Two possible ways to improve the mass resolution for observing hadronic top
quark decay $t \to bW \to 3 jets$ are studied: (1) using fixed cones in the
rest frames of the $t$ and $W$ to define the decay jets, instead of the
traditional cones in the rest frame of the detector; and (2) using the jet
angles in the top rest frame to measure $m_t/m_W$. By Monte Carlo simulation,
the second method is found to give a useful improvement in the mass resolution.
It can be combined with the usual invariant mass method to get an even better
mass measurement. The improved resolution can be used to make a more accurate
determination of the top quark mass, and to improve the discrimination between
$t \bar t$ events and background for studies of the production mechanism.Comment: Revised and expanded. New and better method introduced. Some
  conclusions changed. 17 pages, RevTeX, 4 uuencoded figure

Pumplin, Jon

English

arXiv

The mass resolution for observing hadronic top quark decay t \to b W \to 3 jets can be improved by about 20\% by defining the jets using fixed cones in the rest frames of the t and W, in place of the traditional cones defined in the rest frame of the detector. This improved mass resolution can be used to make a more accurate measurement of the top quark mass, and to improve the discrimination between t \bar t events and background for studies of the production mechanism

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hep-ph/9508267 (MSU-HEP/50808)
Improving the Measurement of the Top Quark Mass
Jon Pumplin
Department of Physics & Astronomy
Michigan State University
East Lansing, MI
Internet address: pumplin@msupa.msu.edu
(August 8, 1995)
Abstract
The mass resolution for observing hadronic top quark decay t ! bW !
3 jets can be improved by about 20% by dening the jets using xed cones in
the rest frames of the t and W , in place of the traditional cones dened in
the rest frame of the detector. This improved mass resolution can be used to
make a more accurate measurement of the top quark mass, and to improve the
discrimination between t

t events and background for studies of the production
mechanism.
Typeset using REVT
E
X
1
The discovery of the top quark in pp collisions at
p
s = 1:8TeV [1,2] represents a mile-
stone for the Standard Model. The next steps beyond that milestone will be to improve the
measurement of the top mass, and to test our understanding of QCD further by studying
the details of its production and decay.
We focus here on the \single-lepton" channel dened by pp ! t

tX with one top de-
caying hadronically (t ! W
+
b ! 3 jets , or its charge conjugate) and the other decaying
leptonically (t ! W
+
b ! `
+

`
+ 1 jet , or its charge conjugate, with ` = e or ). This
channel is experimentally favorable because the high p
?
lepton and large missing p
?
due to
the neutrino provide a clean event trigger that naturally discriminates against backgrounds.
The top mass determination relies on measuring the energies and directions of jets,
and inferring from them the energies and directions of the original quarks. In addition to
instrumental eects and calibrations, this requires corrections for the hard-scale branching of
partons, and for non-perturbative hadronization eects. These QCD eects can be modeled
by event generators such as HERWIG [3], which have been shown to describe the appropriate
physics in Z
0
decay at LEP [4,5], where the original qq energy is precisely known.
Part of the top quark analysis involves measuring the mass of the hadronically decaying
W boson. An eventual goal of the analysis should be to conrm the QCD eects in W decay
that are similar to those seen in great detail for Z
0
decay. In the meantime, we can assume
that the QCD eects are understood, and attempt to use theW mass to help in determining
the correct jet assignments in t

t events. The decay W ! 2 jets is also a potentially useful
tool to calibrate the detector, since the decay jets probe the detector at a wide range of
angles and jet transverse momenta, and the W mass is accurately known. This calibration
is unavailable outside of top quark events, because hadronic W decays are normally obscured
by large QCD backgrounds [6].
The point of this paper can be seen most clearly by considering theW mass measurement.
Since the W is color neutral, its decay products have nothing to do with other jets present
in the t

t nal state. In the W rest frame, the decays will be very similar to Z
0
decays as
observed at LEP. In particular, they will almost always appear as two back-to-back jets that
2
can readily be dened by xed cones of half-angle 
0
oriented in opposite directions. We will
choose 
0
= 30

. This cone size is large enough to contain a good fraction of the energies
of the jets, which are broadened by the collinear radiation singularity of QCD. Meanwhile,

0
= 30

is small enough that it leaves 87% of the full 4 of solid angle outside the two
cones, so not too much \background" is included inside them. We will also try 
0
= 45

,
which leaves 71% of the full 4 solid angle outside the two cones. Although cone algorithms
are not the traditional jet denitions in e
+
e
 
physics, they have been shown to work, and
even to provide superior resolution in jet angle and energy [5].
Our sensible decision to use xed cones in the W rest frame is quite dierent from
the current experimental procedure for top analysis, in which jets are dened by cones
q
()
2
+ ()
2
< R in the Lego variables of the detector. These variables,  =
  log tan =2 = pseudorapidity and  = azimuthal angle, are Lorentz invariant only un-
der boosts along the beam direction; whereas the W can have a rather large component
of boost transverse to the beam. For example, in t

t events with typical acceptance cuts,
the middle 60% of the W transverse momentum distribution extends from 40GeV=c to
110GeV=c. Hence p
W
?
is frequently either large or small compared to m
W
. Lego-dened jet
cones therefore correspond to a wide variety of sizes and shapes of \cone" in the W rest
frame, depending on details of decay angles that have nothing to do with the W decay.
Similarly, the jet cone for the b-quark jet from t ! W b is most reasonably dened by
a xed angle in the t rest frame. Since the b carries color, the argument for a cylindrically
symmetric cone in this rest frame is weaker than in the case of the W decay. But the t rest
frame is nevertheless at least more sensible than using the rest frame of the detector. We
will show by Monte Carlo simulation that dening the b-jet cone in the t rest frame indeed
improves the mass resolution for t signicantly.
It would similarly be logical to dene the jet cone for the b-jet from the other, semilep-
tonically decaying, top in the rest frame of that top. That should be done when the method
advocated in this paper is applied to real data. But we ignore it here, in order to avoid
bringing in the unrelated issue of how best to estimate the longitudinal momentum of the
3
neutrino, which is needed to nd that frame.
The suggestion of this paper can most simply be applied as an add-on to the traditional
top quark analysis with \kinematic reconstruction" [1], i.e., with explicit matching of the
observed jets to decay partons. The procedure for each event is as follows:
1. Identify the four-momenta of the jets corresponding to t! j
1
j
2
j
3
and t! j
4
`, where
j
1
, j
2
come from W decay and j
3
, j
4
are b-quark jets, using the the usual procedures
of CDF or D. b-jet tagging is helpful for this, but not crucial. There may of course
be additional jets observed in the event.
2. Dene j
F
and j
B
to be four-momenta directed in the forward and backward beam
directions: j
F
= (1; 0; 0; 1) and j
B
= (1; 0; 0; 1) for beams in the z-direction.
3. Redene the 4-momentum of j
1
as the sum of the 4-momenta observed in the calorime-
ter detector in all cells (\towers") that lie within 
0
of the old j
1
in the rest frame of
the old j
1
+ j
2
. However, do not include cells that lie closer in angle to j
2
, j
3
, j
4
, j
F
,
or j
B
than to j
1
.
Similarly redene the 4-momentum of j
2
as the sum of the 4-momenta of all cells that
lie within 
0
of the old j
2
in the rest frame of the old j
1
+ j
2
, and are not closer in
angle to j
1
, j
3
, j
4
, j
F
, or j
B
.
Similarly redene the 4-momentum of j
3
as the sum of the 4-momenta in cells that lie
within 
0
of the old j
3
in the rest frame of the old j
1
+ j
2
+ j
3
, and are not closer in
angle to j
1
, j
2
, j
4
, j
F
, or j
B
.
4. Replace the old jet 4-momentum estimates by the new ones and repeat Step 3 a few
times. This iteration converges completely for all but 0:2% of events. A maximum
of 10 iterations is sucient, since the momenta cease changing before that in all but
0:5% of events.
The essential new work is in Step 3, which is very easy to implement in the following
way. Treat the energy in each cell of the detector that receives energy above its noise level
4
as if it came from a zero-mass particle. There will typically be a few hundred such cells
in each event. Make a list of their four-momenta. The appropriate elements of this list to
be added in the various parts of Step 3 can be found without making any explicit Lorentz
transformations, using the exact formula for the angle  between ~p and ~q in the rest frame
of ~r:
cos  =
1  pq r
2
=(pr q r)
q
1   q
2
r
2
=(q r)
2
; (1)
which holds for any massless four-vector p.
To test this idea, HERWIG 5.7 [3] was used to simulate the single-leptonic top channel
at the Tevatron energy
p
s = 1:8TeV. Typical experimental cuts were approximated by
j
`
j < 2:0, p
`
?
> 20GeV=c, p

?
> 25GeV=c. The four hadron jets were required, at the
partonic level, to have p
jet
?
> 20GeV=c and to be isolated in Lego from the lepton by
R 
q
()
2
+ ()
2
> 0:5 and from each other by R > 0:8 . The detector was simulated
by an array of 0:1  0:1 cells in   , covering the region jj < 4:5 . Gaussian energy
resolutions with E=E = 0:55=
p
E for charged hadrons and E=E = 0:15=
p
E for leptons
and photons were assumed (in GeV = 1 units). Constant terms in the energy resolution were
not included. A small amount of shower spreading was included: the energy of each nal
hadron was spread out over a disk of radius 0:1 in the (; ) plane before being deposited in
the appropriate calorimeter cells.
A cone-type jet nding algorithm described previously [7] was used to search for jets in
the simulated calorimeter. A small cone size R = 0:3 was used, as is standard practice in
top quark analysis to reduce the loss of events from overlapping jets, since one is asking
for 4 jets within a small area (frequently  1 <  < +1) in Lego. A minimum \observed"
jet p
?
of 15GeV=c was required. A feature of the jet-nder used is that it nishes with an
iteration in which each cell of the detector that lies within R = 0:3 of at least one jet axis is
assigned to the nearest jet axis. The jet momenta are then recomputed and this procedure is
iterated. This feature tends to improve the mass resolution for objects decaying into jets. It
thus provides an honest comparison to our method, which is similar except that the nearest
5
jets are dened by angles in appropriate rest frames.
To obtain a sample that could be cleanly interpreted, the \observed" jets in the simu-
lation were matched to their original quark partons, using a criterion based mainly on the
agreement in Lego variables  and , together with a small contribution from the agreement
in p
?
. A good unique match was obtained for 84% of the events, which were used to produce
the following Figures.
Figure 1 shows the histogram of the mass observed for W ! jj. The solid curve is for
jets dened in the usual way, by R < 0:3 in the Lego variables of the detector. Note that
the center of the peak is well below the actual m
W
= 82GeV=c
2
assumed in the simulation,
and that the peak is asymmetrical, with a tail toward lower jj masses. This skewing of
the mass distribution toward low mass is a direct result of the soft gluon radiation that is
characteristic of QCD, and which has been veried by the LEP data. It demonstrates that
making signicant QCD corrections is a necessary part of the top quark mass measurement,
especially when a cone size as small as 0:3 is used.
The dashed curve in Figure 1 shows the result of redening the jets in the appropriate
rest frames, using the iteration described above. Note that the qualitative QCD feature of
skewing toward lower mass is still visible, but much less pronounced. Also, the center of the
peak is at a higher mass | much closer to the true partonic mass. This can be understood
from that fact that jet cones dened by  < 30

are generally larger than the traditional
R < 0:3 cones in the detector frame. More importantly, the overall width of the peak is
narrower. For example, if one denes the width of the peak by the range M that contains
the middle 60% of the probability, the new jet denition has reduced M=M by a factor of
0:80 .
Figure 2 similarly shows the histogram of the mass observed for t ! W b ! jjj .
One again sees a QCD skewing of the peak toward lower jjj masses, relative to the value
m
t
= 175GeV=c
2
assumed in the simulation. (This mass shift includes a contribution of
  5GeV=c
2
from losses in jet energy due to neutrinos.) One sees that the new jet deni-
tion again clearly improves the mass resolution | this time reducing M=M for the middle
6
60% of the distribution by a factor of 0:82 .
The choice of cone size 
0
= 30

used in the rest frames is not crucial. In fact, 
0
can be
increased to as much as 45

without spoiling the convergence of the iteration, and with an
even greater improvement in mass resolution. This is shown by the dotted curves in Figs. 1
and 2.
In conclusion, we have seen that redening the jets using xed angle cones in the rest
frames of the decaying t and W systems can be used to improve the mass resolution M=M
by approximately 20% . The new jet denition substantially reduces the shift toward lower
masses of the centers of the peaks, which will reduce the magnitude | and hence also the
associated error | of the QCD corrections needed to infer the peak mass at the partonic
level. This technique should therefore be used to improve the mass measurement of the
top quark. The improved resolution will help by reducing the statistical uncertainty in
the location of the mass peak, which will arise from the relatively small number of events
(100) that will be available in each experiment at the end of the current Tevatron run. It
will help in addition with isolating the signal from the non-top QCD background, since the
narrower peak will produce a larger ratio of signal/background. The improved resolution
for the W ! jj mass measurement will further help to isolate the single-leptonic t

t signal.
Making the hadronic W peak more visible will also help in facilitating a direct calibration
of jet energy measurements using the known W mass.
The improvement in mass resolution described in this paper could also be applied to the
dicult problem of observing doubly hadronic top events t

t! (Wb) (W b)! 6 jets [8].
The procedure advocated here has a solid basis in QCD, and has been shown to be
eective using a reasonable Monte Carlo simulation. The next steps, which can only be
carried out by the experimenters of CDF and D, must be to try the procedure out using
full detector simulations, and then real data.
This work was supported in part by NSF grant number PHY-9507683.
7
REFERENCES
[1] CDF Collaboration (F. Abe et al.), Phys. Rev. Lett. 74, 2626 (1995).
[2] D Collaboration (S. Abachi et al.), Phys. Rev. Lett. 74, 2632 (1995).
[3] G. Abbiendi, I.G. Knowles, G. Marchesini, B.R. Webber, M.H. Seymour and L. Stanco,
Comp. Phys. Comm. 67, 465 (1992).
[4] OPAL Collaboration (R. Akers et al.), Z. Phys. C63, 181 (1994).
[5] OPAL Collaboration (R. Akers et al.), Z. Phys. C63, 197 (1994).
[6] J. Pumplin, Phys. Rev. D45, 806 (1992).
[7] J. Pumplin, Phys. Rev. D44, 2025 (1991).
[8] W. T. Giele et al., Phys. Rev. D48, 5226 (1993).
8
FIGURES
FIG. 1. Simulated dijet invariant mass distribution for W ! jj from semileptonic top events.
The solid curve is for jets dened by the usual
p
()
2
+ ()
2
< 0:3 cones in the rest frame of
the detector. The dashed curve is for jets dened by  < 30

cones in the W rest frame, via the
iteration advocated in this paper. The dotted curve is similar, but for  < 45

cones.
FIG. 2. Simulated trijet invariant mass distribution for t ! W b ! jjj from semileptonic top
events. As in Fig. 1, the solid curve is for jets dened by
p
()
2
+ ()
2
< 0:3 cones, while the
dashed (dotted) curves are for  < 30

( < 45

) cones dened in the t rest frame for the b-jet and
in the W rest frame for the other two jets.
9


Improving the Measurement of the Top Quark Mass

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