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The anomalous magnetic moment of positive and negative muons
Volume
67B, number
PHYSICS
2
THE ANOMALOUS
MAGNETIC
MOMENT
28 March
LETTERS
OF POSITIVE
AND NEGATIVE
1977
MUONS
J. BAILEY’, K. BORER2, F. COMBLEY3, H. DRUMM4, F.J.M. FARLEY’, J.H. FIELD, W. FLEGEL, P.M. HATTERSLEY’ , F. KRIENEN, F. LANGE4, E. PICASSO and W. Von RUDEN4 (CERN Muon Storage Ring Collaboration) Received
15 February
1977
The anomalous g-factor a = (g - 2)/2 has been measured for muons of both charges in the Muon Storage Ring at CERN. The two results, ap+= 1165910(12)X lo-’ and afiL- = 1165936(12) X 10Tg, are in good agreement with each X 1O-g, which is very close to the most recent theoretical predicother, and combine to give a mean n,, = 1165922(g) tion 1165921(10) X lo-‘. For the experimental results, the total statistical and systematic error is given. The measurements thus confirm the remarkable QED calculation plus hadronic contribution, and serve as a precise verification of the CPT theorem for muons.
This letter presents the latest results on the muon g-factor anomaly, a s (g - 2)/2, from the CERN Muon Storage Ring. Measurements of the muon lifetime and electric dipole moment will be reported separately. Preliminary results for the positive muon were presented early in 1975 [l] together with a description of the measurement technique. The experimental has already been discussed in several review articles [3], but nevertheless before detailing the new results for muons of both signs we would like to recall the principles of the method. Owing to radiative corrections, the g-factor of the charged leptons is slightly larger than the value 2 predicted by Dirac theory, and an observable consequence of this is that, in a magnetic field, the particle spin rotates faster than its momentum vector. The angular frequency of this relative ,precession is
wa=-e&, me
2nf,
= lw,l
:
102-146 146-189 189-232 232-275 275-318
‘I’ ,’
,,.,,,
,,,,,,
‘/,~,’ ‘,m,” ,,,‘1’1
I,, ,,
‘,, ,,m,IS,,,ll*ldl,I,,,,, ” /I,, ‘I
405-448 448-491 491-534
IO
20
30 Time
40 tn
50
/
60
microseconds
Fig. 1. Time distribution of decay electrons. The 1.4 X lo8 events shown”are a sum of the three highest electron energy bands in the complete experiment; 120 g- 2 cycles can be seen.
,
and its measurement is used as a means of obtaining the value of the anomaly. The precision with which this can be done depends largely on how well the mean ’ 2 3 4 ’
59-102
Daresbury Laboratory, United Kingdom. Physikalisches Institut, Universitat Bern, Switzerland. Dept. of Physics, University of Sheffield, United Kingdom. Institut fiir Physik der Universitat Mainz, Germany. Royal Military College of Science, Shrivenham, United Kingdom. 6 Dept. of Physics, University of Birmingham, United Kingdom.
magnetic field, seen by the muon population, is known. This requirement is more easily satisfied if the field is uniform, but in such a case the particle ‘motion is not constrained in the direction of the field. Using an electric field to achieve the necessary trapping, shifts the angular frequency of the relative precession to the value
where we have assumed throughout
that both B and E 225
Volume
67B, number
2
Summary
PHYSICS
LETTERS
Table 1 of runs. In this table the errors quoted in
28 March
R include only the statistical
1977
error infa.
Run
Sign
1974 1975 A B C 1976 A B C D E
No. of stops (million)
fa (MHz)
fP (MHz)
11.8 9.5 15.3 11.7 10.9 7.2 10.2 24.8 33.0
0.2327448(56) 0.2327441(52) 0.2327469(69) 0.2327602(54) 0.2327479 (54) 0.2327530(69) 0.2327437(58) 0.2327505 (38) 0.2327535 (32)
62.78297(10) 62.78316(10) 62.78304(10) 62.78356(10) 62.78302(10) 62.78288(10) 62.78287 (10) 62.78294(10) 62.78307(10)
R = faI?p (X 103)
3.707133(89) 3.707110(83) 3.707162(110) 3.707343(86) 3.707179 (86) 3.707269 (110) 3.707121(92) 3.707225 (61) 3.707265 (51) 3.707173(36) 3.707256 (37) 3.707213 (26)
are transverse too. As can be seen, the magnitude of this frequency shift depends upon the particle energy, andiszeroify= [1t(1/a)]1/2=29.3.Atthisenergy and for this combination of fields the spin motion closely approximates to that of the ideal situation in which the particles circulate in a uniform magnetic field. For muons the corresponding momentum is 3.094 GeV/c. In the present experiment, which is the third in a series carried out at CERN [3,4], muons with this momentum travel around a 14 m diameter circle in a 1.47 T magnetic field produced by a polygon of 40 bending magnets. The vertical motion of the muons is constrained by an electric quadrupole field [5]. Injecting a pion beam of momentum slightly higher than the centre of the acceptance of the storage ring produces stored muons with high initial longitudinal polarization. The evolution of this polarizatoin as a function of storage time is observed by selecting highenergy decay electrons (forward-going in the muon rest frame). As the decay electron distribution is asymmetric with respect to the muon spin direction, the count rate is modulated at the angular frequency w, of the relative spin rotation. The time distribution of 1.4 X lo8 decay electrons counted by the energymeasuring shower detectors is shown in fig. 1. The data were fitted by the maximum-likelihood method to the function (3) N(t)=NOIL(t)exp(-t/r){l-Acos(~at+~)} +B], which exhibits the, exponential 226
decay with the muon
time-dilated lifetime 7, the asymmetry A and phase $ of the modulation, and the constant background B. The subsidiary function
L(t) = 1 +ALexp (-t/rL) ,
(4)
represents the small distortion of the data at early times by muon losses and electronic gain changes. All eight parameters NO, T, A, a,, q5,B, A,, q, were allowed to vary and, in order to optimize the precision on c3,, the data from each run were subdivided into four bands of electron energy each with widely different asymmetry A. The values of fa given in table 1 are. the weighted averages of those for the four energy bands. A result of such high statistical precision must be supported by an extensive examination of all parts of the experimental process which may lead to an inaccurate or systematically erroneous measurement. The over-all reliability of the fitting procedure was checked with runs on Monte Carlo simulated data, and the experimental data themselves were subjected to a careful search for systematic effects by dividing them into various subgroups according to different characteristics such as beam intensity, counter position, decay electron energy, and starting time of the fit. No effects were found beyond those consistent with statistical fluctuations. The experiment consists essentially of a measurement of the ratio R of two frequencies: the relative spin precession frequency wa; and the mean magnetic field seen by the muon sample, expressed in terms of the effective mean proton resonance frequency in
PHYSICS LETTERS
Volume 67B, number 2
28 March 1977
Table 2 Corrections and errors to Tp for a particular run.
Weighted mean of map Calibration and diamagnetic shielding Time variation of field and clock drift Vacuum tank, electrodes and inflector Non-ideal phase space muon distribution Electric field effect Pitch correction TP
Frequency (MHz)
Relative error
62.78167(4)
0.7
(pm)
0.00148 (4)
0.6
-0.00004 (6)
1.0
0.00000 (4)
0.6
0.00001(1)
0.1
-0.00010(2) -0.00005 (1)
0.2 0.1
62.78297(10)
1.5
Fig. 2. Values of R = fa/fp for the nine experimental runs. The average values for M+,pL-and the combined average are also shown. The individual runs are statistically consistent with the combined average (x2 = 7.3 for 8 degrees of freedom).
vacuum Op. This latter is related to the mean field B and the mean muon Larmor frequency in vacuum ‘jll, by g
=;q
=f Xijp (h =,wp/wp)
.
(5)
In the following we summarize the findings of an examination of the possible sources of error, dividing them into two groups in so far as they affect these two frequency measurements. Possible sources of systematic effects on the observed muon spin precession frequency derive from the conditions under which the data are collected. In addition to the decay electrons, the 22 shower detectors are exposed to an initial flash (bunch length -1Onsec) of about lo7 particles, which the pion beam transports to the storage ring every PS burst. The photomultipliers are blanked off during this time, thereby largely protecting the fast electronics, but the tubes themselves exhibit subsequent gain changes during the muon storage time. The over-all gain changes of the photomultiplier and fast electronics system, measured by a method using light-emitting diodes, were shown to have no effect on f,. The uniformity of the timing of the arrival of pulses was established with an accurate determination of the time slewing, which was combined with this gain measurement. The effect of time slewing on the frequency fa was shown to be less than 1 ppm. The correct functioning of the time-measuring devices or. digitrons [6] was established by the internal
consistency of the data and by periodical checks with a special test program. The linearity of the digitrons, for time intervals of the order of the stored muon lifetime, was shown to be good to a small fraction of a ppm [7]. In addition to examining the data for ratedependent effects, queuing losses were observed directly by feeding all the pulses handled by the normal five dibitrons also to a single digitron, and comparing the values of o, obtained from the same experimental run treated in these two ways. This information, together with a computer simulation, was used to apply a correction which is included in the results given in table 1, and which for no run exceeded 1.5 ppm. Finally, we should note that the observed (g - 2) frequency can be shifted by spurious periodic variations in intensity, such as ejection from later revolutions of the PS for example. Any portions of data containing these effects were eliminated. As we have indicated above, the second frequency is equivalent to the mean magnetic field seen by the muon sample, and clearly the accuracy with which it is known to the precision of the anomaly. There are essentially three factors which give confidence in this area, and they are: the careful preparation of the magnets by shimming [8] ; periodic full-scale mapping throughout the storage volume with NMR probes; stabilization and monitoring of the field during runs t91* The main purpose of the shimming was to remove the radial dependence of the field; the azimuthal variation due to the fact that the ring was constructed from 40 magnets is less important since its effect on the muon spin is averaged out. The full-scale map of the field consisted of measure227
Volume
67B, number
2
PHYSICS
[ 131 is shown on the second line of table 3, while the effect of weak interactions, calculated within the framework of gauge theories [ 161, is consistent with the entry on the third line. At the bottom of the table we sum these three terms to give the over-all theoretical prediction for the anomaly. Comparing the experimental and theoretical values, we can conclude that the agreement has withstood the factor of 38 improvement in precision which these experiments represent. In particular, it is clear that the theoretical hadronic contribution is essential to this accord. We should note that at the 95% confidence limit these results indicate that the conventional photon propagator cutoff must be A X 18 GeV. At the same level of confidence the electric dipole moment of the muon is 58 X lo-l9 e * cm, but a direct measurement of the electric dipole moment will be reported later. We can examine this agreement in terms of the muon being a composite structure. Then as Brodsky 1171 has pointed out, the coupling to an internal charge current would lead to a modification Au, z O(m,/m*) where m* is a characteristic internal mass, the mass of the first excited state, or continuum threshold. Then the lower limit for m* is of the order of lo5 GeV/c2. Finally, any inequality between the g-factors for positive and negative muons would violate CPT. These measurements yield (g,, - g,+)/g = 0.026 + 0.017 ppm and represent the most accurate test of the CPT theorem applied to muons. contribution
It is a pleasure to express our thanks to those who, over the two-year periode of data-taking, have both helped in the running of the experiment and made specific contributions. We specially thank R.W. Williams and S. Wojcicki for their active participation in many different phases of the experiment. We warmly appreciate the work of E.M. McMillan on the stability of the muon orbits and selection of the field index value. We thank M. Rousseau for his excellent work on the fast electronics, W. Lysenko for his calculation of the effect of the electric field on machine stability, and G. FrCmont and K. Miihlemann for their unfailing technical support. We thank M. Comyn who also cheerfully took part in the runs. Warm thanks are due to G. LebCe and 0. Runolfsson for their crucial con-, tributions regarding the inflector and the field mapping, respectively. 228
LETTERS
28 March
1977
This work was partly supported by the Bundesministerium fur Forschung und Technologie of the German FederalRepublic.
References [l] J. Bailey et al., Phys. Lett. 55B (1975) 420. [2] See, for example: F.J.M. Farley, Contemp. Phys. 16 (1975) 413; F. Combley and E. Picasso, Phys. Reports 14C (1974) 1; J. Bailey and E. Picasso, Progr. Nucl. Phys. 12 (1970) 43; J.H. Field, in: Proc. EPS Internat. Conf. on High-energy physics, Palermo, 1975 (Editrice Compositori, Bologna, 1976) p. 247; F.H. Combley, in: Proc. 7th Internat. Symp. on Lepton and photon interactions at high energies, Stanford University, August 1975, ed. W.T. Kirk (Stanford, 1975) p. 913. [3] G. Charpak et al., Nuovo Cimento 37 (1965) 1241. [4] J. Bailey et al., Nuovo Cimento A9 (1972) 369. [5] W. Flegel and F. Krienen, Nucl. Instrum. Meth. 113 (1973) 549. [6] H.I. Pizer, Nucl. Instrum. Meth. 123 (1975) 461; HI. Pizer, L. van Koningsveld and H. Verweij, CERN 76-17 (1976). [7] J.H. Field, F. Lange and W. von Rtiden, A high-precision method for calibration of digital time recorders, submitted to Nucl. Instrum. Meth. [8] K. Borer et al., Proc. 5th Internat. Conf. on Magnet Technology, Rome, 1975 (Lab. Naz. CNEN, Rome, 1975) p. 133. [9] K. Borer, The nuclear magnetic resonance system for the CERN Muon Storage Ring, submitted to Nucl. Instrum. Meth. [lo] K. Borer and F. Lange, Absolute calibration of the NMR magnetometers and of the magnetic field measuring systerm used in the CERN Muon Storage Ring, submitted to Nucl. Instrum. Meth. [ll] F.J.M. Farley, Phys. Lett. 42B (1972) 66; J.H. Field and G. Fiorentini, Nuovo Cimento 21A (1974) 297. [12] K.M. Crowe et al., Phys. Rev. DS (1972) 2145. [13] J. Calmet, S. Narison, M. Perrottet and E. de Rafael, The anomalous magnetic moment of the muon: a review of the theoretical contributions, Rev. Mod. Phys., to be published; this paper gives an extensive list of references to original contributions and earlier reviews. See also ref. [17]. [14] P. Cvitanovic and T. Kinoshita, Phys. Rev. DlO (1974) 4007; J. Cahnet and A. Peterman, Phys. Lett. 58B (1976) 449; R. Barbieri and E. Remiddi, Nucl. Phys. B90 (1975) 223; M.J. Levine, R.C. Perish0 and R. Roskies, Phys. Rev. D 13 (1976) 997;
Volume
67B, number
Contributions
a,, (QED)
PHYSICS
2
Table 3 to the theoretical
value of a,,.
ap (hadronic) ap (weak)
1165851.8(2.4)X 66.7(9.4) X 1O-9 2(2) x 10-9
a, (theory)
1165921(10)
1O-9
X lo-’
ments at some 250,000 points throughout the storage volume, and from these an average frequency, weighted with an ideal distribution of muon orbits, was obtained. This is given in the first line of table 2, which includes the various contributions which make up the final effective mean field seen by the muon sample for a particular run (1974 data). The NMR probes used in mapping, monitoring, and stabilizing the field were each calibrated against a standard probe of well-defined geometry [lo]. The result of this calibration is included in the second entry of table 2, which also contains the correction necessary in order to obtain the effective proton resonance frequency in vacuum. The crystal clock used for measuring the NMR frequency both when mapping and monitoring the field, was the same as that used for measuring ma; thus the effect of drifts in its frequency largely cancelled out. The absolute values of the magnetic field given by the monitoring probes were consistent with those obtained using the mapping device to better than 0.5 ppm. Thus the slow changes in the magnetic field level between full maps could be reliably followed and corrected for. This correction is combined with that due to the small drift in master clock frequency on the third line of table 2. The next correction arises because the field maps are made in the absence of vacuum tank, electrodes, and inflector. Each of these effects has been measured separately. The actual distribution of the muon orbits can be obtained by observing how the initial bunch of muons circulating round the ring spreads out in time [4]. This information is used in a Monte Carlo orbit tracking program to obtain the correction for the previously assumed ideal phase-space distribution as given in the fifth line of table 1. The spread in the orbits also means that not all the muons have the momentum (3.094 GeV/c) necessary to nullify the effect of the electric field on the muon
28 March 1977
LETTERS
spin. This can be treated as asmall shift in the magnetic field, and the relevant correction is shown in the sixth line, while immediately below, the pitch correction [ 1 I] due to the betatron motion of the particles is given. The result of all these terms is seen in the statement of fP, for a particular run, given at the bottom of table 2. In table 1 we give the individual frequencies fa and fP together with their ratio R for each of the nine experimental runs, and also the weighted average values ofR forp+,p-, and the combined data. The errors quoted are statistical onfa and systematic onfp. All twelve R values are plotted in fig. 2. The over-all average value of R is 3.707213 (27) X 10W3. The error contributions are 7.0 ppm (statistical on f,) and 1.5 ppm (systematic on the magnetic field), over-all 7.2 ppm. From eqs. (1) and (5) we can deduce the expression for the anomaly a&= R/(h -R)
,
(6)
and for a given value of h, the ratio of the muon to proton magnetic moment, our measurements of R can be used to obtain the muon anomaly. For h = 3.1833467 (82) as measured by Crowe and co-workers [12], we get aM+= 1165910(12)X
10Lg
(loppm),
ap-= 1165936(12)X
lbhg
(10ppm)
(7) ,
for muons of each sign, and an over-all weighted value. of ap = 1165922(9)
X 10Wg
(8 ppd
.
(8)
The errors combine those that are both statistical and systematic in origin. The calculation of the value of the anomaly within the theory of quantum electrodynamics (QED) is a triumph in which many theoretical physicists share [ 13 1. Recent contributions include more accurate determinations of the sixth-order terms [ 143 and also the evaluation of several in the eighth order [ 151. The total QED prediction for the muon anomaly is given in line one of table 3. The polarization of the vacuum into virtual hadrons modifies this value in a calculable way since it is related via dispersion relations to the cross-section for efeannihilation into hadrons. The latest estimate for this 229
Volume 67B. number 2
PHYSICS ELTTERS
M.A. Samuel and C. Chlouber, Phys. Rev. Lett. 36 (1976) 442; M.J. Levine and R. Roskies, Phys. Rev. D14 (1976) 2191. [15] B.E. Lautrup and E. de Rafael, N&l. Phys. B70 (1974) 317; M.A. Samuel, Phys. Rev. D9 (1974) 2913; J. Calmet and A. Peterman, Phys. Lett. 56B (1975) 383. [16] R. Jackiw and S. Weinberg, Phys. Rev. D5 (1972) 2396;
230
28 March 1977
W.A. Bardeen, R. Gastmans and B.E. Lautrup, Nucl. Phys. B46 (1972) 319; I. Bars and M. Yoshimura, Phys. Rev. D6 (1972) 374; K. Fujikawa, B.W. Lee and A.I. Sanda, Phys. Rev. D6 (1972) 2923; J. Primack and H.R. Quinn, Phys. Rev. D6 (1972) 3171; and ref. [13]. ]171 S.J. Brodsky, SLACPub-1699 (1975) presented at LAMPF Users’ Group Meeting, Los Alamos (1975).
67B, number
PHYSICS
2
THE ANOMALOUS
MAGNETIC
MOMENT
28 March
LETTERS
OF POSITIVE
AND NEGATIVE
1977
MUONS
J. BAILEY’, K. BORER2, F. COMBLEY3, H. DRUMM4, F.J.M. FARLEY’, J.H. FIELD, W. FLEGEL, P.M. HATTERSLEY’ , F. KRIENEN, F. LANGE4, E. PICASSO and W. Von RUDEN4 (CERN Muon Storage Ring Collaboration) Received
15 February
1977
The anomalous g-factor a = (g - 2)/2 has been measured for muons of both charges in the Muon Storage Ring at CERN. The two results, ap+= 1165910(12)X lo-’ and afiL- = 1165936(12) X 10Tg, are in good agreement with each X 1O-g, which is very close to the most recent theoretical predicother, and combine to give a mean n,, = 1165922(g) tion 1165921(10) X lo-‘. For the experimental results, the total statistical and systematic error is given. The measurements thus confirm the remarkable QED calculation plus hadronic contribution, and serve as a precise verification of the CPT theorem for muons.
This letter presents the latest results on the muon g-factor anomaly, a s (g - 2)/2, from the CERN Muon Storage Ring. Measurements of the muon lifetime and electric dipole moment will be reported separately. Preliminary results for the positive muon were presented early in 1975 [l] together with a description of the measurement technique. The experimental has already been discussed in several review articles [3], but nevertheless before detailing the new results for muons of both signs we would like to recall the principles of the method. Owing to radiative corrections, the g-factor of the charged leptons is slightly larger than the value 2 predicted by Dirac theory, and an observable consequence of this is that, in a magnetic field, the particle spin rotates faster than its momentum vector. The angular frequency of this relative ,precession is
wa=-e&, me
2nf,
= lw,l
:
102-146 146-189 189-232 232-275 275-318
‘I’ ,’
,,.,,,
,,,,,,
‘/,~,’ ‘,m,” ,,,‘1’1
I,, ,,
‘,, ,,m,IS,,,ll*ldl,I,,,,, ” /I,, ‘I
405-448 448-491 491-534
IO
20
30 Time
40 tn
50
/
60
microseconds
Fig. 1. Time distribution of decay electrons. The 1.4 X lo8 events shown”are a sum of the three highest electron energy bands in the complete experiment; 120 g- 2 cycles can be seen.
,
and its measurement is used as a means of obtaining the value of the anomaly. The precision with which this can be done depends largely on how well the mean ’ 2 3 4 ’
59-102
Daresbury Laboratory, United Kingdom. Physikalisches Institut, Universitat Bern, Switzerland. Dept. of Physics, University of Sheffield, United Kingdom. Institut fiir Physik der Universitat Mainz, Germany. Royal Military College of Science, Shrivenham, United Kingdom. 6 Dept. of Physics, University of Birmingham, United Kingdom.
magnetic field, seen by the muon population, is known. This requirement is more easily satisfied if the field is uniform, but in such a case the particle ‘motion is not constrained in the direction of the field. Using an electric field to achieve the necessary trapping, shifts the angular frequency of the relative precession to the value
where we have assumed throughout
that both B and E 225
Volume
67B, number
2
Summary
PHYSICS
LETTERS
Table 1 of runs. In this table the errors quoted in
28 March
R include only the statistical
1977
error infa.
Run
Sign
1974 1975 A B C 1976 A B C D E
No. of stops (million)
fa (MHz)
fP (MHz)
11.8 9.5 15.3 11.7 10.9 7.2 10.2 24.8 33.0
0.2327448(56) 0.2327441(52) 0.2327469(69) 0.2327602(54) 0.2327479 (54) 0.2327530(69) 0.2327437(58) 0.2327505 (38) 0.2327535 (32)
62.78297(10) 62.78316(10) 62.78304(10) 62.78356(10) 62.78302(10) 62.78288(10) 62.78287 (10) 62.78294(10) 62.78307(10)
R = faI?p (X 103)
3.707133(89) 3.707110(83) 3.707162(110) 3.707343(86) 3.707179 (86) 3.707269 (110) 3.707121(92) 3.707225 (61) 3.707265 (51) 3.707173(36) 3.707256 (37) 3.707213 (26)
are transverse too. As can be seen, the magnitude of this frequency shift depends upon the particle energy, andiszeroify= [1t(1/a)]1/2=29.3.Atthisenergy and for this combination of fields the spin motion closely approximates to that of the ideal situation in which the particles circulate in a uniform magnetic field. For muons the corresponding momentum is 3.094 GeV/c. In the present experiment, which is the third in a series carried out at CERN [3,4], muons with this momentum travel around a 14 m diameter circle in a 1.47 T magnetic field produced by a polygon of 40 bending magnets. The vertical motion of the muons is constrained by an electric quadrupole field [5]. Injecting a pion beam of momentum slightly higher than the centre of the acceptance of the storage ring produces stored muons with high initial longitudinal polarization. The evolution of this polarizatoin as a function of storage time is observed by selecting highenergy decay electrons (forward-going in the muon rest frame). As the decay electron distribution is asymmetric with respect to the muon spin direction, the count rate is modulated at the angular frequency w, of the relative spin rotation. The time distribution of 1.4 X lo8 decay electrons counted by the energymeasuring shower detectors is shown in fig. 1. The data were fitted by the maximum-likelihood method to the function (3) N(t)=NOIL(t)exp(-t/r){l-Acos(~at+~)} +B], which exhibits the, exponential 226
decay with the muon
time-dilated lifetime 7, the asymmetry A and phase $ of the modulation, and the constant background B. The subsidiary function
L(t) = 1 +ALexp (-t/rL) ,
(4)
represents the small distortion of the data at early times by muon losses and electronic gain changes. All eight parameters NO, T, A, a,, q5,B, A,, q, were allowed to vary and, in order to optimize the precision on c3,, the data from each run were subdivided into four bands of electron energy each with widely different asymmetry A. The values of fa given in table 1 are. the weighted averages of those for the four energy bands. A result of such high statistical precision must be supported by an extensive examination of all parts of the experimental process which may lead to an inaccurate or systematically erroneous measurement. The over-all reliability of the fitting procedure was checked with runs on Monte Carlo simulated data, and the experimental data themselves were subjected to a careful search for systematic effects by dividing them into various subgroups according to different characteristics such as beam intensity, counter position, decay electron energy, and starting time of the fit. No effects were found beyond those consistent with statistical fluctuations. The experiment consists essentially of a measurement of the ratio R of two frequencies: the relative spin precession frequency wa; and the mean magnetic field seen by the muon sample, expressed in terms of the effective mean proton resonance frequency in
PHYSICS LETTERS
Volume 67B, number 2
28 March 1977
Table 2 Corrections and errors to Tp for a particular run.
Weighted mean of map Calibration and diamagnetic shielding Time variation of field and clock drift Vacuum tank, electrodes and inflector Non-ideal phase space muon distribution Electric field effect Pitch correction TP
Frequency (MHz)
Relative error
62.78167(4)
0.7
(pm)
0.00148 (4)
0.6
-0.00004 (6)
1.0
0.00000 (4)
0.6
0.00001(1)
0.1
-0.00010(2) -0.00005 (1)
0.2 0.1
62.78297(10)
1.5
Fig. 2. Values of R = fa/fp for the nine experimental runs. The average values for M+,pL-and the combined average are also shown. The individual runs are statistically consistent with the combined average (x2 = 7.3 for 8 degrees of freedom).
vacuum Op. This latter is related to the mean field B and the mean muon Larmor frequency in vacuum ‘jll, by g
=;q
=f Xijp (h =,wp/wp)
.
(5)
In the following we summarize the findings of an examination of the possible sources of error, dividing them into two groups in so far as they affect these two frequency measurements. Possible sources of systematic effects on the observed muon spin precession frequency derive from the conditions under which the data are collected. In addition to the decay electrons, the 22 shower detectors are exposed to an initial flash (bunch length -1Onsec) of about lo7 particles, which the pion beam transports to the storage ring every PS burst. The photomultipliers are blanked off during this time, thereby largely protecting the fast electronics, but the tubes themselves exhibit subsequent gain changes during the muon storage time. The over-all gain changes of the photomultiplier and fast electronics system, measured by a method using light-emitting diodes, were shown to have no effect on f,. The uniformity of the timing of the arrival of pulses was established with an accurate determination of the time slewing, which was combined with this gain measurement. The effect of time slewing on the frequency fa was shown to be less than 1 ppm. The correct functioning of the time-measuring devices or. digitrons [6] was established by the internal
consistency of the data and by periodical checks with a special test program. The linearity of the digitrons, for time intervals of the order of the stored muon lifetime, was shown to be good to a small fraction of a ppm [7]. In addition to examining the data for ratedependent effects, queuing losses were observed directly by feeding all the pulses handled by the normal five dibitrons also to a single digitron, and comparing the values of o, obtained from the same experimental run treated in these two ways. This information, together with a computer simulation, was used to apply a correction which is included in the results given in table 1, and which for no run exceeded 1.5 ppm. Finally, we should note that the observed (g - 2) frequency can be shifted by spurious periodic variations in intensity, such as ejection from later revolutions of the PS for example. Any portions of data containing these effects were eliminated. As we have indicated above, the second frequency is equivalent to the mean magnetic field seen by the muon sample, and clearly the accuracy with which it is known to the precision of the anomaly. There are essentially three factors which give confidence in this area, and they are: the careful preparation of the magnets by shimming [8] ; periodic full-scale mapping throughout the storage volume with NMR probes; stabilization and monitoring of the field during runs t91* The main purpose of the shimming was to remove the radial dependence of the field; the azimuthal variation due to the fact that the ring was constructed from 40 magnets is less important since its effect on the muon spin is averaged out. The full-scale map of the field consisted of measure227
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[ 131 is shown on the second line of table 3, while the effect of weak interactions, calculated within the framework of gauge theories [ 161, is consistent with the entry on the third line. At the bottom of the table we sum these three terms to give the over-all theoretical prediction for the anomaly. Comparing the experimental and theoretical values, we can conclude that the agreement has withstood the factor of 38 improvement in precision which these experiments represent. In particular, it is clear that the theoretical hadronic contribution is essential to this accord. We should note that at the 95% confidence limit these results indicate that the conventional photon propagator cutoff must be A X 18 GeV. At the same level of confidence the electric dipole moment of the muon is 58 X lo-l9 e * cm, but a direct measurement of the electric dipole moment will be reported later. We can examine this agreement in terms of the muon being a composite structure. Then as Brodsky 1171 has pointed out, the coupling to an internal charge current would lead to a modification Au, z O(m,/m*) where m* is a characteristic internal mass, the mass of the first excited state, or continuum threshold. Then the lower limit for m* is of the order of lo5 GeV/c2. Finally, any inequality between the g-factors for positive and negative muons would violate CPT. These measurements yield (g,, - g,+)/g = 0.026 + 0.017 ppm and represent the most accurate test of the CPT theorem applied to muons. contribution
It is a pleasure to express our thanks to those who, over the two-year periode of data-taking, have both helped in the running of the experiment and made specific contributions. We specially thank R.W. Williams and S. Wojcicki for their active participation in many different phases of the experiment. We warmly appreciate the work of E.M. McMillan on the stability of the muon orbits and selection of the field index value. We thank M. Rousseau for his excellent work on the fast electronics, W. Lysenko for his calculation of the effect of the electric field on machine stability, and G. FrCmont and K. Miihlemann for their unfailing technical support. We thank M. Comyn who also cheerfully took part in the runs. Warm thanks are due to G. LebCe and 0. Runolfsson for their crucial con-, tributions regarding the inflector and the field mapping, respectively. 228
LETTERS
28 March
1977
This work was partly supported by the Bundesministerium fur Forschung und Technologie of the German FederalRepublic.
References [l] J. Bailey et al., Phys. Lett. 55B (1975) 420. [2] See, for example: F.J.M. Farley, Contemp. Phys. 16 (1975) 413; F. Combley and E. Picasso, Phys. Reports 14C (1974) 1; J. Bailey and E. Picasso, Progr. Nucl. Phys. 12 (1970) 43; J.H. Field, in: Proc. EPS Internat. Conf. on High-energy physics, Palermo, 1975 (Editrice Compositori, Bologna, 1976) p. 247; F.H. Combley, in: Proc. 7th Internat. Symp. on Lepton and photon interactions at high energies, Stanford University, August 1975, ed. W.T. Kirk (Stanford, 1975) p. 913. [3] G. Charpak et al., Nuovo Cimento 37 (1965) 1241. [4] J. Bailey et al., Nuovo Cimento A9 (1972) 369. [5] W. Flegel and F. Krienen, Nucl. Instrum. Meth. 113 (1973) 549. [6] H.I. Pizer, Nucl. Instrum. Meth. 123 (1975) 461; HI. Pizer, L. van Koningsveld and H. Verweij, CERN 76-17 (1976). [7] J.H. Field, F. Lange and W. von Rtiden, A high-precision method for calibration of digital time recorders, submitted to Nucl. Instrum. Meth. [8] K. Borer et al., Proc. 5th Internat. Conf. on Magnet Technology, Rome, 1975 (Lab. Naz. CNEN, Rome, 1975) p. 133. [9] K. Borer, The nuclear magnetic resonance system for the CERN Muon Storage Ring, submitted to Nucl. Instrum. Meth. [lo] K. Borer and F. Lange, Absolute calibration of the NMR magnetometers and of the magnetic field measuring systerm used in the CERN Muon Storage Ring, submitted to Nucl. Instrum. Meth. [ll] F.J.M. Farley, Phys. Lett. 42B (1972) 66; J.H. Field and G. Fiorentini, Nuovo Cimento 21A (1974) 297. [12] K.M. Crowe et al., Phys. Rev. DS (1972) 2145. [13] J. Calmet, S. Narison, M. Perrottet and E. de Rafael, The anomalous magnetic moment of the muon: a review of the theoretical contributions, Rev. Mod. Phys., to be published; this paper gives an extensive list of references to original contributions and earlier reviews. See also ref. [17]. [14] P. Cvitanovic and T. Kinoshita, Phys. Rev. DlO (1974) 4007; J. Cahnet and A. Peterman, Phys. Lett. 58B (1976) 449; R. Barbieri and E. Remiddi, Nucl. Phys. B90 (1975) 223; M.J. Levine, R.C. Perish0 and R. Roskies, Phys. Rev. D 13 (1976) 997;
Volume
67B, number
Contributions
a,, (QED)
PHYSICS
2
Table 3 to the theoretical
value of a,,.
ap (hadronic) ap (weak)
1165851.8(2.4)X 66.7(9.4) X 1O-9 2(2) x 10-9
a, (theory)
1165921(10)
1O-9
X lo-’
ments at some 250,000 points throughout the storage volume, and from these an average frequency, weighted with an ideal distribution of muon orbits, was obtained. This is given in the first line of table 2, which includes the various contributions which make up the final effective mean field seen by the muon sample for a particular run (1974 data). The NMR probes used in mapping, monitoring, and stabilizing the field were each calibrated against a standard probe of well-defined geometry [lo]. The result of this calibration is included in the second entry of table 2, which also contains the correction necessary in order to obtain the effective proton resonance frequency in vacuum. The crystal clock used for measuring the NMR frequency both when mapping and monitoring the field, was the same as that used for measuring ma; thus the effect of drifts in its frequency largely cancelled out. The absolute values of the magnetic field given by the monitoring probes were consistent with those obtained using the mapping device to better than 0.5 ppm. Thus the slow changes in the magnetic field level between full maps could be reliably followed and corrected for. This correction is combined with that due to the small drift in master clock frequency on the third line of table 2. The next correction arises because the field maps are made in the absence of vacuum tank, electrodes, and inflector. Each of these effects has been measured separately. The actual distribution of the muon orbits can be obtained by observing how the initial bunch of muons circulating round the ring spreads out in time [4]. This information is used in a Monte Carlo orbit tracking program to obtain the correction for the previously assumed ideal phase-space distribution as given in the fifth line of table 1. The spread in the orbits also means that not all the muons have the momentum (3.094 GeV/c) necessary to nullify the effect of the electric field on the muon
28 March 1977
LETTERS
spin. This can be treated as asmall shift in the magnetic field, and the relevant correction is shown in the sixth line, while immediately below, the pitch correction [ 1 I] due to the betatron motion of the particles is given. The result of all these terms is seen in the statement of fP, for a particular run, given at the bottom of table 2. In table 1 we give the individual frequencies fa and fP together with their ratio R for each of the nine experimental runs, and also the weighted average values ofR forp+,p-, and the combined data. The errors quoted are statistical onfa and systematic onfp. All twelve R values are plotted in fig. 2. The over-all average value of R is 3.707213 (27) X 10W3. The error contributions are 7.0 ppm (statistical on f,) and 1.5 ppm (systematic on the magnetic field), over-all 7.2 ppm. From eqs. (1) and (5) we can deduce the expression for the anomaly a&= R/(h -R)
,
(6)
and for a given value of h, the ratio of the muon to proton magnetic moment, our measurements of R can be used to obtain the muon anomaly. For h = 3.1833467 (82) as measured by Crowe and co-workers [12], we get aM+= 1165910(12)X
10Lg
(loppm),
ap-= 1165936(12)X
lbhg
(10ppm)
(7) ,
for muons of each sign, and an over-all weighted value. of ap = 1165922(9)
X 10Wg
(8 ppd
.
(8)
The errors combine those that are both statistical and systematic in origin. The calculation of the value of the anomaly within the theory of quantum electrodynamics (QED) is a triumph in which many theoretical physicists share [ 13 1. Recent contributions include more accurate determinations of the sixth-order terms [ 143 and also the evaluation of several in the eighth order [ 151. The total QED prediction for the muon anomaly is given in line one of table 3. The polarization of the vacuum into virtual hadrons modifies this value in a calculable way since it is related via dispersion relations to the cross-section for efeannihilation into hadrons. The latest estimate for this 229
Volume 67B. number 2
PHYSICS ELTTERS
M.A. Samuel and C. Chlouber, Phys. Rev. Lett. 36 (1976) 442; M.J. Levine and R. Roskies, Phys. Rev. D14 (1976) 2191. [15] B.E. Lautrup and E. de Rafael, N&l. Phys. B70 (1974) 317; M.A. Samuel, Phys. Rev. D9 (1974) 2913; J. Calmet and A. Peterman, Phys. Lett. 56B (1975) 383. [16] R. Jackiw and S. Weinberg, Phys. Rev. D5 (1972) 2396;
230
28 March 1977
W.A. Bardeen, R. Gastmans and B.E. Lautrup, Nucl. Phys. B46 (1972) 319; I. Bars and M. Yoshimura, Phys. Rev. D6 (1972) 374; K. Fujikawa, B.W. Lee and A.I. Sanda, Phys. Rev. D6 (1972) 2923; J. Primack and H.R. Quinn, Phys. Rev. D6 (1972) 3171; and ref. [13]. ]171 S.J. Brodsky, SLACPub-1699 (1975) presented at LAMPF Users’ Group Meeting, Los Alamos (1975).