+

0, &iu 1Qu?

- Q 0 1 eixoz 10 0)

or dJ-

u - u”, &a” ) +u>

1~(J-$O/eixQ~/OO).

u’;.,

508

PAVKOVI6 ACKNOWLEDGMENTS

It is a great pleasure to thank Professor C. Fronsdal for his very helpful correspondence.

REFERENCES 1 M. I. PAVKO~, Ann. 2 H. HARARX, “Duality

P&s.

(New

York),

62 (1971),

and Hadron Dynamics,”

1.

Lecture Notes, Brookhaven Summer School,

August, 1969. 8 The properties of crossing of the infinite multiplets were extensively investigated by C. FRONSDAL AND R. WHITE, Phys. Rev. 163 (1967), 1835; C. FRONSDAL, Phys. Rev. 171(1968), 1812.

4 We consider antiparticles as states which can be obtained from the particle states by the application of the CPT operator. A model based on an infinite-component wave equation is per definition CPT symmetric if the associated mass operator anticommutes with the CPT operator. 6 The order in which the models appear in this article is reversed in comparison with (I). The AGN wave equation is considered first because in that model one does not encounter problems with antiparticles. 6 For computational reasons it is more convenient to consider the antiproton as being detected at the end of the process. If the electromagnetic interactions respect charge conjugation symmetry the differential cross sections for detecting a particle or its antiparticle are identical. ’ S. D. DRELL AND J. D. WALECKA, Ann. Phys. (New York) 28 (1964), 18. Note that the metric in this reference is ab = --a& + ab. 8 Our metric is ab = a&, - ab, g,,, = -gij = 1, gfiV = 0 for p f Y. B J. D. BIORKEN, Phys. Rev. 179 (1969), 1547. lo E. ABERS, 1. T. GRODSKY, AND R. E. NORTON, Phys. Rev. 159 (1967), 1222. I1 M. GELL-MANN, D. HORN, AND J. WEYERS, in “Proceedings of the Heidelberg International Conference on Elementary Particles,” (I-I. Filthuth, Ed.), Interscience, New York, 1968. I2 H. BEBFA, F. GHIELMETI-I, V. GORGE, AND H. LEIJT\NYLER, Phys. Rev. 177 (1969), 2146. I8 R. DASHEN, L. ~‘RAIFEARTAIGH, AND SHAU-JIN CHANG, Phys. Reo. 182 (1969), 1819. I4 As in I, we settle for the following representation of y-matrices

The defining equations are: :r#I , YY) = 2&w 3

U@” = ; [Y&i,YYI,

rot = Yo,

Y+= --Y,

P9”=0,1,2,3,

Ys+= Ys ,

ys = imww3 , y4’sa = 1.

Also: oil = o, with i, j, k = 1,2, 3 in cyclic order and ool = iag with I = 1,2, 3. %=(”

J>

at=(ot

“3.

GELFAND, R. A. MINLOS, AND Z. YA. SHAPIRO, “Representations of the Rotation and Lorentz Groups,” Pergamon, New York, 1963. 1e Note that PI(w) is always positive definite while pz(w) can be of both signs.

I6 I. M.

LEPTON

PAIR

ANNIHILATION

509

I7 Y. NAMBU, Progr. Theoret. Phys. Suppl. 37, 38 (1966), 368; Phys. Rev. 160 (1967), 1171; C. FRONSDAL, Phys. Rev. 156 (1967), 1665. See also A. 0. BARUT, D. CORRIGAN, AND H. KLEINERT, Phys. Reo. Letters 20 (1968), 167; Phys. Rev. 167 (1968), 166. ‘” H. BEBIP, F. GHIELMETTI, V. GORGE, AND H. LE~TWYLER, Phys. Rev. 177 (1969), 2133. lb SHAU-JIN CHANG, R. DASHEN, AND L. ~‘RAIFEARTAIGH, Phys. Rec. Letters 21 (1968), 1026; Phys. Rer. 182 (1969), 1805, 1819. ” D. Tz. STOYANOV AND I. T. TODOROV, J. Math. Phys. 9 (1968), 2146. ?’ Note that the definition of V implies antilinearity, i.e., formally, %‘%‘-I = -i.

z From the classification theory of X(2, c) group representations, it is known that the parity operator transforms an irreducible representation g = (k, , c) into its conjugate representation 2 = (--k, , c). The two irreducible representations (k, , c) and (--k,, , -c) are considered equivalent. In the case of Majorana representations, g is therefore equivalent tog. In particular, (-8.0) = ($, 0). In the text, we distinguish g from g in an artificial way, namely, by assigning to 2 a different overall parity. 23 P@,@(x) are Jacobi polynomials as defined in Ref. 20. The explicite values of the Majorana re”presentation matrix elements can be found in Ref. 20 and in Ref. 1. “’ L. MONTANET, in the “Proceedings of the Lund International Conference on Elementary Particles, Lund, June 1969,” (G. van Dardel, Ed.), p. 189, Berlingska Boktryckeriet, Lund, Sweden, 1969. p5 Direct evidence of the existence of the maximal stable hadron mass could come either from experiments designed to find the quark, or from the tendency of Regge trajectories to level off. “’ B. MAGLI~, in the “Proceedings of the Lund International Conference on Elementary Particles, Lund, June 1969,” (G. van Dardel, Ed.), p. 269, Berlingska Boktryckeriet, Lund, Sweden, 1969. ” S. FUBINI AND G. VENEZIANO, Nuoco Cimento 64 (1969), 811; K. BARDAKCI AND S. MANDELSTAM, Phys. Rev. 184 (1969), 1640; L. N. CHANG, P. G. 0. FREUND, AND Y. NAMBU, Phys. Rev. Letters 24 (1970), 628. ‘* By elementary constituents, we have in mind particularly the ‘partons’, as conceived by R. FEYNMAN (unpublished), and J. BJORKEN AND E. PAWHOS, Phys. Rea. 185 (1969), 1975. The field theoretical model which reproduces the results of the parton model in both the electron-nucleon scattering channel and the lepton pair annihilation channel was extensively investigated by S. Drell and collaborators. A sample of the representative literature is given below: S. D. DRELL, D. J. LEVY, AND T. M. YAN, Phys. Rec. Letters 22 (1969), 744; Phys. Rev. 187 (1969), 2159; Phys. Rev. D 1 (1970), 1035; Phys. Rev. D 1 (1970), 1617.