Nonperturbative QCD corrections to electroweak observables. Dru Renner Jefferson Lab (JLab)

Transcription

1 Nonperturbative QCD corrections to electroweak observables Dru Renner Jefferson Lab (JLab) work with Xu Feng (KEK), Grit Hotzel (Humboldt U.) Karl Jansen (DESY) and Marcus Petschlies (Cyprus Institute)

2 Effects of quarks and gluons on electromagnetic properties Dru Renner Jefferson Lab (JLab) work with Xu Feng (KEK), Grit Hotzel (Humboldt U.) Karl Jansen (DESY) and Marcus Petschlies (Cyprus Institute)

3 Discrepancy for muon magnetic moment the muon has the charge of an electron but a larger mass m 106 MeV versus m e MeV it has spin: intrinsic angular momentum (quantum mechanics) L = S, but with quantized values of ± 1/2 generating an intrinsic magnetic moment (gyromagnetic effect) = g e 2m S measurements and theoretical predictions for g differ by 3.6 σ is this new physics or underestimated uncertainties? 1/22

4 Outline motivation: 3.6 σ discrepancy for g and possible new physics focus on the muon g as a concrete and exciting example review the framework for making the theoretical prediction introduce our new method to tackle most pressing uncertainty return to the broader theme of nonperturbative QCD corrections illustrate this with a few examples, depending on the remaining time finish with one quite new result and an outlook for our work 2/22

5 Muon g 2

6 Muon magnetic moment the muon electromagnetic interaction is given by a few functions p p q=p -p γ = ieū(p ) { γ F 1 (q 2 ) + iσν q ν 2m F 2(q 2 ) +... } u(p) the charge and magnetic moment are seen in the q 2 0 limit q = ef 1 (0) g = 2(F 1 (0) + F 2 (0)) F 1 (0) = 1 ensures q = e and it then follows that g = 2 + 2F 2 (0) g = 2 + corrections 2 but not the expected 1 g is calculable in the Standard Model of particle physics 3/22

7 A 3.6 σ discrepancy anomalous magnetic moment due solely to radiative corrections a = g 2 2 = 0 + loop corrections experimental measurement at Brookhaven National Lab (BNL) a ex = (63) 10 3 [0.5 ppm] Standard Model prediction (involves many, many calculations) a th = (49) 10 3 [0.4 ppm] 3.6 σ might indicate physics beyond the Standard Model a ex a th = 2.87(80) 10 9 = 3.6 σ(a ex a th ) 4/22

8 New experiments new experiment at Fermi National Accelerator Lab (FNAL) σ(a ex ) = comparison then dominated by theory errors (σ(a th ) = ) σ(a ex a th ) = assuming the measurement remains consistent (±2 σ) gives a ex a th /σ(a ex a th ) = 3.6 ( ) either way, improvement in Standard Model prediction is desirable 5/22

9 . Standard Model prediction Standard Model describes interactions of elementary particles e,, τ, γ, u, d, s, c, b, t, g, νe, ν, ντ, W +, W, Z 0, h contributions to a arise from all parts of the Standard Model a = aqed + aqcd + aew many contributions are calculated using perturbation theory γ γ ν γ W W γ Ζ γ h at high enough precision, every particle contributes to everything 6/22

10 Beyond the Standard Model a significant discrepnacy in g would be taken to be new physics γ χ χ ~ v [a ] g 2 χ m 2 /m 2 χ agreement contrains any physics beyond the Standard Model M SUSY [GeV] allowed decreasing a excluded SUSY [a ] ( 100 GeV M SUSY ) 2 tan β tan(β) multiple discrepancies are needed to disentangle any new physics 7/22

11 Quarks in the muon QED accounts for % of a, the remainder is given as QCD-VP QCD-LBL EW discrepancy however QED is only 0.3% of the uncertainty, the rest is QCD-VP QCD-LBL EW at this precision a is composed of hadrons (quarks/gluons) 8/22

12 QCD corrections to g 2 QCD contribution is expanded with nonperturbative coefficients a QCD = n=1 α n A (n) qcd = a(2) + a (3,vp) + a (3,lbl) + O(α 4 ) a dispersion relation exists for the leading-order correction a (2) γ γ q q γ γ q 2 theory calculation requires significant experimental input a (2) = α 2 4m 2 π ds s K(2) (s/m 2 )R(s) R(s) = σ(γ hadrons) σ(γ e + e ) this is a significant weakness in the theoretical prediction 9/22

13 Measurement of R(s) R(s) from O(30) final states, O(10) experiments, plus P-QCD improvement in σ(e + e hadrons) claimed to be possible clearly not a proper first-principles theoretical calculation [plot from Jegerlehner, Nyffeler Phys.Rept.477, 2009] 10/22

14 Lattice calculation of a (2) a (2) can also be calculated directly in Euclidean space γ γ q q γ vacuum polarization tensor is a simple two-point function π ν (Q 2 ) = d 4 X e iq (X Y ) J (X)J ν (Y ) = (Q Q ν Q 2 δ ν )π(q 2 ) leading-order QCD contribution [Blum, PRL 2003] a (2) = α 2 0 dq 2 Q 2 w(2) (Q 2 /m 2 ) π R (Q 2 ) π R (Q 2 ) = π(q 2 ) π(0) is finite with R(s) Im π( s + iɛ) 11/22

15 Advantages of Euclidean space no complicated resonance structure, almost boring Q 2 dependence Π(Q 2 ) Q 2 [GeV 2 ] straightforward matching to perturbative QCD at large Q 2 LO QCD correction to every EW observable is due to vac-pol 12/22

16 Lattice details lattice QCD methods are used to define a proper discretized QFT J (X)J ν (Y ) = DUDψDψ exp( S QCD )J (x)j ν (x) integrals computed numerically using Markov Chain Monte Carlo J (X)J ν (Y ) = 1 N U [J (x)j ν (x)] U + O(1/ N) additionally, several limits must be performed numerically a 0 L m u,d m physical u,d m u,d limit is practical, a and L limits approach a critical 4d system ξ = 1/(am proton ) need u and d quarks (N f = 2) and also s and c quarks (N f = 4) 13/22

17 Problems with an external scale a (2) l is made dimensionless by introducing an external scale m l a (2) l = α 2 0 dq 2 Q 2 w(2) (Q 2 /m 2 l ) π R(Q 2 ) the lepton mass is completely unrelated to QCD scales m e GeV m 0.11 GeV m τ 1.8 GeV introduces dep. on lattice spacing in dimensionless quantity Q 2 m 2 l = 1 a 2 a 2 Q 2 m 2 l = 1 a 2 [Q 2 ] latt [m 2 l ] GeV gives strong m P S dep., as seen in vector-meson contribution a l,v g 2 V m 2 l m 2 V 14/22

18 Effective QCD dimension matters d eff captures the dimension of only the QCD scales d eff [X] = a X X a g0 =fixed for a standard QCD mass scale M, d eff is usual mass dimension d eff [M n ] = n however, it differs for a composite observable d eff [ m 2 /m 2 V ] = d eff[ 1/m 2 V ] = 2 for a (2), the nonperturbative result is close to the above 2 d eff [a (2) ] = (5) 15/22

19 Eliminating the external scale this understanding leads to a class of modified observables a (2) = α 2 0 Q 2 w(2) Q2 dq 2 H 2 H2 phys π R (Q 2 ) m 2 H is any hadronic scale and H(m P S m π ) = H phys, so lim a (2) m P S m π = a (2) each a (2) behaves like a proper dimensionless QCD quantity d eff [a (2) ] = 0 each a (2) is composed of hadronic scales only 16/22

20 Q 2 X is well-defined proxy for m V but is still a good choice 17/22. Smart choice for H is important any choice of scale H gives a valid observable a (2),H = α2 0 Q 2 w(2) Q2 dq 2 H 2 H2 phys π R (Q 2 ) m 2 well-established role of vector mesons strongly suggests H = m V a l,v g 2 V m 2 l H 2 phys H2 m 2 V g 2 V m 2 l m 2 ρ for H = m V future calclations will avoid problems with ρ ππ by using Q 2 dπ dq 2 Q 2 =Q 2 X = X

21 Modified approach works well bottom to top: H = 1 (std. method), H = f V and H = m V 6 5 a hvp [10-8 ] a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm m PS [GeV 2 ] each curve is a distinct observable but agreeing at physical point comparing to N f = 2 piece important, full piece is (42) /22

22 Muon g 2 new approach gave arguably the first reliable calculation of a (2) 5.5 [10-8 ] a (2) a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, N f = m PS [GeV 2 ] use of N f = 2 was the only substantially weak part of calculation our error of 2.8% in the ballpark of the 0.8% currently used in SM 19/22

23 Extensions of the new method

24 Many first calculations [10-12 ] a e (2) a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, N f =2 [10-6 ] a τ (2) a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, N f = m PS [GeV 2 ] m PS [GeV 2 ] [10-8 ] a (3,vp) a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, N f =2 α (1) (M Z ) [10-2 ] a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic m PS [GeV 2 ] m PS [GeV 2 ] relevant to: α, new physics, higher precision and Higgs mass 20/22

25 Four-flavor calculation

26 Precision results from N f = 4 QCD N f = 4 calculations can be quantitatively compared to experiment 7 [10-8 ] a (2) a=0.078 fm L=1.9 fm a=0.078 fm L=2.5 fm a=0.061 fm L=1.9 fm a=0.061 fm L=2.9 fm linear quadratic dispersive analysis m PS [GeV 2 ] too earlier to reach conclusions yet, more work is needed, but... a value lower than the dispersive result increases the discrepancy 21/22

27 Conclusions quarks and gluons (QCD) effect electromagnetic properties high-precision experiments reveal these QCD corrections these effects are currently experimentally determined we have a new lattice QCD method to calculate them accounting for all u, d, s and c quarks is underway g e, g, g τ, α(q 2 ) and sin 2 θ W (Q 2 ) and more are within reach 22/22

28 Extra slides

29 Measuring the muon g a magnetic moment precesses in a magnetic field (Larmor) τ = B ω = g e 2m B muon decays in t = 2.2 s at rest, so use time dilation t = γ t γ = 1/ 1 v 2 1 for v 1 use a ring to retain beam and magic γ to eliminate E pollution ω = (e/m )(a B (a (γ 2 1) 1 ) v E) determine muon spin, from the direction of the e + in the decay e + + ν e + ν 23/22

30 QCD dominates SM uncertainty Standard Model error is dominated by the QCD corrections Contribution σ th [10 10 ] QCD-LO [vp] 4.2 QCD-NLO [vp] 0.1 QCD-NLO [lbl] 2.6 EW-LO EW-NLO [had] 0.1 EW-NLO [Higgs] 0.2 QED-[ α 5 ] 0.02 Total 4.9 σ(a ex ) will not probe any higher-order corrections QCD-LO needs factor of 4 imp. and QCD-NLO a factor of 2 at this precision, g 2 is a hadronic quantity 24/22

31 Phenomenological flavor dependence pheno. analysis uses R Nf (s) to extract N f = 2 and 3 contributions R Nf (s) = R(s)( N f Q 2 f )/( N Q 2 f ) 4m2 N s 4m2 N+1 R(E) π π ρ,ω φ J/ψ E [GeV] simple/crude means of estimating importance of strange/charm [R(E) given by F. Jegerlehner s compilation of σ(e + e hadrons)] 25/22

32 Electron g 2

33 Precision α from g e high precision measurement of g e [Harvard, PRL 100: (2008)] g e /2 = (28) [0.3 ppt] extraction of α from g e just sensitive to QCD corrections α 1 = (51) [0.4 ppb] g e probes the extreme low Q 2 of vacuum polarization function a (2) e 4 3 α2 m 2 e dπ R dq 2 Q 2 =0 d eff [a (2) e ] = (1) increased accuracy of α propagates to high-energy SM tests 26/22

34 Electron g 2 test of both the modified approach and the low Q 2 behavior [10-12 ] a e (2) a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, N f = m PS [GeV 2 ] g e is similar to g with our result at 2.8% versus 0.8% used for α 27/22

35 Tau g 2

36 Enhanced BSM sensitivity with the g τ g τ is more sensitive to both QCD and new physics scales Λ Λ [g τ ] m 2 τ /Λ 2 g τ is sensitive to larger Q 2 and is another test of our approach d eff [a (2) τ ] = (13) proposals for measuring g τ at LHC, J-Parc, SuperB, usually by γ τ + τ g τ is more difficult to measure directly but a (2) τ is not 28/22

37 Tau g 2 no QCD perturbation theory, fully nonpertubative calculation 2.6 [10-6 ] a τ (2) a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, N f = m PS [GeV 2 ] for g τ we are already doing better with 2.0% versus 3.3% 29/22

38 Running of α

39 α(q), running of the QED coupling the broad range of scales relevant to QED requires summation = PI this results in a running effective-coupling parameterized by α(q) α(q) = α 1 α(q) nonperturbative physics then propagates to higher energy scales nonperturbative effects impact precision predictions 30/22

40 QCD contributes strongly to α(q) the QCD piece is a series in α with nonperturbative coefficients α qcd (Q) = n=1 α n A (n) qcd = α(1) (Q) + O(α 2 ) for now, am I interested in the leading-order piece α (1) (Q) α (1) (Q) = 4παΠ(Q 2 ) qcd μ ν 2 δ = ( Q Q Q μν ) Π(Q 2 ) NLO piece may be interesting as a simpler light-by-light example qcd 31/22

41 NP-QCD spoils the precision of α QCD corrections [1] are comparable in magnitude to leptonic [2] α qcd (M Z ) = (10) 10 2 α lep (M Z ) = but QCD is solely responsible for the drastic drop in precision σ[α] α = 0.68 σ[α(m Z )] 10 9 α(m Z ) = error on α qcd (M Z ) dominated by nonperturbative contributions α qcd (Q) is measured but should be predicted [1] Davier et. al. arxiv: [2] Steinhauser hep-ph/ /22

42 NP-QCD impacts precision predictions α(m Z ) pushes nonperturbative physics into all high Q predictions α qcd (M Z ) has a strong influence on Standard Model global fits [1] m H = GeV without measured α qcd(m Z ) m H = GeV with measured α qcd(m Z ) the influence of α qcd (M Z ) on m H is second only to that of m t [1] Gfitter arxiv: /22

43 A consistent approach for external scales Q, an external scale unrelated to QCD, spoils dimensional analysis α (1) (Q) = απ(q 2 ) = απ lat ( a 2 Q 2) an exact change of variables Q 2 Q 2 H 2 /H 2 phys a (2) = α 2 0 dq 2 Q 2 w(2) (Q 2 /m 2 ) π R Q 2 in a(2) gives H 2 H 2 phys this suggests a consistent treatment of Q 2 as the external scale Π(Q 2 ) Π Q2 Hphys 2 H 2 lim Π(Q 2 ) = Π(Q 2 ) m P S m π consistent solution for all LO corrections to EW observables 34/22

44 First lattice calculation of α(q) Q 0 = 2.5 GeV is a commonly used matching scale for α (1) (Q 0 ) α (1) (Q 0 ) [10-3 ] a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, N f = m PS [GeV 2 ] calculation is comparable to estimated N f = 2 experimental result first calculation of α(q 0 ) accurate to 2% 35/22

45 Running of α(q) from LQCD lattice artifacts slowly start to become significant for Q 2 7 GeV 2 6 α (1) (Q) [10-3 ] pheno, N f =2 lattice, N f = Q 2 [GeV 2 ] running to larger Q is difficult with standard LQCD methods α (1) (Q) can be calculated nonperturbatively 36/22

46 A hybrid strategy for α(m Z ) LQCD and PQCD can be combined to reach α (1) (M Z ) α(m Z ) = [ α(m Z ) α(q 0 )] P + [ α(q 0 )] L = (40) αs (12) α 1.8 α (1) (M Z ) [10-2 ] a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic m PS [GeV 2 ] current LQCD values for α s would result in α (1) (M Z ) to 0.8% nearly competitive with experimental value at 0.4% 37/22

47 Higher-order corrections

48 . Higher-order QCD vacuum-polarization calculated all three classes of 17 NLO diagrams involving πr(q2) (a) γ γ (b) e γ (c) uncertainty of these not relevant for BNL but is for FNAL [BNL] [FNAL] analytic continuation is significantly more involved Z dq2 (3,vp) 2 2 2) ω (Q /m )π (Q R 2 m2 Q! n 2 X πr (Q ) 2n+1 d + cn m l d(q2)n Q2 n Q2 =0 (3,vp) a = α3 methods developed here may prove useful in other circumstances 38/22

49 Modified approach continues to works a first lattice calculation of a (3,vp), further tests new approach [10-8 ] a (3,vp) a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, N f = m PS [GeV 2 ] complete non-pert. NLO (α 3 ) correction, excluding light-by-light our error of 2.5% is comparable to the 2.1% currently used in SM 39/22

50 Adler function the Adler function eliminates the UV divergence by a derivative D(Q 2 ) = 12π 2 Q 2dπ R dq 2 D(Q 2 ) = D(Q 2 /H 2 phys H2 ) this makes D(Q 2 ) much more sensitive to cut-off effects α (2) s (2 GeV 2 ) = (16) D(Q 2 ) 1 Pheno, N f =2 Lattice, N f =2 LO PQCD, N f =2 Λ (2) = 222 (27) MeV Q 2 [GeV 2 ] can determine α s and Λ at each Q 2 (2 GeV 2 used) without OPE 40/22

51 . Light-by-light contributions light-by-light corrections would be the only missing NLO pieces γ γ external lepton scale again, so modified approach may help here γ γ qcd z warmup with NLO corrections to α and hadronic EW effects 41/22

52 Muonic hydrogen and the 5 σ discrepancy a 5 σ discrepancy in muonic- versus electronic-hydrogen E ex E th = (63) mev the LO QCD corrections to the 2P/2S splitting in p E (2) hfs = 2πα5 3 dπ R dq 2 Q 2 =0 this effect is too small to be relevant to the 5 σ discrepancy but it is closely related to a (2) e and is another intersting test more interesting may be the mixed corrections, like box diagrams box diagrams relevant to Rosenbluth/pol.-transfer and QWeak 42/22

53 Modified approach for muonic hydrogen similar to a (2) e, modified approach works well for E too 0.01 E [mev] a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, N f = m PS [GeV 2 ] rough check sufficient to rule out big mistake for this correction 43/22

54 Definition of a (2) for a > 0 the large Q 2 behavior is parameterized by fitting to π R (Q 2 ) = c + ln Q 2 a n Q 2n n to be precise, we fix the definition at non-zero lattice spacing with 0 dq2 Q 2 uv 0 dq 2 Q 2 uv = 16/a2 the integral is convergent, so this is just a choice of cut-off effects this choice does not require QCD perturbation theory this definition does not force us to introduce a lattice spacing this last point is important given that d eff [a ] 2 44/22

55 Definition of a (2) for L < a n define π R for low Q 2 by including the lowest meson and fitting the π R (Q 2 ) = 5 9 g2 V Q 2 Q 2 + m 2 V + n a n Q 2n fit ensures that π R (Q 2 ) matches lattice calculation for accessible Q 2 extrapolation provides a well-defined finite-volume definition explicit vector-meson term is systematically reabsorbed as L increases 5 9 g2 V Q 2 Q 2 + m 2 V = n b n Q 2 for Q 2 < m 2 V this is not a systematic error but a proper finite-volume definition a practical matter of explicitly verifying controlled finite-size 45/22

56 Details on the effective dimension d eff attempts to capture the dimensionality of only the QCD scales d eff [X] = a X X a g0 =fixed for a standard mass scale M, definition is the usual mass dimension d eff [M n ] = a M n a ( ) 1 a n ˆM n (g 0 ) = a M n ˆM n (g 0 ) a however, it differs for a composite observable d eff m2 m 2 V = d eff [ 1 m 2 V ] = 2 ( ) 1 a n = n for a, we have an expression that must be evaluated on the lattice d eff [a ] = 2 ( dq 2 Q 2 w(q2 /m 2 )Q 2dπ R dq 2 ) / ( dq 2 Q 2 w(q2 /m 2 )π R ) < 0 46/22

57 Vector meson contribution to a the vector-mesons dominate the hadronic contribution to a γ q q γ γ ρ γ on general grounds we expect any model to give a,v c m2 m 2 V tree-level chiral perturbation theory gives a,v = α 2 g 2 V f(m2 /m2 V ) = 2 3 α2 g 2 V m 2 m 2 V + O(m 4 /m4 V ) this allows us to model the vector meson contribution to a (2) 47/22

58 Electromagnetic coupling of vector-meson dimensionless quantities are typically better calculated g V a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear PDG m PS [GeV 2 ] result for g V represents quantitative success for our calculation 48/22

59 Mass of vector-meson dimensionful quantities are sensitive to the overall scale setting 1.2 m V [GeV] a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm model PDG m PS [GeV 2 ] phenomenological fit includes the PDG value of m ρ 49/22

60 Renormalization of QCD + QED introduce a variable muon mass m and quark mass m q m [MeV] 1000 physical point m = m m = m / m q m q a hvp contours m q [MeV] both paths, with m or m /m q fixed, define valid physical limits but m = (m /m q ) m q follows a contour of a (2) in pqcd 50/22

61 Hadronic scheme introduce variable muon mass m and pseudo-scalar mass m P S 200 physical point m = m m = m / m ρ m V m [MeV] 150 a hvp contours m PS [MeV] curve m = (m /m ρ ) m V is implicitly defined so that m m contours from VMD model (ask me) matched to the lattice calc. 51/22

62 Phenomenological description of a (2) can combine model expectations with our calc. of g V and m V a hvp [10-8 ] a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm phenomenological measured m PS [GeV 2 ] apparently strong m P S dependence of m V is reflected in a (2) 52/22

63 Modified method for a (2) bottom to top: H = 1 (std. method), H = f V and H = m V 6 5 a hvp [10-8 ] a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm m PS [GeV 2 ] comparing to N f = 2 piece important, full piece is (42) 10 8 our error of 2.8% in the ballpark of the 0.8% currently used in SM 53/22

64 Standard Model predictions and α had precision of α (σ α /α ) is eroded by QCD corrections σ α(mz ) α(m Z ) σ GF G F σ MZ M Z this impacts many SM predictions, for example m H 54/22

65 Modified definition of α(q 2 ) a change of variables gives a hvp as a hvp = α 2 0 dq 2 Q 2 w(q2 /m 2 ) π R ( Q 2 /H 2 phys H2) this suggests treating Q 2 as an external scale like m 2 and defining α had (Q 2 ) = 4παπ R ( Q 2 /H 2 phys H2) this choice for π R (Q 2 ) then defines all other observables consistently 55/22

66 Running of α includes only the QCD corrections, remember full α 1 (M Z ) α -1 (Q 2 ) Q 2 [GeV 2 ] future work will need matching to pqcd and/or larger Q 2 56/22

67 Isospin violating corrections by varying from m 0 π to m + π, the standard method changes by mu m d = by taking the maximum variation under m 0 π to m+ π and ρ 0 to ρ + mu m d = this suggests isospin violating effects are potentially O(10 10 ) 57/22