QCD + QED studies using Twist-Averaging Christoph Lehner (BNL) May - PowerPoint PPT Presentation

QCD + QED studies using Twist-Averaging Christoph Lehner (BNL) May 1, 2015 – USQCD AHM

(RBC collaboration) 1 / 18

LATTICE QCD AT THE INTENSITY FRONTIER Thomas Blum, Michael Buchoff, Norman Christ, Andreas Kronfeld, Paul Mackenzie, Stephen Sharpe, Robert Sugar and Ruth Van de Water (USQCD Collaboration) (Dated: October 22, 2013) IV. FUTURE LATTICE CALCULATIONS A second advance will be the systematic inclusion of isospin-breaking and electromagnetic (EM) e ff ects. Once calculations attain percent-level accuracy, as is the case at present for quark masses, f K /f π , the K ! π and B ! D ∗ form factors, and ˆ B K , one must study the e ff ects of EM and isospin breaking. A partial and approximate inclusion of such e ff ects is already made for light quark masses, f π , f K and ˆ B K . Full inclusion would require nondegen- erate u and d quarks and the incorporation of QED into the simulations. For some quantities it may su ffi ce to implement this only for the valence quarks (quenched QED), while in gen- eral one must also include mass di ff erences and electrical charges for the sea quarks. One approach for both isospin and unquenched QCD+QED simulations is to reweight pure QCD 2 / 18

LATTICE QCD AT THE INTENSITY FRONTIER Thomas Blum, Michael Buchoff, Norman Christ, Andreas Kronfeld, Paul Mackenzie, Stephen Sharpe, Robert Sugar and Ruth Van de Water (USQCD Collaboration) L ≈ 9 fm BMWc 2014 (Dated: October 22, 2013) IV. FUTURE LATTICE CALCULATIONS 10 ∆ q 2 =-1 A second advance will be the systematic inclusion of isospin-breaking and electromagnetic 5 ∆ M N [MeV] (EM) e ff ects. Once calculations attain percent-level accuracy, as is the case at present for quark masses, f K /f π , the K ! π and B ! D ∗ form factors, and ˆ 0 B K , one must study the e ff ects of EM and isospin breaking. A partial and approximate inclusion of such e ff ects is � already made for light quark masses, f π , f K and ˆ -5 B K . Full inclusion would require nondegen- erate u and d quarks and the incorporation of QED into the simulations. For some quantities -10 it may su ffi ce to implement this only for the valence quarks (quenched QED), while in gen- eral one must also include mass di ff erences and electrical charges for the sea quarks. One 0 20 40 60 80 100 approach for both isospin and unquenched QCD+QED simulations is to reweight pure QCD 1/L[MeV] 2 / 18

LATTICE QCD AT THE INTENSITY FRONTIER Thomas Blum, Michael Buchoff, Norman Christ, Andreas Kronfeld, Paul Mackenzie, Stephen Sharpe, Robert Sugar and Ruth Van de Water (USQCD Collaboration) (Dated: October 22, 2013) Muon anomalous magnetic moment 2 1. The muon anomalous magnetic moment provides one of the most precise tests of the Stan- dard Model of particle physics (SM) and often places important constraints on new theories beyond the SM [1]. The current discrepancy between experiment and the Standard Model has been reported in the range of 2.9–3.6 standard deviations [77–79]. With new experi- ments planned at Fermilab (E989) and J-PARC (E34) that aim to improve on the current 0.54 ppm measurement at BNL [80] by at least a factor of 4, it will continue to play a central role in particle physics for the foreseeable future. 2 / 18

Introduction to the method

QCD setup arXiv:1503.04395 U µ ( x ) U µ ( x ) U µ ( x ) Ψ( x + ˆ L 1 + ˆ Ψ( x + 2ˆ L 1 + ˆ Ψ( x + 3ˆ L 1 + ˆ L 2 ) L 2 ) L 2 ) U µ ( x ) U µ ( x ) U µ ( x ) Ψ( x + ˆ Ψ( x + 2ˆ Ψ( x + 3ˆ L 1 ) L 1 ) L 1 ) Valence fermions Ψ living on a repeated gluon background U µ with periodicity L 1 , L 2 and vectors ˆ L 1 = ( L 1 , 0), ˆ L 2 = (0 , L 2 ) 3 / 18

arXiv:1503.04395 Let ψ θ be the quark fields of your finite-volume action with twisted-boundary conditions ψ θ x + L = e i θ ψ θ x . Then one can show that � 2 π d θ 2 π e i θ ( n − m ) � � Ψ x + nL ¯ x ¯ � � ψ θ ψ θ Ψ y + mL = , (1) y 0 where the �·� denotes the fermionic contraction in a fixed background gauge field U µ ( x ). (4d proof available.) This specific prescription produces exactly the setup of the previous page, it allows for the definition of a conserved current, and allows for a prescription for flavor-diagonal states. 4 / 18

Example: QED mass correction on a lattice in finite volume + 1 1 1 1 1 � + α C ( p ) = p 2 + m 2 p 2 + m 2 ( p − k ) 2 + m 2 2 p 2 + m 2 k k ∈ BZ 4 with p µ = 2 sin( p µ / 2) p ∈ BZ 4 e ipx C ( p ) in finite- Strategy: compute C ( x ) = � volume and perform effective-mass fit 5 / 18

Twist-averaged version: Ψ x + nL ¯ Ψ y + mL ¯ e ik ( y + mL ) � � � � Ψ y + mL Ψ z + lL � 2 π � 2 π d θ ′ d θ 2 π e ik ( y + mL ) e i θ ( n − m )+ i θ ′ ( m − l ) � � � � x ¯ y ¯ ψ θ ψ θ ψ θ ′ ψ θ ′ = , y z 2 π 0 0 Perform sum over m using Poisson’s summation formula yields Ψ x + nL ¯ Ψ y + mL ¯ � e ik ( y + mL ) � � � � Ψ y + mL Ψ z + lL m � 2 π � 2 π d θ d θ ′ 2 π e i θ n − i θ ′ l ˆ � � � � x ¯ ψ θ ′ y ¯ ψ θ ′ = e iky ψ θ ψ θ δ ( k − ( θ − θ ′ ) / L ) , y z 2 π 0 0 with ˆ δ ( k ) = 2 π � n ∈ ◆ δ ( k + 2 π n / L ). L TA yields momentum conservation of twists 6 / 18

Example: QED mass correction on a lattice in finite volume plus TA + � � � 1 1 1 1 1 C ( p ) = + α p 2 + m 2 p 2 + m 2 p 2 + m 2 ( p − k ′ ) 2 + m 2 k ′ 2 k ∈ BZ 4 θ 4 with p µ = 2 sin( p µ / 2) and k ′ µ = k µ + θ µ / L µ p ∈ BZ 4 e ipx C ( p ) in finite- Strategy: compute C ( x ) = � volume and perform effective-mass fit 7 / 18

0.4 O( α 0 ) coefficient O( α 1 ) coefficient, L = 48, k 0 =0 sub 0.3 0.2 m eff,cosh 0.1 0 -0.1 0 5 10 15 20 25 30 35 40 45 t 8 / 18

0 k=0 subtraction k 0 =0 subtraction -0.02 m, O( α 1 ) coefficient -0.04 -0.06 -0.08 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 1/mL 9 / 18

0 TA, 200 twists, mL=4.8 -0.02 m eff,cosh , O( α 1 ) coefficient -0.04 -0.06 -0.08 -0.1 0 2 4 6 8 10 12 14 16 t 10 / 18

0 k=0 subtraction k 0 =0 subtraction TA, 200 twists, mL=4.8 -0.02 m, O( α 1 ) coefficient -0.04 -0.06 -0.08 mL = 4.8 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 1/mL 11 / 18

Proposed studies

Proposed studies ( g − 2) µ HLbL ∆ m π h + + Figure 4: Quark-connected electro-magnetic mass splitting diagrams. f π Figure 7: Light-by-light contribution to ( g − 2) µ u u u � + � + � + � + � + � + Carrasco et al. 2015 � � � � � � d d d (a) (b) (c) u u u � + � + � + � + � + � + � � � � � � d d d (d) (e) (f) q q q u u u � + � + � + � + � + � + � � � � � � d d d (a) (b) (c) q q 1 q 2 (a) (b) (c) u u � + � + � + � + � � � � d d (d) (e) (d) (e) (f) Figure 5: Quark-connected (top) and quark-disconnected (bottom) dia- grams for f π . Figure 6: Soft-photon emission in e ff ective field theory. 12 / 18

Main focus of this proposal 1. Volume-dependence of QCD + QED simulations using the TA method 2. Control stochastic noise introduced by twisting For 1) we propose a study on RBC’s 16c and 24c ensemble for a − 1 = 1 . 73 GeV and m π = 422 MeV (all parameters identical apart from volume). For 2) we propose the computation on the new RBC ensemble 17 (32c, DSDR, zMobius, a − 1 = 1 . 15 GeV, m π = 140 MeV) In the future we hope to complete this study by generating a partner ensemble for the 32c ensemble to study the volume-dependence at physical pion mass. 13 / 18

Strategy Two methods are explicitly spelled out in the proposal: ◮ P+A twist averaging in spatial directions which will be safe regarding 2) but may not achieve the goals in 1) ◮ Full stochastic twist averaging which has a higher probability to achieve the goals in 1) but may suffer from 2) The proposal main text explicitly works out a strategy using stochastic A2A propagators 14 / 18

SPC questions

1. Table 1 only appears to include the cost estimate for a single ensemble (the 32 3 DSDR with mpi 135 MeV and 1/a=1.1 GeV). What is the estimated cost for analyzing the other ensembles? Given that you plan to test multiple methods on the 16 3 ensemble, presumably this cost, although small, is not negligible. The cost for the $m_pi=420 MeV$, 24c ensemble is Lanczos 1.2 hours on 1024 BC1 cores (compared to 28.7 hours for the 32c) Exact solve 0.07 hours on 1024 BC1 cores (compared to 1.13 hours for the 32c) Sloppy solve 0.02 hours on 1024 BC1 cores (compared to 0.21 hours for the 32c) The cost for the 16c ensemble is estimated to be 16^3*32/(24^3*64) \approx 0.15 the cost of the 24c ensemble. Therefore even performing two complete runs (say for full stochastic versus PBC+APBC) on the 16c will only add 0.4 Mio Jpsi-core hours to the total budget. Even very conservatively estimating the cost of extensive experimentation on the 16c ensemble to be 1 Mio Jpsi-core hours, combined with the final-volume study of the best method on the 24c ensemble, will yield a total cost of the 16c and 24c studies that is only 8% of the total requested allocation. 15 / 18