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 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”

4. 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 ?

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 uncertainties can be systematically reduced in time, with the continuously increasing availability of computing power and the development of new theoretical techniques

6. 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

7. 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.

8. 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%

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
A previous estimate
Lattice QCD: Present and Future, Orsay, 2004 and report of the U.S. Lattice QCD Executive Committee ??
4 - 5% %
11%
----
13%
1.2%
(13% on 1-) 2%
(21% on 1-) 4%
(40% on 1-)
 B → D/D*lν %
(9-12% on ξ-1) 3%
(18% on ξ-1) 5%
(26% on ξ-1) ξ
3 - 4% % %
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

11. 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

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. 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 ≃ 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. Minimum lattice spacing
[From 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:

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

17. 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:

18. [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/ ]

19.  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 = fm the number of lattice sites is
Today the typical size is: 323  64
More than 300 times smaller
V  1363  270

20. 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

21. [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/ ]
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/ ]
The B-B mixing
B parameters

22. 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 = fm
[ 1/a = 6.0 GeV ]
[ mπ = 200 MeV ]
Heavy quarks physics 2

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, 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

24. 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):

25. 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 = 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 !!

26. Hadronic matrix element
Estimates of error for 2015
60 TFlop Year
[2011 LHCb]
2 – 3%
4 - 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) %
(9-12% on ξ-1) 3%
(18% on ξ-1) 5%
(26% on ξ-1) ξ
1 – 1.5%
3 - 4% % %
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]

27. 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
(0.6%)
(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

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

29. Sin2β = ± 0.023
α = (91.2 ± 5.4)o
γ = (66.7 ± 6.4)o
Sin2β = ±
α = ( ± 0.45)o
γ = (54.28 ± 0.38 )o

30. 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:

31. 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 = ± 0.031
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. this is the most pessimistic scenario!!
[D’Ambrosio et al., NPB 645]
Today
With a SuperB
Remember:
this is the most pessimistic scenario!!
δS0 = ± 0.32
Λ > %
NP masses > 200 GeV
δS0 = ± 0.059
Λ > 6 95%
NP masses > 600 GeV

34. 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 !!

35. Backup slides

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

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