PH YSI CAL REVIEW VOLUME 110, NUM HER 5 JUNE 1, 1958

p Scattering and the Dispersion Relation"

HONG —YKK CHIULaboratory of Nuclear Studies, Cornell IInipersity, Ithaca, Peso Fork

(Received February 10, 1958)

Theoretical values for D (cu) in the theory of pion dispersion relations have been recalculated with asuitable modi6cation of the experimental values for O.t,& . This modification is supported by the resultsobtained from a recent phase-shift analysis. Agreement with the experimental values for D (or) is achievedfor pion kinetic energies 220 Mev. It may be concluded that the theoretical values for D (co) have astrong dependence on the detailed shape of the peak of o.t,t near the resonance.

I. INTRODUCTION'QUPPI and Stanghellini have shown' that there is a

discrepancy between the dispersion relation forpr —p scattering and the experimental results. Severalauthors' ' have attempted to remove this disagreement,but, although the discrepancy has been reduced, it hasnot disappeared. It is the purpose of this paper toshow that this discrepancy can be further reduced bya suitable modiication of the experimental data forthe pr —p scattering cross section. Such a change was6rst suggested in reference 2 and it was later discussedin a more detailed way by Hamilton. 4

In Sec. II a dispersion relation which is slightlydiferent from Goldberger's original formula' will bederived by carrying our the subtraction procedure at150 Mev (pion kinetic energy). This is used in ourcalculation. We also discuss the new form of the depend-ence of this relation on the value of the coupling con-stant f' In Sec. I.II we discuss possible modificationsof the total cross section, and show that the modifiedcross section is somewhat similar to the values whichwere obtained from a recent phase-shift analysis. ' InSec. IV the results are discussed.

II. DISPERSION RELATION

The dispersion relation for pr —p scattering asderived by Goldberger et ul. ' has the following form:

1 t' coi 1 ( coiD-(~)—

I1+- ID-(t )—-I 1——ID+(t )

2E ti 2& t)1 t "dto'- o o~ f' k'

=—k'P ' + —2—,(1)4s' "„k' o)' —pp co'+co tt' to+tt'/2M

where or is the total pion energy, p, the pion mass, kthe pion momentum, D (&o) and D~(co) the real parts ofthe forward scattering amplitudes for sr —p and pr+ —Pscattering, respectively.

* Supported in part by the joint program of the 0%ce of theNaval Research and the U. S. Atomic Energy Commission.' G. Puppi and A. Stanghellini, Nuovo cimento 5, 1305 (1957).

2 M. H. Zaidi and E.L. Lomon, Phys. Rev. 108, 1352 (1957).' G. Salzman (to be published).'I. Hamilton, Phys. Rev. 110, 1134 (1958), preceding paper.5 Goldberger, Miyazawa, and Oehme, Phys. Rev. 99, 986 (1955).

H. Y. Chiu and E. L. Lomon (to be published).

The dependence of D (cp) on f' in the above formulais rather strong in the region where we are interested,i.e., 0—300 Mev. For example, taking to=2tt, (i.e., 140-Mev pion kinetic energy), k'=3(tie)' and, with f'=0.08,we have

f' tr k'I

= —0.24,tt' E co+tt'/2M)

(1a)

which is of the same order as D (tp) itself.This strong dependence on f' can be avoided by

choosing a subtraction energy co'&tt. Then Kq. (1)becomes

i( co) 1(D-(~)—I 1+—ID-(~') —-I 1—ID+(~')2( M) 2E co)

kpdor p 0 0+(cps —co s)p il +

4m' kp —k top —cp cop+to-

f2 ( k2 kis ) (~2/2M)2 tts—2—I I . (2)

tt' (co+tt'/2M) (tt'/2M)' —co"

We have arranged so that D (co) must pass through theprescribed value D (co') at to=co'. Since (tt'/2M)'= (tt/14)' may be neglected, we write the terms con-taining f' as

f'( k' —k" ) tt'

tt' l'co+ p, '/2M) co"

As compared with (1a), the f' term is decreased bothby the factor' tt'/co"= 1/co", and by the factor (k' —k")/k', which will be small in the region near the resonanceif we choose or' —p = 150 Mev. In this manner we reducethe dependence of D (co) on f'.

Although the dependence on f' is reduced, D (co)becomes more dependent on D (co'). In fact, D (to')must be determined by experiment, just as Lin (1)jD (tt) had to be determined by the 5-state scatteringlength. Since the result of Ashkin et al. at 150 Mev isbelieved to be very accurate, it is convenient to give

~ Units which will be used in this paper are the natural unitsA=c=p, =1.The unit of area is then 20 mb.

s Ashkin, Blaser, Feiner, and Stern, Phys. Rev. 101, 1149 (1956).1140

—p SCATTERING AN D THE DISPERSION RELATION

co' this value. The values of D (co') and D~(oo') are0.26&0.01 and 0.47~0.02, respectively. '

80

80(0)

80-

0„'(in mb)

40-

:t

20-

l00I

200 300

FIG. 1. The total sr cross sections oto& . Curve (a) is that usedin the present calculation. Curve (b) shows Anderson's results. e

Above 300 Mev the results of R. Cool et al )Phys. Rev. .103, 1082(1956)g are used.

m. TOrXL ~-—p CROSS SECTION

In reference 2, Zaidi and I.omon have shown theeffect of a modification of o-t,,~ on the theoretical valueof D (co). However, as was pointed out in reference 4,there may be objections to the actual form of 0&,twhich was used. For example:

(a) It is questionable that there is such a rapid riseof the T=-,' total cross section in the region 100—170Mev. The slope obtained by recent work' of Andersonet Gl. is about 0.75 mb/Mev for sr, and about 2.2mb/Mev for sr+ in this region. Zaidi and Lomon use anaverage s. slope in this region of about 1.0 mb/Mev.It is hard to believe either that the T=-,' cross section

JS 406

kob

20

IOOk

200E (Mev)

500

Fio. 2. Curve (b) is the same as (b) in Fig. 1. Curve (a) is theresult obtained by our interpolation of the phase shifts as dis-cussed in Sec. III.

The integrand of the first term on the right-handside is the slope of the chord connecting the two pointsf(x) and f(x'); when x' is near the peak there is alarge contribution from the vicinity of the peak, andthe principal value of the integral therefore dependsstrongly on the shape of the curve near the peak, andto a lesser extent on f(x') itself.

We therefore suggest that the peak of 0-&.& in refer-ence 9 should be raised by about 7 mb; the resultingvalues are plotted in Fig. 1 together with Anderson'svalues. Justification for this alteration in ot,,t is sup-ported by the values obtained from interpolation ofphase shifts. The phase-shift analysis' is performedunder the assumptions that only S and I' waves con-tribute and that 5» is of the Fermi type. Then Ashkin'saccurate experiments'" give unambiguously positivevalues of 8~3 at 150 and 170 Mev with an M value ofthe order of 10. At 220 Mev there are two solutions for

will rise so rapidly or that charge independence willbreak down at such a low energy.

(b) The large values of o&,& in the region 150—200Mev; these are appreciably greater than the experi-mental values.

Besides the above objections, there seems to be nophysical reason to believe in the "plateau" which occursnear 60 Mev on their cross section curve.

However, Zaidi and I omon's results have shown theimportance of the sharpness of the resonance peak ofo&.& for the theoretical value of D (a&). That thesharpness of the peak gives an important contributionis clear from the way the principal integral is calculated.We have

20,

kp-

(y 0 Q

~0

47 pWQ +

O~40 o~00- IO-

8

I' f(x) f'f(x) f(*')—I' . dx= dx+f(x') ln

x—x'

5—x

x —8I

9 H. L. Anderson, Proceedhngs of the Sixth Annual RochesterConference on High-Energy Physics, 1956 (Interscience Pub-lishers, Inc. , New York, 1956), p. I—22.

=20

t Pion G. M. Momentum)

FzG. 3. The small S and P phase shifts as obtained in reference6. The values of 833 used are similar to those in references 8 and10."Ashkin, 31aser, Feiner, and Stern, Phys. Rev. 105, 724 (1957).

HON 6 —YEE CH I U

0.3-

2- ~ (I)—f a0.08

(2)-—f~0.07

I44li (3)—-f *OD82

h Recolcvloted

o1

a P.l 77a& a-009Io1*0.I 67ops-O. IOI

O 1ap. l65

o &=-O. I pl

0(W) ot I70Mev

O.l

5200

/300 /

//

-0 I-

11220& &330

307'& ..

E"„'s(Mev]

pp s 1 s s 1 rI I

2p

IV. DISCUSSION OF RESULTS

The result of our calculation of D (&o) is plotted in

Fig. 4, together with the results obtained by Salzman, 'who used Anderson's values of 0-~,~ . From a comparisonof Salzman's result with ours, it is clear that the theo-retical value for D (ro) has a strong dependence on thedetailed shape of the peak of 0~,~ . It is seen that thereis, in general, agreement with the experiments up to220 Mev. At higher energies, the only data availableare those by Zinov and Korenchenko"; there the agree-ment is poor. However, their way of calculating D (ro)

may possibly be open to question. While for the forwardscattering cross section they use the differential cross-section value, for the total cross section they use thetransmission measurement value. At lower energies theabsolute value of the differential cross section has beennormalized by comparing the integrated result with thetransmission value. It is not easy to make this compari-son at energies such that pion production can occur.

-0.2 2I7II

V. CONCLUSION

Fro. 4. Curve (1) is the result of our calculation of D (co).Curves {2) and (3) show Salzman's results. The experimenta1 re-sult for D (~) at 170 Mev is also recalculated with o~q takenfrom curve (a) oi Fig. 1.

8~3, and from continuity requirements with respect tothe 150- and 170-Mev experiments the solution which

corresponds to b»=8» is rejected and the remaining

solution, for which 6~3 rises rather rapidly from 170 to200 Mev, is accepted. With these phase shifts, an

interpolation is made and the resulting 0-~,~ is plottedin Fig. 2. The phase shifts which give these values of0~,~ are plotted in Fig. 3.

The peak in 0.~,& at 180 Mev is not as high as thatused in our calculations. However, the steep slope gives

a larger contribution than earlier calculations. At leastthe phase-shift analysis illustrates that it is not un-

reasonable that 0-&,& has a sharper peak than is generallybelieved.

We have used Eq. (2) to calculate the theoreticalvalue of D (ro), with a modification of Anderson'sresults for trr r, . General agreement is obtained for pionkinetic energies up to 220 Mev. This modi6cation of0.&,& is to some extent justified by results from a phase-shift analysis. It seems that the theoretical value ofD (to) has a strong dependence on the shape of theresonance peak of trtoe .1'

ACKNOWLEDGMENT

It is a great pleasure to thank Dr J. Ham. ilton formany fruitful suggestions.

"B.G. Zinov and C. M. Korenchenko, Zhur. Eksptl. i Teoret.Fiz. BB, 1307, 1308 (1957) Ltranslation: Soviet Phys. JETP (tobe published) j.

t Note added irt Proof. In Goldberger's orig—inal formula LEq.(1)j D (cu) shows strong dependence on both D (p) and f'. In ourformula LEq. (2)g D (&o) shows strong dependence on D (ca').However, Eqs. (1) and (2) should yield the same results if (a) thesame o.~& is used, and (b) they both agree at cu =p and ~ =co'. Thebetter agreement achieved here is mainly due to the revised formoi o~~ . It implies a new value of D (tr).