Lattice QCD (Focus on Charm and Beauty Form Factors, R(D*), B- & C

Lattice QCD

(focus on Charm and Beauty form factors, R(D∗), b-& c-quark masses)

Oliver Witzel

18th International Conference on B-Physics at Frontier Machines, Beauty 2019 Ljubljana, Slovenia October 3, 2019 · I Lattice QCD

I Charm and beauty form factors (single pseudoscalar hadronic ﬁnal state)

I R(D∗) (single vector hadronic ﬁnal state)

I b & c quark masses

I In addition Flavor changing neutral currents (FCNC) → Neutral meson mixing → Lattice QCD lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

The Standard Model of Elementary Particle Physics generations gauge Higgs IIIIII forces boson I Quantum Chromodynamics (QCD) describes the strong interactions of quarks and gluons (nonperturbative) u c t g H

I Lattice QCD allows for ﬁrst principle calculations

quarks Also in the nonperturbative regime d s b γ → Systematical procedures to improve uncertainties → 0 Requires large scale computing facilities νe νµ ντ Z →

leptons e µ τ W ±

[DiRAC] [ALCF]

4 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Setting up a lattice calculation Wick-rotate to Euclidean time t iτ I → I Discretize space-time and set up a hypercube of ﬁnite extent L3 T and spacing a × I Use path integral formalism a Z T 1 SE [ψ,ψ,U] E = [ψ, ψ] [U] [ψ, ψ, U] e− hOi Z D D O Uµ(x) Large but ﬁnite dimensional path integral ⇒ ψ(x) Finite lattice spacing a UV regulator L I → Quark masses need to obey am < 1 →

Finite volume of length L IR regulator I → Study physics in a ﬁnite box of volume( aL)3 →

5 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

The Standard Model in a typical lattice calculation generations gauge Higgs IIIIII forces boson I 2+1 dynamical ﬂavor QCD (fully relativistic action) u c t g H dynamical charm is possible: 2+1+1 ﬂavors →

quarks d s b γ

0 νe νµ ντ Z

leptons e µ τ W ±

6 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

quarks d s b γ I c- and b-quarks simulated as valence quarks

0 νe νµ ντ Z

leptons e µ τ W ±

6 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

quarks d s b γ I c- and b-quarks simulated as valence quarks

0 I Weak force carriers and top enter “in” point-like operators νe νµ ντ Z

leptons e µ τ W ±

6 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

quarks d s b γ I c- and b-quarks simulated as valence quarks

0 I Weak force carriers and top enter “in” point-like operators νe νµ ντ Z

I Leptons mainly in post-analysis steps leptons e µ τ W ±

6 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

quarks d s b γ I c- and b-quarks simulated as valence quarks

0 I Weak force carriers and top enter “in” point-like operators νe νµ ντ Z

I Leptons mainly in post-analysis steps leptons e µ τ W ±

I QED and iso-spin breaking required for precision . 1%

6 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Lattice QCD calculation

Simulate Desired result at ﬁnite lattice spacing a take a 0 limit I I → in a ﬁnite volume L3 take L limit I I → ∞ discrete momenta 2π~n/L continuous momenta ~p ⇒ → I lattice regularized I match to some continuum scheme

I bare input quark masses I physical quark masses phys phys phys phys am`, ams , amc , amb ml = mu/d , ms = ms , mc = mc , mb = mb mostly: aM = aMphys π 6 π

I Need to choose gauge and fermion action

I Need to control all limits keeping FV and discretization eﬀects under control u quarks want large volume (large L3) such that M L > 4 → π · b quarks want ﬁne lattice (small a) i.e. am 1 → b 7 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Simulating charm and bottom (schematic) 1 a− > 1.5 GeV charm: RHQ; extrapolations of fully relativistic actions (?) bottom: HQET, NRQCD, RHQ

1 a− > 2.2 GeV charm: fully relativistic action bottom: (guided) extrapolation of fully relativistic action

1 a− > 4.6 GeV bottom: fully relativistic action HQET: static limit, relatively noisy NRQCD: non-relativistic QCD, no continuum limit RHQ or Fermilab: relativistic heavy quark action, complicated discretization errors fully relativistic: (heavy) HISQ, (heavy) MDWF, . . . 8 / 29 Charm and beauty form factors

(single pseudoscalar hadronic ﬁnal state) lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

V from exclusive semileptonic B π`ν decay | ub| → ` ) q2 = M2 + M2 2M E B π − B π ν W b u B π u

I Conventionally parametrized by (B meson at rest)

2 2 2 2 2p 2 2 dΓ(B π`ν) GF Vub (q m` ) Eπ Mπ →2 = | 3 | − 4 2 − dq 24π q MB experiment knownCKM m2 3m2 1 + ` M2 (E 2 M2) f (q2) 2 + ` (M2 M2)2 f (q2) 2 , × 2q2 B π − π | + | 8q2 B − π | 0 | nonperturbative input

10 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Nonperturbative input

I Parametrizes interactions due to the (nonperturbative) strong force

I Use operator product expansion (OPE) to identify short distance contributions

I Calculate the ﬂavor changing currents as point-like operators using lattice QCD

11 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

B π`ν form factors → µ I Parametrize the hadronic matrix element for the ﬂavor changing vector current V 2 2 in terms of the form factors f+(q ) and f0(q ) µ M2 M2 M2 M2 π V µ B = f (q2) p + pµ B − π qµ + f (q2) B − π qµ h | | i + B π − q2 0 q2 ttVµ qu b

t0 tsink u/d l I Calculate 3-point function by Inserting a quark source for a light u/d quark propagator at t → 0 Allow it to propagate to t , turn it into a sequential source for a b quark → sink Use a “light” quark propagating from t and contract both at t with t t t → 0 0 ≤ ≤ sink

12 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Steps of the B π`ν calculation → 1. Extract form factors on each ensemble

[Flynn et al. PRD 91 (2015) 074510] 0.4 2.0 π p

0.2 / 1.0 Bπ 3,0 Bπ 3,i R R s s -1/2 B 1/2 B M

→ M p π = 2π(0,0,0)/L, p = 28% →p = 2π(1,0,0)/L, p = 16% → π p π = 2π(1,0,0)/L, p = 50% 0.0 →p = 2π(1,1,0)/L, p = 9% 0.0 → π p π = 2π(1,1,0)/L, p = 85% →p = 2π(1,1,1)/L, p = 63% → π p π = 2π(1,1,1)/L, p = 89%

0 5 10 15 20 0 5 10 15 20 time slice time slice

13 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Steps of the B π`ν calculation → 1. Extract form factors on each ensemble 2. Perform chiral-continuum extrapolation

[Flynn et al. PRD 91 (2015) 074510] 1.0

3 24 ensemble, aml = 0.005 243 ensemble, am = 0.01 3 l 0.8 32 ensemble, am = 0.004 3 l 32 ensemble, aml = 0.006 323 ensemble, am = 0.008

π l

B continuum || f 0.6 s -1/2 B M

0.4

χ2/dof = 1.39, p = 14% 0.2 0 0.01 0.02 0.03 0.04 ( E / M )2 π Bs

13 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Steps of the B π`ν calculation → 1. Extract form factors on each ensemble 2. Perform chiral-continuum extrapolation Explore other extrapolations, vary inputs, . . . →

[Flynn et al. PRD 91 (2015) 074510] 15 15 varying g varying fπ omitting zero momentum 2 10 10 omitting a term 2 omitting Mπ term 2 2 [%] [%] omitting a and Mπ terms Bπ Bπ 0 + analytic 2 Δ f Δ f analytic omitting a term 5 5

0 0 18 20 22 24 26 18 20 22 24 26 q2 [GeV2] q2 [GeV2]

13 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Steps of the B π`ν calculation → 1. Extract form factors on each ensemble 2. Perform chiral-continuum extrapolation Explore other extrapolations, vary inputs, . . . → 3. Estimate further systematic uncertainties [Flynn et al. PRD 91 (2015) 074510] statistics statistics 300 Bπ 300 Bπ f0 chiral-continuum extrapolation f+ chiral-continuum extrapolation lattice-scale uncertainty lattice-scale uncertainty 16 16 250 renormalization factor 250 renormalization factor HQ discretization errors HQ discretization errors

] LQ and gluon discretization errors ] LQ and gluon discretization errors

2 200 14 2 200 14 other systematics other systematics [% [% 2 150 12 2 150 12 Error [%] Error [%] Error 100 10 Error 100 10

8 8 50 50 6 6 4 4 0 0 18 20 22 24 26 18 20 22 24 26 q2 [GeV2] q2 [GeV2]

13 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Steps of the B π`ν calculation → 1. Extract form factors on each ensemble 4. Kinematical extrapolation 2. Perform chiral-continuum extrapolation z-expansion (BGL, BCL, CLN (if applicable)) → Explore other extrapolations, vary inputs, . . . Compare to other calculations, QCD sum rules → → 3. Estimate further systematic uncertainties Combine with experimental data, extract V → | ub| [Flynn et al. PRD 91 (2015) 074510] 5.0 1.2 f Bπ Bπ 0 f0 1.0 f Bπ 4.0 Bπ + f+ 0 BCL z-fit (K=3 + constraints) BCL z-fit (K=3 + constraints) BCL z-fit (K=4 + constraints) 0.8 3.0 BGL z-fit (N=3 + constraints) and f + f +/0 f ) 2 0.6

2 B* q 2.0 max m / 2 0.4 1- q

( 1.0 0.2 0.0 0.0 0 5 10 15 20 25 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 q2 [GeV2] z

13 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

FLAG: B π`ν [FLAG 2019] →

I Summary table indicating quality of key features

I Combined analysis accounting for correlations (if needed)

I Please cite calculations feeding into FLAG averages [Bailey et al. PRD92(2015)014024] [Flynn et al. PRD91(2015)074510] [Dalgic et al. PRD73(2006)074502] 14 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

[FLAG 2019] FLAG: B π`ν | | + → (BaBar) + (Belle) = (average)

FLAG estimate for = +

+ (BaBar)

= (Belle) (average)

(BaBar) (Belle) = (average) .

HFLAV inclusive (GGOU)

3.0 3.5 4.0 4.5 5.0 5.5 6.0

I Combination of lattice average with experimental results I Many groups are working on updates: 3 Extraction of V : 3.73(14) 10− Fermilab/MILC [Z. Gelzer Lattice X IF 2019] I | ub| · → HPQCD [C. Bouchard Lattice 2019] → Please cite calculations feeding into FLAG averages JLQCD [J. Koponen Lattice 2019] I → [Bailey et al. PRD92(2015)014024] [Flynn et al. PRD91(2015)074510] RBC-UKQCD [R. Hill Lattice X IF 2019] [Dalgic et al. PRD73(2006)074502] → 14 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

V from exclusive semileptonic B K`ν decay | ub| s → `

ν Compared to B π`ν only spectator quark diﬀers W I → b u Bs K s

I Lattice QCD prefers s quark over u quark: statistically more precise, computationally cheaper B factories run mostly at Υ(4s) threshold B mesons I ⇒ LHC collisions create many B and B mesons which decay LHCb I s ⇒ LHCb prefers the ratio (B D `ν)/(B K`ν) V /V → s → s s → ⇒ | cb ub|

15 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

B K`ν s → I HPQCD, RBC-UKQCD, ALPHA [Bouchard et al. PRD90(2014)054506] [Flynn et al. PRD91(2015)074510] [Bahr et al. PLB757(2016)473]

I New 2019: Fermilab/MILC [Bazavov et al. PRD100(2019)034501]

HPQCD 14 (NRQCD) RBC/UKQCD 16 (Fermilab/RHQ) 3 ALPHA 16 (HQET) Fermilab/MILC 18 This work (Fermilab/NR) RBC/UKQCD 16

HPQCD 14

) 2 2 q ( Khodjamirian 17 (LCSR) +,0 f Faustov 13 (RQM) 1 Wang 12 (pQCD)

Duplancic 08 (LCSR)

0 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 2 GeV2 2 q ( ) f+,0(q = 0)

16 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

1.2 reco Bs Ds`ν f (B D ) 1.1 + s s → reco → 1 f0 (Bs Ds ) I HPQCD, Fermilab/MILC → 0.9

[Monahan et al. PRD95(2017)114506] [Bazavov et al. PRD100(2019)034501 (reco)] +,0 f 0.8 I New 2019: HPQCD [McLean et al. arXiv:1906.00701] 0.7 Using heavy HISQ and twisted BC to cover full q2 range 0.6 → 0.5 Simulate an array of “lighter” b-quarks and then extrapolate 0 2 4 6 8 10 12 → 2 GeV2 a 0.09fm, amh = 0.5 q ( ) ' a 0.09fm, am = 0.65 ' h a 0.09fm, am = 0.8 ' h a 0.09fm, amphys, am = 0.5 ' l h NRQCD [1703.09728] a 0.09fm, amphys, am = 0.8 ' l h 1.2 1.2 a 0.06fm, amh = 0.427 1.2 this work ' a 0.06fm, am = 0.525 ' h a 0.06fm, amh = 0.65 s 2 ' f (q ) a 0.06fm, am = 0.8 1.1 + 1.1 ' h 1.1 a 0.045fm, am = 0.5 ' h a 0.045fm, am = 0.65 ' h a 0.045fm, am = 0.8 ' h 1.0 ) ) 1.0 1.0 2 2 q q ( ( s s 0 + f f 0.9 0.9 0.9 0.8 0.8 0.8 f s(q2) 0.7 0 0.7 0.7 0.6 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 2 2 2 2 q [GeV ] q [GeV ] q2[GeV2]

I Updates: RBC-UKQCD [OW Lattice X IF 2019] 17 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

B D`ν →

| |

* (BGL) * (CLN) + =

* ( , ) (BGL) * ( , ) (CLN) = .

HFLAV inclusive

36 38 40 42 44 46

I Please cite calculations feeding into FLAG averages [FLAG 2019] [Na et al. PRD92(2015)054510] [Bailey et al. PRD92(2015)034506]

18 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

D π`ν and D K`ν → → I Only very few published calculations + ( ) + ( ) [Lubizc et al. PRD96(2017)054514]

[Na et al. PRD84(2011)114505] [PRD82(2010)114506] + FLAG average for = + + + = ETM 17D 2 I Full q range can be covered using twisted boundary conditions FLAG average for = +

JLQCD 17B +

= HPQCD 11 / 10B I Updates: FNAL/MILC 04 HPQCD [B. Chakraborty Lattice 2019] → Fermilab/MILC [Li et al. PoS Lattice2018(2019)269] → = ETM 11B JLQCD [Kaneko et al. EPJ WebConf.175(2018)13007] → 0.55 0.65 0.75 0.65 0.75 0.85

[FLAG 2019]

19 / 29 R(D∗)

(single vector hadronic ﬁnal state) lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

B D∗`ν form factors → I Vector ﬁnal state with narrow width approximation

µνρσ µ 2i εν∗kρpσ D∗(k, λ) c¯γ b B(p) =fV h | | i MB + MD∗

µ 2 2MD∗ ε∗ q µ D∗(k, λ) c¯γ γ b B(p) =f (q ) · q h | 5 | i A0 q2 2 µ ε∗ q µ + f (q )(M + M ) ε∗ · q A1 B D∗ − q2 2 2 ε∗ q M M 2 µ µ B D∗ µ fA2 (q ) · k + p −2 q − MB + MD∗ − q

Commonly the HQET variable w = v v 0 > 1 is used with v = p/M and v 0 = k/M I · B D∗

21 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

( ) B D ∗ `ν → (s) I Still only results at zero recoil [FLAG 2019]

New 2019: HPQCD B D∗`ν I s → s at zero recoil [McLean et al. PRD99(2019)114512]

hs (1) (HISQ,HPQCD) A1

hs (1) (NRQCD,HPQCD) A1

hA1(1) (HPQCD)

hA1(1) (NRQCD, HPQCD)

hA1(1) (Fermilab,Fermilab/MILC)

0.75 0.80 0.85 0.90 0.95

[Atoui et al. EPJC74(2014)2861] [Bailey et al. PRD89(2014)114504] [Bailey et al. PRD92(2015)034506] [Na et al PRD92(2015)054510] [Monahan et al. PRD95(2017)114506] [Harrison et al. PRD97(2018)054502] 22 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Update Fermilab/MILC: B D∗`ν [A. Vaquero Lattice X IF 2019] → Results: Chiral-continuum ﬁts I Old double ratio

0.95 Extrapolation 0.95 Extrapolation a = 0.150 fm a = 0.150 fm a = 0.120 fm a = 0.120 fm a = 0.090 fm a = 0.090 fm 0.90 a = 0.060 fm 0.90 a = 0.060 fm Preliminarya = 0.045 fm Preliminarya = 0.045 fm

0.85 0.85 1 1 A A h h

0.80 0.80 I New ratio 0.75 0.75

Blinded Blinded 0.70 0.70 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 w w Left Old fit, Right New fit. Preliminary blinded results. Both plots differ on the accounting of discretization effects, which seem to be large at large recoil

Alejandro Vaquero (University of Utah) B¯ D `ν¯ at non-zero recoil September 24th, 2019 12 / 22 → ∗

23 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Update Fermilab/MILC: B D∗`ν [A. Vaquero Lattice X IF 2019] → Results: R(D∗)

Lattice and joint ﬁts

) Preliminary ν ` ∗ Lattice ` = e, µ Best fit ` = e − , µ − D ` = τ ` = τ → Lattice Best fit B ( Γ

1.0 1.1 1.2 1.3 1.4 1.5 w

Alejandro Vaquero (University of Utah) B¯ D `ν¯ at non-zero recoil September 24th, 2019 21 / 22 → ∗

23 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Update JLQCD: B D∗`ν [T. Kaneko Lattice X IF 2019] →lat D∗ V B h (w) Ratio method: h | µ | i V renormalization factors Z , Z cancel I D Alat B → h (w) ⇒ A V h ∗| µ | i A1 [Hashimoto et al. PRD61(1999)014502] LQCD vs BGL vs CLN vs HQET LQCD vs BGL vs CLN vs HQET

R1 = hV / hA1 R1 = hV / hA1

BGL, CLN results : by courtesy of authors w/ Belle tagged ‘17 BGL, CLN results : by courtesy of authors w/ Belle tagged ‘17 BGL vs CLN, HQET BGL vs CLN, HQET Bernlochner et al. Bernlochner et al. Bigi et al. Bigi et al. Grinstein-Kobach Grinstein-Kobach

our data favor the our data favor the CLN results CLN results

Gambino et al ‘19 tagged+untagged

consistency among LQCD, BGL, CLN, HQET

24 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Update JLQCD: B D∗`ν [T. Kaneko Lattice X IF 2019] → LQCD vs BGL vs CLN shape of hA1 reduced stat.+sys. hAA11( wh) (1) vs w error in the ratio

Belle tag, CLN: 3 |Vcb |×= 10 37.4(1.3) JLQCD

BLRP'17 BGL : 41.5(1.8) Belle untag, CLN: 222 37.4(0.9)

BLRP'17 CLN: 38.2(1.5)

2 2 232 hAA1 ( wh) 1 (1) =−+ 18ρ z(53ρ− 15) z + (231 ρ− 91)z

24 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Further updates: B D∗`ν → HPQCD form factors for B D∗ `ν LANL/SWME form factors at zero recoil I (s) → (s) I [Plenary talk A. Lytle Lattice 2019] [PoS Lattice2018 283]

0.95 HPQCD B D∗ 2 (s) (s) O(λ )-improved, Mπ≈310MeV → 1 O(λ )-improved, Mπ≈310MeV Unimproved, Mπ≈310MeV

| 0.9 (1) Bs Ds∗ B D∗ 1 → A → h |

0.20 0.95 dΓ 2(B− D∗µ−ν)/Γ(B− D∗µ−ν) dq s → s s → s 0.85 dΓ FNAL/MILC ’14 2(B− D∗τ −ν)/Γ(B− D∗µ−ν) dq s → s s → s 0.90 HPQCD ’18

0.15 HPQCD ’18, Mπ≈310MeV

0.85 0 0.03 0.06 0.09 0.12

1 a [fm] A

0.10 h 0.80

0.75

0.05

0.70 a amh 0.427 0.500 0.525 0.600 0.650 0.800 0.\0884 0.0592 0.0441

0.00 0.65 HPQCD0 2 4 Preliminary6 8 10 12 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 q2/GeV 2 w Figs. courtesy Judd Harrison

32 / 35

25 / 29 b & c quark masses lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing b quark mass

( ) I Determinations FLAG average for = + + Experimental values for M , M → Υ ηb + FNAL/MILC/TUMQCD 18 Euclidean time moments of ps correlator for

+ Gambino 17 →

= HPQCD 14B heavy-quark current followed by OPE ETM 16 Ratio of moments of heavy current-current HPQCD 14A → correlator functions FLAG average for = +

+ New FNAL/MILC/TUMQCD determination Maezawa 16 → = HPQCD 13B based on sophisticated ﬁt strategy using HPQCD 10 HQET, HMrASχPT, Symanzik eﬀective theory

PDG [Bazavov et al. PRD98(2018)054517]

4.1 4.3 4.5 4.7 GeV

[FLAG 2019] 2+1: mb(mb) = 4.164(23) GeV 2+1+1: mb(mb) = 4.198(12) GeV

27 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing c quark mass

( ) I Determinations

FLAG average for = + + Experimental values for MD(s) , Mηc HPQCD 18 → + FNAL/MILC/TUMQCD 18 Euclidean time moments of ps correlator for

+ → HPQCD 14A

= ETM 14A heavy-quark current followed by OPE ETM 14 New FNAL/MILC/TUMQCD determination → FLAG average for = + based on sophisticated ﬁt strategy using Maezawa 16

+ JLQCD 16 HQET, HMrASχPT, Symanzik eﬀective theory

= QCD 14 [Bazavov et al. PRD98(2018)054517] HPQCD 10 HPQCD 08B

PDG I Tension with ETMC 2+1+1 determinations 1.25 1.30 1.35 1.40 GeV

[FLAG 2019] 2+1: mc (mc ) = 1.275(5) GeV 2+1+1: mc (mc ) = 1.280(13) GeV

28 / 29 summary lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Summary

I Many calculation for exclusive decays are in progress; calculations are hard, tedious, and take time

I New ideas to compute inclusive decays using lattice techniques [Hashimoto PTEP 2017 (2017) 053B03] [Hansen, Meyer, Robaina arXiv:1704.08993] [Bailas Lattice 2019]

I Interesting new ideas for radiative decays [Kane et al. arXiv:1907.00279] [Martinelli Lattice 2019] [Sachrajda Lattice 2019]

I Not covered: Bc decays, R(J/ψ), etc. [see Poster by J. Harrison]

I Not covered: exclusive baryonic decays Λ Λ `ν and Λ p`ν V / V [Detmold, Lehner, Meinel, PRD92(2015)034503] → b → c b → ⇒| cb| | ub| Λ Λ τν [Datta et al. JHEP08(2017)131] → b → c Λ Λ`ν [Meinel PRL118(2017)082001] → c → 29 / 29 Flavor Changing Neutral Currents lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

B π`` form factor → t ` tVµ qld b Z, γ ` t0 tsink t t W B π l

I If the daughter quark is a d-quark, we have a FCNC decay at loop-level Need to implement additional operators → I Dominant contributions at short distance: f0, f+, and fT

µν pµkν pν kµ 2 π(k) id¯σ b(p) B = 2 − fT (q ) h | | i MB +Mπ

30 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

B π`` form factor → I Fermilab/MILC [Bailey et al. PRD92(2015)014024] χ2/dof = 34.6/36, p = 0.53

4.5 0.5 4 3.5 0.4 3 T h T / f ) 2.5 t (

T 0.3

R 2 1.5 0.2 1 0.5 0.1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 2 4 6 8 10 12 14 16 t Eπr1

I Extract form factor on each ensemble I Perform chiral-continuum extrapolation

I Updates: Fermilab/MILC [Z. Gelzer Lattice X IF 2019], HPQCD [C. Bouchard Lattice 2019] 30 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

t B K`` tVµ → qs b Similar to B π`` but use a strange daughter quark I → t0 tsink l I Fermilab/MILC, HPQCD l

I Please cite calculations feeding into FLAG averages [FLAG 2019] [Bailey et al. PRD93(2016)025026] [Bouchard et al. PRD88(2013)054509]

I Updates: Fermilab/MILC [Z. Gelzer Lattice X IF 2019] 31 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

+ Bs φ` `− form factors → ` Z, γ ` ` ` t t W ν W t W Bs φ Bs φ I Vector ﬁnal state treated in narrow width approximation

I Eﬀective Hamiltonian 10 b s 4GF X ( ) → = V ∗V C O 0 Heﬀ − √ ts tb i i 2 i I Leading contributions at short distance ( ) m e ( ) e2 ( ) e2 0 b µν 0 µ ¯ 0 µ ¯ 5 O7 = 16π2 ¯sσ PR(L)bFµν O9 = 16π2 ¯sγ PL(R)b`γµ` O10 = 16π2 ¯sγ PL(R)b`γµγ ` + + B K ∗` `− and B φ` `− [Horgan et al. PRD89(2014)090501] [PoS Lattice2014 372] I → s → Angular analysis [PRL112(2014)212003] → 32 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

The beauty of seven form factors

µνρσ µ 2 2i εν∗kρpσ φ(k, λ) ¯sγ b Bs (p) =fV (q ) h | | i MBs + Mφ 2M ε∗ q φ(k, λ) ¯sγµγ b B (p) =f (q2) φ · qµ h | 5 | s i A0 q2 2 µ ε∗ q µ + f (q )(M + M ) ε∗ · q A1 Bs φ − q2 " # ε q M2 M2 2 ∗ µ µ Bs − φ µ fA2 (q ) · k + p 2 q − MBs + Mφ − q νµ 2 µρτσ q φ(k, λ) ¯sσ b B (p) =2f (q ) ε∗k p , ν h | | s i T1 ρ τ σ νµ 5 2 µ 2 2 µ q φ(k, λ) ¯sσ γ b B (p) =if (q ) ε∗ (M M ) (ε∗ q)(p + k) ν h | | s i T2 Bs − φ − · " 2 # 2 µ q µ + if (q )(ε∗ q) q (p + k) T3 · − M2 M2 Bs − φ

33 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

−1 sea B φ``: seven form factors (a = 1.784 GeV, aml = 0.005, ams = 0.03224) s →

2.00 2.00 PRELIMINARY PRELIMINARY 0.70 PRELIMINARY 0.45 PRELIMINARY 1.50 1.50 0.50 0.35 1 12 0 V A A A f f f f 1.00 1.00 2 2 2 0.25 2 fV n f n fA1 (n =0) =0.594(12) fA12(n =1) =0.3560(78) ( =1) =1.570(45) A0 ( =1) =1.558(42) 0.30 2 2 2 2 fV n fA n fA1 (n =1) =0.555(13) fA12(n =2) =0.3269(85) 0.50 ( =2) =1.398(45) 0.50 0 ( =2) =1.354(39) 0.15 fV n2 f n2 fA n2 fA n2 ( =3) =1.239(56) A0 ( =3) =1.164(44) 0.10 1 ( =2) =0.513(15) 12( =3) =0.297(10) 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 time slice [lattice unit] time slice [lattice unit] time slice [lattice unit] time slice [lattice unit]

0.70 1.40 1.40 PRELIMINARY PRELIMINARY PRELIMINARY

1.00 0.50 1.00 1 2 23 T T T f f f fT n2 2 2 0.60 1 ( =1) =1.203(30) 0.30 fT2 (n =0) =0.555(11) 0.60 fT23(n =1) =0.989(22) fT n2 2 2 1 ( =2) =1.052(30) fT2 (n =1) =0.536(12) fT23(n =2) =0.911(25) fT n2 f n2 f n2 0.20 1 ( =3) =0.937(37) 0.10 T2 ( =2) =0.480(14) 0.20 T23( =3) =0.829(32) 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 time slice [lattice unit] time slice [lattice unit] time slice [lattice unit]

[Flynn et al. PoS Lattice2016(2016)296]

34 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

+ 2 B φ` `−: seven form factors vs. q s →

2.0 preliminary 1.6 preliminary 0.75 preliminary 0.40 preliminary 1.5 0.70 0.38 1.8 1.4 0.65 0.36 0.60 0.34

1.6 0 1 1.3 12 V A A A f f f 0.55 f 0.32 1.2 1.4 0.50 0.30 1.1 0.28 1.2 + − + − 0.45 + − + − Bs →ϕℓ ℓ 1.0 Bs →ϕℓ ℓ Bs →ϕℓ ℓ Bs →ϕℓ ℓ χ2 /dof =0.64, p =78.7% χ2 /dof =0.95, p =49.0% 0.40 χ2 /dof =1.07, p =37.8% 0.26 χ2 /dof =1.23, p =25.9% 1.0 0.9 0.35 0.24 0.04 0.05 0.06 0.07 0.04 0.05 0.06 0.07 0.04 0.05 0.06 0.07 0.04 0.05 0.06 0.07 E M 2 E M 2 E M 2 E M 2 1.5 ( ϕ / Bs ) 0.70 ( ϕ / Bs ) 1.1 ( ϕ / Bs ) ( ϕ / Bs ) 1.4 0.65 1.0 1.3 0.60 1.2 0.9 1 2 preliminary 0.55 preliminary 23 preliminary T T T f f 1.1 f 0.50 1.0 0.8 aml =0.008 aml =0.010 0.9 0.45 aml =0.006 aml =0.005 B ϕℓ + ℓ− B ϕℓ + ℓ− 0.7 B ϕℓ + ℓ− s → 0.40 s → s → am 0.8 χ2 /dof =0.62, p =80.9% χ2 /dof =8.70, p =1.22% χ2 /dof =1.03, p =41.5% l =0.004 0.7 0.35 0.6 0.04 0.05 0.06 0.07 0.04 0.05 0.06 0.07 0.04 0.05 0.06 0.07 2 2 2 (Eϕ /MBs ) (Eϕ /MBs ) (Eϕ /MBs )

[Flynn et al. PoS Lattice2016(2016)296]

35 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

+ B φ` `−: ﬁrst attempt to use z-parametrization s → [Flynn et al. PoS Lattice2016(2016)296]

2.0 f 0.8 f 0.45 f A0 PRELIMINARY A1 PRELIMINARY A12 PRELIMINARY 1.8 0.40 fA0 (0) =0.56(10) 0.7 fA1 (0) =0.294(84) fA12(0) =0.154(18) 1.6 0.35 1.4 0.6 0 1 12 0.30 A A A f f 1.2 0.5 f 0.25 1.0 0.4 0.8 0.20 0.6 0.3 0.15 0.4 0.2 0.10 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 q2 [GeV2 ] q2 [GeV2 ] q2 [GeV2 ]

2.5 f 1.8 f 1.4 f V PRELIMINARY T2 PRELIMINARY T23 PRELIMINARY 1.6 fT1 fV (0) =0.97(42) 1.2 fT23(0) =0.45(14) 2.0 1.4 fT1 /T2 (0) =0.378(75) 1.0

2 1.2 23 /T V T 1 f f 1.5 T 1.0 0.8 f 0.8 0.6 1.0 0.6 0.4 0.4 0.5 0.2 0.2 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 q2 [GeV2 ] q2 [GeV2 ] q2 [GeV2 ]

36 / 29 neutral meson mixing lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Neutral B meson mixing ¯ ¯ b¯ W d¯ b t d B t t B B WW B d W b d t b

2 2 GF mW 2 2 ∆m = η S M f B V ∗ V , q = d, s q 6π2 B 0 Bq Bq Bq | tq tb| I Experiments measure oscillation frequencies ∆mq extremely precise

I Process is short distance dominated (top quark)

I Theory provides decay constant fBq and bag parameter BBq µ µ 0 q¯γ γ5b Bq(p) = if B p h | | i q Dq 3 B¯ 0 [b¯γµ(1 γ )q][b¯γ (1 γ )q] B0 = f 2 M2 B h q | − 5 µ − 5 | q i 8 Bq Bq Bq

38 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Neutral B meson mixing ¯ ¯ b¯ W d¯ b t d B t t B B WW B d W b d t b

2 2 GF mW 2 2 ∆m = η S M f B V ∗ V , q = d, s q 6π2 B 0 Bq Bq Bq | tq tb| I Experiments measure oscillation frequencies ∆mq extremely precise

I Process is short distance dominated (top quark)

I Theory provides decay constant fBq and bag parameter BBq

2 2 ∆m M V f BB I Advantageous to consider s Bs 2 ts 2 Bs s = ξ | |2 ,ξ = 2 [Bernard, Blum, Soni PRD58(1998)014501] ∆md MB Vtd f BB d | | Bd d

38 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

Basics of the lattice calculation

I Schematic set-up

I Ratio ξ ideal to explore simulating with “heavier than charm” and extrapolating to bottom e.g. [Boyle et al. arXiv:1812.08791] All datapoints shifted to m = mphys and a2 = 0 1.21 π π b c 1.20 C0 t t t C1 1 0 2 M0 1.19 M1 F1 ξ

ξ 1.18 I Only zero momentum ground-states

(bag parameters and decay constants) 1.17

I Fit symmetrized plateau between 1.16 source and operator location 1.15 0.1 0.2 0.3 0.4 0.5 0.6 m 1 [GeV 1] D−s −

39 / 29 lattice QCD charm and beauty form factors R(D∗) b & c quark masses summary FCNC mixing

ξ and V vs. V | td | | ts| I Two latest calculation feature simulations at physical pion masses 1.212(12) 4.4 HPQCD’19 RBC/UKQCD’18 nf = 4 HPQCD’19 4.2 FNAL/MILC’16 ] 2 CKMFitter’18 (tree) nf = 3 FNAL/MILC’16 −

[10 CKMFitter’18

HPQCD’09 | 4.0 UTFit’18 ts V RBC/UKQCD’18 | King et al’19

RBC/UKQCD’14 3.8

1.15 1.20 1.25 6.5 7.0 7.5 8.0 8.5 9.0 9.5 3 V [10− ] ξ | td|

[Dowdall et al. arXiv:1907.01025] [Boyle et al. arXiv:1812.08791] [Bazavov et al. PRD93(2016)113016] [Aoki et al. PRD91(2015)114505] [Gamiz et al. PRD80(2009)014503] Today on the arXiv: dimension 7 operators of B0 B0 mixing I s − s [Davies et al. arXiv:1910.00970]

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