Invited Talk

aµ (had, VP) update

Fred Jegerlehner, DESY Zeuthen/Humboldt University Berlin

' $ Workshop on hadronic vacuum polarization contributions to muon g-2, KEK, Tsukuba, Japan, February 12-14, 2018

& % New updated and expanded edition Jegerlehner F., The Anomalous Magnetic Moment of the Muon. Springer Tracts in Modern Physics, Vol 274 (2017). Springer, Cham (693 pages on one number to 8 digits)

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 Outline of Talk:

OHadronic Vacuum Polarization (HVP) – Data & Status OEffective Field Theory: Resonance Lagrangian Approach OSimpliﬁed: VMD + sQED and ρ0, ω γ Mixing OHVP from lattice QCD − OTheory vs experiment: do we see New Physics? OProspects My focus will be more on less standard issues

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 1 Hadronic Vacuum Polarization (HVP) – Data & Status

had Leading non-perturbative hadronic contributions aµ can be obtained in terms of 2 R (s) σ(0)(e+e γ hadrons)/4πα data via Dispersion Relation (DR): γ ≡ − → ∗ → 3s

2 Ecut Z data Z∞ pQCD αmµ2 Rγ (s) Kˆ (s) Rγ (s) Kˆ (s) ahad = ds + ds

µ 3π s2 s2 ¡

4m2 E2 ¡

π cut G Experimental error implies theoretical uncertainty! G Low energy contributions enhanced: 75% come from region 4m2 < m2 < M2 ∼ π ππ Φ Data: NSK, KLOE, BaBar, BES3, CLEOc had(1) 10 aµ = (686.99 4.21)[687.19 3.48] 10− + ± ± e e−–data based [incl. τ] 2 ∆aµ (δ∆aµ)

ρ, ω ρ, ω 1.0 GeV 0.0 GeV, 0.0 GeV, Υ ∞ ∞ ψ 9.5 GeV 3.1 GeV 3.1 GeV 2.0 GeV φ, . . . 2.0 GeV φ, . . .

1.0 GeV contribution error2 F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 2 γ hard

e+ γ + φ π π−, ρ0 hadrons e− 2 s = M ; s′ = s (1 k), k = E /E φ − γ beam a) b) a) Initial state radiation (ISR), b) Standard energy scan.

Experimental input for HVP: BaBar, BESIII, VEPP-2000 Talks Michel Davier, Christoph Redmer, Ivan Logashenko, Sergey Serednyakov

Recent BES-III vs BaBar and KLOE

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 3 Recent results: + G π π− from BES-III, CMD-3 and CLEOc + 0 G π π−π from Belle + G K K− from CMD-3 and SND G ωπ0 π0π0γ from SND →0 + 0 0 0 G KS K±π π∓, KS K±π∓η, π π−π π , KS KLπ , 0 0 KS KLη, KS KLπ π from BaBar

Energy range ahad[%](error) 1010 rel. err. abs. err. µ × ρ, ω (E < 1 GeV) 540.98 [78.6](2.80) 0.5 % 50.7 % 1 GeV < E < 2 GeV 96.49 [14.0](2.54) 2.6 % 41.5 % 2 GeV < E < incl pQCD 51.09 [ 7.4](1.10) 2.2 % 7.8 % ∞ total 688.65 [100.0](3.94) 0.6 % 100.0 %

+ Additional data besides e e− ones providing improvements:

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 4 τ–decay spectra: good idea, use isospin symmetry to include existing high quality τ–data (after isospin corrections) Alemany et al 1996

+ ¯ + e γ γ u,¯ d bare π π− spectrum after photon VP, + π π−, [I = 1] ··· ISR and FSR subtraction e− u,d

isospin⇑ rotation CVC isospin rotation symmetry ⇓ ↔ u¯ 0 τ − W W 0 bare π π− spectrum after EW radia- π π−, ··· ν¯τ d tive and phase space correction

+ Standard IB corrected data: large discrepancy [ 10%] persists! τ vs. e e− problem! [manifest since 2002 Davier et al, solved∼ 2011] + G pioneered by Alemany, Davier, Hocker¨ 1996 (when e e− ππ data were much less precise that the τ ν ππ data!) → → τ Why is it still a good idea to include τ CC spectra?

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 5 + τ data in many respects are much simpler than e e− ones, because the charged ρ± does not mix with other states in the dominant low energy range below about 1.05 GeV (up to above the φ, below ρ0)!

In contrast understanding NC data requires modeling: CHPT + spin 1 resonances (VMD) Resonance Lagrangian Approach e.g. HLS (massive Yang-Mills) dynamical⇒ widths, dynamical mixing of γ, ρ0, ω, φ ⇒

Commonly forgotten: mixing of ρ0, ω, φ with the photon [ρ0 γ mixing] i.e. effect concerning relation −

⊗ ⊗ ↔ A(x) A(0) j(x) j(0) h i h i photon propagator current correlator

+ + included in NC e e− π π− data, absent in CC τ ντππ data! Also to be accounted→ for in lattice QCD calculations!→

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 6 Effective Field Theory: Resonance Lagrangian Approach Ì Global Fit strategy based on Hidden Local Symmetry (HLS) Benayoun et al Data below E0 = 1.05 GeV (just including the φ) constrain effective Lagrangian couplings, using 45[47] different data sets (6[8] annihilation channels and 10 partial width decays). see Talk Maurice Benayoun Ì Effective theory predicts cross sections: + 0 0 + + 0 0 π π−, π γ, ηγ, η0γ, π π π−, K K−, K K¯ (83.4%), G Missing part: 4π, 5π, 6π, ηππ, ωπ and regime E > E0 evaluated using data directly and pQCD for perturbative region and tail G Including self-energy effects is mandatory (γρ-mixing, ρω-mixing ..., decays with proper phase space, energy dependent width etc) G Method works in reducing uncertainties by using indirect constraints G Able to reveal inconsistencies in data, e.g. KLOE vs BaBar

Remark: HLS supplied by symmetry breakings (isospin, SU(3)) is rather complicated, like EW SM with spin-1 vector bosons as gauge bosons, but with ρ, ω, φ (in place of one Z boson) and parity odd sector WZW type Lagranian terms.

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 7 + J Fit of τ +IB from PDG vs π π−–data

+ Comparing the τ+PDG prediction (red curve) of the pion form factor in e e− annihilation in the ρ ω interference region. Benayoun et al. −

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 8 Simpliﬁed: VMD + sQED and ρ0, ω γ Mixing − Ì Usually using Gounaris-Sakurai ansatz GS s GS GS BWρ(770)(s) 1 + δ 2 BWω(s) + β BWρ(1450)(s) + γ BWρ(1700)(s) · Mω Fπ(s) = , 1 + β + γ

Ì Fit e+e -data for F (s) 2 « δ (complex) and set δ = 0 to obtain FI=1(s) 2 − | π | ρω | π |

2 CMD-2 data for Fπ in ρ ω region together with Gounaris-Sakurai ﬁt. Left: before| | subtraction.− Right: after subtraction of the ω.

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 9 Ì ρ0 γ mixing applied to Breit-Wigner shape and to τ data from ALEPH 2005. −

Effective Lagrangian = + e.m. gauge invariant L Lγρ Lπ + +µ 2 + = D π D π− m π π− ; D = ∂ i e A i g ρ Lπ µ − π µ µ− µ − ρππ µ 2 1 µν 1 µν Mρ µ e µν γρ = Fµν F ρµν ρ + ρµ ρ + ρµν F L −4 − 4 2 2 gρ

Note this is a model most people are using (Gounaris-Sakurai, FSR etc) but neglecting 1–loop γ ρ0 mixing contribution! −

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 10 Ì ρ0 γ mixing applied to Breit-Wigner shape and to τ data from ALEPH 2005. −

Effective Lagrangian as used in GS = + not e.m. gauge invariant L Lγρ Lπ + +µ 2 + = D π D π− m π π− ; D = ∂ i e A i g ρ Lπ µ − π µ µ− µ − ρππ µ 2 2 1 µν 1 µν Mρ µ e Mρ µ γρ = Fµν F ρµν ρ + ρµ ρ ρµA L −4 − 4 2 − gρ

Note this is a model most people are using (Gounaris-Sakurai, FSR etc) but neglecting 1–loop γ ρ0 mixing contribution! −

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 11 Photon is mixing with ρ0: propagators are obtained by inverting the symmetric 2 2 self–energy matrix ×

2 2 2 ! 1 q + Πγγ(q ) Πγρ(q ) Dˆ − = Π (q2) q2 M2 + Π (q2) γρ − ρ ρρ . Effect well known from LEP Z resonance physics: Z γ mixing affects Z lineshape. −

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 12 Self-energies: pion loops to photon-ρ vacuum polarization

i Πµν (π)(q)= + − γγ .

i Πµν (π)(q)= + − γρ .

i Πµν (π)(q)= + − ρρ . Irreducible self-energy contribution at one-loop bare γ ρ transverse self-energy functions − 2 g2 e 2 egρππ 2 ρππ 2 Πγγ = f (q ) , Πγρ = f (q ) and Πρρ = f (q ) , 48π2 48π2 48π2

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 13 Self-energies: pion loops to photon-ρ vacuum polarization

i Πµν (π)(q)= + − γγ .

i Πµν (π)(q)= − γρ .

i Πµν (π)(q)= + − ρρ . Previous calculations, consider mixing term to be constant bare γ ρ transverse self-energy functions − 2 g2 e 2 egρππ 2 ρππ 2 Πγγ = f (q ) , Πγρ = f (q ) and Πρρ = f (q ) , 48π2 48π2 48π2

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 14 where

µ2 2 2 2 2 = / + y y 2 y + , f (q ) q h(q ); h(q ) 2 3 2 (1 ) 2 (1 ) G( ) ln 2 ≡ − − − mπ a scalar self-energy function with y = 2/ and y = 1 1+βπ π , for 2 > 2. Note that 4mπ s G( ) 2βπ (ln 1 βπ i ) q 4 mπ − − all components of the (γ, ρ) 2 2 matrix propagator are proportional to the same function f (q2). × The renormalization conditions are such that the matrix is diagonal and of residue unity at the 2 2 photon pole q = 0 and at the ρ resonance s = Mρ, hence the renormalized self-energies read

ren 2 2 2 2 ren 2 Π (q ) = Πγγ(q ) q Π0 (0) q Π0 (q ) γγ − γγ γγ q2 Πren 2 = Π 2 Re Π 2 γρ (q ) γρ(q ) 2 γρ(Mρ) − Mρ

ren 2 2 2 2 2 dΠρρ 2 Π (q ) = Πρρ(q ) Re Πρρ(M ) (q M ) Re (M ) ρρ − ρ − − ρ ds ρ

2 2 2 where Πγγ(0) = Πγρ(0) = Πρρ(0) = 0 and Πγγ0 (q ) = Πγγ(q )/q , has been used. Note, that the tree (0) 2 level mixing term in the Lagrangian contributes to the bare γρ self-energy as Πγρ = q (e/gρ), which does not affect the renormalized self energies, however.

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 15 The e+e π+π matrix element in sQED is given by − → − = i e2 vγ¯ µu (p p ) F (q2) M − 1 − 2 µ π 2 with Fπ(q ) = 1. In our extended VMD model we have the four terms

e+ π+ γγ ρ ργ ρ + + +

e− π− Diagrams contributing to the process e+e π+π . − → −

F (s) e2 D + eg D g eD g g D , π ∝ γγ ρππ γρ − ρee ργ − ρee ρππ ρρ

Properly normalized (VP subtraction: e2(s) e2): → # h i h i F (s) = e2 D + e (g g ) D g g D / e2 D π γγ ρππ − ρee γρ − ρee ρππ ρρ γγ

" ! F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 16 Typical couplings gρππ bare = 5.8935, gρππ ren = 6.1559, gρee = 0.018149, x = gρππ/gρ = 1.15128.

q q 3 gρππ = 48 π Γρ/(βρ Mρ); gρee = 12π Γρee/Mρ

Note: the γ ρ0 mixing effect −

2 Fπ(s) rργ(s) | 2 | Fπ(s) ≡ | |Dγρ=0 for given Γ g = 0.018149 it is an otherwise parameter free prediction! ρee⇔ ρee The ω induced terms contribute to the pion form factor h i h i ∆F(ω)(s) = e (g g ) D (g g + g g ) D / e2 D , π ωππ − ωee γω − ρee ωππ ωee ρππ ρω γγ which adds to Fπ(s) for the ρ only.

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 17 + ρ γ interference resolves τ (charged channel) vs. e e− (neutral channel) puzzle, F.J.&− R. Szafron, M. Benayoun et al 2011 ρ γ interference (absent− in charged channel) often mimicked by large shifts +5% in Mρ and Γρ -10% i Πµν (π)(q)= + − γρ .

30(s) = rργ(s) RIB(s) 3 (s) −

Ì τ require to be corrected for missing ρ γ mixing! − + Ì results obtained from e e− data is what goes into aµ

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 18 F (E) 2 in units of e+e I=1 (CMD-2 GS ﬁt) | π | − Best “proof”:

Davier et al 2009

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 19 F (E) 2 in units of e+e I=1 (CMD-2 GS ﬁt) | π | − Best “proof”:

Davier et al 2009

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 20 τ decays ALEPH 1997 390.75 2.65 1.94 + IB ± ± no ρ0 γ ALEPH 2005 388.74 4.00 2.07 − ± ± OPAL 1999 380.25 7.27 5.06 ± ± CLEO 2000 391.59 4.11 6.27 ± ± Belle 2008 394.67 0.53 3.66 ± ± τ combined 390.99 0.81 2.01 ± ± + e e− I=1 CMD-2 2006 386.53 2.76 2.59 ± ± SND 2006 383.94 1.40 4.99 ± ± KLOE 2008 380.16 0.34 3.27 ± ± KLOE 2010 377.30 0.71 3.50 ± ± BABAR 2009 389.30 0.37 2.00 ± ± KLOE 2012 375.98 1.12 2.76 ± ± BES3 2015 373.85 1.51 3.36 ± ± CLEOc 2017 390.15 2.79 5.85 ± ± e+e combined 383.93 0.94 2.07 − ± ± 10 aµ[ππ],I =1, (0.592 0.975) GeV 380 390 400 10− − × had I=1 part of aµ [ππ]

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 21 τ decays ALEPH 1997 385.58 1.96 5.40 + IB ± ± + ρ0 γ ALEPH 2005 383.49 1.69 2.30 − ± ± OPAL 1999 375.34 4.73 5.25 ± ± CLEO 2000 386.56 1.60 6.96 ± ± Belle 2008 389.56 0.53 3.66 ± ± τ combined 385.83 0.81 2.00 ± ± + e e− I=1 CMD-2 2006 386.53 2.76 2.59 ± ± SND 2006 383.94 1.40 4.99 ± ± KLOE 2008 380.16 0.34 3.27 ± ± KLOE 2010 377.30 0.71 3.50 ± ± BABAR 2009 389.30 0.37 2.00 ± ± KLOE 2012 375.98 1.12 2.76 ± ± BES3 2015 380.90 1.65 3.46 ± ± CLEOc 2017 390.15 2.79 5.85 ± ± e+e combined 383.93 0.94 2.07 − ± ± 10 aµ[ππ],I =1, (0.592 0.975) GeV 380 390 400 10− − × had I=1 part of aµ [ππ]

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 22 J Is the sQED viable? Look at γγ ππ →

How do photons couple to pions? Di-pion production in γγ fusion. At low energy we have direct π+π production and by strong rescattering π+π π0π0, however − − → with very much suppressed rate. Above about 1 GeV, resolved qq¯ couplings seen. The π0π0 cross section in this ﬁgure is enhanced by the isospin symmetry factor 2, by which it is reduced in reality.

Strong tensor meson resonance in ππ channel f2(1270) with photons directly probe the quarks!

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 23 G Photons seem to see pions below 1 GeV

G Photons deﬁnitely look at the quarks in f2(1270) resonance region

G We apply the sQED model up to 0.975 GeV (relevant for aµ). This should be pretty save (still we assume a 10% model uncertainty)

G Switching off the electromagnetic interaction of pions, is deﬁnitely not a realistic approximation in trying to describe what data we see in e+e π+π − → − G τ data missing γ ρ0 mixing correction Szafron, F.J is substantial part of HLS approach − had G for the I=1 part of aµ [ππ] results in

had 10 δa [ργ] ( 5.1 0.5) 10− . µ ' − ± × as a correction applied for the range [0.63,0.96] GeV. The correction is not too large, but at the level of 1 σ and thus non-negligible.

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 24 Note LQCD simulations are missing this effect!, which is absent in the current- current correlator but shows up in the γ ρ0 2x2 matrix propagator −

Including IB corrected τ data:

had(1) 10 a = (688.07 4.14)[688.77 3.38] 10− µ ± ± ×

+ based on e e−–data [incl. τ-decay spectra].

using ππ scattering phase shifts: the Omnes` representation ﬁxes R δ = i δ(s) = s ∞ (s0) Fπ(s) Fπ(s) e P(s) exp π 4m2 ds0 s (s s) | | π 0 0− where P(s) is a polynomial ﬁxed by appropriate b.c.’s, like P(0) = 1 etc

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 25 J ππ phase shifts data vs theory

The phase of Fπ(E) as a function of the c.m. energy E. We compare the result of the elaborate Roy equation analysis of Leutwyler 02, Colangelo 03, Caprini 16 with the one due to the sQED pion-loop and data Hyams 73, Grayer 74, Protopopescu 73. By analyticity the phase determines the modulus of Fπ

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 26 Caprini et al 2013 contribution to aµ from below 0.63 GeV yields

ππ(γ),LO 10 a [2m , 0.63 GeV] = (133.258 0.723) 10− , µ π ± × + a 40% reduction of the error estimated in a standard calculation in terms of e e− data which yields 132.57(055)(0.93)[1.19] 10 10 . × − One obtains ahad(1) = (689.46 3.25) 10 10 , µ ± × − as a best estimate.

Ì New KLOE combined 0.316 < Mππ < 0.975 GeV Talk Stefan Muller¨ a (π+π ) = (489.8 1.7 4.8) 10 10 . arXiv:1711.03085 µ − ± ± × − Ì New CLEO-c ISR ππ data (CESR storage ring collider) 0.3 < Mππ < 1.00 GeV a (π+π ) = (500.4 3.6 7.5) 10 10 . arXiv:1712.04530 µ − ± ± × − see also Talks Bogdan Malaescu, Alex Keshavarzi, Maurice Benayoun

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 27 Ì Still an issue in HVP

Ì region 1.2 to 2 GeV data; test-ground exclusive vs inclusive R measurements (more than 30 channels!) VEPP-2000 CMD-3, SND (NSK) scan, BaBar, BES III radiative return! still contributes 50% of uncertainty

2012 2017

G illustrating progress by BaBar and NSK exclusive channel data

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 28 excl. vs incl. clash

Present leading uncertainty: hard to improve by direct R(s) measurements

incl. ISR + vs new inclusive dataDHMZ10 by KEDR (e e−) [3.6 σ] 180.2 4.9 ± + DHMZ10 (e e−+τ) [2.4 σ] 189.4 5.4 ±+ JS11 (e e−+τ) [3.4 σ] 179.7 6.0 ± + HLMNT11 (e e−) [3.3 σ] 182.8 4.9 ± + DHMZ10/JS11 (e e−+τ) [3.6 σ] 181.1 4.6 ± # + BDDJ15 (e e−+τ) [4.8 σ] 170.4 5.1 ± + BDDJ15∗ (e e−+τ) [4.2 σ] 175.0 5.0 ± + DHMZ16 (e e−) [3.6 σ] 181.7 4.2 ±+ FJ17 (e e−+τ+ππ phases) [4.3 σ] 178.3 3.5 ± excl. ISR + DHea09 (e e−) [3.5 σ] 178.8 5.8 ± + BDDJ12∗ (e e−+τ) [4.1 σ] 175.4 5.3 ± experiment ∗ HLS global ﬁt BNL-E821 (world average) # HLS best ﬁt 209.1 6.3 ± 150 200 250 a 1010-11659000 µ× Comparison with other Results. Note: results depend on which value is taken for HLbL. JS11 and BDDJ13 includes 116(39) 10 11 [JN], DHea09, × − DHMZ10, HLMNT11 and BDDJ12 use 105(26) 10 11 [PdRV]. × −

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 29 Theory vs experiment: do we see New Physics? Given the CODATA/PDG recommended value of α the theory confronts experiment as follows: Standard model theory and experiment comparison Contribution Value 1010 Error 1010 Reference × × QED incl. 4-loops + 5-loops 11 658 471.886 0.003 Aoyama et al 12,Laporta 17 Hadronic LO vacuum polarization 689.46 3.25 Hadronic light–by–light 10.34 2.88 Hadronic HO vacuum polarization -8.70 0.06 Weak to 2-loops 15.36 0.11 Gnendiger et al 13 Theory 11 659 178.3 4.3 – Experiment 11 659 209.1 6.3 BNL 04 The. - Exp. 4.0 standard deviations -30.6 7.6–

10 Standard model theory and experiment comparison [in units 10− ]. What represents the 4 σ deviation: Ì new physics? Ì a statistical ﬂuctuation? Ì underestimating uncertainties (experimental, theoretical)? Odo experiments measure what theoreticians calculate?

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 30 E Or is it unaccounted for real photon radiation effects? Do experiments measure what theory calculates?

G At the present/future level of precision aµ 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.

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 31 here we are and hope to go:

J-PARC FNAL BNLCERN III CERN II CERN I 201920041976 1968 1961

LO 4th − QED 6th 8th − 10th hadronic VP LO δHVP NLO − NNLO δHLbL hadronic LbL SM predictions SM uncertainty weak LO neg. contribution future ? ∗ HO aµ HVP HLbL − ∗ δaµ /2, δaµ 2/3 New Physics ? ??? SM prediction

3 1 1 3 5 7 9 10− 10− 10 10 10 10 10 11 aµ in units 10−

Past and future g 2 experiments testing various contributions. − ? exp the exp New Physics = deviation (aµ aµ )/aµ . Limiting theory precision: hadronic vacuum polarization− (HVP) and hadronic light-by-light (HLbL) *** digging deeper and deeper ***

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 32 same status for the electron: Harvard Seattle Michigan 2006 1987 1970

δαRb11 LO 4th − QED 6th 8th − 10th hadronic VP LO NLO − 1 NNLO α− (Rb11) = 137.035999037(91) hadronic LbL SM predictions SM uncertainty weak LO neg. contribution future ? ∗ HO a 1 − e ∗ δα− (Rb11)/10 New Physics ? ??? SM prediction

3 1 1 3 5 7 9 10− 10− 10 10 10 10 10 11 ae in units 10−

Status and sensitivity of the ae experiments testing various contributions. The error is dominated by the uncertainty of α(Rb11) from atomic interferometry. ? exp the exp No “New Physics” = deviation (ae ae )/aµ . The blue band illustrates the improvement by the Harvard experiment.− Note the very different sensitivities to non-QED contributions in comparison with aµ. *** still is and remains a QED test mainly ***

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 33 Prospects G a “New Physics” interpretation of the persisting3 to4 σ requires relatively strongly coupled states in the range below about 250 GeV.

G 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.

G 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 aµ constrains BSM scenarios distinctively and at the same time challenges a better understanding of the SM prediction.

The big challenge: two complementary experiments: Fermilab with ultra hot muons and J-PARC with ultra cold muons (very different radiation) to come

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 34 Provided deviation is real and theory and needed cross section data improves the same as the muon g 2 experiments 3σ 9σ possible?! − → Key: more/better data and/or progress in non-perturbative QCD

For muon g 2: − O main obstacle: hadronic light-by-light [data, lattice QCD, RLA]

O progress in evaluating HVP: more data (BaBar, Belle, VEPP 2000, BESIII,...), lattice QCD in reach (recent progress Jansen et al, Wittig et al, Borsanyi et al, Blum et al)

in both cases lattice QCD will be the answer one day , O also low energy effective Resonance Lagrangian and Dispersion Relation approach need be further developed for what concerns HLbL.

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 35 G For future improvements one desperately needs more information from γγ hadrons in order to have better constraints on modeling of the relevant hadronic→ amplitudes. The goal is to exploit possible new experimental constraints from γγ hadrons and crossed channels if possible →

e+ e+ γ γ P,V,S γ q¯q ¯ π¯π,KK,T,Sqq¯q¯q P γ γ γ e− e− + + e P e P γ V γ V γ − γ e e−

e+e− mostly single-tag events: KLOE, KEDR (taggers), BaBar, Belle, BES III; 0 + Dalitz-decays: ρ, ω, φ π (η)e e− Novosibirsk, NA60,JLab, Mainz, Bonn, Julich,¨ BES would be interesting,→ but some buried in the background.

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 36 G the dispersive approach to HLbL(Colangelo et al 2014) 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: 19 independent amplitudes contribute to g 2 vs. data (HVP 1 amplitude vs data) − Last but not least: do theoreticians calculate what experiments measure? (form-factor vs cross-section)

the SM virtual SM real soft γ NP a = a [= F (0)] + ∆aµ [dep. on exp. setup] +∆a µ µ 2 | {z } | {z } µ ??? Fermilab vs J PARC −

exp A lot remains to be done! while a new a µ is approaching!

Thanks you for your attention!

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 37 Supplement

+ Ì τ vs e e− isospin breaking

a) Ratio of F (E) 2 with mixing normalized to the no mixing case. The same ratio based on e+e vs | π | − τ GS ﬁts is also shown. Fee(E) 2 I=1 part only, i.e. ω subtracted, no FSR and Fτ(E) 2 after IB | π | | π | corrections, but without ρ γ mixing correction. b) The same mechanism scaled up by the − branching fraction Γ /Γ(V ππ) for V = ω and φ. In the ππ channel the effects for resonances V → V , ρ are tiny if not very close to resonance

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 38 ee Isospin breaking corrections. Left: The ﬁnal state radiation (FSR) contributing to Fπ (s), τ-decay 1 3 3 related photon radiation GEM− , phase space correction β0/β (s) related to the pion mass difference, − and the product of them rIB. Right: effects related to the shift in the ρ parameters, and from ρ ω ee − mixing, which contributes to the I=0 part of Fπ (s)

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 39 Total IB correction without and including the γ ρ mixing correction −

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 40 1.2

0.8

Isospin Breaking corrections Total corrections (Davier et al.) Total corrections (FJ) Total corrections + ρ-γ mixing (FJ) 0.6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s (GeV2) Total IB correction to be applied to the τ decay spectra. Corrections applied by Davier et al. 2009/15 are compared with the corrections as discussed in the text (labeled FJ). Courtesy of Z. Zhang

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 41 HVP from lattice QCD

had The need for ab initio calculation of aµ is well motivated: need cross checks of standard DR+data based approach

The hope is that LQCD can deliver estimates of accuracy

HVP HVP HLbL HLbL δaµ /aµ < 0.5% , δaµ /aµ 10% > in the coming years.

Primary object for HVP in LQCD: e.m. current correlator in conﬁguration space

J (~x, t) J (~0, 0) , J = 2u¯γ u 1d¯γ d 1 s¯γ s + h µ ν i µ 3 µ − 3 µ − 3 µ ··· In principle, a Fourier transform R Π (Q) = d4xei Qx J (x) J (0) = Q Q δ Q2 Π(Q2) µν h µ ν i µ ν − µν yields the vacuum polarization function Π(Q2) needed to calculate

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 42 R n o aHVP = 4α2 ∞ dQ2 f (Q2) Π(Q2) Π(0) µ 0 − The integration kernel in this representation is

2 2 2 2 16 f (Q ) = w(Q /mµ)/Q ; w(r) = . r2(1+ √1+4/r)4 √1+4/r

2 The integrand Q . Ranges between Qi = 0.00, 0.15, 0.30, 0.45 and 1.0 GeV and their percent had contribution to aµ and the “LQCD sample”

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 43 Ì LQCD lattice in finite box: momenta are quantized Qmin = 2π/L where L is the lattice box length. Q 0 L infinite volume limit min → ⇔ → ∞ Ì Q = 2π/L with m aL 4 for m 200 MeV, such that Q 314 MeV min π π ∼ min ∼ had Ì about 44% of the low x?contribution to aµ is not covered by data yet Π(Q2) − Padé O 2 2 approx. numerical lattice data: Q > (2π/L) 2 × interpolation O extrapolate to Q = 0 via Pade’s´ rs×rs rs × × × rs × × rs × of lattice data O Note need Π(0) ! × rs × × rs × × rs × × O rs required accuracy: needed LQCD × pQCD × rs × × rs 2 × × 2 rs data down to Q 0.1 GeV × ×rs × × min ×rs ≈ ×rs × × ×rs ×rs ×rs ×rs ×rs ×rs ×rs ×rs ×rs Q2 0.1 GeV2 4 GeV2 New: RBC/UKQCD≈ 18 use lattice≈ between 0.1 and 4 GeV2 and R–data for IR and UV tails most precise evaluation so far: aHVP LO = (692.5 2.7) 10 10 [alat = (232.1 1.5) 10 10 ⇒ µ − ± × − µ ± × − =ˆ 33.5% ; adat = (460.4 2.2) 10 10 ] µ ± × − see Talks Christoph Lehner, N.N., Laurent Lellouch, Georg van Hippel, Eigo Shintani

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 44 Summary of recent LQCD results for HVP 10 N f = 2 + 1 + 1 the leading order aµ , in units 10− . ■ RBC/UKQCD 18 I L 692.5 2.67 Labels: marks u, d, s, c, u, d, s ± ■ RBC/UKQCD 18 715.4 18.72 S ± and u, d contributions. Individual ■ BMW 17 711 19 ± ■ HPQCD 16 ﬂavor contributions from light (u, d) 667 13 ± ■ ETM 15 amount to about 90%, strange about 678 29 ± N f = 2 + 1 8% and charm about 2%. ▲ RBC/UKQCD 11 641 46 Budapest, Marseille, Wuppertal, ± ▲ Aubin+Blum 07 748 21 ± Brookhaven, Zeuthen, Mainz, Ed- ▲ Aubin+Blum 07 713 15 ± N f = 2 inburgh, ... The gray vertical band ■ Mainz/CLS 17 represents my evaluation. The 654 38 ± ▲ Mainz/CLS 11 618 64 wheat band represents the HVP ± ❙ ETM 11 572 16 ± required such that theory matches + FJ17 e e−&τ 688.8 3.4 + ± HLMNT11 e e− 694.4 3.7 the experimental BNL result. The ± BDDJ15 HLS ﬁt 681.9 3.2 + ± DHMZ16 e e− 692.3 4.2 + ± very precise RBC/UKQCD point is DHMZ16 e e−&τ 701.5 4.6 ± NP ■ HPV adjusted ∆aµ = 0 obtained by supplementing lattice 720.26 7.01 ± results by R–data. 600 650 700 750 800 aHVP 1010 µ ·

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 45 Neutral channel: muon and electron missing effects in lattice QCD simulations performed in the isospin limit md = mu and without QED effects. Effects have been integrated from 300 MeV to 1 GeV 10 14 δaµ 10 δa 10 × e × Correction type GS fit shift GS fit shift + [1] ? ? I = 1 NC: GS fit of e e− data 489.21 134.49 ω ρ mixing 491.89 +2.68 135.24 +0.75 − 168.39 +0.73 FSR of ee I = 1 + 0 496.11 +4.22 136.41 +1.17 γ ρ mixing 486.47 -2.74 133.99 -0.50 − ? Elmag. shift m 0 m shift of π → π± NC vs. 2 [2] 502.01 +12.81 138.21 +3.72 I = 1 mπ = mπ0 R(s) Fπ 2 [3] | | I = 1 NC mπ = mπ Fπ 455.89 125.76 ± | |2 I = 1 NC mπ = m Fπ 441.97 -13.92 121.85 -3.91 π0 | | Combined 500.91 137.91 mπ = mπ0 [4] Physical mπ = m 489.20 1.12 134.49 0.19 π± Elmag. channels [?] π0γ 4.64 0.04 1.33 0.04 ± ± ηγ 0.65 0.01 0.17 0.00 ± ± π+π π0 missing disconnected ? 5.26 0.15 1.76 0.06 − ± ± [1] [2] 2 [3] [4] ω switched off, [ Fπ fixed], [BW ρ FF], plus e.m. shift in mass&width of the ρ | |

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 46 For the charged channel the corresponding results are collected in Table above.

Charged channel: missing effects in lattice QCD simulations performed in the isospin limit md = mu and without QED effects. Tabulated are the effects δa` (` = µ, e) integrated from 300 MeV to 1 GeV 10 14 δaµ 10 δa 10 × e × Correction type GS fit shift GS fit shift GS fit of τ data 505.32 139.22 +δMρ, +δΓρ 501.44 -3.88 138.16 -1.06 1/GEM 504.62 -0.70 138.94 -0.28 β3 /β3 498.73 -6.59 137.30 -1.92 0 − I = 1, LQCD type 494.15 -11.17 135.96 -3.26

Summing up the various corrections yields the results listed in following Table:

Neutral channel: total shifts for a` (` = µ, e, τ) 10 12 type of correction δaµ 10 δa 10 × e × iso+em from ππ channel : +4.16(4) + 1.42(1) incl e.m. decays π0γ and ηγ: + 5.29(4) + 1.19(4) missing φ π+π π0 ?: + 5.26(15) + 1.35(4) → − sum 14.71(16) 3.96(6) Which of the corrections has to be supplemented depends on the what and whatnot has been included in a given lattice QCD calculation.

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 47 Exclusive channels range [0.3050,1.8000] GeV, ω and φ data 0 π γ 5.34 0.78 0.85 KS K±π∓ 0.78 0.11 0.12 + π π 501.55 73.01 79.58 KS KLη 0.12 0.02 0.02 − + π+π π0 49.15 7.15 7.80 K K−η 0.00 0.00 0.00 − + + ηγ 0.56 0.08 0.09 K K−π π− 0.32 0.05 0.05 + 0 K+K π0π0 0.04 0.01 0.01 π π−2π 17.87 2.60 2.84 − + 0 0 2π 2π− 13.54 1.97 2.15 KS KLπ π 0.11 0.02 0.02 + 0 K K π+π 0.08 0.01 0.01 π π−3π 0.74 0.11 0.12 S L − + 0 K K π+π 0.01 0.00 0.00 2π 2π−π 0.98 0.14 0.15 S S − + K+K π+π π0 0.02 0.00 0.00 2π 2π−η 0.03 0.00 0.00 − − + 0 K K π π0 0.12 0.02 0.02 2π 2π−2π 0.64 0.09 0.10 S ± ∓ + φη 0.01 0.00 0.00 3π 3π− 0.11 0.02 0.02 + 0 ηπ+π π0 0.01 0.00 0.00 π π−4π 0.03 0.00 0.00 − ωπ0 0.83 0.12 0.13 ωη 0.02 0.00 0.00 + K K− 22.26 3.24 3.53 sum no CHPT, no FSR 630.26 91.74 100.00 K0 K0 14.13 2.06 2.24 sum 637.12 92.74 100.00 S L + sum HLS 599.85 0.50 91.74 ωπ π− 0.01 0.00 0.00 + sum non HLS 37.27 2.51 5.43 ηπ π− 0.01 0.00 0.00 + 0 K K−π 0.16 0.02 0.03 0 KS KLπ 0.69 0.10 0.11

RESU2,...+CHPT+FSR: 585.47, 588.07, 589.73 sum includes CHPT tail and full FSR range [0.3050,1.8000]

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 48 Same for HLS channels + π π− 501.55 0.38 2.25 π0γ 5.34 0.03 0.23 ηγ 0.56 0.01 0.06 + 0 π π−π 49.15 0.20 1.91 K0 K0 14.13 0.05 0.31 S+ L K K− 22.26 0.29 0.88 sum 630.26 0.53 3.24 sum HLS 592.99 0.52 3.10 sum non HLS 37.27 0.07 0.93 check sum excl CHPT+FSR 630.26 check sum incl CHPT 632.86 check sum incl CHPT+FSR 637.12 KKPI collected 1.628 KKX collected 2.436 sum R incl FSR 638.13 0.53 6.56 ± ± CHPT= 2.60 FSR= 4.26 CHPT+FSR= 6.86

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 49 Exclusive channels range [0.3050,1.8000] GeV, ω and φ as BW resonances using PDG parameters π0γ 4.23 0.62 0.68 KS K±π∓ 0.78 0.11 0.12 + π π 502.15 73.09 80.53 KS KLη 0.12 0.02 0.02 − + π+π π0 44.20 6.43 7.09 K K−η 0.00 0.00 0.00 − + + ηγ 0.56 0.08 0.09 K K−π π− 0.32 0.05 0.05 + 0 K+K π0π0 0.04 0.01 0.01 π π−2π 17.87 2.60 2.87 − + 0 0 2π 2π 13.54 1.97 2.17 KS KLπ π 0.11 0.02 0.02 − + π+π π0 0.74 0.11 0.12 KS KLπ π− 0.08 0.01 0.01 −3 + + 0 KS KS π π 0.01 0.00 0.00 2π 2π−π 0.98 0.14 0.16 − + K+K π+π π0 0.02 0.00 0.00 2π 2π−η 0.03 0.00 0.00 − − + 0 K K π π0 0.12 0.02 0.02 2π 2π−2π 0.64 0.09 0.10 S ± ∓ + φη 0.01 0.00 0.00 3π 3π− 0.11 0.02 0.02 + 0 ω missing 0.09 0.01 0.01 π π 4π 0.03 0.00 0.00 → − φ missing 0.03 0.00 0.00 ωπ0 0.79 0.12 0.13 → ηπ+π π0 0.01 0.00 0.00 K+K 21.94 3.19 3.52 − − ωη 0.02 0.00 0.00 K0 K0 13.10 1.91 2.10 S L + sum no CHPT, no FSR 623.52 90.76 100.00 ωπ π− 0.01 0.00 0.00 + sum 630.38 91.76 100.00 ηπ π− 0.01 0.00 0.00 + 0 sum HLS 593.03 0.45 90.76 K K−π 0.16 0.02 0.03 0 sum non HLS 37.34 2.50 5.44 KS KLπ 0.69 0.10 0.11

RESU2,...+CHPT+FSR: 581.16, 583.76, 585.42 sum includes CHPT tail and full FSR range [0.3050,1.8000]

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 50 Same for HLS channels + π π− 502.15 0.38 2.27 π0γ 4.23 0.04 0.17 ηγ 0.56 0.01 0.01 + 0 π π−π 44.20 0.39 1.42 K0 K0 13.10 0.16 0.37 S+ L K K− 21.94 0.24 0.56 sum 623.52 0.63 2.92 sum HLS 586.17 0.62 2.76 sum non HLS 37.35 0.07 0.93 check sum excl CHPT+FSR 623.52 check sum incl CHPT 626.12 check sum incl CHPT+FSR 630.38 KKPI collected 1.628 KKX collected 2.44 sum R incl FSR 561.85 0.40 6.43 ± ± CHPT= 2.60 FSR= 4.26 CHPT+FSR= 6.86

F. Jegerlehner muonHVPws@KEK, Tsukuba, Japan, February 12-14, 2018 51