“Lattice QCD and precision flavour physics at a SuperB factory”

1. “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

2. Prepared for:

3. “the dream” 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

4. The EXPERIMENTAL ACCURACY at a SuperB factory will reach the level of 1% or better for most of the relevant physical quantities Can we calculate hadronic parameters with a comparable (~1%)level of precision ?

5. 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 uncertaintiescan be systematically reduced in time, with the continuously increasing availability of computing power and the development of new theoretical techniques

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

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

8. no lattice QCD calculations until 2004. Present error ~ 1% F.Mescia HEP2007 the quenched uncertainty reduced by a factor 1.5 in the last years

9. 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…

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

11. 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 Realistic Assumptions: • I assume thatnon hadronic uncertainties,e.g. N2LO calculations,will be reduced at a level  1%

12. 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

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

14.  Heavy quarks extrapolation: [Now mH≃ mc] 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]  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%

15. Assume simulations at and , and linearly extrapolate in a2. The resulting error is: 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

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

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

18. 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]

19.  If the pion mass is . Thus:  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= 0.01then (assuming c=1): mπL 4.5 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

20. 1/a  20 GeV a  0.01 fm Heavy quark extrapolation  A relativistic b quark cannot be simulated directly on the lattice. It would require amb<< 1. Typically that means: 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

21. Relativistic quarks Static limit (HQET) b quark The B-B mixing B parameters  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]

22. 1 Light quarks physics 2 Heavy quarks physics Nconf = 120 Nconf = 120 a = 0.033 fm [ 1/a = 6.0 GeV ] a = 0.05 fm [ 1/a = 3.9 GeV ] [mπ = 200 MeV] [mπ = 200 MeV] Ls = 4.5 fm [V = 1363  270] Ls = 4.5 fm [V = 903  180] Target simulations to aim at the 1% level precision

23. 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” Berlin plot [Ukawa, Latt’01]

24. 0.05 for improved Wilson Empirical formulae for CPU cost For Nf=2 Wilson fermions: [Del Debbio et al. 06]  Comparison with Ukawa 2001 (the Berlin wall):

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

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

27. UTA in 2015 Table of inputs Precision flavour physics at the SuperB

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

29. 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

30. 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:

31. Example: Bd-Bd mixing Model independent analysis New Physics discovery 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

32. 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

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

34. 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 !!

35. Backup slides

36. 2011 2015 LHCb L=2 fb-1 LHCb L=10 fb-1 Expectations for LHCb from V. Vagnoni at CKM 2006

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