Disconnected beyond 2+2 Thomas Blum Norman Christ Ma sashi Hayakawa - PowerPoint PPT Presentation

1/29 Disconnected beyond 2+2 Thomas Blum Norman Christ Ma sashi Hayakawa Taku Izubuchi UConn/RBRC Columbia Nagoya BNL/RBRC Luchang Jin Chulwoo Jung Christoph Lehner Cheng Tu UConn/RBRC BNL BNL UConn and the RBC/UKQCD collaborations Jun 18, 2018 Helmholtz-Institut Mainz Second Plenary Workshop of the Muon g-2 Theory Initiative

HLbL Connected di agrams 2/29 x op , ν y, σ x, ρ y, σ x, ρ y, σ x, ρ z, κ z, κ z, κ x op , ν x op , ν x src x snk x src x snk x src x snk y ′ , σ ′ x ′ , ρ ′ y ′ , σ ′ x ′ , ρ ′ y ′ , σ ′ x ′ , ρ ′ z ′ , κ ′ z ′ , κ ′ z ′ , κ ′ • Permutations of the three internal photons are not shown. • There should be gluons exchange between and within the quark loops, but are not drawn.

Disconnected diagrams 3/29 • One diagram (the biggest diagram below, referred to as 2 + 2 ) do not vanish even in the SU (3) limit. • We extend the method and computed this leading disconnected diagram as well. x op , ν x op , ν x op , ν z, κ y, σ x, ρ z, κ z, κ y, σ x, ρ y, σ x, ρ x src x snk z ′ , κ ′ x ′ , ρ ′ y ′ , σ ′ x src x snk x src x snk y ′ , σ ′ z ′ , κ ′ y ′ , σ ′ x ′ , ρ ′ x ′ , ρ ′ z ′ , κ ′ x op , ν x op , ν x op , ν z, κ y, σ x, ρ z, κ z, κ y, σ x, ρ y, σ x, ρ x src x snk x src x snk x src x snk z ′ , κ ′ x ′ , ρ ′ y ′ , σ ′ z ′ , κ ′ z ′ , κ ′ x ′ , ρ ′ y ′ , σ ′ x ′ , ρ ′ y ′ , σ ′ • Permutations of the three internal photons are not shown. • There should be gluons exchange between and within the quark loops, but are not drawn. • We need to make sure that the loops are connected by gluons by “vacuum” subtraction. So the diagrams are 1-particle irreducible.

Disconnected diagram beyond 2+2 4/29 x op , ν z, κ x op , ν y, σ x, ρ z, κ y, σ x, ρ x src x snk y ′ , σ ′ z ′ , κ ′ x ′ , ρ ′ x src x snk y ′ , σ ′ x ′ , ρ ′ z ′ , κ ′ • St il lw o r k in g in pr og r ess. • The right loop has been calculated by Christoph Lehner (can also be used to calculated disconnected HVP) and saved to disk. • The left loop can be evaluated by two point source propagators at x and y . We can then randomly sample x and y , similar to the way we evaluted the connected diagrams.

5/29 Pion Tr ansition Form Factor (TFF) on Lattice: RBC results Thomas Blum Norman Christ Masashi Hayakawa Taku Izubuchi UConn/RBRC Columbia Nagoya BNL/RBRC Luchang Jin Chulwoo Jung Christoph Lehner Cheng Tu UConn/RBRC BNL BNL UConn and the RBC/UKQCD collaborations Jun 18, 2018 Helmholtz-Institut Mainz Second Plenary Workshop of the Muon g-2 Theory Initiative

Outline 6/29 We will be working in Euclidean space by default. • Pion TF F f o r mu l a t ion • Model and Lattice results • Contribution to HLbL with pion TFF

Pion TFF formulation 7/29 � 0 | T i J µ ( u ) i J ν ( v ) | π 0 ( p � ) � (1) 2 , q 2 2 ) (X.D. Ji, C. Jung, [hep-lat/0101014]): Momentum space TFF F ( q 1 � i d 4 u e − iq 1 · u − iq 2 · v � 0 | T i J µ ( u ) i J ν ( v ) | π 0 ( p 2 , q 2 2 ) . � ) � = 4 π 2 F π ǫ µ,ν,ρ,σ q 1 ,ρ q 2 ,σ F ( q 1 (2) Coordinate space TFF F c ( x, z 2 ) (previously presented at UConn by Cheng Tu): � 0 | T i J µ ( u ) i J ν ( v ) | π 0 ( p � ) � i u ) ( − i ∂ σ v ) F ′ ( p · ( u − v ) , ( u − v ) 2 ) e ip · v , = ǫ µ,ν,ρ,σ ( − i ∂ ρ (3) 4 π 2 F π Let r = u − v , F c ( x, r 2 ) is the Fourier transformation of F ′ ( p · r, r 2 ) : � ∞ F ′ ( p · r, r 2 ) = dx F c ( x, r 2 ) e ixp · r . (4) −∞ Interestingly, we can prove that: F c ( x, r 2 ) = 0 if x < 0 or x > 1 . (5)

Pion TFF formulation 8/29 � 0 | T i J µ ( u ) i J ν ( v ) | π 0 ( p � ) � i u ) ( − i ∂ σ v ) = 4 π 2 F π ǫ µ,ν,ρ,σ ( − i ∂ ρ 1 � � � � � � dx F c ( x, ( u − v ) 2 ) π 0 ( xu + (1 − x ) v ) � � π 0 ( p × 0 � ) (6) � � � 0 i = 4 π 2 F π ǫ µ,ν,ρ,σ 1 � � � � � u F c ( x, ( u − v ) 2 )] ∂ σ π 0 ( xu + (1 − x ) v ) � � � π 0 ( p × 0 dx [ − ∂ ρ � ) (7) � � � 0 The coordinate space form factor F c ( x, r 2 ) can be interpreted this way: • The dependence on x describe the distribution of the pion source along the segment between the two EM currents. In the r 2 → 0 limit, the function can be factorized into pion d • ist r ibut ion a mpl it u des ) . At tree level, F c ( x, r 2 ) is the same as PDA after normalization. (PD A F c ( x, r 2 ) ∼ x (1 − x ) . • The parameter r = ( u − v ) is the separation between the two EM currents.

Pion TFF formulation: proof for 0 � x � 1 9/29 Define d 3 p � 1 ˆ (2 π ) 3 | π 0 ( p � π 0 ( p π 0 = P � ) � � ) | . (8) 2 E π 0 ,p � d 4 p e ip · ( x − y ) � ˆ � � � � π 0 ( x ) P π 0 π 0 ( y ) � 0 � 0 = G ( x − y ) = (9) p 2 + m π (2 π ) 4 2 ˆ � � π 0 π 0 ( w ) � � 0 � T [ i J µ ( u ) i J ν ( v )] P � 0 � � � ∞ � � i u F c ( x, ( u − v ) 2 )] ∂ σ π 0 ( xu + (1 − x ) v ) P ˆ � � π 0 π 0 ( w ) = 0 ǫ µ,ν,ρ,σ dx [ − ∂ ρ � 0 4 π 2 F π � � � −∞ ∞ � i u F c ( x, ( u − v ) 2 )] ∂ σ G ( x u + (1 − x ) v − w ) . = dx [ − ∂ ρ 4 π 2 F π ǫ µ,ν,ρ,σ (10) −∞ Let w = xu + (1 − x ) v , the above expression should not be singular when x > 1 or x < 0 . Therefore F c ( x, ( u − v ) 2 ) should be zero for x outside of [0 , 1] .

Pion TFF formulation 10/29 Let f ( | r | ) be the function which describes the strength of the π 0 γ γ coupling: 1 2 � � � 2 F π 1 r F c ( x, r 2 )] = 2 z ρ d x [ − ∂ ρ f ( | r | ) . (11) ( r 2 ) 2 3 0 ∞ (2 F π 2 /3) f ( r ) 2 r d r = 1 , ( F (0 , 0) = 1 ). Based on Chiral anomaly, ( π 2 /2) � • 0 Based on OPE, in the r → 0 limit, f ( | r | ) → 1 , ( F ( Q 2 , Q 2 ) → 8 π 2 F π 2 /(3 Q 2 ) ). • For HLbL, the long distance contribution should be dominated by the π 0 exchange process, where the π 0 propagator for a relatively long distance, while the two photons created/anni- hilated the pion are fairly close. Therefore, the x dependence of F c ( x, r 2 ) is less important. Instead, we should focus on the total strength f ( | r | ) .

Outline 11/29 We will be working in Euclidean space by default. • Pion TFF formulation • M od ela n d L a t t ic e r esul t s • Contribution to HLbL with pion TFF

Pion TFF formulation 12/29 Vector Meson Dominance model 2 2 m V m V 2 , q 2 2 ) = F VMD ( q 1 (12) 2 + m V 2 + m V 2 2 q 1 q 2 Two Ends model 2 /2 2 /2 m V 2 + m V 2 , q 2 2 ) = F TE ( q 1 (13) 2 + m V 2 + m V 2 q 1 q 2 Lowest Meson Dominance model 2 2 2 ) = 8 π 2 F π � 1 − 8 π 2 F π � 2 , q 2 2 , q 2 2 ) + 2 , q 2 2 ) F LMD ( q 1 2 F TE ( q 1 F VMD ( q 1 (14) 2 3 m V 3 m V Relation between Momentum space form and Coordinate space form: 1 � � 2 , q 2 2 ) = d 4 z e − iq 1 · r dx F c ( x, r 2 ) e ixp · r F ( q 1 0 1 � � d 4 r e − i ((1 − x ) q 1 − xq 2 ) · r F c ( x, r 2 ) = (15) dx 0

Pion TFF formulation: VMD model 13/29 2 2 m V m V 2 , q 2 2 ) = F VMD ( q 1 2 + m V 2 + m V 2 2 q 1 q 2 1 4 � m V = d x 2 + m V 2 ) + x ( q 2 2 + m V 2 )] 2 [(1 − x )( q 1 0 1 4 � m V = d x (16) 2 − x (1 − x ) m π [[(1 − x ) q 1 − x q 2 ] 2 + m V 2 ] 2 0 Recall 1 � � d 4 r e − i ((1 − x ) q 1 − xq 2 ) · r F c ( x, r 2 ) 2 , q 2 2 ) = F ( q 1 (17) dx 0 4 e ip · r d 4 p � m V VMD ( x, r 2 ) = (18) F c 2 − x (1 − x ) m π [ p 2 + m V (2 π ) 4 2 ] 2 The dependence on x is very weak. Pion is uniformly created/annihilated between the two EM currents.

Pion TFF formulation: TE model 14/29 2 /2 2 /2 m V 2 + m V 2 , q 2 2 ) = F TE ( q 1 2 + m V 2 + m V 2 q 1 q 2 1 2 � d x δ ( x ) + δ ( x − 1) m V = (19) ((1 − x ) q 1 − x q 2 ) 2 + m V 2 2 0 Recall 1 � � d 4 r e − i ((1 − x ) q 1 − xq 2 ) · r F c ( x, r 2 ) 2 , q 2 2 ) = F ( q 1 (20) dx 0 2 e ip · r d 4 p � TE ( x, r 2 ) = δ ( x ) + δ ( x − 1) m V (21) F c p 2 + m V (2 π ) 4 2 2 The value for x is either 0 or 1 . Pion is created/annihilated at the two ends of the segment between the two EM currents location.

Pion TFF formulation: LMD model 15/29 2 2 2 ) = 8 π 2 F π � 1 − 8 π 2 F π � 2 , q 2 2 , q 2 2 ) + 2 , q 2 2 ) F LMD ( q 1 2 F TE ( q 1 F VMD ( q 1 (22) 2 3 m V 3 m V 3 VMD model TE model 2 . 5 LMD model 2 f ( r ) 1 . 5 1 0 . 5 0 0 0 . 5 1 1 . 5 2 2 . 5 3 r (fm)

Lattice results 16/29 RBC/UKQCD 24 3 × 64 Iwasaki+DSDR ensemble: m π = 139 MeV , a − 1 = 1.015 GeV . With z t = 0 , f ( | z | ) can be evaluated with ( t sep = 10 a ) � 0 | T i J µ ( z ) i J ν (0) | π 0 ( p � = 0) � 2 � � 2 F π 1 i = ǫ µ,ν,ρ,σ 2 z ρ i p σ f ( | z | ) , (23) 4 π 2 F π ( z 2 ) 2 3 Using 16 configurations and the point source propagators generated by computing the leading disconnected contribution to HLbL, we obtained:

3 VMD model TE model 2 . 5 LMD model 24D lattice 2 f ( r ) 1 . 5 1 0 . 5 0 0 0 . 5 1 1 . 5 2 2 . 5 3 r (fm)

Outline 17/29 We will be working in Euclidean space by default. • Pion TFF formulation • Model and Lattice results • C on t r ibut ion t o H L bL w it h pion TF F