Lattice QCD and Flavour Chris Sachrajda School of Physics and - PowerPoint PPT Presentation

Lattice QCD and Flavour Chris Sachrajda School of Physics and Astronomy University of Southampton Southampton SO17 1BJ UK Indirect Searches for New Physics at the time of the LHC GGI, Florence, March 23rd 2010 Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 1

1. Introduction There has been a huge improvement in the precision of lattice calculations in the last 3 years or so. There are a number of groups focussing on different aspects on flavour physics. I will talk about progress in kaon physics, particularly from the RBC-UKQCD collaboration using Domain Wall Fermions (set in context). RBC=RIKEN, Brookhaven National Laboratory, Columbia University. UKQCD in this project = Edinburgh and Southampton Universities. We coordinate the generation of (expensive) ensembles and work in subgroups on a wide variety of physics topics. The 2008 paper describing our old ensembles had 33 authors and we are preparing the analogous paper for our new ensembles. A set of references is found at the end of the talk. I also exploit preliminary results of the Flavianet Lattice Averaging Group (FLAG): G. Colangelo, S. Dürr, A. Jüttner, L. Lellouch, H. Leutwyler, V. Lubicz, S. Necco, C. Sachrajda, S. Simula, A. Vladikas, U. Wenger, H. Wittig. Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 2

Introduction Cont. Plan of the Talk 1 Introduction 2 Determination of V us 2.i f K / f π . 2.ii K ℓ 3 decays. 3 B K 4 η and η ′ mesons and mixing. 5 K → ππ Decays 6 Conclusions and Prospects Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 3

RBC-UKQCD Ensembles We use two datasets of DWF with the Iwasaki Gauge Action with a lattice spacing of about 0.114fm: 24 3 × 64 × 16 ( L ≃ 2 . 74 fm) ( 16 3 × 32 × 16 ( L ≃ 1 . 83 fm) ) On the 24 3 lattice measurements have been made with 4 values of the light-quark mass: ma = 0 . 03 ( m π ≃ 670MeV ) ; ma = 0 . 02 ( m π ≃ 555MeV ) ; ma = 0 . 01 ( m π ≃ 415MeV ) ; ma = 0 . 005 ( m π ≃ 330MeV ) . (Using partial quenching the lightest pion in our analysis has a mass of about 240 MeV .) On the 16 3 lattice results were obtained with ma = 0 . 03 , 0 . 02 and 0 . 01 . For the (sea) strange quark we take m s a = 0 . 04 , although a posteriori we see that this is a little too large. We are completing the analysis of an ensemble on a 32 3 × 64 × 16 lattice with a ≃ 0 . 081 fm ( L ≃ 2 . 6 fm) with three dynamical masses ( m π ≃ 310 , 365 and 420 MeV). This will enable us to reduce the discretization errors significantly. Some preliminary results were presented at Lattice 2008, 2009 and elsewhere. Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 4

Global Chiral and Continuum Fits Imagine an idealized situation where simulations are possible at all quark masses for a variety of β s ( β = β i , i = 1 , 2 , ··· , N ). We can choose to fix m ud ( β i ) , m s ( β i ) and a ( β i ) by requiring that 3 physical quantities take their physical values. This defines a Scaling Trajectory . – We use m π , m K and m Ω . We can then calculate other physical quantities ( f π ( β i ) , B K ( β i ) , ··· ). These will have lattice artefacts of O ( a 2 i Λ 2 QCD ) and we imagine extrapolating the results to the continuum limit. At present however, we have to extrapolate to the physical values of m ud (and interpolate to m s ). We have invested considerable effort in defining and performing global fits in which we keep physical Low Energy Constants at all (both) β i and yet treat the artefacts consistently. ALMOST DONE. QCD ) √ , O ( m 2 π / Λ 2 χ ) , O ( a 2 Λ 2 O (( m π / Λ χ ) 4 ) , O ( a 2 m 2 π ) , O (( a Λ QCD ) 4 ) ···× . – We use other ansatz also. Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 5

Lattice Issues - Topology Changing. Although the algorithms used in the generation of field ensembles are formally ergodic, in a finite simulation it may be that the space of field configurations has not been fully sampled. Procedures for calculating autocorrelations exist, but can not be 100% reliable. It has recently been stressed that for fine lattices ( a � 0 . 04 fm), the topological charge does not change (for the actions generally used). Zeuthen and CERN groups, ··· . There is a large amount of algorithmic work being devoted to overcome this problem. Step Scaling Alpha Collaboration . Although the idea of step-scaling and the femto universe have been advocated for a long time by the Alpha collaborations, up to recently they have only been used by a small number of groups. Improved precision in the calculation of physical quantities ⇒ this is becoming a more widely used technique ( B -physics, Non-perturbative renormalization etc.) Match lattices at different β until we end up with a very fine, but small, lattice where connection with continuum QCD can be made reliably. Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 6

Lattice Issues cont. - Reweighting Although we can simulate at m phys , we only know its value a posteriori. s We therefore have to estimate what m s is before performing the simulations. Imagine that we wish to compute (Dirac operator D q = D [ U , m q ] ) � d [ U ] e − S g � det ( D † 2 D 2 ) O ( U ) � O � 2 = � d [ U ] e − S g � det ( D † 2 D 2 ) Imagine also that we performed the simulation with mass m 1 . Now � d [ U ] e − S g � det ( D † 1 D 1 ) O ( U ) w ( U ) � O � 2 = � d [ U ] e − S g � det ( D † 1 D 1 ) w ( U ) where � � D ξ e − ξ † √ � 1 / 2 � D † Ω [ U ] ξ � 2 [ U ] D 2 [ U ] ≡ det − 1 / 2 ( Ω ) = w [ U ] = det � D ξ e − ξ † ξ . D † 1 [ U ] D 1 [ U ] Jointly sampling U and ξ fields ⇒ � O � 2 . One (small) systematic error removed. Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 7

2. V us – f K / f π FLAG Compendium – Preliminary All groups calculate f K / f π . f K /f π 1.14 1.16 1.18 1.2 1.22 1.24 1.26 12 N f = 2 ETM 09 10 N f = 2+1 MILC 09 N f = 2+1 AUBIN 08 8 N f = 2+1 PACS-CS 08 N f = 2 ETM 08A 6 N f = 2+1 BMW 08 N f = 2+1 HPQCD/UKQCD 08 4 N f = 2+1 RBC/UKQCD 07 N f = 2 QCDSF/UKQCD 07 2 N f = 2+1 NPLQCD 07 N f = 2+1 MILC 04 0 our estimate -2 nuclear β decay semi-inclusive τ decay -4 1.14 1.16 1.18 1.2 1.22 1.24 1.26 Flag Compendium – Preliminary: f K / f π = 1 . 190 ( 2 )( 10 ) – Direct N f = 2 + 1 ; f K / f π = 1 . 210 ( 6 )( 17 ) – Direct N f = 2 . Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 8

f K / f π cont. The calculation requires a reliable chiral extrapolation. ⇒ SU(2) ChPT . RBC/UKQCD, arXiv:0804:0473 Is the chiral extrapolation as well under control for all quantities as we think? Very soon, as the simulated masses → m phys the chiral extrapolation will be a π smaller concern. Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 9

Comparison of Results obtained using SU(2) and SU(3) ChPT f PS [MeV] RBC/UKQCD, arXiv:0804:0473 160 Study is performed at NLO in the chiral expansion. 140 black points - partially quenched results with am l = 0 . 01 f π ( m unitary ≃ 420 MeV). π 120 f red points - partially quenched results with am l = 0 . 005 m ll = 331 MeV ( m unitary 100 m ll = 419 MeV ≃ 330 MeV). π f 0 SU(2) fit SU(3) fit We find: m 2 PS [MeV 2 ] f π / f ≃ 1 . 08 , f / f 0 = 1 . 23 ( 6 ) . 80 140 2 250 2 330 2 420 2 0 The corresponding results from the MILC collaboration, who do an NNLO analysis (partly in staggered chiral perturbation theory), with NNNLO analytic terms: � � � � + 6 + 13 f π / f = 1 . 052 ( 2 ) f / f 0 MILC = 1 . 15 ( 5 ) , , − 3 − 3 The large value of f π / f 0 (and even larger values of f PS / f 0 of ∼ 1 . 6 where we have data) lead RBC/UKQCD (and ETMC) to present results based on SU(2) × SU(2) ChPT. Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 10

K ℓ 3 Decays leptons s u ⇒ V us π K s γ µ u | K ( p K ) � = f 0 ( q 2 ) M 2 K − M 2 ( p π + p K ) µ − M 2 K − M 2 � � π π � π ( p π ) | ¯ q µ + f + ( q 2 ) q µ q 2 q 2 where q ≡ p K − p π . To be useful in extracting V us we require f 0 ( 0 ) = f + ( 0 ) to better than about 1% precision. f n = O ( M n χ PT ⇒ f + ( 0 ) = 1 + f 2 + f 4 + ··· where K , π , η ) . Reference value f + ( 0 ) = 0 . 961 ± 0 . 008 where f 2 = − 0 . 023 is relatively well known from χ PT and f 4 , f 6 , ··· are obtained from models. Leutwyler & Roos (1984) Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 11

K ℓ 3 : History – V us from Lattice Simulations, A.Jüttner (Lattice 2007) V us 0.23 0.23 V ud (0 + → 0 + ) F A lavi net Kaon WG f K π + ( 0 ) = 0 . 9644 ( 33 )( 34 ) LATTICE 2007 f + (0) = 0.9644(49) f K /f π = 1.198(10) V us /V ud (K µ 2 ) | V us | = 0 . 2247 ( 12 ) ⇒ fit with unitarity f K 0.225 0.225 fit = 1 . 198 ( 10 ) V us (K l3 ) f π | V us | = 0 . 2241 ( 24 ) ⇒ unitarity A.Jüttner, Lattice 2007 0.97 0.97 0.975 0.975 V ud Our final result from the K ℓ 3 project is f K π + ( 0 ) = 0 . 964 ( 5 ) . P .A.Boyle et al. [RBC&UKQCD Collaborations – arXiv:0710.5136 [hep-lat]] Chris Sachrajda (UKQCD/RBC Collaboration) Florence, 23/3/2010 12