K Meson Leptonic Decays. Progress in Nuclear Physics

77 Pages·1970·1.092 MB·English

by  J. Bailey, E. Picasso, P. B. Jones, D. M. Brink and J. H. Mulvey (Auth.)

PROGRESS IN NUCLEAR PHYSICS Volume 12
EDITORS: D. M. BRINK AND J. H. MULVEY
Part 1
I M E S ON LEPTONIC DECAY
by
P. B. JONES
THE ANOMALOUS MAGNETIC
MOMENT OF THE MUON
AND RELATED TOPICS
by
J. BAILEY and E. PICASSO
P E R G A M ON PRESS
Oxford · New York · Toronto
Sydney · Braunschweig
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First edition 1970
Library of Congress Catalog Card No. 51-984
PRINTED IN GREAT BRITAIN BY PAGE BROTHERS (NORWICH) LTD.
08 015766 1
Κ MESON LEPTONIC DECAY
P. B. JONES
Nuclear Physics Laboratory, Oxford University
Contents
1. INTRODUCTION 1
2. PHENOMENOLOGICAL DESCRIPTION OF THE K MODE 3
i3
2.1. General form of the Ki amplitude 3
3
2.2. Polarization of the muon 6
3. THE K MODE 8
e3
3.1. Radiative corrections 11
3.2. The form of coupling 11
3.3. The form factor /+ (s) 14
3.4. The partial decay rate 15
4. THE Κ MODE: MEASUREMENTS OF f 17
μ3
4.1. The Κ :Κ relative branching ratio 17
μ3 β3
4.2. Density function on the Dalitz plot 20
4.3. Polarization of the muon; μ-e universality 21
5. THE WEAK HADRONIC CURRENT 24
5.1. The 5-wave Κπ interaction 24
5.2. SU(3) violation: the magnitude of/ (0) 30
+
5.3. SU(3) violation: the form factor ratio ξ
35
ACKNOWLEDGMENTS
39
REFERENCES
39
1. Introduction
This article is concerned with the charged K decay modes
l3
which are mediated by the weak strangeness nonconserving vector
1
Β
2 P. Β. JONES
hadronic current. To the extent that the weak semileptonic interaction
satisfies the | Δ/| = \ rule, the amplitudes for the neutral modes
±T ± τ
K°-^en ν,μ π ν
L
are expected to be identical, apart from an overall Clebsch-Gordan
coefficient, with the amplitudes for the charged modes. In sections 3 and 4
the results given in Tables 3, 5, 8, 10, 12 refer to the neutral modes, for
which separate compilations of experimental data have been made.
The general phenomenological form of the K amplitude is described in
l3
section 2.1. If the V-A current-current interaction with local creation of
leptons is assumed, the general amplitude is extremely simple in form.
The only unknown quantities are two form factorsf y which are functions
±
2
of the invariant mass squared of the lepton pair, s = W , and specify
the matrix element of the weak hadronic current. Moreover, in the
K mode, the form factorfX. contributes only to intensity terms which are
e3
2
of order (m lm) relative to unity, and can therefore be neglected.
e K
Owing to the simple form of the hadronic part of the amplitude, the
K mode is well suited to a search for deviations from the local V-A
e3
leptonic coupling. Some experimental limits are summarized in section
3.2. For the Κ mode, the form factor ratio
μ3
can be related to a function of the isospin one half .y-wave Kn phase shift
by a clearly defined procedure which is discussed in section 5.1. Accurate
measurements of the ratio ξ are of interest in connection with the effect
on the weak hadronic current of the SU(3) violating strong interaction.
In sections 5.2 and 5.3 an attempt is made to survey the information about
SU(3) symmetry breaking which can be deduced from present experiments.
Tables 2 to 12 contain compilations of experimental data. Where
experimental data have been averaged a chi-squared value is given as a
test of consistency. For sets of data which lead to unsatisfactory chi-
squared values in relation to the number of degrees of freedom, the error
in the mean has not been adjusted or "stretched" in any way.
The present experimental situation concerning the ratio ξ(0) is
extremely unsatisfactory. The three sets of measurements listed in
Tables 7, 9, and 11 are each self-consistent and lead to fair chi-squared
values. However, the three mean values are not all in good agreement.
Κ MESON LEPTONIC DECAY 3
A number of possible explanations are considered in section 4.3, but do
not seem likely to provide a solution to the problem, which may well have
its origins in systematic experimental errors. For the purposes of section
5, we have chosen to adopt the estimate of section 4.1 which is derived
+
from the Κ : Κ + relative branching ratio measurement.
μ 3 3
2. Phenomenological Description of the K Mode
l3
2.1. General form of the K amplitude
l3
Some of the properties of the amplitude for the K mode can be deter-
l3
mined by simple arguments involving angular momentum conservation.
We assume a two-component neutrino created at the same point in
space-time as the charged lepton. The most convenient frame of reference
is the centre of momentum system of the lepton pair shown in Fig. 1.
FIG. 1. The kinematieal variables and axis of quantization are shown for the
Iv cm system.
If W is the total energy of the lepton pair, the Κ meson momentum k is
defined by the relation
2 2 2
(m + kY = W+{ml + kY
K
The assumption that the leptons are created at the same space-time point
implies, in the absence of derivative couplings, that they will be in a state of
zero orbital angular momentum, and that only their spins and the orbital
angular momentum of the pion contribute to the total angular momentum.
For a neutrino of negative helicity, the possible zero spin final states are
listed in Table 1.
4 P. Β. JONES
TABLE 1. ANGULAR MOMENTUM COMPOSITION OF THE
Κ 12, FINAL STATE
π wave
Coupling S Sz function
Vector 1 1
Υχ-Ι
Scalar 0 0
Tensor 1 0 r ίο
For given values of the lepton spin (S,S ), the dependence of the
Z
amplitude on the angle 0 is therefore completely specified.
π
In order to discuss the couplings shown in column one of Table 1, it is
necessary to write down the most general amplitude which is consistent
with local creation of leptons, a two-component neutrino, and a current-
current form of interaction. The hadronic part of the amplitude can be a
function of three four-vectors P , P, and P + P, two of which are in-
K n t v
dependent. The amplitude can therefore be arranged as the sum of three
terms,
\ G sin 0 ( ^ r) ' {(Ρκ + P.Yfï + (Ρκ - Ρ ) ρ /ή L
π p
s
+ mfL
K
T A
+ — fPiP L J (2.1)
K
J
Μκ
where the scalar, vector and tensor lepton couplings are respectively
L = v-(l+/y )/,
5
L = v(l+/y )y„/,
p 5
L = ivV+iysKsl (2.2)
p&
The wave functions ν and / are four-component spinors. The amplitude
(2.1) assumes the principle of μ-e universality, by which the form factors
fs,f+,f-, and fT are independent of whether the charged lepton created
is a muon or an electron. It is assumed that the leptonic part of the
amplitude does not contain any explicit dependence on P or P and
l v
therefore that there are no derivative couplings of the form proposed
originally by KONOPINSKI and UHLENBECK (1935). The experimental
Κ MESON LEPTONIC DECAY 5
evidence for this is summarized in section 3.2. The scale factor G is the μ
decay coupling constant, and θ is the Cabibbo angle. The hadronic part
of the amplitude is represented by the form factors/ s,/+,/r, and fT, which
are functions of W2, the invariant mass squared of the lepton pair.
Present experimental data are consistent with the prediction/ 5 = fT = 0
of the V-A theory of weak interactions, but small values of f s and fT
cannot be excluded. We shall therefore not consider scalar, vector and
tensor couplings separately but rather rearrange the amplitude (2.1) in
such a way that it depends only on the V-A leptonic current L . For the
p
scalar term we have
L= -—(Pt+PyL,, (2.3)
m
l
because the wave functions ν and / are solutions of the Dirac equation.
For the tensor term, we require the identity
σ„(Ρ,+/> )' = /(7 7·Λ-7^ν7ρ-(Λ)ρ + (Λ.)ρ) (2.4)
ν Ρ
which follows from the anticommutation relation satisfied by the y
matrices. The tensor term in (2.1) can then be written in the form
^ / TP i L — ΡΑΡι - Λ)/7 (Λ + Λ) Lp (2.5)
p P
m mm
K K l
The general amplitude including scalar, vector and tensor couplings is
therefore
G s i n 0l P P
* f A F r \± <iK+«yf+ HPK-P*YÎ-)L (2.6)
P
where
/+ = /i + i— f (2.7)
and
/. = / r- ^ /s-i — l — - p r .
F A P l v ) f 8) ( 2
m m mm
t K K l
The amplitude (2.6) is just the vector coupling amplitude with f v+ and
fY. replaced by the functions /+ and /_.
In the rest frame of the K + meson, the amplitude (2.6) leads to a Dalitz
plot of density
6 P. Β. JONES
Γ(Ε„ Ε)άΕάΕ = sin2 ΟάΕ,άΕ, (2.9)
ι π ι (Ιπγ Am
K
where the Lorentz invariant factor ρ is
2 2
Ρ = | / | K( 8^ ^ v- 2 ^ )
+
+ m] (2m2 + i -2 ±m2, - 4m E)}
K K v
+ Rc(ffl){(m2)(4m E-W2 + m2)}
+ K v
+ \f-\2{(\m2)(W2-m2)} (2.10)
and iy2 = (p + p)2
l v
2 2
= Α ΐ 7 + ^ - 2 ^ £ (2.11)
π
For the AT mode, the terms in (2.10) which are of order mJm relative to
e3 K
unity can be neglected. For this particular mode, it is of interest to
evaluate ρ in the centre of momentum system of the lepton pair. From the
definitions (2.7) and (2.8) of the form factors/+ and/_ we have,
fT 2
ρ = 2\JJ\2 W2k2un20 + \W2 mf*-— Wk cos Θ, (2.12)
n K m
K
where, as in Fig. 1, A: is the pion momentum. The angular dependence of
each term in (2.12) just corresponds to the squared modulus of the
appropriate spherical harmonic in the final column of Table 1. In accor-
dance with the lepton spin assignments given in columns 2 and 3, the
scalar and tensor couplings are incoherent with the vector coupling.
Equation (2.8), which defines/., demonstrates a problem which occurs
for the Κ mode, which is that the contributions from /T and from f s are
μ2)
completely indistinguishable. Any experimental search for a natural
scalar coupling/' must therefore be made using the K mode for which
e3
fü. does not contribute if terms of order mJm relative to unity are
K
neglected.
2.2. Polarization of the muon
Owing to the single helicity state of the neutrino, the charged lepton
created in K decay is in a pure quantum spin state. The μ mesons
l3
created in the Κ mode are therefore completely polarized in a direction
μ3
which depends on the position in the Dalitz plot and on the form factor
ratio ξ(IV2), which is defined by
2
ξ(\Ν) J- (2.13)
J +
Κ MESON LEPTONIC DECAY 7
In order to specify the μ meson polarization, we set up, in the μ rest frame,
right-handed cartesian coordinates defined by the three unit vectors
|p,r
n = n /ln (2.14)
2 3 l!>
constructed from the Ρ and defined in the Κ rest frame. The vectors
π
n, and n lie in the πμν plane in the Κ rest frame. From the amplitude
2
(2.6) we find that the μ polarization vector, in the μ rest frame, is pro-
portional to a vector which can be written down in terms of kinematical
variables defined in the Κ rest frame. The μ polarization vector is
1 = 4τ (2.15)
where
Α = 2^£ Ρ„-«ι1ν
ν
1
-4«2|«-i|2v
2 2
+ ( Re ξ - 1 ) ( W -ηι)Ρ -m £„ Vj
μ μ K
+ m (Im£)P^P, (2.I6)
K
and
f^VJP,
V = Ρ { ^ ^+ (2.17)
π +
A finite value of Im ξ would be the result of a final state interaction, or of
time-reversal noninvariance of the weak or strong interactions. The final
state interaction in the K mode can only be of electromagnetic origin and
l2>
we do not expect the resulting phase difference between /+ and /_ to be
greater than about one milliradian. Any observable finite value of Im ξ
would therefore be clear evidence for time-reversal noninvariance. For
lm{ = 0, the polarization vector η is a linear combination of n and n ,
l 2
8 P. Β. JONES
its direction depending on ξ and on the position in the Dalitz plot. For
2
ξ(1¥) = 0 and — 1, the vector η is shown in Fig. 2 for a number of
positions on the Dalitz plot. For low values of the neutral pion kinetic
energy, the direction of η is very sensitive to variations in ξ.
260
220
111
<
ίΟ-
190
MUON TOTAL ENERGY ( ME V )
FIG. 2. The direction of the muon polarization vector is shown as a function of
position on the K+ Dalitz plot for î = Oand — 1. The vector is referred to
3
the unit vectors n and n .
t 2
For complete descriptions of the Dalitz plot density function, and of the
muon polarization, we refer to the work of NILSSON (1960), BRENE et al.
(1961), and MACDOWELL (1962).
3. The K Mode
e3
If natural scalar and tensor couplings do not exist, the form factor /_
in the general amplitude (2.6) leads to contributions to the K absolute
e3
2
decay rate of order (m lm), which can be neglected. The only quantity
e K
which is unknown is the form factor/ . In the physical region shown in
+