Decay rate of bound muons

ANNALS

OF

PHYSICS:

16:

28%317

(1961)

Decay

Rate ROBERT

Department

Department

of Physics

of Bound W.

and Enrico

of Physics,

Muons*t

HUFF$

Fermi Institute for Nuclear Chicago, Illinois and Carnegie Institute of l’echnology,

Studies,

University

Pittsbwgh,

of Chicago,

Pennsylvania

The total rate and the electron energy spectrum are calculated for t.he decay of a negative muon bound in an atomic 1s state, using relativistic electron and muon wave functions appropriate to an extended nuclear charge distribution, and using a general local, nonderivative, decay interaction. The rate and the spectrum are computed numerically for iron, antimony, lead, and uranium, and the rate is found to be in qualitative agreement with earlier

calculations. The results are compared with experiment after approximate corrections are made for energy loss by the decay electrons in the t,arget, and for the detection threshold for t,hese electrons. These corrections are found t,o explain the great reduction in the experimental ratio of bound to free muon decay rates for heavy elements, t.hus removing the previous theoretical-esperimental discrepancy for these elements. The observed peak in the bound muon decay rate near iron is not, predicted, of possible additional contributions to the esperimental that the large background of low-energy gamma rays

accompanying

but a consideration decay rate suggests associated wit,h the

muon cnptnre might be connected with this peak. I

Negative muons in matter are captured atomically and reach the Is state, from mhich they disappear by t,wo competing processes: nuclear capture, and Recent experiments ( 1-6) 1 have decay into an electron and two neutrinos. measured t,he decay rate of these bound melons as a function of the at,omic number of the binding nucleus, and have found an anomalously high decay rate for the eleme& near iron. Subsequent8 calculations (7, 8) have not, only failed to * A t,hesis submitted to the I>epart,ment of Physics of the University of Chicago in parfulfillment of the requirements for the Ph.D. degree. t This work was support,ed by t,he U. S. Atomic Energy Commission at the Universit? of Chicago, Argonne Nat,ional Laboratory, and Carnegie Institute of Technology; and also by the National Science Foundation, to which the author is indebted for a pre-doctoral fellowship held during the first half of this work. $ Now at the University of California, Lawrence Radiation Laboratory, Berkeley, California. 1 The papers cited in Refs. 2 and 3 use the electron-counting results of Lederman and Weinrich (4). tial

288

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RATE

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MUOSH

explain these results, hut] have suggested an additional discrepancy for the heavy elements. The present, paper is a two-fold attempt to resolve this disagreement1 between t’he experimental resuks and the calculated decay rate. Icirst8, au improved calculation of the decay rat’e of bound muons is performed, in which many of the approximations of the earlier t’reatments are void. Second, various additional processes mhkh might8 contrihutc to the rxperimrntal dewy rate of bound muons are rowidered. A general introdwtion to the suhje(+ of bound muon decay, including some semiclassical estimates of the rate, and a brief description aud cvalunt,ion of earlier calculations, is presenkd in Section II. Theoretical expressions for the decay rate, via several different couplings, are derived in Swtion III. The method of machine ep&ation of these expressions, including an error analysis, is briefly described in Se&on IV. The results of this calculation are presented and discussed in Section Y. Various additional contributions to bound muon decay, particularly those which might explain t,hc observed peak in the decay rate war iron, are investigated in an Appendix. II.

PRELIMINARY

DISCGSSIOiT

OF

BOUND

MUOX

DECAY

It, is the purpose of this se&ion to discuss t’he physics of bound muon decay. This discussion will not, form a basis for the derivations and calculations of subsequent, se&ions of t’his paper, hut, it) is presented only as an aid to understanding t,he pheuomena involved in bound muon decay. There arc three effects which make the rate of the decay p + c + v + W different for bound muons than for free muons. First is the phase space effect,. A hound muon has a distribution of momenta or velocities, given by its wave funrtion iu momentum spaw, but has a single fixed total energy because it is in an energy tigenstat’e. When a muon component with a nonzcro velocity decays, a part, of it,s energy goes into the motion of the center-of-mass of the decay products, and only the remaining part, of the energy will go int,o the motion of the decay products relative to their own center-of-mass system. This is nothing but, a separation of the muon energy int,o a kinetic energy and a “rest cuergy,” which separation depends upon the momentlun of the muon component. When a moment,um component of the bound muon decays, it is the energy which goes into the motion of the decay product’s in their own center-of-mass syst,em, i.e., t,he “rest energy” for that, particular momentum component, which determines the volume of phase space accessible to the decay products. Since t,hr “rest energy” of a muon momentum component is less t,han or equal to the total energy of the bound muon, aud t)his in turn is less than the rest eucrgy of a free muon, t,he accessible volume of phase space in bound muon decay is therefore less than in free muon decay. Thus, the phase space effect, tends to reduce the decay rate of bolmd muons relative to free muons . Arc~ompanging this cff’ect,

290

HUFF

on the decay rate, the muon momentum distribution also produces a Doppler broadening of the energy spectrum of the decay electron. Second is t>he time dilatation effect. The drcay rate of a muon momentum component is less in t-he laboratory frame (i.e., the rpst frame of the binding nucleus) t’han it is in the rest frame of this component. Thus, the time dilat,ation effect, like the phase space effect’, tends to reduce the decay rat,? of bound muons relative to free muons. Third is t’he electron Coulomb effect’. The nucleus that, binds t)he original muon also att,racts the decay electron, which causes the electron wave function t,o have a greater amplitude near the nucleus t#han in t#he case of a plane wave electron. Since the muon wave function is i&elf concmtra.tcd near the nuc’lcus, there is a greater overlap between the muon and elect,ron wave functions, and hence a larger matrix element, than for plane wave electrons. Thus, t’he electron Coulomb effect, as opposed to t’he first t,wo effects, tends to incrcasc the decay rate of bound muons relative to free muons. The phase space and time dilatation &e&s can be treated easily in a cglantitutivc way by classical considerations. Let Q be the bound muon energy eigenvaluc, and q be a particular momentum component of the bound muon, both measured in units of the free muon rest’ mass, with the velocit,y of light taken as unity. For t,he moment, consider only t’his component, and make a Lorentz transformation t,o its own rest frame, where its energy is (Q” - q’)‘“. Thr tot’al rate for a decay into three zero-mass particles via a local nonderivative coupling is proportional to the total volume of phase apace accessible t,o the t’hree part,iclrs, and this is just, proportional to the fift’h power of the availahle energy. Thus, if the electron (and neut8rino) mass is neglectSed, the drcay rate of this rbomponnlt in its own rest frame is (Q” - q*) “’ t’imes the free muon decay rate. Further, the velocity of this component relative to the laboratory frame is q/Q, so that as a result of time dilatation, t,he decay rat,e in the laboratory frame is ( 1 - q’/Q” ) li’ t,imes the decay rate in the rest, frame of t)his component. Therefore, thr ratio of t,he bound muon decay rate to the free muon decay rate is

I?=[ ,q,$p~3qP(q)Q”(l - s”/Q”)“,

(1)

where p(q) is the muon momentum distribut,ion, normalized to Jd”qp(q) = I. Note that the only restri&ion placed on t,he decay interaction is that it, be an essentially local and nonderivative coupling, so that, the above analysis applies t,o any combination of the usual ten-parity conserving and nonconserving int,eractions of Yang and Lee. The above discussion depends upon the validity of represent)ing a hound muon as a distribution of free muons of various momenta. For i q 1 < Q, any muon spinor wave function in momentum space (i.e., the E’ourier transform of the

I)EC.lT

RATE

OF

BOUNJ)

Xi1

MUONS

muon spinor wave function in coordinate space) may be written as a sum of posit’ive and negative energy Dirac spinors for a part’icle of energy +Q, momentum q, and mass ((2’ - qz)‘i2. Because the muon is relat,ively weakly bound, it, may hc cxpect#ed that, the negative energy spinors contribut,c much less to t’hc decay rat,e than do the positive energy spinors, and this conclusion is support,cd by more extensive calculatjionn which show that, neglect of the negat’ive energy spinors causes an error of 3 70 or less, cvm for the heaviest elements. Thus, the essent,ially classical idea of a bound muon as a distribution of free muons of various momenta is certainly a valid concept, in this cont,txt. The next step is the det,ermination of an appropriate momentum distribution for the bound muon. For t#he light, and medium elements, the muon wave function in coordinat,e space is roughly proportional to t,he exponential exp ( -/?r), where fl = (1 - Q’)“’ is obtained from t’hr known muon energy and thus incaludrs much of the effect of the extended nuclear charge distribut,ion. This npproximation yields a momentum distribution p(q)

= sg”.G(/32

(2)

+ q?)“,

and gives a relative bound muon decay rat,e which is plot,tcd as Curve A in Fig. 1. ?uTotethat’ for light, elements, ($).4v s p’ g ( ZOC)‘, where 2 is the nuclear charge and o( is the fine structure constant. Thus an expansion of R as a power series in Z yields I? s Q”(1 - S/3’) g 1 -

(,1.‘&?~

1-

(11)

(Z,)“,

(3)

which result has been obtained previously by several other people (9). For the heavy elements, the muon is wit,hin the nucleus to a large extent, and the wave fnnct,ion may be taken as roughly proporGona1 to the Gaussian, exp ( -$cu?), where w is csscnt’ially t,he nuclear charge density, and may be considered as independent of %. This approximat,ion yields a momentum distribution p(q) = (7rwi-“~? cxp (-q”‘w),

(4)

and gives a relat’ivr bound muon decay rate, K E 0.58 Q’, which is plot,ted as Curve B in Fig. 1. Unfortunately, the electron Coulomb effect can not easily be incorporated into the preceding classical argument. If bound muon decay is looked at, from a different \-iewpointj, however, then the time dilatation, phase space, and electron Coulomb effcct,s may all he included in the same argument, although the latt,er two cffrcats do not) appear explicitly in t,his new argument. If a muon with momentum q and a rest mass of unity is moving in a region with a const,ant, or slowly varying elrctrostat’ic potential, t,hen the decay rate of this muon in its own rest frame will he the same as for a free muon. Rut in the laborat,ory frame, t,he velocity of this ~nunn is ( 1 + qs)P”“q, and thus, as a result, of time dilatation, the

292

R

0.6 -

.

0

UTAH

OlRECT

.

UTAH

COMBINATION

YOVANOWTCH

0

+

YOVANOVITCH

0

GILINSKY -

X

PRESENT

” CALlSRATED MATHEWS

IO

EFFICIENCY’ CALCULATION

CALCULATION

III

III 0

” SANDWICH”

20

II

II 30

40

I 50

III 60

I 70

I 60

I

I

I

90

z

FIG. 1. The ratio of t.he decay rate of bound muons t.o that of free muons, plotted against the atomic number, Z, of the binding nucleus. Curve A is a semiclassical calculation of the phase space and time dilatation effect.s for an exponential muon wave function. Curve B is the same for a Gaussian muon wave function, appropriate for heavy elements. Curve C is a semiclassical calculation of the time dilatat,ion effect alone, for an exponential muon wave function, and coincides with the tfberall calculation (8) for light elements. (See text for details of these semiclassical calcula.tions.) Curve D is interpolated from the calculated values of Gilinslq and Mathews (7). Curve E is interpolated from the values calculated in this paper, and is presented here only for comparison with t.he other calculated curves. Curve E does not include various corrections which must be made before it can be compared with the experimental results. The esperimental values, with errors (“total estimated uncert,ainties” for the Yovanovitch values), are also given: “calibrated efficiency” and “sandwich” methods of Tova.novitch (1); combination, by Utah group, of LedermanWeinrich data with bound muon disappearance rat,e dat,a (Z-4) ; and direct cosmic ray muon dat,a of Utah group (2, S).

decay rate is (1 + q’)-I’? t,imes the free muon decay rat,e. When this analysis is applied to the case of bound muon decay, t#hereis also a further reduction in the decay rate because some of the decay eleckons mill not have sufficient kinct’ic energy to escape to infinity from the region of the elect’rostat’ic potent’ial of the nucleus. This latt,er effect is somcwhatj ambiguous to evaluate explicitly, hut, if bhe

expectation

value

of t,he

nuclear

pot,ential

wit,h

respect

t’o the

init’ial

muon

wave function is bken as a measure of t,he energy required for the decay electron

DECAY

KATE

OF

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Mt-Oh-S

293

to escape t,o infinit,y, then this effect turns out to be much smaller than the time dilatation effect for the light and medium elements, and turns out) tjo be comdarable t,o the time dilat8ation effert for the heavy elements. Thus, for the light and medium elements, the bound muon decay rate should be rat.her accurately given by the time dilatation effect alone, which indicates that the elcctron Coulomb effect nearly compensat8es the phase space etit’tct’ for these elements. For the heavy elrments, however, the decay rate will be somrwhat less than predicted by the time dilatation effect alone, which indicates that the electron Coulomb effect does not compensate the phase space effwt to such a large extent for t,hc heavy elements as it does for the lighter elements. This time dilatation effect gives a ratio of bound muon to free muon decay rates of R = / d”qp(q) (1 + $)-1’s,

(5)

where p(q) is the muon momentum distribution. The muon momentum dist,ribution given earlier (Eel. 2) for the light and medium elements then yields a decay rate ratio which is plotted as Curve C in Pig. 1. As ment8ioned above, this curve is expected to be rather accurat’e for the light and medium elements, but is expected to be an overest’imate of the decay rate for the heavy elements. Unfort,unntely, the exact rst)ent of this overestimate is not’ easy to determine. Xote that) since (q2)AV z ( ZCY)’ for this muon momentum distribution, an cspansion of R as a pomcr series in Z gives

Kotr also that this time dilat,ation effect is independcnt of the part,icular form or structure of the dtcay interaction. However, the additional reduction in the decay rate, caused by the presence of decay clect,rons (in the above classical picture) w&h insufficient energy to escape to infinity, does depend upon the decay intjeraction, because it depends directly upon the shape of the electron energy spect,rum resulting from the decay interaction. The decay interaction which produces the largest number of low-energy electrons is expected to cause the greatest reduct,ion in the decay rate. This effect is clearly seen in the calculated results prrsentjed in Section V. where a scalar interaction (which produces more low-energy elertrons in free muon decay than does a vector interactjion) is found to yield a smaller bound muon decay rate than does a vector interaction. It is also expected, and can be seen in the calculated results of Section V, that this dependence of the decay rate upon the shape of the electron spectrum is much stronger for t,he hea\-y elements. The above argument also prrdicts some of the qualitative feat,urcs of the bound muon decay el&ron spectrum. In racaping to infinity from the region of t,hc nucleus, the dccap electrons will lost some of their kinetic energy, which

291

HUFF

energy loss should he comparable in size to the expectation value of t,he nuclear potential with respect to the initial muon wave function. Therefore, the hound muon decay electron speckurn may be expected to be shifted downward in energy as compared to the free decay spectrum. This shift, should be small fol the light and medium elements, but might be quite pronounced for the heavy elements. In addit,ion, the muon moment,um distrihution also produces a Doppler broadening of the eleckon energy spectrum. Both of these features--the Doppler broadening and the shift t#o lower energies-may be clearly seen in the calculated electron spectra in Section V. The electron Coulomb effect, is more difficult to treat exactly than are the other two effects, and so far it has been considered in only two calculat’ions, by iiberall (8) and by Gilinsky and Mat,hews (7). iiberall &died the expansion of the form R = 1 + aZ + bZ” + . for the light elements. He used the relativistic muon wave function for a point I~IIC~C~S, and terms through Z’ of a Born expansion of the electron Coulomb wave funct,ion for a point nucleus. He obtained the result R = 1 - J,i(20()~, correct through order 2”. This agrees Ah the result of t)he time dilatation effect, and indicates that, the phase space and clectron Coulomb effects do cancel through order Z2. A1 considerat8ion of the extended nuclear charge distribution did not’ suggest, any significant, change in this result. However, t,he whole investigation was based upon an expansion of R in powers of 2 only t,hrough Z’, and, as &rall mentioned, this may not be an appropriate procedure for this problem. Gilinsky and Mathews (7) used a slightly different approach which ar-oided a power series expansion of R. They used a muon wave fun&on proportional to esp ( -fir), where ,8 was determined by a least squares fit to a numerical soMon of t#he Dirac equation with an extended nuclear charge distribution, and they used t#he Sommcrfeld-Maue wave fun&on for the electron. Their result, with interpolat8ion, is plott,ed as Curve D in Fig. 1. Their result, like t:hcrall’s, indicates t,hat the phase space and electron Coulomb effects do cancel to a large extent. Both of these calculations appear to disagree considerably with the experimental resuks (l-6), which are plotted in Fig. 1 for comparison with the various t,heoret,ical predictions. For the heavy elements, t,his disagreement is not so surprising. The Sommerfeld-Maue wave function is an approximation which is least accurate for large Z and for low angular momentum states, which states give an increasingly important. contribution t)o the decay rat)e for heavy elements; and the fitted muon wave functions of Gilinsky and i\lathews are least accurat,e for the heavy elements. Further, the experiments themselves arc least accurate for t,hc heavy elements. The sharp peak in the experimental decay rate for the elements arotmd iron is more dist’urhing, nincc the theoret’ical calculatjions would appear to be more

DECAY

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295

MrOSS

accurate for these elements, and yet t’hey show no indication whatever of such a peak. In fact, the experimental decay rate is seen to be great,er for bound muons t,han for free muons for these clementjs, while the time dilatat,ion argument suggests that the decay rate rat’io should be less than u&y for all elements. However, this t’ime dilatation argument is not t’oo reliable because the quant’um mechanical corrections to it are not known. Further, it might, be possible that the detailed shape of the muon wave function is more import)ant than was suspected in the previous calculations, and that, this might’ explain the observed decay ratme. Thus, it is not, known wit’h any certainty whether or not) t,here is a real discrepancy between theory and experiment regarding this peak in the decay rate for the elements near iron. Because of the present situation, there is a need for a more exact calculation of the bawd muon decay rate, for both the medium and heavy elements. However, since t)he previous calculations do disagree wit,h the experimental results to such a large extent, various other processes which might contribute to the decay of bound muons should also be investigat,ed, and t#his is done in the Appendix. III.

DERIVATION ELECTRON

OF THE ENERGY

EXPRESSIONS SPECTRUM

FOR

FOR THE BOUND

TOTAL MUON

RATE DECAY

AND

In t#his se&on, expressions will be obtained for the tot)al rate and electron energy spectrum for bound muon decay, which expressions are to serve as t,he basis for the machine romput’ations described in the next section. A general local nonderivative decay interaction will be assumed here, and arguments in support of such an interact’ion may be found in the Appendix. It should be not,cd that the treatment of this and subsequent sect,ions is complet,e in itself, and is not, based upon the arguments of Section II in any way. The general local nonderivative interact,ion leading to the decay of a muon into an electron and two neutrinos can be written as a linear combination of the ten parit’y conserving and nonconserving int,eractions of Yang and Lee. It is convenient here t’o use the charge retention order in the particular form

296

HUFF

the representation

70 = (h

-3

y = (-i

i)

i-0 = (-!f

-i)

and t’he witjs fi = c = ~1, = 1 , where w, is t’he free muon rest mass, will he used in this section. It is t,hen t’he coupling constant C’y, which refers to the popular V-A interact,ion. In this calculation, the masses of the elect’ron and neutrinos will be set equal to zero. This will conveniently eliminate all cross terms between different coupling constants in the decay rate, and mill furt’her simplify the computation of the remaining terms. In the free muon decay rate, let the resulting relative error in these latter terms be denoted by A, , and let AZ denote the upper limit on the relative value of t’he cross terms, which upper limit is reached only for certain relations among the various coupling constants. From an analysis of present esperiment’al dat’a, an upper limit of six to eight, electron masses may be placed (10, 11) on the sum of t,he two neutrino masses. In this case calculat,ions will give A, 5 1.2 % and A2 5 7 %. The error in the ratio of bound muon to free muon decay rates is expected t’o be much smaller, and is probably never larger than 1% to 2 %. If the ncut’rino masses were actually zero, then A, = 0.02 % and A2 = 2 %, so that the error in the decay rate ratio would certainly be less than 1% for any set of coupling cow&ant,s, and would bc 0.02 % or less if thcrc mere only one nonvanishing coupling csonstjant, such as for I’-il coupling. Suclear recoil will also be neglected in this calcnlnt’ion, which leads to an estimat,ed error of less than 10P” for all tlements, dur mainly to t,he nuclear recoil energy and the consequent8 reduction in accessible phase space. -1 derivation of t’he esprcssion for t’hr decay rate via the I’-rl interaction will be sufficient t’o show the met,hod used in t!he general case. Then the bound muon decay rat,e bccomcs

where Q is t#he energy of t’hc bound muon, E is thr clcct,ron energy, and 3X=

s

d” r ,-i(k+k’)

‘r [u;,., (r)yX~+

The spinors refer to the particles as follows

indicated

ut"(r)l [&"(k)Yx A+ C,(k’)l. by superscripts,

F u,"(k)t&."(k) = h yx, c z$(k')s"(k')

= X/y',

(9)

and are normalized (lOa)

DE(‘.tY

RATE

OF

BOUXD

s

d” r CG,j.S(r)rO u”,,,je.,, (r)

297

MUOh-S

= 6(E -

6,,, .

E')6jj?

( 1Od 1

Hcrrj = &(.I + $2’) ( a nonzero integer), where J and J F r$ are the total and orbital angular moment8um, with t’otal z-component, s. The Dirac equation in this representation is

[E - V(r) The corresponding

- My0 + iyoy.V]#(r)

= 0.

free muon decay rate is easily shown T, = J,j(8T)-a/

(11)

to be

ct;, 12.

(12)

The next step is the performance of t#he neutrino moment,um integrat,ion and spin summation. The former is simplified by using new four-momentSa k’ = 1;' + 1~ and (I z k’ - k, and the t,ransformation of integrals

with the rcstrict,ion -lie 5 q. 5 ICo. Under this integral, the q-dependent terms resuhing from the neut,rino spin summation may first he replaced by yx ---) 0 and

qhqP--) r&Gk’,

- gx,k’“],

and the p-integration may then be performed, giving

6(Ky)6(K2 +y2) =2a. sd4y

(14)

(15)

The ratio of bound muon to free muon decay rates is t,hen given by

where (1.7a)

and Oip = KA K P- g,xpK2.

(17b)

298 A study

HUFF

of t.hc Diraco equations

in polar form t’hen shows that

d 2 -dr- r - V(r)+ 1 ‘I

I(

k’+

v-

=

0,

(19a)

)

(19b) Here ut is a two-component8 spinor with .titut, = ait,; and uj,s is a two-component spinor function of the unit coordinak vector ?, with the same angular momentum values as sprcificd earlier for the four-component’ spinor function U>j,T , wit,h s The normalizat,ion

dfI II,%, u,.~,S, = 6jjs 6,,, .

has been chosen such that (21)

and such that Wi’ The identit’y

and IV-

approach

unit amplitude

(6. P)?A,,s =

-1IlLj,r

plane waves

at infinity.

(aaj

has been used above, and will also be useful in the angular integration manipulations described below. The absence of a mass term in the electron Dirac eyuat,ion allows t,he choice 6Y~j = &iWj’, as may he seen by a simple rearrangement of Ey. ( 19b). Then %LFE.-j,s= iYS?h>~b>

(23)

and the iys may be brought, to act on the A+ , where &A+ = A+ . Thus, ml, = mtj’, and t.he sum over all j values merely becomes twice the sum over positive j values, so hcncefort,h j will be considered as positive.

DE(‘.4Y

H.4TE

OF

BOCND

The angular integrations involve too many detail here, but the general procedure can be this K-direction is chosen as both the x-axis is nest’ written in two-component, form, and use

ML-OS8

399

manipulations to be followed in described briefly. For a given K, and t,hc quantization axis. mj” is then made of the two relations

and (:3/4a)““F

= 2Yl,“(Q)

-

(.f - iJ)Yl,l(n),~d2

+ (? + iQ)Y1,-1(12),/~2,

where the Y,,,(Q) are the normalized spherical harmonics. t’ions are now seen to involve only the eight integrals

The angular

(35) integra-

where (L, 41) = (0, O), ( 1, 0)) ( 1, - 1)) and ( 1, + I ). E~aluat~ion of t,hese integmls requires the expression of the a&j,, in terms of two-romponent, spinors, utt ) and spherical harmonics; the expansion e-iK’r = LT$ id”[4a(2L

+ 1 )]“‘j,(Kr)

I’I.,,,(Q),

(“7)

where j,(k’r) is t’he Lth order spherical Bessel function; and the expression (12) of a product of two spherical harmonics as a sum of single spherical harmonics. ( Sate that, due to the prior choice of I? as the x-axis and quantizat,ion axis, the rrlation re-iK.r

=

,ivKc-LK.’

(28)

cannot be used to evaluate direct’ly the integrals of the form S~IRC-LK”~*j,~F u t .)

300

HVFF

The s- and t-summations arc somewhat messy, hut we now stjraight8forward. Finally, the integration over the orient&ion of K bwomes a t’rivial multiplication by 47r. The resultIs of the previous paragraphs involve, for each value of E, k’, and j, the following six radial integrals: (30a) (3Ob)

I:{=I mdr&(Kr)Wj+‘C’-,

(3Ocj

I, =

(3Od)

0

L

- dr(Kr)jj(Kr)wj+li-,

.O

Is=s-dr(Kr)jj(KrjWj-V+, (30ej

0

If=

s0

m clr

j

(Kr)

Wj-

(30f )

V-a

In t,erms of t,hese integrals, the decay rate ratio becomes R’ =

Q,jE s0

‘-’ dK 2 3A R,“(E’, k’), j-1 A

s0

(31)

where Rj“(E, K) = {[I1 + f2 - 2(j + 1jIo](Q + ifIs -

Ir + 2(j

-

l)Is](Q

Ej + K[15 - .,I]]’ -

E)

+ K[Il

+ I#”

+ 2[(Q - E:)” - K”]{[T, - 22 + (j + 1)16]” +

115

+ I4 -

(32)

ij - l)S,]” + Cj” - l)[I,? + 1621).

A yuit,e similar procedure for the general coupling yields a bound muon to free muon decay rate ratio of R = (C,/C)K”

+ (C’,/CjK’

+ (C&‘)R’,

(33)

where c, =

c 1Cl,, 12, m,‘I=*

c, = 4

c 1CT:,&12, WL, n=*

(Ma) (Nb)

(36) j=l Q--E R;"“'(E)

=

(32j//a)

dK R;'"'@, K),

(37)

s0

;~iRj"(E, K) = [(Q - I# - K']{ [I, - I, + 2(j + 1)&l’ + [Is + I< - 2(j ( [I, + I? + (j” -

l)l:J’t

+ 3[(Q - fs’)’ + A?]

i.i + 1 )Z,l” + [I5 -

1)[1,2 + I,‘]}

+

.[I, - I, + (j -

I, + cj -

1 )Z,]’

(38)

4K(Q - E){2[1, + 12 - (j + l)I,]

1 )I,] + cj” -

1 ,r,r,\,

R;(F, K) = c,[(Q - 1~')‘)"- K']( [II - L]" + 11, + I,]")

(39)

and Rj“(E, K) and t.hc 1,‘s are given by Eqs. (30) and (133). The corrmponding free muon decay ratt is ,’ ri = [12(8~)“]-’ .c”’ S[C’, R/"(K) + C, R/“(E) + Ct R;(E)] (40) 0

= [la&r):‘]-I(‘, where tht rlwt.ron

(41)

energy specka,

R,"(E) =

16 E2(c, -

12 E),

j&)

RI”(E)

= lo fP(:3 - 4 Ej,

(4%)

R,‘(E)

= l(i B’(2 - ::j

(4%)

Ej

arc given for lat#ervornparison with t,hc R'““‘t(E). A furt,hcr evaluation of these expressionsfor the decay rate rat,io is not feasible analytically, hut must be perfornwcl nurnericnlly. IV.

COMPUTATION

OF SPECTRUM

THE

TOTAL RATE Blil) FOR ROUND MUOX

ELECTROS DECAY

EKERGT

The cornput~atJionof the electron spectra, R“““‘(IS), obtained in the previous sectsion,has been carried out’ on the GEORGE computer at Argonne Sntional

302

HUFF

Laboratory. It, is con\Tenient to divide this work into six secations:I. preparation of the muon functions, 2. construct#ion of t’he Bessel func$ions, 3. conxtructjion of t,he electron functions, 4. computation of the radial integrals, :i. performance of the K-integrat#ions, and 6. performance of the j-summations. Ford and Wills (13) have comput#edthe muon wave functions and binding energies, using nuclear charge distributions which were fitted tjo electron scattering data (l/t-18). The experimental errors in these charge dist,ributions produce an estimated uncert’ainty in t)he binding energies of about, 1 % for the heavy elements and about, f,i % near iron, and the corresponding uncertainty in t)he wave functions probably causes a comparable error in t)he elect’rou specka, R”“‘(E). Ford and Wills give TV+ and rl’- (G and -F in their notat,ion) in tabular form at, intjervals of 0.5 fermi, and these values are used directly as input data, and subsequent~lyinterpolated t,o t#hedesired radial integration mesh. Sext,, all of the required spherical Bessel functions are computed by a rccurrence m&hod, and stored, and are to be recalled later during the evaluation of the radial integrals. In construct,ing the electron wave functions, the nuclear charge distribution is replaced by a uniformly charged sphere, whose total charge and root, mean square radius are chosen to equal t#hoseof t’he charge distribution used by Ford and Wills (IS) in computing the muon wave funct.ions. Inside this sphere, whose radius is r. , and where t,he pot’ent,ial is I’(r)

= -(ZCY/~~)(“,?

-

i-43)

1,ir’,ir,f),

the following series solutjion of the electron radial Dirac equation is used:

A

m +

= ( -c-4,-1

+ w=l,-2) !m

for

111> 0,

A ),L- = ( 64 ?,,+- ~A+,-~),i(m + j + 1.a) for .-lo+ = 1, c = $$(Ero

d,”

= 0 for

+ 3iZa)(r,/r,,),

w 3 >$Za(r/r,,)3,

111< 0,

145a) w 2 0,

(4.5b) (4%) (Ma)

i4Cib)

where the normalizing factor Nj is to be determined. Outside the sphere, the potential is V(,r) = -Z~ol/r, and t,hus, in this region, F,*(r) must, be somelinear combination, B,W,*( r) + B;W,*( r), of the regular and irregular solutions of the Dirac eyuat,ion for a pure Coulomb potential. Let’ W7,*(r) and Wi*(r) approach waves of unit amplitude, but with a phase difference, 6, at infinity. Then TVj’( I’) and m;j-(r) haveunit amplitudo at infinity, provided B,” + 2R,Bi cos 6 +

DECAY

RATE

Bi2 = 1. The correct normalizing is therefore

OF

BOTSD

303

MUOSS

factor for the series solution in the inner region

i4ij

Nj = ; c’r2 + “C’,C, (‘OS 6 + c*;y, where C’, and Ci are determined S,*ird

from the two equat,ions:

= cIrwl*(ro)

The phase difference and the correctly functions are (19) :

s,19’*(r)

+ C$v~~(ro). normalized

0 Ii?

+ij*-z’a-) =

3zj(2t/:r)"

wries

2

(48) used for the Coulomb

)

(50)

(51)

dm*(Er)m,

??I=0

An+ = -[(Za)

&-,

+ (1~ +j

+ s) dL]/m(m

d,- = [(Lx) dmi- + d$-l]s’( m + j + s) do+ = 1 xj = [l$ij

d,* = 0 for

for

+ 2s) 711 2

~1 < 0,

+ s)eTZall’*1 lys + ,iZa) I/l?(% + l),

0,

for

m > 0,

(Xu) (52h) (sac) (53)

where s = +sj(s = --sj) for the regular (irregular) solution. The above series is now used t,o compute t,he properly normalized W,*(r) in the inner region, while W,*(r) is obtained in the outer region by the RungeRutta method (20) of numerically integrat,ing the Dirac equation wit,h the potential -G/Y. The series-computed TVj*(ro) are used as starting values for this int’egration, and the j-dependent quantity, H = h/j (h = 1.2 fermi), is used as the fundamental integration st,ep. The computation error in the elect’ron fun&ions is due mainly to this Runge-Kutta integration process,and leads to an cstimat,ed error of 0.05 % or less in the elect,ron spectra. The replacement of the nucleus by a uniformly charged sphere has Icd t)o a much more convenient calculation, but also results in an error of its own. In lead, the resuks of a few test calculations wit8h a different value of ro indicate that this error in the electron spectra is probably f.2 a/cor less,while in t,hc lighter clemrnts the error is expected to be even smaller. The radial integrals arc performed by Simpson integrat,ion with an interval of h = 1.2 fermi, which leads to an estimated error of 0.1% or lessin the electron specka. Also, t,he radial integrals are terminat,ed at a radius, r, , where t,hc muon wave function I’+ has decreased to a predetermined fract’ion (>,i. 10P3) of its vnluc at the origin. The results of a few test calculations with iwrrased r,. indicate

304

HUFF

TABLE THE

CALCULATED

ELECTRON

IKIll

E/Q 0

0 0 0

0.574 0.424 1.027

FOR

Antimony 0 0 0

0.05

0.10

I

ENERGY SPECTRA, R(E), VARIOUS ELEMENTS"

0.859 O.G69 1.431

0.15

Lead

B~CND

MUON

DECAY

Uranium __~

0 0 0

0 0 0

0.624 0.495 1.010

0.796 0.644 1.253

1.394 1.164 2.083

1.594 1.357 2.304

2.028 1.791 2.740

2.183 1.965 2.837

0.20

1 f%Ci 1.359 2.586

2.067 1.797 2.875

2.498 2.339 2.973

2.584 2.464 2.940

0.25

2.248 1.949 3.144

2.576 2.388 3.135

2.781 2.764 2.824

2.785 2.814 2.689

0.30

2.790 2.593 3.378

2.940 2.919 2.997

2.821 2.971 2.357

2.737 2.922 2.169

0.35

3.228 3.247 3.167

3.052 3.246 2.458

2.552 2.834 1.690

2.393 2.683 1.506

0.40

3.430 3.759 2.436

2.746 3.111 1.637

1.968 2.285 1.002

1.782 2.081 0.868

0.45

2.968 3.523 1.293

1.923 2.290 0.809

1.21!1 1 .‘W 0.470

1.074 1.294 0.401

0.50

1.345 1.678 0.338

0.911 1.121 0.272

0.575 0.708 0.168

0.502 0.619 0.14-l

0.55

0.222 0.282 0.041

0.275 0.345 0.062

0.203 0.25-4 0.046

0.180 0.225 0.040

IS

DECAY

RATE

OF

TABLE Iron

WC) 0.60

BOUND

I-(Contimd) Antimon.;

0.024 0.031 0.001

305

MUONS

0.059 O.Oi5 0.011

0. G5

Led

Uranium

0.055

O.OiO 0.010

0.050 0. OH 0.009

0.0123 0.01% 0.0017

0.0115 O.Oli(j O.oolcl 0.0022 0.0028 0.0002

0.70

0.00036 0.000‘47 0.00004

O.OOli 0.0022 0.0002

0.0023 0.0029 0.00025

0.80

0.00001 o.OQOO1 0.00000

0.00003 0.00004 0.00000

0.00005 0.00006 0.00000

0.98386

0.94940

0.89975

Q=

8 The three values for each energy and element refer lings, in that order. All spectra are normalized relative The electron energy, E, is given in terms of the total Q, in units of t,he free muon rest energy, appears as the

0.88476

to vector, tensor, and scalar coupto a free muon decay rate of unity. energy, Q, of t,he bound muon; and last line of the table.

t.hat, t,his radial cutoff leads to a maximum error of 0.05 % in the electron spectra. The R~'""(li', K) (Eqs. 32, 38, 39) have a peak of widt,h AK M Za, at the point8 K s I?, corresponding to the momentum conservation &function in free muon decay. Because of t,his peak in t,hc K-integrands, this integration is divided int,o several regions, and a separate Gaussian integration (Ref. 20, p. 312) is used in each region. This K-integrat,ion mesh is separately specified as input data for each I3 value, and has been chosen so as to give an error of much less than 0.1 % in the electron spectra. As each succeeding !?4”“‘( E) (E;q. 37) is computed, it and t’hr previous one arc fiitkd to t,hc two-parameter curve, u,je-“, in order to estimate the contribution of the higher j terms, and thus gain an approximation to R"""(h') (IQ. 36). When three successive approximations are equal to within an absolute error of 0.001, t#hej-summat,iou is terminated and the last approsimat,ion is taken as the final value for R"l"*'(I<). Tl us‘: is founcl to lead to a relative error of much less than 0.1 %, except, near the ends of the electron spect’ra, where RS"'t( E) is itself rat,her small. X11 errors which are not mcntionrd in this section may hc assumed to he negligible in comparison wit’h t#hosc which are mentioned. The resulting total c*omputation

error

may

thus

he plactd

at

0.2 Y4 or

ICSN, with

an

additional

in-

HUFF

ANTIMONY 4

/

k

S / --\,

/

2

//

/ //’

/I

,I / //

T

\

\ \

\

\

v:

\ \

\ \

\ \

/

’

\

\ \

/ //

/--

/

/’

/’

2

\ ’ \

// /’

‘\ ‘1

I/

’\ \

nk:;l:;ll ’

\ .

nv-

0

3.1

0.2

0.3

0.4

0.5 E

0.6

0.7

FIG. 2. The calculat,ed electron spectra, R(E), for hound nn~on decay in various elemerits, for vector CT’), tensor (T), and scalar (8) collplings. The electron spectrunl (VP) for free nltlon decay via. vector coupling is included for compnrison. The electron energy, are norrrlnlizrJ relative to ~1 free E, is in mits of t,he free muon rest energy, and all spert.r:t muon decnv rate of unity.

DEC‘IY

0

I

K.1TE:

OF

BOL-h-D

307

MUONS

I

I

0

0.1

0.2

0.3

0.4

0.5 E

0.6

0.7

0

0.1

0.2

0.3

0.4

0.5 E

0.6

0.7

certailky of I % to 2 9:. due to the experimental uncertainties in the nuclear charge distributions, where this latter figure also includes the effect of replacing t,he nucleus by a &iformly charged sphere in computing the clcc+ron KLIT fmictioiis.

308

HUFF

The code for this computation is available, in t#he event’ t’hat any results in addition to those presented in the next section should be desired. The computat’ion of each point of the electron spectra (for all three int#eractions simult,aneously) requires between 15 minutes for t,he heavy elements and 4~5minut,es for the elements near iron. V. RESULTS

AND DISCUSSIOK

The electron spectra, Ral”‘(B) , computed as described in the previous section, are presented in Table I and Fig. 2 for several elements. The shift toward lower energies and the Doppler broadening, which were predicted in Section II, are q&e evident in Fig. 2. It may be seen that R”(E) = 3;R1(E) + f@?(E) for all elements, within t.he estimated computational error. Therefore, t,he elect,ron spectruni may be written in the one-parameter form R(E)

= R”(E) + p[Rf(E) - R”(E)],

(54)

where p = (YiC, + C,)/C. This p is merely t’he familiar Michel p-parameter, which has been used, for example, in analyzing the elect#ron spectrum in the decay of posit’ive muons. Experimentally, it is known (10, 21, 22) that p E /3/4. The decay rat)e ratio is obtained by a Simpson integration of t,he electron sprct#rum, and is presented in Table II. This rat)io also may be writt)en in t,he one-parameter form

R = R" f ,o(Rt - R").

(55)

Cnlikc positive muon decay, however, the bound muon decay rate is not indrpendent, of p, since Rt# R". The vector coupling decay rate ratio, R" (i.e., p = 34) has also been interpolated to give Curve E of Fig. 1, and is seent.o bc somewhat lessthan the value expected from the t)ime dilatation effect, alone (Curve C of Fig. I), as was predicted in Section II. It, is seenthat t,he present’ calculatjion is in rough agreement,wit,h t,he previous calculat#ionsby Gilinsky and Mathews (7) and by iiberall (8). In part)icular, t,he decay rate ratio is seent,o be lessthan unity for all elements, including iron. Se\reral points of t#heelectron spectrum were computed for bouild muon decay iI1 zinc, and these were compared with the elert~ronspectrum values for iron. The results show no significant difference betwrcn the two rlements, and give not the slightest indicat,ion of any peak in the bound muon decay rate near iron. The present calculation is also in rough agreement’ with the earlier cnlculations in predicting only a slight reduction in the decay rate ratio for the heavy element,s. However, because of the great difference bctw-ccn t’he elect’ron spectrum for bourld muon decay and t,hat for free muon decay, the decay rat{> rat,io must be corrected for the following two cticcts heforc a com’parisonwith rxperiment can be made for t#heheavy elements: 1. the energy threshold for detect,ion

DECAY

RA4TE

OF

BOUND

TABLE THE

CALCULATED RATE,

BOUND MUON FOR VECTOR IKlIl

DECAY RATE, (v), TENSOR (“),

309

MUONS

II

R, IN UNITS AND

SCALAR

OF THE FREE MUON (*) COUPLINGS~

Antimony(Iodine)

DECAY

Lead

Uranium

0.828 0.849 0.764

@

0.975

0.914

0.844

R" Rs X0,,. Gp.

0.978 0.966 0.94 1.04 f 0.02

0.924 0.883 0.77b 0.62 zt 0.07c

0.862 0.785 0.50 0.38 + 0.09

a Rwr. includes energy loss in the mental value, R,,,. vitch (f), but which to his results (see b The antimony c Experimental

corrections for t,he lo-Mev threshold for electron detection, and for target (see text for details). R,,,,. is to be compared with the experi, which was obtained by the “calibrated efficiency” method of Yovanodoes not include the 6% “spectral” correction applied by Yovanovitch text for details). spectrum is used to compute this result for iodine. result for iodine.

of the decay electrons, and 2. t’he energy loss by the decay eleckons in the krget. Let w( L, E, E’) be t’he probability that’ an electron wit#h an initial energy E still has an energy E’ or greater after traveling a distance L in a particular element. Then, with an elect’ron det,ect’ion threshold of Eo , the effect,ive decay rate associated with an elcct’ron spectrum R(E) will be Reff = 1 dE R(E)w(L,

E, ISo),

(56)

averaged over the distribution of elect,ron path lengt’hs, L, in the target. For the heavy elemenk, where radiation lossespredominate for t)he electron energies which are of int’erest’ here, it is convenient and appropriate to use the approsimation (23) W(L, E, E,) = r[.r, 111W’,:I&)],‘r(.r),

C.57)

where I’(x, y) = (a - 1, v)! is the incomplete gamma or factorial function (24)) II’ = L/( Lad ln a), and Lrnd is the radiation length for the target material. Further, the average over L may be approximated by setting L cqllal to the mean path length for the given target,. Thr values x = 0.7, x = 1.5, and I( = 2.5, seemt,o be reasonable mean values for t’he iron, iodine, and lead targets, respecGvely, in the Yovanovitch “calibrated efficiency” esperiment (I), whrrc the appropriate threshold energy is Eo = 0.10 (El0 Mcv). These values for x and E. , in conjunct’ion wit#h t’he calculated bound muon decay spect’rum, R’(E), for a vcct)or coupling (the calculated antimony spectrum is used for iodine), lead t,o effective homid muon decay rates of 0.75 (iron), 0.58 (iodine), and 0.106

310

HUFF

(lead). The corresponding free muon decay spectra of Section IV give effective free muon decay rates of 0.78 (iron), 0.49 (iodine), and 0.21 (lead), in the same targets. The ratio, R,,,, , of t#he effective bound muon decay rat)e to the effective free muon decay rate is presented in Table II. Yovanovitch (1 j rstimated these rffeck empirically in iron, and found that t,he free decays were detected wit’h a 6 % greakr cfhciency than were bound decays. His final ralucs (except’ carbon) from thr “calibrated efficiency” method all include a 6%’ increase as an attempt to compensat,e for t,his effect. It is the Yovanovitch “calibrat,ed efficiency” results, prior to the inclusion of this G % “spectral” correct8ion, which are listed as R,,,, in Table II, and which are to be compared to the corrected ratio, R,,,, . It is seen t,hat the difference bet,ween t,hc bound decay and t#he free decay electron spectra are emphasized, rather than reduced, by these effects. h more exact calculation of these energy loss and threshold corrections is not practical at this point, but, t,he above approximate calculation shows t,he importjance of these correct,ions in removing the supposed discrepancy between calculations and experiment for the heavy elements, and t!here is no indicat,ion t,hat, a more exact treatment would cause any qnalit’ative change in these results. Therefore, it, may be concluded t,hat there is no real discrepancy between theory and experiment regarding the bound muon decay rak in heavy elements; although there may be such a discrepancy in regard to the peak in the bound muon decay rate near iron, if this peak were found not to be due to t)he large background of low-energy gamma rays associated with thr accompanying inrlastk muon capkire events.

I would like to express my appreciation to R. H. Da.litz for suggesting this problem, and for his continued advice and encouragement during its completion. ~Many discussions with Y. Nambu, L. Wolfenstein, and H. Primakoff were also quite helpful. I Gsh to thank W. Miller and Ii. :+nith for placing the facilities of the GEORGE computer at Argonne Xational Laboratory at my disposal, md to thank the many persons in thzt computer group for their help during the computational part of the problem. APPEKDIS.

PO,~SIBT,E

There are six prowsses 111u011s

( 1) (2, (3,

ADI~ITIOYAL MUON

CONTRIBTTIOiYS DECAY

TO

BOUSl)

which could, or do, lead to t’hc disappearance

of bound

:

/F + c- + c-~ + C+, /.+e-, cc- +r-- + Y,

(1) (5 ) (6)

P +N+N+E, /m-- + N ----t N’ + v, p - P- + v + ij.

where v and w arc’ neutrinos, and N and N’ are nuclei. kkwpt~ for radiati\.c (‘orrwtions to these six, any additioual processes either would involve new particles

DEC.\T

RATE

OF

BOUKD

311

MUOSS

for which there is no evidence, would involve interactions which couple more than four fermions, or would violate well established principles like charge conservation, angular momentum conservation, etc. Such processes will not be considered here. Of particular concern in this section will be those processes which might’ possibly contribute to t,he observed peak in the hound muon decay rate for elcmrnts near iron. The strong Z-dependence of the observed decay rate in this region suggests that t)he peak could only result from some process which directly involves the stjructure of the nucleus, or perhaps the atomic electrons. With this in mind, the above six proresscs may now he investigated. An upper limit of 10-j has been obtained (25, 26)’ for the rate of t,he process + + c+ + P+ + e- relative to the normal decay mode. Approximately the same /J value should apply to the corresponding process for bound muons, and the rate is not expecated to have a strong %-dtpendenre. In the related process pc- + P- + P- + cm-,the I~LIOI~ does interact, directly with the atomic electrons, hut the rate will be even smaller, in spite of its stronger Z-dependence. Therefore, this procbess need not be considered further. I:or t,he proress p’ + e+ + y, an upper limit of lo@ has been obtained (27,28) for the rate, relative to the normal decay mode. For the related process where t#he photon internally converts to an electron pair, an upper limit of 1O-5 has beeu placed (25, 26) on the relative rate. The corresponding procaesses for hound muons should have much the same rates, wit)h no strong Z-dependence. However, for hound muons, there is a second related process in which the photon of pL + F + y is absorbed by the nucleus. The situation is further complicated by the fact, that the plastic portion (i.e., that portion in which the nucleus iq left in its ground state) of this process is coherent with the elast,ic portions of both the process p + c’- and the process pL- + N + N + t’-. The matrix elements for these three processes arc expected to have similar, but not identical, Z-dependencaci, with possibly an additional dependence on .4 -%, the neutron nlmlhcr, ill the case of the latter process. Conceivably, these three matrix elements caorlld interfere with each other in such a way that the total coherent rate of t’hc three procrsres \zould have a much stronger dependence 011 % and 9 -2 than any single one of the individual processes. E’or example, surh an interference might gi\-e ;I total coherent rate in copper of IO@ or less rclativc to the normal decay, in agreement with esperimcnts ( 29, 30‘) whicah have looked for highc’nergy electrons from hound muons in copper; while for neighboring tlements, such as iron, the interference might be sufficiently incomplete as to allow a total coherent rate of 10% relative to the normal decay, and thcrcby acrount for the observed peak in the hound muon decay rate near iron. Fortunately for the cause 2 The various

paper tlrcny

cited in Ref. 25 also contains modes of t,he positive muon.

n summ:trp

of previous

experimental

work

on

312

HUFF

of simplicity, an experiment.al search for high-energy electrons from bound muons has recently been performed (5’1) in iron, and indicates that t#he t,otal coherent rate in iron is less than 1% relat#ive t,o the normal decay. These results in copper and iron not only show that the peak in the bound muon decay rate near iron can not be accounted for in this manner, but also strongly suggest that these three coherent processes are unimportant for all ot,her element)s as well. This suggestion is supported by an additional objection concerning the inelastic portions of the three processes: If the low rat,e in copper is due to a large interference among large elast’ic matrix elements, then either the inelastic matrix elements are much smaller than expected compared to the elastic matrix elements, or there is also a rather complicated interference among various of the inelast#ic matrix elements. In view of this situation, this question of coherence will not be considered further. A large portion of the nuclear reactions pcL-+ N --f N’ + v will lead to excited nuclei, which will immediately decay by gamma emission. These gammas could constitut’e a large source of background in bound muon decay if they were mistaken for decay electrons, especially in iron and heavier elementIs, where nuclear capture is ten to thirty times as frequent as decay. Most of these gammas are expected to have an energy of 8 Mev or less. Both the Chicago group (32) and the Utah group (53)” have looked for gammas in the energy range of 10 to 30 Mev, and failed to find a sufficient. number t,o explain the enhanced decay rate for iron. At,tempt,s were made to eliminat,e the large background of lower energy gammas, either by count,ing only decay particles (elect#rons or photons) with an energy of 11 Mev or greater (S‘), or by using a triple-coincidcnre electron t,elescope wit’h a low efficiency (less than 10P3 in t,he Yovanovitch experiment) for count)ing gammas of less than 10 Mev (1, 4, 6). However, if t,here were several gammas associated with each capture event,, this would greatly increase t,he probability of a capt,ure event. being seen as a decay event w&h cit,her of t.hesc two detection methods. Since a 1% det,trtion efficiencay for capture events (only one order of magnitude greatt,er t,han Yovanovit,ch’s estimate) would increase the observed decay rat,e by 10 %, in iron, t,his effect should be seriously caonsidcrcd as a possible explanation of the observed peak in t#hc bound muon decay rate near iron. The sharp decrease in t,he observed decay rate in going from nirkel (Z = 28) to zinc (Z = 30 ) would then have to he due to a change in t#hr nuclear gamma rays (such as t,hcir number, energy spectrum, or time spectrum) for these two elements, but t,here is not sufficient experimental information to allow any furt’hcr comments on t,his question to bc made at this time. (Kotc that this sharp dccrease takes place near a magic proton number, 2 = 28.) The decrease in the observed decay rate toward the lighter element,s would then he cxperted, because 3 This result energy gammas

supersedes was first

the earlier suggested.

result

of Beuffel

(94) where

this

background

of high-

DECrlY

RATE

OF

BOUND

MUOSS

313

the ratio of the capture rate (35) to the decay rate is only 5.5 in calcium (2 = 20) and 1.6 in aluminum (2 = 13)) compared to 10 in iron (2 = 26). A repetition of the bound muon decay experiments, using an electron spectrometer or a Cerenkov counter as the electron detector, would greatly decrease the contamination due to these low-energy gamma rays associated with muon capture. The only remaining process to be considered is the normal decay, /.L- * e- + Y + W, and its radiative corrections. The radiative correct,ions, which include both the radiation of real photons (inner bremsstrahlung) and the internal emission and reabsorption of virtual photons, have been found (36) t’o decrease the free muon decay rate by 0.44 %, based on a local V-A interaction. The radiative corrections to the bound muon decay rate are expected to be about the same, with no strong Z-dependence, so that, the effect on the decay rate ratio is probably much less than $5 %. Much the same conclusion should hold for ot,hrr forms of the decay int,eraction, including one which involves an intermediate boson. However, these radiative corrections also change the shape of t’he energy spectrum of the decay electrons. Since t#he bound muon decay rate measurements involve only those electrons above a cert,ain energy (about 10 Mev), these radiat’ive correctionsmay have a significant effect on the experimental decay rate evrn t)hough they do not so affect the true decay rate. A simple esbimate of t,his effect! ran be obtained by a classical treat,ment of the bremsst~rahlung resulting from the acceleration of charge during t,he decay process. Numerical applicat)ion of this treatment to several of t,he electron spectra obt’ainrd in Se&ion V shows that, this inner bremsst~rahlungeffect will changr the experimental decay rate ratio by less than 0.3 % in any element, although the actual effect’ upon the shape of the electron spectra is slightly more import#ant . A much stronger Z-dependence might, be expected for the accompanying inelastic processes in which t,he photon from the radiat,ive decay p- + e- + v + (; + y excites the nucleus or atomic electrons or ejects an el&ron from t#he atom. However, rough calculations suggest that the rates of these nuclear and electronic processes relative t,o the normal decay mode are not’ greater than about lop5 and lo-“, rtsprctively, for any element,, and for clement’s as light as iron, the lat,ter value is reduced t’o lo-“. These processes are therefore too small to have any significant effect, on the experimental holmd muon decay rate, especaially in regard to the ohscrvcd peak near iron. The conclusions of the previous paragraphs arc rather indepeudent of the particular form or structure of the decay interaction, but in the case of the normal decay i&elf, this form and structure have a more significant8 effect,. T’nrious quantit,ies which have been measured for fret muon decay, such as the dcc*ay asymm&ry (,“I, 37‘r, and the helicitly ( 38, 39) and energy spectrum (IO, 21, 22) of thcl dccq electrons, arc consistent with the assumption of a I--;1 decay int,craction, hilt the cspcrimcnts do not rule out the possibilit’y that other forms of

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coupling are also present in the decay intcra&on. In bound muon deray, t,he form of the decay interaction affects the electron energy spectrum, and may also affect the total decay rate. Since the experimental boundmuon decay rate depends upon both of these effects, it, will be desirable t,o compute t,he bound muon decay rate and energy spectrum using a general form for the decay interaction. Muon experiment)s give very lit,tle information at present concerning the struct,ure of the decay interaction (whirh st,ructure could he due to an intcrmediate boson, or to the presence of derivative couplings, for rxamplc ), but the t,ime dilatation argument’ of Section II suggests that t,his structure should have little effect upon the decay ratme ratio. If the currently favorrd universal V-A weak intleractJion is mediated by a charged v&or boson, then more definite conclusions can be drawn. In this case, t,he boson mass must, be at least as large as the K-meson mass in order t,o agree wit’h t,he observed branching ratios for K-meson decay (/to), and this implies that the effect on the bound muon decay ratBeratio will not exceed 1 % or 2 % for t,he heavy elements, and will be even less for the lighter element)s. This charged hoson will also interact wit’h t,hc clcctrostatic field of the nucleus in bound muon decay, and t,his effect, (.41 ) will increase t,he decay rat,e hy less than 2 % and ti % in iron and lead, respectively, under the same condit’ion on the boson mass. Therefore, such a boson can explain n&her the enhanced decay rat,e near iron, nor t#hc reduced decay rat,e for the heavy clemcnts. Furt,hcrmore, it, is not, understood how any reasonable kind of stru&ure of t,he decay interaction could account, for t’he ohserved peak in the decay rnt’e near iron. Therefore, it will he assumed from this point on that the decay proceeds via a local nonderivative intrract’ion. The final step is the determination of suitable initial muon and final c~le&ron and neut,rino stat,cs for t,he decay process. Plane waves are t’o be used for t’he neutrinos, but, the muon and elertron wave funct,ions involve t)he electromagnetic pot,rntial of t,hc nucleus and atomic electrons. Thr appropriat,e electric potential is just, t,hat, which has been determined in recent experiments (24, 1.5) on c&tic scattering of high-energy elertrons by nuclei. In many elements, thcrc is also present a magnet,ic field, due either t,o the nucleus or t,o the at’omic electrons, and it is possible that t,his field may affect) the decay rate of bound muons. In view of t,he enhanced decay rate for the ferromagnet,ic and antiferrom:tgiIcti~ elrmcnts near iron, t,his appears at first. to he an at,tractive possihility. However, nonferromagnet,ic stainless steel shows (1) the same enhanced decay rate as fcrromagnctic iron, and it, is known ( ,$2) that, the average magnetic field at an iron n~~ltus in st,ainleasstcrl is at least an order of magnitude lesst’hnn in frrromagnct iv iron over ptriotls of t,hc order of IO-’ second. Since this time scaleis of t’hc same order as t,hat, which occurs in bound melon deray, it appears t)hat thr rnhanced decay rst,e near iron can not bt at,tribut,ed t,o the magnetic firld of the atomic c,l(,c*trons. Since t,he enhanced decay rate in this region also occurs for srvcral VV~II-~\Y’II

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nuclei, which have no spin and therefore no magnetic moment, the enhancement also can not be attributed to t.he magnet.ic field of the nucieus. Therefore, the magnetic fields of the nucleus and the atomic electrons will not be treated further. It should be noted, however, that a magnetic field effect on the decay rate of as much as 3 % for the medium element,s, and perhaps even more for the heavier elements, would not. be inconsistent with the present experimental evidence on bound muon decay rates. In matter, a negative muon of lOO-Mev kinetic energy will be slowed down and captured atomically, and will reach it’s ground state in the atom, all in a period of less than lo-’ second4-6; and will disappear from this ground state by decay and nuclear capture in a period of lo-’ second or great.er (35). Thus, fewer t,han lo-’ of the muons will decay from other t,han t.he ground state. The decay rate of suc,h muons is expected (e.g., by an applic.at.ion of the time dilatation argument, of Section II j to lie between t’hat of free muons and t,hat of bound muons in t)he ground st#at,e, so t’hat an error of much less than 10V2 will be produced by assuming that all negative muons decay from the ground state. This ground state of t,he muon-nucleus system might be expected to haye the nucleus in its ground st’ate and the muon in the Is-state. However, the muonproton attraction, and the nearness of the muon to the nucleus, may cause a deformation of t,he nucleus for the medium and heavy elements. As a result, the ground statme of the muon-nucleus system would have a lower energy, and would havr an addit~ional component. ront,aining excited nuclear and muon states, and both of these fact.ors may affect, the bound muon decay rate. A nonrelat,ivist.ic variat,ional calculation shows that the greatest effect is generally associat,ed with electric quadrupole excitations of t,ht nucleus, but, on the basis of present cxperimental data (& 1)) the ef?cct, of t)his nuclear deformation upon the bound muon decsay rat.e does not exceed 0.01 70 for any clcmrnt. It will therefore he sufficient to USCa purr Is wave funcbtion for the 11111011. h portion of the decay electrons will be captured into unoccupied electronic le\-rls of the atom, but since sutah decays arc not observed in the bound muon dccsay experiments (and are rspec+rd to be negligible anyway), only t,hose decays which lead to \mbound cllrctrons nerd be considcrrd here. 4 Wightntan (43) computes the time for hydrogen, and gives references to calculations other materials. 6 Holmstrom and Iieuffel (3) fouttd experimentally that, in iron, all mesic s-rays occur within 10-g set or less following the arrival of a cosmic-ray muon, \I-hi& result, is consistent wit,h t,he above statement and conclusion. 6 iTote that. if this period were sufficiently long (say 10e7 set) in a few elements (e.g., elements near iron) such that an appreciable fraction of the muons decayed from other than t.he ground st.at,e, then the n1110n capture rate would also be considerably reduced for these elements. However, sttch a reduction is in disagreement with existing muon capture data (35).

for

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In summary, then, bou~~d muon decay is essentially the decay of a Is-state muon into two plane wave neutrinos and an unbound electron, via a local nonderivative interaction, with muon and electron wave funct’ions appropriat,e t,o the actual electrostatic potential of the binding nucleus. W&h the possibleexeept,ions of this decay proc.essitself, or experimental cont,amination from the lowenergy gamma rays associated with muon capture, no explanation has been discovered in this Appendix for the observed peak in t,he bound muon decay rate near iron. RECEIVED:

,July

14,

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cimento