Particle Physics 3 nd Handout Feynman Graphs of

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Particle Physics 3 nd Handout Feynman Graphs of QFT • QED • Standard model

Particle Physics 3 nd Handout Feynman Graphs of QFT • QED • Standard model vertices • Amplitudes and Probabilities • Forces from particle exchange • QCD http: //ppewww. ph. gla. ac. uk/~parkes/teaching/PP/PP. html Chris Parkes

Quantum Electro. Dynamics (QED) u u Developed ~1948 Feynman, Tomonaga, Schwinger Feynman illustrated with

Quantum Electro. Dynamics (QED) u u Developed ~1948 Feynman, Tomonaga, Schwinger Feynman illustrated with diagrams Photon emission e- Pair production e- e- Time: Left to Right. e+ Anti-particles: backwards in time. annhilation e- e+ c. f. Dirac hole theory M&S 1. 3. 1, 1. 3. 2 Process broken down into basic components. In this case all processes are same diagram rotated We can draw lots of diagrams for electron scattering (see lecture) Compare with

Orders of u The amplitude T is the sum of all amplitudes from all

Orders of u The amplitude T is the sum of all amplitudes from all possible diagrams Feynman graphs are calculational tools, they have terms associated with them Each vertex involves the emag coupling ( =1/137) in its amplitude So, we have a perturbation series – only lowest order terms needed More precision more diagrams There can be a lot of diagrams! N photons, gives n in amplitude c. f. anomalous magnetic moment: After 1650 two-loop Electroweak diagrams Calculation accurate at 10 -10 level and experimental precision also!

The main standard model vertices At low energy: Strong: All quarks (and Weak neutral

The main standard model vertices At low energy: Strong: All quarks (and Weak neutral current: EM: anti-quarks) All charged particles All particles No change of flavour Weak charged current: All particles Flavour changes

Amplitude Probability |Tfi|2 (1) The Feynman diagrams give us the amplitude, c. f. in

Amplitude Probability |Tfi|2 (1) The Feynman diagrams give us the amplitude, c. f. in QM whereas probability is | |2 So, two emag vertices: e. g. e-e+ - + amplitude gets factor from each vertex And xsec gets amplitude squared (2) for e-e+ qq with quarks of charge q (1/3 or 2/3) • Also remember : u, d, s, c, t, b quarks and they each come in 3 colours • Scattering from a nucleus would have a Z term If we have several diagrams contributing to same process, we much consider interference between them e. g. (b) e(a) eee+ e+ e+ ee+ Same final state, get terms for (a+b)2=a 2+b 2+ab+ba

Massive particle exchange Forces are due to exchange of virtual field quanta ( ,

Massive particle exchange Forces are due to exchange of virtual field quanta ( , W, Z, g. . ) E, p conserved overall in the process but not for exchanged bosons. You can break Energy conservation as long as you do it for a short enough time that you don’t notice! i. e. don’t break uncertainty principle. Consider exchange of particle X, mass mx, in CM of A: B X A Uncertainty principle Particle range R For all p, energy not conserved So for real photon, mass 0, range is infinite For W (80. 4 Ge. V/c 2) or Z (91. 2 Ge. V/c 2), range is 2 x 10 -3 fm

Virtual particles This particle exchanged is virtual (off mass shell) e. g. (E, p)

Virtual particles This particle exchanged is virtual (off mass shell) e. g. (E, p) symmetric Electron-positron (E, -p) collider e- + e+ (E , p ) Yukawa Potential Strong Force was explained in previous course as neutral pion exchange Consider again: • Spin-0 boson exchanged, so obeys Klein-Gordon equation See M&S 1. 4. 2, can show solution is R is range For mx 0, get coulomb potential Can rewrite in terms of dimensionless strength parameter

7. 1 M&S Quantum Chromodynamics (QCD) QED – mediated by spin 1 bosons (photons)

7. 1 M&S Quantum Chromodynamics (QCD) QED – mediated by spin 1 bosons (photons) coupling to conserved electric charge QCD – mediated by spin 1 bosons (gluons) coupling to conserved colour charge u, d, c, s, t, b have same 3 colours (red, green, blue), so identical strong interactions [c. f. isospin symmetry for u, d], leptons are colourless so don’t feel strong force • Significant difference from QED: • photons have no electric charge • But gluons do have colour charge – eight different colour mixtures. Hence, gluons interact with each other. Additional Feynman graph vertices: Self-interaction 3 -gluon 4 -gluon These diagrams and the difference in size of the coupling constants are responsible for the difference between EM and QCD

Running Coupling Constants - QED + + + - - - + +Q +

Running Coupling Constants - QED + + + - - - + +Q + - - Charge +Q in dielectric medium Molecules nearby screened, At large distances don’t see full charge Only at small distances see +Q + + Also happens in vacuum – due to spontaneous production of virtual e+e- pairs e+ e - And diagrams with two loops , three loops…. each with smaller effect: , 2…. QED – small variation e. As a result coupling strength grows with |q 2| of photon, 1/128 1/137 higher energy smaller wavelength gets closer to bare charge 0 |q 2| (90 Ge. V)2

Coupling constant in QCD • Exactly same replacing photons with gluons and electrons with

Coupling constant in QCD • Exactly same replacing photons with gluons and electrons with quarks • But also have gluon splitting diagrams g g g This gives anti-screening effect. Coupling strength falls as |q 2| increases Grand Unification ? g LEP data Strong variation in strong coupling From s 1 at |q 2| of 1 Ge. V 2 To s at |q 2| of 104 Ge. V 2 Hence: • Quarks scatter freely at high energy • Perturbation theory converges very Slowly as s 0. 1 at current expts And lots of gluon self interaction diagrams

Range of Strong Force Gluons are massless, hence expect a QED like long range

Range of Strong Force Gluons are massless, hence expect a QED like long range force But potential is changed by gluon self coupling Qualitatively: QED + - Standard EM field Form of QCD potential: QCD q q Field lines pulled into strings By gluon self interaction QCD – energy/unit length stored in field ~ constant. Need infinite energy to separate qqbar pair. Instead energy in colour field exceeds 2 mq and new q qbar pair created in vacuum This explains absence of free quarks in nature. Instead jets (fragmentation) of mesons/baryons NB Hadrons are colourless, Force between hadrons due to pion exchange. 140 Me. V 1. 4 fm Coulomb like to start with, but on ~1 fermi scale energy sufficient for fragmentation