Pion Transition Form Factor from Lattice QCD in Position Space Cheng - PowerPoint PPT Presentation

Pion Transition Form Factor from Lattice QCD in Position Space Cheng Tu Lattice 2018, Michigan State University July 24, 2018 Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 1 / 16

Collaborators Thomas Blum (UConn), Norman Christ (Columbia), Masashi Hayakawa (Nagoya), Taku Izubuchi (BNL/RBRC), Luchang Jin (UConn/RBRC), Chulwoo Jung (BNL), Christoph Lehner (BNL) and the RBC/UKQCD collaborations Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 2 / 16

Outline Construct Coordinate Space Formula 1 Matrix Element Proof F c ( x , ( u − v ) 2 ) in x ∈ [0 , 1] Convert F c to Momentum Space Form Factor Parametrization 2 OPE analysis Parametrization Resent Results 3 Study r Dependence of the Coordinate Space Formulation Point Source Propagator and Contraction Ensembles Plots Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 3 / 16

Construct Coordinate Space Formula Matrix Element Matrix Element Neutral Pion Transition Form Factor [Ji and Jung, 2001, Feng et al., 2012, Grardin et al., 2016] � d 4 ue − iq 1 · u − iq 2 · v � 0 | T { iJ µ ( u ) iJ ν ( v ) } � π 0 ( p ) � � M µν ( q 1 , q 2 ) = i ǫ µνρσ q 1 ,ρ q 2 ,σ F ( q 2 1 , q 2 = 2 ) 4 π 2 F π Coordinate Space Formulation i � F ′ ( p · ( u − v ) , ( u − v ) 2 ) e ipv � � π 0 ( p ) � � ǫ µνρσ ( − i ∂ u ρ )( − i ∂ v � 0 | T { iJ µ ( u ) iJ ν ( v ) } = σ ) 4 π 2 F π Give the Result with the Pion Operator: � 1 i − ∂ u ρ F c ( x , ( u − v ) 2 ) ∂ σ π 0 ( xu + (1 − x ) v ) � π 0 ( � � = � 0 | ǫ µνρσ � � p ) � dx 4 π 2 F π 0 where � ∞ F ′ ( p · ( u − v ) , ( u − v ) 2 ) = dxF c ( x , ( u − v ) 2 ) e ixp · ( u − v ) −∞ F c ( x , ( u − v ) 2 ) = 0 if x < 0 or x > 1 Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 4 / 16

Construct Coordinate Space Formula Proof Proof When u t = v t (also p · ( u − v ) is a real number), define F c ( x , ( u − v ) 2 ) to be the Fourier transformation of F ′ ( p · ( u − v ) , ( u − v ) 2 ): � ∞ F ′ ( p · ( u − v ) , ( u − v ) 2 ) = dxF c ( x , ( u − v ) 2 ) e ixp · ( u − v ) −∞ Then, we have: � ∞ i � π 0 ( p ) ǫ µνρσ ( − i ∂ u ρ )( − i ∂ v dxF c ( x , ( u − v ) 2 ) e ip · ( xu +(1 − x ) v ) � � � 0 | T { iJ µ ( u ) iJ ν ( v ) } = σ ) 4 π 2 F π −∞ Consider the following three point function (assuming u t , v t > w t ) by inserting the pion d 3 p projection operator ˆ � π 0 ( � � 1 π 0 ( � � � � � P π 0 = p ) p ) � : (2 π ) 3 2 E π 0 ,� p � 0 | T { iJ µ ( u ) iJ ν ( v ) } ˆ P π 0 π 0 ( w ) | 0 � � ∞ i − ∂ u ρ F c ( x , ( u − v ) 2 ) � � = ǫ µνρσ dx ∂ σ G ( xu + (1 − x ) v − w ) 4 π 2 F π −∞ Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 5 / 16

Fc ( x , ( u − v )2) in x ∈ [0 , 1] Construct Coordinate Space Formula F c ( x , ( u − v ) 2 ) in x ∈ [0 , 1] Select ω = x ′ u + (1 − x ′ ) v + ε , where ε is a very small distance, we get: � 0 | T { iJ µ ( u ) iJ ν ( v ) } ˆ P π 0 π 0 ( ω ) | 0 � � ∞ i ρ F c ( x , ( u − v ) 2 ) ∂ σ G (( x − x ′ )( u − v ) − ε ) � − ∂ u � = ǫ µνρσ dx 4 π 2 F π −∞ In the area of x ′ > 1 or x ′ < 0, � 0 | T { iJ µ ( u ) iJ ν ( v ) } ˆ P π 0 π 0 ( ω ) | 0 � should be finite when ε → 0, but the above formula suggest a singularity behavior of ∂ σ G (( x − x ′ )( u − v ) − ε ). This implies that: F c ( x , ( u − v ) 2 ) = 0 if x < 0 or x > 1 Then the integral becomes from 0 to 1: � π 0 ( � � � � 0 | T { iJ µ ( u ) iJ ν ( v ) } p ) � 1 i − ∂ u ρ F c ( x , ( u − v ) 2 ) ip σ e ip · ( xu +(1 − x ) v ) = ǫ µνρσ � � dx 4 π 2 F π 0 or, in pion operator expression: � 1 i − ∂ u ρ F c ( x , ( u − v ) 2 ) ∂ σ π 0 ( xu + (1 − x ) v ) � π 0 ( � � � � � = � 0 | ǫ µνρσ dx p ) 4 π 2 F π 0 Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 6 / 16

Construct Coordinate Space Formula Convert Fc to Momentum Space Form Factor Convert F c to Momentum Space Form Factor and the Property of F c Let v = 0 and p = q 1 + q 2 , we get mapping from F c to momentum space form factor: � 1 � F ( q 2 1 , q 2 d 4 u e − iq 1 · u dx F c ( x , u 2 ) e ixp · u 2 ) = 0 p = 0, q 1 = 0, F ( q 2 1 → 0 , q 2 In the Chiral limit and � 2 → 0) = 1, and: � 1 � d 4 u dx F c ( x , u 2 ) = 1 0 Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 7 / 16

Parametrization OPE analysis OPE analysis When u and v are very close, we have such approximation [Gerardin et al., 2016]: d 4 p e ip · ( u − v ) � ( u − v ) ρ γ ρ T ψ ( u ) ¯ ψ ( v ) ≈ ip ρ γ ρ + m = (2 π ) 4 2 π 2 (( u − v ) 2 ) 2 i ¯ i ¯ � � � � T ψ ( u ) γ µ ψ ( u ) ψ ( v ) γ µ ψ ( v ) ( u − v ) ρ ( v − u ) ρ 2 π 2 (( u − v ) 2 ) 2 ¯ 2 π 2 (( v − u ) 2 ) 2 ¯ ≈ − ψ ( u ) γ µ γ ρ γ ν ψ ( v ) − ψ ( v ) γ ν γ ρ γ µ ψ ( u ) = − ǫ µνρσ ( u − v ) ρ � ¯ ψ ( u ) γ σ γ 5 ψ ( v ) + ¯ � ψ ( v ) γ σ γ 5 ψ ( u ) 2 π 2 (( u − v ) 2 ) 2 Consider the definition of pion form factor: = F π p σ e ip · u = � 0 | − iF π ∂ σ π 0 ( u ) � π 0 ( � � π 0 ( � � � � � � 0 | ¯ u ( u ) γ σ γ 5 u ( u ) p ) p ) Analogously, the two current operator can be shown as follow: � 2 F 2 i 1 � ∂ σ π 0 ( u + v µ , ν are close π T { iJ µ ( u ) iJ ν ( v ) } − − − − − − − − → ǫ µνρσ 2( u − v ) ρ ) 4 π 2 F π (( u − v ) 2 ) 2 3 2 Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 8 / 16

Parametrization Parametrization Parametrization Do Fourier series expansion of F c ( x , ( u − v ) 2 ) (or other parametrization), using the fact that 0 ≤ x ≤ 1: � 2 F 2 1 � − ∂ u ρ F c ( x , ( u − v ) 2 ) =2( u − v ) ρ π (( u − v ) 2 ) 2 3 ∞ f n ( | u − v | ) (2 n + 1) π � × sin ((2 n + 1) π x ) 2 n =0 � π 0 ( � � � � 0 | TiJ µ ( u ) iJ ν ( v ) p ) � 2 F 2 i 1 � π = ǫ µνρσ 2( u − v ) ρ ip σ 4 π 2 F π (( u − v ) 2 ) 2 3 � 1 ∞ f n ( | u − v | ) (2 n + 1) π � sin ((2 n + 1) π x ) e ip · ( xu +(1 − x ) v ) dx × 2 0 n =0 Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 9 / 16

Resent Results Study r Dependence of the Coordinate Space Formulation Study r Dependence of the Coordinate Space Formulation Define f ( | r | ): � 2 F 2 � 1 1 � − ∂ u ρ F c ( x , r 2 ) π � � dx = 2 r ρ f ( | r | ) 3 ( r 2 ) 2 0 that is: ∞ � f ( | r | ) = f n ( | r | ) n =0 Let u = r / 2, v = − r / 2 and r t = 0, � p = 0, we have: � 2 F 2 i 1 � π � � π 0 ( � � 0 | TiJ µ (0 , � r / 2) iJ ν (0 , − � r / 2) p = 0) � = ǫ µνρσ 2 r ρ ip σ f ( | r | ) 4 π 2 F π 3 ( r 2 ) 2 Remember we also have the constrains from the pion decay width, which imply f ( | r | ) should satisfy: � ∞ π 2 2 F 2 π 3 f ( | r | )2 rdr = 1 2 0 Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 10 / 16

Resent Results Point Source Propagator and Contraction Point Source Propagator and Contraction Propagator: 1024 point source propagator in each configuration 256 random area group chosen from the lattice 4 random points per group Three-Point Correlation Function: C conn � � 0 | J µ (0 , � r ′ / 2) J ν (0 , − � r ′ / 2) P + ( � x , t π ) | 0 � e i � p · � x µν ( | � r | , � p , t π ) = | � r ′ | = | � r | ,� x Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 11 / 16

Resent Results Ensembles Ensembles 24c64 Ensemble Figure: pion mass Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 12 / 16

Resent Results Plots Plots 24c64 Lattice, 1024 Point Source Propagator 1 . 2 f ( r ) t sep = 6 f ( r ) t sep = 10 1 0 . 8 f ( r ) 0 . 6 0 . 4 0 . 2 0 0 1 2 3 4 5 6 7 8 9 r � 2 F 2 � 1 1 � − ∂ u ρ F c ( x , r 2 ) π � � dx = 2 r ρ f ( | r | ) ( r 2 ) 2 3 0 Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 13 / 16

Resent Results Plots Partial integration f ( r ) t sep = 6 Partial integration f ( r ) t sep = 10 1 0 . 8 0 . 6 f ( r ) 0 . 4 0 . 2 0 0 1 2 3 4 5 6 7 8 r � 2 F 2 � 1 1 � − ∂ u ρ F c ( x , r 2 ) π � � dx = 2 r ρ f ( | r | ) ( r 2 ) 2 3 0 Cheng Tu (University of Connecticut) Pion Transition Form Factor from Lattice QCD in Position Space July 24, 2018 14 / 16