and supersymmetry: status and prospects
The experimental determination of the muon magnetic moment and its theoretical prediction within the Standard Model and the MSSM are reviewed. A 3 deviation between experiment and Standard Model prediction has been established, and supersymmetry could provide a natural explation of this deviation. Possible future improvements and the case for a new experiment are discussed.
pacs:13.40.EmElectric and magnetic moments and 14.60.EfMuons and 12.60.JvSupersymmetric models
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For decades, the main arguments in favour of supersymmetry at or below the TeV-scale have been of a theoretical nature and have been related to e.g. naturalness in the Higgs sector or unification of gauge couplings. Now, the muon magnetic moment has developed into one of the strongest and most robust observational indications for the existence of supersymmetry at or below the TeV-scale. Independent of theoretical naturalness or unification arguments, a 3 deviation between the experimental and Standard Model (SM) theory value of has been established, and this deviation can be well explained by TeV-scale supersymmetry but is very hard to accomodate within many other scenarios for physics beyond the SM. In these proceedings111These proceedings are based on review (); WhitePaper (). we briefly review the current status of within the SM and the minimal supersymmetric Standard Model (MSSM) and we discuss possible future improvements.
2 Deviation between the experimental value and the SM theory prediction
The muon magnetic moment has been measured at the recent E821 experiment at Brookhaven. The final result of this experiment reads BNLfinal ()
The success of this experiment has inspired tremendous progress also on the SM theory evaluation of , particularly on the hadronic vacuum polarization contributions and hadronic light-by-light contributions, the two contributions with the by far largest theory uncertainties.
The hadronic light-by-light contributions are tiny but important at the current level of precision. They cannot be evaluated from first principles. In the 1990’s these contributions have been evaluated by two groups BPP (); HayakawaKinoshita (). Since then, major progress has been made in two directions: first a sign error in these calculations was uncovered in KnechtNyffeler () (and confirmed by the original authors), the correction of which shifted the SM theory prediction by more than . Second, new short-distance constraints on the relevant light-by-light correlator were studied and incorporated into the computation in MelnV (), which again shifted the result by about . The recent developments are reviewed in more detail in BijnensPrades (); MillerRR (); CzarneckiProc (), and current estimates for the hadronic light-by-light contributions vary between
The hadronic vacuum polarization contributions are currently the dominant source of the SM theory uncertainty. Via the optical theorem, they can be related to the cross section for hadrons, which can be measured. The recent progress is due to refined ways to combine existing experimental data on hadrons, obtained from different experiments and for different energies, and to improved measurements of hadrons. In the last 10 years, new results on this cross section have become available from BES-II BES (), CMD-2 CMD2a (), and most recently from SND SND () and CMD-2 CMD2b (), both in Novosibirsk, and from KLOE KLOE () and BaBar. The KLOE measurement is particularly interesting since it is the first one using radiative return measurements. Three major groups Davier06 (); Teubner06 (); Jegerlehner () have presented updated evaluations that incorporate the latest measurements222They differ e.g. in the way they incorporate the KLOE data., with results in the range
In principle, part of the hadrons cross section could be obtained in an alternative way from hadronic decays Alemanyetal (). This was particularly useful when the data was rather imprecise and dominated by only the CMD-2 data. Now, several data sets are available, and a disagreement between based and data based analyses of has led most groups to a preference of the theoretically cleaner based analyses (see e.g. the discussions in Davier06 (); MillerRR ()).
After the most recent progress the SM theory prediction for has reached a very mature state. The full prediction is obtained by adding the QED and electroweak to the hadronic contributions. The review MillerRR () obtains
a deviation! The results obtained in Davier06 (); Teubner06 (); Jegerlehner () differ slightly but all obtain deviations of more than . Therefore a 3 deviation between the experimental and the SM theory value of has been firmly established.
3 Muon magnetic moment and supersymmetry
If the observed deviation is not due to an error or a statistical fluctuation, where could it come from? If supersymmetry (SUSY) exists, the superpartner particles would give rise to a contribution to of approximately
where denotes the common superpartner mass scale, the ratio of the two Higgs vacuum expectation values, and the Higgsino mass parameter. Hence, supersymmetry could easily be the origin of the observed deviation of , e.g. for SUSY masses of roughly 200 GeV and or SUSY masses of 500 GeV and .
Although this result is very well known and has been stressed many times, see e.g. CzMarciano (), it is quite non-trivial and singles out supersymmetry among many extensions of the SM. One should note that the deviation of is almost twice as high as the SM electroweak contributions, i.e. diagrams with , , Higgs exchange etc.,
Likewise, a generic extension of the SM with weakly interacting particles and characteristic mass scale will be suppressed by and lead to contributions of the order
which is far too small except for very small , which is typically already ruled out.
Supersymmetry has two advantages compared to such generic extensions of the SM: First, masses for the relevant supersymmetric particles, mainly smuons and charginos as small as GeV are still experimentally allowed. Second, the parameter can provide an enhancement by a factor of up to about 50.
The behaviour can be easily explained on a diagrammatic level. Each diagram contributing to must contain a chirality flip between a left- and a right-handed (s)muon. The -enhancement arises in diagrams where the necessary chirality flip occurs at a muon Yukawa coupling, either to a Higgsino or Higgs boson, because this coupling is enhanced by compared to its SM value. The -parameter mediates the transition between the two Higgs/Higgsino doublets , and this transition enhances diagrams because only couples to muons while has the larger vacuum expectation value, . Therefore, all -enhanced terms are also proportional to sign(). This behaviour is not restricted to the one-loop level but repeats itself in higher orders.
3.2 Status of the MSSM prediction for the muon magnetic moment
The fact that supersymmetry is potentially the origin of the observed deviation justifies a precise analysis of the prediction for within the MSSM (for a review see review ()). The MSSM prediction is given by the SM prediction plus the genuine SUSY contributions, arising from diagrams with SUSY particle loops.
The SUSY one-loop contributions consist of diagrams with chargino/sneutrino or neutralino/smuon loops. These diagrams have been known for a long time oneloop (). The full expression is not repeated here. The approximation (6) can serve as a guideline. The mass parameters governing the one-loop SUSY contributions are mainly the left-handed smuon mass and the gaugino mass , while and the right-handed smuon mass have a smaller influence. If all mass parameters are equal to , (6) is an excellent approximation. If there are mass splittings, (6) still provides a reasonable estimate if is identified with a value between and . Contrary to the other mass parameters, increasing can lead to enhancements, e.g. if due to a diagram with bino exchange, which is directly .
At the two-loop level two kinds of SUSY contributions are known. QED-logarithms arising from SUSY one-loop diagrams with additional photon exchange have been evaluated in twoloopB () and amount to of the one-loop contributions. Two-loop diagrams involving closed loops of either sfermions (stops, sbottoms, etc) or charginos/neutralinos have been evaluated in twoloopA (). They amount to about of the one-loop contributions if all SUSY masses are degenerate but can be much larger, e.g. if smuon masses are very heavy but stops and/or charginos and Higgs bosons are light.
A very important question regards the remaining theory error of the SUSY contributions to . This theory error arises from unknown two-loop and higher order contributions. It has been estimated in review () to
which is smaller than the current SM theory error and the experimental uncertainty.
3.3 Implications on SUSY phenomenology
Fig. 1 summarizes the current status of and SUSY. A scan of the MSSM parameter space has been performed (for and taking into account experimental constraints from e.g. Higgs searches and -physics; for further details see review ()), and the resulting values for , including all known one- and two-loop contributions, are plotted as a function of the mass of the lightest observable SUSY particle. Fig. 1 confirms again that SUSY can easily explain the observed deviation if is below about 600 GeV.
Apart from , significant information on SUSY parameters can be inferred from the measured dark matter density, if it is assumed to consist of the stable, lightest SUSY particle. The two observables tend to constrain orthogonal directions in the multi-dimensional SUSY parameter space and are thus complementary. Several recent comprehensive studies Rosk (); All (); Ellis (); Sfitter () have shown that the MSSM is able to simultaneously accomodate all existing data from , dark matter, -physics and electroweak precision observables. This is even possible, in spite of some slight tensions, in the constrained MSSM (CMSSM), a model with only 4 input parameters. One result of these studies is that rather low SUSY masses are preferred as a consequence of the deviation.
4 Future prospects and the case for a new experiment
The present deviation is one of the strongest observational hints for the existence of supersymmetry at or below the TeV-scale. However, although the deviation is tantalizing it is not quite large enough to be regarded as a proof of physics beyond the SM. Fortunately, there are good prospects that the current uncertainty of of the deviation (5) can be reduced significantly in the near future.
The current theory error of of the SM prediction (4) will soon decrease due to currently ongoing more precise determinations of the hadrons cross section. Both KLOE and BaBar will soon release data on the most important channel using radiative return measurements. If these data are in agreement with the Novosibirsk data, they will not only reduce the error but also significantly increase our confidence in the data. The new data will immediately improve our knowledge of the hadronic vacuum polarization contributions to , which currently are the dominant source of error.
The second most important source of theory error are the hadronic light-by-light contributions. These are notoriously difficult to evaluate, but they have moved into the centre of attention, and several groups are currently investigating these contributions, using both established and novel approaches. A determination of these contributions with a relative accuracy of about seems possible. In combination, a reduction of the theory error of the SM prediction down to within the next few years seems likely.
The tantalizing status of the current deviation, together with the prospect for an improvement of the SM theory prediction, highlights the need for and the potential of a new, better experimental measurement of WhitePaper (). A corresponding experiment, E969 at Brookhaven E969 (), with the goal of a final uncertainty of , has been proposed and received scientific approval at Brookhaven.
In an optimistic scenario, where the theory error is reduced to and the magnitude of the deviation between SM theory and experiment remains the same, this new measurement would lead to
This more than 7 deviation would dramatically sharpen the case for new physics. The impact it would have on SUSY phenomenology is illustrated in figs. 2, 3 WhitePaper (). Fig. 2 shows the same scan of the possible SUSY contributions to as in fig. 1, versus the future deviation. The precision of (10) would lead to strong upper and lower mass bounds on SUSY particles which could complement mass measurements from LHC.
Fig. 3 illustrates how might complement even comprehensive LHC measurements. The analysis in SfitterOld (); Sfitter () shows that using a global fit of the MSSM to LHC data one can determine SUSY masses rather precisely but the parameter rather poorly. If the benchmark point SPS1a SPS () is realized, the LHC-analysis of SfitterOld () yields , the improved analysis of Sfitter () yields . Since is directly proportional to , a precise determination as in (10) would provide an invaluable complement to LHC in the determination of . Fig. 3 shows the value of as a function of . In all parameters except for have been fixed to the SPS1a values, which are accessible well at the LHC.
A tantalizing deviation of more than between the SM theory prediction and the experimental value of has been established. Supersymmetry with rather light masses and moderate to large could easily be the origin of this deviation. The near future is very promising if the proposed E969 experiment E969 () is realized. The SM theory uncertainty will soon further decrease and a new experiment could push the significance of the deviation up to more than 7.
Acknowledgments: It is a pleasure to thank the organizers of SUSY07 for this enjoyable conference.
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