Hadron Structure and the Parton-Parton Interaction in a Partial

ATKINSON,

1082

CONTOGOURIS,

K. J. M. Moriarty, Phys. Rev. D 3, 2425 (1971). G. Cohen-Tannoudji, F. Henyey, and R. Lacaze, Saclay Report No. D. Ph. —T/71-29 (unpublished). ITwo other recent papers on dual models with finitewidth resonances are as follows: R. Ramachandran and and

PHYSICA L REVIEW

VOLUME

D

Hadron Structure and the Parton-Parton

AND

GASKELL

M. O. Taha, Phys. Rev. D 5, 1015 (1972); L. Gonzales Mestres, Lett. Nuovo Cimento 2, 251 (1971). Neither of these articles deals with the problem of the asymptotic behavior in the right-half s plane.

6, NUMBER

15 AUGUST 1972

4

Interaction in a Partial Bootstrap Model~

Fred Cooper and Edmond Schonberg Belfex Graduate School of Science, Yeshiva University, Nehru York, Neu York 10033 (Received 21 March 1972) We present a model where hadrons are infinitely composite objects made up of pointlike constituents (partons) . The requirement that the hadron wave function be power-behaved determines the nature of the parton-parton interaction, In particular o(parton-parton) const at high energies. If this is implemented by vector-gluon exchange between partons, the following simple picture of electromagnetic form factors and large-angle hadron-hadron scattering arises: (a) The electromagnetic form factor E(t) has the same power behavior as the hadron wave function. (b) The. -large-angle scattering cross section is proportional to ~E(t)(4. (c) The effective interaction of the vector gluon with the composite hadron is described by the same form factor E(t) as the electromagnetic interaction of the hadron.

-

I. INTRODUCTION The structure of hadrons has been explored along three complementary lines: (a) Elastic e-P scattering has shown that the electromagnetic form factor of the proton falls off as (q') ' or faster, for large q'. This is seen to be strong evidence for the composite nature of the proton, and various bootstrap models have been proposed to describe it. (b) Inelastic e-p scattering has shown that the hadronic structure functions Wy and vW, are functions of the dimensionless variable v = v/q' alone, for v, q'-~. The simplest model that reproduces this scaling is the parton picture, which describes the proton as "made up" in some fashion of bare pointlike constituents which interact locally with the electromagnetic field. ' Furthermore, the fact that W, -~vW, as v/q'-~ indicates that partons, if they exist, have spin ~. Finally, with the use of sidewise dispersion relations one can relate the wave-function renormalization constant Z, to an integral over the structure functions W, and vW, and obtain thus rigorous bounds for Z, . Present data are consistent with Z, = 0, which is the field-theoretical criterion for compositeness. (c) Elastic P-P scattering at large angles is reasonably well described by'

'

d,

.

- I&. (t)l'f(s, t),

f(s, t) is some slow-varying function for which various forms have been suggested. This seems to indicate that hadronic charge and electric charge have the same spatial distribution in the

where

hadron. It also indicates that each hadron "probes" the wave function of the other in the same fashion that a photon does. ' This points in the direction of a vector interaction between the constituents of the hadrons. Results in current algebras and the analysis of light-cone singularities also suggest that partons (or quarks) interact among themselves via vector "gluons. " In this paper we would like to propose a simple model to obtain a unified picture of (a), (b), and (c). We use an off-shell partial-bootstrap approa. ch to describe the proton as an infinitely composite object, i.e. , containing an infinite number of pointlike partons. We assume that the force that binds the parton to the hadron is the parton-parton force itself. The bootstrap is partial in the sense that the parton-parton interaction itself is considered as primitive. Only the composite hadron structure is bootstrapped. The material is organized as follows: In Sec. II we develop our partial-bootvertex strap model for the hadron-hadron-parton

HADRON

STRUCTURE AND THE PARTON-PARTON a

FIG.

1.

We

FIG. 3. The bootstrapped Bethe-Salpeter equation in the ladder approximation.

first treat the case of spin-0 hadrons

The vertex function satisfies a nonlinear integral equation of the type proposed by Cutkosky, ' and which has been studied in detail by Harte. ' We find that power-behaved wave functions are asymptotic solutions of this integral equation if the parton-parton scattering amplitude -s for large s. This is consistent with vectorgluon exchange. In Sec. III we treat the case of spin--,' partons interacting with spin- —, and spin-0 composite hadrons. We show that both cases require the same parton-parton interaction. In Sec. IV, following the work of Ball and Zachariasen' we show that power-behaved wave functions and vertex functions lead to a power-behaved electromagnetic form factor, and in particular we can obtain from this model the usual dipole behavior F(t) - t In Sec. V we examine the considerable simplification to our bootstrap that arises if we assume that the parton-parton force is mediated by a single-vector-meson exchange. Our integral equations become linear, and the interaction kernel that describes parton-hadron binding is simply related to the electromagnetic form factor. In Sec. VI we discuss large-angle hadron-hadron elastic scattering and we show that the Born term of our theory leads to the Wu-Yang formula' and partons.

and that higher-order graphs are negligible, thus justifying the Wu- Yang conjecture. Section VII summarizes our results and con-

clusions. Throughout this paper, we use "hadron" to mean "infinitely composite hadron" and "parton" to mean "pointlike constituent of hadron. " In fieldtheoretical language, the wave-function renormalization constant of the hadron vanishes, that of the parton does not. Whether our model is indeed such a limit of a bona fide field theory is an open question.

Nevertheless we shall use, whenever it is convenient, the terminology and some results of field theory. We shall also consider our diagrams as connected in some fashion with Feynman graphs. This may not be the most consistent approach to a bootstrap theory, but seems unavoidable in the context of this model, with both composite and elementary particles. II. SPINLESS HADRONS

FIG. 2. The Bethe-Salpeter equation in the ladder

AND PARTONS

The Bethe-Salpeter (BS) equation' provides an elegant framework for a bootstrap model. A bound state may be depicted symbolically by Fig. 1, in which I is the sum of all connected diagrams. In the simplest case, one takes I to be a oneparticle exchange, and the BS equation is equivalent to a sum of ladder graphs, as shown in Fig. 2. A bootstrap picture is obtained when the constituents a are made to be identical to the bound state B, and the coupling g is replaced by a full vertex function I'. In Harte's model, ' both a and V are taken to be the same hadron as B, so that he obtains the result shown in Fig. 3. I' is damped exHe finds that, asymptotically, ponentially when any one of its legs is far off the mass shell. A similar result is obtained in other models that are related to the bootstrap, e.g. , the Veneziano model' and the thermodynamical model. This indicates that in a complete bootstrap the binding forces are too weak (i.e. , lead to wave functions that are too spread-out in configuration space) to produce power-behaved wave functions. We therefore choose a Partial bootstrap, by choosing one of the constituents to be elementary (the parton). Furthermore, we assume that it is bound to the composite object by interacting with one of the elementary constituents contained in it, as described by Fig. 4. Here double lines represent hadrons, single lines partons.

"

p

FIG. 4. The bootstrap equation for the hadron-parton state. Double lines represent hadrons, single lines partons. The hatched circle represents the partonverparton interaction. I is the hadron-hadron-parton tex. bound

approximation.

1083

s

The Bethe-Salpeter equation for a bound state.

function.

INTERACTION. . .

F.

1084

COOPER AND

Following the same picture, the 3-compositehadron vertex will satisfy the symbolic equation of Fig. 5. Now we write the equations corresponding to the diagrams above. Our notations shall be as follows: The hadron of momentum P and mass M is a bound state of a parton of momentum P, and mass m, and a composite hadron identical to the original one, of momentum P, =P —P, and mass M. In a field-theoretical framework the bound-state wave function y(x„x,) is described as y

(x„x,) =(0~T((I),(x,) (t), (x)}~ P ),

where Pj, =M'. It obeys the homogeneous

Xx(x„x)=

fd

equation

,x'G.'(x'„,x)G,'(x„x)

x-d x'd ,x'd

xK(x„x„x„x,) y(x7, xs) . Here G,',

t",'

are the full propagators for partiand K is the kernel which includes all

cles a, b, irreducible graphs. and separating the

On going to momentum c.m. motion we find:

space

G.'-'(p, ) G,'-'(p. )X(M', p, ', p.')

I p, p„qX

M',

'-~

'-~,

-m') -g„'[II(q') —II(m')]}

',

G„'(q')

'.

- [ln(q'/m')]

(5b)

The fact that the propagator for a composite particle decreases slower than that of an elementary particle for high q' seems to be a general feature of field theories with Z, =0. In what follows we shall neglect possible logarithmic dependences, and assume from now on that G„(q') goes as a constant for large q'. It will be convenient occasionally to rewrite Eq. (2) in terms of the vertex function'2 I'(p, ~, p, ', p, ') in order to obtain an analytic continuation in M . When P2 = M2

I (P' =M'

p

'

p ') = G'-'(p,

')G'-'( p ') y(M' p

'

p ')

(6)

I' satisfies the equation

f&(&q„-q)G'(q', )G'((p —q)*) &&

I'(M', q', (P —q)') dqq.

We now examine the behavior of y(M', p, ', p, ') when p, (the mass squared of the hadron) is fixed, and p, From Eqs. (2), (4), and (5) we have

'

'- ~.

x(M', (x,

'-,

q, ')

,

f

I(q', d„q)—

x y(M, q', (P —q)~} d'q

We see that if

. .I'(P, q),

1 1

I'

does not depend on P,',

x-(p') " ' (4)

For simplicity we shall omit Z, in what follows.

P-q FIG. 5. The bootstrap equation for the three-composite-hadron vertex.

. (8)

where asymptotically then

.

'.

(5a)

In the above simple model we therefore find

I(P, P„q)-(

Thus for a parton (Z, q(: 0) we have

G,'(q') - [ Z, q']

'.

(3)

self-energy part. Using as a guide a scalar field theory with two particles (g, (I)) with an interaction g, g (t) we find in perturbation theory that

- In(q'/m')

[a' '(11(q') —11(m'))]

G'(q')

q, P —q' dq.

where m is the physical mass and II(q~) the proper

II (q') —lI (m')

All we require is that Z, (parton) x 0 whereas for the hadron Z, =O; thus

&(M', q, *, (x*')

Since we are interested in the asymptotic behavior of X(M', P, ', P, ') as P, and/or P, we need to know the behavior of the full Green's function in those limits. In this section we consider only spinless particles. These Green's functions obey the Dyson equation" with the solution b, '(q') = fZ, (q'

E. SCHONBERG

FIG 6. Kinematics for the kernel of the BS equation.

(9b)

HADRON

STRUCTURE AND THE PARTON-PARTON

I'(P, q) y( M', q', (P —q)~) d4q Our model for the interaction

INTERACTION. . .

1085

(10)

&

kernel is given by the graph of Fig. 6. In terms of the vertex function (6)

we have

I'(q', k', (q —k)')G„(k')r( p, ', k', (P —p, —k)')

[( f(P»

q,

P2y

- k)'-m'][(P

-P

&) describes the parton-parton

- k)'-m']

interaction.

As we are keeping the hadron near its mass shell,

P =M

(11) in terms of the rightmost wave function:

we can rewrite

x~,' ~, P (

'

2

k)2

»P»a

If we now require that (9a) and (9b) be self-consistent, lhaJ

k

d'k.

(12)

then

(13)

P

f

It is easy to see that (13) obtains if corresponds to vector-meson exchange in two channels (Fig. 7). We have

f(P

p

q k)

(p

p +

2/)

+ (P —2 —k)

(P

~ the first term dominates

for

' Pg p, as desired.

Similarly a contact interaction (P

Pl q) ( Pg (P -P, --q)'-mv'

g

V&

2

k)2

and goes

as

V" behaves as

' P, +q-2k) (-P+P, +q)-P,

wave functions are consistent Eq. (2) if there is a fundamental parton-parton interaction of the above type. Before verifying that the convergence requirements of Eq. (10} obtain, let us examine the physical meaning of our bootstrap picture, by iterating the graph of Fig. 4, looking at it when P'- ~ in a timelike direction, as in Fig. 8. We see a cascade 3" partons —2 hadrons. of the parton: 1- 3This displays clearly the infinitely composite nature of the hadron. One might hope that just as the

Thus power-behaved with the bootstrap

'q)

(P

/

v

2P 1

]

(p p

)

k) U

dense fishnet graphs are connected with the Veneziano model" and explain the high-energy smallangle scattering, this class of graphs which become dense as n- ~ is responsible for large-angle behavior at high energies. If the vertices in Fig. 8 were those of a p4 theory, one would expect that any iterated graph in the sequence would behave as (p, ') " where n is the number of iterations. ' The solution of the bootstrap equation corresponds to the limit n- ~, and therefore we would expect the wave function to decrease faster than any power of P, ', as was

~ ~ ~

q-k p„

FIG. 7. The gluon-exchange parton-parton

approximation amplitude.

(14)

to the

FIG. 8. Iteration of the graph of Fig. 4.

F.

COOPER AND

found by Harte.

If on the other hand the partonparton interaction is mediated by a vector meson, the additional powers of P,' in each vertex compensate for the powers of (P,') ' from the parton propagators, and a power-behaved solution exists. Observe that this implies a weak momentum-transfer dependence of the parton-parton interaction, and a constant total cross section. Again, this results from taking the parton-parton interaction to be fundamental, and not bootstrapped. Let us now return to the convergence of Eq. (10). We see from (11) that the answer depends on the behavior of the vertex function I'(q', k, (q —k)') when all three. legs are off their mass shell. It is clear that the homogeneous Bethe-Salpeter equation is of little use here, as it is just the residue of the inhomogeneous Bethe-Salpeter equation at the position of the physical pole P'=M'. In order to have some hint of the behavior of X and I' when one of the h. adrons is off the mass shell, w'e use the same approach as before, plus our knowledge of the hadron Green's function (5). As P, '=m' (the parton is close to its mass shell) and P, we find from (7)

'-~,

x(M', P, ', P,

'--)

I I, P„P„qx I', q', I -q'd'q

=

(18) and if

(P 2)-n

(17a)

x-(P2') "

(17b)

[compare with (9)]. To verify that this is compatible with our equation for I, we rewrite (11) in terms of P, (our asymptotic var iable): k2

SCHONBERG This is similar to the BS ladder-approximation

wave function used by Ball and Zachariasen:

(

) k)2) X(P2 s r(P2 (q —k)'-m'

x f(P, P„q, k)d'k.

)

We

shall assume that this form is valid asymptotically for P, (P, —k)'-~, and k'+M'. It is now easy to verify that (19) is compatible with (17) and (9). In terms of the off-shell parton hadron masses

'-~,

P,'

and

P, ' we have then

I,( P', P ', P ')-

1

+

P2)

(20)

That is, if the parton is far off the mass shell, I'-(P, ') ', while if the hadron is off the mass shell, then I'- (P, ') 2. This additional damping in the off-mass-shell composite leg seems very reasonable. It is now readily verified that (19) and (20) guarantee the convergence of all our integrals and the correctness of exchanging the order of limits and integrals. The wave function (19) is an asymptotically self-consistent solution of our bootstrap. then

III. SPIN-2 PARTONS

spin--,' partons our fundamental vertex has to be described differently. We describe now a spin--,' hadron as a bound state of a spin--,' parton and a spinless hadron. The latter can be thought of as two quarks with l=0. The parton is either a quark or a "bare proton. " Our vertex is a matrix in spinor space, and we can write in general In order to introduce

I (P', P,', P, ') = I, + r, (P, —m)+1, (P —M) . (21)

r„ I'„and

then

2

E.

1, are

invariant functions of the mo-

m enta. When the proton is on the mass shell (P' =M') I', is missing. Since our integral equation does not bootstrap I', we shall ignore it in w'hat follows.

Following the discussion of Sec. II [see Eq. (4)], the Green's function involved will have the asymptotic behavior

- (P' —m) ' for spin- — ', partons, ', hadrons, S„(P')- const for spin- — G„(P') - const for spinless hadrons . S', (P')

(18)

We want to see whether (17b) and (18) lead to (17a). This will be the case if the behavior of y(P, ', lP,, (P, —k)') in the Parton mass (P, —k)' compensates for the behavior of f(P, P„q, k), which

The kinematics

(22)

are given in Fig. 9. I'(1.), I'(2),

is seen from (14) to be

This leads (though not unambiguously) lowing ansatz for X:

to the fol-

P

1

P

'(P —k)'[aP '+ &(P —k)'] FIG. 9. Kinematics for the kernel with spin-2 partons.

STRUCTURE AND THE PARTON- PARTON INTERACTION. . .

HADRON

and I'(3) will designate the vertex functions at the respective vertices. We work in the frame where

p|O=M

P = p| + pa = (M, 0) . We are interested In that limit

I

in the limit p, ' =

F(M', (1, , P, ') + ()(, —m)I', (M', ),',

t-~,

f

),')

-p20=2t

piI = IP. = l & I

(23)

.

f as a V'„V'" intera, ction.

For simplicity we write

P, finite.

The equation corresponding

[I', ()) + (P' —g —m)I',

1087

(l)],

to Fig. 9 is

, y„

[r,(2) + (j+ p —m) r, (2)]

x Tr

(24a)

this simplifies to

After a little algebra,

r, +(p, -m)r,

d'qd'k,—

]f + m

=

, r, (1)+r,(1)

Ap'2+

Bg+ C q

(24b)

where

r, (2)r, (3)

[(q+ k)' —m'] [(p, + k)' —m']

q

~

r, (2) r, (s) (k' —p, [(q+ k)'-m'][(P, + k)'-m'] r, (2)r, (3)

[(q+ k)' —m'][( p, + k)' We can separate

I',

and

I',

p, 'r, (M', p, ', p, ')

by taking the

=, ',

q

(

-(

M

k',

q+k

+

r, (3)r, (2) '-m'

p, +k

)

2

r,',(1)

](

, +r, (l) (p,

+

r, (2) f', (3)

(q+ k)' —m'

trace of (24b), before and after multiplying

p,

(Ap,

With the kinematics of (23) it is easy to verify that the coefficient of I"(2)I (3) is a constant in P, ', so that the asymptotic behavior of (26) is determined by that of I'(3) in its parton leg of momentum (P, +k). If

r, (3) =r,(p, '=M',

', ', ', ',

r, (2) r, (s) '-m'

—2mM+m')+(m —M)

k. k-

- m']

r, (M', P,', P, ') —m r, (M', P, ', P, ')

Mr, (s}r,(2)

k+k -mM

by P', . We obtain

((P —q) (AP, + Bk+ C q) j d'kd'q,

(26a)

~

)

+Bk+Cq)'id'kd'q.

(26b)

Furthermore, with an explicit expression for I" similar to (20) all integrals are convergent and a power-behaved vertex function is a self-consistent equation to our bootstrap. From now on we shall consider the explicit form (19) and (20).

(p, +k)')

=[(p, +k)'] "

then

r, (M', p, ', p.') - (t) -"

(27)

(28)

FIG. 10. The triangle approximation electromagnetic vertex.

to the

F.

1088

COOPER AND

E.

SCHONBERG P2

p+4, I

FIG. 11. Iteration of Fig. 10(b) in terms of the

FIG. 12. Kinematics for the electromagnetic

vertex.

hadron wave function.

how Yamada's

IV. ELECTROMAGNETIC FORM FACTOR OF COMPOSITE HADRONS

Yamada" has shown in the framework of the Bethe-Salpeter equation, that the leading contribution to the high-q2 behavior of the electromagnetic (em) form factor is given by the triangle diagrams as long as the wave function of the composite hadron goes as 1/p', where p is the relative momentum of the components of the hadron, when both components are off the mass shell. This is the case for our wave function (19). It is easy to see

(P +P

)

E()) )

f

d

))((M, 8(p

—k)

result follows from our bootstrap. Consider first spinless hadrons, say z' and z', and spinless partons P' and P'. The simplest graphs that contribute to the em vertex function are those of Fig. 10, and the iteration of the graph of Fig. 10(b) leads to Fig. 11. This last graph falls faster in q2 than the graph of Fig. 10(a) because it depends on two hadron wave functions. lit goes essentially as the square of Fig. 10(a).] With the kinematics of Fig. 12 the vertex satisfies the following equation:

)G„((p -k) }(q -2k) y(M, (0+)'), (0

+)')

).

(29)

q'-~.

In that limit We see that the behavior of E(q') is determined by that of X(M', (q+ k)', (Pm+)1)') as Furthermore the behavior of both wave functions as k'-~ guarantees y-(q') ' and therefore F(q')- (q') the convergence of the remaining integrals. This is identical to the result of Ball and Zachariasen. The case of spin--,' hadrons is treated in exactly the same fashion and we refer to Ball and Zachariasen ', as before. for the details. We assume that the parton has spin —

'.

Setting

(p. If'

I

pg

=r„&,(q')+(p, +p. )„F.(q')

=,

and using (21) for the hadron-hadron-parton

y„F,(q')+(p, +p, )„F,(q') Again,

(30)

vertex, we have

G„((p, -P)') F,(1) „,

the behavior of

„,

, +1,(2)

(31)

(lnq')' 1

(a+q)', (p, —u)') determines that of F(q') as q'-~.

r, (2) = I', (m',

If we use the explicit form (20) for 1 „and (28)

for p2,

, +r, (1) y„r,(2)

i.e.,

q4

&2(q')

- —..

(34)

V. SIMPLIFIED INTEGRAL EQUATIONS

1 ' (a(&+ q)'+ &(pl —&)'](pl —&)'

r,

2)=

1

(32)

Ia(@+q)'+&(p, —&) P(p, —A')' '

the integrals finds

can be evaluated explicitly and one

(lnq )

- —.. &.(q')2-1

If instead of (30) one chooses the more usual form

factors y„F,(q') +io„,(P, —P, )„F, one finds also

If we make the approximation that the partonparton amplitude is given just by the vector gluon exchange, then considerable simplifications occur in our equations. Our integral equation for the hadron-hadron-parton vertex is given by Fig. 13, and we see that the gluon-hadron vertex I." is similar to the electromagnetic vertex F, i.e., given by the triangle of Sec. IV. F and I' satisfy now the set of coupled equations of Figs. 14(a) and 14(b). The equation corresponding to Fig. 14(a) is now a standard (i.e. , not a bootstrap) Bethe-Salpeter equation in the ladder approximation:

HADRON

INTERACTION. . .

STRUCTURE AND THE PARTON-PARTON

1089

(35)

For p, on the mass shell, F(q', (q —p, )') is related to the elastic and inelastic electroproduction structure functions. Thus in principle it is a known function and (35) is a linear integral equation whose solution can be explored numerically. Asymptotically it shows again that F and g have the same power behavior in P, ', which just restates the consistency conditions we found in the preceding sections. VI. HADRON PROCESSES AT LARGE s AND t

In Sec. II it was found that the hadron-hadronparton vertex function damps faster in the off-shell hadron momentum than in the off-shell parton momentum. The explicit expression (20) gives

(36) where P, and P2 are the momenta of the parton and hadron, respectively. This indicates that large momentum transfers to the composite hadron will result in off-shell partons rather than off-shell hadrons. Then, large-angle hadron-hadron scattering will be dominated by interactions among individual partons from each hadron. Hadron exchange will be considerably damped by several powers of momentum transfer. Therefore, the simplest graphs one could consider would be iterations of one-parton exchange. But these are important only if they lead to constant cross sections of high energy. added in Proof. This statement is incorrect At larg. e s and t, s/t fixed, a two-quark exchange mechanism seems to provide not only the correct energy dependence at 90' but also the correct angular dependence for large angles. We unfortunately ignored these diagrams by only considering the large-s, fixed-t behavior of these graphs. See Gunion, S. Brodsky, and R. Blankenbecler,

[¹te

Phys. Letters (to be published). ] This requires spin-1 exchange, as in the eikonal model, or in our case, gluon exchange (Fig. 15). lvVe saw in the preceding section that the gluon coupling to the hadron for large momentum transfers is described by the triangle graph. Therefore Fig. 15 is just a Special case of Fig. 16, where the parton-parton interaction reappears. This implements our description of hadron-hadron scattering for large s and t as an interaction among individual partons in each hadron. Vfe notice that for large t the parton legs are far off the mass shell, and from our discussion of the electromagnetic form factor, we see that the I; dependence of this amplitude will be essentially governed by one of the hadron-hadron-parton vertex functions on each hadron leg, i.e., when the momentum transfer flows mostly through one of the vertices. Then if g(i) - t ', the amplitude of Fig. 16 will go as A

-t

"f(s, t) + "f(s, u-), u

(37)

f(s, t) is the off-mass-shell parton-parton scatter ing amplitude. In the case where the hadrons are different

where

(quasielastic scattering) the amplitude will depend wave function in the upper half and lower half of the diagram falls slowest in t = q', the off-shell momentum of the parton line. Depending on the quantum numbers of the hadrons involved, the power dependence of the hadron wave function with an off-shell paz ton might be slightly different. Equation (3V) states that the t dependence of the hadron-hadron scattering amplitude is proportionon whichever

J.

I

In

V,

v

(b) FIG. 13. The bootstrap equation for the hadron-parton approximation to the vertex in the single-gluon-exchange parton-parton amplitude.

FIG. 14. Coupled integral equations for the hadron wave function and the electromagnetic form factor in the single-gluon-exchange

approximation.

F.

1090

COOPER AND

E. SCHONBERG P3

p)

P&

+ P

p

FIG. 15. Hadron-hadron gluon-exchange

scattering in the singleappr oximation.

al to the square of that of the wave function, and from the discussion of the preceding section, to the square of the t dependence of the electromagnetic form factor. If f(s, t) is given by a vector gluon exchange, then for large s, t, f(s, t)- s/t, and

W(s, t) = —, [F(t)]'.

If instead

(38)

f(s, t) is a V„V" interaction

A(s, t) = s[F(t)]', which

the

"Born

ZM', p, -t' ', u'G„u' y M, y-p, ', I'

(kP)Q(P+Pk)2)(P,

for

g

)-P2

~

(41)

P, q-

With the explicit expression (41), we can evaluate (40) explicitly. We find that this box diagram is suppressed by a factor of s ' at high energies with respect to single gluon exchange. Our result here parallels that of Harte, i.e., that for vertex functions that are sufficiently damped when any leg goes off the mass shell, only tree graphs are important at high s and t. VII. CONCLUSIONS

Let us summarize briefly what we have done: Starting from a "hybrid" bootstrap picture where the hadron is considered as a bound state of an infinitely composite object and an elementary ob-

F

p

't

3

k-p3

p'I

2 F

F

p

FIG. 17. The first iteration of the "Born term" of the hadron-hadron

term" of the scattering amplitude in our model, and one could ask whether the above results are not radically altered when one includes the iterations of Fig. 16. Fortunately, the damping of the vertex function (20) when the hadron is off-massshell ensures that graphs as those of Fig. 17 are unimportant at high s, t. For simplicity, let us take to be a vector-gluon exchange. Then each triangle graph can be replaced by a form factor F(s, t), which is equal to the electromagnetic form factor F(t) when both hadron lines are on the mass shell, but goes as (k') ' when one of the hadron momenta k'-~. The iterated gluon exchange of Fig. 17 will have the expression

M', p,

-k',

p, +p, -&'

G„p, +p, -&'

~

1

p

FIG. 16. The simplest contributions to the hadronhadron scattering amplitude in terms of the partonparton interaction.

+k) (P, +2P, +k)(P, +k) (P, +2P4dk (k- p, )'-m~' (p, —k)'-mv'

where

F(M, p,

scattering amplitude.

P

f

we get

simply

is the Wu-Yang formula. The diagram of Fig. 16 is essentially

P

(40)

ject ("parton") we have determined the asymptotic behavior of the corresponding bound-state wave function. This asymptotic behavior is determined by the nature of the parton-parton interaction, and if this is chosen appropriately, one obtains power-behaved wave functions. These in turn lead to power-behaved electromagnetic form factors. In particular, the usual "dipole" form factors are found to be asymptotic solutions to our

equations. Further examination of the bound-state wave functions when the composite leg is off the mass shell shows that processes where off-shell composite hadrons are present, are damped considerably with respect to those where partons are off shell. This provides a simple picture for high-s and -t hadron-hadron scattering and the muchdiscussed connection between hadronic and electromagnetic form factors follows in a simple and graphic fashion, The parton-parton interaction, which is at the core of this model, has the same high-energy behavior as would be given by single vector-gluon exchange and the interaction of this gluon with a composite hadron is similar to that of the photons with hadrons. If we approximate the full parton-parton amplitude by single gluon exchange, a simple reciprocal bootstrap between hadron wave functions and electromagnetic form

HADRON

STRUCTURE AND THE PARTON- PARTON INTERAC TION. . .

!

factors emerges which should be investigated more deeply. A large number of questions remain that have to be explored. At the phenomenological level, deep-inelastic scattering and many-particle production have a simple description in this model. The numerical predictions that follow from it are being examined. At a more fundamental level, the connection of this model with a bona fide field theory is unclear. As in any other bootstrap model, the condition Z, =0 seems to play a very important role. At the same time, partons seems to be necessary to understand a large number of high-energy phenomena, so that we require a field theory with some composite and some elementary objects, i. e. , with

*Work supported in part by the National Science Foundation and a Frederick Cottrell Grant in Aid. D. Bjorken, SLAC ~For a review and references see Report No. SLAC PUB 905 (unpublished). 2F. Cooper and Heinz Pagels, Phys. Rev. D 2, 228 (1970) . T. T. Wu and C. N. Yang, Phys. Rev. 137, B708 (1965). 4J. V Allaby etal. , Phys. Letters 24B, 181 (1970). 5R. E. Cutkosky and M. Leon, Phys. Rev. 138, 8667 (1965) .

J.

6J. Harte, Phys. Rev. 171, 1825 (1968); 184, 1936 (1969); J. Harte and K. Ong, iMd. 184, 1948 (1969); J. Harte, iMd. 188, 2372 (1969). 7J. Ball and F. Zachariasen, Phys. Rev. 170, 1541 (1968) .

For a general review and references see N. Nakanishi, Progr. Theoret. Phys. (Kyoto) Suppl. 43, 1 (1969). Y. Nambu, Phys. Bev. D 4, 1193 (1971). R. Hagedorn, Nuovo Cimento Suppl. 3, 147 (1963) . 11F. Dyson, Phys. Rev 75, 1736 (1946). We use the field-theoretic definition of the vertex function, as given, e.g. , in G. Barton, Introduction to Advanced I'ield Theory (Interscience, New York, 1966). ~3H. Nielsen and P. Olesen, Phys. Letters 32B, 203

(1970) . ~4G, Tiktopoulos, Phys. Rev. 131, 480 (1963). ~ M. Yamada, Progr. Theoret. Phys. (Kyoto) 40, 848

Z, ~ 0 also.

1091

"'" At this

stage the model requires a number of results from field theory, such as the behavior of Green's functions for large momenta, and relies heavily on a literal interpretation of Feynman graphs. Hopefully this borrowing can be shown to be a true derivation from field theory. Work in this direction is in progress.

"

ACKNOWLEDGMENTS

We would like to thank Professor L. Susskind for many stimulating conversations and suggestions and Professor J. Harte for discussing his work with us. We would also like to thank Cathy Schonberg for the care and patience expended in the preparation of this manuscript.

(1968) . 6S. Weinberg, in Bxandeis University Summer Institute in Theoretical Physics, edited by K. Ford and S. Deser (Wiley, New York, 1965), Vol. 2, shows how one might add known bound states into the Lagrangian of an underlying field theory (say quarks with gluons) by using the condition Z3 =0, and without affecting the underlying dynamics. It will be interesting to see whether this idea will lead to our equations for the vertex function in some approximation. ~VKen Johnson, this issue, Phys. Rev. D 6, 1101 (1972), has been able to derive an equation related to Eq. (24) from an underlying quark-gluon field theory. He suggests that in such a theory the spectrum of hadrons has Regge behavior, and that the quarks will not be seen. '~We have been able to show that if there is an underlying parton field theory which is conformally invariant and has fields of anomalous dimension, then our integral equation is conformally invariant. In such theories, the form of the 2- and 3-point functions is specified. If one supplements our integral equation by a similar one for the hadron Green's function, then one determines via these integral equations the anomalous dimensions of the hadrons in terms of the dimension of the parton field. Thus the power behavior of the wave function (and the electromagnetic form factor) is a function of the dimension of the parton.