Particle Physics Design Group Studies Worksheet. Particle Physics Experiments: Colliders vs Fixed Targets

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1 January Particle Physics Design Group Studies Worksheet Introduction In this worksheet a number of topics are introduced which are relevant to the design study you will be performing in group studies. Some subjects are reminders of material you have met in lectures, others are topics which you may not have met in any detail in your lectures so far. In each case only a brief introduction is given; you are expected to follow up each subject in textbooks or other sources before you will be able to complete the problems given. Pointers to appropriate reference sources are given in each section; you may have to follow up further references within those sources, or elsewhere, to gain in-depth knowledge that you will need for the project work. There is a list of all the references at the end, plus some additional possible starting points. Collaborative work and discussions within the group to address these issues and problems are very much expected and encouraged. However, written and numerical answers to questions should be your own individual work, as always. Note that the convention h/2π = c = 1 is normally used in particle physics, including in this worksheet. Particle Physics Experiments: Colliders vs Fixed Targets In particle physics there are two basic kinds of experiments. One uses a beam hitting a stationary target in the laboratory, a so-called fixed target experiment. The second collides two beams, usually head on, and often with equal and opposite momentum; this is a particle collider. The instantaneous interaction rate in a collider is usually much less than that in a fixed target experiment, and so colliding-beam machines are often built as storage rings to allow the beams to circulate and collide over several hours. Q1. (5 points) Colliders Consider the interaction of a high-energy proton beam (p) hitting protons (p) which are essentially at rest in a liquid hydrogen fixed target. Calculate relativistically, from first principles, the proton energy needed to create a Higgs boson (H) with mass m H = 125 GeV via the reaction pp pph. What is the minimum proton beam energy needed to produce a Higgs boson with the above mass in a pp colliding-beam experiment? Compare this value with the beam energy of the LHC. Comment briefly on why the LHC beam energy is so much different. Event Rates and Luminosity At a particle collider the number of events produced per second dn/dt(s 1 ) for a process with cross section σ(m 2 ) depends only on the luminosity L(m 2 s 1 ), where dn/dt = Lσ. Rare

2 January processes have low cross sections and so it is essential to keep the collider luminosity high to observe them. This is achieved by having large numbers of particles in each beam, colliding these as frequently as possible and making the beam sizes small. Of course these quantities do vary and so L is the instantaneous luminosity. In particle and nuclear physics the barn is frequently used as the unit of cross section and this is equal to10 28 m 2. The cross sections for many interesting processes at high-energy colliders are much smaller than this, e.g. picobarns, pb (10 40 m 2 ). The instantaneous luminosity is usually quoted in units of cm 2 s 1, but may also be quoted in units of pb 1 s 1. Depending on the magnitude of the cross section it may sometimes be more convenient to work with e.g. femtobarns, fb (10 43 m 2 ), or nanobarns, nb (10 37 m 2 ), rather than pb. When running an experiment over a long period the instantaneous luminosity will vary and the total number of events recorded will depend on the average instantaneous luminosity multiplied by the total time of the experiment (t). This product is known as the integrated luminosity (L INT ) and is frequently used in describing the sensitivity of an experiment. It is often quoted in inverse picobarns (pb 1 ). We need high L INT and this is achieved by maximising both the instantaneous luminosity L and the running time t as far as possible. An experiment with an integrated luminosity of500pb 1 would expect to observe, on average, 500 events of a process with a cross section of 1pb, if it were fully efficient. As the events occur at random, this is an average expectation and the actual numbers observed in a series of independent experiments of this type follow a Poisson distribution with this mean. Unless this number N is very small, it is reasonable to approximate the spread (standard deviation) in the number of events to be N. Q2. (15 points) Event rates The LHC will deliver proton-proton collisions to the LHCb experiment with a centre-of-mass energy of s = 14 TeV, an instantaneous luminosity of cm 2 s 1 and with proton bunches crossing in the experimental interaction region every 25 ns. The total cross section for pp interactions at this energy is 120 mb, and the inclusive b b production cross section, pp b bx, is 1 mb. (a) Calculate the average number of pp interactions per bunch crossing, and the average number of b b pairs produced each second. (b) The LHCb Collaboration wants to study the rare decay modeb 0 s φγ, which has a branching fraction of (i.e. this fraction of all B 0 s mesons decay in this way). Assume that 10% of all the b produced form B 0 s mesons, and that the overall efficiency of the LHCb experiment to detect and reconstruct B 0 s φγ decays is 1%. For how long will the LHCb experiment need collect data, running at this luminosity, in order to expect to reconstruct an average of 10 B 0 s φγ decays. What is the probability that less than 5B 0 s φγ decays are reconstructed in this data sample? (c) The LHCb Collaboration is also searching for another, previously unobserved, very rare decay mode of the B 0 s meson with a predicted branching fraction of Assuming the same detection and reconstruction efficiency as in part (b), and that the LHC can continue to deliver proton collisions at an average instantaneous luminosity of cm 2 s 1 with an operational efficiency of 25%, for how long would the experiment need to run, in order to expect to reconstruct an average of one of these very rare decays? What is the probability that none of these very rare decays would be reconstructed in such a data sample? For how long would the

3 January LHCb experiment need to run under these conditions, so that the probability to reconstruct none of these very rare decays is less than 5%. Q3. (10 points) Statistics of discovery The LHCb experiment is used to search for the production of new particles among the decay products of known, heavier and copiously produced particles. Consider the search for a new particle that is produced in the decay of the Λ b baryon in a data sample corresponding to an integrated luminosity of 3 fb 1. Assuming the visible production cross section of the Λ b to be 1 nb, and the branching fraction for decays into the new particle to be 10 6 : a: What is the probability that at least one of these new particles has been produced within this data sample? b: Assume that after further data-taking, 30 fb 1 of data are collected. What is the probability that no new particle has been produced? c: A measure of the statistical significance of a signal may be given in a counting experiment by the number of signal events divided by the uncertainty on the number of expected background events. The Standard Model background in the mass range relevant to the above new particle has an effective branching fraction of Estimate how much integrated luminosity the experiment requires to observe a significance of 3σ? Accelerator Physics The motion of a particle with charge e and velocity v passing through a volume containing a magnetic fieldb and an electric fielde is determined by the Lorentz force F = e(v B +E). (1) The magnetic force acts in a direction perpendicular both to the direction of the particle and the direction of the magnetic field, whereas the electric force acts along the direction of the electric field. In the absence of an electric field, a particle travelling at right angles to a constant magnetic field will describe a circular trajectory. The formula that relates the radius R of the orbit to the momentum p of the particle and the magnetic field is p = BeR. (2) An electric field can be used to accelerate the particle along its direction of motion. In a fixed magnetic field, the radius R will then increase in proportion to the momentum p as in the cyclotron. Alternatively, the radius may be kept constant by increasing the magnetic field in proportion to the momentum as the particle accelerates, and this is the principle of the synchrotron. The advantage of the synchrotron is that the volume of the magnet is greatly reduced as the field is only needed along a fixed trajectory, resulting in an annular ring of magnets for a practical device. High-energy protons and electrons are accelerated using synchrotrons. If, after acceleration to a given momentum, the magnetic field is held fixed and the particles are allowed to circulate at constant momentum, the synchrotron becomes a storage ring. A colliding-beam machine can be formed by allowing two beams of particles to circulate in opposite directions along

4 January approximately the same trajectory and causing them to collide at specific locations at which the beams are focussed to a small size to increase the rate of collision (L σ). This can only be done for particles of opposite charge, the same energy but opposite momentum, e.g. electrons and positrons in LEP at CERN, protons and antiprotons in the Tevatron at Fermilab. In order to collide two particle beams of the same charge, two intersecting rings of magnets are required with fields in opposite directions. The LHC at CERN is of this type. Q4. (2 points) Bending power of LHC dipole magnets One of the factors limiting the maximum beam energy at the LHC is the strength of the dipole magnets. Calculate the radius of an ideal LHC, using only dipole magnets all the way around a circular circumference. Use typical numbers for the magnetic field, B = 8 T, and the beam momentum, p = 7 TeV. How does this result compare to the actual radius of the LHC? The ultimate momentum (or energy) that can be achieved in a synchrotron is obviously proportional to the product of the magnetic field and the radius of the orbit. However, at sufficiently high energy, or more strictlyγ = E/mc 2, particles that are accelerated emit a significant amount of radiation. The transverse acceleration in a synchrotron gives rise to synchrotron radiation which, at a fixed radius, increases as the fourth power of the particle energy and also as the inverse of the fourth power of the particle mass, making it much more serious for electrons. In a colliding-beam machine such as LEP, the energy lost in each revolution due to synchrotron radiation has to be replaced by re-accelerating the electrons and positrons with a radio-frequency electric field. Otherwise the particles would spiral inwards and be lost. This effect limits the maximum energy at which it is practical to operate a circular collider. The instantaneous luminosity, L, in a colliding-beam experiment can be calculated from several equivalent expressions. One useful form for circular colliders, assuming that the particle densities are Gaussian transverse to the beam direction, is the following: L = I 1 I 2 /(e 2 kf rev 4πσ x σ y ), (3) wheref rev is the revolution frequency,k is the number of bunches of particles per beam,σ x and σ y are the RMS beam sizes in the transverse direction, and I 1,I 2 are the two beam currents. To achieve very high luminosities some present and future machines have many bunches to overcome space-charge limitations. LEP was operated with 4, 8 and 12 bunches per beam, and the LHC is designed for 2800 bunches spaced apart by just25ns. To increase the luminosity, it is important to keep the transverse beam size small at the interaction point; this is done by a process known as squeezing the beams. More information on accelerator physics can be found in the books by Tigner and Chao, Wilson, and Wille. Q5. (8 points) Accelerator luminosities a) Convert the above equation for L into an expression containing instead N, the number of particles per bunch. Calculate the minimum value for N at a 500 GeV (250 GeV on 250 GeV) collider (k = 1) to produce an average of 10 events per day for a process with a cross section of100fb, if the beam dimensions areσ x = σ y = 1mm, andf rev is10 3 Hz. Convert your value for N into a beam current (A) and an instantaneous luminosity (in units ofcm 2 s 1 ). b) Redo the calculations assuming σ x = σ y = 0.01 mm(10 µm). This question should make you appreciate some of the real problems in studying these rare reactions (tiny beam sizes with

5 January lots of particles in them). At LEP, typical instantaneous luminosities were cm 2 s 1. At LHC the design is cm 2 s 1. Interaction of Radiation with Matter: Particle Detectors The detection of particles in experiments depends on the characteristics of the interactions of the particles with the detector material. Several different types of interactions are important, the main ones being: ionisation and excitation of atoms by charged particles; electromagnetic interactions in nuclear electric fields; and hadronic interactions of strongly-interacting particles with nuclei. Many standard particle and nuclear physics textbooks have an introduction to the interactions of radiation with matter, and particle detectors. Good, more advanced, textbooks are those by Green and by Kleinknecht. The Review of Particle Physics contains many useful formulae. Energetic charged particles cause ionisation and excitation of atoms as they move through material. The average energy loss of charged particles per unit length, de/dx, through ionisation and excitation is described by the Bethe-Bloch equation. The energy spectrum of ionised electrons falls steeply with increasing energy, and energetic primary ionisation electrons may produce secondary ionisation. The total number of ions produced per unit length when a particle passes through the material depends both on the energy loss (given by the Bethe-Bloch equation, on average) and on the specific properties of the material, particularly the average energy needed to create each ion. This is as high as 40 ev for helium, but effectively as low as 3.6 ev in silicon, where the charge is carried by electron-hole pairs rather than electrons and ions. In addition to interaction with atomic electrons, charged particles may also interact in the strong electric fields around nuclei. All charged particles are affected by Coulomb scattering, which gives rise to multiple scattering when passing through matter. Electrons may give up a significant fraction of their energy in inelastic collisions with nuclei, via the so-called bremsstrahlung (radiation loss) process, where the energy is radiated off as a photon. Incident photons with sufficient energy can also interact in the nuclear electric field to produce electron-positron pairs, so-called conversions. High-energy electrons or photons impacting on dense materials thus produce electromagnetic showers of electrons and photons via multiple bremsstrahlung and conversion processes. Eventually all the energy of an incoming particle may be absorbed, if the material is thick enough. The length scale of these electromagnetic processes is the radiation length, denoted X 0. Strongly interacting particles (hadrons) can also interact via collisions with atomic nuclei; typically for high-energy particles several collision products will be produced, the charged ones going on to ionise surrounding material. Analogously to the electron/photon case, this in turn gives rise to an hadronic shower of particles, which may be absorbed if the material is thick enough. In this case, the typical length scale is characterised by the nuclear interaction length λ I.

6 January Tracking Detectors The basic idea of a tracking detector is to detect the passage of electrically charged particles by ionisation energy loss, but to include only a little material so that the processes of showering do not start, and to minimise multiple scattering. In this way, particle trajectories can be followed over distances of metres or more. Most particle physics experiments employ magnetic fields in the region around the interaction point so that moving charged particles are bent by the Lorentz force on them. In colliding-beam experiments this is typically provided by a solenoid wound around the beam direction, providing a uniform magnetic field along the beam axis. The measurement of several points along a particle trajectory with tracking detectors allows the curvature of a particle track to be measured, and hence the momentum can be deduced. Typical tracking detectors in use today are constructed with large gas volumes used for detection of ionisation. Ions drift over distances varying from a centimetre or two to several metres. Best established are drift chambers, where the ionisation drifts a few centimetres before being amplified at wire planes. Another technology, the time-projection chamber, uses much longer drift distances of up to a few metres. An alternative technology for tracking detectors is to use thin silicon sensors, and to measure electron-hole pairs produced by passing particles. These devices may be made very precisely, as sensor tracks may be etched at the level of tens of microns. They are particularly useful for measuring particle trajectories close to the primary interaction point, to look for particles which travel only short distances (tens of microns or more) before decaying. Q6. (10 points) Momentum resolution A tracking detector surrounding a colliding-beam interaction point makes 40 equally spaced measurements of the particle trajectories in the coordinate plane perpendicular to the beam direction, over a distance of 1.5 m. The precision of each measurement is 200 µm in the azimuthal direction (i.e. perpendicular both to the beam direction and to a line drawn from the measurement to the beam axis). The detector is immersed in a uniform magnetic field along the beam direction of2t. a) Ignoring multiple scattering effects, what is the expected momentum resolution for muons with 1 GeV, 100 GeV and 1 TeV momentum, respectively, travelling out at 90 to the beam direction? b) If the detector is filled with argon at20 C and atmospheric pressure, estimate approximately the size of the error on the momentum measurements from multiple scattering, for the three momentum values given in a). How does this compare with the errors from the intrinsic point resolution as a function of momentum? Calorimetric Detectors Tracking detectors have two disadvantages for measuring the particles produced in a collision; they see only electrically charged products, and also the resolution for very energetic particles, which are bent little by magnetic fields, is quite poor. The measurements may therefore be complemented by calorimeters, which are large detectors typically made out of dense solid

7 January materials such as lead and iron. Calorimeters are designed to absorb the energy of particles hitting them via the showering mechanism, and to be thick enough that the full shower is contained inside them. Lead and iron in themselves are not sufficient, as although incident particles shower inside them there is no way to see the resultant energy deposition. A standard technique to overcome this is to place thin layers of a detector material, such as scintillator slabs or gas detectors, between thick layers of the dense absorber. In this way, the charged particles in the shower are sampled at various points in the shower. Alternative techniques exist by using a dense active medium, such as crystal calorimeters, lead-glass, etc. The characteristic length scales of electromagnetic and hadronic showers are different, but a significant fraction of the products of collision events can be produced in the form of photons and electrons. Calorimetry is therefore usually divided into two parts: a front part (nearest the interaction point) is thin and highly instrumented to catch the electromagnetic showers, and it will also catch the start of some of the hadronic showers; a thicker, less well-instrumented, back part catches the rest of the hadronic showers. These are conventionally called electromagnetic and hadronic calorimeters, although the functions actually overlap. Q7. (20 points) Calorimeter resolutions (a) Describe the main features of homogeneous and sampling electromagnetic calorimeters and the differences between them. What are the advantages and disadvantages of each type of calorimeter when used as part of a general-purpose particle physics detector? (b) The energy resolution of a sampling electromagnetic calorimeter can be written in the form σ(e) E = a E b E c, where the symbol denotes addition in quadrature, E is the energy in GeV and a,bandcare constants. Explain the physical origin of each of the three terms in this expression and give typical values of a,bandc. Multi-layer Detectors and Particle Identification A typical particle physics experiment is a hybrid consisting of multiple layers: precise tracking close to the interaction point; larger tracking detectors next; then electromagnetic and hadronic calorimeters. On the outside of the detectors are placed extra tracking detectors called muon detectors. Muon particles interact only minimally with the material of the detector and are, therefore, more likely than other charged particles to travel unhindered to the outside of the detector. In addition to measuring particle momenta and energies, the information from the different detectors may also be combined to give further information about the type of the particles, socalled particle identification. For example, by matching up electromagnetic calorimeter energy deposits with charged particle tracks, it may be possible to identify individual neutral particles. For charged particles, the rate of energy loss information, de/dx, may allow protons, electrons and charged pions, for example, to be separated.

8 January Further specialised detectors are used in some experiments to give extra information for the purposes of particle identification. Examples are Cherenkov detectors and transition radiation detectors. References - to get you started D. H. Perkins, Introduction to High Energy Physics, 4th edition 2000, Camb. Univ. Press. A. Das and T. Ferbel, Introduction to Nuclear and Particle Physics, 1993, Wiley. G. Kane, Modern Elementary Particle Physics: The Fundamental Particles and Forces, 1993, Perseus Books. B. R. Martin and G. Shaw, Particle Physics, 2nd edition 1997, John Wiley. I. Hughes, Elementary Particles, 3rd edition 1991, Camb. Univ. Press. M. Tigner and A. Chao, Handbook of Accelerator Physics and Engineering, 2nd edition 2013, World Scientific (online access to full text). E. Wilson, An Introduction to Particle Accelerators, 2001, Oxford Univ. Press. K. Wille, The Physics of Particle Accelerators: an Introduction, Oxford Univ. Press. Hands-on CERN web pages: v21en/index.html (in particular the detectors part) D. Green, The Physics of Particle Detectors, 2000, Camb. Univ. Press. K. Kleinknecht, Detectors for Particle Radiation, 2nd edition 1998, Camb. Univ. Press. The Review of Particle Physics, K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, (2014). W. R. Leo, Techniques for Nuclear and Particle Physics Experiments - A How-to Approach, 2nd edition 1994, Springer Verlag. R. K. Bock and A. Vasilescu, The Particle Detector BriefBook, R. K. Bock and W. Krischer, The Data Analysis BriefBook, H. Murayama and M. Peskin, Ann. Rev. Nucl. Part. Sci. 46 (1996)