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Lifetime determinations in the presence of cuts
Nuclear Instruments and Methods in Physics Research A274 (1989) 557-559 North-Holland, Amsterdam
557
LIFETIME DETERMINATIONS IN THE PRESENCE OF CUTS Louis LYONS
Nuclear Physics Laboratory, Keble Road, Oxford, UK Peter CLIFFORD
Mathematical Institute, St. Giles, Oxford, UK Received 14 March 1988 and in revised form 22 August 1988 We discuss the determination of the lifetime of a short lived particle from a set of N observed decay times t when for each observation there is a minimum and a maximum observable time t,"'" and t,"ax . By incorporating information concerning the expected distribution of tin" and tmax, we obtain a more accurate estimate than by using the individual t,"'" and tlmax values . Thus is a specific example of the more general point that extra theoretical input can sometimes be used to improve the accuracy of estimates. One of the basic experimental problems in atomic, nuclear and elementary particle physics is the determination of the lifetime of an exponentially decaying state. In one class of this type of experiment, we observe the individual times t, (i = 1, 2, . . ., N), at which N particles decay. Experimental conditions are usually such that for each particle, decays can be observed over the range of times between tin and tmax . In the traditional approach, the lifetime T is estimated by maximising the likelihood Y= n `
1e
T ^ax
1m,
-r,/T
1 - e-`/Tdt T
We consider an improved approach where, by using information concerning the expected distributions of t `"`n and/or tmax, we obtain a more accurate estimate than by using their individual values in eq . (1) above. We discuss two separate cases. (1) tmax cuts . We consider the determination of the lifetime of our decaying particles, in the situation where the t7n are all zero, but the t," are finite . We achieve the improvement by using the extra information that the production distribution along the length L of the fiducial region is expected to be uniform . Thus the probability of having a potential length l for decay is and hence that of having a potential length greater than 1 0 is
Y(lo)LX to
(1)dl=1-la/L .
(The modifications required to allow for the attenuation of the beam along the target and for nonforward production angles are easy and trivial respectively .) In terms of times, this becomes y
( ) = 1 -t /t Ôax , t
where to ax is the time taken for a particle of momentum p, to cross the fiducial volume, i.e . (5)
tô a= mL/P,
Then, given the uniform distribution of production positions, the probability of observing a decay at time t, and with a potential decay time t,max greater than t, is dt
ax). e',/T(1 t,/t0_ T
We finally write a likelihood function for the observed set of events as Y=
dn d t,
II ft~" dn r
dt,
where the integration in the denominator can be performed analytically . We note that the likelihood function (7) does not contain the individual potential times t,", but only IÔ ax .
We have compared the performance of the two likelihood functions (1) and (7) by Monte Carlo calculations. We have generated samples of decays corresponding to a lifetime to , for a set of particles produced uniformly along a fiducial region of length L, and all travelling with the same momenta. Decays occurring beyond the end of the fiducial region were rejected .
558
L. Lyons, P. Clifford / Lifetime determinations m the presence of cuts 30% r
Conventional approach Improved method
Or/T o
20°îo
4
10
40
L (cm) -"
100
400
022 0`/To
020
0.18
016
014
2
4
6 ymm
8
10
12
14
16
(cm)
Fig. 1. (a) The fractional error a/-ro on the estimate of the lifetime for "experiments" of 500 events, as a function of the length L of the visible region of the detector, when the mean decay length is 6 cm. (b) The fractional error a/,ro for a sample of 25 events with impact parameters y greater than yin , . The hfetime of the K° is taken as 10 -1° s. The triangles are for the conventional treatment of the data ; the circles are for the improved technique. Each sample contained a large number (500) of accepted decays, in order to reduce the effect of bias in small samples with a t max cut. Then 100 samples were produced, and the spread of the maximum likelihood estimates used to determine the accuracy of each method . This procedure was repeated for several lengths L of the visible volume . In fig. la, we plot the accu-
racies of the two approaches, as a function of L. We see that our new approach does give a smaller error than the conventional method, although the improvement is small for L larger than about 3 times the decay length . We have also checked that the bias of the improved method is if anything slightly smaller than that of the standard approach.
L. Lyons, P. Clifford / Lifetime determinations in the presence of cuts (2) tmin Cuts . Decays that occur too close to the production position are difficult to recognise as such . Thus experiments have some region at small times in which decays are detected with poor efficiency . The standard procedure, which we copy, is to reject decays with times below some cutoff, such that the efficiency for larger times is constant. This minimum time cutoff tm'n can be event dependent. The way it varies with the event configuration depends on the specific experimental details of how such decays are observed. We discuss a situation in which a decay is recorded if at least one of its impact parameters y (i .e., the distance by which a decay track, when extrapolated backwards, misses the production vertex) is larger than some specified value Ymm . For simplicity we consider our decay particle to be a Ko , which decays isotropically in its rest system, and into a single two-body charged-particle decay mode - , TT + m . Ko (8) For a given K o momentum p and for a specified decay distance d, we can calculate the range of decay angles in the K ° rest system such that the impact parameters of the decay pion will be larger than yn,n . Since the K ° decay distribution is known (isotropic), this immediately gives us the probability g(p, d) that a K o will satisfy the selection cut, for these values of d and p . For values of p greater than 4 GeV/c, g is to a good approximation a function of p/d only, being zero up to dlP _ 1 .1Y_,, [GeV/c ] -1 and then rising steadily towards unity . In analogy with the previous example, we now make use of the known distribution of K o decay angles (and hence oft mm) in order to improve the precision of our lifetime estimate. The probability of observing a decay at time t, and with ,,,n less than t, is dn_1 di, 'r
(9)
Our likelihood function now becomes =~ `
dn d t,
f oo dt,
dt,
(10)
where the integral of the denominator extends from zero to infinity . We note that, in contrast to the likelihood function (1), eq. (10) does not contain the individual tm`" values . It instead makes use of the tm`" distribution, derived from the isotropy of the K o decays . It is worth stressing that our approach does not result in our accepting events with impact parameters below the defined cutoff (or equivalently with t G t`° ` ° ). Indeed the samples for the two different likelihood functions (10) and (1) are identical . The improvement
559
resulting from the use of the likelihood function (10) thus arises from the way in which the events are treated, and not from different or larger numbers of events . (Similar remarks apply to the tmax situation considered above .) We have again performed Monte Carlo simulations to test the performance of our likelihood function (10) . We have generated a series of 100 "experiments", where each "experiment" contains 25 decaying K o s of 10 GeV/c momentum, and with impact parameters satisfying the ymin cut . The spreads of the resultant distributions of best values are used as estimates of the accuracy of each technique. These are plotted as a function of the impact parameter cutoff ym , in fig. 1(b) . As expected, the fractional error of the conventional method is consistent with 1/ 25 . The improved method, however, has an error which, for large values of ym is some 20% lower . A reduction in a as ymin increases is possible because each point corresponds to requiring a constant number of Monte Carlo decays surviving the cut . We have also checked that there is no discernable bias of the estimates based on the likelihood function (10) . The above method can be fairly easily extended to cases where the particle whose lifetime we wish to determine has several decay modes to multibody final states, and decays with a known nonuniform angular distribution . The presence of multibody final states will probably necessitate the use of Monte Carlo calculations to obtain the probability function g, rather than the analytical one we used for the K ° . This procedure has in fact been applied by the NA27 Collaboration for multibody D ° and D t decays, as part of the method of determining charm particle lifetimes [1] . The motivation there was not so much to reduce the error on the estimates, but rather to provide a formalism that did not require t ,,n for each individual event. We conclude that the conventional method of determining lifetimes in the presence of cuts is not optimal, and improvements can be achieved by incorporating uncontroversial theoretical information concerning the distribution of tmax or t min . Finally we note that the ideas presented here have wider application than just for lifetime determinations . Acknowledgement We would like to express our thanks to Christian Defoix, who developed the formalism of the g functions as part of the procedure mentioned above for determining charm particle lifetimes . Reference [11 M. Bengalli et al ., submitted to Nucl . Instr. and Meth .
557
LIFETIME DETERMINATIONS IN THE PRESENCE OF CUTS Louis LYONS
Nuclear Physics Laboratory, Keble Road, Oxford, UK Peter CLIFFORD
Mathematical Institute, St. Giles, Oxford, UK Received 14 March 1988 and in revised form 22 August 1988 We discuss the determination of the lifetime of a short lived particle from a set of N observed decay times t when for each observation there is a minimum and a maximum observable time t,"'" and t,"ax . By incorporating information concerning the expected distribution of tin" and tmax, we obtain a more accurate estimate than by using the individual t,"'" and tlmax values . Thus is a specific example of the more general point that extra theoretical input can sometimes be used to improve the accuracy of estimates. One of the basic experimental problems in atomic, nuclear and elementary particle physics is the determination of the lifetime of an exponentially decaying state. In one class of this type of experiment, we observe the individual times t, (i = 1, 2, . . ., N), at which N particles decay. Experimental conditions are usually such that for each particle, decays can be observed over the range of times between tin and tmax . In the traditional approach, the lifetime T is estimated by maximising the likelihood Y= n `
1e
T ^ax
1m,
-r,/T
1 - e-`/Tdt T
We consider an improved approach where, by using information concerning the expected distributions of t `"`n and/or tmax, we obtain a more accurate estimate than by using their individual values in eq . (1) above. We discuss two separate cases. (1) tmax cuts . We consider the determination of the lifetime of our decaying particles, in the situation where the t7n are all zero, but the t," are finite . We achieve the improvement by using the extra information that the production distribution along the length L of the fiducial region is expected to be uniform . Thus the probability of having a potential length l for decay is and hence that of having a potential length greater than 1 0 is
Y(lo)LX to
(1)dl=1-la/L .
(The modifications required to allow for the attenuation of the beam along the target and for nonforward production angles are easy and trivial respectively .) In terms of times, this becomes y
( ) = 1 -t /t Ôax , t
where to ax is the time taken for a particle of momentum p, to cross the fiducial volume, i.e . (5)
tô a= mL/P,
Then, given the uniform distribution of production positions, the probability of observing a decay at time t, and with a potential decay time t,max greater than t, is dt
ax). e',/T(1 t,/t0_ T
We finally write a likelihood function for the observed set of events as Y=
dn d t,
II ft~" dn r
dt,
where the integration in the denominator can be performed analytically . We note that the likelihood function (7) does not contain the individual potential times t,", but only IÔ ax .
We have compared the performance of the two likelihood functions (1) and (7) by Monte Carlo calculations. We have generated samples of decays corresponding to a lifetime to , for a set of particles produced uniformly along a fiducial region of length L, and all travelling with the same momenta. Decays occurring beyond the end of the fiducial region were rejected .
558
L. Lyons, P. Clifford / Lifetime determinations m the presence of cuts 30% r
Conventional approach Improved method
Or/T o
20°îo
4
10
40
L (cm) -"
100
400
022 0`/To
020
0.18
016
014
2
4
6 ymm
8
10
12
14
16
(cm)
Fig. 1. (a) The fractional error a/-ro on the estimate of the lifetime for "experiments" of 500 events, as a function of the length L of the visible region of the detector, when the mean decay length is 6 cm. (b) The fractional error a/,ro for a sample of 25 events with impact parameters y greater than yin , . The hfetime of the K° is taken as 10 -1° s. The triangles are for the conventional treatment of the data ; the circles are for the improved technique. Each sample contained a large number (500) of accepted decays, in order to reduce the effect of bias in small samples with a t max cut. Then 100 samples were produced, and the spread of the maximum likelihood estimates used to determine the accuracy of each method . This procedure was repeated for several lengths L of the visible volume . In fig. la, we plot the accu-
racies of the two approaches, as a function of L. We see that our new approach does give a smaller error than the conventional method, although the improvement is small for L larger than about 3 times the decay length . We have also checked that the bias of the improved method is if anything slightly smaller than that of the standard approach.
L. Lyons, P. Clifford / Lifetime determinations in the presence of cuts (2) tmin Cuts . Decays that occur too close to the production position are difficult to recognise as such . Thus experiments have some region at small times in which decays are detected with poor efficiency . The standard procedure, which we copy, is to reject decays with times below some cutoff, such that the efficiency for larger times is constant. This minimum time cutoff tm'n can be event dependent. The way it varies with the event configuration depends on the specific experimental details of how such decays are observed. We discuss a situation in which a decay is recorded if at least one of its impact parameters y (i .e., the distance by which a decay track, when extrapolated backwards, misses the production vertex) is larger than some specified value Ymm . For simplicity we consider our decay particle to be a Ko , which decays isotropically in its rest system, and into a single two-body charged-particle decay mode - , TT + m . Ko (8) For a given K o momentum p and for a specified decay distance d, we can calculate the range of decay angles in the K ° rest system such that the impact parameters of the decay pion will be larger than yn,n . Since the K ° decay distribution is known (isotropic), this immediately gives us the probability g(p, d) that a K o will satisfy the selection cut, for these values of d and p . For values of p greater than 4 GeV/c, g is to a good approximation a function of p/d only, being zero up to dlP _ 1 .1Y_,, [GeV/c ] -1 and then rising steadily towards unity . In analogy with the previous example, we now make use of the known distribution of K o decay angles (and hence oft mm) in order to improve the precision of our lifetime estimate. The probability of observing a decay at time t, and with ,,,n less than t, is dn_1 di, 'r
(9)
Our likelihood function now becomes =~ `
dn d t,
f oo dt,
dt,
(10)
where the integral of the denominator extends from zero to infinity . We note that, in contrast to the likelihood function (1), eq. (10) does not contain the individual tm`" values . It instead makes use of the tm`" distribution, derived from the isotropy of the K o decays . It is worth stressing that our approach does not result in our accepting events with impact parameters below the defined cutoff (or equivalently with t G t`° ` ° ). Indeed the samples for the two different likelihood functions (10) and (1) are identical . The improvement
559
resulting from the use of the likelihood function (10) thus arises from the way in which the events are treated, and not from different or larger numbers of events . (Similar remarks apply to the tmax situation considered above .) We have again performed Monte Carlo simulations to test the performance of our likelihood function (10) . We have generated a series of 100 "experiments", where each "experiment" contains 25 decaying K o s of 10 GeV/c momentum, and with impact parameters satisfying the ymin cut . The spreads of the resultant distributions of best values are used as estimates of the accuracy of each technique. These are plotted as a function of the impact parameter cutoff ym , in fig. 1(b) . As expected, the fractional error of the conventional method is consistent with 1/ 25 . The improved method, however, has an error which, for large values of ym is some 20% lower . A reduction in a as ymin increases is possible because each point corresponds to requiring a constant number of Monte Carlo decays surviving the cut . We have also checked that there is no discernable bias of the estimates based on the likelihood function (10) . The above method can be fairly easily extended to cases where the particle whose lifetime we wish to determine has several decay modes to multibody final states, and decays with a known nonuniform angular distribution . The presence of multibody final states will probably necessitate the use of Monte Carlo calculations to obtain the probability function g, rather than the analytical one we used for the K ° . This procedure has in fact been applied by the NA27 Collaboration for multibody D ° and D t decays, as part of the method of determining charm particle lifetimes [1] . The motivation there was not so much to reduce the error on the estimates, but rather to provide a formalism that did not require t ,,n for each individual event. We conclude that the conventional method of determining lifetimes in the presence of cuts is not optimal, and improvements can be achieved by incorporating uncontroversial theoretical information concerning the distribution of tmax or t min . Finally we note that the ideas presented here have wider application than just for lifetime determinations . Acknowledgement We would like to express our thanks to Christian Defoix, who developed the formalism of the g functions as part of the procedure mentioned above for determining charm particle lifetimes . Reference [11 M. Bengalli et al ., submitted to Nucl . Instr. and Meth .