# Energy Flow in CMS Calorimetry

**Dan Green**

**September, 2001**

**Introduction**

The Compact Muon Solenoid (CMS) is one of the two general purpose
detectors being built to perform experiments at the CERN Large Hadron
Collider (LHC). The calorimetry in CMS consists of an electromagnetic
compartment of PbWO4 crystals \[1\]. Thus, there is only a single
longitudinal segment which contains the photons and electrons. However,
there is a very fine transverse segmentation. A crystal tower subtends a
dη x dϕ \~ 0.0174 x 0.0174 element of solid angle. This segmentation
corresponds roughly to the Moliere radius in the crystal.

The hadronic compartment is also a single longitudinal segment \[2\].
The transverse segmentation is five times coarser, 0.087 x 0.087.
Indeed, it is well matched to the physical size of an hadronic shower.

In this note, we examine the possibility of performing particle
identification using the longitudinal and transverse segmentation of the
CMS calorimetry. In addition, we consider a mixed measurement for jets,
consisting of a mixture of neutral particle calorimeter clusters and
charged particle momenta measured in the CMS magnetic tracking system.
This hybrid, or "energy flow" set of measures should be a better measure
of the kinematics of a jet because the tracking error is much smaller
than the calorimetric energy error for low momentum charged particles.
At issue is whether the jet errors are dominated by energy measurement
errors or by other mechanisms, such as gluon radiation.

**Transverse Shape in ECAL**

The electromagnetic calorimeter, ECAL, has a fine transverse
segmentation, as mentioned above. However, there is no longitudinal
segmentation, so that the detector is blind to the details of the shower
development in depth. Test beam data with a setup close to that proposed
fro CMS has been taken \[3\]. In particular, a 7 x 7 array of crystals
was exposed to beams of pions and electrons. The data for four single
pions of 100 GeV beam momentum are shown in Fig. 1. Note that the pions
which interact in ECAL typically deposit energy in at least two
crystals.

The analogous deposits for 100 GeV electrons are shown for four
individual events in Fig.2. Note that the showers are , indeed, narrower
than the pion showers. However, it is clear that the electron can always
have an impact point near to the crystal boundary. Therefore, electrons
depositing energy in two crystals are not rare.

![](media/image1.wmf){width="2.5688976377952755in"
height="1.9645669291338583in"}![](media/image2.wmf){width="2.606299212598425in"
height="1.994488188976378in"}![](media/image3.wmf){width="2.505511811023622in"
height="1.9173228346456692in"}![](media/image4.wmf){width="2.4181102362204725in"
height="1.85in"}

Figure 1: Energy deposition in a 7 x 7 crystal ECAL array for 100 GeV
pions which have interacted in ECAL.

![](media/image5.wmf){width="2.486220472440945in"
height="1.5374015748031495in"}![](media/image6.wmf){width="2.70748031496063in"
height="1.6744094488188976in"}![](media/image7.wmf){width="2.486220472440945in"
height="1.5374015748031495in"}![](media/image8.wmf){width="2.5964566929133857in"
height="1.605511811023622in"}

Figure 2: Energy deposition in a 7 x 7 crystal ECAL array for 100 GeV
electrons.

The distribution of the energy weighted transverse radius for electrons
and pions is shown in Fig. 3. Clearly, there is some discrimination
between pions and electrons. However, incisive particle identification
cannot be made. A clean particle identification requires additional
data, such as a fine grained shower maximum detector or much more
information of the development of the shower in depth.

![](media/image9.wmf){width="2.9161417322834646in"
height="2.025196850393701in"}![](media/image10.wmf){width="2.9161417322834646in"
height="2.2681102362204726in"}

Figure 3: Distribution of the energy weighted transverse distance of
energy deposits for pions and electrons.

**Calorimeter Clusters**

In order to explore the energy flow concept, a sample of "Z" with 120
GeV mass was generated. Initial and final state radiation was turned on
in the PYTHIA Monte Carlo program. The resulting final state particles
were passed through the CMS detector simulation. The output which was
available was the energy deposited in the towers of the calorimetry and
the generator level charged particle tracks at the production vertex.
The first task was to combine contiguous calorimeter hits into
"clusters" and to look at the longitudinal information available for
that cluster.

The mean number of hits with transverse energy above 0.1 GeV in the
electromagnetic compartment is 324, while the hadronic compartment has
344 hits on average. Since there are roughly 5000 hadronic towers in the
CMS barrel plus endcap ( spanning \|η\| \< 3 with size 0.087 x 0.087),
the occupation of the calorimeter is fairly sparse.

The calorimeter hits were clustered into groups of towers. The
clustering began in the electromagnetic calorimetry, ECAL, as it had the
finest granularity. Hits were ordered in transverse energy, Et. The
largest Et tower served as a seed for the cluster. Additional hits in
the ECAL were added to the cluster in a 3 x 3 ECAL array centered on the
seed tower. All hits above 0.1 GeV were accepted. The ECAL clustering
was terminated when the seed tower transverse energy fell below 0.2 GeV.
All towers that participate in a cluster are removed from further
clustering.

The information in depth was used to sort the ECAL clusters into two
categories. Energy in the hadronic calorimeter compartment (HCAL) behind
the ECAL seed was added in a 3 x 3 HCAL tower array if the found HCAL
transverse energy is greater than 30% of the ECAL cluster energy. This
value was picked using test beam data \[3\] for pions interacting in the
ECAL compartment. If there was no HCAL deposit, then the cluster was
considered to be a neutral particle, flag = 0. If there was a match, the
cluster was considered to be a charged pion which interacted in the ECAL
compartment.

When the ECAL energy is exhausted, the remaining HCAL energy is ordered
in Et. The seed in HCAL has adjacent 3 x 3 HCAL towers clustered
similarly to the ECAL clustering. The process is terminated when the
seed tower Et falls below 0.2 GeV.

The result of this clustering is a cluster energy, Et weighted centroid
in (η,ϕ), and a particle flag. The flag is 0 if the ECAL cluster has no
associated HCAL energy. It is 1 if there is associated HCAL energy of at
least 30% of the ECAL energy. Finally, the flag is 2 if there is only
HCAL energy without sufficient associated ECAL energy. This last
category is assumed to be a charged pion which interacted only in the
HCAL.

There are 241 calorimeter clusters on average. The reduction with
respect to calorimeter tower hits is due to the large fraction of ECAL
events with Et slightly above 0.1 GeV which fall below the seed
threshold. As noted below, there is a rough agreement between the number
of clusters and the number of charged tracks.

The average cluster transverse energy summed over all flags is 0.94 GeV,
with 0.2 GeV in the ECAL and 0.75 in the HCAL. This scale of energy is
that of a typical single particle from a minimum bias event. Since the
calorimetry covers \|η\| \< 5, the number of clusters corresponds to a
particle density of 2.4 per unit of rapidity. The number of towers is
0.4 in the ECAL and 1.15 in the HCAL, again summed over all flag types.
The distribution of the number of HCAL towers in all clusters is shown
in Fig. 4.

![](media/image11.wmf){width="3.75in" height="2.8125in"}

Figure 4: Distribution of the number of HCAL towers in a cluster for all
clusters. The maximum number in the 3 x 3 allowed array is 9. Clusters
with zero HCAL towers are neutral particles ( flag = 0 ).

The clustering procedure was used to organize the "raw" calorimeter
tower hits. The resulting Et of all clusters in one Z(120) event as a
function of the cluster ϕ is shown in Fig. 5. The two jet structure
resulting from the Z(120)  qq decay is very evident.

![](media/image12.wmf){width="3.875in" height="2.90625in"}

Figure 5: Calorimeter clusters -- one event with Et of the cluster
plotted against the azimuthal angle of the cluster. The back to back dϕ
\~ π dijet structure is clearly seen.

**Track -- Cluster Matches**

The tracks used in this study were not actual CMS tracks but the PYTHIA
generated charged particles. Given that the tracking momentum resolution
is much better than the calorimeter energy resolution, this
approximation should not alter the qualitative conclusions which are
made.

For events with a produced Z of 120 GeV mass, there are 233 charged
tracks on average, with a standard deviation of 144. The primary vertex
is on the coordinate origin with a standard deviation in z of 5.2 cm.
There are 32% of the tracks which never reach the barrel calorimetry ,
i.e. loopers.

These tracks are swum in a uniform axial magnetic field to a radius
corresponding to the front face of the HCAL. The event by event z
position of the vertex is used in the swim. Only tracks with \|η\| \< 3
and transverse momentum, Pt, greater than 1 GeV are considered. These
cuts approximate the angular coverage of the CMS tracker and the lowest
transverse momentum not curled up by the magnetic field (loopers). The
"looper" transverse momentum to strike ECAL is 0.78 GeV.

Deviations of the swum tracks from the calorimeter clusters are
calculated taking the cluster error to be 0.087 (HCAL tower size) for
both ϕ and η. Matching in energy assumes a dominant calorimetric error
with a 100% stochastic term and a 5% constant term \[3\]. Tracks that
are within the calorimetric resolution (tower size) in η and ϕ at the
HCAL radius are first merged and their momenta added when considering a
match to a calorimeter tower. This procedure allows for matches in the
dense core of jets where charged tracks overlap within a calorimeter
tower.

A match between the calorimeter cluster and the track is defined to
exist if \|dη\| is less than 1 (deviation normalized to HCAL tower size)
if \|dϕ\| is less than 2 and if the difference between track momentum
and calorimeter cluster energy, normalized to calorimeter resolution, is
between --1 and 3. The offset is due to the observation that the
calorimeter energies are systematically lower than the track momenta. In
the case of multiple matches, the match with the lowest "chi squared" is
taken.

The difference in η between tracks and clusters is shown in Fig. 6 for
matches which pass the cuts mentioned above. Note that the match
improves as the cluster Et increases as we expect due to reduced bending
in the magnetic field and improved multiple scattering. There is clearly
a band of "accidental" matches which occur at low Et. A similar
situation obtains for matches in ϕ except the cut limits are wider
because this is the bend plane in CMS.

![](media/image13.wmf){width="4.797244094488189in"
height="3.598031496062992in"}

Figure 6: Match in pseudorapidity between charged tracks and calorimeter
clusters (flag \> 0) as a function of cluster Et.

![](media/image14.wmf){width="4.922047244094488in"
height="3.691732283464567in"}

Figure 7: Match in azimuthal angle between charged tracks and
calorimeter clusters (flag \> 0) as a function of cluster Et.

The match of track momentum to cluster energy is shown in Fig. 8. Note
that the calorimetry is systematically under calibrated. Therefore, when
clusters are replaced by tracks, we expect the jet Et will be increased.
Note also that the error scale set by the calorimeter energy resolution
is reasonable. A cut of +- 2 standard deviations about a mean of one is
efficient. The CMS calorimeter is nonlinear. This behavior is seen in
Fig. 8 in that very high momenta are well matched, while lower momenta
are underestimated by the calorimetry. Note that these energy matches
are not "accidentals". They survive increasingly hard cuts on matching
in η and ϕ.

![](media/image15.wmf){width="4.922047244094488in"
height="3.691732283464567in"}

Figure 8: Match in energy between charged tracks and calorimeter
clusters (flag \> 0) as a function of cluster Et.

If a match exists in all three kinematic variables, the calorimeter
cluster is replaced by the matching track. A new flag is defined to be =
3 if this replacement is made. Tracks which would never reach the
barrel, "loopers", are added back into the "energy flow" list and a flag
= 4 is set. The end result is an energy flow list consisting of
calorimeter clusters (flag = 0, 1, or 2), matched (merged) tracks (flag
= 3), and tracks which would not register in the calorimetry ( flag =
4).

**Jet Properties**

The data set was 120 GeV "Z" bosons decaying into light quarks with
initial state and final state radiation turned on. The data used was
either calorimeter clusters or a hybrid energy flow set consisting of
unmatched clusters plus matched tracks and unobserved track loopers.

The largest Et cluster served as the seed for the first jet. The next
highest Et cluster a distance \> 2R from the first seed was defined to
be the seed of the second jet. All clusters within a cone of radius R
about the two seeds were taken to be fragments of the jet. The vector
sum of all the fragments defined the momentum of the jet. The clusters
were assumed to be massless. A plot of the Et of the first seed cluster
is shown in Fig. 9.

![](media/image16.wmf){width="4.4222222222222225in"
height="3.316666666666667in"}

Figure 9: Transverse energy of the first seed cluster found by the jet
algorithm. Note that the leading cluster is not soft, but has an Et \~
20 GeV.

The number of clusters found within a cone of R = 0.8 is shown in Fig
10. Note that there are \~ 300 distinct HCAL towers within this cone, so
the cluster density even within the jet is not very high.

The transverse energy of the first found jet is shown in Fig. 11. Note
the prominent Jacobean peak at \~ half the Z(120) mass.

![](media/image17.wmf){width="4.375in" height="3.28125in"}

Figure 10: Number of calorimeter clusters found within a cone of radius
R = 0.8 for the first found jet. The mean number is \~ 23.

![](media/image18.wmf){width="4.5472222222222225in"
height="3.410416666666667in"}

Figure 11: The transverse energy of the first found jet in Z(120)
events.

The clusters are treated as massless particles which are summed to find
the jet momentum vector. The mass of the jet was, then, non -- zero. The
distribution of mass for the first found jet is shown in Fig. 12. It is
, typically, small on the scale of the parent mass of 120 GeV. We have
assumed that this procedure does not add to the error in the dijet mass
determination.

![](media/image19.wmf){width="4.9222222222222225in"
height="3.691666666666667in"}

Figure 12: Distribution of jet mass for the first found jet in Z(120)
events.

**Dijet Mass**

The dijet mass was calculated for the two largest transverse momentum
clusters. The calculation was done both for pure calorimeter clusters
and for the energy flow list consisting of clusters, matched charged
tracks, and barrel loopers. The distributions shown here require that
both jets have an axis within \|η\| \< 3, roughly where tracking exists
in CMS.

The event by event correlation between the mass calculated from clusters
and from the energy flow list is shown in Fig. 13. Note that in most
cases the latter calculation gives the larger mass. This fact is simply
a reflection of the offset in the track momentum -- calorimeter cluster
energy match, Fig. 8.

![](media/image20.wmf){width="4.547244094488189in"
height="3.410236220472441in"}

Figure 13: The correlation between dijet masses for Z(120) events
calculated from calorimeter clusters and from clusters plus tracks (
energy flow ).

The distribution in dijet mass computed using only calorimeter clusters
is shown in Fig. 14. The distribution calculated using all, the
available information from calorimetry and tracking is shown in Fig. 15.
Note that the latter mass is systematically larger than the former.

Gaussian fits were made to the histogrammed data. The mean mass is 84
GeV for the calorimeter clusters. The fractional mass error is 39%. For
the case where matched tracks are used and loppers are added the mean
mass is 108 GeV and the fractional mass error is 29 %. In both cases the
Gauss Ian fit has a chi squared \~ 1, indicating an adequate fit. It
appears that the energy flow technique improves the mass measurement.
That indicated that the energy error on the charged hadrons in a jet
makes a non-negligible contribution to the overall mass resolution of
dijets.

Note that this study makes as complete a simulation of the CMS detector
as is presently possible. It is of interest to compare the results of
this study to the conclusions of earlier and simpler models of jet
resolution.

![](media/image21.wmf){width="4.6722222222222225in"
height="3.504166666666667in"}

Figure 14: Mass distribution for Z(120) events using calorimeter
clusters. The line is a Gaussian fit to the data over the mass range
Z(50, 150) GeV.

![](media/image22.wmf){width="4.7972222222222225in"
height="3.597916666666667in"}

Figure 15: Mass distribution for Z(120) events using the energy flow
list. The line is a Gaussian fit to the data over the mass range (50,
150) GeV.

**Discussion**

An early study \[4\] of the dijet mass resolution for W,Z masses at low
transverse momentum revealed several competing effects. Using an assumed
calorimetric resolution with a stochastic term of 60% and a constant
term of 3% \[2\], the contribution of the energy error to the mass
resolution was \~ 7.2 %. The contributions due to fragmentation and the
underlying event depend on the cone size. For R = 0.6 the fragmentation
error is 5.5 %, while the underlying event contributes 9.4 %. At a
larger cone size, R = 1.0, the fragmentation outside the cone has a
reduced 3.6 % contribution, while the fluctuations on the underlying
event within the cone rise to 13.8%.

These conclusions were made in the absence of initial state or final
state radiation. The classic early treatment of the emission of a single
gluon by quarks \[5\] indicated that the problems caused by radiation
could be severe. Consequently, a simple Monte Carlo program was written
to look at gluon showers from quarks in the loading log approximation
(LLA).

An Et = 60 GeV jet was generated and the effect of gluon showering could
be isolated. The total quark plus gluon energy within a cone of radius R
= 1.0 is shown in Fig. 16. Note that there is a substantial fraction of
the events with observed energies \< 50 GeV, indicating large effects
due to radiation.

![](media/image23.wmf){width="4.5472222222222225in"
height="3.410416666666667in"}

Figure 16: Distribution of the energy within a cone of radius R = 1.0
starting with a quark of energy 60 GeV and allowing it to shower gluons.

Events were studied for containment of energy within a cone. In Fig. 17
is shown the fraction of 60 GeV Et jets which have a fraction of that
energy within a cone of R = 1.0 and 0.6. For example, with R = 1.0, only
\~ half the jets have 80% containment of the initial energy. This full
shower result is somewhat worse than the simple estimate \[5\] based on
single gluon emission.

![](media/image24.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

Figure 17: Fraction of jets where the initial energy is contained within
a cone of R = 0.6 and 1.0 for different energy containment fractions.

The severity of the effect of radiation prompted a more complete study.
This study has been made of the effects of initial state and final state
radiation on the dijet mass resolution \[7\] for Z bosons. For a cone
size of R = 0.7 the fractional mass error was 11% without radiation,
rising to 19 % with radiation turned on. Unfolding in quadrature,
radiation appears to contribute 15% to the mass error by itself. Note
that in this study a toy detector was used.

The error without radiation compares well with the effects quoted above
\[4\] folded in quadrature due to energy resolution, underlying event
fluctuations, and fragmentation out of cone. These amount to 13% for R =
0.6. Folding in the effect of radiation, we expect a fractional mass
resolution of 22% in the absence of the effects of the detector itself,
such as inert material in the tracker, etc.

The Z mass resolution with and without radiation is shown in Fig. 18.
The effect of radiation is to broaden the peak and induce a long, low
mass, radiative tail into the mass distribution. The shape is reasonably
consistent with that shown in Fig.15.

![](media/image25.png){width="2.65625in" height="5.083333333333333in"}

## Figure 18: Z mass distribution with and without initial and final state radiation. Note that the distribution is the reconstructed mass divided by the generated mass. Radiation shifts the mean to lower values, broadens the peak, and induces a low mass radiative tail.

## 

## 

## References

1.  ECAL Technical Design Report, CERN/LHCC 97-33, December 1997

2.  HCAL Technical Design Report, CERN/LHCC 97-31, June 20, 1997

3.  V. Abramov et al., Nucl. Inst. Meth. In Physics Res., A 457 (2001)
    75-100

4.  A. Beretvas et al., Fermilab FN-626, October 1994

5.  G. Sterman and S. Weinberg, Phys. Rev. Lett., 39, 23, 1977

6.  D. Green, Fermilab, FN-666, February, 1998

7.  K. Maeshima et al., CMS Note, 1998/026, February 28, 1998

**New file from Pal Hidas**

**Z(120) with Etj cut and in barrel, 360 events**

**\<nt\> = 145 +- 68**

**sigma(zv) = 5.4 cm**

**\<nhcal\> = 234 +- 86**

**\<necal\> = 295 +- 58**

**clustering**

**\<ncl\> = 151 +- 52**

**do not make e/h corrections -- no help in Mjj or missing Et**

**swim tracks -- merge into 3 x 3 in HCAL (resolution)**

**\<nt\> = 90 - no loopers and merged**

**match tracks \<nmatch\> = 6.5 +- 3.2, iflg \> 0, deta = +-1, dphi =
+-2, dE = -1,+3**

**correct e flow by sum(Et) of match tracks**

**\<Etmatch\> = 41 +- 23 GeV, Mjj-\> Mjj -- (\<Etmatch)-40)**

**\<Et\> = 142 +- 47 GeV -- sum Et in the event**

**Missing Et = 11.9 +- 7.2**

**Missing Et not helped by match because only high Et tracks match**

**Find Mjj correlated with sum(Et) in event. Cut latter at 250, 200 GeV.
This gets rid of third jet in the event typically**

**Mean J mass is 8.6 GeV, massless clusters**

**Do full fits to pure Gausian**

**Optimize cone size -- R = 1.0, no pileup**

**Fit with 309 is 81.7(1.1) +- 17.1(1.0) (20.9% chi/dof = 2.5/12) -- cal
only**

**303 is 105.5(1.4) +- 17.8(1.1) (16.9% chi/dof = 9.6/12) -- e flow**

**24% improvement**

![](media/image26.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image27.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image28.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image29.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image30.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image31.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image32.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image33.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image34.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image35.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image36.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image37.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image38.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image39.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image40.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image41.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image42.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image43.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image44.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

**Mean = 81.7 +-1.1 GeV, Sigma = 17.1 +-1 GeV, chisq/DOF = 2.5/12**

![](media/image45.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

![](media/image46.wmf){width="5.600694444444445in"
height="4.200694444444444in"}

**Mean = 105.5 +- 1.35 GeV, Sigma = 17.81 +- 1.11 GeV, chisq/DOF =
9.6/12**

![](media/image47.png){width="2.875in" height="2.15625in"}

![](media/image49.png){width="2.75in" height="2.0694444444444446in"}

![](media/image50.png){width="2.875in" height="2.1618055555555555in"}

![](media/image51.png){width="3.0in" height="2.251388888888889in"}

![](media/image52.png){width="3.125in" height="2.3534722222222224in"}

![](media/image53.png){width="3.25in" height="2.4506944444444443in"}

![](media/image54.png){width="3.25in" height="2.4458333333333333in"}