Schwinger-Dyson equations and Ward identities in NC QFT on the - PowerPoint PPT Presentation

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät ▼ü♥st❡r Schwinger-Dyson equations and Ward identities in NC QFT on the example of scalar models ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ ❲❲❯▼ü♥st❡r Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 2 /18 ▼ü♥st❡r Noncommutative space (Moyal) with a noncommutative product ( f ⋆ g )( x ) � = ( g ⋆ f )( x ) for f , g ∈ S ( R 2 ) The vector space of Schwarz functions has a matrix basis → "dynamical" matrix model Schwinger-Dyson equations + Ward-Takahashi identities → closed integral equation (for φ 3 D and φ 4 D in the large V , N limit) ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ ❲❲❯▼ü♥st❡r φ 3 was solved for D = 2 , 4 , 6 by Grosse, Sako, Wulkenhaar (’17,’18) φ 4 was solved for D = 2 by Panzer, Wulkenhaar (two weeks ago) φ 3 and φ 4 are in the same class of models Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 3 /18 ▼ü♥st❡r Example φ 3 2 The action is   N N N H nm φ nn + λ � � �  , S [ φ ] = V 2 φ nm φ mn + κ φ nm φ ml φ ln )  3 n , m =0 n =0 n , m , l =0 V + µ 2 where φ Hermitian matrix, H nm = E n + E m , E m = m 2 energy distribution, V deformation parameter of the Moyal plane and κ ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ renormalization parameter. ❲❲❯▼ü♥st❡r � D φ exp � � , φ nm ↔ 1 ∂ Z [ J ] := − S [ φ ] + V � n , m J nm φ mn ∂ J mn V Z [ J ] Z [0] ≈ 1 + V � n G | n | J nn � � � � V 2 V 2 G | n | G | m | + 1 + � 2 G | nm | J nm J mn + 2 G | n | m | J nn J mm + .. n , m Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 4 /18 ▼ü♥st❡r Schwinger-Dyson equation + Ward-Takahashi identity N � � G p = 1 − κ − λ G pm − λ � V 2 G p | p − λ G 2 p H pp V m =0 & � � 1 + λ G p 1 − G p 2 1 G p 1 p 2 = E p 1 − E p 2 H p 1 p 2 ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ Ward identity: Z should be invariant under φ → U φ U † , U ∈ U ( N ) ❲❲❯▼ü♥st❡r ∂ 2 � � � N � N ∂ ∂ V ∂ J nk ∂ J km Z [ J ] = ∂ J km − J mk Z [ J ] J kn k =0 k =0 ( E n − E m ) ∂ J nk Limit V , N → ∞ with N V = Λ 2 � Λ 2 lim 1 � N m =0 f ( m f ( x ) with x = m V ) = 0 V V → closed integral equations Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 5 /18 ▼ü♥st❡r 3-colour scalar model in 2D 3 scalar fields φ a for a ∈ { 1 , 2 , 3 } with the same energy distribution, where φ a is a Hermitian matrix. The action is definite 1   3 N 3 N H nm mn + λ � � � � 2 φ a nm φ a σ abc φ a nm φ b ml φ c S [ φ ] = V  ,  ln 3 a =1 n , m =0 a , b , c =1 n , m , l =0 ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ where H nm = E n + E m , E n = µ 2 2 + n ❲❲❯▼ü♥st❡r V , σ abc = | ε abc | and V as deformation parameter of the Moyal space. � �� 3 a =1 D φ a � � � − S [ φ ] + V � 3 � N n , m =0 J a nm φ a Z [ J ] := exp a =1 mn nm ↔ 1 ∂ φ a ∂ J a V mn 1 arXiv:1804.06075 with R. Wulkenhaar Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 6 /18 ▼ü♥st❡r Correlation functions are expansion coefficients of log Z [ J ] Z [0] . The first terms are 3 N Z [ J ] � V mn + 1 � 2 G a | a � � 2 G aa | nm | J a nm J a | n | m | J a nn J a Z [0] =: 1 + mm a =1 n , m =0 3 N � ln + 1 V 2 G a | bc � � 3 G abc | nml | J a nm J b ml J c | n | ml | J a nn J b ml J c + σ abc lm ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ a , b , c =1 n , m , l =0 ❲❲❯▼ü♥st❡r � + 1 6 V G a | b | c | n | m | l | J a nn J b mm J c + ... ll Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 7 /18 ▼ü♥st❡r Example of a Schwinger-Dyson equation 3 1 λ G aa � p 1 p 2 = + σ abc 1 + p 1 + p 2 H p 1 p 2 b , c =1 � 1 N p 1 p 2 n + 1 p 1 | p 1 p 2 + 1 p 2 | p 1 p 2 + δ p 1 p 2 � V 2 G a | bc V 2 G a | bc V 3 G a | b | c � G abc × . p 1 | p 1 | p 2 V ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ n =0 ❲❲❯▼ü♥st❡r One gets a tower of D-S equations. Try to decouple the tower in the large N , V limit with Ward-Takahashi identities Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 8 /18 ▼ü♥st❡r Ward-Takahashi identities � � � 3 � N mn + λ � 3 � N H nm 2 φ a nm φ a n , m , l =0 σ abc φ a nm φ b ml φ c S [ φ ] = V a =1 n , m =0 a , b , c =1 3 ln The identity is obtained by the invariance of Z [ J ] under U ( N ) transformation ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ ❲❲❯▼ü♥st❡r Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 8 /18 ▼ü♥st❡r Ward-Takahashi identities � � � 3 � N mn + λ � 3 � N H nm 2 φ a nm φ a n , m , l =0 σ abc φ a nm φ b ml φ c S [ φ ] = V a =1 n , m =0 a , b , c =1 3 ln The identity is obtained by the invariance of Z [ J ] under U ( N ) transformation ( φ → U φ U † ). For the φ 3 / 4 model ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ D ❲❲❯▼ü♥st❡r N � �� ∂ 2 V � ∂ ∂ � − − J mk Z [ J ] = 0 . J kn ( E n − E m ) ∂ J nk ∂ J km ∂ J km ∂ J nk k =0 Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 9 /18 ▼ü♥st❡r Ward-Takahashi identities � � � 3 � N mn + λ � 3 � N H nm 2 φ a nm φ a n , m , l =0 σ abc φ a nm φ b ml φ c S [ φ ] = V a =1 n , m =0 a , b , c =1 ln 3 The identity is obtained by the invariance of Z [ J ] under U ( N ) transformation ( φ a → U φ a U † , ∀ a ). Simultaneous transformation of all 3 fields ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ ❲❲❯▼ü♥st❡r 3 N � ∂ 2 � �� V ∂ ∂ � � J a − J a − Z [ J ] = 0 . kn mk ∂ J a nk ∂ J a ∂ J a ∂ J a ( E n − E m ) km km nk a =1 k =0 Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 10 /18 ▼ü♥st❡r Ward-Takahashi identities � � � 3 � N mn + λ � 3 � N H nm 2 φ a nm φ a n , m , l =0 σ abc φ a nm φ b ml φ c S [ φ ] = V a =1 n , m =0 a , b , c =1 3 ln The identity is obtained by the invariance of Z [ J ] under U ( N ) transformation ( φ a → U φ a U † , fix a ). Transformation of one field ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ N N ∂ 2 � � V ∂ ∂ � � J a − J a + ❲❲❯▼ü♥st❡r mk kn ∂ J a nk ∂ J a ∂ J a ∂ J a E n − E m km nk km k =0 k =0 N 3 � � ∂ 3 ∂ 3 λ � � = σ acd − σ acd V ( E n − E m ) ∂ J a nk ∂ J c kl ∂ J d ∂ J c nk ∂ J d kl ∂ J a k , l =0 c , d =1 lm lm Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 11 /18 ▼ü♥st❡r Ward-Takahashi identities � � � 3 � N mn + λ � 3 � N H nm 2 φ a nm φ a n , m , l =0 σ abc φ a nm φ b ml φ c S [ φ ] = V a =1 n , m =0 a , b , c =1 3 ln The identity is obtained by the invariance of Z [ J ] under U ( N ) transformation ( φ a → φ ′ a ( φ a , φ b ) , φ b → φ ′ b ( φ b , φ a )). Mixed Transformation ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ N N ∂ 2 � � V ∂ ∂ � � J b − J a + ❲❲❯▼ü♥st❡r mk kn ∂ J a nk ∂ J b ∂ J a ∂ J b E n − E m km nk km k =0 k =0 N 3 � � ∂ 3 ∂ 3 λ � � = σ bcd − σ acd V ( E n − E m ) ∂ J a nk ∂ J c kl ∂ J d ∂ J c nk ∂ J d kl ∂ J b k , l =0 c , d =1 lm lm Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 12 /18 ▼ü♥st❡r ◮ Using the Ward identities twice ◮ Performing the large V , N limit ◮ Closed integral equations are derived for this special model ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ ❲❲❯▼ü♥st❡r Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 13 /18 ▼ü♥st❡r Nonlinear closed integral equation of the 2-point func- tion for the 3-colour model λ 2 1 G aa p 1 p 2 = + 1 + p 1 + p 2 (1 + p 1 + p 2 ) ( p 1 − p 2 ) � Λ 2 � � � 3 G aa G aa qp 2 − G aa × dq p 1 p 2 p 1 q 0 � Λ 2 � Λ 2 dq G aa p 1 q − G aa dq G aa p 2 q − G aa ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ � p 1 p 2 p 1 p 2 − + ❲❲❯▼ü♥st❡r q − p 2 q − p 1 0 0 Can be used perturbatively ∞ G aa � λ n G aa p 1 p 2 = n , p 1 p 2 n =0 Alexander Hock

❲❡st❢ä❧✐s❝❤❡ ❲✐❧❤❡❧♠s✲❯♥✐✈❡rs✐tät 14 /18 ▼ü♥st❡r 3-Colour model, 2-point function perturbative result (Λ 2 = ∞ ) 2 λ 2 log( 1+ p 1 1+ p 2 ) 1 G aa p 1 p 2 = + (1 + p 1 + p 2 ) 2 ( p 1 − p 2 ) 1 + p 1 + p 2 3 log( 1+ p 1 1+ p 2 ) 2 2 λ 4 � + (1 + p 1 + p 2 ) 2 ( p 1 − p 2 ) (1 + p 1 + p 2 )( p 1 − p 2 ) ❧✐✈✐♥❣ ❦♥♦✇❧❡❞❣❡ ❲❲❯▼ü♥st❡r 2 log(1 + p 1 ) 2 log(1 + p 2 ) p 1 (1 + 2 p 1 )(1 + p 1 ) − + p 2 (1 + 2 p 2 )(1 + p 2 ) � π 2 � � π 2 � p 1 p 2 (1 + 2 p 2 ) 6 + 2 Li 2 ( 1+ p 1 ) (1 + 2 p 1 ) 6 + 2 Li 2 ( 1+ p 2 ) � − + (1 + 2 p 1 ) 2 (1 + p 1 + p 2 ) (1 + 2 p 2 ) 2 (1 + p 1 + p 2 ) Alexander Hock