NPR step-scaling across the charm threshold Julien Frison - PowerPoint PPT Presentation

Intro. Strat. Results BK Concl. NPR step-scaling across the charm threshold Julien Frison University of Edinburgh For the RBC-UKQCD collaboration 32nd International Symposium on Lattice Field Theory Lattice’14 - June 24th, 2014

Intro. Strat. Results BK Concl. RBC Ziyuan Bai (Columbia) UKQCD Thomas Blum (UConn/RBRC) Rudy Arthur (Odense) Norman Christ (Columbia) Peter Boyle (Edinburgh) Xu Feng (Columbia) Luigi Del Debbio (Edinburgh) Tomomi Ishikawa (RBRC) Shane Drury (Southampton) Taku Izubuchi (RBRC/BNL) Jonathan Flynn (Southampton) Luchang Jin (Columbia) Julien Frison (Edinburgh) Chulwoo Jung (BNL) Nicolas Garron (Dublin) Taichi Kawanai (RBRC) Jamie Hudspith (Toronto) Chris Kelly (RBRC) Tadeusz Janowski (Southampton) Hyung-Jin Kim (BNL) Andreas Juettner (Southampton) Christoph Lehner (BNL) Ava Kamseh (Edinburgh) Jasper Lin (Columbia) Richard Kenway (Edinburgh) Meifeng Lin (BNL) Andrew Lytle (TIFR) Marina Marinkovic Robert Mawhinney (Columbia) (Southampton) Greg McGlynn (Columbia) Brian Pendleton (Edinburgh) David Murphy (Columbia) Antonin Portelli (Southampton) Shigemi Ohta (KEK) Thomas Rae (Mainz) Eigo Shintani (Mainz) Chris Sachrajda (Southampton) Amarjit Soni (BNL) Francesco Sanfilippo (Southampton) Sergey Syritsyn (RBRC) Matthew Spraggs (Southampton) Oliver Witzel (BU) Tobias Tsang (Southampton) Hantao Yin (Columbia) Jianglei Yu (Columbia) Daiqian Zhang (Columbia)

Intro. Strat. Results BK Concl. Outline Introduction 1 Strategy 2 NPR Results 3 Consequences on B K 4 Conclusion 5

Intro. Strat. Results BK Concl. Motivations The story so far LQCD has made huge progresses, especially with chiral extrapolation NPR allows us to get Z factors with high precision for many operators Perturbative matching introduces the dominant error in B K What more can we do? Claim PT is not our job? Increase the scale ! If we get PT to higher order the effect of this increase will be even stronger. Then we should treat the charm quark accordingly

Intro. Strat. Results BK Concl. General strategy Take 0 . 8 GeV ∼ µ 0 < µ 1 . . . < m SMOM < . . . µ n ∼ 5 GeV c Define threshold step scaling functions: � − 1 Λ 2+1+1 ( a , µ n , m c ) � Λ 2+1+1 ( a , µ n +1 , m c ) σ ( µ n , µ n +1 , m c ) = lim a → 0 Then �O ( µ 1 , m c ) � 2+1+1 = Π n σ ( n , n + 1) �O ( µ 0 ) � 2+1 ren ren Choose scale from W 0 at suff. IR Wilson flow time that we match the IR limit of 2+1+1 flav theory to the 2+1f theory. For µ 0 >> m s , m u , m d this is equivalent to matching massless mu,d,s. Fix m c to its physical value, defined by NPR in a small volume by taking hierarchy of scales: µ d / s < µ 0 < m c < µ n Run from off-shell amplitudes in approx massless 3f theory to off shell amplitudes in approx massless 4f theory. Treat charm threshold effects treated non-perturbatively, and the charm at its physical mass at all stages. Mass independence of Z m in RI schemes is satisfied if p , a − 1 ≫ Λ , m q Do not need m q → 0

Intro. Strat. Results BK Concl. Ensembles N f = 2 + 1 Ensembles B K has been computed on a wide set of (M)DWF ensembles, including two ensembles at the physical quark masses, and lattice spacing going up to 3 GeV. N f = 2 + 1 + 1 Ensembles L 3 × T × L 5 a − 1 β m l m c 32 3 × 64 × 12 5 . 70 0 . 0047 0 . 243 3 . 0 GeV 32 3 × 64 × 12 5 . 70 0 . 002 0 . 243 3 . 0 GeV 32 3 × 64 × 12 5 . 70 0 . 0047 0 . 01 3 . 0 GeV 32 3 × 64 × 12 5 . 77 0 . 0044 0 . 213 3 . 6 GeV 32 3 × 64 × 12 5 . 84 0 . 0041 0 . 183 4 . 3 GeV 32 3 × 64 × 12 5 . 84 0 . 002 0 . 183 4 . 3 GeV

Intro. Strat. Results BK Concl. RI-SMOM scheme Kinematics p 1 p 2 Non-exceptional schemes avoid π pole p 2 1 = p 2 2 = ( p 1 − p 2 ) 2 2 q no � p i combination cancels out p 1 p 2 many orientations satisfy this condition but cont. limit is universal Renormalisation condition Z Tr [ P ijkl G ijkl ] = Tr [ P ijkl G ijkl ] | tree P ijkl = γ i δ ij γ k δ kl or P ijkl = / q ij / q kl different schemes allow us to evaluate the truncation error Very versatile method, with many knobs to turn With five 4-volume factors plus HDCG it is very cheap

Intro. Strat. Results BK Concl. Step-scaling and ratios Z lat → RI / SMOM ( p 2 ) / Z lat → RI / SMOM ( p 2 0 ) has an universal cont. limit Even if you use Wilson, Twisted, Staggered or anything, you can use our result As a corollary we can form other interesting ratios: Z dir 1 ( p 2 ) / Z dir 2 ( p 2 ) is 1 up to discr. effects Z ens 1 ( p 2 ) / Z ens 2 ( p 2 ) is constant up to discr. effects Those ratios have several advantages: No dependance on p 0 nor ( ap 0 ) 2 contamination Correlated through a − 1 (often main src of error) Allow an easy study of p 2 dependance of discr. effects, instead of working slice-by-slice and throwing away a lot of information

Intro. Strat. Results BK Concl. Chiral extrapolation systematics light quark mass dependance of Z B K b=5.70 RI-SMOM γγ 1 ZBK(ml=0.0047)/ZBK(ml=0.002) 0,9998 0,9996 0,9994 dZBK 0,9992 0,999 0,9988 0,9986 0 0,5 1 1,5 2 2,5 3 2 (ap)

Intro. Strat. Results BK Concl. Charm effects Z B K ratio between two different m c 1,001 ratio 1 ZBK 0,999 0,998 0 2 4 6 8 2 (ap)

Intro. Strat. Results BK Concl. Looking at different orientations O(4) breaking terms comparison on different ensembles 0 dZBK b=5.70 (lagrange interp) dZBK b=5.84 (spline interp) -0,001 dZBK b=5.84 (lagrange interp) -0,002 dZBK -0,003 -0,004 -0,005 0 1 2 3 ap2 0 )(1 + α ( p )( ap ) 2 + β ( p )( ap ) 4 ), but In principle Z a ( p 2 ) = Z 0 s ( p 2 , p 2 p dependence small after Λ QCD

Intro. Strat. Results BK Concl. Discretisation effects substraction ratio Z B K (5.70) over Z B K (5.84) 0,978 dZBK (lagrange) fit dZBK (spline) fit dZBK corrected 0,976 dZBK 0,974 0,972 10 20 30 40 2 p

Intro. Strat. Results BK Concl. Step-scaling results ( γγ ) N f =2+1+1 B K step-scaling from 3GeV RI-SMOM γγ scheme 1,05 b=5.70 b=5.77 b=5.84 b=5.70 sub b=5.77 sub b=5.84 sub 1-loop PT ZBK/ZBK(3GeV) 1 0,95 0 20 40 60 80 100 2 (GeV 2 ) p

Intro. Strat. Results BK Concl. Step-scaling results ( / q ) q / N f =2+1+1 B K step-scaling RI-SMOM qq scheme 1,1 b=5.70 b=5.77 b=5.84 b=5.70 sub b=5.77 sub b=5.84 sub 1,05 1-loop PT ZBK/ZBK(3GeV) 1 0,95 0 20 40 60 80 100 2 (GeV 2 ) p

Intro. Strat. Results BK Concl. The 3 GeV starting point 0 . 61 32I 0 . 56 0 . 60 24I B K ( SMOM ( q, q ) 3 GeV) 48I 0 . 59 64I 32Ifine 0 . 55 B K ( SMOM ( q, q ) 3 GeV) 0 . 58 32ID 0 . 57 0 . 54 0 . 56 32I 0 . 55 24I 0 . 53 48I 0 . 54 64I 32Ifine 0 . 53 32ID 0 . 52 Unitary extrapolation 0 . 52 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 000 0 . 005 0 . 010 0 . 015 0 . 020 a 2 (GeV − 2 ) m l (GeV) PRELIMINARY B K ( / q / q , 3 GeV) = 0 . 5343(29) B K ( γγ, 3 GeV) = 0 . 5168(28) ⇒ B K ( MS , 3 GeV) = 0 . 5296(29) stat (20) FV (2) χ (107) NPR

Intro. Strat. Results BK Concl. Running to 5 GeV and higher 2 ) scheme B K in MSbar(p using only central value of B K (3GeV) 0,56 b=5.70 sub gg b=5.77 sub gg b=5.84 sub gg 0,54 b=5.70 sub qq b=5.77 sub qq b=5.84 sub qq B K 0,52 0,5 Y R A N I M 0,48 0 I 20 40 60 80 100 L E 2 (GeV 2 ) R p P B K ( MS , 5 GeV) = 0 . 5103(28) stat (20) FV (2) χ (45) NPR B K ( MS , 9 GeV) = 0 . 4913(28) stat (20) FV (2) χ (3) NPR ??

Intro. Strat. Results BK Concl. The discr. errors, which are the main challenge for increasing the scale, are well under control This is also a strong evidence that, more generally, our action is well-behaved Our strategy of getting discr errors from the p 2 dependence seems payful We have presented a very promising preliminary result at 5 GeV, and more than halfed the error bar Our strategy seems to be valid up to 9 GeV, however one has to be careful about the systematics we’ve presented, in particular charm effects A FV study would be necessary to complete those results. For the moment we can only extrapolate our previous experience Our results confirm quite impressively something we have always observed: the convergence is much faster in RI / SMOM / q Generalisation to Z m , B K BSM, K → ππ ∆ I = 1 / 2 or 3 / 2, ...

Intro. Strat. Results BK Concl. Thanks for your attention!