“Lattice QCD and precision flavour physics at a SuperB factory” V. Lubicz Outline Estimates of uncertainties of Lattice QCD calculations in the SuperB factory era Precision studies of flavor physics at the SuperB: impact of experimental and theoretical constraints Padova, 23 ottobre 2007

Prepared for:

The goal of a SuperB factory: Precision flavour physics for indirect New Physics searches An important example: - Test the CKM paradigm at the 1% level Today With a SuperB in 2015 “the dream”

The EXPERIMENTAL ACCURACY at a SuperB factory will reach the level of 1% or better for most of the relevant physical quantities Central Value Current error SuperB (75 ab-1) sin2β 0.680 0.026 (4%) 0.005 (0.7%) α 105o 7o (7%) 1-2o (1-2%) γ 54o 20o (37%) 1-2o (2-4%) |Vcb| (10-3) 41.7 2.2 (5%) 0.2 (0.5%) |Vub| (10-4) 36.4 2.0 (5%) 0.7 (2%) Δmd (ps-1) 0.507 0.005 (1%) 0.002 (0.4%) Δms (ps-1) 18.06 0.12 (0.7%) 0.05 (0.2%) BR(B→τν) (10-4) 0.83 0.48 (64%) 0.03 (4%) ASL(Bd) [10-3] - 0.7 5 0.1 Can we calculate hadronic parameters with a comparable (~1%) level of precision ?

Why Lattice QCD Lattice QCD is the theoretical tool of choice to compute hadronic quantities It is only based on first principles It does not introduce additional free parameters besides the fundamental couplings of QCD All systematic uncertainties can be systematically reduced in time, with the continuously increasing availability of computing power and the development of new theoretical techniques

Present theoretical accuracy ?? 11% |Vub| B → π/ρ l ν 13% |Vtd/Vts| B → K*/ρ (γ,l+l-) 4% (40% on 1-)  B → D/D*lν |Vcb| B → D/D* l ν 5% (26% on ξ-1) ξ Δmd /Δms 14% |Vtd| Δmd fB B → l ν εK 0.9% (22% on 1-f+) |Vus| K → π l ν Estimated error in 2015 Current lattice error Hadronic matrix element CKM matrix element Measurement

History of lattice errors 1.23(6) 5% 262(35) 13% 189(27) 14% Hashimoto Ichep’04 1.24(4)(6) 6% 276(38) 193(27)(10) 15% Lellouch Ichep’02 1.16(5) 4% 267(46) 17% 200(30) Bernard Latt’00 ---- 175(25) Flynn Latt’96 1.21(2)(5) 246(16)(20) 10% 223(15)(19) 11% Tantalo CKM’06 Uncertainties have been dominated for many years by the quenched approximation. Unquenched calculations still have relatively large errors.

the quenched uncertainty reduced by a factor 1.5 in the last years Sharpe Latt’96 0.90(3)(15) 17% Lellouch Latt’00 0.86(5)(14) Hashimoto Ichep’04 0.73(5)(5) 10% Tantalo CKM’06 0.78(2)(9) 12% F.Mescia HEP2007 the quenched uncertainty reduced by a factor 1.5 in the last years no lattice QCD calculations until 2004. Present error ~ 1%

Estimates of Lattice QCD uncertainties in the SuperB factory era: WARNING  Uncertainties in Lattice QCD calculations are dominated by systematic errors. The accuracy does not improve according to simple scaling laws Predictions on the 10 years scale are not easy. Estimates are approximate In many cases, experiments have been more successful than expectations/predictions… Is that also true for theoretical results ?  I have tried to be conservative…

Hadronic matrix element A previous estimate S.Sharpe @ Lattice QCD: Present and Future, Orsay, 2004 and report of the U.S. Lattice QCD Executive Committee ?? 4 - 5% 5.5 - 6.5% 11% ---- 13% 1.2% (13% on 1-) 2% (21% on 1-) 4% (40% on 1-)  B → D/D*lν 1.5 - 2 % (9-12% on ξ-1) 3% (18% on ξ-1) 5% (26% on ξ-1) ξ 3 - 4% 2.5 - 4.0% 3.5 - 4.5% 14% fB 0.4% (10% on 1-f+) 0.7% (17% on 1-f+) 0.9% (22% on 1-f+) 1-10 PFlop Year 60 TFlop Year 6 TFlop Year Current lattice error Hadronic matrix element

Assumptions: I neglect the impact of algorithmic improvements and of the development of new theoretical techniques. I only take into account the increase of precision which is expected by the increase of computational power. Very conservative I assume that non hadronic uncertainties, e.g. N2LO calculations, will be reduced at a level  1% Realistic

Strategy: Determine the parameters of a “target” lattice simulation (i.e. lattice spacing, lattice size, quark masses…) aiming at the 1% accuracy on the physical predictions Evaluate the computational cost of the target simulation Compare this cost with the computational power presumably available to lattice QCD collaborations in 2015

Estimate of computational power 2007 2015 The Moore’s Law Today ~ 1 – 10 TFlops 2015 ~ 1 – 10 PFlops For Lattice QCD:

Sources of errors in lattice calculations  Statistical - O(100) independent configurations are typically required to keep these errors at the percent level  Discretization errors and continuum extrapolation: a→0 [Now a ≲ 0.1 fm]  Chiral extrapolation: [Now mu,d ≳ ms/6]  Heavy quarks extrapolation: [Now mH ≃ mc]  Finite volume [Now L ≃ 2-2.5 fm]  Renormalization constants: Ocont(μ) = Z(aμ,g) Olatt(a) - In most of the cases Z can be calculated non-perturbatively: accuracy can be better than 1%

Minimum lattice spacing [From S.Sharpe @ Lattice QCD: Present and Future, Orsay, 2004] Rough estimate:  Assume O(a) improved action: - Improved Wilson: n=3. Staggered, maximally twisted, GW: n=4 - For light quarks: Λ2 ~ Λn ~ ΛQCD. For heavy quarks: Λ2 ~ Λn ~ mH Assume simulations at and , and linearly extrapolate in a2. The resulting error is:

Minimum lattice spacing (cont.) We require: Λn ≈ mHad ≈ 0.8 GeV amin  0.056 fm , n=3 amin  0.065 fm , n=4 Simulations with light quarks only: Simulations with heavy quarks: Λn ≈ mc ≈ 1.5 GeV amin  0.030 fm , n=3 amin  0.035 fm , n=4 Today: a ~ 0.06 - 0.10 fm ( cost ~ a-6 )

Minimum quark mass (mπ/mρ)min  0.27  Chiral perturbation theory (schematic): - c1 ~ c2 ~ O(1)  Assume simulations at two values of mπ/mρ. The resulting error is  If we require ε = 0.01 then (assuming c2=1): (mπ/mρ)min  0.27 Physical value: Today:

[Becirevic, Villadoro, hep-lat/0311028] Minimum box size Finite volume effects are important when aiming for 1% precision. The dominant effects come from pion loops and can be calculated using ChPT. E.g: [Becirevic, Villadoro, hep-lat/0311028]

 For matrix elements with at most one particle in the initial and final states finite volume effects are exponentially suppressed: with c ~ O(1)  If we require ε = 0.01 then (assuming c=1): mπ L  4.5  If the pion mass is . Thus: L  4.5 fm  With a = 0.033 fm the number of lattice sites is Today the typical size is: 323  64 More than 300 times smaller V  1363  270

Heavy quark extrapolation  A relativistic b quark cannot be simulated directly on the lattice. It would require a mb<< 1. Typically that means: 1/a  20 GeV a  0.01 fm This lattice is too fine, even for PFlop computers.  Two approaches to treat the b quark: - HQET 1) Use an effective theory on the lattice: - NRQCD (no continuum limit) - “Fermilab” 2) Simulate relativistic heavy quark in the charm mass region and extrapolate to the b quark mass  The most accurate results can be obtained by combining the two approaches

[Becirevic et al., hep-lat/0110091] Relativistic quarks Static limit (HQET) b quark  Besides the static point, lattice HQET also allows a non-perturbative determination of (Λ/M)n corrections [Heitger, Sommer, hep-lat/0310035] The point interpolated to the B meson mass has an accuracy comparable to the one obtained in the relativistic and HQET calculations [Becirevic et al., hep-lat/0110091] The B-B mixing B parameters

to aim at the 1% level precision Target simulations to aim at the 1% level precision Nconf = 120 Ls = 4.5 fm [V = 903  180] a = 0.05 fm [ 1/a = 3.9 GeV ] [ mπ = 200 MeV ] Light quarks physics 1 Nconf = 120 Ls = 4.5 fm [V = 1363  270] a = 0.033 fm [ 1/a = 6.0 GeV ] [ mπ = 200 MeV ] Heavy quarks physics 2

Estimates of CPU costs  The cost depends on the lattice action: Wilson - Standard - O(a)-improved - Twisted mass Staggered Ginsparg-Wilson - Domain wall - Overlap Cheap, but affected by uncontrolled systematic uncertainty [det1/4]. Not a choice for the PFlop era. Good chiral properties, 10-30 times more expensive than Wilson Tremendous progress of the algorithms in the last years. “The Berlin wall has been disrupted” [Ukawa, Latt’01] Berlin plot

Empirical formulae for CPU cost For Nf=2 Wilson fermions: [Del Debbio et al. 06] 0.05 for improved Wilson  Comparison with Ukawa 2001 (the Berlin wall):

Cost of the target simulations: Affordable with 1-10 PFlops !! Nconf = 120 Ls = 4.5 fm [V = 903  180] a = 0.05 fm [ 1/a = 3.9 GeV ] [ mπ = 200 MeV ] Light quarks phys. 0.07 PFlop-years Wilson 1-2 PFlop-years GW Nconf = 120 Ls = 4.5 fm [V = 1363  270] a = 0.033 fm [ 1/a = 6.0 GeV ] [ mπ = 200 MeV ] Heavy quarks phys 0.9 PFlop-years Wilson Overhead for Nf=2+1 and lattices at larger a and m is about 3 Affordable with 1-10 PFlops !!

Hadronic matrix element Estimates of error for 2015 60 TFlop Year [2011 LHCb] 2 – 3% 4 - 5% 5.5 - 6.5% 11% 3 – 4% ---- 13% 0.5% (5% on 1-) 1.2% (13% on 1-) 2% (21% on 1-) 4% (40% on 1-)  B → D/D*lν 0.5 – 0.8 % (3-4% on ξ-1) 1.5 - 2 % (9-12% on ξ-1) 3% (18% on ξ-1) 5% (26% on ξ-1) ξ 1 – 1.5% 3 - 4% 2.5 - 4.0% 3.5 - 4.5% 14% fB 1%  0.1% (2.4% on 1-f+) 0.4% (10% on 1-f+) 0.7% (17% on 1-f+) 0.9% (22% on 1-f+) 6 TFlop Year Current lattice error Hadronic matrix element 1-10 PFlop Year [2015 SuperB]

Precision flavour physics at the SuperB Central Value Current error Error in 2015 sin2β 0.680 0.026 (4%) 0.005 (0.7%) α 105o 7o (7%) 1o (1%) γ 54o 20o (37%) 1o (2%) λ 0.2258 0.0014 (0.6%) 0.0008 (0.4%) |Vcb| (10-3) 41.7 2.2 (5%) 0.2 (0.5%) |Vub| (10-4) 36.4 2.0 (5%) 0.7 (2%) Δmd (ps-1) 0.507 0.005 (1%) 0.002 (0.4%) Δms (ps-1) 18.06 0.12 (0.7%) 0.05 (0.2%) mt (GeV) 163.8 3.2 (2%) 1.5 (1%) fBs√Bs (MeV) 262 35 (13%) 2.5 (1%) ξ 1.13 0.06 (5%) 0.006 (0.5%) fB (MeV) 189 27 (14%) 1.9 (1%) BR(B→τν) (10-4) 0.83 0.48 (64%) 0.03 (4%) BK 0.90 0.09 (11%) 0.009 (1%) εK 2.280 0.013 (0.6%) ASL(Bd) [10-3] - 0.7 5 0.1 UTA in 2015 Table of inputs

UTA in the SM: 2007 vs 2015 σ() /  = 20% σ() /  = 1.3% σ()/  = 4.7% σ()/  = 0.8%

Sin2β = 0.690 ± 0.023 α = (91.2 ± 5.4)o γ = (66.7 ± 6.4)o Sin2β = 0.6749 ± 0.0043 α = (104.55 ± 0.45)o γ = (54.28 ± 0.38 )o

Experimental error in 2015: SM prediction for Δms Δms = (17.5  2.1) ps-1 Δms = (17.93  0.25) ps-1 Δms = (XX.XX  0.05) ps-1 Experimental error in 2015:

Model independent analysis New Physics discovery Example: Bd-Bd mixing Model independent analysis CBd = 1.04 ± 0.34 φBd = (-4.1 ± 2.1)o With present central values, given the Vub vs sin2β tension, the Standard Model would be excluded at > 5σ CBd = 0.997 ± 0.031 φBd = (0.02 ± 0.51)o

Minimal Flavor Violation The most pessimistic scenario for indirect NP searches in flavour physics No new sources of flavour and CP violation NP contributions controlled by the SM Yukawa couplings Ex: Constrained MSSM (MSUGRA), …. 1HDM / 2HDM at small tanβ Same operator as in the SM NP only modifies the top contribution to FCNC and CPV NP in K and B correlated 2HDM at large tanβ New operator wrt the SM Also the bottom Yukawa coupling can be relevant NP in K and B uncorrelated

this is the most pessimistic scenario!! [D’Ambrosio et al., NPB 645] Today With a SuperB Remember: this is the most pessimistic scenario!! δS0 = -0.16 ± 0.32 Λ > 2.3 Λ0 @ 95% NP masses > 200 GeV δS0 = 0.004 ± 0.059 Λ > 6 Λ0 @ 95% NP masses > 600 GeV

Let’s work on that !! Conclusions  The performance of supercomputers is expected to increase by 3 orders of magnitude in the next 10 years (TFlop → PFlop)  Even without accounting for the development of new theoretical tools and of improved algorithms, the increased computational power should by itself allow lattice QCD calculations to reach the percent level precision in the next 10 years  If this expectation is correct, the accuracy of the theoretical predictions will be on phase with the experimental progress at the Super B factory.  The physics case for a SuperB factory is exciting… Let’s work on that !!

Backup slides

Expectations for LHCb 2011 2015 from V. Vagnoni at CKM 2006 LHCb L=2 fb-1 2011 2015 LHCb L=10 fb-1

The agreement is spectacular! BK = 0.75 ± 0.09 ^ UTA BK = 0.79 ± 0.04 ± 0.08 ^ Lattice [Dawson] 2% ! fBs√BBs = 261 ± 6 MeV UTA Lattice [Hashimoto] fBs√BBs = 262 ± 35 MeV ξ = 1.24 ± 0.08 UTA Lattice [Hashimoto] ξ = 1.23 ± 0.06 The agreement is spectacular!