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 I II III 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 [ ; ;U] E = [ ; ] [U] [ ; ; 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 I II III 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 The Standard Model in a typical lattice calculation generations gauge Higgs I II III 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 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 The Standard Model in a typical lattice calculation generations gauge Higgs I II III 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 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 The Standard Model in a typical lattice calculation generations gauge Higgs I II III 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 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 The Standard Model in a typical lattice calculation generations gauge Higgs I II III 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 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 ! 1 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 j ubj ! ` ) 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 = j 3 j − 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 j + j 8q2 B − π j 0 j 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 j j 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 π 0.2 / 1.0 B 3,0 π B 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 → 0.0 → p π = 2π(1,1,0)/L, p = 9% p = 2π(1,1,0)/L, p = 85% → π 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 π π B B 0 + analytic f f 2 Δ Δ 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.