Coulomb multiple scattering signal in vertex
Mon Nov 29 19:21:24 CST 1999
The alternative measurements of particle momentum are highly important for
kink reconstruction. As usual, several kink solutions for several hyperon
masses and different Z positions of decay vertices would pass reasonable
criteria.
The alternative method can be useful to find wrong momenta due to wrong track
link. For example, analyzing Monte Carlo data on Lambda_c decays into sigma
hyperons I observed 1200 from 2900 events when hyperon decaying upstream m1
spectrometer was linked to foreign m1 segment (see Table 5).
The measurements based on Coulomb multiple scattering (Cms) signal have
very long history. One can measure a track radius induced by Cms or track angle
after scatterer or track position relative to some known vector etc. This is
good to know average momentum of a data sample.
Numerous measurements of track angle at short distances when a distribution
is about of Gaussian and sigma is proportional to 1/p can result in a very good
momentum resolution limited only by statistics.
I used this technique and 12 correlated quantities to measure particle
momentum in vertex spectrometer. We have 20 planes of silicon (Table 1).
They produce Cms and measure track position with accuracy of about few um.
I combined 3 plane measurements to obtain one measurement of Cms angle as
theta = (d3-d2)/(z3-z2) - (d2-d1)/(z2-z1)
where di are hit positions relative to vertex track segment vector and zi are
plane positions. The usage of relative hit measurements is important only if
we combine different views (e.g. 12-th combination in Table 2). The hit
distributions in vertex track segment for 6 planes are shown in fig.1
The distributions of theta(1)*theta(1), theta(4)*theta(4) and correlated
term theta(1)*theta(4) at 4 values of particle momentum are in fig.2
These distributions are scaled by factors printed in histogram titles. Those
factors were calculated in first alignment run and then were used to make 4
sigma cut in second run. The mean values of the above distributions for all
combinations at 20 values of particle momentum are the subject of alignment.
Some data are presented in Tables 1-4. The combinations have various
sensitivity. The sensitivity was estimated as the ratio of signals at 10 GeV
and 250 GeV (infinity) as well as in terms of P* - a momentum at which a Cms
error is equal to measured error (learned from Peter Cooper).
I don't know why they are so different but the Monte Carlo data show just
the same quantities. I was surprised how good MC does the work in vertex. So,
this is software product. For myself I learned that best measurements were
done at shortest distances. In future one should try to add or replace some
combination to increase the sensitivity of the method.
Using alignment data we constrain 12x12 correlation matrices S(m) at all 20
momentum points sigma vectors Sg(m). The determinants of those matrices are
given in Table 4. Then at any of 20 points we can write Likelihood Function:
chi2= 0.5*(REAL(ncmb)*alog_twopi+alog(abs(Det)))
do i=1,ncmb
chi2=chi2+alog(Sg(i))
do j=1,ncmb
k=(i-1)*ncmb+j
chi2=chi2+0.5*dS(i)*dS(j)*B(k)/(Sg(i)*Sg(j))
enddo
enddo
Lh=-chi2
where dS(i) is measured signal in combination (i) and B is inverse S.
In the intermediate momentum points the subroutine p_from_s recalculates
vector and correlation matrix using 1/p approximation. The order of matrix
can be less then 12 but must be greater then 5.
Using this subroutine p_from_s I processed several 1M files. I selected
pfit tracks with 20 hit in vertex and 4 momentum interval:
(10+-1) (20+-1.5) (30+-2) and (95+-15) GeV
The resulting momentum spectra are presented in fig.3
and corresponding likelihood distributions in fig.4
The momentum spectra in left part were obtained using all 12 combinations,
the spectra in right part are the combined result of the Fit with Ndof=12
and 11 when the smallest signal was rejected. As usual the difference in
the predictions is less than 1 GeV, but sometimes it is huge!
The likelihood function in Fig.4 on left are the values calculated for known
input momentum and to the right the maximum values from the fit.
I don't understand why the average value is not a constant, but practically
I know that if I make a cut on 2 than the momentum spectra in Fig.3 will be
more narrow and without long tail to high values.
This might be a case when we should cut very large Chi2 as well as very small?
The example, how Cms signal can be used to kink selection is given in fig.5
This is Monte Carlo sample for Lambda_c decays with alignment run 1820.
Short resume is printed in Table 5. Two histograms on top of figure show
all kink solutions for a pair of tracks assigned to Monte Carlo sigma decay.
Left picture shows the solutions accepted by Cms cut and right picture shows
rejected solutions. The choice was done as 1 standard deviation around the
solution with maximum likelihood value (+-0.5). Two bottom histograms show
what was happened with solution which has nearest to Monte Carlo momentum.
For example, in sip case (sigma+ -> proton) the ratio of accepted/rejected
solutions is equal to 1470/1723 and the ratio of accepted/rejected sigmas
is equal to 616/373. Do we have a positive balance?
The code will be placed into off781 soap area.
The subroutine: p_from_s.F must be in source directory,
common block: strk_bk.inc in include directory,
alignment file: cms_10783.con.v01 and user_msp.F - example how to
get Fig.3 and Fig.4 in example directory.
They can be found in ~vmatveev/V0/ as well.
data from file :cms_10783.con.v01
-------------------------------------
3h msc table 1 : planes
plane view Z res_plcon w_plcon angle degr.
1 181 1 3.9670 0.115E-02 0.750E+06 0
2 182 2 5.4370 0.577E-03 0.300E+07 -90
3 183 3 8.5390 0.577E-03 0.300E+07 -135
4 184 4 10.0160 0.577E-03 0.300E+07 +135
5 185 1 12.9580 0.577E-03 0.300E+07 0
6 186 2 14.4270 0.577E-03 0.300E+07 -90
7 187 3 17.6330 0.577E-03 0.300E+07 -135
8 188 4 19.1230 0.577E-03 0.300E+07 +135
9 192 1 21.9450 0.722E-03 0.192E+07 0
10 195 2 23.4750 0.722E-03 0.192E+07 -90
11 198 3 26.6030 0.722E-03 0.192E+07 -135
12 201 4 27.9820 0.722E-03 0.192E+07 +135
13 204 1 30.9450 0.722E-03 0.192E+07 0
14 207 2 32.4130 0.722E-03 0.192E+07 -90
15 210 3 35.5210 0.722E-03 0.192E+07 -135
16 213 4 37.0470 0.722E-03 0.192E+07 +135
17 216 1 39.7880 0.722E-03 0.192E+07 0
18 219 4 41.3800 0.722E-03 0.192E+07 -15
19 222 3 44.5180 0.722E-03 0.192E+07 -165
20 225 1 46.0850 0.722E-03 0.192E+07 +180
3h msc table 2 : combinations
combi pla1 pla2 pla3 vi scatters norm dZ1 dZ2 Z
1 181 185 192 111 4.500 4.500 3.2404 8.9910 8.9870 3.97
2 185 192 204 111 4.500 4.500 3.2404 8.9870 9.0000 12.96
3 192 204 216 111 4.500 4.500 3.2404 9.0000 8.8430 21.94
4 204 216 225 111 4.500 3.500 3.0822 8.8430 6.2970 30.94
5 182 186 195 222 4.500 4.500 3.2404 8.9900 9.0480 5.44
6 186 195 207 222 4.500 4.500 3.2404 9.0480 8.9380 14.43
7 183 187 198 333 4.500 4.500 3.2404 9.0940 8.9700 8.54
8 187 198 210 333 4.500 4.500 3.2404 8.9700 8.9180 17.63
9 198 210 222 333 4.500 4.500 3.2404 8.9180 8.9970 26.60
10 184 188 201 444 4.500 4.500 3.2404 9.1070 8.8590 10.02
11 188 201 213 444 4.500 4.500 3.2404 8.8590 9.0650 19.12
12 201 213 219 444 4.500 2.500 2.9155 9.0650 4.3330 27.98
3h msc table 3 : sensitivity
comb entry sig infinity ratio at 10 GeV P* correlation with
1 15239 0.0576 0.9975 1.319 0.936 8.597 -0.3928 2
2 21545 0.0550 0.9907 1.471 0.872 10.793 -0.4459 3
3 19325 0.0531 0.9628 1.784 0.813 14.772 -0.4378 4
4 16339 0.0571 0.9367 5.229 1.031 51.329 -0.4631 12
5 20744 0.0437 0.9868 1.412 0.948 9.966 -0.4139 6
6 21491 0.0508 0.9843 1.467 0.884 10.733 0.1774 12
7 20189 0.0446 0.9926 1.443 0.918 10.402 -0.3948 8
8 21136 0.0538 0.9821 1.493 0.872 11.088 -0.4257 9
9 18515 0.0519 0.9657 2.160 0.863 19.142 -0.3396 12
10 20598 0.0449 0.9932 1.477 0.911 10.865 -0.4025 11
11 21424 0.0521 0.9872 1.535 0.877 11.646 -0.4490 12
12 19256 0.0781 0.9399 4.473 1.027 43.597 0.0000 0
3h msc table 4 : points
mom P dP min sta max sta Det
1 5.00 0.50 31171. 59981. 0.01436
2 10.00 1.00 53249. 99443. 0.01482
3 15.00 1.50 57628. 105269. 0.01155
4 20.00 2.00 62712. 112075. 0.00898
5 25.00 2.50 70931. 123073. 0.00600
6 30.00 3.00 69838. 119459. 0.00447
7 35.00 3.50 60359. 102324. 0.00378
8 40.00 4.00 51317. 87232. 0.00320
9 45.00 4.50 42411. 71729. 0.00287
10 50.00 5.00 35966. 61309. 0.00285
11 60.00 6.00 57073. 97768. 0.00253
12 70.00 7.00 42445. 73266. 0.00230
13 80.00 8.00 31849. 55699. 0.00206
14 90.00 9.00 24478. 44010. 0.00195
15 100.00 10.00 19369. 35436. 0.00172
16 120.00 12.00 27456. 51877. 0.00154
17 150.00 15.00 21428. 42788. 0.00153
18 170.00 17.00 9945. 20572. 0.00127
19 200.00 20.00 10019. 21136. 0.00144
20 250.00 50.00 10031. 21545. 0.00145
Table 5 : Monte Carlo Kinks in Lambda_c decays.
Total 50000 CUT 1 killed 2253 no Lambda_c
Total 47747 CUT 2 killed 16266 no Sigma
Total 31481 CUT 3 killed 10068 Z kink < 46
Total 21413 CUT 4 killed 3999 Z kink > 400
Total 17414 CUT 5 killed 249 no kink daughter ?
Total 17165 CUT 6 killed 10056 Z end daughter < 1100
Total 7109 CUT 7 killed 425 sigma track lost by tracking
Total 6684 CUT 8 killed 2443 kink daughter lost by tracking
Total 4241 CUT 9 killed 0 sigma track was linked to other (not kink)
Total 4241 CUT 10 killed 1307 lost in V0
Total 2934
--------------------------------
sip-sta 1147 in one track 307 foreign tail 354
sin-sta 562 in one track 11 foreign tail 247
sim-sta 1225 in one track 18 foreign tail 612
Events with SV 1729 Total sv 19235
vmatveev@fnal.gov