... E. Klempt, K. Merle, K. Neubecker, C.Sabev, H. Schwenk, V.H. Walter, R.D. .... 74] Jungmann, P.E.G. Baird, J.R.M. Barr, C. Bressler, P.F. Curley, R. Dixson, ...
Aspects of Fundamental Muon Physics Klaus P. Jungmann Physikalisches Institut, Universitat Heidelberg Philosophenweg 12, D-69120 Heidelberg, Germany Abstract
The muon is a second generation fundamental fermion which belongs together with the electron and tauon to the group of charged leptons. To our present knowledge it behaves as a point-like particle which is subject to electromagnetic, weak and gravitation interactions. It di ers from electron and tauon in its behaviour and properties solely through its mass which is called lepton universality. The muon itself and the muonium atom, its bound state with an electron, have been widely used to investigate fundamental laws and symmetries in physics. Since all their known interactions can be calculated very accurately, they have been employed also to determine very accurate values of fundamental constants. New results on the muon magnetic anomaly, the muonium hyper ne structure and a muonium 1s-2s laser excitation are discussed in detail as well as a new accurate determination of the ne structure constant.
1 Introduction In particle physics all unambiguous experimental observations to date can be described by a theoretical framework known as the "standard model". In this theory matter in nature is composed of fundamental point-like fermions which are called quarks - up (u), down (d), strange (s), charm (c), top (t) , bottom (b) - and leptons - electron (e), electron neutrino (e), muon (), muon neutrino ( ), tauon ( ), tauon neutrino ( ) and their respective antiparticles (Fig. 1). Two quarks carrying either -1/3 or 2/3 electric charge units and a charged lepton carrying one electric charge unit and a corresponding electrically neutral neutrino form one generation. Three particle generations are known to date. All ordinary matter consists of rst generation fundamental fermions (Fig. 2). Three di erent interactions - gravitation, electromagnetic and weak - are experienced by all these particles. A fourth fundamental force - the strong interaction - a ects only the quarks (Fig. 3). The basic interactions between the fundamental fermions are mediated by gauge bosons - the graviton, the photon, the W- and Z-bosons and eight gluons (Fig. 4). A major step forward in the recognition of the basic laws in physics was the successful description of electricity and magnetism in a single theory by Maxwell in the last century. A second step followed in the last thirty years by unifying electromagnetic and weak interaction in the electroweak standard model by Glashow, Salam and Weinberg. At present there are various attempts to continue the search for a common description of fundamental forces in one single theory. Such e orts are known, e.g. as grand uni cation theories (GUT's). It should, however, be noted that besides the great successful description of all particle physics, the standard model leaves many open questions about the physical nature of observed processes and the parameters describing particles and interactions. For example, interaction strengths and particle masses as well as the origin for parity violation in weak interaction are put 'by hand' into the theory. Accurate experiments can stringently tests the theoretical description of processes in physics and explore the limits of the physical picture. The standard model of the uni ed electroweak interaction has been con rmed with an impressive precision in high energy electron-positron collider experiments in the LEP storage ring at CERN [1]. On the other end of the energy scale Scottish Summer School on Muon Pysics, SUSSP51, St. Andrews, 1998
1
FERMIONS
Spin=1/2, 3/2, 5/2, ...
Leptons, Spin=1/2 Mass [GeV/c -2 ]
Flavour νe
electron neutrino
e
electron
νµ
muon neutrino
µ
muon
ντ
tau neutrino
τ
tauon
Quarks, Spin=1/2 Electric charge [e]
Flavour
approximate Mass [GeV/c-2 ]
Electric charge [e]
< 7 x 10-6
0
u
up
0.005
2/3
0.511
-1
d
down
0.01
-1/3
< 0.2
0
c
charm
1.5
2/3
106
-1
s
strange
0.2
-1/3
0
t
top
-1
b
bottom
< 30 1777.1
2/3
173 4.7
-1/3
Figure 1: Fundamental fermions of the standard model. Examples of bosonic hadrons
Examples of fermionic hadrons --Baryons qqq and antibaryons qqq
Mesons qq Symbol
Name
π+
Pion
K-
Kaon
ρ+
Rho
D+
D+
ηc
Eta-c
Quark content ud su ud cd cc
Electric charge
Mass [GeV/c -2 ]
Spin
Symbol
+1
0.140
0
-1
0.494
0
p p
+1
0.770
1
+1
1.869
0
0
2.979
0
Quark content
Electric charge
Mass [GeV/c -2 ]
Spin
+1
0.938
1/2
Antiproton
uud --uud
-1
0.938
1/2
n
Neutron
udd
0
0.940
1/2
Λ
Lambda
uds
0
1.116
1/2
Ω
Omega
sss
-1
1.672
3/2
Name Proton
Figure 2: Some hadrons. electroweak theory can be confronted with the material furnished by atomic spectra. Information can be extracted which is complementary to the results from high energy physics [2]. It is, for example, a remarkable feature of nature that the weak mixing angle sin2 W appears to be constant over 10 orders in the square of the four-momentum transfer Q2, from the experiments at LEP to the measurements of parity violation in atomic Cesium. Although the e ects are small in atomic systems, they can be carefully surveyed with highly precise spectroscopy thanks to modern microwave and laser technology. In general, crucial and fundamental tests require systems where the contribution from well known interactions, particularly the electromagnetic, are well understood. Here single particles like electron, muon and their neutrinos o er excellent opportunities. Among the important experiments are measurements of the parameters like , e.g. masses and magnetic moments, as well as investigations of decay modes. Next to single particles, one-electron atoms like hydrogen (pe? ) and muonium (+ e? ) are the simplest systems available (Fig. 5). In 1932 when a muon was rst observed in the cloud chamber of P. Kunze in Rostock, Germany, [3], the discovery remained rather unspectacular, because the standard model of the time had only protons, electrons and photons. The new particle was unforeseen. Years later, after Yukawa had predicted a particle exchange process for the nuclear force, the rediscovery by Anderson and Neddermeyer became more famous, although the muon's identity was mistaken for the particle 2
Properties of Interactions Interaction
Weak Electromagnetic (Electroweak)
Strong Fundamental Rest
mass - energy
flavour
electric charge
colour charge
strong "rest" interaction
particles affected
all
quarks, leptons
electrically charged
quarks, gluons
hadrons
interaction mediating particle
graviton
γ
gluons
mesons
Gravitation
Property acts on
strength for 2 quarks at: (relative to electromagn.)
W ,W ,Z
(not yet observed) -18
10 {3x10
-41
10 -41 10
m
-17
m
10
for 2 protons in nucleus
-
+
0
0.8 -4 10
-36
10
-7
1 1
25 60
-
1
-
20
Figure 3: Properties of fundamental interactions.
BOSONS Electroweak Spin = 1 γ
Photon
WW+ Z
0
Mass [GeV/c -2 ]
Electric charge [e]
0
0
80.22
-1
80.22
+1
91.187
0
Spin = 0, 1, 2, ... Strong or Colour Spin = 1 g
gluon
Mass [GeV/c-2 ]
Electric charge [e]
0
0
Figure 4: Fundamental (gauge) bosons of the standard model. we now know to be the pion (). Since then, the muon with its accurately known properties, its decays and its bound states were of crucial importance for model building in basic physics as well as they served as tools in applied sciences. Atomic hydrogen has played an important role in the history of modern physics. The successful description of the spectral lines of the hydrogen atom by the Schrodinger equation [4] and especially by the Dirac equation [5] had a large impact on the development of quantum mechanics. Precision measurements of the ground state hyper ne structure splitting by Nafe, Nelson and Rabi [6, 7] in the late 1940's were important contributions to the identi cation of the magnetic anomaly of the electron. Together with the observation of the \classical" 22S1=2 ? 22P1=2 Lamb shift by Lamb and Retherford [8] they pushed the development of the modern theory of Quantum Electrodynamics(QED). Today the anomalous magnetic moment of the electron, which is de ned as ae = (ge ? 2)=2, H of where ge is the electron g-factor, and the ground state hyper ne structure splitting HFS atomic hydrogen are among the most well known quantities in physics. Experiments in Penning traps with single electrons [9] have determined ae=1 159 652188.4(4.3)10?12 to 3.7 ppb. QED calculations have reached such a precision that the ne structure constant can be extracted to H = 1 420:405 751 766 7(9) 3.8 ppb [10]. Hydrogen hyper ne structure measurements yield HFS MHz (0.006 ppb) [11, 12] and hydrogen masers even have a large potential as frequency standards. However, the theoretical description of the hyper ne splitting is limited to the ppm level by the fact that the proton's internal structure is not known well enough for calculations of nearly similar accuracy [16, 17]. Neither can experiments supply the necessary data on the mean square charge 3
2
2 P3/2 74 MHz 2
F=2 F=1
9875 MHz
2 S1/2
F=1 558 MHz
1047 MHz
F=0
F=1 2
2 P1/2
λ = 244 nm
187 MHz F=0
∆ν1s2s = 2455 THz
λ = 244 nm
2
1 S1/2
Figure 5: The energy levels of muonium for principal quantum numbers n=1 and n=2. The level scheme is analogous to atomic hydrogen. The energy scale for gross structure is somewhat smaller due to reduced mass. The hyper ne larger because of the larger muon magnetic moment F =splitting 1 gross,is ne compared to the proton. The indicated and hyper ne structure transitions were studied yet. The most accurate measurements are the ones involving the n=1 ground state in which the = 4463 MHz atoms can be produced HFS eciently.
∆ν
F=0
radius and the polarizability, for example from electron scattering, nor is any theory, for example low energy quantum chromodynamics (QCD), in a position to yield the protons internal charge structure and the dynamical behavior of its charge carrying constituents. The situation is similar for the 2S1=2 - 2P1=2 Lamb shift in hydrogen, where atomic interferometer experiments nd 2HS1=2 ?2P1=2 = 1 057.8514(19) MHz (1.8 ppm) [13, 14, 15]. The knowledge of the proton's mean square charge radius limits any calculations to the 10 ppm level of precision [16, 17, 18, 19]. The 1s-2s level separation in atomic hydrogen has reached 1s?2s;H = 2 466 061 413 187:34(84) kHz - a fascinating accuracy for optical spectroscopy - and one can expect even further signi cant improvements [20, 21]. The Rydberg constant has been extracted by comparing this transition with others in hydrogen and amounts to R1 = 10 973 731:568 639(91)cm?1. It is todays best known fundamental constant. However, the knowledge of the mean square charge radius limits comparison between experiment and theory again at the 40 kHz level. In fact, in the case of deuterium the measurements have been employed to infer deuteron parameters [22]. In addition to the information obtainable from the natural isotopes hydrogen (pe? ), deuterium ? (de ) and tritium (te? ) or hydrogen-like ions of natural elements ((ZA Xe? )(Z ?1)+), exotic hydrogen-
like atoms can provide further information about the interactions between the bound particles and the nature of these objects themselves [23]. Such systems can be formed by replacing the electron in a natural atom by an exotic particle (e.g. ? ; ?; K ? ; p), or by electron capture of an exotic 4
Table 1: The ground state hyper ne structure splitting HFS and the 1s-2s level separation 1S ?2S of hydrogen and some exotic hydrogen-like systems o er narrow transitions for studying the interactions in Coulomb bound two-body systems. In the exotic systems the linewidth has a fundamental lower limit given by the nite lifetime of the systems, because of annihilation, as in the case of positronium, or because of weak muon or pion decay. The very high quality factors (transition frequency divided by the natural linewidth = ) in hydrogen and other systems with hadronic nuclei can hardly be utilized to test the theory because of the insuciently known charge distribution and dynamics of the charge carrying constituents within the hadrons. Positronium Muonium e+ e? + e?
Hydrogen pe?
Muonic Helium4 ( ? )e?
Pionium + e?
Muonic Hydrogen p?
2466.1
2468.5
2458.6
4.59105
1S?2S
1233.6y
2455.6
1S ?2S
1.28y
.145
1.310?6
.145
1S?2S 1S?2S
9.5108
1.71010
1.91015
1.71010
HFS
203.4
4.463
1.420
4.466
--
HFS
1200
.145
4.510?22
.145
--
.145
HF S HF S
1.7102
3.1104
3.21024
3.1104
--
3.1108
[THz]
[MHz]
[GHz]
[MHz]
12.2 2.0108
.176 2.61012 4.42107
y Only the 1S-2S splitting in the triplet system of positronium is considered in this table.
"nucleus" (e.g. e+ ; + ; + ). Even atoms consisting of two exotic particles have been produced occasionally, e.g. + ? and ? + were found in in- ight decays of neutral kaons KL0 [24]. Some of the systems are compared in Table 1 with respect to the possible spectroscopic resolution for the 1S-2S and the ground state hyper ne structure transition. Muonic atoms ((? AZ X)) and ions ((? AZ X)n+ ) are of particular interest for spectroscopy, since the Bohr radius a is about 207 times smaller for muons than for electrons, a = (me =m )a0 , where me and m are the masses of the electron and the muon and a0 = h2 =(me e2 ) = 0:529 10?10 m, with the electric charge unit e and Planck's constant h. Bound muonic states are therefore more sensitive to the properties of the nuclei. This has been widely applied for determinations of nuclear charge moments and for examining nuclear polarization [25, 26, 27]. The muon orbit for higher nuclear charges Z is signi cantly smaller than the electron Compton wavelength C = h=(me c) = 3:86 10?13 m, which is the typical spatial dimension of vacuum uctuations. In contrast to electronic systems, the vacuum polarization contributions in muonic atoms are substantially larger than the self energy, since the Uehling potential, which describes to lowest order the modi cation of the nuclear potential due to vacuum uctuations, scales approximately with the cube of the particle mass and the self energy is inversely proportional to it. Higher order vacuum polarization contributions are needed for a satisfactory description. There is special interest in muonic hydrogen (p? ), since, in principle, one could obtain from a measurement of its ground state hyper ne 5
splitting and the 2S-2P Lamb shift more detailed information on the proton's charge radius and polarizability [28, 29, 23, 30, 31, 32]. The electroweak standard model can be critically tested by a measurement of the four coupling constants describing the parity odd neutral current interaction in the two isotopes of muonic boron [2, 33, 34]. The existence of additional neutral vector bosons and leptoquarks and composite quark contact interaction could be surveyed with a sensitivity beyond the possibilities of LEP. The 2S and 2P states are mixed by the weak interaction. An observable for a future measurement could consist of directional correlation between the muon decay electron and a single photon emitted from the 2S-1S transition. The muonic helium atom (( ? )+ e? ) [35, 36, 37] consists of a pseudo-nucleus ( ? )+ , which is itself a hydrogen-like system, and an electron. It is the simplest atomic system involving both a negative muon and an electron. A precise value for the magnetic moment of the negative muon has been deduced from the Zeeman splitting of the ground state hyper ne structure as a test of the CPT theorem, which is the basic assumption that all physics remains the same, if the charges, the parity and the time direction (described by their operators C,P and T) involved in a process under consideration is reversed in its descriptions. Already in the 1970's the muonic helium ion ( ?)+ was the rst exotic atomic system for which a successful laser experiment has been reported [38, 39]. Allowed electric dipole transitions between the n=2 ne structure levels have been induced. A precise test of QED vacuum polarization could be established and the rms charge radius of the -particle was extracted. The experiment needed as a prerequisite the metastability of the 2S state at high pressures (40 bar), which could not be con rmed in several independent approaches [40, 41, 42] and a second attempt of a laser experiment could not con rm the existence of a signal [43], which leaves us with an open puzzle. In purely hadronic systems like pionic hydrogen (p? ) [44] and antiprotonic hydrogen [45], called protonium (pp) [46, 47], the transition frequencies are dominantly due to Coulomb interaction. They bear additional line shifts and broadenings in the spectral lines due to strong interaction between the constituents. They o er the possibility to study the strong interaction between the particles at zero energy. From the transition frequencies in pionic atoms (or possibly in far future from the pionium atom (+ e? ) [48]) one can derive a value for the pion mass [49]. At present the determination of the best upper limit for the muon neutrino mass from the muon momentum from pion decays at rest [50, 51, 52] is spoiled by the uncertainties of the pion mass. For leptons no internal structure is known so far. Scattering experiments have established that electron (e), muon() and tauon ( ) behave like point-like particles down to dimensions of less than 10?18m [53, 54] which is three orders of magnitude below the proton's rms charge radius [55, 56, 57]. Purely leptonic hydrogen-like systems like positronium (e+ e? ) [58, 59] and muonium (+ e? ) [62, 63, 64] have interesting perspectives for testing bound state QED, for searching for deviations from present models and for testing fundamental symmetry laws. Positronium is a particle anti-particle system and signi cantly di ers from muonium and hydrogen-like systems with di erent constituent masses. The Furry picture, where the electronic states are in zeroth order solutions of the Dirac equation in an external electrostatic eld and which is successfully applied for the heavier systems, is not appropriate. A theoretical description must start from the fully relativistic Bethe-Salpeter formalism or an approximation to it [65]. Depending on the C-parity of the state, the ground state of the positronium atom annihilates into two (11 S0 ) or three (13 S0 ) photons. Standard theory was con rmed in various searches for rare decay modes which have been carried out [59] in order to nd unknown light particles, e.g. axions, or violations of fundamental laws, e.g. C-parity symmetry. For the 3 S-states annihilation into a single virtual photon causes signi cant shifts at the ne structure level. Microwave spectroscopy experiments on ne structure transitions in the n=2 state of the triplet system (parallel e? and e+ spin) are in moderate agreement with each other and with theoretical calculations [60]. The ground state hyper ne splitting presents a theoretical challenge and would call for more precise measurements [61]. 6
Positronium was the second exotic system in which laser excitation could be achieved [66]. The 1S-2S transition frequency has been measured to be 1PS S ?2S (exp) = 1233 607216.4(3.2) MHz (2.6 ppb)[67], which agrees with theory 1PS (theory) = 1 233607221.0(1.0) MHz (0.8 ppb)[60], S ?2S within two standard deviations. This constitutes a test of the QED contributions to 10 ppm. The theoretical uncertainty of 1 MHz is the estimated size of uncalculated higher order (higher than 4R1 ) terms [68]. Their evaluation will be cumbersome, because of the in ation of the number of Feynman diagrams due to virtual annihilation. From the result one can conclude that electron and positron masses are equal to 2 ppb, the best test of mass equality for particle and antiparticle next to the K 0 K 0 system [86], where the relative mass di erence is 10?18. Precision experiments on muonium o er a unique opportunity to investigate bound state QED without complications arising from nuclear structure and to test the behavior of the muon as a heavy leptonic particle and hence the electron-muon(-tau) universality, which is fundamentally assumed in QED theory. Of particular interest is the ground state hyper ne structure splitting, where experiment [69] and theory [16, 17] agree at 300 ppb which is at a higher level of precision than in the case of atomic hydrogen, where the limitation in theory arises from the knowledge of the muon mass. An accurate value for the ne structure constant can be extracted to 140 ppb. The Zeeman e ect of the ground state hyper ne sublevels yields the most precise value for muon the magnetic moment with an accuracy of 360 ppb. Preliminary results from a new experiment will be discussed in sect. 5.1.1 which should improve these gures by about a factor of three. Signals from the "classical" 22 S1=2-22P1=2 Lamb shift in muonium have been observed [70, 71]. However, with 1.4% precision they are not yet in a region where they can be confronted with theory. The sensitivity to QED corrections is highest for the 1S-2S interval in muonium due to the approximate 1=n3 scaling of the Lamb shift. Compared to hydrogen, the radiative recoil and the relativistic recoil e ects are larger by a factor of mp =m 8.9, where mp is the proton mass and m the muon mass. A hydrogen-muonium isotope shift measurement in this transition as well as a technically only slightly more dicult measurement of the 1s-2s transition frequency in muonium can lead to a new and accurate gure for the muon mass m . The transition has recently been excited successfully in two independent experiments [72, 73, 74, 75]. In a series of three (g-2) experiments at CERN the magnetic anomaly a of the free muon itself has been carefully studied [76, 77] by observing the di erence (!a )of the spin precession frequency and the cyclotron frequency in the homogeneous magnetic eld of a storage ring. The relative contributions from heavier virtual particles to the magnetic anomaly are larger for the muon by a factor of (m =me )2 ) 4:2 104 compared to the electron. This makes the muon a better probe for new physics. A new muon storage ring experiment at BNL, Brookhaven, USA, aims for a precision of 0.35 ppm for a [78]. The production of virtual particles can be explored into the TeV/c2 mass range and tighter constraints for new physics beyond the standard model [54] will be set. The sensitivity is similar to that of experiments at the present and future high energy machines (LEP 200, Fermilab pp. LHC) and partly beyond the scope of those. The experiment will test the renormalizability of the weak interaction in a cleaner way than from the masses of Z- and W-bosons, where the e ect is clouded by radiative corrections involving top quarks and Higgs bosons, or from nuclear -decays, where complications arise from nuclear structure. In order to reach the desired goal, the muon mass must be known to at least 0.1 ppm. At present, muonium spectroscopy is the only viable way to obtain such a precision. The spectroscopic experiments in muonium are closely inter-related with the determination of the muon's magnetic anomaly a through the relation = (1 + a ) eh =(2m c). The results from all experiments establish a self consistency requirement for QED and electroweak theory and the set of fundamental constants involved (Fig. 6) The constants ; m ; are the most stringently tested important parameters. The only necessary external input are the hadronic corrections to a which can be obtained from a measurement of the ratio of cross sections (e+ e? ! + ? )=(e+ e? ! hadrons). Although, in principle, the system could provide the relevant electroweak constants, the Fermi coupling constant GF and sin2 W , the use of more accurate values from independent measurements may be chosen for higher sensitivity to new physics. As a 7
Figure 6: Relation between measurements of the muon and electron anomalous magnetic moments and spectroscopy of the energy levels in muonium. Each experiment needs precise results obtained in the other ones. The whole set is a stringent consistency test for theory and the set of fundamental constants involved. The only external input required is the hadronic correction to a . matter of fact, an improvement upon the present knowledge of the muon mass at the 0.35 ppm level is very important for the success of a new measurement of the muon magnetic anomaly presently under way at the Brookhaven National Laboratory. The relevance of the experiment, which aims for 0.35 ppm accuracy, arises from its sensitivity to contributions from physics beyond the standard model and the clean test it promises for the renormalizability of electroweak interaction [76, 77, 78]. The decay of the muon itself is an interesting process to study. From precision measurements of its lifetime one can extract the best value for the important Fermi coupling constant describing the coupling strength in weak interaction. Also the muon analogon to inverse -decay, the muon capture process can reveal insights in fundamental symmetries, for example in the possiblity of an asymmetry in nature against reversing the time direction in the theoretical description of a process [79]. Rare decays of muons and decays forbidden by lepton number conservation have been studied very carefully since the late 1940's in order to nd hints on the muon's nature. Today there are various attempts to identify in forbidden muon decays physics beyond the standard model. Of interest in this connection are searches for the decay mode ! e and for -e conversion in a muon-nucleon reaction. Particularly the muonium atom, which consists of two leptons from two di erent lepton generations, o ers important possibilities. Since in the bound state the particles are close to each other for relatively long interaction times compared to scattering experiments, one can expect the system to be very sensitive to exotic interactions between electron and muon. A possible muonium to antimuonium conversion has rst been considered by Pontecorvo (in 1958) in analogy to the K0 K0 oscillations [80]. The process violates the separate additive conservation of lepton avour numbers. Although not provided in the standard model, it would be allowed in many extensions to it like left-right symmetry, supersymmetry and technicolor [81]. 8
2 Magnetic anomaly of the muon Trapping of elementary particles in combined magnetic and electric elds has been very successfully applied for obtaining properties of the respective species and for determining most accurate values of fundamental constants. The magnetic anomaly of fermions a = 21 (g ? 2) describes the deviation of their magnetic g-factor from the value 2 predicted in the Dirac theory. It could be determined for electrons and positrons in Penning traps by Dehmelt and his coworkers to 3.7 ppb [9]. Accurate calculations involving almost exclusively the "pure" Quantum Electrodynamics (QED) of electron, positron and photon elds allow the most precise determination of the ne structure constant [10] by comparing experiment and theory in which appears as an expansion coecient. The high accuracy to which calculations in the framework of QED can be performed is demonstrated by the satisfactory agreement between this value of and the ones obtained in measurements based on the quantum Hall e ect [82] as well as the ac-Josephson e ect and the gyromagnetic ratio of protons in water [83], or the number extracted from the very precisely known Rydberg constant [21] using an accurate determination of the neutron de Broglie wavelength [84, 85] and well known relevant mass ratios [86]. 1
2
3
e+ e− l
l
µ+ µ− l
l
l
l
Figure 7: QED Feynman diagrams of lowest order contributing to muon g-2. They are very similar to the ones responsible for the lowest order Lamb shift in atomic hydrogen. (a)
(b)
(c)
W ν µ
φ
Z0 µ
µ
µ
µ
µ
Figure 8: Lowest order weak processes contributing to muon g-2. The anomalous magnetic moment of the muon a has a (m =me )2 4 104 times higher sensitivity to heavier particles and other than electromagnetic interactions. Those can be investigated carefully, as very high con dence in the validity of calculations of the dominating QED contribution arises from the success of QED describing this quantity for the electron. For the muon a has been measured in a series of three experiments at CERN [77] all using magnetic muon storage. Contributions arising from strong interaction amount to 60 ppm and could be identi ed in the last of these measurements. At the Brookhaven National Laboratory thr new dedicated experiment has started measurements to determine the muon's magnetic anomaly. It aims for 0.35 ppm relative accuracy meaning a 20 fold improvement over previous results. At this level it will be particularly sensitive to contri9
1
2
3
H l
111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111
11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11
0000 1111 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111
e l
H l
l
l
l
Figure 9: Strong interaction diagrams contributing to muon g-2. Whereas the processes 1 and 2 can be obtained from measurements of the cros section e+ + e? ! any hadrons as a function of center of mass energy, process 3 needs to be calculated and causes right now one of the largest uncertainties of the hadronic contribution to muon g-2. Table 2: Sensitivity to new physics of the g-2 experiment at BNL aiming for 0.35 ppm relative accuracy. new physics sensitivity other experiments
a
Muon substructure excited muon W -boson substructure W anomalous magnetic moment Supersymmetry right handed WR -bosons heavy Higgs boson Muon electric dipole moment for substructure a m2 =2
m aW
5 TeV 400 GeV 400 GeV 0.02
mWe mW 0 mH D
130 GeV 250 GeV 500 GeV 4 10?20ecm
LHC similar LEP II similar LEP II 100-200 GeV LEP II 0.05, LHC 0.2 Fermilab pp similar Fermilab pp similar
butions arising from weak interaction through loop diagrams involving W and Z bosons (1.3 ppm). The experiment promises further a clean test of renormalization in weak interaction. The muon magnetic anomaly may also contain contributions from new physics [89, 90, 91]. A variety of speculative theories can be tested which try to extend the present Standard Model in order to explain some of its not yet understood features. This includes muon substructure, new gauge bosons, supersymmetry, an anomalous magnetic moment of the W boson and leptoquarks. Here this measurement is complementary to searches carried out in the framework of other high energy experiments. In some cases the sensitivity is even higher (Table 1). In the BNL experiment [88] polarized muons are stored in a magnetic storage ring of highly homogeneous eld B and with weak electrostatic focussing using quadrupole electrodes around the storage volume. The di erence frequency of the spin precession and the cyclotron frequencies, (1) !a = a me c B; is measured, with m the muon mass and c the speed of light, by observing electrons/positrons from the weak decay ! e + 2. For relativistic muons the in uence of a static electric eld 2 ? 1) which corresponds to = 29:3 and a muon momentum of 3.094 vanishes [92], if a = 1=( p GeV/c, where = 1= 1 ? (v =c)2 and v is the muon velocity. The momentum needs to be accurate at the 10?4 level for a corresponding correction to be below the desired accuracy for a . For a homogeneous eld the magnet must have iron ux return and shielding. To meet this, the particular momentum requirement and to avoid magnetic saturation of the iron a device of 7 m 10
Figure 10: The observed muon spin precession signal in the new muon g-2 experiment at Brookhaven. radius was built. It has a C-shaped iron yoke cross section with the open side facing towards the center of the ring. It provides 1.4513 T eld in a 18 cm gap. The magnet is energized by 4 superconducting coils carrying 5177 A current. The storage volume inside of a Al vacuum tank has 9 cm diameter. The magnetic eld is measured by a newly developed narrow band magnetometer system which is based on pulsed nuclear magnetic resonance (NMR) of protons in water. It has the capability to measure the eld absolute to 50 ppb [93]. The eld and its homogeneity are continuously monitored by 366 NMR probes which are embedded in the Al vacuum tank and distributed around the ring. Inside the storage volume a trolley carrying 17 NMR probes arranged to measure the dipole eld and several important multipole components as well a fully computerized magnetometer built from all nonferromagnetic components is used to map the eld at regular intervals. The accuracy is derived from and related to a precision measurement of the proton gyromagnetic ratio in a spherical water sample [94]. The eld homogeneity at present is about 25 ppm. It will be improved to the ppm level using mechanical shimming methods and a set of electrical shim coils. The eld integral in the storage region is known at present to better than 1 ppm in absolute terms at any time and eld drifts of a few ppm/hour were observed. Due to thermal insulation for the magnet yoke and active feedback from the readings of the xed NMR probes the eld path integral can be kept constant within 0.1 ppm. The weak focussing is provided by electrostatic quadrupole eld electrodes with 10 cm separation between opposite plates. They cover four 39 sections of the ring. The electric eld is applied by pulsing a voltage of 24:5 kV for a few ms duration to avoid trapping of electrons and electrical breakdown [95]. Due to parity violation in the weak muon decay process the positrons are emitted preferentially into the muon spin direction causing a time dependent variation of the spatial distribution of decay particles in the muon eigensystem which translates into a time dependent variation of the energy distribution observed by detectors xed inside the ring. 11
Results from muon g-2
CERN pos. muons CERN neg. muons CERN combined BNL pos. muons All combined
±0.35ppm
Theory
1165900
1165920
1165940
magnetic anomaly of the muon [10-9]
Figure 11: Results from the startup run of the new muon g-2 experiment. The improvements over previous experiments include an azimuthally symmetric iron construction for the magnet with superconducting coils, a larger gap and higher homogeneity of the eld, an electron/positron detector system covering a larger solid angle and using segmented detectors and improved electronics. A major advantage is the two orders of magnitude higher primary proton intensity available at the AGS Booster at BNL. A new key feature will be the direct injection of muons into the storage volume using an electromagnetic kicker as compared to lling the ring with decay muons from injected pions which has been employed so far. In order for the new muon g-2 experiment to reach its design accuracy besides the eld also the muon mass respectively its magnetic moment needs to be known to 0.1 ppm or better. An improvement beyond the present 0.36 ppm accuracy of this constant can be expected from both microwave spectroscopy of the muonium atom's (+ e? ) hyper ne structure and from laser spectroscopy of the muonium 1s-2s transition [23]. Provided the systematic uncertainties, which mainly arise from the knowledge of the convolution of the muon distribution in the storage volume and the magnetic eld there as well as from imperfections of the electronics and muon losses, can be controlled at the 10?7 level, only the statistical error remains of concern. It is given for the asymmetry frequency !a by p (2) !a = (2) !a !a A p(Ne ) where A is the observed decay asymmetry and Ne the recorded number of decays. Therefore besides the number of particles in particular a large decay asymmetry is desired. 12
The experiment achieved in its startup run 1997 a measurement of value for the muon magnetic anomaly to 13 ppm. It is in agreement with previous CERN results and with theory. The near future will show whether there is new physics to be expected, when the statistical uncertainty can be signi cantly lowered. The main means to achieve this, the electromagnetic magnetic kicking device, could be shown in summer 1998 to allow muon injection yields of more than one order of magnitude larger than with pion injection. According to the standard theory an elementary particle is not allowed to have a nite permanent electric dipole moment (edm) as this would violate time reversal symmetry. An edm of the muon would manifest itself in the Brookhaven g-2 experiment in a time dependent up down asymmetry of decay positrons which can be searched for along with the muon g-2 measurements [77]. The new g-2 experiment will be sensitive to yield here a 10 fold improvement over the present limit at 3:7(3:4) 10?19 e cm. A very promising approach seems to be the proposed much more sensitive search for a muon edm in a dedicated succeeding experiment [96]. Major modi cations of the present g-2 magnet setup would be required which involve the application of a radial electric eld to compensate g-2 precession and the switching from electrostatic to alternating gradient focussing by replacing the pole tips of the magnet with appropriately shaped pieces of iron. In case of a nite electric dipole moment a time dependent asymmetry in the muon decay rates counted above and below the storage region is expected as a signature. Such an experiment may achieve up to four orders of magnitude improvement and could reach a level of sensitivity at which several theoretical models, particularly such involving supersymmetry, could be tested. For some models the sensitivity to new physics would be higher than for edm searches of the electron or the neutron, particularly if the e ect would scale nonlinear with the mass of the leptons.
3 Muon Decays
3.1 Allowed Muon Decays
Muons can originate from pion () decays. This process is dominant in the production of cosmic ray muons after pion production through the bombardment of nuclei in the gas molecules of the atmosphere by energetic protons of cosmic origin. Cosmic muon uxes at sea level are about 180 particles/(m2 s). Pion creation in nuclear reactions with proton beams is also employed in todays meson factories - laboratories which muon uxes of up to a few times 106/sec can be obtained. The positive (negative) muon itself decays mostly into a (positron) electron and two neutrinos. + ! + +
+ ! e+ + e +
;
? ! ? + ; ? ! e? + e + : These equations re ect the fact that the negative muon is a particle, the positive muon an antiparticle as antineutrinos are indicted by overlining. Both pion and muon decays are due to the weak interaction and violate parity-conservation [97, 98]. The non-conservation of parity in pion decay causes the spin of the positive (negative) decay muon to be directed against (in) its propagation direction, because neutrinos only have negative helicity. In a positive (negative) muon decay the positrons (electrons) are preferentially emitted in (against) the direction of the muon spin. This e ect is widely used experimentally to determine the average spin of a muon ensemble after their decay. The maximum kinetic energy of the decay positrons or electrons is Emax = 52:3 MeV corresponding to half the muon mass and the assumption of negligible muon neutrino mass. The probability with which the positron is emitted in a certain direction within a certain energy range can be calculated by integrating over the neutrinos' momenta, because they usually cannot be observed. The double di erential distribution W(; ) of the decay positrons with respect to the spin of the muon is given by d2 W = W(; ) d d cos 13
Figure 12: Feynman diagram describing the weak process of muon decay. G2m5 (2 ? 1) 2 (3) = 1923 (3 ? 2) 1 + 3 ? 2 cos d d cos ; where = E=Emax and is the angle between the muon spin and the momentum of the emerging positron. The asymmetry in the angular distribution is a consequence of parity violation in the weak interaction. The energy spectrum of the positrons or electrons follows from eq. (3) by only integrating over cos dW() = W()d = 2(3 ? 2)2= d ; (4) with the muon lifetime G 2 m5 = 1923 ; (5) the experimental value of which is = 2.19703(4)10?6 s [86]. The asymmetry factor preceding the cosine function in eq. (4) depends on the positron energy and for an ensemble of muons with polarization P the asymmetry is (6) a() = 32??21 P : For decay positrons in the same plane as the muon spin gives dNe+ (1 + a cos ) (7) d and for the decay electrons in the ? decay dNe? (1 ? a cos ): (8) d -integrated over all energies a = 1=3. For the highest energies a = 1 (see gure 14). The spectral shape of the muon decay energy spectrum re ects the V-A nature of the decay process. Using a parametrization of Michel other interaction types can be excluded in precision experiments determining the so called Michel parameters [99, 86]. A 1.4(0.4)% fraction of all muons decays with the additional release of a photon according to + ! e+ + e + + : Due to a radiative correction process involving an intermediate photon which vanishes into a electron positron pair, there is a rare decay mode + ! e+ + e + + e+ + e? ; which has a 3:4(0:4) 10?5 branching ratio [86]. Throgh eq.(5) a strong relation between , m and GF is established. Since radiative corrections due to QED e ects to this equation are known well, a precise measurement can yield an accurate value of the GF , which is one of the most fundamental constants in the electroweak model. Improvements are desired, particularly since other important weak constants like the Z boson mass are precisely known from experiments at LEP, for example. There are at present three e orts at BNL, RAL and PSI to improve to the 1 ppm level [100, 101, 102].
14
(a)
(b) sνµ νµ
π+
µ
+
e
spin - direction sµ
sµ
θ
+
µ+
π+-decay
µ +-decay
νµ νe
Figure 13: a) pion decay in the pion rest frame, the spin directions ~s are indicated. b) muon decay in its rest frame.
(a)
(b) Sµ
Sµ
a = 1/3
θ
a=1
Figure 14: Muon decay. The rate of the decay positrons is given in a polar diagram with an angle with respect to the muon spin. (a) Averaged over all positron energies the asymmetry parameter a is 1=3. For negative muons a is ?1=3 and the electron is emitted predominantly against the direction of the muon spin. (b) For the highest energy positrons a equals 1.
3.2 Lepton Number Violating Muon decays
To date experiments indicate the conservation of lepton quantum numbers. This fact can be described in several di erent solely empirical laws [103, 104, 105, 106, 107, 108], some of which follow additive and some obey multiplicative, parity-like, schemes. Experiments have given no indication yet for favouring any of them. The standard model states for every lepton avour a separate additively conserved quantum number. However, such lepton numbers have no status, unless their conservation can be associated with a local gauge invariance [109]. Mixings between di erent generations are well known in the quark sector and the Cabbibo-Kobayashi-Maskawa matrix [110] relates the weak quark eigenstates with their mass eigenstates. A familiar example are the K0 -K0 oscillations. At present we are left puzzled why leptons do not show any similar mixing. Recent experimental hints for neutrino oscillations, which have a potential for changing this situation, are not covered here (see e.g. [111]). Many extensions to the standard model have been proposed and are presently discussed which try to explain further some of its not well understood features like e.g. parity violation in weak interaction or particle mass spectra. They are put by hand into this remarkable theoretical framework 15
rel. number W(ε)⋅τµ/2 / asymmetry a(ε)
1 0.8 0.6
W(ε)⋅τµ/2=(3-2ε)ε2
0.4 a=(2ε-1)/(3-2ε)
0.2 0 -0.2 -0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 ε=E/Emax
Figure 15: The relative number W() and the asymmetry a of the spatial distribution versus the energy of the decay positrons. For low energy positrons the asymmetry is negative. which appears to serve as an extremely robust description of all con rmed particle physics. Lepton
avor violation (LFV) appears naturally in such models which include Left-Right-Symmetry, Supersymmetry, Technicolor, Grand Uni cation, String Theories, Compositeness, and many others. Such models continue to stimulate experimental searches in a large range of energies. With some low energy experiments new physics can be probed at mass scales far beyond the reach of present accelerators or such planned for the future. Highest sensitivity has generally been reached in dedicated search experiments particularly on Kaons (K) and muons () (Table 3), where also a high discovery potential for new physics exists [112], as well as in non accelerator experiments searching for neutrinoless double -decay. The decays of heavier objects created in high energy collisions, however, can be observed less accurately. The progress in the K and eld is indicated in Fig.16 which shows more than 10 decades of improvement since the rst experiments in the late 1940's. The highest recent gain in sensitivity is for muonium (M=+ e? ) to antimuonium (M=? e+ ) conversion due to a new, yet unused signature (see sec. 3.5). The decay ! e was the rst being searched for shortly after the muon's nature as a heavy electron-like particle became apparent. Searches for rare and forbidden muon decays have been among the most precise experiments in physics since and have always been of special interest in the context of uni ed gauge theories, as they can provide accurate tests of speculative models and because of the achievable experimental precision they may be able to discriminate between such [127]. Recently forbidden muon decays have attracted attention, when their possible sensitivity to e ects arising in minimal supersymmetry (SUSY) were discussed in theoretical studies [128]. It was pointed out that for values of tan ( the ratio of the vacuum expectation values of the two Higgs elds involved) which exceed about 3, the branching ratio should be above 10?14 for a decay ! e and above 10?16 for ! e conversion on a Ti nucleus, almost independent of all other parameters in the model. This has stimulated a letter of intent to the Paul Scherrer Institute (PSI), Switzerland, and a proposal to BNL to search for the respective processes. 16
Figure 16: Dedicated searches for lepton number violating processes involving muons () and kaons (K). Recent K experiments and + e? { ? e+ conversion exhibit the most signi cant gains in sensitivity. Points in 1998 and beyond are projections of possibilities by the respective experimenters. In the eld of searching for SUSY e ects in low energy experiments rare decay experiments are in some competition with the just started new precision measurement [129] of the muon magnetic anomaly a where the contribution from SUSY is of order a (SUSY ) = 140 10?11tan (100GeV=m) ~ 2 with m~ the mass of the lightest SUSY particle (see [130]). The measurement goal is a (exp) = 40 10?11 and should be reached around the year 2001.
3.3
! e decay The signature of a ! e event is a 52.8 MeV positron emitted back to back with a 52.8 MeV photon. The MEGA experiment at the late Los Alamos Meson Physics Facility (LAMPF) consisted of a magnetic spectrometer to observe the charged nal state particle and three pair spectrometers for detecting the photon through its e+ e? pair creation in lead converters. Random coincidences at high rates are reported as major background. Data taking is completed and 16% of the data could be analyzed leading to an upper limit on the branching ratio of 3:8 10?11 [122] which slightly improves the value of 4:9 10?11 established in a crystal box detector also at LAMPF [131]. At PSI new e orts are being discussed to reach a sensitivity of about 5 10?14 for this decay mode within the next couple of years. The suggested instruments include solutions like a large solid angle magnetic spectrometer for the e+ surrounded by a crystal calorimeter for the , or liquid Xe calorimeters for the and others [132]. It should be noted that the tightest bounds on bileptonic gauge bosons, which are common to many speculative standard model extensions, come from ! e , if avour democracy is assumed [133].
17
Table 3: Recently obtained upper limits on lepton number violating processes (90% C.L.). decay limit decay limit Z0 ! e 2:5 10?6 [113] KL ! e 2 10?11 [117] Z0 ! e 7:3 10?6 [113] KL ! 0e 3:2 10?9 [118] Z0 ! 1:0 10?5 [113] K+ ! + e 4 10?11 [119] 0 ? 5 + + D ! e 1:9 10 [114] ! e e 2:5 10?3 [120] D0 ! 0 e 8:6 10?5 [114] ! eee 1 10?12 [121] D0 ! e 3:4 10?5 [114] ! e 3:8 10?11 [122] 0 ? 6 ? ? B ! e 5:9 10 [115] Ti ! e Ti 6:1 10?13 [123] B0 ! e 5:3 10?4 [115] ? Ti ! e+ Ca 1:7 10?12 [124] B0 ! 8:3 10?4 [115] + e? ! ? e+ GMM < 3 10?3GF [125] B0 ! K e 1:8 10?5 [115] 76Ge !76Se e? e? T1=2 > 1:2 1025y [126] ! e 2:7 10?6 [116] me (Maj:) < 0:45eV [126] ! 3:0 10?6 [116]
3.4
! e conversion Many constraints for speculative models arise from the present experimental bound on the conversion process +Z ! e+Z (Table 3), which is the tightest for all studied lepton number violatimg (LNV) decays. The variety of theoretically possible processes that can be tested includes, e.g. supersymmetric loop graphs, heavy neutrinos, leptoquarks, compositeness, Higgs bosons and heavy Z 0 bosons with anomalous couplings. Generally it is more sensitive to new Physics than ! e in a wide class of models where the process is generated at the one loop level [134]. The process needs to involve a nucleus to assure elementary conservation laws. If the nucleus is left in its ground state, a conversion event is signaled through the release of a 105 MeV electron, which is uniquely distinguishable from normal muon decay electrons ranging up to 53 MeV. Among the physically relevant intrinsic background processes is decay in the atomic orbit after a muonic atom has been formed, which can release much higher energetic electrons, and radiative muon capture, where the photokinematic end point can be close to the signal electron energy. The ongoing SINDRUM II experiment uses the worldwide brightest continuous muon channel E5 at PSI. Their new results limit the branching ratios ? Ti ! e+ Cags to below 1:7 10?12 for the Ca nucleus in the ground state [124], ? Ti ! e+ CaGDR to below 3:6 10?11 leaving Ca with giant dipole resonance excitation [124], and ? Ti ! e? Ti to below 6:1 10?13 for Ti in the ground state [123]. For the ground state processes the nucleons interact coherently which enhances the possible e ect. In order to boost accuracy in the near future the SINDRUM II collaboration wants to take advantage in the gain of muon ux through a ? converter, a novel superconducting device in the beam line which collects 's and releases only 's with very low contamination. The latter point is essential as 's are a source of potential background due to nuclear reactions. The projections of the collaboration for the achievable limit in the coherent ? Ti ! e? Ti case are in the 10?14 region. The new Muon Electron Conversion (MECO) experiment proposed at BNL [135] (see Fig.17) is very close in its design to a proposal by Lobashev and collaborators for the Moscow Meson Factory. The setup consists of a target station for / production which uses a proton beam from the AGS accelerator, an S-shaped transport and puri cation section and a detector the basic idea of which is to let electrons from normal muon decay pass without being seen and to observe only the 105 MeV signal electrons. The goal is the 10?16 level in sensitivity, which will stringently test supersymmetric models; there is an anticipated ultimate capability for 10?18 .
3.5
+ e? ! ? e+ conversion The hydrogen-like muonium atom consists of two leptons from di erent generations. The close con nement of the bound state o ers excellent opportunities to explore precisely fundamental electron-muon interactions [136, 23]. Since the e ect of all known fundamental forces in this 18
1
Figure 17: The MECO experiment planned at BNL (see [135]). system are calculable very well mainly in the framework of quantum electrodynamics (QED), it renders the possibility to search sensitively for yet unknown interactions between both particles. A M-M-conversion would violate additive lepton family number conservation and is discussed in many speculative theories (see Fig. 18). It would be an analogy in the lepton sector to K0 -K0 oscillations.
+
p p p p p
e?
p p p
++
p p
? e+
(a)
+
p p p p p
e?
p p p p p
N
e+
j j
+ e?
L
p
p
p
WR
p
p
p
p
p
p
+
p
p
p
p
p
p
p
p p p p p
?
e?
p
p
p
R
p
W
j e jL p
R (b)
p
p
p p p p p
p
p
p
p
p
p
L
X ++
? e+ ? e+
(d)
(c)
Figure 18: Muonium-antimuonium conversion in theories beyond the standard model. The interaction could be mediated by (a) a doubly charged Higgs boson ++ [137, 138], (b) heavy Majorana neutrinos [137], (c) a neutral scalar N [139], e.g. a supersymmetric -sneutrino ~ [109, 140], or (d) a dileptonic gauge boson X ++ [141]. The setup at PSI (Fig. 19) [142] is designed to employ the signature developed in a predecessor experiment at LAMPF, which requires the coincident identi cation of both particles forming the antiatom in its decay [143, 144]. Muonium atoms in vacuum with thermal velocities, which are produced from a SiO2 powder target, are observed for antimuonium decays. Energetic electrons from the decay of the ? in the antiatom can be observed in a magnetic spectrometer at 0.1 T magnetic eld consisting of ve concentric multiwire proportional chambers and a 64 fold segmented hodoscope. The positron in the atomic shell of the antiatom is left behind after the decay with 13.5 eV average kinetic energy [145]. It can be accelerated to 7 keV in a two stage electrostatic device and guided in a magnetic transport system onto a position sensitive microchannel plate detector (MCP). Annihilation radiation can be observed in a 12 fold segmented pure CsI calorimeter around it. The relevant measurements were performed during in total 6 month distributed over 4 years during which 5:7 1010 muonium atoms were in the interaction region. One event fell within a 99% con dence interval of all relevant distributions (Fig. 20). The expected background due 19
pump iron MCP CsI
+
iron collimator
µ
+
Figure 19: Top view of the MACS (Muonium - Antimuonium - Conversion - Spectrometer) apparatus at PSI to search for M?M - conversion [142].
magnetic field coils hodoscope MWPC annihilation beam counter photons SiO2-target e accelerator
1m
separator e
to accidental coincidences is 1.7(2) events. Thus an upper limit on the conversion probability of PMM 8:2 10?11=SB (90% C.L.) was found, where SB accounts for the interaction type dependent suppression of the conversion in the magnetic eld of the detector due to the removal of degeneracy between corresponding levels in M and M. The reduction is strongest for (VA)(VA), where SB =0.35 [146, 147]. This yields for the traditionally quoted upper limit on the coupling constant in e ective four fermion interaction GMM 3:0 10?3GF (90%C:L:) with GF the weak interaction Fermi constant. This new result, which exceeds bounds from previous experiments [143, 148] by a factor of 2500 and the one from an early stage of the experiment [142] by 35, has some impact on speculative models. A certain Z8 model is ruled out with more than 4 generations of particles where masses could be generated radiatively with heavy lepton seeding [149]. A new lower limit of mX 2.6 TeV/c2 g3l (95% C.L.) on the masses of avour diagonal bileptonic gauge bosons in GUT models is extracted which lies well beyond the value derived from direct searches, measurements of the muon magnetic anomaly or high energy Bhabha scattering [141, 133]. Here g3l is of order 1 and depends on the details of the underlying symmetry. For 331 models this translates into mX 850 GeV/c2 which excludes their minimal Higgs version in which an upper bound of 600 GeV/c2 has been extracted from an analysis of electroweak parameters [150, 151]. The 331 models may still be viable in some extended form involving a Higgs octet [152]. In the framework of R-parity violating supersymmetry [140, 109] the bound on the coupling parameters could be lowered by a factor of 15 to j 132231 j 3 10?4 for assumed superpartner masses of 100 GeV/c2 . Further the achieved level of sensitivity allows to narrow slightly the interval of allowed heavy muon neutrino masses in minimal left-right symmetry [138] (where a lower bound on GMM exists, if muon neutrinos are heavier than 35 keV) to 40 keV/c2 up to the present experimental bound at 170 keV/c2 . In minimal left right symmetric models, in which MM conversion is allowed, the process is intimately connected to the lepton family number violating muon decay + ! e+ + + e . With the limit achieved in this experiment this decay is not an option for explaining the excess neutrino counts in the LSND neutrino experiment at Los Alamos [153, 154]. The consequences for atomic physics of muonium are such that p the expected level splitting in the ground state due to M ? M interaction is below 1:5 Hz= SB reassuring the validity of fundamental constants determined in muonium spectroscopy. A future M ? M experiment could take particularly advantage of high intense pulsed beams. In contrast to other LNV muon decays, the conversion through its nature as particle - antiparticle oscillation, has a time evolution in which the probability for nding M in the ensemble remaining after muon decay increases quadratically in time, giving the signal an advantage growing in time over major exponentially decaying background [144]. 20
Rdca [cm]
Rdca [cm]
5 2.5
5 2.5
0
0
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-2.5 -5
-5 -20
-10
0
10
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20
TOF - TOFexpected [ns]
-10
0
10
20
TOF - TOFexpected [ns]
Figure 20: Time of ight (TOF) and vertex quality for a muonium measurement (left) and the same for all data of the nal 4 month search for antimuonium (right). One event falls into the indicated 3 standard deviations area.
4 Theory of the Muonium Atom 4.1 One-electron Atoms
4.1.1 Dirac Theory of the Gross and Fine Structure
The total energy of an electron bound in an one-electron atom [155] can be expressed as Etot(n; j; l; F) = ED (n; j) + ERM (n; j; l) + EQED (n; j; l) +EHFS (n; j; l; F; I) + Estrong + Eweak + Eexotic ; (9) with the principal quantum number n, the electron angular momentum j, the the orbital angular momentum l, the total angular momentum F, and the nuclear Spin I. The dominant part is the Dirac energy ED (n; j) which is an eigenvalue of the Dirac equation for an electron of rest mass me bound to a point-like nucleus of in nite mass and with a charge of Z e which creates a potential V = ?Z =r, ED (n; j) = me c2 (f(n; j) ? 1) ; (10) where me represents the electron mass and c is the speed of light. The function f(n; j) is given by v u 1 (11) f(n; j) = u 2 ; t 1 + nZ ?" with
" = j + 12 ?
s
2 1 j + 2 ? (Z )2 :
(12)
The Dirac theory describes the system to the ne structure level. The nite nuclear mass is taken into account by a reduced mass ERM correction to ED . This contribution is composed of two parts ERM (n; j; l) = ENRRM + ERRM (n; j; l): (13) The rst one describes the classical nonrelativistic reduced mass term m r ENRRM = m ? 1 ED (14) e and the second one is the relativistic reduced mass e ect which arises in the full relativistic treatment of the two-body problem 4 m3r c2 1 2r c2 (Z ) 1 m 2 ERRM (n; j; l) = ? 2(m + m ) (f(n; j) ? 1) + 2n3m2 j + 21 ? l + 21 (1 ? l0 ); (15) e N N 21
and which depends also on the orbital angular momentum l. Higher order terms are neglected. The reduced mass is de ned by the expression mr = me mN =(me + mN ): (16) Additional corrections will be discussed below. They arise from quantum electrodynamical e ects (EQED (n; j; l)), the hyper ne interaction (EHFS (n; j; l; F)), the strong interaction (Estrong , which enters indirectly through the QED e ect of vacuum polarization), the weak interaction (Eweak , due to Z boson exchange), and possibly from exotic processes (Eexotic), which are so far not provided in standard theory.
4.1.2 QED Corrections - Lamb Shift
The deviations of the real atomic levels from the ne structure level energies as predicted in the Dirac theory and corrected for a nite nuclear mass can be calculated to high precision within the framework of QED. They are referred to as the Lamb shift and originate mainly from radiative corrections to the electron propagator, from vacuum polarization, from nuclear motion and from the e ects of nuclear motion in radiative corrections (combined e ects). Nuclear motion can be taken into account in a rst approximation using the reduced mass concept. Further corrections (in higher than rst order of the mass ratio) due to the dynamics of the nuclear motion in the system are traditionally called recoil corrections. The QED corrections depend most of all on the probability density of the electron wave function at the origin (important for the for s-states) and the expectation value for r?3 which both are proportional to 1=n3 . The theoretical work over the last four decades has been reviewed recently by Sapirstein and Yennie [16, 17]. Therefore we quote the major results. For electronic atoms the dominating contribution arises from one-photon radiative corrections to the electron propagator (self energy and anomalous magnetic moment) and from vacuum polarization e ects to the potential. 1. In the non-recoil limit (derived for in nite nuclear mass, and with the additional inclusion of the reduced mass dependence, which for example describe the scaling of the wave functions, and mass corrections for the magnetic moments) they amount to 2 (Z )4 Fn(Z ) (17) Eone?loop (n; j; l) = me c n 3 where Fn(Z ) = A40(n) + A41(n) ln ((Z )?2 ) + (Z )A50(n) +(Z )2 G(n; Z ) + A61(n) ln ((Z )?2 ) + A62(n) ln2 ((Z )?2 ) (18) with = me =mr . The coecients Apq from expansions in r (Z )p lnq (Z )?2 , where r corresponds to the number of emitted and re-absorbed virtual photons, have been calculated analytically: 3 10 m 4 4 r A40(nS) = 9 ? 15 ? ln(k0 (nS)) m e VP 3 2 3 m m 4 C r r lj A40(n; l 6= 0) = 2(2l + 1) m + ? 3 lnk0 (n; l 6= 0) m e e 3 m 4 A41(nS) = 3 mr e 3 1 5 m 139 r ? ln2 m A50(nS) = 4 128 + 192 e VP 2 3 21 2 mr A61(1S) = 28 3 ln2 ? 20 ? 15 me 1 In the case of muonic atoms the situation is reversed: The vacuum polarization dominates over the self energy part.This makes muonic atoms particularly interesting for studying vacuum polarization e ects.
22
3 2 m 67 16 r (19) A61(2S) = 3 ln2 + 30 ? 15 V P me mr 3 A61(2P1=2) = 103 180 me 3 29 m r A61(2P3=2) = 90 m e 3 r ; A62(nS) = ? [1] m me where ln k0(n; l) denotes the Bethe logarithm with the values [156] lnk0(1S) = 2:984 128 5559(3) lnk0(2S) = 2:811 769 8932(5) (20) ln k0(2P) = ?0:030 016 708 9(3) : The expressions arising from vacuum polarization are indicated in fgV P brackets. The nonlogarithmic terms G(n; Z ) of order me (Z )6 and higher orders are available numerically:
Figure 21: Lowest order contributions to the Lamb shift. (a) Electron self energy. (b) Vacuum polarization correction to the potential. The heavy lines represent the electron in an external static nuclear eld. 3 m r G(n; Z ) = [GSE (n; Z ) + GV P (n; Z )] m e with the self energy contributions GSE (1S) = [?30:3 1:5] GSE (2S) = [?32:1 1:5] GSE (2P1=2) = [?0:8 0:3] GSE (2P3=2) = [?0:5 0:3]
and the vacuum polarization terms
4 ln2 ? 1289 + 19 ? 2 + O(Z ) 15 1575 45 27 19 ? 2 + O(Z ) GV P (2S) = ? 743 + 900 45 27 9 + O(Z ) GV P (2P1=2) = ? 140 1 + O(Z ) ; GV P (2P3=2) = ? 70 GV P (1S) =
23
(21)
(22) (23)
(24)
where Clj = 2(j ? l)=(j + 1=2). Recently analytical expressions for some of the lowest order contributions to G(n; Z ) for the 1S state were presented by Pachucki [157, 61]. He obtains G(1S) = ?31:04309(1) in agreement with the results in eqn.(24). The Lamb shift is largest for s-states, since they have a nite probability density at the origin. Not included in the expressions of eqn.(19,24) are vacuum polarization loops formed by particles di erent from the electron ( muons, taus, hadrons). In general they contribute proportional to the inverse square of their masses. For example muons contribute (me =m )2 2:3 10?5 as much as electrons. For the classical Lamb shift this is of the order of 600 Hz. The hadronic contribution to the s-states is of relative order (Z )4 (me =m )2 [158], where m is the pion mass. Fortunately it can be calculated in detail without knowledge of the underlying strong interaction theory using input data from the measured ratio of the total cross sections (e+ e? ! hadrons)=(e+ e? ! + ? ). So far the expressions are valid for small me =mN ratios only. In the case of positronium one needs to take into account the self energy of the positron as well as the spin orbit coupling e ects caused by the anomalous magnetic moment of the positron [16]. The positronium atom further needs corrections for the hyper ne splitting of order 2R1 =n3. For the triplet states the annihilation into a single virtual photon causes an upward shift of the order 2R1 =2n3. The relativistic recoil (also known as Salpeter correction) contributes to relative order (Z )5 m3r =(me mN ) and can be derived from the Bethe-Salpeter equation [159]. The Breit equation [160] used formerly is not appropriate since it misses terms of that order. We have 3r c2 (Z )5 2 1 ) ? 8 lnk (n; l) ? 1 ? 7 a m (25) Erel?rec (n; l) = m m n3 3 l0 ln( Z 3 0 9 l0 3 n e N e 2 mN ? m2 2? m2 l0 m2N ln m m ? me ln m N
r
e
r
m2e c2 (Z )6 (4 ln2 ? 7 ; mN n3 2 where the state dependent function an [161] is 2 1 1 1 ? l0 an = ?2 ln n + (1 + 2 + ::: + n ) + 1 ? 2n l0 + l(l +11)(2l (26) + 1) : Recoil terms of order (Z )6 m2e =mN have been recently presented by Erickson, Grotch and Doncheski [162, 163]. The precision achieved both in experiment and theory has reached a level where the separate treatment of radiative and recoil corrections does not provide results of sucient accuracy. Combined e ects have to be taken into account. The e ect of nuclear recoil in radiative corrections (radiative-recoil) has been calculated by Bhatt and Grotch [164] to be 5m2 c2 35 7333 ? 0:415 0:004 : e Erad?rec (n; l) = (Z ) ln2 ? (27) l0 n3mN 4 960 The inverse mN dependence of the relativistic-recoil and the radiative recoil terms causes a significant di erence in the Lamb shift for hydrogen and muonium where they are larger by the ratio mp =m 8.9 of the proton to the muon mass. Higher order corrections of order 2 can be summarized in a way analogous to the one-loop terms as 2 4 (28) Etwo?loop (n; l; j) = me 2(Z ) n3 Hn(Z ) : The functions Hn(Z ) are: Hs?states(Z ) =
10 2 3 2 9 mr ? 4358 1296 ? 27 + 2 ln2 ? 4 (3) me 24
3
2 m 1 3 C 197 r lj 2 2 (29) Hnon s?states(Z ) = + 72 + 6 ? ln2 + 2 (3) 2(2l + 1) m e P x. with the Riemann -function (x) = 1 =1 1= 2 Two-loop binding corrections of order (Z )5 contribute - 296.90(4) kHz for the 1s and 37.112(5) kHz for the 2s state [165, 175, 166]. Some additional higher order radiative corrections have been recently calculated [167], they contribute at the few kHz and below level to the 1s and 2s Lamb shifts and are of no concern for muonium yet. The nite charge distribution within the nuclei of hydrogen-like systems with hadronic nuclei give rise to a shift mainly for the s-states since they have the largest probability density within the nuclei. Assuming a spherical charge distribution of radius rN it amounts for nonrelativistic systems to
3 2 i : r hrN Efinite?size(n) = 3n2 3 (Z )4 m 2 me c
(30)
Table 4: Fundamental constants necessary for the calculations of bound two body level structures and transition frequencies. constant Rydberg constant R1 ne structure constant electron mass me muon mass m proton mass mp deuteron mass md electron g-factor ge muon g-factor g proton g-factor gp deuteron g-factor gd muon/electron mass ratio m =me proton/electron mass ratio mp =me deuteron/electron mass ratio md =me proton sizey rp deuteron size rd speed of light c Planck constant h = 2 h Fermi coupling constant GF =(h c)3 weak mixing angle sin2 W (MS) muon lifetime
[cm?1] [MeV/c2 ] [MeV/c2 ] [MeV/c2 ] [MeV/c2 ]
[fm] [fm] [m/s] [eVs] [GeV?2]
value 10 973 731.568 639(91)
reference [20, 21]
1 / 137.0360037(33) 0.51099906(15) 105.658389(34) 938.27231(28) 1875.61339(57) 2.002319304386(20) 2.002331846(17) 5.585694772(126) 1.714876460(48)
[82, 170] [171] [171] [171] [171] [171] [171] [171] [171]
206.768262(30)
[171]
1836.152701(37)
[171]
3670.483014(75) 0.862(12) 2.116(12) 299 792458 4:1356692(12) 10?15
[171] [56] [172] [171] [171]
1:16639(1) 10?5
[86]
0.23124(24) 2.19703(4)
[86] [86]
[s]
y For the mean square charge radius of the proton the results of the most recent electron scattering experiment were taken. An earlier published value [57] has been shown to be wrong due to an error in the extrapolation of experimental data to low momentum transfer. A re-analysis yields agreement with the new results [173].
In principle, an in uence of deviations from the spherical nuclear shape must be expected. For most practical cases, however, the mean square charge radius can be chosen for < rN2 >. Combined 25
e ects of nite nuclear size with vacuum polarization and radiative corrections are of the order of (Z )5 (mr =me )3 (< rN2 > =2c ) and can be neglected. 2 .
4.1.3 Hyper ne Structure
The interaction of the magnetic moment N of the nucleus with spin I with the electrons magnetic moment e gives rise to the hyper ne structure splitting which is to a good approximation given by the Fermi formula [168] : 3 m m e r 1 F(F + 1) ? I(I + 1) ? j(j + 1) : 2 EHFS (n; j; l; F; I) = (Z ) ZgI m (1 + "QED ) m hcR 3 n j(j + 1)(2l + 1) N e (31) Here the term "QED takes into account relativistic e ects, radiative and recoil corrections as well as the nite nuclear size and nuclear polarizability: "QED = "rad + "rec + "rad?rec + "nucl?size + "nucl?pol (32) For the muonium atom, where nuclear structure e ects are absent, the ground state splitting between the F=0 and F=1 levels can be expressed as ?3 2 R1 1 + me M (Z ) (33) HFS = 16 3 B m (1 + "rad + "rec + "rad?rec ) +strong + weak + exotic : Using 2 e c2 (34) R1 = 2 m h we have ?3 8 m m e e M 4 2 HFS = 3 (Z ) c ( h ) 1 + m (1 + "rad + "rec + "rad?rec ) (35) B +strong + weak + exotic ; which is a powerful relation for an accurate determination of the ne structure constant, as the quantity me =h can be determined accurately from measurements of the neutron de Broglie wavelength and well known mass ratios. The radiative corrections to the hyper ne splitting of one-electron atoms are in low orders 5 3 2 "rad = (1 + a ) 1 + 2 (Z ) + ae + (Z ) ln2 ? 2 2 2 8 (Z ) 281 ? 3 ln(Z ) ln(Z =4) + 480 + (Z ) (16:9042 0:11) 2 + (Z ) (0:7717 4) : The nuclear recoil contributes 2 3Z m m m (m Z ) 1 65 e r "rec = ? m2 ? m2 ln m + m m 2 ln 2Z ? 6 ln2 + 18 : e e e The radiative recoil correction to the hyper ne interval is e ?2 ln2 m + 13 ln m 21 + (3) + (2) "rad?rec = (Z )m 2 m me 12 me 2 ? 4 ln3 m + 4 ln2 m + O(ln m ) : + + 35 9 3 m 3 m m
(36) (37)
e e e 2 For muonic atoms the nuclear-size e ects are more important due to the larger overlap of the muon wave functions with the nuclear charge density compared to corresponding electronic states.
26
From the hadronic vacuum polarization there is a shift due to strong interaction. me ?3 strong = 83 (Z )4 c2( mhe ) 1 + m B (Z )m 13 m 21 m e 2 1:91(0:26) 2 m ?2 ln m + 12 ln m 2 + (3) + (2) ;
e
(38)
e
which amounts to 250 Hz.
4.2 Weak Interaction
The weak interaction through the exchange of a Z boson gives rise to an additional contribution to the level energies. Because of the very short range of the weak interaction due to the large mass ( 91 GeV=c2) of the Z boson the interaction can be modeled as a point current-current interaction with an e ective vector-axial vector type Hamiltonian [182, 183, 184, 185]: ?
P
V V e e N N + C V A e e N 5 N CeN eN AA e 5 e N 5 N : AV +CeN e 5 e N N + CeN
Heff = GpF2 N
(39)
Here GF is the Fermi coupling constant of the weak interaction, e describes the electron wave function and N runs over all the nucleons in the system. Although the fundamental weak coupling of the electron is to the quarks within the nucleus (in case of hadronic nuclei), at low energies the nucleons may be considered as fundamental [184]. In the case of muonium N represents the muon. The coupling constants for di erent nuclei are listed in table 5. They are valid in the limit of low momentum transfer and they can be expressed in terms of the electroweak mixing angle (Weinberg angle) W and the ratio of the axial vector to vector amplitude gA =gV with the experimental value gA =gV = 1:2573(28) [86]. Table 5: Coupling constants for Z boson exchange between electron e and \nucleus" N in hydrogenlike systems in the limit of low momentum transfer [184]. W is the Weinberg angle (weak mixing angle). gA =gV is the ratio of the axial vector to vector coupling amplitude from neutron -decay. \nucleon" N
VV CeN
VA CeN
AV CeN
AA CeN
proton p
? (1?4sin22 W )2
gA (1?4sin2 W ) gV 2
(1?4sin2 W ) 2
? 2ggAV
neutron n
(1?4sin2 W ) 2
? ggVA (1?4sin2 W )
? 21
gA 2gV
muon
? (1?4sin2 W )
(1?4sin2 W ) 2
(1?4sin2 W ) 2
? 21
2
2
2
2
The Parity violating vector - axialvector (V A) and axialvector - vector (AV ) terms will not shift atomic energy levels to rst order perturbation theory. Only the parity conserving weak interactions can contribute signi cantly to the level energies. Theoretical work has been performed for the s-states and for the hyper ne structure intervals of hydrogen and hydrogen-like atoms by Beg and Feinberg [182]. Independent calculations by Starshenko and Faustov [183] on the weak e ects on the hyper ne structure yield agreeing results. 27
4.2.1 Weak E ects on s-States
The vector-vector (V V ) coupling causes a common shift of the s-state sublevels, since the vectors current N N is independent of nuclear spin. For the nS states in hydrogen-like atoms with natural nuclei there is a signi cant shift [182] 3 p2GF m2 2me c2 Z Z;A e V V + C V V ) + (A ? Z)(C V V + 2C V V ) : Eweak (nS) = ( n3 ) Z(2C (40) ep en ep en 2 The expression re ects the basic coupling to the up- and down-quarks in the protons and neutrons. For atomic hydrogen the weak interaction shift amounts to H (nS) = 7:2 (2C V V + C V V ) kHz ; (41) Eweak en n3 ep and for deuterium to D (nS) = 21:6 (C V V + C V V ) kHz: (42) Eweak en n3 ep In the muonium atom the coupling is between one electron and one muon. One nds p 2 2 2 M (nS) = ?( 1 ) 2GF me me c C V V = ? 7:2 (C V V ) kHz : (43) Eweak e 3 n 2 n3 e The contributions to the 1S-2S energy separations are 190 Hz, 613 Hz and 24 Hz for hydrogen, deuterium and muonium respectively. The accurate most recent 1S-2S measurements in atomic hydrogen [21] combined with the uncertainty of the QED theory establish an upper bound of V V j 16 . j2CepV V + Cen
4.2.2 Weak E ects on Hyper ne structure
The axialvector - axialvector (AA) term contributes to the ground state hyper ne structure splitting, since the axial current N 5 N is proportional to the nuclear spin. The hyper ne splitting [182, 183] for hydrogen and for muonium are a ected signi cantly di erent by the weak interaction H due to the di erent muon - electron and nucleus - electron couplings. For the shift HFS;weak in hydrogen there there is a relation H HFS;weak GpF MN2 me C AA (1 + gA ) + C AA (1 ? gA ) =(1 + ) (44) = 12 p en H gV gV HFS 16 22 2 mp ep AA ) ; 2:2 10?6(CepAA ? 19 Cen
where MN is the nucleon mass and p =1.79. Since the experimental error is far below the theoAA j 0:5 . Unfortunately, no improvement retical uncertainty of 1 ppm one nds jCepAA ? 19 Cen can be expected in the future, unless there will be a better understanding of the proton's internal structure from both the experimental and the theoretical side. In the case of muonium one nds for the parity conserving weak e ect p 2e 2 me c2 2 M AA ; HFS;weak = 2GF m Ce (45) 2 AA from table 5 which yields with Ce M AA Hz = ? 65 Hz : HFS;weak = 130 Ce
(46)
M and the newest experimental The present uncertainty arising from the QED calculations for HFS AA result (sect. 5.1.1) xes an upper limit at jCe j 4 . Signi cant improvements in theory are possible with an improved muon mass and would reduce this limit further . Note that the signs of the weak e ects in hydrogen and muonium are opposite. This was discovered only recently by Eides [185]: If both particles in the atomic system are di erent components
28
of the same SU(2) doublet, as it is the case for hydrogen, the signs of the electromagnetic and the weak interaction are the same. Since for the upper and lower components of weak doublets the weak neutral current has opposite sign and the neutral current interaction is identical for particles and antiparticles, the levels of muonium, which consists of the antiparticle + and e? , are shifted with an opposite sign compared to hydrogen.
4.3 Exotic Interactions
Exotic interactions between electron and "nucleus" can contribute to the level energies. Therefore precise measurements can provide stringent bounds on the strength of such processes. Of particular interest in this connection is for the muonium atom the possibility of a spontaneous muonium to antimuonium conversion. Various possibilities have been discussed in theory which could explain muonium-antimuonium conversion. All proposed mechanisms of interest can be described by an e ective V-A Hamiltonian [137, 138] (47) HMM = GpMM (1 + 5 ) e (1 + 5 ) e + H:C: ; 2 where GMM is the coupling constant. In a eld free environment the probability of a system which starts as pure muonium to decay as antimuonium is PMM = 2:6 10?5
GMM 2 : GF
(48)
Figure 22: Possible splitting of muonium hyper ne levels due to muonium-antimuonium conversion. The ground state of the coupled muonium-antimuonium system has eight energy eigenstates which are di erent from the four eigenstates of the uncoupled muonium and antimuonium atoms. Their energies have been calculated by Schafer [188], Matthias [189], as well as Hou [147] and Sasaki [146] and coworkers. In the absence of external elds the muonium-antimuonium oscillations cause a splitting of the hyper ne levels of s-states of (49) MM (nS) = hM jHMM jM i = 519 n3 (GMM =GF ) Hz: The e ect is shown qualitatively in Fig. 22. For the present experimental upper limit on GMM of 3 10?3GF (90%C.L.) [125] this amounts to MM (1S) 1:5 Hz in the ground state. 29
4.4 Theoretical values for muonium transition frequencies
In muonium the 1s-2s level separation can be calculated using the fundamental constants of table 4 and is 1s?2s(theory) = 2 455 528 934:1(1:4) MHz(0:6ppb): (50) For the hyper ne splitting the latest compilation of all theoretical results [174, 175, 176, 177, 178, 179] yields HFS (theory) = 4 463 302 649(517)(34)(< 100)Hz(120ppb) (51) [180]. The rst uncertainty arises from the muon magnetic moment, the second one from the ne structure constant (from quantum Hall e ect) and the last one is due to uncalculated higher order terms. For systems with hadronic nuclei the structure of these particles has to be taken into account. The in uence on the hyper ne structure splitting consists of two components: the e ects of the charge distribution and e ects of the dynamics of the charge carrying constituents within the proton. In hydrogen the size e ects together with additional recoil contributions amount to 33(1) ppm [17]. The nuclear polarization e ects are not well known. Their contribution is estimated to be less than 4 ppm [181]. Since muonium is free of these nuclear complications, the hyper ne structure splitting of the atom is ideally suited for testing bound state QED, for investigating contributions from the weak interaction and for searching possible exotic interactions. With con dence in the theoretical calculations one can extract accurate values for the fundamental constants that appear in the equations.
5 Microwave Spectroscopy of Muonium
5.1 Ground State Hyper ne Structure Splitting 5.1.1 Ground State in external magnetic Fields
The perturbation Hamiltonian describing the n=1 ground state in an external magnetic eld B is [190] me g0 I B ; (52) HZ = AHFS I J + B gJ J B ? m B M , I is the muon spin operwhere AHFS is the ground state hyper ne structure splitting hHFS ator, J is the electron total angular momentum, B is the Bohr magnetron, and gJ and g0 are the gyromagnetic ratios of the electron and the muon in the muonium atom which di er from the free particle values ge and g due to relativistic binding corrections [191], 2 2 me 3 2 2 me (53) g ` = g 1 ? 3 + 2 m ; gJ = ge 1 ? 3 + 2 m + 4 : The behaviour of the magnetic sublevels of the F=1 and F=0 hyper ne states in an external magnetic eld B is expressed by the Breit-Rabi formula [190] F hAHFS p1 + 2MF x + x2 ; (54) E(n=1; F; MF ) = ?g0 (me =m )B MF B ? hAHFS ? ( ? 1) 4 2 where x = (gJ + g0 (me =m ))B B=(h AHFS ) ; x = 1 , B 0:1585 T (55) is the magnetic eld parameter. The Breit-Rabi equation does not incorporate the e ect of higher quantum states. Such corrections of relative order (HFS =R1)2 10?12 are negligible. O diagonal elements of the Hamiltonian eqn.(52) to higher n states would cause corrections of the size (B B=R1 )2 and can be neglected for any magnetic eld strength of practical interest.
30
Energy / h [ ∆νHFS ]
10 (1/2, 1/2)
8 6
(1/2,-1/2)
( F , MF )
4
(1, 1)
2
1
ν12
2
(1, 0)
0
( MJ , Mµ )
(1,-1)
-2
(0, 0)
-4
ν34
3 4
(-1/2,-1/2)
-6 -8 -10
(-1/2, 1/2)
0
0.5
1
1.5
2
2.5
Magnetic Field [ T ]
Figure 23: Muonium ground state Zeeman levels in an external magnetic eld. The two transitions indicated were measured in the latest experiments.
5.1.2 Muonium Formation in Gases
Muonium atoms can be produced in reasonable quantities only at accelerator sites. The most ecient way of forming muonium is by stopping positive muons in suitable gases and by electron capture according to the reaction + + G ! M + G+ ; (56) where G stands for the particular gas atom used. Chemical reactions and depolarization e ects can be avoided by using noble gases. Muonium can be principally formed in any quantum state. However, the process is only possible, if it is energetically allowed. Muonium has in its ground state a binding energy of EI (M) = 13:54 eV . For gases with higher ionization potentials EI (G) the energy defect has to be brought up from kinetic energy Et . The energy di erence EI (G) ? EI (M) in the muonium-gas center of mass system transforms for room temperature gases with atom masses mG with negligible kinetic energies into the laboratory frame as a threshold energy of Et = (EI (G) ? EI (M))(m + mG )=mG : (57) For Xe we have Et(Xe) = ?1:41 eV and muonium formation is always possible. Experimentally a muonium formation fraction of 100% is found [192]. Of practical importance are the cases for Argon (Et(Ar) = 2:22 eV ) and Krypton (Et (Kr) = 0:46 eV ) with formation fractions of 65(5)% [192] and 80(10)% [193]. The slowing down of the muons and the electron capture process are mainly due to Coulomb interaction. Magnetic interactions are of negligible importance. Therefore the muon polarization is not destroyed. The formation probabilities fi ; i = 1; 2; 3; 4 of the ground state sublevels have been worked out using the density matrix formalism [194]. The index i refers to the four di erent states for which the assignment of (F; MF ) quantum numbers at low elds and of (Me ; M ) quantum numbers in strong elds is shown in Fig.23. For a muon polarization P in the z direction and for unpolarized electrons in the target material one nds in a magnetic eld along the z axis ? f1 = 41 (1 + P) ; f2 = 41 1 + P(s2 ? c2 ) ; ? f3 = 14 (1 ? P) ; f4 = 41 (1 + P(c2 ? s2 ) ; (58) 31
where
s = sin( 21 arc cotx) and c = cos( 12 arc cotx) with the dimensionless eld parameter x as de ned in eqn.(55).
(59)
5.1.3 Experiments in Magnetic Fields
It is a consequence of the Breit-Rabi equation, eqn.(54), that with the labeling of levels as shown in Fig.23 we have in magnetic elds M 12 + 34 = HFS (60) h i M (1 + x2)1=2 ? x : 12 ? 34 = 2 hg ` B + HFS (61)
The sum of the frequency 12 for a transition between level 1 and 2 and the frequency 34 for M , the hyper ne splitting at zero a transition between level 3 and 4 is always equal to the HFS elds. From the di erence frequency one can obtain a value for the magnetic moment of the positive muon + . In an experimental situation one can assume similar absolute experimental errors for the transition frequency determination and constant relative experimental precision for the magnetic eld at low and high magnetic eld values. Therefore, it appears that high eld M can measurements are better suited for measurements of the magnetic moment, whereas HFS be measured in both eld regimes accurately. In a series of experiments both strong and weak eld experiments with steadily increasing accuracy [195, 196, 197, 198, 199, 200, 201, 202, 203, 69] have been carried out. The latest and most accurate strong eld experiments were carried out at LAMPF [69, 169] in Kr gas at pressures around one atmosphere. In order to be able to correct for density shifts in the Kr gas, data were recorder for di erent pressures between 0.5 and 1.5 atm. The values corresponding to zero pressure were found by extrapolation. Polarized surface muons are stopped inside of a microwave resonator in the gas target, where they form polarized muonium in a strong magnetic eld. According to eqn.(58) the atoms are predominantly in the states (Me = + 21 ; M = s) and (Me = ? 12 ; M = s) , where s is either + 21 or ? 12 depending on the relative orientation of the muon spin direction and the B eld. The direction of the muon spin can be detected by the spatial anisotropy of the decay positron distribution from the muon decay + ! e+ + +e [97, 98]. The microwave transitions 12 and 34 involve a change of the muon spin orientation which can be detected as a change of the spatial forward/backward asymmetry of the decay positrons in scintillation counters. A precision experiment M = 4 463 302:88(16) kHz in B 1:36 T eld was completed in 1982 at LAMPF and found HFS for the muonium ground state hyper ne splitting and =p = 3:183 346 1(11) for the ratio of the muon magnetic moment to the proton's magnetic moment. This can be converted into a value for the muon mass m using me from [86] m = me g2 p Bp = 105:658 386(44) MeV=c2 . With con dence in the QED calculations as quoted in section 4.1.3 one can extract from the measurements a precise value for the ne structure constant . A most recent calculation yields ?1(M; HFS) = 137:035994(18) [87]. Table 6: The ne structure constant can be extracted from di erent measurements in atomic physics and in condensed matter. Experimental Method Value of ? 1 Accuracy Reference Muonium hyper ne structure 137.036 996 3(80) 58 ppb [169] Electron g-2 137.035 999 93(52) 3.8 ppb [10] Quantum Hall e ect 137.036 003 7(33) 24 ppb [82] ac-Josephson e ect 137.035 977 0(77) 56 ppb [83] neutron de Broglie wavelength and R1 137.030 011 4(50) 37 ppb [85] A new and recently completed experiment used a homogeneous magnetic eld of 1.7 Tesla. The experiment employed the technique of "old muonium" which allowed to reduce the linewidth of 32
Figure 24: The LAMPF muonium ground state hyper ne structure experiment. (a) An electrostatic chopping device in the beam line provides the pulses necessary for the old muonium technique. (b) Polarized muons are stopped in Kr gas inside of a microwave resonator located in a strong magnetic eld. The microwave transitions are observed as a change in the spatial asymmetry of the decay positrons by the scintillation counters. [69]. the signals can be reduced below half of the "natural" linewidth nat = ( )?1 =145kHz, where is the muon lifetime of 2.2 . For this purpose the basically continuous beam of the LAMPF stopped muon channel was chopped by an electrostatic kicker device into 4 s long pulses with 14 s separation. Only atoms which were interacting with the microwave eld for periods longer than several muon lifetimes were detected [204]. Although less particles contribute to the signal, there is a gain in accuracy, as for short times, when most muons decay, only a few have undergone a transition. Depending on details of the experiment the is an optimum waiting time around 4-6 s. The basically statistically limited results improve the knowledge of both zero eld hyper ne splitting and muon magnetic moment by a factor of three [169]. The results of the two directly measured frequencies at LAMPF experiment in 1.7 T magnetic eld are 12 = 1 897 539 800(35)Hz(18ppb) and 34 = 2 565 762 965(43)Hz(17ppb). This yields HFS = 4 463 302 765(53)Hz(12ppb)
(62)
in good agreement with the theoretical value of eq. (51). For the magnetic moment one nds =p = 3:183 345 13(39)(120ppb) which translates into a muon-electron mass ratio of m =me = 206:768 277(24)(120ppb). 33
(63)
(NMW-on-NMW-off)/NMW-off ∈%∉
(NMW-on-NMW-off)/NMW-off ∈%∉
40
old
conven-
20
0 251
40
old
conven-
20
0 600
949
500
(νMW - 1897*10 ) ∈kHz∉
1000
(νMW - 2565*10 ) ∈kHz∉
3
3
Figure 25: Conventional and 'old' muonium lines. The narrow 'old' lines are also higher. The frequencies 12 and 34 correspond to transitions between (a) the two energetically highest respectively (b) two lowest Zeeman sublevels of the n=1 ground state. As a consequence of the Breit-Rabi equation describing the behaviour of the levels in a magnetic eld, the sum of these frequencies equals at any xed eld the zero eld splitting and the di erence yields for a known eld the muon magnetic moment m . As the hyper ne splitting is proportional to the fourth power of the ne structure constant , the improvement in will be much better than for HFS and the value is comparable in accuracy with the value of determined with the Quantum Hall e ect [82]. Here the value h=me as determined with the help of measurements of the neutron de Broglie wavelength [84, 85] can be bene cially employed to gain higher accuracy compared to the traditionally used determination based on the very well known Rydberg constant. In future this quantity may be obtained very accurately through photon recoil measurements under way at various atomic physics laboratories. At present the ne structure constant from muonium hyper ne splitting from eq. (35) is ?1 = 137:036 004 7(48)(35ppb) :
(64)
If extracted in the traditional way from eq.(33), one nds ?1 = 137:035 996 3(80)(58ppb). The two standard deviation agreement of values determined from the electron magnetic anomaly and from the hyper ne splitting in the muonium atom may be interpreted as the most precise reassurance of the internal consistency of QED, as the rst case involves QED of free particles whereas in the second case distinctively di erent bound state QED approaches need to be applied [205]. The limitation of from muonium HFS arises mainly from to the muon mass. Therefore any better determination of the muon mass, respectively its magnetic moment, e.g. through a precise measurement of the reduced mass shift in the muonium 1s-2s splitting, will result in an improvement of this value of . It should be noted explicitly that already at present the good agreement between the ne structure constant determined from muonium hyper ne structure and the one from the electron magnetic anomaly is considered the best test of internal consistency of QED, as one case involves bound state QED and the other case free particles. It should be mentioned that a factor of ve improvement both in the hyper ne splitting of the ground state and of the muon magnetic moment, which would lead to a well improved ne structure 34
constant value from electromagnetically bound states, appears possible at present at PSI. With a dispersion corrected (see [206]) electrostatic chopper in the E5 area one can expect muon rates of order 5-7107+ /s in a chopped beam with a 1:3 on/o ratio. This is one order of magnitude above the particle ux at the LAMPF stopped muon channel with the additional bene t of a better beam spot.
5.2 Classical Lamb Shift in Muonium 5.2.1 Beam Foil Production of Muonium
Muonium in vacuum can be obtained from a beam neutralization technique analogous to methods in proton beam foil spectroscopy [207, 208]. Positive muons passing through metal foils in vacuum may capture to some fraction an electron thus forming muonium [209, 210], and negative muonium ions [211]. The electron capture depends on the velocity, but not on the mass of the projectile particle [212]. The rate is maximal for muon velocities equal to the Fermi energy of the electrons in the metal which is of the order of c. There is no strong dependence on the particular material of the metal foil observed [213]. For optimizing the muonium rate one has to take into account that due to multiple scattering in the muon production target the beam rates are proportional to the 3.5th power of the beam momentum. In addition, at low energies losses due to muon decay in ight become important. Therefore the optimal muonium production momentum is about 7.3 MeV/c at which a yield of 12% muonium atoms per incident muon was observed at LAMPF for a 200 g/cm2 Al foil [213]. The muonium atoms have typical kinetic energies of 5 keV. The angular distribution has been found to be dominated by multiple scattering of the muon in the target foil [209]. A fraction of 10% of the atoms is in the metastable 22S1=2 state [213] in accordance with a 1=n3 law for the distribution between quantum states as observed for atomic hydrogen [214]. The negative muonium ion (+ e? e? ) has been observed with a production yield of 1:5 10?5 per incident muon.
5.2.2 Classical 2S1=2 ? 2P1=2 Lamb Shift
In independent experiments at TRIUMF [70] and LAMPF [71, 215, 213] both the 22S1=2 ? 22P1=2 classical Lamb shift transitions and 22 S1=2 ? 22P3=2 ne structure transitions could be induced. Beam foil produced fast muonium atoms in the metastable 22 S1=2 state were passed through a section of a rf transmission line (22S1=2 ? 22P1=2) respectively a tuneable microwave cavity (22 S1=2 ? 22 P3=2). A transition to a P state is followed by the rapid decay into the ground state within the natural P state lifetime of P =1.6 ns. The occurrence of a transition was observed as a reduction of the 22S1=2 population which was probed further downstream by quenching the 22S1=2 state through admixing of the close lying short lived 22P1=2 state in an external electrostatic eld and by observing Lyman- radiation in CsI coated microchannel plates (TRIUMF) or UV sensitive photomultipliers (LAMPF). For small electric elds E (Stark energy small compared to the ne structure splitting, i.e. E < 3:5kV=cm), the lifetime of the 22S1=2 state is shortened to [155] 2
V 475 cm 2S ' 2P 41 + E
!23
5:
(65)
The apparatus of the TRIUMF experiment is shown in Fig.26 together with the signals obtained. The hyper ne structure is resolved. The ne structure measurement at LAMPF [213] has yielded a 22 S1=2 ? 22P3=2 transition frequency of 9 895+35 ?30 MHz which can be converted into a Lamb shift value by subtracting it from the calculated 22P1=2 ? 22P3=2 energy di erence of 10 921:833(3) MHz [216]. The results for the classical Lamb shift in muonium are listed in Table 7. In the experiments an accuracy of 1.4% could be reached. They are in good agreement with QED calculations [16]. But they are not yet an as severe test for theory as the measurements in hydrogen, where the classical Lamb shift is measured in a fast beam to be 2H2S1=2 ?22 P1=2 (exp)=1 057.845(9) MHz [217, 218] 35
Figure 26: Measurement of the muonium n=2 Lamb shift at TRIUMF [70]. (a) A schematic view of the experimental setup. Fast metastable 2S muonium is produced by a beam foil technique and Lamb shift transitions are induced in a rf transmission line. The 2S population is probed by an electric quench eld and Lyman- photon detection. (b) Signals at di erent power levels. Table 7: Classical 22 S1=2 ? 22P1=2 Lamb shift in muonium. Comparison between experiment and theory. Experiment 22S1=2 ?22 P1=2 [MHz] experimental method Ref. TRIUMF 1 070(+12)(-15) direct 22 S1=2 ? 22P1=2 trans. [70] LAMPF 1 042(+21)(-23) direct 22 S1=2 ? 22P1=2 trans. [215] LAMPF 1 027(+30)(-35) extracted from 22 S1=2 ? 22P3=2 trans. [213] Theory 1047.49(1)(9) { [16] and in an atomic interferometer to be 2HS1=2 ?2P1=2 (exp)=1 057.8514(19) MHz [13] in agreement with the theory which nds 2HS1=2 ?2P1=2 (theor) = 1 057:873(11) MHz [17]. Signi cant di erences other than reduced mass e ects between hydrogen and muonium are caused by the nuclear size e ect in hydrogen of 145(4) MHz. The di erences in the QED contributions are due to radiative recoil (+24.7 kHz) and higher order recoil (-19.5 kHz). This is new and untested physics which requires for an experimental accuracy of several kHz.
5.3 Ground State Hyper ne Structure Splitting in Vacuum
The precision measurements of the muonium hyper ne splitting performed so far su er from the interaction of the muonium atoms with the foreign gas atoms. With the discovery of polarized thermal muonium emerging from SiO2 powder targets and other materials into vacuum (see table 8) [219, 220] measurements of the HFS are now feasible in vacuum in the absence of a perturbing foreign gas. Corrections for density e ects are obsolete. In a pioneering experiment at PSI the transitions between the 12S1=2 ; F = 1 and the 12S1=2 ; F = 0 hyper ne levels could be induced in zero magnetic eld (see Fig. 1). The atoms were formed by electron capture after stopping positive muons close to the surface of a SiO2 powder target. A fraction of these di used to the surface and left the powder at thermal velocities (7.43(2) mm/s) 36
Table 8: Thermal muonium in vacuum production. Di erent methods and target materials are compared which produce thermal vacuum muonium with signi cantly di erent eciencies. The beam momentum has been optimized for maximum e ect. SiO2 powder targets are evidently at present the optimal choice as converter material. target material target density target thickness opt. muonium fraction Ref. [mg/cm3] [mg/cm2] + e? =stop [%] SiO2 powder 32 4.6 17(1) [219] SiO2 powder 32 2.8 15.9(3.6) [220] SiO2 powder 32 9.0 8.27(31) [221] SiO2 aerogel 5 7.5 2.32(13) [221] SiO2 aerogel 18 9 1.57(20) [221] W foil (2130 K) 19.3 96.5 4(2) [222] C60=70 fullerens 1400 210 1.85(23) [221] Cotton 10 3.6 2.25(46) [221] Cotton coated with SiO2 powder 17 5.8 11.43(31) [221] Microchannel plate 2000 400 2.44(34) [221]
Figure 27: Principle of the setup for a measurement of HFS in vacuum. for the adjacent vacuum region. A rectangular rf resonator operated in TEM301 mode was placed directly above the muonium production target. The atoms entered the cavity with thermal velocities through a wall opening. The e+ from the parity violating muon decay + ! e+ + e + were registered in two scintillator telescopes which were mounted close to the side walls of the resonator. The telescope axes have been oriented parallel respectively antiparallel to the spin of the incoming muons which had been rotated transverse to the muon propagation direction by an ExB separator (Wien lter) in the muon beam line. At the resonance frequency of 4.46329(3)GHz a reduction of the muon polarization of 16(2)% has been observed as a signal from the hyper ne transitions [227]. Certainly the method needs a lot of development work in order to improve the signal strength. However, ultimately one expects higher precision from experiments on muonium in vacuum than from measurements in gases.
37
Figure 28: First observed muonium HFS signal in vacuum.
6 Two-photon Spectroscopy of Muonium
The 12S1=2 and the 22S1=2 states in hydrogen-like systems are both of odd parity. Single photon electric dipole transitions are parity forbidden. Two-photon transitions [228, 229, 230, 231, 232, 233], however, are allowed. Indeed, they are the dominant radiative decay channel for the metastable 2S state leading to a lifetime of 2S =1/7 s [155] in natural hydrogen. Two-photon transitions are second order in perturbation theory. The strength of an absorption signal for two photons of equal energy depends strongly on the level structure of the atom under investigation. As a general rule, however, one needs very high intensities for achieving signi cant transition probabilities. Such intensities are available from pulsed laser sources only. Since muonium atoms can be produced at moderate quantities (as compared to natural hydrogen which is available from gas bottles) an experiment bene ts largely from a pulsed accelerator muon source. In addition, it is of importance for experiments that accidental background in the detectors can be suppressed in a pulsed experiment by suitable gating. The two-photon transition is Doppler-free to rst order [229], if it is induced by the absorption of one photon from each of two counter-propagating laser beams, because a moving atom sees the two beams Doppler shifted with opposite sign and to rst order by the same amount (see g. 29). Doppler free excitation of the 1s-2s transition in muonium has been achieved in the past at KEK in Tsukuba, Japan, [72] and at the Rutherford Appleton Laboratory (RAL) in Chilton, Didcot, UK [74, 75]. The accuracy of the last experiment was limited by ac Stark e ect (the e ect of the strong electric eld in the laser beams on the atomic level structure) and a frequency chirp (time dependent laser frequency variation) caused by rapid changes of the index of refraction in the dye solutions of the laser ampli er employed in the laser system. The experiment yielded 1S?2S = 2 455529002 (33)(46) MH z for the centroid (corrected for hyper ne structure) 1S-2S transition frequency. This value is in agreement with theory within two standard deviations [17]. The Lamb shift contribution to the 1S-2S splitting has been extracted to LS = 6 988 (33)(46) MHz; this the most precise experimental Lamb shift value for muonium available today. From the isotope shifts between the muonium signal and the hydrogen and deuterium 1S-2S two-photon resonances we deduce the mass of the positive muon as m = 105.65880 (29)(43) MeV/c2 . An alternate interpretation of the result yields the best test of the equality of the absolute value of charge units in the rst two generations of particles at the 10?8 level [75]. A new measurement of the 1S-2S energy splitting of muonium by Doppler-free two-photon spectroscopy has been performed at RAL where the worldwide brightest pulsed surface muon channel exists. At RAL protons are accelerated in the ISIS synchrotron to 800 MeV. The extracted 38
Figure 29: Principle of the muonium 1S-2S experiment. (a) The transition between the 12S1=2 and the 22 S1=2 state is induced by the simultaneous absorption of two photons from two counterpropagating laser beams at = 244:2 nm wavelength. The metastable 2S state is photoionized by a third photon from the same light eld and the muon set free by the ionization is detected as a signal for the two-photon transition. (b) The transition is Doppler-free to rst order for the absorption of two photons from each of the beams. proton beam shows a double pulse structure with 330 ns separation between pulses and has a repetition rate of 50 Hz. At a muon beam momentum of 26.5 MeV/c the DEVA port of the channel delivers about 3000 + to the experiment in a 10 % momentum band with almost 100% polarization, if a 5 mm carbon target is used for pion production [234]. An E B separator in the beam line (Wien lter) reduces positron background eciently to below 1.5%. The muon beam re ects the time pulse structure of the primary proton beam with a width of 82 ns for every single pulse component. For the experiment thermal muonium atoms in vacuum are produced by the SiO2 -powder technique (see sect. 5.3) [219]. The muonium production was optimized and monitored by observing the decay positrons from the decay + ! e+ + +e in a wire chamber telescope. The equipment has been optimized for the high instantaneous decay positrons rates after stopping a muon pulse in the target. More than one positron passes on average through the telescope within a muon lifetime . Fast wire chamber readout electronics has been especially developed which is based on VME multi-hit TDC modules. The muonium atoms interact close above the target ( 8 mm) with two counterpropagating high power laser beams at = 244:2 nm wavelength. The pulsed laser light arrives tL = 1:8 s after the muons in the interaction region. It has a diameter of 2rL = 2.5(5) mm and propagates in dL =8(1)mm distance parallel to the target. The laser beam crosses the muonium cloud 10 times between two mirrors and is then re ected back onto itself in order to enhance the overlap of the cloud and the laser light. Numerical estimates under the assumption of a cos angular distribution, where Theta is the angle between the atom velocity and the powder surface normal direction, and the Maxwell-Boltzmann velocity distribution yield about 1 atom in the laser beam for every laser pulse. 39
+ Ar Laser
Ti:sapphire Laser
FM Saturation Spectroscopy
AOM
Optical Fibers
Alexandrite Laser
Heterodynemeasurement
Frequency tripling
Fast beam diagnostics
Wavemeter + Interferometer
To Vacuumchamber
Figure 30: Laser system employed in the new muonium 1s-2s experiment.
Figure 31: Schematics of the detector setup for the RAL muonium 1s-2s laser spectroscopy experiment. The necessary high power UV laser light was generated in a newly developed laser system by frequency tripling the output of an alexandrite ring laser ampli er in crystals of LBO and BBO. Typically UV light pulses of energy 3 mJ and 80 nsec (FWHM) duration were used. The alexandrite laser was seeded with light from a continuous wave Ti:sapphire laser at 732 nm which was pumped by an Ar ion laser. Fluctuations of the optical phase during the laser pulse were compensated with an electro-optic device in the resonator of the ring ampli er to give a frequency chirping of the laser light of less than about 5 MHz. The laser frequency was calibrated by frequency modulation saturation spectroscopy of a hyper ne component of the 5-13 R(26) line in thermally excited iodine vapour. The frequency of the reference line is about 700 MHz lower than 1/6 of the muonium transition frequency. The cw light was frequency up-shifted by passing through two acousto-optic modulators (AOM's). The muonium reference line has been calibrated preliminarily to 3.4 MHz at the Institute of Laser Physics in Novosibirsk. An independent calibration at the National Physics Laboratory (NPL) at Teddington, UK, was performed to 140 kHz (0.35 ppb) [235]. The 1S-2S transition was detected by the photoionization of the 2S state by a third photon from 40
Figure 32: The 12S1=2 (F=1)!22S1=2(F=1) transition signal in muonium, not corrected for systematic shifts due to frequency chirping and ac Stark e ect. the same laser eld. The slow muon set free in the ionization process is accelerated to 2 keV and guided through a momentum and energy selective path onto a microchannel plate particle detector (MCP). Background due to scattered photons and other ionized particles can be reduced to less than 1 event in 5 hours by shielding and by requiring that the MCP count falls into a 100 nsec wide window centered at the expected time of ight for muons and by additionally requiring the observation of the energetic positrons from the muon decay. On resonance an event rate of 9 per hour was observed. The number of MCP events as a function of the laser frequency is displayed in Fig. 6. The line shape distortions due to frequency chirping were investigated theoretically using density matrix formalism to nd numerically the frequency dependence of the signal for the time dependent laser intensity and the time dependent phase of the laser eld which were both measured for every individual laser shot using fast digitzed amplitude and optical heterodyne signals [236, 237]. The model was veri ed experimentally at the ppb level by observing resonances in deuterium and hydrogen in the same apparatus. A careful analysis is in progress. Preliminary results indicate that the accuracy will be of order 10 MHz for the 1s-2s level splitting. It can be expected that future experiments will reach well below 1 MHz in accuracy promising an improved muon mass value. The theoretical value at present is basically limited by recoil terms at the 0.6 MHz level. Even cw laser excitation, which would totally avoid any chirping problems, may be possible at future high intensity muon sources under construction or in planning (see 7). At KEK a group was successful in producing a beam of slow muons, i.e. muons of energies in the eV region, by rst exciting muonium into the 2P state and the by photoionization with a second laser [238]. This may open a path leading to ultra slow muon beams for studies of thin lms or condensed matter surfaces.
7 Long term future possibilities It appears that the availability of particles limits the ability to derive much better values for fundamental constants, to nd nonstandard physical processes or to impose signi cantly improved limits in continuation of the search programs of existing dedicated experiments. Therefore any measure to boost the particle uxes is a very important step forward. The ? converter at PSI or the dedicated tailored muon production of the planned MECO experiment at BNL are examples of novel attempts to overcome this problem. In principle, we need signi cantly more intense 41
accelerators, such as they are presently discussed at various places. For some experiments, like improvements in the muon lifetime, studies of muon capture in light nuclei or precision muonium spectroscopy, one could have very valuable improvements on a short time scale from an intense chopped muon beam. Such a facility would be possible at PSI in the E3 or even better in the E5 beam area with some well designed chopping device added to the existing facilities [239]. For the mentioned experiments a pulsed beam structure is already at present more important than highest integral particle uxes. In the intermediate future the Japanese Hadron Facility (JHF) or a possible European Spallation Source (ESS) are important options. Also the discussed Oak Ridge neutron spallation source could in principle accommodate intense muon beams. The most promising facility would be, however, a muon collider [240], the front end of which could provide muon rates 5-6 orders of magnitude higher than present beams (see Table 9). Table 9: Muon uxes of some existing and future facilities, Rutherford Appleton Laboratory (RAL), Japanese Hadron Facility (JHF), European Spallation Source (ESS), Muon collider (MC). RAL(+ ) PSI(+ ) PSI(? ) JHF(+ )y ESS(+ ) MC (+ , ? ) 6 8 Intensity (/s) 3 10 3 10 1 108 4:5 107 4:5 107 7:5 1013 Momentum bite pm/p[%] 10 10 10 10 10 5-10 Spot size (cm cm) 1.22.0 3.32.0 3.32.0 1.52.0 1.52.0 fewfew Pulse structure 82 ns 50 MHz 50 MHz 300 ns 300 ns 50 ps 50 Hz continuous continuous 50 Hz 50 Hz 15 Hz
y Recent studies indicate that the 1011 particles/s region might be reachable [241].
At such facilities low and high energy experiments come close together. It was noted, for example, already in the early 60ies that, e.g. the process e? e? ! ? ? is closely related to muonium-antimuonium conversion [242]. Indeed such scattering experiments were carried out at the Princeton-Stanford storage rings at Stanford yielding the at the time best limit on the coupling constant GMM [243]. Today, similar proposals have been made for scattering of high energy e? on e? , e? on e+ , ? on ? and ? on + [244, 245, 246]. They were mainly discussed in connection with bileponic gauge bosons. Even a lower limit for the cross section of the process e? e? ! ? ? was found, provided the sum of the light neutrino masses exceeds 90 eV [246]. Pronounced resonances have been predicted particularly for such experiments at the Next Linear Collider or the high energy end of a muon collider. Although the nature of the muon, the reason for its existence ( as well as of other higher generation particles) remains a mystery, both the theoretical and experimental work in fundamental muon physics have contributed to a deeper understanding of particle interactions in general. One particular value of the experiments in the eld are their continuos contributions towards guiding theoretical developments by excluding various speculative models. It will remain a privilege of the future to nd - maybe - an answer to Prof. Rabi's important question, which he asked after learning about the leptonic nature of the muon:
"Who ordered that?"
8 Acknowledgments It is a pleasure to thank the organizers of this muon summer school for the opportunity to include a few pieces of the rich and exciting eld of fundamental muon physics. Part of the work reported was supported by the German Bundesminister fur Bildung und Forschung (BMBF) and by a NATO research grant. The author is grateful to his collaborators on several muon and muonium experiments, who all contributed to increasing our knowledge about the muon. 42
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