Minimum Bias Pileup

**and**

**Missing Et at CMS**

**Dan Green**

**Fermilab**

**September, 1999**

**Introduction**

The missing Et trigger was constructed by the UA1 experiment in
recognition that a "hermetic" detector could trigger on produced
neutrinos by asking for an imbalance in the transverse energy in the
final state \[1\].

Clearly, the ability to trigger on missing Et implies that the effects
of neutrinos dominate over other effects such as energy leakage due to a
lack of hermeticity of the detector. In fact, as the high luminosity
hadron collider runs are being prepared for, Run II at the Fermilab
Tevatron and the ATLAS and CMS detectors at the LHC, one must re-examine
this assumption. In particular, the pileup from multiple ordinary,
"minimum bias", events may limit the effectiveness of such a trigger.

**Minimum Bias Event Model**

In order to get a grasp on the possible effects of pileup, a simple
model was written. Only pions were generated in the final state. There
were no leading baryons generated, although this effect is known to
exist. \[2\] The secondary pions were generated as an exponential in
Pt^2^, with an adjustable parameter which is chosen to fix \<Pt\>. A
value of \<Pt\> = 0.75 GeV was used here. The distribution is shown in
Fig.1.

![](media/image1.png)

Figure 1: Distribution of Pt for secondary pions in the simple Monte
Carlo model. The mean has been adjusted to be \<Pt\> = 0.75 GeV.

The azimuthal angle was chosen uniformly. The polar angle was chosen
such that the rapidity of the pion was uniformly distributed for \|y\|
\< yo and which was distributed as a power law decrease from yo to zero
at the kinematic limit y~max~. For an incident proton momentum of po,
the maximum pion, mass m, rapidity is po \~ m~t~sinh(y~max~) where
m~t~^2^ = m^2^ + Pt^2^. In this note, a linear falloff in rapidity was
used, although other dependencies were examined. The rapidity
distribution had 2 adjustable parameters, the value of yo and the power
law falloff behavior. The single particle distribution is given in
Fig.2.

![](media/image2.png){width="5.8in" height="4.375in"}

Figure 2: The single particle rapidity distribution for secondary pions.
Note the rapidity "plateau" for \|y\| \< 5 and the linear falloff to the
kinematic limit.

Note that the single pion distributions follow the inclusive
distributions seen in hadron collider experiments \[3\]. In fact, one
body relativistic phase space, without kinematic limits, is d^4^P(p\*p
-- m^2^) \~ dPxdPydPz/E \~ dydPt^2^. Therefore, a uniform rapidity
distribution is simply a statement that we pick from a single particle
phase space distribution.

The model for a minimum bias event was extremely simplified. Basically,
there were no correlations. Pions were picked from the inclusive
distributions in Pt and y. The "generation" of pions continued until all
the initial state energy and momentum were consumed. The last pion was
chosen to exactly conserve the 3 components of momentum. Energy was not
explicitly conserved.

The resulting pion multiplicity is shown in Fig. 3. The mean is \<n\> \~
88 or \~ 59 changed pions in the event on average. Comparing to a
compilation of data at lower energies \[4\], this value for the mean
multiplicity is quite reasonable

![](media/image3.png){width="6.1in" height="4.538888888888889in"}

Figure 3: The total charged plus neutral pion multiplicity in a minbias
event. The mean multiplicity is, \<n\> = 88.

The density of particles in rapidity space can be extracted from Fig.2.
For all particles it is \~ 6 particles per unit of rapidity on the
plateau. Therefore, the density of 4 for charged particles is obtained.
This value is reasonable \[3\] assuming a ln(s) behavior in
extrapolating data from lower energies.

The resulting minimum bias event is shown in a "lego plot" in Fig.4. The
cell sizes are those appropriate to the CMS hadronic calorimeter which
has 72 divisions in azimuth and covers out to \|y\| = 5 in
pseudorapidity with \~ "square" cells in and .

![](media/image4.png){width="5.9in" height="4.411111111111111in"}

Figure 4: Lego plot for a single minimum bias event for the CMS HCAL.
The (x,y) axes are ( ), and Pt is the vertical axis.

Note that in a single event with n \~ \<n\> \~ 88, there is 66 GeV of
total transverse energy, on average. If the total Et were distributed
statistically, one might expect \~ 8 GeV of Et in a minimum bias event.

**Monte Carlo for Pileup**

The beam bunch crossings are populated by a number of minimum bias
events which are Poisson distributed and , for a given luminosity,
characterized by a mean number of minimum bias events per crossing. For
this study, a fixed number of events was put into a crossing, in order
to cleanly isolate the effects of a given number of events per crossing.

The distribution of the total Et for a crossing when there was one and
only one event per crossing is shown in Fig.5 The mean transverse energy
is \<Et\> \~ 5 GeV. There are 2 major contributors to this value. First,
the single particle energies are smeared by a parameterized calorimeter
resolution as, dE = aE, where a = 1.0 with E in GeV. Second, the
calorimetric coverage is truncated at \|y\| \< 5 as is the present CMS
design. Note that, from Fig. 2, a substantial fraction of the pions
falls outside the CMS coverage. In the simple model used here, with no
short range correlations, the effect of truncation of angular coverage
is comparable to that of calorimetric energy resolution in inducing a
spurious total transverse energy into the event. Therefore, even perfect
calorimetric energy resolution would not substantially alleviate the
effects of pileup in the context of the present model. Note also that
the effects of magnetic field, e.g. "loopers", have not been taken into
account in the present treatment.

![](media/image5.png){width="5.0in" height="3.75in"}

Figure 5: Distribution of Et for a crossing containing one minimum bias
event. The mean is \<Et\> = 5.0 GeV.

Note that there is a \> 1 % chance to have \> 12 GeV of "missing energy"
in the crossing due simply to fluctuations in a single minimum bias
event. If we take the inelastic cross section to be 100 mb, then there
is effectively a 1 mb trigger cross section for "neutrinos" with
transverse energy \> 12 GeV.

The total transverse energy in a crossing is a global variable. As such,
the fluctuations in transverse energy are difficult to calculate
analytically, Et = (E~xi~)^2^ + (E~yi~)^2^. The result of propagating
the errors due to energy resolution does not lead to a transparent
expression. Suffice it to say that dEt \~ a/Et is a rough estimator of
the effect of energy measurement errors. Taking Et \~ 88 GeV per event,
we estimate dEt \~ 9 GeV, which is at least comparable to the observed
mean of 5 GeV. Note, however that energy measurement is not the
predominant contributor to Et.

The mean value of Et for a crossing as a function of the number of
events in the crossing is shown in Fig.6. Note that the scaling as n
appears to hold, as would be expected on statistical grounds.

![](media/image6.png){width="5.9in" height="4.425in"}

Figure 6: Mean Et per crossing as a function of the number of minbias
events per crossing. The basic functional dependence is \<Et(n)\> \~ 5.0
GeVn.

One strong indication that the behavior of the Et of the crossing is not
naively statistical, is that the r.m.s. of the Et distribution divided
by the mean, /\<Et\>, is roughly constant for from 1 to 20 events per
crossing with a value of \~ 50%. This behavior is seen in D0 Run I data
\[5\] also.

The probability for a crossing to have an Et above thresholds of 20, 30
and 40 GeV are shown in Fig.7 for numbers of events per crossing from 2
to 20. Note the rapid rise with n. The behavior is very approximately a
rise as n^3^. Note that with 20 events per crossing, the total particle
Et in the crossing is \~ 1.32 TeV, with a naïve statistical fluctuation
of 36 GeV.

![](media/image7.png){width="6.1in" height="4.208333333333333in"}

Figure 7: Probability for a crossing to have a total transverse energy
above a threshold of 20, 30 and 40 GeV as a function of the number of
minbias events per crossing.

Note that there is a 1 % probability per crossing at \~ 20 GeV for n = 2
events/crossing, \~ 30 GeV for n = 6 events/crossing, and \~ 40 GeV for
9 events per crossing. Therefore, if the present model has any validity,
triggering at moderate to high luminosity in CMS on low missing Et will
not be particularly useful in SUSY and other searches.

The distribution in the total Et of the crossing for n = 2, 5, 10 and 20
events per crossing is given in Fig.8. Note the rapid increase in the
high Et tail at fixed threshold as n increases.

![](media/image8.png){width="6.4in" height="5.05in"}

Figure 8: Distribution of total transverse energy in a beam crossing,
Et, for a) 2 events/crossing, b) 5 events/crossing, c) 10
events/crossing and d) 20 events/crossing.

Data from Run I of D0 \[5\] display similar behavior. In Fig. 9 is shown
the probability per crossing to pass a threshold of 35 GeV for D0
trigger data with from 1 to 9 events/crossing. A sharp rise with n is
seen, similar to that observed in Fig.7. However, for n = 1 event per
crosing, the mean Et is \~ 10 GeV with a r.m.s. of 5.2 GeV. The r.m.s.
divided by the mean is roughly the same as in the present study, but the
value of Et for 1 event is almost twice for D0 as it is for this study.
Given the extremely simplified model used in the present study and the
large number of experimental effects left out, this disparity is ,
perhaps, not surprising.

![](media/image9.png){width="5.2in" height="3.9in"}

Figure 9: Do data on the probability to exceed a 35 GeV threshold in a
crossing as a function of the number of events per crossing.

**Conclusions**

The transverse energy in a beam bunch crossing induced by truncation of
the angular coverage and by calorimetric energy resolution has been
studied. For low LHC luminosity, with 1 event per crossing there is \~ 5
GeV transverse energy on average, with a 1% chance to exceed a threshold
of 12 GeV for the crossing. The mean Et increases as the square root of
n, with a constant r.m.s./mean. At a luminosity with 20 events/crossing,
there is a 1% chance per crossing to pass a missing Et cut of 40 GeV.

The effect of placing cuts on the entries put into the Et global sum was
studied. An angular restriction of \|y\| \< 3 and 2 was compared to the
basic \|y\| \< 5 cut. Assuming that poorly measured particles were at
fault, a cut of E \> 10 and 20 GeV was also made on single pions.
Assuming that low Pt entries were fluctuating, a cut of Pt \> 1.0 and
2.0 GeV on single particles was studied. None of these cuts made any
significant improvement in the Et distribution of a crossing in the case
of 20 events per crossing. The distribution seems to be almost
"holographic"; no matter how it is cut the same distribution, mean and
r.m.s. is obtained. Clearly, more incisive cuts, perhaps sorting offline
on the primary vertex, must be studied.

**References**

1\. G. Altarelli and L. Di Lella, Proton-Antiproton Collider Physics,
World Scientific, 1989. "The Physics of Intermediate Vector Bosons", p.
177

2\. D. Horn and F. Zachariasen, Hadron Physics at Very High Energies,
Benjamin, 1973, p. 41,42

3\. G. Altarelli and L. Di Lella, Proton-Antiproton Collider Physics,
World Scientific, 1989., "properties of Soft Proton-Antiproton
Collisions", p. 85

4\. Review of Particle Physics, 3,(1998)

5\. A. Boehnlein, private communication