CComplete experiments in pseudoscalar mesonphotoproduction
Yannick WunderlichHelmholtz-Institut f¨ur Strahlen- und Kernphysik,Universit¨at BonnNussallee 14-16, 53115 Bonn, GermanyOctober 14, 2015
Abstract
The problem of extracting photoproduction amplitudes uniquely from so calledcomplete experiments is discussed. This problem can be considered either for theextraction of full production amplitudes, or for the determination of multipoles.Both cases are treated briefly. Preliminary results for the fitting of multipoles, aswell as the determination of their error, from recent polarization measurementsin the ∆-region are described in more detail.
For the photoproduction of a single pseudoscalar mesons, i.e. γN −→ P B , it canbe shown that the most general expression for the reaction amplitude, with spin andmomentum variables specified in the center of mass frame (CMS), reads (cf. the workby CGLN [1]) F CGLN = i(cid:126)σ · ˆ (cid:15) F + (cid:126)σ · ˆ q (cid:126)σ · ˆ k × ˆ (cid:15) F + i(cid:126)σ · ˆ k ˆ q · ˆ (cid:15) F + i(cid:126)σ · ˆ q ˆ q · ˆ (cid:15) F . (1)Each spin-momentm structure in this expansion is multiplied by a complex functiondepending on the total energy W and meson scattering angle θ in the CMS. The 4functions { F i ( W, θ ) ; i = 1 , . . . , } are called CGLN-amplitudes and contain all infor-mation on the dynamics of the reaction.Since all particles in the reaction except for the meson P have spin, the preparation ofthe spin degrees of freedom in the initial state as well as the (generally more difficult)1 a r X i v : . [ nu c l - t h ] S e p able 1: The 16 polarization observables accessible in pseudoscalar meson photopro-duction (for a more elaborate version of this Table, cf. [2]).Beam Target Recoil Target + Recoil- - - - x (cid:48) y (cid:48) z (cid:48) x (cid:48) x (cid:48) z (cid:48) z (cid:48) - x y z - - - x z x z unpolarized σ T P T x (cid:48) L x (cid:48) T z (cid:48) L z (cid:48) linearly pol. Σ H P G O x (cid:48) T O z (cid:48) circularly pol. F E C x (cid:48) C z (cid:48) measurement of the polarization of the recoil baryon B facilitate the experimental de-termination of 16 polarization observables, summarized in Table 1. All observables aredefinable as asymmetries among different polarization states (see [2]). They containthe unpolarized differential cross section σ , the three single spin observables { Σ , T, P } (corresponding to beam, traget and recoil polarization), as well as twelve double po-larization observables which are divisible into the distinct classes of beam-target (BT),beam-recoil (BR) and target-recoil (TR) observables.Once the equations connecting the measurable observables to the model independentproduction amplitudes are worked out (reference [2] contains instructions on how to dothis), it becomes apparent that all of these relations can be summarized by the relationˇΩ α = qk (cid:88) i,j =1 F ∗ i ˆ A αij F j = qk (cid:104) F | ˆ A α | F (cid:105) , α = 1 , . . . , . (2)The 16 real profile functions ˇΩ α , connected to the polarization observables via ˇΩ α = σ Ω α , are bilinear hermitean forms in the CGLN amplitudes and can be represented bythe generally complex hermitean matrices ˆ A α (cf. [5] for a listing of those).A change of the basis of spin quantization for the photoproduction reaction allows forthe definition of different systems of spin amplitudes.2elicity amplitudes H i ( W, θ ) or transversity amplitudes b i ( W, θ ) are possible choices(cf. [4]). The different kinds of amplitudes are all related among each other in alinear and invertible way. Therefore, they can be seen as fully equivalent regardingtheir information content. The expressions for the polarization observables in the aforementioned different systems of spin amplitudes retain the mathematical structure ofequation (2), while the observables are now represented by different matricesˇΩ α = qk (cid:104) H | Γ α | H (cid:105) = qk (cid:104) b | ˜Γ α | b (cid:105) . (3)The Γ α (or ˜Γ α in case of transversity amplitudes) are a set of 16 hermitean unitaryDirac Γ-matrices (cf. [4, 5]). They have useful properties, the exploitation of whichfacilitates the identification of complete experiments. Since photoproduction allows access to 16 polarization observables but needs 4 complexamplitudes for a model independent description (constituting just 8 real numbers), thefact can be anticipated that measuring all observables would mean an overdeterminationfor the problem of extracting amplitudes.This issue has triggered investigations on so called complete experiments (cf. [3, 4]),which are subsets of a minimum number of observables that allow for a unique extractionof the amplitudes. Here one generally means unique only only up to an overall phase,since equations (2, 3) are invariant by a simultaneous rotation of all amplitudes by thesame phase. Also, the complete experiment problem is first of all a precise mathematicalproblem disgregarding measurement uncertainties.Chiang and Tabakin have published a solution to this problem (cf. [4]) that shall bedepicted here. First of all it was noted that, using the fact that the ˜Γ-matrices are anorthonormal basis of the complex 4 × b ∗ i b j = 12 (cid:88) α (cid:16) ˜Γ αij (cid:17) ∗ ˇΩ α . (4)This relation allows for the determination of the moduli | b i | and relative phases φ bij ofthe b i and therefore fully constrains them up to an overall phase. Generalizations ofequation (4) for helicity and CGLN amplitudes are possible (see [5]) but shall not bequoted here. 3nother important property of the ˜Γ is that they imply quadratic relations amongthe observables known as the Fierz identities (see [4])ˇΩ α ˇΩ β = (cid:88) δ,η C αβδη ˇΩ δ ˇΩ η , (5)where C αβδη = (1 /
16) Tr (cid:104) ˜Γ δ ˜Γ α ˜Γ η ˜Γ β (cid:105) .Equations (4) and (5) are all that is needed to prove that 8 carefully chosen observablessuffice in order to obtain a complete experiment ([4]). Among those should be theunpolarized cross section and the three single polarization observables. The remainingquantities have to be picked from at least two different classes of double polarizationobservables, with no more than two of them from the same class. The word ’prove’means in this case that for all cases mentioned in reference [4], equation (5) was usedto express the missing 8 observables in terms of the measured ones.In practical investigations of photoproduction data, the goal is not to determine thefull reaction amplitudes, but rather the partial waves, in this case called multipoles. The expansions of the full amplitudes F i into multipoles are known (cf. eg. [2]). In casethese expansions are truncated at some finite angular momentum quantum number (cid:96) max ,an approximation that is justified for reactions with supressed background contributions(eg. π photoproduction), then the profile functions defined in equation (2) can bearranged as a finite expansion into associated Legendre polynomialsˇΩ α ( W, θ ) = qk (cid:96) max + β α + γ α (cid:88) k = β α ( a L ) αk ( W ) P β α k (cos θ ) , (6)( a L ) αk ( W ) = (cid:104)M (cid:96) max ( W ) | ( C L ) αk |M (cid:96) max ( W ) (cid:105) . (7)The parameters β α and γ α defining this expansion are given in Table 2 (the wholenotation is according to [6]).The real Legendre coefficients ( a L ) αk are given as bilinear hermitean forms in terms ofthe 4 (cid:96) max multipoles, which are gathered in the vector |M (cid:96) max (cid:105) . Therefore, the problemof multipole-extraction from a set of fitted coefficients4able 2: The parameters defining the TPWA problem, equations (6) and (7).Type ˇΩ α α β α γ α Type ˇΩ α α β α γ α I ( θ ) 1 0 0 ˇ O x (cid:48)
14 1 0S ˇΣ 4 2 − O z (cid:48) − T
10 1 − C x (cid:48)
16 1 0ˇ P
12 1 − C z (cid:48) G − T x (cid:48) − H − T z (cid:48)
13 1 0ˇ E L x (cid:48) F
11 1 − L z (cid:48)
15 0 +1( a L ) αk leads to a similar mathematical structure compared to the equation (2) en-countered in the investigation of complete experiments in section 2. The question forsuch complete sets can now be asked again, but in the context of a truncated partialwave analysis (TPWA).It is a very interesting fact that in this case, the number of observables that is needed forcompleteness reduces as compared to the case with full production amplitudes. This istrue at least in the mathematically precise situation, without measurement uncertainty.The algebra that is needed to prove this result was first worked out by Omelaenko [7](for a recent and more detailed account, cf. [8]).It is sufficient to investigate the discrete ambiguities allowed by the group S observables,i.e. { σ , Σ , T, P } . It is then seen that the latter are invariant under one mathematicalambiguity transformation, called the ’double ambiguity’, which is present in principlefor all energies. There may also be additional pairs of solutions, called accidential am-biguities, depending on the numerical confguration of the Legendre coefficients. It canhowever be shown that those play no role for the mathematically exact case. The abovementioned double ambiguity on the other hand can be resolved by either the F or G observable, as well as every observable from the BR and TR classes. Therefore one islead to mathematically complete sets containing just 5 observables, for example { σ , Σ , T, P, F } . (8)5 TPWA fits using the bootstrapping method
Here we will describe preliminary results of a TPWA fit to actual data comprising theset of observables (8). The observables σ and Σ are taken from the works [9] and [10].Recent measurements of T and F were performed at MAMI [11]. For P we take theKharkov data [12].The fit procedure proceeds as follows, using a truncation at (cid:96) max = 1 ( S - and P -waves).First, Legendre coefficients are determined by fitting the angular distributions (6) tothe data. The index set for the fitted observables is α F ∈ { , , , , } (cf. Table 2)in this particular case here. In the next step, we minimize the functional (up to nowomitting correlations)Φ M ( M (cid:96) max ) = (cid:88) α F ,k (cid:32) (cid:0) a Fit L (cid:1) α F k − (cid:104)M (cid:96) max | ( C L ) α F k |M (cid:96) max (cid:105) ∆ ( a Fit L ) α F k (cid:33) , (9)using the results from the angular fit and varying the real and imaginary parts of themultipoles (the FindMinimum routine of MATHEMATICA is employed). The overallphase of the multipoles is constrained to Re [ E ] ≥ E ] = 0, since thisphase can never be obtained from a truncated fit to the data alone.In order to exclude any kind of model dependencies, the start parameters for the fit arenot taken from a prediction, but are determined randomly by using a Monte Carlo sam-pling of the relevant, (8 (cid:96) max −
1) = 7 dimensional multipole space (the space spannedby the real and imaginary parts). This sampling is simplified by the fact that the totalcross section ˆ σ , being a sum of moduli-squared of multipoles, already constrains therelevant part of the multipole space to a 6 dimensional ellipsoid. ReE C (cid:43) (cid:64) mFm (cid:68) P r e li m i n a r y (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) ReE C (cid:43) (cid:64) mFm (cid:68) P r e li m i n a r y (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) ImE C (cid:43) (cid:64) mFm (cid:68) P r e li m i n a r y
31 32 33 34 350.00.20.40.60.8
ReM C (cid:43) (cid:64) mFm (cid:68) P r e li m i n a r y (cid:45) (cid:45) (cid:45) ImM C (cid:43) (cid:64) mFm (cid:68) P r e li m i n a r y ReM C (cid:45) (cid:64) mFm (cid:68) P r e li m i n a r y (cid:45) (cid:45) ImM C (cid:45) (cid:64) mFm (cid:68) P r e li m i n a r y Figure 1: Histograms resulting from the bootstrapping at E LAB γ ≈
80 300 320 340 360 380 400 42002468 R e E C (cid:43) (cid:64) m F m (cid:68) P r e li m i n a r y
280 300 320 340 360 380 400 420 (cid:45) R e E C (cid:43) (cid:64) m F m (cid:68) P r e li m i n a r y
280 300 320 340 360 380 400 420 (cid:45) (cid:45) (cid:45) (cid:45) I m E C (cid:43) (cid:64) m F m (cid:68) P r e li m i n a r y
280 300 320 340 360 380 400 4200102030 R e M C (cid:43) (cid:64) m F m (cid:68) P r e li m i n a r y E LAB γ [MeV]
280 300 320 340 360 380 400 420 (cid:45) (cid:45) (cid:45) I m M C (cid:43) (cid:64) m F m (cid:68) P r e li m i n a r y
280 300 320 340 360 380 400 4200246810 R e M C (cid:45) (cid:64) m F m (cid:68) P r e li m i n a r y E LAB γ [MeV]
280 300 320 340 360 380 400 420 (cid:45) I m M C (cid:45) (cid:64) m F m (cid:68) P r e li m i n a r y Figure 2: Results of the bootstrapping procedure for S - and P -wave multipoles (red:Kharkov P data; blue: SAID-predictions for P ). The colored model curves are forcomparison taken from MAID [14] (green), SAID [15] (brown) and Bonn-Gatchina [16](cyan).The amount of N MC = 1000 Monte Carlo start configurations was chosen. It hasto be reported that using this procedure, it was possible to find a pronounced bestminimum for the dataset under investigation.In addition one would wish to have an estimate for the errors of the resulting multi-poles, as well as a check whether the data allow any ambiguities caused by their finiteprecision. To achieve both tasks, a method known as ’bootstrapping’ was chosen ([13]).In this approach, the data are resampled using a gaussian distribution function centeredat µ = ˇΩ α having a width σ = ∆ ˇΩ α for each datapoint. In this way, an ensemble of 250additional datasets was generated, each time starting at the original datapoints. Theabove mentioned TPWA fit procedure was then applied to each ensemble member. If agood minimum is found in each case, one can histogram the results and extract meanand width for each parameter (cf. Figure 1).The bootstrap did not show any indications of ambiguities allowed by the data. There-fore, the results for mean and width of the single solution found can be plotted againstenergy, the result of which is shown in Figure 2.7ecause the errors of the Kharkov dataset are very large, additional fits were per-formed replacing these data by a SAID-prediction for P which has been endowed with a5%-error at each datapoint. The results indicate that the uncertainty of the multipoles,especially for M − , is quite sensitive to this replacement (Figure 2). Mathematically complete sets of observables contain a minimum number of 8 in thecase of spin amplitude extraction and 5 for a TPWA. First investigations of a particu-larly simple fit in the ∆-region confirm the latter result.Bootstrapping methods were proposed in order to get a good estimate for the error ofthe fitted multipoles. This error is seen to shrink in case more precise pseudodata forthe recoil polarization observable P are introduced. Acknowledgments
The author wishes to thank the organizers for the hospitality, as well as for provid-ing a very relaxed and friendly atmosphere during the workshop.This work was supported by the
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