Primer on ILC physics and SiD software tools

Abstract

We first outline the standard model of particle physics, particle production and decay, and the expected signal and background at a Higgs factory like the international linear collider (ILC). We then introduce high-energy colliders and collider detectors and briefly detail the ILC and the silicon detector (SiD), one of the two detectors proposed for the ILC. Next, we review the available software tools for ILC event generation, SiD detector simulation, and event reconstruction. Finally, we suggest open avenues in research for detector optimization and physics analysis. The pedagogical level is suitable for advanced undergraduate and beginning graduate students in physics and research scientists in related fields.

Appendices

Appendix

Exercises

1.1 Higgs factory physics

• 1. Show that, for scattering from a hard sphere with radius R, the impact parameter and scattering angle are related by $b=R\cos \frac{1}{2}\theta$. Calculate the total cross section for $b<R$. What is the luminosity?

• 2. The Rutherford potential is given by $V(r)=q_1q_2/r^2$, from which it can be shown that $\cot \frac{1}{2}\theta =(2bE/q_1q_2)^2$, where E is the energy of the incident particle.

  • (a) Calculate the classical differential cross section for Rutherford scattering.

  • (b) Calculate the total cross section for Rutherford scattering. Explain.

• 3. Show that the nth Born Approximation can be written $\varPsi _n = \varPsi _0 + \sum _{i=1}^{n} \int g^i V^i \varPsi _0$ using the recursion relation in Eq. 12.

• 4. The Yukawa potential is given by $V(r)=\beta \exp (-\mu r)/r$ where $\beta ,\mu$ are constants.

  • (a) Calculate the Yukawa differential cross section in the first Born approximation.

  • (b) Calculate the Rutherford differential cross section in the first Born approximation.

• 5. Apply the Euler–Lagrange equation

  $$\begin{aligned} \partial _{\mu } (\partial \mathcal {L}/\partial (\partial _{\mu } \phi ))-\partial _{\phi } \mathcal {L}= & {} 0 \end{aligned}$$

  to $\mathcal {L}_0$, $\mathcal {L}_{1/2}$, $\mathcal {L}_1$ to obtain the Klein–Gordon, Dirac, and Maxwell equations.

• 6. Show that $\mathcal {L}_{1/2}+\mathcal {L}_1$ is invariant under U(1) transformation $\psi \rightarrow \exp (-i\alpha (x)) \psi$ if $A \rightarrow A+ \nabla \alpha$.

• 7. The only leptons which are unstable against decay are the $\mu$ and the $\tau$.

  • (a) Sketch the tree-level Feynman diagrams for each decay, using vertices from Fig. 3.

  • (b) Obtain their lifetimes $\tau _{\mu },\tau _{\tau }$ from the PDG and give an expression for their time-dilated values.

• 8. Of all the hadrons (mesons and baryons), only the proton is stable against decay. Sketch the Feynman diagram for neutron decay $n \rightarrow e^- \bar{\nu }_e p$, neutral pion decay $\pi ^0 \rightarrow \gamma \gamma$, and charged pion decay $\pi ^+ \rightarrow \mu ^+ \nu _{\mu }$.

• 9. The coupling of the Higgs boson to a gauge boson V is $2m_{V}^2/v$, yet the decays $H \rightarrow gg$ and $H \rightarrow \gamma \gamma$ occur despite $m_{g}=m_{\gamma }=0$. Sketch the loop-level Feynman diagrams for these decays using vertices from Fig. 3.

• 10. Calculate the decay rate ratios $\varGamma _{H \rightarrow f\bar{f}}/\varGamma _{H \rightarrow b\bar{b}}$ for leptons $f=e,\mu ,\tau$ and quarks $f=u,d,s,c$.

• 11. Using CKM matrix elements from the PDG, estimate the decay rate ratios below.

  • (a) $\varGamma _{W^+ \rightarrow u\bar{q}_d}/\varGamma _{W^+ \rightarrow u\bar{d}}$ for $q_{d}=s,b$

  • (b) $\varGamma _{W^+ \rightarrow c\bar{d}}/\varGamma _{W^+ \rightarrow c\bar{s}}$

• 12. Sketch the tree-level diagram for $\tau$ decay through a virtual W. Based on the decays of the virtual W allowed by energy conservation and the W decay rates in Table 4, estimate the $\tau$ branching ratios. Compare to branching ratios in the PDG and comment.

• 13. From the tree level Z and W boson partial widths given in this section:

  • (a) Calculate each partial width numerically, and the total width from their sum.

  • (b) Calculate the branching ratios for each decay. Compare your results to the PDG measured values. Explain.

• 14. From the tree level (and loop level $gg,\gamma \gamma$) H boson partial widths given in this section:

  • (a) Calculate each partial width numerically, and the total width from their sum.

  • (b) Calculate the branching ratios for each decay. Compare your results to the PDG measured values. Explain.

• 15. Show that any four-vector contraction $p_{\mu }p^{\mu }$ is a relativistic invariant.

• 16. Consider the Mandelstam variables s, t, u.

  • (a) Show that $s-t-u=\sum _i m_{i}^2$.

  • (b) Write s, t, u in terms of $(E_i,\mathbf {p}_i)$ and the cosine of angles between incident particles.

  • (c) Show that in two-body scattering $\sqrt{s}=E_1+E_2$ if 1 and 2 collide at $\theta =\pi$ with $\mathbf {p}_1=\mathbf {p}_2$.

• 17. Define $R(\sqrt{s})\equiv \sum _q \sigma _{e\bar{e} \rightarrow q\bar{q}}/\sigma _{e\bar{e} \rightarrow \mu ^+ \mu ^-}$, where $N_c=3 \rightarrow N_c=1$ in the cross section for muon pair production. Sketch Rvs.$\sqrt{s}$ for $\sqrt{s} \in [0,20]$ GeV.

• 18. The Higgs boson decay rate to hadrons dominates all others. Explain why there is no Higgs peak in Fig. 6 near $m_{H} \approx 125$ GeV.

• 19. Sketch the Feynman diagrams for the 1f, 2f, 3f backgrounds with initial states $\gamma e$ and $\gamma \gamma$ discussed in this section.

• 20. Calculate the number of ILC Higgstrahlung events produced at $\sqrt{s}=250, 350, 500$ GeV for the projected integrated luminosities in Table 10 using the cross sections in Table 6. Repeat for all backgrounds.

1.2 ILC: accelerators and detectors

• 1. The Tevatron accumulated $\mathcal {L}=11$ fb$^{-1}$ during Run2, from 2001 to 2011. The LHC accumulated $\mathcal {L}=23$ fb$^{-1}$ during 2012.

  • (a) If the Tevatron ($\sqrt{s}=1.96$ TeV) cross section for Higgs boson production is 1pb, how many Higgs bosons are in the Run 2 dataset?

  • (b) If the LHC ($\sqrt{s}=8$ TeV) cross section for Higgs boson production is 22pb, how many Higgs bosons are in the 2012 dataset?

• 2. Read the ATLAS and CMS papers announcing discovery of the Higgs boson at the LHC, [7] and [8]. Compare and contrast their analyses. How are the detectors different?

• 3. Calculate the magnetic field B used in Lawrence’s 27 in tabletop proton cyclotron if the applied RF frequency was $\nu _0=27$MHz. What was the proton’s momentum at maximum radius? What was the relativistic correction factor $\gamma$?

• 4. Calculate the magnetic field B used in Lawrence’s 60 in proton cyclotron if the maximum momentum was 16 MeV. What was the cyclotron frequency $\nu _0=qB/2\pi m$? What was the relativistic correction factor $\gamma$?

• 5. Plot the orbital frequency vs.relativistic energy of a proton in a cyclotron with magnetic field B=0.01, 0.1,1 T. At what energies are the frequencies within tolerances of 1,10,100% of the cyclotron frequency $\nu _0=qB/2\pi m$?

• 6. For each collider in Table 8, compare the cross-sectional area A of the beams to that of SPEAR. Obtain the bunch populations and frequency $n_1,n_2,f$ from the PDG and elsewhere.

• 7. For each collider in Table 8, determine

  • (a) the beam energy required for a fixed target accelerator to produce the same energy necessary for new particle creation.

  • (b) the magnetic field required to hold the accelerated particle in circular orbit.

  • (c) the power loss of one accelerated particle to synchrotron radiation.

• 8. Derive an expression for the square of the center of mass energy for head-on collision of particles with energies $E_1$ and $E_2$. Derive the factor by which this is reduced for a crossing angle $\theta$. What is this factor for an ILC crossing angle $\theta =40$ mrad?

• 9. Derive a with units in Eq. 36 from the Lorentz law.

• 10. Calculate the instantaneous luminosities for each ILC $\sqrt{s}$ in Table 10. Then, calculate the number of years of continuous running at each instantaneous luminosity required to obtain the projected integrated luminosities for scenario H-20 in the same Table.

• 11. Consider elements C, Si, Fe, W, U for calorimetry.

  • (a) To contain 95% of electrons and photons, how deep (in m) must an ECal be for each element? 98%? 99%?

  • (b) To contain 95% of hadrons, how deep (in m) must an HCal be for each element? 98%? 99%?

• 12. Modern collider detectors place detectors in order of smallest to largest radius: tracker, ECal, HCal (TEH). For all five other possible configurations (THE, ETH, EHT, HTE, and HET), describe the performance for electrons, photons, charged hadrons, and neutral hadrons. Why is the muon detector placed at largest radius?

• 13. Plot the tracking momentum resolution vs.$p_T$ at $\theta =\pi /2$ for each detector listed in Table 12 on the same plot. Comment.

• 14. Plot the electromagnetic calorimeter energy resolution vs.energy for each detector listed in Table 12 on the same plot. Comment.

• 15. Plot the hadronic calorimeter energy resolution vs.energy for each detector listed in Table 12 on the same plot. Comment.

• 16. Calculate the ionization energy loss for a minimum ionizing particle ($\beta ^2=0.9$) in the SiD vertex detector and compare it to the energy loss due to electromagnetic showering. Do the same for the SiD tracker.

• 17. How many nuclear interaction lengths $\lambda$ are in the SiD vertex detector? Tracker? ECal? What fraction of its initial energy will a hadron lose before it enters the HCal?

• 18. Muon energy loss in SiD.

  • (a) Calculate the ionization energy loss of a muon traversing the entire SiD detector assuming minimum ionization.

  • (b) Calculate the synchrotron energy loss of a muon traversing the SiD solenoid field. Compare to the ionization energy loss.

• 19. Calculate the minimum $p_T$ required for a charged particle to reach the inner and outer radii of the SiD

  • (a) Vertex detector and tracker

  • (b) ECal and HCal

  • (c) Muon detector

• 20. Read the descriptions of SiD and ILD in [4]. How does ILD differ from SiD? How is it the same?

1.3 SiD: simulation and reconstruction

• 1. For area of the unit circle $\int _{\mathbb {S}^2} dA$,

  • (a) Write a Python program estimate the integral, sampling from $(x,y) \in [-1,1] \times [-1,1]$.

  • (b) Now go to $(r,\theta )$ coordinates, perform the $\theta$ integration, and sample from $r\in [0,1]$.

  • (c) Plot the approximation vs. N for both and compare.

• 2. Consider the differential cross section for $e^+ e^- \rightarrow \gamma ^{\star } \rightarrow q\bar{q}$ with $\beta \approx 1$,

  $$\begin{aligned} \frac{\mathrm{d}\sigma }{\mathrm{d}\varOmega }= & {} 3 Q_{f}^2 \frac{\alpha ^2}{3s} \left( 1+ \cos ^{2} \theta \right) \end{aligned}$$

  • (a) Integrate analytically to obtain the total cross section.

  • (b) Write a Python program to generate N events which reproduce $\mathrm{d}\sigma /\mathrm{d}\varOmega$ for large N.

• 3. Consider muon decay $\mu \rightarrow e \nu _{e} \nu _{\mu }$ and let E represent the energy of the electron. The partial width is given by

  $$\begin{aligned} \frac{\mathrm{d}\varGamma }{\mathrm{d}E}= & {} \left( \frac{g_W}{m_W} \right) ^4 \frac{m_{\mu }^2 E^2}{2 (4 \pi )^3} \left( 1-\frac{4E}{3m_{\mu }} \right) \end{aligned}$$

  • (a) Calculate the total width $\varGamma _{\mu }=\int \mathrm{d}E \mathrm{d}\varGamma /\mathrm{d}E$.

  • (b) Write a Python program to generate N decays which reproduce $\mathrm{d}\varGamma /\mathrm{d}E$ for large N.

• 4. Install Whizard2 on your local computer. Use the script in Table 15 (left) to generate Higgstrahlung events at $\sqrt{s}=250$ GeV. What is the reported cross section?

• 5. Repeat the above exercise, but with ISR turned off. What is the reported cross section?

• 6. Install MG5 aMC@NLO on your local computer. Use the script in Table 15 (right) to generate Higgstrahlung events at $\sqrt{s}=250$ GeV. What is the reported cross section?

• 7. Switch the beam polarizations for the Higgstrahlung events from +30%,-80% to -30%,+80% for $e^+,e^-$. What are the reported cross sections in Whizard2 and MG5 aMC@NLO?

• 8. Using either Whizard2 or MG5 aMC@NLO, what is the reported cross section for $e^+e^- \rightarrow W^+ W^-$ with unpolarized beams at $\sqrt{s}=250$ GeV? Repeat with beam polarizations +30%,-80% and -30%,+80% for $e^+,e^-$. What are the reported cross sections?

• 9. Radiative return to the Z with Whizard2.

  • (a) Generate $e^+ e^- \rightarrow q \bar{q}$ events at $\sqrt{s}=250$ GeV with ISR turned off. What is the reported cross section?

  • (b) Now turn ISR on and repeat. What is the reported cross section? Explain why there is such a large difference.

• 10. Install Geant4 on your local computer.

  • (a) Build a simple model of the SiD Tracker, ECal, and HCal where each is a simple rectangular slab.

  • (b) Histogram the tracker momentum, ECal energy, and HCal energy left by mono-energetic electrons, photons, charged pions, neutral kaons, and muons after passing through your simulation

• 11. Python fast simulation.

  • (a) Write a Python program which implements a simple fast detector simulation with the SiD Tracker, ECal, and HCal. Use the parametrizations in Table 12.

  • (b) Histogram the tracker momentum, ECal energy, and HCal energy left by mono-energetic electrons, photons, charged pions, neutral kaons, and muons after passing through your simulation.

• 12. Install Delphes on your local computer. Obtain the DSiD detector card from HepForge. Run over the Whizard2 files of Higgstrahlung events at $\sqrt{s}=250$ GeV made in the exercise above. Histogram the kinematic distributions of electrons, photons, charged hadrons, and neutral hadrons.

• 13. Install ILCSoft on your local computer. Run the ddsim executable with the DD4hep compact description of SiD over the Whizard2 files of Higgstrahlung events at $\sqrt{s}=250$ GeV made in the exercise above. Then, run the Marlin executable with the nominal SiD reconstruction XML file. Histogram the kinematic distributions of electrons, photons, charged hadrons, and neutral hadrons.

• 14. Repeat the previous exercise, but vary the SiD detector in the DD4hep XML file in the following ways and report on any changes.

  • (a) Add one (two, three) more layer(s) to the Tracker.

  • (b) Take one (two, three) layer(s) away from the ECal.

  • (c) Use tungsten for the absorber in the HCal rather than steel.

  • (d) Omit the vertex detector.

• 15. Track parameters.

  • (a) Derive expressions for the $p_T$ and $p_z$ of a charged particle starting from a subset of the five track parameters.

  • (b) Derive an expression for the transverse impact parameter $d_0$ starting from the circular fit parameters $(x_0,y_0)$ and R.

• 16. A particle flow algorithm must extrapolate tracker tracks to calorimeter barrels and endcaps. For a track with circular fit parameters $(x_0,y_0)$ and R, solve for (x, y) where the track intersects a barrel at radial coordinate r.

• 17. For SiD, calculate the points (x, y) of intersection with tracker, calorimeter and muon detector inner and outer radii for

  • (a) an electron with $p_T$=20 GeV and a photon with E=25 GeV (assume $\phi _0=\pi /4$)

  • (b) a $K_L$ with $p_T$=50 GeV and a charged pion with $p_T$=45 GeV (assume $\phi _0=\pi /2$)

  • (c) a muon with $p_T$=30 GeV (assume $\phi _0=\pi$)

• 18. Prepare a set of $10^4$ simple events, $e^+ e^- \rightarrow b\bar{b}$ at $\sqrt{s}=250$ GeV, in Whizard2 or MG5 aMC@NLO. Run the nominal ILCSoft SiD simulation and reconstruction on the events. Identify the configurable parameters in the following processors and vary them around the nominal values in the Marlin XML. How do the results compare to the nominal results?

  • (a) Tracking Processor: minimum hits for track, maximum $\chi ^2$, seed finding

  • (b) Particle Flow Processor: tracks inputs, maximum track/cluster association distance

  • (c) Jet Finding Processor: jet finding algorithm and algorithm parameters

• 19. Write an analysis in Root C++ to select Higgstrahlung events simulated with Delphes and DSiD. For $\int \mathrm{d}t L=1$ ab at $\sqrt{s}=250$ GeV, how many signal events are recovered by your analysis?

• 20. For the analysis of Higgstrahlung events in the previous problem, run the code over background events simulated with Delphes and DSiD. How many background events survive your analysis?

ILCSoft installation and use

It is a truism that technical software documentation is obsolete almost as soon as it is written. Hopefully, the instructions below will still be useful at the time the reader tries them. They assume a user in a bash shell on a Linux operating system connected to the Internet.

1.1 ILCSoft from CVMFS

ILCSoft version v02-00-02 is current at the time of writing, but make sure to use the most recent version. Presumably, the reader is in a directory like /home/potter/ilcsoft/v02-00-02 and will make a new directory under the ilcsoft directory for other versions. In what follows, bash$\texttt {>}$ is the shell prompt. The backslash ${{\backslash }}$ indicates to continue on the same line with no space and should not be typed.

These instructions assume that the CernVM Filesystem is mounted on your local computer. If it is not, your system administrator can mount it with the instructions in sec. B.3. First put the following two lines in a file called setup_v02-00-02.sh:

Source these setups, get the lcgeo package from GitHub, and build it. Note that in the cmake command the you need . These should be returning paths to the compiler binaries in /cvmfs/sft.cern.ch to the cmake command and should use the character (keyboard top left) rather than the ’ character (keyboard middle right), despite the typography below:

You may get an error indicating that you need to use a more recent version of cmake than you have installed. If so consult your system administrator. Now do some cleanup:

The which command should return the executable from /cvmfs/ild.desy.de rather than from your local install. Now test the simulation executable ddsim:

Check the outputs to make sure they make sense. Now let us install some reconstruction software and run Marlin on the simple simulation file we just generated (simple_lcio.slcio). First, we need to get the SiDPerformance package and update the file SiDPerformance/gear_sid.xml:

Make sure to use the right detector description. At the time of writing, this is SiD option 2 version 3, but use the version appropriate to your task. The line in SiDPerformance/gear_sid.xml should read $\langle$global detectorName="SiD_o2_v03"$/\rangle$. for SiD option 2 version 3. Now we run:

1.2 LCIO files with python (pyLCIO)

The output LCIO files generated with ddsim and Marlin can be read by a Python program using the pyLCIO package. For Python reconstruction, track and calorimeter hits can be read in and used as input to reconstruction algorithms. For Python analysis of reconstruction objects created with Marlin, one first needs to know what objects are available in the LCIO file.

LCIO objects are stored in collections. The following Python code imports a reader from pyLCIO, uses it to read an LCIO file named ’infile.lcio’, then cycles through the events in the file, and prints the names and types of all collections in the file:

It is frequently useful to use this code to find out the exact names used for collections in an LCIO file since these names are configurable by XML for both ddsim and Marlin and will vary from LCIO file to LCIO file.

We now give an example of code which obtains collections of Monte Carlo truth objects, track objects, and PFO objects and performs some manipulations on the objects:

For a Monte Carlo truth object, the getPDG() method returns the Particle Data Group ID. In the case of a PFO, the getType() method returns not the truth PDG ID, but the PFO hypothesis: 11 for electrons, 22 for photons, 211 for charged hadrons (not just charged pions), 2112 for neutral hadrons (not just neutrons), and 13 for muons.

Fig. 13

Available getters for an event in an LCIO file. MCParticle objects are Monte Carlo truth objects, ie truth information provided by the event generator. Track and Cluster objects are reconstructed from detector simulation hit objects. A PFO is an example of a ReconstructedParticle. Credit: LCIO [50]

See Fig. 13 for the available getters in an LCIO event.

1.3 CernVM filesystem (CVMFS)

You will need the root password to install CVMFS. If you do not have it, ask your system administrator. Do the following:

In file /etc/cvmfs/default.local put these lines:

Now obtain the desy.de.pub key

and place it in the file /etc/cvmfs/keys/desy.de.pub. In file /etc/cvmfs/domain.d/desy.de.conf put these lines:

Then, enter the following commands to set up CVMFS with auto-mounting and check the configuration:

The necessary mounts from CERN and DESY should be at /cvmfs/sft.cern.ch and /cvmfs/ilc.desy.de.