A critical test of models for the low energy πN amplitude

Nuclear Physics B39 (1972) 237-266. North-Holland Publishing Company

A CRITICAL TEST OF MODELS FOR THE LOW ENERGY 7rN AMPLITUDE * G. HOHLER, H.P. JAKOB ** and R. STRAUSS ** Institut fur Theoretische Kernphysik, Universitat Karlsruhe, Germany Received 19 October 1971 Abstract: Starting from fixed-t dispersion relations, phase shifts and total cross sections, the coefficients of an expansion of the invariant 7rN-amplitudes at the crossing symmetry point are calculated, considering the nucleon pole term and the A (1232)-exchange contribution separately. A similar calculation is made for an expansion at threshold. The results are compared with predictions from m3dels based on Ward identities and phenomenological Lagrangians. Limitations and ambiguities of treating A-exchange by a pole term are discussed and the magnitude of effects is estimated, which are not included in the models. In many cases our conclusions differ from those of earlier authors.

1. INTRODUCTION Starting from Ward identities [ 1 - 7 ] or from phenomenological Lagrangians [ 8 - 1 2 ] , many authors have recently tried to describe the 7rN scattering amplitude at low energies as a sum o f baryon and meson exchange terms. Predictions for scattering lengths, effective ranges, and, in some cases, for the partial waves at low energies were calculated and the more or less good agreement with the experimental information was considered to be an argument in favour o f a particular approach. If one compares the results o f different authors, one gets the impression that the experimental information is compatible with the predictions from quite different models. For instance a A(1232) exchange term is taken into account by some authors and ignored by others [8]. The pseudovector coupling for the nucleon exchange term seems to be as good as the pseudoscalar coupling [10] and even the sign o f the p-exchange contribution turns out to be different in different approaches [ 11]. Furthermore Peccei [9] found the unexpected result that the agreement with experiment is improved, if the p-exchange term is replaced by a direct interaction.

* Preliminary results were reported at the Lund Conference (1969). ** Supported by Bundesministerium fttr Bildung und Wissenschaft.

238

G.H6hler et al., L o w energy ~rN amplitude

It is the aim of the present paper to show that the test of the above mentioned models can be considerably improved. (i) Primarily the models lead to predictions for the g N a m p l i t u d e at low energies. The authors calculated scattering lengths in order to be able to compare their results with "experimental values". However one should notice that accurate values of scattering lengths cannot be obtained directly by extrapolating phase shifts. This is not only due to the difficulties of low energy experiments, but also to unsolved theoretical problems in connection with the separation of the strong and electromagnetic contributions to the amplitude. The scattering lengths of Hamilton and Woolcock [ 13], which are often quoted as the most accurate "experimental values", have not been derived from the low energy data alone. The accuracy was considerably improved by using a variety of dispersion relations together with the experimental information at all energies, as far as it was available in 1962. Since dispersion relations are used anyhow, it is favorable to calculate directly the amplitudes, which are needed for the test of the models. These amplitudes are "experimental" in the same sense as the scattering lengths of Hamilton and Woolcock. The uncertainties are not larger than for the scattering lengths, since the input in the dispersion integrals is the same and the amplitudes are considered only in the kinematical region near threshold. The only rapidly varying contribution is the nucleon pole term. Since it occurs in the dispersion relation as well as in all models, it can be considered separately, the rest of the amplitudes being slowly variable. It is sufficient to consider the first terms of an expansion in powers of p = ( s - u ) / 4 M and t, the variable v being chosen because of crossing symmetry. The models predict directly the coefficients of this expansion and therefore a comparison with our table 1 gives more information than a comparison with the smaller number of only six s- and p-wave scattering lengths. Furthermore each scattering length depends on several contributions and it is difficult to locate the origin of a discrepancy. Some authors discussed in addition s-wave effective ranges. However in this case not even a crude approximation can be determined directly from phase shifts, since the higher terms of the effective range expansion are important already at very low energies [ 14]. (ii) Another advantage of our method is the possibility to compare a certain part of the prediction, for instance the contribution of A(1232)-exchange, with the corresponding term o f the dispersion relation calculation. In this way it is possible to estimate the errors of the narrow resonance approximation, to determine coupling constants and to discuss the magnitude of effects, which have been neglected in the simple models. (iii) The best possibility to test the models as far as resonance exchanges in the t-channel are concerned, is to compare the predictions for the T = J = 0 and T = J = 1 mr NN partial waves f~+ and f l with the projections from f'Lxed-t dispersion relations.

G. Hb'hler et al., L o w energy *rN amplitude

239

Table 1 C o e f f i c i e n t s o f l o w e n e r g y e x p a n s i o n (2.5). i=

1 1.6 +_ 3 - 23.9 24.6

-

--

0.9 - 2d'

- 3.28 3.88 0.13

-

0.73 4 M.f'

- 1.6 +- .3 - 23.9 24.6 C +. l -

-

-

b,:

-

3

1.13+-.1 4 . 4 0 -0.73 -4.65 0.26 0.66 d'

-0.25 16 M 2 e '

4

5

0.00 0.00

1.12 -1.11

0.00

0.01

0

-0.92 0.96

0.09 -0.09

-0.28 0.28

0.01 0

0.04 0

0.00 0

0.00

1.13 -0.73 0.26

1.12 -0.77 0.13

0.17 -0.16

0.20 -0.15

0.01

0.05

0.9

0.66

d'

0.48

-0.29 0.29

0.00

0.00

a: z Osypowski E x p - N, ps N, ps - N, pv -A,p -A, - ETC

-0.04

0

0

0.3+-0.2

0.99

-0.06

0.25

-0.94

0.06

-0.25

0.44 -+ 0 . 0 2 2.04 0.90 1.15 - 1.16

0.25

Exp -Zx, p -&,np - ETC

n

0.1

0

0.05

0.00

0

0

-0.156

0.19

0

-

0

-0.30

-0.11 0.01 0.17 -0.03 -0.08

-

c-*. O~ypowski

0.02 -0.02

-0.12

4.4

E x p - N, pv -A, p - A, n p

-1.15 1.11

4.6

4 Mc'

b.*. l Osypowski

-Mr'~45

-0.45 0.08

0.11

E x p - N, ps -A, p -A, np

4M(4Me'-f')

8.4 5.1 3.6 4.3

8.0 ± 0.4 2.0 4.2 4.8 5.4

E x p - N, pv -£x, p - A, np

Osypowski

0.19 -0.18

- 2d'

4 Mc'

-

2

0.00

-0.04

-0.034

0.17

0.04

0.035

-0.04

0.01

0.00

0.00

Me'~45

0

0

p

b-i Osypowski E x p - N, ps N, PS - N, pv -&, p -ix, np - ETC

e,: c i Osypowski

240

G. HO'hler et al., L o w energy nN amplitude

(iv) In our method the amplitudes are considered not only near threshold but also in the unphysical region ( M - 1) 2 <~ s <~ (M+ 1) 2 between the threshold of the s- and u-channel reactions. Adler [15] conjectured that for a certain point in this region the transition to zero pion mass gives a smaller correction than at the physical threshold. Furthermore the narrow resonance approximation for A(1232)-exchange is expected to be better in the center of this region than at low physical energies. Dispersion methods allow to calculate the cusp-like behaviour of the real parts of the amplitudes at threshold, which is neglected in the above mentioned models together with other effects of unitarity. The extrapolation of the "experimental amplitude" into the unphysical region does not introduce an important additional error, since the difference between an amplitude in the unphysical region and, at the same t-value, in the physical region at low energies, is given by a strongly convergent integral.

2. EXPANSION OF THE AMPLITUDES AT LOW ENERGIES In addition to the usual invariant amplitudes A + and B -+ [16] we consider S--t,/

C=A + M - 4M2-t

V

B=A + 1-

(2.1)

B, t

4M 2 where

S--it U- 4 M - w +

t 4--M-'

(2.2)

C and B being the helicity no-filp and flip amplitudes in the t-channel. M = nucleon mass, co = pion lab. energy, h= c = mTr+ = 1, unless stated otherwise. At low energies and in the unphysical region between the thresholds ( M - 1)2 < s < 014+ 1) 2 the nucleon pole term N,p = 0,

N,p

M~ s

M 2- u

= ~

PB- - u

I~B-+P '

- 4M 2 4 M y B = t - 2,

g2 47r - 4 M 2 f 2 = 14.6,

(2.3)

of the fixed-t dispersion relations [ 16] has a rapid energy dependence. Therefore we introduce the amplitudes A'=A,

if= B-BN,p ,

C'= C - C N , p ,

(2.4)

G. HOMer et al., Low energy ~rN amplitude

241

which are expected to be slowly variable functions of 1) and t as long as the distance to the s, t, u-channel poles and thresholds is not too small. Taking into account crossing symmetry, we write the expansions * +

+

+

+

A+(1), t) = (a 1 + a2t ) + (a 3 + a4t ) 1)2 + a5 v4 + .... l)-1 A - ( p , t) = (a 1 + a2t ) + (a 3 + a4t )1)2 + a51)4 + .... +

1)-1~+(1),t)= (b 1 + b2t) + (b 3 + b4t)1)2 + b51)4 + .... B'-(1), t ) = (b 1 + b2t) + (b 3 + b4t)1)2 +051)4 + ....

+ 2 +c51) + 4 +..., C+(1),t) = (Cl+ + C 2 t ) + ( c 3+ +c4t)1) 1)-1 C-(1), t) = (¢1 + ¢2 t) + (¢3 + ¢4 t) 1)2 + ¢51)4 + ....

(2.5)

where the coefficients are real numbers. The dispersion relations show that the amplitudes have a branch point singularity at v = 1)1 = 1 + t/4M, which corresponds to s = (M+ 1) 2. Therefore the expansion (2.5) does not converge for v 2 >_ 1)2. Fig. 1 shows the cusp-like behaviour at threshold and a comparison between the exact amplitude and the first terms of the expansion (2.5) at t = 0. Higher terms in t are negligible at least up to I tl ~- 2. In order to determine the coefficients o f eq. (2.5) we have evaluated the fixed-t dispersion relations

(1

oo

ImA ±(v,t)

Re A ± (1), t) = 7rl f

~ +

~

1)

dr',

v1

(2.6)

-~r f

I m B -+(v',t)

,1- v

T-

dr'.

v1 The dispersion relations for ~± follow from (2.6) 2v

t) = 7 f

Re

=

dV

I m C - ( v ' , t)

(2.7)

V2_1)2

Vl ~

ReC*(v,t)=A÷(O,t)+

2

-~

r' v2f ~ d~-1)

Im C÷(v'; t) /,,,2 _ t)2

(2.8)

v1 * Of course only two of the amplitudes are independent, but it is useful to consider all three of them.

242

G. Hbhler et aL, Low energy ~rN amplitude

g2/

,1 u71ReC-(u) -0.z \.N. \.

fReC+(w)- ~ -0~

\.\ -0.5

.

,.///

, \,\.

-0.[6 N~,~,~.~ ~2

o'2 o'~ ~

-1.0~ -

/'//~/

\

o18 ~o

L

1.2

-0.1 0

02

0.4

0.6

0.8

1.0 1.1

Fig. 1. Real parts of C -+in the unphysical region. Solid lines: Re C+ - g2/M and Re C-/w as calculated from the dispersion relations. The cusp of Re C-/to is larger by a factor 1.7, if a++ = 0. Dot-dashed lines: real parts minus the last term in (A.4), which is responsible for the cusp effect. Dashed lines: the first two terms of the expansion (2.5), The zero of Re C + occurs at to 2 = -0.085 at the intersection of the solid line and the curve BT = - Born term + g2/M . It is of interest in the phase representation [56]. The curves belong to the evaluation in ref. [57] and differ slightly from table l. Eq. (2.8) includes the possibility that a subtraction is required in the relation for A ÷, eq. (2.6). A t present there is no reason to d o u b t the validity o f the u n s u b t r a c t e d relation. This leads to the sum rule for the " s u b t r a c t i o n f u n c t i o n " [17]

oo A÷(0't)=~ 2

d v i m A + ( v ,, t ) . -~-

f

(2.9)

v1 The d e t e r m i n a t i o n o f the coefficients in (2.5) is an extension o f our earlier w o r k [ 1 8 - 2 0 ] . Details are given in the appendix. The expansion is simply related to the first terms o f the partial wave series in the t-channel

C+(v, t) f ° ( t) 5 + - - - ( 3 M 2 v 2 _ p _2 q _2) f + (2 t ) + .... 47r p2_ 2p2

B+(v, t) -4rc~-

-

(2.10)

1 5 M f2_ x/-6(t) + ....

C-(v, t) _ 3M 1 7M 2 3 4nv ~ f+(t) + ( 5 M 2 v 2 - 3 p 2 q _ ) f + ( t ) + ... , p_

B- v , ~ = ~ 2 f l ( t ) + 4re

2p 2

~__(5M2v2_p2_

-

q _2 ) f _3 (t) +...

(2.11)

243

G. HOhler et al., Low energy ~rNamplitude

where the notation of Frazer and Fulco [21] was used for the rnrNlq partial waves f J ( t ) and p 2 = M 2 _ 1 t, q 2 = 1 - 1 t. For the comparison with (2.5) one has to take the real parts of the partial wave amplitudes, which are complex-valued functions of t. One should notice that because of the cusp-phenomenon at w = 1 there is no simple connection between our coefficients in (2.5) and the lowest terms of an expansion of the amplitude at threshold in powers of q 2 and t, q being the c.m. momentum (see appendix).

4n

-

I+

ao++

1+ + l2a 1- + 8M2]

+ ( a0+ + 9 3 1 1__ a ) t 2 \32M 4 8 a 2 + + ~-a 2_ + 8M 2 a 1_ + 16M2 1+

+

b +al-

ReB(v,t) 47r

a0÷ q2 + . . . .}

+2al+ + 2M] {(a

- 2M

2.12)

a 0 + ~ 3(a al+\ 1- - al+ + 4 M 2 ] + ~ 2 - - a2+ + ~ - ~ ) t

+ ~15- ( a2+ + a3 - _ a3+) t2 + . . . .} \4M 2 al+- denotes the scattering lengths and b the s-wave effective range, which is defined by

Re f0+ = a0+ + b q 2 , or q ctg a =~01+ + l r q 2 ,

The higher terms in (2.12) depend on p-wave effective ranges etc. Subtracting the nucleon pole term part, our evaluation leads to *

'{

Re

1 Re B'÷ 4rr

-

= 1 . 1 5 ( a •o + + 0 . 0 1 0 5 ) + ( 0 . 1 2 + - O . 0 1 ) t + ( O . 1 6 7 + O . 0 0 8 ) q 2

,

= - 0 . 3 4 + 0.04

* The errors are estimated from a comparison between different calculations. They do not inelude possible systematic errors of the phase shifts.

244

G. H&hler et al., Low energy ~rN amplitude

Table 2 Isospin even and odd scattering lengths and effective ranges following from eq. (2.14). N, p: nucleon pole term contribution (f2 = 0.081). Errors are not well defined, since phase shifts are not quite consistent with the dispersion relation. 1 _+

+ p aN,

0+

-l.906

1.886

-0.02 52-0.02

1

-0.098 0.055

0.042±0.007 0.078±0.008

-0.056 0.133

1+

0+

~* a

+ p bN,

3"*

0.14

-0.19

1 Re C 4n

a*

aN, p

0.1418 -0.0422±0.0005 -0.044 -0.055

b÷

b~q,p

-0.05±0.02

= -(0.0485 + 0.0005)

a'-

0.048

0.031 50.003 -0.026 ±0.002 b'-0.039

a0.100 -0.013 -0.081 b0.009±0.005

(0.012 +- 0.002) t-- (0.041 -+ 0.007)q 2

1 Re B'- = 0.76 -+ 0,03 . 4ir

(2.14)

The scattering lengths and effective ranges following from (2.14) are given in table 2. Their values depend on the rrN coupling constant for which f 2 = 0.081 was assumed ( g 2 / M = 27.36,g2/47r = 14.63). Our discussion is preliminary insofar as charge-dependent effects are neglected (for instance the 7r° - lr 5 mass difference or the dependence of the A(1232) mass and width on the charge state), for which a model-independent theory does not exist. This leads to uncertainties, if the predictions from our low energy amplitudes are compared with the nN data and if one attempts to improve the accuracy of the determination o f f 2 (see the remarks in sect. 7 of ref. [20]).

3. THE A(1232)-EXCHANGE POLE TERMS It was one of the main points in the fundamental paper by CGLN [16] that at low energies the integrands of the dispersion integrals are dominated by the contributions of the 33-partial wave 1 imA~(v,,t)=(21) 4-~

Imf33 '2 ' + (oq + a 2 t ) , 3q (E M)

1 i m B ~ (u,, t) = ( 21 ) Imf33 4~ 3 q , 2 ( E , + M ) (HI + / 3 2 0 , where

(3.1)

G. HUhler et al., L o w energy ~rN amplitude

, t v' = o2 + 4 M '

2iqf33 = r~33

245

e 2i833 -- 1

(3.2)

'

~1 = 3 ( W + M ) q 2 + ( W - M ) ( E + M ) 2 '

a2 = 3 ( W + M ) ,

131 = 3q2 - ( E + M ) 2 ,

3 • /32 = ~-

(3.3)

In (3.1) ai,/3i have to be taken at a pion lab. energy w ' . W = V/-s and q denote the total energy and the m o m e n t u m in the c.m.s., E 2 = M 2 + q2. Inserting these expressions into the dispersion relations (2.6) one obtains in a narrow resonance a p p r o x i m a t i o n the "pole terms" for A-exchange [22]

B~ (v, t) =

~

(,6~'+/3~ t)

1- - p :1;.

1 VA + V ]

/

(3.4) '

where g , 2 and uA are defined b y Wm

g,2 _ 3 4rr 7r

f

Imf33 - dco

2M

E;-M

q2

(3.5)

'

1 2Mva=s*-M

2-

l +~ t = 2 M w * +

l t.

(3.6)

If a kinematic variable has a star as an upper index, it has to be taken at the resonance position. In order to test the validity of the pole a p p r o x i m a t i o n we have calculated

X ( v ) = - ~1 rd V - - V

_,

ImA~(v,t),

v1

lf

. . . .

V--V

Im B• (u, t)

(3.7)

vl using the P33 phase shift and a cut-off at T~r = 700 MeV. It is e x p e c t e d that X -1 , y - 1 are linear in u and also in s at fix6d t, as long as s - s * is large in c o m p a r i s o n with the w i d t h X_ 1 _

9(s-s*)

g.2 (,~, + ~ t)'

(3.8)

246

G. H6hler et al., L o w energy lrN amplitudes Ty-1

0.5 0.4 "~l.x x. x.

0.3 0.2

xXX x

0.1 i xKxx

i

0.5

10

15

20

0.5

1.0

1.5

2.0

toe~

-Q20~ Fig. 2. Determination of the ~NA coupling parameters from eq. (3.8). The parameters of the fit are given in (3.9), (3.10).

and similarly for y - 1 . It turns out that the "best values" for s* and g , 2 obtained from X -1 and y - 1 differ by 3% and 11% respectively. Furthermore they have a slight t-dependence. We choose an overall average g , 2 = 3.22,

g47r , 2 _ 0.264 -~ 13.5 GeV -2 ,

s* = 76.4,

(3.9)

which gives a reasonable fit according to fig. 2. These values lead to the following parameters of the pole term formula (3.4) IV* = 8.74 -"- 1219 MeV, w* = 2.25,

ct~ = 4 8 5 . 8 ,

a~ = 23.19,

3 /32 = ~

E* = 6.90, /37 = -

q , 2 = 2.40, 178.2, (3.10)

It is remarkable that the attempt to include the effect of the resonance shape leads to a lV*-value somewhat below the resonance position (1232 MeV). The exact 33-contribution to the dispersion integrals is compared with the pole terms in fig. 3. We conclude that the A-pole approximation can only be used, if one is willing to accept an uncertainty of the order of 10% in the range ( M - 1) 2 <~ s <~ ( M + I ) 2, Itl<~5.

247

G. H6hler et al., Low energy rrN amplitude

201 ReAA(s,t=O) ,

~\

10 50 I

70 i

I

-20

s i

I

I

I

~

' ReB~(s,t=O) I 5 ,

sd ,

'

,

90

/ -io

110

---

/

I

Fig. 3. Validity of the pole approximation for Re A ~ and B~x. Solid lines: exact calculation, dashed lines: pole approximation.

In a Lagrangian approach g , 2 is determined by the width of the A ~ NTr - decay

g,2 F(W*) - 4rr

q,3(E,+M ) 3 I4,'*

(3.11)

Many authors (for instance in refs. [1, 5, 9, 10, 12] ) used the zrNA- coupling constant, which follows from this relation. Even if an energy-dependent width is taken in the Breit-Wigner fit to th~ P33-amplitude, one obtains a value g-~2

4n

0.37 - ~ 1 9 G e V -2 from F ( W * ) = l l 5 M e V

(3.12)

which is larger than our result (3.9) by 40%. Therefore the authors found a 33-contribution to the 7rN-amplitude which is 1.4 times larger than the amplitude following from the dispersion theory~ Since this ]s not a physical effect b.ut a deficiency of the isobaric model, we think'that one must take the value (3.9) o f g .2 in quantitative investigations. The evaluation of (3.5) gives 0.264 in good agreement with (3.9). Some authors def'me another coupling constant f * 2 which amounts to

248

G. Hdhler et al., L o w energy ,rN amplitude

f * 2 - g*2 M 4rr 3 W* - 0.068

(3.13)

if (3.9) is inserted. Amati and Fubini [22] mentioned not only a relation similar to (3.5), but also another determination of the coupling constant. "Minimizing the violation of unitarity" they found f , 2 = 3 f 2 ~ 0.12,

(3.14)

which is too large by almost a factor of two. The discrepancy should not be taken seriously, since nucleon recoil effects have been neglected at very high energies. Some of the conclusions of Pradham et al. [23] are based on the large value (3.14). The A-pole term contributions to the coefficients in eq. (2.5) are

A2-

9M

co*

l+t~

4Moo* t + - -co.2 + ~

v_l B~X - 2g .2 9M

/3~ { (/3~ w.2 1+ ~-~

+ 60.2 + '/3~

M60 '~ co*2

4~/~o*]~J

1 ) 2M~w* t

The formulas for v-1 A ; and B~x follow from those for v-1 B~x and A~x, if~3* is replaced by - ~ at.* and a* by 1 / 3 * respectively. The numerical values in table 1 were calculated from the dispersion integral (2.6), inserting the P33-phase shift. The i = 1, 2 coefficients deviate from those calculated with eqs. (3.10), (3.15) only by a few %. However the deviation is much larger for some of the i = 3, 4 coefficients, for instance 20% for b 3 and 35% for b 4. A comparison between the values for "Exp. - N, ps" and "A, p" (table l) shows to what extent an improved version of the CGLN-approximation [16, 24] agrees with the experimental amplitude. The i -- 3, 4 and 5 coefficients follow essentially from a subtracted dispersion integral. Correspondingly they are very well approximated by A, p except for c 3, where the convergence is slower than in the other cases. There are deviations of 20-50% for the i = 1 coefficients, with the exception o f a 1, where one is surprised to find a rather good agreement. The i = 2 coefficients show the influence of t-channel exchanges and will be discussed later. Kottler and Willey [25] have recently derived "threshold theorems" which essentially correspond to the statement that the improved CGLN-approximation is good at threshold. It is not clear to us, why these authors preferred crude estimates instead +

÷

+

G. HOhler et al., Low energy 7rN amplitude

249

of inserting phase shifts into the dispersion relation. In fact this was already done for the 33- and non-33-contributions in refs. [18, 24, 26]. In particular it is not true that the N- and A-contributions to C-(1, 0) = A - ( l , 0) + B-(1, 0) "nearly cancel", although similar statements have been made several times in the literature. The evaluation from phase shifts shows that the A-contribution amounts to -57% of the N-contribution in this case. Furthermore we think that the antisymmetry under crossing cannot be used in order to show that the corrections are small in the case of A-, since one could as well consider A - / ( s - u). In fact, table 1 shows a large correction in this case.

4. t-CHANNEL-EXCHANGE CONTRIBUTIONS We have seen in sect. 3 that the sum of the nucleon and A-exchange pole terms gives good approximations for the i = 3, 4 and 5 coefficients of our expansion. Of course it is possible to improve the i = 1 and 2 coefficients by considering higher nucleon resonances and the high energy behaviour of the imaginary parts [24]. It is certainly of interest to know the magnitude of the different contributions, but this does not help directly to construct a simple model for the nN amplitude at low energies. One of the basic ideas for an improvement of the CGLN approach is to express the high energy contributions, including part of the higher resonances, by contributions of the nearest t-channel singularities. Mandelstam [27] pointed out that this can be done by subtracting the lowest n n N N partial waves from the fixed-t dispersion relation. This proposal was taken up by Frazer and studied in detail by Dietz. It turned out that an independent method of Cini and Fubini leads to closely related results. References and more details are given in ref. [7]. Unfortunately this method is of limited applicability for two reasons: (i) one has to use assumptions for the t-channel amplitudes (for instance p-dominance) and values of the coupling constants, (ii) t-channel exchange terms occur only together with left hand cut terms of the n n N N partial wave dispersion relations, which cannot be calculated and therefore have to be introduced as unknown parameters. Other attempts to derive improved results for the i = 1 and 2 coefficients are based on Ward identities, current algebra and PCAC [ 1 - 9 ] . In this approach an important part of the J = T = I t-channel exchange contribution is expressed by isovector nucleon form factors, for which experimental data can be inserted. Furthermore instead of the unknown left hand cut contributions one has "non-pole terms" belonging to N- and A-exchange, which can be calculated. The main difficulties of this method follow from extrapolations in pion mass and from the neglection of unitarity effects. An improved treatment was developed by Fubini and Furlan [28]. It is remarkable that their corrections to the amplitudes at threshold are given by the same total cross sections, which determine the magnitude of our cusp effect (A.4).

250

G. HOMeret al., Low energy lrN amplitude

The superposition of s, t, and u-channel exchanges leads to questions about double counting, which were investigated in Oehme's papers on duality [29] (see also ref. [7] ).

5. THE MAIN TERMS IN WARD IDENTITY APPROACHES In the following we shall give formulas and numerical values for the most important terms which occur in on-shell Ward identity approaches [ 3 - 5 ] . 5.1. The equal-time commutator term In all papers on this subject an important contribution follows from an equaltime commutator, which can be expressed by the isovector nucleon form factors p-1 AETC (u, t) = - - - 2 F V ( t ) = --4.28 -- 0.25 t ,

(5.1)

BET C (v, t) = ~2' GV(t) = 5.44 + 0.30 t ,

v -1 C~T c (v, t) -

2

f~r = 0.93 is the pion decay constant. The numerical values were calculated with the following experimental results [30] G V(t) = 0.5 + 0.033 t ,

GV(t) = 2.35 + 0.13 t ,

FV(t) = 1.85 + 0.11 t ,

FV(t) = 0.5 + 0.023 t .

(5.2)

5.2. The nucleon pv-Born term In Ward identity approaches one finds the prescription that nucleon exchange has to be calculated with pv-coupling. These terms differ only by constants in A ± and B + from the pole terms (2.3) of the dispersion relation A* _ g2 N, pv M '

BN,PV = BN, p ,

C-~N,pv -

(/) 2_v2)(~ _ 4 _ ~ )

4M 3

÷

+

,

(5.3)

G. HOMeret al., Low energy IrNamplitude

A ~q,pv = 0,

g2 2M 2 ,

B~q,pv = BN, P

g2 v Iv2 - 1 - ( 4 ~ - ) 2 + 1 tl 2M 2 ( v 2 _ v 2 ) ( 1 4M 2t )

CN,pv -

251

(5.4)

In contrast to Ci~,p the pv terms C N,pv ± have the property to be very small at threshold. 5.3. A-exchange terms A-exchange terms are calculated from 2nd order Feynman graphs. The results can be written

A~=A~,p+A~,np,

B~=B~,p+B~,np,

(5.5)

where Aa,p and Ba,p agree with the narrow resonance approximation (3.4) of the P33"contribution to the dispersion relation, if the coupling constant is defined by (3.5). The "non-pole terms" depend on the choice of the propagator and of the interaction term in the Lagrangian. Taking the most general propagator [31], which depends on a parameter c, and the simple interaction term

"/2I = g* ~

(5.6)

~N aU ~b~r+ h.c. ,

one obtains the following expressions for the non-pole terms [ 14] *

a,np

9 g* 2M ~

--

+ 4W *21--[(Rec+lcl2)( 6 + 1 8 W * [M

+lcl2(1---~t) + 1 + 4

v-1 B + _ ~,np

4 g*2M 9 S*

~----*112C+11-2)

2c~

* Eq. (A.4) of ref. [14] had to be corrected.

W*~Mt) +

2

'

(5.7)

(5.8)

252

G. H6hler et aL, Low energy ~rNamplitude

v-lA~,np

w*

}

_ 4 g*2M2 {3 9 ~ I c l 2 + R e c + ~-- ( R e c + l c l 2) 12c+ii - 2 ,

B2,np = l g .2 { ( W ~ * + M ) 2 +4 W*+M [,cl2 +Rec] 2M 1 [51c12 + 2 R e c _ ~1 iclZt] } 12c+i1_ 2 s* 2 Icl2 +s-~

(5.9)

(5.1o)

Special cases of eqs. (5.7)-(5.10) were discussed in the literature. In the first paper on this subject, Rarita and Schwinger [32] considered c = -31-. Recently many authors used c = -1 (for instance Schnitzer [2], Chang [33], Carreras et al. [3], Osypowski [5] ). Petersson [34] chose c = 0. The parameter c determines the spin 1 admixture to the spin 3 field. At low energies the pole terms C~,p have about the same magnitude and the opposite sign as the non-pole terms C ~. . This leads to a cancellation, which turns A'nP _1. out to be exact at threshold for the choice of Rarita and Schwinger, c = In this case the A-exchange non-pole contributions to the scattering amplitudes are given by +

8 *2" / E * + M ~ 2 _ M iV/ [ ~ \ ~

1

A~,np(V,t)=-79g +

v-1 BA,np(V, t) -

v-1 A~x,np(V, t ) -

B~x,np(V,t) = l g * 2

4 g*2M

9

s*

9

s*

-~

-

(5.12)

'

+

M 1 + 10W-~+7

and

,

W*2

(5.13)

W*2

C~x,p(1, 0 ) = ~8g "2" m E*+M( - ~ 2 - ~M ) = -C+A,np (1,0),

4 .2- E*+M

C~x,p(1, 0) = - ~ g

lVl

2M

5'.11)

1_

W*

C~,nP (1' 0) "

(5.15)

(5.16)

Table 3 gives the numerical values of the non-pole terms in our expansion (2.5). It is seen that the ambiguity is negligible in many cases. In particular C~,np depends only very weakly on c

G. HOMeret al., Low energy ~rNamplitude

253

Table 3 Coefficients of low energy expansion for A non-pole terms. +

+

+

+

c

a1 = c1

a) = c 2

b] = c 3

_ 1 3

- 24.6

- 0.26

- 0.13

- 1

- 25.2

0.03

- 0.13

0

- 25,1

Peccei

- 25.2

0.05

- 0.03

Pantin

- 25,4

0.04

0.21

ai

bi

ei

c~

- 3.60

4.75

1.15

0.03

0.44

0.73

1.17

0.00

1.16

1.16

0.01

0.61

0.55

1.16

0.00

0.2

0.86

1.1

0.00

lg,21(W*+M]2 v-lC-~,np(t=O)=9 [\ W* ]

+

5 Icl2__+ 2 Re c } W,212c+112 '

(5.17)

as long as c is n o t near to its singular value - 1. The total A - c o n t r i b u t i o n +

A +A,p (0, 2) + A,a,np (0, 2) =

= ~2g . 2

2 W+M + - = 0.0075 ~ 3 × 1 0 - 4 AA,p W2(W 2 - M 2)

(5.18)

is i n d e p e n d e n t o f c and very small, ifA~x p is taken in the pole a p p r o x i m a t i o n *. H o w e v e r a 1 and b 1 have a rather strong i - d e p e n d e n c e and remarkably large values for c = - ~ (see sect. 7). Peccei [9] used a more general interaction term ~° I = g* xItz~ 0uu qt N 8 v ~brr + h.c.

(5.19)

and i m p o s e d a subsidiary c o n d i t i o n in order to ensure that the Lagrangian is invariant under a point transformation, which mixes only the spin 1 c o m p o n e n t s o f qsAu. His results for the non-pole terms are also shown in table 3. N a t h et al. [35] f o u n d that Peccei's subsidiary c o n d i t i o n is unnecessary and that invariance o f the Lagrangian under the point t r a n s f o r m a t i o n is valid for a one-para-

* We are grateful to Prof. Schnitzer for calling our attention to this fact, which is important for the discussion of the a-commutator term. The r.h.s, of (5.18) is much smaller than our number in ref. [39], which belongs to A ~, p as calculated from the phase shift.

254

G. H6hter et al., Low energy nN amplitude

meter class of interaction matrices 0uv, The parameter Z is fixed by demanding that the interaction should be consistent with the principles of field quantization, which gives Z = 1 . It is amusing to see that the most recent result for the AA and Bzx amplitudes as given by Nath et al. for Z = ½ agrees exactly with eqs. (5.11)-(5.14), which follow from the old Rarita--Schwinger prescription c = 1 and the simple coupling (5.6). Of course the earlier methods were not satisfactory and there is no direct relation between the parameters c and Z. Nevertheless it could be interesting to took for an explanation of the agreement. A difficulty in the treatment of Nath et al. was recently pointed out by Hagen [36]. Pantin [37] proposed another prescription, in which the s-wave has no contribution from A-exchange in the direct channel even at energies different from threshold. But unfortunately his A-exchange amplitude has a factor t/u, which leads to a pole in the physical region *

6. COMPARISON WITH THE RESULTS OF OTHER AUTHORS Our result for the coefficients of the expansion (2.5) is given in table 1. The coefficients belonging to exp -- A N,pv - A z~ ' N,pv

BA '

.4- = Aex p -, A ~ - AETC ,

(6.1)

B- = Bexp - BN,pv - B 2 - BETC ,

show that part of the amplitude, which remains after the main effects are subtracted. The numbers for the total A-exchange contributions A/,, BA belong to the prescription of Nath et al. [35] (or c = ~). It is easy to calculate the coefficients for the other cases by adding the difference of the non-pole terms (table 3). A detailed investigation of the Ward identities for rrN scattering was recently performed by Osypowski [5]. His 2nd order expansion can easily be compared with table 1. The main differences are (i) Our table includes the t-dependence o f the electromagnetic nucleon form factor, which is neglected by Osypowski as a higher order effect, (ii) his z~-exchange terms are 45% too large because of his choice o f the rrN2xcoupling constant, (iii) he uses c = - 1 in the non-pole terms.

* We are grateful to Dr. A.S. Macfarlane for calling our attention to Hagen's work and for a discussion on Pantin's method.

G. Hi~hler et al., Low energy ~rN amplitude

255

The comparison leads to the following conclusions: + + -7 (i) The terms a 1 , a 2. These numbers are closely related to Osypowskl s o-commutator term Z. A detailed discussion including an estimate of the errors and a comparison with the recent work of Cheng and Dashen [38] was recently given elsewhere [39]. It was shown that the Cheng-Dashen result is not compatible with the dispersion relation for (3/3 t ) C ÷ at t = 0. In general it was assumed that the dispersion relation for A ÷ does not converge without a subtraction. However a detailed investigation led to the result that all data are compatible with the validity of the unsubtracted relation, which has schannel helicity conservation at s --* oo as a consequence [17]. Table 1 shows that ÷ the unsubtracted integral for a 1 = 27.4 - 1.6 = 25.8 is almost saturated by the 33contribution (23.9). The small value for ~ is not in contradiction with the large T = J = 0 exchange effect in n N backward scattering (ref. [40], earlier references are given there). The situation is similar as in the cases of N and A-exchange contributions to C ± ( 1 , 0 ) in the Ward identity approach: the coupling is strong and the pole terms are large, but they are completely or approximately cancelled by non-pole terms. In order to see the influence of the poles, one has to look at the derivatives. Table 1 shows that 2 ÷ of the large a2-value follows from A-exchange, but the rest is still large enough to be compatible with a strong T = J = 0 exchange interaction. In the Ward identity approach T = J = 0 meson exchange effects are described by two terms: the o-commutator term ~ and an axial-vector nucleon scattering term, which has a kinematical factor t - 2 * (cf. eq. (5.17) in ref. [5] and appendix C in ref. [ l ] ). We think that the second term is large at the e-pole and gives an ira+ portant contribution to a 2. Unfortunately it would be too simple to work with an ansatz for the exchange of a single meson. Considering a dispersion relation, one of us was able to show [43] that the dispersion integral over the T = J = 0 nTrNN amplitude

f 4

imfO(t,)

t'd-tl

M2 --It'

(6.2)

has a zero at t = - 2 5 , indicating that Im f o has at least two contributions of comparable magnitudes and opposite signs. Of course all fits in which two or more T =

* In a Lagrangian framework Achuthan and Steiner [41 ] emphasized the need for a derivative enn-coupling, which also has a linear t-dependence, Genz and Steiner [42] investigated the hypothesis that e couples to the trace of the energy momentum tensor. From this requirement, presence of both nonderivative and derivative czrTr-couplingfollows and their relative strength is predicted. A similar t-dependence follows from the dispersion approach [7]. We are grateful to Drs. Genz and Steiner for a discussion.

256

G, Hi4hler et aL, Low energy *rN amplitude

J = 0 mesons are involved are highly speculative. One should notice that the convergence o f the integral in (6.2) is not secured without the term M 2 - 1 t', which becomes large only fbr very large t'. (ii) The terms b 1, c 3, s ~ , b~. Contributions to these terms are expected from form factors at the nNN and nNA-vertices, corrections to the nucleon propagator and from the approximation of the axial-vector nucleon scattering amplitude by a /',-exchange term [51]. However there is no explanation, why these corrections are so large, In fact, except for b~, they are comparable with the A-exchange pole terms. it is true that the coefficients depend strongly on the parameter c in the A-propagator and can be made smaller, if one chooses for instance c = - t. But we think that it is not satisfactory to treat c as an adjustable parameter and to ignore the result of Nath et al. [35]. The discussion of a i will be continued in the next paragraph. (iii) The terms a i , ~ 2- These coefficients would have been considerably larger, if we had not included the t-dependence of the electromagnetic nucleon form factors, We conclude that the systematic second order expansion of Osypowski [5] led to the neglection of an important effect. * (iv) The term c ~. The numerical value shows that there is an additional contribution which is of the same magnitude as the correction following from A-exchange. Again one of the neglected higher order terms in Osypowski's expansion is important. (v) The terms with i = 3, 4, 5. The success of the Ward identity approach follows from that of CGLN [ 16], except tbr a~, where a correction is expected. The large value of c ; follows mainly from b ~. Nath et al. [35] came to the conclusion that their model led to scattering lengths "in good agreement with the experiment". From our point of view the situation looks different. First of all the approximate agreement with the s-wave scattering lengths is not a merit of the specific model, but a consequence of the low energy theorems of Adler [144] and Adler-Weisberger [15]. A model is necessary for the corrections due to the extrapolation in pion mass, as given for instance by Fubini-Furlan [281. However the &-exchange term of Nath et al. does not contribute, since it vanishes at threshold. The only difference between the approach o f Nath et al. and that of earlier authors is the magnitude o f the A-non-pole terms, which are listed in table 3. It is clear from our table 1, that the prediction for a i" and b~ is quite wrong, if our ETC term or a p-exchange term is taken into account. Eq. (2.12) suggests that this should lead to a discrepancy in the p-wave scattering length combination a~_ - ai+ and indeed, Nath et al, found 0.120, whereas the compilation [45] recommends 0,066 -+ .003. Even if the error estimate is considered to be too optimistic, this is a big discrepancy. It is somewhat obscured in the table of Nath et al., since these authors

* The neglection of the pionic form factor of the nucleon in the Goldberger-Treiman relation is also questionable, since it differs appreciably from 1 (K2(0) = 0~85).

G. HUhler et aL, Low energy nN amplitude

257

include "experimental" values, which are based on the rather poor experimental knowledge in 1 9 6 3 - 6 5 . If Weinberg's "direct scattering term" is used instead of p-exchange, Nath et al. find a better result (Peccei [9] made a similar remark). This term does not contribute to A - and our table 1 shows that this is a great improvement. There is a similar improvement in b~, which follows from the fact that the situation in c~ must be unchanged because of the low energy theorem. Of course one cannot obtain a good description of 7rN scattering by using N and A-exchange and a direct term only, since a model of this type is just the improved CGLN approximation [24], normalized to the low energy theorem by an additional constant parameter. Table 1 shows that one needs in addition an appreciable contribution to a~, b~, which according to (2.11 ) strongly suggests an effect of T = J = 1 meson exchange in the t-channel. This is discussed within the framework of dispersion relations in ref. [7]. The above mentioned difficulty can be described as a "double counting" due to the superposition of resonance exchanges in all 3 channels. Double counting is avoided in the C -+-amplitudes because of the low energy theorems. But the attempts to derive similar theorems for the A- and B-amplitudes have shown that one cannot eliminate a contribution of an axial-vector nucleon scattering amplitude. (see sect. 7). From this point of view the Lagrangian calculations represent an estimate of the axial-vector term, which follows from the assumption of A-exchange dominance. If the prescription of Nath et al. is used, the result is not even a zero order approximation. Table 1 shows that constants have to be added to a i and bi, which cancel most of the/x-nonpole terms, but the physical origin is unclear and the exact magnitude is not known. A possible way for an investigation of the difficulty is to continue our study of the relation between the Ward identity and dispersion relation approaches [7], since double counting cannot occur in the second case. Several points of our discussion above are also valid for Peccei's approach [9]. We do not go into the details, but mention only that the study of the electromagnetic form factor of the nucleon [46] and of the backward dispersion relation [47] indicate that a pole term ansatz for p-exchange is only a crude approximation (see also ref. [7] ). Dutta-Roy et al. [10] started from a Lagrangian for N, A, O, e, n in which the NNn coupling is pseudoscalar. The simplest type of coupling is taken in the other cases and a p N N tensor coupling is included. The calculation is not made according to the Feynman rules, since only the pole term is taken into account for A-exchance. The authors concluded that this rather arbitrary prescription leads to a "very good agreement" with the observed s- and p-wave scattering lengths. This result is surprising, since our investigation in ref. [7] shows that essentially the same N, A, p-terms can be obtained from a fixed-t dispersion relation, but there is a further large contribution from the left hand cut o f f +1 (see also Oehme's lecture [29] and our discussion in ref. [47] in connection with the Isaev-Meshcheryakov sum rule). It turns out that Dutta-Roy et al., overestimated the 33-contribution by

G. Hdhler et al., L o w energy n N amplitude

258

40% as a consequence of their choice of the coupling constant. Furthermore there exist so many values for the p-coupling constant that it is almost an adjustable parameter in this method. As far a s A - and B- are concerned, the proposal of these authors is essentially to omit the non-pole terms and this is certainly an improvement according to table 1. We think that prescriptions as those of Dutta-Roy et al. should not be taken seriously unless the authors give a justification from a physical point of view. If the above remarks are taken into account, the "agreement with the experiment" cannot be accepted as an argument. Other investigations can be discussed along similar lines, for instance the (N, A, p) model of Schwarz [12], the work of Carreras et al. [3], Wray [4], Pantin [37] etc. We think that in general the accuracy of the models was overestimated. Most of the authors treated "experimental results" in a rather uncritical way, ignoring the fact that conclusions from the most recent phase shift analyses are more reliable than those from the pioneer work 5 or l0 years ago. Limitations in space do not allow to discuss details, but this can easily be done by a comparison with the present paper and, in some cases, with ref. [7]. Usually the authors of resonance exchange models assumed the validity of Sakurai's universality relation [48] for the p-couplings. Only Kramer [11] considered a more general case and he even found the opposite sign f p ~ f ~ N N < 0. As a consequence he has no difficulty with the negative slope of the nnNN partial wave Re ~ l at t = 0, which cannot be explained in Sakurai's p-dominance model [49] (cf. c~ in table l), however Kramer's model leads to discrepancies in two other places. (i) The study of the 7rN backward dispersion relation allows to determine the nnNN-cut contribution from nN phase shifts. Assuming that the cut is dominated by p-exchange, one finds for the contribution of the vector coupling (notation as in ref. [47] ) Re F 1p -

1 4M2 { R e R ~ 3 4M 2_ t

t ReSt}, 4M 2

(6.3)

where F1 -

4M /f 1 4M 2 -- t

t fl} 4Mx/~

(6.4) "

On the other hand Re Flp is given by

• Re I lo

1 Io..IoNN 3

4n

1 m2 - t

(6.5)

0

Fig. 4 shows the result derived from figs. 5a, b of ref. [47]. It is not in good agreement with a pole term formula, but a negative sign offp~r~fpN N is clearly excluded.

G. H6hler et al., L o w energy ~rN amplitude

259

Re I~lp ' 10 2

-4 I |

-20 I 5

-40 I 10

-80 20

-60 I 15

~ ~

t q2

-2 -3 -4 -5

(L67

ooo>o.f-~----° ,U

L68

-6 Fig. 4. Re P l p ( t ) according to eq. (6.3) L 68 refers to the CERN experimental solution, L 67 to the CERN theoretical solution, ref. [53].

(ii) The Frazer-Fulco theory of the electromagnetic nucleon form factor should give a reasonable approximation for the slope at t = 0. If the sign offpTrTrfoNN differs from the usual one, the spectral function of the Dirac form factor F~ ( or that of the pion form factor) has a dip instead of the p-peak and one would get a negative (r2). [Cf. ref. [46] and ref. [50], p. 279].

7. THE SUM RULE OF GOLDBERG AND GROSS Several authors investigated the question, whether a relation analogous to the Adler-Weisberger relation exists for the coefficient:a~ of the expansion of the A-amplitude (references are given in ref. [51 ] ). It was found that the situation is different insofar as one has an additional term, which cannot be expressed by rrN scattering amplitudes ai -

2 4M ~- F V (0) + ~ - A 11 (0)',

[,,

£.

(7. I)

A 11(w) is an invariant axial vector-nucleon scattering amplitude, which belongs to the tensor [3,ta, 7v] in the complete tensor basis I of Goldberg and Gross [51 ]. A comparison with the Ward identity approaches shows that the first term on the r.h.s, of eq. (7.1) is just the "ETC-term" (5.1). The second term is usually approximated by considering A-exchange only 4M A f2 11(0)~ w~olimco-1 [A~x,p +A~x,np] .

(7.2)

260

G. Hbhler et al., Low energy nN amplitude

It depends appreciably on the choices for the A-propagator and the interaction term. We calculate the left hand side of (7.2) from (7.1) and table 1 * and the right hand side from different prescriptions for A-exchange 8.1 --5.1 4.1 + - 0 . 4 = - 4 . 7 -4.5 4.9 -

c =-lorNathetal. c=0 c=- 1 Peccei Pantin.

(7.3)

The comparison shows that the prescription of Nath et al. leads to a serious discrepancy. Another aspect of this problem can be seen, if we follow Goldberg and Gross [51], who derived an interesting sum rule from the assumption that A 11 (co) and A-/co fulfil unsubtracted dispersion relations

F v (0) -

M2 7r

f

lm A 1 (co) d co.

(7.4)

1

A 1 (co) is the only axial-vector nucleon scattering amplitude which contributes at t = 0 in the soft pion limit. It is related t o A - a n d A l l by

A-(co) _

2

F2V(0) + w M _ 2

A'i (co) + 4M ~ (co), f2 Ah

(7.5)

where the tilde means that the nucleon pole terms is subtracted. Eq. (7.5) is an exact consequence of the basic assumptions of these authors. In the high energy limit the first two terms on the r.h.s, cancel each other. It is interesting to compare the A-contributions on both sides of (7.5). From (7.2) and the imaginary part of (7.5) we conclude that the A-contribution to the sum rule (7.4) is given by the n o n - p o l e part of the A-exchange term

M2 I m A l ( c o ) = -12f2 A~'nP(CO)

~=06(co*-co)"

(7.6)

CO

* The determination of an "experimental" value for A 11(0) is closely related to that of if-(0). See ref. [19].

261

G. H6hler et aL, Low energy ~rN amplitude

The A-contribution to (7.4)

2 F222V(0)IA = al Ia,np

I7

0.61 0.44 0 -3.6 0.2

Peccei if c = - 1 if c = 0 ifc = 1,Nathetal. Pantin

(7.7)

is strongly model dependent and differs much from the total value of the left hand side: 2 F V ( 0 ) / f 2 = 4.28. Goldberg et al. suggested that the missing large contribution in the sum rule (7.4) could be due to o-exchange. Our evaluation which is based on phase shifts and the Regge model fit of ref. [52] confirms that the p-exchange contribution to Re A - l e o at co = 0 is remarkably large, A - being related to A 1 by (7.5) See table 4. Table 4 Contributions to Re A / w at 03 = 0.

P33 = A -

5.1

Pll

D13

DI5

-

-

(1.3

0.5

0.5

F15 0.6

o-exchange k > 2 GeV/c -

1.3

rest -

0.7

sum 8.4

However the p-exchange contribution to (7.4) is still too small, if co A 11 + 0 for w ~ oo as assumed by Goldberg and Gross. It would be interesting to check, whether o-exchange is really forbidden in A 11' Otherwise there could be a large O-contribution to A 1, which is the average over large non-pole terms of high spin resonances, if the notion of duality is applicable.

8. CONCLUSION (i) Coefficients of a low energy expansion of the invariant amplitudes can be calculated with an accuracy, which is comparable to that of the scattering lengths. It is an advantage to consider these coefficients as the "experimental information", for which predictions from models are derived, since each resonance exchange term gives characteristic contributions to some o f the coefficients and their number is larger than that of the scattering lengths. (ii) In a realistic model for the isospin odd amplitude at low energies one needs at least N, A and o-exchanges, p-dominance models (N, P) and the CGLN approximation (N, A) give a wrong s and t-dependence. But even a N, A, 0-exchange model cannot be expected togive accurate results, since the usual description o f the exchanges by pole terms is a crude approximation (see below) and further contributions to the amplitude are not negligible.

262

G. Hiihler et al., Low energy nN amplitude

(iii) The treatment of A-exchange has two difficulties: (a) There exist many different prescriptions for calculating 2nd order Feynamn graphs for this case, but all of them are not satisfactory. This leads to a considerable uncertainty of the prediction for some amplitudes. (b) If the nNA-coupling constant is calculated in the usual way from the decay width, the pole term at low energies is 40% larger than the analogous term in the dispersion theory, which is calculated from the phase shift. This defect is a consequence of the large width. It is possible to improve the pole term by using an effective coupling constant (3.5). But even then one cannot expect an accuracy better than 10% in the region ( M - I ) 2 ~ s ~ (M+ 1)2, I tl < 5. (iv) If p-exchange is described by pole terms, one cannot expect accurate results. Originally one has equal-time commutators, which are expressed by nucleon form factors. It is well-known that the approximation of isovector form factors by p-pole term formulas is bad, for instance the nucleon radius is too small by a factor of two in the case of G v . Formally this can be corrected by considering a "dipole fit", but at least part of this effect is a consequence of the finite o-width [46]. (v) Corrections are expected from unitarity effects, which are neglected in the usual Lagrangian appraoches, since they belong to Feynman graphs of higher order. The most important case is probably the influence of the nucleon singularity just below t = 4 on p-exchange, which enhances the finite width effects mentioned in (iv). Unitarity is also responsible for the singularity of the real parts of the amplitudes at threshold, which is neglected in all models (except ref. [28] ). Because of the cusp-like behaviour, there is no simple and accurate relation between the scattering lengths and the coefficients of the expansion at the crossing symmetry point (t-channel unitarity is treated in ref. [7] ). (vi) At present Ward identities and phenomenological Lagrangians for 7rN scattering have no predictive p o w e r beyond the low energy theorems of current algebra and even these have an appreciable uncertainty, because they are valid for an unphysical amplitude. All other predictions follow from assumptions, which are qualitatively convincing (A, o-exchange), but to a large extent arbitrary as far as the quantitative formulation is concerned. It is misleading to say that a certain set of assumptions is justified by the "very good agreement with experiment", since (a) there are many ways to construct models which agree with experiment, since one can choose different sets of exchanged particles, in some cases different values of masses, widths, coupling constants, coupling terms, propagators etc., and (b) some of the experimental amplitudes are not yet well known. It is also misleading to compare predictions with experimental values, which were obtained long time ago, although different groups presented results derived from phase shift analyses of much more accurate and complete data [45] (unless one gives an argument, why the old values are preferred). We conclude that the main problems in this field are at present to find a physical argument, which leads to a unique expression for A-exchange, to improve the treatment of p-exchange and to study the mechanism of the T = J 0 t-ct-annel exchange, which probably cannot be described by the exchange of a single particle. =

G. HOhler et al., Low energy nN amplitude

263

We are grateful for discussions with Profs. S. Coleman, L. Maiani, B. Renner, H.J. Schnitzer, P. Stichel, J. Wess and B. Zumino.

APPENDIX Determination o f the coefficients in eqs. ( 2 . 5 ) a n d (2.14) The simplest determination of the coefficients would be to evaluate the dispersion integrals, calculating the imaginary parts from total cross sections and phase shifts. In order to improve the accuracy and to get results independent o f specific high energy assumptions, we applied another method, in which the real parts o f the amplitudes are also calculated from phase shifts and charge-exchange forward cross sections. Then our coefficients are treated as unknown parameters, which are chosen in such a way that the dispersion relation is fulfilled as good as possible. This method is well-known and has for instance been used in our earlier work [ 18 - 2 0 ] . The accuracy is limited by the fact that the phase shifts are not quite consistent with the dispersion relations. + + + The determination of c 1 and c 2 is described in ref. [39]. The uncertainty o f c 1 is closely related to that ofa~+, since Re C*(0 0) -- Re C ÷ ( 1 , 0 ) is given by an integral over well-known total cross sections. For the same reason c~ is known as well as a~+ (aside from an error due to that o f f 2 ) . In this case we relied on our work in ref. [20]. The other i = 1 and 2 coefficients in table 1 were determined in an analogous way (see also refs. [18, 19] for bi). The "CERN experimental" phase shift set [53] was preferred, but calculations with other sets were also performed and used as well as the inconsistencies with the dispersion relations in order to estimate the errors. The i = 3, 4, 5 coefficients were determined by evaluating the rapidly converging integrals. In the case of the "A, p" contribution the numbers belong to an evaluation of the dispersion integral over the 33-contribution to the integrand ( T , < 700 MeV). The result is somewhat different from the pole approximation, a fact which should be noticed if cancellations occur as for instance in eq. (5.18). The coefficients in (2.14) were determined as the sum o f the coefficients in (2.5) at co = 0 and the rapidly converging integral for the difference between ~o = 1 and 0. Approximately they also follow by summing the first term of the series (2.5), using table I. However this is not a reliable method, since the series does not converge at threshold. *

* This difficulty was ignored in the recent paper by Altarelli et al. [54]. It has the consequence that their new sum rule cannot be used for reliable calculations, unless the error is estimated, for instance from our evaluation of the dispersion integrals.

G. H6hleret al., Low energy 7rNamplitude

264

The q2-terms in (2.14) follow from

dk 2

~

ReC 1

2n2

k2=O= M(1-1/4M2) 2 dk {o+(w)_ o+(1)}

(A.1)

f 0

dk2[.4 rr Re C-] k2=O- (1-1/4M2) --2f2 2 d Foo-1

2n 2

-~

-(co)

a-(l)

.

0 Denoting the integrals by I ±, one obtains the effective ranges of the s-wave amplitudes (2.13) from

•

+ , a0+ b+=-al_-2al+-~M-+~

f2 {M(l~l/4M2)2~I+ }

M+ 1

(A.2)

b - = - a ~ _ - 2ai+ + l a~ M+ 1 [

+M

( 1 + ~ /,+

-2f 2

I

((1-1/4M2)2 +I-].

The integrals I ± are dominated by the 33-contribution. If they are evaluated in the narrow resonance approximation and inserted into (A.2), one obtains in a very simple way the result of an investigation by Hengeltraub et al. [55], who attempted to extend the Fubini-Furlan method [28]. The discrepancy between their result for g*2/4n and our eq. (3.9) is a consequence of their approximations and the use of old experimental values. One could think that the determination of the effective ranges b ± is difficult, since these quantities occur also in the total cross sections in the integrand of/±

~--(a+o+)2t2(a~+)2+2q 2 a+o+b*+2a~+b-+ a-

4n

+ (a~_) 2 + 2ao+ao+ + 2q 2 \

1 (~41 _ a4)~ + )

( a 4 + 2 a 4) + .... (A.3)

ao+b ao+b +ao+b -

- +

-

+

-

265

G. HOhler et al., Low energy ~rN amplitude

but this is not serious, since the higher terms of the effective range expansion are important already at the lowest experimentally accessible energies (ref. [26] and fig. l in ref. [14] ). F o r an estimation o f the errors of b +- one should better start from Re f0+ as expressed by the partial wave projection of the fixed-t dispersion relation [241. The value of the total cross section or- at threshold is not well known because of its dependence on a~+. Taking - 0 . 0 2 and 0 for this scattering length, one obtains o - ( 1 ) = 1.5 mb, and 2.5 mb respectively. The cusp effect is described by

Re ~ + ( w ) = Re ~+(1) + k2 2 rr

f dk' 0 k'2-k2

_ o÷(1) X / ~ _ ~ - 2 0 @ k 2 )

I v + ( ° ° ' ) - °+(1)]

, (A.4)

cam Re ~ - ( c o ) = Re C ( 1 ) + k 2 2 0

k'2-k2

- o-(l) N/1-oo2 0(-k2) .

NOTE ADDED IN PROOF A new determination of the coefficients " a +1, a2+ from 1971 CERN phase shifts led to a~ - g2/M = - 1.53 +- 0.2, a~ = 1.11 + 0.02 (H.P.Jakob, CERN preprint TH 1446, Dec. 1971).

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

K. Raman, Phys. Rev. 164 (1967) 1736 and references given there. H.J. Schnitzer, Phys. Rev. 158 (1967) 1471. C. Carreras and A, De Rujula, Nuovo Cim. 64 (1969) 1014. D.A. Wray, Ann. of Phys. 51 (1969) 162. E.T. Osypowski, Nucl. Phys. B21 (1970) 615. J.C. Huang and J.R. Urani, Phys. Rev. D1 (1970) 877. G. Hb'hler and P. Stichel, Z. Phys. 245 (1971) 387. J. Wess and B. Zumino, Phys. Rev. 163 (1967) 1727. R.D. Peccei, Phys. Rev. 176 (1968) 1812. B. Dutta-Roy, I.R. Lapidus and M.J. Tausner, Phys. Rev. 177 (1969) 2529, ibid. 181 (1969) 2091. l l l ] G. Kramer, Phys. Rev. 177 (1969) 2515. [12] J.H. Schwarz, Phys. Rev. 175 (1968) 1852.

266 [13] [14] [15] [16] [17] [18] [191 [20] [21] [22] [23] [24]

[25] 26] 27] 28] 29] 301 31] 32] 331 [34] [35] [36] [37] [381 [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]

G. HOhler et al., L o w energy nN amplitude

J. Hamilton and S. Woolcock, Rev. Mod. Phys. 35 (1963) 737. P. Achuthan, G.E. Hite and G. H~3hler, Z. Phys. 242 (1971) 167. S. Adler, Phys. Rev. 140 (1965) B736. G.F. Chew, M.L. Goldberger, F.E. Low and Y. Nambu, Phys. Rev. 106 (1957) 1337. G. H/3hler and R. Strauss, Z. Phys. 232 (1970) 205. G. H6hler, H.G. Schlaile and R. Strauss, Z. Phys. 229 (1969) 217. G. Hohler and R. Strauss, Z. Phys. 204 (1967)419. G. ttohler and R. Strauss, Z. Phys. 240 (1970) 377. W.R. Frazer and J.R. Fulco, Phys. Rev. 117 (1960) 1603. D. Amati and S. Fubini, Ann. Rev. Nucl. Sci. 12 (1962) 359. T. Pradham, E.C.G. Sudarshan and R.P. Saxena, Phys. Rev. Lett. 20 (1968) 79. J. Baacke, G. HOhler and F. Steiner, Z. Phys. 221 (1969) 134; J. Baacke and F. Steiner, Fortschr. Phys. 18 (1970) 67 and unpublished work by these authors and Drs. H.G. Schlaile and R. Strauss. H. Kottler and R.S. Willey, Phys. Rev. 178 (1969) 2146. G. Hohler and K. Dietz, Z. Phys. 160 (1960) 453. S. Mandelstam, Phys. Rev. 115 (1959) 1741. S. Fubini and G. Furlan, Ann. of Phys. 48 (1969) 322. R. Oehme, Nucl. Phys. B16 (1970) 161, Springer Tracts in Mod. Phys. 57 (1971) 119. E. kohrmann, Proc. of the Lund Int. Conf. on elementary particles, (1969) 11. C. Fronsdal, Nuovo Cim. Suppl. 9 (1958)416. W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61. L.W. Chang, Phys. Rev. 162 (1967) 1497. B. Petersson, Lecture notes, Int. Summer School at Karlsruhe (1968). L.M, Nath, B. Etemadi and J.D. Kimel, On the uniqueness of the interaction of spin 3/2 particles, Florida State University, Tallahassee preprint (1971). C.R. ttagen, New inconsistencies in the quantization of spin 3/2 fields, Rochester preprint (1971). C.F.A. Pantin, Higher spin baryons and chiral Lagrangian calculations, Cambridge preprint DAMTP 71/3, England. T.P. Cheng and R. Dashen, Phys. Rev. Lett. 26 (1971) 594. G. Hohler, H.P. Jakob and R. Strauss, Phys. Lett. 35B (1971) 445. G. Hohler and H.P. Jakob, Nuovo Cim. Lett. 2 (1971) 485. J. Engels and G. HOhler, Int. Conf. on elementary particles at Kiev (1970); J. Engels, Nucl. Phys. B25 (1971) 146. P. Achuthan and F. Steiner, unpublished calculation, University of Karlsruhe. tt. Genz and F. Steiner, Nuovo Cim. Lett. 1 (1971) 355. R. Strauss, Thesis, University of Karlsruhe (1970). S. Adler, Phys. Rev. 137B (1965) 1022. G. Ebel et al., Springer Tracts in Mod. Phys. 55 (1970) 239, Compilation of coupling constants and low energy parameters. G. Hohler, R. Strauss and H. Wunder, Calculation of isovector nucleon form factors, Karlsruhe preprint (1969). J. Engels, G. Hohler and B. Petersson, Nucl. Phys. BI5 (1970) 365. J.J. Sakurai, Currents and mesons (The university of Chicago Press, 1969). G. H6hler, J. Baacke and F. Steiner, Z. Phys. 214 (1968) 381. J. Hamilton in High energy physics I, Ed. Burhop (Academic Press New York, 1967). H. Goldberg and F. Gross, Phys. Rev. 162 (1967) 1350. G. HOhler, J. Baacke and G. Eisenbeiss, Phys. Lett. 22 (1966) 203. rrN Partial Wave Amplitudes, Berkeley report UCRL-20030 (Feb. 1970). G. Altarelli, N. Cabibbo and L. Maiani, Phys. Lett. 35B (1971) 415. A. Hengeltraub abd J. Vasconcelos, Nuovo Vim. 64A (1969) 649. M. Sugawara and A. Tubis, Phys. Rev. 138 (1965) B242. G. Hi~hler and R. Strauss, Tables of pion-nucleon forward amplitudes. Karlsruhe preprint (Oct. 1971).