Simulating Quantum Chromodynamics coupled with Quantum Electromagnetics on the lattice Yuzhi Liu Indiana University liuyuz@indiana.edu Fermilab Lattice and MILC Collaborations The 36th Annual International Symposium on Lattice Field Theory East Lansing, MI, USA July 22-28, 2018 1/17
Motivation ◮ Many lattice-QCD calculations are now reaching a precision for which electromagnetic (EM) and isospin-breaking effects may enter near the level of current lattice uncertainties. ◮ Current dominant errors for the calculation of the hadronic contributions to the muon anomalous magnetic moment (g - 2) are from omission of EM and isospin breaking, and from quark-disconnected contributions. (HPQCD, PRD 96(2017) no.3, 034516) u, d, s, c sea MILC 18 ◮ The calculation of EM and Fermilab/MILC 17 isospin-violating effects in the kaon RM123 17 ETM 14 and pion systems is a long-standing problem and is crucial for determining u, d, s sea BMW 16 the light up- and down-quark masses. QCDSF 15 (MILC, arXiv:1807.05556, and Blum et al. 10 MILC 09 Fermilab Lattice, MILC, and TUMQCD Collaborations u, d sea RM123 13 arXiv:1802.04248) RBC 07 0 . 35 0 . 4 0 . 45 0 . 5 0 . 55 0 . 6 m u /m d 2/17
QCD + QED action In the continuum, the QCD Lagrangian density (in Minkowski space) for one spin-1/2 field without interacting with the EM field is 1 � ψ f ¯ i ( i γ µ D f µ − M f ) ψ f 4 g 2 G a µν G µν L QCD = L QCD F + L QCD G = a . (1) j − ij f The Euclidean QCD + QED Lagrangian density is 1 1 � ψ f ¯ i ( γ µ D f µ + M f ) ψ f 4 g 2 G a µν G µν 4 e 2 F µν F µν , L = j + + (2) a ij f with D f µ = ∂ µ + iA µ ( x ) + iq f A ′ µ ( x ) , (3) q f = 2 / 3 � for u quark , e ≈ 4 π/ 137 , (4) G a µν = ∂ µ A a ν ( x ) − ∂ ν A a µ ( x ) + f abc A b µ ( x ) A c ν ( x ) , (5) F µν = ∂ µ A ′ ν ( x ) − ∂ ν A ′ µ ( x ) . (6) The QCD + QED action becomes � dx 4 L = S F + S G QCD + S G QED . S = (7) 3/17
QCD + QED action ◮ The lattice QCD ( SU ( 3 ) ) gauge action S G QCD is a function of ◮ the link variable U µ ( n ) = e iA µ ( n ) and the QCD coupling g . ◮ The lattice QED ( U ( 1 ) ) gauge action S G QED is a function of µ ( n ) = e iqA ′ µ ( n ) for compact QED; ◮ the link variable U ′ q or ◮ the real valued vector potential of an EM field A ′ µ ( x ) for non-compact QED. and ◮ the QED coupling e . ◮ The lattice fermion action S F is a function of ◮ the link variables U µ ( n ) and U ′ q µ ( n ) (i. e., S F has both SU(3) and U(1) components). 4/17
QCD + QED action ◮ The naive QCD+QED lattice fermion action is ψ ( x )[ M ( U eff )] xy ψ ( y ) , S naive � ¯ = (8) F x , y where ψ ( x ) is the charged spin 1/2 particle field. ◮ The staggered fermion classical Hamiltonian is obtained by changing the ψ ( x ) field to the staggered field χ ( x ) , introducing the pseudo-fermion filed Φ (on even sites only) and the canonical momentum h and h ′ conjugate to A µ and A ′ µ , 1 1 2 h ′ 2 H [Φ q e ; A ′ ; U ; U ′ q ; g ; e ] = � 2 h 2 � + + S PF + S G QCD + S G QED . (9) i i i i ◮ The staggered pseudo-fermion action with n f degenerate fermion flavors is � � � − n f / 4 � � � M † [ U eff ] M [ U eff ] � � S PF = Φ � Φ , (10) � � � � U eff � � U eff � � � x ,µ δ x , y − µ − U eff † � U eff M x , y = 2 m δ x , y + D x , y = 2 m δ x , y + η x ,µ x − µ,µ δ x , y + µ . µ (11) 5/17
Non-compact QED ◮ The non-compact U(1) lattice gauge action is defined as µ ( n )) = 1 � S NC G QED ( A ′ F 2 µν ( n ) , (12) 4 e 2 n ,µ,ν = 1 µν ( n ) = β u 1 � F 2 � F 2 µν ( n ) , (13) 2 e 2 2 n ,µ<ν n ,µ<ν with F µν ( n ) = [ A ′ µ ( n ) + A ′ µ ) − A ′ ν ) − A ′ ν ( n + ˆ µ ( n + ˆ ν ( n )] . (14) ◮ The U(1) momentum is defined via dU ′ q µ ( n ) A ′ µ ( n ) q f U ′ q = i ˙ µ ( n ) ≡ iH ′ q µ ( n ) U ′ q µ ( n ) , (15) d τ with µ ( n ) = e iqA ′ U ′ q µ ( n ) , (16) µ ( n ) q f . H ′ q µ ( n ) = h ′ (17) Since ˙ A ′ µ ( n ) = h ′ µ ( n ) , h ′ µ ( n ) is a conjugate field to A ′ µ ( n ) , we can consider A ′ µ ( n ) as coordinate and h ′ µ ( n ) as momentum conjugate to the corresponding coordinate. ◮ The kinetic part of the Hamiltonian can then be written as 1 1 2 h ′ 2 � � [ H ′ q µ ( n ) 2 ] . µ ( n ) = (18) 2 q f 2 n ,µ n ,µ 6/17
Non-compact QED: gauge force ◮ U(1) gauge field update: Since ˙ A ′ µ ( n ) = h ′ µ ( n ) , the A ′ should be updated according to A ′ → A ′ + h ′ d τ. (19) ◮ U(1) momentum update: The U(1) gauge force contributing to the U(1) momentum change is dS NC dh ′ G QED d τ = − , (20) dA ′ with µ ( n ) = 1 dS NC G QED / dA ′ � [ A ′ µ ( n ) + A ′ ν ( n + µ ) − A ′ µ ( n + ν ) − A ′ � ν ( n )] e 2 ν − [ A ′ µ ( n − ν ) + A ′ ν ( n − ν + µ ) − A ′ µ ( n ) − A ′ � ν ( n − ν )] , (21) � = β u 1 [ F µν ( n ) − F µν ( n − ν )] . (22) ν 7/17
Fermion forces ◮ The fermion force has contributions from both SU(3) and U(1). ◮ Taking the MC simulation time τ derivative of the Hamiltonian and requring it to be zero, one gets the SU(3) and U(1) fermion forces. ◮ The SU(3) contribution (QCD force) is � � ∂ S ∂ S i ˙ U † H µ ( n ) = U µ ( n ) ∂ U µ ( n ) − µ ( n ) ∂ U µ † ( n ) � � − 1 ∂ S ∂ S U † U µ ( n ) ∂ U µ ( n ) − µ ( n ) , Tr (23) ∂ U µ † ( n ) N c � ∂ S � = 2 U µ ( n ) , (24) ∂ U µ ( n ) AT where the operation AT stands for taking the anti-Hermitian and traceless part of the matrix M AT = 1 1 2 ( M − M † ) − Tr ( M − M † ) . (25) 2 N c 8/17
Fermion forces ◮ The fermion force has contributions from both SU(3) and U(1). ◮ Taking the MC simulation time τ derivative of the Hamiltonian and requring it to be zero, one gets the SU(3) and U(1) fermion forces. ◮ The U(1) contribution (QED force) is � � ∂ S ∂ S q f Tr i ˙ � U † h ′ µ ( n ) = U µ ( n ) ∂ U µ ( n ) − µ ( n ) , (26) ∂ U † µ ( n ) q or � ∂ S � q f ImTr ˙ � h ′ µ ( n ) = 2 U µ ( n ) . (27) ∂ U µ ( n ) q f ◮ In Eqs. (23, 24, 26, and 27), U µ ( n ) is the product of SU(3) U µ ( n ) and U(1) U ′ q µ ( n ) . 9/17
Pure gauge U(1) test ◮ The pure gauge U(1) Hamiltonian is 1 2 h ′ 2 H [ A ′ ; e ] = � + S NC G QED . (28) i i ◮ The non-compact U(1) gauge action S NC G QED is only a function of the dimensionality d and lattice volume V on the finite periodic lattice G QED = ( d − 1 )( V − 1 ) S NC . (29) 2 ◮ One can use this to check the correctness of the pure gauge U(1) code. 1.0010 expected measured G QED / (( d − 1)( V − 1) / 2) 1.0005 1.0000 0.9995 S NC 0.9990 0.15 0.2 0.25 0.3 0.35 β u 1 10/17
U(1) with fermions test ◮ The U(1) with fermion Hamiltonian is 1 H [Φ q � 2 h ′ 2 e ; A ′ ; U ′ q ; e ] = + S PF + S NC G QED . (30) i i ◮ Time history of the U(1) gauge action 4 3 G QED / ( β u 1 V ) 2 S NC 1 0 0 200 400 600 800 1000 # traj 11/17
SU(3) with fermions test ◮ The SU(3) with fermion Hamiltonian is 1 H [Φ q e ; A ′ ; U ′ q ; e ] = � 2 h 2 + S PF + S G QCD . (31) i i ◮ Time history of the SU(3) Plaquette 2.6 2.4 Re ( PLAQ SU (3) ) 2.2 2 1.8 0 200 400 600 800 1000 # traj 12/17
SU(3) + U(1) with fermions test ◮ The SU(3) + U(1) with fermion Hamiltonian is 1 1 H [Φ q 2 h ′ 2 e ; A ′ ; U ′ q ; e ] = � 2 h 2 � + S PF + S G QCD . + S NC + G QED . (32) i i i i ◮ Time history of the SU(3) Plaquette 2.6 2.4 Re ( PLAQ SU (3) ) 2.2 2 1.8 0 200 400 600 800 1000 # traj 13/17
Integration algorithms ◮ The integration algorithm is based on decomposing the Hamiltonian in exactly integrable pieces H ( φ, h ) = H 1 ( φ ) + H 2 ( h ) , (33) i h 2 with H 1 ( φ ) = S ( φ ) and H 2 ( h ) = � i / 2 for example. The algorithm consists of repeated applying the following two elementary steps I 1 ( ǫ ) :( h , φ ) → ( h , φ + ǫ ∇ h H 2 ( h )) , (34) I 2 ( ǫ ) :( h , φ ) → ( h − ǫ ∇ φ S ( φ ) , φ ) . (35) ◮ The leap-frog algorithm corresponds to the following updates I ǫ ( τ ) = [ I 1 ( ǫ/ 2 ) I 2 ( ǫ ) I 1 ( ǫ/ 2 )] N s , (36) with τ = N s ǫ the length of the trajectory. The leading violation due to the finite step-size ǫ is O ( ǫ 2 ) . ◮ The Omelyan integrator [ I 1 ( ξǫ ) I 2 ( ǫ/ 2 ) I 1 (( 1 − 2 ξ ) ǫ ) I 2 ( ǫ/ 2 ) I 1 ( ξǫ )] N s , (37) reduces the coefficient of the ǫ 2 term and improves the scaling behavior. ◮ In the MILC code for the HISQ fermion related calculations, an Omelyan based “3G1F” integrator is used. ◮ The algorithm can be made exact by a Metropolis acceptance step: Hybrid Monte Carlo algorithm. 14/17
SU(3) + U(1) with fermions test ◮ The change of the Hamiltonian during the trajectory is expected to scale with ǫ 2 for the integrators used. ◮ Scaling of the the change of action | ∆ H | with the step sizes ǫ . The upper blue points are from the leap-frog integrator and the lower red points are from the “3G1F” integrator. 4 3 | ∆H | 2 1 0 0 0.01 0.02 0.03 ǫ 2 15/17
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