The incompleteness of complete pseudoscalar-meson photoproduction

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1 The incompleteness of complete pseudoscalar-meson photoproduction 13 th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon Rome, September 3 - October 4, 213 Tom Vrancx Ghent University, Belgium Phys. Rev. C 87, 5525 (213) (arxiv: ) tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

2 Complete sets in meson photoproduction Theoretical versus experimental complete sets Introduction The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

3 Complete sets in meson photoproduction Theoretical versus experimental complete sets Complete sets in meson photoproduction Complete sets Quantum mechanics: observables can be expressed as bilinear combinations of complex amplitudes Pseudo-scalar meson photoproduction e.g. γp K + Λ 2 kinematical degrees of freedom, e.g. invariant mass W and meson scattering angle θ c.m. 8 spin degrees of freedom (photon, target, and recoil each have 2), so 8 possible but only 4 independent complex amplitudes Quantum states only determined up to constant phase factor, so 4 moduli and 3 relative phases of the amplitudes can be extracted Complete set: set of minimum number of observables from which the moduli and relative phases can be determined unambiguously tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

4 Complete sets in meson photoproduction Theoretical versus experimental complete sets Theoretical versus experimental complete sets Theory Barker, Donnachie, and Storrow: complete sets require 9 observables of a specific type Contested in 1996 by Keaton and Workman, and by Wen-Tai Chiang and Tabakin Wen-Tai Chiang and Tabakin: 8 well-chosen observables define a complete set Need for additional observable (8 observables required to solve for 7 variables) due to phase ambiguity Experiment Observables suffer from experimental uncertainty 8 observables still sufficient to reach situation of complete knowledge about amplitudes? tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

5 Observables in the transversity basis Inferring the amplitudes from the asymmetries Merits of the transversity basis The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

6 Observables in the transversity basis Inferring the amplitudes from the asymmetries Merits of the transversity basis Observables in the transversity basis Transversity amplitudes Unnormalized transversity amplitudes b 1 = y + J y + y b 2 = y J y y b 3 = y + J x y b 4 = y J x + y Normalized transversity amplitudes a j = b j b1 2 + b b b = 4 2 rjeiα j Assymetries Single asymmetries: 1 polarized state, 2 unpolarized states Double asymmetries: 2 polarized states, 1 unpolarized state A = dσ(b 1,T 1,R 1 ) dσ (B 2,T 2,R 2 ) dσ (B 1,T 1,R 1 ) + dσ (B 2,T 2,R 2 ) tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

7 Observables in the transversity basis Inferring the amplitudes from the asymmetries Merits of the transversity basis Observables in the transversity basis (B1, T1, R1) (B2, T2, R2) This work Literature Σ (y,, ) (x,, ) r1 2 + r2 2 r3 2 r4 2 T (, +y, ) (, y, ) r1 2 r2 2 r3 2 + r4 2 P (,, +y) (,, y) r1 2 r2 2 + r3 2 r4 2 Cx (+,, +x) (+,, x) 2Im(a1a 4 + a2a 3) 2Im(a1a 4 a2a 3) Cz (+,, +z) (+,, z) +2Re(a1a 4 a2a 3) +2Re(a1a 4 + a2a 3) Ox (+ π 4,, +x) (+ π 4,, x) +2Re(a1a 4 + a2a 3) +2Re(a1a 4 a2a 3) Oz (+ π 4,, +z) (+ π 4,, z) +2Im(a1a 4 a2a 3) +2Im(a1a 4 + a2a 3) E (+, z, ) (+, +z, ) +2Re(a1a 3 a2a 4) 2Re(a1a 3 + a2a 4) F (+, +x, ) (+, x, ) 2Im(a1a 3 + a2a 4) +2Im(a1a 3 a2a 4) G (+ π 4, +z, ) (+ π 4, z, ) 2Im(a1a 3 a2a 4) +2Im(a1a 3 + a2a 4) H (+ π 4, +x, ) (+ π 4, x, ) +2Re(a1a 3 + a2a 4) 2Re(a1a 3 a2a 4) Tx (, +x, +x) (, +x, x) +2Re(a1a 2 + a3a 4) +2Re(a1a 2 a3a 4) Tz (, +x, +z) (, +x, z) +2Im(a1a 2 + a3a 4) +2Im(a1a 2 a3a 4) Lx (, +z, +x) (, +z, x) 2Im(a1a 2 a3a 4) 2Im(a1a 2 + a3a 4) Lz (, +z, +z) (, +z, z) +2Re(a1a 2 a3a 4) +2Re(a1a 2 + a3a 4) Inconsistency between this work and literature Substitution a 3 a 3 required to obtain agreement with literature Obtained helicity and CGLN representations consistent with literature tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

8 Observables in the transversity basis Inferring the amplitudes from the asymmetries Merits of the transversity basis Inferring the amplitudes from the asymmetries Normalization condition for the a i Inferring the moduli r r2 2 + r2 3 + r2 4 = 1 Moduli can be inferred unambiguously from normalization condition and single asymmetries r 1 = Σ + T + P r 2 = Σ T P Only 3 independent moduli r 3 = Σ T + P r 4 = Σ + T P No elimination of dependent modulus though Dependent modulus will be of importance when finite precision is involved tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

9 Observables in the transversity basis Inferring the amplitudes from the asymmetries Merits of the transversity basis Inferring the amplitudes from the asymmetries Inferring the relative phases 6 possible combinations of relative phases α ij = α i α j (i j), only 3 independent Independent phases: δ i, δ j, δ k (reference phase α l, i j k l) Dependent phases: ij, ik, jk For example, α 4 as reference phase: {δ 1, δ 2, δ 3} and { 12, 13, 23} Specific complete set (3 single, 4 well-chosen double asymmetries) gives access to {δ i, δ j, ik, jk } Two kinds of complete sets: 4 or 8 possible solutions for {δ i, δ j, ik, jk } Actual solution satisfies (trivial) relation Third independent phase δ i ik δ j + jk = δ k = δ i ik or δ k = δ j jk For infinite precision, relative phases can be determined unambiguously tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

10 Observables in the transversity basis Inferring the amplitudes from the asymmetries Merits of the transversity basis Inferring the amplitudes from the asymmetries Finite precision issues Two distinct estimators for third independent phase: δ k = δ i ik and δ k = δ j jk Estimates of independent phases depend on choice of reference phase δ α j i δ α k i δ α k j Consistent set of estimators { δ α l i Consistent estimators δ α j i, δ α l j, δ α l k } for independent phases = δ α k i δ α k j For example, α 4 as reference phase and complete set yields { δ 1, δ 2, 13, 23 } δ 1 = 3 δ ( δ ) 23 δ 2 = 3 δ ( δ ) 13 δ 3 = 1 2 ( δ1 ) ( δ2 ) 23 tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

11 Merits of the transversity basis Observables in the transversity basis Inferring the amplitudes from the asymmetries Merits of the transversity basis Why the transversity basis? Moduli are extracted from single asymmetries Single asymmetries database generally has better statistics compared to double asymmetries Determination of relative phases from double asymmetries requires knowledge of moduli The higher the moduli s precision, the higher the phases precision For example, in helicity basis moduli are extracted from {C z, E, L z } Less precise moduli, hence less precise phases Asymmetry statistics for γp K + Λ 226 single asymmetries (Σ, T, P ) 456 double asymmetries (C x, C z, O x, O z ) tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

12 The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

13 r1 6 cos θ c.m..75 GRAAL.513 cos θ c.m cos θ c.m..154 r2 r3 r tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

14 r1 6 cos θ c.m..75 GRAAL RPR cos θ c.m cos θ c.m..154 r2 r3 r tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

15 .145 cos θ c.m cos θ c.m cos θ c.m. 61 r r r r tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

16 .145 cos θ c.m cos θ c.m cos θ c.m. 61 r r r r tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

17 Simulation strategy Example reaction: γp K + Λ Example complete set: {Σ, T, P ; C x, O x, E, F } Input model: Regge-plus-Resonance (RPR) model Reggeized t-channel background Discrete number of resonances: S 11(1535), S 11(165), F 15(168), P 13(172), P 11(19), P 13(19), D 13(19), and F 15(2) Measured asymmetry simulated by generating events from Gaussian distribution Mean: RPR-211 prediction at certain (W, cos θ c.m.) Standard deviation: desired experimental resolution σ exp Standard error propagation, including correlations σ 2 (f) = ( ) f 2 σ 2 (A i) + f A i i A i i,j i j f A j σ(a i, A j), tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

18 Simulated measurement of complete set Simulated measurement at (W = 17 MeV, cos θ c.m. =.5) for σ exp =.5 Simulation A Simulation B Model Σ.543 ± ± T 95 ± ± P.111 ± ± C x 89 ± ± O x.5971 ± ± E.5655 ± ± F 411 ± ± Extracted moduli Simulation A Simulation B Model r 1.36 ± ± r 2 58 ± ± r 3 42 ± ±.18 4 r 4 48 ± ± tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

19 Simulated measurement of complete set Calculated phase constraints Solution Simulation A Simulation B δ 1 13 δ n σ ci (%) δ1 13 δ n σ ci (%) 1.29 ± ± ± ± ± ± ± ± Most likely solution for each data set Simulation A Simulation B Model δ 1 24 ± 1.74 ± 78 δ ± ±.348 δ ± ± 34 tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

20 Simulated measurement of complete set Calculated phase constraints Solution Simulation A Simulation B δ 1 13 δ n σ ci (%) δ1 13 δ n σ ci (%) 1.29 ± ± ± ± ± ± ± ± Most likely solution for each data set Simulation A Simulation B Model δ 1 24 ± 1.74 ± δ ± ± δ ± ± For data set B, most likely solution is not the correct solution tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

21 Sources of incompleteness Incorrect solutions for the relative phases Imaginary solutions Moduli: when argument of square root becomes negative, inevitably leads to imaginary phases Phases: when discriminant of quadratic equation becomes negative Quantifying incompleteness Introduce insolvability η(w, cos θ c.m.) Fraction of simulated (complete) data sets at (W, cos θ c.m.) that cannot be solved, or are solved incorrectly η = η imaginary + η incorrect Construct insolvability maps for various experimental resolutions σ exp tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

22 12 1 η (σ exp =.1) η incorrect (σ exp =.1) 1.5 cos θcm η (σ exp =.1) η incorrect (σ exp =.1) 1.5 cos θcm FIG. 5: (Color online) The {Cx, Ox, E, F } insolvabilities η = ηimaginary + ηincorrect and ηincorrect as a function of W and cosθcm for two values of the input experimental resolution, namely σexp =.1 and σexp =.1. tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

23 Eliminating incorrect solutions Marking most likely solution as correct solution not statistically sound More conservative approach: imposing tolerance level on confidence intervals Most likely solution only accepted when ci ci tolerance Mean ci values for 1 correct and 1 incorrect solutions at random kinematical points (W < 2.5 GeV) Imposing tolerance level not effective CI (%) σ exp Correct Incorrect Entire elimination of incorrect solutions would lead to rejection of vast majority of correct solutions tom.vrancx@ugent.be The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21

24 Conclusions most advantageous from experimental viewpoint Determination of moduli is unambiguous, however estimates can become imaginary Multiple solutions for relative phases Estimates can become imaginary In absence of imaginary estimates, incorrect solutions cannot be told apart from correct ones Complete sets are experimentally incomplete Study overcomplete sets Outlook Investigate whether one or more additional double asymmetries could help resolve the phase ambiguity The incompleteness of complete pseudoscalar-meson photoproduction MeNu / 21