Order a 4 Intro. Strat. Results Concl. Backup slides Nf=2+1+1 renormalisation of four-quark operators Julien Frison University of Edinburgh For the RBC-UKQCD collaboration 33rd International Symposium on Lattice Field Theory Lattice’15 - Kobe - July 15th, 2015
Order a 4 Intro. Strat. Results Concl. Backup slides RBC-UKQCD collaboration BNL and RBRC Edinburgh University Tomomi Ishikawa Peter Boyle Taku Izubuchi Luigi Del Debbio Chulwoo Jung Julien Frison Richard Kenway Christoph Lehner Ava Khamseh Meifeng Lin Brian Pendleton Shigemi Ohta (KEK) Oliver Witzel Taichi Kawanai Azusa Yamaguchi Christopher Kelly Plymouth University Amarjit Soni Nicolas Garron Sergey Syritsyn University of Southampton CERN Jonathan Flynn Marina Marinkovic Tadeusz Janowski Columbia University Andreas Juettner Ziyuan Bai Andrew Lawson Norman Christ Edwin Lizarazo Xu Feng Antonin Portelli Luchang Jin Chris Sachrajda Bob Mawhinney Francesco Sanfilippo Greg McGlynn Matthew Spraggs David Murphy Tobias Tsang Daiqian Zhang York University (Toronto) University of Connecticut Renwick Hudspith Tom Blum
Order a 4 Intro. Strat. Results Concl. Backup slides Introduction 1 Motivations Operators and chiral sectors Strategy 2 Step-scaling Charm threshold Ensembles RI-SMOM scheme Numerical Results 3 Wilson flow and topological charge Continuum limit Including O ( a 4 ) discretisation terms 4 Method B K Perspectives for other sectors Conclusion 5
Order a 4 Intro. Strat. Results Concl. Backup slides Motivations The story so far The Rome-Southampton method with momentum sources has proven to be very efficient We have presented last year a N f = 2 + 1 + 1 strategy to treat the charm threshold ... and preliminary results for B K What more can we do? All systematics were not fully under control We need generalisation to SUSY B K and K → ππ
Order a 4 Intro. Strat. Results Concl. Backup slides Operators and chiral sectors sector dim SUSY B K K → ππ I = 2 K → ππ I = 0 B K (27 , 1) 1 � � � � (8 , 8) 2 � � � (8 , 1) 4 � (6 , 6) 2 � With our chiral action, we can consider each chiral sector separately. Q (27 , 1) = [¯ s γ µ (1 − γ 5 ) d ] [¯ s γ µ (1 − γ 5 ) d ] (1) Q (8 , 8) = [¯ s γ µ (1 − γ 5 ) d ] [¯ s γ µ (1 + γ 5 ) d ] (2) 1 Q (8 , 8) = [¯ s (1 − γ 5 ) d ] [¯ s (1 + γ 5 ) d ] (3) 2 Q (6 , 6) = [¯ s σ µν (1 − γ 5 ) d ] [¯ s σ µν (1 − γ 5 ) d ] (4) 1 Q (6 , 6) = [¯ s (1 − γ 5 ) d ] [¯ s (1 − γ 5 ) d ] (5) 2
Order a 4 Intro. Strat. Results Concl. Backup slides Step-scaling If one chooses a scaling trajectory and compute at constant physical masses the continuum limit Z ( p | m ) σ ( p , p 0 | m ) = lim Z ( p 0 | m ) , (6) a → 0 this is universal, i.e. only dependent on the continuum RGEs from one scale to another. Does not depend on your scaling trajectory, your lattice action, the O (4) orbit of momenta, and so on. Therefore those ratios are convenient building-blocks that we can combine with as many ratios as we want into a telescopic product.
Order a 4 Intro. Strat. Results Concl. Backup slides Charm threshold m p
Order a 4 Intro. Strat. Results Concl. Backup slides Ensembles N f = 2 + 1 Ensembles Matrix elements have 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, 0 . 1, 0 . 0186 3 . 0 GeV 32 3 × 64 × 12 5 . 70 0 . 002 0 . 243 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, 0 . 0146 4 . 3 GeV 32 3 × 64 × 12 5 . 84 0 . 002 0 . 183 4 . 3 GeV
Order a 4 Intro. Strat. Results Concl. Backup slides 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 Cheap once we have configs, excellent signal/noise
Order a 4 Intro. Strat. Results Concl. Backup slides Wilson flow and topological charge b5.70_2p2_mob2 flow history b5.84_2p2 flow history (first configuration is b5.70_2p2, with b+c=5) excluding first configurations 5.5 sqrt(t 0 ) sqrt(t 0 ) w 0 w 0 3.4 5 3.2 4.5 3 2.8 4 2.6 3.5 2200 2300 2400 2700 2800 1700 1800 1900 2000 2100 2200 2300 2500 2600 3 2 2 1.5 1 1 0 0.5 -1 0 -2 -0.5 -3 -1 2100 2150 2200 2250 2300 1000 1500 2000 2500 3000
Order a 4 Intro. Strat. Results Concl. Backup slides Continuum limit (intro) The following results are produced through this process: 1 Compute step-scaling from 3 GeV for each ensemble (and orientation) 2 Interpolate them to a common set of p 2 values 3 For each interpolated p 2 and each orientation, extrapolate independently in a 2 4 The difference between two orientations is an estimate of the systematics from higher order discretisation terms
Order a 4 Intro. Strat. Results Concl. Backup slides Continuum limit I (27,1) γγ 1,1 σ cont[1,1,0,0] σ fine[1,1,0,0] σ coarse[1,1,0,0] 1,05 σ cont[1,1,1,1] σ fine [1,1,1,1] σ coarse[1,1,1,1] NLO LO 1 σ 0,95 0,9 0 20 40 60 80 100 120 2 p
Order a 4 Intro. Strat. Results Concl. Backup slides Continuum limit II (27,1) qq 1,15 σ cont[1,1,0,0] σ fine[1,1,0,0] 1,1 σ coarse[1,1,0,0] σ cont[1,1,1,1] σ fine[1,1,1,1] σ coarse[1,1,1,1] 1,05 NLO σ 1 0,95 0,9 0 20 40 60 80 100 120 psq
Order a 4 Intro. Strat. Results Concl. Backup slides Continuum limit III (8,8) 12 gg σ cont[1,1,0,0] 0,2 σ fine[1,1,0,0] σ coarse[1,1,0,0] σ cont[1,1,1,1] 0,1 σ fine[1,1,1,1] σ coarse[1,1,1,1] σ (p,3GeV) 0 NLO LO -0,1 -0,2 -0,3 -0,4 0 20 40 60 80 100 120 2 p
Order a 4 Intro. Strat. Results Concl. Backup slides Continuum limit IV (6,6) 12 gg 0,015 σ cont[1,1,0,0] σ fine[1,1,0,0] σ coarse[1,1,0,0] 0,01 σ cont[1,1,1,1] σ fine[1,1,1,1] σ coarse[1,1,1,1] σ (p,3GeV) 0,005 0 -0,005 0 20 40 60 80 100 120 2 p
Order a 4 Intro. Strat. Results Concl. Backup slides Method Because most of the running comes from low-order PT, discretisation effects might be well-described by PT However we do not need any explicit perturbative formula At high- p 2 , Λ QCD becomes irrelevant, dimensional analysis suggest discr is mostly ( ap ) n -dependent. More precisely: the boosted coupling constant only grows logarithmically (see hep-lat/1412.0834) When combined with this all-order theoretical argument, the p 2 -dependence gives additional information Of course it doesn’t replace a third ensemble, but the latter would be much more expensive than this project
Order a 4 Intro. Strat. Results Concl. Backup slides B K B K in RGI scheme using only central value of B K (3GeV) b=5.70 sub gg 0.78 b=5.77 sub gg b=5.84 sub gg b=5.70 sub qq b=5.77 sub qq b=5.84 sub qq 0.76 B K 0.74 PRELIMINARY 0.72 0 20 40 60 80 100 2 (GeV 2 ) p
Order a 4 Intro. Strat. Results Concl. Backup slides Perspectives for other sectors I coarse fine Λ 11 / Λ 11 1,015 gg [1,1,0,0] gg [1,1,1,1] 1,01 qq [1,1,0,0] qq [1,1,1,1] ratio of interpolations 1,005 1 0,995 0,99 0 10 20 30 40 50 60 70 2 p
Order a 4 Intro. Strat. Results Concl. Backup slides Perspectives for other sectors II coarse fine Λ 12 / Λ 12 gg [1,1,0,0] gg [1,1,1,1] 1,2 qq [1,1,0,0] qq [1,1,1,1] 1,1 ratio of interpolations 1 0,9 0,8 0,7 0 10 20 30 40 50 60 70 2 p
Order a 4 Intro. Strat. Results Concl. Backup slides Perspectives for other sectors III coarse fine Λ 34 / Λ 34 gamma-gamma scheme direction [1,1,0,0] direction [1,1,1,1] 2,2 2 ratio of interpolations 1,8 1,6 1,4 1,2 0 10 20 30 40 50 60 70 2 p
Order a 4 Intro. Strat. Results Concl. Backup slides We have presented preliminary results for the 5 × 5 SUSY B K step-scaling A tiny “ Nf = 3 / Nf = 4” has not been included here With O ( a 2 ) extrapolations the discretisation errors are huge A third ensemble would be most useful Generating new ensembles, with stats for the low-energy scale-setting, is painful It is important to extract as much info as possible from ensembles we already have discr. effects in SUSY B K have a very smooth p 2 -dependence, this looks like a relatively operator-independent observation.
Order a 4 Intro. Strat. Results Concl. Backup slides Thanks for your attention!
Recommend
More recommend
