Quark and Gluon Angular Momentum in the Nucleon PDF
Preview Quark and Gluon Angular Momentum in the Nucleon
Quark and Gluon Angular Momentum in the Nucleon Matthias Burkardt [email protected] New Mexico State University QuarkandGluonAngularMomentumintheNucleon–p.1/32 Motivation polarized DIS: only 30% of the proton spin due to quark spins ∼ ֒ ‘spin crisis’ ‘spin puzzle’, because ∆Σ much → −→ smaller than the quark model result ∆Σ = 1 ֒ quest for the remaining 70% → quark orbital angular momentum (OAM) gluon spin gluon OAM ֒ How are the above quantities defined? → ֒ How can the above quantities be measured → QuarkandGluonAngularMomentumintheNucleon–p.2/32 example: angular momentum in QED ~ 3 ~ ~ 3 ~ ~ ~ J = d r ~r E B = d r ~r E A γ × × × × ∇ × Z Z (cid:16) (cid:17) h (cid:16) (cid:17)i ~ ~ ~ ~ ~ ~ ~ use E A = Ej Aj E A × ∇ × ∇ − · ∇ (cid:16) (cid:17) (cid:16) (cid:17) 3 ~ ~ ~ and integrate d r ~r E A by parts × · ∇ (cid:16) (cid:17) R ~ 3 j ~ j ~ ~ ~ ~ ~ ֒ J = d r E ~r A + ~r A E + E A γ → × ∇ × ∇ · × Z h (cid:16) (cid:17) (cid:16) (cid:17) i replace 2nd term (eq. of motion ~ E~ = ej0 = eψ ψ), yielding † ∇ · ~ 3 ~ j ~ j ~ ~ J = d r ψ ~r eAψ + E ~r A + E A γ † × × ∇ × Z h (cid:16) (cid:17) i ~ ~ ψ ~r eAψ cancels similar term in electron OAM ψ ~r (p~ eA)ψ † † × × − ~ ֒ decomposing J into spin and orbital also shuffles angular γ → momentum from photons to electrons! QuarkandGluonAngularMomentumintheNucleon–p.3/32 example (cont.) total angular momentum of isolated system uniquely defined ~ ambiguities arise when decomposing J into contributions from different constituents gauge theories: changing gauge may also shift angular momentum between various degrees of freedom ֒ decomposition of angular momentum in general depends on → ‘scheme’ (gauge & quantization scheme) does not mean that angular momentum decomposition is meaningless, but one needs to be aware of this ‘scheme’-dependence in the physical interpretation of exp/lattice/model results in terms of spin vs. OAM and, for example, not mix ‘schemes’, e.t.c. QuarkandGluonAngularMomentumintheNucleon–p.4/32 Outline Ji decomposition Jaffe decomposition recent lattice results (Ji decomposition) model/QED illustrations for Ji v. Jaffe Chen-Goldman decomposition QuarkandGluonAngularMomentumintheNucleon–p.5/32 The nucleon spin pizza(s) Ji Jaffe & Manohar L q 1 L 1 ∆Σ ∆Σ q 2 2 J g L ∆G g ‘pizza tre stagioni’ ‘pizza quattro stagioni’ 1 1 only ∆Σ ∆q common to both decompositions! 2 2 q ≡ QuarkandGluonAngularMomentumintheNucleon–p.6/32 P Angular Momentum Operator angular momentum tensor Mµνρ = rµTνρ rνTµρ − ∂ Mµνρ = 0 ρ ֒ J˜i = 1εijk d3rMjk0 conserved 2 → R d 1 1 ˜i ijk 3 jk0 ijk 3 jkl J = ε d r∂ M = ε d r∂ M = 0 0 l dt 2 2 Z Z Mµνρ contains time derivatives (since Tµν does) use eq. of motion to get rid of time derivatives integrate total derivatives appearing in T0i by parts yields terms where derivative acts on ri which then ‘disappears’ ֒ Ji usally contains both → ‘Extrinsic’ terms, which have the structure ‘~r Operator’, × and can be identified with ‘OAM’ ‘Intrinsic’ terms, where the factor ~r does not appear, and × can be identified with ‘spin’ QuarkandGluonAngularMomentumintheNucleon–p.7/32 Ji-decomposition L q 1∆Σ 2 J g ~ 3 1 ~ ~ ~ ~ ֒ J = d r q Σq + q ~r iD q + ~r E B q 2 † † → × × × h (cid:16) (cid:17) i (cid:16) (cid:17) or, wRith SµP= (0, 0, 0, 1) 1 1 = J + J = ∆q + L + J q g q g 2 2 q q (cid:18) (cid:19) X X 1 1 3 3 3 1 2 ∆q = d r P, S q (~r)Σ q(~r) P, S Σ = iγ γ † 2 2 h | | i Z 3 3 ~ ~ ~ ~ L = d r P, S q (~r) ~r iD q(~r) P, S iD = i∂ gA q † h | × | i − Z (cid:16) (cid:17) 3 3 ~ ~ J = d r P, S ~r E B P, S g h | × × | i Z h (cid:16) (cid:17)i QuarkandGluonAngularMomentumintheNucleon–p.8/32 Ji-decomposition L q 1∆Σ 2 J g ~ 3 1 ~ ~ ~ ~ ֒ J = d r q Σq + q ~r iD q + ~r E B q 2 † † → × × × applies to eachh vector com(cid:16)ponent (cid:17)of niucleon(cid:16)angular(cid:17)momentum, R P but usually applied only to zˆ component where at least quark spin has parton interpretation as difference between number densities ∆q from polarized DIS 1 J = ∆q + L from exp/lattice (GPDs) q 2 q L in principle independently defined as matrix element of q ~ 1 q ~r iD q, but in practice easier by subtraction L = J ∆q † q q 2 × − (cid:16) (cid:17) J in principle accessible through gluon GPDs, but in practice g 1 easier by subtraction J = J g 2 q − Ji makes no further decomposition of J into intrinsic (spin) and g extrinsic (OAM) piece QuarkandGluonAngularMomentumintheNucleon–p.9/32 L for proton from Ji-relation (lattice) q lattice QCD moments of GPDs (LHPC; QCDSF) ⇒ ֒ insert in Ji-relation → i i J = S dx [H (x, ξ, 0) + E (x, ξ, 0)] x. q q q Z (cid:10) (cid:11) ֒ Lz = Jz 1∆q q q 2 → − L , L both large! u d present calcs. show L + L 0, but u d ≈ disconnected diagrams ..? 2 m extrapolation π parton interpret. of L ... q QuarkandGluonAngularMomentumintheNucleon–p.10/32
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