Determining Momentum and Energy Corrections for g1c Using Kinematic Fitting

Transcription

1 CLAS-NOTE 4-17 Determining Momentum and Energy Corrections for g1c Using Kinematic Fitting Mike Williams, Doug Alegate and Curtis A. Meyer Carnegie Mellon University June 7, 24 Abstract We have used the CLAS Kinematic Fit and the exclusive reaction γ π + π to derive energy and momentum corrections for the g1c run eriod. By selectively leaving articles out of the fit, and then comaring fit and measured quantities, we have been able to derive corrections for the hoton energy and the final state articles. Contents 1 Introduction 2 2 The Procedure The π Sectrum The π + sectrum The sectrum The Photon Beam Energy Comarison of Results Mass eaks for known resonances Summary and Conclusions 18 A π Corrections 19 B π + Corrections 27 C Alying the π + corrections to the roton 3 1

2 1 Introduction In late 23, a Kinematic Fitting ackage for the CLAS exeriments was released by the authors. The details of this ackage can be found in an earlier CLAS-Note [1], and background information on the mathematics of the fitting rocedure are described in a book by Brandt [2]. The urose of this study was to understand what if any momentum and energy corrections should be alied to the data from g1c. 2 The Procedure The γ π + π data set is an excellent environment in which to study the CLAS detector. The reaction is fully reconstructed, which in turn allows one to carry out a 4-constraint (4-C) fit to the data. The ull [3] and confidence-level distributions from this data can then oint one in the correct direction to small roblems in the data set. These small systematic errors can then be studied in detail by selectively leaving out articles in a series of 1-C fits erformed on the data. This study started by examining the momentum resolution of the π by using the reaction: γ π + (π ) missing. Treating the π as an undetected article in a 1-C fit yields a momentum, π (fit), that is not biased by the measured value. The difference, π = π (fit) π (meas) (1) can then be studied as a function of ion angles, momentum and sector number. Following this work, the π is corrected, and one can then study the correction to the π + momentum. Because of the asymmetric bending of ositive and negative articles, these two corrections are not exected to be the same. With this comlete, the roton is examined. While we nominally exect the roton and the π + corrections to be the same, there may be residual issues in the energy loss corrections of rotons versus ions that introduce small differences. All energy loss corrections are erformed using the ELOSS [6] ackage and are erformed before any fits or comarisons. Finally, we can look at the deviation in the tagged hoton energy. This rocess can then be iterated tbtain the best set of corrections. We note here that the changes to the ELOSS corrections along with the corrections to the tagged hoton energy were found to be significant enough to warrant restarting the rocess with first aroximations for these already inserted. The results resented in the following sections were obtained from the final iteration. As a reference oint, we resent the confidence-level and ull distributions obtained by erforming a 4-C fit on the data rior to making any of the corrections resented in this aer (standard ELOSS corrections have been made). Figure 1 shows the confidence level distribution, while Figure 2 shows the ull distributions, the means and widths of the ull distributions are given in Table 1. Both the confidence level and ull distributions look quite reasonable, while it could be argued that some of the ulls are skewed, they are within accetable limits for carrying out kinematic fits. 2

3 C Fit No Corrections Figure 1: The confidence level distribution for the 4-C fit to γ π + π before any momentum corrections have been made(standard ELOSS corrections have been made). Quantity Mean Sigma λ φ π λ π φ π π λ π φ π E γ Table 1: The means and widths of the ull distributions, shown in Figure 2, before momentum corrections have been alied to the data. 3

4 2 x1 2 x1 2 x λ 12 1 φ x x x π + 1 λ π + 1 φ π x π - 2 x λ π - 2 x φ π x E γ Figure 2: The ull distributions for the 4-C fit to γ π + π before any momentum corrections have been made. The ulls have been fit with a Gaussian function which is overlaid as the solid line. The means and σ s are given in Table 1. 4

5 2.1 The π Sectrum When the π is treated as a missing article in a 1-C fit, we can form the quantity π (see Eq. 1). To get an idea of where roblems might be, we made 2-D histograms of vs., θ and φ, then fit each bin to a gaussian and lotted the means. Figure 3 shows the mean of this difference as a function of the π momentum, while Figures 4 and show it as a function of θ and φ resectively. One striking feature seen in all of these lots is the large offset in sector 2 relative to the other sectors in CLAS. Figure 4 shows that this aears to be concentrated at forward angles (θ), and that the deviation can get as large as 3 MeV/c. All other sectors show a significantly smaller deviation. This effect was actually first observed by Kim [4] and is attributed to an alignment issue in the Region-3 drift chambers. Clearly any corrections that we make are going to have to be sector deendent. It is also clear that the corrections have an aarent comlicated deendence on θ, φ and, something noted by Taylor []. Rather than trying to correct the momentum in terms of its magnitude,, we chose to use the, where Bdl B d l is the total amount of magnetic field the article travels through in CLAS. This deendence was seen in earlier work by Kim [4]. The quantity Bdl is comuted by crudely swimming the articles through the CLAS magnetic field. Figure 6 shows a lot of the result of several of these tracks. quantity which is actually measured, Bdl We can then correct Bdl by dividing each sector u into 1 φ bins giving us a correction of the form, Bdl Bdl = α ij + β ij Bdl = a i + b i θ + c i θ 2 + d i θ 3 + e i θ 4 (3) (2) where i = 1, 2,.., 6 is the sector number and j = 1, 2,...1 is the φ bin. The arameters α, β, a, b, c, d and e are constants which are fit to the available data. Terms where Bdl deends only on θ, φ and sector number are due to drift chamber misalignments. Thus, Equation (3) can be interreted as the θ-deendence of the misalignments, while α ij in Equation (2) can be interreted as the φ-deendence Bdl of the misalignments averaged over all θ. The term β ij in Equation (2) accounts for roblems in calculating B d l that could be caused either by inaccuracies in the CLAS magnetic field ma or by drift chamber misalignments leading to systematic shifts in the articles trajectories with resect to the field ma. Figures in Aendix A show the results of lotting Bdl, dividing each sector into 1 φ bins, for the π. Each bin is, to first order, linear in Bdl Bdl vs. as exected. We fit each bin to Equation(2), then aly this correction and lot Bdl vs. θ for each sector (see Figure 24 in Aendix A). We do not correct the measured angles θ and φ. We found the systematic shifts in these angles to be less than 1mrad, which is not enough to ose any roblems during the kinematic fitting rocess. It is imortant to kee in mind that EXCEPT for some forward angle regions in sector 2, the tyical size of these momentum corrections are a few MeV. These are NOT large corrections. Figure 17 shows the corrections to the π momentum as a function of, θ and φ resectively summed over all sectors.

6 Sector 1 Sector Sector 3 Sector Sector Sector Figure 3: vs. : The change in momentum of the π as a function of the π momentum for each of the six sectors in the CLAS detector. 6

7 Sector 1 Sector Sector 3 Sector Sector Sector Figure 4: vs. θ: The change in momentum of the π as a function of the π θ angle for each of the six sectors in the CLAS detector. 7

8 Sector 1 Sector Sector 3 Sector Sector Sector Figure : vs. φ: The change in momentum of the π as a function of the π φ angle for each of the six sectors in the CLAS detector. 8

9 3 2 q = 1 P = 1 GeV/c o Θ = q = 1 P = MeV/c o Θ = q = -1 P = 1 GeV/c o Θ = q = -1 P = MeV/c o Θ = Figure 6: Outut of the mini tracking to comute the B d l along a article trajectory through the magnetic field. 9

10 2.2 The π + sectrum We followed the same rocedure for the π + as we did for the π. We treated the π + as the missing article in a 1-C fit and used the quantity Bdl to determine the corrections to the magnitude of the π + momentum. Figures 26-31, in Aendix B show the results of lotting Bdl Bdl vs., dividing each sector into 1 φ bins, for the π +. Each bin is, to first order, linear in Bdl just as for the π. We fit each bin to Equation(2), then aly this correction and lot Bdl vs. θ for each sector (see Figure 32 in Aendix B). Again, excet for some forward angle regions in sector 2, the tyical size of these momentum corrections are a few MeV. Figure 2 shows the corrections to the π + momentum as a function of, θ and φ resectively summed over all 6 sectors. 2.3 The sectrum If all of the roblems in the measured momentum are due to either magnetic field or alignment issues in CLAS, then one would exect that the corrections for all ositive articles (π +, K + and ) would be the same, and all corrections for negative articles, (π and K ) will be the same. Initially, we were seeing differences between and π +. However, we soon realized that it aeared the ELOSS corrections were still somewhat too small. In Figure 7 we show the ELOSS correction ( ) as a function of the article momentum for the roton. Figure 8 shows the correction to ELOSS needed on the roton as a function of the size of the alied correction. The corrections needed are linear with sector deendent sloes corresonding to about 18%. We attribute this additional correction to energy loss in traversing the tracking region of CLAS. Note that this is only really relevant for low momentum (< 3 MeV/c) rotons. If we aly these sector deendent scale factors to the ELOSS ackage corrections, then the corrections that we derived for the π + in the revious section also work for the rotons. Thus, our corrections deend only on which sector in CLAS the article traveled through and the article s charge. This is consistent with our icture of how these corrections should work. (GeV) P eloss P measured (GeV) Figure 7: Eloss Correction vs. : Correction alied to the magnitude of the roton momenta by the ELOSS ackage. 1

11 ..4.3 Sector Sector Sector Sector Sector..4.3 Sector Figure 8: P vs. Eloss Correction : Correction needed for the roton momentum, after the π + momentum corrections and standard ELOSS corrections have been made, versus the energy loss correction obtained from the ELOSS ackage. 11

12 Sector ELOSS Scale Factor Table 2:. Scale Factors this analysis shows need to be alied to the energy loss corrections obtained from the ELOSS ackage. 2.4 The Photon Beam Energy In the final round of analysis, all final state articles in the detector have had their momentum corrected, and we then want to look at the measured hoton energy. Because it is not trivial to modify the kinematic fit to leave the hoton energy free, we took a somewhat different aroach here. We have used the target mass and the detected final state articles to redict what the hoton beam energy, E γ(fit), should be and then comared it to the measured energy, E γ(measured). So E γ(fit) is defined as, E γ(fit) = E + E π + + E π m. (4) We can then form the difference: E γ = E γ(fit) E γ(measured). () Tbtain a correction for the hoton energy, we first histogrammed E γ vs. E γ. We then fit each bin of this histogram to a Gaussian then lotted the mean versus the hoton energy. Figure 9 shows this histogram fitted to a iecewise quadratic function in E γ. It is interesting that this function is not constant. The double hum structure is similar in shae to the results of Steanyan [7]. (As an aside, there is an absolute offset between the values in this study and those of Steanyan). Since this observation, the shift can be exlained by a calculable sage in the tagger [9] E γ E γ (GeV) Figure 9: Correction to the tagged hoton beam energy as a function of the beam energy. 12

13 3 Comarison of Results After all of the corrections described in this note have been erformed, the resulting confidence-level is shown in Figure 1. Clearly, some of the errors going into the fit were slightly too large. A result that should be exected from the fact that we have corrected 4 of the 1 measured quantities. In our earlier work [1], the errors on the energy-loss were fit to the data. After this study, we refit those and come u with numbers that are about 6% of the earlier error. After we have adjusted the errors to comensate, the resulting confidence level distribution is shown in Figure 11 and the ull distributions are shown in Figure 12. The means and σ s of the ull distributions are shown in Table 3. It should be noted that the error that we adjusted is only one of several comonents that feed into the total error. The effect on the total error is much smaller than imlied by the 6% number, but it is the one degree of freedom that is not a-riori totally constrained by hysics C Fit With Corrections Figure 1: The confidence level distribution for the 4-C fit to γ π + π after all momentum corrections have been made. The rise near 1 demonstrates a need to scale down our errors slightly now that we ve imroved the data being inut to the fit C Fit With Corrections With Adjusted Errors Figure 11: The confidence level distribution for the 4-C fit to γ π + π after all momentum corrections have been made and the errors have been adjusted. 13

14 λ 12 1 φ π λ π φ π π λ π φ π E γ Figure 12: The ull distributions for the 4-C fit to γ π + π after all momentum corrections have been made. 14

15 Quantity Mean(initial) Mean (final) Sigma(initial) Sigma(final) λ φ π λ π φ π π λ π φ π E γ Table 3: The means and widths of the ull distributions before and after all momentum corrections have been alied to the data. 1

16 3.1 Mass eaks for known resonances In order to demonstrate that the kinematic fit is robust against these small errors in the momentum and energy, we have examined the reaction: γ π + π X miss. (6) where the missing article can be either a π or an η. In the case of the π, there are two three ion resonances of interest, η π + π π and ω π + π π. In the case of the missing η, the main resonance of interest is the η ηπ + π. We select the data for dislay by lacing a 1% confidence level cut on the 1C fit to the two channels. In Figure 13, we show the three mass eaks as roduced using unfit and uncorrected momentum. This is effectively the missing mass off the γ X miss sectrum. The eaks have been fit using a Voigtian function [8] with the natural width of the resonance as an inut arameter. Figure 14 shows the same sectrum after all momentum corrections have been made, but again without fitting. There is significant imrovement in the accuracy of masses and a slight inrovement in the widths. In Figure 1 we show the kinematically fit sectrum starting from the uncorrected momentum, while in Figure 16 we show the fits starting from the corrected momentum. In Table 4 we summarize the results of all of these fits. It should be ointed out that the value of the fit mean can vary by about ±.2 MeV/c 2 due to assumtions about background shae and variations in fit ranges. Excet for the case of uncorrected measured data, all of the masses are extremely accurate. The slight imrovements in the width from the momentum corrections translates into a narrower resolution in the fit data as well. Particle Mass [1] Uncor., Measured Cor., Measured Uncor., Fit Cor., Fit Mean σ Mean σ Mean σ Mean σ η ω η Table 4: The mass and widths of known resonances fit using a Voigtian function to derive the mass and resolution width. The four blocks of data corresond to uncorrected momentum using measured values, corrected measured momentum, uncorrected momentum that has been kinematically fit and corrected momentum that has been kinematically fit. All quantities are given in MeV/c Measured M = 1.9 MeV σ = MeV M(π + π - π ) GeV Measured M = MeV σ = MeV M(π + π - π ) GeV Measured M = 96.2 MeV σ =.81 MeV M(π + π - η) GeV Figure 13: The invariant mass sectrum for the η (left), ω (center) and η (right) made using the unfit and uncorrected momentum. 16

17 Measured with Corrections M = 48.4 MeV σ = MeV M(π + π - π ) GeV Measured with Corrections M = MeV σ = 6.32 MeV M(π + π - π ) GeV Measured with Corrections M = 97.2 MeV σ =.88 MeV M(π + π - η) GeV Figure 14: The invariant mass sectrum for the η (left), ω (center) and η (right) made using the unfit but corrected momentum. 22 Kinematically Fit 2 M = 48.7 MeV 18 σ = 4.9 MeV M(π + π - π ) GeV 2 Kinematically Fit 18 M = MeV 16 σ =.869 MeV M(π + π - π ) GeV Kinematically Fit M = 96.6 MeV σ = 4.18 MeV M(π + π - η) GeV Figure 1: The invariant mass sectrum for the η (left), ω (center) and η (right) made using the kinematically-fit uncorrected momentum Kinematically Fit with Corrections M = 47.8 MeV σ = 3.21 MeV M(π + π - π ) GeV 2 Kinematically Fit with 18 Corrections 16 M = MeV 14 σ =.62 MeV M(π + π - π ) GeV Kinematically Fit with Corrections M = 97.3 MeV σ = 4.7 MeV M(π + π - η) GeV Figure 16: The invariant mass sectrum for the η (left), ω (center) and η (right) made using the kinematically-fit and corrected momentum. 17

18 4 Summary and Conclusions We have carried out a detailed study of momentum and energy corrections for the CLAS g1c data. The result of this work is three-fold. First, the hoton beam energy needs to be corrected by the interesting function shown in Figure 9. If the sloe and offset are removed from this function, the wiggles are almost certainly within the design goals of the tagger. However, using this function, we can actually imrove uon that slightly. Second, we believe that the energy loss of the articles as they travel through the tracking region is not accurately handled in CLAS. We have identified a sector deendent effect that is on average about 18% of the ELOSS correction that needs to be added. This correction is needed if we want the residual momentum corrections for all articles of the same sign to be identical. Finally, we have develoed residual momentum corrections for articles that deend only on the sector in CLAS and the charge of the articles. All of these functions will be checked into the CLAS CVS reository. Alying these corrections will rovide imroved reconstructed quantities in CLAS data. We also oint out that even before these corrections, the error functions worked out in our kinematic fit study [1] were of high enough quality that a good kinematic fit could be made. While slight imrovements are achieved by alying the corrections before using a fit, they are NOT necessary tbtain good results kinematically fitting the data. References [1] Mike Williams and Curtis A. Meyer, Kinematic Fitting in CLAS, CLAS-Note 3-17, November 23. [2] Sigmund Brandt, Statistical and Comutational Methods in Data Analysis, North Holland Publishing Comany, (197), (see Chater 9). [3] A. G. Frodesen and O. Skjeggestad, Probability and Statistics in Particle Physics, Columbia University Press, (1979), (see Chater 1 on stretch functions). [4] K. Y. Kim, H. Denizli, J. A. Mueller and S. A. Dytman, General momentum correction form for CLAS: A study of momentum corrections for the Prod 1-9 cooking of e1c data, CLAS-Note 1-18, October 21. [] S. Taylor and G. Mutchler, Analysis udate for E89-24 (radiative decays of excitedstate hyerons), CLAS Analysis Note 23-18, February 24. [6] Eugene Pasyuk, The CLAS ELOSS Package. [7] Stean Steanyan, Evidence for an Exotic Pentaquark Baryon State Θ + (13), in the γd K + K n reaction, CLAS Analysis Note 23-1, July 23. [8] Curtis A. Meyer, Mike Williams and Robert Bradford, Comuting Invariant Masses and Missing Masses, CLAS-Note 4-8, February 23. [9] Dan Sober, rivate communication. [1] The 22 version of the PDG. 18

19 A π Corrections This aendix shows detailed lots of the π momentum corrections. Figures 18, 19,2,21,22 and 23 show the quantity Bdl versus Bdl in sectors 1-6. The lots are broken into ten bins in φ in each sector. Figure 17 shows the corections alied to data from one run in g1c summed over all 6 sectors. These corrections are small, tyically only a few MeV. One interesting feature of these lots is that the sloe decreases as φ increases for each sector. We interret this as a rotation of the drift chambers with resect to the torus coils. Thus, one side of each sector is closer to the coils while the other side is farther away than what is inut to the tracking code. This results in a reorted Bdl that is too large on the side of the sector rotated away from the coils and too small on the side of the sector rotated towards the coils. While the decreasing sloe with resect to φ is common to all sectors, the actual values of the sloes are different in each sector. One interretation for this effect is that the current isn t the same in all six torus coils. In fact, to cause the affect that we see in the data the differences in the current would have to be.2%, or a few ams (GeV) θ φ Figure 17: Corrections to the π momentum as a function of, θ and φ summed over all 6 sectors. 19

20 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 18: Bdl Bdl vs. : The change in Bdl, for the π Bdl, in Sector 1. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 2

21 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 19: Bdl Bdl vs. : The change in Bdl, for the π Bdl, in Sector 2. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 21

22 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 2: Bdl Bdl vs. : The change in Bdl, for the π Bdl, in Sector 3. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 22

23 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 21: Bdl Bdl vs. : The change in Bdl, for the π Bdl, in Sector 4. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 23

24 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 22: Bdl Bdl vs. : The change in Bdl, for the π Bdl, in Sector. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 24

25 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 23: Bdl Bdl vs. : The change in Bdl, for the π Bdl, in Sector 6. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 2

26 Sector Sector Sector Sector Sector Sector Figure 24: Bdl units of kg cm GeV. vs. θ : The change in Bdl, for the π Bdl, in each sector as a function of θ. has 26

27 B π + Corrections This aendix shows detailed lots of the π + momentum corrections. Figures 34, 3,36,37,38 and 39 show the quantity Bdl versus Bdl in sectors 1-6. The lots are broken into ten bins in φ in each sector. Figure 2 shows the corrections made tne run from the g1c data summed over all 6 sectors. The corrections are small, tyically only a few MeV. The π + corrections have a decreasing sloe, in the Bdl Bdl vs. lots, as φ increases. Recall that in Aendix A we also saw this affect in the π corrections and attributed it to a rotation of the drift chambers with resect to the torus coils. A rotation of the drift chambers would be seen in both ostive and negative articles (GeV) θ φ Figure 2: Corrections to the π + momentum as a function of, θ and φ summed over all 6 sectors. 27

28 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 26: Bdl Bdl vs. : The change in Bdl, for the π+ Bdl, in Sector 1. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 28

29 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 27: Bdl Bdl vs. : The change in Bdl, for the π+ Bdl, in Sector 2. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 29

30 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 28: Bdl Bdl vs. : The change in Bdl, for the π+ Bdl, in Sector 3. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 3

31 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 29: Bdl Bdl vs. : The change in Bdl, for the π+ Bdl, in Sector 4. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 31

32 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 3: Bdl Bdl vs. : The change in Bdl, for the π+ Bdl, in Sector. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 32

33 2 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 31: Bdl Bdl vs. : The change in Bdl, for the π+ Bdl, in Sector 6. has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. 33

34 Sector Sector Sector Sector Sector Sector Figure 32: Bdl units of kg cm GeV. vs. θ : The change in Bdl, for the π+ Bdl, in each sector as a function of θ. has 34

35 C Alying the π + corrections to the roton The lots in this aendix show the affects of alying the π + corrections along with the extra eloss correction to the roton. Figures 34, 3,36,37,38 and 39 show the quantity Bdl versus Bdl in sectors 1-6. The lots are broken into ten bins in φ in each sector. The red circles were calculated using the measured momentum with the standard eloss correction alied. The blue triangles were calculated after alying the π + corrections to the roton momentum and using the increased eloss correction. These corrections work quite well, in almost every bin Bdl is consistent with zero. Figure 33 shows the results of alying the π + momentum corrections along with the increase to the ELOSS correction (not the correction itself, just the extra 18%) to the roton momentum for one run in the g1c data. Excet for rotons with mometum less than about 3 MeV/c 2, these corrections are small, tyically less than MeV/c (GeV) θ φ Figure 33: Corrections, including our increase to the eloss correction (but not the standard ELOSS correction), to the roton momentum as a function of, θ and φ summed over all 6 sectors. 3

36 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 34: Bdl Bdl vs. : The change in Bdl, for the roton, in Sector 1. Bdl has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. The red circles were calculated using the measured momentum with the standard eloss correction. The blue triangles were calculated after alying the π + momentum corrections to the roton and using the increased eloss correction. 36

37 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 3: Bdl Bdl vs. : The change in Bdl, for the roton, in Sector 2. Bdl has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. The red circles were calculated using the measured momentum with the standard eloss correction. The blue triangles were calculated after alying the π + momentum corrections to the roton and using the increased eloss correction. 37

38 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 36: Bdl Bdl vs. : The change in Bdl, for the roton, in Sector 3. Bdl has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. The red circles were calculated using the measured momentum with the standard eloss correction. The blue triangles were calculated after alying the π + momentum corrections to the roton and using the increased eloss correction. 38

39 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 37: Bdl Bdl vs. : The change in Bdl, for the roton, in Sector 4. Bdl has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. The red circles were calculated using the measured momentum with the standard eloss correction. The blue triangles were calculated after alying the π + momentum corrections to the roton and using the increased eloss correction. 39

40 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 38: Bdl Bdl vs. : The change in Bdl, for the roton, in Sector. Bdl has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. The red circles were calculated using the measured momentum with the standard eloss correction. The blue triangles were calculated after alying the π + momentum corrections to the roton and using the increased eloss correction. 4

41 φ (-2,-2 ) φ (-2,-1 ) φ (-1,-1 ) φ (-1,- ) φ (-, ) φ (, ) φ (,1 ) φ (1,1 ) φ (1,2 ) φ (2,2 ) Figure 39: Bdl Bdl vs. : The change in Bdl, for the roton, in Sector 6. Bdl has units of kg cm GeV and φ is the sector deendent azimuthal angle, with φ = corresonding to the center of the sector. The red circles were calculated using the measured momentum with the standard eloss correction. The blue triangles were calculated after alying the π + momentum corrections to the roton and using the increased eloss correction. 41

42 Sector Sector Sector Sector Sector Sector Figure 4: Bdl units of kg cm GeV vs. θ : The change in Bdl, for the roton, in each sector as a function of θ. Bdl has. The red circles were calculated using the measured momentum with the standard eloss correction. The blue triangles were calculated after alying the π + momentum corrections to the roton and using the increased eloss correction. 42