THE UNIVERSITY OF MICHIGAN
INDUSTRY PROGRAM OF THE COTTEGE OF ENGINEERING
STUDY OF POSITIVE TAU MESON DECAYS
IN A PROPANE BUBBLE CHAMBER
Theodore Francis Zipf
January, 1958
IP 269
Doctoral Committee:
Assistant
Professor
Professor
Associate
Professor
Professor Martin L. Perl, CoChairman
Donald A. Glaser, CoChairman
Wayne E. Hazen
Professor Maxwell 0. Reade
George E. Uhlenbeck
ii
ACKNOWLEDGMENTS
The author wishes to express his
gratitude to Professors M. L. Perl and
D. A. Glaser who provided friendly advice and guidance whenever it was needed
or requested.
The research was supported in part
by the United States Atomic Energy Commission.
iii
TABLE OF CONTENTS
Page
PRE GE........................................................ iii
LIST OF FIGURES................................................ v
LIST OF TABLES................................................. vi
I. INTRODUCTION.............................................
1.1 Survey of the Heavy Mesons......................
1.2 Further Consideration of the Particles and the
Aim of the Present Investigation................
II. THEORETICAL CONSIDERATIONS IN T+ MESON DECAY............. 11
2.1 The Kinematics of T+ Meson Decay................ 11
2.2 Comparison of the SpinParity Properties of the
G+ and T....................................... 18
2.3 Calculation of the T+ Decay Spectrum for Certain
SpinParity Combinations........................ 22
2.4 The T+ Decay Spectrum When Parity is not Considered in the Decay............................... 31
III. EXPERIMENTAL APPARATUS AND PROCEDURE.................... 37
3.1 The K+ Meson Beam............................... 37
3.2 The Detecting Device............................ 38
3.3 The Scanning Procedure.......................... 44
3.4 Reconstruction of Events........................ 45
3.5 Analysis of the Events.......................... 51
3.6 Experimental Errors............................. 54
IV. DISCUSSION OF EXPERIMENTAL RESULTS AND CONCLUSIONS....... 61
4.1 Description of the Data......................... 61
4.2 Comparison of the Data with the Theoretical
Distribution Functions.......................... 62
4.3 Resume of Results.............................. 77
BIBLIOGRAPHY................................................... 79
APPENDIX....................................... 81
iv
LIST OF FIGURES
Figure Page
1 Momentum Diagram..................................... 13
2 Phase Space for T Meson Decay........................ 16
3 Energy Spectra for SpinParity Combinations (S < 2)
as Predicted by DalitzFabri model 2  not shown...... 32
4 Angular Spectra for SpinParity Combinations (S < 2)
as Predicted by DalitzFabri model 2  is not shown. 33
5 Plan View of Experimental SetUp..................... 39
6 Plot of Beam X,Y, D Distributions.................... 40
7 End and Top Views of Bubble Chamber.................. 42
8 Stereo Views of an Event............................. 4 43
9 Stereo Camera  Bubble Chamber System................ 46
10 Diagram of T+ Meson Decay............................ 50
11 Measured +p Lengths and Corresponding Gaussian....... 57
12 Plot of K+  p Scattering Coplanarity Angles......... 58
13 Experimental Energy Distribution for 118
Unambiguous Events.................................. 63
14 Experimental Cos G Distribution of 118 Unambiguous
Events............................................... 64
15 Experimental Energy Distribution Corrected According
to Fo_.......................................... 73
16 Experimental Cos 9 Distribution Corrected According
to Fo..................... 75
v
LIST OF TABLES
Table Page
I Decay Modes of the K and K~ Particles and Per
Cent of Total K Particles the Decay Mode Represents.. 2
II Measured Masses and Standard Deviations for K+ Mesons
from Primary and Secondary Methods................... 4
III Allowed (2, 2') Values for a T Meson of Given J
and P............................................... 28
IV F's Calculated from Equation 226.................. 28
JP
V Normalized Distribution Functions.................... 29
VI Expressions for the Angular and Energy Spectra....... 31
VII Expected Values of Probability Ratios................ 68
vi
CHAPTER I
INTRODUCTION
1.1 Survey of the Heavy Mesons
In 1949 Brown et al. observed, in a photographic emulsion
exposed to the cosmic radiation, a particle whose mass was intermediate
between that of the ~ meson and that of the proton. This particle
came to rest in the emulsion and then decayed into three charged
particles. Mass measurements showed that the secondary particles
were very probably t mesons. Since that time particles of similar
mass as that of the particle observed by Brown et al. have been observed. These particles not only have exhibited positive and negative
values of the electron charge but also have appeared in the neutral
charge state. One of the striking properties of these particles is
the large number of ways in which they are observed to decay. It has
become customary to refer to these particles as K particles or heavy
mesons and to further classify them according to their decay scheme.
When only one of the secondaries is charged and the remaining one or
more secondaries are neutral the particle is represented by the letter
K with two subscripts. The first subscript indicates the type of
charged secondary and the second subscript indicates the total number
of secondaries involved in the decay. In Table I the known decay
modes of the positive and neutral heavy mesons and the percentage
of the total number of K's represented by a particular decay mode
are listed. The K decay modes and the relative abundances of these
1
2
TABLE I
DECAY MODES OF THE K+ AND K~ PARTICLES AND PER CENT
OF TOTAL K PARTICLES THE DECAY MODE REPRESENTS
Per Cent of Total K's
Represented by Decay Mode
Species
Decay Mode
K+
x2
K+2
T
K+ (T 1+)
iO
It+ + ItO
+' + v
2r + + t
2it~ + Ti+
I + i0 + v
+
e + A+
+
x + em + v
+ _+ e
_ +
~ + 4 + V
it+ + it + Ao0
27%
57%
6%
2%
4%
4%
0
T
modes are not as well established as are those for the K+ case.
Until 1954 the K particles which were observed were those
which had been produced in the cosmic radiation. Thus, primarily because it is not possible to produce these particles copiously using
cosmic ray intensities, extensive studies of the properties of heavy
mesons were not feasible. In 1954, however, it became possible, by
3
using the high energy, high intensity primary particle beams of the
Cosmotron, and later the Bevatron, to produce sufficient numbers of
these heavy mesons to be able to determine, accurately their elementary properties. It is the purpose of the following paragraphs in
this section to review briefly the results of those investigations.
The mass appears, within the limits of experimental error,
to be the same for all of the K particles. Two types of experiments
are used to determine the mass. In the primary mass type of measurement the beam of charged particles is first momentum selected by means
of a magnetic field and then is brought to rest in an absorber. The
knowledge of the range and the momentum allows one to assign a value
to the mass since for particles of the same charge in the same absorber
the range is a function of velocity only. The other type of mass measurement which has been employed extensively is the socalled "secondary mass measurement." This method consists of measuring the kinetic
energy of the secondaries and then, employing the known mass of the
secondaries, equating the mass of the primary (in energy units) to the
sum of the secondary masses (in energy units) plus the sum of the kinetic energies of the secondaries. Since it is very difficult to measure the energy of an uncharged secondary this method is not applicable
to the particles which decay via the K 3(T'), K 3, and Ke3 modes. It
is a very important result that the masses obtained by the secondary
method agree well with those obtained by the primary method since this
tends to rule out the possibility of the primary K meson undergoing an
undetected energy transition to a lower state (that of the immediate
parent) which then decays according to the observed mode. 23 Table II
4
shows a comparison of recent mass measurements by means of the two
methods.
TABLE II
MEASURED MASSES AND STANDARD DEVIATIONS FOR
K+ MESONS FROM PRIMARY AND SECONDARY METHODS
Primary Mass Secondary Mass
T 966.6 + 1.9 966.1 + 0.7
IC+42 967.2 + 2.2 964.8 + 2.8
K+2 966.7 + 2.0 964.2 + 2.0
K3 966. + 6
Ke3 963. + 10
The lifetimes of the T. K 2 and K have been measured to a
high degree of accuracy by Fitch and Motley using counter techniques.5'6
They obtain the following values for the lifetime:
+0.0o8 8
T(T) = (1.170.07 ) X10 sec.
T (K2) = (1l.210 +0.11) x 108 sec.
T(Ki2) = (1.17.0 (.708) x 108sec.
These values lie well within the experimental errors of the somewhat
less accurate emulsion results.7,8 The lifetimes of the Ko3 and Ke3
have been measured only by means of the emulsion technique.9 The lifetimes so obtained are
T(K 3) = (0.88 + 0.2.) x 108 sec.
T(Ke3) = (1.44 + 0.46) x 108 sec.
5
which are in agreement with the T, K 2 and K 3 lifetimes. Among the
neutral K mesons it has been well established recently that there are
two types of particles. One of these, the G01, is short lived with a
lifetime of 10
T(G) = (0.74.15 0 ) x 10 sec.
8 11
The other, the 902, is long lived with a lifetime of roughly 10 sec.
At present, the data regarding the scattering and the production of K+ mesons is rather scant. Thus it is difficult to state whether
or not there exists a coorelation between the interactions of the particle and the mode by which it decays.
Thus it might appear that one is led to the tentative conclusion
that the K+ particles should be considered as a single parent particle
with several modes of decay. The difficulty in this approach arises when
an attempt is made to assign nonclassical properties to the parent K
particle. For some time it has been assumed that among the set of
simultaneous observables unique to a given nuclear particle, are the
intrinsic spin and the intrinsic parity (i.e, the behavior of the wave
function of the particle under inversion of the space coordinates) of
the particle. In other words, it is required that for any transition
between states the total angular momentum of the system and the behavior
of the wave function of the system under the parity operation remain
invariant. Then using the laws of conservation of parity and angular
momentum it can be shown that the K 2 cannot have either spin zero and
odd parity, or spin one and even parity. It can also be shown that the
T cannot have spin zero and even parity but for spin one it can have
6
either even or odd parity. Thus if the K 2 and T are the same particle
then this particle cannot have spin zero. Experimentally no spin effects
have been observed for the K,12 while several emulsion experiments have
indicated that the spin of the T may be zero and also that the T does not
have spin one and odd parity (denoted by 1). So from the viewpoint of
mass, lifetime, and interaction properties all of the K+ particles seem
to be identical, but if spin and parity are taken into account one finds
that, if the spin is zero or one, the particles may not be identical.
Furthermore, there appears to be no means, within the framework of
present day physics, of explaining this difference if parity is conserved in the decay. This presents a problem, often referred to popularly as the "TG puzzle," which is one of the major difficulties in the
understanding of high energy nuclear phenomena.
1.2 Further Consideration of the K Particles and the Aim of the
Present Investigation.
The problem which was introduced in the previous section
suggests two possible areas of further research. The first, and perhaps most obvious, approach is a more exhaustive experimental study of
the K particle properties, while the second requires a careful reexamination of the presently accepted theoretical concepts of contemporary
nuclear physics. Clearly, these two types of research are not independent,
for the correctness of a proposed theory can only be judged by reference
to accurate experimental results, and useful experimentation derives its
direction from theoretical conjecture.
A great deal of experimental research remains to be done in the
study of the behavior of the secondaries emitted in K meson decay. That
7
such work can yield information regarding the primary follows from the
fact that it is possible to predict the behavior of the secondaries if
certain properties of the primary are known. I:a most cases the information, concerning the primary, which must be known are quantum mechanical parameters. If the laws of conservation of angular momentum and
parity are assumed to be valid then the relevant parameters are the
spin and the intrinsic parity. The method which is used in decay studies
then is to predict the behavior of a given decay mode for different values of the spin and parity. The results of experiment are then compared
with these theoretical predictions. The spin and parity which lead to
calculated results consistent with experiment are then taken to be the
correct values. The T meson is a particularly good subject for this
type of study since it decays into three charged secondaries which,
because of their charge, are easily observed. Dalitz 15 and Fabri6
have predicted the energy and angular distributions of the secondaries.
All of these calculations have assumed the validity of the law of conservation of parity. Recently several experimenters have examined the
energy and angular distributions from large samples of T mesons decaying
from rest in nuclear emulsions.13'17 The data collected from all of
these experiments are compatible with 0 and 2 and several sets have been
compatible with 1+. Perhaps the most salient feature about all of these
experiments is that none of them, either individually or when combined, give
close agreement with any one of the predicted distributions. Thus there is
some justification for the existance of a certain amount of doubt as to
the correctness of the assumptions used in calculating the theoretical
spectrum.. Lee and Yangl8 have examined in detail the validity of the
8
law of conservation of parity. Although they have not been able to
resolve the uncertainties surrounding the nature of the K particles,
they have been successful in that they have provided the impetus for
experiments whose results contain extremely important physical information. Wu et al. 1920 and Garwin, Lederman and Weinrich21 have established that parity is not conserved in those weak interactions (i.e.,
slow processes) associated with: decay, and the decay of the T and
the [ mesons. As yet it has not been ascertained experimentally whethei
or not this result extends to other weak interactions which do not
22
involve the neutrino. With this in mind Lomon2 has calculated the
g energy and angular spectrum for T decay assuming the nonconservation of parity. He finds some disagreement with the world data for
a spin zero T meson but by adjusting certain constants, which is allowed, he finds that he can obtain fair statistical agreement with a spin
two T meson. It is important to note that Lomon's results depend heavily
on the assumption that no experimental bias exists in the world data.
The present investigation was undertaken in an attempt to
collect and analyze data on T meson decay by means of an experimental
method which differs somewhat from the photographic emulsion technique
which has been used exclusively by previous investigators. In this experiment the T mesons are brought to rest and then decay in a propane
bubble chamber.
Although a detailed discussion of nuclear emulsions is beyond
the scope of this work a brief comparison of this technique with bubble
chamber methods is of value. In emulsions tracks resulting from a large
number of beam pulses are observed in the field of view of the microscope
9
used for the scanning. The scanner must follow a track of correct blob
density to its end and further must identify the decay moOde by following
each decay product to its end. In the bubble chamber technique, on the
other hand, each pulse of the beam is photographed with a stereo camera.
The greater range of the secondaries in propane as compared to that in
emulsion "enlarges" the decay event to an extent such that the scanning
of the bubble chamber photographs may be done either with the naked eye
or with low magnification viewers. The two stereo photographs give the
scanner views of two spatial orientations of the event and this tends
to increase the efficiency with which the events may be identified. In
the emulsion technique, the scanner, in most cases, is limited to one
orientation of the event, and this condition may give rise to a geometrical bias in this type of experiment. It is one of the purposes
of this experiment to ascertain whether or not such a geometrical bias
exists in the emulsion data on T+ meson decays. A rough estimate of the
length of scanning time required for emulsion and bubble chamber experiments, each with the same number of events, shows that the bubble chamber
scanning time would be approximately onethird of that required for emulsions. Since the time required to measure and analyze the events is
approximately the swame for both techniques this factor of three represents the increase in speed of experimentation made possible by the
bubble chamber technique.
CHAPTER II
THEORETICAL CONSIDERATIONS IN T+ MESON DECAY
2.1 The Kinematics of T+ Meson Decay
The effects of the laws of conservation of momentum and energy
on the reaction T+ + + + + A will now be considered. The positive
and negative pions will be assumed to have the same mass.
First consider a three body decay in the laboratory coordinate system for the case in which the center of mass of the three
particle system is in motion. At any instant in time the positions of
the three particles may be represented by the vectors rl, r2, and r3,
where the origin of the coordinate system is taken to be an arbitrary
point in space. In this problem all three particles have the same mass
4 4
and the particles whose positions are given by rl and r2 are the identical pions. The momenta conjugate to rl, r2, and r3 are the individual
particle momenta pi, P2, and p3. The kinematics of T decay can be given
for this completely general case in terms of the coordinates and momenta
introduced above, but for investigations, such as this, in which the spin
angular momentum of the decaying particle is to be determined the coordinate system which is chosen is that system in which the decaying particle
is at rest, that is, the center of mass system. In the following the
center of mass system will be used exclusively and the vertex of the
event will be taken as the origin. In this system the law of conservation of momentum provides the constraint that the vector sum of the
momenta of the decay products must equal zero. This requires that the
momentum vectors of the three pions lie in the same plane. Thus the
11
12
problem now requires the knowledge of only two momenta and it is therefore convenient to define a new system of variables.
Since particles 1 and 2 are identical no generality is lost
if these two momenta are combined to give a single new momentum. The
new system of variables defined for the general case of T+ decay will
consist of three momenta and three space variables. The new momenta
will be: P, which is proportional to the sum of the three pion momenta,
P
p, which is proportional to the relative momentum of the two like pions,
and p', which is proportional to the momentum of the odd pion, in this
case the r, with respect to the center of mass of the system consisting
of the two positive pions. The new space variables will be the coordinates conjugate to P, p, and p', namely R, r, and r'. In particular,
these new variables are
 = X (( + P+ )
=12(P2 ) (21)
^'2 2 1 P2]
and R (r1 + 2 + r)
r =
r = (r2 ) r(22)
r' = [3 — 1 (r + 2)]
This system of variables was first introduced by Fabri6 and the trans4 4 4  an 4 P 2 
formation from r l, r2, r and p1, P2,? to R, r, r' and P, P P is referred to as the "Fabri Transformation". Figure 1 is a momentum diagram
13
P2
t 3
FIGURE I Momentum Diagram
showing the quantities introduced above. From (21) it follows that
the center of mass system is defined by P = O. The choice of constant
coefficients is such that the total angular momentum in this system is
J = j + j' (23)
where
j r x p
e > > (24)
j' =r' x p'
In later sections it will be found convenient to separate
the moduli of the vectors p and p' from the unit vectors u and u
which give their direction. From equation (21) it follows that
P2 = P2 2 1/2 p2
= 1 2 5
P = 2 P3 (25)
A(P2 2
cos ~ = u. = (u2 2)
[P32(p12+ p22 1/2 P2)]1/2
where G is the angle between p and'.
The law of conservation of energy can be expressed for the
case of T decay as
E + E + E = M = m+ Q (26)
where Ei is the total energy of the ih particle, M and m are the rest
masses (in energy units) of the T and A mesons respectively, and Q is
the total kinetic energy given to the pions on decay of the T from
rest. For a three body decay in the center of mass system two parameters are sufficient to specify completely thekinematics. Thus all
possible kinematical states can be represented in a two dimensional
15
space. More specifically, it is possible to establish a one to one
correspondence between tihe two parameters which determine the decay
configuration and the points of a plane region. The conservation of
energy and momentum will limit the allowed area. The conservation of
energy requires that the sum of three pion energies be equal to the Q
value of the decay, approximately 75 Mev. A geometrical representation
of this is provided by the equilateral triangle which has the property
that the algebraic sum of the perpendicular distances from the three
sides of the triangle to any point interior to the triangle is equal
to a constant, the altitude of the triangle. Thus all possible states
allowed by the conservation of energy will be represented by points
lying in the interior of an equilateral triangle of altitude Q. This
triangle has been drawn in Figure 2. The perpendicular distance from
a given side of the triangle to a point representing the state will be
equal to the kinetic energy of one of the particles appearing in the
decay. Using a polar coordinate system, (p, c), with the pole at the
geometric center of the triangle and measuring cp from an axis passing
through the pole and one of the vertices, the following expressions
for the total energies of the particles are obtained
E1= M [1 + p K cos(t  2c)]
E2 M= [ p K cos(p + 2T) (27)
3 3
E M 1 + p K cos ip
where K = Since E = + the p and cos can be
written as
16
0
I
6:
x
x =
FIGURE
2 Phase Space For
Meson Decoy
17
2 M2 1 2
p =  K [(1  p cos c)  K (1  p2 sin2 p)]
p2 M K (l + p cos (p) [1  K (1  p cos c)] (28)
3 2
p sin q (1  2 K cos cp)
[(1p cos p)  K(lp cos2q)l /2[(lp cos p) K(lp sin) ]/2
It is clear that all the points contained in the interior of
the triangle cannot represent allowed states of the system. For example,
the vertices of the triangle are ruled out because this would correspond
to one particle receiving all the energy and this, of course, is not compatible with the conservation of momentum. When K is small the nonrelativistic
energymomentum relations may be employed and then it can be shown that
the maximum energy which any particle can receive in the decay is equal
to 2/3 Q. Furthermore, it can be shown that, under this condition, the
effect of conservation of momentum is to limit the allowed states to the
interior of a circle which is tangent to each side of the triangle at
its midpoint as shown in Figure 2. When K is not small, the situation
is somewhat more complicated. Using the conservation of momentum and
Ei2 2 + m2, where Ei is given in terms of p and p in (27), it is
possible to show that the allowed region is limited by a cubic of the
form
+ (l + ) 2 1(29)
p.. p.. = 0, (29)
a cos 35C a cos 35
where a = 1/2 K (1  1/2 K)2 = 0.09. Note that in the nonrelativistic
limit (K  0)( and therefore C0O) this equation reduces to the equation
of a circle. Equation (29) is represented by the dotted line in Figure
1, and the nonrelativistic region is the solid circle. Reference to this
figure shows that the nonrelativistic approximation is a rather good one for
this problem. The results obtained when it is not possible to consider
K as a small quantity simply restrict the permitted region somewhat,
which in turn changes the maximum energy received by any one of the
pions from the nonrelativistic result of 2/3 Q to approximately 0.96(2/3)Q.
2.2 Comparison of the SpinParity Properties of the Q+ and T+
In this section the effects of the laws of conservation of
parity and angular momentum on T+ meson decay will be considered. The
14,15
first investigation of these effects was carried out by Dalitz.'
This was followed by the work of Fabri.6 The presentation given here
follows closely the work of these two authors.
Before proceeding with the analysis it is first necessary to
make certain definitions. An arbitrary direction in space is taken as
the axis of quantization, and this is denoted by the unit vector uo.
The eigenvalues of the square of the total angular momentum J12 are
J(J + 1) and those of J uo are mj. Similarly J2 and jj'l have
eigenvalues i(I + 1) and'(I' + 1) respectively and j uo and j' ~ uo
have the eigenvalues m and m'. The final state can be written as a
function of p, u, u' and will be an eigenstate of I J and J * uo relative to the J and mJ corresponding to the spin of the T+ and its orientation in space with respect to the axis of quantization.
The law of conservation of angular momentum applied to T
decay requires the total angular momentum of the three pion final state
to be equal to the initial spin angular momentum of the T. The law of
conservation of parity states that the behavior of the final state wave
function under inversion of coordinates must be the same as that of the
initial state wave function under the same operation. This latter rule
19
can be applied to a hypothetical decay process such as A  B + C +... in
the following way: assume that the final state can be represented by a
wave function which is proportional to the product of the wave functions
of the individual particles B, C, etc. It is assumed that these wave
functions can be written in such a way as to exhibit a definite symmetry
with regard to inversion of the space coordinates. Then the symmetry of
the product wave function under spatial inversion is the same as the combination of the individual symmetries under the same operation. It is
customary to represent the symmetry by (+1l) depending upon whether or
not the wave function changes sign upon inversion. Then the combination
of the individual symmetries takes the form (+l)(1) = 1 etc. Thus in
the two body decay A  B + C, if the wave function for A has odd parity
and the wave function for B has even parity then the wave function for
C must have odd parity. The parity of the individual wave function is
the product of the intrinsic parity of the particle and the parity of the
state. If the wave function is an eigenfunction of the angular momentum
then the parity of an orbital angular momentum state goes as (1).
As an example of a simple application of the angular momentum
and parity conservation rules the case of the G~ meson will be considered. The 9~ meson is known to have a decay of the form
~ ,r + A (a)
and may undergo the decay
Go0 go + go~. (b)
The it meson has odd intrinsic parity so the final state wave functions
for both (a) and (b) contain a term (_1)2 upon inversion, that is, there
is no sign change of the final state wave function due to the intrinsic
20
parity of the A mesons. The G~ has an angular momentum and the conservation of angular momentum requires the final state for (a) and (b)
to have an angular momentum equal to that of the initial state. The
orbital angular momentum for a two pion system is just r x p = 2
where r is the relative position vector and p is the relative momentum.
Since the spin of the pion is zero the only angular momentum which can
enter into the final state is that arising from an orbital state. Thus
12 4
the final state will be an eigenstate of I and the projection of 2
along the axis of quantization, I * uo. Thus the conservation of
4 4
angular momentum is simply J = 2. The parity of the final state wave
function will then depend upon whether 2 is even or odd. If the spin
of the G~ is odd, then the parity is odd, and, if the spin of the G~
is even, then the 9~ has even parity. Process (a) can take on all
values of 2 and thus the knowledge of the existence of (a) yields
little information about the spin of the 9~. The existence of process
(b), however, would restrict the possible spins of the 9~. This restriction arises from the fact that two neutral pions are identical
bosons, which requires that the wave function representingthe system
formed by the two must not change under the interchange of the two particles. Such a wave function must have a value of 2 which is even.
Thus if the two A~ decay mode of the 9~ exists then the 9~ has even
spin and even intrinsic parity.
Next consider the case of the K or 9 which decays as
+ +
 A + i~. Here the I of the final state can be odd or even with
corresponding parity. Therefore, the 9+ must have spinparity combinations of the type: (0+), (1), (2+), etc.
21
Finally consider the somewhat more complicated case of T
decay. This particle is known to undergo the decays
T + + t + K (a')
T+, + + 0 (b')
In both (a') and (b') the parity of the final state wave function due
to the intrinsic parity of the three pions is (1)3, or odd. Thus the
parity of the final state wave function is (1)P' where P' is the parity
of the final space state. This will depend upon 2, which is proportional to the angular momentum of the two r+ system, and on i', proportional to the orbital state of the A with respect to the center of mass of
the two t+ system. The positive pions are identical bosons and the
same argument that was applied to 0~  2~0, requires that 2 be even.
Thus the parity of the final state depends only upon the wave function
with respect to u'. It follows that if the parity of the v+ is odd
then P' must be even which requires 2' to be even and, conversely, if
the parity of the T+ is even then P' must be odd and 2' odd. Since
the orbital angular momentum must satisfy the condition
  2 tl J K< 2 +
it is possible to conclude that for a particle of even parity and zero
spin there is no allowed three itmeson decay. For, in order to have
J = O, 2' must be equal to 2, but 2 must be even and therefore 2' must
be even which leads to a spin zero particle of odd parity. Therefore,
the T+ must have one of the following spinparity combinations: (0),
(1+), (1), (2+), (2) etc.
Now the original assumption about the nature of the 0+ and the
T+ was that they are different decay modes of the same particle and that
22
particle has a definite parity and further that parity is conserved in
both decay modes. For the 9+ and r+ to be a single particle of definite
parity with conservation of parity in the decay the T+ decay spectrum
must indicate a spinparity combination of (1), (2+), (3) etc.
If such a T+ decay spectrum is not found then it is necessary
to assume that either the 9+ and T+ are different particles, or that
they are the same particle but that the particle does not have a definite parity, or that the parity is not conserved in the decay. If
the parity is not conserved in the T decay then it might be possible
to find further evidence for this in the T+ decay spectrum by itself
without regard to the 9+  T relationship. This is discussed in a
later section.
2.3 Calculation of the T+ Decay Spectrum for Certain SpinParity
Combinations
The starting point of the detailed theoretical study of T
meson decay is the general quantum mechanical expression for the transition rate. In this case it is written as
dR(pF'cC/ ) =  r n ( II,, > C(p epcs) (210)
where M(p', cos 9) is the matrix element of the transition and p(p',
cos 9) is the density in phase space. It is in calculating M(p', cos 9)
that the physical assumptions of the model are introduced.
M(p', cos 9) is calculated by considering the matrix elements
for the production of momenta k and k', where = ph and k' p'/h
23
respectively. Neglecting final state interactions, the wave function
ik'r rfor k is proportional to e and that for k' is proportional to
ik'r' +
e. Then the matrix element for the decay of the T meson in
the state J, mJ into the state described by momenta k and k' is
d( )' (211)
Where, in the gJm there appear terms representing the decay interaction and the initial state wave function which have been integrated
over the coordinates of the T meson. Furthermore, the gJm (r, rF) in
the integral of (211) shall be such that not only is angular momentum
conserved but also such that if the integral is expanded in powers of
the momentum the series will converge. It is assumed that gJm is not
explicitly momentum dependent and also that it is rotationally invariant.
Then
6~mJ~t)P) = s fr^rJ Q,(V.2mwl,N Jms)
21,,
x^r, 7( e) (212)
where fJ2r, is independent of mj. Substituting this into equation
(211) yields
m77 f /A = Jd)d e=/ ei
(213)
m, m'
6 8, ~' (8~
Next the exponentials are expanded using the expansion
47 4
eiW~  8; Xalo3 Y (&?) 1 (214)
where Gk and cpk are the spherical angular coordinates of k. Then
x,' A, (S *> q9+.)(215)
X r 77 J) ) (0)(f)
Integration over the space angular coordinates leads to
()  (47)4CF dY'7 f (Yr.,+*
x (7fr)f ('r) j (YP(nm ^Jm) (216)
where the orthonormal property of the spherical harmonics has been
utilized. Equation (216) can be written in a more convenient form as
wf(7tXX)J? = Y \a (iui)J Y~ ^\ L ( r,)(
A,2 (,  I)
7I) 9rmIL7Jr2)(217)
where
a (J,')J,, = ('' iR"  ((*T)J;;''' ) (218)
To evaluate a(k, k'), it is necessary to make an assumption about
the form of f(r, r')Jg,. This function is defined in such a way that
Jaa
25
it is zero for either r > 1 or r' > I where * = f/mTc. This corresponds
to the introduction of a momentum cutoff k = 1/. and k' = 1/X. Thus
the aJ, (k, k') will have a contribution only for r < K. In r+ decay
C
a<f mW (^30'
always. This corresponds to
r.,
ri< i n/m )0(3m^)'^Oi;i0.2
so it is possible to use only the leading term in the expansion for
small kr
* fr)  (ir)/l)!!
To this approximation
a (4)), (~ Zi ~'/drd'(rr
XII~ I am, a(219)
or
a(i' ~ ^' I (r,.  i' (220)
where a(r, r') J, is independent of k and k'. Then,;'~,"' ~* >'lt (221)
XZc~,,~~, (e',, )i
For a T decay in which the angle 9 between k and k' and the
magnitude of the momentum, XI, are given then the kinematics of the final
state are, of course, completely specified. The matrix element for decay
into this state is then
26
J= (222)
( eR'^)'7 C;)) (U'coS)
where the mJ dependence is removed by summing over all mj compatible
with mJ = m + m'. Since the axis of quantization is arbitrary it
can be taken to lie along a line with spherical angular coordinates
(Gk', cpk). Then (221) becomes
w7 di'J= ; <(r l)~r, rl,,;n,o0 P P'Jmo~if
= o< (r,rM, " i (I'' nOIO S7n) (225)
str
So the integration over k gives
IM (K, cos s)il =(4 tS) f tr,r 8)(; i c n )
r X', P I
x (IP,OI(P'J. M) ('')( l ) (1
X costa s) e(k'klS(cositos&)d*'
=(8d C r) 117x)6' A ^ fl (224)
X (Vr M Q)(I^l)! r (coswl
Replacing k and X' by p/t and p'/fi respectively and dividing by the
constants yields the distribution function
27
F(pcoAe)  2o(rr)i.'p"(i2+ OhS2L%).s, 4,P
(225)
P
A (C)! I ( os
where i; '_< J _< $ + $'. The assumptions made in the derivation
above are such that the lowest (1,T) pairs make the largest contribution to F. Thus, neglecting highe r angular momentum terms
in______ (226)
In obtaining equation (225) the only conservation law which
has been used explicitly is the law of conservation of angular momentum.
This restricts the values of I and il which are allowed by requiring
that they satisfy the triangle condition,   i' < J < 2 + 2'. The
conservation of parity places a further condition on i and i'. This is
that the states of I and i' which are summed over must be compatible with
parity conservation in the sense described in Section 2.2. In Table III
those values of (, 2') which are allowed by both the conservation of
angular momentum and parity are listed. This table shows for each value
of J and P of the T meson the allowed (2, 2') states which may occur in
the final state. To the left of the dotted line are the pairs (2, 2')
corresponding to the minimum value of the quantum mechanical combinations
of I and 2' compatible with the choice of J and P. Examples of the higher order terms lie on the right. In Table IV the FJp's calculated by
JP.
28
taking only the first term, that is by using equation (226), are listed.
TABLE III
Allowed (., Q') Values for a T Meson of Given J and P
J, P
Possible (i,')
+1
1
1 +1
1
2 +1
1
(0, 0)
(o, 1)
(2, 2)
(2, 1)
(0, 2); (2, 0)
(2, 2);
(2, 1);
(4, 4);
(2, 3);
(2, 4);
(4,
(4,
(6,
(4,
(4,
4)...
3)...
6)...
3)...
2)...
TABLE IV
FJp's Calculated from Equation 226
J
0
1
1
2
2
P
+
+
1
f2
p,
p4p 4sin2G cos2G
p4p 2sin2G
120 12p4 + Io212 2p'4 + 21ao211'2olip2p'2(3COs2 1)
It is convenient to write the FJp in terms of new parameters
E J
= Emax, where Emax = 2/3 Q, and x = 173 (E1  2) where Ei = Eci+/Emax
x Fmx 1 C  2.
and e1 > c2. From equation (25) it follows that cos 9 = :q e
El > E2 i Xm''N35",e(1  E)
29
For comparison with experimental data the functions Fjp (e, x) must
be normalized. That is
 — h  1
J =
where xm =v(  e). The normalized F p(E, x) are given in Table V.
Note that for the 2 case the values of A and B, which represent the
weight of the contribution of the (2, 0) and (0, 2) states, are unknown.
TABLE V
Normalized Distribution Functions
JP
J P Fjp(E, x)
0  rt/8
1  24w2(l  E)2 [l(x/x)2] (x/xm)2
2 + 2nt (1  e)2 [l(x xm)2]
2  (2/5) [/(A2+ B2)] [A2e2 + B2(le)2 + 2ABe(1e)(3 cos 21)]
where
1 Jo F,,Z )ds Jx
, dc c1
0 / 0
[I
30
The transition rate in terms of (E, x) can now be written as
ddx FP(Ex) E(,
As a function of (e, cos 9) it is
de ~1)  F (E Cos ) f(r cos C ).
dlE J(c F)) cP5 F c )^o
The factor p is the volume in phase space, and it is in calculating this
quantity that the conservation of energy and momentum are used explicitly. It is also assumed that the probability of a decay leading to a
specified accessible volume of phase space is directly proportional
to that volume. It can be shown that p(e, x) is a constant and that
p(e, cos 9) is proportional to vI(. Thus the expression of the
energy spectrum is
a fr tr) x 
and that for the angular spectrum is
d ()co7)co ) E io )d4
The results of these integrations are given in Table VI. Figure 3
and 4 are the resulting curves.
31
TABLE VI
Expressions for the Angular and Energy Spectra
J P dR/d(cos 9)
0 1 1
1 +1 1
1 1 15/2 cos2G sin2G
2 +1 3/2 sin2G
dR
de
o 1 8/t \/(1E)
1 +1 6/t e)
1 1 192 x 16/15t e2(1e)2 E~(1~)
2 +1 (16)2/3t ~(1e)2 4 (1E)
2.4 The T Decay Spectrum When Parity is not Conserved in the Decay
When the conservation of parity is not assumed in the T+
decay, equation (225) is still valid and the only restriction placed
on the (2, 2') pairs is that which arises from the conservation of
angular momentum. The distribution function will now be a sum of contributions from (2, 2') substates of both parities. The amount with
which each of the two parity states contributes to the distribution
function obviously depends on the form of the decay interaction. However, if the DalitzFabri model is to be preserved, it must still be
assumed that the lower (2, 2') terms constitute the major contribution
to the spectrum. Since nothing is known about the form of the parity
nonconserving decay interaction the spectrum will only be discussed
32
1.00
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E
FIG. 3.
Energy spectra for spinparity combinations(S < 2)
as predicted by Dalitz  Fabri theory.
2 is not shown.
33 
C+ 01.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Cos &
FIG.4.
Angular spectra for spinparity combinations (S, 2)
as predicted by Dalitz  Fabri theory.
2 is not shown.
on the basis of the following two assumptions:
(1) The form of the interaction is such that no final
substate is favored by virtue of its parity, that is,
the interaction is parity independent.
(2) The magnitude of the contribution from the various
(i, V') terms in a given distribution function decreases
as (2, 2') increases.
The requirement that 2 be even is not relaxed, of course.
The spin zero distribution function will be the same as that
given previously for (0) since only one parity state can be formed for
this value of the spin. The major contribution to this distribution
function arises from the term in which the (, 2') pair is (0, 0).
Reference to Table III shows that the next term corresponds to (2, 2)
and is assumed to be very small. For spin one, the spectrum will be
the same as that previously given for (1+) since the lowest term is
(0, 1) and the next lowest term (2, 1) is from the same parity state.
It is not until the third term that an (2, 2') pair corresponding to
an opposite parity state can contribute. For spin two the (2, 1) pair
might be included with the (0, 2) and the (2, 0) pair to given a spectrum different from the 2+ or 2 spectrum. This leads to a function of
the form
F(pCoC )  tI ioJp + 2 Jll.lI pf"if'(3co'')
+ CeYS'@) I 4pC' CS0 )
where the o's are unknown and must be adjusted to fit the experimental
22
distribution. Lomon has considered this spectrum and the spectrum
35
from other spins with parity nonconservation. The agreement of the
spectrums with the experimental data is not improved for odd spins or
for spin zero. For spin two better agreement can be attained but
this is because there are so many arbitrary constants to adjust. However, unless the precision and statistics of the spectrum measurements
are considerably improved it is doubtful that it can be shown to be
necessary to introduce parity nonconservation in the T+ decay on the
basis of the T spectrum itself without considering the G+ decay mode.
CHAPTER III
EXPERIMENTAL APPARATUS AND PROCEDURE
3.1 The K+ Meson Beam
The K mesons for this experiment were obtained by bombarding
a 3 inch thick copper target with the 3 Bev external proton beam of the
Cosmotron at the Brookhaven National Laboratory. If only K mesons
were produced in this bombardment an experimentally useful beam of
such particles could be formed rather easily. A magnetic system incorporating vertical and horizontal focussing in addition to momentum
separation could be placed so as to accept a certain solid angle of the
charged reaction products and those products of correct momentum could
then be focussed at the detector chosen for the experiment. In fact,
among the charged reaction products the x+ mesons are more plentiful
than the K's by a factor of about one hundred. Thus, since the A+
meson has a lifetime comparable to that of the K+, a beam formed by
such a magnetic lens system would be mostly j+ mesons. Since the method
of detection chosen for this experiment is essentially a visual one it
is necessary to keep the number of particles which arrive at the detector
per Cosmotron pulse belowa maximum of about 20. This can be achieved by
an ordinary analysisfocussing system by decreasing the number of protons
in the initial external beam. Such a resolution of this difficulty has
the disadvantage that it results in only one K meson in about 40 Cosmotron pulses, i.e., about 3.33 minutes. It should be pointed out that
although protons are among the collision products the range of those
reaching the detector via the analysis system is such that they are
37
38
stopped in the wall of the detector and therefore do not constitute
a background problem.
The problem introduced above was solved by using a two magnet
system with an absorber between the magnets. This system was designed
by Dr. Donald I. Meyer and is illustrated in Figure 5. In this system
the momentum analyzed beam of A+ and K+ mesons is brought to a focus
at a 3 inch Be moderator. In traversing this moderator the K's lose
more momentum than do the t+'s. The second analyzing magnet is placed
after the moderator, and the detector is placed at the focal point of
this second magnet which corresponds to the post moderator momentum.
In such a system it is necessary to keep the distances as short as
possible so that the number of K mesons lost from the beam is kept at
a minimum. This was accomplished by combining strong focussing with
momentum analysis in the same magnet.23 By using a large gradient
both horizontal and vertical focussing were obtained with a 3 meter
object to image distance for each magnet. In Figure 6 a plot of the
beam obtained from this system is drawn. The historgrams were obtained
by plotting the end points of the stopping T meson tracks, excluding
those which suffered large angle scatterings. In all three histograms
90% of the beam lies between the dashed lines. Using this system it
was possible to obtain one K+ meson in about 6 Cosmotron pulses, i.e.,
about 30 seconds and, in all, about 3500 K+ mesons were observed in the
course of the experiment.
3.2 The Detecting Device
A propane (C3H8) bubble chamber with sensitive volume 12 x 12 x 30
cm was the detecting device utilized in this experiment. The operating
EXTERNAL
PROTON
BEAM
A A. l A
oI^ rj MESON;
I
ANALYSING
FOCUSING
MAGNET
ANALYSING 1
FOCUSING
MAGNET o <
KMESONS
PROPANE
12"X5"X5" BUBBLE
CHAMBER
I
I
FIGURE 5 
PLAN VIEW OF EXPERIMENTAL SETUP
suo!inq!iS!i a IX wuoe8 o0 4Od 9 3ln91l
(Wo) a
1____ 01 0
I, ~ Of m
I
C
cn
+ (w) Z  
Cl 9 0 90
0
z
01 1~ S O S 1O~~0
n
I: I O ^0?
m
OZ
or
a,
09
, (W3) X _
I=^^^  ^ — __.rLj'  1  t h o'
OZ m
~ ~I } I J~<
~~~I I _ m~~~ O
I,~~~~: ~z
(I)
0 
41
temperature of the chamber was 52.8~C. Figure 7 shows end and top views
of the chamber. The main body of the chamber is of aluminum and the compressed propane is expanded by means of a diaphragm located in a throat
at the bottom of the chamber. Full details of bubble chamber construction and operation are discussed adequately elsewhere and will not be
considered here.
The events occurring in the chamber are photographically
recorded on 70 mm film by two cameras set at approximately a 15~ stereo
angle. The film was held flat by means of pressure plates and vacuum
backs in these cameras. Two Xenon filled flash tubes act as a light
source for bright field photography. The duration of the flash was
approximately 40 Is. The iris diaphragms of the cameras were set at
f45 and this provided depth of field sufficient to yield fairly good
focus in all parts of the chamber. The camera was supported rigidly
by an I beam structure after focussing and levelling. The arrangement
of the cameras and the illumination system is illustrated in Figure 7.
In order that events may be reconstructed from the stereo
views it is necessary to establish a coordinate system in the chamber.
This was accomplished by a regular rectangular grid of fiducial marks
printed on the front and back windows. These were carefully aligned by
means of a surveyor's transit prior to the experiment.
In Figure 8 the two stereo views of an event are shown. The
top picture is the view of the chamber as seen by the upper camera,
and the bottom is that of the bottom camera.
BEAM WINDOW
COSMOTRON BEAM.
I
ro
i
 I
CAMERA
HAMBER
DIFFUSING SCREEN
8 FRESNEL LENS.
FIGURE 7. END AND TOP VIEWS OF BUBBLE CHAMBER
0:c:
4i%
JrC7?F
A:~t
cv
0
C,
<I
*c
C,I
t;I
ccc
C
Irl
r",
Ct~
rrIt
c~~~ 5.*~u~.i:  ~":';':"~~..
" " lx Lr ~:C ~.2:j I: ~
* "+... %
~sz :~:
a.,1.'~., y c* ~ * ~~~:~ ~r2:~r j:.I~
,:::: r: ~ :: ~: ~'':::~s
~~:~.J****LU*WIr*rr
4 : ":.:' *c:..~ a:t~":ISiu5iB
3.3 The Scanning Procedure
About 25,000 pictures were taken and of these 20,400 were
scanned. The remainder of the pictures were not scanned because of
excess beam intensity, no beam being present, or damage to the film
in the developing process. The large size of the film made it possible
to scan the pictures with the aid of only a hand magnifier of 6.5x magnification. Since the tracks in the chamber are produced by particles
of the same momentum and since the only beam particles reaching the
chamber are A, A or K mesons it follows that the heaviest particles,
that is, the K's, will produce the tracks with the highest bubble density.25 The scanning was done by looking first along the entrance of
the chamber. Those tracks having higher bubble density were then
followed into the chamber. Decay into a lightly ionizing secondary
or into the characteristic T+ mode served to identify the K+ meson.
Since K proton and K+nucleus27 scattering crosssections were to
be obtained from these pictures great care was taken to identify all
of the K particles. However, except for the T+ decay mode no attempt
was made to identify the various types of K+ mesons. It should be emphasized that in the scanning both stereo views were used. This gives
the scanner the ability to observe an event from two viewing points.
This tends to increase the scanners' efficiency for certain types of
events. In particular, T+ decays in which the r meson has a very
short track are often detected only after a look at both stereo views.
A check on the reliability of the scanning was made by carefully rescanning about 3,000 of the pictures. In this rescan only one
new T event was found. This event was unusual in that it scattered
through a large angle immediately after entering the chamber, and the
decay occurred near the bottom of the chamber. The important result of
the rescanning appears to be that it is unlikely that T'S were missed
if they came to rest in the chamber without suffering a large angle
scattering while traversing the first few centimeters of the chamber.
It is estimated that better than 90% of the T mesons present have been
found in the scanning.
In all, 189 possible T mesons were found in the scanning of
the pictures. For a portion of the run. the top camera did not function
properly and stereo views were not obtained for all of the pictures.
At the end of the run the beam intensity increased intermittently
making accurate identification and measurement difficult in a large
number of cases. To avoid sample bias and error, therefore, all of
these events were discarded. This corresponded to approximately 5000
pictures, or about 28 T events. Thus there remained 161 possible T
mesons which could be subjected to measurement and analysis.
5.4 Reconstruction of Events
In Figure 9 the geometry of the stereo camerabubble chamber
system is illustrated. The two film surfaces are plane and parallel and
the distance between optic axes of the two camera lenses is 2W. The
coordinate system on film 1 is (x, y) and that on film 2 is (x' y').
S and S' are the distances between the lenses, 1 and 2, and the front
window of the bubble chamber, which has thickness d, while v and v1 are
the distances between the lenses and their respective film planes. The
FIGURE 9 Stereo Camera  Bubble Chamber System
47
coordinates of a point P in the bubble chamber are taken to be X, Y,
and D, where D is the perpendicular distance from the inside plane of
the front window to P. The origin of the coordinate system is taken
to lie on the optic axis of lens 1. X, Y, and D can be written in
terms of the film coordinates as28
X = (x/v)(S + dT + UD) = (x'/v')(S' + dT' + DU')
Y = (y/v)(S + dT + UD) = 2W (y'/v')(S' + dT' + DU')
= 2W (y/v)(S + dT)  (y'/v')(S' + dT")
(y/v)U + (y'/v) U' (31)
where
T = nl(a2 + 1)  a2
t =[n12(12 I + 1),a2]1/2
U = [n22(C + 1)  (a2
U' = n22(c'2 + l) ?,22
in which nl and n2 are the indices of refraction of glass and propane
respectively and where
2 2 2
2 x + y 2' +
a  2  a
v v,
The quantity a (or C') entering here is actually the tangent of the angle
between the optic axis and a ray passing through the center of the lens
and then intersecting the film plane at the point (x, y) or (x', y').
For the chamber used in the present experiment a and a' have maximum
values of about 1/5.
48
These equations can be put in a more useful form by rewriting
them as
X = x(P + DQ) = x'(P + DQ')
(32)
Y = y(P + DQ) = 2W  y'(P' + DQ')
where
S + dT u
P = I = V
S' + dT' u'
VI V'
Expanding T, T', U and U' gives, to first order in a or a', the constants
T = T'= 1/nl
U = U' = 1/n2.
Making use of the accurately known distances between the inside surfaces
of the front and back windows and between the grid marks on each of
these surfaces it is now possible to determine the values of P, P',
28
Q, and Q'. This evaluation was carried out by C. Graves. It is
important to note in equations (31) and (32) that while the y coordinates on both films, that is, y and y', need to be measured, it is
mecessary to measure the x coordinate on only one of the films to determine the coordinates (X, Y, D). Furthermore, it should be borne in mind
that these equations represent the special case in which the optic axis
of lens 1 passes through the origin of coordinates. If another origin
is to be chosen in the measuring process then a correction must be applied.
The measurement of the events is accomplished by means of a twin
objective traveling microscope. In this instrument the film is clamped
securely in place on a movable stage. The stage and the objective each
49
have one degree of freedom, and their directions of travel are at
right angles. Thus by moving both the stage and the objective it
is possible to view any point on the film. The eyepieces of the
microscopes have cross hairs and the measurement of a point is
accomplished by setting these cross hairs on the bubble of interest.
The position of the cross hairs can then be read from a scale graduatel to 1/1000 of an inch.
A schematic diagram of T+ meson decay is given in Figure
10. In this hypothetical decay the three types of tracks which occur
most frequently have been drawn. These are the N track with a one
prong star at its ending, the curved track arising from multiple
coulomb scattering and the single large angle scattering. In the
case of a straight track only two points on the track are measured.
When a star occurs the length of its prongs are always measured. In
the situation where a large angle scattering occurs the vertex point
and the end point of the track were measured. If the prescattering
portion of the track did not appear to be straight an intermediate
point was also measured. The badly curved tracks represented the most
difficult measuring problem. In this case the angles which are calculated may not be reliable so it is extremely important that an accurate
length measurement be available. The method used was to measure points
along the track sufficiently close together so that the angle between
the chord connecting two adjacent points and the tangent to the track
at the first point is extremely small. On some tracks it was necessary to measure as many as six points along the track. When A+ 4 t+ e+
50
one prong stor
7r meson
t meson /
FIGURE 10
Diagram of T Meson Decay
51
decays occurred in the chamber the coordinates of the beginning and
of the end of the i+ track were also measured.
To reduce the chance of a measurement error occurring the
events were measured two times before any reconstruction was attempted.
These measurements were made at times fairly close together and thus
the possibility of two repeated errors through "memory" of the event
is not ruled out. As a check for such an effect about onehalf of
the events were remeasured a third time at a later date. No discernable effect was observed. If the results of the first two measurements
did not agree with each other then two more measurements were made. In
all cases it was possible, by careful checking, to resolve the difference.
The coordinates obtained by the measurements are then punched
on IBM cards. To check against error in this process the cards were
run through an IBM Card Verifier. The IBM 650 computer was programmed
to calculate the coordinates of the points by means of equations (32).
In addition to calculating the coordinates the machine was also programmed to compute the lengths of all tracks, the angles G123 and G126
shown in Figure 10, and the degree of coplanarity of the three pion
tracks.
3.5 Analysis of the Events
It was pointed out in Chapter II that in a 3 body decay from
rest, such as T+ meson decay, the momentum vectors of the decay products must be coplanar. The use of this fact and the knowledge of the
Q value reduces the requirements of momentum and energy conservation
to three equations in five variables. These can be the individual
52
momenta of the three n mesons and the angles 0123 and @126 shown in
Figure 10. Thus the knowledge of any two of these variables can completely determine the kinematics of the event.
For use in the analysis the values of pi, P2, and p3 were
computed by choosing the values of 123 and 126 In this computa125 126'
tion 123 was held fixed and 126 was varied in 3~ increments, to include all possible decay configurations for the value of G123, then
9123 was increased by 3~ and 126 varied again. This process was repeated until all possible configuration had been considered. The Q
value assumed in these calculations was 75 Mev. The results were
then plotted and the resulting curves provided the kinematical information used in analyzing the data.
In the analysis of a particular event the first requirement
used was that of coplanarity. The angle of coplanarity is obtained
by taking the arcsin of the inner product of a line in the decay plane
and the normal to the decay plane. In the several measurements that
were made of an event the average value of this angle was required
to be less than 5~ for the event to be considered a decay from rest.
When all three decay products were such that they stopped in the
chamber the average measured value of the length of these tracks was
sufficient to specify the event by means of the range energy relations.
However, the further requirement was added that these values of the
energy must be consistent with those obtained from measured average
values of the angles using the kinematics curves. When one of the A
mesons produced in the decay escapes from the chamber the range of the
two remaining pions is sufficient to determine the kinematics of the
53
event, but again, the energy values so obtained were required to be
consistent with those obtained using the angular measurements. In
the third case, that is, when two of the pions escape leaving the
third to come to rest in the chamber, the analysis must be done
solely by means of the kinematics curves with the measured range
of the pion acting only as a consistency check. In this experiment
the majority of the events fall into the second category, that is,
when two pions come to rest in the chamber.
When the T meson comes to rest in the propane it interacts
29
predominantly with the carbon nucleus. 9 The reaction which takes
place is
12
+ + C1 > 2 + lp + 3n
The experimental work of Ammiraju and Lederman3 has shown that the
majority of the a particles have energy less than 5 Mev. An a particle of this energy does not produce a track of sufficient length
to be observable in a propane bubble chamber. The protons, however,
are observed to have high energy tracks somewhat more frequently, and
these can be observed. The secondary particles resulting from iinteractions with propane were in all cases consistent with the
results of Ammiraju and Lederman for hydrocarbons. It should be
emphasized that the knowledge of the angles 9125 and 9126 serves to
distinguish, in most cases, the single prong star from a pion scattering
since the kinematics give the range of the pion. Thus, if the prong begins at the limit of the pion range then the prong is definitely an
interaction product. On the other hand, if the pion track length
54
plus the prong length are greater than the pion range then the event is
most probably an interaction in flight. In all cases which were observed
in this experiment only one ambiguous situation arose and in this case
the t probably interacted in flight. A further point to be mentioned
is that although it is not always possible to distinguish a meson from
a nucleon by bubble density and scattering it is almost always possible
to distinguish an electron by means of these techniques.
3.6 Experimental Errors
In most treatments of experimental errors the various errors
which may occur in the process of collecting and analyzing the data are
grouped into two categories. The first type are referred to as the
systematic errors while the second are the random errors. The systematic error is defined as an influence of approximately constant magnitude on the experimental data which is inherent in the process of
obtaining the data. In an experiment such as this one systematic
errors might be introduced if, for example, there was an error in the
graduated scale of the measuring instrument or perhaps if the range
energy curves were improperly computed. Random errors are those errors
introduced into the data by a large number of irregular and fluctuating
causes. Examples of these might be the thermal expansion or contraction of the film due to temperature variations during the measurement,
or vibrations of the measuring instrument due to mechanical noise in
the building.
In point of fact, most experimental errors are neither purely
systematic nor purely random, but rather constitute a mixture of the
two types. As an illustration of this point consider the fundamental
55
type of measurement involved in an experiment of this sort, that is, the
measurement of the coordinates of a bubble. For purposes of illustration the geometric center of the image of the bubble on the film will
be assumed to be the ideal point to measure. Now, if the measureer
consistently chooses some point other than the center there will be
a systematic error which will be equal to the distance from the "correct"
point to the measured point. The random error arises from the fact that
in a large number of measurements the experimenter is not able to set
the cross hairs of the microscope exactly on the point of the bubble
that he desires to measure but rather will obtain a distribution of
measured coordinates about the ideal point.
The random error will fluctuate from observation to observation and hence cannot be precisely evaluated. Thus errors of this type
may be regarded as random variables and treated by the methods of probability theory. Systematic errors, on the other hand, are approximately constant in their influence on the data and thus must either
be eliminated or precisely evaluated.
In an experiment of this sort a check for systematic errors
can be made by measuring a known constant which is not connected with
the analysis to be carried out in the experiment. The data in this
experiment contains two such independent checks. One comes from the
reaction x+ , + + v, where the + is at rest and the L+ has a constant range corresponding to 4.17 Mev kinetic energy. Measurements
were made on these,u+'s which appeared in almost all orientations and
positions in the chamber. Each i+ track was measured at least twice
and the average value of these measurements was plotted. In all,
56
132 i+ tracks were measured. The results of these measurements is
plotted in Figure 11. The value of the 4+ length obtained by taking
the mean from this plot is
}>(cm) = 0.29 + 0.03 cm.
If the density of propane is assumed to be 0.45 gm/cm3 then
R (gm/cm2) = 0.13 + 0.1 gm/cm2.
From the rangeenergy curves this is seen to correspond to
E = 4.2 + 0.2 Mev
which is consistent with the known value of the A+ kinetic energy.
The second check is found in the elastic scattering
K+ + p  K + p. These events should also be coplanar since the
proton is initially at rest in the propane. The data on these
events were measured by Drs. M. L. Perl and D. I. Meyer, Thirtytwo elastic scatterings were observed and, of these 23 of them had
measured coplanarity angles of less than 5~. These data are plotted
in Figure 12. It seems to be possible to explain the existence of
the remaining nine events with coplanarity angle greater than 5~ as
being due to poor photography, extremely short recoil tracks, or an
orientation of the scattering plane which makes measurement exceedingly
difficult.
The existence of strong internal consistency in this experiment provides somewhat weaker evidence for the absence of systematic
error. In particular, no event was found in which the results obtained
by the use of the inematics curves andthe angular measurements were
different from those obtained by using the rangeenergy curves and
the length measurements.
W
UL)
Z.
z
w
w
U 10
ILO
0
0
z 5
0 1 I I,, III
0.20 022 0.24 0.26 0.28 1/, 0.30 032 0.34 036
4e/L (cm)
FIGURE II Measured.L+ Lengths And Corresponding Goussion
I

8
z
LU
LL
0
z
<) OF COPLANARITY
FIGURE 12
Plot Of K p Scattering Coplonarity Angles
59From the amount of consistency among the energy values from
several measurements and between the values obtained by two different
methods of analysis it is estimated that the error in the energies of
the pions is approximately + 1 Mev.
CHAPTER IV
DISCUSSION OF EXPERIMENTAL RESULTS AND CONCLUSIONS
4.1 Description of the Data
In Section 3.3 it was stated that 161 possible T+ decay events
were found in the scanning. Fine scanning and preliminary analysis
showed that six of these events were not v+ mesons. There were six
cases of possible r+ mesons scattered throughout the data in which
either the film was badly damaged or the photography was extremely
poor. In these cases reliable measurement and analysis was not possible so the events were discarded. Since these events represent isolated instances at intervals of always at least 3,000 good pictures it is
very improbable that the discarding of these events introduces any
appreciable bias. Measurement showed that 13 of the events were
decays in flight, and these also were not analyzed. Thus there remained 136 T+ meson decays from rest which could be analyzed.
The analysis of the decays from rest produced 111 cases of T+
mesons in which the energy of the 35mesons was determined by both the
rangeenergy and the angularkinematics method. The remaining 25 cases
represent situations in which the identity of the C is uncertain. In
these cases one of the t+ mesons comes to rest and undergoes decay in
the chamber while the other two particles escape through the walls.
The knowledge of the angles gives the energies of the three pions, and
this assignment is confirmed by the knowledge of the range of one of the
X mesons. In seven of these twentyfive cases the energies of the two
unidentified pions are within three Mev of each other. The average of
61
62
these energies was taken to be the n energy. This leaves 118 cases in
which the  energy and the cos 9 are determined. The energy spectrum
of these 118 cases is plotted in Figure 13, and the angular spectrum
appears in Figure 14. Note that since cos G is a derived quantity
determined from c, c1 and ~2 it has an error associated with it which
is equal to the combined errors in these quantities. This error in
cos 9 is approximately 0.06.
The 18 cases in which the K is not identified will be treated
in the next section. In the appendix the complete data on all 136 cases
is given in tabular form.
4.2 Comparison of the Data with the Theoretical Distribution Functions.2
In Chapter II the distribution functions for T+ decay were
introduced. In this section these distribution functions will be compared with the experimental data. These comparisons will be made by
means of certain statistical procedures which will now be introduced.
The first type of statistical test which will be applied to
the data will be a relative probability test. This test employs the
fact that the distribution function F(e, x) is simply a probability
density, that is, the probability of a set of N random events is proN
portional to II Fj(~i, xi). If, however, there is no a priori reason
to prefer one of the two distribution functions it may be possible to
conclude that the observed data is more probable on the basis of one
distribution function than on the basis of the other by considering
the relative probability of the one to the other. For example, if
FJ(e, x) and FJ,(c, x) are the distribution functions in question then
z
IL
0
0 IIII I II
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E — 1 
FIGURE 13 Experimental Energy Distribution For 118 Unambiguous Events
N
35
30
20
I0
0
d
z
0  a — I a I  I I pA
0.0 0.2 0.4 0.6 0.8 1.0
cos 
FIGURE 14 Experimental Cos 8 Distribution of 118 Unambiguous Events
65
the relative probability of Fj(E, x) to Fj,(e, x) is
N
p' ill "J i i
J N (41)
( FJ(ic xi)
i=l Ji i
Now, as an example, suppose that there is some region on the (e, x)
surface where F, << FJ. That is, in this region, by hypothesis J'
there should be few events while by hypothesis J there should be a
J
large number of events. It follows then that P can become a very
small number if a large number of events occur in this region of the
(e, x) surface. Similarly if few events occur in this region but many
occur in a region where FJ, > F the quantity PJ can become very
J
large. Thus one would tend to accept hypothesis J is PJ were less
T 3
than 1 and reject it if P were greater than 1. This type of statisJ
tical test therefore has the advantage that it provides a shape dependent comparison of the data with two alternative hypotheses. It has the
disadvantage that it can be extremely sensitive to a single point of
the data. This latter remark is of special importance in the present
investigation. It must be recalled that the distribution functions
FJ(c, x) are obtained by an approximation which discards higher order
terms. Thus the existence of one event which has zero probability on
the basis of a given distribution function does not necessarily rule
out the possibility of the distribution function being the true one.
It should be pointed out further that the existence of experimental
error in the quantities e and x prevents the assignment of the value
66
2+ 1zero of FJ(c, x). In the calculation of PO and P0 from the data
obtained in this experiment there were three instances in which
FJp(Ei, xi) could have been zero had not the existence of experimental error been recognized. The procedure adopted was to use an
F p(Ei, xi) calculated assuming ci and xi to have the maximum experimental error. In the case where there exist more than two possible
distribution functions the most nearly correct distribution function
will, in a large sample, be the one which consistently has a high
probability relative to all the others.
Before proceeding it should be remarked that in the actual
computation of the relative probability exceedingly large numbers are
encountered. Thus it becomes convenient to consider the logarithm of
relative probability, which is
I N F,(CiF Xi)
oglog P = F log F i
J i=1 FJ (i' Xi)
The mean value of this quantity is then
F N Fjt(ci, x.)
(log ) = (E log )
Fj expt. i=i Fj (ei, Xi)
As a means of comparison between the theory and the experimental data is is useful to calculate the expected values of the mean
F
log  assuming e and x to be distributed according to the various
proposed distribution functions. For example, the expected value of
the mean log  if FJ is assumed to be the correct distribution funcF
tion is
1 FT>\ F [ XlogJ 
og Fj> l Fj (lo x)Fj(E, x)dedx. (42)
As a measure of dispersion about the expected value of the mean it
is necessary to calculate the theoretical second moment. This is
0{ \ 2 FJ >F} (43)
J
where
<lfog2  > Fj(e, x)dedx.
FiJ~~ Frj (e, x)
F
J
The theoretical standard deviation is perhaps the more familiar quantity
and this is defined as the square root of the second moment. Then if
FJ is the correct distribution function it is highly probable that the
experimental result will lie within several standard deviations of the
expected value. It is clear that if both (log J) and the theoretiFj expt.
cal quantities defined in equations (42) and (43) are multiplied by the
factor N then the experimental and theoretical quantities can still be
compared. As a matter of convenience and also in an attempt to follow
conventions established by previous authors this will be done here.
Then the expected values of the mean and the second moment become
(log P) N <1og
J ^ v ^J/ F
68
and
C2 =N(2)
Fj = N4F
respectively. The expectation values and their standard deviations h
have been calculated using the parity conserving distribution functions.
The results are listed in Table VII, and these values will be compared
with the experimental results.
Probability Ratic
Fl+
0plF
/p+\
0/F1+
( 0 F0
<1).
dp2+
< F2
TABLE VII
Expected Values of Probability Ratios
As a Function of N
100.085N + 0.52 4N
10+0.059N + 0.20 NN
100.59N + 0.51 N
+0.98N + 2.55 TN
100.36N + 1.13 $N
10+0.29N + 0.68 fN
For N = 156
o111.7 + 5.8
o1 8.5 + 2.5
1055.0 + 5.8
1+133.0 + 29.6
o049o + 13.2
10+39.4 + 8.0
It is also necessary to introduce some quantitative measure
of the degree of deviation of the experimental results from a given
theoretical distribution function. In the limit of large N the deviation of the data from the correct distribution function will be Gaussian
distributed. The wellknown X2 test3 is then applicable. In this test
69
the quantity
r (fi  Npi)
iM = L ~ — Np(4_4)
i=l Npi
is X2 distributed with rl degrees of freedom. Here NPi is the expected number of events in the ith interval and f. is the number of
events in this interval which actually occur in the data. The X2
distribution will then give the probability that this or a larger
value of M arise from N random events distributed according to a
particular distribution function. It is important to realize that
the X test cannot distinguish between two distribution functions
which predict the same values of Npi for every interval. The X2
test suffers from a disadvantage similar to that of the relative
probabilities test. That is, a large deviation from the expected
value in one interval, which could be the result of experimental
error, can lead to a very low X2 probability.
In the previous section it was mentioned that the data
contained 18 decays in which one of the A+ mesons and the A meson
leave the chamber before they come to rest. Thus it is not possible
to identify the A meson. Kinematically, however, these events are
completely specified so that the Ax energy is known to be one of the
two escaping pion energies. The existence of this situation introduces a certain amount of bias into the data, and therefore it is
necessary to adopt a consistent method of evaluating this bias. The
method which has been adopted is to first calculate all possible repJ'P'
lative probabilities, P. Among these numbers one distribution
JP
70
function will have the highest relative probability. In this case it is
the distribution function for the 0 spinparity combination. The relative probabilities are then written in the form PJP and now they will
0be seen to all have values between 0 and 1. Now the effect of the 18
ambiguous events can be estimated by computing the relative probability
for the two possible choices of the nA energy. Of these two possibilities
the one which tends to make the relative probability larger, that is,
least favorable to 0, is then multiplied by the value of the relative
probability which was obtained for the first 118 events. This procedure
carried out 18 times. If at any time the relative probability PJP
0approaches closely the expected value (PJ F obtained by assuming
jp
FJp to be the correct distribution function then the data is also consistent with the JP hypothesis. It should be noted that, for the statistical procedures chosen, this is the most severe assignment of the
ambiguous events that could be made. It will be shown below that, except for 2, in no case does the data appear consistent with any spinparity combination other than 0.
The relative probabilities for the 118 unambiguous events are
1+ 14.9
01 27.7
P = 10
0and
P2+ = 10.2
P = 10
0If the 1 abiuous events are theven distributed in favor of the 1+ spinparity combination the resulting relative probability is
71
(l+) 1 9
0 corr. 1+
This is 8.8 standard deviations from the expected value (Pi+ while
it is within one standard deviation of \P ). For the 1distribu\ Fo0
tion the largest value of the relative probability which can be obtained
is
(P1) = 10233
0 corr. 1which is 5.5 standard deviations from (P ) and 5.0 standard
deviations from P. Similarly, for tie 2+ case the corrected
0relative probability is
(p2+) = 8.2
0 corr. 2+
This result is 6.0 standard deviations from the expected value for F
and 3.1 from that for F. Thus even when the 18 ambiguous events are
allowed to assume values of e and x which make the various relative probabilities least favorable for 0 the data cannot be made to be consistent with any one of the distribution functions for other spinparity
combinations. Furthermore, and this is perhaps the most important result, the relative probabilities obtained by this procedure are not inconsistent with the 0 distribution function.
The next step in the analysis is the testing of the experimental
data for consistency with the 0 distribution. To do this the value of e
and cos 9 for the 18 ambiguous events are distributed in accordance with
the probabilities obtained from the 0 distribution function. The procedure is to divide the energy or angular range into a definite number
of equal intervals and then to calculate the probabilities for each of
72
these intervals. That is, the area under the curve representing the
appropriate spectrum is computed. The events are then distributed with
respect to the weights given by the probabilities concerned. For example, if a certain interval was twice as probable as another and there
were three uncertain events which could fall in either of the two intervals then the events would be distributed with two to the most
probable and one to the least probable interval. This procedure must,
of course, be done separately for the X and cos 9 distribution. When
the values of e for the uncertain events are distributed according to
F and added to the other 118 events an energy spectrum results which
is shown in Figure 15. The relative probabilities calculated for this
spectrum are
(l+) = 1013 03
0 corr. 0(e)
1(p ) 10 2750,
0 corr. 0(E)
and
2+ 110.88
(P ) ~
0 corr. 0(e)
Here, the relative probability for 1+ is within one standard deviation
of <Pl+ while (P1) are 4.4 and (2.8
\ 00 0 corr. 0 0 corr. 0standard deviations from (Pl 0_ and (P2+)  respectively. The
X2 test of this distribution yields a value of M with a 25% probability.
When the cos G distribution is treated in the same way one obtains
z
w
U.
0
6
z
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
FIGURE 15 Experimental Energy Distribution Corrected According To FFIGURE 15 Experimental Energy Distribution Corrected According To Fo
I
1
A)
I
(P+)
0 corr. 0(cos 9)
pl)
0 corr. 0(cos G)
2+
(Po)
0 corr. 0(cos 9)
= 0135.20
1025.9g
12.64
= 10
and
Where (PO+). 00 corr. 0(cos 9)
is again within one standard deviation of (P1+ >
and (P ) and (Po ) 0 are 4.7 and 2.9 standard deviations
0 corr. 0 ~ corr. 0(cos 9) (cos 9)
from (p O and (P ) respectively. This spectrum is plotted in
Figure 16. The X2 probability for the cos G distribution is 7%. It
should be noted that this quantity is largely influenced by the interval
from 0.8 to 1.0. If this interval is excluded a probability of 25% is
obtained.
It is of interest to consider the best and worst values of
the X2 probability which can be obtained by arrangement of the uncertain
portion of the data. For the energy distribution one can obtain a X2
probability as high as 40% and as low as 7%. For the cos 9 distribution
the highest probability is 20% and the lowest is 2%.
The final test which shall be considered is one which was
52
proposed recently by Orear. Orear has pointed out that the low energy
end of the T+ decay spectrum for spin zero is independent of the momentum
while that for spin 1 has at least a p2 dependence. This can be seen by
reference to equation (225). He assumes that the distribution functions
I
20,U
z
LLi
0
Z — I I I I
0.0 0.2 0.4 0.6 0.8 1.0
cos e 
I
\J1
I
FIGURE 16 Experimental Cos 8 Distribution Corrected According To F_
76
for spin 1 and spin zero are identical for values of the At energy
above 10 Mev. Below 10. Mev the distribution functions are normalized
according to
o0.1.'ff( )5 ~ () ^)E<~ (45)
where p(e) is the density in phase space. This test still rests heavily
on the original assumptions of the DalitzFabri mode. First, the assumption of the validity of the conservation of angular momentum is, of
course, not relaxed. Secondly the assumption that the higher powers
of the momentum occurring in the distribution function do not contribute still must be included. If the second assumption is correct then
the test is to a certain extent independent of whether or not parity
is conserved in the decay. This follows from the fact that the 1 term
has at least a dependence on the fourth power of the momentum, and thus
even though a mixing of the two parity states did occur in the decay
the contribution from the state of negative parity would be negligible.
This is the reson that the factor p(e) is the same on both sides of the
equation. The test proposed by Orear then consists of calculating the
relative probabilities P1 for only these low energy pions. The reason
for the sensitivity of this test in distinguishing between spins 1 and
0 lies in the marked difference in the shape of the two energy spectra
in this region. Thus the quantity
77
24
11 fl(Ri)
pi =11
Pi
0 24
n f ()
i=l ~ 1
has been calculated using the data obtained in this experiment. The
result is
1 = 1.9
P = 101,
0
that is, the relative probability is in favor of spin zero.
4.3 Resume of Results
The results of the analysis carried out in the previous
section indicate that the data is inconsistent with spinparity combinations 1+, 1, and 2+. The 2 spectrum cannot be determined since
it depends on the values of certain constants which are not known.
With this exception the data indicate that if the spin of the r+ is
less than or equal to 2 then it probably has zero spin. The experimental value of P gives the least agreement with theoretical pre0diction on the 0 hypothesis but in no way is it possible to arrange
the uncertain portion of the data such that the results which would
be expected if 1 were the correct spinparity combination are obtained.
2+
The value of P is in fair agreement with the predicted number, and
0the results for P+ are excellent. If parity is not conserved in the
0decay then the results are inconsistent with spin 1 for the T meson
but are consistent with spin 0. There appears to be, at present, no
means of testing for consistency with spin 2 if parity is not conserved.
Comparison of the data obtained in this experiment with the
emulsion data13'17 reveals no striking differences in the data obtained
78
by the two experimental techniques. In particular, the number of low
energy pions found in this experiment is roughly the same per cent of
the total sample as that found in some of the more recent emulsion
experiments.32 This tends to confirm the argument that the absence
of low energy  events in earlier experiments was due to bias of some
33
sort. Also, the relative probabilities calculated from the data in
this experiment have about the same amount of agreement with expected
values as those obtained from the emulsion data. Thus the important
result of this experiment appears to be that it confirms, within the
limits of experimental accuracy, the results of the emulsion experiments.
BIBLIOGRAPHY
1. Brown, R., U. Camerini, P. H. Fowler, H. Muirhead, C. F. Powell,
and D. M. Ritson, 1949, Nature, Lond., 163, 82.
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on High Energy Physics, (Interscience Publishers Inc., New York,
1956) p. V25.
5. V. Fitch and R. Motley, Phys. Rev. 101, 496 (1956).
6. V. Fitch and R. Motley, Phys. Rev. 105, 265 (1957).
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278 (1957).
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on High Energy Physics, (Interscience Publishers, Inc., New York,
1956), p. V9.
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15. Dalitz, R., Phys. Rev. 94, 1046 (1954).
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M. Merlin, G. Vanderhaege, Nuovo Cimento (To be published).
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79
80
21. Garwin, R. L., Lederman, L. M., and Weinrich, M., Phys. Rev.
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APPENDIX
List of the Data
Frame No. 1 2 cos 9
39316 0.77 0.54 0.20 o.48
39881 o.60 0.83 0.08 0.88
41772 0.26 0.75 0.49 0.34
42041 0.20 0.89 0.41 0.70'
42047 0.15 0.90 0.44 0.75
42064 0.68 0.68 0.14 o.66
42112 o.68 0.65 0.18 0.57
42526 o.86 0.52 0.12 o.66
42668 o.49 0.52 0.50 0.02
42734 0.40 0.83 0.28 0.65
42894 0.41 0.67 0.45 0.27
42911 0.89 0.49 0.15 o.63
42963 0.06 0.72 0.72 0.00
43209 0.75 0.75 0.00 0.98
43262 0.14 0.75 o.65 0.26
43305 0.61 0.81 0.08 0.86
43671 0.21 0.86 0.48 0.54
44012 0.70 0.41 0.38 0.02
44091 0.74 0.59 0.16 0.57
44191 0.36 o.64 o.48 0.19
44855 0.24 0.79 o.48 o.43
81
82
List of the Data Cont'd
Frame No.
44954
45028
45792
45862
46120
46196
46260
46429
46544
46585
46875
46966
47149
47416
47422
47904
48579
48624
48996
49143
49313
49419
49647
e
0.07
0.45
0.44
o.46
0.13
0.85
0.20
0.28
0.73
0.92
0.40
0.17
0.50
0.12
0.88
0.81
0.16
0.41
0.21
o.o8
0.13
0.69
0.14
e1
0.76
0.78
0.68
0.78
0.84
0.47
0.75
0.74
0.73
0.31
0.69
0.91
0.62
0.72
0.32
0.47
o.68
0.87
0.76
0.80
0.79
o.68
0.81
2
0.68
0.27
0.38
0.26
0.53
0.17
0.54
o.46
0.03
0.27
0.40
0.41
o.38
0.65
0.30
0.24
0.67
0.22
0.53
0.61
0.57
0.13
0.54
cos G
0.19
0.58
0.58
0.60
0.53
0.47
0.30
0.36
0.91
0.07
0.35
0.76
0.28
0.12
0.03
0.33
0.03
0.78
0.52
0.41
0.38
0.70
0.46
83
List of the Data Cont'd
Frame No. e1 2 cos 0
49873 0.12 0.85 0.57 o.48
49893 0.45 o.68 0.36 0.36
50017 0.71 0.60 0.28 o.4o
50304 0.75 0.75 0.00 1.00
50511 0.17 0.72 0.62 0.16
50582 0.75 0.75 0.00 1.00
50737 0.61 0.79 0.10 0.81
50798 0.14 0.79 0.58 0.34
51125 o.56 0.62 0.31 0.36
51194 o.48 o.85 0.17 0.78
51453 0.71 0.80 0.09 0.89
51565 o.68 0.70 0.13 0.70
52390 o.84 0.44 0.22 0.36
52596 0.41 o.80 0.28 0.61
52896 0.53 0.58 0.38 0.24
53176 o.66 o.68 0.15 o.66
53181 0.69 o.6o 0.22 0.48
53272 o.66 0.47 0.36 0.13
53297 0.34 0.87 0.27 0.74
53369 0.30 0.90 0.31 0.74
55275 o.64 0.62 0.56 0.07
55436 0.76 0.41 0.33 0.12
55526 0.11 0.79 o.63 0.29
84
List of the Data Cont'd
Frame No.
55637
56189
56386
56412
56569
56670
56700
56978
57042
57204
57428
57486
57516
57839
57880
58102
58435
58592
58819
58851
59180
59326
59499
0.50
0.71
0.26
0.66
o.68
0.32
0.67
0.21
0.26
0.31
0.52
0.13
0.18
0.15
0.46
o.48
o.58
O.11
0.81
0.47
0.32
0.13
E1
0.83
0.47
o.84
0.65
0.59
0.67
0.57
0.75
0.71
0.87
0.75
0.74
0.78
0.92
0.74
0.64
0.82
0.72
o.88
0.56
0.76
0.71
0.91
~2
0.16
0.32
0.44
0.19
0.23
0.50
0.26
0.55
0.52
0.33
0.23
0.63
0.52
0.40
0.31
0.38
0.33
0.34
0.51
0.12
0.26
0.45
0.49
cos 9
0.78
0.18
0.52
0.57
0.45
0.21
0.36
0.24
0.25
0.67
0.60
0.18
0.38
0.83
0.50
0.30
0.57
0.44
0.66
0.64
0.58
0.32
0.73
85
List of the Data Cont'd
Frame No.
59591
59677
60128
60169
60288
60549
60610
60683
60691
60770
60799
60848
60937
61492
61549
61922
62015
62053
62345
62737
63457
52335
53270
E
0.51
0.43
o.48
0.12
0.07
0.57
0.21
0.26
o.84
0.85
0.24
0.93
0.06
0.56
0.69
0.44
0.14
0.50
0.49
0.64
0.90
0.74
o.64
61
0.63
o.65
0.63
0.71
0.78
0.74
0.75
0.72
0.49
0.52
0.92
0.44
0.81
0.74
0.69
0.68
0.89
0.76
0.70
0.48
0.30
0.74
0.64
62
0.36
0.41
0.43
0.67
0.66
0.19
0.54
0.52
0.17
0.15
0.35
0.14
0.62
0.21
0.09
0.56
0.47
0.23
0.22
0.58
0.50
0.02
0.25
0.32
0.28
0.24
0.o06
0.26
0.58
0.29
0.27
0.50
0.58
0.77
o.65
0.46
0.62
0.76
0.36
0.69
0.62
0.56
0.13
0.00
0.96
0.45
cos 9
86List of the Data Cont'd
Frame No.
cos G
56369
56457
59615
63421
0.58
0.53
0.63
0.71
0.58
0.53
0.63
0.71
0.33
0.44
0.23
0.o8
0.31
0.33
0.48
0.81
87
Datafor the 18
Ambiguous Events
Frame No.
41856
43245
43564
44194
47263
48442
50297
50562
53308
57100
59673
60543
60997
62065
44389
49261
51531
52908
CA
0.54
0.70
0.45
0.80
0.86
0.76
o.84
0.63
0.56
0.72
0.76
0.39
0.47
0.55
0.90
0.77
0.74
o.65
~B
0.78
0.65
0.71
0.24
0.31
0.53
0.50
0.56
0.75
0.59
0.61
0.62
0.81
0.70
0.36
o.69
0.55
0.47
eStopping
0.15
0.13
0.33
0.45
0.31
0.23
0.15
0.37
0.16
0.19
0.13
0.50
0.21
0.22
0.24
0.07
0.22
0.34
cos QA
0.72
0.65
0.44
0.30
0.00
0.39
0.54
0.23
o.68
0.51
0.65
0.14
0.70
0.58
0.23
0.78
0.43
0.17
cos OB
0.54
0.69
0.17
0.47
0.69
0.61
0.74
0.30
0.54
0.63
0.87
0.12
0.39
0.42
0.80
0.89
o.6o
0.36
UNIVERSITY OF MICHIGAN
3 9015 03529 7400