SUPPLEMENTO AL V()LU~IJd I I , ~ERIE X N. 1, 1955 ])EL NUOVO CIMlgNTO 2 ~ Seme.~tre

On the Measurement of the Mean Life Time of Strange Particles.

U . A S ' I A L D I

I s t i t u t o di F i s i ea de l t 'U~dcers i t5 - R o m a

I s t i t u t o _Va:ionale di F i s i c a ~Vucleare - Sez io~e di R o m a

1 . - I n t r o d u c t i o n .

I t is well known that one of the more peculiar properties of heavy mesons

and hyperons consists iu the fact that their mean life appears to be extremely

long if ( 'ompared with their rate of production.

The simple knowledg'e of the order of magnitude of their mean life is suffi-

cient to prove the existence of such a difficulty whose solution probably needs

the diseovery of some very import~mt feature of these new objects.

However, one ('an hope to learn something more about the structure of

these particles and to g'et a hint to the solution of the above mentioned problem

by establishing the values of the mean lives and the branching ratios of the

various possible decay processes for ea('h one of the species of particles.

I t follows that the problem of the experimenta.1 determination of the mea.n

life of strange particles has to be carefully considered with the double aim of

extending the interval of time on which the measurements are possible, and

of improving the precision of the corresponding results.

Starting from this point of view, we have devoted, during' the last few

months in Rome, a certa.in time to this problem, and I am reporting' here our

considerations and conclusions which refer mainly to the cause of charged pa.rt-

icles observed in nuclear emulsions.

The more direct method of measuring the mean life of a. charged unstable

particle is obviously that bused on fast counters technique.

A first step in this direetion has been made recently by MgZZETTI and

I(EUFFEL (1), who have given some evidence for the existence of K-particles with a mean life of al)out 8.7.10 9 s.

(1) I,. MEZZET'r~ and J. W. KeU~,U,'EL: P h y s . L'ec., in press.

254 E. AMALDI

This value however, is a l ready very close to the l imit than can be a t ta ined by such a technique: in fact , the resolving power of the exper imenta l set-up

of the preceding authors is of abou t 3-10-9 s and it appears ve ry difficult, a t least in the immedia te future, to improve it by more t han a factor 2 or 3.

On the other hand, we know tha t m a n y of the s trange particles have life- t imes appreciably shorter than 10 -9 s. For instance we know tha t the mean life of A ~ is

= (3.7_+~:s~)'10 -~~ s ,

and t ha t of 0 ~ is T = (1,5200:~3)*10 10 S .

I t is well known tha t the measurements of these short life-times have been per formed by means of cloud chamber observations.

On the other hand, no systematic work has been done until recently to

explore the possibility of life-time determinat ions based on emulsion technique measurements .

Before going into the details of the methods tha t we have developed, I

would like to draw at tent ion to the fact tha t with emulsions still more than with cloud chambers and almost as well as with counters technique, the main

difficulty is due to the uncer ta in ty in the identification of the particles. There- fore, the lower limit of the life-time of charged particles t ha t can be deter-

mined in emulsions, is imposed by the requirements for their identification. This last depends, of course, on the propert ies of the considered type of par t - icles, in par t icular on the type of their decay process. I f the decay process is so typieM as to ensure the identification of the p r imary particle and to allow the determinat ion of its velocity, one can push the measurements down to very short t imes: wi thout going into the details of such a hypothe t ica l ease,

i t m a y be sufficient to notice tha t a mean life of 10-14 s irr the l abora to ry f rame

of reference, corresponds to a mean free pa th of 30 ~m, if the particle has a veloci ty close to the velocity of light.

Such a ease however, is purely hypothet ica l and probab ly does not exist

in practice. I n general, the identification of the particle mus t be based on

the determinat ion of its mass, charge and velocity, apar t f rom its decay process. This means t ha t in general, i t will be necessary to dispose of t racks of a t least

1 m m lengths. I t follows tha t for a particle of mass not very different f rom

tha t of the p ro ton and kinetic energy of about 1 GeV, the lower l imit of the

life-time tha t we can measure by means of emulsion observations, is of the

order of 10 -15 s. The existence of more than one type of particle of equal charge and ve ry close masses, can make the identification very difficult also

if their mean pa th is longer than 1 mm. As regards long times, we will see in the following tha t in principle any life

could be measured with emulsion technique, provided we dispose of a suffi-

ON THE 3 I E A S U R E 3 I E N T OF T I l E ~[EAN L I F E TIME OF S T R A N G E P A R T I C L E S ")55

ciently great number of identified pa.rticles. In practice, however, the in tens i ty at disposal at present is so snmll tha t already life-times of the order of :lO-S--]O -'~ s are too long to be measured with ~my acceptable precision. This

upper limit is of secondary import'~nce because, as a l ready s ta ted above, other techniques can be used in these eases.

2. - T h e M e t h o d of M a x i m u m L i k e l i h o o d for C loud C h a m b e r T e c h n i q u e .

Before describing tile methods tha t we propose to use with emulsion tech- nique, i will recall very briefly one of the methods used in the case of cloud elmmbers, which appears to be by far the best one.

This method, developed by ]~Al~TLETT (2), c~ln be presented as follows.

Let us assume tha t we have ~ identified p~trticles of the same species, for e~wh one of which the following quantit ies have been measured (~): the length l~

of the pa th crossed b y the particle~ between its appearence in the i l luminated

region and its decay, within the region where the observat ion of its (tetchy is possible; the m a x i m u m length of pa th L, available for the same particle, af ter

its appearence in the i l luminated region, within the region where the obser- va t ion of its decay is possible, and finally the velocity fi~.

F rom these data , one can immedia te ly derive their observed t ime of flight t~ which they spent in the f rame of reference in which they are at rest, to cross the distance l~, and their so called potential t ime I'~ t ha t they would have spent

to cross, in the same frame of reference, the pa th L~ if they had not decayed. If v is the mean life of the species of particle under consideration, one can

say tha t the probabi l i ty tha t the r-th particle is observed to decay between t,.

and t r+dt , , having at our dispos~fl for observat ion a potent ia l t ime T~, is s imply o'iven by

cxp [ - - tr/T] dt,. (1) 1- - exp [--- T~/'T] T "

Therefore the probabi l i ty tha t for fixed v our n particles decay just a t the observed moments , is given by the expression

(~) G(r; t~, t,,. . . . , # , . . . , t,,: 1', , T,,, ..., T , , )d t~dt . , . . . . . . . d # dt~ ----

f l exp [ - - tr/'g] d # - - ~ 1 - - e x p [ - Tr/~ ] T

The function (; is called the (~ likelihood function ~,; according to R. A. FIS[4EI~

(2) ~L S. BARTLETT: Phi l . 3iag., 44, 249, 1407 (1953). (a) j . G. ~VrLSOX and C. C. BUTLER: -Phil. ,flag., 43, 993 (1952).

256 E. AMALD1

the best es t imate of ~ is the value tha t makes G a m a x i m u m for given values of # and T~ (r--~l, 2, ..., n). This method calls for m a n y comments and cri-

ticisms (4). Bu t I will not go into any details nor will I give the final expres-

sion for the est imate of ~. In fact, discussing the case of emulsion technique,

we will have the oppor tuni ty of writ ing down an expression of the mean life which includes, as par t icular case, tha t relative to the cloud chamber technique.

3. - The Method of Maximum Likel ihood for the Emuls ion Technique.

The simplest thing to do passing f rom cloud chambers to emulsions, con-

sists in t rying to extend the method previously described. This has been done by CASTAG]NOLI, CORTIN[ and FRANZINETT[ who have presented their repor t

a t the Padua Conference (5). The main difference between these two cases is obviously due to the fact

tha t while in a cloud chamber the particles fly keeping their velocity pract ical ly

constant , in emulsions, the particles are rapidly slowed down on account of the much higher density of the medium. The t ime (in the f rame of reference

in which the particle is at rest) necessary for a particle of residual range R to be slowed down to thermal energies, is obviously given b y the expression

R

(3) t j cfi 0

Dividing both sides of cq. (3) by the mass M of the particle and recalling t h a t

RIM is a function only of the velocity fi, one can write

(4) M - - ! _~ = F(f l ) .

Making use of well known empirical expressions valid in the emulsion for the

range-veloci ty relation, one obtains (6)

. 10 ~ (5) ~ -

for fi < 0.4 (namely R/M ~ 2-10 ~ ,zm), where t is in seconds, M in protonic

masses and R in microns, and

t _ 2 .76.10-~(R_t ~ (6) i

(4) M. ANNIS, W. CHESTON and H. PRIMAKOFF: Rev..Mod. Phys., 25, 818 (1953). (5) C. CASTAGNOLI, G. CORTINI and C. FRANZINETTI: Suppl. Nuovo Cimento, 12,

297 (1954). (6) E. AMALDI, G. BARONI, G. CORTINI, C. FRANZINETTI and A. MANFREDI~-I:

Suppl. Nuovo Cimento, 12, 181 (1954).

ON THE M E A S U R E M E N T OF THE MEAN L I F E TIME OF S T R A N G E P A R T I C L E S 257

for f i ) t ) . 4 , where t and M are in tile same units and R in cm. By means of

these two equations we can easily pass f rom range measurements to t ime

measurements . I t is now clear t ha t we c~n repeat the same stat is t ical considerations tha t

are valid for cloud chambers. Therefore equat ion (2) will be valid also in the case of emulsion obserw~tions.

Also in this case the lengths lr and L, and the corresponding t imes t, and

Tr refer to the region where the observat ion of the decay process is possible i.e., in this ease, the volume actual ly scanned. Fur the rmore one has to distin-

guish between three types of events (5):

a) particles which ~re brou~'ht to rest in the emulsion and then decay ;

b) particles which decay in flight inside the emulsion in such a posi t ion

and with such a (vector) veloci ty tha t they would have been brought to rest in the emulsion, if they had not decayed;

c) particles which decay in flight in the emulsion and ~re not to be included in group b).

Both the events a) and b) are obviously associated with an infinite po- tential t ime; fur thermore the events of type a) supply us different information

f rom tha t supplied by events of type b) and c). While for these last groups

of events we know at wha t ins tant the corresponding particles have decayed,

for the particles of groups a) we can only s tate tha t the decay happened at a t ime not smaller t han the corresponding modera t ion t ime t o (r~--l, 2, ..., no) given by eq. (3).

I t follows tha t we have to integrate equations (2) with respect to dt~, tit.2, ..., dt,~ f rom the corresponding t o to infinity.

One can so find a new likelihood function whose m a x i m u m defines the best es t imate of r. One gets

We note tha t in the case of cloud chambers only events of group c) exist

a n d therefore the corresponding expression of v reduces to the last t e rm ap-

pearing a t the right hand side of equat ion (7).

For the expression of the stat ist ical error by which the es t imated value of r is affected, we refer to the original paper is).

We prefer to discuss here the sys temat ic error which could affect our results _in a substant ia l way.

In any measuremen t of the type of tha t considered here there are two

1 7 . S u p p l e m e ~ t o a l N u o v o C i m e n t o .

2 5 8 :E. AM&LDI

different sources of systematic experimental errors which are usually indicated as experimental bias.

The first one is of geometric nature, the second of kinematic or ion . With the expression geometric experimental bias we indicate the fact t ha t

as a consequence of the finite dimensions of the stack of emulsions, the per- centage of particles which do not decay before excaping from the emulsion depends on their range.

The interesting feature of the method of maximum likelihood described

above is tha t the geometric bias is eliminated by the introduct ion of the po- tential t ime T, .

With the expression kinematic experimental bias we indicate the fact tha t the probabil i ty p of observation of a decay process, will be in general, a function of the velocity of the particle at the instant at which it decays. In fact, the faster i s a particle, the smaller is the corresponding probabil i ty of detection of its decay. Tha t is due, in par t to the composition (in the laboratory frame of reference) of the velocity of the particle with the velocity of the corres- ponding decay products in its frame of reference, in par t to the lower ioniz- ation of the particle itself which again is a function of the velocity.

I f we could determine in some way the probabil i ty Pr of detection relative to each one of the considered events, we could obviously correct the est imated values of ~ also from this source of systematic error.

In this case eq. (7) has obviously to be substi tuted by the following

expression

(8) T - - / l~,it~ + l~.,k~t~ ,4- ~ tt -]- erl/~__ 1 P~

~b 1 nc

where we have made the reasonable assumption tha t the probabil i ty of de-

tection of particles decaying at rest is equal to 1.

4. - The Method of Residual Time.

After the Conference of Padua where the preceding method had been pre- sented, 'we tried to develop some method which would allow the evaluat ion and possibly the elimination of the kinematic experimental bias. This new method has the disadvantage tha t it does not allow a simple correction for

the geometrical experimental bias. The principle of the method is the following (7): let us consider for a mo-

(7) E. AMALDI, C. CASTAGNOLI, G. CORTINI and C. FRANZINETTI: NUOVO Cimento,

12, 668 (1954).

O N T H E M E A S U R E M E N T O F T I l E 5 l E A N L I F E T I M E O F S T R A N G E P A R T I C L E S 259

ment a stack of emulsions of infinite dimension containing ~t source of mono- energetic particles of :~ well defined species. To the k inemat ic energy Ko of

the emit ted particle will correspond a range R0, a veloci ty fi0 and, according to equat ion (-l), a moderat ion t ime to.

I t is now obvious tha t in order to get some informat ion on the mean life of the pa r tMes considered, it is convenient to describe their mot ion th rough the emulsion by means of the variable t rehtted to the ins tantaneous residual range R by equat ion (4).

So we can say tha t the number of particles which decay between t and t + d t is given by

I dt (9) d n = S e x p - - r

where S is an inessential constant expressing the to ta l number of particles

emit ted at the ins tant to by our source. Equa t ion (9) shows tha t the number of particles decaying per unit of t ime varies exponential ly with t (Fig. 1). All particles which decay at instants between 0 and to decay in flight, while the area enclosed by the exponential at the left of the ordinate

axis (negative values of t) corresponds to par- ticles tha t decay a t rest. This simple export-

ential law holds not only if the particle source is monoenergetic, bu t also when a value to of t exists, which leaves at its r ight all sources

of particles. I t is now evident tha t making use of repre-

, / pU) ~ i i

I i t - "

i i I

Fig. 1.

t O t

sentat ions of this type, we can deduce the mean life of the particles conside- red. For instance, we can divide the t ime to in a certain number of intervals, all of length 0 (in Fig. 1 0= l t o ) and call N1, N2, ... the numbers of particles decaying in each one of them.

Then one finds immedia te ly tha t the following relations hold

N1 0 (10) 1 +~v~ = e x p - ~ ,

Ni 0 (11) iv"i 1 ~ exp -r i ~ 2

where No is the number of particles decaying at rest.

In the ideal case of complete absence of any type of exper imenta l bias,

equations (10) and (11) can obviously be used to deduce T. Bu t also in the

260 n. AMALDI

case of a non negligible k inemat ic exper imenta l bias we will be able not only to establish the value of v bu t also to determine the probabi l i ty of detections defined above in connection with the me thod of the m a x i m u m likelihood.

F r o m wha t we have said, we have to expect t ha t the probabi l i ty of de-

tect ion is a function of the ionization of the type

(12) p ( i ) = l for i > i o

P ( i ) = io for i < i o

where io and ~ are two convenient empirical constants. Considering t h a t i

is a function of the velocity, one can express p as a function of t. We will get a curve of the tpye of t ha t given in Fig. 1 (dotted line).

Le t us assume now tha t we can choose such a small value of 0 t ha t all

(or a lmost all) the particles decaying in the in terval 0 ~ t ~ 0 have a pro-

babi l i ty of detect ion equal to 1. Then equat ion (10) will be correct (or a lmost correct) and we can use it

in order to calculate e ~ and v. Once we know e ~ we can compare it with

the exper imenta l ratios hi~hi-1 (n~ -~ v]~:Vi) and deduce the average efficiency of detect ion relative to the successive intervals

~ ) i n~ e x p [ 0I i ~ 2 , ~]i-1 n ~ - i

(13)

where

(14) ~ i =

iO

f e x p It~v] p(t) dt (i-1)0

90

fexp [t/v] dt (~-1~0

The knowledge of the ~ is sufficient to give some informat ion on the experi- men ta l bias. B u t one can go fur ther and introduce in equations (14) some

well chosen expression for p(t) containing a few parameters which can be

adjus ted to give the exper imenta l values of the ~i. For such a purpose one can make use of the expression (12) or, in order

to have an expression still simpler in the sense t ha t it contains a single adjust -

able constant T,

(14) p ( t ) = l for t ~ 0 ; p( t ) -=exp[ - - t /T ] for t > 0 .

Once we have determined the values of the constants appear ing in the

chosen expression of p(t) we drop the assumpt ion ~1 = 1, we calculate ~

and repea t the preceding procedure.

ON T I1E M E A S U R E M E N T OF T I l E M E A N L I F E T I M E OF S T R A N G E P A R T I C L E S 261

In such a way, it is possible to get, by successive approximations, a value

of T which has been corrected for the kinematic experimental bias.

The above described procedure of successive approximations gives a correc~

result only if all the ranges of the parti('les involved in it are short with respect

to the dimensions of the stack of emulsions. If this condition is not satisfied

the geometrical experimental bias will affect the results.

In this case the best procedure consists in applying the method of resid-

ual time so as to determine by successive approximations p(t), and once

known the pk -- p(/~,~) for ea('h one of the available events, estimate v by means

of equation (8).

Finally, 1 would like to note that, as long as the assumption that p is a

fnn(.tion only of the velocity and therefore of the residual time t, the effect

of the kinematic experimental bias on the estimate of v with the method of

maximum likelihood is in part compensated by the fact tha t in equ,ttions (7)

and (8) appears only tile observed time of flight and not the residual times.

5. - A few Examples of Application of the Preceding Methods.

We ('an llOW try to apply tile preceding methods to the observations collected

up to now on the strange particles.

The only case which allows an a t tempt to be made is tha t of the hyperons.

By making use of 20 cases in which particles of mass larger than tha t of

protons have been observed to decay or to interact according to the scheme,

y+ ~ p q _ = o q_117 MeV

Y • =~-~-115 MeV

Y- -,'- nucleus -~ star

and assuming that they are all due to the same type of particle, we obtain

3 " 1 0 10 ~-~" T ~ 0 . 5 " 1 0 lO S ,

where the upper limit is dedu(,ed by means of the method of maximum like-

lihood, and the lower limit by means of the method of residual time estimating

generously the kinematic experimental bias. The data available up to now

are too scarce to allow any a t tempt at successive approximations. The value

given above has to be taken, keeping in mind tha t we are not sure tha t all

the events used are due to the same particle: therefore it must be considered

only as a first a t tempt while the correct value will be deduced later when many

more examples of well identified hyperons will be available.

In the case of =-mesons it is not possible to apply either of the two methods

2 62 s. AMALD]:

described above. Out of 43 well established examples of v-mesons, no one has

been observed, up to now, to decay in flight. On the other hand, the sum of

all the corresponding moderat ion times is about 0.9.10 -s s. Therefore we can

conclude tha t probably the mean life of v-mesons is not much shorter than

10 -s s, provided we can exclude tha t the lack of observations of z-mesons

decaying in flight might be due to the kinematic experimental bias. This

seems to be the case because the decay process of v-mesons is so typical tha t

the probabili ty of observation of decay processes in flight is expected to be

close to 1. One can add that also cloud chamber observations seem to indicate a mean

life of v-mesons not shorter than 10 s s. The situation is quite different for all other types of K-mesons known today,

whose process of decay consists in the emission of a single charged particle.

In these cases, the kinematic experimental bias can completely alter our

observations and therefore any considerations of the type of tha t made for

z-mesons loses significance. In spite of that , in occasion of a discussion held with B. RossI at Varenna,

Mrs. ScARsI and VITALE have calculated for various groups of K-mesons, the

sum of the corresponding moderation times. Their data are collected in the

following Table I.

TA:DLE I.

Number of events

82

l l

13

9

Total 115

Identification of the secondary

unidentified L-meson =-meson ~-meson ~-nleson

p~ of the secondary

50 MeV/c 50 MeV/c

Total moderation time

11.4.10-9

1 . 4 - 1 0 - 9

1 . 1 �9 1 0 -9 0.6.10 -9

14.5.10 -9

These 115 processes of decay at rest have to be compared with a single

case of decay in flight. However, from this data no conclusion can be derived

on the mean life of K-mesons unless at least some rough estimate of the

kinematic experimental bias has been established.