P H YS I CAL R EVI EW VOLUME 111, NUM BER 6 SEPTEMBER 15, 1958

Fourth-Order Nucleon-Nucleon Dispersion Relations

M. T. GRis&Rv t'

Palmer Physical Laboratory, Princeton University, Princeton, lY'ev Jersey

(Received May 21, 1958)

The fourth-order contribution to the nucleon-nucleon scattering amplitude is shown to satisfy the condi-tions necessary for writing dispersion relations. The imaginary part is explicitly calculated and shown toagree with the absorptive part obtained from general principles.

A T the present time there exists no derivation ofthe dispersion relations for nucleon-nucleon

scattering. A proof of their validity, making use ofgeneral requirements such as causality, invariance, etc. ,has been given recently, ' but only under the unphysicalcondition

M ) (K2—1)Mii.

Nambu, ' using a perturbation theoretic approach,has been able to show the validity of the dispersionrelations for the vertex function, but the situation isunclear for the four-point function.

Dispersion relations for nucleon-nucleon scatteringhave been given' and work is in progress to apply themto a treatment of experimental data and a derivation ofa nucleon-nucleon potential. It is of some interest toverify that, at least in low orders of perturbation theory,the nucleon-nucleon scattering amplitude does satisfydispersion relations of the usual variety, not only toallay fears of the possibility that it does not, but alsoin the hope that one might 6nd what specific propertiesof the interaction could be used to supplement theconditions employed in the general approach of refer-ences 1 and 2.

In this paper we propose to study the fourth-ordercontribution to the nucleon-nucleon scattering ampli-tude. The Born term, by which we mean the familiarsecond-order amplitude, presents little difficulty and ishardly worth discussing, although it is important indefining the renormalized coupling constant (see G.N.O.for a detailed discussion of this point) . On the other handit is hopeless to study the sixth or higher order terms bymeans of the explicit calculations we wish to carry outhere, although in principle there seems to be no basic

difhculty.Our notation is essentially that of G.N.O. Incoming

and outgoing nucleons are denoted by p, k, and p', k',respectively. The following combinations are useful:

(m, the nucleon mass). The 5-matrix is defined by

S=1+iR,

(p', k'~

R~k, p) = (2~)'b(p'+k' —p —k)

X F(p', k'; p, k). (2)-(2m')'ko Po kopo-

We study F, the Feynman amplitude. For simplicitywe take a neutral theory; the presence of isotopic-spinoperators does not change the character of the problembut complicates writing. Also for ease of writing we set

~(p') v»(p) =s(p', p),~(p') v.~(p) = I'.(P',P),

~(p') v~v.v.~(p) = 2'"(P',P).

1 t" ImM„(v', Q')

M, (v,Q') =—)

dv')

P P 26

We work with A= c= i, and omit the coupling constantthroughout.

The dispersion relations are usually discussed in termsof the retarded amplitude M„which, as is well known,equals the Feynman amplitude for values of themomenta corresponding to a physical scattering act.I et us, for the moment, talk about scattering of scalarnucleons, in order to avoid technical complicationsassociated with the presence of Dirac spinors andy matrices. Then M„(or F) depends on the two inde-pendent scalars v and Q'. M„as a function of v forfixed Q' is defined for physical values of v, v) v„=m+Q'/4', corresponding to real momenta. In order toobtain dispersion relations, we must prove that, as afunction of the complex variable v, M „can be extendedover the whole u plane and that it has the requiredanalytic character, so as to enable us to write

E= (k+k')/2,r = —(I' E)/m,

1 i" A (v', Q2)

M„(v,Q') =— dv'7l QQ P P Z6

&= (P+P')/» or, with M=D+iA (D and A, the dispersive and

Q= k —k'= p' —p absorptive parts of M„, real functions of the momenta))

* Charlotte Elizabeth Procter Fellow.f Present address: Department of Physics, University of Illinois, (4)

Urbana, Illinois.'Bremermann, Oehme, and Taylor, Phys. Rev. 109, 2178

{1958). Our task then, in this fourth-order calculation, is to' Y. Nambu, Nuovo cimento 6, 1064 (1957). ~ ~

3 Go]dberger, Nambu, and Oehme, Ann. phys. 2, 226 (1957) exhibit F and 3f„show that M, does satisfy disPersionThis paper will be referred to hereafter as G. N. O. relations, calculate the imaginary part of M„, and to

17i9

1720 M. T. GR ISA RU

P~

P k

{D)

FIG. 1. Fourth-order Feynman diagrams.

show that it agrees with the conventional form of Awhich we shall write below.

Since we are concerned with physical nucleons, wemust settle the question of how to treat the spinors and

y matrices appearing in our amplitudes. The standardway, as given in G.N.O., consists in writing

M=u (p')up. (k')m p, pu (p)up(k),

S(p' p) T-p(k'») I ft(~ Q')~.p+fs(»Q')P-&p3

Strictly speaking, we should investigate fr and fsseparately, but it is more convenient, and it certainly

expanding 5K into a suitable set of y matrices multipliedby I' or E, and studying the coefFicients which arescalar functions of v and Q'. We shall not follow thisprocedure here. Instead we will take terms resultingfrom Feynman diagrams and study them individually.A typical term might have the form

does not acct our results, to study the combinationf,b,p+fsP~Ep, keeping in mind that where vectorcomponents appear explicitly they should be regardedas Axed parameters.

Ke shall write explicit forms for the various contri-butions from the Feynman diagrams, their sum makingup F in fourth order; then, indicate how to obtain thecontributions to 3f„and show that dispersion relationscan be written for the individual terms. Following this,we will calculate the imaginary part of the terms andshow that their sum is equal to the fourth-order contri-bution to the absorptive part A. The form of A whichwe shall use is derived in G.N.O. from general principlesand we reproduce it below:

A (p', k'; p, k) =s-(2s-) 'Lp»ps'/m'j*'u. (k')

xE-L&p'If-I )& If Ip»(p- —P—&)+&I'I fpl ~)&~l f-I p»(p- —P+&)3up(k), (~)

where f is the right-hand side of the Dirac equation,and p„ is the momentum of the intermediate state u.

I. THE I'EYNMAN AMPLITUDE

We exhibit here the fourth-order Feynman dia-grams (Fig. 1) and the various contributions to F.They have been given before' but we write them in aform which is more convenient for our purpose.

I A+A +Br+Br +B2+B2+++c+D+D +Pl+ ' '1+4 ]&(2s)4

A =4S(p'&p)S(k'&k)i d'qrd'qs (qt+qs p'+p)—I:(p p)'+p'j' (q

—'+m') (q '+m')

(qs p')-(qt+ p)—pBr =T p(p', p)S(k', k)i d'q, d'q, 8(qr+qs —p'+p)

(p' p)'+I" (p+—q )'+~' (qt'+m') (q '+m')

I (qs+k'). (qr —k) pB,=S(p',p)T p(k', k)i ~' d'qtd'qstI(qt+qs —p'+p)(p' —p)'+p' (k —qr)'+p' (qts+m') (qss+m')

(qt —p)-(qs —k') pC= V-(P', P) Vp(k—',k)i J~d'qtd'qs ~(qt+qs P k)——(p —qt)'+p' (k' —qs)'+y' (q +mr') (qt'+m')

(6)

(qt p)-(qs+k) p—

1 (qs —p')-(qt —p) pEt T p(P P)S(k k)s

1

d qrd qs 5(qs P)(p' —p)'+p' (p q,)'+I" (q,'+m') —(q,'+m')

1 j.D= V (p', p) Vp(k', k)s I d'qrd'qs 8(qt+qs p'+k)—

(p qr)'+~' (k+qs)'+I" (q—t'+m') (q +ms')

etc.

The primed quantities are obtained from the unprimedones by interchanging p' and k' and changing the over-all sign,

We must discuss the question of renormalization.AI1 but the terms C, D, C', and D' are infinite, and we

should separate their Rnite parts before we discussanalyticity or calculate the imaginary parts. We canshow however that for our purpose this is not necessary

' K. M. Watson and J. V. Lepore, Phys. Rev. 76, 1157 (1949).

NUCLEON —NUCLEON DISPERSION RELATIONS 1721

since all the information we desire can be obtainedfrom a study of the divergent expressions themselves.

Let us illustrate this by looking at the term A',which can be written

I(v) —I(—v„) c)I(v)

BV v= —v~

(v+ v„),

the diBerentiation being carried out under the integralsign. This term is finite and the second-order pole atv= —v„has been removed. One must of course thinkof the subtracted term I'(—v„) (v+ v„) ' as being addedto and treated together with the Born contribution andsubtractions from higher orders which exhibit the samesingularity. The resulting coeKcient of (v+ v„) ', up totrivial factors, is defined as the square of the renormal-ized coupling constant. The other subtracted termcorresponds to meson mass renormalization and needsnot concern us here.

Carrying out the integration over q2, we can write

I(v) = d'q, q (q, , v).

If the integral converged, the analytic character ofI(v) could be determined by studying y(qi, v). Wewould find that I(v) is certainly analytic wherever

&p(qi, v) is, for all qi. It turns out that p is analytic atv= —v„ for any q&. Therefore

V (qi, v) —V (qi, —v.) ~V (qi, v)

Bv v= —v~

has the same analytic character as p(qi, v) and further-more the integral over q& converges. Thus, the an-alyticity properties of the renormalized term aredetermined by &p(qi, v) alone and, if we are willing totalk of the analytic character of a divergent expression,they are the same as those of I(v). In what follows weshall study I(v), but it will be clear that only thoseproperties of I(v) will be needed, which follow fromproperties of the integrand, so that all our statementswill really be statements about the renormalized term.

In calculating the imaginary part of the renormalizedterm the same ideas apply. Using the fact that p(qi, v)is analytic at v= —v„, one can easily show that theimaginary part of the renormalized term is equal to[ImI(v)$/(v+v„)'. As we shall see, ImI(v) is finite.

1d qid qg 5(qi+qg k +p)

[(&' p)'—+~'3' "qy qg

—mI(v)

(q '+m') (q '+m') (v+v )'

where —v„=m —p, '/2m+Q'/4m. I(v) is a quadraticallydivergent integral. According to Watson and Lepore, 4

the renormalized term is given by

F(v,Q') = lim M(v+iri, Q'). (7)

We shall write dispersion relations for M, and deducerelations for Ii.

Let us first treat the terms C and D. We carry outthe integration over one of the q's by means of the6 function, introduce a suitable number of Feynmanparameters x,, integrate over the remaining q, andobtain expressions of the form

( r', x)vlim

p+ ~ [)i(x,)v+o (x,)—i» j4

X(5 e or P Ep) (8)

where r(x, ,v) is a polynomial in x, and is at most quad-ratic in v; X(x;) is a quadratic function in the x;, andfor the range of variation of the variables is bounded,Xi &~)i &X2, a(x;) is also quadratic but has the furtherproperty 0 o.

& ~& o ~& o.2, at least for sufficiently smallQ'. The property o.)0 is of interest since it determinesthe existence of a gap in the cut v plane, for the analyticcontinuation of F.

We write the integral as

r(v) = lim~~0+ ~) )p

dX2[)iv+0 'L»)

7+" dx, . (9)~i,(o P.v+o.—i»]'

' Y. Nambu, Phys. Rev. 98, 803 (1955); 100, 398 (1955).6 R. J. Eden, Proc. Roy. Soc. (London) A210, 388 (1932), who

discusses similar questions, shows in what sense this is equivalentto working with the renormalized quantities.

In general, the imaginary parts of all the divergent termswe have to treat turn out to be finite; the bearing thishas on the connection between dispersion relations andrenormalization has been discussed elsewhere. '

The other infinite terms can be treated in a similarfashion. For 8 and 8' one has to carry out only cou-pling constant renormalization. On the other hand, Eand E' require nucleon mass renormalization; once thisis done, the result is either a constant or else of the forme(v+ v„) ' (e being an infinite constant) and contributesonly to the coupling constant renormalization term.We can therefore drop these terms completely.

We may then (formally) study the divergent integralsin order to determine the analytic character of therenormalized terms. Alternatively, we may think ofusing a cuto6. ' The divergent integrals will also supplyus with the correct imaginary parts.

II. ANALYTICITY AND THE DISPERSIONRELATIONS

The Feynman amplitude Ii(v,Q') is defined forv) v„=m+Q'/4m. In this section we wish to show that,as a function of the complex variable v, it can be ex-tended to a function M(v, Q'), analytic in. the cut plane,such that for v& v„

1'722 M. T. GRISARU

Let us now define the functions

OR, (v) = lim3.)0

T

P.v+0-+zo]'

f 7

+ dx,'

&i,&0 P,v+0 —ioj4(1O)

OR„(v) = lim ~~ dx,x)0 Pv+0 zoj

+~

dx4~~&0 P,v+~+ioj'

OR(v) = dx;P.v+o.)4

Furthermore, in the region where it is defined, F(v)is the boundary value of this analytic function,

They are analytic in the lower and upper half plane,respectively, and equal and real at least in the interval

(~i/Xo) (v ( (o i/Xi)

on the real axis. Therefore they define a function OR(v)analytic in the cut ~ plane, with cuts running from —~to some a «(oi/bio, and from some fi &~ oi/Xi to + oo:

We first note that for v) —'v(2m) = m—+Q'/4m, wecan rotate in the usual fashion the path of integrationin the qo plane, and obtain a real expression. It ispossible then to define, as for the terms C and D,advanced and retarded amplitudes and an amplitude5R, analytic in the cut plane, with the cut runningfrom —~ to —v(2m). The usual type of dispersionrelation can then be written for the renormalized terms.

We have been somewhat vague about the location ofthe various branch points. Actually they are dificultto calculate from the forms given above. However, it isobvious that they are located at points where theamplitudes develop imaginary parts. Since we plan tocalculate these imaginary parts, we shall be able toidentify them.

One other questionable point is the type of dispersionrelations we wrote. The form we wrote for 5K does notindicate whether it vanishes for large v or not. If it doesnot, one should supply, in the usual manner, a sufhcientnumber of powers of v in the denominator to insurevanishing as ~y~~oo. We have not investigated thisquestion in detail, since it would require a detailedstudy of the integrals, but in any event, it does notaffect the main conclusions of this paper. For the fourth-order contribution to meson-nucleon scattering, it turnsout that the various terms in the amplitude behave aslnv or (lnv)'-/v, for large v, z and a similar, althoughpossibly worse behavior might be expected in our case.

O(y) = lim OR(v+ig) =OR, (v). (12)III. THE IMAGINARY PART

For OR(y) we can write, taking a suitable contour

1 p" ImOR(v'+io)OR(v+ig) =—

Idv'

P P Zg

(13)

(however, see end of this section) so that

1 t." ImOR„(v')OR„(y) =—

I dv'7I ~ oo P P Z'g

(14)

This gives the desired dispersion relation for OR„(v).It remains to calculate ImOR, (v) and to show that it isidentical with the previously defined absorptive part.

Note that we have assumed, in claiming that 0-&0,that Q'(Q, „',where by actual calculation, Q,„'=4zz'.

It should not be too difficult to extend the proof tolarger values of momentum transfer. We will return tothis point later.

We turn now to the divergent terms, and treat themin accordance with previously made remarks. Thusfor A' we have to study

y(y)= lim i dx~~ ~~

~~0+

q' —2m(v —v„) (x—x')+m'X de

t q'+2m(v —y,) (x—x')+m' —ioj'

We wish to evaluate the imaginary part of M„(v)in a form which allows easy comparison with the ab-sorptive part. Let us start by formally calculating theimaginary parts of the various contributions to J'(v),without any restriction on v. It is evident that we have

ImM„(v) =ImF(v) for v) a,

ImM, (v)= —ImF(v) for v«a,

where a is the branch point for the cut running to —~ .Let us look now at the first term A. We require theimaginary part of

gy' g2—SS

8 =i d4qid4qz 6(qi+qo —Q)—,(1$)(qiz+m') (qoo+ m')

as follows from remarks we made concerning the han-dling of divergent terms. We write this as

. l f8 =z d qid qo 8(i14+ifz Q)

414 ' ilo m +qio(qio Qo)

Lqio' —4oi'+zojL(qio —Qo)' —coo'+zo'j

where we have set oo=+(q'+m')'. We carry out the

7 E. Kazes, Phys. Rev. 108, 123 {19S7).

NUCLEON —NUCLEON DISPERSION RELATIONS 1723

integration over q&0 by evaluating the residues at thepoles in the upper half plane )the integrand behaves as(qio) ' for large qio]. We assume first of all that thepoles do not coincide, Mi/ —Qo+M2. Writing theintegrand as

f(gi Q2+qio(qio QO)7)

Lq10 Ml +20]L(q10 QO) M2 +20 ]we obtain

not contribute to the imaginary part of 8. We noteincidentally that in Imd the 6 function excludes suchvalues of the energies.

The same approach may be used in evaluating theimaginary parts of the other terms. For the finite termsthe evaluation is somewhat more laborious but thefollowing rule emerges: Given an expression of the form

2J d'qid'q2 b(qi+q2 8)f—(qi, q2)

8 = 22rJ

d qid q2 5(iIi+212—Q)

1 1 1X

2Mi Mi+QO+M2 —'&/ Mi —QO—M2+&J

X,(19)', i (qi —h,)'+m 2

(q2—h, )2+m, 2

its imaginary part is

P fdQmr(qi)dQmr(q2)r, s

XfLtli'il2+Mi(Mi+Qo)]+2M2 Mi QO+M2 2Y/

1

. fL& il+ ( —Q)] (1&)QO+M2+itl

Since we have assumed Mi —M2+QO&0 the imaginarypart of 8 comes from just one factor in each term of thesum and gives

1d'q, d'q2 b (tii+ q2 —Q)

CO jC02

X&(qi+h, +q2+h, H) f(q,+—h„q2+h, )

, (20)s~r (q,+h, —h,)2+m, 2 (q2+h, —h,)2+m/.jets

where we have set

dQm„(q) =d'q 8(q2+m, 2)0(qo).

(We remark that this result is not true in higher orders,because of difhculties associated with the handling ofcoincident poles; although coincident poles do occur infourth order, they contribute to the real part alone. ')

We exhibit now the various contributions to

d'q, d'q, b (qi'+m') 5(q,'+m')2~

&«(qio) tl(q20) f(qi q2) b(at+a —Q)

XLb(q10+q20+QO)+b(q10+q20 QO)]1

8(qO) = 1 qO)0

=0 qo(0.

Actually the two 8 functions in the time componentsgive equal contributions as an invariance argumentshows (e.g. , by going to the frame where QO=O).Therefore, the imaginary part of 8 is

ImF = 4 L1/(22r) 2](a+ a'+ bi+ bi'+ +d') ) (21)

a=4S(p', p)S(k', k)

f qy qg—m'

XJ

dQ„(q,)dQ„(q2) 8 (q,+q2—P'+ P)

L(P' —P)'+~']'

bi Tp(p', p) S(k', k——)

X dQ„(qi)dQ„(q2)8(qi+q2 —P'+P)

(q2—P )-(qi+P) ~

XL(P' P)'+~'X(p+q )'—+~']

Imp= —2r2 I d4q, d'q25(qi2+m2)b(q22+m2)

Xe(qis)8(q20)&(qi+q2 Q)f(ql q2)' (Ig)

It remains to settle the question of coincident poles,Mi —M2+Qo ——0. There is no difficulty in evaluating theresidue; however, in this case the result is real and does

It should be pointed out that the coincidence of poles we areconsidering is different from that discussed for instance in refer-ence 6. In (16), the integrand has poles at rd~+2r. " and-Qp 07g+z6, in the upper half plane, and coI —ie' and Qp+cd2 —$Q",in the lower half plane. We are concerned with the coincidence ofpoles in the upper half plane, —coI=Qp —cog. On the other hand,in reference 6, the discussion centers around th, e coincidence, inthe limit e', e"~0, of a pole in the upper half plane and a pole inthe lower half plane Qp

—co2=co& or Qp+cv&= —co&. It is this type ofcoincidence of poles which actually gives rise to branch points inthe amplitude, and indicates the appearance of imaginary parts.

1724 M. T. GR I SA RU

bo=~(p' P) 2'-p(k', k) to write

X «(qi)«(qo)8(qi+qo —p'+p)Im3f, (i) = 4I 1/(2or)'5I —a' —b, '—b, '

+c (first term)+c' (first, term)—c' (second term) —d (first term)

(qo+k')-(qi —k) p —d' (second term)5.XL(p' —p)'+p'5L(k —qi)'+p'5 It remains to show that this is equal to the absorptive

part.c= —V.(p', p) Vp(k', k)

IV THE ABSORPTIVE PART

«„(qi)«„(q,)8(q,+q, p —k)—

(qi-P)-(q -k) pX

L(P q)'+~—'5:(k' q)'+~—'5

+ "«.(qi)«. (qo)b(qi+q2 —P'+P)

glagaPX

I (P+q,)'+m'5L(k'+q, )'+mo5

d = U (p', p) Up(k', k)

X «~(qi)«„(qo)8(qi+q, —p'+k)

(qi —P)-(qo —k) pX

L(P qi)'+~'X—(k+qo)'+J"5

%e rewrite here the absorptive part

2 (P', O', P,k) = sr(2or) oLPoPo'/m'5-'*u. (k')

X~.L&p'I f„l~&&~l fpl p&b(p„—r—If)+(p'I fp I ~&&~l f.I p&b(p I'+—If)5~p(k)

Note that in the first term the state lm& has nucleonnumber E=2, while in the second E=O. The fourth-order contribution to A(i) is obtained by calculatingeach matrix element in second order. (The one mesonintermediate state, with one matrix element calculatedin third order and the other in first order also con-tributes. However, this is just a correction to the second-order term of the absorptive part, and should be treatedtogether with it, in renormalizing the coupling constant. )We may then restrict ourselves to the following inter-mediate states: E=2: Two nucleons of momenta q'~, q~,E=O: Two mesons, or a nucleon-antinucleon pair ofmomenta q~, q2.

We give the details of the calculation for the two-nucleon intermediate state:

+ "do.(q )«.(q )b(q+q P'+P)—qlnqop

X s

C(p+qi)'+m'5L(k —qo)'+m'5

pf

Ao~=7r(2')o

X&P'I f-I q qo&&qiqol Apl p&

po 0I d'qid'q2 un(k')

m snins J

Again the primed quantities are obtained from theunprimed ones by interchanging p' and k' and changingthe over-all sign.

VVe observe immediately that some of the terms areactually zero; in fact, this is the case for all those con-taining b(qi+qo+Q), since Q is kept space-like while

q, +q, is time-like. This eliminates a, bi, bo, the secondterms of c and d, and the first term of O'. Of the re-maining terms u', bj.', b2', and the first term of d aredifferent from zero only for —no (i (—i (2m) = —m+Q'/4m, the second terms of c' and d' are differentfrom zero for —oa (i (—i (2p) = m —2p'/m+Q'/4m; asshown by a previous remark, their sign should bechanged when computing the imaginary part of 3I„(p).Finally, the first terms of c and c' contribute only form, —Q'/4m= i (2m) (i (no. All these statements followfrom a study of the conditions imposed

bye�(qi+

qo+ . ).This allows us to identify the various branch points and

Xe (kp)8(q, +q, P K) . (22—)—Using standard methods we relate each matrix elementto a suitable scattering amplitude, oR the energy shellif necessary. (This is not the case for do~, but for theother contributions we have to consider amplitudescorresponding to scattering of particles, some of whichhave negative energy. We calculate the amplitudes forpositive energies, then extend them analytically intothe unphysical region; since only second-order termsare needed, this extension is quite trivial. ) Thus

12

&q q I fpl p&~p(k)=, ~(q,q; P,k), (»)(2~) 'qioqoopo

where E is the Feynman amplitude for the processp+k —+q,+qo. Then

NUCLEON —NUCLEON DISPERSION RELATIONS 1725

1 m2

A2W Z2 (2~)'»i~" qio q20

dgydg2F*(qi,q2, p', k')

1 m'~

dQ (q,)dQ (q,)2 (2~)' &

X6(qi+q2 —P—E) Q F*F.

XF(qi,q2; p,k)6(q, +q,—P—E)(24)

1 1-U-(k', P) Up(P', k)

4 (2m.)'

XJ dQ„(qi)dQ„(qg)b(q&+qg k'+—p)

g 1ag2PX

L(P+qi)'+~'7C(p'+ q )'+~'7

Now, in second order,

F=S(qi, p)S(q2, k)(P qi)'+—~'

spin1 1

+— V (k', p) Up(p', k)4 (2~)'

fX dQ„(qi)dQ„(q2)8(qi+q2 —k'+p)

and

—S(p', q,)S(k', qi)(P' q~)'+~'—

~' 2 F*F= kV-(P', —P) Up(k', k)

(qi —P)-(q2 —k) pX

C(k —q2)'+~'7C(P' —qi)'+v'7

+-', V (k', p) Vp(p', k)

(qi —P)-(q2 —k) px, . (26)L(p —qi)'+~'7C(p' —

q )'+~'7

Inserting this expression into the integral gives the two-nucleon contribution, and it is easy to see that it isidentical to the first terms of c and c'. We write belowthe full contribution to A (p', k', p, k) in fourth order:

1A pic

———— V„(p',p) Vp(k', k)4 (2m)'

X dQ„(qi)dQ„(q2)5(qi+q2 —p —k)

—S(qg, p) S(qi, k)(k—qi)'+p"

(»)1

F*=S(p',qi)S(k', q2)(k' —q2)'+ p,

'

g lag 2gX

C(p+ qi)'+no'7C(k —qg)'+m'7

Aii g= S(k',p)S(p', k)(2~)'

XJdQ (qi)dQ (q~)8(qi+q2 —k'+p)

gy' g2—S2 1 1

L(k' —)'+~'7' 4 (2~)'

X dQ (qi)dQ (q2)5(qi+q2 —k'+p)

(q2—k')-(qi+P) p

XL(k' P)'+~'7C(P—+qi)'+~'7

1+- S(k' P) 2'-p(p-'»)4 (2')'

X dQ. (qi)dQ. (q,)~(q,+q,—k'+P)

(q2+P')-(qi k7p—X

C(k' —P)'+~'7C(k —q )'+~'7

1 1V (p', p) Vp(k', k)

4 (2m)'

(qi P)-(q k)p— —X

C(P—qi)'+~'7C(k' —q2)'+~'7

+— U (k', p) Up(p', k)4 (2m)'

dQ„(qi)dQ„(q2)b(qi+q2 —p —k)

(qi —P)-(q2 —k) p

C(P—qi)'+u'7C(p' —q2)'+u'7

X dQ (q&)dQ„(q2)8(qi+qp —k'+p)

(qi-P)-(q~-k) pX . (27)

C(p q)'+I '7C(k+q )'—+~'7

Comparing with the contribution to ImM„we findthe correspondence

A2~c (first term)+c' (first term),

A2~~ —c' (second term) —d' (second term),

Azr~~ ~' &i' &2' d(firs—t term—). —

1726 M. T. GRI SARU

(Note that it may be necessary, in some of the terms,to change the sign of the external momenta in the in-tegrals. However, since the integrals are scalars orsecond-order tensors in the external momenta, theyare not affected by this transformation. )

With this, our identification is complete. We haveproven that in fourth order ImM„(v) =A (v), and that

A (v'))

P P Zc

or, since for v)vv, A(v)=ImF(v),

the standard form of the dispersion relations.

CONCLUSION

We have shown that the dispersion relations arevalid for the fourth-order term in the nucleon-nucleonscattering amplitude. The restriction to small momen-tum transfers is not serious. We have used it to insurethe existence of a gap in the v plane for individualterms, but once the dispersion relations have beenwritten it should be possible to extend them by analyticcontinuation in the variable Q . Alternatively, instead

of studying each individual term one could, by suitablemanipulation, get combinations of terms for whichoA)0 or 0%&0 always. For such terms one wouldalways have a region on the real v axis, extending to+~ or —ao where no branch cut appears.

If one tries to extend the proof to higher orders, oneencounters the difficulty that, although each Feyn-man diagram still gives a denominator of the formP,v+0 —ie]" (X and 0 rational functions of the x,'s),nothing can be said about X and 0-, except that they arebounded. It should be possible here also to write com-binations of contributions for which X or 0. have adefinite sign, and then discuss analyticity in the samemanner as we have done above. A calculation of theimaginary part along the lines outlined above is dificult,unless one has some knowledge about the appearanceof coincident poles. If some way of handling themcould be found, it seems very likely that the imaginarypart could be written in such a form that one wouldhave little difhculty in identifying it with the absorp-tive part. This would then give a complete proof of thedispersion relations in perturbation theory.

ACKNOWLEDGMENTS

The author wishes to thank Dr. M. L. Goldbergerand Dr. R. Oehme for suggesting the problem and forvaluable discussions.