PARITY NON-CONSERVATION Violation of Mirror Symmetry in the Production and Decay of Cosmic Ray Muons and Measurement of the Muon Magnetic Moment PREPARATORY QUESTIONS

January 12, 2001

Massachusetts Institute of Technology Physics Department 8.13/8.14 2000/2001

Junior Physics Laboratory Experiment #47 PARITY NON-CONSERVATION Violation Of Mirror Symmetry In The Production And Decay Of Cosmic Ray Muons And Measurement Of The Muon Magnetic Moment PREPARATORY QUESTIONS

1. What are the primary cosmic rays, and how do they give rise to polarized muons at sea level?

2. What are the decay schemes of positive and negative muons?

3. Why is a non-zero expectation value of the dot product of the momentum polar vector ~p of the decay electron and the magnetic moment axial vector ~µ of the muon proof that parity conservation has been violated in both the production and decay of the muon?

4. How do the energies of the electrons produced in the decay of stopped muons compare with the energies lost by muons passing through the plastic scintillator and the brass slab?

5. Compute the Larmor precession period of a muon in a magnetic ﬁeld of 100 gauss. (Assume the muon is simply a ”heavy electron.”)

6. Derive formulas for the magnitude B of the ﬁeld inside a solenoid and the power P required to maintain it under the following conditions: current = I,length=L,width =w,height=h<

7. Estimate the rate at which muons traverse the apparatus (”straight-through rate”), and the rate of muons stopping in the brass plate (”stopping rate”). (See the apparatus description below for the parameters of the equipment. According to Rossi (1948), the 2 directional intensity of muons at sea level is approximately I(θ)=Iνcos θ,where −2 −2 −1 −1 Iν =0.83 × 10 cm s sr ,andθis the angle from the zenith. The diﬀerential range spectrum of muons at sea level from 0 to 300 g cm−3 is approximately constant and equal to 5 × 10−6g−1s−1sr−1.

8. If random, uncorrelated, rectangular-shaped pulses of width t arrive at rates and at the inputs of a coincidence circuit, what will be the rate of output pulses due to accidental coincidences? If random pulses from the (S1+S2) coincidence circuit arrive at the rate of 10 s−1 at the start input of the TAC (time-to-amplitude converter) and random January 12, 2001 2 47. Parity

(uncorrelated) pulses from the (P1+P2) coincidence circuit arrive at a rate of 50 s−1 at the stop input of the TAC, what will be the rate of recorded intervals between random (uncorrelated) start and stop pulses with durations between zero and 3 times the mean life of muons?

1 INTRODUCTION

The violation of mirror symmetry (parity non-conservation) is a fundamental characteristic of all processes mediated by the weak force. This experiment is a search for evidence of mirror symmetry violation in the decay of cosmic-ray muons. The method is a measurement of the distributions in duration of the time intervals between the stoppings of cosmic-ray muons in a magnetized brass plate and the detections of their decay electrons in the upward and the downward directions. If the duration distributions show periodic oscillations about the expected exponential decay curves, then one can conclude that the direction distribution of the decay electrons rotates about the direction of the magnetic field. This implies both that the muon has an intrinsic magnetic moment associated with its spin angular momentum which precesses under the influence of the magnetic field, and that the stopped muons are polarized. If, in addition, the phases of the oscillations in the upward and downward decay curves differ by π(180◦), then the period of oscillations must be the Larmor period. This, in turn, implies that the muon decay process is not mirror symmetric. Moreover, if cosmic-ray muons are indeed polarized, then mirror symmetry must be violated in the decay of their parent particles which are mostly pions produced high in the atmosphere by primary and secondary cosmic ray hadrons. The Larmor period yields a measure of the muon magnetic moment, a result first obtained by Garwin, Lederman & Weinrich in 1957 in an experiment that proved parity violation in the decay of muons produced be the Nevis cyclotron of Columbia University.

To obtain statistically significant evidence of mirror asymmetry in this experiment, data must be accumulated during a total running time of many weeks. Special arrangements are therefore necessary for performance of this experiment. During the period of your sched- uled laboratory sessions, you will be expected to understand, calibrate, and monitor the performance of the equipment, and to add additional data to the archive of the experiment. Meanwhile, you will be expected to develop software to test for the presence of a statistically significant parity-violating effect in the data that has been accumulated over many weeks in the data archive, and to demonstrate the validity of your software by the generation and analysis of simulated data. Milestones for this work are set to insure that you will be prepared to analyze the accumulated data set in a timely manner and report the results of your work.

Those who undertake this experiment should have sufficient programming skill to develop a code of several hundred lines, most conveniently in LabVIEW on Windows PC’s or C on Athena. Each team is required to develop its own software. Use of packaged subroutines such as those in ”Numerical Recipes” (Press et al. 1989) or in the text of Bevington & Robinson (1992) is encouraged. The analysis will require non-linear curve fitting, evaluation January 12, 2001 3 47. Parity of statistical significance, and generation of simulated data sets with realistic statistical fluctuations. (Your performance will be evaluated on the basis of your understanding of the experiment and the quality of your data accumulation and analysis and not on whether your results actually prove the presence of a parity-violating effect.)

The experiment should be run continuously and the data should be captured periodically (every 2-4 days) and added to the PC data archive maintained in the archive folder ”Muon Polarization Data” (accessible from the Windows desktop). A log sheet in the log book should be filled out at the start of each run and identified with each batch of data transferred to the archive. Cooperation among different groups running the experiment at the same time will be essential in matters of setting the photomultiplier voltages and the discriminator levels, maintaining the timing calibration, monitoring the magnetic current supply, monitoring the various counting rates, and the systematic accumulation and sharing of data. At the same time, each team will be responsible for development and demonstration of their own analysis software and presentation of their results.

To insure steady progress toward the goals of the experiment each team should adhere to the following schedule:

Day 1. Explore the apparatus, calibrate it, update the log book, and start a run.

Day 2. Turn in a documented code that generates simulated data sets with random Poisson errors in tabular form in a ﬁle of two columns of lifetime vs counts. The parameters of the generator should be based on the rates you measured during the ﬁrst day and values of p (see Equation (1) below) from 0.0 to 0.05). A method of generating random Poisson variates is described in the Introductory Experiment on Poisson Statistics. Include plots of several simulated data sets.

Day 3. Turn in a documented decay analysis code based on the Levenberg-Marquardt method (see Chapter 8 of Bevington & Robinson and/or Chapter 14 of Press et al.) and results on the mean lives in brass of positive and negative muons based on the real data accumulated to date. You will also need to develop a code to add the various data sets with possible allowance for a shift in the zero channel. (It is wise to add the sets ﬁrst in groups of ten, and then all together.)

Day 4. Turn in a full analysis of your simulated data sets with an evaluation of the distribution of the variances of the ﬁtted parameters.

The oral or written report(s) should present at least the following:

1. the meaning and physical signiﬁcance of parity violation in beta decay,

2. a theoretical topic relevant to an understanding of the experiment ,

3. a description of the experiment,

4. the calibration procedures and your software, January 12, 2001 4 47. Parity

5. an analysis of the real data,

6. an evaluation of the statistical signiﬁcance of the result based on studies of simulated data sets.

2 DISCOVERY OF PARITY VIOLATION IN WEAK INTERACTIONS

Before 1956 it was generally believed that every process in nature that could, in principle, be exactly duplicated in the form of its mirror image. In other words, except for the accidents of biological evolution and industrial standards, nature was thought to make no distinction between left and right ”handedness.” In particular, this principle of mirror symmetry implied that any experiment could also be performed in its mirror-image version with identical results. For example, the mirror image of an experiment on optical rotation in a solution of dextrose looks exactly like the same experiment carried out with a solution of levulose. In a perfectly mirror-symmetric world, it would be impossible to tell an alien on Mars in words how to duplicate exactly a conventional right-handed corkscrew without refering to some mutually observable celestial conﬁguration or communicating by circularly polarized radiation.

The discovery of parity non-conservation began with the observation in cosmic-ray experiments of certain unstable particles that appeared to be identical in mass, charge and mean life which nevertheless decayed into ﬁnal states with opposite parity (either two or three pions). Confronting that fact, Lee and Yang (1957) proposed that the parent particles in the two and three pion decay processes are, in fact, identical (and subsequently called kaons), and suggested a general violation of parity conservation in weak interactions as an explanation of the kaon decay phenomenon. They further suggested that experimenters examine other processes involving the weak force for parity-violating asymmetries, in particular the beta decay of muons and of magnetically aligned nuclei. Within the year Wu (1957) detected the asymmetry in the direction of ejection of electrons in the beta decay of magnetically aligned Co60 atoms. Friedman and Telegdi (1957) detected asymmetry in the decay of muons arising from the decay of pions produced at the University of of Chicago cyclotron and stopped in a nuclear detection emulsion. Garwin, Lederman and Weinrich (1957) measured the asymmetry in the decay of muons at the Nevis cyclotron of Columbia University and determined the muon magnetic moment from the Larmor precession in a magnetic ﬁeld. Lee and Yang received the Nobel prize for their work.

The polarization of cosmic-ray muons at sea level was first demonstrated by Clark and Hersil (1957) in a measurement of the effect of a magnetic field on the probability of downward decay of muons stopped in a brass plate. Similar experiments were soon reported by several groups in the Soviet Union. Turner, Ankenbrandt and Larsen (1971) measured the Larmor precession of the up-down asymmetry in the decay of cosmic-ray muons stopped in a magnetized aluminum plate. Amsler (1974) described a pedagogical experiment, similar to the one now operating in the MIT Junior Lab, in which the Larmor precession of the up-down asymmetry of stopped muons is measured. January 12, 2001 5 47. Parity

In the Wu experiment a paramagnetic compound of cobalt containing the radioactive (beta decay) isotope Co60 was cooled to 0.01 K in the magnetic field of a solenoid so as to cause an alignment of the magnetic moments of the Co60 nuclei. The rates of ejection of decay electrons in the forward and backward directions relative to the direction of the field were measured. The data proved that there was a correlation between the direction of the magnetic field B~ and the momentum p~ of the decay electron, i.e. a non-zero expectation value of dot product . To see how this phenomenon violates the principle of mirror symmetry, imagine a mirror image of the Wu experiment in which one end of the solenoid, with axis parallel to the mirror, is labeled ”up” and the other end ”down.”. The current in the solenoid, circulating in a particular sense, gives rise to a magnetic field represented by an axial vector. Suppose the preferred electron ejection direction, represented by a polar vector, is up. Obviously the mirror image of the preferred ejection direction will also be up. On the other hand, application of the right-hand rule to the current in the mirror image of the solenoid defines a reversal in the direction of the axial vector representing the magnetic field. Thus the correlation between the ejection direction and magnetic field direction in the mirror image is the reverse of the observed correlation in the real world, i.e., the quantity p~ · B~ reverses sign on reflection. Since there is no way to mimic the mirror image of the Wu observation in the real world, it follows that the beta-decay of Co60, mediated by the weak force, is not symmetric with respect to a mirror reflection of spatial coordinates.

The parity-violating implication of a cosmic-ray muon decay experiment is a little more sub- tle than that of the Wu experiment. There is no practical way to align muons by an external magnetic field. What one actually observes in this experiment is periodic variations in the frequency of detection of electrons ejected in the upward and downward directions in the decay of cosmic-ray muons that stop in a horizontal brass plate permeated by a uniform horizontal magnetic field. The existence of a periodic variation demonstrates that the probability distribution of the momentum p~ of a decay electron rotates under the influence of the magnetic field. From this one can conclude that the muons must be precessing under the influence of the magnetic field and must therefore possess a magnetic moment m~ associated with an intrinsic angular momentum (spin), and that the stopped muons are polarized to some degree. If the observed variations in the upward and downward directions differ in phase by π(180◦), then the observed period must be the Larmor period of the precession (not twice the Larmor period), and the decay process must violate mirror symmetry, i.e., the expectation value of the dot product ~p · m~ is not zero. Since the dot product of an axial vector and a polar vector changes sign under mirror reflection, the non-zero value implies mirror symmtry violation in muon decay. The question then arises as to how the stopped cosmic-ray muons get polarized in the first place.

3 THE ORIGIN AND PROPERTIES OF COSMIC-RAY MUONS

Primary cosmic rays, arriving from outer space, consist of bare nuclei with elemental abun- dances similar to that of the sun, i.e. about 90% protons, 9% helium nuclei, and 1% nuclei of the higher-Z elements. The energies of primary nuclei arriving at Boston range from the geomagnetic cutoﬀ energy (about 1010 eV for protons) up to more than 1020eV ≈ 10 joule. January 12, 2001 6 47. Parity

When a primary cosmic-ray nucleus interacts with an air nucleus near the top of the atmosphere it initiates a cascade of nuclear interactions among whose products are positive and negative pions (mass = 273.12 me, spin = 0) in approximately equal numbers with identical mean lives at rest of 0.026030 µs. Some of these interact with air nuclei causing additional cascades of nuclear interactions, and some decay in ﬂight into positive and negative muons.

A charged pion (mass = 273.12 me, spin = 0) decays into a muon (mass = 206.7me), spin = 1/2) and a muon neutrino (mass = 0, spin = 1/2) in a weak-force parity-nonconserving process in which the angular momentum and associated magnetic moment of the muon are parallel to the direction of motion at birth, i.e., the muon is 100 percent polarized. In its rest frame a pion decays isotropically giving rise to a muon with an energy of approximately 4.2 MeV, unique because of the two-body decay. In the earth frame the energy spectrum of muons that were ejected by their parent pions in the downward direction is shifted to slightly higher energies relative to the energy spectrum of muons ejected in the upward direction. Since the muon intensity is a steeply decreasing function of energy, the shift to higher energy of the spectrum results, at any given energy or residual range, in an increase in intensity of downward-produced muons relative to the intensity of upward-produced muons. (Residual range of a charged particle of a given energy is the average value of the thickness, measured in grams per square centimeter, to its stopping point.)

Muons have a mean life at rest of 2.19714 µs. They do not interact strongly with air nuclei and, due to the relativistic dilation of their mean lives, many survive to sea level, losing energy by multiple ionizing collisions with air atoms at the rate of about 2 MeV g−1 cm2 during their passage through the atmosphere (total thickness ≈ 1000 g cm2). Hayakawa (1957) showed that ionizing collisions do not cause appreciable depolarization. Thus the population of muons that stop in a given thickness of material (i.e. that have residual range at sea level in the interval from zero to the thickness of the brass plate), has a slight excess of those that were produced in the downward direction in the parent pion rest frames. The population of stopped muons is therefore partially polarized to a degree that depends on the energy spectrum of the parent pions and the energy of the pion decay. This fact, in itself, demonstrates that pion decay, with its correlation between the momentum and magnetic moment of the daughter muon, violates mirror symmetry. The stopped muons, both positive and negative, produce decay electrons with an (parity-violating) excess of upward over downward ejections in the processes

− µ → e +¯ν+νµ

+ µ→e +νe+¯νµ

The spectrum of energies of the electrons from these three-body decays is a continuum from zero to 55 MeV, with a maximum near 50 MeV. A horizontal magnetic ﬁeld in the brass causes the muons to precess about the direction of the ﬁeld with the result that the distribution in duration of the time intervals, P(t), between the arrivals of the stopped muons and the emission of decay electrons in the downward direction oscillates about an exponential decay curve. Under these conditions the distribution of observed time intervals can be represented to good accuracy by the equation January 12, 2001 7 47. Parity

−t/τ+ −t/τ− P (t)=[1−acos ωt][A+e + A−e ]+B (1)

where a is a constant that depends on the polarization of the stopped muons, the intrinsic asymmetry of muon decay, and the depolarizing eﬀects of coulomb scattering on the directions of the decay electrons, is the angular velocity of Larmor precession, and are the number and mean life of the positive muons respectively, and are the corresponding quantities for negative muons (which disappear rapidly by nuclear absorption as well as by decay), and B is the rate of background events. is slightly larger than due to the fact that the primary cosmic rays which initiate the cascades of interactions that give rise to the muons are almost all positively charged nuclei.

4 APPARATUS

Figure 1 is a schematic diagram of the apparatus. S1 and S2 are plastic scintillation ”paddles ”(density=1.032gcm−3) with dimensions in centimeters of 60 x 60 x 2 cm, each with a Lucite light guide optically connected to a single photomultiplier (PM).

They are operated in coincidence (S1+S2) to signal the passage of a penetrating cosmic-ray particle, most probably a muon, and to provide ”start” signals for the Time-to-Amplitude Converters (TAC). The brass (density=8.38 g−3 ) plate in which some of the muons stop is 1.91 cm thick, 45 cm wide, and wound with 642 turns of 1.29 mm diameter copper wire over a length of 89.0 cm. In operation the wire carries a current of 10.0 amperes and is cooled by air driven by fans. S3 is a plastic scintillator with dimensions in centimeters of 90 x 45 x 5.08. It is viewed from below by two high-gain PMs operated in coincidence (P1+P2) to provide to the TAC reliable ”stop” signals that are free from contamination by ”after-pulses” generated in the separate PMs. Since both the start and stop signals for recording upward decay events come from the same measurement chain, the start signal is delayed by a 28-m delay cable so that the pulse produced by an arriving muon doesn’t terminate the timing sequence that it initiates.

The most common (S1+S2) +(P1+P2) event is a straight-through traversal of all three scintillation counters and the brass plate by an energetic muon. The resulting output of the DOWN TAC is a signal whose amplitude measures the artiﬁcial delay in the 78-m cable introduced in the circuit to shift the start of the distribution of delays comfortably away from the zero channel of the Multi-Channel Analyzer (DOWN MCA). The peaked distribution of amplitudes from straight-through events recorded by the MCA provides a convenient marker of the zero-delay position in the distribution of delays. The stop signal from a straight-through event arrives at the UP TAC before the start signal, so the TAC automatically resets after its timing limit is exceeded.

Traveling through the atmosphere at nearly the speed of light, singly charged particles like muons lose approximately 2 GeV and arrive at sea level with energies averaging about 4 GeV. Occasionally (about 5 times per minute) a muon near the end of its range causes a (S1+S2) January 12, 2001 8 47. Parity

Figure 1: Schematic diagram of the apparatus for investigating the polarization of cosmic-ray muons. coincidence signal and stops in the brass plate, having initiated a timing sequence in both the UP and DOWN TACS. If the stopped muon is negative, it is most probably absorbed by a nucleus of the brass plate before it decays. Only rarely does a negative decay electron emerge from the plate to trigger (P1+P2) or (S1+S2) and stop the timing sequence. If the muon is positive, it will decay into an electron neutrino and a muon anti-neutrino and an energetic positron (anti-electron) which may escape from the plate. If the muon decays after the timing sequence is started in the UP TAC, and the electron travels upward, a second (S1+S2) signal is generated which stops the timing sequence. If it travels downward, the resulting (P1+P2) signal will stop the timing sequence of the DOWN TAC. The amplitude of the TAC output signal is recorded by the MCA as one event in a channel whose number is proportional to the delay between the start and stop signals. The distribution of delay between the stopping and decay of muons is a function of their lifetime in brass. It can be represented by the sum of two exponentials, one with a mean life of ≈ 2µsec representing the decay of positive muons, and the other with a much shorter mean life representing the combined effects of nuclear absorption and decay of negative muons. With a magnetic field in the brass plate, the combined exponential decay curve is modulated by the precession of the muons which causes the asymmetric, parity-violating decay patterns of the positive and negative muons to rotate in synchronism about the horizontal direction of the field.

4.1 CALIBRATION

To get useful data from the setup it is essential that the scintillation detectors be operated at levels of sensitivity that insure eﬃcient detection of muons and their decay electrons while, at January 12, 2001 9 47. Parity

Figure 2: Schematic illustration of a calibration data curve of the (S1+S2) coincidence rate versus the counting rate of one of the detectors (S1). Typical operating points for S1 and S2 are about 300 cts/s. After proper adjustment, the coincidence rate should be about 14 cts/s. the same time, do not cause excessive rates of accidental coincidences between ”background” pulses that will dilute the precession-modulated muon decay data and obscure the desired eﬀect under a high level of random background events. (Background pulses are generated by Compton interactions of environmental gamma-rays in the scintillators and ”dark-current” pulses caused by thermionic emission of electrons from the PM cathodes.) Two controls are available for each of the three scintillation counters - the high voltage (HV) supplied to the PMs, and the discriminator settings of the constant fraction discriminators. The latter work well at discrimination levels of 0.1 volts which can be set with the dials and locked. The high voltages supplied to the PMs can then be adjusted to achieve the desired sensitivity of the detectors.

The desirable counting rates of S1 and S2 are the lowest values for which the efficiency for the detection of straight-through muons is nearly 100%. To find the proper operating condition you should measure the (S1+S2) coincidence counting rate as a function of the singles rates in the individual counters. The latter can be adjusted by adjustment of their respective high voltages. You will find that the coincidence rate increases rapidly as the singles rates rise to several thousands of counts per second, and then more slowly on a ”plateau” on which the efficiency for detection of ”straight-through” muons approaches 100% , as illustrated in Figure ??.

In a similar manner, adjust the P1 and P2 rates (P1 and P2 share a common high voltage so that their relative rates must be adjusted by changing the discriminator levels) to obtain eﬃcient detection of straight-through muons without an excessive accidental rate. With P1 and P2 rates of about 400 cts/s, the ﬁnal coincidence rate should be about 55 cts/s.

Calibrate the TAC/MCA combination with the time calibrator. To do this, connect the START pulse to the discriminator input for S1; connect the STOP pulse to the discriminator input for P1. Set the period of the time calibrator to 1.28 µsec. Switch the coincidence circuits to require only single pulses from the S1 and P1 discriminators. Record the timing pulses on 1/8 of the MCA display (256 channels). With the MCA cursor, determine the January 12, 2001 10 47. Parity

number of channels for 10*1.28 µsec. compute the time interval per channel.

Note that in the data archived prior to July 2000, the time per channel was adjusted (via MCA gain) to be 0.0465 µsec/channel. The new MCA card does not have adjustable gain and the time per channel is now 0.04 µsec/channel. This requires that the two data sets (before and after July of 2000) be analyzed separately, or combined appropriately.

This procedure will assure that your data are compatible with those in the archive, and can be added to all the other data without adjustment for diﬀerent timing calibrations.

Coordinate with any other teams that are doing the experiment. Start a run with data recorded in a fresh MCA display. After 10 or more minutes check that ”straight-through” events are accumulating at a high rate in the zero delay channel of the DOWN MCA, and muon-decay counts are accumulating near the left end of the display at a rate of about 1/minute.

The current, which determines the ﬁeld intensity, should be maintained at a constant value of 10.0 amperes.

5 DATA ANALYSIS

In preparation for the data analysis many sets of simulated data should be prepared with correct Poisson ﬂuctuations and realistic values of the expected number of decay events and background. For this you will need to use a subroutine to compute random Poisson variates (see Press et al. 1989) for each of the expectation numbers of events computed according to equation (1).

You will find that the zero-delay channel in the DOWN data has changed over the course of the experiment due to changes in the DOWN delay cable. To prepare the DOWN data for the real analysis you will have to combine the archive files, taking care to offset the channel numbers as needed so the zero-delay channel of the combined data is number 1. All the UP data have been taken with the 28-m delay cable ( 0.06 µsec) in the start line.

The following is a possible approach to the analysis of the data and the search for a significant modulation: A function consisting of the sum of two exponentials plus a constant background is fitted by least squares to the raw data by adjustment of the two amplitudes, the two mean lives of the component exponentials representing the mean decay curves of the positive and negative muons, and the background rate. The ratios of the raw data to the fitted curve are then calculated. The reduced data, representing the modulation of the decay curve about the fitted smooth exponential curve, is then subjected to Fourier analysis, and the resulting power density spectrum is examined for evidence of a periodicity corresponding to the Larmor precession of the muons.

Another approach is to fit equation (1) to the raw data by the method of least squares. The best strategy in the use of a curve-fitting program is first to fix the polarization and frequency parameters to zero, and to fit only the four parameters of the exponentials and January 12, 2001 11 47. Parity the background rate. Then the frequency can be given an initial value close to the expected value, and a new fit computed with all seven parameters free to vary.

The Levenberg-Marquardt method of non-linear least-squares curve fitting and evaluation of errors in the fitted parameters is described in Bevington and Robinson (1992) and also in Press et al. (1989). A straight forward approach to evaluating the statistical significance of an apparent detection of a modulation is to generate many statistically equivalent sets of pseudo data with no modulation and appropriate random Poisson errors, subject the data sets to the same analysis as the real set, and then to determine how frequently you obtain by chance an apparent result of magnitude equal to or greater than what you actually did obtain.

Ideal results of this experiment consist of determinations of the mean lives of positive and negative muons in brass, statistically convincing evidence of modulation indicative of parity violation, and a determination of the muon magnetic moment and gyromagnetic factor.