The partial wave amplitude for high-energy pion-pion scattering (III)

Nuclear Physics 64 (1965) 685--693; (~) North-Holland Publishing Co., Amsterdam Not to

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T H E PARTIAL WAVE A M P L I T U D E FOR HIGH-ENERGY P I O N - P I O N SCATTERING (HI) P. OLESEN

Universitetets lnstitut for Teoretisk Fysik, Copenhagen Received 21 August 1964. Abstract: In the previous papers an approach to the high-energy behaviour o f the partial wave amplitudes A~l(s) has been developed, and it was found that A~l(s) decreases logarithmically with the energy. In the present paper the high-energy parameter ~,~I, defined through Azl(s) --~ i b,~I log s] -1 for s --~ oo, is investigated as a function o f I and L It is possible to express ~,z! entirely in terms of low-energy parameters (which in principle can be obtained from experiments). The behaviour for 1 --~ oo o f ~zI is investigated in a simple model, where the low-energy region is so small that it only contains the S-wave resonance COABC.It turns out that 1/~I decreases exponentially and that ~ I is entirely due to isospin zero exchange; ~zl is independent o f L

1. Introduction

In two previous papers a, 2) (henceforth to be referred to as I and II, respectively) an approach to high, energy strong interaction physics has been developed. The basic assumptions are maximal analyticity in the energy-momentum variables and the diffraction picture, i.e. the assumption of a purely absorptive high-energy amplitude. For simplicity we have considered pion-pion scattering. The method used is an N/D approach (see I and appendix 2 of II), which yields an integral equation for the asymptotic behaviour of the inelasticity Rtt(v) (the ratio of the inelastic to the elastic partial wave cross section). From this equation it was possible to find the asymptotic behaviour of RI(v) from Re A[(v) and N[(v) in the asymptotic region. The quantity Re A[(v) was found by an argument of the Pomeranchuk type and N[(v) was treated by an effective range approximation, assuming a "sufficient peripheral" interaction. The resulting expression for the partial wave amplitude is (see eq. (II.33))

A[(v) =

{~/(7~)2v -1 log 2 v - i7~ log v}- 1.

(1)

Here, ~ and ~[ are some parameters depending on the angular momentum l and the isobaric spin L Thus, what we have reached so far is the determination of the dependence of A~(v) on the energy variable v (the square of the magnitude of the centreof-mass momentum). For v sufficiently large eq. (1) becomes i A~(v) ~ - 7[ log v The parameter of most interest in eq. (1) is therefore 7[. 685

(2)

686

P. OLESEN

In the present paper we want to study the dependence of ~[ on the variables l and L The general philosophy behind this calculation has been outlined in I; what we want is to express 7[ in terms of low-energy input parameters (we assume that we have a sufficient knowledge of the low-energy region from experiments). In II we proposed to use the N/D equations (see sect. 5 of II) in the practical calculations. However, since the N[D equations with the boundary condition N[(v) ~ N[(oo) ~ 0 for v ~ co are rather complicated (see appendix 2 of II) it is much more convenient to work directly from the partial wave dispersion relations themselves. In doing so one has to specify precisely what one means by the concept "low-energy resonance region". Here we adopt the following picture. There exists at least one resonance at each value of I for l in the range from 0 to L. For l > L there are no resonances in the region v < vz, where A[(v) simply vanishes. A resonance is described completely by a Breit-Wigner formula. It is of course obvious that such a model is somewhat crude. On the other hand, since we are calculating the high-energy behaviour of A[(v), we do not expect that small errors in the low-energy input information can have any great influence. The general calculations are carried out in sect. 2. Unfortunately, the resulting expressions are still rather complicated; in order to get an insight into the structure of our results we have therefore considered the mathematical details in a very simple model of the low-energy resonance region. These calculations and estimates are given in sect. 3.

2. A General Expression for the Parameter y/ The unsubtracted partial wave dispersion relation for AI(v) is

lf-

PfiO

Re Al(v) = ~ _~odv' [AAl(v;)]rav,_v+ ~

dr' Imv,_vA~(v'),

(3)

where [AA[(V)]LH is the discontinuity in A[(v) across the left-hand unphysioal cut. In the following we shall split up the range of integration in eq. (3) into different pieces according to our actual knowledge of the weight functions in eq. (3). We start by considering the quantity 1 ~ -I

EI(O : - |

dr'

71:4 --VL

(4) VP~V

Here, vL is a separation point which can be taken as the point where the existence of non-vanishing double spectral functions forces the partial wave expansion of [dA[(v)lra to diverge. We can therefore use the weU-known formula 3)

- YJO

dv'P~ 1 + 2

Im~z 1,'

(._,1, v', x + 2 - 7 - )

(5)

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687

with 2

Im ~x(v, x) = ~ (21'+ 1) E ctn' Im ls=O

A[:(v)Pr(x ).

(6)

I'=O

Here, a n, is the crossing matrix

~w

1

=

.

--1

(7)

~--

For convenience, we use the zero-width approximation in the form given by Balks 4) Im

Al(v) = 7r(vl)tF[,~(v- vl),

(8)

where vl is the position of the resonance with the quantum numbers l and I, and Fl is the half-width. From eqs. (5) and (6) we obtain L

[AA[(v)]L. = _n ~] (2/'+ 1) V 1'=0

x ~ ctn,Ps 1 + 2 I'=0

V

( v+q

× (~:)Vrl:Pv 1+ 2 vl: / '

(9)

where v > -VL. The quantity L is the highest angular momentum for which a resonance can occur. If we insert this expression into eq. (4) we obtain L

2

El(v) = E (2I'+ 1) E =,,,(vl:)"rl:t,,,(v, ~l:) 1'=0

(lO)

I'=0

-t

Pt ( 1 + 2 Y+I~

Iw(v,y) = fv-,L dr'

+ l1/ . v'(v'--v)v' ] P,, (1 + 2 v'Y

(10a)

The integral Iu,(V, y) can be written in the form

Iw(v, y) = l fl-2fr+t)/'I'dzPl(z) Pr ( z ( y + 2 ) + 3y+2~ J1-2c,+1,

x-z

(11)

7'

x = 1+2 y+I

(lla)

V

Eqs. (10) and (11) provide us with an expression for the contributions from the lowenergy part of the left-hand cut. In deriving these results we have assumed that vL is so large that the range of integration includes the L resonances. Otherwise, we have to replace the upper limit in the sum over 1' in eq. (10) by a lower number than L.

688

P. OLESEN

We next investigate the contributions from the high-energy parts of the left- and right-hand cuts. We have to be sure that we get the right threshold behaviour. Let us consider the function A[(,)

=

1--

+1f~(1-~) \

v'-v

(12)

For v ~ +__oo the function A[(v) behaves at most like v1-1. In addition, Al(v) has no branch cuts, and we assume that it has no poles. Thus, A[(v) is holomorphic in the entire plane, and from Cauchy's inequality it follows that A[(v) is a polynomial of the maximal degree I - 1 . Thus we have Re

A[(v) = E~(v)+ F[(v)+ v-zf - , L [~V')]LH

+

1~1 l " o o

dr'

Al(v') +A[(v),

(13)

F[(v)= P [~"dv' Im A/(v')

(14)

-/

dr' Im

wh~e 7~JO

V~--V

For v ~ 0 the correct threshold behaviour is that Re A[(v) is proportional to vz; we assume that E[(v) and F[(v) have this behaviour. Thus the polynomial A[(v), which is of degree l - 1, must be cancelled by the other terms in eq. (17). Thus, if we keep only the terms of the degree vt the function A[(v) can be omitted from eq. (17). In this manner we get

Re A[(v) ~ El(v) + F[(~) + -'1-

v"(v'-v)

v'/,~o Im A / ( v ' ) , ,

--|

a,.

- - Q V ,

¢'(v'-v)

(lS)

where the equality sign should be understood in the sense described above. The remaining contribution to Re A~(v), namely F~(v) defined in eq. (14), is treated in a similar way. Thus we replace eq. (14) by the equation

F[(v) ~ v'~Jor', dr' Im A[(v') '

(16)

where the "cancelling polynomial" is emitted. In the following we are interested in

PION-PION

SCATTERING

(IH)

689

the threshold behaviour, and it is therefore reasonable to use the zero-width approximation (8) in eq. (16). In this way we obtain

Fl(v) ,~ v'F[/v[,

v ~ 0.

(17)

The remaining work is then straightforward. In the last term of eq. (15) we use the result (2) obtained in II and interpolate in such a way that we get the right expression for Im A~(v) at v = vx. Thus we write Im

A[(v) = f l(v) Im A[(VR)

[

l

1 1 + Y7 log v

/2(v) lo-~x_i '

(18) v > YR.

Here f~(v) are some "convergence factors" which go to 1 for v ~ vR and which go to zero faster than 1/logv for v --, oo. Since it was shown in II that [AA[(V)]Ln has the same behaviour on the high-energy part of the left-hand cut as the behaviour o f Lm A[(v) on the high-energy part of the right-hand cut, we use as a first approximation

1{ I

EaA;(~)],.,, -- r-I ioglvl

f3(v) 1

(19)

1o-0-ff-~d+A(@EaAI(--VL)],.,,,

where the quantities f3, 4 are defined in the same way as the f l , 2. We emphasize that this expression is only a first-order expression; actually we know (see II) that there are strong medium-range forces. However, the specific form given in eq. (19)is not crucial for the following arguments. For the high-energy part G[(v) we thus obtain . G~(v) I f -- "Ldv,[AAtt(V')]LH + _1~ oOdv, ImA~(v') 7CJ -o0

Vt -

{f +

-

V

7~J vR

V

dv' ( 1 v'~+ x \log I¢1

v~_ Im Arty "~( dv'fl(v -- ) " "J,R ¢,+1

{

Vr~

+[

AA",(--VL)]LH

L

'f*( )


¢o d~ ~ , ~

.

(20)

The last term in eq. (20) depends in a very specific way on how the interpolation is performed; there are other (and reasonable) possibilities. However, we assume that vx is so large that eq. (2) is a good approximation for v a little above vx. Therefore fl(v) and f4(v ) are defined in such a way that the last term in eq. (20) vanishes. Physically, this implies that the parameter y[ is the only parameter of interest in the high-energy domain. We assume this to be a good first approximation. The remaining

690

P. OLESEN

integrals can then be estimated and one finds *

Gl(v)

C

1,1 '

(21)

where C is a constant which depends only slightly on v and 1; in the following we take C to be a "universal" constant. In eq. (21) we have assumed vR < VL; otherwise vR should be replaced b y vi.. Finally we note that Re Al(v ) ,~ air' for v -+ 0, (22) where al is the scattering length. The partial wave dispersion relation (15) n o w becomes

a~v' ,~ El(v ) + v' F[ v--~+vt T~xCIVR z

(23)

for v ~ O, where El(v) is given by eq. (10). Since El(v) is expressed in terms o f lowenergy parameters and since C can be calculated (in practice b y numerical integration), we have reached our purpose, n a m e l y to calculate the high-energy p a r a m e t e r ?l in terms o f low-energy input parameters.

3. The S-Wave Resonance Model One o f the most interesting questions is to investigate h o w Tl behaves for l --* oo. It is rather involved f r o m a mathematical point o f view to use the formalism given in sect. 2. We shall therefore investigate a simple model, where the behaviour o f Yl can be found directly for l ~ oo. We consider a value o f l > L. There are thus no resonances (for v < VR) and the simplest assumption a b o u t the low-energy amplitude is that it vanishes. In eq. (22) this implies that the scattering length vanishes. Thus, as our first simplification, we use a I = 0, l > L. (24) Furthermore, in order to simplify the integral Iw(v, y) in eq. (10a) or (11) we assume that the only resonance which exists is the S-wave resonance. Physically, tkis means that we take the low-energy region to be as small as possible; such a starting point is perhaps not too unrealistic in our a p p r o x i m a t i o n scheme (a discussion o f this point will be given in section 4). t The integrands containingfa andfs in eq. (20) have been omitted since, according to our assumptions about the high-energy behaviour, they are expected to be of no significant importance (f~ and f, go to zero so fast that the high-energy behaviour is essentially given by eq. (2)). It can then be estimated that the logarithmic factors in the remaining integrals only have a slight influence, so that the integrals in a first approximation behave like VR ~tl+l

"

PION-PION SCATTERING(IIl)

691

Thus the o n l y term which occurs in the sum over l' in eq. (10) is l ' = obtain 2 El(v) = E ~ . , v o r1"o t ,I'( v ) ,

I'=0

0, and we

(25)

where

dz.

1 II-2(v0 I'+ 1)/VL PI(Z) vJ *-2(roy+*) x--z

It(v)

(26)

Here x is given in eq. (1 la). An approximate explicit expression for the integral I,(v) can be found if we take Vlo'= 0 and VL = m in the limits of integration. Then we obtain

I'(v) ~ 2 Q' ( l + 2 v~°'+ v

'

(27)

where the integral representation

Q,(x) =½ f+l P,(z) de d - 1 X--Z

(28)

has been used. Thus eq. (25) becomes

E o~H,voFoQt " "

El(v) ~ - V

I+2

I'=0

•

.

(29)

V

Here we let v ~ 0. Using

~/~r(/+ 1) z_l_ , Q~(z) ,~ 2,+, r(/+~)a •~ Z

-l-1 for z ~ o o ,

(30)

we obtain

E,'(v)

2

-~

2

, F~" e_l,og(~oz,+, )

~,Eo~,,,v'o ,:o'+--i

(31)

Using eq. (21) or

Gl(v) = v'c e_,,o,..

(32)

where vR < VL, we obtain 78

~

I' I' voFo e-tlog(~ov+l)+

1½22'+* I'=00gll' VgO ""~----~1

1 C e_,,o~vR = a[ = 0, ]~'~17

(33)

that is, V0/"0 e - l log 4(vo/' + *)/VR

~,[

2c r=o

v~'+l

(34)

692

r. OLDEN

Since it is necessary t h a t y~ b l o w s u p for l -~ oo in o r d e r to have a convergent slim o f p a r t i a l waves, is m u s t be r e q u i r e d t h a t

vR <

4(v~"+ 1).

(35)

Actually, there is one S-wave resonance c%nc. Thus O ~ c will provide us with an upper bound on vR. From the inequality (35) we find that the upper bound on the centre-of-mass energy corresponding to the maximal value of vR is E[.m. =az = 2~/v~az+ 1 = 690 MeV.

(36)

This limit is well above the to^, c mass 317 + 6 MeV, with a width below 16 MeV *, but well below the first weU-established P-wave resonance p with mass 757 + 5 MeV and half-width 60_+ 10 MeV. Thus the model appears to be self-consistent if the existing experimental information is used. It should be noted that since the only existing S-wave resonance has isospin zero, eq. (34) shows that the parameter y~ depends entirely on the exchange of isospin zero particles. This result is partly due to the approximate nature of the model. However, a preliminary investigation of the influence of the P-wave resonances shows that they will not affect eq. (34) very much. This result, if it is correct also with P and D-waves included, is consistent with a theorem stated by Foldy and Peierls 6) (see also refs. 7' s)). Since al o = ~ for all values of I we get 1

x/M voFo ° ° e-i log 4(v00+ I)/VR

r[

3C Vo °+ 1

(37)

We thus obtain the result that yr is independent o f L Thus, for high l the energy dependence and the isobaric spin dependence areindependent of the angular momentum L 4. Discussion Our S-wave model can be viewed either as a rather unrealistic model or as a first approximation in an iterative procedure with the purpose o f obtaining all the higher resonances. The last point of view has been put forward in sect. 6 of II, and we shall not repeat the motivation here. However, we would like to point out what the next step in the iterative procedure is. The calculations performed so far have given us the following picture. Below the point vR we have the S-wave resonance and nothing else, while above the point vR we have a behaviour of the type given in eq. (18) with Ira A~(VR)= 0, and where the parameter y[ is expressed in terms of well-known quantities. If we then use the N/D equations we can take into account the high-energy behaviour, in particular its influence on the low-energy region around v = vR, in a more accurate way and we can then hope to obtain the P-wave resonances above vR (effectively we increase Vl0. t All data are taken from ref. 6).

PION-PION SCA'IWERING (HI)

693

Proceeding in this way we can hope that it is possible to obtain at least a qualitatively correct picture of pion-pion scattering in terms o f the S-wave resonances. Finally, we would like to point out that there is not so much point in evaluating V[ numerically. The reason is that the quantity of primary experimental interest is the total amplitude, not the partial wave amplitude. As pointed out in I I the sum of the asymptotic partial wave amplitudes is not the same as the asymptotic total amplitude. Further work concerning how one can obtain, if it is possible at all, the asymptotic total amplitude from the asymptotic partial wave amplitude, is required. It is not impossible that this problem can be solved by considering the contributions from the partial wave amplitudes in the crossed channels to the amplitude in the direct channel.

Note added in proof: Since the present paper was written, important new experimental information has become available. This information is of much interest in connection with the approach carried out in I, II; therefore a short discussion will be added here. The Brookhaven experiment on small-angle pion-proton scattering (and the C E R N experiment on proton-proton scattering) reported at the D u b n a Conference 1964, has shown that at machine energies the real part o f the scattering amplitude is an appreciable fraction o f the imaginary part. This could be interpreted as a breakdown of the diffraction picture. However, if one analyses the situation by use o f the forward dispersion relations it turns out that the diffraction picture can only be violated if the total cross sections vanish in the asymptotic region 9). This is, however, not in accordance with the cosmic ray data. Therefore, we conclude that the diffraction picture is very likely to be correct. The experiment only shows that the machine energy region is not the asymptotic region. A more detailed discussion can be found in ref. 9). At the D u b n a Conference it was also reported that the C0ABCis probably not a resonance. Instead, it should be considered as an S-wave enhancement. However, the considerations in sect. 3 are actually based on the assumption that there is a large S-wave interaction in pion-pion scattering. The experimental result does therefore not contradict our assumption in sect. 3. References I) 2) 3) 4) 5) 6) 7) 8) 9)

P. Olesen, Nuclear Physics 60 (1964) 641 P. Olesen, Nuclear Physics 62 (1964) 593 G. F. Chew and S. Mandelstam, Phys. Rev. 119 (1960) 467 L. A. P. Bal~tzs, Phys. Rev. 129 (1963) 872 M. Roos, Nuclear Physics 52 (1964) 1 L. L. Foldy and R. F. Peierls, Phys. Rev. 130 (1963) 1585 D. Amati, L. L. Foldy, A. Stanghellini and L. Van Hove, Nuovo Cim. 32 (1964) 1685 R. F. Peierls and T. L. Trueman, Phys. Rev. 134 (1964) B 1365 P. Olesen, Phys. Lett. 13 (1964) 175