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Particle Physics Nikos Konstantinidis

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3 Practicalities (I) Contact details Contact details My office: D16, 1 st floor Physics, UCLMy office: D16, 1 st floor Physics, UCL My e-mail: n.konstantinidis@ucl.ac.ukMy e-mail: n.konstantinidis@ucl.ac.uk Web page: www.hep.ucl.ac.uk/~nk/teaching/PH4442Web page: www.hep.ucl.ac.uk/~nk/teaching/PH4442 Office hours for the course (this term) Office hours for the course (this term) Tuesday 13h00 – 14h00Tuesday 13h00 – 14h00 Friday 12h30 – 13h30Friday 12h30 – 13h30 Problem sheets Problem sheets To you at week 1, 3, 5, 7, 9To you at week 1, 3, 5, 7, 9 Back to me one week laterBack to me one week later Back to you (marked) one week laterBack to you (marked) one week later

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4 Practicalities (II) Textbooks Textbooks I recommend (available for £23 – ask me or Dr. Moores)I recommend (available for £23 – ask me or Dr. Moores) Griffiths Introduction to Elementary Particles Griffiths Introduction to Elementary Particles AlternativesAlternatives Halzen & Martin Quarks & Leptons Halzen & Martin Quarks & Leptons Martin & Shaw Particle Physics Martin & Shaw Particle Physics Perkins Introduction to High Energy Physics Perkins Introduction to High Energy Physics General reading General reading Greene The Elegant UniverseGreene The Elegant Universe

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5 Course Outline 1. Introduction (w1) 2. Symmetries and conservation laws (w2) 3. The Dirac equation (w3) 4. Electromagnetic interactions (w4,5) 5. Strong interactions (w6,7) 6. Weak interactions (w8,9) 7. The electroweak theory and beyond (w10,11)

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6 Week 1 – Outline Introduction: elementary particles and forces Introduction: elementary particles and forces Natural units, four-vector notation Natural units, four-vector notation Study of decays and reactions Study of decays and reactions Feynman diagrams/rules & first calculations Feynman diagrams/rules & first calculations

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7 The matter particles All matter particles (a) have spin ½; (b) are described by the same equation (Diracs); (c) have antiparticles All matter particles (a) have spin ½; (b) are described by the same equation (Diracs); (c) have antiparticles Particles of same type but different families are identical, except for their mass: Particles of same type but different families are identical, except for their mass: m e = 0.511MeV m =105.7MeV m =1777MeV Why three families? Why they differ in mass? Origin of mass? Why three families? Why they differ in mass? Origin of mass? Elementary point-like (… but have mass/charge/spin!!! ) Elementary point-like (… but have mass/charge/spin!!! )

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8 The force particles All force particles have spin 1 (except for the graviton, still undiscovered, expected with spin 2) All force particles have spin 1 (except for the graviton, still undiscovered, expected with spin 2) Many similarities but also major differences: Many similarities but also major differences: m = 0 vs. m W,Z ~100GeV Unlike photon, strong/weak mediators carry their own charge The SM provides a unified treatment of EM and Weak forces (and implies the unification of Strong/EM/Weak forces); but requires the Higgs mechanism ( the Higgs particle, still undiscovered!). The SM provides a unified treatment of EM and Weak forces (and implies the unification of Strong/EM/Weak forces); but requires the Higgs mechanism ( the Higgs particle, still undiscovered!).

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9 SI units not intuitive in Particle Physics; e.g. SI units not intuitive in Particle Physics; e.g. Mass of the proton ~1.7×10 -27 kgMass of the proton ~1.7×10 -27 kg Max. momentum of electrons @ LEP ~5.5×10 -17 kgmsecMax. momentum of electrons @ LEP ~5.5×10 -17 kgmsec Speed of muon in pion decay (at rest) ~8.1×10 7 msecSpeed of muon in pion decay (at rest) ~8.1×10 7 msec More practical/intuitive: ħ = c =1; this means energy, momentum, mass have same units More practical/intuitive: ħ = c =1; this means energy, momentum, mass have same units E 2 = p 2 c 2 + m 2 c 4 E 2 = p 2 + m 2E 2 = p 2 c 2 + m 2 c 4 E 2 = p 2 + m 2 E.g. m p =0.938GeV, max. p LEP =104.5GeV, v =0.27E.g. m p =0.938GeV, max. p LEP =104.5GeV, v =0.27 Also:Also: Time and length have units of inverse energy! Time and length have units of inverse energy! 1GeV -1 =1.973×10 -16 m1GeV -1 =6.582×10 -25 sec1GeV -1 =1.973×10 -16 m1GeV -1 =6.582×10 -25 sec Natural Units

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11 4-vector notation (I) 4-vector: An object that transforms like x between inertial frames E.g. Invariant: A quantity that stays unchanged in all inertial frames E.g. 4-vector scalar product: 4-vector length: 4-vector length: Length can be: >0timelike <0 spacelike =0lightlike 4-vector Lorentz Transformations

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12 4-vector notation (II) Define matrix g: g 00 =1, g 11 =g 22 =g 33 =-1 (all others 0) Define matrix g: g 00 =1, g 11 =g 22 =g 33 =-1 (all others 0) Also, in addition to the standard 4-v notation (contravariant form: indices up), define covariant form of 4-v (indices down): Also, in addition to the standard 4-v notation (contravariant form: indices up), define covariant form of 4-v (indices down): Then the 4-v scalar product takes the tidy form: Then the 4-v scalar product takes the tidy form: An unusual 4-v is the four-derivative: An unusual 4-v is the four-derivative: So, a is invariant; e.g. the EM continuity equation becomes: So, a is invariant; e.g. the EM continuity equation becomes:

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13 What do we study? Particle Decays (A B+C+…) Particle Decays (A B+C+…) Lifetimes, branching ratios etc…Lifetimes, branching ratios etc… Reactions (A+B C+D+…) Reactions (A+B C+D+…) Cross sections, scattering angles etc…Cross sections, scattering angles etc… Bound States Bound States Mass spectra etc…Mass spectra etc…

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15 Study of Decays ( A B+C+… ) Decay rate : The probability per unit time that a particle decays Decay rate : The probability per unit time that a particle decays Lifetime : The average time it takes to decay (at particles rest frame!) Lifetime : The average time it takes to decay (at particles rest frame!) Usually several decay modes Usually several decay modes Branching ratio BR Branching ratio BR We measure tot (or ) and BRs; we calculate i We measure tot (or ) and BRs; we calculate i

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16 as decay width as decay width Unstable particles have no fixed mass due to the uncertainty principle: Unstable particles have no fixed mass due to the uncertainty principle: The Breit-Wigner shape: The Breit-Wigner shape: We are able to measure only one of, of a particle We are able to measure only one of, of a particle ( 1GeV -1 =6.582×10 -25 sec ) M0M0M0M0 N max 0.5N max

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17 Study of reactions ( A+B C+D+… ) Cross section Cross section The effective cross-sectional area that A sees of B (or B of A)The effective cross-sectional area that A sees of B (or B of A) Has dimensions L 2 and is measured in (subdivisions of) barnsHas dimensions L 2 and is measured in (subdivisions of) barns 1b = 10 -28 m 2 1b = 10 -34 m 2 1pb = 10 -40 m 2 Often measure differential cross sections Often measure differential cross sections d /d or d /d(cos )d /d or d /d(cos ) Luminosity L Luminosity L Number of particles crossing per unit area and per unit timeNumber of particles crossing per unit area and per unit time Has dimensions L -2 T -1 ; measured in cm -2 s -1 (10 31 – 10 34 )Has dimensions L -2 T -1 ; measured in cm -2 s -1 (10 31 – 10 34 )

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18 Study of reactions (contd) Event rate (reactions per unit time) Event rate (reactions per unit time) Ordinarily use integrated Luminosity (in pb -1 ) to get the total number of reactions over a running period Ordinarily use integrated Luminosity (in pb -1 ) to get the total number of reactions over a running period In practice, L measured by the event rate of a reaction whose is well known (e.g. Bhabha scattering at LEP: e + e – e + e – ). Then cross sections of new reactions are extracted by measuring their event rates In practice, L measured by the event rate of a reaction whose is well known (e.g. Bhabha scattering at LEP: e + e – e + e – ). Then cross sections of new reactions are extracted by measuring their event rates

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19 Feynman diagrams Feynman diagrams: schematic representations of particle interactions Feynman diagrams: schematic representations of particle interactions They are purely symbolic! Horizontal dimension is (…can be) time (except in Griffiths!) but the other dimension DOES NOT represent particle trajectories! They are purely symbolic! Horizontal dimension is (…can be) time (except in Griffiths!) but the other dimension DOES NOT represent particle trajectories! Particle going backwards in time => antiparticle forward in time Particle going backwards in time => antiparticle forward in time A process A+B C+D is described by all the diagrams that have A,B as input and C,D as output. The overall cross section is the sum of all the individual contributions A process A+B C+D is described by all the diagrams that have A,B as input and C,D as output. The overall cross section is the sum of all the individual contributions Energy/momentum/charge etc are conserved in each vertex Energy/momentum/charge etc are conserved in each vertex Intermediate particles are virtual and are called propagators; The more virtual the propagator, the less likely a reaction to occur Intermediate particles are virtual and are called propagators; The more virtual the propagator, the less likely a reaction to occur Virtual: Virtual:

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20 Fermis Golden Rule Calculation of or has two components: Calculation of or has two components: Dynamical info: (Lorentz Invariant) Amplitude (or Matrix Element) MDynamical info: (Lorentz Invariant) Amplitude (or Matrix Element) M Kinematical info: (L.I.) Phase Space (or Density of Final States)Kinematical info: (L.I.) Phase Space (or Density of Final States) FGR for decay rates (1 2+3+…+n) FGR for decay rates (1 2+3+…+n) FGR for cross sections (1+2 3+4+…+n) FGR for cross sections (1+2 3+4+…+n)

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21 Feynman rules to extract M Toy theory: A, B, C spin-less and only ABC vertex 1.Label all incoming/outgoing 4-momenta p 1, p 2,…, p n ; Label internal 4-momenta q 1, q 2 …,q n. 2.Coupling constant: for each vertex, write a factor – ig 3.Propagator: for each internal line write a factor i /(q 2 –m 2 ) 4.E/p conservation: for each vertex write (2) 4 4 (k 1 +k 2 +k 3 ); ks are the 4-momenta at the vertex (+/– if incoming/outgoing) 5.Integration over internal momenta: add 1/(2) 4 d 4 q for each internal line and integrate over all internal momenta 6.Cancel the overall Delta function that is left: (2) 4 4 (p 1 +p 2 –p 3 …–p n ) What remains is:

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22 Summary The SM particles & forces [1.1->1.11, 2.1->2.3] The SM particles & forces [1.1->1.11, 2.1->2.3] Natural Units Natural Units Four-vector notation [3.2] Four-vector notation [3.2] Width, lifetime, cross section, luminosity [6.1] Width, lifetime, cross section, luminosity [6.1] Fermis G.R. and phase space for 1+2–>3+4 [6.2] Fermis G.R. and phase space for 1+2–>3+4 [6.2] Mandelstam variables [Exercises 3.22, 3.23] Mandelstam variables [Exercises 3.22, 3.23] -functions [Appendix A]-functions [Appendix A] Feynman Diagrams [2.2] Feynman Diagrams [2.2] Feynman rules for the ABC theory [6.3] Feynman rules for the ABC theory [6.3] d/d for A+B–>A+B [6.5] d/d for A+B–>A+B [6.5] Renormalisation, running coupling consts [6.6, 2.3] Renormalisation, running coupling consts [6.6, 2.3]

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