ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Report NOTE ON THE PAIS-PICCIONI EXPERIMENT K. M. Case Project 2457 OFFICE OF NAVAL RESEARCH, U. S. NAVY DEPARTMENT CONTRACT NO. Nonr-1224-(15) March 1956

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ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE OF CONTENTS Page ABSTRACT iii I. INTRODUCTION 1 II. PHENOMENOLOGICAL DESCRIPTION 1 III. SOLUTIONS 3 IV. VALUES OF THE PARAMETERS 4 V. CLOUD-CHAMBER EXPERIMENT 5 VI. BUBBLE-CHAMBER EXPERIMENT 7 VII. CONCLUSION 13 VIII. REFERENCES 14 ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ABSTRACT Two experiments to check the Gellmann-Paisl "particle mixture" suggestion are analyzed, using a detailed, if crude, phenomenological description. A cloud-chamber experiment of the type proposed by Pais and Piccioni2 is compared to an experiment using a liquid-xenon bubble chamber.3 It is found that for likely values of the parameters involved, both experiments are considerably more difficult than envisaged by Pais and Piccioni. For intensity reasons, the bubble-chamber experiment seems preferable. However, unless quite accidentally the relevant cross sections have rather special values, it would seem that experiments of this type would have to wait until accelerators are available capable of producing considerably greater intensities than those in operation at present. -_____ - ~~~~~~iii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN I. INTRODUCTION Recently Pais and Piccioni2 have proposed a cloud-chamber experiment to verify the Gellmann-Paisl suggestion that the 0o should be considered as a "particle mixture." In view of the availability of a xenon-filled bubble chamber,3 it is of some interest to compare the feasibility of a bubble-chamber experiment with the Pais-Piccioni experiment. Below, a set of phenomenological equations describing a crude model of these experiments is given. Solutions for the two types of experiments are obtained. From these it follows that for likely values of the parameters involved: a. Both experiments are considerably more difficult than anticipated by Pais and Piccioni; b. Bubble-chamber experiments are somewhat more feasible; c. Unless the 90 absorption cross section and the Go G, mass difference are miraculously just right, experiments of this sort must wait until greater numbers of Go's are available. II. PHENOMENOLOGICAL DESCRIPTION Our model is the following. The wave functions describing g0 and o5 particles are to be linear combinations with prescribed phases of functions 0 0 describing G1 and G2 particles. Specifically, we have (o) = 1 + i2 (la) J2 = 1 - i2 (lb) J2 The G1 is to undergo the familiar two it decay with lifetime TI '- 1.5 x 10-10 seconds. The 91 has a completely different set of decay modes with a lifetime T2 >> Ti. (To obtain detailed numerical results, we will idealize this and put T2 =. ) In passing through matter, the 0~ can be absorbed while the 90 cannot. (For simplicity we follow Reference 2 and omit consideration of all other processes.) This absorption will be described by an effective lifetime T. Clearly, 1

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1/T = Npvaa, (2) where v is the velocity of the o0, p the density of absorber material, N the number of absorbing particles per gram, and aa the absorption cross section per particle. Let the wave function describing the state at time t after a G0 is produced be (t =C(t 1 (t) = () + ()2 (3) We have d(t) _d(t) (t)1 [(t)] (4) dt ph describes the phase change due to the kinetic energies. Thus, as in Reference 2, we have \dtph where i = [C^2p22 + m2 c4]1/2/ (i=1,2). (6) The expression d (t) dt Id is due to the G0 decay. Again, as in Reference 2, we can write it as 1 1 2 T2 - ()d 2T= - - C - 2 2 2 (7) The expression L d (t) dt is due to the absorption of v~'s. Phenomenologically it is 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN dI(t)J = - 1 [Component of ~0's in {(t)] (8) dt a 2T Using equations (1) and (3) we obtain 1 [il(t) + i O2(t)] k(t) = / [Ol(t) - i C2(t)] ( ) + (-t). (9) Hence, [dj(t)] = 1 [cl(t) + i C2(t)] () (la) dt ~, 2T 1JL i (t- + [c1(t) + i ca(t)] 2 (lOb) Since d~(t) do= l + dC 2 (11) dt dt dt We obtain, on inserting (5), (7), and (10) into (4) and equating coefficients of 1L and i2, --- ( + 1) -z i (12) dt T '+7 dt 4 " ^ - (2+ 2 2 (13) where o 1 xi = ii+ (i=l,2) III. SOLUTIONS For initial values o1(0) and 0C2(0), the solution of (12) is c(t) ( 21(t) - ) R )2 et (R) + ( (0) -R (0) e 4) \w2(t 1 - 1 R where ~ 3 ~~~~~~~~~~~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1 2 + - + (X1 XO 2 X = 2 T - 2 ( 2 + /2} (15) and R= - i/2T - x) +1(7 2)2T + T 1/4T (16) The probability of the 2x1 decay at time t is p2(t) - (t) (17) T1 Hence, the ratio of the intensity of decays at time t after production to thos at time zero is I j[(O ) - R 2(0)] e-)I t + [a2(0) - R l(O)] R e- 2t - R- 1 -2 1 -R2 (18) ~ ~1c(o) 12 IV. VALUES OF THE PARAMETERS Measuring time in units of T1, we see that the relative intensities given by (18) depend on the two parameters D = T1/T and y = TAau. Here A) = 0 0 (l1-D2 arises from any difference in the masses of 91 and 92. Since such differences presumably would originate from the same interactions which occasion the difference in lifetimes, it seems reasonable that y l. To see the effects of differing values of y, we have computed the results to be expected for y = 0, 1, and 2:. While the decay lifetime T1 is known to be #1.5 x 10-1 seconds, comparably accurate information for the 0o absorption lifetime T does not seem to be available at present. To obtain some idea of T. let us assume each nucleon can absorb ~ 's with a cross section aa. Then T =, (19) Nopvaa where No is Avogadro's number and p is the absorber density. Thus, 1 () (v) (A) 6.2 x 106. (20) " 'I~e W ^0

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Assuming v src and the absorption cross section per nucleon as 1 millibarn, we have Xe) 6.2 x 10-3 (21a) (Pb) 3 5.0 x 10-2. (2lb) With either substance we are certainly in a situation of little absorption. To obtain an extreme upper limit for 5, let us assume aa \5 x 10-26 cm2 (i.e., roughly geometrical per nucleon). Then B(Xe) e 0.3 (22a) (Pb) f 1-5. (22b) From (21) and (22) one can conclude that with either xenon or lead we do not have a situation in which { is very large. Upper limits in the two cases are of the order of 1/3 and 1, respectively, with somewhat lower values being likely. V. CLOUD-CHAMBER EXPERIMENT By this we mean an experiment of the type proposed by Pais and Piccioni. A beam of go's produced in a lead plate are to decay in a cloud chamber until all go's are removed. The remainder then passes through a lead plate which removes the 0o component. Since, effectively, G~'s are produced by removing g0's, the familiar 2i decay can be expected to reappear below the second plate. As indicated in Reference 2, the intensity of decays below the second plate may be expected to be of the order of 1/4 of that below the first. However, implicit in the derivation of this result is the assumption that T(pb) << T1 (i.e., >> 1). In the other limit of 3 << 1, qualitatively different results will occur. If T(pb) > TL, most GO's arising from G0 absorption will decay within the lead. The number of decays occurring below the second plate will be very small. To analyze the situation in detail, we can use equations (12) and (18). On production in the first plate, we have c02(0) = ial()0. After passing through a region with T Joo for a time T long compared to T1, we will have ca(T) = O (2 ) and c~2(T) -= 02(0) - iao.(O). (24) 5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Using these as the initial values when the beam enters a second plate, the ratio of 27i decays occurring at a time t later to those occurring just below the first plate is I(t) - IR12 e2t - e t12 I(O) l1-R2 2 = R 2 -(l+ )t/2T [cosh(6lt/T1) - cos(62t/T1)] (25) where 51-+ i82 = ( i27y), (26) and R (_____________27) = - [(I + i2y) +(1 + i2)2- ] (27 Let us first consider some limiting cases. 1) 3>> Here 51 ~, 02 2V O, R ^-i, and hence, I(t) 1 -(1+)t/2lC I() - e cosh - (28) I(O) 2 L \ J The maximum number of G0 decays occurring below the plate is clearly obtained by choosing the thickness of the plate so that the time of traversal tt satisfies T < tt <Ti. (29) In this case we have I(O) (which is just the Pais-Piccioni prediction). 2) f3 << 1 _=___=__1iB, 2=7, B = - 2(14427) (51) _________________________________6~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN -~ t) ~ ~2 et/2Tt 1(t ), l" D |z 1e (cosh t/21 - cos 7t/Tl). (32) I(o) 211+i 2yl Now the maximum number of decays occurring below the plate is of the order of magnitude of the coefficient in (32), i.e., (t) 2525 [ ]max 211+i2H27 (33) Thus, for 3 = 0.1 and choosing the most favorable case of = 0, we have [1}x].005, (34) L J(0 max as compared with the value.25 of (30). 3) In Fig. 1 the maximum ratio of the number of decays occurring below the second plate to that below the first is shown as a function of P for y = 0, I, 2Xg. (Maximum means having chosen the thickness of the second plate to make this ratio greatest.) It is striking how slowly the value 0.25 is reached as a function of 3. For y = 27t and P = 1.5 [which we have seen is a probable upper limit for B(pb)], we have (I)max = 0.007 \o/max It would appear doubtful whether a recurrence of 2iT decays with such a low intensity is observable. VI. BUBBLE-CHAMBER EXPERIMENT It should be possible to produce G0's directly within a liquidxenon bubble chamber. Since one would be able to observe 0~'s moving in all directions from the point of production, a considerable increase in the number of G0's observed as compared with producing them in a lead plate above a cloud chamber would result. The experiment we have in mind consists of measuring the distance of the G0 decay from the point of production. From a knowledge of the velocity of the 90 the time of decay is known. Equation (18) [with a2(0) = iac1(0)] shows that the relative number of decays at time t after production is given by 7

.30 I z Cloud-chamber experiment rn.25 z (I/1o)m =Max. emerging relative ~ o intensity u B= Decay lifetime | m.20 Absorption lifetime l ~ 1 (!/ o)m X = Fi, A.05 ~ — I 0. 0.2 0.4 06 0.8 1 2 4 6 8 10 20 406080100 I Fig. 1.0___________________________________________________________________________15 m.1

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN I(t) e-t/2Tl I(o) - [(1 R- +R)]2 + 4R2R { } = [(+R2)2 + R2] e( + /2)t/T -(35) + [ (R + R2)] [(1 - R2)2 + R21 e-(/2 - 81)t/T1 + e-t/2TL-2[R2(2-R2) + R2(l-R2)] cos 62 t/T -2Rl(1 - R - R2 2R2) sin 82 t/Tj. Here R = R1 + iR2 is given by (27) and 61 and 62 are given by (26). Now, of course, 3 refers to the xenon. 1) >> Equation (35) becomes I(t) =, et/2. (6) In this limit the G0's would behave as if their lifetime in xenon was double that found in cloud-chamber studies. This would certainly be easy to observe. Unfortunately the estimates of Section IV indicate that we are far from this limit. 2) << I(t) _ e(t/1T 42 +e [cos(yt/T1) + 2y sin (yt/T1)]. (37) I(0+ 4(1+4y2) ++4y2 Thus, only when the number of decays has dropped quite low will there be any deviations from what one could expect for a particle with lifetime T1. 3) In Figs. 2, 3, and 4 decay curves for various values of ( are given, corresponding to y = 0, 1, and 2t, respectively. From Fig. 2 (7=0) we see that for 3 small (P < 1) significant deviations from 3 = 0 (no 0o absorption) occur only when the number of decays is quite small. Thus, for f = 1/3 there are twice as many decays as for 5 = 0 only when the latter is approximately 1/70 of its initial value. 9

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN _8 __ _ Bubble-chamber experiment X \\. Tl=o:0. | I I I \,^ \ i __ I/Ids Relative no. of 80 decays.~^4 L \ \ \l Decay lifetime '~.~ l '~ e= Absorption lifetime.2 tS.,.08 006 I/Io.04 p8a Fig. 2.004 ~~~~~2~10 On' 1/3.01.o-008 a ' -.,,....- ~- 1/10.002 ~~~V.001..... 0 2 3 4 5 6 7 t/TI Fig. 2 10

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN,86 __ ' XBubble-chamber experiment.6 \__ ~.4~ ~ ^ \ l,, IoRelative no. of 80 deca.ys.4 ~\ \\ Q\ Decoy lifetime - L Nt Absorption lifetime.04.08 \\\.006 /I I 0 I I \h \\ D\.04.02.008 -.006 \.004 ~~~\ \ \ \, \.001 0 1 2 3 4 5 6 7 t/tr Fig. 3 ~~08 II \~,\ \

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN -.,',. ~ Bubble-chamber experiment \.^4 \E\\ |:I/Io' Relative no. of 8~ decays ^ V^^Xk^^~~~ Decay lifetime LI ^ \~\k\ \ ' =Absorption lifetime. 0~.06\\ ~t\ \\\ ___/Z o__\\ \\ __\ ~~looo. ' I/lo ~ X \ 0 I\ it 0,.02i\ \ \ 200 ~008.006 ~\ \ 1/3.004 C 110 T 1\/ \ \ '.002 ~~ ~~ \ Fig. 4

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Figure 3 (7=l) leads to similar conclusions, except for one salient feature. For 3 1/3 there is a tremendous decrease in the number of 2t decays expected in the region where, for no absorption of GO's, we would expect 1% of the decays to occur. While the observation of this effect would be difficult, it is hard to imagine that a decrease by a factor of 100 could not be seen. It should be pointed out that this phenomenon only occurs for a region of values y - 1, vx 1/3. However, the discussion in Section IV by no means rules out this possibility. The curves of Fig. 4 for y=2t are quite remarkable. Increasing { from zero to infinity does not change the curves smoothly from the straight line describing the decay e-t/T1 to that for e-t/2TI. As P increases from zero, the decay curves develop characteristic oscillations [due to the sine and cosine terms in (35)]. Since these oscillations probably would be averaged out in an experiment, we have also given curves for this average. These oscillations are most pronounced for C = 1. The decay occurs increasingly rapidly with increasing 3 until 3 -12.5. In this region the decay curves look as if the G0 lifetime is considerably shorter than 1.5 x 10-10 sec. Increasing 3 further the decay curves swing back up, becoming essentially straight. Two phenomena should be observed. a. Extremely large values of: are required before the curve for B = o is even approximately realized. Thus, 3 = 200 is clearly far from this limit. b. The decay curves for 12.5 <,60 coincide with the averaged curves for 12.5 < 3 < 0. For example, the curve for: = 40 falls, within the accuracy with which one can read the figure, exactly on the averaged curve for - = 1. Thus, if one did see an effect of the kind indicated —namely, an apparent increased decay rate-one could only conclude that there are two possible absorption lifetimes T compatible with the results. VII. CONCLUSION We have seen that, in general, for reasonable values of the parameters the "mixture property" of Gellmann and Pais will affect only a small proportion of the G0's-in either a bubble chamber or a cloud chamber. Under such circumstances the possibility of observing more G0 decays makes the bubble chamber preferable for experiments of this type. At a time when G0's are available in sufficient abundance that experiments on 1% of them are feasible, this check on the Gellmann-Pais hypothesis could well be done and would be of great interest. One proviso to the above general conclusion holds. If, accidently, T 1, { 3 1/3 for xenon, the bubble-chamber experiment would be practical even now. 1 1 R

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN It is a pleasure to thank Professors D. A. Glaser and G. E. Uhlenbeck for helpful discussions. VIII. REFERENCES 1. M. Gellmann and A. Pais, Phys. Rev., 97, 1387 (1955). 2. A. Pais and 0. Piccioni, Phys. Rev., 100, 1487 (1955). 3. J. L. Brown, D. A. Glaser, and M. L. Perl, Phys. Rev. (to be published). 14

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