RTJ's     PWA Logbook

Richard Jones
started May 20, 1999
last updated Nov 15, 2000

The partial wave analysis of the Jetset data sample has been carried out in collaboration by Antimo and me. I was responsible for laying the theoretical groundwork and producing the amplitude generator, and Antimo carried out the fits. The summary of Antimo's work and discussions with collaborators regarding his results can be found here. Before we go to press with our results, we thought it would be best to do a cross-check of each others' work. This logbook summarizes the results of my own independent pwa that I am carrying out as a part of this cross-check. It is not meant to be exhaustive, but rather to be as quick as possible. We just want to be sure that the major conclusions of our paper survive an independent analysis. The event selection criteria are not being examined. Rather, I obtained the event sample for this analysis directly from Antimo, so we can be sure we are working from the same data. This applies both to experimental data and to Monte Carlo. We are sharing the same subroutine that generates the partial wave amplitudes, that I wrote some time ago, and cross-checked again during the month of March. Other than that, we are not sharing analysis tools.
 

 

Table of Contents

 

Which waves need to be included in the fit?

[RTJ] May 20, 1999

    The somewhat arbitrary rule that we adopted when we started the PWA was that we would include all allowed waves up to and including J=4 and L = 4 (G-wave) where L is the orbital angular momentum of the   final state. The list of all allowed waves under this restriction is shown in Table 3 taken from an earlier draft of the PWA article.

    The symmetries of the strong interaction of the   final state have been very useful in reducing the total number of waves we need to include in the pwa to these 23. The symmetries further simplify the problem by reducing the number of interference terms that contribute to the differential cross section. Cross terms between two waves in the amplitude sum are zero unless the two waves satisfy both of the following criteria.

    1. Total spin S must be equal for the two waves. This turns out to be the same as the statement that two waves of opposite parity cannot interfere. This rule follows from the fact that our initial state is unpolarized.
    2. If J=0 for one of the waves then the other wave must have J=even. This follows from the general properties of Clebsh-Gordon coefficients.
    The vanishing of these interference terms is more than a computational simplification, it also has consequences for the fit. Rule (1) above divides the waves into two groups, those with positive and those with negative parity. Since there are no interferences between the two groups, there is an overall phase for each group that is unobservable. The second rule does not break the waves into two separate groups, and so does not reduce the dimension of the fit. Expressed in terms of real fit coefficients, 23 waves gives 46 parameters, minus two unobservable phases gives 44 for the dimension of the space in which the fit must search for a maximum likelihood.

    In order for the fit to return a reliable error estimate on the parameters it is important that the waves supplied as input be linearly independent. This means that not only must they be mathematically different functions but their orthogonality must survive to some approximation after the experimental acceptance is applied. The partial wave sum has been constructed out of a set of orthogonal functions. This means that when integrated over angles all of the cross terms vanish. This guarantees that the functions are linearly independent, that is, that the angular distribution corresponding to a given set of amplitudes is unique, up to the well-defined set of ambiguities discussed above. This statement is only rigorously correct in the case of unit acceptance.

    To see how the orthogonality of the above waves survives the acceptance, I used the full Monte Carlo sample of events generated according to phase space to numerically integrate over angles every term in the pwa sum. These numbers can be viewed as a matrix with one wave index labeling the row and the other labeling the column. Each element of the matrix was computed together with its error (from Monte Carlo statistics) and is shown in Table 1. In the case of unit acceptance this would have come out as the unit matrix, within errors. Note that the waves have been renormalized so that the diagonal terms are unity. I did this to make it easy to evaluate the effects of acceptance on the distinguishability of waves. If the acceptance were uniform, the off-diagonal elements would be zero. If one of the cross terms were unity, it would mean that those two waves are indistinguishable given our acceptance. The fact that all of the cross terms are at the level of 10% or smaller indicates that even with our large acceptance corrections, from a statistical point of view our apparatus is quite capable of telling all of these waves apart. Of course systematics are the bugger.

    A priori all of these waves are important. I see no reason why any of them can be eliminated from the fit by general arguments or why any of pair of them should be considered redundant. According to our Monte Carlo, they are all distinguishable even after acceptance is applied.

Do we agree on the normalization integrals?

[RTJ] May 21, 1999
[AP] May 24, 1999

    I think there is some problem on some of the integrals. I have plotted my values with fits in the usual web page. They can also be found here 1-9, 10-18 and 19-23.

[RTJ] May 24, 1999

    Yes, I have found the difference. In calculating my normalization integrals I am including all solutions within a certain radius of the peak in the Monte Carlo Goldhaber plot, and then dividing by the total number of solutions. You must be somehow taking just one solution per Monte Carlo event, and then dividing by the number of   events generated. How do you chose which solution to count? The one that lies closest to the peak? We agree at momenta above 1.5GeV because that there is only one solution in the region per event, so you get the same one no matter how you chose it (count all hits within a given radius, chose the nearest to the mass peak per event, or something else). What is happening at low momenta is that 4K phase space has shrunk far enough that my circle around the peak now overlaps with the ridge of "reflection" solutions that lies along the upper diagonal of all of the Goldhaber plots. The angular distribution of these reflection solutions must be radically different from phase space in order to drive some of these normalization integrals crazy as soon as they start to get included in the integral.

    In my opinion, we should stay away from the reflection region of the Goldhaber plots in this analysis. Even at the lowest momentum there is a clear dip between the peak and the reflection ridge. So what I have done is to decrease the size of my circle around the peak for the lowest momentum points. My values for the mass cut radius are given, together with the integrated luminosity, in Table 2 for each data point in our sample. The revised normalization integrals with these modified cut radii are shown in Fig. 2. The agreement with Antimo's fits is now excellent. I do still see regions where the fit deviates from the Monte Carlo data by more than the statistical errors, and so I will continue to use the MC data themselves instead of a fit. But from these plots it is apparent that the normalization integrals cannot give rise to any signficant disagreements between this work and that of Antimo.

[AP] June 3, 1999

    The way the I have selected the solution is to take the one closest to the peak. Here is the slide as a PS file that shows how this works.

[RTJ] June 4, 1999

    I have no real reason to prefer my method of selecting solutions over yours, provided that we restrict ourselves to a region close enough to the peak to exclude reflections. We are now doing essentially the same thing to select our solutions. Now, on to the fit...

Are we using the same resonance lineshape for the phi?

[RTJ] June 6, 1999
[AP] June 8, 1999

    I think you are right concerning the use of the weights from channel likelihood. In this way we also fix the phi phi contribution with respect to background. I presume the method is similar to your way of removing the background. I am repeating the fits using this method. Preliminary results show that the D0 amplitude gets larger and errors became smaller.

[RTJ] June 8, 1999

    Can we compare directly the functions that we are using for the function w(m)? It is a 2D function, but I have made a radial projection around the central mass of the and then cut a slice through the origin to show the shape of the peak. It is plotted in Fig. 1. Note that because it is a ratio of two distributions B(m) / P(m) both the shape and the normalization are significant. We have to agree on this function if our fits are going to be compared. Antimo, can you plot yours in a similar way, so we can compare them?

    The way I obtained this shape was by taking the Monte Carlo Goldhaber plot and making a radial projection to reduce it to one dimension. I defined the range of the radius such that it cut off before reaching the reflection region of the Goldhaber plot. I normalized the MC data to have a unit integral over the Goldhaber plot. This is my Monte Carlo estimator function for B(m). I then did the same for the 4K Monte Carlo, but this time I had to be very careful about the normalization. 4K MC looks like a flat distribution over the Goldhaber plot, but only because you have included all three solutions from every event. Depending on how you chose your solutions, you can get very non-uniform mass distributions for 4K. To keep things simple and smooth, I included all solutions as a part of P(m) and normalized the 4K Monte Carlo to 3 over the whole Goldhaber plot. This is my Monte Carlo estimator for P(m).

    In the end I only consider the region centered around the peak, but in this way my normalization of P(m) is consistent with that described above for B(m). I could have defined P(m) as the distribution of only the nearest solution per event to the peak, and normalized to unity. This will give the same answer as my method (it had better!) provided that the normalization integral for the nearest-solution approach stays away from all of the reflection regions. Since I do not know where all of the reflection areas from the lower-left region lie, I decided my procedure was easier. It probably doesn't matter. Anyway, since Antimo is using the nearest-solution approach, this alternative approach gives us a cross check.

    To obtain the function w(m) I first smooth the P(m) radial plot by a polynomial fit, and then divide the B(m) plot by it. The resulting Monte Carlo estimate for w(m) I then fit to a double-gaussian with 4 parameters: two heights and two sigmas. The mean values are fixed at the mass. This double-gaussian fit provides an analytic function that gets evaluated for every event and provides the values for w(m) that go into the likelihood.

    Let's compare our w(m) functions and make sure we agree before we go on and do the angular distributions. I now have results for the partial wave analysis, by the way, but don't want to show anything until we have set up the problem in the same way and can make an intelligent comparison of our respective results.

[AP] June 10, 1999

    I agree with the equations you are using. I have attached the plot you are asking for.

[RTJ] June 11, 1999

    The scale and resolution of this plot make the shape of the peak hard to read. I have plotted the same data, but with better resolution, for a couple of points in our momentum scan. Here they are for point 7 and point 10. These data are the values of w / w' in the language of Eqn.7. On these plots I have superimposed my weight function w(m) from Eqn.3. With the higher resolution in these plots, you can see several different trajectories. These come from the different runs that were combined together to make the one data point. They all have the same shape, but different normalizations that reflect the different admixtures of background that channel likelihood saw for each run. Variations of a factor of 2-3 like this are reasonable between runs, seeing how the conditions of the trigger (especially the Cerenkov) changed from year to year. By comparison, my w(m) function has a fixed normalization that is derived from Monte Carlo and the relative sizes of the background and signal contributions to the Goldhaber plot are adjusted (parameter c) during the pwa fit. The comparison should address the shape only, and not the relative scale. The conclusion from these two plots is that there is very little difference between the my peak shape that Antimo is now using. We are making real progress.

[RTJ] August 16, 1999

    In the procedure I am following, both the fraction and the angular distribution parameters emerge from the same fit. So the first check on the validity of the results is to look at the quality of the fit on the mass spectra. This amounts to integrating over all angles and plotting the distribution in masses. The fit only covers a limited region of the Goldhaber plot that lies within a circle whose radius varies with energy and is chosen to exclude the reflection band. The results of the fit are shown in Table 4 below. The corresponding results from channel likelihood, labeled as Antimo in the plot, are also shown. Channel likelihood provides an analysis of the entire Goldhaber plot and so the column labeled fit radius does not apply to Antimo's results.

Table 4: summary of results from the maximum likelihood fit to all of the Jetset 4K data.
point
no.
momentum
range (MeV)
integrated
luminosity (/nb)
fit
radius (MeV)
N
(this work)
N
(this work)
N
(Antimo)
1 1180-1200 86.566 25 355 ± 24 66 ± 16 379 ± 28
2 1220-1246 94.778 30 451 ± 27 188 ± 22 489 ± 32
3 1260-1280 89.767 35 587 ± 30 309 ± 25 647 ± 35
4 1300-1330 100.170 40 777 ± 36 432 ± 30 844 ± 39
5 1345-1390 103.494 45 938 ± 41 656 ± 38 1006 ± 42
6 1390-1404 94.122 50 1121 ± 43 726 ± 38 1241 ± 46
7 1405-1430 104.093 55 1549 ± 51 1119 ± 47 1724 ± 54
8 1435-1465 88.509 60 1217 ± 47 1026 ± 45 1321 ± 48
9 1465-1500 80.625 60 1152 ± 45 1043 ± 44 1211 ± 47
10 1505-1650 73.836 60 903 ± 43 1096 ± 45 860 ± 42
11 1700-1800 70.218 60 668 ± 32 1860 ± 47 926 ± 46
12 1900-2000 66.883 60 640 ± 47 1733 ± 57 739 ± 42
    The x and y phi-band plots showing the quality of the fits can be seen by clicking on the underlined values in the column labeled N. The red curve overlayed on the data in the plots is the sum of the function B(m) and a flat background, each normalized to their respective yields from the fit. A flat background is shown rather than the phase-space function P(m) because the data actually reflect the product P(m)A(m,) of phase space times acceptance. Over most of the energy range of this experiment the product is strikingly close to uniform across the Goldhaber plot. The phi-band plots are only shown to suggest the quality of the agreement between data and fit.

How do we know when we have a good fit?

[RTJ] August 25, 1999

    By a good fit we mean one that is statistically compatible with the data. With binned data the ² provides the classic means for judging goodness of fit. In the case of our angular distributions, however, binning the data is not appropriate. A single measurement is represented by 6 angles (3 pairs): the production angles, and the two sets of decay angles. One of these, the production azimuth, can be discarded because of symmetry. Even so, if one were to divide each axis up into 30 divisions, the full histogram would contain in excess of 24 million bins, most of whom would be empty. Reducing the number of bins substantially would introduce a bias to our PWA by filtering out waves in the data that vary more rapidly in angle than others. Thus we need a method for judging goodness of fit that does not require binned data, for which the classic choice is the likelihood ratio test.

    The likelihood ratio test works as follows. Suppose one has a Monte Carlo generator that generates according to a distribution controlled by M parameters. He can then use this generator to create any number of samples of a given size, each of which are statistically consistent with the parent distribution, but independent of each other. These samples can then be submitted to a fit, and maximum likelihood estimates of the original parameters can be obtained for each sample generated. For each sample one then has two numbers, L0 and Lmax. Lmax is the value of the likelihood at the maximum, the value that is returned from the fit. L0 is the value of the likelihood for the sample if one calculates it using the true parameters that were used by the generator. These two are different numbers because the fit will prefer parameters slightly away from the true ones in the general case for finite samples. It can been proved that the quantity 2ln(Lmax/L0) is distributed as a ² with M degrees of freedom. The conditions for this proof are fairly general, but it is strictly only true in the limit of large sample size.

    This test can be turned around to function as a goodness-of-fit test in the following way. The "true" values of the parameters (PWA amplitudes) are unknown to us. But the maximum likelihood test tells us that if a given set of parameters might be the true ones (statistically compatible) then the likelihood L0 must be close enough to Lmax to give a reasonable value for the log-ratio statistic given above.

[RTJ] August 25, 1999

    First I want to test the MLT on Monte Carlo. To do this, I produced 100 different sets of initial PWA amplitudes, chosen at random. For each of the 12 data points I then generated 100 Monte Carlo data sets, with similar statistics in and 4K events to what is shown in Table 4. In this test we are looking for two deviations. The test statistic is expected to behave like a ² with M degrees of freedom, 45 in the present case. If the test statistic comes out systematically low then it means that the effective number of degrees of freedom that are controlling the shape of the data is less than we think. This is possible if, for example, the same or similar shape can be made by different combinations of waves. Note that it is only events within the acceptance of our apparatus that are contributing to the likelihood; changes in the shape of the parent distribution that take place outside the acceptance do not affect the likelihood analysis. On the other hand, if the test statistic comes out systematically higher than expectation then it means that there are systematic errors in the model that are biasing the maximum likelihood fit away from the true parameters. One example of such a systematic is detector angular resolution, whose effects are not taken into account in our PWA model. Such problems are expected to show up in the fits to Monte Carlo as well as in real data.

Table 5: Monte Carlo check of the Likelihood Ratio Test applied to 100 random samples of similar composition to the real data
point
no.
momentum
range (MeV)
N in N out N in N out 2 ln(Lmax/L0)
1 1180-1200 346 ± 44 370 ± 46 66 ± 0 42 ± 12 32 ± 10
2 1220-1246 411 ± 87 415 ± 85 186 ± 0 183 ± 12 27 ± 11
3 1260-1280 576 ± 46 563 ± 47 315 ± 0 328 ± 17 32 ± 9
4 1300-1330 731 ± 104 671 ± 95 449 ± 0 513 ± 21 46 ± 12
5 1345-1390 881 ± 170 835 ± 158 637 ± 0 681 ± 28 39 ± 11
6 1390-1404 1072 ± 122 970 ± 113 740 ± 0 840 ± 27 56 ± 13
7 1405-1430 1486 ± 220 1366 ± 202 1137 ± 0 1261 ± 41 66 ± 11
8 1435-1465 1202 ± 84 1062 ± 84 1044 ± 0 1179 ± 34 63 ± 14
9 1465-1500 1148 ± 32 991 ± 41 1063 ± 0 1211 ± 34 80 ± 16
10 1505-1650 902 ± 30 802 ± 39 1101 ± 0 1210 ± 28 62 ± 13
11 1700-1800 668 ± 26 325 ± 40 1414 ± 0 1754 ± 37 178 ± 27
12 1900-2000 639 ± 26 527 ± 35 1223 ± 0 1339 ± 27 64 ± 12
[AP] October 14, 1999

    If you go on my pwa web page you will find the information you asked for. In particular, the list of waves is explained in the text dated 26 May 1999. The amplitudes are (13,11,12,x,y) where x,y may change from one bin to the other and are written in the minuit output. The list of amplitudes is always in my JETSET NOTE 96-01.

[RTJ] October 14, 1999

    I found the results indicated above from Antimo here. Without changing anything, I simply copy the final amplitudes that Antimo has provided and plug them into my fit. I want to emphasise that this is an important test of the robustness of these results, because of the following differences between my and Antimo's approach.

    1. Antimo has a strict cut around the peak to select those solutions that are admitted to the pwa. From the 1996 draft of the PWA paper he records that only mass bins with over 50% from channel likelihood are included. I estimate that this results in a cut radius of about 20MeV around the center of the peak. For my analysis, I have drawn as large a circle as I can have and still avoid the kinematic reflection band in the Goldhaber plot. The radii I am using are shown in column 4 of Table 4.
    2. For the shape of the peak, Antimo is using the weights from channel likelihood, whereas I am using a mass- dependent fit to projections of the Goldhaber plot to a sum of two Gaussians.
    3. We use different ways of normalizing the angular distribution. Antimo fixes the amplitude of the leading wave to unity and then divides by the normalization integral at each point. I use the extended likelihood procedure. Extended likelihood treats all amplitudes as independent variables, but automatically constrains the total expected yield to equal the total number of observed events through the use of something like a Lagrange multiplier.
Point 3 above states that we have different definitions for the likelihood, which means that a comparison is only possible by translating Antimo's amplitudes from his into my language (or vice-versa) and calculating the likelihood again. Point 2 above implies that Antimo and I will come up with somewhat different numbers for the total counts of and background events. This is really orthogonal to what we are after in this study, however, which is to determine the goodness-of-fit of the angular distributions to the data. Hence any decrease in the likelihood that I calculate for Antimo's fit parameters that comes from his shift in the total cross section away from my best-fit value should be removed before we do the comparison. That way any likelihood decrease that is observed can be associated with the angular fit alone.

Both of these things are actually easy to do. First I wrote a small script to input Antimo's parameters from his results file and write input data files in the format required for my fit function. Then I fixed the angular amplitude coefficients in MINUIT and told it only to treat the total yields of and background events as fit variables. Following this two-parameter fit, I come up with a new likelihood that I call Antimo's likelihood. This can be compared directly with my Lmax from the full 23-wave fit. The results are shown in Table 6 below.
Table 6: comparison of Antimo's pwa solution with that obtained with a full 23-wave fit.
point
no.
momentum
range (MeV)
N
(full fit)
N
(full fit)
N
(Antimo)
N
(Antimo)
2 ln(Lmax/L0)
1 1180-1200 355 ± 24 66 ± 16 332 ± 22 89 ± 15 35.7
2 1220-1246 451 ± 27 188 ± 22 427 ± 23 212 ± 18 42.2
3 1260-1280 587 ± 30 309 ± 25 570 ± 27 326 ± 21 50.2
4 1300-1330 777 ± 36 432 ± 30 765 ± 30 444 ± 24 63.6
5 1345-1390 938 ± 41 656 ± 38 922 ± 33 672 ± 29 55.0
6 1390-1404 1121 ± 43 726 ± 38 1096 ± 31 751 ± 25 49.6
7 1405-1430 1549 ± 51 1119 ± 47 1511 ± 37 1157 ± 32 80.0
8 1435-1465 1217 ± 47 1026 ± 45 1175 ± 32 1068 ± 31 58.2
9 1465-1500 1152 ± 45 1043 ± 44 1105 ± 30 1090 ± 30 86.2
10 1505-1650 903 ± 43 1096 ± 45 854 ± 26 1145 ± 31 88.0
11 1700-1800 668 ± 32 1860 ± 47 643 ± 16 1885 ± 39 62.6
12 1900-2000 640 ± 47 1733 ± 57 576 ± 16 1797 ± 38 85.4

    The full fit shown in Table 6 is the same as that reported in Table 4. It contains 23 waves plus one parameter for background for a total of 45 degrees of freedom. Antimo's solution is restricted to 5 waves per point, so it carries 11 degrees of freedom. This means that the hypothesis entailed in Antimo's solution is equivalent to the statement that the numbers in the last column are distributed as a with 34 degrees of freedom. When compared with the last column in Table 5, these data are in fair agreement with the hypothesis.

    The small errors reported for the resonant and non-resonant yields from Antimo's solution is a reflection of the fact that these were obtained from my own refit where I froze the angular distributions according to Antimo's solution but allowed the fit to reapportion the total yields, as I explained above. Freezing the angular distributions reduces the statistical errors on the yields because the errors do not include the statistical fluctuations coming from uncertainty in the detector acceptance that arises from errors in the angular distribution.

[RTJ] October 15, 1999

    Antimo's solution can be taken at two levels:

    1. a minimal set of waves that gives an acceptable fit
    2. the best-fit values for the amplitudes of the selected waves
    Given the differences in the way that Antimo and I chose the data for the pwa and fit the peak shape, it may be that we will find different values for best-fit amplitudes even if we use the same waves. To check this out, I redo the fit to find my best-fit values for (b) above, given (a). Table 7 below is a copy of Table 6 except that I have refit the amplitudes in addition to the channel yields.

Table 7: comparison of a full 23-wave fit with that obtained by restricting the fit to the waves from Antimo's solution.
point
no.
momentum
range (MeV)
-ln(Lmax) (full fit) -ln(L0) (Antimo) -ln(L5) (5-wave refit) 2 ln(Lmax/L5) correlation factor
1 1180-1200 3239.9 3261.2 3254.2 28.6 0.978
2 1220-1246 4997.7 5021.3 5018.2 41.0 0.985
3 1260-1280 6851.0 6879.0 6875.9 49.8 0.990
4 1300-1330 8898.8 8933.5 8930.4 63.2 0.986
5 1345-1390 11842.4 11871.4 11866.1 47.4 0.984
6 1390-1404 12755.6 12783.5 12774.0 36.8 0.993
7 1405-1430 18297.1 18341.1 18319.3 44.4 0.988
8 1435-1465 15671.4 15702.8 15690.4 38.0 0.982
9 1465-1500 14897.9 14941.6 14922.9 50.0 0.988
10 1505-1650 13653.8 13697.9 13684.4 61.2 0.990
11 1700-1800 15948.2 15995.1 15974.8 53.2 0.968
12 1900-2000 14294.1 14334.7 14318.6 49.0 0.933

    The hypothesis (a) above is sustained if the values in the next to last column of Table 7 are compatible with a with 34 degrees of freedom. I would argue that these data support the hypothesis, but the matter should be discussed. The last column was included to give an idea of how different the angular distributions coming from the 5-wave refit are from those of Antimo. The correlation is defined as the expectation value of (xy - <x><y>) where x is the angular density function from Antimo's fit and y is the one coming from my 5-wave refit. The expectation value is evaluated on Monte Carlo data. The high degree of correlation shows that, the two solutions have essentially the same shape, at least over that part of phase space where we have data. The agreement is impressive, in view of the fact that we admit different events to the pwa, and gives us confidence that the shapes we extract are not too sensitive to these kind of details.

How can we relate the fit parameters of AP to those used by RJ?

[RTJ] February 25, 2000
[AP] April 10, 2000

    From values of x and the amplitudes c in my fit, how do I calculate the partial cross sections for each wave?

[RTJ] April 13, 2000

How do the results compare with the total cross section we published?

[RTJ] March 7, 2000
Table 8: comparison of channel apportioning from the full pwa fit with the channel likelihood results, and between the cross section using phase-space acceptance and that obtained using the acceptance from the full pwa fit.
point
no.
momentum
range (MeV)
N
(full fit)
N
(full fit)
N
(ch.likelhd.)
N
(ch.likelhd.)
(PRD)
(µb)
(full fit)
(µb)
1 1180-1200 381 ± 28 40 ± 21 379 ± 28 42 ± 28 0.69 ± 0.11 1.45 ± 0.70
2 1220-1246 488 ± 32 151 ± 26 489 ± 32 150 ± 32 0.82 ± 0.07 1.05 ± 0.70
3 1260-1280 665 ± 35 231 ± 28 647 ± 35 249 ± 35 0.84 ± 0.07 1.10 ± 0.60
4 1300-1330 854 ± 49 355 ± 44 844 ± 39 365 ± 39 0.85 ± 0.04 3.00 ± 2.00
5 1345-1390 1011 ± 36 583 ± 30 1006 ± 42 588 ± 42 0.92 ± 0.06 0.77 ± 0.50
6 1390-1404 1251 ± 46 596 ± 38 1241 ± 46 606 ± 46 0.86 ± 0.04 3.50 ± 1.80
7 1405-1430 1738 ± 43 931 ± 32 1724 ± 54 944 ± 54 0.92 ± 0.06 1.45 ± 0.30
8 1435-1465 1358 ± 46 885 ± 41 1321 ± 48 922 ± 48 0.82 ± 0.05 2.50 ± 1.20
9 1465-1500 1214 ± 34 981 ± 30 1211 ± 47 984 ± 47 0.74 ± 0.04 1.30 ± 0.50
10 1505-1650 888 ± 38 1111 ± 40 860 ± 42 1139 ± 42 0.54 ± 0.04 2.40 ± 1.40
11 1700-1800 937 ± 23 1591 ± 34 926 ± 46 1602 ± 46 0.48 ± 0.03 1.10 ± 0.50
12 1900-2000 730 ± 21 1643 ± 37 739 ± 42 1634 ± 42 0.38 ± 0.04 2.00 ± 1.10

[PER] March 21, 2000
[RTJ] March 23, 2000

    The two columns sigma(PRD) and sigma(full fit) represent the same number, somehow. But we don't measure a total cross section in Jetset because of our limited acceptance. One way to get a total cross section is to make an assumption about the shape of the angular distribution, say just uniform phase-space. Then the yields just scale into a cross section with no inflation of errors. Another way is to get the shape from a fit to the angular distributions, and then integrate the shape to get the cross section. That way you get inflation of errors because the fit is unconstrained in regions where we have no data, but those regions contribute to the integral. So if the fit is underconstrained then you should see the errors blow up because of the uncertainty in how the shape extrapolates to regions of zero acceptance. When errors get larger than the estimated value and the estimator is positive definite by construction then there is a systematic bias on the high side of the best value. The conclusion from this is that phase-space underestimates our errors, a zillion-wave fit goes to the other extreme, and we are seeking a few-wave fit that is essentially as good but incorporates reasonable assumptions about the convergence of the expansion in angular momentum instead of an ad-hoc assumption of phase-space.

    N is the yield after the total cross section has been multiplied by acceptance. Antimo's number N(ch.likelhd.) is just a fit of the Goldhaber plot. To get something to compare with him, I took the angular shape from the zillion-wave best fit for each point, froze the shape coefficients for the angular distributions and asked the fit to just tell me how much of the sample was phi,phi and how much background (parameterized as phase-space). The agreement shows that the channel apportioning gets the same answer independent of assumptions about angular distributions. This result had better be the case because the two factorize in the physical cross section. The agreement also shows that two different people using different tools on two different continents can do the same calculation and get the same answer.

[RTJ] March 24, 2000

    Seeing Paul's reaction to Table 8, I think it would be more enlightening to look at similar quantities but where the 5-wave fit is being compared with the PRD values, where the total cross section should be better constrained. These results are shown below in Table 9.

Table 9: comparison of channel apportioning from the 5-wave pwa fit with the channel likelihood results, and between the cross section using phase-space acceptance and that obtained using the acceptance from the 5-wave pwa fit.
point
no.
momentum
range (MeV)
N
(5-wave fit)
N
(5-wave fit)
N
(ch.likelhd.)
N
(ch.likelhd.)
(PRD)
(µb)
(flat)
(µb)
(5-wave fit)
(µb)
1 1180-1200 371 ± 22 50 ± 13 379 ± 28 42 ± 28 0.69 ± 0.11 0.37 ± 0.06 0.52 ± 0.10
2 1220-1246 468 ± 19 171 ± 7 489 ± 32 150 ± 32 0.82 ± 0.07 0.40 ± 0.04 0.32 ± 0.09
3 1260-1280 652 ± 27 244 ± 18 647 ± 35 249 ± 35 0.84 ± 0.07 0.50 ± 0.04 0.36 ± 0.07
4 1300-1330 840 ± 29 369 ± 19 844 ± 39 365 ± 39 0.85 ± 0.04 0.62 ± 0.03 0.41 ± 0.06
5 1345-1390 1005 ± 31 589 ± 23 1006 ± 42 588 ± 42 0.92 ± 0.06 0.53 ± 0.03 0.38 ± 0.02
6 1390-1404 1245 ± 32 602 ± 20 1241 ± 46 606 ± 46 0.86 ± 0.04 0.86 ± 0.04 0.57 ± 0.07
7 1405-1430 1732 ± 38 936 ± 25 1724 ± 54 944 ± 54 0.92 ± 0.06 0.92 ± 0.06 0.63 ± 0.05
8 1435-1465 1333 ± 33 910 ± 26 1321 ± 48 922 ± 48 0.82 ± 0.05 0.83 ± 0.05 0.58 ± 0.07
9 1465-1500 1206 ± 29 989 ± 25 1211 ± 47 984 ± 47 0.74 ± 0.04 0.72 ± 0.04 0.52 ± 0.05
10 1505-1650 876 ± 24 1123 ± 28 860 ± 42 1139 ± 42 0.54 ± 0.04 0.49 ± 0.04 0.41 ± 0.04
11 1700-1800 916 ± 21 1612 ± 33 926 ± 46 1602 ± 46 0.48 ± 0.03 0.43 ± 0.03 0.46 ± 0.04
12 1900-2000 701 ± 16 1672 ± 35 739 ± 42 1634 ± 42 0.38 ± 0.04 0.39 ± 0.04 0.51 ± 0.05

    The last three columns in Table 9 require explanation. The PRD cross section is just what we published in 1998. The 5-wave fit cross section in what comes back from the PWA fit for the total resonant cross section when I put in the 5 waves recommended by Antimo. There are three reasons for the differences between the PRD and 5-wave fit values.
     

    1. Data selection:
      Certain data sets were excluded from the analysis that was used for our PRD paper because there were normalization problems. Notably, most of the 1994 data were excluded. These data are almost a third of our total statistics, and are included in the PWA because we considered that the normalization problem was only an issue of scale and not of the shape of the angular distributions. Also, the PWA data had a cut placed on the RICH to select 4K, whereas the RICH was not used in the PRD analysis.

    2. Acceptance:
      The net acceptance depends on how the events are distributed in angles. The acceptance functions obtained from the fit are compared in Fig. 3. There are 15% jumps between points because of running under different trigger conditions, but generally the momentum-dependence is smooth. The total cross section comes from dividing the yield by acceptance, so increased acceptance results in a decreased cross section.

    The cross section column labeled flat in Table 9 was made to enable the separation of these two effects. To obtain the flat cross section values, I took the yields from my fit and corrected them with an acceptance calculated from the Monte Carlo I am using for the PWA, but assuming flat angular distributions. This takes out the acceptance effects, so any remaining discrepancy wiht the PRD value must come from item (1) above.

    Several things emerge from looking at Fig. 3. The solid curve I obtained by digitizing the image from our PRD paper, Fig. 6a. I obtained the black points from the Monte Carlo used in this analysis without any weighting of angular distributions (i.e. events generated with uniform angular distributions). The departure of points 1-5 from the curve is interesting, and correspsonds to the discrepancy between the PRD and flat values for those points in Table 9. In drawing the curve the way we did in the PRD Fig. 6 I am guessing that we judged the acceptances for points 1-5 were too high. Is this correct Antimo?

    In what follows, I am going to ignore any putative normalization corrections. We can discuss that later, as regards what we put into the paper for the sake of consistency with the former publication, but for the purposes of the PWA I am not going to fiddle. The red points in Fig. 3 follow a smooth curve, apart from wiggles that mimic those in the black dots. The question of resonances in our data set are independent of broad smooth deformations of the acceptance function, or at least had better be if we are to believe our conclusions. This is why we decided to include these data in the PWA in the first place. My conclusions are (1) that the change in the acceptance function due to angular distribution effects is real and large, but smooth as a function of beam energy, and (2) normalization questions affect the magnitude of the cross section at the low end of our energy range, but in examining the overall behavoior of the partial waves the acceptances from Monte Carlo will be taken at face value. Lets now go on and look at the results.


 

Do we have evidence for a resonance at 2.230?

[RTJ] March 24, 2000

    The results of the PWA are given in Table 10. The two different sets of solutions correspond to slightly different parameterizations of the resonance lineshape in the Goldhaber plot. Set B was obtained using a double-Gaussian fit to the Monte Carlo Goldhabber plot. The shape makes a good visual fit to the real mass spectrum in the -band plot, but gives a systematic bias towards low yields when tested on known mixes of resonant and 4K Monte Carlo as shown in Table 5. Set A was obtained by tweaking the resonant lineshape in the Goldhaber plot until this bias is removed. Set A tends to have longer tails to higher masses than appears in the Monte Carlo mass plots, which explains why it finds more in the data than set B. Visually, it is not possible to distinguish the two lineshapes by examining the Goldhaber plot of the real. Hence I consider the differences between sets A and B indicative of the systematic errors implicit in the parameterization of the resonance lineshape.

Table 10: results from PWA analysis of Jetset data, using a 5-wave ansatz for the angular distributions and two slightly different parameterizations (A and B) for the shape of the in the Goldhaber plot. The PWA for each point is completely independent of the others. For an index of the quantum numbers for each wave, see Ref. [2].
wave
no.
Set A Set B
cross section relative phase cross section relative phase
11 xs11a.eps ph11a.eps xs11b.eps ph11b.eps
12 xs12a.eps ph12a.eps xs12b.eps ph12b.eps
13 xs13a.eps (reference wave) xs13b.eps (reference wave)
15 xs15a.eps ph15a.eps xs15b.eps ph15b.eps
16 xs16a.eps ph16a.eps xs16b.eps ph16b.eps
20 xs20a.eps ph20a.eps xs20b.eps ph20b.eps
tot xsta.eps   xstb.eps  

    The consistency between sets A and B is excellent and shows that we are not sensitive to the details of the resonance lineshape. The phases have been plotted with repeated data points every 2 so that the natural flow from point to point is not interrupted by corssing through some arbitrary principal value cut. What is not shown on the plot is that the overall sign of the phases is not determined in the fit; there is an overall sign ambiguity in the phase. That is unfortunate because we cannot tell whether it is the 3++ wave or the 2++ waves that is responsible for the resonant structure at 1.450 that is seen in the relative phase between waves 15 and 13. I would argue from the shapes of the partial cross section that the 2++ waves are responsible; they all show some evidence of a peak around 1.450. Although no one of them is overwhelming, the sum shows a clean enhancement there, and the peak in the total cross ection is clearly coming from them. Whatever you think of that argument, I think the answer to the question, do we see a resonance at 2.230 with a narrow width decaying to . I think the answer is yes!

    So what's next? One thing anyone looking at these data is going to ask is, did you try extending wave 16 to the entire data set. If the resonance is 3++ you might see it in that wave as well. As it is, changing waves from 16 to 20 right in the vicinity of our peak might raise some questions. Antimo, did you every try a 6-wave hypothesis and use the same set of waves across the entire energy range?

[RTJ] April 26, 2000
Table 11: comparison of the 5-wave fit (Antimo's solution) to a 6-wave fit that employs all of the waves from the 5-wave fit for all of the points instead of switching waves midway across the mass range.
point
no.
momentum
range (MeV)
-ln(Lmax) (full fit) -ln(L5) (5-wave fit) -ln(L6) (6-wave fit) 2 ln(Lmax/L5) 2 ln(Lmax/L6) correlation factor
1 1180-1200 3239.9 3254.2 3251.5 28.6 23.3 0.969
2 1220-1246 4997.7 5018.2 5008.9 41.0 22.5 0.925
3 1260-1280 6851.0 6875.9 6873.2 49.8 44.4 0.970
4 1300-1330 8898.8 8930.4 8924.4 63.2 51.2 0.950
5 1345-1390 11842.4 11866.1 11863.8 47.4 42.8 0.958
6 1390-1404 12755.6 12774.0 12771.7 36.8 32.2 0.994
7 1405-1430 18297.1 18319.3 18316.0 44.4 37.8 0.992
8 1435-1465 15671.4 15690.4 15684.7 38.0 26.6 0.982
9 1465-1500 14897.9 14922.9 14920.4 50.0 45.0 0.997
10 1505-1650 13653.8 13684.4 13681.3 61.2 55.0 0.993
11 1700-1800 15948.2 15974.8 15968.2 53.2 40.0 0.992
12 1900-2000 14294.1 14318.6 14311.8 49.0 35.4 0.926

[RTJ] November 15, 2000

 

Useful links

[1] Antimo's logbook on partial wave analysis
[2] Wave table listing the quantum numbers of the waves that are considered in the PWA.

This page is maintained by Richard Jones.
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