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Towards the continuum limit with improved Wilson Fermions employing open boundary conditions Part 1 Wolfgang Sldner for RQCD Regensburg University Lattice 2016 The 34rd International Symposium on Lattice Field Theory July 29th, 2016

  1. Towards the continuum limit with improved Wilson Fermions employing open boundary conditions –Part 1– Wolfgang Söldner for RQCD Regensburg University Lattice 2016 The 34rd International Symposium on Lattice Field Theory July 29th, 2016 QCD QCD W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 1 / 36

  2. Introduction Motivation Lattice QCD today more computing power and better algorithms → statistically more precise results increasingly important to control systematics ⇒ obviously, very important: controlled continuum limit Problem when lattice spacing a → 0 ⇒ freezing of topology lattice simulations get stuck in topological sectors problems start at a � 0 . 05 fm ⇒ simple solution: lattice simulations with open boundary conditions [Lüscher and Schaefer 2011] QCD → topology can flow in and out through the boundary W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 2 / 36

  3. Introduction Lattice QCD with Open Boundaries Open Boundaries 0.80 0.75 F 0 k ( x ) | x 0 = 0 = F 0 k ( x ) | x 0 = T = 0 plaquette action density 0.70 P + ψ ( x ) | x 0 = 0 = P − ψ ( x ) | x 0 = T = 0 , 0.65 � � ¯ � x 0 = 0 = ¯ � ψ ( x ) P − ψ ( x ) P + x 0 = T = 0 0.60 0.55 P ± = 1 2 ( 1 ± γ 0 ) 0.50 0 20 40 60 80 100 120 t Major N f = 2 + 1 CLS effort CLS: HU Berlin, CERN, TC Dublin, Mainz, UA Madrid, Milano Bicocca, Münster, Odense/CP3-Origins, Regensburg, Roma I, Roma II, Wuppertal, DESY Zeuthen QCD W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 3 / 36

  4. Simulation Details Simulation Overview Lattice Action Two degenerate light quarks and one strange quark Non-perturbatively improved Wilson action (clover) Tree-level improved Symanzik gauge action ∃ three different quark mass plane trajectories (1) m = m symm κ ℓ + 1 2 3 m = 2 m ( ℓ ) ight + m ( s ) = const. ↔ κ s = const. → renormalized 2 � m ℓ + � m s = const. + O ( a ) . (2) � m s = � m s , ph Strange AWI mass � m s = const. → renormalized � m s = � m s , ph , up to tiny O ( a ) effects. (3) m s = m ℓ (Mainz/Regensburg) For joint non-perturbative renormalization program QCD → simulations with anti-periodic boundary conditions (for a > 0 . 05 fm) W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 4 / 36

  5. Simulation Details Overview of the simulation strategy → 1606.09039] 0.30 f m s = f m s, ph physical point 0.25 m l = m s m = m symm π ) ∼ m s 0.20 K − m 2 0.15 t 0 (2 m 2 0.10 0.05 0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 t 0 m 2 π ∼ m l 1 generate the m = m symm trajectory, starting from the m s = m ℓ point where m ≈ m ph . 2 add points along the symmetric trajectory ( m ℓ = m s ). 3 fit AWI masses (with O ( a ) -improvement) to a known parametrization, using both trajectories. 4 determine the “physical” point on the m = m symm line, imposing � m s / � m ℓ = 27 . 46 ( 44 ) [FLAG 2] → � m s , ph QCD 5 predict κ ℓ , κ s pairs for which � m s = � m s , ph from the parametrization in order to add � m s = � m s , ph simulation points. W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 5 / 36

  6. Simulation Details How to predict κ s as a function of κ ℓ for � m s fixed I m jk = ∂ 4 � 0 | A jk 4 | π jk � m j + � � m k = � Average AWI masses: 2 2 � 0 | P jk | π jk � � 1 � m j = 1 1 − Lattice quark masses: 2 a κ j κ crit The Point along the symmetric line ( m 1 = m 2 = m ℓ = m s = m 3 ) where � m jk = 0 defines κ j = κ crit . Problem: Different renormalization of flavour-singlet and non-singlet quark mass combinations: � 1 � �� � Z m ( m s − m ℓ ) = 1 − 1 m ℓ = Z A = � m s − � m 13 − � 2 m 12 2 a κ s κ ℓ Z P � 2 � 2 m ℓ + m s = 1 + 1 3 = Z A � but: Z m r m m = Z m r m − m κ ℓ κ s κ crit 3 6 a Z P QCD NB: Due to r m > 1 m ℓ < m s can become negative, away from the symmetric line. W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 6 / 36

  7. Simulation Details How to predict κ s as a function of κ ℓ for � m s fixed II � � 1 � � 1 �� �� � m = Z − 1 + 2 3 3 � m s − � + 3 � m s = 2 m ℓ 2 + r m − , 2 a κ s κ ℓ κ s κ ℓ κ crit where Z = Z m Z P / Z A . Setting � m s = � m s , ph gives � 3 a � 1 2 m s , ph + ( 1 − r m ) 1 + 3 r m 1 Z � = κ s 2 + r m κ ℓ κ crit 2 Subtracting the physical point result from both sides of the equation gives: � 1 � 1 1 + 2 ( 1 − r m ) 1 = − , κ s κ s , ph 2 + r m κ ℓ κ ℓ, ph while the target κ ℓ that corresponds to a given � m ℓ value can be obtained through 1 1 + 2 a ( 2 + r m ) ( � m ℓ − � = m ℓ, ph ) QCD κ ℓ κ ℓ, ph 3 Zr m W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 7 / 36

  8. Simulation Details How to predict κ s as a function of κ ℓ for � m s fixed III Z and κ ℓ, ph can be obtained from � m 13 − � m 12 as a function of κ ℓ along the m = const. line. Then κ s , ph is automatically determined too. Zr m (and κ crit if needed) can be obtained from � m as a function of 1 /κ along the symmetric m = m = m s = m ℓ line. We carry out full order a improvement. In this case four combinations of improvement coefficients ( A , B 0 , C 0 and D 0 ) appear. Does the κ s prediction strategy work? To be addressed later: Scale setting/tuning; we assumed that the physical point is on the m = m symm trajectory that we simulate (at least up to O ( a 2 ) corrections). But is this true? QCD W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 8 / 36

  9. Simulation Details m s = � m s , ph : prediction vs. simulation Predicted and simulated value of physical � m s , ph [hep-lat 1606.09039] β = 3 . 4, � m s = � m s, ph 1 . 02 β = 3 . 55 m s, ph 1 . 01 m s / � � 1 . 00 0 . 99 1 2 3 4 5 6 7 8 m ℓ / � � m ℓ, ph QCD Mismatch at β = 3 . 4 due to shift of c A value but still very constant! W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 9 / 36

  10. Simulation Details N f = 2 + 1 CLS simulations Great visibility at Lattice 2016 plenaries: Coordinated Lattice Simulations : HU Berlin, CERN, TC Dublin, Mainz, UA Madrid, Milano Bicocca, Münster, Odense/CP3-Origins, Regensburg, Roma I, Roma II, Wuppertal, DESY Zeuthen QCD W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 10 / 36

  11. Simulation Details CLS ensemble overview → JHEP 1502 (2015) 043 [hep-lat 1411.3982] m = m symm m s = � � m s , ph 450 450 U103 N300 B450 J500 H200 H101 400 N202 400 H107 U102 S400 350 350 N204 N203 H102 J501 N302 m π [MeV] m π [MeV] 300 300 N401 S201 N201 U101 H106 N200 H105 J303 N101 250 250 S100 C102 C101 200 D200 D101 200 D201 150 D150 150 physical physical D100 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 a 2 [fm 2 ] a 2 [fm 2 ] U: 128 × 24 3 S: 128 × 32 3 D: 128 × 64 3 B: 64 × 32 3 C: 96 × 48 3 J: 192 × 64 3 H: 96 × 32 3 N: 128 × 48 3 QCD W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 11 / 36

  12. m = m symm line Tuning details and results for the m = m symm trajectory QCD W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 12 / 36

  13. m = m symm line Tuning strategy: m = m symm 1 . 20 β = 3 . 4 1 . 18 β = 3 . 46 β = 3 . 55 1 . 16 β = 3 . 7 β = 3 . 85 1 . 14 φ 4 1 . 12 1 . 10 1 . 08 1 . 06 1 . 04 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 φ 2 φ 2 = t 0 m 2 φ 4 = 8 t 0 ( m 2 K + m 2 π / 2 ) ∼ m π ∼ m l , At fixed β match lattices with different lattice spacings at flavor symmetric point (i.e. m ud = m s → m π = m K ≈ 415 MeV ) The (small) slope of φ 4 as a function of φ 2 was determined at β = 3 . 4 from a set of preliminary runs: � � φ 4 mud = ms = 1 . 15 QCD � Physical target (yellow bands): t 0 = 0 . 1465 ( 21 )( 13 ) fm [BMW], m π = 134 . 8 ( 3 ) MeV, m K = 494 . 2 ( 4 ) MeV [FLAG 2] W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 13 / 36

  14. m = m symm line Chiral extrapolation: m = m symm → hep-lat 1606.09039 1 . 6 1 . 6 1 . 4 1 . 4 1 . 2 1 . 2 1 . 0 1 . 0 φ 4 0 . 8 φ 4 0 . 8 m = m symm m = m symm 0 . 6 0 . 6 m s = � � m s, ph m s = � � m s, ph 0 . 4 0 . 4 m s = m ℓ m s = m ℓ 0 . 2 0 . 2 phys. point phys. point 0 . 0 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 φ 2 φ 2 β = 3 . 4 a ≈ 0 . 085 fm β = 3 . 55 a ≈ 0 . 064 fm Combination shown is constant to NLO χ PT along m = const. Corrections are of higher order or O ( a ) . Dependence on φ 2 becomes weaker towards smaller a → mostly lattice artefact? At the physical point we are still within the target range! QCD W. Söldner, G. Bali (Regensburg) RQCD results on CLS open BC ensembles Lattice 2016 14 / 36

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