lattice QCD a personal view in 20 minutes. . . Nazario Tantalo INFN sez. “Tor Vergata” Centro Ricerche e Studi “E. Fermi” Naples, 11-04-2007
introduction lattice QCD is the only reliable way to perform a precision test of the standard model at the non-perturbative level ≤ 2006: lattice QCD has been used in order to estimate the hadronic inputs to the CKM Unitarity Triangle Analysis (UTA) since these where unknowns . the UTA can be now performed without the need of lattice inputs ( UT fit Collaboration JHEP 0610 (2006) 081 ) ≥ 2007: to the lattice it is asked to provide, let’s say, ∆ M s within the standard model with an error of the same order of the one quoted by the CDF collaboration: i.e. all the systematics have to be under control beyond any reasonable doubt! different groups are following different strategies to reach the same goal and in the end, hopefully, we will have the best we can ask: the same result confirmed by different calculations presently we still have numbers affected by largely different systematics . . .
UNC: quenching quenching is definitively bad • the effect of the quenching is observable dependent and impossible to quantify without unquenched simulations in the same range of parameters at the same time quenched simulations have been able to predict sin 2 β sin ( 2 β ) = 0 . 650 ± 0 . 120 1995 Ciuchini et al Z. Phys. C 68 (1995) 239 2000 sin ( 2 β ) = 0 . 698 ± 0 . 066 Ciuchini et al JHEP 0107 (2001) 013 exp sin ( 2 β ) = 0 . 687 ± 0 . 032 the effectiveness of quenching may be due to a sort of “matching”: one has to tune the quark masses and the lattice spacing by using experimental inputs
UNC: rooted staggered staggered fermions are introduced on the lattice by simulating the following quark action (actually in its improved version): 2 3 η n ,µ “ ” X 4X U n ,µ χ n + µ − U † χ D stag χ = ¯ χ n ¯ n − µ,µ χ n − µ + m 0 χ n 5 2 µ n affected by doubling, i.e. it has 2 4 = 16 one-component fermions rooting means that gauge configurations are generated according to the following partition function: Z DUe − Sg n o 1 / 4 Z root Nf = 3 = det [ D stag ( m u )] det [ D stag ( m d )] det [ D stag ( m s )] S. R. Sharpe@LATTICE 2006 [ PoS LAT2006 (2006) 022 ]: Q: “Rooted staggered fermions: Good, bad or ugly?” A: ugly ! in the sense that are affected by unphysical contributions at regulated stage that need a complicate analysis to be removed •
other sources of systematics chiral pathologies ( χ E ) f B , for example, is expected to diverge in the quenched chiral limit • use of NLO χ PT formulas to extrapolate results gives complete control or large(reliable) errors • cutoff effects ( a E ) single and coarse lattice spacing • extrapolations with 3 or more lattice spacings gives complete control or large(reliable) errors • finite volume effects ( L E ) may be the dominant source of uncertainty in unquenched calculations of M π , f π , etc. •
lattice calculations of heavy–light systems, issues on currently affordable lattice sizes (at least in unquenched simulations) one has am b > 1 Lm d > 1 am b < 1 Lm d < 1 we are able to simulate “relativistic” beauty–light systems on big volumes with big cutoff effects small volumes with big finite volume effects so we have to devise smart strategies to cope with this two-scales problem . we can divide the different approaches in big volume strategies finite volume strategies
lattice calculations of heavy–light systems, big volume strategies HQET one can resort to the static approximation that can also be non–perturbatively renormalized • E Eichten et al Phys. Lett. B 234 (1990) 511 J Heitger et al JHEP 0402 (2004) 022 HQET EXT one can simulate the relativistic theory with heavy masses around the physical charm mass and extrapolate (or interpolate with the static) to the beauty mass relying on HQET predictions • FERMILAB the FERMILAB approach consists in simulating the following action with am 0 > 1 " # aD 2 a � D 2 i σ ij F ij i σ 0 i F 0 i X ¯ γ · � 0 S = ψ n m 0 + γ 0 D 0 + ζ� D − r t − r s + c B + c E ψ n 2 2 4 2 n i.e. the Symanzik effective action for quarks with | a � p | ≪ 1 with mass dependent coefficients usually computed perturbatively • A X El-Khadra et al Phys. Rev. D 55 (1997) 3933 S Aoki et al Prog. Theor. Phys. 109 (2003) 383 N H Christ et al hep-lat/0608006
lattice calculations of heavy–light systems, big volume strategies NRQCD another possibility consists in simulating one among the possible lattice discretizations of the NRQCD lagrangian at a given order in v 2 and α s ; for example − ∆ 2 H = + δ H 2 m b (∆ 2 ) 2 ig ( � E · � ∆ · � E − � δ H = − c 1 + c 2 ∆) 8 m 3 8 m 2 b b g ∆ × � E − � σ · ( � E × � ˜ ˜ − c 3 � ∆) 8 m 2 b a 2 ∆ ( 4 ) a (∆ 2 ) 2 g σ · � ˜ − c 4 � B + c 5 − c 6 8 m 2 16 m 2 24 m b b b A Gray et al Phys. Rev. D 72 (2005) 094507 the theory is non–renormalizable, can be matched to QCD only perturbatively, the continuum limit cannot be taken (heavy–light?) •
lattice calculations of B K ≤ 2006 the bag parameter is defined as the matrix element of the following operator i ¯ j = O VV + AA − O VA + AV ¯ i γ µ ( 1 − γ 5 ) q j γ µ ( 1 − γ 5 ) q O 1 = ψ ψ 8 � ¯ 0 | ˆ 0 � = � ¯ 0 | ˆ 0 � = 2 2 O 1 | K O VV + AA | K K B K ( µ ) K K M K f 3 ˆ ˜ ˆ O [ VV + AA ] = Z 11 O [ VV + AA ] + ∆ 12 O [ VV − AA ] + ∆ 13 O [ SS − PP ] + ∆ 14 O [ SS + PP ] + ∆ 15 O [ TT ] Wilson fermions: M Guagnelli et al JHEP 0603 (2006) 088 ALPHA 06: P Dimopoulos et al. Nucl. Phys. B 749 (2006) 69 D Becirevic et al Phys. Lett. B 487 (2000) 74 SPQCDR 02: D Becirevic et al. Nucl. Phys. Proc. Suppl. 119 (2003) 359 SPQCDR 05: F Mescia et al. PoS LAT2005 (2006) 365 HPQCD-UKQCD 06 RBC-UKQCD 06 CP-PACS 01 SPQCDR 02 SPQCDR 05 Becirevic 00 UKQCD 04 BosMar 03 JLQCD 97 Babich 06 ALPHA 06 MILC 03 RBC 03 RBC 05 1.3 the results do not show a significant dependence 1.2 1.1 upon the number of dynamical flavors 1 0.9 upon the renormalization procedure 0.8 upon the finite size effects 0.7 0.6 0.5 0.4
lattice calculations of B B q ≤ 2006 SPQCDR 01 UKQCD 00 Gadiyak 05 HPQCD 06 JLQCD 02 JLQCD 03 APE 00 1.2 B Bs ( m b ) : 1.1 the observable does not seem to depend upon 1 the number of dynamical flavors 0.9 the renormalization systematics 0.8 the heavy quark “technology” 0.7 0.6 SPQCDR 01 UKQCD 00 Gadiyak 05 JLQCD 02 JLQCD 03 APE 00 1.2 1.1 B B ( m b ) : 1 there isn’t a sizable dependence even on the quark mass all the “dependencies” are factorized in the vacuum 0.9 saturation approximation: decay constants. . . 0.8 0.7 0.6
lattice calculations of f D q ≤ 2006 FNAL-MILC-HPQCD 05 FNAL-MILC-HPQCD 05 CP-PACS 00 CP-PACS 00 UKQCD 00 ROMEII 03 HPQCD 03 ALPHA 03 FNAL 97 MILC 98 MILC 02 MILC 02 350 1.5 1.4 300 1.3 MeV 250 1.2 200 1.1 150 1 fDs CLEO: fDs = 274 ( 13 )( 7 ) MeV = 1 . 27 ( 14 ) BaBar+CLEO: fD BaBar: fDs = 283 ( 17 )( 7 )( 14 ) MeV fDs CLEO: = 1 . 23 ( 11 )( 4 ) fD B. Aubert et al hep-ex/0607094 M. Artuso et al arXiv:0704.0629 M Artuso et al Phys. Rev. Lett. 95 (2005) 251801
lattice calculations of f B q ≤ 2006 ROMEII-ALPHA 06 CP-PACS 00 CP-PACS 01 Fermilab 97 UKQCD 00 ROMEII 03 UKQCD 04 Gadiyak 05 HPQCD 05 JLQCD 99 ALPHA 03 JLQCD 03 ALPHA 06 MILC 98 MILC 02 APE 00 400 f Bs : 350 the quenched results are very precise but significantly 300 smaller than the unquenched ones MeV 250 this may signal a sizable dependence of this observable upon the number of dynamical flavors 200 150 100 0 2 4 6 8 10 12 14 16 18 CP-PACS 00 CP-PACS 01 UKQCD 04 Gadiyak 05 HPQCD 05 JLQCD 03 MILC 02 1.5 1.4 f Bs / f B : 1.3 the static numbers by Gadiyac et al 06, UKQCD 04 are identical and one sigma higher than the others 1.2 1.1 1
”un-rooted” unquenched algorithms in the last few years there has been a dramatic improvement in non-staggered unquenched algorithms M. L¨ uscher Comput. Phys. Commun. 165 (2005) 199 C. Urbach et al Comput. Phys. Commun. 174 , 87 (2006) M. A. Clark et al Phys. Rev. Lett. 98 , 051601 (2007) all these algorithms are based on the HMC and make use of multiple time step integrators S. Duane et al Phys. Lett. B 195 , 216 (1987) J. C. Sexton et al Nucl. Phys. B 380 , 665 (1992) Z “ ” dU e − SG [ U ] det D † [ U ] D [ U ] = Z Z P 2 R d 4 x φ † ( x ) D † [ U ] D [ U ] φ ( x ) dP X dUdP d φ † d φ e 2 − SG [ U ] − F i [ M − 1 = = − F G [ U ] − ] i d τ i Y D † [ U ] D [ U ] = M i [ U ] dU = P U i d τ Z P 2 X 2 − SG [ U ] − P R d 4 x φ † ( x ) M i [ U ] φ ( x ) d φ † = i d φ i e i dUdP i
light Wilson(-like) quarks: unquenching effects large scale projects of unquenched simulations with light Wilson quarks have been started genuine unquenching effects show up by studying the large time behavior of two-point correlation functions L. Del Debbio, L. Giusti, M. Luscher, R. Petronzio and N. T. JHEP 0702 (2007) 056 m val = 43 MeV ^ m val = 44 MeV ^ 630 m sea = 49 MeV m sea = 24 MeV 600 570 540 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.6 0.8 1.0 1.2 1.4 1.6 1.8 + + M eff ( t ) = M 0 + c e − 2 M π t + . . .
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