DESY 18-057, HU-EP-18/11 April 2018
The Muon g-2 in Progress††thanks: Presented at the XXIV Cracow EPIPHANY Conference on Advances in Heavy Flavour Physics,
9-12 January 2018, Crakow, Poland. Dedicated to the memory of Maria Krawczyk.To appear in Acta Physica Polonica B.
Two next generation muon experiments at Fermilab in the US and at J-PARC in Japan have been designed to reach a four times better precision from 0.54 ppm to 0.14 ppm and the challenge for the theory side is to keep up in precision as far as possible. This has triggered a lot of new research activities. The main motivation is the persisting 3 to 4 deviation between standard theory and experiment. As Standard Model predictions almost without exception match perfectly all other experimental information, the deviation in one of the most precisely measured quantities in particle physics remains a mystery and inspires the imagination of model builders. Plenty of speculations are aiming to explain what beyond the Standard Model effects could fill what seems to be missing. Here very high precision experiments are competing with searches for new physics at the high energy frontier lead by the Large Hadron Collider at CERN. Actually, the tension is increasing steadily as no new states are found which could accommodate the discrepancy. With the new muon experiments this discrepancy would go up at least to 6 , in case the central values do not move, up to 10 could be reached if the present theory error could be reduced by a factor of two. Interestingly, the new from Berkeley by R. H. Parker et al. Science 360, 191 (2018): gives an prediction such that shows a deviation now.
PACS numbers: 14.60.Ef,13.40.Em
A particle with spin like the muon exhibits a magnetic moment :
Its Dirac value is modified by radiative corrections known as the muon anomaly. The electromagnetic lepton vertex tested in the static limit here is the simplest object you can think of.
The muon anomaly is responsible for the Larmor spin precession and for its tracking one needs polarized muons orbiting in a homogeneous magnetic field. To this end one is shooting protons on a target producing pions which decay by the parity violating weak process into polarized muons of negative helicity which are injected into a storage ring where they decay producing positrons flying preferably in direction of the spin of the decaying muon. For helicity and electron flight direction are reversed. Indeed the two parity violating weak decays perfectly transport the needed spin precession information.
The Larmor precession frequency developing in the beam of polarized spinning muons injected into a homogeneous magnetic field is detected by counting the positrons or electrons ejected by the decaying muons preferably along the spin vector.
In storage ring type experiments as the CERN, Brookhaven and Fermilab experiments the muon beam has to be focused by electric quadrupole fields , but the beam dynamics can be kept simple by running at the “Magic Energy” where is directly proportional to . At magic energy at about 3.1 GeV indeed we have
First lepton magnetic moment measurements were by Stern and Gerlach in 1922 revealing the famous factor and much later by Kusch and Foley in 1948 who first observed the anomaly for the electron.
A crucial point is that at 3.1 GeV the muons life-time in the lab frame is by times longer than in the rest frame. This makes it possible to store and let muons circle in a storage ring.
A precise experimental determination of has to be based on measurements of ratios of frequencies. From and and using or and eliminating the muon mass one obtains in terms of 3 frequency measurables: the free proton NMR frequency , the muon Larmor precession frequency , and from the muonium hyperfine splitting experiment at LAMPF . The actual result from BNL ( updated) is 
To come are two complementary experiments: the magic improved muon experiment at Fermilab, tuning , and a novel cold muons experiment at J-PARC using a small storage ring at . Both experiments attempt to improve the error by a factor 4. Most importantly, the ultra relativistic muons (CERN, BNL, Fermilab) and the ultra cold muons (J-PARC) experiments exhibit very different systematics and the latter will provide an important cross check of the magic gamma ones (see  and references therein). More on the experimental aspects and status the reader may find in the contribution by Lusiani , in these Proceedings.
The muon anomalous magnetic moment is a number represented by an overlay of a large number of
individual quantum corrections of different sign, which depend on a
few fundamental parameters. In any renormalizable theory like the SM
it is an unambiguous prediction of that theory. It is an ideal monitor
for physics beyond the SM. The muon is about a
factor 19 or 46 (if theory uncertainties included) more sensitive to
New Physics (NP) than the electron as we expect . The new muon search
for NP will take place as usual by confronting the new experiments
with SM theory
and the goal is to reach a precision in experiments and in theory. The coming round of “digging deeper” into the virtual quantum world is based on an improvement of the 5 numbers that have relevant uncertainties. These are , and , experimentally limited at 120 ppb. The expected experimental improvement will increase to if theory as today and to to if the SM prediction is improved by a reduction of the hadronic uncertainty by a factor 2, which concerns and That’s what we hope to achieve! A case that promises New Physics to be seen with high significance.
In the following I will focus on the parts of the SM prediction which are limiting its precision, the leading order hadronic photon vacuum polarization (LO-HVP) and the hadronic light-by-light (HLbL) contributions.
2 Evaluation of the Leading Order
The hadronic contribution to the vacuum polarization can be evaluated, with the help of dispersion relations (DR), from the energy scan of the ratio which can be measured up to some energy above which we can safely use perturbative QCD (pQCD) thanks to asymptotic freedom of QCD. Note that the DR requires the undressed (bare) cross–section . The lowest order HVP contribution is given by
where is a known kernel function growing form at the thresholds to 1 as . The integral is dominated by the resonance peak shown in Fig. 2. The –data are displayed in Fig. 3. I apply pQCD from 5.2 GeV to 9.46 GeV and above 11.5 GeV.
Figure 4 shows the distribution of contributions and errors between different energy ranges. One of the main issues is in the region from 1.2 GeV to 2.0 GeV (see Fig. 5), where more than 30 exclusive channels must be measured and although it contributes about 14% only of the total it contributes about 42% of the uncertainty.
In the low energy region, which is particularly important for the dispersive evaluation of the hadronic contribution to the muon , data have improved dramatically in the past decade for the dominant channel (CMD-2 , SND/Novosibirsk , KLOE/Frascati [10, 11, 12, 13, 14], BaBar/SLAC , BES-III/Beijing ), CLEOc/Cornell  and the statistical errors are a minor problem now. Similarly, the important region between 1.2 GeV to 2.4 GeV has been improved a lot by the BaBar exclusive channel measurements in the ISR mode [18, 19, 20, 21]. Recent data sets collected are: , and from CMD-3 [22, 23], and , , , , , and from SND [24, 25, 26].
Above 2 GeV fairly accurate BES-II data  are available. A new inclusive determination of in the range 1.84 to 3.72 GeV has been obtained with the KEDR detector at Novosibirsk  (see figures 3 and 5). Recent new experimental input for HVP has been obtained by CMD-3 and SND at VEPP-2000 via energy scan and by BESIII at PEPC in the ISR setup. In Fig. 6 I show a collection of results obtained by various groups since 2009. Figure 6 illustrates the progress as well as the major uncertainties of SM predictions.
Remarkable progress has been achieved by lattice QCD groups in calculating . Primary object for HVP in LQCD is the electromagnetic current correlator in Euclidean configuration space, which yields the vacuum polarization function needed to calculate The integrand and the need for lattice size extrapolation is illustrated in Fig. 7.
Results are shown in Fig. 8. The major part of LQCD uncertainties comes from the need of extrapolations (finite volume, lattice spacing and physical parametrers if not simulated at the physical point). In fact the momentum region below ( the lattice size) which for presently accessible accounts for about 40% of the integral can only be obtained by extrapolation. The very precise RBC/UKQCD point is obtained by combining the directly accessible lattice results only (33.5%) with R–data (66.5%) where the latter are more precise.
3 Hadronic Light-by-Light Contribution: Problems, Results
Key object is the hadronic contribution to the full rank-four
light-by-light scattering tensor ( denoting the photon field)
which embodies the four electromagnetic current amplitude
The hadronic part with shares the following characteristic properties: 1) it is a non-perturbative object, 2) the covariant decomposition involves 138 Lorentz structures (43 gauge invariant), 3) 28 amplitudes can contribute to , by permutation symmetry 19 thereof are independent, 4) fortunately HLbL is dominated by the pseudoscalar exchanges described by the effective Wess-Zumino Lagrangian, 5) generally, pQCD is used to evaluate the short distance (S.D.) tail, 6) the dominant long distance (L.D.) part must be evaluated using some low energy effective model which includes the pseudoscalars as well as the vector mesons ( ). The latter mediate the vector meson dominance mechanism which is providing the necessary damping of the high energy behavior. More recently, is has been shown that a data driven dispersion relation approach is possible and very promising  and a number of improvements have been obtained already [50, 51].
One usually applies appropriate low energy effective hadron theories,
like Hidden Local Symmetry (HLS), Extended Nambu-Jona-Lasinio (ENJL)
models, examples of the Resonance Lagrangian Approach (RLA), or large
QCD inspired ansätze and other QCD inspired modelings which
amount to calculate the following type of diagrams
The non-perturbative L.D. contributions is dominated the exchange and requires the knowledge of the off-shell form-factor (see Fig. 9).
A basic problem in estimating the HLbL scattering contribution we have
because in contrast to the one-scale HVP, HLbL exhibits 3 different
energy scales. Fig. 10 illustrates the –plane
of the general –domain of the form-factor
|??? multi-scale regions|
|– Data + Dispersion Relation,|
|– OPE, QCD factorization,|
|– Brodsky-Lepage approach|
|– Models constrained by data|
Lets focus on the leading exchange contribution. What do we know?
Constraint I: decay
The constant is well determined by the decay rate (from Wess-Zumino (WZ) Lagrangian).
Information on come from experiments as shown in Fig. 11.
Constraint II: by the VMD mechanism the related Brodsky-Lepage behavior provides the necessary damping (cutoff) in order to obtain finite integrals (the constant WZ form factor leads to a divergent result). Variants of models satisfying the constraints I and II yield similar answers. But ambiguities remain as only single tag data are available (one photon real) so far, as displayed in Fig. 11.
Recently the leading pseudoscalar meson exchange matrix element
has been evaluated beyond the single tag case in lattice QCD [52, 53]. For the first time could be measured on the lattice and clearly discriminates all simple VMD model ansätze! What remains is the large– QCD (OPE constrained) LMD+V ansatz 
which for the pion-pole approximation is well constrained now, i.e. parameters are rather well under control by QCD asymptotics and experimental and lattice data. QCD + constraints by data fixes , , are absent in chiral limit such that only and remain as essential parameters if one adopts the VMD mechanism and identifies with masses.
One important issue concerns the need of analytic continuation, as illustrated in Fig. 12.
In principle this should be answered within the dispersive approach to
HLbL or in lattice QCD (see
e.g. [55, 56, 53]). So far most estimates
adopt the pion-pole approximation (except [57, 36])
and apply VMD dressing at external vertex (except ). Adopting a LMD+V fit, my
estimation for the leading LbL contribution from PS mesons is
Table 1 lists a number of results for the -exchange contribution using very different approaches.
|Non-local quark model|||
|Dyson-Schwinger Eq. Approach|||
|Form-factor inspired by AdS/QCD|||
|Chiral quark model|||
|Magnetic susceptibility constraint||[57, 36]|
Besides the pseudoscalar contributions one
similarly can estimate the axial-mesons , the
scalars , -loops and residual quark-loop
contributions. Tensor mesons 
and a NLO  contribution are also to be included. I then estimate
I have scaled up the quadratically combined error on the l.h.s. by a factor 2 on the r.h.s. to account for uncertainties which are difficult to be quantified more precisely. For details I refer to Sect. 5.2.10 of my book .
4 Theory vs. Experiment: do we see New Physics?
Table 2 compares SM theory with the BNL experimental result.
|QED incl. 4-loops + 5-loops||11 658 471.||886||0.||003||[70, 71, 72]|
|Hadronic LO vacuum polarization||689.||46||3.||25|||
|Hadronic HO vacuum polarization||-8.||70||0.||06|||
|Weak to 2-loops||15.||36||0.||11|||
|Theory||11 659 178.3||4.3||–|
|Experiment||11 659 209.||1||6.||3|||
|The. - Exp. 4.0 standard deviations||-30.||6||7.||6||–|
What may the 4 deviation be: new physics? a statistical fluctuation? underestimating uncertainties (experimental, theoretical)? Do experiments measure what theoreticians calculate? Could it be unaccounted real photon radiation effects? For possible effects related to lepton flavor violation see e.g. [36, 74, 7] and references therein.
At the present/future level of precision depends on all
physics incorporated in the SM: electromagnetic, weak, and strong
interaction effects and beyond that all possible new physics
we are hunting for. Figure 13 illustrates past and
expected progress in “the closer we look the more there is to
see”. Here we are and hope to go. It contrast with the same status
for the electron, Fig. 14 shows that still is
and remains a QED test mainly.
Note added: a new more precise value of from atomic interferometry with Cesium has been obtained at the University of California Berkeley : giving an prediction such that a deviation. Previously with we had and with a 1.1 deviation. Although the central value moved closer to experimental value the deviation has increased owing to the more precise value of . Note that the “discrepancy” is of opposite sign of the one!
A “New Physics” interpretation of the persisting 3 to 4 gap requires relatively strongly coupled states in the range below about 250 GeV. Search bounds from LEP, Tevatron and specifically from the LHC already have ruled out a variety of Beyond the Standard Model (BSM) scenarios, so much hat standard motivations of SUSY/GUT extensions seem to fall in disgrace. There is no doubt that performing doable improvements on both the theory and the experimental side allows one to substantially sharpen (or diminish) the apparent gap between theory and experiment.
In any case constrains BSM scenarios distinctively and at the same time challenges a better understanding of the SM prediction. The two complementary experiments on the way, operating with ultra hot muons  and with ultra cold muons , repsectively, especially could differ by possible unaccounted real photon radiation effects. Provided the deviation is real and theory and needed hadronic cross section data can be improved as expected the muon experiments could establish at about standard deviations.
A remark concerning HVP issues in the standard data based time-like
approach is in order here:
i) How to combine a pretty large number of data-sets to a truly reliable -function. What is the true uncertainty? What part is reliably taken from pQCD? Including or excluding outdated (=older less precise) data-sets? Bare versus physical cross sections, how reliable is VP subtraction?
ii) Radiative corrections specifically for the ISR method, sQED issues etc. The ISR method requires one order in more precise RC calculation relative to the SCAN method, at least full 2–loop Bhabha and/or as well as ISR–FSR interference in the channel. What about RC to other more complicated channels (see e.g. [77, 78]). What about disentangling 30 channels and recombining them in the 1 to 2 GeV region (quantum interference, missing parts, double counting issues)?
iii) What precisely do we need in the DR? The 1PI “blob”, which is not a measurable quantity. Need undressing from QED effects, photon VP subtraction, FSR modeling, mixing? Do we do this at sufficient precision?
vi) Non-convergence of Dyson series for OZI suppressed narrow resonances (see e.g. ).
v) Missing data compatibility among different experiments. Here, global fit strategies (see e.g. [34, 32]) can help to learn more about possible problems.
Of course, I think we are doing the best to our knowledge. However, there is no unambiguous method to combine systematic errors. Uncertainties are definitely squeezed beyond what can be justified beyond doubts, I think.
Therefore, the very different Euclidean approaches, lattice QCD and the proposed alternative direct measurements of the hadronic shift , in the long term will be indispensable as complementary cross-checks.
For future improvements of the HLbL part one desperately needs more information from (see e.g. ) in order to have better constraints on modeling of the many relevant hadronic amplitudes. The dispersive approach to HLbL [49, 51] is able to allow for real progress since contributions which were treated so far as separate contributions will be treated “rolled into one” (as entirety). Note that HLbL depends on 19 independent amplitudes which contribute to while HVP depends on a single one. Last but not least: do theoreticians calculate what experiments measure (form-factor vs cross-section)?
A lot remains to be done while a new is in sight.
Many thanks to the organizers for the kind invitation to the 2018 Cracow EPIPHANY Conference and for giving me the opportunity to present this talk. I gratefully acknowledge the support by DESY.
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