HVP lattice status report RBC/UKQCD

Christoph Lehner (BNL)

June 20, 2018 – Mainz

Collaborators in the RBC/UKQCD g − 2 effort

Tom Blum (Connecticut)Peter Boyle (Edinburgh)Mattia Bruno (BNL)Norman Christ (Columbia)Vera Gulpers (Southampton)Masashi Hayakawa (Nagoya)James Harrison (Southampton)Taku Izubuchi (BNL/RBRC)

Luchang Jin (Connecticut)Chulwoo Jung (BNL)Andreas Juttner (Southampton)Christoph Lehner (BNL)Kim Maltman (York)Aaron Meyer (BNL)Antonin Portelli (Edinburgh)Tobi Tsang (Edinburgh)

Outline

1.

Calculation of the hadronic vacuum polarization contribution to the muon anomalousmagnetic moment

T. Blum,1 P.A. Boyle,2 V. Gulpers,3 T. Izubuchi,4, 5 L. Jin,1, 5

C. Jung,4 A. Juttner,3 C. Lehner,4, A. Portelli,2 and J.T. Tsang2

(RBC and UKQCD Collaborations)1Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA

2School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, UK3School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK

4Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA5RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA

(Dated: January 22, 2018)

We present a first-principles lattice QCD+QED calculation at physical pion mass of the leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. Thetotal contribution of up, down, strange, and charm quarks including QED and strong isospin break-ing e↵ects is found to be aHVP LO

µ = 715.4(16.3)(9.2) 1010, where the first error is statisticaland the second is systematic. By supplementing lattice data for very short and long distances withexperimental R-ratio data using the compilation of Ref. [1], we significantly improve the precision ofour calculation and find aHVP LO

µ = 692.5(1.4)(0.5)(0.7)(2.1) 1010 with lattice statistical, latticesystematic, R-ratio statistical, and R-ratio systematic errors given separately. This is the currentlymost precise determination of the leading-order hadronic vacuum polarization contribution to themuon anomalous magnetic moment. In addition, we present the first lattice calculation of thelight-quark QED correction at physical pion mass.

PACS numbers: 12.38.Gc

INTRODUCTION

The anomalous magnetic moment of the muon aµ is de-fined as the deviation of the Lande factor gµ from Dirac’s

relativistic quantum mechanics result, aµ =gµ2

2 . It isone of the most precisely determined quantities in parti-cle physics and is currently known both experimentally(BNL E821) [2] and from a standard model theory cal-culation [3] to approximately 1/2 parts per million.

Interestingly, the standard model result aSMµ deviates

from the experimental measurement aEXPµ at the 3–4

sigma level, depending on which determination of theleading-order hadronic vacuum polarization aHVP LO

µ isused. One finds

aEXPµ aSM

µ = 25.0(4.3)(2.6)(6.3) 1010 [4] ,

31.8(4.1)(2.6)(6.3) 1010 [5] ,

26.8(3.4)(2.6)(6.3) 1010 [6] , (1)

where the quoted errors correspond to the uncertaintyin aHVP LO

µ , aSMµ aHVP LO

µ , and aEXPµ . This tension

may hint at new physics beyond the standard model ofparticle physics such that a reduction of uncertainties inEq. (1) is highly desirable. New experiments at Fermilab(E989) [7] and J-PARC (E34) [8] intend to decrease theexperimental uncertainty by a factor of four. First resultsof the E989 experiment may be available before the endof 2018 [9] such that a reduction in uncertainty of theaHVP LO

µ contribution is of timely interest.In the following, we perform a complete first-principles

calculation of aHVP LOµ in lattice QCD+QED at physi-

cal pion mass with non-degenerate up and down quarkmasses and present results for the up, down, strange, andcharm quark contributions. Our lattice calculation of thelight-quark QED correction to aHVP LO

µ is the first suchcalculation performed at physical pion mass. In addition,we replace lattice data at very short and long distancesby experimental e+e scattering data using the compila-tion of Ref. [1], which allows us to produce the currentlymost precise determination of aHVP LO

µ .

COMPUTATIONAL METHOD

The general setup of our non-perturbative lattice com-putation is described in Ref. [10]. We compute

aµ = 4↵2

Z 1

0

dq2f(q2)[(q2) (q2 = 0)] , (2)

where f(q2) is a known analytic function [10] and (q2)is defined as

Px eiqxhJµ(x)J(0)i = (µq

2 qµq)(q2)with sum over space-time coordinate x and Jµ(x) =iP

f Qf f (x)µ f (x). The sum is over up, down,strange, and charm quark flavors with QED chargesQup, charm = 2/3 and Qdown, strange = 1/3. Forconvenience we do not explicitly write the superscriptHVP LO. We compute (q2) using the kernel functionof Refs. [11, 12]

(q2) (q2 = 0) =X

t

cos(qt) 1

q2+

1

2t2

C(t) (3)

arX

iv:1

801.

0722

4v1

[hep

-lat]

22

Jan

2018

arXiv:1801.07224, accepted by PRL

2.

Improved methods for reduced statistical and systematic errors

1 / 24

Time-Moment Representation

Starting from the vector current Jµ(x) = i∑

f Qf Ψf (x)γµΨf (x) we maywrite

aHVP LOµ =

∞∑

t=0

wtC (t)

with

C (t) =1

3

∑

~x

∑

j=0,1,2

〈Jj(~x , t)Jj(0)〉

and wt capturing the photon and muon part of the HVP diagrams(Bernecker-Meyer 2011).

The correlator C (t) is computed in lattice QCD+QED at physical pionmass with non-degenerate up and down quark masses including up,down, strange, and charm quark contributions. The missing bottomquark contributions are computed in pQCD.

2 / 24

Diagrams – Isospin limit

2

with C(t) = 13

P~x

Pj=0,1,2hJj(~x, t)Jj(0)i. With appro-

priate definition of wt, we can therefore write

aµ =X

t

wtC(t) . (4)

The correlator C(t) is computed in lattice QCD+QEDwith dynamical up, down, and strange quarks and non-degenerate up and down quark masses. We compute themissing contributions to aµ from bottom quarks and fromcharm sea quarks in perturbative QCD [13] by integratingthe time-like region above 2 GeV and find them to besmaller than 0.3 1010.

We tune the bare up, down, and strange quark massesmup, mdown, and mstrange such that the 0, +, K0, andK+ meson masses computed in our calculation agree withthe respective experimental measurements [14]. The lat-tice spacing is determined by setting the mass toits experimental value. We perform the calculation as aperturbation around an isospin-symmetric lattice QCDcomputation [15, 16] with two degenerate light quarkswith mass mlight and a heavy quark with mass mheavy

tuned to produce a pion mass of 135.0 MeV and a kaonmass of 495.7 MeV [17]. The correlator is expanded inthe fine-structure constant ↵ as well as mup, down =mup, down mlight, and mstrange = mstrange mheavy.We write

C(t) = C(0)(t) + ↵C(1)QED(t) +

X

f

mfC(1)mf

(t)

+ O(↵2,↵m,m2) , (5)

where C(0)(t) is obtained in the lattice QCD calculationat the isospin symmetric point and the expansion termsdefine the QED and strong isospin-breaking (SIB) correc-tions, respectively. We keep only the leading correctionsin ↵ and mf which is sucient for the desired precision.

We insert the photon-quark vertices perturbativelywith photons coupled to local lattice vector currents mul-tiplied by the renormalization factor ZV [17]. We useZA ZV for the charm [22] and QED corrections. TheSIB correction is computed by inserting scalar operatorsin the respective quark lines. The procedure used fore↵ective masses in such a perturbative expansion is ex-plained in Ref. [18]. We use the finite-volume QEDL

prescription [19] and remove the universal 1/L and 1/L2

corrections to the masses [20] with spatial lattice size L.The e↵ect of 1/L3 corrections is small compared to ourstatistical uncertainties. We find mup = 0.00050(1),mdown = 0.00050(1), and mstrange = 0.0002(2) forthe 48I lattice ensemble described in Ref. [17]. The shiftof the mass due to the QED correction is significantlysmaller than the lattice spacing uncertainty and its e↵ecton C(t) is therefore not included separately.

Figure 1 shows the quark-connected and quark-disconnected contributions to C(0). Similarly, Fig. 2shows the relevant diagrams for the QED correction to

FIG. 1. Quark-connected (left) and quark-disconnected(right) diagram for the calculation of aHVP LO

µ . We do notdraw gluons but consider each diagram to represent all ordersin QCD.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 10 20 30 40 50 60 70r

Resulting two-point p(d) from p(r)=(1.5 + r)-5

Figure 6: Displacement probability for 48c run 1.

(a) V (b) S (c) T (d) D1 (e) D2

(f) F (g) D3

Figure 7: Mass-splitting and HVP 1-photon diagrams. In the former the dotsare meson operators, in the latter the dots are external photon vertices. Notethat for the HVP some of them (such as F with no gluons between the twoquark loops) are counted as HVP NLO instead of HVP LO QED corrections.We need to make sure not to double-count those, i.e., we need to include theappropriate subtractions! Also note that some diagrams are absent for flavornon-diagonal operators.

8

FIG. 2. QED-correction diagrams with external pseudo-scalaror vector operators.

the meson spectrum and the hadronic vacuum polariza-tion. The external vertices are pseudo-scalar operatorsfor the former and vector operators for the latter. Werefer to diagrams S and V as the QED-connected and todiagram F as the QED-disconnected contribution. Wenote that only the parts of diagram F with additionalgluons exchanged between the two quark loops contributeto aHVP LO

µ as otherwise an internal cut through a singlephoton line is possible. For this reason, we subtract theseparate quantum-averages of quark loops in diagram F.In the current calculation, we neglect diagrams T, D1,D2, and D3. This approximation is estimated to yield anO(10%) correction for isospin splittings [21] for which theneglected diagrams are both SU(3) and 1/Nc suppressed.For the hadronic vacuum polarization the contribution ofneglected diagrams is still 1/Nc suppressed and we adopta corresponding 30% uncertainty.

In Fig. 3, we show the SIB diagrams. In the calcu-

x

x

x

(a) M

x

x

x

(b) R

x

x

x

(c) O

Figure 8: Mass-counterterm diagrams for mass-splitting and HVP 1-photondiagrams. Diagram M gives the valence, diagram R the sea quark mass shifte↵ects to the meson masses. Diagram O would yield a correction to the HVPdisconnected contribution (that likely is very small).

9

FIG. 3. Strong isospin-breaking correction diagrams. Thecrosses denote the insertion of a scalar operator.

3 / 24

Diagrams – QED corrections

2

with C(t) = 13

P~x

Pj=0,1,2hJj(~x, t)Jj(0)i. With appro-

priate definition of wt, we can therefore write

aµ =X

t

wtC(t) . (4)

The correlator C(t) is computed in lattice QCD+QEDwith dynamical up, down, and strange quarks and non-degenerate up and down quark masses. We compute themissing contributions to aµ from bottom quarks and fromcharm sea quarks in perturbative QCD [13] by integratingthe time-like region above 2 GeV and find them to besmaller than 0.3 1010.

We tune the bare up, down, and strange quark massesmup, mdown, and mstrange such that the 0, +, K0, andK+ meson masses computed in our calculation agree withthe respective experimental measurements [14]. The lat-tice spacing is determined by setting the mass toits experimental value. We perform the calculation as aperturbation around an isospin-symmetric lattice QCDcomputation [15, 16] with two degenerate light quarkswith mass mlight and a heavy quark with mass mheavy

tuned to produce a pion mass of 135.0 MeV and a kaonmass of 495.7 MeV [17]. The correlator is expanded inthe fine-structure constant ↵ as well as mup, down =mup, down mlight, and mstrange = mstrange mheavy.We write

C(t) = C(0)(t) + ↵C(1)QED(t) +

X

f

mfC(1)mf

(t)

+ O(↵2,↵m,m2) , (5)

where C(0)(t) is obtained in the lattice QCD calculationat the isospin symmetric point and the expansion termsdefine the QED and strong isospin-breaking (SIB) correc-tions, respectively. We keep only the leading correctionsin ↵ and mf which is sucient for the desired precision.

We insert the photon-quark vertices perturbativelywith photons coupled to local lattice vector currents mul-tiplied by the renormalization factor ZV [17]. We useZA ZV for the charm [22] and QED corrections. TheSIB correction is computed by inserting scalar operatorsin the respective quark lines. The procedure used fore↵ective masses in such a perturbative expansion is ex-plained in Ref. [18]. We use the finite-volume QEDL

prescription [19] and remove the universal 1/L and 1/L2

corrections to the masses [20] with spatial lattice size L.The e↵ect of 1/L3 corrections is small compared to ourstatistical uncertainties. We find mup = 0.00050(1),mdown = 0.00050(1), and mstrange = 0.0002(2) forthe 48I lattice ensemble described in Ref. [17]. The shiftof the mass due to the QED correction is significantlysmaller than the lattice spacing uncertainty and its e↵ecton C(t) is therefore not included separately.

Figure 1 shows the quark-connected and quark-disconnected contributions to C(0). Similarly, Fig. 2shows the relevant diagrams for the QED correction to

FIG. 1. Quark-connected (left) and quark-disconnected(right) diagram for the calculation of aHVP LO

µ . We do notdraw gluons but consider each diagram to represent all ordersin QCD.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 10 20 30 40 50 60 70r

Resulting two-point p(d) from p(r)=(1.5 + r)-5

Figure 6: Displacement probability for 48c run 1.

(a) V (b) S (c) T (d) D1 (e) D2

(f) F (g) D3

Figure 7: Mass-splitting and HVP 1-photon diagrams. In the former the dotsare meson operators, in the latter the dots are external photon vertices. Notethat for the HVP some of them (such as F with no gluons between the twoquark loops) are counted as HVP NLO instead of HVP LO QED corrections.We need to make sure not to double-count those, i.e., we need to include theappropriate subtractions! Also note that some diagrams are absent for flavornon-diagonal operators.

8

FIG. 2. QED-correction diagrams with external pseudo-scalaror vector operators.

the meson spectrum and the hadronic vacuum polariza-tion. The external vertices are pseudo-scalar operatorsfor the former and vector operators for the latter. Werefer to diagrams S and V as the QED-connected and todiagram F as the QED-disconnected contribution. Wenote that only the parts of diagram F with additionalgluons exchanged between the two quark loops contributeto aHVP LO

µ as otherwise an internal cut through a singlephoton line is possible. For this reason, we subtract theseparate quantum-averages of quark loops in diagram F.In the current calculation, we neglect diagrams T, D1,D2, and D3. This approximation is estimated to yield anO(10%) correction for isospin splittings [21] for which theneglected diagrams are both SU(3) and 1/Nc suppressed.For the hadronic vacuum polarization the contribution ofneglected diagrams is still 1/Nc suppressed and we adopta corresponding 30% uncertainty.

In Fig. 3, we show the SIB diagrams. In the calcu-

x

x

x

(a) M

x

x

x

(b) R

x

x

x

(c) O

Figure 8: Mass-counterterm diagrams for mass-splitting and HVP 1-photondiagrams. Diagram M gives the valence, diagram R the sea quark mass shifte↵ects to the meson masses. Diagram O would yield a correction to the HVPdisconnected contribution (that likely is very small).

9

FIG. 3. Strong isospin-breaking correction diagrams. Thecrosses denote the insertion of a scalar operator.

For diagram F we enforce exchange of gluons between the quark loops as otherwise acut through a single photon line would be possible. This single-photon contribution iscounted as part of the HVP NLO and not included for the HVP LO.

Diagrams T, D1, D2, D3 are not included for the central value of the currentcalculation. They are suppressed by SU(3), 1/Nc , or both and we estimate theircontribution in our uncertainty. Diagrams V, S, and F are included.

4 / 24

Diagrams – Strong isospin breaking

2

with C(t) = 13

P~x

Pj=0,1,2hJj(~x, t)Jj(0)i. With appro-

priate definition of wt, we can therefore write

aµ =X

t

wtC(t) . (4)

The correlator C(t) is computed in lattice QCD+QEDwith dynamical up, down, and strange quarks and non-degenerate up and down quark masses. We compute themissing contributions to aµ from bottom quarks and fromcharm sea quarks in perturbative QCD [13] by integratingthe time-like region above 2 GeV and find them to besmaller than 0.3 1010.

We tune the bare up, down, and strange quark massesmup, mdown, and mstrange such that the 0, +, K0, andK+ meson masses computed in our calculation agree withthe respective experimental measurements [14]. The lat-tice spacing is determined by setting the mass toits experimental value. We perform the calculation as aperturbation around an isospin-symmetric lattice QCDcomputation [15, 16] with two degenerate light quarkswith mass mlight and a heavy quark with mass mheavy

tuned to produce a pion mass of 135.0 MeV and a kaonmass of 495.7 MeV [17]. The correlator is expanded inthe fine-structure constant ↵ as well as mup, down =mup, down mlight, and mstrange = mstrange mheavy.We write

C(t) = C(0)(t) + ↵C(1)QED(t) +

X

f

mfC(1)mf

(t)

+ O(↵2,↵m,m2) , (5)

where C(0)(t) is obtained in the lattice QCD calculationat the isospin symmetric point and the expansion termsdefine the QED and strong isospin-breaking (SIB) correc-tions, respectively. We keep only the leading correctionsin ↵ and mf which is sucient for the desired precision.

We insert the photon-quark vertices perturbativelywith photons coupled to local lattice vector currents mul-tiplied by the renormalization factor ZV [17]. We useZA ZV for the charm [22] and QED corrections. TheSIB correction is computed by inserting scalar operatorsin the respective quark lines. The procedure used fore↵ective masses in such a perturbative expansion is ex-plained in Ref. [18]. We use the finite-volume QEDL

prescription [19] and remove the universal 1/L and 1/L2

corrections to the masses [20] with spatial lattice size L.The e↵ect of 1/L3 corrections is small compared to ourstatistical uncertainties. We find mup = 0.00050(1),mdown = 0.00050(1), and mstrange = 0.0002(2) forthe 48I lattice ensemble described in Ref. [17]. The shiftof the mass due to the QED correction is significantlysmaller than the lattice spacing uncertainty and its e↵ecton C(t) is therefore not included separately.

Figure 1 shows the quark-connected and quark-disconnected contributions to C(0). Similarly, Fig. 2shows the relevant diagrams for the QED correction to

FIG. 1. Quark-connected (left) and quark-disconnected(right) diagram for the calculation of aHVP LO

µ . We do notdraw gluons but consider each diagram to represent all ordersin QCD.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 10 20 30 40 50 60 70r

Resulting two-point p(d) from p(r)=(1.5 + r)-5

Figure 6: Displacement probability for 48c run 1.

(a) V (b) S (c) T (d) D1 (e) D2

(f) F (g) D3

Figure 7: Mass-splitting and HVP 1-photon diagrams. In the former the dotsare meson operators, in the latter the dots are external photon vertices. Notethat for the HVP some of them (such as F with no gluons between the twoquark loops) are counted as HVP NLO instead of HVP LO QED corrections.We need to make sure not to double-count those, i.e., we need to include theappropriate subtractions! Also note that some diagrams are absent for flavornon-diagonal operators.

8

FIG. 2. QED-correction diagrams with external pseudo-scalaror vector operators.

the meson spectrum and the hadronic vacuum polariza-tion. The external vertices are pseudo-scalar operatorsfor the former and vector operators for the latter. Werefer to diagrams S and V as the QED-connected and todiagram F as the QED-disconnected contribution. Wenote that only the parts of diagram F with additionalgluons exchanged between the two quark loops contributeto aHVP LO

µ as otherwise an internal cut through a singlephoton line is possible. For this reason, we subtract theseparate quantum-averages of quark loops in diagram F.In the current calculation, we neglect diagrams T, D1,D2, and D3. This approximation is estimated to yield anO(10%) correction for isospin splittings [21] for which theneglected diagrams are both SU(3) and 1/Nc suppressed.For the hadronic vacuum polarization the contribution ofneglected diagrams is still 1/Nc suppressed and we adopta corresponding 30% uncertainty.

In Fig. 3, we show the SIB diagrams. In the calcu-

x

x

x

(a) M

x

x

x

(b) R

x

x

x

(c) O

Figure 8: Mass-counterterm diagrams for mass-splitting and HVP 1-photondiagrams. Diagram M gives the valence, diagram R the sea quark mass shifte↵ects to the meson masses. Diagram O would yield a correction to the HVPdisconnected contribution (that likely is very small).

9

FIG. 3. Strong isospin-breaking correction diagrams. Thecrosses denote the insertion of a scalar operator.

For the HVP R is negligible since ∆mu ≈ −∆md and O is SU(3) and 1/Nc

suppressed. Therefore we do not include R and O for the current calculation and onlyestimate their contribution in our uncertainty. The leading diagram M is included.

5 / 24

Regions of precision (R-ratio data here is from Fred Jegerlehner 2017)

3

0 100 200 300 400 500

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

× 10

-10

t / fm

R-ratioLight+Strange (64I)

FIG. 4. Comparison of wtC(t) obtained using R-ratio data[1] and lattice data on our 64I ensemble.

lation presented here, we only include diagram M. Forthe meson masses this corresponds to neglecting the seaquark mass correction, which we have previously [17] de-termined to be an O(2%) and O(14%) e↵ect for the pi-ons and kaons, respectively. This estimate is based onthe analytic fits of (H7) and (H9) of Ref. [17] with ratiosC

m, K

2 /Cm, K

1 given in Tab. XVII of the same reference.For the hadronic vacuum polarization the contribution ofdiagram R is negligible since mup mdown and di-agram O is SU(3) and 1/Nc suppressed. We thereforeassign a corresponding 10% uncertainty to the SIB cor-rection.

We also compute the O(↵) correction to the vectorcurrent renormalization factor ZV used in C(0) [17, 18]and find a small correction of approximately 0.05% forthe light quarks.

We perform the calculation of C(0) on the 48I and 64Iensembles described in Ref. [17] for the up, down, andstrange quark-connected contributions. For the charmcontribution we also perform a global fit using additionalensembles described in Ref. [22]. The quark-disconnectedcontribution as well as QED and SIB corrections are com-puted only on ensemble 48I.

For the noisy light quark connected contribution, weemploy a multi-step approximation scheme with low-mode averaging [23] over the entire volume and two levelsof approximations in a truncated deflated solver (AMA)[24–27] of randomly positioned point sources. The low-mode space is generated using a new Lanczos methodworking on multiple grids [28]. Our improved statisti-cal estimator for the quark disconnected diagrams is de-scribed in Ref. [29] and our strategy for the strange quarkis published in Ref. [30]. For diagram F, we re-use point-source propagators generated in Ref. [31].

The correlator C(t) is related to the R-ratio data[11] by C(t) = 1

122

R10

d(p

s)R(s)sep

st with R(s) =3s

4↵2(s, e+e ! had). In Fig. 4 we compare a latticeand R-ratio evaluation of wtC(t) and note that the R-ratio data is most precise at very short and long dis-tances, while the lattice data is most precise at interme-diate distances. We are therefore led to also investigatea position-space “window method” [11, 32] and write

aµ = aSDµ + aW

µ + aLDµ (6)

with aSDµ =

Pt C(t)wt[1 (t, t0,)], aW

µ =Pt C(t)wt[(t, t0,) (t, t1,)], and aLD

µ =Pt C(t)wt(t, t1,), where each contribution is

accessible from both lattice and R-ratio data. We define(t, t0,) = [1 + tanh [(t t0)/]] /2 which we find tobe helpful to control the e↵ect of discretization errorsby the smearing parameter . We then take aSD

µ and

aLDµ from the R-ratio data and aW

µ from the lattice.In this work we use = 0.15 fm, which we find toprovide a suciently sharp transition without increasingdiscretization errors noticeably. This method takes themost precise regions of both datasets and therefore maybe a promising alternative to the proposal of Ref. [33].

ANALYSIS AND RESULTS

In Tab. I we show our results for the individual as wellas summed contributions to aµ for the window methodas well as a pure lattice determination. We quote sta-tistical uncertainties for the lattice data (S) and the R-ratio data (RST) separately. For the quark-connectedup, down, and strange contributions, the computation isperformed on two ensembles with inverse lattice spacinga1 = 1.730(4) GeV (48I) as well as a1 = 2.359(7) GeV(64I) and a continuum limit is taken. The discretizationerror (C) is estimated by taking the maximum of thesquared measured O(a2) correction as well as a simple(a)4 estimate, where we take = 400 MeV. We findthe results on the 48I ensemble to di↵er only a few per-cent from the continuum limit. This holds for the fulllattice contribution as well as the window contributionsconsidered in this work. For the quark-connected charmcontribution additional ensembles described in Ref. [22]are used and the maximum of the above and a (amc)

4

estimate is taken as discretization error. The remain-ing contributions are small and only computed on the48I ensemble for which we take (a)2 as estimate of dis-cretization errors.

For the up and down quark-connected and discon-nected contributions, we correct finite-volume e↵ects toleading order in finite-volume position-space chiral per-turbation theory [34]. Note that in our previous pub-lication of the quark-disconnected contribution [29], weadded this finite-volume correction as an uncertainty butdid not shift the central value. We take the largest ratioof p6 to p4 corrections of Tab. 1 of Ref. [35] as systematicerror estimate of neglected finite-volume errors (V). Forthe SIB correction we also include the sizeable di↵erenceof the corresponding finite and infinite-volume chiral per-turbation theory calculation as finite-volume uncertainty.For the QED correction, we repeat the computation us-ing an infinite-volume photon (QED1 [36]) and includethe di↵erence to the QEDL result as a finite-volume er-ror. Further details of the QED1 procedure are providedas supplementary material.

The precision of lattice data deteriorates exponentially as we go to large t, however, is precise at intermediatedistances. The R-ratio is very precise at long distances.

Note: in this plot a direct comparison of R-ratio and lattice data is not appropriate. Continuum limit,infinite-volume corrections, charm contributions, and IB corrections are missing from lattice data shown here.

6 / 24

Window method

We therefore also consider a window method. Following Meyer-Bernecker2011 and smearing over t to define the continuum limit we write

aµ = aSDµ + aW

µ + aLDµ

with

aSDµ =

∑

t

C (t)wt [1−Θ(t, t0,∆)] ,

aWµ =

∑

t

C (t)wt [Θ(t, t0,∆)−Θ(t, t1,∆)] ,

aLDµ =

∑

t

C (t)wtΘ(t, t1,∆) ,

Θ(t, t ′,∆) = [1 + tanh [(t − t ′)/∆]] /2 .

In this version of our calculation, we useC (t) = 1

12π2

∫∞0

d(√s)R(s)se−

√st with R(s) = 3s

4πα2σ(s, e+e− → had)to compute aSD

µ and aLDµ .

7 / 24

How does this translate to the time-like region?

Supplementary Information – S1

SUPPLEMENTARY MATERIAL

In this section we expand on a selection of technical de-tails and add results to facilitate cross-checks of di↵erentcalculations of aHVP LO

µ .

Continuum limit: The continuum limit of a selec-tion of light-quark window contributions aW

µ is shown inFig. 8. We note that the results on the coarse lattice di↵erfrom the continuum limit only at the level of a few per-cent. We attribute this mild continuum limit to the fa-vorable properties of the domain-wall discretization usedin this work. This is in contrast to a rather steep contin-uum extrapolation that occurs using staggered quarks asseen, e.g., in Ref. [42].

The mild continuum limit for light quark contribu-tions is consistent with a naive power-counting estimateof (a)2 = 0.05 with = 400 MeV and suggests thatremaining discretization errors may be small. Since wefind such a mild behavior not just for a single quantitybut for all studied values of aW

µ with t0 ranging from 0.3fm to 0.5 fm and t1 ranging from 0.3 fm to 2.6 fm, wesuggest that it is rather unlikely that the mild behav-ior is result of an accidental cancellation of higher-orderterms in an expansion in a2. This lends support to ourquoted discretization error based on an O(a4) estimate.In future work, this will be subject to further scrutiny byadding a data-point at an additional lattice spacing.

Energy re-weighting: The top panel of Fig. 9 showsthe weighted correlator wtC(t) for the full aµ as well asshort-distance and long-distance projections aSD

µ and aLDµ

for t0 = 0.4 fm and t1 = 1.5 fm. The bottom panel ofFig. 9 shows the corresponding contributions to aµ sep-arated by energy scale

ps. We notice that, as expected,

aSDµ has reduced contributions from low-energy scales and

aLDµ has reduced contributions from high-energy scales.

In the limit of projection to suciently long distances, we

120

140

160

180

200

220

240

260

280

300

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

x 10

-10

a2 / fm2

t0 = 0.4, t1 = 0.9, Lightt0 = 0.4, t1 = 1.0, Lightt0 = 0.4, t1 = 1.1, Light

FIG. 8. Continuum limit of light-quark aWµ with t0 = 0.4 fm

and = 0.15 fm.

0

50

100

150

200

250

300

350

400

450

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10

-10

t / fm

C(t) wtC(t) wt θ(t,1.5fm,0.15fm)C(t) wt [1-θ(t,0.4fm,0.15fm)]

1E-03

1E-02

1E-01

1E+00

1E+01

1E+02

1E+03

1E+04

1E+05

0.1 1 10 100sqrt(s) / GeV

Σt C(t) wtΣt C(t) wt θ(t,1.5fm,0.15fm)

Σt C(t) wt [1-θ(t,0.4fm,0.15fm)]

FIG. 9. Window of R-ratio data in Euclidean position space(top) and the e↵ect of the window in terms of re-weightingenergy regions (bottom).

may attempt to contrast the R-ratio data directly withan exclusive study of the low-lying states in the latticecalculation. This is left to future work.

Statistics of light-quark contribution: We use animproved statistical estimator including a full low-modeaverage for the light-quark connected contribution in theisospin symmetric limit as discussed in the main text.For this estimator, we find that we are able to saturatethe statistical fluctuations to the gauge noise for 50 pointsources per configuration. For the 48I ensemble we mea-sure on 127 gauge configurations and for the 64I ensem-ble we measure on 160 gauge configurations. Our resultis therefore obtained from a total of approximately 14kdomain-wall fermion propagator calculations.

Results for other values of t0 and t1: In Tabs. S I-S VII we provide results for di↵erent choices of windowparameters t0 and t1. We believe that this additionaldata may facilitate cross-checks between di↵erent latticecollaborations in particular also with regard to the upand down quark connected contribution in the isospinlimit.

Supplementary Information – S1

SUPPLEMENTARY MATERIAL

In this section we expand on a selection of technical de-tails and add results to facilitate cross-checks of di↵erentcalculations of aHVP LO

µ .

Continuum limit: The continuum limit of a selec-tion of light-quark window contributions aW

µ is shown inFig. 8. We note that the results on the coarse lattice di↵erfrom the continuum limit only at the level of a few per-cent. We attribute this mild continuum limit to the fa-vorable properties of the domain-wall discretization usedin this work. This is in contrast to a rather steep contin-uum extrapolation that occurs using staggered quarks asseen, e.g., in Ref. [42].

The mild continuum limit for light quark contribu-tions is consistent with a naive power-counting estimateof (a)2 = 0.05 with = 400 MeV and suggests thatremaining discretization errors may be small. Since wefind such a mild behavior not just for a single quantitybut for all studied values of aW

µ with t0 ranging from 0.3fm to 0.5 fm and t1 ranging from 0.3 fm to 2.6 fm, wesuggest that it is rather unlikely that the mild behav-ior is result of an accidental cancellation of higher-orderterms in an expansion in a2. This lends support to ourquoted discretization error based on an O(a4) estimate.In future work, this will be subject to further scrutiny byadding a data-point at an additional lattice spacing.

Energy re-weighting: The top panel of Fig. 9 showsthe weighted correlator wtC(t) for the full aµ as well asshort-distance and long-distance projections aSD

µ and aLDµ

for t0 = 0.4 fm and t1 = 1.5 fm. The bottom panel ofFig. 9 shows the corresponding contributions to aµ sep-arated by energy scale

ps. We notice that, as expected,

aSDµ has reduced contributions from low-energy scales and

aLDµ has reduced contributions from high-energy scales.

In the limit of projection to suciently long distances, we

120

140

160

180

200

220

240

260

280

300

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

x 10

-10

a2 / fm2

t0 = 0.4, t1 = 0.9, Lightt0 = 0.4, t1 = 1.0, Lightt0 = 0.4, t1 = 1.1, Light

FIG. 8. Continuum limit of light-quark aWµ with t0 = 0.4 fm

and = 0.15 fm.

0

50

100

150

200

250

300

350

400

450

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10

-10

t / fm

C(t) wtC(t) wt θ(t,1.5fm,0.15fm)C(t) wt [1-θ(t,0.4fm,0.15fm)]

1E-03

1E-02

1E-01

1E+00

1E+01

1E+02

1E+03

1E+04

1E+05

0.1 1 10 100sqrt(s) / GeV

Σt C(t) wtΣt C(t) wt θ(t,1.5fm,0.15fm)

Σt C(t) wt [1-θ(t,0.4fm,0.15fm)]

FIG. 9. Window of R-ratio data in Euclidean position space(top) and the e↵ect of the window in terms of re-weightingenergy regions (bottom).

may attempt to contrast the R-ratio data directly withan exclusive study of the low-lying states in the latticecalculation. This is left to future work.

Statistics of light-quark contribution: We use animproved statistical estimator including a full low-modeaverage for the light-quark connected contribution in theisospin symmetric limit as discussed in the main text.For this estimator, we find that we are able to saturatethe statistical fluctuations to the gauge noise for 50 pointsources per configuration. For the 48I ensemble we mea-sure on 127 gauge configurations and for the 64I ensem-ble we measure on 160 gauge configurations. Our resultis therefore obtained from a total of approximately 14kdomain-wall fermion propagator calculations.

Results for other values of t0 and t1: In Tabs. S I-S VII we provide results for di↵erent choices of windowparameters t0 and t1. We believe that this additionaldata may facilitate cross-checks between di↵erent latticecollaborations in particular also with regard to the upand down quark connected contribution in the isospinlimit.

Most of ππ peak is captured by window from t0 = 0.4 fm to t1 = 1.5 fm,so replacing this region with lattice data reduces the dependence onBaBar versus KLOE data sets.

8 / 24

Results (Fred’s alphaQED17 results used for window result)

4

a ud, conn, isospinµ 202.9(1.4)S(0.2)C(0.1)V(0.2)A(0.2)Z 649.7(14.2)S(2.8)C(3.7)V(1.5)A(0.4)Z(0.1)E48(0.1)E64

a s, conn, isospinµ 27.0(0.2)S(0.0)C(0.1)A(0.0)Z 53.2(0.4)S(0.0)C(0.3)A(0.0)Z

a c, conn, isospinµ 3.0(0.0)S(0.1)C(0.0)Z(0.0)M 14.3(0.0)S(0.7)C(0.1)Z(0.0)M

a uds, disc, isospinµ 1.0(0.1)S(0.0)C(0.0)V(0.0)A(0.0)Z 11.2(3.3)S(0.4)V(2.3)L

a QED, connµ 0.2(0.2)S(0.0)C(0.0)V(0.0)A(0.0)Z(0.0)E 5.9(5.7)S(0.3)C(1.2)V(0.0)A(0.0)Z(1.1)E

a QED, discµ 0.2(0.1)S(0.0)C(0.0)V(0.0)A(0.0)Z(0.0)E 6.9(2.1)S(0.4)C(1.4)V(0.0)A(0.0)Z(1.3)E

a SIBµ 0.1(0.2)S(0.0)C(0.2)V(0.0)A(0.0)Z(0.0)E48 10.6(4.3)S(0.6)C(6.6)V(0.1)A(0.0)Z(1.3)E48

a udsc, isospinµ 231.9(1.4)S(0.2)C(0.1)V(0.3)A(0.2)Z(0.0)M 705.9(14.6)S(2.9)C(3.7)V(1.8)A(0.4)Z(2.3)L(0.1)E48

(0.1)E64(0.0)Ma QED, SIB

µ 0.1(0.3)S(0.0)C(0.2)V(0.0)A(0.0)Z(0.0)E(0.0)E48 9.5(7.4)S(0.7)C(6.9)V(0.1)A(0.0)Z(1.7)E(1.3)E48

a Rratioµ 460.4(0.7)RST(2.1)RSY

aµ 692.5(1.4)S(0.2)C(0.2)V(0.3)A(0.2)Z(0.0)E(0.0)E48 715.4(16.3)S(3.0)C(7.8)V(1.9)A(0.4)Z(1.7)E(2.3)L(0.0)b(0.1)c(0.0)S(0.0)Q(0.0)M(0.7)RST(2.1)RSY (1.5)E48(0.1)E64(0.3)b(0.2)c(1.1)S(0.3)Q(0.0)M

TABLE I. Individual and summed contributions to aµ multiplied by 1010. The left column lists results for the window methodwith t0 = 0.4 fm and t1 = 1 fm. The right column shows results for the pure first-principles lattice calculation. The respectiveuncertainties are defined in the main text.

We furthermore propagate uncertainties of the latticespacing (A) and the renormalization factors ZV (Z). Forthe quark-disconnected contribution we adopt the addi-tional long-distance error discussed in Ref. [29] (L) andfor the charm contribution we propagate uncertaintiesfrom the global fit procedure [22] (M). Systematic errorsof the R-ratio computation are taken from Ref. [1] andquoted as (RSY). The neglected bottom quark (b) andcharm sea quark (c) contributions as well as e↵ects ofneglected QED (Q) and SIB (S) diagrams are estimatedas described in the previous section.

For the QED and SIB corrections, we assume domi-

nance of the low-lying and states and fit C(1)QED(t)

as well as C(1)mf

(t) to (c1 + c0t)eEt, where we vary c0

and c1 for fixed energy E. The resulting p-values arelarger than 0.2 for all cases and we use this functionalform to compute the respective contribution to aµ. Forthe QED correction, we vary the energy E between thelowest and energies and quote the di↵erence as ad-ditional uncertainty (E). For the SIB correction, we takeE to be the ground-state energy.

For the light quark contribution of our pure lattice re-sult we use a bounding method [37] similar to Ref. [38]and find that upper and lower bounds meet within errorsat t = 3.0 fm. We vary the ground-state energy that en-ters this method [39] between the free-field and interact-ing value [40]. For the 48I ensemble we find Efree

0 = 527.3MeV, E0 = 517.4 MeV + O(1/L6) and for the 64I en-semble we have Efree

0 = 536.1 MeV, E0 = 525.1 MeV+ O(1/L6). We quote the respective uncertainties as(E48) and (E64). The variation of ground-state en-ergy on the 48I ensemble also enters the SIB correctionas described above.

Figure 5 shows our results for the window method witht0 = 0.4 fm. While the partial lattice and R-ratio contri-butions change by several 100 1010, the sum changesonly at the level of quoted uncertainties. This providesa non-trivial consistency check between the lattice and

680 690 700 710 720 730 740

× 10

-10 aµ

100 200 300 400 500 600 700

aµ, Latticeaµ, R-ratio

0 1 2 3 4 5 6 7

0.5 1 1.5 2 2.5t1 / fm

Lattice ErrorR-ratio Error

Total Error

FIG. 5. We show results for the window method with t0 = 0.4fm as a function of t1. The top panel shows the combinedaµ, the middle panel shows the partial contributions of thelattice and R-ratio data, and the bottom shows the respectiveuncertainties.

the R-ratio data for length scales between 0.4 fm and2.6 fm. We expand on this check in the supplementarymaterial. The uncertainty of the current analysis is min-imal for t1 = 1 fm, which we take as our main resultfor the window method. For t0 = t1 we reproduce thevalue of Ref. [1]. In Fig. 6, we show the t1-dependenceof individual lattice contributions and compare our re-sults with previously published results in Fig. 7. Ourcombined lattice and R-ratio result is more precise thanthe R-ratio computation by itself and reduces the ten-sion to the other R-ratio results. Results for di↵erentwindow parameters t0 and t1 and a comparison of indi-vidual components with previously published results areprovided as supplementary material.

For the pure lattice number the dominant errors are (S) statistics, (V)finite-volume errors, and (C) the continuum limit extrapolationuncertainty. Updates for S,V,C in second part of talk.

For the window method there are additional R-ratio systematic (RSY)and R-ratio statistical (RST) errors.

9 / 24

Window method with fixed t0 = 0.4 fm

4

a ud, conn, isospinµ 202.9(1.4)S(0.2)C(0.1)V(0.2)A(0.2)Z 649.7(14.2)S(2.8)C(3.7)V(1.5)A(0.4)Z(0.1)E48(0.1)E64

a s, conn, isospinµ 27.0(0.2)S(0.0)C(0.1)A(0.0)Z 53.2(0.4)S(0.0)C(0.3)A(0.0)Z

a c, conn, isospinµ 3.0(0.0)S(0.1)C(0.0)Z(0.0)M 14.3(0.0)S(0.7)C(0.1)Z(0.0)M

a uds, disc, isospinµ 1.0(0.1)S(0.0)C(0.0)V(0.0)A(0.0)Z 11.2(3.3)S(0.4)V(2.3)L

a QED, connµ 0.2(0.2)S(0.0)C(0.0)V(0.0)A(0.0)Z(0.0)E 5.9(5.7)S(0.3)C(1.2)V(0.0)A(0.0)Z(1.1)E

a QED, discµ 0.2(0.1)S(0.0)C(0.0)V(0.0)A(0.0)Z(0.0)E 6.9(2.1)S(0.4)C(1.4)V(0.0)A(0.0)Z(1.3)E

a SIBµ 0.1(0.2)S(0.0)C(0.2)V(0.0)A(0.0)Z(0.0)E48 10.6(4.3)S(0.6)C(6.6)V(0.1)A(0.0)Z(1.3)E48

a udsc, isospinµ 231.9(1.4)S(0.2)C(0.1)V(0.3)A(0.2)Z(0.0)M 705.9(14.6)S(2.9)C(3.7)V(1.8)A(0.4)Z(2.3)L(0.1)E48

(0.1)E64(0.0)Ma QED, SIB

µ 0.1(0.3)S(0.0)C(0.2)V(0.0)A(0.0)Z(0.0)E(0.0)E48 9.5(7.4)S(0.7)C(6.9)V(0.1)A(0.0)Z(1.7)E(1.3)E48

a Rratioµ 460.4(0.7)RST(2.1)RSY

aµ 692.5(1.4)S(0.2)C(0.2)V(0.3)A(0.2)Z(0.0)E(0.0)E48 715.4(16.3)S(3.0)C(7.8)V(1.9)A(0.4)Z(1.7)E(2.3)L(0.0)b(0.1)c(0.0)S(0.0)Q(0.0)M(0.7)RST(2.1)RSY (1.5)E48(0.1)E64(0.3)b(0.2)c(1.1)S(0.3)Q(0.0)M

TABLE I. Individual and summed contributions to aµ multiplied by 1010. The left column lists results for the window methodwith t0 = 0.4 fm and t1 = 1 fm. The right column shows results for the pure first-principles lattice calculation. The respectiveuncertainties are defined in the main text.

We furthermore propagate uncertainties of the latticespacing (A) and the renormalization factors ZV (Z). Forthe quark-disconnected contribution we adopt the addi-tional long-distance error discussed in Ref. [29] (L) andfor the charm contribution we propagate uncertaintiesfrom the global fit procedure [22] (M). Systematic errorsof the R-ratio computation are taken from Ref. [1] andquoted as (RSY). The neglected bottom quark (b) andcharm sea quark (c) contributions as well as e↵ects ofneglected QED (Q) and SIB (S) diagrams are estimatedas described in the previous section.

For the QED and SIB corrections, we assume domi-

nance of the low-lying and states and fit C(1)QED(t)

as well as C(1)mf

(t) to (c1 + c0t)eEt, where we vary c0

and c1 for fixed energy E. The resulting p-values arelarger than 0.2 for all cases and we use this functionalform to compute the respective contribution to aµ. Forthe QED correction, we vary the energy E between thelowest and energies and quote the di↵erence as ad-ditional uncertainty (E). For the SIB correction, we takeE to be the ground-state energy.

For the light quark contribution of our pure lattice re-sult we use a bounding method [37] similar to Ref. [38]and find that upper and lower bounds meet within errorsat t = 3.0 fm. We vary the ground-state energy that en-ters this method [39] between the free-field and interact-ing value [40]. For the 48I ensemble we find Efree

0 = 527.3MeV, E0 = 517.4 MeV + O(1/L6) and for the 64I en-semble we have Efree

0 = 536.1 MeV, E0 = 525.1 MeV+ O(1/L6). We quote the respective uncertainties as(E48) and (E64). The variation of ground-state en-ergy on the 48I ensemble also enters the SIB correctionas described above.

Figure 5 shows our results for the window method witht0 = 0.4 fm. While the partial lattice and R-ratio contri-butions change by several 100 1010, the sum changesonly at the level of quoted uncertainties. This providesa non-trivial consistency check between the lattice and

680 690 700 710 720 730 740

× 10

-10 aµ

100 200 300 400 500 600 700

aµ, Latticeaµ, R-ratio

0 1 2 3 4 5 6 7

0.5 1 1.5 2 2.5t1 / fm

Lattice ErrorR-ratio Error

Total Error

FIG. 5. We show results for the window method with t0 = 0.4fm as a function of t1. The top panel shows the combinedaµ, the middle panel shows the partial contributions of thelattice and R-ratio data, and the bottom shows the respectiveuncertainties.

the R-ratio data for length scales between 0.4 fm and2.6 fm. We expand on this check in the supplementarymaterial. The uncertainty of the current analysis is min-imal for t1 = 1 fm, which we take as our main resultfor the window method. For t0 = t1 we reproduce thevalue of Ref. [1]. In Fig. 6, we show the t1-dependenceof individual lattice contributions and compare our re-sults with previously published results in Fig. 7. Ourcombined lattice and R-ratio result is more precise thanthe R-ratio computation by itself and reduces the ten-sion to the other R-ratio results. Results for di↵erentwindow parameters t0 and t1 and a comparison of indi-vidual components with previously published results areprovided as supplementary material.

For t = 1 fm approximately 50% of uncertainty comes from lattice and 50% ofuncertainty comes from the R-ratio. Is there a small slope? More in a few slides!Can use this to check experimental data sets; see my KEK talk for more details

10 / 24

No new physicsKNT 2018

Jegerlehner 2017DHMZ 2017DHMZ 2012

HLMNT 2011RBC/UKQCD 2018RBC/UKQCD 2018

BMW 2017Mainz 2017

HPQCD 2016ETMC 2013

610 630 650 670 690 710 730 750aµ × 1010

BMW and RBC/UKQCD pure lattice are compatible both with nonew physics and R-ratio, need more precision!

11 / 24

Consolidate continuum limit

Adding a finer lattice

Add a−1 = 2.77 GeV lattice spacing

I Third lattice spacing for strange data (a−1 = 2.77 GeV withmπ = 234 MeV with sea light-quark mass corrected from global fit):

50

52

54

56

58

60

-0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

x 10

-10

a2 / fm2

LL SinLL

LC SinLC

Published

In this figure, we have attempted a linear fit in a2. The p value of all shownfits is good and does not resolve the a4 or a2 log(a2) coecients from zero. Wecan, however, allow them to be included in the fit (for now just a4), whichsignificantly increases the uncertainty of the extrapolation

50

52

54

56

58

60

-0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

x 10

-10

a2 / fm2

LL SinLL

LC SinLC

Published

A better way to study the quality of agreement of di↵erent discretizationsis to look at correlated di↵erences between the di↵erent methods on the sameensemble. In these di↵erences virtually all statistical noise cancels

4

I For light quark need new ensemble at physical pion mass. Proposedfor early science time at Summit Machine at Oak Ridge later thisyear (a−1 = 2.77 GeV with mπ = 139 MeV).

12 / 24

Statistical noise

Improved bounding method

Bounding Method BMW/RBC/UKQCD 2016

Our correlator in finite volume

C (t) =∑

n

|〈0|V |n〉|2e−Ent .

We can bound this correlator at each t from above and below bythe correlators

C (t;T , E ) =

C (t) t < T ,

C (T )e−(t−T )E t ≥ T

for proper choice of E . We can chose E = E0 (assumingE0 < E1 < . . .) to create a strict upper bound and any E largerthan the local effective mass to define a strict lower bound.

13 / 24

Improved Bounding Method RBC/UKQCD 2018

Therefore if we had precise knowledge of the lowest n = 0, . . . ,Nvalues of |〈0|V |n〉| and En, we could define a new correlator

CN(t) = C (t)−N∑

n=0

|〈0|V |n〉|2e−Ent

which we could bound much more strongly through the largerlowest energy EN+1 E0. New method: do a GEVP study of FVspectrum to perform this subtraction.

For more details on how to determine the energies, matrixelements, and the new bounds see discussion contributiontomorrow by Aaron Meyer!

14 / 24

Improved Bounding Method – Update for a−1 = 1.73 GeVensemble

Results for light-quark isospin-symmetric connected contribution:

I Original bounding method: 631.4(10.0)× 10−10

I Improved bounding method: 625.7(3.9)× 10−10

I Lower end of error bars still touch, more statistics under way(factor 2 more in a few weeks)

I For a−1 = 2.359 GeV ensemble, will start generating data forthis method on July 1st, for a−1 = 2.77 GeV we are aiming atthis fall.

15 / 24

Finite-volume errors

Beyond finite-volume scalar QED

Compute finite-volume effects from first-principles

Study QCD at physical pion mass at three different volumes:

L = 4.66 fm, L = 5.47 fm (published data), L = 6.22 fm

Results for light-quark isospin-symmetric connected contribution:

I aµ(L = 6.22 fm)− aµ(L = 4.66 fm) = 12.2× 10−10 (sQED),21.6(6.3)× 10−10 (lattice QCD)

I Improved bounding method crucial for reduced statistical noise toresolve the FV effect clearly

I First time this is resolved from zero in a first-principles calculationat physical pion mass (previously bound in E. Shintani 2018)

I Need to do better than sQED in finite-volume

16 / 24

Gounaris-Sakurai-Luscher method [H. Meyer 2012, Mainz 2017]

I Produce FV spectrum and matrix elements from phase-shift study(Luscher method for spectrum and amplitudes, GS for phase-shiftparametrization)

I This allows for a prediction of FV effects beyond chiral perturbationtheory given that the phase-shift parametrization captures allrelevant effects (can be checked against lattice data)

17 / 24

First constrain the p-wave phase shift from our L = 6.22 fmphysical pion mass lattice:

0

1

2

3

4

5

6

7

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

sqrt(s) / GeV

Gounaris-Sakurai Phase-Shift Parametrization32ID lattice data (6.2fm box at physical pion mass)

Eρ = 0.766(21) GeV (PDG 0.77549(34) GeV)Γρ = 0.139(18) GeV (PDG 0.1462(7) GeV)

18 / 24

Predicts |Fπ(s)|2:

0

10

20

30

40

50

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

sqrt(s) / GeV

|Fπ|2

We can then also predict matrix elements and energies for ourother lattices; successfully checked!

19 / 24

GSL finite-volume results compared to sQED and lattice

Results for light-quark isospin-symmetric connected contribution:

I FV difference between aµ(L = 6.22 fm)− aµ(L = 4.66fm) = 12.2× 10−10 (sQED), 21.6(6.3)× 10−10 (lattice QCD),20(3)× 10−10 (GSL)

I GSL prediction agrees with actual FV effect measured on the lattice,sQED is in slight tension, two-loop FV ChPT to be compared nextBijnens and Relefors 2017

I Use GSL to update FV correction of arXiv:1801.07224:aµ(L→∞)− aµ(L = 5.47 fm) = 16(4)× 10−10 (sQED),22(1)× 10−10 (GSL); sQED error estimate based on Bijnens andRelefors 2017, table 1.

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Outlook for errors

In the next 6 months we can expect:

I Statistical error reduction from improved bounding method byfactor of 3 (16.3× 10−10 → 5× 10−10)

I Better control of finite-volume correction (target error ofsmaller than 2.5× 10−10)

I Consolidate continuum limit (no error reduction at this pointexpected but more confidence)

These improvements would reduce our current error from18.7× 10−10 → 7.5× 10−10; may be able to distinguish “no newphysics” and “R-ratio” scenarios.

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Further work in progress on similar time-scale:

I Much better statistics for diagram M (strong-isospin breaking)

I Use HLbL data to improve QED precision, computesub-leading diagrams M. Bruno

I Update for our 2015 disconnected diagrams result with morestatistics

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No new physicsKNT 2018

Jegerlehner 2017DHMZ 2017DHMZ 2012

HLMNT 2011RBC/UKQCD 2018RBC/UKQCD 2018

BMW 2017Mainz 2017

HPQCD 2016ETMC 2013

610 630 650 670 690 710 730 750aµ × 1010

Error bar of RBC/UKQCD 2018 pure lattice result may be halvedby end of year.

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Conclusions

I We now have a lattice calculation at physical pion mass with QCD+QED andnon-degenerate light quark masses.

I We have results both for a pure lattice and a combined lattice/R-ratio analysis.This can cut out a significant fraction of ππ data sets from R-ratio(BaBar/KLOE).

I We have a new method to tame statistical noise (Improved Bounding Methodcombined with GEVP study, see A. Meyer discussion tomorrow)

I We have for the first time resolved from first-principles QED the finite-volumeeffects on two boxes and cross-checked against sQED and GSL.

I We have a third lattice spacing for the strange quark contribution and by theend of the year hopefully also for the light quarks

I Possible that by end of the year the pure lattice result can distinguish between“no new physics” and “R-ratio” scenarios

I Similarly by end of the year we may have resolution on ππ BaBar/KLOEdifference

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Backup

We perform the calculation as a perturbation around anisospin-symmetric lattice QCD computation with two degenerate lightquarks with mass mlight and a heavy quark with mass mheavy tuned toproduce a pion mass of 135.0 MeV and a kaon mass of 495.7 MeV.

The correlator is expanded in the fine-structure constant α as well as∆mup, down = mup, down −mlight, and ∆mstrange = mstrange −mheavy.We write

C (t) = C (0)(t) + αC(1)QED(t) +

∑

f

∆mf C(1)∆mf

(t)

+O(α2, α∆m,∆m2) .

The correlators of this expansion are computed in lattice QCD withdynamical up, down, and strange quarks. We compute the missingcontributions to aµ from charm sea quarks in perturbative QCD (RHAD)by integrating the time-like region above 2 GeV and find them to besmaller than 0.3× 10−10.

We tune the bare up, down, and strange quark masses mup, mdown, andmstrange such that the π0, π+, K 0, and K+ meson masses computed inour calculation agree with the respective experimental measurements.The lattice spacing is determined by setting the Ω− mass to itsexperimental value.

We perform the lattice calculations for the light quark contributions usingRBC/UKQCD’s 48I and 64I lattice configurations with lattice cutoffsa−1 = 1.730(4) GeV and a−1 = 2.359(7) GeV and a larger set ofensembles with up to a−1 = 2.774(10) GeV for the charm contribution.

From the parameter tuning procedure on the 48I we find∆mup = −0.00050(1), ∆mdown = 0.00050(1), and∆mstrange = −0.0002(2).

The shift of the Ω− mass due to the QED correction is significantlysmaller than the lattice spacing uncertainty and its effect on C (t) istherefore not included separately.

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