1. Lattice QCD calculations in the SuperB Era
Vittorio Lubicz
A status report 3 years after
A. Stocchi 2006
V. Lubicz 2006

2. The theoretical accuracy must compete with the experimental one
The goal of the SuperB factory is precision flavour physics for indirect New Physics searches
An important example:
- Test the CKM paradigm at the 1% level
“the dream”
Today
With a SuperB in 2015
The theoretical accuracy must compete with the experimental one

3. Can we reach a comparable accuracy in Lattice QCD ?
The EXPERIMENTAL ACCURACY at a SuperB factory will reach the level of 1% or better for most of the relevant physical quantities A.
Can we reach a comparable accuracy in Lattice QCD ?

4. Hadronic matrix element
V.Lubicz @
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
[2009]
Current latt. error (2006)
Hadronic matrix element
1-10 PFlop Year
[2015 SuperB]
Lattice QCD: Present and Future, Orsay, 2004 and report of the U.S. Lattice QCD Executive Committee

5. History of lattice errors (before 2006)
1.23(6) 5%
262(35)
13%
189(27)
14%
Hashimoto
Ichep’04
1.24(4)(6) 6%
276(38)
193(27)(10)
15%
L.Lellouch
Ichep’02
1.16(5) 4%
267(46)
17%
200(30)
C.Bernard
Latt’00
----
175(25)
J.Flynn
Latt’96
1.21(2)(5)
246(16)(20)
10%
223(15)(19)
11%
N.Tantalo
CKM’06
For many years, uncertainties have been dominated by the quenched approximation.
QUENCHED
UNQUENCHED

6. Increasing of computational power Unquenched simulations
THE “PRECISION ERA” OF LATTICE QCD: WHY NOW
Increasing of computational power Unquenched simulations
For Lattice QCD today: ~ 5–30TFlops
(like the # 500 in the TOP500 list)
The Moore’s Law
TeraFlops machines are required to perform unquenched simulations. Available only since few years.
[Del Debbio et al. 2006]
CPU cost for Nf=2 Wilson fermions:

7. 2) Algorithmic improvements:
Light quark masses
in the ChPT regime
“The Berlin wall has been disrupted”
Ukawa 2001 (The Berlin wall):
CPU cost (for Nf=2 Wilson fermions):
Del Debbio et al. 2006:
Today:
Few years ago:
ChPT

8. KAON AND B PHYSICS ON THE LATTICE
Collaboration
Quark action Nf
a [fm]
(M)min [MeV]
Observables
MILC
+ FNAL, HPQCD,…
Improved staggered
2+1
 0.045
230
fK, BK, fB, BB, B→D/ lν
PACS-CS
Clover (NP)
0.09
156 fK
RBC/UKQCD
DWF
 0.08
290
f+(0), fK, BK,
K→
BMW
Clover smeared
 0.07
190
JLQCD
Overlap 2
[2+1]
0.12 BK
ETMC
Twisted mass
[2+1+1]
260
f+(0), fK, BK, fB
QCDSF
 0.06
300
f+(0), fK

9. Cost of the “SuperB” lattice simulation
Simulation parameters
The SuperB
is running
Nconf = 120
Ls = 4.5 fm
[V = 1363  270]
a = fm
[ 1/a = 6.0 GeV ]
[ Mπ = 200 MeV ]
~ 3 PFlop-years
Affordable with
1-10 PFlops avilable for Lattice QCD in 2015 !

10. Hadronic matrix element
THE 2009 STATUS REPORT
Hadronic matrix element
Lattice error in 2006
Lattice error in 2009
6 TFlop Year
60 TFlop Year
1-10 PFlop Year
0.9% ?
0.7%
0.4%
 0.1%
11% 5% 3% 1% fB
14% % %
1 – 1.5%
13%
4 - 5%
3 - 4% ξ %
0.5 – 0.8 %
 B → D/D*lν 4% 2%
1.2%
0.5% %
2 – 3%
----
3 – 4%
[2009]
[2011 LHCb]
[2015 SuperB]

11. Vus from kaon decays: f+ (0) and fK/fπ
[ ]
NEW K π
Vus
|Vus|Kl3 = (13)
|Vus| = (10)
Using unitarity and |Vud| from nuclear b decays
f+(0) = (3) (4)
0.5%
Error in 2006: 0.9%
Predicted error with 6 TFlops
0.7%

12. The accuracy is comparable to the one reached on f+(0) [0.5%]
Vus from kaon decays: f+ (0) and fK/fπ Kπ
[ ]
NEW K
Vus
[Marciano 04]
|Vus|Kl3 = (13)
|Vus| = (10)
Using unitarity and |Vud| from nuclear b decays
|Vus|Kl2 = (19)
0.8%
fK/f = (1) (10)
The accuracy is comparable to the one reached on f+(0) [0.5%]

13. K0-K0 mixing: BK ^ BK= 0.90 ± 0.03 ± 0.15 ^ BK= 0.86 ± 0.05 ± 0.14 ^
VqsVqd *
All unquenched calculations until last year at fixed (and rather large) lattice spacing
[VL, C.Tarantino ] ^
BK= 0.90 ± 0.03 ± 0.15
17% ^
BK= 0.86 ± 0.05 ± 0.14
17% ^
BK= 0.79 ± 0.04 ± 0.08
11% ^
BK= ± 0.037 5%

14. K0-K0 mixing: BK ^ BK = 0.731 (7) (35) [ V.Lubicz@LATT’09 ] VqsVqd * K
NEW
[ ] K
VqsVqd *
From the UT fit, assuming the validity of the Standard Model
(with Kε ≈ 0.92,
A.Buras, D.Guadagnoli, )
BK = 0.81 (8) ^
BK = (7) (35) ^ 5%
Predicted error with 6 TFlops 5%
Error in 2006: 11%

15. K-K AND B-B MIXING BEYOND THE SM
[M.Ciuchini et al., hep-lat/ ]
APE 99
Babich et al 06
CP-PACS 06 
K - K
NEW CALCULATIONS ARE NEEDED !!
B - B
JLQCD 02
HPQCD 06
APE 01
The full operator basis only in the quenched approximation
For K-K mixing results quite in disagreement

16. B-mesons decay constants: fB,fBs
2009 lattice averages from J.Laiho, E.Lunghi, R.Van de Water,
Predicted error with 6 TFlops 4%
Error in 2006: 14%
fB = ± 9.9 MeV
fBs= ± 9.5 MeV
4-5%
Predicted error with 6 TFlops 3%
Error in 2006: 5%
fBs/fB = ± 0.027 2%

17. Combining with fB and fBs:
B-B mixing: BBd/s B
VtbVtq *
BBd = ± 0.11 ^
BBs = ± 0.06
Only one modern calculation
HPQCD [ ]
Combining with fB and fBs:
Error in 2006: 13%
Error in 2006: 5%
fBs√BBs= 275 ± 13 MeV ^
ξ = ± 0.028 5% 2%
Predicted error with 6 TFlops 4%
Predicted error with 6 TFlops 3%

18. Is there an “fDs puzzle” ?
A lattice result by HPQCD, which claims a 1.2% precision on fDs, shows a discrepancy of about 3σ with the PDG’08 average:
fDs= 241 ± 3 MeV HPQCD 07
fDs= 273 ± 10 MeV PDG 08
B.Dobrescu, A.Kronfeld,
“Evidence for nonstandard leptonic decays of Ds mesons”
This year, after a new CLEO-c result, the tension is reduced at the 2σ level:
fDs= ± 6.8 MeV HFAG 09

19. Predicted error with 6 TFlops
Vcb and Vub exclusive
Lattice averages from VL, C.Tarantino
Error in 2006: 4%
Error in 2006: 11%
F(1) = ± 0.022
G(1) = ± 0.035 2%
|Vub|excl.= (35.0 ± 4.0) 10-4
11% 3%
Predicted error with 6 TFlops 6%
Predicted error with 6 TFlops 2%

20. Good agreement: a success for the Standard Model
OF LATTICE PARAMETERS
UT-angles
UT-lattice
Assuming the validity of the Standard Model one can perform a fit of the hadronic parameters:
2%!
from Δms BK
fBs√BBs (MeV) ξ
UTA
0.81 ± 0.08
265 ± 4
1.25 ± 0.06
Lattice
0.73 ± 0.04
275 ± 13
1.24 ± 0.03
Good agreement: a success for the Standard Model

21. The expected accuracy has been reached! (except for Vub)
THE 2009 STATUS REPORT
Hadronic matrix element
Lattice error in 2006
Lattice error in 2009
6 TFlop Year
60 TFlop Year
1-10 PFlop Year
0.9%
0.5%
0.7%
0.4%
 0.1%
11% 5% 3% 1% fB
14% % %
1 – 1.5%
13%
4 - 5%
3 - 4% ξ 2% %
0.5 – 0.8 %
 B → D/D*lν 4%
1.2% %
2 – 3%
----
3 – 4%
[2009]
[2011 LHCb]
[2015 SuperB]
The expected accuracy has been reached! (except for Vub)