PHVSI CAL REVIEW VOLUMLI i32, NUM BLIR 4 i S NOVKM n KR l ~C13

Formulation anti Numerical Solution of a Set of Dynamical Equations for theRegge Pole Parameters*

HUNG CHENG AND DAVID SHARPt

Caljforrtia Irtstetttte of Technology, Pasadena, Caleforrtsa

(Received 8 July 1963)

In a previous paper, the principles of analyticity and unitarity were shown to lead to a set of couplednonlinear integral equations for the Regge pole parameters. In this paper, we demonstrate, for both bosonand fermion trajectories, that these equations can be written in a very simple form which makes many oftheir mathematical properties transparent and permits their numerical solution by iteration. We thenproceed to carry out their numerical solution in a number of interesting cases. Because our equations areapproximate, we 6rst solved the equations in the potential-theory case, where our results could be com-pared with those obtained from the Schrodinger equation. The agreement in most cases is good. Then weturn to the determination of the Regge pole parameters which describe relativistic 2fw scattering at highenergies. Neglecting the inelastic contributions, we calculate the Pomeranchuk trajectory, the p-mesontrajectory, and the second vacuum trajectory 2". One notable result of this set of calculations is that thefunction Re n(t) for the Pomeranchulc trajectory, as determined by our equations, agrees well with theresults obtained by Foley et al from an. analysis of the s. p angular distributions in the range —0.8(BeV/c)'&t& —0.2(BeV/c)'. No spin-2 resonanace is found to lie on this trajectory. As for the p trajectory, we findthat Ir, (t), —0.8(BeV/c)'&t&0& is larger than 0.9 for a wide range of input parameters. The width of thep resonance, as determined from our equation, is several times larger than the experimental width. Thisprobably means that inelastic contributions must be included to obtain a correct value for the width.Finally, we outline various problems which remain to be investigated.

I. INTRODUCTION

'F Regge poles are to play an important role in under-- standing the properties of high-energy scattering

cross sections and of the many newly observed reso-nances, it appears essential to have a method for thedynamical determination of the Regge pole parameters.This belief is based on the following considerations:

(1) Recent measurements of the angular distributionsin srp and pp scattering' ' at high energies (15&s/2srtAI'(25) have been analyzed on the basis of a Reggepole model. The constancy of the total cross sections inthe two systems at these energies at first suggested thatone can assume that the dominant contribution to thecross sections comes from the Pomeranchuk trajectory.That this assumption cannot be correct in both cases,at least as far as the differential cross sections are con-cerned, is shown by the facts that almost no diffractionshrinking is observed in the srp system while consider-able shrinking is observed in the pp system. If thehypothesis that Regge poles dominate the high-energyscattering is still valid, it must mean that in the presentenergy range the analysis of the cross sections is compli-cated by the presence of several trajectories contributingin an important way. If this is the case, it would seem

*Work supported in part by the U. S. Atomic Energy Commis-sion. Part of the work reported here is included in a thesis to besubmitted by David H. Sharp to the California Institute of Tech-nology in partial ful6llment of the requirements for the degreeof Doctor of Philosophy.

)National Science Foundation Predoctoral Fellow, 1960—63.'K. J. Foley, S. J. Lindenbaum, W. A. Love, S. Ozaki, J. J.

Russell, and L. C. L. Yuan, Phys. Rev. Letters 10, 376 (1963).' C. C. Ting, L. W. Jones, and M. L. Perl, Phys. Rev. Letters9, 468 (1962).

~ A. N. Diddens, E. Lillethun, G. Manning, A. E. Taylor, T. G.Walker, and A. M. Wetherell, Phys. Rev. Letters 9, 108, 111(1962).

that reasonably clear cut experimental tests of the Reggepredictions about total cross sections and diffractionpeaks would be possible only if the Regge pole param-eters involved were known functions.(2) There is some reason to believe' that when multi-particle states are included in the analysis of relativisticscattering processes, the analyticity properties of the Smatrix in the J plane will be complicated by the pres-ence of cuts in addition to simple poles. This circum-stance would result in further ambiguities in the in-terpretation of experimental data, which would besomewhat alleviated if the pole parameters were known.

(3) It is a consequence of the Regge formalism that aset of resonances or bound states, all having the samequantum numbers including J parity, but having differ-ent values of J and occurring at different energies, willall lie along the same Regge trajectory' ' Ir(t). Theexistence of Regge cuts should not lead to any ambigu-ities in experimentally establishing the existence andproperties of any such resonances. For this reason,the possibility of grouping the new resonances in Reggefamilies, and of correlating a set of resonance parameterswith each other and with the observed total cross sec-tions and angular distributions remains as an interestingapplication of the Regge theory. To make good use ofthis possibility, however, it seems essential to have amethod with which to determine the Regge poleparameters.

In a previous paper, ' the authors made use of the

'S. Mandelstam, lecture given at the California Institute ofTechnology, January 1963, and private communication.' R. Blankenbecler and M. L. Goldberger, Phys. Rev. 126, 766(1962).

6 G. F. Chew and S. C. Frautschi, Phys. Rev. Letters 8, 41(1962).

I H. Cheng and D. Sharp, Ann. Phys. (N. Y.) 22, 481 (1963).

i854

DYNAMICAL EQUATIONS FOR REGGE POLE PARAMETERS 1855

analytic properties of the Regge pole parameters n(t)and r(t) plus the unitarity condition satisfied by thepartial-wave amplitude to derive a coupled set of inte-gral equations which determine the location n(t) andthe residue r(t) of a Regge pole as functions of t. Theequations obtained are approximate in that: (i) Onlytwo-body scattering processes are included, and (ii)the unitarity condition is employed in a form which isvalid only when Imn(t) is small. This latter conditionimplies that the influence of the coupling of one Reggepole ta another is neglected. Many aspects of theseequations were not understood at the time, in particular,the circumstances under which a unique solution mightexist were not known. Moreover, numerical solutionshad not been obtained and a quantitative idea of theusefulness or range of validity of the approximationsmade had not been arrived at.

It is our purpose in thi:s paper to discuss the propertiesof the above mentioned equations in considerably moredetail and to obtain numerical solutions of them inseveral interesting cases.

In Sec. Il, we show how to transform our originalset of equations so as to obtain an integral equationinvolving the single unknown function Imn(t). OnceImn(t) is obtained by solving this equation, we obtainRen(t) and the residue r(t) by performing simple integraltransforms. The derivation has been carried out forboson and fermion Regge trajectories.

Because the equations we use are approximate, it isvery desirable to compare our results for the Reggeparameters with those obtained in some rigorous way.This is possible only in potential theory. Consequently,in Sec. III, we specialize the equations derived in Sec.II to their nonrelativistic form. We also make in Sec.III a number of comments on the more formal mathe-matical properties of these equations, especially thoserelated to the uniqueness question.

In Sec. IV, we present our calculations of the Reggeparameters in the case of scattering in a single Yukawapotential of unit range. A wide variety of potentialstrengths are considered. These results are criticallycompared to those obtained by Ahmadzadeh, Burke, andTate' and by Lovelace and Masson. '

In Sec. V, we solve the equations for the case of rela-tivistic m~ scattering. In the case of mx scattering, wehave obtained the positions of the poles describingthe Pomeranchuk trajectory, the p-meson trajectory,and the second vacuum trajectory introduced byIgi.' The properties of the I' trajectory, as com-puted from our equations, agree well with those as-certained by Foley et al 'from an a.nalysis of m pangular distributions. We use our results ori the p-mesontrajectory to obtain n, (t), t &0, which governs the energy

' A. Ahmadzadeh, P. Burke, and C. Tate, Phys. Rev. 131, 1315(1963).' C. Lovela'ee and D. Masson, Nuovo Cimento 26, 472 (1962).

'P K. Igi,' Phys. Rev. Letters 9, 76 (1962).

dependence of cr -„—'o +„and of the correspondingangular distributions.

Finally, in Sec. VI, we summarize the conclusionsreached in this paper and outline a number of interestingproblems which remain to be investigated.

(2.1)

The situation is not so simple for the functionr(t)/q' &'&. The difficulty is that, because n(t) presumablyapproaches a negative quantity as t ~ &~, we cannotalways write a dispersion relation for r(t)/q' i'& in theonce-subtraced form of Eq. (2.1). We can avoid this

difhculty in the case of equal mass scattering by dealingwith the function r(t)e '~ &" But if w.e consider thescattering of particles of unequal mass, a dispersionrelation for r(t) would be complicated by the presenceof kinematic cuts coming from the factor q' &'~.

We have found that, for the purpose of obtainingequations for the Regge pole parameters from the princi-ples of analyticity and unitarity, it is wholly adequatesimply to know that r(t)/q' i'& is real analytic, and nooccasion will arise where it is necessary to have a dis-persion relation for r(t)/q'~"&. Therefore, 'we can avoidthe complications mentioned above.

We shall use the following kinematic variables:

t= 4''= total c.m. energy squared in t channel=m, '+mes+2(L(mo'+q') (mss+q') jl+q ); (2.2a)

and

"H. Cheng, Phys. Rev. 130, 1283 (1963).

II. FORMULATION OF A SET OF INTEGRALEQUATIONS FOR THE REGGE POLEPARAMETERS ' RELATIVISTIC CASE

Let us consider the relativistic scattering of twospinless particles a and b with masses m and mq. Weshall discuss the Regge poles of the partial-wave ampli-tude in the t channel for the reaction a+9 ~ a+b. Ourpurpose in this section is to derive a set of equationswhich will allow an approximate dynamical deterrnina-tion of the position n(t) of the Regge pole and of itsresidue r(t), which is equal to Res LA(n(t), t)j.

The authors have recently suggested' that the Reggeparameters can be determined from. .the principles ofanalyticity and unitarity. If crossing" of trajectories isneglected& then both n(t) and r(t)/q'~{'& are real analyticfunctions of t with branch cuts from Tp to ~ where Tpis the threshold value of t for the reaction a+b~a+b.

The function n(t) is assumed to have a behavior atinfinity which permits us to express its real analyticityby means of a dispersion relation of the 'simple form

H. CHENG AND D. SHARP

where q=c.m. momentum of an incoming or outgoingparticle.

The unitarity condition shall be written as

r(t) = Imn(t) (&o/q), t& Tp. (23)

Equation (2.3) is an approximate form of the unitaritycondition

LA (l, t) —A*(P,t) j/2i= (q/~)A*(l*, t)A(t, t) (2.4)

One point is worth noticing. We know that Imn(t) isalways real, but no may be complex, and at first sight,the right side of (2.10) may appear to be complex.However, we can easily see that we can replace ~p byRenp and the integral by its Cauchy principle part, andthen the right side of (2.10) is actually real.

Now let us determine the function F(t). We shall as-sume that r(t) has no poles, in which case F(t) is entirein t We. obtain from Eq. (2.10) that

which is valid when l=n(t) and Ren(Tp) & —-', .F. Zachariasen" has pointed out to us that we can

use Eq. (2.1), (2.3), and the real analyticity of r(t)/q' &o

to derive a very simple integral equation for Imn(t).This can be done as follows:

Since we know that the function r(t)/q' i'& ianalytic, we have

lnq' "Imn(t')dt'Imn(t) —~ liF(t)q' ' exp

faceTo t —to

=XF(t)q' &"', (2.11)

r*(t)=r(te)e "-&"& t& Tp.

s realwhere P is a constant. If we now require that Imn(t)vanishes as t ~ ~, then F(t), being entire, is a poly-

(2.5a) nomial of order rs satisfying the inequality

According to Eq. (2.3), r(t+) is real. Therefore,

r(t+) =r(t-)e-"-&'-i,

where t+=t+ie. Let us write

I( n(~—)

(2 5b) Moreover, from (2.9), we find

qp"'F (tp) =«(tp) .

(2.12)

(2.13)

Thus, we can infer that the general form of F(t) is

where F(t) is a rational function of t, and U(t) is ananalytic function of t cut from Tp to ~.The discontinu-ity of U(t) across the branch cut can be obtained from(2.5) and (2.6);

r(t,)-F(t) = II

go~~' '-~ to —t(2.14)

where the t; specify the location of the zeroes of r(t).We shall go into the question of zeroes, and the connec-tion between the number of zeroes and the asymptoticbehavior of Imn(t) more fully in Sec. III which treatsthe otential theor case. If we suppose that the tra-

or 1 zero, for example, then thee the form:

U(t+) U(t )=—2i Im—n(t) lnq'. (2.7)

Consequently, we can apply Cauchy's theorem to theanalytic function U(t) to find

t —tp" ln(q's/q') Imn(t')

dt (2.15)Xexp(t—tp)

"ln(q"/q') Imn(t')dt' , (2.9)

(t' —t) (t' —t,)

t' —topXexp)t t, q t qq'p-

=~(to)lGl Etp —tiJ Eqp)where the dispersion relation for n(t), Eq. (2.1), has

been used to replace n(t) in (2.6) by the right side of(2.1). Equations (2.3) and (2.9) then give t—tp

" ln(q's/q') Imn(t')Xexp- —dt'

t' —tp0

p(t—tp)

"ln(q") Imn(t') jectory of interest has 0resultant equations tak„(t'—t) (t'-t, )

where we have normalized U(t) so that U(tp)=0.Equations (2.1), (2.6), and (2.8) give

r(t) =F(t)q"

Imn(t) =—F(t)qs~o

t—tp"ln(q"/q') Imn(t')

dt'7r TQ t —) t —to

Xexp—

t& T,. (2.10)"F.Zachariasen (private communication). Professor Zacharia-

sen's observation has proven to be of decisive importance inextracting useful information from our equations.

t & Tp. (2.16)

Equations (2.15) and (2.16) are the desired results.What we have achieved is a decoupling of Eqs. (2.1)and (23) so as to obtain an integral equation involvingthe sirsgle unknown function Imn(t). Once we havesolved for Imn(t), we can obtain Ren(t) by performinga simple Hilbert transform. For t& Tp, r(t) is obtainedalgebraically from the unitarity condition (2.3) and,

DYNAMICAL EQUATIONS FOR REGGE POLE PARAMETERS

andrr+( —W)=n (W)

r ( W)= ——r (W).

(2.18)

(2.19)

The function n+(W) is real analytic and satisfies adispersion relation"

Imn+ (W') d W'O' —H/ p

(W) = (Wo)+(W' —W) (W' —W,)

Imn 5"' dW( )(2.20)

u r (W'+W)(W'+Wp)

~h~~e Wr= (m,+m )iss the total c.m. threshold of thesystem and 5'p the energy at the point of subtraction.In writing this dispersion relation, we have ignored thebranch cuts arising from the crossing" of the trajectorieso.+ and~ .

The unitarity condition satisfied by these amplitudesis of the form

W—TVO

Lf+(J W)-f+*(J*W)3/»= qf~*(J*,W)f~(J,W),W) Wr . (2.21)

for other values of t, it can be obtained from the dis-persion relation for r(t)e '~ &'& if m, =ms, and from Eq.(2.9) in the general case.

Equation (2.15) has many attractive features. Itincorporates the known threshold behavior of Imn(t),it exhibits the possible zeroes of r(t) explicitly, it has areasonable asymptotic behavior, and it is in a formwhich suggests the possibility of a solution by someiteration procedure. If this is the case, it is plausiblethat the solution is unique if Renp, r (tp) and the locationof any possible zeroes in r(t) is given. These and otherproperties of the integral equation (2.15) for Ima(t)are discussed in Sec. III. Here we shall proceed directlyto a derivation of the integral equations which governImn(t) in case a fermion Regge pole is exchanged in thet channel.

We shall consider Regge poles in the partial-waveamplitudes f~(J,W), W=gt, which describe transi-tions in states of definite total angular momentumJ=l+ ~~ and orbital angular momentum L. These ampli-tudes have the following symmetry" ":

f+(J, —W) = f (J,W—) . (2.17)

Accordingly, the Regge pole parameters connected withfg(J,W) satisfy" "

We approximate the unitarity condition (2.21) by

r~(W+ie) = (1/q) Imn~(W+ie), W) Wr. (2.22)

The functions f+(J,W) have kinematic singularitiesand, as a result, the functions r~(W) have kinematicsingularities. However, the functions

W f~(J,W)hp(J, W) =

E+m. (q')~ &

(2.23)

where E is the energy of the particle u

W'+m, '—m ps

do not have kinematic singularities. ' Therefore, thefunctions (W/E&m, )r~(W)/(q') &~& & are real analyticin the TV plane, with branch cuts from 8'z to andfrom — to —8'~. Consequently, we have

r~(W —ie) =r~(W —ie)e'~i&~+&ir ~~& &&. (2.24)

I.et us consider the amplitude h+(J,W). As before, wewrite

W r (W)

8+m, (q') +i~»=F(W) e~'~&, (2.25)

where F(W) is an entire function of W and U(W) isanalytic in 8' cut from W& to ~ and from —~ to—W~. We obtain

8'—8'pU(W) =— Im~ (W') lnq'-'

dW'r (W' —W) (W' —Wp)

Imer (W') lnq"d W'. (2.27)

w, (W'+ W) (W'+ Wp)

From (2.25), (2.26), and. (2.27) we get

U(W+ie) U(W——ie) = 2i Imo+(W+ie) —lnq',

W) Wr (2.26a)and

U(W+ie) —U(W —ie)=2i Imn (—W+ic) ln~q'~,W - —Wr. (2.26b)

The above equations give

r (W)=ln (q "/q')

Imrr+ (W') d W'

r (W' —W) (W' —Wp)

W—Wp " In(q"/q')

u r (W'+ W) (W'+ Wp)

8+m, ~'—~o(q') '~wo& —&F (W) exp—

'r w

Imn (W')dW', (2.28)

"S.W. MacDowell, Phys. Rev. 116, 774 (1960)."' W. Frazer and J. Fulco, Phys. Rev. 119, 1420 (1960)."V.N. Gribov, Zh. Eksperim. i Teor. Fiz. 43, 1329 (1962) Ltranslation: Soviet Phys. —JETP 16, 1080 (1963)."V. Singh, Phys. Rev. 129, 1889 (1963);N. Dombey (private communication).

H. CHENG AND D. SHARP

andpE+m,

Imu+(W) =I (q )I+&~»F (W) exp

W

ln (q"/q')

s „(W'—W) (W' —Wp)Imn+ (W') d W'

Also, we have

r (W)= —r (—W)

8'—8'p ln (q"/q')Imn (W')dW', W& Wr. (2.29)

sr (W'+W)(W'+Wp)

(E m.)—~(q')~+&~» —&F (—W) exp(W)

ln(q"/q')

r (W'+W)(W' —Wp)Imn+ (W') dW'

and

W+ Wp " ln(q'P/q')Imn (W')dW', (2.30)

sr (W' —W)(W'+Wp)

E—m, )Imn (W)= — j(q') +&~»F(—W) exp

W )ln (q "/q')

r (W'+W) (W' —Wp)Imn+ (W') dW'

ln(q'P/q')Imn (W')dW', W& Wr. (2.31)

r (W' —W) (W'+ Wp)

The function F(W) is a polynomial of order I in W, and satisfies

n+2n (~)(0,and, hence, can be written as

S'p 1 ~ t'W —Z, )F (W)=,r+(Wo) IIIEp+m, (qp')~+&~pi —'* ~=i gawp —Z,)

Wp E+m, ~ W—Z)Imu+(W) = q(q'/qp')"' " 'r+(Wo) II

W E,+m. '=i Wp —Z,)

where Z, are the zeroes of r+(W).We thus obtain the following set of coupled integral equations to solve for Imn+(W):

Xexp—W—Wp " q'P) Imn~(W') Imu (W')

ln —/

w q') ( ' —W) (W' —Wp) (W'+ ) (W'+ Wp)dW', W& W~, (2.32)

Wp E m. — ~ (Z;+W)Imn (W)= — q(q'/qp')~+&~» Ir~(wp)III

W E,+m.' .= &Z,—W,)

XexpImnp(W') Imn (W')W+ Wp " q")ln—

q') (W'+ W) (W' —Wp) (W' —W) (W'+ Wp)dW', W& Wr. (2.33)

Given Imn+(W), we may obtain the functions r+(W) from (2.28) and (2.30).

III. FORMULATION AND DISCUSSION OF A SET OF INTEGRAL EQUATIONSFOR THE REGGE PARAMETERS: POTENTIAL THEORY CASE

In this section we shall turn to the formulation of a set of integral equations for the Regge parameters in thecase when the scattering may be described by a superposition of Yukawa potentials. This topic is of interestbecause the most clear-cut check on the validity of our approximate form of the unitarity condition comes from acomparison of our results for the Regge parameters, computed for a single Yukawa potential, with the existingresults found by numerically solving the Schrodinger equation. Secondly, we can establish, in this case, severalrather precise theorems regarding the properties of the integral equation which we shall derive for Imn(v). Here,s =k' is the energy.

p O I E P A R A M E T E R SATlONS FOR REGGE POLDYNAM ICAI. EQUATI

(3 1)Writing

f 0 to ~ when crossing of trajec-r v P ("' are both real analytic functio f'ons of v cut from o ~,f th it it o diti dThe a roximate form or e u

'tories is neglected. The app

r(v)=Imn v v.

g the p

v —vpV a0~ V—V;

expvp

ln(v'/v)

P P P —Vo

ln(v'/v)

V—P V —

Vp

andv «v —v;

r v =r vo — — expvp

P—Po

r(v)/v =F(v)e &"'

in the relativistic case, w~ ~ ~

we obtain1'

same rocedure as inand appiyin

(3.2)

(3.3)

(3.4)and

v denotes the point of subtraction and v;, z—,f the eroes of r( ).~ m gives the location o

th elationship betwee nWe would now liklike to s ow t e red th asymptotic

e

the number of zeroes of r, v) an ebehavior of the Regge parameters.

) a (~)+n+~zImn(v) ~ p

(3.12)

is real, then the cut isw ic is a finite number. If v, is5,9,11from v, to ~. Now»"

Imn(v) ~ —(g'/2& v)

«(v) ~ —(g'/2 v)

and r(v) is real for v real. Thus,

I ( )~ v "'"i v~0. (3 5)IQlo,'v ~ p

E uation q. s o(34) shows that the nummber of zeroes is

r(v )v

and therefore,2

v lnv as v~ ~2 3.13DU(v)+AW(v) ~ —2i Imn(v) lnv as v —+ pp, .13—+0.

qbounded by

(3.6)

3.5) gives.5)'

the familiar thresho ld behavior ofh

'ht-hand planet eh Regge poles in t e rig

(3.7)Ren(0)) —-', .

actually independent of our approxi-'o( )e this statement, we reca a

'h f 0ywell as those coming from the crossin

rite

Thus, we have

1 AW(v')U()+W(.)-— "AU(v')

dvP —Pp

v ~ ~, (3.14)

s 3.8 (3.11), andwhich is a ni e nufi 't number. Equations ( . ),(3.14) together give

~ pa(~)+n V~rv~pWe can therefore w

r(v) = v &"&Ii(v)e~&"&eU(2 ) W'(V) 3.8

anal tic in v cut from 0 to pp and W(v)f i ofith cuts arising romis analytic in v wit

nd W'(v) can be wrjectories. The functions U(v) an v cin the form

(

w e'

b r of zeroes of r(v). Comparingwhere e is the num er o z3 12 and (3.15), we conclude that

(3.15)

«i= —n(~) —1. (3.16)

v —vp" 1 d U(v')

0 P —P V —Pp

U(v) =

andv —vp 6W(v') d v'

2

o)

nd BW(v) are the discontinuitieshe cut, and c is the contour of the() ()

rossin of trajectories.'g o o 'gstraight line" going from v, to v, an

1 0 W(v')W(v) - —— dv',

7l c P —Vo

and( )/ p]v p+'*e~&"), v)0. 3.17Imn(v)=L« vp vp p v

at v= 0, then we obtainIf we take the subtraction point at v=

' iPi '"e ("'dv' (3.17a)U(v) = —X— v' iPi e'

v(3.11)

itten'

ctor having n ( pa )= —1, there-

3 2 f r the lead'ng t a' ct rwhich has no zero. Then we m yma write

r v v —vp)" ln(v'/v)Vo V —Vo

H. CHEN 6 AN D D. SHARP

and

where

Imn(v) =7 v &o&+&e~&"', v&0,

X= lim0 v~(")

Xt and Xs but the same subtraction constant n(0). Thena change of variable shows that

U&&&~ (= U&s)~[ (3 19)

(Ly J &oi+t/sj t, D& ] &o)+t/s]

And, if we take the subtraction point at v= ~, then we

have "in(v'/v)U(v) = —— v' &"'+"e~&"'/dv' (3.17b)

7l 0 V v

and as a result

(&x"'~ (=&s&'&~ . (3.20)

(p g~io)+&/s// (p Ja&o)+t/s

In particular, we have

Imn(v) =Xv &"~+ie " v)0, o, i&) (eo ) —/xi&) (eo ) (3.21)where

r(v)X= lim

V(v)

We should like to point out several interesting conse-quences of Kqs. (3.17a) and (3.17b). We requireImn(0)=0 and Imn(eo)=0. Thus, the following in-

equalities have to be satisfied:

~(0)& —s

A

(3.18a)

(3.18b)

"M. Jell-Mann and F. Zachariasen (private communication).

If we take n(~) =—1, which is correct for the leadingtrajectory, then (3.17b) shows that Imn(v) has the cor-rect asymptotic form as v~ eo, providing X=g'/2.The solution of (3.17b), which is the equation havingthe subtraction point at v= ~, should thus be expectedto give a good approximation to n(v) and r(v) at largev. For the same reason, the solution of (3.17a), which

gives the correct threshold behavior, should approxi-mate n(v) and r(v) accurately at small v. It has beenpointed out to the authors'~ that (3.17b) is dependenton the coupling constant g' only and is independent ofthe range p of the potential. But (3.17b) is good onlyfor v large, and when the energy is large the mass canusually be neglected. In fa.ct, the asymptotic forms foro. (v) and P(v) have been shown to be independent ofp, . It is therefore natural that the range of the potentialdoes not enter in (3.17b). On the other hand, if wemake a subtraction at v=o, or at some point vo nearzero, then the solution will be accurate at low energyif the subtraction constants o/(vo) and r(vo) are bothsupplied. It should be noticed that if we make a subtrac-tion at some finite point vo, then the solution of (3.17)would not automatically give n(~)= —1, in disagree-ment with the known behavior of the trajectory. How-ever, in this case, we expect the solution to be accurateonly at low energy, and its behavior at v= ~ cannot, ingeneral, be expected to be given in a precisely correctway using our approximate equations.

Suppose we have two functions U&'&(v) and U&s&(v)

satisfying (3.17a) with different subtraction constants

Thus, we see that n (~ ) is determined by the subtractionconstant n(0) and is independent of X. Similarly, thesolutions of (3.17b) give the same a(0), if n(ee) is6xed and X is varied, and equalities similar to (3.19)and (3.20) hold.

Now let us turn to the question of the existence ofa solution of Eq. (3.17a) or (3.17b). First, it is clearthat because of Kqs. (3.19) and (3.20), if there is a solu-tion of Eq. (3.17a) for a certain 'A and n(0), then thereis always a solution of Eq. (3.17a) for an arbitrary X

and the same n(0). The same is true for (3.17b). Thequestion of existence and uniqueness of a solution de-pends on the subtraction constant &s(0) )or a(eo )j only.Secondly, (3.17a) does not have a solution for an arbi-trary n(0). A necessary condition for the existence of asolution of Eq. (3.17a) is Kq. (3.18a). For, if there is asolution of Eq. (3.17a), then U(0) =0, and the integralon the right side of (3.17a) does not converge at theend point v =0 unless (3.18a) is satisfied. Similarly, anecessary condition for the existence of a solution ofEq. (3.17b) is (3.18b).

Some precise theorems on the existence and unique-ness of the solutions of Eqs. (3.17a, b) can be proved"if certain conditions on the subtraction constants aresatisfied.

IV. REGGE POLE PARAMETERS FOR A SINGLEYUKAWA POTENTIAL. PRESENTATION

AND DISCUSSION OF RESULTS

The Regge pole parameters associated with a singleVukawa potential of unit range have been obtained byAhmadzadeh, Burke, and Tate~ and by Lovelace andMasson, ' for several potential strengths. Ahmadzadehet aL.' obtained their results by solving the Schrodingerequation numerically, while Lovelace and Masson' useda continued fraction technique applied to the known' ' "form (in potential theory) of the asymptotic (k' —+ ~)expansions of the Regge parameters.

A comparison of the Regge parameters as calculatedusing Eq. (3.2) of the preceding section with the resultsof Ahmadzadeh et aL.' and Lovelace and Masson'provides an important test of the accuracy of our ap-

' H. Cheng (to be published).

L4oig

D&NAMpc AL QUATfPNS FFOR RF

I

E pOL

, I

ARAM F IERS &86&

t l

-Io o.o ooI ~

OA

Rs .O2

l

0,25

0.2I

OA o,o oo

A*5

Fn. 2. Re&(v v

~ l.oo

v 2—2

8) Swith those of Ah madzadeh t al.e . (Ref.

AST

CS, v~0ICS, V~ Ip

- "0.50

0 o.o

1+v

ohio o~

Fro 1 Im (

O.IO-

0,05

2.0—

For th e case A=3 Fiate

ut as eveloped by D G.OIl-

n-Ramo-Vfooldrid eof particular value

'

erne F i e escription of th

tory re orto VAx)ldrid

( b1' h d).

0.5—

0 . 0.10 . 0 50

V

0.20 0.50

I+V

0,40 0.50

is workrsiis "/ (1+with those of

v, 0&v

g t gt A-D . ere, v=P

= 15.

o,so

I

compare

l

gle attract'h those of 'Ah

" ~ The result

A=0.05 03e &ukawa pote

' madzadeh et l.ts of this

case A —0o05and 0.~ for th

quat~ons, ~~bt~~~t'n stren

nd 03 He case A —'5 &ons have be

v, =0

proximation Th

ion obtain d he solutiofor the and 6 the 1

Ouris sectipn cpn

t fpuild jn thean accurac

ur procedure wpntains such a

.A ~ 8 F.

e strong-coup];y comparable

tp supPly thas to use the r 1

comparispn. p

. , igs. 7 and 8 w &P»g case. For th

e value pfiesu ts pf R f

' ur popre n a case

' ~

e case

point v, . pyo o(v) and p

„es &and 9 theagreement, hut

e'nwhichwe oht.

tho e actua]] p

~ at a subt'

e cprrect ual'&

ut the solutio

«nitar, 't~ d. . )' o t»n p(v, I f»ction cur~t

. q itative shap» still po~~e

and (2 g)iti» (31) Th

' o Imo(v~) and

a region aro~

d is quantitat'

q numericall ~9 fhenwesolve E

case A=) shoound the subtr

'tively ac-

e+(v) as funct') or the functio

qs. (3.2) and 6)ows the same

ction point. Thn ral featuress ]gs

ions if we used hy e obtained af

~ '0 and 0.30o the weaI

p functionse t eaveraast enex

a ter a few,

weland

a -couplin

O. , F . 1, th ~Pent much . n

Aa

or the Imn curve

ing ca s ith th

th'

is region from the co

Let us omconsider som'

u n t

men

stronual curves. In thome individu . n t

einte raccuracy with

h

g j. 4 010itat' 10 .T

togo

program could not h andle such ra'

t' ll th

rapid chang

that if we have oy e same.

' il

ra es ac-

t Iob g:()"""( is a real an

t

h rajectpry c dna ytic

0.10 O.RO

j ctones; andonsi ered

1

b

/

error.without si n'

I

Next, we c id

h signi6cant™

e consid

si n

I

g™f~

l

1y

—p

~p

and3oe o intermed' I.O

H. CHENG AND D. SHARP

n 2.0

'& '&

-0,4 -O.R

Aa I5

LIH——CS, v~ I

- 0.5

i+v~ \

0 ' O.R 0.4 0.6

- -05

- -1.0

0.8 IO

za e, Burke, and Tate. ' The curvbl o t hc s ape, and there is q

e ow-energy region.To summarize, we feel that the resul ts presented here

potential theory.e o owing general concluclusions in the case of

(I) Dure uatiq ons provide a dynamical deterfhR hhme ers w ic always gives the correct

Fro. 4. Ren(v) versus v/(1+v), —2(v( ~.tial strength A = 3.5 compared with those(R 1. 9). S io ofF' 3 Rec

curately, although this situation c

We do not understand wh our app o"'muc etter in the stron - an

n in t e intermediate cou line intermediate cou 1

p g ~ o ~ po-

in which the fie rst and second Re e tra'eoupling region is one

gg

th d '1 t' f

'g s rengt cross. If this is the ca

hf hu'ua ions or the Re e awe ave use themh m here are not correct. '"

&Aoj

.0

Fxo. 6. Ren(v) versus v/(1+v), —2(vtial strengths A =1 and 3 compare t ose of ovelace and

e cap ion of Fig. 5.

CS, vsvl

Av5

shape of the curves, and

(2) In case the coupling of the Re e

consider able ge goo uantitative

i era e range of energy.tive agreement over a

Fina 1ina~ly, we emphasize that our et t d fo th 1 d y ~h~~h hav

wi e interesting to see if our equation yields

A= I.8

ABT

aso 0.40 040 O,eo aro aeo aeo

Fxo. 5. Im». (v) versus v/(1+v), 0(v ( &0

d 'thth f L lYukawa potentials of unit ran e andTh o tof bt t

g ~ grac ion was vo=i. Here y= 2

) ~

—-- CS& v~=0 l

However we b, we have no evidence that such croin fact responsible for th d'

n interesting fact is that the curves Rc negative values fairl c

R () hhh hic as een proven rigorously. ' ' " ~A=5,., (.) 0~5'=' "' '

en(~) —2.66.g This result was

0 alo 0go O.30 0,40' ~~ 0.50 ' ILeo ohio OPO OPO GQ

Fxo. 7. Ima(v) versus v/(1+v), 0(v(»compared with those of Ahmadzadeh e

werYukawa potential of unit A~=uni rangeandstren thA=

ing e

d tth ' 01 g,nd p()=0.4. Here, v= ~

D YNAM I CAL E QUATIONS FOR REGGE PREGGE POLE PARAMETERS i863

j I are Oure procedure was to suppl as in uPPY 'P P

I I I I I I

Oe

A= },8

ABT

—-—CSI u*O4

—aso

ReaL

0 C12

V

}+2I I I I I

.e I.o

Ir..(oo )= —(4»'/3)(2n, (0)+1)r, 0,and solve the equation for Intn b an

'

output functia est eavera evg value of the input and

unctions as the next in ut. Thobt i dfo th di

Wee tspersion relation (2.1).

e have obtained solutions for 0-

30 —+30 mb. Thions or Ir (co) in the range

ig. , our results for Ren„(t), for —0.8(BeV/c '

FIG.ro. 8. Ren(v) versus v/(1+v1 —2potential strength 9=1 8 compared with th

. (R f. 8). S t o of F'

s or a

~ ~

cr»» 20 mb

accurate solutions f hzeroes and f h'

or ot er tra ecor w Ic n(0)) ——j ctories which have

0,80—

V. THE REGGE PRE

OLE PARAMETERS INELATIVISTIC me SCATTERING

In this section, we shall a l t

W..;ll,...;d„h.o iscusse asticmw scatterin athi h

t 'this scattering ofe contri utions to

nc u trajector whichyo e vacuum and n t'0~=i

jectory whichic gives a 2» (J= 1 I= 1'„~ ~=, and the p tra-

e s a also briefly discuss the second

(2

0.40

I

l.00I

0.50I

I I

090 tDO

I

I

2.00 4.00

t (BeVj)e

FzG. 10. omeranchuk trajectory. Ima t. Th th h g the in turves s own were calc

7 ~ C

e caption of Fig. 9.inpu

"0.&

- 0.4

Re aPL

P, tr»» v 20 mb

P, cr»» 15 mb

P,a~» v lo mb

(f&P

et.0, are compared with th

from an analysis of the ~ an uose o tained bP g

It will be noted that Ra ove-mentioned ran e of /

error Gags around thour e~„curves fal within the

p ye experimental oine ee that this agreement is reasonabl y

9.0 08I I

|L& -0.4I 'I I I

I I-a4

-0.2I I I

I I

a2 0

25+ v

0.4 0.6

IA

0 02I I

I

og 40

I I I I I I I I I I I I I I I I I I I

FIG. 9. PPomeranchuk trajector .X(B V/ )'P&t& . Th t}1

K

t.h' t t:() „(0)=( )=15 b d (c) '0'=c nv( )=1, » (~)=20mb.

vacuum trajectory I" introduced b I i."t of th e xm-scatterin cr

i a e, we ave concentrated hereth o itio (t) of th j

en occur in all reactions hquantum numbers.

s aving the proper

It'We shall 6rst discuiscuss the Pomeranchuk tra'e

ts Regge pole parameters will be de

the Pomeranchukw ic, as we have meentioned, couple

c u trajectory only to itself.

FiG. 11. Compari-son of Ren(t) for thePomeranchuk tra-jectory as computedin this paper (inputparameters; n„(0)= 1,a (~)=10, 15, 20mb) with Ren(t) asdetermined by Foleye$ al. (Ref. 1) froman analysis of 2r pangular distributionsfor incident momentain the range 7 BeV/cto 17 BeV/c and—0.80(BeV/c)'&t

& —0.20 (8eV/c)'.

t, 00

~ P

0.90

Re@po.eo

o~~ = 20 mb

CT7rg -" l5 mb

cr~~ = IO mb

ExperimentolPoints

I I I I I I I I I I I I I I I I I I I

I.O O.9 O.e Og O.6 O50.5 0.4 0.3 0.2 0 I

~-t (Bevj )

0.70

0.60

20 ' s relation is derived'

th'

e in e ppendix.

1864 H. CHENG AN D D. SHARP

TABLE I. A list of values of P~ and n~(0) for input parameterscI, (m, ')=1 and ImIs, (m, ')

Imn p(m, s)

0.0050.0100.0250.100

Ima p(m, s)r ——

P

m...(m,2)

3.79 m4.45 m6.35 m

189 m

0.9900.9830.9660.913

FIG. 12. P' trajectory and p-meson trajectory. Renp (t) versus tsReIsI, (t) versus t, —O.g(BeV/c)'&t& ~. The input parameter;were: (a) For the P'; u„(0)=0.50, o +'(s) (s= 20(BeV)s) = 5 mb;(b) for the p, Ren~(mp) =1, Imcx~(m~') =0.10.

significant because the region where the comparison ismade is very close to the subtraction point (f=0), whichis, of course, the region in which our results are mostreliable. Secondly, the results are not extremely sensi-tive to the value of the input parameter o (~).

The fact that our result for Reo,„agrees with that ofFoley et uI. ' naturally implies that it disagrees withReo.„as it has so far been determined from an analysisof E.V scattering data. ' '

We do not have a resolution of this puzzle. However,we do feel that it is more likely that the m V rather thanthe ES angular distributions are dominated by thePomeranchuk trajectory. The reason for this is thatthe statement that the Pomeranchuk trajectory domi-nates XX scattering, which depends on the assumptionof a cancellation of large contributions from the I" andoI (or perhaps p) trajectories, ""is much more model-dependent than the conjecture that it dominates ~Ãscattering.

We note from Fig. 9 that Ren„(f) does not passthrough 2 for any value of f. This implies that there isno spin-2 resonance on the Pomeranchuk trajectory.However, it may well be that the inclusion of inelasticstates could change this conclusion. Moreover, the re-gion where the curves peak Lt 2 or 3 (BeV/c)'] israther far away from the subtraction point, which mayresult in further inaccuracies.

We have also obtained solutions for the I"trajectory, 'assuming n„(0)= rsand, quite arbitrarily, that at f=0and s 20 (BeV)' it contributes 5 mb to the total srsr

cross section. Results are shown in Figs. 12 and 13. Itis of interest to note that Re+„ falls off considerablyfaster for negative t than does the Pomeranchuk tra-jectory, and that it reaches its peak value at a muchlower energy Lf 0.15 (BeV/c)'].

Lastly, we have obtain ed solutions for the p tra-jectory.

"F.Hadjioannou, R. J. N. Phillips, and W. Rarita, Phys. Rev.I.etters 9, 183 (1962)."D, H. Sharp and W. G. Wagner, Phys. Rev. 131,2224 (1963).

In this case, we solved for the trajectories in thefollowing way. We used the fact that Ren, (sr', ') = 1 andthen we chose a reasonable corresponding value ofImu, (ssr, '). We then obtained a set of solutions corre-sponding to these parameters, computed e„(str,') andchecked to see if the width as given by

wherests pI"

p Imrs——,(srs p')/e p (stp'),

e p (SSSp') =d ReIr, /df~

I

(5.1)

(5 2)

I 40/

I

0.50I

t.00I

2.00f (BeVjc)s

I

4.00

FIG. 13.P' trajectory and p-meson trajectory. Im~„.(t) versus t;Imn„(t) versus t, 0.0g(BeV/c)'&t& ~. These curves were calcu-lated with the input parameters listed in the caption of Fig. 12.

sI S. J. Lindenbaum (private communications).

came out correctly. Using this trial and error procedure,we were not able to find a set of parameters which gavea precisely correct value for the p width.

In Figs. 12 to 15, we display our results for ReIr, (t)and Imn, (t) for several values of the input parameterImn, (mp'). The corresponding values of the width andn, (0) are summarized in Table I.

It is to be noted that we obtain a very large value ofIr, (0), and that we find Ir, (t))0.90 for —0.80 (Be V/c)'&f:0 (Fig. 14). This fact is quite insensitive to themagnitude of the input parameter n, (sIr,'). Thus, wefeel that the numbers we obtain for n, (t), f =0 or f &0,may not be modified greatly by the inclusion of inelasticstates. The value of Ir, (t), —0.80 (BeV/c)'&t 0, thatwe find seems to be consistent with the recent observa-tions of Lindenbaum et al.23 who find little or no energydependence of the sr' angular distributions. This sug-

QUATIONS FOR REGGE POI F PARAMF TERS 1865

-0.6

—-- Imac aO.OIO

Imao 0.025—"—Imac ~0.005

0.6I

-0.6I

I

OP

-014I

I-0.4

-O.RI

I I-02 00 OR

I

0.5

v25+ v

Oee 0.6 OAI I I

I.O R.O 4.0—f (BeV/c)

Fio. 14. p-meson trajectory. Ren, (t) versus t, —0.8(BeV/e)'(t & ~. The three curves shown were calculated from the inputParameters: (a) Ren, (mrs) =1, Imn~(mrs) =0.005; (b) Renr(mre)=1, 1mn~(mre)=0. 010; (c) Renr(m )=1, Imnr(m )=0.025.

gests that 0.80 (u, (D) 1, while we find typicallypica yn, (0) 0.98. An analysis of earlier data" on the ~+ptotal cross sections, restricted to incident momentagreater than 10 BeV/c may also support the conclusionthat n, (0) is 0.80 or larger. "Moreover, one shouldbear in mind that the 0-„„data is so poor that a determi-nation of n, (0) 0 4from. that data is without muchstati. stical significance. "

The width of the p meson comes out too large b ayafactor of 5, assuming F, 100 MeV. This, no doubt,indicates that inelastic states must be included in orderto obtain the p width correctly. This is probably notsurprising in view of the results of other attempts todetermine the p width dynamically. "d

ere is an additional complication that enter ths eetermination of the widths from our Regge paramete

Thime ers.

is is the fact that e, (ttz, ') is a small difference of largequantities, and a very small percentage error 'n

R.Ren, ( s%) may result in very large errors in

ep ( 100%). This may account for some of the error inour value of the p width.

Finally, we would like to record that we foundRen~ (ao ) —0.66 (o = 15 mb); Ren„( eo ) —0.63and Ren, (oo ) —0.56 LImn, (m, ') =0.10$. We havemade no explicit assumption about the asymptoticbehavior except that Imn(t) ~ 0 as t ~ ~.

In the potential theory case, where a comparison withan exact solution is possible, the agreement is gratifyingin most instances.

We do not understand why, in the nonrelativisticcase, the accuracy of the solution obtained appears tobe poorest when the potential strength is in the range

A 3. It may mean that, for A in this range, theone-pole approximation is not adequate. Alternatively,this trajectory may cross another, in which case theequations must be formulated differently. "

In the relativistic case, the solutions obtained forthe Pomeranchuk trajectory agree quite well with theexperimental results of Foley et al.' Our solution fso u ions ort e p trajectory give a value of rrr( 0) which seems to beconsistent with recent measurements of I.indenbaumet a/. 23 However, we find that the width of the p reso-nance comes out too large. The inclusion of inelasticchannels should improve the results. But whether wecan achieve quantitatively accurate solutions by in-cluding just the two-body inelastic channels remains tobe seen.

The work carried out in this paper suggests a numbero interesting problems, both analytical and numerical,for further investigation.

We have mentioned the problem of including theinelastic channels in the equations, and finding theireffect on, for example, the p width.

~ ~

A critical test of our equations can come from adetermination of the fermion trajectories, using theequations derived in Sec. II. For example, if we supplythe mass of the nucleon and the vcr coupling strength,can we predict the position and width of the f» reso-nance that is believed to lie on the nucleon trajectory'If so, the same method can be used to discuss all themeson-baryon resonances.

We have noted (Sec. V) that the Pomeranchuk tra-jectory Ren~(t) that we obtain is in agreement with thatobtained from m.E scattering, but not with the resultsfrom EE scattering. This probably means that severalRegge poles contribute in an important way to EE scat-

0.50

VI. CONCLUSION

We have presented in this paper an approximatemethod for the dynamical determination of the Reggepole parameters. The equations we have derived f rh't is purpose are simple in structure and rather easy to

solve numerically.

~ S. J. Lindenbaum, W. A. Love, J. A. Niederer, S. Ozaki, J.J.Russell, and L. C. L. Yuan, Phys. Rev. Letters 7, 352 (1961).

» V. I. Lendyel and J. Mathews (private communication).'6 See, for example: F. Zachariasen and A. C. Zemach, Phys.

Rev. 128, 849 (1962), who Gnd 1'r 400 MeV after including thecontribution of w~ intermediate states.

I

0.50 4.00 —t (BeV/c)

Fio. 15. p-meson trajectory. Imn, (t) versus t, 0.08(BeV/e)'&t& cc. The three curves shown were calculated using the inputparameters listed in the caption of Fig. 14.

i866 H. CHENG AN D D. SHARP

tering at presently explored energies. To achieve acorrect understanding of high-energy .VA' and 3;Escattering, which, because of the spin structure of theamplitudes will involve the application of our equationsin the many-channel case, forms another interestingand important problem.

Turning now to analytical problems, it is clear thatan improvement of the one-pole approximation for thepartial-wave amplitude is very desirable. By includingthe correct contribution of a few nearby poles in thepartial-wave amplitude, one could probably obtainsatisfactory solutions in al/ instances for the potentialcase. A representation of the partial-wave amplitudessolely in terms of Regge pole parameters should helpsuch a formulation.

It would be interesting to learn if the zeroes of theresidue functions, which appear as input parameters inour equations in their present formulation, can bedetermined if several poles are coupled together. If thisis not the case, how can one determine the number andlocation of the zeroes af a given trajectory? The residuefunctions of the Pomeranchuk trajectory have a zerowhen n„passes through zero. Since we have not takenaccount of this fact in the numerical work carried outhere, it will be interesting to see how the solutions aremodi6ed if a zero is supplied.

Finally, we wish to repeat that one feature of disper-sion theory, the crossing symmetry, has so far beentotally neglected in our method. An application of thecrossing theorem may enable one to determine many ofthe subtraction constants in a self-consistent manner.Work in this direction is still lacking.

ACKNO%'LED GMENTS

The authors wish to thank Professor F. 7achariasenfor a number of very stimulating and fruitful discussions.

We are deeply grateful to Dr. G. J. Culler and Dr.B. D. Fr'ied of the Thomson-Ramo-Wooldridge Cor-poration for their generosity in making available tous the TRW on-line computing system, with the aid ofwhich all the numerical work reported here was carriedout. We also wish to thank them for a number of discus-sions on the numerical aspects of our problem.

It is also a pleasure to thank George Boyd for muchassistance in programming and for helpful discussionson the numerical analysis. The aid of Robert Bolmanwith the computer is also acknowledged.

Finally, we wish to thank Professor S.J.I.indenbaumand Dr. A. Ahmadzadeh, Dr. J. M. Cornwall, Dr. B.J.Kayser, Dr. V. Singh, and Dr. W. G. Wagner for manyinteresting conversations pertaining to this work.

2 sinn. n„(t)

where P„(&)= —~L2u„(&)+1jr„(t);ri, (/) =ResA(l, t)i,„(o.As $, u —+ ~, we have

4$ "~{t) e '~4M ~~(t)

„,„„,„„,, (. ..,.A ($,t) -+—

3 ~'"I'(cx, (&)+1) 2 sin~n„(t)

APPENDIX: RELATIONSHIP OF THE TOTAL CROSS SECTION TO THE RESIDUE FUNCTION"

The contribution of the Pomeranchuk trajectory to the ~~-scattering amplitude A ($,t) is

2$ —/ 2@

I1+ — +P~~(ii I

1+4m' — ( i—4m'

(A1)

(A2)

(A3)

This formula is valid for all $ and u. Now let us consider the physical region in the $ channel; $ ~ +~, t—4m '(0and fixed. Then u= ~u~e+' . Equation (A2) then gives

r(& (~)+1) — 4$ -a~(ti —e (+a~(t)+1——

+ ($ ])—1p (()e (~a@(l)—m'"I'(n, (t) +1)

~

t—4m. '~

2 sin7rn„(t)

The amplitude A ($,t) is related to the total or~-scattering cross section by

Now0 ($) = (16m./$) ImA ($,0) . (A4)

r( „(t)+-',)ImA ($A= ~P„(t) e

x-'('I'(n, (t)+1)Evaluating (A5) at t= 0 and setting n~(0) = 1, we find

/

t —4m. '/

or~-(~)= (4~/3) p. (o)

= —4n'r„(0). (A6)

A similar derivation, of course, applies to any other trajectory.27 We wish to thank Dr. W. G. Wagner for a helpful discussion of the points covered here,