Nucleon Form Factor

11. n f . F R I E D

and therefore

m i l dx,+l X1-2. . . xn+,2(X,z2)3[(.1)l12+(~z)1/2]-6. . . (xn+1x1)"(.~1)112+(~n+1)1~21--G

which is the lower bound of (20).

P H Y S I C A L R E V I E W D V O L U M E 2 , N U M B E R 1 2 1 5 D E C E M B E R 1 9 7 0

Threshold Theorem for the Radius of the F2v Nucleon Form Factor*

JEFFREY PAUL ROYER Stanford Linear Accelerator Center, Stanford University, Stanford, Calqorwia 94305

(Received 17 K'ovember 1969)

We develop a method for expanding the absorptive part of the sidewise dispersion relation for the nucleon Fzv radius. Vtre assume that a threshold expansion is valid and we are able to obtain the exact first-order terms in this expansion in the limit of the pion mass going to zero. We are able to prove that the first-order terms in this expansion come solely from the T - N intermediate state. Thus, the expansion provides justification for the retention of only the T-N intermediate state in a first attempt a t describing the radius, as in Drell and Silverman. Furthermore, the expansion provides a handhold for attempting to estimate the corrections to the first-order term coming from the 2 ~ - i V and higher intermediate states. I t must be emphasized that we are not providing justification for the assumption of threshold dominance. We are trying to follow that assumption to its logical conclusion. That is, if the threshold dominance idea is good, then a threshold expansion provides a logical way to calculate low-energy parameters, Wc have provided such an expansion for the Fzv radius.

I. INTRODUCTION

I N this paper we develop a method for expanding the absorptive part of the sidewise dispersion relation1

for the nucleon Fzv radius. ITe assume that a threshold expansion is valid and we are able to obtain the exact first-order terins in this expansion, in the liinit of the pion mass going to zero. ~ ' e are able to prove that the first-order terms in this expansion come solely from the a-S intermediate state. Thus the expansion provides justification for the retention of on1 the T-S intermediate state in a first attempt a t describing the radius

* \.l ork supported bl the IT S Atomic Energy Commission. A. M Bincer, Phj s. Rev. 118, 855 (1960).

as in Drell and Silverinan.2 Furthermore, the expansion provides a handhold for attempting to estimate the corrections to the first-order term conling froin the 2a-N and higher intermediate states.

lye now repeat the preceding in a more n~athematical way.

The sidewise dispersion relation1 for the FzV nucleon form factor is

S. D. Drell and D. J. Silverman, Phys. Rev. Letters 20, 1325 (1968).

2 T I I R E S H O T , D T H E O R E h l F O R R L \ 1 ) I U S O F F ~ V N U C L E O N . . . 3041

where M is the nucleon mass, p is the pioil mass, and I is the 4-inomenturn of the electromagnetic current. The radius3 is the slope at 12=0, so the dispersion relation for the radius is

where the prime denotes differentiation with respect to 12.

In the limit of ,u --t 0, we expand Abs F2v'(0,1V2) in powers of (11'2-Aif2)/LW2:

We show that Co, C1, and Cz vanish and we find that CS is given entirely by the n-.IT intermediate state.

I t must be emphasized that we are not providing justification for the assunlption of threshold dominance. \Te are trying to follow that assumption to its logical conclusion. That is, if the threshold dominance idea is good, than a threshold expansion provides a logical way to calculate low-energy parameters. We have provided such an expansion for the F2v radius.*

The method of expanding the absorptive part depends crucially on the limit p -+ 0. The details of the expansion may be expressed simply as follows. We first find the leading terms in the absorptive part as p * 0. We then expand these leading terms in powers of (1172--M9/1M2.

Drell and Silverman2 provided a hint leading to the method of isolating the rnost important contributioil as p -+ 0. In their analysis, which retained only the x-*IT intermediate state (Fig. I), i t was observed that the dominant contribution as p -+ 0 comes from the pion- pole term of the time-reversed electroproduction amplitude. The explanation is simply that this term is most singular in the pion mass. The algebraic reason is seen as follows. he pion propagator in Fig. 2 carries the 4-momentum I of the electromagnetic current. When we differentiate with r e s~ec t to LZ to find the radius, we must differentiate the'propagator, which gives us a term more singular in the pion mass than any of the other

3 I t is customary to define the form-factor radius as follows:

where 12 is the square of the electromagnetic 4-momentum and F is the form factor in question. For general definitions, see S. D. Drell and F. Zachariasen, Electromagnetic Structure of Nucleons, (Oxford U . P., Fair Lawn, N. J., 1961).

4 The reasons we write F28 are: (a) The sidewise dispersion relation for F1 contains a subtraction constant which may or may not be a function of 12. Without knowledge of whether this is true, we cannot hope to calculate the radius of F1. For a more extensive discussion of this, see D. J. Silverman, Ph.D. dissertation, Stanford University Physics Department, 1968 (unpublished). (b) We show explicitly in the appendix that the leading terms in the threshold expansion do not contribute to Fpg.

FIG. 1. n-iV intermediate state contribution to the absorptive part.

1 -1 at threshold. (4)

Thus, the obvious auestion arises: Can we use an argument based on th'is observation to pick out the leading terms, as ,u-0, in the contribution to the absorptive part from higher-mass intermediate states? We are able to answer this question in the affirmative by using partial conservation of axial-vector current (PCAC) and current algebra to isolate, at threshold, the pion-pole terms which carry the 4-momenturn I of the electromagnetic current. These terms give the rnost singular contribution to the absorptive part of the radius as p -+ 0. In this manner, we isolate the dominant terms at threshold as p -+ 0. U'e then expand these terms in (TT72-iM2)/M2 to obtain the announced expansion.

In Sec. I1 we undertake the explanation of the graphical methods employed to handle the multipion, nucleon intermediate state.

In Sec. I11 we apply the graphical techniques to an analysis of the relevant rnatrix elements needed for the proof of the threshold theorem. Section I11 A discusses the matrix element of the electromagnetic current and Sec. I11 B discusses the matrix element of the nucleon source. We complete the proof of the theorem in Sec. IV. Concluding remarlis are contained in Sec. V.

11. DEVELOPMENT OF GRAPHICAL PROCEDURES

This section undertalies the explanation of the graphical methods employed in the discussion of the multipion, nucleon intermediate state. The graphs function as numonics for complicated analytical expressions much the same way as Feynnian graphs function in quantum electrodynamics (QED). We use the graphs first to avoid cumbersome analytic expressions and second, to help the reader develop an intuition for the manner in which we handle the multipion, nucleon

FIG. 2. Pion-pole term of the time-reversed electroproduction amplitude.

I

d k P

3042 J E F - F R E Y P A U L R O I ' E R 2

FIG. 3. Graphical represe~ltation of Eq. (5).

intermediate state. The graphs we define are essentiallj identical to those defined in Abarbanel and Kuss i i~ov.~

The basic object which we want to represent graphically is the Fourier transform of the illatrix elenlent of the time-ordered product of a given set of operators. For example,

or J"(x) should be associated with the solid line. Some operators mill arise by using the following SU(2 ) current commutation relations:

To represent this graphically we define the eleinents in Fig. 5 . The lowerlnost leg will enter the blob. Again, it will be clear from the graph whether the solid line is V,s(x) 01. Jp(x).

Many soft-pion techniques take advantage of PCAC to relate the pion field to the divergence of the non-

J l d4X1d4X2e-z~ .~1~-ivz.72 ( f I TQl (x i )Q2(~*) I4 (5)

is some is such an object; ji) is some initial state, I J ) ' final state, and Q,(x,) is some local operator carrying momentum qi. The graphical representation of this object ~vould be that shown in Fig. 3. The two heavy lines are the initial and final states and the two wavy lines are the operators Ql(xl) and Q2(rz). The blob represents our ignorance of what is going on in the interaction.

Our discussion will contain four different lrinds of operators.

A&) : The non-strangeness-changing strong- interaction axial-vector current. (6)

Dm= d,A,@(x) : The divergence of 6. ( 7 )

Va*(x) : The non-strangeness-changing strong- interaction vector current. (8)

J ~ ( x ) : The strong-interaction electronlagnetic current. (9) FIG. 5. Graphical elements arising from current commutators.

The graphical elements for these operators are shown strangeness-changing axial-vector current; that is, in Fig. 4. I t will be clear froin the graph whether V,&(x)

d,A ,"(2) = P ~ J ~ + , ( % ) , (13)

AL(x) where &(z) is the pion interpolating field and J, is the pion decay constant. The derivative is then removed

2 frorn the tiine-ordered product and the resulting ex-

Dol(x) pression is examined in various low-energy limits. / FIG. 4. Graphical elements We will depend strongly on the graphical techniques

for the currents.

/ to expend the matrix elements of products of D,'s; that $@I is, we will give a graphical algorithm for pulling the

derivatives out of the time-ordered product. We will ?(XI / find that by pulling the derivatives off of the D,'s, we -- expand our original graph into a set of graphs, each of

6 N, D. 1. *barbanel and Shmuel Nussillov, Ann, phys. (N, y,) which is a silnple arrangelnent of chains of axial-vector 42, 467 (1967). and vector currents.

2 T H R E S H O L D T H E O R E M F O l i R A D I I J S O F F a v N U C L E O N . . . 3043

To make the preceding statement clear, we next illustrate the use of the graphs in expanding the matrix elements of the Da1s. We will expand the expression

which is represented graphically in Fig. 6. The first step is to take the derivative of the first operator outside of the time-ordered product. When this is done, we obtain two terms. One term is just the derivative acting on the whole matrix element and the other term is an equal-time commutator coming from the action of the derivative on the 0 function of the time-ordered product. Explicitly, expression (14) equals

FIG. 7. Graphical representation of Eq. (16).

ing the derivative to a momentuin. This is justified as long as the matrix elements of the operators do not grow as x,t - + x . The result then is that

The graphs which correspond to Eq. (16) are shown in Fig. 7. Tl'e see two conventions implicit in Fig. 7.

(1) Wc will assume all currents flow into the blob. (2) An axial current connected to a blob is assumed

to be dotted with iq,, its momentum.

We now proceed in the same manner to remove the derivative from the second current. This time however, the equal-time commutator is non-negligible. Using the commutation relation ( lo) , we obtain the following expression:


I t would serve us well a t this point to remark that we =iql,,iq2v d4xld4.2.2e-inl .zle-inz will neglect all comnlutators of the type arising in the / / " "

secondpart of expression (15). This conlmutator is the well-known a term. We neglect this term because x ( f 1 TAcilp(x1)Aa~(x2) 1 i) d , A ~ a p%nd we will ultimately be interested in the limit 3 0. As long as there is no pion pole in the a field, . .

-ZYlpl&azalaz / d4x2e-"~n+a) .i2(fl YaI*(x2) i). (1 j ) this u term will go to zero faster than the singular ternis which we show to exist in this limit. So, neglecting the u term, we are left with just the first term in expression Equation (17) is represented graphically in Fig. 8. w e (15). R e now perform an integration by parts, chang- note the following conventions.


3044 J E F F R E Y P A U I , R O Y E R 2

(1) There is a negalive sign for each eq~~al-t ime commutator.

(2) The indices on the antisjmmetric e are represented clockwise on the graph.

(3) The solid line is V,"(x), not Jp(x). J ~ ( x ) can only appear when it is in the matrix elenlent to begin with.

Il'hat we need to know is how to get the graphs without writing the anal) tic expressions. This is done by simply changing the sj-mbol for the divergence into an axial and adding the syn~bols for the various equal-time

commutators which arise frorn the other operators in the matrix eleruent. 'I'o ensure correct 01 dering of the t,,k symbols, we make the convention of attaching the symbol representing Amo(%) into the left-hand side of the other two operators in the vertex.

\ire now work out a second example to illustrate the graphical method. Figure 9 shows a step-by-step reduction of a graph into a sum of graphs by the method stated above. I n Fig. 9, we leave ofi the lines representing the initial and Gn'il states to simplifj- the graphs.

The analytic expression for this reduction is

There are two more things to be mentioned before leaving this section. Since we are going to be dealing with pions, via PCAC, we will want the amplitude to be symmetrized in the pion indices. This allows us to drop the current labels in the graphs. We now just write out the expression for a particular graph, assuming sonle

convention for the current labels, then we symmetrize the expression and multiply it by the number of tirlles that graphs was repeated in the reduction. The set of graphs in Fig. 9 would then be shown as in Fig. 10.

Finally, we remove the explicit pion poles frolu the single axial-vector currents by defining a new operator, which has no pion poles, via the relation6

By using this relation, we can elinlinate the type of graph where Ap enters the blob and replace it by Atp. l n doing so, it is found that the matrix elenlent containing the divergence of the eliminated axial-vector current is

F I G . 10. Simplification of the graphs in Fig. 7. - --

The nucleon matrix element of A' call easily he calculated in a terms of the usual pseudovector and induced pseudoscalar form

C H A N G E + + factors. Using the definition SECOND CURRENT

( f i z I A t f i ( ~ ) I P I ) = (P21Aw(x) Ifi1)+i(~~/~~)(P2la~A~(x) / P I ) , where q = p 2 - p 1 , we obtain

( P z / A ' ' ( ~ ) I P l ) = ~ ( ~ 2 ) { g ~ ( 4 ~ ) ~ ' y j f q"/p2 X [ / Z A ( @ ) (p2-q2) - 2 , 1 . 1 g ~ ( ~ ~ ) ] 7 ' 4 ) U ( f i 1 ) .

+ If we use PCAC and the Goldberger-Treiman relation, me can use / l , i ( q 2 ) ~ 2 F g A ( p 2 ) /p2-q2 when q2=p2. By doing this, me see that the coefficient of qfi vanishes when q 2 , p 2 . So with q2 in this region,

FIG. 9. Graphical reduction corresponding to Eq. (18). (fi21A'C(0) l ~ l ) ~ O ( f i 2 ) g ~ ( q ~ ) y f i y : l ~ ( f i l )

2 T H R E S H O L D T H E O R E M F O R R A D I U S O F F l v N U C L E O N . - . 3045

FIG. 11. Representation of Fig. S after the explicit pion poles have been removed. i. 2 ‘ 7 2 \ / . 2 , z \ u - I

nlultiplied by the factor (1-q2/p2). That is, the graphs of Fig. 10 would be changed to those of Fig. 11. The prime on the wiggly line indicates A'. The result of the replacement is to add a prime to all single wiggly lines and to multiply the divergences by (l-q,2/p2), where q, is the momentum of the respective divergence.

We can now use the reduction formula and PCAC to relate the divergences to pion in-states. The rules for writing out the graphs are summarized :is follows.

(1) Successively change each divergence to an axial vector plus commutators.

(2) Put primes on all single wiggly lines and put factors of (1 -q,2/p2) for each divergence.

(3) Put a factor on each graph where n is the number of vertices in the graph.

(4) Put factors of i q P on the axial vectors which have not been used to coml-nute something.

( 5 ) Symmetrize in the isospin indices.

A short cut to step one is just to list the different waq-s of combining the currents into chains and then multiply each possibi1it~- b>- its repetition number R:

where .I\' is the total number of original axial vectors in matrix element, Ci is the nunlber of axial vectors in chain of type i, and Ri is the number of chains of type i .

111. ANALYSIS OF ABSORPTIVE PART

Now we turn to the task of examining in detail the absorptive part of the FsV for111 factor. f f e will use the graphical many soft-pion method to discover the dominant contributions to the absorptive part. The sense in which we use the word dominant was mentioned in the Introduction and will be made explicit in the following. Roughly, we want to think of the absorptive part as being expanded simultaneously in the two parameters M2/dPand (TT72-M2)/M2. I t is in this way that we will obtain a threshold expansion for the absorptive part as p2 -+ 0.

We have made some changes from the definitions in Bincer's paper (Ref. 1). The most notable change is a change of variables from Bincer's W to W2. A complete discussion of this change and the resulting change in the deiinitions of the off-shell form factors is unnecessary for our purposes, For a detailed discussion see D. J. Silverman, Ph.D. dissertation, Stanford University Physics De- partment, 1968 (unpublished).

The Fs nucleon form factor satisfies an unsubtracted sidewise dispersion relation,'

where l2 is the invariant nlass of the photon and LV2 is the invariant mass of the off-mass-shell nucleon.

The absorptive part of the dispersion relation is given by7

2, represents both the intermediate state sum and the corresponding phase space, q(0) = &(O) ($+I--M), where &(0) is a nucleon field operator, and v,"s a projection operator which projects out the Fz part. This expression for the absorptive part becomes clearer when we look a t Fig. 12.

TABLE I. Iiesonances contributing to the absorptive part.

The absorptive part is given by the product of two matrix elements. l i r e see the off-shell nucleon (solid line) enter a strong interaction blob and emerge into an intermediate state In). The intermediate state then goes into another blob and turns into an off-shell photon and a nucleon. The total absorptive part is then given by summing over all possible intermediate states and pro- jecting out the part belonging to F2.

The lowest-mass intermediate state which can contribute is that containing one pion and one nucleon. The next most important states are those obtained b>- adding Inore pions. Adding four pions brings us up to a threshold of -1500 MeV where resonances such as those in Table I, begin to make contributions to the absorptive part.

J

fi~~d--Xj = xrl =-:e-- OFF-SHELL

P" I

NUCLEON ON-SHELL INTERMEDIATE NUCLEON STATES

FIG. 12. Graphical representation of the absorptive part.

J E F F R E Y P A U L K O Y E R 2

FIG. 13. n-N intermediate-state cotltributio~l to the absorptive part.

TVe remind the reader a t this point that the object of this work is to follow the threshold-dominance idea to its logical conclusion. If the threshold-donlinance idea is valid, then it should be best implenlented by providing an expansion about the threshold. Lye provide a method for determining the coefficient of the leading term and for estimating the higher-order terms of this expansion. If the nuillerical results obtained fro111 this expansion are in wild disagreement with experiment, this would lead us to suspect that threshold dominance does not apply here and that contributions froin resonances and higher intermediate states farther along the cut must be taken into account.

We will be working from the point of view that the resonant contributions are really enhancements of some manv-nion nucleon state and it is onlv the contribution

4 L

to the threshold expansion as ,u2 -+ 0 which will interest us. Thus, we will be considering only states with one nucleon and n pions.

The first step toward the threshold expansion is to examine each of the matrix elements in the absorptive part. We do this by expanding the matrix element according to the graphical method in Sec. 11. For example, we would apply the graphical expansion to the matrix elements represented in Fig. 13 for the one-pion, nucleon intermediate state and to those represented in Pig. 14 for the two-pion, nucleon intermediate state.

After we expand the inatrix elements according to the graphical description, we will examine the graphs to determine which graphs give the nlost singular (in p 2 ) contribution to the radius a t threshold. We will then make a threshold expansion of those most singular graphs.

Through the graphical description we are able to deal - - A

with the case of n pions almost as easily as with the cases of one or two pions. This results from the fact that there are basically only four different t ~ p e s of chains which enter into the strong-interaction blob of the graphs. \Ye can examine each of these in detail near threshold and actually determine their singularity structure as w2 + 0.

FIG. 14. 2n-N intermediate-state contribution t o the absorptive part.

I n the following two subsections, we will work first with the matrix elenlent of the electronlagnetic current and second with the matrix element of the nucleon source. Before enlbarliing on this, however, we must remark that we are making an expansion in powers of (TIr2--M2)/M2 and that we will set ,u2 + O everyplace where it does not result in a singularity. In particular, this means that threshold is a t &f2 and that the q,, -+ 0 as we approach threshold (all four components). There- fore we will set q, ,= 0 everywhere except in those graphs where the sing~ilnl-it occurs.

A. Electromagnetic Current Matrix Element

I n this section. we study the matrix elements of the electromagnetic current

U'e show that certain types of graphs dominate the graphical expansion as p2-7'0, Our search for the dominant graphs was strongly inotivatecl by the clue given by Drell and S i l ~ e n n a n . ~ This clue indicated that the strongest singularities would occur in those graphs which had an axial-vector current carrying the electro-

)ESIRED TYPE )F CHAIN

FIG. 15. Example of a dominant graph.

magnetic nlonlentunl into the blob. That is, the dominant graphs would have a chain which started with the electromagnetic current and ended with an axial-vector current connected to the blob (see Fig. 15 for an example). [Ye show explicitly that it is only the above type of chain which can lead to strong singularities in the pion mass a t threshold.

Suppose we start with the inatrix element shown in Fig. 16. \Then we perforln the graphical expansion of the graphs in Fig. 16, we obtain a sum of graphs, each of which is a blob with various chains attached (see Sec. I1 on graphical expansion). The important fact to be noted is that there are only four different types of chains. These are classified as follows.

(A) Chain without the electromagnetic current, ending in a vector.

(B) Chain without the electron~agnetic current, ending in an axial vector.

((') Chain containing the electromagnetic current, ending in a vector.

(D) Chain containing the electromagnetic current, ending in an axial vector.

2 T I - I R E S H O L D T H E O R E f i I F O R I I . 4 D I U S O F F ~ v N U C L E O S 3047

Examples of these four types of chains are shown in Fig. 17.

Every graph in the graphical expansion is just a blob with various conlbinations of the above types of chains attached. The chapter on graphical expansion explains the details of the expansion. We need be concerned only as far as knowing what the general tern1 loolis lilie. Each blob will contain one chain-of either type C or D. The remainder of the chains will be made of conlbinations of types A and B.

IVhat we ProDose to do is to examine the behavior of A

the above chains as we near threshold to see whether singularities in the pion mass occur. We first consider chains of type A. IL7e suppose that a chain of type A is attached to a blob along with other unspecified types of chains (see Fig. 18).

The analytic expression for such a blob can be written as

(ps 1 T A , , ( ~ ) d I ks') , (24)

where the qi, are the momenta of the axial vectors connected to the chain, qj, is the momentum of the axial

n D I V E R G E N C E S - 2% i

FIG. 16. Typical graph to which the graphical analysis is applied.

yector which has not been commuted with anything, 0 represents all other operators in the time-ordered product, and the dR represents all other exponenti~l factors and space-time integrals for the operators in 0. We have omitted the e t j k symbols and factors which are not essential to our argument.

We want to examine this expression when p2-+ 0 nears the threshold, which means that all four components of q;, --+ 0. I n this case the entire expression vanishes except for contributions from nucleon pole terms.

The contribution from the pole term in the qi, -+ 0 limit is that calculated by attaching the chain to the external nucleon legS (see Fig. 19). In this limit

G ' ( k + i ~ l ) y ~ y b ~ ~ r , U(R,SI) +iql , / d ~ - - - .. . , ( 2 5 )

2fz .C qz To examine the qp 4 0 limit, we use an analysis similar to that

used by Adler to derive strong-interaction consistency conditions. See S. L. Adler, Phys. Rev. 139, 1638 (1965).

A B C D

FIG. 17. Evamples of the four different types of chains.

where 6' is-essentially the time-ordered product of the operators 0. We see fro111 this that A is finite in this linlit and that the chain of type A cannot lead to singu- . .

larcies in the pion mass. Any singularities must occur in 0'.

The analysis of chains of type B proceeds similarly to the analysis of chains of type A, with the exception that the form factors are changed from axial-vector form factors to vector form factors, and the y5isomitted:

with the same conventions as before. In the limit p2= 0, q,, -+ 0, B becomes

Again, as in A, this is finite as qia--f 0 and the only place singularities can occur is in 0'. The analysis of chains C and D differs slightly from that of A and B. We will assume that all of the chains of types A and B have already been analyzed before proceeding with the analysis of C and D. Since each chain of type A or B is attached to the external nucleon legs as we approach threshold, the result after the analyses of all A and B chains is that C or D is sandwiched directly between nucleons. This is shown symbolically in Fig. 20.

The contribution of a C - or D-type chain then comes clown to its matrix elenlent between nucleons. Type C gives the vector forin factors of the nucleon and type D

TYPE A

\J FIG. 18. Graph discussed in the analysis of a chain of type A.

3048 J E F 1 ; R E Y I ' A U L R O k ' E l <

gives the axial-vector form factors of the nucleon. I t is easy to see that the type-C chain is nonsingular as p2-+ 0 because there are no pion poles in the vector form factor^.^ There is, however, a pion pole in the induced pseudoscalar form factor which is of the t j pe we would expect to be important from the clue in Drell and S i l ~ e r m a n . ~ This pion pole will carry the momentum Cq,+l, where the q; are the momenta entering the chain.

I ts contribution will have the form

which will give l/p4 when we differentiate to find the radius.

We have analyzed the matrix element of the electromagnetic current by anal) zing separately each of the four possible t lpes of chains which can be attached to the strong-interaction blob coming from the graphical expansion. Of these chains, only that of type D, an axial-vector current entering the blob and carrying the electromagnetic momentum 1, can contribute a singularity as p2 -+ 0. Since there can be only one chain of type D attached to any particular blob, the order of the singularity cannot be greater than l/p4. The contributions from the chains A, B, and C are finite as p2 -+ 0 a t threshold and are easily calculated as discussed in the preceding. The upshot is that we will keep only those

graphs in the expansion which contain chains of type D. In Fig. 21, we show the expansion of the one, two, and three pion cases in terms of graphs, and underline the dominant graphs.

As we a ~ ~ r o a c h the threshold in the u2=0 limit, the A A

underlined graphs simplify further to the graphs shown in Fig. 22, where the A-, B-, C-, and D-type chains are attached as indicated from the preceding analysis.

We might mention a t this point that if we considered a Dure u2= 0 theorv. the radius would in fact be infinite d i e to the graphswhich had a pion pole carrying the electromagnetic momentum.10 That is, the underlined graphs above would behave like 1/12 at threshold in- stead of 1/(12-p2). Thus in a pure p2=0 theorj-, the underlined graphs would indeed dominate the others; in fact, the finiteness of the other graphs would rrlalie them negligible compared to those underlined. 1%-hen we give the pion a finite nonzero mass, we are increasing the importance of the neglected graphs compared to those underlined. This importance is measured then by the size of the ion mass relative to the nucleon mass. which is the expansion parameter in this problem.

B. Matrix Element of Nucleon Source

Sow, we turn to the matrix element of the nucleon source,

(ksf,qlcul,. . . , qnanj ~ ( 0 ) 10). (29)

Our first concern is to determine whether or not this inatrix element can be singular in the pion mass as we

TYPES

ANALYSIS OF A and 8

C H A I N S

TYPES A and B

FIG. 20. Symbolic representation of the matrix elements after the analyses of chains of types A and B.

Although the vector form factor does not have a singularity in p2, its derivative may be singular depending on the model assumed. However, no model gives more than a l/p2 for the derivative of the form factor. So the terms containing chain C ~vould still be small compared to the l/p4 singularity in the derivative of the induced pseudoscalar form factor.

'OThat the nucleon radius is infinite when p is equal to zero can also easily be seen using the dispersion relation in the photon niomentum:

When pZ-t 0, the lower limit on the integral is 0, Near qrz=O, the absorptive part is proportional to the T-T scattering phase shift which theoretical evidence indicates is finite. Therefore the integral blows up a t the lower limit.

2 T H R E S H O L D T H E O R E 1 1 F O R R A D I U S O F F z v N U C L E O L 3049

- - -

FIG. 21. Graphical expansion of the one-, two-, and three-pion cases.

near threshold. If we attempt to apply the graphical analysis of the previous sections to the above matrix element, the following unknown comrnutator arises:

If we assume that this commutator vanishes outside the light cone so as to he consistent with nlicroscopic causality, then Eq. (30) must be proportional to a 4-dimensional 6 function or derivatives of such a 6 function. I t nlust also be proportional to a field which can create and destroy a nucleon. The sinlplest expression which has the correct transformation properties and is consistent with the above requirenients is

Lye will use the following relation, as a model, although the analysis of Eq. (29) does not depend on this particular model; we choose this form for the sake of convenience :

IVe could replace the $(x) on the right bl- some arbitrary function which transforms like dix) and which can , \

create and destroy a nucleon. We could also add terins which are proportional to derivatives of 64(x-y). None of these changes in themselves col~ld rnake the matrix element singular as p"0. The key assumption we have made is that the matrix elenlent of the cominutator itself does not blow up in the p 2 = 0 limit.

This assumption is very reasonable in view of the fact that the terms which arise because of the above eaual-time commlitators onlv contribute additions to the on-shell matrix element, proportional to some power of (lk72-M2)/M2. This fact follows by setting the off- shell invariant mass equal to M" and using the reduction fonnula to get rid of the nucleon field operator by changing it to a nucleon in-state. The absence of the field operator prevents any terms froin arising in the

= .p, + PERMUTATIONS

f LA ,

w d p , = 1 f--- + ~p, Y' +$< j.2 .y1\

+ PERMUTATIONS

FIG. 22. 'The dominant graphs in the one-, two-, and three-pion cases in the p2 -+ 0 limit.

graphical reduction which contain the above corn- mutator. Thus, as the off-shell nucleon mass goes from FY2 to M2, the contributions from the graphs containing colnmutators vanish independently of the pion mass. Therefore, these terms can not be singular in the pion mass.

Now, if we apply the graphical techniques to Eq. (29), using the model [Eq. (32)], we will end up with a sum of graphs, each of which is of the general type shown in Fig. 23. That is, the matrix elements represented by the graphs will he the time-ordered product of a nucleon field with a number of chains of types A and B, sandwiched between a final nucleon 1 k,s') and the vacuum.

We examine these chains exact11 as we did the similar chains in the preceding section. The chains conle out of the time-ordered ~ r o d u c t and are attached to the external nucleon leg. Their contriblition is linite and nonsingular as q , , -+ 0 and p+ 0. The matrix elenlent left after the removal of all of the chain is the matrix eleinent of the nucleon source connecting I ks') to the vacuum. This final matrix eleinent is just the wave function and does not have any pion poles. Therefore, there is no possibility here of a pion-pole singularity and all we must be concerned with is the expansion of Eq. (29) in (IV2-M2)/M2.

IV. COMPLETION OF PROOF

We have studied the two matrix elements and have determined the following program.

NUCLEON, K , S ' - FIG. 23. Typical graph appearing in the analysis o f

the nucleon-source matrix element.

3050 J E F F R E Y P A U I , R O Y E R 2

TABLF, 11. Powers of (W2-M2)/i1P. -- - - - ---

Int Matrix Phase state element space Total

(1) UYth regard to (ps 1 J@(O) [Bstqlal. . . qna,), we-will keep only those graphs of the graphical expansion which are most singular in the pion mass, i.e., terrns which contain a chain of type D. These singular graphs will be expanded in powers of (IP2-M2)/M2.

(2) IVith regard to (ks'glal. . . q7,ct,& 1 11(0) / 0), we have shown that we obtain no pion inass singularities; therefore, all we need to do is expand this in powers of ( I ~ ~ ~ - M ~ ) / M ~ .

(3) We will evaluate the phase-space integral a t threshold, assuming the matrix elernents to be constant a t their threshold value. The threshold behavior of the phase space for 1 massive and n massless particles can be calculatedl1 and the result is that the phase space behaves like [(W2-M2)/M2]27L-1. This behavior of phase space is crucial in our final argument. We can count on an extra factor of [(T1'2-M2)/M2]2 for each additional pion.

By combiniilg (1)-(3) above, we can show that the leading contribution to the threshold expansion of the absorptive part of the radius comes from only the a-l\r intermediate state. This result is obtained by merely counting powers of the threshold expansion parameter.

For the intermediate state with one pion and a nucleon, the product of the matrix elements gives two powers of (W2-M2)/M2. This, multiplied by the ap- propriate phase space, gives a total of three powers of (ITT2-M2)/M?.'2

For the intermediate state with two pions and a nucleon, the product of the matrix elernents gives a t least one power of (W2-M2)/M2,13 The phase space gives three powers which yields a total of four powers, or a t least one inore than the one-pion nucleon state gives.

The phase space with three pions is already five powers of (?1/72-M2)/M 2, so we do not have to calculate the product of the inatrix elements. This information is summarized in Table 11.

So we see this result: The lowest-order term in the threshold expansion of the absorbtive part of the radius of F ~ v , in the p2 --+ 0 limit, comes entirely froin the n-N state.

V. CONCLUSION

The theorem proves that the a-N intermediate state is the major contributor in the calculation of the FPV

l1 See Appendix D. See Appendix B.

la See Appendix C.

radius with the sidewise dispersion relation. This fact has been, of course, conjectured, assumed, hoped for, and used with success since the advent of the sidewise dispersion relation.14 However, the words "major contributor" remained vague and unqualified, somehow meaning that we did need to consider the intermediate states containing more pions. The proof of this theorem now provides a precise statement of the meaning of the wolds "rnajor contributor." This precise stateinent aiises frorn the anal> sis of the contribution of the various intermediate states to the terrns of an expansion in the parameter (ITT2--M2)/M2 as ,L/M 3 0.

The utility of this statement is not coinpletely clear because p / A f - 117, which ineans that corrections of order 15% maj corne by just consideii~lg the next terms in ,u/M. Also, we have no guarantee that the threshold expansion converges. As with inost expan- sions in physics, we must calculate the expansion coefficients to determine whether or not the expansion is useful.

The results of Drell and Silverman2 indicate that the threshold expansion for the F2v radius does converge rapidly and that the n-1' intermediate state gives most of the radius.

The questioil arises as to whq we cannot make a similar theorem for thc calculation of the anomalous moment. lye probably could produce such a theorern but results show that the contributions from the Sll and P11 resonances inalie significant contributions to the anonlalous moment which means the threshold expansion would converge very slowly and would be of little utilit) .

n r e take the opportunity here to mention that we tried to use the formalism developed here to understand the laige radii of the K ~ B forin factors implied by the work of Roy.16 LVe failed to provide any- dynarnical reason for the large radii and we conclude that if it exists, it must be attributed to a subtraction constant.

ACKNOWLEDGMENTS

1 would like to thank Dr. S. D. Drell for the initial suggestion ol this problem and for his continuing guidance and encouragement during the resulting research.

APPENDIX A

The on-shell vertex function I',(P, $+I) is defined by

The reduction formula is used to continue analytically the vertex function in the (mass)2 of the initial nuc1eon.l

I4 S. D. Drell and H. K. Pagels, Phys Rev. 140, B397 (1965) ; R. G. Parsons, ibid. 168, 1562 (1968); H. R. Pagels, ibid. 140, B999 (1965); D. U. L. Yu and L. Grunbaum, Bull. Am. Ph1 s SOC. 13, 24 (1968).

l6 P. Roy, Ph.D. dissertation, Stanford Universit~ I'hqsics Department, 1968 (unpublished).

2 T H R E S H O L D T H E O R E M F O R R A D I U S O F F ~ v N U C L E O N . . . 3051

TABLE 111. Projection operator coefficients. -- -- --

rn12+ IM (4tiT212)-2[6i4M2+314 (LV2 - M 2 ) - 312 (bV2- hf2)2 ] a 2 2 + ~~f//3(4W21~)-2[414+812~f2+412(W2-M2) - 2 ( W 2 - M 2 ) 2 ] a,'+ (4H7212)-2(W2 - M 2 ) (- 312J$f2) PI2+ M(4W212)-2(W2-iU2) [$14- f12(W2-MZ)] Pi2+ 144 (4 W212)-2 (W2 - M 2 ) [612M2] Pa2+ M2(4W212)-2(W2-M2)[-3 ( W 2 - M 2 ) ]

The most general form for the off-shell vertex, which transforms like a vector under proper Lorentz trans- formations and is restricted by the Dirac equation on the final spinor, is16

The projection operators vPi*(W2j are defined ns follows:

For the purposes of this paper, we need to know only vr2+(l,V2), which is derived in Ref. 7:

a and p are given in Table 111.

APPENDIX B

I n this appendix, we discuss the contribution to the absorptive part from the one-pion, nucleon intermediate

l6 S. D. Drell, and H. R. Pagels, Phys. Rev. 140, 33397 (1965).

state. We show this contribution gives three powers oE (IV2-M2)/M2 as stated in Sec. 1V above.

Using the graphical methods developed in Sec. 11, we quickly work out the following expressions for the needed matrix elements. We start with the electromagnetic matrix element and use the reduction formula on the pion, plus PCAC, to obtain

( p s I J@(O) / ks',qa) = ( i , / fT) - ("?,")

Then we apply the graphical analysis to the right- hand side of Eq. (37), keeping only the pion-pole term, to obtain the following:

IVe use the reduction formulas in the following forin to obtain an expression for the matrix element of the nucleon source, near threshold, i.e., TIr2 -+ M2:

we obtain

Putting these two n~atrix elements in the expression for the absorptive part of F2, we obtain

3052 J E F F R E I - 1 ' " iUL R O Y E I Z 2

TABLE IV. Contributions to the absorptive part coming carrying the elcctroinagnetic momentum. The result is f ~ o l n the r - N intermediate state.

- -

11'2 - M~ (ps I Jfi(01 I ks',qlal,qza~'l a12+

M 2 1v2- , I42 2

azZ+ (2) ( ~ + 2 1 1 ~ j (angles]

The last trace is to project us onto either the isovector or isoscalar part. By doing the isospin trace, we see that there is no contribution to the isoscalar part in this order. To do the remaining trace we refer to Appendix A where the explicit forms of the projection operators are written out.

In doing the trace, we keep in mind that we are only concerned with finding out the order of the leading term. The calculation of these traces is straightforward but long and tedious.

We present the results of the traces in Table IV. Table IV contains six rows each labeled by an a or a P. These a's and p's correspond to those in the projection operator vW2f and thus label the contribution from those corresponding terms in the projection operator. The bracket labeled "angles" signifies a function which depends on the angles to be integrated over, but which is independent of (LV2-M2)/&f2.

The results in Table IV show that the first-order contribution cancels and so the product of the matrix elements is of second order in (ItT2-M2)/M2 as stated in Sec. IV. The phase-space integral gives us another power of (W2-M2)/M2. Thus the contribution to the absorptive part is of order 3 in (W2-M2)/M2.

APPENDIX C

I11 this appendix, we discuss the contribution to the absorptive part from the two-pion, nucleon intermediate state. We show that this gives greater than or equal to four powers of (W2-M2)/M2 as stated in Sec. IV.

K e will follow closely the analysis contained in Appendix B, leaving out some of the pedagogical steps included there.

Using the graphs, we can immediately isolate the dominate parts of the electromagnetic current matrix element, i.e., those graphs which have a pion pole

+(terms with 1 ++ 2) . (43)

Similarly to the method in Appendix R , we obtain the following:

+(terms with 1 ++ 2 ) . (44)

LVe will omit writing out the total expression for the absorptive part because i t is long and cumbersome and it does not serve to elucidate anything. The isospin traces yield the same results as in the single-pion case. There is no contribution to the isoscalar part of the F z form factor in this order. The statement which was made in Sec. IV requires that we obtain a t least one power of (II-2-1W2)/1ML from the product of the matrix elements. 15'e can see that this is a fact in the following way. If we count powers of q, in the traces to be done, we see that there is alwa\s one power. IVhen we evaluate the phase-space integral a t threshold, all the q,'s are set to zero by the 6 function, and the contribution to the absorptive part vanishes. As we move away froin threshold M2, we can have a nonzero contribution but it must be proportional to a t least the first power of (IP2 -1M2)/11,12.

The phase space for one massive and two massless particles gives three powers of (W2-M2)/M2. Com- bining these results, we get a t least four powers of (1V2-M2)/M2 as stated.

APPENDIX D

In this appendix, we prove the following statement: The phase space for one massive and n rnassless particles goes like [(W-M)/MI2"-I a t threshold, where 1;1' is

2 T I I R 1 : S H O L U T L I E O I I E M F O R R A D I U 5 O F F s v N U C L E O N . . . 3053

the total c.m. energy oi the systeal. K e follow the p(Q,I',E) is worked out by finding the Jacobian of this method of Stapp.17 transformation of variables.

We first make seine definitions: n hi;

n d3k d3q, P=n -. n - -- =d3PdEdQdr p(CL,I',E), i=1 E <=I E k Ei

The limits of the integrals over the 37 are established n by using the definition

P = k + C qi=(total momentum in c.m. frame), (45) i= I En^lk>3nk-1> M

E = (total c.m. energy). So

The dQ and d r are defined as follows. First, make a 1i31z d3q1 d3PdB fi / ---64(il+~ q.-Q) = / T63(~)6(E-A7) Lorentz transformation to a frame where k'+qll=O. ,=, E~ E, i

The superscript indicates that we have made one Lorentz transformation. Define

xfi 2-1 /n..i,: .i:11tn-1J,"7~-1 - 2 J , n2 L E ~ I JJ i= n 1 kt%. 3 x 1 ~ (M2+k112)112+ (k112)112

= (energy of the massive particle and the (50) first lnassless particle in their c.m. frame), (46) In order to make the threshold dependence clear, we

kll= qll. define new variables

Now, we make a second 1,orentz transformation to @lti -3%-1) lzi= , E ' r E - M .

a frame where k2+q12+q22=0, and define the c.m. E' energy of these three particles in the second frame, Then

31i2= (31i12+kz22)1/z+ (k222)1/2, k 2 ~ = q 2 ~ . d3k d3q, (li) fi / - --h4(k+z q. -Q)

Likewise, frame 12 is defined by %=I Ek E , i

k

k k + Z $=I qSic=0: = / ~ ~ 6 3 ( ~ ) 6 ( E - J V ' )

t m k = ( t m k - ~ ~ + k k k ~ ) " ~ + ( k k k ~ ) ~ ' ~ , kkk=qkk, (48)

mn--E=(total c.m. energy of all particles).

Our new integration vaiiables are

(i)dP (3 variables)

(ii) angles of the k,, (2n variables)

(iii) d'X, (n variables).

This is a total of 3n+3 variables, or the correct number. 'IZ'e now define

dQ= (angle differentials of the kt,) ,

We now need to see how IT h;i depends on E'. i = l

Solving explicitly for kkk,

n we see that each kkk gives us an additional power of E'. iW=U d w i . When we count the total number of powers of El, we

i=l get n-1 powers from the differentials and n powers l7 H. P. Stapp, LRL Report NO. UCRL-10261, 1962 (unpub- froin the K k k . The total is then (E')~"-'. since

lished). E'= W - M , we have [(W-M)/M]2n-1.

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Nucleon Form Factor