1 Methods of Experimental Particle Physics Alexei Safonov Lecture #2

2 Particle Physics and the Origin of Universe time 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

3 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/  bosons Responsible for radioactive decays and electromagnetic interactions Strong force: Only quarks participate, transmitted by gluons Holds proton and other composite hadrons together Dark Matter is unexplained at all Discovery of neutrino oscillations makes Standard Model at least “not quite right” Gravity is not included

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: This is nothing but E=p 2 /2m The trouble is this equation is non-relativistic 5

Dirac’s Equation Need to go from So you write something like this: 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 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 A μ is the covariant four-potential of the electromagnetic field generated by the electron itself; B μ 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: 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