I've spent quite a bit of time trying to understand the significance of Cyrus'
RTWIDDLE, which I'll just call R for simplicity. I find the products of various
moments of the multiplicity distributions rather unintuitive, so I hoped to get
some feeling for what it all means in terms of charged and neutral particle dis-
tributions. I think the exercise has been very educational for me.
First of all some notation: Most of the time I'll use C for the number of
charged tracks in an event and N for the number of neutrals (gammas or pizeros).
I use M = C + N for the total charged+neutral multiplicity. My approach was
to Monte Carlo the DCC and binomial distributions starting with an experimental
distribution for M, which I took from a long-forgotten run where I used an
early version of Ken's tracker for the charge mult. distribution and gamma counter
hits >1 GeV for the neutrols. It turns out the M distribution does not matter at
all, at least in the answers for R, so this is non-controversial, but for the
record, here is the dist. of N+C=M from 0 to 20:
DATA NPLUSC/19309,20967,14906,8092,3955,1883,883,446,
+ 206,119,74,38,19,13,6,4,0,2,1,0,0/ !OBSERVED C+N
I neglect events with M=N+C=0 (which was a fictional number anyway). For M=1,
for example, I generate 20967 events; for N=2, 14906 events, etc. For each M the
distribution of N is chosen to follow a Poisson or DCC distribution, as prescrib-
ed by Cyrus. In this notation we define f=N/(N+C), and for the binomial dist.,
the prob. of getting a particular f is
P(f) = M! f**N (1 - f)**(M-N)/(N! (M-N)! )
For the DCC, P(f) is simply prop. to 1/sqrt(f) . [I normalize everything in
the program so prop. constant doesn't matter.[
In this notation, R = <NC><C>/(<C(C-1)><N>), Cyrus "predicts" that for a
binomial distribution, R =1.0 and for DCC, R=0.50. He also predicts <N>/<C>=
<f>/(1-<f>) for the binomial and 0.5 for the DCC.
Since this is likely to be a long note, let me summarize here. I find essential-
ly perfect agreement with Cyrus for the binomial for any <f> and any M distribu-
tion. Thus I am completely confident that a generic (binomial) distribution gives
R approx. 1.0. For the DCC distribution I also get very good agreement for R
(0.506). However, if I do the MC in what I consider to be the correct way I get
a much lower value for <N>/<C>. I believe the difference lies in the way the
events with N=0 are treated. Since P(f) = 1/sqrt(f) for the DCC distribution,
events with f=0 are highly probable!
Generally I required M>0 (since otherwise we would not get a trigger), and
I allowed N and C to range from 0 to M. I tried to generate the two distribu-
tions is essentially the same way, so that the comparison would be more reliable.
For the DCC distribution I truncated the N=0 probability by replacing N and M by
N+epsilon, M+epsilon where epsilon=.001 usually. I did verify that the results
for R were essentially independent of epsilon.
Now for some graphs. Here is N/(N+C),N vs C, N, C, and N for events with M=10
for the DCC case. Note the high probability for N=0 as expected for a 1/sqrt(f)
probablility dist.
TOT. EVENTS EXPECTED WITH M>0= 51614
N/(N+C)
3300 -
3200 I
3100 I
3000 I
2900 I
2800 I
2700 I
2600 I
2500 I
2400 I
2300 I
2200 I
2100 I
2000 I
1900 I
1800 I
1700 I
1600 I
1500 I
1400 I
1300 I
1200 I
1100 I
1000 I
900 I
800 I
700 I
600 I
500 I
400 I
300 I
200 I
100 I ----- -- --- - ---- - --- - - --- --- - -
CHANNELS 10 0 1 2 3 4 5
1 123456789012345678901234567890123456789012345678901
CONTENTS1000 3
100 2
10 8 112 7 1 2 41 1 3 2 1 3 1 4
1. 7 16129 11 811 0 2202 9 643 2 0 121 743 1 4
LOW-EDGE -
*10** 1 1. 111112222233333444445555566666777778888899999
0 113579135791357913579135791357913579135791357913579
* ENTRIES = 3711 * ALL CHANNELS = 0.3711E+04 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.2000E-01 * MEAN VALUE = 0.5578E-01 * R . M . S = 0.1829E+00
N vs C
CHANNELS 10 U 0 1 2 O
1 N 12345678901234567890 V
****************************
OVE * * OVE
18.5 * * 20
17.5 * * 19
16.5 * * 18
15.5 * * 17
14.5 * * 16
13.5 * * 15
12.5 * * 14
11.5 * * 13
10.5 * + * 12
9.5 * 2 + * 11
8.5 * + * 10
7.5 * ++ * 9
6.5 * 622 * 8
5.5 * C4+2+ + * 7
4.5 * NH93 + + * 6
3.5 * *UK6422 * 5
2.5 * **XDA2++ * 4
1.5 * ****JA8+ * 3
.5 * *****TC7442 + * 2
- .5 * **********WGB6+ 2+ * 1
UND * * UND
****************************
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
N
5000 -
4800 I
4600 I
4400 I
4200 I
4000 I
3800 I
3600 I
3400 I
3200 I
3000 I
2800 I
2600 I
2400 I
2200 I
2000 I
1800 I
1600 I
1400 I
1200 I
1000 I
800 I
600 I
400 I
200 I-----------
CHANNELS 10 0 1 2
1 12345678901234567890
CONTENTS1000 4
*10**- 1 100 81
10 95621
1. 45172521
0 14775410213100000000
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
* ENTRIES = 51606 * ALL CHANNELS = 0.5161E+05 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.1000E+01 * MEAN VALUE = 0.9013E-01 * R . M . S = 0.4684E+00
C
2150 -
2100 I
2050 I
2000 I
1950 I
1900 I
1850 I
1800 I
1750 I
1700 I
1650 I
1600 I
1550 I
1500 I-
1450 II
1400 II
1350 II
1300 II
1250 II
1200 II
1150 II
1100 II
1050 II
1000 II
950 II
900 II
850 II
800 II-
750 I I
700 I I
650 I I
600 I I
550 I I
500 I I
450 I I
400 I I-
350 I I
300 I I
250 I I
200 I I-
150 -I I
100 I I-
50 I I--------- --
CHANNELS 10 0 1 2
1 12345678901234567890
CONTENTS1000 21
*10**- 1 100 114731
10 1057778311
1. 61770509805311
0 65531525689272610210
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
* ENTRIES = 51606 * ALL CHANNELS = 0.5161E+05 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.1000E+01 * MEAN VALUE = 0.2103E+01 * R . M . S = 0.1456E+01
N for N+C=10
58 -
56 I
54 I
52 I
50 I
48 I
46 I
44 I
42 I
40 I
38 I
36 I
34 I
32 I
30 I
28 I
26 I
24 I
22 I
20 I
18 I
16 I
14 I
12 I
10 I
8 I
6 I
4 I-
2 II ---- - -
CHANNELS 10 0 1 2
1 12345678901234567890
CONTENTS 10 5
1. 74 1211 1 2
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
* ENTRIES = 69 * ALL CHANNELS = 0.6900E+02 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.1000E+01 * MEAN VALUE = 0.7826E+00 * R . M . S = 0.2179E+01
R = 0.5064200 TOTGOOD= 51606.00
MEAN N,C,NC,C*(C-1) 9.0125181E-02 2.103496 9.6384138E-02 4.442119
Entries= 51606.00
____________________________
The lines just above give R and the means used to calculate it. The means were calculated
explicitly, but note that they agree with those in the relevant histograms. As I mentioned
above, the value of <N>/<C>=0.043, while Cyrus "predicted" 0.50, as I understand his notes.
I will comment further at the end, but note also that the f and N distributions for DCC
don't look at all like the experimental ones, at least from any run that I've ever looked at.
Below are the same histograms for the Binomial case with <f>=0.333333. As I mentioned
previously I checked explicitly that R does not depend on <f> or the M distribution.
TOT. EVENTS EXPECTED WITH M>0= 51614
N/(N+C)
660 - I
640 I I
620 I I
600 I I
580 I I
560 I I
540 I I
520 I I
500 I I
480 I I
460 I I
440 I I
420 I I
400 I I
380 - I I
360 I I I -
340 I I I I
320 I I - I I
300 I I I I I
280 I I I I I
260 I - I I I I
240 I I I I I I
220 I I I I I - I
200 I I I I I I I
180 I I I I I I I
160 I I I - I I I I
140 I I I I I I I I
120 I I I I I I- I I
100 I -I I I I II I I -
80 I II I -I I -II I I I -
60 I -II I II I I I I -I I I
40 I I I I- II- I I I I II I - I
20 I --I I-II-I I I-I I-- I --II--I I --I -- -
CHANNELS 10 0 1 2 3 4 5
1 123456789012345678901234567890123456789012345678901
CONTENTS 100 3 12 6 1 3 61 2 3
10 6 1504 52 752 1 7712 1 541 8 2 6 1
1. 5 121048971651 8933906 6 4971915 6 224 83 4
LOW-EDGE -
*10** 1 1. 111112222233333444445555566666777778888899999
0 113579135791357913579135791357913579135791357913579
* ENTRIES = 3789 * ALL CHANNELS = 0.3789E+04 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.2000E-01 * MEAN VALUE = 0.3313E+00 * R . M . S = 0.1943E+00
N vs C
CHANNELS 10 U 0 1 2 O
1 N 12345678901234567890 V
****************************
OVE * * OVE
18.5 * * 20
17.5 * * 19
16.5 * * 18
15.5 * * 17
14.5 * * 16
13.5 * * 15
12.5 * * 14
11.5 * * 13
10.5 * * 12
9.5 * * 11
8.5 * * 10
7.5 * + * 9
6.5 * 2 +++ * 8
5.5 * 324334+ + * 7
4.5 * 4IQI98542+ ++ * 6
3.5 * *****KJ76+ * 5
2.5 * ******YJD6+ + * 4
1.5 * *******RI723 * 3
.5 * ********C93 + * 2
- .5 * ******NB + + * 1
UND * * UND
****************************
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
N
2500 -
2400 I
2300 I
2200 I
2100 I
2000 I-
1900 II
1800 II
1700 II
1600 II
1500 II
1400 II
1300 II
1200 II
1100 II
1000 II
900 II
800 II
700 II
600 II-
500 I I
400 I I
300 I I
200 I I-
100 I I-----
CHANNELS 10 0 1 2
1 12345678901234567890
CONTENTS1000 21
*10**- 1 100 4951
10 18843
1. 9246492
0 21115715100000000000
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
* ENTRIES = 51784 * ALL CHANNELS = 0.5178E+05 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.1000E+01 * MEAN VALUE = 0.7323E+00 * R . M . S = 0.8518E+00
C
2250 I
2200 I
2150 I
2100 I
2050 I
2000 I
1950 I
1900 I
1850 I
1800 I
1750 I
1700 I
1650 I
1600 I
1550 I
1500 I
1450 I
1400 I
1350 I
1300 I
1250 I
1200 I-
1150 II
1100 II
1050 II
1000 II
950 -II
900 I I
850 I I
800 I I
750 I I
700 I I
650 I I
600 I I
550 I I
500 I I-
450 I I
400 I I
350 I I
300 I I
250 I I
200 I I-
150 I I
100 I I-
50 I I---------
CHANNELS 10 0 1 2
1 12345678901234567890
CONTENTS1000 21
*10**- 1 100 92141
10 08898731
1. 2226730262
0 64655970246522100000
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
* ENTRIES = 51784 * ALL CHANNELS = 0.5178E+05 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.1000E+01 * MEAN VALUE = 0.1470E+01 * R . M . S = 0.1213E+01
N for N+C=10
19.5
19 --
18.5 II
18 -II
17.5 I I
17 I I
16.5 I I
16 I I
15.5 I I
15 I I
14.5 I I
14 I I
13.5 I I
13 I I
12.5 I I
12 I I
11.5 I I
11 I I
10.5 I I
10 I I
9.5 I I
9 -I I
8.5 I I
8 I I-
7.5 I I
7 I I
6.5 I I
6 I I
5.5 I I
5 I I
4.5 I I
4 I I
3.5 I I
3 I I-
2.5 I I
2 I I
1.5 I I
1 I I
.5 I I
CHANNELS 10 0 1 2
1 12345678901234567890
CONTENTS 10 111
1. 989983
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
* ENTRIES = 76 * ALL CHANNELS = 0.7600E+02 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.1000E+01 * MEAN VALUE = 0.3105E+01 * R . M . S = 0.1314E+01
R = 0.9934390 N/(C+N) 0.3333333 TOTGOOD= 51784.00
MEAN N,C,NC,C*(C-1) 0.7322725 1.470319 1.070292 2.163216
Entries= 51784.00
__________________________
The values of R and the means are again given just above. Note that <N>/(<C>+<N>)
is very close to 0.3333 as advertised and R is very nearly 1.00. Again the means
agree with those in the relevant histograms. Note also that the N distribution is a
lot more like the experimental ones (see below).
To get a better idea of what we are seeing, I looked at "neutrals" from a few
recent runs. Here I define N simply as the number of G counters with energy > 1 GeV.
This is fairly arbitrary but should be within maybe a factor of 2 of pizeros. I would
expect <f> to be considerably larger with this definition than the real f, since there
are of course 2 gammas per pizero and the gammas generally dump their energy in more
than one counter. I have not looked at charged particles at all. Here's the N
distributions for events with 2 or more pbar counters with the right time range.
In the second graph I require a Kicker counter with the right timing. This is for Run
939 which I believe is the run for which Cyrus found the anomalously low value of
R when a Kicker tag was required. Note that the N distributions as measured by
gamma counter hits looked very similar with and w/o the Kicker tag. I looked at
other runs and saw little difference in the N distributions.
N
6400 -
6200 I-
6000 II
5800 II
5600 II
5400 II-
5200 I I
5000 I I
4800 -I I
4600 I I
4400 I I
4200 I I
4000 I I-
3800 I I
3600 I I
3400 I I
3200 I I
3000 I I-
2800 I I
2600 I I
2400 I I
2200 I I
2000 I I-
1800 I I
1600 I I
1400 I I
1200 I I-
1000 I I
800 I I-
600 I I-
400 I I-
200 I I--------
CHANNELS 10 0 1 2
1 12345678901234567890
CONTENTS1000 46653211
100 731298917421
10 8075592564227311
1. 2233472123052170211
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
* ENTRIES = 34118 * ALL CHANNELS = 0.3412E+05 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.1000E+01 * MEAN VALUE = 0.2922E+01 * R . M . S = 0.2375E+01
N with K>0
47
46 -
45 I
44 I
43 I
42 I
41 I
40 -I
39 II
38 II
37 II
36 -II
35 -I I
34 I I
33 I I
32 I I
31 I I-
30 I I
29 I I
28 I I
27 I I
26 I I
25 I I
24 I I
23 I I
22 I I
21 I I
20 I I
19 I I
18 I I
17 I I
16 I I
15 I I-
14 I I
13 I I
12 I I -
11 I I-I
10 I I
9 I I
8 I I-
7 I I
6 I I
5 I I
4 I I
3 I I
2 I I --
1 I I-II -
CHANNELS 10 0 1 2
1 12345678901234567890
CONTENTS 10 33443111
1. 560615128122 1
LOW-EDGE -
10 111111111
1. 123456789012345678
0 55555555555555555555
* ENTRIES = 240 * ALL CHANNELS = 0.2400E+03 * UNDERFLOW = 0.0000E+00
* BIN WID = 0.1000E+01 * MEAN VALUE = 0.3050E+01 * R . M . S = 0.2469E+01
____________________
To my eye at least the experimental N dists for run 939 and nearby look a lot more
like the binomial than the DCC. There is little sign of the large number of events with
N=0 which seems to be predicted for DCC.
I don't have any good explanation for the anomalous R that Cyrus got for Run 939
with the K tag required. I also don't have a good explanation as to why my DCC Monte
Carlo gives the right R, but much lower <N> than Cyrus expects. I suspect it has
something to do with the treatment of events with N=0. I believe I have treated these
sensibly by requiring C+N>0 but allowing C, N individually to be zero.
Comments please!!!
--Mike