We chose not to go into any details with regard to the issues associated with the constant
term as we felt it was beyond the scope of the paper under consideration and due to the
fact that we would like to include those aspects as part of another comprehensive paper
on digital calorimetry. The paper does contain brief statements which answer your
questions and which we expand on here.
1) “This is not of …… spectrum of hadrons inside a jet…… mean around 9-10 GeV
(Fig. 2).”
While single particle resolutions are an essential ingredient, the jet energy
resolution is the primary quantity of interest from the physics point of view. We
evaluate the implications of the single particle resolutions shown in Fig. 3 on the
final jet resolution in ZZ events at s = 500 GeV by following the recipe outlined
below:
a) following the energy-flow paradigm assume perfect momentum
measurement for charged hadrons from tracker
b) for photons smear the generated 4-vector with the nominal 17%/E
sampling term
c) for neutral hadrons smear the generated 4-vector with the resolutions
shown in Fig. 1 (this is Fig. 3 in the paper) by fitting /E as function of E
with a polynomial.
d) pT order stable MC particles ignoring neutrinos (see Fig. 2).
e) Start with the highest pT particle and cluster within a 0.7 simple cone (see
Fig. 3)
f) Repeat for remaining unclustered particles
g) Add individual smeared energies to get jet energy
Figure 1:Energy resolution as a function of the incident energy for single charged pions.
Figure 2: Stable MC particles in ZZ events at s = 500 GeV.
Figure 3: Fraction of energy carried by neutral hadrons (top,  ~ 11%) and photons
(bottom,  ~ 24%) in 0.7 simple cone jets.
The smeared jet energies are compared to the corresponding unsmeared jets to
estimate the resultant energy resolution. Fig. 4 shows the /E as a function of the
generated jet energy using the analog and digital parameterization of the single
particle resolutions for 3cm x 3cm cells.
Figure 4: Energy resolution of 0.7 cone jets in ZZ events as a function of the generated (unsmeared)
jet energy.
As can be seen from Fig. 4 the ‘digital’ resolutions are competitive with analog
resolutions even for high energy jets with the rising constant term seen in Fig. 1 not
making any significant impact on the resultant jet resolutions.
2) “Simulations indicate that……using multiple thresholds…..instance).”
Simulations clearly indicate that for cell sizes we are considering (3 cm x 3 cm) semidigital (two bits or three thresholds) readout, instead of the digital (one bit or single
threshold) readout discussed above, can redress the concerns associated with the
degradation of resolution at the high energy end of the single particle spectrum. Let
us start by revisiting the problem. Fig. 5 and 6 show the (E)/E and (N)/N plots for
10 and 50 GeV charged pions respectively. While cell counting works very well for
the 10 GeV sample, Fig. 6 clearly shows that by 50 GeV it is definitely performing
worse than the analog measurement. The reason for this is the fluctuations in the
electromagnetic content of hadron showers (see Fig. 7). Fig. 7 shows the total number
of cells above a minimal threshold (taken here to be 0.25 MIP) for 10 and 50 GeV
charged pions as a function of the fraction of the shower energy deposited in cells
with energy greater than 10 MIPs (or roughly as a function of the amount of energy of
the hadron shower being deposited mainly through s). As is well known the fraction
of energy being deposited by electromagnetic means and hence its fluctuations
increases with the hadron momentum leading to a larger spread in the number of cells
E=0.183
N=0.166
Figure 5: Distributions of live energy (left) and number of cells above threshold (right) for 10 GeV
charged pions.
E=0.101
N=0.153
Figure 6: Distributions of live energy (left) and number of cells above threshold (right) for 50 GeV
charged pions.
Figure 7: Total number of cells above a minimal threshold of 0.25 MIP as a function of the energy
fraction contained in cells with energy greater than 10 MIPs for 10 GeV (left) and 50 GeV (right)
charged pions.
above threshold for the 50 GeV sample. This can be clearly seen by comparing the 2d
distributions for 10 and 50 GeV shown in Fig. 7. Improving the resolution at the high end
thus amounts to changing the slope of the 50 GeV distribution so as to make the “no. of
hits above 0.25MIP” independent of “fraction of energy deposited in cells with
E>10MIP”.
This can be achieved if there are multiple thresholds available. By attaching a threshold
dependent weight to each cell the slope change can be accomplished. Simply put a cell
may be counted more than once depending on the threshold it passes. For the purposes of
our simulation studies we used three (equivalent to a two-bit readout) thresholds. The
thresholds were placed at (this has not fully been optimized) at 0.25, 10 and 20 MIPs. We
arbitrarily chose the weight to be assigned to the cells to be 2n where n goes from 0 to 2.
Thus a cell passing a threshold of 0.25 MIP was counted once, that passing 10 MIP twice
and the one above 20 MIP four times. The results of the cell no. distribution and the
resultant resolution using this ‘semi-digital treatment’ can be seen in Fig. 8. It can be
concluded from Fig. 8 that with multiple thresholds (3 in our case) the competitiveness of
the (semi)digital treatment can be maintained even at high energies thus eliminating
concerns related to the prospect of an increasing constant term.
E=0.101
E=0.092
Figure 8: Distributions of live energy (left) and number of cells above threshold (right) for 50 GeV
charged pions. The “semi-digital” treatment has been applied to the number of cells distribution.