Logarithmic corrections Yang Mills theory Nikolai Husung with Peter Marquard, Rainer Sommer Wuhan, China, June 17th, 2019 to a 2 scaling in lattice
Motivation Figure: Deviation of the step scaling function from its continuum counterpart, O g g − g const. a O(3) model: “Worst case” example [Balog et al., 2009, 2010] as an example, in the 2-dimensional O(3) model [Balog et al., 2009, 2010]. Page 2 0 . 03 0 . 02 δ Σ( a ) = Σ( a ) − Σ(0) 0 . 01 0 0 0 . 05 0 . 1 0 . 15 0 . 2 a / L | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Motivation Figure: Deviation of the step scaling function from its continuum counterpart, O(3) model: “Worst case” example [Balog et al., 2009, 2010] as an example, in the 2-dimensional O(3) model [Balog et al., 2009, 2010]. 2 parameter fjt Page 2 8 6 L 2 Σ( a ) − Σ(0) 4 a 2 2 0 1 . 6 1 . 8 2 1/ g 2 0 0 ) − 2 + O (( g 2 δ Σ = const. a 2 � ( g 2 0 ) − 3 − 1 . 1386( g 2 0 ) − 1 ) � | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Symanzik efgective theory I (2) (4) i (3) i Idea: Parametrise lattice artifacts originating from the lattice action where a is the lattice spacing and Page 3 Symanzik efgective theory [Symanzik, 1980, 1981, 1983a,b] (1) [and for a fjeld Φ ] by a minimal basis of operators living in a continuous L + a 2 δ L + O ( a 4 ) , L eff = Φ + a 2 δ Φ + O ( a 4 ) , Φ eff = � δ L = b i O i , � δ Φ = c i Φ i , with free coeffjcients b i and c i . | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Symanzik efgective theory II the lattice pure gauge action > Local SU( N ) gauge symmetry, > discrete rotation and translation invariance (at least in infjnite volume) Require minimal basis for physical energies and matrix elements (“on-shell”) Georgi, 1991]. Page 4 Occurring operators O i and Φ i must comply with symmetries, i.e. for > C -, P - and T -symmetry, ⇒ broken O(4) symmetry. ⇒ use EOMs to reduce set of operators [Lüscher and Weisz, 1985; | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Symanzik efgective theory III EOM i (5) Page 5 The only operators O i left are [Lüscher and Weisz, 1985] O 0 = 1 O 1 = 1 � tr ( D µ F νρ D µ F νρ ) , tr ( D µ F µρ D µ F µρ ) , g 2 g 2 0 0 µ O 2 = 1 tr ( D µ F µν D ρ F ρν ) = 0 . g 2 0 We then fjnd for a fjeld Φ in analogy to [Balog et al., 2009, 2010] 1 � �� � � 1 + a 2 c i δ Φ � b j δ O + O ( a 4 ) � Φ � latt ( a ) = � Φ � cont i ( a ) − j ( a ) . (6) j =0 | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Symanzik efgective theory IV i Leading lattice artifacts Need to understand (leading) lattice spacing dependence of b j . Page 6 � � 1 � Φ � latt ( a ) ¯ � Φ � cont = 1 + a 2 c i δ Φ b j δ O × [1 + O ( α (1/ a ))] + O ( a 4 ) � � ¯ i ( a ) − j ( a ) j =0 c i and ¯ with tree-level coeffjcients ¯ � � � Φ O j ( x ) � cont ; R � δ O d 4 x j ( a ) = − �O j ( x ) � cont ; R , � Φ � cont ; R µ =1/ a � � Φ i � cont ; R � δ Φ i ( a ) = . � � Φ � cont ; R � � µ =1/ a ⇒ Renormalisation Group | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Symanzik efgective theory V Leading lattice artifacts artifacts from observables. 1-loop might be needed to obtain leading logarithms. require 1-loop coeffjcients. However, if tree-level coeffjcient is zero expansion in the lattice spacing. Remarks: Page 7 i � � 1 � Φ � latt ( a ) ¯ � Φ � cont = 1 + a 2 c i δ Φ b j δ O × [1 + O ( α (1/ a ))] + O ( a 4 ) � ¯ � i ( a ) − j ( a ) j =0 c i and ¯ > Tree-level coeffjcients ¯ b j can be obtained from classical > We limit ourselves to the leading behaviour as a ց 0 , i.e. we do not > We consider only the case c i = 0 to all orders, i.e. no additional | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Renormalisation Group I Use Renormalisation Group Equations (RGEs) to determine Page 8 (8) (7) renormalisation scheme independent renormalisation scale dependence µ 2 d δ O i ( µ ) β ( α ) = µ 2 d α ( µ ) = γ ij δ O = − α 2 � j ( µ ) , β n α n , d µ 2 d µ 2 n ≥ 0 where γ is the anomalous dimension matrix γ ij = µ 2 d ln ( Z ) ij = ( γ 0 ) ij α + O ( α 2 ) , O i ; R = Z ij O j ;0 . d µ 2 | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
i D i Renormalisation Group II (9) i . the RGI with leading power determined by Note: The renormalisation scale dependence is only in the prefactor of i O (10) ij d x Page 9 the Renormalisation Group Invariant (RGI) We choose a basis such that γ 0 = diag { ( γ 0 ) 1 , . . . , ( γ 0 ) n } and introduce γ i = ( γ 0 ) i γ i δ O D O µ →∞ [2 β 0 α ( µ )] ˆ i (Λ) = lim i ( µ ) , ˆ , β 0 with RGI scale Λ . This allows us to rewrite α ( µ ) � � γ ( x ) � β ( x ) + γ 0 δ O − ˆ D O γ i Pexp i ( µ ) =(2 β 0 α ( µ )) j (Λ) β 0 x 0 | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Renormalisation Group II (9) Note: The renormalisation scale dependence is only in the prefactor of (10) ij d x Page 9 the Renormalisation Group Invariant (RGI) We choose a basis such that γ 0 = diag { ( γ 0 ) 1 , . . . , ( γ 0 ) n } and introduce γ i = ( γ 0 ) i γ i δ O D O µ →∞ [2 β 0 α ( µ )] ˆ i (Λ) = lim i ( µ ) , ˆ , β 0 with RGI scale Λ . This allows us to rewrite α ( µ ) � � γ ( x ) � β ( x ) + γ 0 δ O − ˆ D O γ i Pexp i ( µ ) =(2 β 0 α ( µ )) j (Λ) β 0 x 0 � γ i � =(2 β 0 α ( µ )) − ˆ γ i D O [ α ( µ )] 1 − ˆ i (Λ) + O . the RGI with leading power determined by ˆ γ i . | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Computing leading anomalous dimensions I Renormalise operator basis at 1-loop by computing connected Green’s Chetyrkin et al., 1998]. contributions following e.g. the procedure from [Misiak and Münz, 1995; To obtain the anomalous dimension we extract only the UV-pole q q Tools: QGRAF, FORM and momenta p k , q . A (gauge fjxed), con A Z ij functions with operator insertion in continuum theory Page 10 � � ˜ A 1; R ( p 1 ) . . . ˜ A n ; R ( p n ) ˜ O i ; R ( q ) con = p 1 p 2 � � ˜ A 1;0 ( p 1 ) . . . ˜ A n ;0 ( p n ) ˜ = Z n O j ;0 ( q ) with fundamental gauge fjelds ˜ p 1 p 3 p 2 | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Computing leading anomalous dimensions I Renormalise operator basis at 1-loop by computing connected Green’s Chetyrkin et al., 1998]. contributions following e.g. the procedure from [Misiak and Münz, 1995; To obtain the anomalous dimension we extract only the UV-pole q q Tools: QGRAF, FORM and momenta p k , q . A (gauge fjxed), con A Z ij functions with operator insertion in continuum theory Page 10 � � ˜ A 1; R ( p 1 ) . . . ˜ A n ; R ( p n ) ˜ O i ; R ( q ) con = p 1 p 2 � � ˜ A 1;0 ( p 1 ) . . . ˜ A n ;0 ( p n ) ˜ = Z n O j ;0 ( q ) with fundamental gauge fjelds ˜ p 1 p 3 p 2 | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Obtain relevant part of mixing matrix via > Can keep p k arbitrary and Common approach with class of “redundant” operators Q . (11) Q Z QQ Q Z Z R Q on-shell). operators contribute (unless , but EOM-vanishing q Computing leading anomalous dimensions II relevant for renormalisation. > Only gauge-invariant operators > “on-shell” momenta avoid gauge-variant contributions from gauge-fjxing. operators contribute, but 2- and 3-point functions are accessible for renormalisation. Background fjeld method [’t Hooft, 1975; Abbott, 1981, 1982; Lüscher and Weisz, 1995] Page 11 > If q � = 0 total divergence | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
Obtain relevant part of mixing matrix via Computing leading anomalous dimensions II > Only gauge-invariant operators with class of “redundant” operators Q . (11) Q Z QQ Q Z Z R Q on-shell). operators contribute (unless Common approach relevant for renormalisation. 1982; Lüscher and Weisz, 1995] 1981, > “on-shell” momenta avoid gauge-variant contributions from gauge-fjxing. operators contribute, but 2- and 3-point functions are accessible for renormalisation. Background fjeld method [’t Hooft, 1975; Abbott, Page 11 > If q � = 0 total divergence > Can keep p k arbitrary and q = 0 , but EOM-vanishing | Logarithmic corrections to a 2 scaling in lattice Yang Mills theory | Nikolai Husung | Wuhan, China, June 17th, 2019 DESYª
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