Phenomeological Determination of the Orbital Angular Momentum

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1 Phenomeological Determination of the Orbital Angular Momentum Gordon P. Ramsey Loyola University Chicago and Argonne National Lab Collaboration with Y. Binder & D. Sivers SPIN 2008, University of Virginia-Oct. 9, 2008

2 Outline 1. Proton structure status 2. Modeling the gluon asymmetry a. Physical constraints b. DGLAP evolution 3. Results for the gluon asymmetry 4. Implications for orbital dynamics 5. Phenomenology/conclusions

3 Proton Structure Quark spin (helicity) ΔΣ 0.30 ± 0.03 Transverse components Δ T Σ Gluon spin ΔG Orbital motion L z (quark and gluon) J z sum rule: ½ = ½ΔΣ + ΔG + L z

4 Asymmetry Model Approach Model the gluon asymmetry A(x,t) ΔG(x,t)/G(x,t) where t is the evolution variable t ln[α s LO (Q 02 )]/ln[α s LO (Q 2 )]. Use the models for the asymmetry and the data on G(x,t) to extract ΔG. Combine data on ΔΣ with the J z SR to give insight on the relative size of L z The models of A(x,t) give a range of possible values of L z

5 Motivation Uses the gluon degrees of freedom to access L z, complementary to Skyrme or chiral quark models Provides a theoretical basis for determining ΔG and ΔG/G instead of χ 2 fits to data (which are still limited in kinematic range) Can be combined with other approaches for a cross check of results

6 Definitions Gluon asymmetry: A(x,t) ΔG(x,t)/G(x,t) Split A(x,t) into t-dependent and t-independent parts: A(x,t) = A 0 (x) + (x,t) where A 0 (x) [ ΔG/ t]/[ G/ t] is calculable via DGLAP evolution. Thus, ΔG(x,t) = A 0 (x) G(x,t) + Δg (x) where Δg (x) = G accounts for the difference in ΔG and G evolution at large t.

7 Calculating the asymmetry Choose a suitable model for Δg and use the definition of A 0 (x) to determine the asymmetry. A 0 = [ΔP Gq Δq+ΔP GG (A 0 G+Δg )]/ [P Gq q+p GG G] Due to zeros in the denominator, the equation is transformed into. A 0 [P Gq q+p GG G] - ΔP GG [A 0 G] = [ΔP Gq Δq+ΔP GG (Δg )]

8 Modeling the gluon asymmetry Physical constraints on A 0 Endpoints: A 0 (0) = 0, A 0 (1) = 1 Positivity: A 0 (x) 1 (all x) Monotonicity To satisfy these assume A 0 has the form A 0 Ax α (B 1)x β + (B A)x β+1 and generate Ansätze for Δg : (positivity) Δg dx 0.25

9 Models of Δg Using integral constraints to satisfy positivity, assume Δg has a form: Δg = ± N x a (1-x) b, where the normalization is chosen to model different integral values one model assumes negative correction at small x and positive at large x: Δg = x (x-0.25) (1-x) 5 with <Δg > = 0

10 Asymmetry results Asymmetry extremes as a function of x Middle curve is for Δg = 0, all x. A 0 These are not Constrained by data.

11 Nature of L z Total Start with J z sum rule: ½ = ½ ΔΣ + ΔG + L z (A 0 (x) G + Δg ) + L z L z <(A 0 (x) G + Δg )> ± 0.04 Evolution: L z / t - A 0 (x) [ G/ t] at LO & NLO

12 Constraints as a function of Δg The range of A 0 is near linear in x and satisfies all physical constraints. These are not constrained by data. The models of Δg giving these asymmetries leads to constraints on ΔG and L z Values of Δg satisfying physical constraints: <Δg > 0.25 Constraint on ΔG: without data constraints ΔG dx 0.59 ± 0.04 Constraint on L z : <L z > 0.44 ± 0.04

13 Results with data constraints The three models of A 0 that best satisfy the limited existing data (within one sigma, without the large negative result from charm production) are: Δg <Δg > <ΔG> L z -2(1-x) x 2 (1-x) x(1-x) The corresponding constraints on ΔG and L z are: ΔG dx 0.25 ± L z 0.17 ± 0.04

14 Asymmetry results Asymmetry models as a function of x that are within 1σ of existing data from HERMES and COMPASS. A 0 These are constrained by data.

15 Asymmetry results Asymmetry models as a function of x that are within 1σ of existing data from HERMES and COMPASS. A 0 These are constrained by data. Note that the term =Δg G must be subtracted from the curves to match data.

16 Asymmetry results Asymmetry models as a function of x that are within 2σ of existing data from HERMES and COMPASS. A 0 These are somewhat constrained by data. The blue curve gives <ΔG>=0 and L z = 0.35 consistent with CQPM

17 Other asymmetry models Other models include modifying A 0 with an additional small correction, δa 0, independent of Δg, which truncates at poles of the DGLAP calculation: δa 0 = 0 for x>x c where x c is at the pole (i.e. small x 0.30) tests the stability of the non -linear equations used to calculate the asymmetry These give the unconstrained limits: 0.00 < L z < 0.18

18 L z (x)-speculative Although the J z =½ sum rule holds strictly for the integrals of the distributions, assume as a first approximation that: J z =½ ½ΔΣ(x) + ΔG(x) + L z (x) Using ΔΣ(x) based on CTEQ5 and the GGR polarized distributions and the models of ΔG(x) generated here, we get a general behavior of the x-dependence of the orbital angular momentum of the constituents.

19 L z (x) with <Δg >=0 model Plot of L z (x) vs x. L z (x) Small-x constituents have negative OAM & Large-x constituents have positive OAM x

20 Conclusions 1. The gluon asymmetry ΔG/G has been modeled to satisfy theoretical constraints 2. The LO asymmetry models cluster in a range around the line A 0 = x. NLO calculations add about 10% to the asymmetry at small x. 3. Without data constraints: <ΔG> 0.59 ± <L z > 0.44 ± 0.04

21 Conclusions continued 4. With 1σ data constraints: 0.18 <ΔG> 0.25 ± <L z > 0.17 ± Two σ constraints allow models that give larger L z, more consistent with Covariant QPMs 6. Our results for L z are consistent with ChQMs and provide an alternative way of calculating the OAM using gluon degrees of freedom 7. Modeling the OAM x-dependence using the J z sum rule as a 1 st approximation gives negative OAM at small-x and positive at large-x

22 Final conclusion 8. Measurements of ΔG/G (extension of present experiments) and ΔG alone (jet production and prompt photon production) over a wide kinematic range is very important in narrowing the allowable models of the gluon asymmetry and determination of the OAM of the nucleon constituents 9. Determining transversity properties of the proton can add additional valuable information on the orbital angular momentum of its constituents.

POSSIBL-E EXPERIMENTS ON THE 200-GeV ACCELERATOR. A. D. Krisch University of Michigan. R. Serber Columbia University.

FN-68 POSSIBL-E EXPERIMENTS ON THE 200-GeV ACCELERATOR A. D. Krisch University of Michigan R. Serber Columbia University August 23, 1967 We will describe a number of experiments that might be dcne on.,he