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UNIVERSITY OF LONDON

MSci/BSc EXAMINATION 2012

For Internal Students of Royal Holloway

DO NOT TURN OVER UNTIL TOLD TO BEGIN

PH3520 : PARTICLE PHYSICS

Time Allowed: TWO hours

Answer THREE Questions

Approximate part-marks for questions are given in the right-hand margin

The total available marks add up to 120

No credit will be given for attempting any further questions

College Calculators are provided

Royal Holloway University of London 2012

PH3520

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GENERAL PHYSICAL CONSTANTS

Permeability of vacuum 0 = 4 10-7 H m-1

Permittivity of vacuum 0 = 8.85 10-12 F m-1

01/ 4 = 9.0 109 m F-1

Speed of light in vacuum c = 3.00 108 m s-1

Elementary charge e = 1.60 10-19 C

Electron (rest) mass em = 9.11 10-31 kg

Unified atomic mass constant um = 1.66 10-27 kg

Proton rest mass pm = 1.67 10-27 kg

Neutron rest mass nm = 1.67 10-27 kg

Ratio of electronic charge to mass / ee m = 1.76 1011 C kg-1

Planck constant h = 6.63 10-34 J s

/ 2h = = 1.05 10-34 J s

Boltzmann constant k = 1.38 10-23 J K-1

Stefan-Boltzmann constant = 5.67 10-8 W m-2 K-4

Gas constant R = 8.31 J mol-1 K-1

Avogadro constant AN = 6.02 1023 mol-1

Gravitational constant G = 6.67 10-11 N m2 kg-2

Acceleration due to gravity g = 9.81 m s-2

Volume of one mole of an ideal gas at STP = 2.24 10-2 m3

One standard atmosphere 0P = 1.01 105 N m-2

MATHEMATICAL CONSTANTS

2.718e 3.142 log 10 2.303e

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PART MARKS

1. (a) The tau lepton was discovered at the SPEAR storage ring by observing events such as

,

where represents unseen particles. For this event type, draw a Feynman diagram that shows the production of the tau leptons and also diagrams for their subsequent decays. Label all particles and indicate the coupling strengths for all vertices. [8]

(b) Show that if the particles are all highly relativistic so that one can neglect their rest masses, the invariant mass squared of the unseen particles in the reaction above (the missing mass squared) can be found from

,

where is the centre-of-mass energy, and are the energies of the

electron and muon in the c.m. frame and is the angle between their momentum vectors.

Describe how the distribution of missing mass in the events observed led one to conclude that there was more than one unseen particle in the final state. [12]

(c) The total cross section for the reaction can be written in particle physics units as

2

tot 2cm

43E = ,

where is the fine structure constant. Suppose an accelerator runs at for seconds at an average luminosity

of . Find the expected number of events of this type. Use . [6]

(d) The distribution of the angle between the and the incoming follows

.

Estimate the fraction of events one would expect with the direction more than 15 degrees from the beam line (i.e., ). [8]

(e) Explain how by measuring the cross section of the reaction one can determine the mass of the tau lepton. Illustrate your answer with a sketch. [6]

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PART MARKS 2. (a) Describe the first observation of on-shell Z bosons in proton-antiproton

collisions, including information on the relevant decay modes of the Z and on how one was able to recognise events of this type experimentally. Sketch a Feynman diagram that indicates both production and decay of the Z.

What is meant by the term "underlying event"? Indicate in your Feynman diagram what particles are connected with this. [10]

(b) Sketch the cross section as a function of the centre-of-mass energy from 10 to 100 GeV. Indicate on the sketch the mass and total width of the Z resonance.

Draw two lowest-order Feynman diagrams that contribute to and which contain different intermediate bosons. State for roughly what energy ranges each of the diagrams makes a significant contribution to the total amplitude. [8]

(c) The total decay width of the Z is found to be = 2.5 GeV. Find the mean lifetime of the Z in seconds and evaluate numerically. You may use

and .

Describe how a measurement of can be used to determine the number of light neutrino types . [10]

(d) Consider the reaction at a centre-of-mass energy . Suppose the Z is produced on shell and that it decays into a neutrino-antineutrino pair.

Draw the lowest order Feynman diagram for this reaction, including the Z decay to .

What would such an event look like in one of the LEP detectors such as ALEPH?

Show that the energy of the photon in the centre-of-mass frame is

. [12]

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PART MARKS 3. (a) At the Large Hadron Collider (LHC), protons collide head-on at a centre-of-

mass energy of 7 TeV.

Suppose one wanted to achieve the same centre-of-mass energy in a fixed-target experiment. Find the required beam energy and evaluate in GeV. Use the proton mass and state any approximations made. [8]

(b) The LHC is housed in the same tunnel used earlier by the LEP collider. The LHC collides protons with a beam energy of 3.5 TeV, whereas the beam energy at LEP was only 100 GeV. The energy emitted as synchrotron radiation is, however, very low at the LHC, but at LEP it was 3.6 GeV per electron per turn. Find the energy emitted per proton per turn at the LHC for a beam energy of 3.5 TeV. Use the electron mass as well as the proton mass given above. [10]

(c) Explain how a measurement of momentum and ionization energy loss can be used to distinguish between charged pions, kaons and protons.

Illustrate your answer with a sketch of versus for particles of these types.

Describe how this method can be used to search for fractionally charged particles such as quarks with charge . What would one expect for the minimum produced by such particles relative to the minimum found for charged pions. Have searches of this type indicated evidence for fractionally charged particles? [10]

(d) Describe how in a tracking chamber one can measure the momentum of a charged particle, stating any needed relations between the momentum and other relevant quantities. State how the relative accuracy of the momentum measurement depends on the momentum and explain qualitatively why the accuracy becomes worse for increasing momenta.

Describe how one can measure the energy of a high-energy photon. Draw Feynman diagrams for the fundamental interactions involved and indicate with a sketch what happens when a high-energy photon interacts with matter. State how the relative accuracy of the energy measurement depends on the photon's energy and explain qualitatively why the accuracy of such a measurement improves with increasing energy. [12]

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PART MARKS

4. (a) Consider the reaction . Draw a Feynman diagram for this decay indicating all particles (including quarks inside the hadrons) and all coupling strengths, including CKM matrix elements.

State how the amplitude for the decay depends on the couplings you have given and on the mass of the W boson.

The maximum invariant mass of the system occurs when, in the rest

frame of the , the meson is also at rest and the emerge back to

back. In this configuration, show that the invariant mass of the system is equal to the mass difference between the and .

Explain how this fact may be used to express the decay rate in terms of the Fermi constant, . [16]

(b) The mean proper lifetime of the meson is 1.64 ps and its mass is 5.3 GeV. Suppose a is produced with a momentum of 100 GeV. Find the mean distance in mm that it will travel before decaying. [8]

(c) Explain what it means for a jet of hadrons to be "b-tagged". Describe a method by which b-tagging is done and illustrate it with a simple sketch. [8]

(d) Consider a hadron jet initiated by a b quark from produced at a centre-of-mass energy , and suppose the b quark hadronizes to

become a meson, which subsequently decays to . Describe how to find such a decay experimentally, including information on how to identify the and , making reasonable assumptions about whether and how these decay. [8]

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PART MARKS 5. (a) At the LEP Collider, an electron and positron could collide to form a quark-

antiquark pair. Make a sketch of roughly what one would expect to see in a detector for an event of this type.

Describe how events of this type can be used to test the hypothesis that quarks are spin- particles. [8]

(b) Sketch the strong coupling constant, , as a function of the energy scale, , and comment on how this curve suggests that quarks cannot exist as free particles. [6]

(c) Draw a Feynman diagram for the reaction , where q stands for any one of the quark flavours g is a gluon. Label all particles and indicate the coupling strengths for all vertices.

Sketch roughly what an event of this type would look like in a detector, supposing a centre-of-mass energy near the Z resonance.

Describe qualitatively how events of this type can be used to estimate the strong coupling constant . [12]

(d) In reactions such , large numbers of hadrons are produced, including some mesons. Describe how in practice it is possible to measure the average number of mesons produced in this reaction by exploiting the decay .

Describe how one can determine the mean lifetime of the meson.

Draw a Feynman diagram for the decay that includes a single gluon. Explain why this diagram alone would not be expected to provide an accurate estimate for the rate of this decay. [14]

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PART MARKS

6. (a) List all of the kinematically allowed decays of a boson to quarks or to leptons. For the decays to quarks, state for each mode whether it is Cabibbo allowed, singly Cabibbo suppressed, or doubly Cabibbo suppressed. [8]

(b) Consider the reaction .

Draw three lowest-order Feynman diagrams for this reaction.

By neglecting Cabibbo suppressed modes, show that the fraction of such events where both W bosons decay to hadrons is 4/9.

How would an event of this type look in a detector if both W bosons decay to hadrons? What other type of event in collisions could result in a similar final state?

Explain two methods by which can be used to estimate the mass of the W boson. Explain how the decay modes of the W bosons influence what steps must be carried out to determine the W mass. [12]

(c) Draw one of the important Feynman diagrams for production of the Higgs boson in a proton-proton collision and its subsequent decay to . Justify your choice of particles that participate in the diagram.

Suppose the mass of the Higgs boson is 125 GeV, and suppose we look for events with with both W bosons decaying leptonically. Discuss briefly the advantages and disadvantages of this decay mode relative to the case where both Ws decay to hadrons. [10]

(d) Suppose the mass of the Higgs boson is 125 GeV, which implies that its branching ratio to is approximately 0.15. The cross section for Higgs Boson production at the LHC at a centre-of-mass energy is 20

pb. For an integrated luminosity of , find how many events of the type will be produced and evaluate numerically. [10]

END

MSci/BSc EXAMINATION 2012