Consequences of analyticity and unitarity for partial-wave amplitudes

ANNALS

OF

PHYSICS:

30,

446475

Consequences

A. P. The Enrico

Fermi Institute

(1964)

of Analyticity and Partial-Wave Amplitudes*

BALACHANDRAN~ for

Nuclear

AND Studies,

FRASK The

Unitarity

VON

University

for

HIPPEL$ of Chicago,

Chicago,

Illinois

Asymptotic and integral relationships between the functions associated with the unphysical and physical spectral functions of partial-wave amplitudes with I 2 1 for the scattering of spinless particles are developed. The results are used to show that: (a) The function associated with the unphysical singularities must have a closely limited asymptotic behavior if the dispersion relation is to be consistent. (b) The unphysical singularities determine precisely the leading asymptotic term in the function associated with the physical cut and in the absence of unphysical singularities, there can be no scattering. (c) In many cases, the unphysical singularities determine the asymptotic limits of the transmission factor and the real part of the phase shift. (d) If the transmission factor goes to zero (maximal inelasticity), the rate at which it approaches this limit is sometimes bounded by a known function. In these cases, the unphysical singularities and the transmission factor together determine the leading asymptotic term of the phase shift. (e) In some situations, a sign inconsistency in the asymptotic constraints implies the existence of poles in any N/D solution with residues whose signs prevent their interpretation as bound states. These poles are then “ghosts.” (f) In an exact partial-wave dispersion relation, at least when I 2 2, there is no simple way by which to determine, from the sign of the unphysical function at threshold or at infinity, whether the corresponding “force” is attractive or repulsive. This sign is fixed at these energies by general arguments. As a result of these constraints, one can conclude that the usual approximations of the unphysical cut in terms of the singularities of pole diagrams have a chance of satisfying consistency requirements only for the s- and p-waves with spin zero and one particle exchanges. For p-waves, the exchange of a spin one particle is required. Certain rapidly convergent integral identities are developed which can be used to test for the presence of ghosts or of bound states in N/D solutions or to test the accuracy of solutions. Finally, the relativistic Chew-Low model is used to demonstrate numerically the importance of some of the restrictions obtained. * This work supported by the t Present address: Department $ Present address: Laboratory York.

U. S. Atomic Energy of Physics, Syracuse of Nuclear Studies, 446

Commission. University, Syracuse, Cornell University,

New Ithaca,

York. New

PARTIAL-WAVE

DISPERSION

RELATIONS

447

I. INTRODUCTION

A partial-wave amplitude has two sets of singularities, “physical” and “unphysical.” The physical singularities are associated with the elastic and inelastic scattering cross sections of the two interacting part,icles and the unphysical singularities may be interpreted as providing t’he “forces” underlying their interaction. This cause-effect interpretation is incomplete, however, because the unphysical singularities themselves are functions of the physical cross sections in the crossed channels of the full scattering amplitude and for these, the physical singularities of the direct channel provide, in turn, the forces. Consequently, as can be seen in the Mandelstam representation for instance, there is no fundamental dist,inction in origin bet,ween physical and unphysical singularities. In the partial-wave amplitude, the distinction in the int#erpretation of the singularities is simply due to the asymmetry of the partial-wave projection operation. It leaves the discontinuities across one set of cuts, aside from kinematical factors, identifiable as partial-wave cross sections, while the remaining singularit#ies have no such simple relationship to the physical cross sections in bhe crossed channels. Although the partial-wave amplitudes do not explicitly preserve the symmetric relationship between the channels, the ease of the techniques available for the solution of their dispersion relations, when the unphysical singularities and inelasticities are given, has encouraged calculations aimed at finding simple models which produce qualitatively the bound states and resonances observed in certain angular momentum states. Ordinarily, an attempt is made in these calculations to replace the unphysical cut by a cut associated with simple particle exchange diagrams in the crossed channels which produce an attractive force for the appropriate quantum numbers. The Chew-Low model (1) for the I = ,3$,J = $4 pion-nucleon resonance was the first of many such model calculations. Recently, “bootstrap” calculations have attempted a development of this approach by taking a small set of partial waves and approximating their unphysica,l singularities by the projection of the physical singularities and poles of another small set of partial waves in a crossed channel. The reciprocal causeeffect relationship between the channels then results in self consistency requirements which may be solved for the coupling constants and ratios between particle and resonance masses (2). Thus partial-wave dispersion relations are continuing to be actively developed into useful analytical tools with which to test our understanding of elementary particle processes and this is perhaps an appropriate time to attempt to understand just how sophisticated more quantitative models of elementary particle interactions will have to become before general analyticity and unitarity proper-

448

BALACHANDRAN

AND

VON

HIPPEL

ties of partial-wave amplitudes can be preserved in their solutions. The purpose of this paper is to contribute to such an attempt’ with special emphasis upon the high energy and threshold properties of the terms associated with the different sets of singularities in the amplitude. Except in the last section, we will discuss only the scattering of spinless particles, although the generalization of these considerations to the situation where the particles have arbitrary spin seems to involve no serious difficulty. Also, only partial waves with 1 2 1 will be considered since the presence of a possible subtraction constant for s-waves makes their analysis more cumbersome. At the present level of sophistication, however, this limitation may be of no great consequence as few resonances occur with 1 = 0 due to the absence of a centrifugal barrier. For higher spin systems such as the pion-nucleon system, even s-wave amplitudes decrease sufficiently fast at infinity to require no subtraction and can therefore be included in this analysis. The paper is divided into seven sections: In Section II, the notation and formalism of partial-wave dispersion relations are summarized. In Section III, it is shown that there is always a precise cancellation of the leading asymptotic terms as s --$ co of what we shall call the “physical” and “unphysical” functions associated with the unitarity and unphysical cuts respectively. This cancellation requirement shows that there must always be an unphysical cut or poles for I 2 1 and that it is impossible to obtain nonzero scattering behavior for these partial waves using only an absorption paramet’er. That is, there must always be forces present. Since, as will be shown, the unitarity condition fixes the sign and within limits, the form of the leading asymptotic term of the physical function, the cancellation requirement fixes the sign and within corresponding limits, the leading asymptotic term of the unphysical function. These results increase our insight into the role of the cutoff in many model calculations where the unphysical function otherwise has a more singular behavior than that allowed by unitarity. It appears, in one phase shift analysis of n-N scattering at Ieast (5), that some lower partial waves may be approaching their asymptotic limits already at energies of the order of a Bev with the transmission factor going rapidly to zero. If this is true for general scattering situations, a phase shift analysis up to a few Bev scattering energy can be used to fix the leading asymptotic term of the exact unphysical function by emplo.ying t.he constraints which we derive in Section III.’ r A very careful analysis of partial-wave dispersion relations has already been made by Frye and Warnock (3) who give, in particular, a thorough discussion of the N/D formalism which is not studied here. Some of our results overlap theirs: these are indicated as such and are included here for completeness. Further developments are contained in ref. 4. 2 It may be that high energy elastic scattering experiments can also provide information

PL4RTIBL-W-4VE

DISPERSIOK

RELATIOXS

449

Section III concludes with a more detailed analysis of the asymptotic properties of the transmission factor and the real part of the phase shift. We find that there are cases in which the transmission factor cannot vanish too rapidly or approach zero at high energies and that in these cases, the leading asymptotic term of the phase shift is precisely fixed. In Section IV, we use the asymptotic constraints of Section III to show how, in the N/D method of solution, a partial-wave amplitude may be produced whose unphysical function satisfies our asymptotic bounds obtained from consistency requirements in Section III even though the input function in t,he initial dispersion relation does not. The N/D solution then has poles, not initially assumed, associated with the zeros of D. An initially inconsistent equation which may be corrected in this way can occur when the inelastic cross section becomes maximal or the real part of the phase shift approaches an integral multiple of T sufficiently rapidly at high energies. These new poles correct an asymptotic inconsistency in sign or a too rapid rate of decrease of the input function in the dispersion relation so that it becomes consistent with the conditions found necessary in Section III. For certain Z, the new poles may have the properties of bound states produced by the forces associated with the unphysical cut; we show, however, that for the remaining I, some of these poles would necessarily have residues of the wrong sign and would therefore be “ghosts” with no physical meaning. In Section V, the consequences of the requirement that a partial-wave amplitude have the correct threshold behavior are examined. It turns out that, for 1 1 2, the unphysical function is required to have the opposite sign at threshold to that which it must have asymptotically. This function in t,he neighborhood of the threshold is therefore nonzero and negative and cannot change to reflect the attractive or repulsive nature of the interaction. Since it is nonzero at threshold, if the unphysical hmction is used as a first “Born approximation” to the partial-wave amplitude, the associated phase shift does not go to zero sufficiently fast at threshold for Z 2 2. The distinction between attractive and repulsive interactions for systems without bound states seems to be related to the magnitude of the unphysical function at threshold and its oscillations above threshold, but no quantitative criterion has been obtained. In Section VI, we introduce a test for solutions of partial-wave dispersion relations. It may be used either to see whether singularities not assumed in the original dispersion relation have been produced in the solution (due to the about the asymptotic behavior of the transmission factor. Results obtained by Yamamoto (6) indicate that the rapid decrease with energy in fixed angle p-p scattering cross section observed by the Cornell-Brookhaven collaboration (7), if fitted by Orear’s exponential in the transverse momentum (8), can be explained if all the partial waves with impact parameters within a certain radius go to the same limit at high energies. The most plausible ccnnmon limit would be that of maximal inelasticity for which the partial-wave amplitudes approach an imaginary value independent of the real phase shifts.

450

BALACHANDRAN

AND

VOX

HIPPEL

presence of zeros of D in the N/D method for instance) or to test the accuracy of numerical solutions in different regions. In Section VII, we discuss a numerical solution of the relativistic Chew-Low model for the 3-3 pion-nucleon resonance. We analyze the procedure of Frautschi and Walecka (9) who altered the D-wave threshold behavior in order to satisfy the asymptotic consistency requirements discussed in Sections III and V. We illustrate the importance of the consistency requirements developed in the previous sections, even for such a simple model, by showing that the threshold alteration is equivalent to the introduction of a pole in the amplitude. Although the residue and location of this pole have been chosen by them to simplify the kinematical structure of the amplitude, this choice is in fact arbitrary and the position of the resonance depends sensitively upon the pole location. II.

THE

FORMALISM

OF

PARTIAL-WAVE

DISPERSION

RELATIONS

We restrict the following discussion to the scattering of two spinless particles of equal mass m. An example with spin and unequal mass is treated in Section VII. It is assumed that the inelastic threshold does not lie below the elastic threshold. The isospin index is omitted throughout since we do not require the scattering to occur in a definite isospin state. The partial-wave amplitude jr(s) is defined as the projection of the full scattering amplitude A[s, -2q’( 1 - cos 0 j] by the formula fi(S> = ; s:

d cos BPI (cos e)A[s, -2q2(1

Here, 9 is the center-of-mass momentum, energy related to q through

- cos e)l

(2.1)

s is the square of the center-of-mass

s = 4(q2 + m”)

and 8 is the scattering angle. For real s 2 4m2, the partial-wave amplitude fi(s) describes the physical scattering of the two particles. It may be expressed in terms of the related scattering matrix element X,(s) through the equation fi(S> = &

(Sds)

-

1)

(2.2a)

where (2.2b) Due to unitarity, 1Sl(s)l 5 1 for s 2 4m2. The matrix elemen SE(s) can therefore be parametrized by its magnitude r]l(s) and its phase 26r(.s) as

PARTIAL-WAVE

DISPERSIOK

St(s)

451

RELATIOBS

= m(s) exp P&(s)1

la.31

where, for s 2 4m2, 5 1

(2.4)

and 61(s) is real. The inelastic cross section a?(s) mission factor 7jt( s) :

is simply related to the trans-

0 6 m(s)

o;“(s)

= (21 + 1)(7r/q2)[1

- ?j?(s)]

(2.5)

lvhile the elastic cross section a?‘(s) depends upon both VI(S) and al(s) : &(s)

= (2Z + 1)(7r/q”)l

rll(s) exp [%S~(s>l -

11’

(2-G)

Equations (2.2) through (2.6) show that fl(s) must, in the presence of scattering, be complex for s >= 4m2. Since fl( s) is assumed to be a real analytic function, this implies that it has a cut P from 4m2 to 00along which fl(S - ie) = ji*(s + ie) where, as E-+ O+, s f it refer to either side of the cut andfl(s + &) corresponds to the fl(s) defined in Eq. (2.2a). In addition to this physical cut P, fl(s) will in general have an unphysical cut and poles. The location of this set of singularities will be referred to as U. For an amplitude obeying the Mandelst,am representation, the unphysical cut runs from - 00 to 4m2 - mo2where rno is the mass of the lightest system exchanged in the crossed channels. The only property of U which we shall use, however, is that it does not intersect the physical cut. From Eq. (2.3), it follows that as s --f CCalong the positive real axis,3’4 fds)

= O(1)

(2.7)

We assumethat this bound is in fact valid as s ---) w in any direction, an assumption which can be proved using the Phragmen-Lindeliif theorem if fi (s) is bounded by a polynomial as s -+ CCin any direction and is O(1) as s -+ 00 along the unphysical cut. This assumption implies the existence of a once substracted dispersion relation forfr(s). For 1 2 1, it is possible to avoid even one subtraction, however, if use is made of the threshold zeros of fl(s): .fl(S) = O(q”l) cut

as q2-+ 0

(2.8)

3 In what follows, the limit s ---) m refers to s tending to infinity along the unitarity unless otherwise stated. 4 The symbols 0, o, and - will be used in this paper in the usual way. Thus: (i) fl(s) = 0(4(s)) for some 4(s) as s + a if / fc(s) [ $ M 1 $(s) 1 for some fixed M as s 4 a. (ii) St(s) = 0(+(s)) as s + a if [f,(s) j < e / e(s) 1 for any e > 0 as s ---) a. (iii) j”~(.s) - @(s) as s ---f a if f~(s)/+(s) --f 1 as s ---f a.

452

BALACHANDRAN

AND

VON

HIPPEL

We assume this threshold behavior which holds unless the full amplitude A[s, t = -22p2(1 - cos 0)] has a singularity at t = 0 as s t 4m2. The s-wave amplitude is not affected by Eq. (2.8) ; it may still have one subtraction constant in its dispersion relation and therefore constitutes a special case. In this paper, the 1 = 0 partial wave will not be treated. Our results therefore apply directly only for I 2 1. The partial waves with I 2 1 are the most interesting for the study of resonances, anyway, as resonances seldom occur with 1 = 0.5 For 1 2 1, we construct a new amplitude hi(s) which decreasessufficiently rapidly at infinity so that it requires no subtraction: h(s) = flWh2

(2.9)

Then, by assumption, hi(s) = O(l/ s ) as s -+ 00 in any direction. Use of Eq. (2.8) shows that hl( s) has no new pole at s = 4m2. It has therefore the dispersion relation Re

hi(s)

=

W(S)

sjn

2&(s)

2p(s)qZ

1

&’

= a su

cbds’)

s’ -

s

(2.10a) s hlu(s) + Re hlP(s),

s 2 4m”

where, clearly, Im

hlP(S)

=

1

-

w(s)

cc@

a(s)

(2.10b)

2P(Sh?

and is equal to Im hi(s) for s 2 4m2. Here we have chosen Re hlP(s) and hi’(s) to represent the functions associated with the physical and unphysical singularities respectively. We have labeled the unphysical spectral function by c#J~( s). It will have d-functions if there are poles in hlu(s). The form of the physical spectral function is obtained from Eqs. (2.2a), (2.3), (2.9) and the real analyticity of h(s).

A decomposition of Re hlP(s) Frye and Warnock (3) :

which we will find useful has been suggested by

= Re hi’(s)

+ Re hlE(s),

s 2 4mP

5 For s-waves, quite special mechanisms seem necessary to replace the trapping effect of the centrifugal barrier. Dalitz and Tuan (IO) have discussed one such mechanism in which the s-wave resonance is dynamically interpretable as a bound state in a closed twobody channel which is weakly coupled to the open channel.

PARTIBL-WAVE

DISPERSIOS

453

RELATIONS

where Q is the inelastic threshold.6 This form is useful because we regard the spectral function &(s> for the unphysical cut and the transmission factor am as given. Therefore, both hi’ and Re h:(s) serve as input functions in the problem. The unknown function 82(s) has been isolated in the positive definite spectral function of hlE( s). For later convenience, we also define here the function Re hLv(s) : Re hlv(s) = hi’

+ Re hi’(s) (2.12)

where V runs over U as well as from sr to = and #l(s) takes on appropriate values in these ranges: h(s)

= 42(s),

s e u,

= 11- m(s)1 ’

%(s)f22 III.

CONSEQUENCES

OF

UNITARITY

FOR

(2.13)

s 2 SI. THE

ASYMPTOTIC

BEHAVIOR

In this section we will assume that hlu(s) and the spectral functions 41(s) and Im hlP(s) have, as s 3 cc, asymptotic forms which do not oscillate indefinitely. While there is no general proof for this assumption, it should be valid at least for ordinary model calculations. Most of the limitations which we shall obtain are simply due to the unitarity bounds. (3.la) and 0 s Im h2(s) = ImhiP(s)

5 -4 + 0 0 -1 s S

as s + =. By Eq. (3.lb), Im h,(s) = 0( l/ s) as s -+ m along P. If its derivative exists, is continuous and does not oscillate indefinitely as s --f m, Frye and Warneck (S) and Warnock (11) have shown that’ 6 Although, as a mnemonic device, the superscripts E and I have been attached to Re hr”(s) and Re hi’(s), it should be emphasized that their spectral functions do not correspond to the physical cross sections defined in Eqs. (2.5) and (2.6). 7 Professor Warnock has pointed out to us that there is a further assumption necessary to complete the proofs of this formula in refs. 3 and 11. It is necessary to assume essentially the absence of indefinite oscillations in the derivative of Im hi. See R. L. Warnock (to be published).

4.54

BAWCHANDRAN

Re h:(s)

= -i

.4ND

$ l’,

VON

HIPPEL

(3.2)

m ds’ Im h2(s’) + 0(1/s)

as s --f co along P. The assumptions about the behavior of Im hl( s) can probably be weakened and in any case should again be valid for model calculations. Since Im hi(s) in Eq. (3.2) is positive definite, the first term on the right side of Eq. (3.2) is never zero in the presence of scattering and is therefore the leading term of Re hlP(s).* Inserting Eq. (2.10b) into Eq. (3.2), we find, for the leading term

m’(s)

of Re hP(s),

(3.3) which has the bounds9 --- 4 In s 5 XIP(S) $ -a S’ Ii- s The left hand bound is attained when right hand bound is attained when the The parameter a is equal to zero if and the amplitude hi(s) is identically zero THE

UNPHYSICAL

a>0

vr( m ) = 1, cos 261( w ) = - 1 and the integral in Eq. (3.3) converges as s -+ 00. only if there is no scattering, in which case by analytic continuation.

FUNCTION

We now show how the asymptotic behavior of the unphysical function mines that of the physical function and of the partial-wave cross sections. The sum of hlu(s) and Re hlP(s) is, of course, Re hi(s) which cannot the unitarity bound (3.la). Consequently, the leading asymptot(ic term of hlu(s) must be small enough to be cancelled by Re hip(s) down to this This means, by the bound (3.4), that”

and x2’(s)

2 -2/s.

deterviolate XL’(S) order.

It will be shown below that XL~(S) has, in fact, the lower bound

8 When the integral in Eq. (3.2) converges as s --+ PJ, the assertion that the term eontaining it is the leading term of Re hlP(s) is equivalent to the assertion that the limit s ---f 00 and the integral over s’ in the defining equation (2.10a) changed. This can be justified with some mild restrictions on Im in Eq. (3.2) gets replaced by a term 0(1/s). (See Eq. (2.9) and cock’s review article (18) for a discussion of this point.) QHere, as in what follows, such inequalities and equalities terms are to be understood as accurate up to terms of lower 10 This bound is originally due to Chew and Mandelstam (13). to that used in the Pomeranchuk theorem (14).

for Re hlP(s) can be interhi(s) when the term 0(1/s) ff. in Hamilton and Woolinvolving order. The proof

the

asymptotic

is quite

similar

PA4RTIAL-WA4VE

DISPERSION

RELATIONS

455

YIU(S) 2 a/s, a>0 that xl”(s) has bounds of the same magnitude but with the opposite signs:

SO

as those of XI’(S)

as s ---f oc,

(3.5) This is just the condition for it to be always possible for xl’(s) and xIp(s) to cancel each other. Thus, unitarity limits the form of the leading asymptotic term of hi’(s) so that it decreases neither too slowly nor too rapidly. Among simple particle exchange diagrams Eq. (3.5) can be satisfied only for spin one particle exchange. In the following paragraphs, we examine in detail the consequences of particular asymptotic behaviors of xzu(s) within the range specified by Eq. (3.5) : (a) XI’(S)

= c In s/s, c # 0

Here, c must be positive since the necessarily negative leading term xIP(s) of Re hip(s) must precisely cancel xl”(s) in order for Re h,(s) to satisfy the bound (3.la). By Eq. (3.3), c = (2/7r)[l

- q1( a) cos 2&(m)]

and therefore, c attains its maximum value when vi( a, ) = 1, S1( P ) = (n + ],&)T. For this case, by Eqs. (2.5) and (2.6), we see that a;‘(s) reaches its maximum c$(s)

--+ (2Z + 1)47r/q2

(3.6)

as s -+ ‘13 and

Gys)/&(s)

+ 0.

(3.7)

The total partial-wave cross section also reaches its maximum. There is no experimental support for the possibility suggested in Eqs. (3.6) and (3.7). A more plausible ansatz than maximal elasticity is that for which the transparency factor 7$(s) goes to zero and the inelastic cross section becomes maximal :

g?(s) + Then, independently

(21

+ lM22

of 61( co ) , cT$(s) + (2Z + 1>d42.

This situation corresponds to c = 2/7r in case (a). Its plausibility is based upon the statistical effect of the opening of increasing numbers of inelastic channels at high energies.

456 We now consider Eq. (3.5) : (b) xi”(s)

BALACHANDRAN

AND

the asymptotic

behaviors

= o(ln s/s),

VOX

HIPPEL

lying between

but does not decrease so rapidly

the two

limits

in

as constant/s

An example is &s)

= c In)

c # 0,

S

O
Since, here too, xlP(s) must cancel xLU(s) in order that (3.la) be satisfied, x~~I(s) i 0. Further, since, by (3.3), ql( m ) cos 2&(( CQ) # 1 would imply that xl’(s) = c In s/s, c > 0, we must have adw 1 = 1,

al( 00) = n7r

where n is an integer. The leading asymptotic term of 1 - rll( a) cos 2&(s) will be determined by the requirement that xIu( s) be cancelled by xEp(s). For instance, if xl”(s) = c In as/s, the appropriate asymptotic form of 1 - ~]~(a) cos 2h1(s) which gives xl’(s) = --xl”(s) is, according to Eq. (3.3), 1 - Al

cos 261(s) N (c?ra/2) In”-‘s

If we write %d.s) = 1 - rbl(S), cos 2&(s) = 1 - @z(s),

h(s)

2 0

42(s)

2

(3.3) 0

both +1(s) and +2(s) must go to zero as s -+ CQ , and, as we saw in the example, the asymptotic form of XL’(S) determines the leading asymptotic t.erm of CJQ( a) + &(s). If an assumed +1(s) in ql( s) is too small to produce a term which will cancel xl’(s), the leading asymptotic term of +2(s) and therefore of c$~(s)is exactly fixed. If, on the other hand, &(s) is too large, the dispersion relation is inconsistent. For the cross sections, we have, q2&(s> + 0

If the leading asymptotic terms in this case(and also in case(c) below) are Z-independent, then there is a simple optical model interpretat.ion in which the transparency and refractive index go to unity at high energies. We finally consider the lower limit of Eq. (3.5) and prove at the same time that it is indeed a lower limit. This will also prove the result, quoted in the introduction, that if the unphysical function is zero, there can be no scattering. (c) $(s)

= a/s, a B 0

PARTIAT,-~AVE

DISPERSION

m3LL4mocx

457

In this case since, at least for small enough a, x1’(s) by itself satisfies the unitarity bound in (3.la), it is not immediately obvious that it must precisely cancel x1’(s). To prove that such a cancellation does indeed occur, it is necessary to consider the relationship between the asymptotic behavior of Re hl(~) and Im h,(s). The asymptotic form of Re hl( s) may be expressed, according to Eqs. (2.2), (2.3), and (2.9) as Re h(s)

-

27jl(s) sin 261(s)/s.

If cancellation does not force it to go to zero more rapidly than as constaut/‘s, then TJ~(‘L, ) sin Z&1( = ) # 0. This means that the necessary condition for Re hl’( s) to be 0( l/s), namely, vl( x: ) cos 261( 00) = 1, does not hold and a contradiction with (3.la) must result. Therefore, as in (a) and (b), Xl”(S)

= -xm

(3.9)

and since, except for the trivial case in which hi(s) E 0, the positive definiteness of Im hlP(s) results in a xlP(s) 2 -a/s, a > 0, the lower bound in Eq. (3.5) has also been proved. In Section VII, it is shown that the relativistic Chem-Low model violates the corresponding lower bound for spin 0 - spin 44 P-wave scattering if proper threshold behavior is assumed. Equation (3.9 ) means that the leading asymptotic term of the unphysical function fixes exactly the leading asymptotic term of the physical function. It would, of course, not make much sense if instead these leading terms associated with the cross sections were independent of the ‘Lforces.” It may be seen by Eq. (3.2) that case (c) requires the integral over Im hlP(s) to be bounded and hence implies that Im hlP(s) = 0(1/s In s). Here again sl( m) = 1, 61(m) = 12~. In terms of the parametrization in Eq. (3.8), &(s) and &.(s) are both o(l/ln s). From Eq (2.10), it may be seen that in case (c) where hi”(s) = 0( l/s), for any 4,1(s) which is sufficiently well-behaved,

Inconsistent asymptotic behavior can arise in a model in which all but one isospin state in the crossed channels are ignored. An example is the Chew-Low model of pion-nucleon interactions where the forces between the pion and nucleon are approximated by the exchange of a nucleon in the crossed channel and inelasticity is ignored. Since the relevant, elements of the crossing matrix have opposing signs in the I’ = $5 and T = ai channels, hlU(s) is guaranteed to have the incorrect sign asymptotically for one of these channels if J # $5’. (If J = vi+, the situation is complicated by the presence of the additional nucleon pole in the T = $$ channel.) The consequences of such an inconsistency when hlu(s) = 0( l/s) asymptotically will be discussed in Section IV.

458 THE

BALACHANDRAN TRANSMISSION

AND

VON

HIPPEL

FACTOR

Somewhat more insight into the role of the transmission factor may be obtained if the relationship between the functions Re hi”(s) and Re hlv(s) defined in Eqs. (2.11) and (2.12) are considered. The decomposition of Re hi(s) into these components has the advantage that the term containing the unknown function a,(s) has a positive definite spectral function and we have also separated out some of the dependence upon the transmission factor (which, by assumption, is known) through the positive definite spectral function of Re hlz(s). We earlier denoted by xL”( s) and xLP(s) the leading asymptotic terms of hl’( s) and Re hlP(s). The form of xLp(s) was actually written out for casesin which Im hlP( s) satisfies general continuity conditions in Eq. (3.3). We now denote the leading asymptotic term of Re hlE(s) and Re hlv(s) by xl”(s) and xl”(s). Because of the positive definiteness of Im hi”(s), xIE(s) may be written out in analogy to Eq. (3.3) as

XlE(s) =.-IIs8

dsf r]~(s’) sin2 &(s’)

s n-

4m=

(3.10)

PW)d2

From Eq. (3.10), it is clear that

XIE(S)1 -; !y

(3.11)

while the positive definiteness of the Im hlE(.s) requires the conjugate bound xm The unitarity

5 -a/s,

a>0

(3.12)

bound (3.la) requires that x?(s)

= -xl”(s)

(3.13)

where the argument producing this result for the case in which xlE(s) = 0( l/s) is again a two-step one according to which failure of the cancellation condition (3.13) would result in a Re hi(s) inconsistent with the assumed convergence of the integral in xzE(s) as s -+ ~0. Equation (3.13) results in the following bound for x~~(s)ll:

We have, for the leading asymptotic form of Re hi’(s), the negative definite function x:(s) : Xl’(S) = -11 The lower is also implicit

bound on x?‘(s) in that reference.

11 s &I [l - ds’>l s n-s4m2 2Pmi2

in Eq. (3.14) is contained in ref. 9 while the upper (Cf. Eq. (111.15) and the ensuing remarks there.)

bound

PARTIAL-WAVE

The definition

Re hi’(s)

DISPERSION

= hi’(s)

+ Re hi(s)

XIU(S) 2 (a/s)

459

RELATIONS

-

then yields, with

Eq. (3.14),

Xl’(S)

which, because of the negative definiteness of xl’(s), is a stronger constraint than that in Eq. (3.5) for xlU(s). Equations (3.10) through (3.14) can be used to explore the relationship between the asymptotic transmission factor and phase shift from a slightly different point of view than that explored above where the results were analyzed in terms of xLu(s). We will again consider separately the consequences of assuming different asymptotic behaviors for Re hi’(s) consistent with Eq. (3.14). (a)

xlV(s)

= c In s/s, c # 0

Equation that

(3.14) gives c > 0 and from Eqs. (3.10) and (3.13), it can be shown

This results

in the mutual

limitations

of c and vl( co ) : (3.15)

as well as al( m ) # nn where n is any integer. Thus, in this case, Levinson’s theorem (15) (proved for a class of potentials) which requires az( 00) = M necessarily does not hold.12 (b)

xl”(s)

= o(ln s/s),

but does not dewease so rapidly

as constant/s

A simple example here is J(s)

= c-

In OLs s ’

c > 0,

O
where the positiveness of c follows from Eq. (3.14). If Eq. (3.13) is to be satisfied, Eq. (3.10) shows that VI(S) sin’ 61(s) + 0 at a rate fixed by the rate of decrease of XI”(S). In our example, CTff

sin2 f%(s) w---n 4 1

w(s) This implies a constraint

as s + I2 It

cc, which might

corresponding

means, in particular,

be, however,

that

the

theorem

a-1~

to that found in Eq. (3.15) for case (a) :

that )72(s) cannot decrease still

holds

for

the

sum

of the

more

rapidly

eigenphases.

460

BALACHANDRAN

AND

VON

HIPPEL

than l/in ‘--Ors.13This type of constraint holds in general for any xl”(a) in case (b). If r]l(s) decreases more slowly than required by the bound sin* al(s) $ 1, then Levinson’s theorem may hold as sin* al(s) -+ 0.14 cc> xm

= 4s

Here a should again be greater than zero. In this case, Eq. (3.10) shows that ~~(8) sin* al(s) obeys the bound q)(s)

sin* 61(s) = o(l/ln

s)

as s -+ 00. Therefore, if ~~(a) decreases more slowly than l/In s or if vl( CC) # 0, 6r( co ) is constrained to be equal to na.14 We believe that the asymptotic constraints discussed in this section have not been emphasized sufficiently in the literature, We have shown that for calculations in which a particular set of unphysical singularities and inelasticities are assumed, their asymptotic behaviors are strongly correlated by consistency requirements which in turn put strong constraints upon the asymptotic behavior of al(s). Frye and Warnock (5) also have emphasized such constraints in their work. We close this section by pointing out that if consistency is violated only by terms of the order of constant/s, the N/D solution may still exist. It must, however, then contain new poles corresponding to zeros of D which will correct the inconsistency. If the inconsistency is by terms of an order larger than constant/s, however (as, for example, by a term c In OIs/s, cx > 0, c # 0), such terms cannot be compensated by pole contributions and the N/D solution will certainly not exist. IV.

RESTRICTIONS

UPON

BOUND

STATE

PARAMETERS

In Section III, we have studied some asymptotic constraints relating the phase shift and transmission factor to the unphysical function. In this section, we separate the bound state poles from the rest of the unphysical function in order to obtain a useful insight into the properties of the poles produced by the zeros of D in an N/D calculation. These poles are not included among the singularities in the initial dispersion relation and, therefore, can force Re hi” or hi”(s) to obey the consistency constraints found in Section III if consistency can be obtained merely by the addition of terms of the order of constant/s. The representation of hi(s) as such a ratio produces this flexibility because it preserves all the assumed properties of the amplitude except that it allows for the possibility of 13 This result may be compared with the result of Frye and Warnock (8) who also find restrictions upon the rate of decrease of vi(s) as conditions for the integral equation for LV in the N/D method to be of the Fredholm type. 14 See refs. 3 and 11 for a more detailed discussion of the conditions on qr(s) under which 61(m) = n7r.

Pe4RTI~L-WAVE

DISPERSION

461

RELATIOSS

new poles not assumed originally through zeros of D. The circumstances under which such zeros arise are not known in general, but they must certainly occur when the N/D method produces an amplitude consistent with unitarity from a dispersion relation not initially satisfying the associated constraints studied in Section III. An example of such a situation is that in which Re hi”(s) or hL”(s) is 0( l/s) asymptotically, but is either negative or o( l/s), thereby violating Eq. (3.14) or (3.5) respectively. The case for which Re hlV(.s) is 0( l/s) is the more inclusive and interesting since, by Eqs. (3.10) and (3.13), it requires only that rlr(s)sin261(s)beo(l/lns)ass~ 00 whichisguaranteedifeithervl(s)orsill’?61(s) or both go to zero sufficiently rapidly. When hl”(s) = 0( l/s), we are confined to a situation where both ~~(8) and cos 261(s) are 1 - 0(1/h s), a special case of the assumption Re hL”(s) = 0( l/s). As mentioned in Section III, inconsistent asymptotic behavior may arise in a model in which all but one isospin state in the crossed channels are ignored, since then, the elements of the crossing matrix and therefore the unphysical function can change sign for different isospin states in the direct channel. If the Rehl” is 0(1/s), poles will always be produced in some of t’he isospin states in the N/D solutions of such models in order to maintain consistency wit’h unitarity. In our approach to the question of poles produced in the N/D method when xl’(s) = 0( l/s), we separate out the pole contributions to xl’-(s). If the initially assumed hi’.(s) has n poles with residues xi (i = 1, 2, . . . , n), then the associated xlV( s) may be written as

Xl”(S) = XY’W + ; $I 2 xi where x;‘(s) is obtained from xl’(s) by removing all the pole contributions. When an N/D solution exists with no poles in addition to those initially assumed, Eqs. (3.13) and (3.10) give

or

2 xi + &s) = 1 s i=l

a

mclsr WCS’> sin’

4m2

Pw

&(s’) d2

It may be that the left hand side of Eq. (4.2), that is xl”(s), is not positive. Then, the equation is inconsistent and any N/D amplitude, if it exists, will necessarily contain m additional simple poles with residues Xi (i = n + 1, n + 2, .‘. ) n + m). WTe may rewrite the condition (4.2) so as to include the new poles which produce asymptotic consistency in the partial-wave dispersion relation asI I5 If poles of order tribute to Eq. (4.3).

higher

than

one are present

in the AT/D

solution,

they

will

not

con-

462

BALACHANDRAR’

AND

VOX

HIPPEL

?l+Wl jT+, Xi=.-~xlv(s) +; J,:,ds’ ‘ds’) sinz *‘(“) PWM2

(4.3)

The second term on the right side of Eq. (4.3) is positive definite. Therefore, ?Z+??Z JF+, Ai > -sxm (4.4) Thus, if sxl’( s) is negative definite or o( l/s), the new poles are obviously necessary and the sum of their residues must be positive. Therefore, at least for one i, equal to j say, Xj

> 0,

*iIn+

(4.5)

These poles may, in general, lie anywhere in the complex plane except along the physical cut where the unitarity condition prevents the presence of poles. However, they have a possible physical significance only if they lie along the positive s-axis and below the physical cut. If they lie in this interval, they might be interpretable as bound states produced as a result of the forces and their presence in an N/D solution may be permissible and interesting. The criterion for their physical interpretation is that their residues have the correct sign. What this correct sign of the residues is for a particular 1 may be most easily studied by examining the Feynmann diagram in which a bound state of the appropriate angular momentum occurs as an intermediate state in the direct channel. It turns out that the contribution to hi(s) coming from this diagram consists essentially of the product of the pole denominator [ml2 - s]-’ and the threshold factor p2”-l) multiplying a positive product g2 of numerical factors. Since the mass mi of the bound state is restricted to the interval 0 S rn? 5 4m2

(4.6)

and q2 is negative in this interval for the scattering of equal mass particles, the sign of the residue, which is equal to -g2p2’1-1) evaluated at s = mi2, is fixed as (-1)’ (16). Th is means that, for even 1, poles corresponding to physical bound states could add asymptotic terms to a Re hlV(s) inconsistent by terms 0( l/s), bringing it into consistency with (3.14) by satisfying the relationship (4.3). Of course, poles with residues of the correct sign, but not located in the interval (4.6), might also be produced. Furthermore, Eq. (4.4) does not guarantee that there are no poles with residues of the incorrect sign. For odd 1, if poles are produced which bring about the consistency of Re hi”(s) with Eq. (3.14) through Eq. (4.3), then Eqs. (4.4) and (4.5) show that some of the Xi must necessarily have a sign inconsistent with their interpretation as bound states. Such poles, as well as the ones not located in the interval (4.6), are ordinarily called “ghosts”

PARTIAL-WAVE

DISPERSIOK

463

RELATIONS

and would contribute to the analytic properties in a way having no interpretation in terms of physical processes.16 Further sum rules of the type (4.2) and (4.3) for the pole residues can be obtained for 1 2 2 and will be discussed briefly in the next section. T’. THRESHOLD

PROPERTIES AND

OF THEIR

PARTIAL-WAVE CONSEQUENCES

DISPERSION

RELATIONS

In defining hz(s), use was made of the fact that for E 2 1, .fl(s) (defined in Eq. (2.2a)) has an Z-fold zero at the eIastic threshold s = 4m2. Only one of these zeros was removed in the definition of hi(s) however. For 1 2 2, the remaining zeros put precise integral constraints upon Re hi(s), namely that hlu(s) must cancel Re hlP(s) to the necessary order just at threshold. Frye and Warnock (3) have observed that the alternative where hi’(s) and Re hi’(s) have these zeros separately is in contradiction with the positive definiteness of Im hlP(s). This can be easily seen as due to the fact that for s = 4m2, the integral in Eq. (2.10a) defining Re hlP(s) becomes positive definite. Consequently, at threshold, since hzu(4m2) = -Re

hlP(4m2),

122

(5.la)

hLu(4wz2) must be negative definite and must have just the opposite sign to that which by Eq. (3.5), it must have asymptotically. Similarly, Re hlv(s) defined in Eq. (2.12) is also negative definite at s = 4m2 since by Eqs. (2.11) and (2.12), it may be seen that Re hlT’(4m2) = -Re

hlE(4m2)

(5.lb)

and Re hlE(4m2) > 0. This sign of Re h{“(s) is also the negative of its sign as s -+ m by Eq. (3.14). The negative definiteness of hi’(s) or Re hlV(s) in the vicinity of the threshold allows an interesting insight into the role of the Born approximation in partialwave dispersion relations and its relation to these functions for syst,ems without bound states. The first Born approximation hlB( s) for a potential always reflects the sign of the force in the sense that if we set Re

hz(s)

=

ds)

sin

2&(s)

=

hzw,

s 2 4m2

(5.2)

2Pa12

the phase shift St(s) will be positive or negative according to whether the net force is attractive or repulsive. Since hlB(s) also has the threshold behavior 2(z-1), the phase shift 61(s) has the correct threshold behavior q2’+’ and the scat4 16 In case the reader becomes worried about an a priori distinction between even and odd 1 here, we should point out that the signs of the respective residues for the physical bound states and ghosts are functions of the amplit.ude used. For instance, these signs would be reversed for the amplitude hl(s)/pz for I 2 2.

464

BALACHANDRAN

AND

VON

HIPPEL

tering length defined as lim,z+O 6~(s)/qzE+’ exists and reflects by its sign the nature of the force. The function h?(s) has a left hand cut just as hl”(s) does. Therefore, hr”(s) has sometimes been interpreted as playing a role similar to the first Born approximation and the sign of the force has been associated with its sign. However, we have seen that hl”(s) cannot vanish at threshold. Therefore, the phase shift calculated by the formula (5.3) can vanish no faster than q3 as q t 0 and the corresponding scattering length does not exist for 1 2 2. Further, we also saw that for 1 2 2, hlu(s) < 0 at threshold. So, if we ignore the threshold behavior of al(.s) and just calculate its sign from Eq. (5.3), it will always be negative in the neighborhood of threshold and indicate a repulsive interaction. This is, of course, quite unlike the behavior of the first Born approximation of a potential. In models such as the Chew-Low model, the first Born approximation is often used for hlu(s). The above arguments show the inconsistency of this approach for 1 L 2. The identities (5.la, b) do suggest that a possible interpretation of the sign of the force may be based upon the magnitude of hlu(s) and Re hLv(s) at s = 4m2 for 2 L 2. Thus, for instance, as hl”(4m2) becomes increasingly negative, Re hlP(4m2) will become increasingly positive due to Eq. (5.la) or there will be increasing amount of scattering. There is a complication in interpreting this phenomenon as due to an increasingly attractive force, however, as a repulsive force of increasing range will also lead to precisely the same behavior of the scattering cross section. It is, however, probably true that a long range repulsive force is associated with an hlu( s) which oscillates along the unitarity cut. We can make this plausible by pointing out that the sign of the contribution to Re hip(s), from an isolated peak in lm hlP(s) caused by the phase shift going through a/2, changes from positive to negative as the energy increases through the peak. This is the behavior appropriate to the real part of the amplitude h,(s) = tl(s) sin 2Sl(s)/2p(s)q2 near a resonance for which the phase 61(s) increases through g/2. When the forces are repulsive, on the other hand, the phase decreases through ~/2 and the amplitude Re hi(s) must change from negative below to positive above the peak. Therefore, the contribution of the peak in Im hlP(s) to Re h*(s) must be cancelled by an even stronger oscillation of opposite sign of hl’( s) which “forces” the amplitude to reflect the repulsive nature of the interaction. Let us return to the discussion of threshold behavior. Proper threshold behavior for 1 2 2 may be guaranteed by working with a new amplitude g!‘)(s) from which all the threshold zeros have been removed:

PARTIAL-WAVE

DISPERSION

465

RELATIOSS

where fl(s) has been defined in Eq. (2.2a). However, while this.removes the necessity of cancellation of integrals at threshold, new integral cancellations must be introduced to maintain the unitarity bound upon go’): gp(s)

= 0(1/J)

(5.5)

as s --+ CC. In general, if we define also the amplitudes d”‘(s) then,ass-+

= fWlq2”,

nSl

(5.6)

00, g;“‘(s)

The dispersion

relation

for g$“‘(s)

= 0(1/s”)

(5.7)

reads, according to Eqs. (2.8) through

(2.13),

where V and $J~(s) have been defined in Eqs. (2.12) and (2.13). Equation (5.7) means that the leading term in the asymptotic expansion of Re g:“‘(s) is 0( l/s”). Then, since the spectral functions of g!“‘(s) are 0( l/s”), if they have a reasonable behavior at infinity, it is clear that Re g!“‘(s) may be expanded as Re g!“‘(s) where M?’

Equation ments

n-2 M(n) = c -&- + 0(1/s”) u=o

are the moments

(5.7) may therefore

1dslsIY 4G’) Q(n-1) 7r s v 4

be expressed in terms of the cancellation

I ; S,12 d;Sty ~4s’)

sin2 Us’)

require-

= o, (5.10)

Pw)q’2n

v = 0, 1, . . * ) n - 2,

n = 2,3, . a. ,1

There is also a condition for 1 = n - 1 which may be proved in precisely the same way as Eq. (3.13). Thus, when the integrals in Eq. (5.10) exist also for I = n - 1, we have,

466

BALACHANDRAN

AND

VON

HIPPEL

(5.11) n = 2,3, . . . , 1 It is readily verified that the sum rules for n < I are actually just linear combinations of the sum rules for n = 1. In view of the positive definiteness of the second integral in Eqs. (5.10) and (5.11), we have,

These inequalities are necessary conditions for the existence of solutions with the correct threshold behavior.17 They also contain sum rules for bound state parameters which are generalizations of those given in Eqs. (4.2) and (4.3). Thus, assuming that all the poles in N/D are simple and denoting the residue of the ith pole in the amplitude gi”’ (s) by kin’ and its position by sa, we find, corresponding to Eq. (4.3) the identities,

(5.13) v = 0, 1, . * * ) n - 1,

n = 2,3, * . - ,1

where p is the total number of new poles in the solution. IMead (4.4), we now get,

of the inequality

(5.14) v = 0, 1, . *. , n - 1,

n = 2,3, -es ,

These constraints may be used to elaborate the test, discussedin connection with Eq. (4.4), for the interpretation of the poles introduced by the N/D method into a partial-wave amplitude. According to our discussionthere, the sign of the bound state residue in h,(s) is (-1)’ and therefore, by Eq. (5.6), the sign of At”’ must be (-l)‘-“+’ if they are associated with physical states. (If, for example, n = 2, Xi”’ is related to the &defined in Eq. (4.1) through X2!‘)= Xi/q: where q: is the value of q2when evaluated at s = .si. Since qi2 < 0 and the sign of Xi was found to be ( - 1)I, the sign of X?’ should be ( - 1) Et’ which is just ( - 1) ‘--nf1 when n = 2.) Thus, unless the right hand side of Eq. (5.14) is negative for even 1 - n, the presence of ghosts in the N/D solution is guaranteed. There are also some constraints on the positions s; of the poles implicit in Eq. (5.14). I7 A more complete

analysis of these equations

is contained

in ref. 4. See also ref. f 7.

PARTIAL-WAVE

VI.

DISPERSION

A TEST

FOR

467

RELATIOXS

SOLUTIONS

We wish to describe here a numerical test which will fail only if a “solution” to a partial-wave dispersion relation contains singularities not assumed in the original dispersion relation or if the numerical solution itself is poor. The test has the advantage that the integrals which it involves can be made to converge as rapidly as one wishes in the high energy region where accuracy may not’ be of interest. The derivation of these equations rests on the observation that the function [h,(s)]” for an integer n is a real analytic function with singularities at the same locations as those of hz(.s).” For simplicity, we will assume that all the poles in h,(s) have been subtracted from it and call the resulting amplitude h,‘(s). The discontinuities across the cuts of [/~~‘(a)]” (which we assume are integrable) are given in terms of h,‘(s) by Disc [h,‘(s)]”

= [&‘(a + ie)]” -

[h,‘(s

- in)]”

(6.1)

If hi’(s) has complex cuts, s f in of course refer to the appropriate cuts. The dispersion relation for [h;(s)]” reads

sides of the

(6.2)

Since Re [hi’(s)]” must vanish : 1-

n-s lr

is O(C)

dStS~YDisc [hl’(s’)l” 2i

for large s, the first IL -

1 moments on the right side

+ f lI2 czs~s~y Disc b+;‘(s)Y = o (6.3) Y =

O,l,

...

,n

-

2

For large n - V, the integrals in Eq. (6.3) converge very rapidly. With decreasing n - V, the equations become more sensitive to regions where s is large. Equations (6.3) are thus quite flexible and should prove useful in testing the goodness of a numerical solution in different regions. Further, they can be used as analytic tools to test for the presence of unknown poles in an N/D solution not assumed to be present in the original equation. A way in which such poles might be produced by an inconsistency of the original dispersion relation has been discussed in Sections IV and V. Since Eq. (6.3) has been written down assuming the absence of these poles, the equalities will be destroyed in their presence. The sensitivity of I* The modulus

consideration theorem (18).

of such

a function

was suggested

by Landau’s

proof

of the maximum

468

BALACHANDRAN

AND

VON

HIPPEL

the moments to ‘1~- v may also give some idea of the distance of these poles from s = 0. Further equations of the form (6.3) can also be written down by considering functions such as [hl(~)/q”~-‘)I” for 1 2 2. VII.

APPROXIMATIONS

TO

THRESHOLD EXAMPLE

BEHAVIOR-A

NUMERICAL

In this section, we outline some results from a numerical solution of the relativistic Chew-Low model for the 3-3 pion-nucleon partial-wave dispersion relation. This calculation is relevant to the questions discussed in this paper because, as a result of the approximation of the forces by the nucleon pole in the crossed r--N channel, some of the consistency requirements discussed in Sections III and IV, which apply to this model, are violated. This inconsistency has been removed in the formulation of Frautschi and Walecka (9) by sacrificing the proper threshold behavior at the threshold of the 03,s unitarity cut which appears in the same amplitude as the P 3,2 cut when the dispersion relations are written in the w = s1’2 plane. In our treatment, we have slightly generalized Frautschi and Walecka’s method by making explicit the fact that they have introduced a pole in ignoring part of the D-wave threshold behavior and treating the position of this pole as a parameter. We test quantitatively in this way whether or not the position of the 3-3 resonance is insensitive to the incorrect treatment of the distant threshold. Our numerical method, matrix inversion by computer, is capable of considerably greater accuracy than the approximation made by Frautschi and Walecka, but we find that their approximation of the unphysical cuts by a small number of poles gives a surprisingly accurate solution to their equations for the position of the 3-3 resonance. Our conclusion about their analytic approximation to the amplitude is that the solution is quite sensitive to the location of the movable pole which we substitute for their fixed pole at the D-wave threshold. When this pole is moved by about 10 % of its distance from the Pa/2 threshold, the location of the resonance moves by as much as one pion mass. It thus appears that they have introduced a concealed parameter into the calculation as difficult to interpret as the cutoff in the static Chew-Low model, and have arbitrarily fixed its position to simplify the kinematical structure of the amphtude. The sensitivity of the results to this parameter should not be too surprising: the results of the static Chew calculation too were found similarly sensitive to the position of the cutoff even when this was also quite “distant” from the location of the resonance (IS). Thus, consistency requirements must be taken seriously and it appears that they can only be sacrificed by the introduction of explicit or hidden parameters. Once consistency requirements are satisfied, it may be that arguments about “distant singularities” will be more valid. Before discussing our results in detail, it will be necessary to summarize the formalism and the method of solution.

P.~IITI.Uz-WAVE

DISPERSION

469

RELkTIOXS

THE FORMALISM We follow the forrnulation of Frazer and Fulco (20) and Frautschi and Walecka (9). The kinenlatical singularities of the amplitude have been fully discussed by these authors and we will nlerely state here that the appropriate variable in which all the cuts and poles are associated with physical processes is w = sl’*, the total center-of-mass energy of the T-N system. We then work in ternis of t’he PJ,? anlplitude hl+(w

+ in) = (2w2/q3)

which has the MaeDowel tude : h1+(-w

exp [i&+(w)]

synlnletry

- iE> = -ii-(w

sin&+(w),

property

w 2 IIf + 1

(21) relating it to the D3,2 ampli-

+ ic)

= - ( 2w2/q3) exp [i&-(w)]

(7.1)

sin L( w ),

(7.2)

wZm.+l

Here we have taken the pion niass as our unit so that the nucleon niass becornes 6.72. The subscripts 1-t and 2 - denote the P3,2 and Dzj2 aniplitudes respectively. The isospin subscript has been suppressed. The elastic unitarity condition is expressed in terrns of the real phase shifts. The factor 2w2/q3 renioves the centrifugal barrier zeros at the P-wave threshold and corresponds to the l/p(s)q” of the hi(s) in Section II. Because the pion and nucleon masses are unequal, y2 is quite a complicated function of w: 2 = [w” -

(112+ l)‘][w?

-

(111 -

I)‘]

(7.3)

4W’

It nlay be seen that, besides the zeros at w = f (m + l), q2 has additional zeros atw = ~(m1)andadoublepoleatw = 0. The partial-wave projection of the pole tern1 in the crossed T-N channel gives a cut in the w-plane running along the entire imaginary axis, and two short cuts, one in the interval m

-

(l/m)

and the other along its reflection across t,hese cuts is Disc hl+(w)

= -2&g

5 w 5 (m” + 2)“’ through

[(w + m)’ -

(7.4)

the in~aginary axis. The discontinuity

l](w

-

m)z (7.5)

-

[(w - ?I$ -

l](w

where X=

w* -

d - 2 - 1, 2y*

+ 111); (3z2 -

1)

470

BALACHANDRAN

AND VON HIPPEL

and IJ’ is the pion-nucleon coupling constant. The symbol Eis equal to +l on the cuts along the upper imaginary axis and in the interval m - (l/m) s w s (m” + 2)l’* and - 1 along the reflections of these cuts through the origin. There are altogether five cuts: two physical and three unphysical. The dispersion relation rea,ds:

(7.6) wlm+l where (7.7) ZEROS OF THE AMPLITUDE

It may easily be verified that (7.6) is asymptotically a perfectly consistent equation. The unphysical function has an asymptotic behavior constant/w which, in analogy with our discussionin Section III, can be easily shown to imply that &+( = ) and SZ-( a > = nr. However, in defining $+( w ), we only took care of the threshold behavior at the P3p threshold. Strictly speaking, another zero at w = -m - 1 at the D-wave threshold of the crossed McDowell cut should be divided out if the coupled D3,2 partial wave is to be treated consistently. This threshold factor, as well as a zero at w = -m + 1 is divided out in the formalism of Frazer and Fulco by an extra factor [(w + m)” - 11.We now show that the resulting dispersion relation has no solution. In Section V, we noted that becauseof the positive definitiveness of Re hlP( s) at threshold, hl”(s) must be negative definite there for 1 2 2 in order to have a solution with the correct threshold zeros. Frye and Warnock (3) have pointed out, however, that in models such as this one, which take into account the contributions of only a finite number of partial waves in the crossed channels, hl”(s) has a threshold zero of order 1 - 1. Since we discussedin Section V only the spinless case, we will here recapitulate the argument briefly as it applies to the S-N scattering. We call the projection of the pole term in the crossed channel h:+(w). This function has the threshold behavior which the full amplitude $+(w) should have. Consequently the remaining term in hi+(w), namely h:+(w) must also have the proper threshold zeros. All of these zeros except for the extra zero at the D-wave threshold have been divided out in the definition of hi+(w), however. If this remaining zero is divided out, the dispersion relation for h:+(w) becomes

PARTL4L-WAVE

Re hr+;(w) [(w + n2)2 -

l]

DISPERSION

RELATIONS

471

= p --m-l dw, Im h2-( -w’ + ie) H s--m [(w’ - m)’ - l][w’ - w]

+ ; /I,.,

dw’

Im hl+(w’ + &) [(w’ + my - l](w’ - w] ’

(7.8) w 1 ‘717. + 1

Since Re h:+(w) is O(ln w,/w) as w 3 0~, the left-hand side of (7.8) is 0( In w/w”), But, by t,he positive definiteness of Im h2-(w + ie) and Im hl+(w + &) along the physical cuts, it is obvious that the first moment of the right-hand side of (7.8) does not vanish in the presence of scattering and this side is therefore 0( l/w). Consequently the equation for hi+(w) is inconsistent when all the zeros are taken into account. Erautschi and Walecka therefore used the amplitude hr+(w) for which only the Y3/2threshold zeros are taken into account. This assumption is equivalent to assumingpoles at the D,:, threshold and at w = -m + 1 in the exact amplitude. As has been mentioned in the introduction to this section, in our calculation, we have slightly generalized the amplitude used by Frautschi and Walecka so that it is consistent with exact threshold conditions and with unitarity limitations upon the asymptotic behavior. We did this by multiplying the amplitude assumed by Frazer aud Fulco by the factor ](w + w,)’ - 11, thereby cancelling the consequencesof their added threshold factor at infinity. Our factor corresponds to the assumption of a pair of poles at the points w = -w, & 1 in Frazer and Fulco’s amplitude. Explicitly, the amplitude which we used is + g1+(w + ie>= 2w2 -43 ‘(w [(w +

wp)’ - ‘I exp [i&+(w)] sin 61+(w), my - l] wlm+l

with g1+(-w

2w* [(w - w,)’ - l] - $6) = - exp [i&-(w)] p3 [(w - ?72)’- 11

sin L(w), (7.9b) wznz+l

METHOD

OF SOLUTION

We have used the customary N/D method of solution and set a+(w)

(7.10)

= N(w)lD(w)

where N(w) contains the unphysical cut U and D(w) contains the unitarity cuts. This leads to the coupled integral equations, N(w)

= ; /-,dw

I Disc hy+(w’) [(w’ + w,)* - 11 D(w’) 2i [(w’ + rn)” - l] GTZi

’

(7.11a)

472

D(w)

BALACHANDRAN

w--Qsm wQ)(d

Cl--

IF

-

+

-

w)

VON

HIPPEL

[(w’ + 972)” [(w’ + WJ” -

dw’

d-1

NW')

(w'

AND

[(w’ [(w’

- my - wp)’ -

11 11

I] N(-w’) I] (w’ +

(7.11b)

1

(w’ + w>

WQ)

where advantage has been taken of the elastic unitarity form of the amplitude along its physical cuts. The subtraction in D which normalizes it to unity at w = is permissible since only the ratio N/D is physically meaningful. When (7.11a) is substituted in (7.11b) and the order of integrations reversed (which is permissible because all the integrals are absolutely convergent), (7.11b) becomes

wQ

D(w)

= 1 +y/

u [K(w,

dw

w’)

, Disc h:+(w’) 2i

- H(w,

[(w’ + w,)” - 11 [(w’ + m)” - 11

w’>]D(w’>

where

K(w,w’)= ill ,jw”p”3 [cwn+ d” - l1 ’ 2Wu2 [(w”+ wp)’ - l] (w”-

m H(w,w’)= s dw”L2?y2!I3 [(WV [(w” mfl

wg wp)2 -

- w’), WQ) (wn - w)(wU

l] l]

1 . (W,, + wQ) (w”

(7.12) + w>(w”

+ w’>

Equation (7.12) was solved using the University of Chicago Computer Institute IBM 7090 digital computer. The functions K(w, w’) and H(w, w’) were evaluated at a grid of points along the cuts by replacing the integrations in Eq. (7.12) by summations up to wn = 120 and using the analytic forms N (w” H(w,

w’) E &

for wN > 120. Equation K(w,

w’) -

(7.13a)

H(w, w’) w -

s 120

’ 16(w - w’)

= 8(1202 - w”) ’ for D(w)

(w"

ln

+

w)(w"

(12OY - w* C (12O)Z - w“J w

now becomes a matrix D(wi)

= 1 + c

- w’)

1

gives, upon integration

W

The equation

- w)iwY

c-2 dwN

L(w<,

=

+

(7.13a)

w')

1 w#w’, ’

(7.13b)

w’

equation: wj)D(wj)

(7.14)

PSRTIAL-WAVE

where the summation L(Wi,

DISPERSIOS

is along the unphysical

Wj) = !I%$?

FIG.

circle

I

I

II 2

II 4

[(wi + w,)’ - 11 [(Wj + fn)2 - l]

7 Wj)l

I

tt ,, II 6m 8 wp (pion mosses) position

I 8

2. Dependence of the resonance position open circle is the result obtained in ref. 7.

FIG.

The

H(Wi

I

1. Dependence of the resonance is the result obtained in ref. 7.

I 6

cut and

Awj Disc hI+(wj) 2i

x K(WL,Wj)-

473

RELATIONS

upon

I

II IO

the

I 12 upon

the

pole

parameter

wP . The

open

I 14 pion-nucleon

coupling

constant.

4’74

BALACHANDRAN

AND VON HIPPEL

w. = m

0” J 1 m 7

’

8

Threshold CM Energy

I IO

9 ( pion

masses

)

FIG. 3. The dependence of the P3,2 phase shift upon w for different coupling strengths. For the experimental strength I$ = 15, the resonance becomes a just barely bound state.

Equation (7.14) was solved for D(w) aIong the unphysical cuts by matrix inversion and D(w) along the physical cut was then obtained directly by insertion in Eq. (7.14). For sets of (wi , wj) of increasing size, the convergence was rapid and gave a solution in good qualitative agreement with that obtained by Frautschi and Walecka who replaced each of the short cuts by one pole and the entire imaginary axis by four poles. Our convergence was essentially complete for ten poles when two of these poles were placed on each of the short cuts and the rest of the poles were placed at distances of 2.8, 10, 20.4, and 110.5 from the origin along the positive and negative imaginary axes. All poles were centered in their associated intervals. In this approximation, about 3 min. of actual machine running time was involved. RESULTS

Our most interesting result, the variation of the resonance position with w, is plotted in Fig. 1. The strong dependence of this position upon w, near the value w, = m used by Frautschi and Walecka is obvious. Their result is indicated by the open circle. It will be noticed that for w, = m, the resonance becomesa bound state with energy 7.7 in our calculation. For completeness we give a few of the other results which we have obtained. In Fig. 2, the variation of the resonance position when 20, is set equal to m and

PARTIAL-%VAVE

I~ISPEI~SIOS

475

IlELATIONS

the square of the coupling constant g2 (experimentally equal to 15) is weakened is shown and in Fig. 3 the phase shift 6,+(w) is plotted as a function of w with w, = m and for various g2. It was found that the singularities along the imaginary axis had a great effect in producing the zero of Re D associated with the resonance. ACKNOWLEDGMENTS We wish Corporation

to thank the for providing

Mathematical Analysis Department us with a matrix inversion program.

RECEIVEL):

April 28, 1964

of the

Lockheed

Aircraft

REFERENCES I. G. F. CHEW AND F. E. Low, Phys. Rev. 101, 1570 (1956). 2. See for example: G. F. CHEW, Proc. 1962 High-Energy C’onj. CERN, (Geneva, la&?), p. 525; Phys. Rev. Letters 9, 233 (1962); F. ZACHARIASEN AND C. ZEMACN, Phys. Rev. 128, 849 (1962); R. E. CUTKOSKT, Ann. Phys. (N. Y.) 23, 415 (1963); V. SINCH AND B. UDGAONKAR, Phys. Rev. 130, 1177 (1963); E. ABERS, F. ZACHARIASEN, AND C. ZEMACH, Phys. Rev. 132, 1831 (1963); R. H. CAPPS, Phys. Rev. Letters 10, 312 (1963); H. M. CHAN, P. C. DE CELLES, AND J. E. PATON, Phys. Rev. Letters 11, 521 (1963); P. CARRUTHERS, Phys. Rev. 133, B497 (1964). 8. G. FHYE AND R. L. WARNOCK, Phys. Rev. 130,478 (1963). 4. A. P. BALACHANDRAN, J. Math. Phys. 6, 614 (1964); “Application of the Method of &lomerits to Partial-wave Dispersion Relations I,” EFINS-64-U); Ann. Phys. (n;. Y.) (in press). 5. I.,. D. ROPER, Phys. Rev. Letters 12, 340 (1964). 6. K. YAMAMOTO, EFINS-64-11; Phys. Rev. 136, B567 (1964). 7. W. F. BAKER, E. W. JENKINS, A. L. READ, Ci. COCCONI, V. T. COCCONI, A. D. KRISCH, J. OREAR, R. RUBINSTEIN, D. B. SCARL, AND B. T. ULHICH, Whys. Rev. Letters 12, 132 (1964). &. J. OREAR, Phys. Rev. Letters 12, 112 (1964). g. 8. C. FRAUTSCHI AND J. 13. WALECKA, Phys. Rev. 120,1486 (1960). 10. R. H. DALITZ AND S. F. TUAN, Phys. Rev. Letters 2, 425 (1959). 11. R. L. WARNOCK, Phys. Rev. 131, 1320 (1963). 12. J. HAMILTON AND W. S. WOOLCOCK, Rev. Mod. Phys. 36, 737 (1963). 13. C;. F. CHEW AND S. MANDELSTAM, ~Vuovo Cimento 19, 752 (1962). 14. I. POMEKANCHUK, Zh. Experim. i Teor. Piz. 34, 725 (1958), (Translation: Soviei Phya.JETP 7, 499 (1958)). 15. N. LEVINSON, Kgl. Dan&e Videnskab. Selskab, Mat. Pys. Medd. 26 No. 9 (1949). See also R. L. WARNOCK (11) and the papers referred to therein. 16. See, for example, R. H. DALITZ, “Strange Particles and Strong Interactions,” pp. 154, 158. Oxford Univ. Press, London, 1962. 17. J. A. SHOHAT AND J. D. TAMARKIN, “The Problem of R,loments.” American filathemati. cal Society, New York, 1943. 18. E. T. COPSON, “An Introduction to the Theory of Functions of a Complex Variable,” p. 162. The English Language Book Society and Oxford Univ. Press, London, 1961, 19. G. F. CHEW, Phys. Rev. 96, 1669 (1954). 20. W. R. FRAZER AND J. R. Fc~co, Phys. Rev. 119, 1420 (1960). $1. S. W. MACI>OWELL, Phys. Rev. 116, 774 (1959).