Non-perturbative determination of improvement coefficients using - PowerPoint PPT Presentation

Non-perturbative determination of improvement coefficients using coordinate space methods in N f = 2 + 1 lattice QCD Piotr Korcyl work done in collaboration with Gunnar Bali for the RQCD collaboration based on arXiv: 1607.07090 34rd International Symposium on Lattice Field Theory, Southampton, UK Piotr Korcyl Non-perturbative determination of improvement coefficients 1/ 17

Introduction Improvement of LQCD with Wilson fermions Wilson fermions introduce cut-off effects linear in the lattice spacing a . One can account for them following Symanzik improvement programme � � = S continuum + aS 1 + a 2 S 2 + . . . S QCD a ( β ) Improvement of the action requires knowledge of the c SW coefficient and of b g and b m which are proportional to the quark mass. Apart of the action itself we want to use improved operators. We usually define µ ( x ) = ¯ A jk , I ψ j ( x ) γ µ γ 5 ψ k ( x ) + ac A ∂ sym P jk ( x ) µ and � � 1 + ab A m jk + a 3˜ A jk , R A jk , I ( x ) = Z A b A m µ ( x ) µ ⇒ full, non-perturbative improvement of Wilson fermions needs c SW , b g , c J , b J . Piotr Korcyl Non-perturbative determination of improvement coefficients 2/ 17

Introduction CLS ensembles The CLS initiative is currently generating ensembles with N f = 2 + 1 flavours of non-perturbatively improved Wilson Fermions and the tree-level L¨ uscher-Weisz gauge action at β = 3 . 4 , 3 . 46 , 3 . 55 and 3 . 7. This corresponds to lattice spacings of a ∈ [0 . 05 , 0 . 09] fm. Improvement coefficients c SW → Bulava, Schaefer, ’13 c A → Bulava, Della Morte, Heitger, Wittemeier ’15 b Γ , ˜ b Γ → this talk Mass dependent improvement coefficients We use coordinate space method proposed by Martinelli et al. (Phys. Lett. B 411, 141 (1997)) to determine b J , ˜ b J for flavour non-singlet scalar, pseudoscalar, vector and axialvector currents. Piotr Korcyl Non-perturbative determination of improvement coefficients 3/ 17

Notation We denote quark mass averages as m jk = 1 2( m j + m k ) , where � 1 � m j = 1 1 − . 2 a κ j κ crit The mass dependence of physical observables can be parameterized in terms of the average quark mass m = 1 3 ( m s + 2 m ℓ ) . We define connected Euclidean current-current correlation functions in a continuum renormalization scheme R , e.g., R = MS , at a scale µ : � � � � R � � ( jk ) (0) � T J ( jk ) ( x ) J G R J ( jk ) ( x , m ℓ , m s ; µ ) = Ω � Ω . Piotr Korcyl Non-perturbative determination of improvement coefficients 4/ 17

Method Two observations 1) The continuum correlation function differs from that of the massless case by mass dependent terms G R J ( jk ) ( x , m ℓ , m s ; µ ) = G R J ( jk ) ( x , 0 , 0; µ ) � � m 2 x 2 , m 2 � FF � x 6 , m � ψψ � x 4 , m � ψσ F ψ � x 6 �� × 1 + O , 2) The continuum Green function G R above can be related to the corresponding Green function G obtained in the lattice scheme at a lattice spacing a = a ( g 2 ) as follows: � � 2 (˜ G R Z R g 2 , a µ ) J ( jk ) ( x , m ℓ , m s ; µ ) = J � � 1 + 2 b J am jk + 6¯ G J ( jk ) , I ( n , am jk , am ; g 2 ) , × b J am Piotr Korcyl Non-perturbative determination of improvement coefficients 5/ 17

Method � jk , am ( ρ ) ; g 2 � n , am ( ρ ) � � G J ( jk ) − m ( ρ ) m ( σ ) � � = 1 + 2 b J a rs jk n , am ( σ ) rs , am ( σ ) ; g 2 G J ( rs ) � m ( σ ) − m ( ρ ) � � a 2 , x 2 � + 6˜ b J a + O , We can construct two useful observables: � 12 , am ( ρ ) ; g 2 � n , am ( ρ ) G J (12) R J ( x , δ m ) ≡ � � = 1 + 2 b J a δ m n , am ( ρ ) 13 , am ( ρ ) ; g 2 G J (13) � n , am ( ρ ) , am ( ρ ) ; g 2 � G J (12) � = 1 + (2 b J + 6˜ � � R J ( x , δ m ) ≡ b J ) a δ m n , am ( σ ) , am ( σ ) ; g 2 G J (12) ⇒ Let’s see how this works in practice! Piotr Korcyl Non-perturbative determination of improvement coefficients 6/ 17

Overview of CLS ensembles at β = 3 . 4 0.30 m s = const. physical point 0.25 m l = m s TrM = const. 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 Piotr Korcyl Non-perturbative determination of improvement coefficients 7/ 17

List of ensembles β name κ l κ s # conf. step 3.4 H101 0.136759 0.136759 100 10 3.4 H102 0.136865 0.136549339 100 10 3.4 H105 0.136970 0.13634079 103 5 3.4 H106 0.137016 0.136148704 57 5 3.4 H107 0.136946 0.136203165 49 5 3.4 C101 0.137030 0.136222041 59 10 3.4 C102 0.137051 0.136129063 48 10 3.4 rqcd17 0.1368650 0.1368650 150 10 3.4 rqcd19 0.13660 0.13660 50 10 3.46 S400 0.136984 0.136702387 83 10 3.55 N203 0.137080 0.136840284 74 10 3.7 J303 0.137123 0.1367546608 38 10 Piotr Korcyl Non-perturbative determination of improvement coefficients 8/ 17

Example: R P − 1 for n = (0 , 1 , 1 , 1) 0 . 05 H102 0 . 04 H105 H106 0 . 03 R P − 1 H107 C101 0 . 02 0 . 01 0 . 00 0 . 000 0 . 005 0 . 010 0 . 015 0 . 020 0 . 025 1 /κ s − 1 /κ l Piotr Korcyl Non-perturbative determination of improvement coefficients 9/ 17

Example: � R P − 1 for n = (0 , 1 , 2 , 2) 0 . 15 rqcd17/rqcd21 0 . 10 rqcd17/H101 rqcd21/H101 0 . 05 R P − 1 � 0 . 00 − 0 . 05 − 0 . 10 0.0 0.002 0.004 0.006 0.008 1 /κ ( σ ) − 1 /κ ( ρ ) Piotr Korcyl Non-perturbative determination of improvement coefficients 10/ 17

Tree-level cut-off effects 2 . 5 S 2 . 0 P V 1 . 5 A b tree J 1 . 0 0 . 5 0 . 0 0 1 2 3 4 5 6 x/a Piotr Korcyl Non-perturbative determination of improvement coefficients 11/ 17

Corrections at medium distances Applying OPE to the ratio of correlation functions one gets G J (12) ( x ) G J (34) ( x ) = 1 + ( A J 12 − A J 34 ) x 2 �� � � 2 − A J x 4 + · · · , A J 12 A J 34 + B J 12 − B J + 34 34 with the mass dependent coefficients � � jk = − 1 k + m j m k A J m 2 j + m 2 , 4 s J m 2 j m 2 jk = π 2 + π 2 2 + s J k B J 32 N � FF � + ( m j + m k ) � ψψ � . 16 8 N s J We correct our observables R J and � R J by subtracting the leading continuum corrections. Piotr Korcyl Non-perturbative determination of improvement coefficients 12/ 17

Improved observables � � ( x , δ m ) + π 2 � � 2 + s J R J ( x , δ m ) − R tree M 2 π − M 2 F 2 0 x 4 B J ( x , δ m ) ≡ J K 8 N s J � 1 � − 1 − 1 = b J + O ( x 6 ) + O ( g 2 a 2 ) + · · · , × κ s κ ℓ � � ( x , δ m ) + π 2 � 2 � 2 + s J � R J ( x , δ m ) − � � R tree F 2 0 x 4 B J ( x , δ m ) ≡ δ M π J 8 N s J � 1 � − 1 1 = b J + 3˜ b J + O ( x 6 ) + O ( g 2 a 2 ) + · · · . × κ ( σ ) − κ ( ρ ) Piotr Korcyl Non-perturbative determination of improvement coefficients 13/ 17

Example of numerical data 4 3 2 B S ( x ) 1 H102 0 S400 N203 − 1 0.0 0.1 0.2 0.3 0.4 x [fm] Piotr Korcyl Non-perturbative determination of improvement coefficients 14/ 17

Results in the scalar channel 3 . 5 β = 3 . 46 β = 3 . 55 β = 3 . 7 β = 3 . 4 3 . 0 2 . 5 b S 2 . 0 1 . 5 1 . 0 H102 H105 H106 H107 C101 C102 S400 N203 J303 Piotr Korcyl Non-perturbative determination of improvement coefficients 15/ 17

Final results: rational parametrizations 3 . 0 2 . 5 2 . 0 b S 1 . 5 1 . 0 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 g 2 Final results b S ( g 2 ) = 1 + 0 . 11444(1) C F (1 − 0 . 439(50) g 2 )(1 − 0 . 535(14) g 2 ) − 1 b P ( g 2 ) = 1 + 0 . 0890(1) C F (1 − 0 . 354(54) g 2 )(1 − 0 . 540(11) g 2 ) − 1 b V ( g 2 ) = 1 + 0 . 0886(1) C F (1 + 0 . 596(111) g 2 ) b A ( g 2 ) = 1 + 0 . 0881(1) C F (1 − 0 . 523(33) g 2 )(1 − 0 . 554(10) g 2 ) − 1 Piotr Korcyl Non-perturbative determination of improvement coefficients 16/ 17

Conclusions ˜ b J improvement coefficients at β = 3 . 4 ˜ b S = 2 . 0 (1 . 3) stat (0 . 3) sys ˜ b P = − 3 . 4 (1 . 3) stat (0 . 6) sys ˜ b V = − 0 . 1 (0 . 4) stat (0 . 1) sys ˜ b A = 1 . 4 (0 . 4) stat (0 . 9) sys Conclusions We implemented a coordinate space method to determine improvement coefficients proportional to quark mass With a negligible numerical effort we can achieve a 5% - 10% precision on b J Improvement coefficients proportional to the trace of the mass matrix are accessible, but need a better statistical precision Piotr Korcyl Non-perturbative determination of improvement coefficients 17/ 17