Electromagnetic and spin polarisabilities from lattice QCD

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1 Lattice Hadron Physics 2006 Electromagnetic and spin polarisabilities from lattice QCD William Detmold [ WD, BC Tiburzi and A Walker-Loud, PRD73, ]

2 I: How to extract EM and spin polarisabilities from lattice QCD using external fields II: How to relate lattice measurements to the polarisabilities of the real world

3 Hadron polarisabilities Hadron polarisabilities describe the deformation of a particle in an external (EM) field Quadratic energy shifts from effective Hamiltonian: H = H 0 µ B 2πα E 2 2πβ H 2 2πγ 1 σ E E +... Magnetic moment Electric pol Magnetic pol First spin pol Electric and magnetic polarisabilities: ability to align with or against the applied field Spin and higher order polarisabilities are less intuitive: more detailed view of EM structure

4 Compton scattering Experimentally measured in the low frequency limit of real Compton scattering γ(ω, q) γ(ω, q ) p T γn p Thomson limit and Low Gell-Mann Goldberger LET determined by Born terms (charge and magnetic moment) T γn = f(ω, q, q, ɛ, ɛ, σ; Z, µ, α, β, γ 1,...,4 ) + O(ω 4 ) Kinematics Next order given in terms EM and spin polarisabilities

5 Experiment MAMI, Saskatoon, JLab, OOPS, ELSA, HIγS EM and 2 combinations of spin polarisabilities are measurable for the proton but difficult experiments Neutron accessed via (quasi-)elastic Compton scattering on the deuteron - even more difficult α p = 12.0(6), β p = 1.9(6), α n = 13(2), β n = 3(2) 10 4 fm 3 γ π (p) = 39(2), γ (p) 0 = 1.0(1), γ π (n) = 59(4), 10 4 fm 4 [de Jaeger & Hyde-Wright 05] Sign indicates diamagnetic nature of nucleon Small size of polarisabilities indicates tightly bound relativistic system - hard to deform

6 Further polarisabilities Higher orders in the frequency expansion gives higher order polarisabilities [Holstein et al. 99] Virtual and doubly virtual Compton scattering leads to generalised polarisabilities [Guichon, Liu & Thomas 95] de Jaeger & Hyde-Wright

7 Lattice approaches 1. Four point correlators Analogous to experimental measurement Difficult - many disconnected contractions 2. Energy shifts in two point correlators in external U(1) field Quenched QCD: external field can be added after gauge configurations are generated QCD: external field must be known during gauge field generation - costly but multipurpose

8 External field method Quenched external fields simple to apply: U a µ(x) U a µ(x) U ext µ (x) E.g.: magnetic field B = (0, 0, B) Quantised for periodic links U ext 0 = U ext 2 = U ext 3 = 1, U 1 (x) = e iebx 2 Look for shift in energy quadratic in B C (τ, B) = x 0 χ ( x, t)χ (0) 0 Magnetic polarisability = exp [ (M µ B + 2πβ B 2 )τ ] + O( B 3 ) Magnetic moment

9 Field constraints Field values are restricted by a number of constraints Perturbative in EFT: eb, ee < m 2 π Periodicity of box: e.g. magnetic field U µ (x + Lˆν) = U µ (x) a 2 eb = 2πn L, n Z Landau levels well represented Existing calculations do not satisfy these constraints

10 External field method Can study more than energy shifts - hadronic correlator analysis effective field theory matching behaviour of QCD correlator to EFT correlator (not just in ε-regime) E.g.: charged particle in constant electric field C ss (τ; E) = x = δ s,s exp 0 χ s ( x, t)χ s (0) 0 [ (M + 2πα E 2 )τ q2 E 2 ] 6M τ 3 Electric polarisability Acceleration of proton at large times +... Valid for L 1 < m π, ee < m 2 π

11 External field method All six polarisabilities can be calculated utilise all information in hadron correlators including spin-flip matrix elements Spin polarisabilities require space/time varying U(1) fields: E.g. γ E1E1 ( Uµ ext = e iaea µ(x), A µ (x) = a 6t 2 2a, ib ) 6t, 0, 0 2 C ( p, τ; A) C ( p, τ; A) = exp [ 2π a a 6 b 6 γ E1 E 1 τ ] +...

12 Quenched lattice polarisabilities External field calculations of magnetic moments and EM polarisabilities have a long history Martinelli et al., Bernard et al.: µ for n, p, [83] Fiebig et al.: α for neutron [89] Christensen et al.: α for uncharged particles [05] Lee et al.: µ for baryons [05] Lee et al.: β for many baryons and mesons [05] Preliminary work on spin polarisabilities

13 Quenched magnetic polarisabilities Use four field values (average over +/-) [Lee et al., hep-lat/ ]

14 Quenched magnetic polarisabilities [Lee et al., hep-lat/ ] Calculated for many hadrons n, p, Σ ±,0, Ξ 0,, ++,±,0 Σ ±,0, Ξ 0,, Ω, π ±,0, K ±,0, ρ ±,0, K ±,0

15 Quenched electric properties [Christensen et al., hep-lat/ ] Also do calculations with four field values (pos/neg) Neutral particles n, Σ 0, Ξ 0, 0 Σ 0, Ξ 0, π 0, K 0, ρ 0, K 0 n Ξ 0

16 Chiral perturbation theory Many studies of nucleon polarisabilities in the context of chiral perturbation theory (χpt) Extended to partially-quenched χpt at finite volume using heavy baryon formalism No undetermined LECs at NLO - loops are more important than counter-terms Functional form similar to χpt, but couplings change

17 PQχPT contributions Anomalous π 0 γγ Δ-pole graphs L O O P S

18 Wess-Zumino-Witten L P Q Chiral anomaly contributes through π 0 γγ L P Q π 0 γγ = 3e2 16π 2 f tr [ ΦQ 2] ɛ µνρσ F µν F ρσ Extension to partially quenched QCD nontrivial π 0 γγ ɛµνρσ F µν F ρσ [a 1 str [ ΦQ 2] + a 2 str [ΦQ] str [Q] L P Q π +a 3 str [Φ] str [Q] 2 + a 4 str [Φ] str [ Q 2] ] 0 γγ = 3e2 [ 16π 2 f str ΦQ 2] ɛ µνρσ F µν F ρσ No need to extend Witten s global quantisation construction to graded Lie groups

19 Infinite volume results Proton electric polarisability Involve axial couplings and quark charges α p = e2 4πf 2 [ 5GB 192π 1 m π + 5G B 192π 1 + G ] T m uj 72π 2 F α(m π ) + G T 72π 2 F α(m uj ) Singular in chiral limit Non-analytic function involving isobar F α (m) = 9 2 m m 2 2( 2 m 2 ) 3/2 ln [ ] 2 m 2 + iɛ + 2 m 2 + iɛ Results for other polarisabilities similar but also have contributions from anomaly and poles

20 Finite volume effects Polarisabilities are very sensitive to infrared scales Expect large FV effects in lattice calculations Easily included in EFT for p-regime Momentum integrals mode sums where d d k (2π) d 1 L 3 k = 2π L n 10% FV effects even at d k0 m π for 2π k n i Z = 500 MeV

21 Electric polarisability again α(l) = e πf 2 0 [ dλ 3G B F α (M uu ) + 3G BF α (M uj ) ] +8G T F α (M uu) + 8G T F α (M uj) M ab = m 2 ab + 2λ + λ2 F α (m) = 180λ 2 I 7 2 (m) + 190J 7 2 (m) 280λ2 J 9 2 (m) 455K 9 2 (m) +315λ 2 K I β (M) = E 1 β ( n 2 + x 2 ) L 3 [ ] L β n k 3 + π 3 2 Γ(β) 2 + M 2 Γ(β)L 3 k (m) + 252L 11 2 (m) dt t β 5/2 t x2 e n 0 e π2 n 2 L β (M) = I β 3 (M) 3M 2 I β 2 (M) + 3M 4 I β 1 (M) M 6 I β (M) K β (M) = I β 2 (M) 2M 2 I β 1 (M) + M 4 I β (M) J β (M) = I β 1 (M) M 2 I β (M) t + 1

22 Volume Dependence: Proton Αp Γ 1 p m Π 250 MeV m Π 350 MeV m Π 500 MeV L fm X = L fm Β p Γ 2 p L fm X(L) X( ) X( ) L fm

23 Volume Dependence: Neutron Αn Γ 1 n m Π 250 MeV m Π 350 MeV m Π 500 MeV L fm L fm Β n Γ 2 n L fm L fm

24 Volume Dependence: Quenched Αp L fm Β p L fm Γ 1 p Γ 2 p L fm L fm m π = 500 MeV

25 Thomson Limit (ω = 0) Thomson limit for photon-neutron scattering Vanishes at infinite volume! 0.5 m Π 250 MeV 0.4 m Π 350 MeV m Π 500 MeV A 1 QCD Ω 0,L M N e L fm

26 Other external fields Not limited to physical EM fields Can also use for any other quark bilinear twist-two matrix elements (PDFs, GPDs), spin content (momentum injection) EMC effect from lattice QCD (nuclear effects in parton distributions) neutrino breakup of the deuteron (flavour) twisted boundary conditions

27 EMC effect EMC 1983: Modification of PDFs in nuclei F A 2 (x) A F N 2 (x) Was a surprise since ɛ/m 1%

28 EMC on the lattice Simplest manifestation: R d (x, Q 2 ) = F d 2 (x, Q 2 ) F p 2 (x, Q2 ) + F n 2 (x, Q2 ) 1 Lattice methods can be used to investigate the EMC effect Measure two-particle energy levels in external field coupled to twist-two operators Determine 2-body coefficients in d O µ 1...µ n d Leading medium modification of moments [WD Phys Rev D , WD & ]W Chen Phys Lett B625,165]

29 Future Prospects All EM and spin polarisabilities can be measured with external fields Preliminary lattice calculations underway for spin polarisabilities Large volume effects and strong mass dependence require large volumes and small masses! Higher order and generalised polarisabilities [(doubly)- virtual Compton scattering] are also measurable Parity violating polarisabilities??

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31 To be more specific... T γn = A 1 (ω, θ) ɛ ɛ + A 2 (ω, θ) ɛ ˆk ɛ ˆk + i A 3 (ω, θ) σ ( ɛ ɛ) + i A 4 (ω, θ) σ ( ˆk ˆk) ɛ ɛ [ +i A 5 (ω, θ) σ ( ɛ ˆk) ɛ ˆk ( ɛ ˆk ) ɛ ˆk ] [ + i A 6 (ω, θ) σ ( ɛ ˆk ) ɛ ˆk ( ɛ ˆk) ɛ ˆk ] A 1 (ω, θ) = Z 2 e2 M N + A 3 (ω, θ) = e2 ω 2M 2 N e2 4M 3 N A 2 (ω, θ) = ( µ 2 (1 + cos θ) Z 2) (1 cos θ) ω 2 + 4π(α + β cos θ)ω 2 + O(ω 4 ) e2 4M 3 N (µ 2 Z 2 )ω 2 cos θ 4πβω 2 + O(ω 4 ) ( Z(2µ Z) µ 2 cos θ ) + 4πω 3 (γ 1 (γ 2 + 2γ 4 ) cos θ) + O(ω 5 ) A 4 (ω, θ) = e2 ω 2M 2 N A 5 (ω, θ) = e2 ω 2M 2 N A 6 (ω, θ) = e2 ω 2M 2 N µ 2 + 4πω 3 γ 2 + O(ω 5 ) µ 2 + 4πω 3 γ 4 + O(ω 5 ) Zµ + 4πω 3 γ 3 + O(ω 5 )

32 EFT correlators Pionless effective field theory: cutoff p < m π Lagrangian [ ( ) L eff ( x, τ; A) = Ψ ( x, τ) τ + i q A 4 + ( i q A) 2 µ σ 2M H ( +2π α E 2 β H 2) ( 2πi γ E1 E 1 σ E E +γ M1 M 1 σ H H + γm1 E 2 σ i E ij H j + γ E1 M 2 σ i H ij E j)] Ψ( x, τ) +... Resum interactions with external field