On the limits of the hadronic energy resolution of calorimeters

Dual-readout and compensation are the principles to resolve the problem of hadron calorimeter which is the poor hadronic energy resolution caused by the fluctuations of the electromagnetic shower component and the binding energy loss. These principles have been proved experimentally for last several decades. However, the theoretical limits of the hadronic energy resolution of calorimeters which adopt the compensation or dual-readout is an intriguing question. In this talk, I explain the hadron shower physics for the better appreciation of the main reasons which cause the poor hadron performance of calorimeter, and present the limits of the hadronic energy resolution of the dual-readout and compensation calorimeters


Introduction
For last 40 years, the poor performance for detecting hadrons have been understood in the level of fundamental shower physics, and physicists have been proved experimentally that hadron calorimeters designed on the basis of compensation and dual-readout methods can measure the energy of hadrons and jets with very good energy resolution. Also, it is an interesting question what is the theoretical limit of the hadronic energy resolution that can be achieved by means of the two methods. Using GEANT4 [1], the limits of the hadronic energy resolution for compensation and dual-readout are calculated, and I present the results in this talk.

Hadron shower
When a high-energy hadron enters an absorber, it ionizes the atoms of the absorber, and then, at some depth, it breaks up a nucleus. From the nuclear reaction, the types of produced particles are mesons, baryons, soft γ's and nuclear fragments. In this interaction, to break up the nucleus, some portion of the energy of an incoming hadron which amounts to the binding energy of released nucleons should be supplied to the nucleus, which is called binding energy loss. It does not contribute to the calorimeter signals and has large non-Gaussian fluctuations. Nuclear fragments and slow charged particles are stopped by losing their energies through the ionization of medium. Energetic mesons and baryons induce another nuclear reactions. Neutrons loose their energies by nuclear reactions. In this hadronic cascade, π 0 's and η's are produced and immediately decay into two γ's which develop the electromagnetic shower. This is the electromagnetic (em) component of hadron shower, which fluctuates in non-Gaussian. The others except the em component are the non-electromagnetic (non-em) or hadronic component. The fraction of the binding energy loss reaches up to 40% of the non-electromagnetic component. 2 will see, in hadronic showers a certain fraction of the dissipated energy is fundamentally undetectable.
When discussing em showers (Section 2.1), we saw an important difference between the absorption of photons and electrons. Electrons lose their energy in a continuous stream of events, in which atoms of the traversed medium are ionized and bremsstrahlung photons are emitted. On the other hand, photons may penetrate a considerable amount of matter without losing any energy, and then interact in a manner that may change their identity (i.e., the photon may turn into a e + e pair). When a high-energy hadron penetrates a block of matter, some combination of these phenomena may occur (Figure 2.22). When the hadron is charged, it will ionize the atoms of the traversed medium, in a continuous stream of events, in much the same way as a muon of the same energy would do (Section 2.2). However, in general, at some depth, the hadron encounters an atomic nucleus with which it interacts strongly. In this nuclear reaction, the hadron may change its identity dramatically. It may, for example, turn into fifteen new hadrons. Also the struck nucleus changes usually quite a bit in such a reaction. It may, for example, lose ten neutrons and three protons in the process and end up in a highly excited state, from which it decays by emitting several -rays.
Neutral hadrons do not ionize the traversed medium. For these particles, nuclear reactions are the only option for losing energy. This is in particular true for neutrons, which are abundantly produced in hadronic shower development. As a result, neutrons deposit their kinetic energy in ways very different from those for the charged shower particles, with potentially very important implications for calorimetry.
The particles produced in the first nuclear reaction (mesons, nucleons, s) may in turn lose their energy by ionizing the medium and/or induce new (nuclear) reactions, thus causing a shower to develop. Conceptually, this shower is very similar to the em ones discussed in Section 2.1. Initially, the number of shower particles increases as   The response to the non-em component is considerably smaller than that to the em component and its fluctuation is much broader than the em one due to the fluctuations of the binding energy loss. The calorimeter is characterized by the ratio of the response to the em and that to the non-em component ("e/h"). The calorimeter in Figure 2 has e/h = 1.8, which is called noncompensating. In this figure, we can notice that minimum ionizing particle has larger response than the em component because the shower sampling process for the em component is less efficient than for mips.
The left in Figure 3 is the distribution of em shower fraction of hadron showers initiated by 150 GeV pions in the Pb-fiber calorimeter. It clearly shows event-to-event fluctuations of em fraction of hadron showers are large and non-Poissonian. Shown in the right of Figure 3 is the average em shower fraction as a function of pion energy. This was measured with calorimeters whose absorbers were copper and lead. The relation between the average em fraction and the energy of incoming pion can be described with Eq. 1: where E 0 is the average energy to produce a pion in the shower, and k is the average multiplicity in the nuclear reaction. The average em fraction increases as the pion energy rises, and depends  Figure 3. The fluctuations of the em fraction of hadron showers (left), and the average em fraction as a function of pion energy (right).
on the type of materials. The consequence of this increase of the average em fraction appears in the non-linear calorimeter response to hadrons and a poor hadronic energy resolution. The eventby-event fluctuations of em fraction shown in Figure 3 (left) is also the cause of a poor hadronic energy resolution. The fundamental reason of the poor performance of hadron calorimeters is the fluctuations of the invisible energy. Typically, the non-em component has smaller response than the em component since the invisible energy doesn't contribute to the calorimeter signal. Fluctuations of the invisible energy give rise to the fluctuations of energy sharing between the em and non-em components of hadron showers. This phenomenon is responsible for the poor hadronic energy resolution, signal non-linearity, and non-Gaussian response function.

Compensation and Dual-Readout
Compensation and dual-readout are the successful methods to mitigate the effect caused by fluctuations of the invisible energy. The former is achieved by boosting the response to MeVtype neutrons and reducing the response to charged shower particles. The representative compensating calorimeter is SPACAL, which was built with Pb and plastic scintillation fibers. The special volume ratio of Pb and scintillation fibers to achieve e/h = 1 is 4:1. As a result of this special calorimeter design, the signal distributions for hadrons have narrow and Gaussian response function as shown in Figure 4. Shown in Figure 4 are the signal distributions of SPACAL for the different pion energies such as 10, 40, and 150 GeV [2]. SPACAL accomplishes the good hadronic energy resolution 30%/ √ E. But the fixed volume ratio requires a small sampling fraction, which results in the limited em energy resolution. The need for efficient neutron detection implies a large detector volume and a long signal integration time.
Dual-readout method is to measure the em fraction event-by-event by means of comparing scintillation and Cerenkov signals, and to correct the measured energy. Since the em shower particles move relativistically, Cerenkov signal is produced mainly by these particles since the non-em components don't contribute to the Cerenkov signal because they are dominated by non-relativistic shower particles. Using this dual-readout method, the effect of fluctuations in the em fraction can be eliminated, and the correct hadron energy can be reproduced. In dual-readout calorimeters, e/h = 1 can be achieved without the limitations such as the small sampling fraction, a large detector volume, and a long signal integration time. The performance the small sampling fraction. This fact implies the long signal integration time and the limited em energy resolution.
Dual-readout method is to measure the em shower fraction event-by-event by means of comparing scintillation and Cerenkov signals, and to correct the measured energy. Since the em shower particles move relativistically, Cerenkov signal is produced mainly by them while the nonem components don?t contribute to the Cerenkov signal because they are non-relativistic shower particles. Using this dual-readout method, fluctuations in the em fraction can be eliminated and the correct hadron energy can be reconstructed. e/h = 1 can be achieved without the limitations such as the small sampling fraction, a large detector volume, and a long signal integration time. The upper left plot illustrates the 2D dual-readout signal distribution. Y-axis is Cerenkov and Figure 5. The average signal per GeV as a function of pion energy (a), the signal distribution for 60 GeV pions (b), and /E as a function of 1/ p E for pion and proton (c). All these results are obtained with the RD52 Pb-fiber calorimeter [3].
X-axis is scintillation signals divided by the beam energy. Events are distributed around the line connected between fem=0 and 1. The responses to S channel can be expressed as the sum of the electromagnetic shower fraction and the response of the absorber-scintillation fiber structure to the non-em component. The response to C channel can be found in the same way. By solving these two equations, we can obtain the corrected energy as functions of measured S and C signals and also the em shower fraction. This procedure corresponds to moving the events to the C=S line represented with the green dashed line. The other method is rotating the event ensemble to this way and project them to scintillation signal axis.
Using the rotation method, we could obtain very narrow and Gaussian hadron signal distribution. For example, if you see the upper right plot, we got Gaussian response function, and 3.9Also, we could reconstruct the beam energy. If you see the upper left plot, the DR calorimeter responses to pions and protons are linear within 1If you see the lower plot, the fractional widths are scaled as 30Dual-readout calorimeter has the same hadronic performance for both pions and protons. mall sampling fraction. This fact implies the long signal integration time and the limited nergy resolution. ual-readout method is to measure the em shower fraction event-by-event by means of aring scintillation and Cerenkov signals, and to correct the measured energy. Since the em er particles move relativistically, Cerenkov signal is produced mainly by them while the nonomponents don?t contribute to the Cerenkov signal because they are non-relativistic shower cles. Using this dual-readout method, fluctuations in the em fraction can be eliminated and orrect hadron energy can be reconstructed. e/h = 1 can be achieved without the limitations as the small sampling fraction, a large detector volume, and a long signal integration time. upper left plot illustrates the 2D dual-readout signal distribution. Y-axis is Cerenkov and re 5. The average signal per GeV as a function of pion energy (a), the signal distribution 0 GeV pions (b), and /E as a function of 1/ p E for pion and proton (c). All these results btained with the RD52 Pb-fiber calorimeter [3].
is is scintillation signals divided by the beam energy. Events are distributed around the line ected between fem=0 and 1. The responses to S channel can be expressed as the sum of the romagnetic shower fraction and the response of the absorber-scintillation fiber structure to on-em component. The response to C channel can be found in the same way. By solving two equations, we can obtain the corrected energy as functions of measured S and C signals also the em shower fraction. This procedure corresponds to moving the events to the C=S represented with the green dashed line. The other method is rotating the event ensemble is way and project them to scintillation signal axis. sing the rotation method, we could obtain very narrow and Gaussian hadron signal ibution. For example, if you see the upper right plot, we got Gaussian response function, 3.9Also, we could reconstruct the beam energy. If you see the upper left plot, the DR imeter responses to pions and protons are linear within 1If you see the lower plot, the ional widths are scaled as 30Dual-readout calorimeter has the same hadronic performance oth pions and protons. signal, the hadronic energy resolution that can be obtained with the dual-readout method is already superior to what has been achieved by the best compensating calorimeters.

Conclusions
The hadronic performance of the calorimeter systems currently used in experiments at high-energy particle colliders is dominated by fluctuations in the energy fraction used to break up atomic nuclei in the shower development. Two different methods have been proposed and tested to mitigate these effects: compensation and dual-readout. Both methods have been demonstrated to be very effective in improving the hadronic calorimeter performance. Calorimeters based on these methods have achieved hadronic signal linearity, Gaussian response functions, very good hadronic energy resolutions and a correct reconstruction of the hadronic energy in instruments calibrated with electrons. We have investigated and compared the principles on which both methods are based and concluded that dual-readout calorimetry provides somewhat superior performance, combined with fewer practical restrictions, than compensation.
The performance results of the RD52 dual-readout calorimeter shown in Fig. 4 were obtained with a method described in this Appendix. As pointed out by D. Groom in the Review of Particle Physics [27], the (S, C) data points from a dual-readout calorimeter are clustered around a straight line in the scatter plot (Fig. 12a). This line links the point for which f em = 0 with the point for which f em = 1. The latter point is located on the diagonal (C = S), where also the data points for em showers are located. The angle ✓ is only determined by the e_h values of the Éerenkov and scintillation calorimeter structures and is independent of energy and of the particle type. It is also the same for single hadrons and jets. Its value is related to the parameter in Eq. (7), as: = cot ✓. This equation represents a transformation in which each data point is moved up along the red line until it intersects with the diagonal, as illustrated in Fig. 12b. The data points obtained in this way are thus clustered around the same energy value as the data points for electrons of the same energy. Projecting these data points on the horizontal (S) and vertical (C) axes leads to Gaussian signal distributions centered around the correct particle energy. This is true for both pions and 156 d) Figure 5. The average signal per GeV as a function of pion energy (a), the signal distribution for 60 GeV pions (b), and the fractional width of signal distribution (σ/E) as a function of 1/ √ E for pion and proton (c) obtained with the RD52 Pb-fiber calorimeter [3]. The rotation of event ensemble for hadrons around the point that the C=S line intersects the line connected between f em = 0 and 1 [4].
of dual-readout calorimeters was established with the so-called "rotation method". Shown in Figure 5 (d), hadron events are distributed around the red line connected between f em = 0 and 1 in C (Cerenkov) vs S (Scintillation). The rotation method rotates this event ensemble over an angle 90 • − θ around the point that the C=S line intersects the line connected between f em = 0 and 1, and then, project the rotated events onto the x-axis. Using the rotation method, Figure 5 (a), (b) and (c) are obtained. In Figure 5 (b), we obtain the very narrow Gaussian signal distribution for 60 GeV pions of which σ/E is 3.9%, and the correct pion beam energy is reproduced. Figure 5 (a) shows that the dual-readout calorimeter are linear within ±1% uncertainty for both pions and protons. In Figure 5 (c), the fractional widths (σ/E) are scaled as 30%/ √ E and negligible constant term for both pions and protons. Dual-readout calorimeter has the same hadronic performance independent of the type of hadrons.

The Limits of the Hadronic Energy Resolution
For the prediction of the limits of the hadronic energy resolution, we simulate a very large absorber to contain the entire hadron showers with GEANT 4.10.3 and FTFP BERT physics list. The simulations were carried out by sending pions into the Cu and Pb absorbers. The pion beam energies are 10, 20, 50, 100, 200, 500 and 1000 GeV. From these simulations, we extract the information of the em shower fraction, total nuclear binding energy loss and the total kinetic energy of the neutrons. The em shower fraction is determined by summing the energy carried by π 0 's and γ's, and dividing it by the total pion energy. The binding energy loss is found by multiplying the number of released nucleons by the binding energy of nucleon in each nuclear reaction. The total kinetic energy of the neutrons is obtained by taking all neutrons produced in nuclear reactions into account.   Figure 6 shows the em shower fraction and the neutron kinetic energy are correlated with the binding energy loss. The em shower fraction appears to be better correlated. This fraction is the quantity that is measured with the dual-readout method. By subtracting the em shower fraction from 1, the fraction of the non-em component can be obtained. In real experiment, for dual-readout calorimeters, the non-em component can be found with the measured em shower fraction and the corrected energy using dual-readout formula. This fact indicates that the non-em component has better correlation with the binding energy loss than the total neutron kinetic energy.
The left and right in Figure 7 are the distributions of the ratios of binding energy loss to total non-em energy and the neutron kinetic energy to binding energy loss. From Figure 7, the correlations between the binding energy loss and the non-em energy, and total neutron kinetic energy can be found by dividing the rms by mean values. The non-em energy has the better correlation to the binding energy losses. Figure 8 (a) is the average em shower fraction and the average fraction of the binding energy losses as a function of the pion energy. The red is copper and the blue is lead. As the pion energy increases, the average em shower fraction increases and the average fraction of the binding energy losses decreases. Figure 8 (b) shows the limits of the hadronic energy resolution in absence of DR or compensation. It is the energy resolution as a function of 1/ √ E. The resolution was found by dividing the rms of the binding energy loss distribution by the visible energy. The visible energy was obtained by subtracting the average binding energy loss from the beam energy. The energy resolution improves with energy, but does not scale with 1/ √ E.    Figure 9 predicts the limits of the hadronic energy resolutions for dual-readout and compensation in the cases of Cu and Pb absorbers. The energy resolution for each pion energy was found by multiplying the average fraction of the binding energy losses by the correlation between the total non-em energy and the binding energy losses or the total kinetic energy of neutrons and the binding energy loss. Black is compensation, and both red and blue are dualreadout. The hadronic energy resolution scales with 1/ √ E for both compensation and dualreadout. The limits of the hadronic energy resolution for dual-readout are 12% and 13%/ √ E for Cu and Pb, respectively. For compensation, the limits are 19% and 21%/ √ E for Cu and Pb, respectively. Dual-readout has lower limits of the hadronic energy resolution than compensation. This conclusion is supported by experimental results. Figure 10 (a) is the distribution of the Cerenkov signal for 100 GeV pions measured with the prototype Cu-fiber dual-readout calorimeter [5]. This event sample is divided into the three subsamples based on the three em shower fraction ranges shown in Figure 10 (b). Figure 10 [5]. The distribution of the Cerenkov signal for 200 GeV multiparticle events (c) and the distributions of the subsamples selected by the neutron fractions in the scintillation signal (d) [6]. the signal distributions derived on the basis of the neutron fraction are obviously broader than those obtained by the em shower fraction. This fact is consistent with the results introduced in Figure 9, which can be interpreted as dual-readout method mitigate the effect of fluctuations of the invisible energy loss more effectively.

Conclusion
In conclusion, dual-readout and compensation approaches remedy the poor hadronic performance caused by fluctuations of the invisible energy loss. The limits of the hadronic 8 energy resolution of dual-readout and compensation are introduced in Section 4. This GEANT4 simulation study suggests that dual-readout method can achieve a better hadronic energy resolution than compensation. However, hadron calorimeters built with both methods can obtain the good energy resolution, signal linearity, Gaussian response functions and the same calorimeter response to electrons, pions and protons.