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 Pergamon Press Ltd., Headington Hill Hall, Oxford Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523 Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1 Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright © Pergamon Press Ltd 1970 All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo- copying, recording or otherwise, without the prior permission of Pergamon Press Ltd. 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 +