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 −

0 0 x3 L 1 0 − cl Electric Field on a Lattice: Uµ (x) = exp( iq x4 δµ3) − E 0

T 1 −

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 −

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 −

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 − 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. neutron in electric field µ = N † /∂ + E( ) σ F N L E − 4M µν µν Boost projected correlator! " µ Tr[ G(τ)] = Z 1 E exp( τEeff) P± ± 2M 2 − ! " 1 µ2 E = M + 2 α + ... eff 2E E − 4M 3 ! " Simultaneous fit can determine magnetic moment & electric polarizability Simulataneous Two-state fits

... six parameters

µ Tr[ G(τ)] = Z 1 E exp( τEeff) P± ± 2M 2 − ! " 1 µ2 E = M + 2 α + ... eff 2E E − 4M 3 ! " Simultaneous fit can determine magnetic moment & electric polarizability I II excludes largest field 4 3 αE [10− fm ]: 3.13(8) 3.29(24)

µ [µ ]: 1.37(5) 1.34(6) N − − statistical uncertainty only... 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. proton in electric ﬁeld 2 EE µ˜ = P † D/ + E( ) σ F P L E − 4M µν µν ! "

charge => acceleration anomalous moment Dirac moment 1 µ˜2 E = M + 2 α + ... Boost projected correlator eff 2E E − 4M 3 µ˜ ! " Tr[ G(τ)] = Z 1 E (τ,Q ,Eeff) P± ± 2M 2 D E ! " Simultaneous fit can determine anomalous moment & electric polarizability Simulataneous Two-state fits

... six parameters

1 µ˜2 E = M + 2 α + ... Boost projected correlator eff 2E E − 4M 3 µ˜ ! " Tr[ G(τ)] = Z 1 E (τ,Q ,Eeff) P± ± 2M 2 D E ! " Simultaneous fit can determine anomalous moment & electric polarizability I II excludes largest field 4 3 αE [10− fm ]: 2.44(11) 2.38(26) 1.32(3) 1.32(8) µ˜ [µN ]: statistical uncertainty only... Summary

Lattice QCD in Background Fields: Study hadron correlation functions in external ﬁelds, deduce EM structure. Field quantization: ideal for multipole moments and multipole polarizabilities. Demonstrated for pions and nucleons in electric ﬁelds.

Challenges: continuum & inﬁnite volume limits, chiral extrapolation, sea quark charges

Prospect: Quark orbital angular momentum from lattice: momentum extrapolation 2-point function in background gravitational ﬁeld

Outlook: Spontaneous chiral symmetry breaking has fundamental consequences for low-energy structure of hadrons. Conﬁrmation from both experiment & lattice will be a milestone.