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Electromagnetic Polarizabilities from Lattice QCD

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Electromagnetic Polarizabilities from Lattice QCD Electromagnetic Polarizabilities from Lattice QCD W. Detmold (College of William and Mary) * B. Tiburzi (University of Maryland) A. Walker-Loud (College of William and Mary) Outline Physics Motivation: Low-energy hadron structure chiral symmetry electromagnetic polarizabilities Lattice QCD: Background field method Results: Electric Polarizabilities Pions Nucleons Physics Motivation Spontaneous chiral symmetry breaking has important consequences on low energy hadron structure In external EM fields, hadrons polarize against strong chromodynamic forces p! = α E! − E Chiral dynamics: charged pion cloud 1 ∆H = α E" 2 deforms (any hadron) −2 E Hadronic polarizabilities are tightly constrained by chiral dynamics 2 2 π e p e αE = N 2 αE = N ! 2 mπΛ mπΛ Electromagnetic Polarizabilities One-loop result + (Holstein) π Two-loop result (Gasser, Ivanov & Sainio) Mainz extraction (Ahrens, et al.) N Compton Scattering Theory: include delta resonance? magnetic isovector? New data 60, 100 MeV Extraction: deuteron ( 3 He plans) ∼ taken! ! 100 MeV 65 MeV Extraction: ∼ approved Lattice QCD Use first principles numerical techniques to + calculate polarizabilities directly from QCD...? Stable hadrons! neutron, pion, etc. Lattice 4-point functions: not enough time! Not enough space! 2πn ψ(x + L)=ψ(x) k = 450 MeV → L ∼ Also limitation for 3-point functions: magnetic moment, quark orbital angular momentum Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action E.g. neutral pion in electric field Aµ = x4 δµ3 −E 1 0 2 2 0 (p! =0,x4)= π ∂4∂4 + mπ0 + mπ0 αE π L 2 ! − E " G(τ)= χ(#x, τ)χ†(#0, 0) = Z exp( Eτ) ! " − !!x 1 2 E = m 0 + α π 2 E E E.g. neutral pion in electric field Anisotropic 2+1 clover lattices (HadSpec) Detmold, Tiburzi, Walker-Loud Pion mass: 390 MeV Phys.Rev.D79: 094505, 2009 200, 20c 128 lattices, 12 src/cfg (EigCG) M (t)= log G(t + 1)/G(t) eff − cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 T 1 − x4 0 0 x3 L 1 0 − cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 T 1 − x4 0 0 x3 L 1 0 − U (x)U (x + 3)ˆ U †(x + 3+ˆ 4)ˆ U †(x + 4)ˆ = exp(iqF ) = exp(iq ) 3 4 3 4 34 E cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 T 1 − x4 0 E 0 x3 L 1 0 − U (x)U (x + 3)ˆ U †(x + 3+ˆ 4)ˆ U †(x + 4)ˆ = exp(iqF ) = exp(iq ) 3 4 3 4 34 E cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 T 1 − x4 0 E E E E E 0 x3 L 1 0 − U (x)U (x + 3)ˆ U †(x + 3+ˆ 4)ˆ U †(x + 4)ˆ = exp(iqF ) = exp(iq ) 3 4 3 4 34 E cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 T 1 − E E E E E E E E E E x 4 E E E E E 0 E E E E E 0 x3 L 1 0 − U (x)U (x + 3)ˆ U †(x + 3+ˆ 4)ˆ U †(x + 4)ˆ = exp(iqF ) = exp(iq ) 3 4 3 4 34 E cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 (1 T ) (1 T ) (1 T ) (1 T ) (1 T ) E − E − E − E − E − T 1 − E E E E E E E E E E x 4 E E E E E 0 E E E E E 0 x3 L 1 0 − U (x)U (x + 3)ˆ U †(x + 3+ˆ 4)ˆ U †(x + 4)ˆ = exp(iqF ) = exp(iq ) 3 4 3 4 34 E cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 (1 T ) E − T 1 − E E E E E E E E E E x 4 E E E E E 0 E E E E E 0 x3 L 1 0 − Uµ⊥(x) = exp(iq Tx3 δµ,4δx4,T 1) E − cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 E T 1 − E E E E E E E E E E x 4 E E E E E 0 E E E E E 0 x3 L 1 0 − Uµ⊥(x) = exp(iq Tx3 δµ,4δx4,T 1) E − cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 E E E E T 1 − E E E E E E E E E E x 4 E E E E E 0 E E E E E 0 x3 L 1 0 − Uµ⊥(x) = exp(iq Tx3 δµ,4δx4,T 1) E − cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0 (1 LT ) E E E E E − T 1 − E E E E E E E E E E x 4 E E E E E 0 E E E E E 0 x3 L 1 0 − Uµ⊥(x) = exp(iq Tx3 δµ,4δx4,T 1) E − cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E Uµ⊥(x) = exp(iq Tx3 δµ,4δx4,T 1) E − 2πn (1 LT ) exp(iqF34) q = E − −→ E TL Quantization Condition 2πn cf. Momentum Quantization k = L 24c 64 lattices π0 tsrc = 52 Hadron Structure in Strong QED Backgrounds Small lattices, strong fields Tiburzi, Nucl.Phys.A814: 74, 2008 2 2 2 ( /mπ) =0.18 n E ∆E(x)/E(0) Strong field ChPT power counting /m2 1 /Λ2 1 E π ∼ E χ " ∞ 2n Eπ0 (x)= αnx n=0 ! x +1 log Γ 2 ∼ 2 x = /mπ " # E Magnetic fields even larger (B/m2 )2 =0.18n2(T/L)2 (T/L)2 3 4 π ∼ − Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action E.g. charged pion in electric field Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action E.g. charged pion in electric field 2 2 2 (x4)=π† ∂4∂4 + x4 + E ( ) π L ! − E E " G(τ)= χ(#x, τ)χ†(#0, 0) = Z exp( Eτ) ! " − !!x 2 G(τ)=Z E2+ (√2 τ ) 2 E D− E E Detmold, Tiburzi, Walker-Loud, Phys.Rev.D73: 114505, 2006 E.g. charged pion in electric field 2 2 3 Detmold, Tiburzi, Walker-Loud NR Q t G(t) exp Et E Phys.Rev.D79: 094505, 2009 −→ − − 6M ! " M (t)= log G(t + 1)/G(t) eff − E.g. charged pion in electric field 2 2 3 Detmold, Tiburzi, Walker-Loud NR Q t G(t) exp Et E Phys.Rev.D79: 094505, 2009 −→ − − 6M ! " M (t)= log G(t + 1)/G(t) eff − E.g. charged pion in electric field 2 2 3 Detmold, Tiburzi, Walker-Loud NR Q t G(t) exp Et E Phys.Rev.D79: 094505, 2009 −→ − − 6M ! " M (t)= log G(t + 1)/G(t) eff − E.g. charged pion in electric field 2 2 3 Detmold, Tiburzi, Walker-Loud NR Q t G(t) exp Et E Phys.Rev.D79: 094505, 2009 −→ − − 6M ! " M (t)= log G(t + 1)/G(t) eff − E.g. charged pion in electric field 2 2 3 Detmold, Tiburzi, Walker-Loud NR Q t G(t) exp Et E Phys.Rev.D79: 094505, 2009 −→ − − 6M ! " M (t)= log G(t + 1)/G(t) eff − Nucleon (preliminary) Nucleon: Compare momentum vs. coordinate space Compton scattering tensor T = A (θ, ω)#ε ! #ε + A (θ, ω) #ε ! kˆ #ε kˆ! γN 1 · 2 · · Low-energy expansion (Qe)2 e2 A (θ, ω)= + [µ2(1 + cos θ) Q2](1 cos θ)ω2 1 − 2M 8M 3 − − 1 + (α + β cos θ)ω2 + (ω4) 2 O Background field method Thomson term = acceleration Frequency expansion akin to field strength expansion Anticipate: account for magnetic moment Born terms Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action 1 E = M + α 2 + ... E.g. neutron in electric field 2 EE µ = N † /∂ + E( ) σ F N L E − 4M µν µν ! " Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action E.g. neutron in electric field µ = N † /∂ + E( ) σ F N L E − 4M µν µν Unpolarized correlator! " Tr[G(τ)] = Z exp( τEeff) − 1 µ2 E = M + 2 α + ... eff 2E E − 4M 3 ! " Unlike experiment, cannot measure electric polarizability! Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action E.g. neutron in electric field µ = N † /∂ + E( ) σ F N L E − 4M µν µν ! " Magnetic field σ F = S" B" Electric field σ F = K" " µν µν · µν µν · E 1 1 Spin Projection (1 S3) Boost Projection = (1 K3) 2 ± P± 2 ± Background Field Method Measure lattice correlation functions in external fields Study field dependence to determine parameters in the single particle effective action E.g.
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