Methods of Experimental Particle Physics Alexei Safonov Lecture #2 1 Particle Physics and the Origin of Universe • One example of an open question is the baryon asymmetry • Lots of protons, very few antiprotons • Why? Shouldn’t there be equal numbers? • Something must have happened in the very early Universe at the level of basic interactions that shifted things there 2 Standard Model of Particle Physics • Physical content: • 12 basic particles • Each has an antiparticle • Interact via force carriers called gauge bosons • Higgs boson giving mass to all other particles • Includes 2.5 forces: • Electroweak=“electromagnetic + weak” combined force • All basic particles participate, transmitted by W/Z/g bosons • Responsible for radioactive decays and electromagnetic interactions • Dark Matter is unexplained at all • Discovery of neutrino Only quarks participate, oscillations makes Standard transmitted by gluons Holds proton and other composite Model at least “not quite right” hadrons together • Gravity is not included • Strong force: • • 3 MATH IN PARTICLE PHYSICS 4 Special Relativity and QM • Particle physics deals with very small objects where quantum mechanics is the only way to describe things • These things tend to move very fast, so special relativity is equally important • Need to combine both • Start with good old QM equation for a free particle: 1 2 • i H t 2m • This is nothing but E=p2/2m • The trouble is this equation is non-relativistic 5 Dirac’s Equation p2 E p 2 m2 • Need to go from E 2m • So you write something like this: 2 ( ) 2 m 2 t • But the problem is that the density is not always positively defined in this case • • Dirac took this equation: • • And tried to write the part under the square root as a square of an operator: • But you need to remove cross-terms, which is only possible if AB+BA=0 • Too bad numbers don’t work that way 6 Dirac’s Equation • But it works with matrices • That’s great because electron has a spin, you actually want these wave functions to be 2-component spinors like in QM • The strange thing was that one needs 4x4 matrices to satisfy all the requirements • Write as • Apply the same operator again (from left): • That looks familiar if k=m and so • Can write in an explicit 4-dimensional form: • Where matrices 7 . . Solution of Dirac’s Equation • Solutions: • Fourier transformation for general solution: • Where s is spin +/- ½ and a and b can be thought of as operators creating/annihilating a fermion/anti-fermion • Lagrangian: • This is because one should be able to get equations of motion from the Lagrangian as in classic mechanics: • In field theory you do L L 0 ( ) 8 Propagator • Take particle from point x to point y: • Where T is “time ordering”: • Plug in our solutions from before: • where 9 Fermion Propagator • We just learnt how to calculate the amplitude (related to probability) for a particle to travel from point x to point y • In momentum space: • This is something we will need to know how to do to calculate probabilities for scattering processes • Can propagate other particles too 10 Anything Else? • Yeah, we only know how to describe a free electron and know how to propagate it from point A to point B • Electromagnetism is not about that, it’s about interactions with electromagnetic field – need to add photons that can interact with electrons • Where D is the covariant derivative • Aμ is the covariant four-potential of the electromagnetic field generated by the electron itself; • Bμ is the external field imposed by external source; • is electromagentic field tensor 11 QED Lagrangian and Feyman Rules 12 QED Lagrangian and Feyman Rules 13 Calculating Amplitudes • This is what we need to do at the end: • Calculate quantum mechanical probability of a specific process happening • First calculate amplitude, amplitude squared is the probability • For scattering processes it is called “cross-section” 14 Other Processes • What else can happen? • There could be more complex diagrams that include more interconnections and intermediate particles 15 Good vs “Not So Good” Theories • Renormalizability • Once you start adding more and more complex diagrams, all sorts of ugly things start happening • Infinite amplitudes and cross-sections • Renormalization takes care of it • Or can be an “effective” theory, then no requirements 16 References • Historical overview and derivation of Dirac’s equation: • http://en.wikipedia.org/wiki/Quantum_electrodynamics • A formal (and lengthy) introduction to QED: • Peskin, Schroeder, “An Introduction to Quantum Field Theory”, section 3 • Feynman rules for QED: • Peskin, Schroeder, “An Introduction to Quantum Field Theory”, section 4 17

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