Muon g-2 Theory: the Hadronic Part

I present a status report of the hadronic vacuum polarization effects for the muon $g-2$, to be considered as an update of [1]. The update concerns recent new inclusive $R$ measurements from KEDR in the energy range 1.84 to 3.72 GeV. For the leading order contributions I find $a_\mu^{\mathrm{had}(1)}=(688.07\pm 4.14)[688.77\pm3.38]\times 10^{-10}$ based on $e^+e^-$data [incl. $\tau$ data], $a_\mu^{\mathrm{had}(2)}= (-9.93\pm 0.07) \times 10^{-10}$ (NLO) and $a_\mu^{\mathrm{had}(3)}= (1.22\pm 0.01) \times 10^{-10}$ (NNLO). Collecting recent progress in the hadronic light-by-light scattering I adopt $\pi^0,\eta,\eta'$ [$95 \pm 12$] + axial--vector [$8 \pm ~3$] + scalar [$-6\pm ~1$] + $\pi,K$ loops [$-20\pm 5$] + quark loops [$22\pm ~4$] + tensor [$1\pm ~0$] + NLO [$3\pm ~2$] which yields $ a^{(6)}_\mu(\mathrm{lbl},\mathrm{had})=(103 \pm 29) \times 10^{-11}.$ With these updates I find $a_\mu^{\rm exp}-a_\mu^{\rm the}=(31.3\pm 7.7)\times 10^{-10}$ a 4.1 $\sigma$ deviation. Recent lattice QCD results and future prospects to improve hadronic contributions are discussed.

This review of the hadronic vacuum polarization (HVP) contributions to the muon g − 2 is to be considered as a complement to the theory reviews by Marc Knecht and Massimiliano Procura which focus on the hadronic light-by-light (HLbL) part and the reviews on hadronic cross sections by Graziano Venanzoni, Simon Eidelman and Achim Denig in these Proceedings.
The present experimental muon g − 2 result from Brookhaven (BNL) a exp µ = (11 659 209.1 ± 5.4 ± 3.3 [6.3]) × 10 −10 [2] soon will be improved by the new muon g − 2 experiments at Fermilab and J-PARC. The Fermilab experiment will be able to reduce the error by a factor 4, the J-PARC experiment will provide an important cross check with a very different technique [3]. It means that the new muon g − 2 experiments are expected to establish a possible new physics contribution at the level ∆a µ = a exp µ − a the µ = 6.7 σ provided theory remains as it is today and the central value does not move significantly. If we achieve a reduction of the hadronic uncertainty by factor 2 we would arrive at ∆a µ = 11.6 σ. That's what we hope to achieve. Figure 1  Virtual effect form low energy hadronic excitations are the standard problem in electroweak precision physics. At a certain level of precision predictions are hampered by nonperturbative effects, which technically are not under desirable control on the theory side. For the muon g − 2 the leading hadronic effects are related to the diagrams in figure 2 and concern (a) . Light quark loops appear as non-perturbative hadronic "blobs". The evaluation of the corresponding non-perturbative effects relies on hadron production data in conjunction with Dispersion Relations (DR), or on low energy effective modeling by the Resonance Lagrangian Approach (RLA), specifically by the Hidden Local Symmetry (HLS) model [4], or the Extended Nambu-Jona-Lasinio (ENJL) model [5], large-N c QCD inspired methods [6] and on lattice QCD. Different strategies apply for the different kinds of contributions: (a) HVP one evaluates via a dispersion integral over e + e − → hadrons data. Here 1 independent amplitude is to be determined by one specific data set. Global fits based on the RLA (like HLS) allow to improve the data-driven evaluations [7]. Lattice QCD is the ultimate tool to get QCD predictions in future.
(b) HLbL so far has been evaluated by modeling via the Resonance Lagrangian Approach (RLA) (chiral perturbation theory (CHPT) extended by vector meson dominance (VMD) in accord with chi-  ral structure of QCD) or by large-N c inspired methods and operator product expansions (OPE). A data driven approach based on dispersion relations [8] is attempting to exploit γγ → hadrons − data systematically (here 19 independent amplitudes are to be determined by as many independent data sets, fortunately not all are equally important numerically). Also in this case lattice QCD for me is the ultimate approach, although tough to be achieved with limited computing resources.
(c) HEW corrections due to quark triangle diagrams: since triple vector amplitudes vanish VVV = 0 by Furry's theorem only VV A (of ff Z -vertex) contributes. Thus it is ruled by the ABJ anomaly, which is perturbative and non-perturbative simultaneously, i.e. the leading effects are calculable. The anomaly cancellation condition intimately relates quark and lepton contributions and the potentially large leading corrections cancel [9][10][11] such that hadronic corrections are well under control. The hadronic contribution to the vacuum polarization can be evaluated, with the help of dispersion relations, from the energy scan of the ratio R γ (s) ≡ σ (0) (e + e − → γ * → hadrons)/ 4πα 2 3s which can be measured up to some energy E cut above which we can safely use perturbative QCD (pQCD) thanks to asymptotic freedom of QCD. We apply pQCD from 5.2 GeV to 9.46 GeV and above 11.5 GeV (see figure 5 below). Note that the DR requires the undressed (bare) cross-section σ (0) (e + e − → γ * → hadrons) = σ(e + e − → γ * → hadrons) |α(0)/α(s)| 2 . The lowest order (LO) VP contribution is given by whereK(s) is a known kernel function growing form 0.63 · · · at the 2m π threshold to 1 as s → ∞.    Figure 4 shows the distribution of contributions and errors between different energy ranges. One of the main issues is R γ (s) in the region from 1.2 GeV to 2.0 GeV, where more than 30 exclusive channels must be measured and although it contributes about 20% only of the total it contributes about 50% of the uncertainty. In the low energy region, which is particularly important for the dispersive evaluation of the hadronic contribution to the muon g − 2, data have improved dramatically in the past decade for the dominant e + e − → π + π − channel (CMD-2 [12], SND/Novosibirsk [13], KLOE/Frascati [14][15][16], BaBar/SLAC [17], BES-III/Beijing [18]) 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 [19][20][21][22]. Recent data sets collected are: e + e − → 3(π + π − ), e + e − →pp and e + e − → K 0 S K 0 L , K + K − from CMD-3 [23,24], and e + e − →nn, e + e − → ηπ + π − , e + e − → π 0 γ, e + e − → ωηπ 0 , e + e − → ωη, e + e − → K + K − and e + e − → ωπ 0 → π 0 π 0 γ from SND [25][26][27].
Above 2 GeV fairly accurate BES-II data [28] are available. Recently, a new inclusive determination of R γ (s) in the range 1.84 to 3.72 GeV has been obtained with the KEDR detector at Novosibirsk [29] (see figure 5). A big step in improving low energy cross section measurements has been

NLO and NNLO HVP effects
The next-to-leading order (NLO) HVP is represented by diagrams in figure 7. With kernels from [30], the results of an updated evaluation are presented in table 1. The next-to-next leading order (NNLO)  contributions have been calculated recently [31]. Diagrams are shown in figure 7 and corresponding contributions evaluated with kernels from [31]   The first direct measurement of the timelike complex VP function in the ρ resonance region by KLOE [33] (see figure 8) nicely confirms dispersion relation calculation and demonstrates the importance of including the imaginary part in vacuum polarization subtraction in obtaining the undressed σ (0) (s) version of the physical hadronic cross-sections σ(s). The complex running fine structure con- σ(e + e − →µ + µ − ) pt as well as R(s) = σ(e + e − →π + π − ) σ(e + e − →µ + µ − ) , which determines Im α(s) = − α 3 R(s), and knowing the modulus |α(s)| one can extract Re α(s) as well (see G. Venanzoni's Contribution fort details).
The imaginary parts in the perturbative regions usually are small relative to the leading logarithms which govern the running couplings (renormalization group approach). In the hadronic shift however, resonances are accompanied by imaginary parts which may be huge in particular near resonances which can decay via OZI suppressed channels only (see Sect. 5 of [1]). Figure 9. Comparing the τ+PDG prediction (red curve) of the pion form factor in e + e − annihilation in the ρ − ω interference region. Reprinted from [7], with kind permission of The European Physical Journal (EPJ).

Low energy effective Lagrangian theory
A low energy effective field theory description of hadronic data requires an extension of CHPT towards higher energies, which mainly requires to include spin 1 resonances ρ, ω, φ, · · · in accord with the symmetries of QCD. Principles to be included are the chiral structure of QCD, VMD and electromagnetic gauge invariance. The spin 1 resonances are important in the evaluation of HVP as well as of HLbL effects. One possible implementation is the HLS model, which for what concerns HVP can also be seen as a generalized Gounaris-Sakurai model. In the neutral channel e + e − → hadrons γ, ρ 0 , ω, φ mixing makes the channel rather complicated, while in the charged channel of τ → ν τ hadrons [34][35][36][37][38] is much simpler as the ρ ± do not mix with other hadrons. It is thus tempting to start with the isospin rotated τ ± → ν τ π ± π 0 decay spectra and supplement them with appropriate isospin breaking and mixing effects to predict e + e − → π + π − , with the result shown in figure 9. It shows that there is no τ vs. e + e − conflict and actually simultaneous fits allows one to reduce uncertainties   . The HLS best fit BDDJ15 # (NSK+KLOE10+KLOE12) does not include BaBar ππ data [39], while BDDJ15 * does. JS11 [40], FJ16 [1] is updated and includes the BES III and KEDR data. Further points are BDDJ12 [7], DHMZ10 [41], DHMZ16 [22,44], HLMNT11 [43] and DHea09 [42]. The DHMZ10 (e + e − +τ) result is not including the ρ − γ mixing correction, i.e. it misses important isospin breaking effects. In contrast, DHMZ10/JS11 is obtained by including this correction, which brings the point into much better agreement with standard analyses based on e + e − data alone, as for example the DHMZ10 (e + e − ) result. (see also [21,45] of HVP by using indirect constraints. The global fit strategy followed in [7] takes into account data below E 0 = 1.05GeV (just above the φ ) to constrain the effective Lagrangian couplings. Used are 45 different data sets, 6 annihilation channels and 10 partial width decays. The effective theory then allows us to predict cross sections for the channels π + π − , π 0 γ, ηγ, η γ, π 0 π + π − , K + K − , K 0K0 , which account for 83.4% of a had µ . The missing channels 4π, 5π, 6π, ηππ, ωπ and the higher energy tail E > E 0 is evaluated using data directly and pQCD for the perturbative region and tail. All mixing effects, as γρ -mixing, ρω -mixing, · · · , as well as the decay branching fractions are dynamically generated by including self-energy effects of the spin 1 mesons. One thus is taking into account proper phase space, energy dependent widths etc. Such fit strategy is able to shed light on incompatibilities in the data, e.g. KLOE vs BaBar, by comparing the fit qualities, but also reveals the compatibility of τ-decay spectra with e + e − -data after accounting for the mixing effects like including γ − ρ 0 mixing. HLS estimates are included in table 10 together with other recent results.

HVP from lattice QCD (following H. Wittig at LATTICE 2016)
The need for ab initio calculation of a had µ is well motivated: -the problems to determine nonperturbative contributions to the muon g − 2 from experimental data at sufficient precision persists and is not easy to improve, -a model-independent extension of CHPT to the relevant energies ranges up to 2 GeV is missing, while the new experiments E989 FNAL and E34 J-PARC require an improvement of the hadronic uncertainties by a factor two to four. (3) in the coming years. Primary object for getting HVP in LQCD is the e.m. current correlator in configuration space In principle, a Fourier transform yields the vacuum polarization function Π(Q 2 ) needed to calculate The integration kernel in this representation is As LQCD per se has to work on a lattice in a finite box, momenta are quantized Q min = 2π/L, where Figure 11. Left: the integrand of (6) as a function of Q 2 . Also shown are ranges between Q i = 0.00, 0.15, 0.30, 0.45 and 1.0GeV and their percent contribution to a had µ and a possible "LQCD sample". Right: typical ranges for lattice data and their extrapolation to low Q 2 .
A promising alternative method to determine a had µ is possible by a dedicated measurement of α(t) at spacelike momentum transfer as advocated in [62] and [63]. Given α(−Q 2 ) and the fact that the leptonic contribution is well under control in perturbation theory one can extract the hadronic shift and determine a had µ via the representation where Q 2 (x) ≡ x 2 1−x m 2 µ is the spacelike square momentum-transfer. In the Euclidean region the integrand is highly peaked around half of the ρ meson mass scale (see figure 13). The method is very different from the standard approach based on (1): radiative corrections are very different (much simpler) as no hadronic final states need to be understood, no VP subtraction is to be performed, no exclusive channel collection etc. So, even a 1% level measurement can provide important independent information. This in view of the problem to get accurate hadronic total cross-section in the range The possible processes to measure α(t) are Bhabha scattering e + (p + ) e − (p − ) → e + (p + ) e − (p − ) or muon electron scattering (see figure 14). The Bhabha process has two tree level diagrams a t-and Figure 14. Measuring the running charge in the spacelike regime. Left: VP dressed tree level Bhabha scattering in QED, for small t (small angels) the s channel is suppressed. Right: getting α(t) from the µ − e − → µ − e − process.
a s-channel one. With the positive c.m. energy square s = (p + + p − ) 2 and the negative momentum transfer square t = (p − − p − ) 2 = − 1 2 (s − 4m 2 e ) (1 − cos θ) , θ the e − scattering angle, there are two very different scales involved which helps to isolate the t channel of interest. The VP dressed lowest order cross-section is where A ik are tree level helicity amplitudes, i, k =L,R denote left-and right-handed electrons. The dressed transition amplitudes in the massless limit (m e ≈ 0 ) read Preferably one uses small angle Bhabha scattering (small |t|) as a normalizing process which is dominated by the t-channel ∼ 1/t, however, detecting electrons and positrons along the beam axis often has its technical limitations. Care also is needed concerning the ISR corrections because cuts for the Bhabha process (e + e − → e + e − ) typically are different from the ones applied to e + e − → hadrons. Usually, experiments have included corresponding uncertainties in their systematic errors, if they have not been explicitly accounted for by applying appropriate radiative corrections. For details I refer to [62] and the Contribution by Luca Trentadue.
While the Bhabha process requires to sort out the s channel from the t channel, the pure t channel reaction of µ − e − scattering µ − (p − ) e − (q − ) → µ − (p − ) e − (q − ) provides a much simpler setup and could be realized as a fixed target experiment [63] at existing facilities. The leading order cross-section in this case has the simple form exhibiting the effective charge as an overall factor. For details see Luca Trentadue's Contribution. Such an experiment would provide data for the Euclidean electromagnetic current correlator Π γ (Q 2 )− Π γ (0) = −∆α had (−Q 2 ) = α α(−Q 2 ) + ∆α lep (−Q 2 ) − 1 and would allow for a directly check of lattice QCD data. In addition one could determine ∆α had −Q 2 at Q ≈ 2.5GeV by this method (one single number!) as the non-perturbative part of ∆α had M 2 Z = ∆α had −Q 2 + pert. when evaluated in "Adler function" approach advocated in [64].  Table 3 summarizes the present status of the SM prediction for a µ in comparison with the experimental value [2]. For a recent update of the weak contribution see [70]. As an estimate based on [4][5][6]10

Theory confronts experiment
The result differs little from the "agreed" value (105 ± 26) × 10 −11 presented in [47] and (116 ± 39) × 10 −11 estimated in [46]. Both included a wrong, too large, Landau-Yang theorem violating axial-vector contribution from [10], correcting for this we obtain our reduced value relative to [46]. 14 ppm] the next years. Next generation experiments require a factor 4 reduction of the uncertainty optimistically feasible should be a factor 2 we hope.
In view of the upcoming two complementary experiments, one at Fermilab working with ultra hot muons and the other at J-PARC operating with ultra cold muons (very different radiation effects), the big challenge is to keep up on the prediction side as much as possible. The deviation between theory and experiment can be scrutinized provided theory and the needed cross section data improves the same as the muon g − 2 experiments. Primarily we need more/better data and/or progress in non-perturbative QCD, where the main obstacle (data, lattice QCD, RLA) is the hadronic light-bylight scattering contribution. Progress in evaluating HVP also depends on more data (BaBar, Belle, VEPP-2000, BESIII,...) and lattice QCD where recent progress is very promising (see figure 12). In both cases HVP as well as HLbL, lattice QCD will provide answers one day, but also low energy effective RL and DR approaches need be further developed. One has also to keep in mind that progress in calculations of radiative corrections [71][72][73] is mandatory in precision measurements of hadronic cross sections.
For future improvements of HLbL one urgently needs more information from γγ → hadrons physics [74,75] in order to have better constraints on modeling hadronic amplitudes (see [76][77][78] for theoretical studies). Some sample processes are collected in figure 15. Mostly experiments at e + e − - facilities investigate single-tag events (higher rates, lower background). New data are expected from KLOE, KEDR exhibiting taggers and from BaBar, Belle, BES III which have high luminosity. More information is also expected from Dalitz-decay studies ρ, ω, φ → π 0 (η)e + e − possible at Novosibirsk, CERN NA60, JLab, Mainz, Bonn, Jülich and BES. Unfortunately some of the interesting processes seem to be buried in the background. The background is a general problem in γγ → hadrons physics. The dispersive approach [8,66] is able to allow for real progress since contributions which we have treated so far as separate contributions will be treated in an integral manner. An example is the γγ → ππ process which includes contributions attributed to the two-pion channel , the pion-loop, the scalar contribution as well as the tensor contribution. All-in-one can be gotten from the experimental data (see e.g. figure 3 of [79]). This also will settle such issues as the pion polarizability. A lot remains to be done while new a exp µ is expected soon. For details see the Contribution by M. Procura.
Thanks to the organizers for the kind invitation and support to the "KLOE-2 Workshop on e + e − collision physics at 1 GeV" at Frascati, and for giving me the opportunity to present this talk.