Lattice QCD inputs for the SM: Select Highlights - Aida X. El-Khadra University of Illinois - CERN Indico

Lattice QCD inputs for the SM:
 Select Highlights

 Aida X. El-Khadra

 Standard Model at the LHC 2021
 (Online)
 26-30 April 2021

Outline

 Lattice QCD Introduction

 Semileptonic B meson decay form factors

 • | Vub | and | Vcb |

 • LFU
 muon g-2
 Summary and Outlook

A. El-Khadra SM@LHC, 26-30 Apr 2021 2

Lattice QCD Introduction
 X 1
 LQCD = ¯f (D
 / + mf ) f + trFµ⌫ F µ⌫
 4
 f

 discrete Euclidean space-time (spacing a)
 derivatives ➙ difference operators, etc…
 L x
 finite spatial volume (L)
 a finite time extent (T)

 adjustable parameters
 lattice spacing: a➙0
 finite volume, time: L ➙ ∞, T > L
 quark masses (mf): MH,lat = MH,exp
 tune using hadron masses mf ➙ mf,phys mud ms mc mb
 extrapolations/interpolations

A. El-Khadra SM@LHC, 26-30 Apr 2021 3

Lattice QCD Introduction
 X 1
 LQCD = ¯f (D
 / + mf ) f + trFµ⌫ F µ⌫
 4
 f

 discrete Euclidean space-time (spacing a)
 derivatives ➙ difference operators, etc…
 L x
 finite spatial volume (L) Integrals are evaluated
 a finite time extent (T) numerically using monte
 carlo methods.

 adjustable parameters
 lattice spacing: a➙0
 finite volume, time: L ➙ ∞, T > L
 quark masses (mf): MH,lat = MH,exp
 tune using hadron masses mf ➙ mf,phys mud ms mc mb
 extrapolations/interpolations

A. El-Khadra SM@LHC, 26-30 Apr 2021 3

Lattice QCD Introduction
 X 1
 LQCD = ¯f (D
 / + mf ) f + trFµ⌫ F µ⌫
 4
 f

 discrete Euclidean space-time (spacing a)
 derivatives ➙ difference operators, etc…
 L x
 finite spatial volume (L) Integrals are evaluated
 a finite time extent (T) numerically using monte
 carlo methods.
 L
 adjustable parameters L
 lattice spacing: a➙0 a (fm)

 finite volume, time: L ➙ ∞, T > L a (fm)
 quark masses (mf): MH,lat = MH,exp
 tune using hadron masses mf ➙ mf,phys mud ms mc mb
 extrapolations/interpolations

A. El-Khadra SM@LHC, 26-30 Apr 2021 3

L x
 Lattice QCD Introduction
 a

 The State of the Art

 Lattice QCD calculations of simple quantities (with at most one stable
 meson in initial/final state) that quantitatively account for all systematic
 effects (discretization, finite volume, renormalization,…) , in some cases
 with
 • sub percent precision.
 • total errors that are commensurate (or smaller) than corresponding
 experimental uncertainties.
 Scope of LQCD calculations is increasing due to continual development of
 new methods:
 • nucleons and other baryons
 • nonleptonic decays ( K ! ⇡⇡, …)
 • resonances, scattering, long-distance effects, …
 • QED effects
 • radiative decay rates …

A. El-Khadra SM@LHC, 26-30 Apr 2021 4

Lattice QCD: Overview
 aHVP
 µ
 LO
 aHLbL
 µ [inspired by A. Kronfeld]
 gA, gT , gS
 hB̄q0 |Oi B=2
 |Bq0 i MEs for light nuclei
 hD̄0 |Oi C=2
 |D0 i
 B ! K ⇤ `` ! K⇡ `` …
 B̂K … Λb → p, Λc, Λ MK , ✏ K
 nucleon form factors, .. B → Xcℓν,
 B!D 2
 f+,0 (q ), . . .
 K + ! `+ ⌫ ( ) … other inclusive
 S=1
 h⇡⇡(I=0) |H |K 0 i decay rates,
 (q…
 K!⇡
f+ (0) B!⇡
 f 2
 ) K + ! ⇡ + `+ ` …
 …
 +,0,T

fK ± fB(s) … h⇡⇡(I=2) |H S=1 0
 |K i K + ! ⇡ + ⌫ ⌫¯

 Complexity
 ✓
 LQCD Complete First results, new methods, new ideas,
 flagship LQCD results, physical params, pilot projects, first studies
 results large(ish) errors incomplete unphysical
 systematics kinematics
A. El-Khadra SM@LHC, 26-30 Apr 2021 5

Form factors for B → π ℓνℓ and | Vub |
 µ+
 Vub
 W ⌫µ
 b̄ ū

 B0 ⇡

 d

 d (B!⇡`⌫) 2 2 2
 = (known) ⇥ |Vub | ⇥ f+ (q ) q 2 = (pB p⇡ ) 2
 dq 2

 ★ calculate the form factors in the low recoil (high q2) range.
 ★ use model-independent parameterization of q2 dependence.

 ★ calculate the complete set of form factors, f+ (q 2 ), f0 (q 2 ) and fT (q 2 ) .
 for f+ (q ), fcompare shape between experiment and lattice.
 2 2
 ★ 0 (q )

A. El-Khadra SM@LHC, 26-30 Apr 2021 6

Form factors for B → π ℓνℓ and | Vub |

 S. Aoki et al
 FLAG 2019 review, 1902.08191
 webupdate: flag.unibe.ch/2019/

 RBC [arXiv:1501.05373, PRD 2015]
 FNAL/MILC [arXiv:1503.07839, PRD 2015]

 Ongoing work by
 HPQCD, FNAL/MILC,
 JLQCD, RBC/UKQCD,…

 shape of f+ agrees with experiment and uncertainties are commensurate
 fit lattice form factors together with experimental data to determine
 | Vub | and obtain form factors (f+, f0) with improved precision…
 similar analysis for | Vub /Vcb | from Λb decay with LHCb [arXiv:1503.01421,
 PRD 2015; arXiv:1504.01568, Nature 2015].

A. El-Khadra SM@LHC, 26-30 Apr 2021 7

Form factors for Bs → K ℓνℓ

 S. Aoki et al
 FLAG 2019 review, 1902.08191
 webupdate: flag.unibe.ch/2019/

 HPQCD [arXiv:1406.2279, PRD 2014]
 RBC/UKQCD [arXiv:1501.05373, PRD 2015]
 FNAL/MILC [arXiv:1901.02561, PRD 2019]

 Lattice results for Bs → K and Bs → Ds form factors can be combined
 with new LHCb results for Bs decay rates
 Ongoing work by FNAL/MILC, RBC/UKQCD, JLQCD, HPQCD

A. El-Khadra SM@LHC, 26-30 Apr 2021 8

Form factors for B → D ℓνℓ and | Vcb |

 S. Aoki et al
 FLAG 2019 review, 1902.08191
 webupdate: flag.unibe.ch/2019/

 HPQCD [arXiv:1406.2279, PRD 2014]
 FNAL/MILC [arXiv:1505.03925, PRD 2015]

 ★ The form factors obtained from the combined exp/lattice fit are well determined
 over entire recoil range.
 ★ Canbe used for an improved SM prediction of R(D).
 ★ Ongoing work by FNAL/MILC, JLQCD, RBC/UKQCD, HPQCD

A. El-Khadra SM@LHC, 26-30 Apr 2021 9

fac

 (
 Form factors for B → D ℓν
 0.0
 0 E.
 in
 Comparison
 2 4 with
 6 prior
 8 results
 10 12

 FIG. 10: Results for s
 f0,+ (q 2 ) s s ℓ for a number of
 We very briefly compare our results
 against q 2 at the physical
 2 2
 q [GeV ] qui
 point, comparing the ratio method (from Appendix B) and FIG. 12: Di↵erential decay rates for the Bs ! Ds µ⌫µ and
 the direct method (from Section III 2). quantities
 Bs ! Ds ⌧ ⌫calculated in previous subsections with those
 ⌧ decays, calculated using the form factors deter-

 based
 minedon thework.
 in this form factors calculated by the HPQCD
 HPQCD [arXiv:1906.00701, PRD 2020] FNAL/MILC [arXiv:1901.02561, PRD 2019]
 Here we provide a new SM prediction for the quantity
 NRQCD [1703.09728] B(Bs ! Ds ⌧ ⌫⌧ )
 1.2 this work R(Ds ) = , (35)
 B(Bs ! Ds l⌫l )
 1.1 f+s (q 2)
 where l = e or µ (the di↵erence between e and µ is neg-
 ligible in comparison to our precision on R(Ds )). Our
 1.0
 result is
 0.9 R(Ds )|SM = 0.2987(46), (36)

 0.8 in which we averaged over the l = e and l = µ cases.
 Note that |Vcb | and ⌘EW cancel in this ratio. We give
 f0s(q 2)
 0.7 an error budget for this result in terms of the uncertain-
 ties from our lattice QCD calculation in Table VII. Our
 0.6 result agrees with, but is more accurate than, the previ- FIG
 0 2 4 6
 q 2[GeV2]
 8 10 12 ous lattice QCD value of R(Ds ) (0.301(6)) from [35]. An fun
 experimental result for R(Ds ) would allow a new test of
 FIG. 11: Our final result for f0,+ s Reconstructed from B→D reco form factors
 (q 2 ) compared to form fac-FIG. 20. The reconstructed form factors f þ;0 ðBs → Ds Þ ob-
 lepton universality.
 tors calculated using an NRQCD action for the b quark [35].tainedWe
 Part of the NRQCD band is shaded darker than the rest
 from Eq.
 the analogous (7.12). R(D)and
 [1505.03925]
 quantity B /B
 expect very little di↵erence between R(Ds ) and
 ratio
 because sthe mass [1403.0635]
 of the
 (q 2 ' 9.5GeV2 ) to signify the region where lattice results spectator quark has little e↵ect on the form factors [34].
 were directly calculated. The NRQCD form factors in the Lattice QCD calculations that involve light spectator
 ★
 BCL parameterization. be used to prediction R(D )
 rest of the q 2 range are the result of an extrapolation using a
 Can . the process Bs ! Ds is under better control. Pre-
 quarks have larger statistical errors, however, which is
 swhy
 ★ New: experimental measurements of differential decay rate by LHCb
 vious lattice QCD results for R(D) are 0.300(8) [23] and
 0.299(11) [22], in which any di↵erence with our result for
 ★ Ongoing
 tion (1) from [28] and ⌘EW work
 = 1.011(5) by FNAL/MILC,
 [23]. The distribu- JLQCD,
 R(Ds ) is too RBC/UKQCD,
 small to be visible withHPQCD
 these uncertainties.
 2 2
 tion in the ⌧ case is cut o↵ at q = m⌧ and so, although
 there is enhancement from m2` /q 2 terms in Equation (1)
 that reflect reduced helicity suppression, the integrated IV. COMPARISON TO HQET
 branching fraction for the ⌧ case is smaller than for the
A.µ.El-Khadra SM@LHC,In26-30FigureApr
 13 we show our form factor results at two
 2021 10
 The ratio of branching fractions for semileptonic B de- key values of q , the zero recoil point and q 2 = 0, as a
 2

Form factors for B → D* ℓνℓ and | Vcb |

 d 2 2 1/2 2
 = (known) ⇥ |Vcb | ⇥ (w 1) ⇥ (w)|F(w)|
 dw
 w = vB · vD ⇤

 ★ F(w) = f [hA1 (w), hV (w), hA2 (w), hA3 (w)]
 ★ results for form factor at zero recoil:

 FNAL/MILC [J. Bailey et al, arXiv:1403.0635, 2014 PRD] F(1) = 0.906(4)(12)
 HPQCD [Harrison et al, arXiv:1711.11013, 2018 PRD] F(1) = 0.895 (10)(24)
 AAACCHicbZDLSgMxFIbPeK31NurShcEitCBlplSsC6EoiMsK9gKdoWTStA3NXEgyQhm6dOOruHGhiFsfwZ1vY9rOQlt/CHz5zzkk5/cizqSyrG9jaXlldW09s5Hd3Nre2TX39hsyjAWhdRLyULQ8LClnAa0rpjhtRYJi3+O06Q2vJ/XmAxWShcG9GkXU9XE/YD1GsNJWxzxKHII5uhmjvF1Al8gqVi7OnFN9swr5UrnQMXNW0ZoKLYKdQg5S1Trml9MNSezTQBGOpWzbVqTcBAvFCKfjrBNLGmEyxH3a1hhgn0o3mS4yRifa6aJeKPQJFJq6vycS7Es58j3d6WM1kPO1iflfrR2rXsVNWBDFigZk9lAv5kiFaJIK6jJBieIjDZgIpv+KyAALTJTOLqtDsOdXXoRGqWhrvivnqldpHBk4hGPIgw3nUIVbqEEdCDzCM7zCm/FkvBjvxsesdclIZw7gj4zPH0WnlP8=

 ⇤
 ★ result for F B s !D s (1) : HPQCD [McLean et al, arXiv:1904.02046]
 ★ Non-zero recoil form factors: ongoing efforts by

 FNAL/MILC [A. Vaquero @ IPPP workshop ``Beyond Flavor Anomalies’’]
 JLQCD [T. Kaneko @APLAT 2020 conference, arXiv1912.11770]
 LANL/SWME [Bhattacharya et al, arXiv:2003.09206]

 A. El-Khadra SM@LHC, 26-30 Apr 2021 11

Implications for |Vub|

 [LHCb, arXiv:2001.03225]
 CLN
 Bs ! Ds , Ds⇤ `⌫
 BGL

A. El-Khadra SM@LHC, 26-30 Apr 2021 12

Implications for | Vub /Vcb |
 PHYSICAL
 LHCb [Aaij et al, arXiv:2012.05143, REVIEW LETTERS 126, 081804 (2021)
 2021 PRL]

 LHCb
 INFN (Italy), SURF (Netherlands), P
 First(United
 observation by LHCb!
 Kingdom), RRCKI and Yan
 − +
 CSCS (Switzerland), IFIN-HH (Roman
 Bs → K µ ν µ
 0
 LCSR (Khod.& Rus.2017) Measured
 PL-GRID rates in two
 (Poland), andlarge
 OSC bins
 (US). We
 2 4
 q < 7 GeV / c
 2
 2 2
 high q > 7 GeV
 communities behind the multiple op
 Bs → K −µ +ν µ
 0
 LQCD
 LQCD (MILC2019)
 (FNAL/MILC2019) packages
 2 on which
 2 we depend. In
 2
 q2 > 7 GeV / c4 low q < 7 GeV
 members have received support from
 Λ0b → pµ −ν µ LQCD (Detmold2015) (Germany), EPLANET, Marie
 q2 > 15 GeV2/ c4 Need smaller
 Actions, andbins
 ERCfor shape Union),
 (European
 comparison between
 Labex P2IO, experiment
 and OCEVU, and Régio
 Vub / Vcb (PDG)
 excl excl andAlpes
 LQCD(France), Key Research Pro
 Sciences of CAS, CAS PIFI, CAS CC
 0 0.1 0.2 Research Funds for Central Universitie
 Vub / Vcb Program of Guangzhou (China), RFBR
 LLC (Russia), GVA, XuntaGal, and GE
FIG. 2. Measurements of jV ub j=jV cb j in this Letter and in the Royal Society and the Leverhu
Ref. [7] and ratio inferred from the Particle Data Group (PDG) Kingdom).
[26] averages of exclusive jV ub j and jV cb j measurements, where
the Λ0b → pμ− ν̄μ result is not included. The form factor calcu-
lation used in each case is mentioned [31–33].
A. El-Khadra SM@LHC, 26-30 Apr 2021 [1] N. Cabibbo, Unitary symmetry and
 13 l

Exclusive vs. inclusive |Vcb| and |Vub|

 HFLAV (spring 2019)

 B ! D`⌫
 B ! D⇤ `⌫

 zero recoil

 B ! ⇡`⌫

 ⇤ c ` ⌫
 /⇤ b !
 ! p ` ⌫
 ⇤b

 ~3 tension between inclusive and exclusive |Vcb| and |Vub|

 A. El-Khadra SM@LHC, 26-30 Apr 2021 14
 

Exclusive vs. inclusive |Vcb| and |Vub|

 HFLAV (spring 2019)

 B ! D`⌫
 B ! D⇤ `⌫
 Need LQCD
 zero recoil calculations of the
 B → D* form
 B ! ⇡`⌫ factors at nonzero
 recoil

 ⇤ c ` ⌫
 /⇤ b !
 ! p ` ⌫
 ⇤b

 ~3 tension between inclusive and exclusive |Vcb| and |Vub|

 A. El-Khadra SM@LHC, 26-30 Apr 2021 14
 

Chiral-continuumForm
 fits factors for B → D* ℓνℓ
 A. Vaquero @ IPPP workshop ``Beyond Flavor Anomalies’’ [FNAL/MILC, in preparation]

 0.95 Extrapolation
 MILC 2014
 fm
 fm
 0.90 fm
 fm
 fm
 Available lattice data and simulations
 0.85

 0.80
 Using 15 Nf = 2 + 1 MILC ensembles of sea asqtad quarks
 The heavy quarks are treated using the Fermilab action
 0.75
 0.50

 0.70 0.40
 1.000 1.025 1.050 1.075 1.100 1.125 1.150 1.175

 0.30
 Preliminary results, combined fit p value = 0.96

 ml/mh
 hA1 (1) = 0.909(17) 0.20

 ★ Results for hA1(w), hA2(w), hA3(w), hV (w). 0.10

 ★ Can be used to calculate R(D*) (lattice-only)
 0.00
 Alejandro Vaquero (University of Utah) B̄ ! D ⇤ `¯
 ⌫ at non-zero recoil April 21st , 2021 11 / 22 0.0 0.06 0.09 0.12 0.15 0.18
 a (fm)

 ★ Can be used in joint fits with experimental data from BaBar and Belle
 to determine | Vcb | and R(D*) (lattice + exp) Alejandro Vaquero (University of Utah) B̄ ! D ⇤ `¯
 ⌫ at non-zero recoil

A. El-Khadra SM@LHC, 26-30 Apr 2021 15

BSM phenomenology: LFU τ/

 (⇤)
 (⇤) B(B ! D ⌧ ⌫⌧ )
 R(D ) =
 B(B ! D(⇤) `⌫)

A. El-Khadra SM@LHC, 26-30 Apr 2021 16
 

BSM phenomenology: LFU τ/
 R(D⇤ ) results in context

 No constraint wMax : R(D⇤ )Lat = 0.266(14) R(D⇤ )Lat+Exp = 0.2484(13)
 W/ constraint wMax : R(D⇤ )Lat = 0.274(10) R(D⇤ )Lat+Exp = 0.2492(12)
 Phys.Rev.D 100 (2019), 052007; Phys.Rev.D 103 (2021), 079901; Phys.Rev.Lett. 123 (2019), 091801

 Alejandro Vaquero (University of Utah) B̄ ! D ⇤ `¯
 ⌫ at non-zero recoil April 21st , 2021 21 / 22

A. El-Khadra SM@LHC, 26-30 Apr 2021 17
 

Outline

 Lattice QCD Introduction

 Semileptonic B meson decay form factors

 • | Vub | and | Vcb |

 • LFU
 muon g-2
 Summary and Outlook

A. El-Khadra SM@LHC, 26-30 Apr 2021 18

Muon anomalous magnetic moment
 e ~
 The magnetic moment of charged leptons (e, µ, τ): µ
 ~ =g S
 2m

 At leading order, g = 2: = ( ie) ū(p0 ) µ
 u(p)

 Quantum effects (loops):
  µ⌫
 0 µ 2 i q⌫
 = ( i e) ū(p ) F1 (q ) + F2 (q 2 ) u(p)
 2m

 Note: F1 (0) = 1 and g = 2 + 2 F2 (0)

 g 2
 Anomalous magnetic moment: a⌘ = F2 (0)
 2

A. El-Khadra SM@LHC, 26-30 Apr 2021 19

Muon g-2: history of experiment vs theory

 Dam (Expt – Thy) x 10-10 plot by David Hertzog
 Precision
 BNL E821 Goals
 BNL
 FNAL E989

 Dam ~3.7 s

 Thy Initiative
 SM Theory Evaluations

 YEAR
A. El-Khadra SM@LHC, 26-30 Apr 2021 20

Muon g-2: SM contributions
 aµ = aµ (QED) + aµ (Weak) + aµ (Hadronic)

A. El-Khadra SM@LHC, 26-30 Apr 2021 21

Muon g-2: SM contributions
 aµ = aµ (QED) + aµ (Weak) + aµ (Hadronic)
 QED
 aQED
 µ+…(↵(Cs)) = 116 584 718.9 (1) ⇥ 10
 11
 0.001 ppm

 EW
 +… aEW
 µ = 153.6 (1.0) ⇥ 10 11
 0.01 ppm

 Hadronic…
 …Vacuum Polarization (HVP)
 6845 (40) × 10−11 0.34 ppm
 α2 [0.6%]
 +…

 …Light-by-Light (HLbL)
 92 (18) × 10−11 0.15 ppm
 α3 +… [20%]

A. El-Khadra SM@LHC, 26-30 Apr 2021 21

Muon g-2: SM contributions
 aµ = aµ (QED) + aµ (Weak) + aµ (Hadronic)
 QED
 aQED
 µ+…(↵(Cs)) = 116 584 718.9 (1) ⇥ 10
 11
 0.001 ppm

 EW
 +… aEW
 µ = 153.6 (1.0) ⇥ 10 11
 0.01 ppm

 Hadronic…
 …Vacuum Polarization (HVP)
 6845 (40) × 10−11 0.34 ppm
 α2 [0.6%]
 +…

 …Light-by-Light (HLbL)
 92 (18) × 10−11 0.15 ppm
 α3 +… [20%]

A. El-Khadra SM@LHC, 26-30 Apr 2021 21

Muon g-2: Hadronic Corrections
 1. Dispersive data-driven approach:
 Use experimental data together with dispersion theory. For example:
 e+
 ➠ hadrons
 HVP: e−
 Many experiments (over 20+ years) have measured the e +e −cross sections for
 the different channels over the needed energy range with increasing
 precision. The combined data + dispersion theory yield HVP with a current
 error ~ 0.6%.

 HLbL:
 New dispersive approach now also allows for data-driven evaluations of HLbL,
 currently ~20% error ➠ theory error is (almost) completely quantified.
 Replaces previous results obtained using simplified models of QCD.

A. El-Khadra SM@LHC, 26-30 Apr 2021 22

Muon g-2: Hadronic Corrections
 2. Euclidean Lattice QCD:
 ab-initio method to quantify QCD effects
 already used for simple hadronic quantities with high precision
 requires large-scale computational resources
 allows for entirely SM theory based evaluations

 Lattice HVP: ~2% error
 • Complete calculations by ~6 different lattice collaborations
 • Uncertainties are still larger than data-driven approach, but first lattice
 result with 0.8% uncertainty [Borsanyi et al, arXiv:2002.12347, 2021 Nature]
 • Improved calculations a high priority for the lattice community

 Lattice HLbL: ~45% error
 • first complete calculation by RBC/UKQCD [T. Blum et al, arXiv:1911.08123, PRL2020]
 • New: complete calculation by Mainz [E.H. Chao et al, arXiv:2104:02632]
 • expect improvements from continued computational effort

A. El-Khadra SM@LHC, 26-30 Apr 2021 23

HLbL: Comparison
 HLbL
 aµ

 Glasgow consensus (09) models of QCD +EFT,
 N/JN09
 large Nc
 J17
 Mainz21 (+ charm-loop)
 not used in WP20 Lattice QCD + QED
 RBC/UKQCD19
 (+ charm-loop)
 WP20 data-driven
 data + dispersive
 dispersive approach
 WP20

 0 20 40 60 80 100 120 140 160
 HLbL 11
 aµ × 10

 Now well-determined in two approaches, systematically improvable

A. El-Khadra SM@LHC, 26-30 Apr 2021 24

HVP: Comparison
 BMW20 ⇥ ⇤ SM
 aHVP
 µ + aQED
 µ + aWeak
 µ + aHLbL
 µ aexp
 a
 µ µ

 HVP from: BNL+FNAL
 LM20
 BMW20 Lattice QCD + QED
 ETM18/19
 Mainz/CLS19
 FHM19
 PACS19
 RBC/UKQCD18
 BMW17
 RBC/UKQCD hybrid: combine data &
 Fermilab uncertainty goal
 data/lattice
 lattice
 BDJ19
 J17
 not used in WP20

 DHMZ19 data driven
 KNT19
 + unitarity/analyticity
 WP20
 constraints
 -60 -50 -40 -30 -20 -10 0 10 20 30
 SM exp 10
 (aµ -aµ ) x 10

A. El-Khadra SM@LHC, 26-30 Apr 2021 25

Muon g-2: experiment vs theory

 [B. Abi et al (Muon g-2 Collaboration), Phys. Rev. Lett. 124, 141801 (2021)]

A. El-Khadra SM@LHC, 26-30 Apr 2021 26

Summary Table
 Contribution Section Equation ⇥1011 References Value ⇥1011
 Section ValueEquation Ref
 Experiment (E821)
 Experimental Eq. (8.13)
 average (E989+E821) 116 592 089(63)
 Eq. (8.13)Phys.Rev.Lett.
 116592061(41) Ref. [1] 116 592
 124,089(63)
 141801 Ref
 HVP LO (e+ e ) Sec. 2.3.7 Eq. (2.33) Sec. 2.3.7 6931(40)
 Eq. (2.33)Refs. [2–7] 6931(40) Ref
 HVP NLO (e+ e Sec.
 ) 2.3.8 Eq. (2.34) Sec. 2.3.8 98.3(7)
 Eq. (2.34)Ref. [7] 98.3(7) Ref
) HVP NNLO (e+Sec.e ) 2.3.8 Eq. (2.35) Sec. 2.3.8 12.4(1)
 Eq. (2.35)Ref. [8] 12.4(1) Ref
 dsc) HVP LO (lattice,Sec.
 udsc)
 3.5.1 Eq. (3.49) Sec. 3.5.17116(184)
 Eq. (3.49)Refs. [9–17] 7116(184) Ref
 logy) HLbL (phenomenology)
 Sec. 4.9.4 Eq. (4.92) Sec. 4.9.4 Eq.
 92(19)
 (4.92)Refs. [18–30] 92(19) Ref
omenology)HLbL NLO (phenomenology)
 Sec. 4.8 Eq. (4.91) Sec. 4.8 Eq.2(1)
 (4.91)Ref. [31] 2(1) Ref
) HLbL (lattice, uds)
 Sec. 5.7 Eq. (5.49) Sec. 5.7 Eq.
 79(35)
 (5.49)Ref. [32] 79(35) Ref
 logy + lattice)
 HLbL (phenomenology
 Sec. 8 + lattice)
 Eq. (8.10) Sec. 8 Eq.
 90(17)
 (8.10)Refs. [18–30, 32] 90(17) Ref
 QED Sec. 6.5 Eq. (6.30) 116
 Sec.
 5846.5
 718.931(104)
 Eq. (6.30)Refs.
 116[33,
 58434]
 718.931(104) Ref
 Electroweak Sec. 7.4 Eq. (7.16) Sec. 7.4 153.6(1.0)
 Eq. (7.16)Refs. [35, 36] 153.6(1.0) Ref
NLO + NNLO)
 HVP (e+ e , LO Sec.
 + NLO8 + NNLO)
 Eq. (8.5) Sec. 8 6845(40)
 Eq. (8.5) Refs. [2–8] 6845(40) Ref
logy + lattice
 HLbL + (phenomenology
 NLO) Sec. 8 + lattice Eq. +
 (8.11)
 NLO) Sec. 8 Eq.
 92(18)
 (8.11)Refs. [18–32] 92(18) Ref
 Total SM Value Sec. 8 Eq. (8.12) Sec.116
 8 591 810(43)
 Eq. (8.12)Refs. [2–8,
 11618–24,
 591 810(43)
 31–36] Ref
 aexp
 µ aSM
 µ Di↵erence: aµ Sec.
 := a exp
 8
 µ a SM
 µ Eq. (8.14) Sec. 8 279(76)
 Eq. (8.14)
 251(59) 279(76)

e contributions aSM
 Table 1:toSummary
 µ . Afterof website:
 the contributions
 experimental to aSM
 https://muon-gm2-theory.illinois.edu
 number
 µ . from
 AfterE821,
 the experimental
 the first block
 number
 givesfrom
 the main
 E821,
 results
 the first
 for block
 the hadronic
 gives the
2 to 5 ascontributions
 well as the combined
 from Secs. result
 2 to for
 5 asHLbL
 well as
 scattering
 the combined
 from phenomenology
 result for HLbL scattering
 and latticefrom
 QCDphenomenology
 constructed in Sec.
 and 8.
 lattice
 The Q
s the quantities
 second block
 entering
 summarizes
 our recommended
 the quantities
 SM value,
 entering
 in our
 particular,
 recommended
 the totalSMHVP value,
 contribution,
 in particular,
 evaluated
 the total
 from
 HVPe+ econtributi
 data,
 A. El-Khadra SM@LHC, 26-30 Apr 2021 27
 er. The and
 construction
 the total HLbL
 of the number.
 total HVP Theand
 construction
 HLbL contributions
 of the total
 takes
 HVP into
 andaccount
 HLbL contributions
 correlations among
 takes into
 the terms
 account
 at di↵erent
 correlation

Summary and Outlook

Lattice QCD calculations of semi-leptonic B(s) meson form factors are
very mature, including (almost) complete sets for π, K, D, Ds final states
-also true for rare decay form factors (e.g. B → Kℓℓ )
-4 groups working on B(s) → D* (s)
 form factors
➠ meeting the growing precision needs of the experimental program
➠ more information on | Vub | , | Vcb | incl. vs excl. puzzle
 aµ = 251 (59) difference between exp and SM at 4.2σ
precision will improve in experiment and theory
scope of LQCD calculations continues to increase (new methods, new
formulations, new quantities)

 The next few years will be very exciting!

 28

Thank you!

 Farah Willenbrock
 29

Appendix

 30

Heavy Quarks
 • For light quark (mq ≪ ΛQCD) quantities, the leading discretization errors
 ∼ (aΛ)2 — if the fermion action is O(a) improved.
 • Using the same action for heavy quarks (mQ > ΛQCD) results in leading
 discretization errors ∼ (amQ)2. The effects are large, if amq ≮ 1, which is true
 for b quarks on most available ensembles.
 ➠Two classes of solutions:

 1. avoid ∼ (amQ)2 effects using EFT (HQET, NRQCD)
 but: nontrivial matching and renormalization
 • rel. heavy quarks (Fermilab, Columbia,..): matching rel. lattice action via
 HQET to continuum
 • lattice NRQCD, HQET: use EFT to construct lattice action
 2. brute force: use the same lattice action for heavy quarks as for light quarks
 • generate gauge ensembles with a small enough so that (amb) < 1
 • supplement with HQET inspired extrapolation and/or static limit

A. El-Khadra SM@LHC, 26-30 Apr 2021 31

The z-expansion
 t = q2 z
 t p
 t+ t
 p
 t+ t0
 z(t, t0 ) = p p
 t+ t + t+ t0
 t± = (mB ± m⇡ )2
 2
 qmax =t for kinematic
 range: |z| < 1.
 kinematic range [m2` , qmax
 2
 ]

 Bourrely at al (Nucl.Phys. B189 (1981) 157)
 The form factor can be expanded as: Boyd, Grinstein, Lebed (hep-ph/9412324, PRL
 95; hep-ph/9504235, PLB 95; hep-ph/9508211,
 NPB 96; hep-ph/9705252, PRD 97)
 1 X Lellouch (arXiv:hep- ph/9509358, NPB 96)
 f (t) = ak (t0 )z(t, t0 )k Boyd & Savage (hep-ph/9702300, PRD 97)
 P (t) (t, t0 ) Bourrely at al ( arXiv:0807.2722, PRD 09)
 k=0

 • P(t) removes poles in [t-,t+]
 • The choice of outer function affects the unitarity bound on the ak.
 • In practice, only first few terms in expansion are needed.

A. El-Khadra SM@LHC, 26-30 Apr 2021 32
 

Muon g-2 Theory Initiative
 Maximize the impact of the Fermilab and J-PARC experiments
 ➠ quantify and reduce the theoretical uncertainties on the hadronic
 corrections
 summarize the theory status and assess reliability of uncertainty estimates
 organize workshops to bring the different communities together:
 First plenary workshop @ Fermilab: 3-6 June 2017
 HVP workshop @ KEK: 12-14 February 2018
 HLbL workshop @ U Connecticut: 12-14 March 2018
 Second plenary workshop @ HIM (Mainz): 18-22 June 2018
 Third plenary workshop @ INT (Seattle): 9-13 September 2019
 Lattice HVP at high precision workshop (virtual): 16-20 November 2020
 Fourth plenary workshop @ KEK (virtual): 28 June - 02 July 2021

 White Paper posted 10 June 2020:
 [T. Aoyama et al, arXiv:2006.04822, Phys. Repts. 887 (2020) 1-166.]
 132 authors, 82 institutions, 21 countries

A. El-Khadra SM@LHC, 26-30 Apr 2021 33

Muon g-2 Theory Initiative
 Steering Committee

 Gilberto Colangelo (Bern)
 Michel Davier (Orsay)
 Simon Eidelman (Novosibirsk)
 Aida El-Khadra (UIUC & Fermilab)
 Martin Hoferichter (Bern)
 Christoph Lehner (Regensburg University & BNL)
 Tsutomu Mibe (KEK) J-PARC Muon g-2/EDM experiment
 Lee Roberts (Boston) Fermilab Muon g-2 experiment
 Thomas Teubner (Liverpool)
 Hartmut Wittig (Mainz)

A. El-Khadra SM@LHC, 26-30 Apr 2021 34

Lattice HVP: results from BMW
 Article
 [Borsanyi et al, arXiv:2002.12347, 2021 Nature]
 214 no reason to
 660

 Aubin et al.20
 need furthe

 This work
 212
 that the ten
 640
 210 is usually co
 (3σ) and mu
 620

 Blum et al.19
 SRHO(>0.4fm) 208 SRHO improvement discovery (5
 alight

 SRHO(>1.3fm) As a first s
 600

 ,win )iso
 !

 R-ratio/lattice
 SRHO(0.4-1.3fm)+NNLO(>1.3fm) 206
 ified observ

 (alight
 none
 580 204 interval by a
 denote as aµ
 560 202 Its shorter-d
 No improvement noise and to
 540 200
 fermions, it
 198 Fig. 4, wher
 200k 150k 100k 50k 0 0.005 0.01 0.015 0.02 0.000 0.005 0.010 0.015 0.020 a function o
 #fits 2 2
 a [fm ] a2 (fm2) not require
a Fig. 3 | Example continuum limits of a lµight. The light-green blue points, but correspond to SRHO taste improvement for Fig. 4 |fm
 t ≥ 0.4 Continuum
 and no extrapolation of the isospin-symmetric, light, other lattic
 Small statistical errors and large discretization effects (before corrections)
ed ‘none’ correspond to our lattice results with no taste
The blue squares repesent data that have undergone no taste
 improvement for smaller t. The purple histogram results from fits using the
 connected component of the window observable a µ,win, (a lµ,win
 SRHO improvement, and the corresponding central value and error is the
 purple band. The darker grey circles correspond to resultspoints
 correctedare
 ight
 ) iso. The data
 withextrapolated to the infinite-volume limit. Central values are
 ours19,20. Th
 this challe

 Intermediate window aμW:
or t < 1.3 fm and SRHO improvement above. The blue curves
example continuum extrapolations of improved data to SRHO in the range 0.4–1.3 fm and with NNLO SXPT for larger t. These latter fits
 medians; error bars are s.e.m. Two different ways to perform the continuum
 serve to estimate the systematic uncertainty of the SRHO improvement. The
 Our alight
 µ,win d
 a2, up to and including a4. We note that extrapolations in
 α s (1/a) the strong coupling at the lattice scale, are also grey band includes this uncertainty, and the correspondingextrapolations
 histogram is shown are shown: one without improvement, and another with and ref. 19, re

 -3.7 σ tension with data-driven evaluation
 show linear, quadratic(KNT)
 with grey. Errors are s.e.m.
 corrections from a model involving the ρ meson (SRHO). In both cases the lines
 ur final result. The red circles and curves are the same as the
 R-ratio app
 2
 and cubic fits in a with varying number of lattice authors of r
 spacings in the fit. The continuum-extrapolated result is shown with the results
 -2.2 σ tension with RBC/UKQCD18 from Blum et al. and Aubin et al. . Also plotted is our R-ratio-based 19 20
 result.
 To conclu
 determination, obtained using the experimental data compiled by the authors
 Need to quantify the differences between data-driven
 of ref. and our lattice evaluations
 results for the non-light-connected and
 contributions. This
 plot is convenient for comparing different lattice results. Regarding the total
 4
 tributions (s
 hadronic co
 ‐H
 muon, aLO
 the BMW results for the various energy/distance
 a
 scales
 , for which we must also include the contributions of flavours other than
 µ,win

 light and isospin-symmetry-breaking effects, we obtain 236.7(1.4) on the tot
 µ
 crepancy be
 lattice and 229.7(1.3)tot from the R-ratio; the latter is 3.7σ or 3.1% smaller than the and can be s
 lattice result. R-ratio dete
 confirmed—
 A. El-Khadra SM@LHC, 26-30 Apr 2021 35
 of QCD. Tho

Lepton moments summary
 SM Exp
 a` a`
 Harvard’08 FNAL+BNL +1.3 × 105
 200
 average
 Cs
 Sensitivity to heavy
 100

 2.4σ new physics:
 0
 1.6σ m2`
 Rb aNP ⇠ 2
 -100 4.2σ `
 ⇤
 2
 -200 (mµ /me ) ⇠ 4 ⇥ 104
 WP SM
 -300
 −5.2 × 105

 1014 ⇥ ae 1011 ⇥ aµ 107 ⇥ a⌧
 Cs: α from Berkeley group [Parker et al, Science 360, 6385 (2018)]
 Rb: α from Paris group [Morel et al, Nature 588, 61–65(2020)]
A. El-Khadra SM@LHC, 26-30 Apr 2021 36