Application of modulus-phase dispersion relations to πN backward scattering

Nuclear Physics B88 (1975) 442-450. © North-Holland Publishing Company

APPLICATION OF MODULUS-PHASE DISPERSION RELATIONS TO IrN BACKWARD SCATTERING

J.L. SANCHEZ-GOMEZ and L. PARAMIO Departamento de F[sica Teor[ca, Universidad Autdnoma de Madrid and G.L F. T., Madrid Received 2 October 1974 (Revised 13 December 1974)

We derive modulus-phase dispersion relations for the ~rN backward scattering amplitudes. The convenience of these dispersion relations in treating ~rN backward scattering is pointed out. A numerical evaluation of the ~r-p backward scattering phase is also given.

1. Introduction In this paper modulus-phase dispersion relations (MPDR) [1] are introduced as a practical tool to study ~N backward scattering, especially in the intermediate (s ~ 16 GeV 2) energy region where the knowledge of the backward amplitudes seems to be of decisive importance in order to understand the (apparently) complicated 7rN backward dynamics. In sect. 2 we write the MPDR for the ~±p backward elastic scattering amplitudes (BSA). A numerical procedure to evaluate these MPDR is developed in sect. 3. In sect. 4 numerical results of the ~ - p backward phase are given for energies s ~ 16 GeV 2. The treatment here is rather analogous to that previously applied by Alvarez-Estrada et al. to pion photoproduction [2]. The type of reactions a + b -~ c + d (m b = rod) have been thoroughly studied by Hite and Jacob [3] both in forward and backward directions, generalizing Atkinson's method for nN backward scattering [4]. The treatment in this paper is admittedly less general than that of [3]. The aim, though, is different and we believe we will show the application of MPDR to nN backward scattering to be of practical interest.

2. MPDR for ltN BSA Let us start by defining the ~±p BSA in the following way --~t)2 A~(t) + IM(s - u)(M 2 - ~t) 2 B ±(t)),

(1)

J.L. Sanchez-Gomez, L. Paramio / 7rN back scattering

443

where A and B are the standard zrN invariant amplitudes. Notice that according to this definition C is dimensionless. Kinematics in the backward direction states s = M 2 +/l 2 -

u=M 2 +/2-

~1 t - - 2 P t q t ,

(2a)

~t + 2Ptqt ,

(2b)

where

Pt

=

(¼t - M2) ~ ,

(3a)

qt = ( i t - / / 2 ) ~ ,

(3b)

and the definition of the square roots is, as usual, Ptqt > 0, t >4M2; Ptqt < 0, t < 4#2; P t q t = +i IPtl Iqtl, 4# 2 < real t < 4 M 2 and t = real t -+ ie. As already discussed in [4], C±(t) are analytic functions in the complex t-plane cut along the real axis from t = 0 to _oo and t = 4/~2 to oo. C_ has also a pole at t~

p_

=

4g 2

#4 M2

-

0.077 GeV 2 ,

(C+ has no poles). This analytic structure is shown in fig. 1. Now upon making use of the analytical properties just mentioned and by following a method very similar to that developed in [2] (see also [1] for more details) we are able to write down the following MPDR for the functions C±(t)(_oo < t <~0): qJ+(t) -= arg C(t + ie) = arg D(t + ie) + I[t(t - 4/a2)]-~ I -

/1"

X

(;2

4#

dt'

In IC(t') I ) lntC(t')l - Jf 0 dt' (t' - t) [t'(t' - 4/22)] ( t ' - t) [ - t ' ( 4 / ~ 2 - t')]~ ' -~

(4)

t)

p--~+ J,',5'/,"//,;',"//%'/////V/,; rc-p~

p~ _~7"

~

~.~ . , , , , . . . . , , , , , , , , , , , , , , , , , l l , l , l , , , ~ , l l l l l l l l l l / l l l l l l l l l l l / l l l / l l FIII//II/ilIIII,'Iil/I//IIIII/////(~II///II////Ill//I////////I//II" 4~ a ~ M 2

Fig. 1. Analytical structure of C_ in the complex t plane.

~I

444

J.L. Sanchez-Gomez, L. Paramio /lrN back scattering

Table 1 Real zeros of the n±p ~ n±p BSA in the complex t-plane (units GeV2) CERN III

SACLAY

zero lr+p

0.063

0.055

zero n-p

0.075

0.074

where the symbol f stand for integral principal value. D ( t ) is given by

(s)

O(t) = Dr(t ) Dc(t) ,

where [ t ( p - 4#2)] ½ + [ p ( t - 4#2)] ~ [t(r - 4#2)] ½ + [r(t - 4#2)] ~

Dr(t) = [t(p - 4#2)] ½ - [p(t - 4#2)] ½ [t(r - 4#2)] ~ - [r(t - 4#2)] 2x '

(6)

Dc(t) = 1-I [t(zi - 4#2)]-~ - [zi(t - 4p2)]~ [ t ( z t - 4p2)]~ - [z~.(t - 4p2)] ~ i

[t(z i _ 4#2)]~ + [zi( t _ 4#2)]½ [t(z~. - 4#2)] ½ + [zT(t - 4#2)] -~ '

(7) p, r, z i being the positions of the pole (rr- case) real zero and complex ones, respec-

tively. We have written "real zero" because there exists just one in each amplitude. Their positions have been previously determined [5] by means of fixed t dispersion relations using CERN and Saclay phase shifts and the results are given in table 1. The usefulness of eq. (4) lies in the fact that ICI is related to experimental data over most of the integration region, i.e., for t outside the interval (4# 2, 4M 2) which is unphysical, through the following equations 1

do

±

t~<0,

iC±(t)12 = 4E2s (M2 _ gt)_d__~(7r p ~ p

t > ~ 4 M 2,

Pt d o . _ IC±(t)12 =-~t t - - ~ p p ~ n ± T r ± ) ,

±

±

7r ) ,

(8a)

(8b)

where E = (s + M 2 - #2)/2s½. Notice that, because of its definition, C has not any kinematical singularity (whereas A' has one at t = 4M2). Notice also that the rr±p BSA phase is n o t ~± but (see fig. 1) q~± = 2rr - ff~. 3. The discrepancy function 4± Due to the presence of an unphysical interval and also to the possible existence

445

J.L. Sanchez-Gomez, L. Paramio / 7rN back scattering

of complex zeros of C± it is convenient to introduce the following quantity (discrepancy)

A(t)=tb(t)_ argDr(t) I[t(t-41~2)]{l[SM

dr' 2 t'-t

7r

×

in IC(t')l

2 1- f 0

(t' - t) [t'(t' - 4/1 )]5

_~

dt'

lnlC(t')l

t' - t [ - t ' ( 4 / l ~ - t ' ) ] { J

q "

(9)

We will compute A± by using nN phase shifts in the region where the last are available (and reliable) i.e. s ~ 4 GeV 2. (Notice that although the calculation is performed taking t as variable the results are expressed in terms of s for obvious convenience.) Then we will attempt extrapolating A to higher energies which will allow us to get the backward phases there. In order to compute A± we have used (i) Experimental data in the scattering channel [ 6 - 8 ] (see below for further comments about the data). (ii) Experimental data on the annihilation reactions p p -+ 7r+rr- [9,10] and a naive Regge-like extrapolation for the high energy part in this channel. As we are calculating in the intermediate energy region of the scattering channel, that part of the crossed one should be of little importance. (iii) The backward phases 45 have been calculated by means of the CERN [11] and Saclay [12] sets of phase shifts. We should comment, at this point, that these two sets give rather different values of the backward phases at certain energy points (a well-known fact since the backward phases are very sensitive to interferences among different partial waves). So that to be sure as much as possible of the reliability of the computed phases we here considered only those points where both sets give practically the same results. The discrepancy functions A± are shown in fig. 2a,b respectively. We see that a rather strong structure appears in A+ (fig. 2a) around s = 2.5 GeV 2. This should reflect the presence of a complex zero in the neighborhood of that energy point. (The rrp -~ rrp backward d.c.s, has a dip in the proximity o f s = 2.5 GeV). Therefore, extrapolating A+ to higher energies is not so simple and a detailed numerical treatment is in order. For this reason we will forget about the 7r+ case in this first paper. Concerning A one gets, in principle, an appreciable structure near threshold (s = 1.2 GeV2). This should be due to the existence of a complex zero of C_ in the vicinity of threshold. The zero position (in fact there are two complex conjugate zeros) has been determined as shown in the Appendix to be s --- (1.2713 -+i 0.0038) GeV 2. The complex zero has been taken into account in the new computation of A which is the one shown in fig. 2b.

446

J.L. Sanchez-Gomez, L. Paramio

~rN back scattering

A + ( Oeg )

18C

15C

80-

t t tt t t t t s(Gev 2 ) f/ll2

IBO

l

l

115

4

I q 5

'

l

l

l

~

p

I

,

I

,

~

,

q

,

,

,

~

. . . .

,

,

I

,

I

.

25

~!

,

A_{Deg )

150

I0{ s

, 1.3

,

,

,

! 25

.

.

.

(Gev 2) I. 35

Fig. 2. The discrepancyfunction (a) A+, (b) A_.

4. Phase of the lr-p backward scattering amplitude First of all, we will comment about the 7r-p backward scattering data: (a) Low-energy region (s <~ 2 GeV2). The angular distributions measured by several groups [13] have been extrapolated to 0 = 180 °. This region presents no problem because all groups pracitcally agree. (b) Resonance region ( 2 - 4 GeV2). The data of Rotschild et al. [14] and Binnie et al. [7] have been used. There exist other sets of data which disagree with those just mentioned in the region 2.4-3.2 GeV 2 where they find a backward cross section systematically bigger [15]. Our election has been motivated mainly because of

447

J.L. Sanchez-Gomez, L. Paramio /~rN back scattering

the apparent consistency of the results presented in ref. [7] and their agreement with those of Rotschild et al. However, new experimental results in the resonance region would be valuable in order to eliminate this incertitude with the data. (c) "Intermediate" ( 4 - 1 6 GeV 2) region. There are several measurements [16] in this region, although the data start to be rather scarce from s = 12 GeV 2 onwards. (d) High energy region (s > 16 GeV2). There are just a few points where the backward cross section has been measured [17,18]. Consequently, we have parametrized the BCS according to an "effective GORE" model [19] in the following way (da) ~-~

=A(s+M2_~2)2s~

(10)

'

180 °

A and ~ being obtained through a fit to the experimental data (mentioned above) with the results A = 46.934 ~b/GeV 4,

a -- - 2 . 8 9 4 .

This fit is shown in fig. 3. Notice that in our parametrization o(180 °) ~ s -0"9 for large s whereas the GORE model gives o(180 °) ~ s - I , if ~ 0 = _ 0.5. After having discussed the way of dealing with the data, we come now to the extrapolation of the discrepancy. As already mentioned, the best procedure to extrapolate A should be a modelindependent one. However, to do this one needs much more (and more accurate) data. For this reason, in this first paper we shall present a much simpler (though model=independent) method of performing the extrapolation of A_ taking advantage of its ~moothness. The experience gained in dealing with pion photoproduction tells us that what is important is to know 4CI in the unphysical region just in some average sense so that no detailed knowledge of ICl is necessary [2]. Consequently, we have param-

S (Gev z } eo

z

30

Fig. 3. Fit to the high energy data (see the text). Squares, ref. [17] circles, ref. [18].

ZL. Sanchez-Gomez, L. Paramio /~rN back scattering

448

etrized C_ in the interval (4/~2, 4M 2) as: C = C (Born) + (4/~2 - t)~ -

-

aM +b(t_ t - Mp 2- iFpMp

4/.t2) I + d ( a M 2 _ t) ~, (11)

where M p , Pp are the p-mass and width and the parameter a plays the role of "effective" coupling constant, b and d are the (real) background parameters. All other possible resonances are not taken into account due to the above-stated reasons. We will presently see that this parametrisation accounts reasonably well for the discrepancy in the region where the last is known. The parameters are not quite free but they are restricted so to have C_(t = 4/a 2) = 350 -+ 50,

C _ ( t = 4M 2) = 38 + 8 .

(12)

The first condition is imposed in order to get C_(4# 2) in agreement (within "generous" error) with the theoretical calculation of Nielsen et al. [20] whereas the second one comes from an extrapolation to physical threshold of experimental data in the annihilation channel [9]. (Also we have allowed large errors in this case.) The Born term is given by C (Born) = M Gr2/4rr (1a2 -

-

1 ~-

~t)2

M 2 - s

(13) '

_(Deg.) 90

70

50

30

I0

•

S (Oev 2 ) /"

, ,

I

i

S

6

I

~ 8

I

I I0

I

I 12

I

I 14

i

I 16

Fig. 4. Phase of the ~r-p backward scattering amplitude. The circles show our results. The continuous line is the prediction of the GORE model [19]. The broken line represents the results with the model of ref. [22].

8

J.L. Sanchez-Gomez, L. Pararnio / ~rN back scattering

449

with 1Gr2 = 14.6. In fig. 2b we show A_ and the best fit to it using eq. (10). The resulting parameter values are a =-1.5,

b = 18,

d = 185.

Now upon using C_ as given by eq. (11) with the preceding parameter values one can proceed to calculate the ~r-p BSA phase. The results are shown in fig. 4. Again we can observe the presence of a remarkable structure near s = 5 GeV 2. This is probably due to the existence of a complex zero in the vicinity of this energy. The backward zr- cross section shows a dip in s = 4.5 GeV 2 [6]; also the existence of a zero (or at least an "almost zero") in the backward amplitude can be deduced from the work of Buttimore and Spearman [21 ]. As for higher energies, we have stopped the calculation at s = 16 GeV 2 because of the already mentioned difficulties with the high energy data. In fig. 4, we also represent the backward phase predicted by the GORE [19] and Donnachie-Thomas [22] models. Notice the flatness of the phase calculated by means of these models in contraposition of the structure shown in our calculation (even at s > 8 GeV2). Whether the structure comes from inadequacy of the data or is more fundamental is difficult to discern now. To finish with, we would like to point out that the method expressed here can also be employed for dealing with the annihilation reaction NN ~ rrrr where no acceptable theoretical treatment seems to exist. The only inconvenience in doing this now is the lack of enough experimental data to calculate the first integral in eq. (4) (which is this case would be the principal part integral) and hence the necessity of a good set of experimental data. Should there exist in the future, the present treatment would be a convenient one. We thank Drs. R.F. Alvarez-Estrada, Dr. B. Carreras and Professor F.J. Yndurain for comments and suggestions and also Dr. J.A. Rubio for information on the experimental situation in ~p ~ zr+Tr- .

Appendix In this appendix the position of the complex zero (near threshold) of the l r - p BSA is determined. Recalling the definition of C_ [eq. (1)] and using the CERN phase-shift [12] we parametrize C_ as C

= 0.319 - 0.1309 q2 _ 0.4784 q4 + iq(0.1822 - 0.3006 q2 + 0.1087 q4), (A.1)

where q is the c.m.s, momentum in units of#. This Tayler expansion is supposed to be valid in a region of the complex q-plane defined as 0~
Imq~l

.

450

J.L. Sanchez-Gomez, L. Paramio / 7rN back scattering

U p o n solving the e q u a t i o n C = 0 we find q = 0.834 + i 0 . 0 1 6 , which lies well within the appropriate region. With this value we obtain the zero position in the s and t c o m p l e x planes t = ( - 0 . 0 5 4 2 T- i 0.0021) GeV 2, s = (1.2713 -+ i 0.0038) G e V 2 . U n f o r t u n a t e l y these values c a n n o t be checked by using the Saclay, phase shifts because t h e y are n o t k n o w n at e n o u g h low energy points.

References [1] R.F. Alvarez-Estrada, Ann. of Phys. 68 (1971) 000 and references therein. [2] R.F. Alvarez-Estrada, B. Carreras and J.L. S~nchez-G6mez Nucl. Phys. B48 (1972) 141; B58 (1973) 254. [3] G.E. Hite and R. Jacob, Phys. Rev. D5 (1972) 422. [4] D. Atkinson, Phys. Rev. 128 (1962) 1908. [5] L. Paramio and J.L. S~nchez-G6mez, Anales de Fisica, to appear. [6] Particle Data Group, Report LBL-63 (1973). [7] D.M. Binnie et al., Paper presented at the 2nd. Aix-en-Provence Conf., September 1973. [8] W.F. Baker et al., Phys. Rev. Letters 32 (1974) 251. [9] F. Eisenhandler et al., Phys. Letters 47B (1973) 531. [10] A. Brabson et al., Phys. Letters 42B (1972) 287. [11] S. Almehed and C. Lovelace, Nucl Phys. B40 (1971) 157. [12] R. Ayed, P. Bareyre and Y. Lemoigne, Saclay preprint (1972). [13] D.V. Bugg et al., Nucl. Phys. B58 (1973) 378; H.R. Rugg~ et al., Phys. Rev. 129 (1963) 2300; L.K. Goodwin et al., Phys. Rev. Letters 3 (1959) 512; P.M. Ogden et al., Phys. Rev. 137 (1965) 1115; A.D. Brady et al., Phys. Rev. D3 (1971) 2619. [14] R.E. Rotschild, Phys. Rev. D5 (1972) 499. [15] P.J. Duke et al., Phys. Rev. 149 (1966) 1077; J.A. Helland et al., Phys. Rev. 134 (1964) B1062. [16] S.W. Kormanyos et al., Phys. Rev. 164 (1967) 1661; R.J. Ott et al., Phys. Letters 42B (1972) 133; D.P. Owen et al., Phys. Rev. 181 (1969) 1794; E.W. Anderson et al., Phys. Rev. Letters 20 (1968) 1529; W.F. Baker et al., Nucl. Phys. B25 (1971) 305. [17] E.W. Anderson et al., ref. [16]. [18] D.P. Owen et al., ref. [16]. [19] E.L. Berger and G.C. Fox, Nucl. Phys. B16 (1971) 1. [20] H. Nielsen, J. Lyng Petersen and E. Pietarinen, Nucl. Phys. B22 (1970) 525. [21] N.H. Buttimore and T.D. Spearman, Dublin preprint TCD 1972-2. [22] A. Donnachie and P.R. Thomas, Nuovo Cimento 19A (1974) 279.