VOLUME 6, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S MARCH 15, 1961

ANALYSIS OF THE ANOMALY IN DOUBLE MESON PRODUCTION IN p+d COLLISIONS AND THE S-WAVE PION-PION INTERACTION*

Tran Nguyen Truong Laboratory of Nuclear Studies, Cornell University, Ithaca, New York

(Received February 20, 1961)

The purpose of this note is to show how the anomaly in double pion production in p + d colli­sions observed by Abashian, Booth, and Crowe1

may be explained by a nonresonant final-state interaction of the produced pions in the isospin state T = 0, instead of assuming the existence of a neutral oo° particle.

The separation of the production process into two separate mechanisms, that of the primary interaction in which the particles are produced, and the final-state interaction in which the pro­duced particles interact with each other, is well known.2 It is assumed in this note that the final -state interaction is responsible for any deviation from simple phase-space predictions. In the following calculation we neglect the effects of the deuteron and He3 wave functions and the pion-nucleon final-state interaction, since they tend to smear out the spectrum and therefore are unlikely to produce a sharp peak. We limit our­selves to an S-wave (T = 0) ir-ir interaction3; the next angular momentum D state, which is per­mitted by 7T-7T isospin T = 0, is very unlikely to contribute.4 A straightforward analysis based on charge independence yields

o[p + d - He3 +?r+ + 7T~) + o(p + d - H e 3 + ir° + 7r°)

H l / i l 2 + l / o l 2 ,

a(/? + ̂ -H 3+7r+ + 7r°)=l/1l2,

(1)

(2)

where f j is the production amplitude in a state in which the two pion charge states couple to­gether to make an isospin T. Experimentally (2) is much smaller than (1), which is what one expects for the production of the two pions in the nonresonant P state at this energy.4 In the following analysis we neglect the contribution of the / x production amplitude to (1). The cross section is proportional to

d3pL d3p2 d*pz~ * s HJ 2EX 2E2 2E

1 ^ ( p i +P0+Po " P j

X&(E1+E2+E3-ET)\Tfi(q/2)\2. (3)

The subscripts 1, 2, and 3 refer, respectively, to the two pions and the He3; q/2 is the momen­tum of each pion in their own center-of-mass system. We have put the primary interaction

matrix to be constant, and T^{q/2) is the part which is due to the final-state interaction. After carrying out the integration in (3), we have

d2o/dp3dtt3 •-c(P32/u3)(q/uJ\Tf.\

2, (4)

where oog2- q2 +4m7r2 and the He3 dynamical quan­

tities are evaluated in the laboratory system; [Ps'/w3)(q/wq) is the phase-space volume ele­ment. To obtain I Tfi 12 empirically, we divide the measured spectrum by the normalized phase space. Using the data given in reference 1, the plot of ITfjl2 is shown in Fig. 1. For a very small radius of the primary interaction, it is the square of the ratio of the pion-pion wave function with and without ir-ir interaction taken at the origin, and is nothing but the usual en­hancement factor.5 We use an exponential poten­tial well of range d- \fi/\ic to calculate this en­hancement factor; we also use an asymptotic 7T-7T wave function with exponential cutoff for the range of the primary interaction to obtain the energy dependence of the matrix element, and

2.0

1.6

1.4

1.2

1.0

0.8 h

/ do do do do

= 0.54 (S=0.28) = 0.72 (S=0.35) = 1.00 (S*0.42) * 1.27 (S=0.49)

0.6

FIGc lo The empirical probability density function | Tfil2 as a function of the He3 momentum. Experi­mental data are taken from reference 1 at incident proton energy of 743 Mev. Fits to the data by the theory of pion-pion final-state interactions are shown*

308

VOLUME 6, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S MARCH 15, 1961

find that to fit the data one needs a larger scat­tering length. A reasonable fit to the data is obtained with the well depth parameter s = 0.3 to 0.5 corresponding to a scattering length a0« |#//xc to \K/iic (attractive), a0 being the scattering length for T = 0. Since we take into account only the effect of 7T-7T interaction, the result should only be regarded as evidence for the T = 0 TT-TI interaction being attractive and for its scatter­ing length being not too large. It is interesting to note that this conclusion is in agreement with that obtained by Ishida et_aL6 and Efremov e^aL7

in their work on the 631 and 613 phase shifts in pion-nucleon scattering.

Mitra found that it is possible to fit the r-decay data spectrum with either or both T = 0 and T = 2 resonances, and with the position of the T = 0 resonance at co^2 « 12m7r

2 which is quite far away from the energy region we are consider­ing.8 A resonance at uniJ

2 = Sm^2 would not fit the r-decay data. A low-energy T = 0 TT-TT reso­nance would also imply a too large inelastic cross section for the process IT +N-+2ir +N near the pion-production threshold.9 The analysis of the r decay by Lomon, Morris, Irwin, and Truong5 shows that the ir-ir interaction is a t t rac­tive for the state T - 2 and probably repulsive for the state T = 0; with both T = 2 and T = 0 at­tractive, and T = 2 more attractive than T = 0, a fit to the data was also obtained although it was not as good.

It is worthwhile noticing that the previous r e ­mark is not in disagreement with the results ob­tained by the dispersion relation methods of Khuri and Treiman.10 Combining the present analysis of the anomaly in the p+d experiment and the T -decay data, it may be possible to infer that the S-wave pion-pion interaction in the states T = 2 and T = 0 are both attractive with T - 2 more at­tractive.

We would like to mention that our analysis does not exclude the possibility of the existence of the o>° particle. Accurate experimental data at dif­ferent incident proton energies and He3 angles, could be used to tell whether the observed peak is due to the final-state interaction of the two pions or to the existence of the o>° particle.

We would like to thank Professor H. A. Bethe for his advice, encouragement, and interest in

this work. A number of helpful discussions with Dr. R. Schult, Professor T. Kinoshita, and Pro ­fessor E. Lomon are very much appreciated.

T h i s work was supported by the joint p r o g r a m of the Office of Naval R e s e a r c h and the U0 S» Atomic Energy Commission 0

iA. Abashian, N. E. Booth, and K0 MD Crowe, Phys . Rev, Le t t e r s 5>, 258 (1960). More recen t data on H3

show that t h e r e i s no peak in the spec t rum of the H3„ We would l ike to thank Dr . J . Iwadare , D r . Go M c -Cauley, and Dr0 How-sen Wong for the i r communica ­tion of the r e su l t s of the Berkeley Conference on Strong In terac t ions , December , 1960 (unpublished).

2K0 M. Watson, Phys , Rev0 8£9 1163 (1952); K. Brueckner , Phys . Rev. 82, 598 (1951).

3 P-wave 7r-7r in teract ion was cons idered by A. Tubis and J . L. Uretsky, Phys* Rev0 L e t t e r s 15, 513 (1960). This calculation is not consis tent with the p re sen t da ta .

4 F r o m the usual centrifugal b a r r i e r a rgument , one can s ee that the production of two pions in P s t a te ( re la t ive to each other) should be much l e s s than in S s ta te , i . e . , the H3 c r o s s sect ion should be much l e s s than the He3 c r o s s sect ion at this energy . The probabili ty of the two pions being emit ted in the r e l a ­t ive D s ta te should be negligible.

5 E . Lomon, S. M o r r i s , J . Irwin, and T . Truong, Ann. Phys . (to be published).

6K. Ishida, A. Takahashi , and Y. Ueda, P r o g r . Theore t . Phys . (Kyoto) 23, 731 (1960).

7A. V. Efremov, V. A. Meshcheryakov, D. V. Shirkov, and H. -Y. Tzu, Proceedings of the 1960 Annual International Conference on High-Energy P h y s ­ics at Roches ter ( In tersc ience Pub l i she r s , I n c . , New York, 1960), p . 278.

8A. N. Mitra , in Proceedings of the 1960 Annual International Conference on High-Energy Phys ics a t Roches te r ( In tersc ience Pub l i she r s , I n c . , New York, 1960), p . 525.

9L 0 S. Rodberg, Phys . Rev, Le t t e r s 3 , 58(1959) . 10No No Khuri and S. Bc T r e i m a n , Phys* Rev. 119,

1115 (1960). To fit the r - d e c a y data, Khuri and T r e i m a n obtained the re la t ion a2 - aQ = 0.7#//uc (with the convention a > 0 for an a t t rac t ive in te rac t ion) . This relat ion could be sat isf ied by taking both T = 2 and T = 0 repuls ive with T = 0 m o r e repu ls ive , a0<a2<0; or T = 2 weakly a t t rac t ive and T = 0 repu ls ive , a2>0, a0 < 0; o r T = 2 m o r e a t t r ac t ive than T = 0, 0 < <z0 < <z2° If it was requ i red that a0 >0, then only the l as t so lu ­tion holdso T h e s e solutions could be understood qual i ­tatively from the enhancement factor a rgument , A s i m i l a r calculation to that in r e fe rence 5 is being c a r r i e d out under l e s s s e v e r e r e s t r i c t i o n on the cu rva tu re .

309