Extraction of Azimuthal Asymmetries using Optimal Observables
EExtraction of Azimuthal Asymmetries using OptimalObservables ∗ J¨org Pretz and Fabian M¨uller
Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany III. Physikalisches Institut B, RWTH Aachen University, 52056 Aachen, Germany JARA-FAME (Forces and Matter Experiments), Forschungszentrum J¨ulich andRWTH Aachen University
Abstract
Azimuthal asymmetries play an important role in scattering processes with polar-ized particles. This paper introduces a new procedure using event weighting to extractthese asymmetries. It is shown that the resulting estimator has several advantages interms of statistical accuracy, bias, assumptions on acceptance and luminosities com-pared to other estimators discussed in the literature. keywords: data analysis, polarisation, likelihood, parameter estimation, event weight-ing, analyzing power, minimal variance bound, Cram´er-Rao bound, optimal observ-ables, generalized method of moments
This paper describes the extraction of an azimuthal asymmetry (cid:15) occurring in an eventdistribution given by N ( ϑ, ϕ ) = 12 π L a ( ϑ, ϕ ) σ ( ϑ ) (1 + (cid:15) ( ϑ ) cos( ϕ )) . (1)The variables in eq. 1 are defined in table 1. Event distributions of this type appear forexample in scattering processes of a transversally polarised beam on a spin 0 target [1].The parameter (cid:15) is the product of the polarisation and an analyzing power, (cid:15) = P A . Once (cid:15) is determined one can either determine the polarisation P if the analyzing power A isknown, or vice versa. To cancel systematic effects, one usually takes two data sets withopposite polarisations, e.g. polarisation up ( P ↑ ) and down ( P ↓ ). The acceptance factor a ( ϑ, ϕ ) may have an arbitrary dependence on the ϕ and ϑ . The only assumption is thatthe acceptance is the same for the two data sets. ∗ accepted for publication in EPJC a r X i v : . [ phy s i c s . d a t a - a n ] J a n n this paper a new estimator using event weights and a χ -minimization is introduced.The method is an application of optimal observables discussed in refs. [2, 3], but it alsotakes into account luminosity and acceptance effects. The paper is organized as follows.In section 2 several estimators to determine (cid:15) (i.e. P or A ) are discussed and compared.Section 2.2 introduces the new method. Possible extensions of this new weighting/fittingmethod are discussed in Sec. 3. In general one can distinguish two classes of estimators: estimators using event counts,discussed in subsection 2.1 and estimators using event weights, discussed in subsection 2.2.
Here events around ϕ = 0 and ϕ = π as indicated by the dark region in figure 1 enterthe analysis. The expectation value for the number of events in the left ( L ) part of thedetector is given by: (cid:68) N ↑ L (cid:69) = 12 π (cid:90) ϕ max − ϕ max L ↑ a ( ϕ ) σ (cid:16) P ↑ A cos( ϕ ) (cid:17) d ϕ (2)= L ↑ a L σ (cid:16) (cid:104) cos( ϕ ) (cid:105) L P ↑ A (cid:17) (3)with a L = 12 π (cid:90) ϕ max − ϕ max a ( ϕ )d ϕ and (cid:104) cos( ϕ ) (cid:105) L = (cid:82) ϕ max − ϕ max a ( ϕ ) cos( ϕ )d ϕ (cid:82) ϕ max − ϕ max a ( ϕ )d ϕ . To simplify the notation the ϑ -dependence is dropped. Similar equations exist for (cid:68) N ↑ R (cid:69) , (cid:68) N ↓ L (cid:69) and (cid:68) N ↓ R (cid:69) .In the cross ratio δ = (cid:68) N ↑ L (cid:69) (cid:68) N ↓ R (cid:69)(cid:68) N ↑ R (cid:69) (cid:68) N ↓ L (cid:69) = (1 + (cid:104) cos( ϕ ) (cid:105) L P ↑ A )(1 + (cid:104) cos( ϕ ) (cid:105) R P ↓ A )(1 + (cid:104) cos( ϕ ) (cid:105) R P ↑ A )(1 + (cid:104) cos( ϕ ) (cid:105) L P ↓ A ) (4)introduced in ref. [4], the usually unknown luminosities, acceptances and unpolarized crosssection cancel. Replacing the expectation values by the actual measured event counts, thefollowing estimator for the analyzing power A can be derivedˆ A = X − (cid:112) X − Y ( δ − Y , with (5) X = P ↓ ( (cid:104) cos( ϕ ) (cid:105) R − (cid:104) cos( ϕ ) (cid:105) L δ ) + P ↑ ( (cid:104) cos( ϕ ) (cid:105) L − (cid:104) cos( ϕ ) (cid:105) R δ ) and Y = 2 (cid:104) cos( ϕ ) (cid:105) L (cid:104) cos( ϕ ) (cid:105) R P ↓ P ↑ ( δ − . y ϕϕ max − ϕ max π − ϕ max π + ϕ max R L
Figure 1: Definition of azimuthal angle and accepted events. The beam moves in z -direction, i.e. out of the plane. 3ote that to evaluate (cid:104) cos( ϕ ) (cid:105) L,R information on the acceptance is needed. This methodwas for example applied in ref. [5]. Here bins of ∆ ϕ = ±
30 degrees were used.Another possibility is to consider estimators of the typeˆ A = 1 P (cid:104) cos( ϕ ) (cid:105) N ↑ ( ↓ ) L − N ↑ ( ↓ ) R N ↑ ( ↓ ) L + N ↑ ( ↓ ) R or ˆ A = 1 P (cid:104) cos( ϕ ) (cid:105) N ↑ L ( R ) − N ↓ L ( R ) N ↑ L ( R ) + N ↓ R ( L ) (6)where various corrections have to be applied in order to compensate for acceptance andluminosity difference between the two data sets. These type of estimators were used inrefs. [6, 7].Common to these estimators is that they reach the same statistical error σ . In generalit is more convenient to work with the figure of merit (FOM) defined by FOM = σ − .To evaluate the FOM we make a few assumptions to simplify the notation: First, P ↑ = − P ↓ , in addition we assume that one takes roughly the same number of events in bothpolarisation configurations. We also assume a uniform acceptance in ϕ . It is straightforward to derive formulas dropping these assumptions but the expressions are gettingcumbersome. These assumptions do not change the overall conclusions comparing differentestimators. Instead of discussing the FOM on A , we will discuss the FOM of (cid:15) .Error propagation from eqs. 5 or 6 leads toFOM counts (cid:15) = N t ot (cid:104) cos( ϕ ) (cid:105) − (cid:104) cos( ϕ ) (cid:105) (cid:15) (7)where N t ot is the total number of events entering the analysis. Details of the calculationare given in app. B.1. Neglecting the term with (cid:15) , one finds:FOM counts (cid:15) = N t ot (cid:104) cos( ϕ ) (cid:105) (8)= N ϕ max π (cid:32) (cid:82) ϕ max − ϕ max cos( ϕ )d ϕ (cid:82) ϕ max − ϕ max d ϕ (cid:33) = N ( ϕ max ) πϕ max (9)where N = (cid:82) π aσ ( L ↑ + L ↓ )d ϕ is the total number of events available in both polarisationstates. Thus N t ot = N (2 ϕ max ) / ( π ) is the total number of events entering the analysis.The full line in figure 2 shows the FOM calculated according to eq. 9 for different ϕ -ranges. Increasing ϕ max , the FOM increases first. Around ϕ max ≈
65 degrees it startsto decrease. The reason is that one adds more and more events where cos( ϕ ) is small.These events carry less information on (cid:15) and dilute the sample in the way the analysis isperformed. This clearly shows that this cannot be the optimal strategy. In the next sectionestimators will be discussed where the FOM reaches the dashed line, which correspondsto the Cram´er-Rao bound. In this section estimators are discussed which use event weights instead of event countsas in the previous subsection. In ref. [8] weighted sums (cid:80) i cos( ϕ i ) are introduced in4ariable meaning N ( ϑ, ϕ ) number of events observed (cid:104) N ( ϑ, ϕ ) (cid:105) expectation value of number of events σ ( ϑ ) unpolarized cross section ϑ polar angle ϕ azimuthal angle, ϕ = 0 corresponds to positive x -direction (cid:15) = P A asymmetry parameter to be determined P beam polarization A ( ϑ ) analyzing power L luminosity a ( ϑ, ϕ ) acceptanceTable 1: Definitions of variables used in eq. 1. max j m a x F O M / F O M /degree max j FOM counts (MC simulation)FOM weighting (MC simulation)FOM counts (analytic)FOM weighting (analytic)
Figure 2: Figure of merit (FOM) for estimators using event counts and event weighting,calculated analytically (lines) and from MC simulation (symbols).5rder to extract (cid:15) . To cancel acceptance effects the authors propose to combine the eventdistributions from the two polarisation states. They do not address the question how todeal with different luminosities in the two different polarisation states. The method wasapplied in ref. [9] where an azimuthal symmetry of the detector is assumed. It is alsoshown in ref. [8] that with this weighting procedure the FOM reaches the Cram´er-Raobound as does the unbinned likelihood method. An unbinned likelihood method was usedin ref. [10]. It is not straight forward to apply because the probability density functionis not completely known. Acceptance effects have to be verified using a Monte Carlosimulation.Now a new method, reaching the Cram´er-Rao bound as well, is introduced. Theadvantage is that no knowledge about the acceptance is required (as long as it is the samefor both data sets, as in any other method) and no corrections concerning the luminositieshave to be applied. On the contrary, information on the acceptance and luminosity factor L σ a are obtained in parallel to (cid:15) in this method.We consider the following six observables N ↑ (cid:88) i =1 cos n ( ϕ i ) and N ↓ (cid:88) i =1 cos n ( ϕ i ) , with n = 0 , , . The sums run over the number of events in the given polarisation state including allazimuthal angels. Note that n = 0 corresponds just to the number of events observed, n = 1(2) are higher moments and correspond to the sum over events weighted withcos( ϕ )(cos ( ϕ )).For an arbitrary acceptance in ϕ we can write the following Fourier series: a ( ϕ ) = a + ∞ (cid:88) n =1 a n cos( nϕ ) + b n sin( nϕ ) . (10)The expectation values of these observables are given by (cid:10) N ↑ (cid:11) = 12 π L ↑ σ (cid:90) π (cid:34) a + ∞ (cid:88) i = n a n cos( nϕ ) + b n sin( nϕ ) (cid:35) (cid:0) P ↑ A cos( ϕ ) (cid:1) d ϕ = L ↑ σ a (cid:16) a a P ↑ A (cid:17) , (11) (cid:42)(cid:88) ↑ cos( ϕ i ) (cid:43) = 12 π L ↑ σ (cid:90) π cos( ϕ ) (cid:34) a + ∞ (cid:88) n =1 a n cos( nϕ ) + b n sin( nϕ ) (cid:35) (cid:0) P ↑ A cos( ϕ i ) (cid:1) d ϕ = 12 L ↑ σ a (cid:16) P ↑ A (cid:16) a a (cid:17) + a a (cid:17) , (12) (cid:42)(cid:88) ↑ cos ( ϕ i ) (cid:43) = 12 π L ↑ σ (cid:90) π cos ( ϕ ) (cid:34) a + ∞ (cid:88) n =1 a n cos( nϕ ) + b n sin( nϕ ) (cid:35) (cid:0) P ↑ A cos( ϕ ) (cid:1) d ϕ = 12 L ↑ σ a (cid:16)(cid:16) a a (cid:17) + 14 3 a + a a P ↑ A (cid:17) . (13) (cid:68) N ↓ (cid:69) , (cid:68)(cid:80) ↓ cos( ϕ i ) (cid:69) , (cid:68)(cid:80) ↓ cos ( ϕ i ) (cid:69) of the second polarisation state by replacing P ↑ with P ↓ . The integrals extend over allazimuthal angles from 0 to 2 π . It is also possible to apply the method for a limitedrange as in the previous section. In this case the integrals would extend over [ − ϕ, ϕ ] and[ π − ϕ, π + ϕ ] (dark region in figure 1).Assuming that the polarisations P ↑ and P ↓ are known, using a χ minimization com-paring the expectation values with the observables, one can determine the following 6unknown parameters: ( L ↑ σ a ) , ( L ↓ σ a ) , a a , a a , a a , A . The χ is given by: χ = ( (cid:126)y o bs − (cid:126)y m odel ) C − ( (cid:126)y o bs − (cid:126)y m odel ) T (14)with (cid:126)y o bs = N ↑ , (cid:88) ↑ cos( ϕ i ) , (cid:88) ↑ cos ( ϕ i ) , N ↓ , (cid:88) ↓ cos( ϕ i ) , (cid:88) ↓ cos ( ϕ i ) ,(cid:126)y m odel = (cid:68) N ↑ (cid:69) , (cid:42)(cid:88) ↑ cos( ϕ i ) (cid:43) , (cid:42)(cid:88) ↑ cos ( ϕ i ) (cid:43) , (cid:68) N ↓ (cid:69) , (cid:42)(cid:88) ↓ cos( ϕ i ) (cid:43) , (cid:42)(cid:88) ↓ cos ( ϕ i ) (cid:43) . The covariance matrix C of the observables is given in app. A. The easiest way to obtainvalues for the parameters is to minimize eq. 14 numerically although analytic, but cum-bersome, expressions exist for the parameters. The numerical solution is also preferredin view of possible extensions of the method discussed in Sec. 3, where analytic solutionsmay not exist.The FOM, calculated using the same conditions as used for FOM counts (cid:15) in eq. 9, isderived in app. B. The final result is:FOM weighting (cid:15) = N t ot (cid:10) cos ( ϕ ) (cid:11) (cid:104) cos ( ϕ ) (cid:105) − (cid:104) cos ( ϕ ) (cid:105) (cid:15) . (15)Neglecting the term with (cid:15) one finds:FOM weighting (cid:15) = N t ot (cid:68) cos ϕ (cid:69) (16)= N ϕ max π (cid:82) ϕ max − ϕ max cos ( ϕ )d ϕ (cid:82) ϕ max − ϕ max d ϕ = N ϕ max + sin( ϕ max ) cos( ϕ max ) π . (17)It is shown as a dashed line in figure 2. At small ϕ max the FOM of counting and weightingestimators coincide, at larger ϕ max , FOM weighting (cid:15) keeps increasing.7 .3 General discussion on the figure of merit In this subsection we make some general remarks about the FOM reachable for eventdistributions of the type n ( ϕ ) = α ( ϕ ) (1 ± β ( ϕ ) (cid:15) ) . (18)As shown in ref. [11] the estimatorˆ (cid:15) = (cid:80) ↑ w ( ϕ i ) − (cid:80) ↓ w ( ϕ i ) (cid:80) ↑ w ( ϕ i ) β ( ϕ i ) + (cid:80) ↓ w ( ϕ i ) β ( ϕ i ) (19)is bias free, where w ( ϕ ) is an arbitrary weight function. The FOM is given byFOM w(cid:15) = N t ot (cid:104) wβ (cid:105) (cid:104) w (1 − (cid:15) β ) (cid:105) . The choice w = 1, or to be more precise w = 1 if the event enters the analysis and w = 0else, results in FOM w =1 (cid:15) = N t ot (cid:104) β (cid:105) (cid:104) (1 − (cid:15) β ) (cid:105) . (20)The choice w = β leads to the largest FOM (in the limit (cid:15) (cid:28)
1) reaching the Cram´er-Raobound: FOM w = β(cid:15) = N t ot (cid:10) β (cid:11) (cid:104) β (1 − (cid:15) β ) (cid:105) . (21)Translated to azimuthal asymmetries the factor β ( ϕ ) equals cos( ϕ ). The two FOMs givenin eq. 7, sec. 2.1 and eq. 15, sec. 2.2 are identical to the FOMs of eqs. 20 and 21,respectively. In this subsection we crosscheck the results of the previous subsections and discuss possiblebias with the help of Monte Carlo simulations. A Monte Carlo simulation with 10 eventsin total was performed by generating data according to eq. 1 for two polarizations stateswith P ↑ = 0 . P ↓ = − . A = 0 .
2. The acceptance was once assumed to beuniform in ϕ and once the following parameterization a ( ϕ ) = 1 + 0 . ϕ ) − . ϕ ) − . ϕ ) + 0 . ϕ )+ 0 . ϕ ) + 0 . ϕ ) − . ϕ ) + 0 . ϕ ) (22)was used. In the analysis it is assumed that a ( ϕ ) is unknown. Table 2 summarizes theresults found using a MINUIT minimization in ROOT [12] to minimize χ in eq. 14. Onesees that with the weighting/fitting method, one recovers the input analyzing power andthe acceptance factors. No bias is observed. The cross ratio method, using events in the8arameter input value cross ratio, counting weighting/fituniform acceptance A . ± . . ± . a /a − . ± . a /a − . ± . a /a − . ± . A . ± . . ± . a /a . ± . a /a − . − . ± . a /a . ± . − . < ϕ max < . A only in the case of uniform ϕ acceptance as expected, since (cid:104) cos( ϕ ) (cid:105) was calculatedunder this assumption.The circles in figure 2 show the FOM obtained from the RMS of 1000 simulations wherethe analyzing powers was calculated according to eq. 5 for various values of ϕ max . Thesquare symbol is the FOM obtained from MINUIT using the weighting/fitting procedure.There is perfect agreement between the simulations and analytic formulas. This subsection discusses some extensions which can be applied to the weighting/fittingmethod but in general not easily to the other methods.If the polarisation vector points for example in an arbitrary unknown direction (cid:126)P = P (cos( ϕ ) , sin( ϕ )) in the x - y plane, the observed signal is N ( ϕ ) ∝ (1 + (cid:15) c cos( ϕ ) + (cid:15) s sin( ϕ )) . (23)In this case, in the analysis one has to include also the sums (cid:88) ↑ sin( ϕ i ) n and (cid:88) ↓ sin( ϕ i ) n for n = 1 , . This gives in total 10 equations for 10 unknowns. The unknowns are( L ↑ σ a ) , ( L ↓ σ a ) , a a , a a , a a , b a , b a , b a , (cid:15) c and (cid:15) s . Including also tensor polarisation for a spin 1 particle, the event distributions reads N ( ϕ ) ∝ (1 + (cid:15) c cos( ϕ ) + (cid:15) s sin( ϕ ) + (cid:15) c cos(2 ϕ ) + (cid:15) s sin(2 ϕ )) . N, (cid:88) i sin n ( ϕ i ) , (cid:88) i cos n ( ϕ i ) , for n = 1 , , , . for now in total three polarisation states. The number of equations increases to 27 for 19parameters ( L ↑ σ a ) , ( L ↓ σ a ) , ( L σ a ) , a a , a a , a a , a a , a a , a a , b a , b a , b a , b a , b a , b a ,(cid:15) c , (cid:15) s , (cid:15) c and (cid:15) s . Looking at eq. 11 to 13, one observes that the parameter a appears only once andeven suppressed with respect to a by a factor 3. One could set a to zero resulting in afit with 6 equations for 5 unknowns, which makes a χ test possible. It is also possibleto add a data set with unpolarized beam to the fit. This is for example useful if the twopolarisations P ↑ and P ↓ are different and not known.It is interesting to note that the method introduced here, especially for the case were thenumber of equations exceeds the number of parameter is a special case of the “GeneralizedMethod of Moments” (GMM) widely used in economics (e.g. see ref. [13, 14]). Two types of estimators to extract azimuthal asymmetries have been compared. One isbased on event counts and one on event weighting. It was shown that estimators just usingevent counts do not use the full information contained in the data. This is reflected in thefact that the figure of merit is smaller than in methods where events are weighted withan appropriate weight. The optimal weight for azimuthal asymmetries is cos( ϕ ). It canalso be shown that using this weight, the FOM is the same as in a maximum likelihoodmethod reaching the Cram´er-Rao limit of the lowest possible statistical error.Among the estimators using event weights the method introduced in this paper hasthe advantage that no knowledge about the acceptance is required and no correction dueto possible difference in luminosity has to be applied. On the contrary, the method evenprovides information on the azimuthal dependence of the acceptance. The method is easilyextendable to more observables. Acknowledgements
The authors would like to thank M. Hartmann for comments and discussions on the paper.This work was triggered by discussions on polarimetry for a storage ring electric dipolemoment (EDM) measurement pursued by the JEDI collaboration and was supported bythe ERC Advanced Grant (srEDM http://collaborations.fz-juelich.de/ikp/jedi/ Covariance matrix of observables
The covariance matrix for the observables (cid:126)y o bs = N ↑ , (cid:88) ↑ cos( ϕ i ) , (cid:88) ↑ cos ( ϕ i ) , N ↓ , (cid:88) ↓ cos( ϕ i ) , (cid:88) ↓ cos ( ϕ i ) is C = (cid:32) C ↑ C ↓ (cid:33) with C ↑ ( ↓ ) = N ↑ ( ↓ ) (cid:80) ↑ ( ↓ ) cos( ϕ i ) (cid:80) ↑ ( ↓ ) cos ( ϕ i ) (cid:80) ↑ ( ↓ ) cos( ϕ i ) (cid:80) ↑ ( ↓ ) cos ( ϕ i ) (cid:80) ↑ ( ↓ ) cos ( ϕ i ) (cid:80) ↑ ( ↓ ) cos ( ϕ i ) (cid:80) ↑ ( ↓ ) cos ( ϕ i ) (cid:80) ↑ ( ↓ ) cos ( ϕ i ) . A derivation of the correlation between sums over events for different weights used herecan be found in ref. [11] (appendix A).
B Figure of merit for cross ratio counting and weighting/fittingmethod
B.1 FOM of counting methods
Assuming P ↑ = − P ↓ and (cid:104) cos( ϕ ) (cid:105) L = − (cid:104) cos( ϕ ) (cid:105) R =: (cid:104) cos( ϕ ) (cid:105) , eq. 5 simplifies toˆ (cid:15) = 1 (cid:104) cos( ϕ ) (cid:105) √ δ − √ δ + 1 . Applying standard error propagation, one finds σ (cid:15) = d (cid:15) d δ σ δ with d (cid:15) d δ = 1 (cid:104) cos( ϕ ) (cid:105) √ δ ) √ δ and σ δ = (cid:115) N ↑ L + 1 N ↑ R + 1 N ↓ L + 1 N ↓ R δ . Using N ↑ L,R = N t ot ± (cid:104) cos( ϕ ) (cid:105) (cid:15) , and N ↓ R,L = N t ot ± (cid:104) cos( ϕ ) (cid:105) (cid:15) , with N t ot = N ↑ L + N ↑ R + N ↓ L + N ↓ R , one finds σ δ δ = 4 √ N t ot (cid:113) − (cid:104) cos( ϕ ) (cid:105) (cid:15) . √ δ = 1 + (cid:104) cos( ϕ ) (cid:105) (cid:15) − (cid:104) cos( ϕ ) (cid:105) (cid:15) we finally arrive at σ (cid:15) = 1 (cid:104) cos( ϕ ) (cid:105) √ δ ) √ δ √ N t ot (cid:113) − (cid:104) cos( ϕ ) (cid:105) (cid:15) δ (24)= 1 (cid:104) cos( ϕ ) (cid:105) (cid:115) − (cid:104) cos( ϕ ) (cid:105) (cid:15) N t ot . (25)The FOM is given by FOM (cid:15) = N t ot (cid:104) cos( ϕ ) (cid:105) − (cid:104) cos( ϕ ) (cid:105) (cid:15) which agrees with eq. 7. For the estimators in eq. 6 the FOM is obtained by a similarprocedure. B.2 FOM of weighting methods
Defining the luminosity factor (cid:96) = L σ a , equations 14 can be linearized around (cid:96) ↑ = (cid:96) , (cid:96) ↓ = (cid:96) , (cid:15) = (cid:15) for a = a = a = 0. Resulting in a system of linear equations y m odel = Ax p ara + y with x p ara = (∆ (cid:96) ↑ , ∆ (cid:96) ↓ , ∆ (cid:15) ) T , y = (cid:96) (cid:18) , (cid:15) , , , (cid:15) , (cid:19) T A = (cid:15) c (cid:96) c c − (cid:15) c − (cid:96) c c and C = (cid:96) c (cid:15) c c (cid:15) c c (cid:15) c c (cid:15) c − c (cid:15) c − c (cid:15) c − c (cid:15) c − c (cid:15) c , with c n = (cid:104) cos( ϕ ) n (cid:105) . The covariance matrix C is the same as in app. A except thatwe used here the expectation values instead of sum over events to arrive at an analyticexpression. The covariance matrix for the parameters (∆ (cid:96) ↑ , ∆ (cid:96) ↓ , ∆ (cid:15) ), which is identical12o the covariance matrix for the parameters ( (cid:96) ↑ , (cid:96) ↓ , (cid:15) ) since they just differ by a constantvector, is given by: C p ara = ( A T C − A ) − = (cid:96) (cid:96)
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