document: over the past 83 years , the study of dipole moments of elementary particles has provided a wealth of information on subatomic physics . from the pioneering work of stern@xcite through the discovery of the large anomalous magnetic moments of the proton@xcite and neutron@xcite 
   , the ground work was laid for the discovery of spin , of radiative corrections and the renormalizable theory of qed , of the quark structure of baryons and the development of qcd . 
   a charged particle with spin @xmath2 has a magnetic moment @xmath3 where @xmath4 is the gyromagnetic ratio , @xmath5 is the anomaly , and the latter expression is what one finds in the particle data tables.@xcite    the dirac equation tells us that for spin one - half point - like particles , @xmath6 for spin angular momentum , and is unity for orbital angular momentum ( the latter having been verified experimentally@xcite ) . for point particles , 
   the anomaly arises from radiative corrections , two examples of which are shown in fig . 
   [ fg : aexpan ] . 
   the lowest - order correction gives the famous schwinger@xcite result , @xmath7 , which was verified experimentally by foley and kusch.@xcite the situation for baryons is quite different , since their internal quark structure gives them large anomalies . 
   in general @xmath5 ( or @xmath8 ) is an expansion in @xmath9 , @xmath10 with 1 diagram for the schwinger ( second - order ) contribution , 5 for the fourth order , 40 for the sixth order , 891 for the eighth order . 
   the qed contributions to electron and muon 2 have now been calculated through eighth order , @xmath11 and the tenth - order contribution has been estimated.@xcite    .,scaledwidth=45.0% ]    .transformation properties of the magnetic and electric fields and dipole moments . [ cols="^,^,^,^",options="header " , ]     the magnetic and electric dipole moments can be represented as the real and imaginary parts of a generalized dipole operator @xmath12 , and the interaction lagrangian becomes @xmath13 \mu f_{\alpha \beta}\ ] ] with @xmath14 and @xmath15 . 
   the electron anomaly is now measured to a relative precision of about four parts in a billion ( ppb),@xcite which is better than the precision on the fine - structure constant @xmath16 , and kinoshita has used the measured electron anomaly to give the best determination of @xmath16.@xcite the electron anomaly will be further improved over the next few years.@xcite    the muon anomaly is measured to 0.5 parts per million ( ppm).@xcite the relative contributions of heavier particles to @xmath5 scales as @xmath17 , so the muon has an increased sensitivity to higher mass scale radiative corrections of about 40,000 over the electron . at a precision of @xmath18 ppm , 
   the muon anomaly is sensitive to @xmath19 gev scale physics . 
   the standard model value of @xmath20 has measurable contributions from three types of radiative processes : qed loops containing leptons ( @xmath21 ) and photons;@xcite hadronic loops containing hadrons in vacuum polarization loops;@xcite and weak loops involving the @xmath22 and @xmath23 weak gauge bosons ( the standard model higgs contribution is negligible),@xcite @xmath24 a significant difference between the experimental value and the standard model prediction would signify the presence of new physics . 
   a few examples of such potential contributions are lepton substructure , anomalous @xmath25 couplings , and supersymmetry.@xcite    the cern experiment@xcite observed the contribution of hadronic vacuum polarization shown in fig . 
   [ fg : had](a ) at the 8 standard deviation level . 
   unfortunately , the hadronic contribution can not be calculated directly from qcd , since the energy scale is very low ( @xmath26 ) , although blum@xcite has performed a proof of principle calculation on the lattice . 
   fortunately dispersion theory gives a relationship between the vacuum polarization loop and the cross section for @xmath27 , @xmath28 where @xmath29 and experimental data are used as input . 
   the factor @xmath30 in the dispersion relation , means that values of @xmath31 at low energies ( the @xmath32 resonance ) dominate the determination of @xmath33 . in principle 
   , this information could be obtained from hadronic @xmath34 decays such as @xmath35 , which can be related to @xmath36 annihilation through the cvc hypothesis and isospin conservation.@xcite however , inconsistencies between information obtained from @xmath36 annihilation and hadronic tau decays , plus an independent confirmation of the cmd2 high - precision @xmath36 cross - section measurements by the kloe collaboration,@xcite have prompted davier , hcker , et al , to state that until these inconsistencies can be understood only the @xmath36 data should be used to determine @xmath33.@xcite         conversion , showing the relevant slepton mixing matrix elements . 
   the mdm and edm give the real and imaginary parts of the matrix element , respectively . ] 
   the hadronic light - by - light contribution ( see fig . 
   [ fg : had](e ) ) has been the topic of much theoretical investigation.@xcite unlike the lowest - order contribution , it can only be calculated from a model , and this contribution is likely to provide the ultimate limit to the precision of the standard - model value of @xmath20 . 
   one of the very useful roles the measurements of @xmath20 have played in the past is placing serious restrictions on physics beyond the standard model . with the development of supersymmetric theories as a favored scheme of physics beyond 
   the standard model , interest in the experimental and theoretical value of @xmath20 has grown substantially . 
   susy contributions to @xmath20 could be at a measurable level in a broad range of models . 
   furthermore , there is a complementarity between the susy contributions to the mdm , edm and transition moment for the lepton - flavor violating ( lfv ) process @xmath37 in the field of a nucleus . 
   the mdm and edm are related to the real and imaginary parts of the diagonal element of the slepton mixing matrix , and the transition moment is related to the off diagonal one , as shown in fig . 
   [ fg : susy ] . 
   this reaction , along with the companion lfv decay @xmath38 , will be searched for in `` next generation '' experiments now under construction.@xcite    from neutrino oscillations we already know that lepton flavor is violated , and this violation will be enhanced if there is new dynamics at the tev scale . 
   this same new physics could also generate measurable effects in the magnetic and electric dipole moments of the muon as well.@xcite 
   the method used in the third cern experiment and the bnl experiment are very similar , save the use of direct muon injection@xcite into the storage ring,@xcite which was developed by the e821 collaboration . 
   these experiments are based on the fact that for @xmath39 the spin precesses faster than the momentum vector when a muon travels transversely to a magnetic field . 
   the spin precession frequency @xmath40 consists of the larmor and thomas spin - precession terms . 
   the spin frequency @xmath40 , the momentum precession ( cyclotron ) frequency @xmath41 , are given by @xmath42 the difference frequency @xmath43 is the frequency with which the spin precesses relative to the momentum , and is proportional to the anomaly , rather than to @xmath8 . 
   a precision measurement of @xmath20 requires precision measurements of the muon spin precession frequency @xmath44 , and the magnetic field , which is expressed as the free - proton precession frequency @xmath45 in the storage ring magnetic field . 
   the muon frequency can be measured as accurately as the counting statistics and detector apparatus permit . the design goal for 
   the nmr magnetometer and calibration system was a field accuracy of 0.1 ppm . 
   the @xmath46 which enters in eq . 
   [ eq : omeganoe ] is the average field seen by the ensemble of muons in the storage ring . in e821 we reached a precision of 0.17 ppm in the magnetic field measurement . 
   an electric quadrupole@xcite is used for vertical focusing , taking advantage of the `` magic ''  @xmath47 at which an electric field does not contribute to the spin motion relative to the momentum . with both an electric and a magnetic field 
   , the spin difference frequency is given by @xmath48 , \label{eq : tbmt}\ ] ] which reduces to eq . 
   [ eq : omeganoe ] in the absence of an electric field . for muons with @xmath49 in an electric field alone , the spin would follow the momentum vector . 
   the experimental signal is the @xmath50 from @xmath51 decay , which were detected by lead - scintillating fiber calorimeters.@xcite the time and energy of each event was stored for analysis offline . 
   muon decay is a three - body decay , so the 3.1 gev muons produce a continuum of positrons ( electrons ) from the end - point energy down . 
   since the highest energy @xmath50 are correlated with the muon spin , if one counts high - energy @xmath50 as a function of time , one gets an exponential from muon decay modulated by the @xmath52 precession . 
   the expected form for the positron time spectrum is @xmath53 $ ] , however in analyzing the data it is necessary to take a number of small effects into account in order to obtain a satisfactory @xmath54 for the fit.@xcite the data from our 2000 running period is shown in fig . 
   [ fg : wig00 ]    the experimental results from e821 are shown in fig . 
   [ fg : amu ] , with the average @xmath55 which determines the `` world average '' . 
   the theory value@xcite @xmath56 is taken from hcker et al.,@xcite , which updates their earlier analysis@xcite with the kloe data;@xcite and from hagiwara , et al.,@xcite who use a different weighting scheme for the experimental data when evaluating the dispersion integral but do not include the kloe data . 
   when this theory value is compared to the standard model value using either of these two analyses@xcite for the lowest - order hadronic contribution , one finds @xmath57 or a discrepancy of 2.7 standard deviations .    . 
   the strong interaction contribution is taken from references @xcite and @xcite . ]    to show the sensitivity of our measurement of @xmath20 to the presence of virtual electroweak gauge bosons , we subtract off the electroweak contribution of @xmath58 from the standard model value , compare with experiment and obtain @xmath59 a 4.7 standard deviation discrepancy . 
   this difference shows conclusively that e821 was sensitive to physics at the 100 gev scale . at present , it is inconclusive whether we see evidence for contributions from physics beyond the standard - model gauge bosons .    with each data set , 
   the systematic error was reduced , and for the final data set taken in 2001 the systematic error on @xmath60 was 0.27 ppm with a statistical error of 0.66 ppm . given the tantalizing discrepancy between our result and the latest standard - model value , and the fact that the hadronic error could be reduced by about a factor of two over the next few years,@xcite we submitted a new proposal to brookhaven to further improve the experimental measurement . 
   the goal of this new experiment is @xmath61 ppm total error , with the goal of controlling the total systematic errors on the magnetic field and on the muon frequency measurement to 0.1 ppm each . 
   our proposal@xcite was given enthusiastic scientific approval in september 2004 by the laboratory , and has been given the new number , e969 . 
   negotiations are underway between the laboratory and the funding agencies to secure funding .    a letter of intent ( loi ) for an even more precise 2 experiment was also submitted to j - parc.@xcite in that loi we proposed to reach a precision below 0.1 ppm . since it is not clear how well the hadronic contribution can be calculated , and whether the new brookhaven experiment e969 will go ahead 
   , we will evaluate whether to press forward with this experiment at a later time . 
   our loi at j - parc@xcite was predicated on pushing as far as possible at brookhaven before moving to japan . 
   while the mdm has a substantial standard model value , the predicted edms for the leptons are unmeasurably small and lie orders of magnitude below the present experimental limits given in table [ tb : edm ] . 
   an edm at a measurable level would signify physics beyond the standard model . 
   susy models , and other dynamics at the tev scale do predict edms at measurable levels.@xcite    a new experiment to search for a permanent edm of the muon with a design sensitivity of @xmath62 @xmath63-cm is being planned for j - parc.@xcite this sensitivity lies well within values predicted by some susy models.@xcite feng , et al.,@xcite have calculated the range of @xmath64 available to such an experiment , assuming a new physics contribution to @xmath20 of @xmath65 , @xmath66 where @xmath67 is a _ 
   cp _ violating phase . 
   this range is shown in fig . 
   [ fg : phicp ] . 
   available to a dedicated muon edm experiment.@xcite the two bands show the one and two standard - deviation ranges if @xmath20 differs from the standard model value by @xmath68 . ]    of course one wishes to measure as many edms as possible to understand the nature of the interaction . 
   while naively the muon and electron edms scale linearly with mass , in some theories the muon edm is greatly enhanced relative to linear scaling relative to the electron edm when the heavy neutrinos of the theory are non - degenerate.@xcite     and @xmath69 . ]    with an edm present , the spin precession relative to the momentum is given by @xmath70   \nonumber \\ & \qquad \ \   + & { e \over m}\left [ { \eta \over 2 } \left ( { \vec e \over c } + \vec \beta \times \vec b \right ) \right ]   \label{eq : omegawedm}\end{aligned}\ ] ] where @xmath71 and @xmath72 . for reasonable values of @xmath73 , the motional electric field 
   @xmath74 is much larger than electric fields that can be obtained in the laboratory , and the two vector frequencies are orthogonal to each other . 
   the edm has two effects on the precession : the magnitude of the observed frequency is increased , and the precession plane is tipped relative to the magnetic field , as illustrated in fig . 
   [ fg : omegaeta ] . 
   e821 was operated at the magic @xmath75 so that the focusing electric field did not cause a spin precession . in e821 
   the tipping of the precession plane is very small , ( @xmath76 mrad ) if one uses the cern limit@xcite given in table [ tb : edm ] . 
   this small tipping angle makes it very difficult to observe an edm effect in e821 , since the 2 precession ( @xmath44 ) is such a large effect . 
   we have recently introduced a new idea which optimizes the edm signal , and which uses the motional electric field in the rest frame of the muon interacting with the edm to cause spin motion.@xcite the dedicated experiment will be operated off of the magic @xmath75 , for example at @xmath77 mev / c , and will use a radial electric field to stop the @xmath52 precession.@xcite then the spin will follow the momentum as the muons go around the ring , except for any movement arising from an edm . thus the edm would cause a steady build - up of the spin out of the plane with time . 
   detectors would be placed above and below the storage region , and a time - dependent up - down asymmetry @xmath78 would be the signal of an edm , @xmath79 a simulation for @xmath80 cm is given in fig . 
   [ fg : edmsig ] .     or @xmath81 ] 
   the figure of merit for statistics in the edm experiment is the number of muons times the polarization . in order to reach @xmath82 cm , 
   the muon edm experiment would need @xmath83 , a flux only available at a future facility . 
   while progress can still be made at brookhaven on @xmath20 , a dedicated muon edm experiment must be done elsewhere . 
   measurements of the muon and electron anomalies played an important role in our understanding of sub - atomic physics in the 20th century . 
   the electron anomaly was tied closely to the development of qed . 
   the subsequent measurement of the muon anomaly showed that the muon was indeed a `` heavy electron '' which obeyed qed.@xcite with the sub - ppm accuracy now available for the muon anomaly,@xcite there may be indications that new physics is beginning to appear in loop processes.@xcite    the non - observation of an electron edm is becoming an issue for supersymmetry , just as the non - observation of a neutron edm implies such a mysteriously ( some would say un - naturally ) small @xmath84-parameter for qcd . 
   the search for edms will continue , and if one is observed , the motivation for further searches in other systems will be even stronger . 
   the muon presents a unique opportunity to observe an edm in a second - generation particle , where the _ cp _ phase might be different from the first generation , or the scaling with mass might be quadratic rather than linear .      if susy turns out to be _ the _ extension to the standard model , then there will be susy enhancements to @xmath20 to the muon edm and also to the amplitudes for lepton flavor violating muon decays . 
   once the susy mass spectrum is measured , @xmath20 will provide a very clean measurement of @xmath85.@xcite    if susy or other new dynamics at the tev scale are not found at lhc , then precision experiments , which are sensitive through virtual loops to much higher mass scales than direct searches for new particles , become even more important . 
   experiments such as edm searches , 2 and searches for lepton flavor violation , all carried out at high intensity facilities , may provide the only way to probe these higher energy scales . 
   opportunities at future high intensity facilities are actively being pursued , and both the theoretical and experimental situations are evolving . 
   it is clear that the study of lepton moments and lepton flavor violation , along with neutron edm searches , will continue to be a topic of great importance in the first part of the 21st century . 
   _ acknowledgments : i wish to thank my colleagues on the muon 2 experiment , as well as m. davier , j. ellis , e. de rafael , w. marciano and t. teubner for helpful discussions . 
   special thanks to y. semertzidis for critically reading this manuscript . _                                  j.l . 
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   carey , a. gafarov , i. logashenko , k.r . 
   lynch , j.p . 
   miller , b.l . 
   roberts ( co - spokesperson ) , g. bunce , w. meng , w.m . 
   morse ( resident spokesperson ) , y.k . 
   semertzidis , d. grigoriev , b.i . 
   khazin , s.i . 
   redin , yuri m. shatunov , e. solodov , y. orlov , p. debevec , d.w . 
   hertzog ( co - spokesperson ) , p. kammel , r. mcnabb , f. mlhauser , k.l . 
   giovanetti , k.p . 
   jungmann , c.j.g . 
   onderwater , s. dhamija , t.p . 
   gorringe , w. korsch , f.e . 
   gray , b. lauss , e.p . 
   sichtermann , p. cushman , t. qian , p. shagin , s. dhawan and f.j.m . 
   farley , which can be found at : http://g2pc1.bu.edu/@xmath88roberts/
summary: from the famous experiments of stern and gerlach to the present , measurements of magnetic dipole moments , and searches for electric dipole moments of `` elementary '' particles have played a major role in our understanding of sub - atomic physics . in this 
   talk i discuss the progress on measurements and theory of the magnetic dipole moment of the muon . 
   i also discuss a new proposal to search for a permanent electric dipole moment ( edm ) of the muon and put it into the more general context of other edm searches . 
   these experiments , along with searches for the lepton flavor violating decays @xmath0 and @xmath1 , provide a path to the high - energy frontier through precision measurements .