Section VII The Standard Model TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

The Standard Model Spin ½ Fermions Charge (units of e) LEPTONS -1 QUARKS PLUS antileptons and antiquarks. Spin 1 Bosons Mass (GeV/c2) Gluon g 0 STRONG Photon g 0 EM W and Z Bosons W, Z0 91.2/80.3 WEAK The Standard Model also predicts the existence of a spin 0 HIGGS BOSON which gives all particles their masses via its interactions. Charge (units of e) -1 2/3 -1/3

Theoretical Framework Macroscopic Microscopic Slow Classical Mechanics Quantum Mechanics Fast Special Relativity Quantum Field Theory The Standard Model is a collection of related GAUGE THEORIES which are QUANTUM FIELD THEORIES that satisfy LOCAL GAUGE INVARIANCE. ELECTROMAGNETISM: QUANTUM ELECTRODYNAMICS (QED) 1948 Feynman, Schwinger, Tomonaga (1965 Nobel Prize) ELECTROMAGNETISM: ELECTROWEAK UNIFICATION +WEAK 1968 Glashow, Weinberg, Salam (1979 Nobel Prize) STRONG: QUANTUM CHROMODYNAMICS (QCD) 1974 Politzer, Wilczek, Gross (2004 Nobel Prize)

ASIDE : Relativistic Kinematics Low energy nuclear reactions take place with typical kinetic energies of order 10 MeV << nucleon rest energies ) non-relativistic formulae OK. In particle physics kinetic energy is of O (100 GeV) >> rest mass energies so relativistic formulae usually essential. Always treat -decay relativisticly (me ~ 0.5 MeV < 1.3 MeV ~ mn-mp). WE WILL ASSUME UNDERSTANDING OF FOUR VECTORS and FOUR MOMENTA from the Part II Relativity Electrodynamics and Light course ... in natural units: Definitions: mass = m velocity= v = βc(= β) mom = p Consequences:

ASIDE cont: (Decay example) Consider decay of charged pion at rest in the lab frame Conservation of Energy and Momentum says: Appropriate choice of axes gives, and noting pion is at rest (pπ=0) gives: So have two simultaneous equations in two unknowns & . Once you know p you know E etc via etc …. (Simplify by taking mν=0 )

ASIDE cont : Relevance of Invariant mass In Particle Physics, typically many particles are observed as a result of a collision. And often many of these are the result of decays of heavier particles. A common technique to identify these is to form the “invariant mass” of groups of particles. e.g. if a particle X!1+2+3…n where we observe particles 1 to n, we have: J/y In the specific (and common) case of a two-body decay, X!1+2, this reduces to the same formula as on a preceding slide, i.e. Typical invariant mass histogram:

ASIDE cont : Colliders and √s Consider the collision of two particles: The invariant quantity  is the angle between the momentum three-vectors is the energy in the zero momentum (centre-of-mass) frame; It is the amount of energy available to the interaction e.g. in particle-antiparticle annihilation it is the maximum energy/mass of particle(s) that can be produced

Fixed Target Collision For e.g. 450 GeV proton hitting a proton at rest: Collider Experiment For then and  = e.g. 450 GeV proton colliding with a 450 GeV proton: In a fixed target experiment most of the proton’s energy is wasted providing forward momentum to the final state particles rather than being available for conversion into interesting particles.

ASIDE over Back to the main topic of combining Quantum Mechanics with Special Relativity…..

Problems with Non-Relativistic Schrodinger Equation To describe the fundamental interactions of particles we need a theory of RELATIVISTIC QUANTUM MECHANICS. Schrodinger Equation for a free particle comes from classical or equivalently promoted to an operator equation using energy and momentum operators: giving which has plane wave solutions: Schrodinger Equation: 1st Order in time derivative 2nd Order in space derivatives Schrodinger equation cannot be used to describe the physics of relativistic particles. ħ =1 natural units Not Lorentz Invariant!

Klein-Gordon Equation (relativistic) From Special Relativity: From Quantum Mechanics: Combine to give: KLEIN-GORDON EQUATION Second order in both space and time derivatives and hence Lorentz invariant. Again plane wave solutions but this time requiring allowing Negative energy solutions required to form complete set of eigenstates ANTIMATTER

Antimatter Feynman-Stückelberg interpretation: The negative energy solution is a negative energy particle travelling forwards in time. And then, since:  Interpret as a positive energy antiparticle travelling forwards in time. Then all solutions describe physical states with positive energy, going forward in time. Examples: e+e- annihilation pair creation e+ g g e+ T T All quantum numbers carried into vertex by e+, same as if viewed as outgoing e-.

More particle/anti-particle examples In first diagram, the interpretation easy as the first photon emitted has less energy than the electron it was emitted from. No need for “anti-particles” or negative energy states. In the second diagram, the emitted photon has more energy than the electron that emitted it. So can either view the top vertex as “emission of a negative energy electron travelling backwards in time” or “absorption of a positive energy positron travelling forwards in time”. g T g T

Normalization Can show solutions of Schrodinger Equation (non-relativistic) satisfy: and solutions of Klein-Gordon Equation (relativistic) satisfy: Both are of the form: and the divergence theorem tells us that this means that D is conserved, where: If we interpret D as number of particles, and  as number of particles per unit volume, then: For plane wave of arbitrary normalization find that: Non-relativistic: particles per unit volume. Relativistic: particles per unit volume.

Interaction via Particle Exchange Consider two particles, fixed at and , which exchange a particle of mass m. Calculate shift in energy of state i due to this process (relative to non-interacting theory) using 2nd order perturbation theory: Work in a box of volume L3, and normalize s.t. i.e. use 2E particles per box! Space Time state i state j state i Sum over all possible states j with different momenta.

Consider ( transition from i to j by emission of m at ) represents a free particle with Let g be the “strength” of the emission at . Then: Similarly is the transition from j to i at and is Substituting these in, we see the shift in energy state is Original 2 particles

Normalization of source strength “g” for relativistic situations Previously normalized wave-functions to 1 particle in a box of side L. In relativity, the box will be Lorentz contracted by a factor g i.e. particles per volume V. (Proportional to 2E particles per unit volume as we already saw a few slides back) Need to adjust “g” with energy (next year you will see “g” is “lorentz invariant” matrix element for interaction) Conventional choice: (square root as g always occurs squared) Source “appears” lorentz contracted to particles of high energy, and effective coupling must decrease to compensate. The presence of the E is important. The presence of the 2 is just convention. The absence of the m (in gamma) is just a convention to keep g dimensionless here. Rest frame 1 particle per V Lab. frame 1 particle per V/g

Beginning to put it together Different states j have different momenta . Therefore sum is actually an integral over all momenta: And so energy change:

Final throes The integral can be done by choosing the z-axis along . Then and so Write this integral as one half of the integral from - to +, which can be done by residues giving Appendix D

Can also exchange particle from 2 to 1: Get the same result: Total shift in energy due to particle exchange is ATTRACTIVE force between two particles which decreases exponentially with range r. Space Time state i state j state i

Yukawa Potential YUKAWA POTENTIAL Characteristic range = 1/m (Compton wavelength of exchanged particle) For m0, infinite range Yukawa potential with m = 139 MeV/c2 gives good description of long range interaction between two nucleons and was the basis for the prediction of the existence of the pion. Hideki Yukawa 1949 Nobel Prize

Scattering from the Yukawa Potential Consider elastic scattering (no energy transfer) Born Approximation Yukawa Potential The integral can be done by choosing the z-axis along , then and For elastic scattering, and exchanged massive particle is highly “virtual” q2 is invariant “VIRTUAL MASS”

Virtual Particles Forces arise due to the exchange of unobservable VIRTUAL particles. The mass of the virtual particle, q2, is given by and is not the physical mass m, i.e. it is OFF MASS-SHELL. The mass of a virtual particle can by +ve, -ve or imaginary. A virtual particle which is off-mass shell by amount Dm can only exist for time and range If q2 = m2, then the particle is real and can be observed. ħ =c=1 natural units

For virtual particle exchange, expect a contribution to the matrix element of where Qualitatively: the propagator is inversely proportional to how far the particle is off-shell. The further off-shell, the smaller the probability of producing such a virtual state. For m  0 ; e.g. single g exchange q2 0, very low energy transfer EM scattering COUPLING CONSTANT STRENGTH OF INTERACTION PHYSICAL (On-shell) mass VIRTUAL (Off-shell) mass PROPAGATOR

Feynman Diagrams Results of calculations based on a single process in Time-Ordered Perturbation Theory (sometimes called old-fashioned, OFPT) depend on the reference frame. The sum of all time orderings is not frame dependent and provides the basis for our relativistic theory of Quantum Mechanics. The sum of all time orderings are represented by FEYNMAN DIAGRAMS. Space Time Time + Space = FEYNMAN DIAGRAM

Feynman diagrams represent a term in the perturbation theory expansion of the matrix element for an interaction. Normally, a matrix element contains an infinite number of Feynman diagrams. But each vertex gives a factor of g, so if g is small (i.e. the perturbation is small) only need the first few. Example: QED Total amplitude Fermi’s Golden Rule Total rate

Anatomy of Feynman Diagrams Feynman devised a pictorial method for evaluating matrix elements for the interactions between fundamental particles in a few simple rules. We shall use Feynman diagrams extensively throughout this course. Represent particles (and antiparticles): and their interaction point (vertex) with a “ “. Each vertex gives a factor of the coupling constant, g. Spin ½ Quarks and Leptons Spin 1 g, W and Z0 g

External Lines (visible particles) Internal lines (propagators) Spin ½ Particle Incoming Outgoing Antiparticle Spin 1 Spin ½ Particle (antiparticle) Spin 1 g, W and Z0 g Each propagator gives a factor of

q e p u n d p e Examples: ELECTROMAGNETIC STRONG Electron-proton scattering p Quark-antiquark annihilation q STRONG WEAK u d e n p Neutron decay

Summary Relativistic mechanics necessary for most calculations involving Particle Physics and the Standard Model Anti-particles arise from requiring wave equation be relativistically invariant Calculating Matrix Elements from Perturbation Theory from first principles is a pain – so we don’t usually use it! Need to do time-ordered sums of (on mass shell) particles whose production and decay does not conserve energy and momentum ! Feynman Diagrams / Rules manage to represent the maths of Perturbation Theory (to arbitrary order, if couplings are small enough) in a very simple way – so we use them to calculate matrix elements! Approx size of matrix element may be estimated from the simplest valid Feynman Diagram for given process. Full matrix element requires infinite number of diagrams. Now only need one exchanged particle, but it is now off mass shell, however production/decay now conserves energy and momentum.

Section VIII QED

QED QUANTUM ELECTRODYNAMICS is the gauge theory of electromagnetic interactions. Consider a non-relativistic charged particle in an EM field: given in term of vector and scalar potentials Maxwell’s Equations g Change in state of e- requires change in field  Interaction via virtual g emission

DEMAND that QED be a GAUGE THEORY Appendix E Schrodinger equation is invariant under the gauge transformation  LOCAL GAUGE INVARIANCE LOCAL GAUGE INVARIANCE requires a physical GAUGE FIELD (photon) and completely specifies the form of the interaction between the particle and field. Photons (all gauge bosons) are intrinsically massless (though gauge bosons of the Weak Force evade this requirement by “symmetry breaking”) Charge is conserved – the charge q which interacts with the field must not change in space or time. DEMAND that QED be a GAUGE THEORY Appendix E

The Electromagnetic Vertex All electromagnetic interactions can be described by the photon propagator and the EM vertex: The coupling constant, g, is proportional to the fermion charge. Energy, momentum, angular momentum and charge always conserved. QED vertex NEVER changes particle type or flavour i.e. but not or QED vertex always conserves PARITY STANDARD MODEL ELECTROMAGNETIC VERTEX g +antiparticles

Pure QED Processes Compton Scattering (ge-ge-) e- g g e- Bremsstrahlung (e-e-g) Pair Production (g e+e-) e- g e- g e- g e- g e- Nucleus The processes e-e-g and g e+e-cannot occur for real e, g due to energy, momentum conservation. Ze Nucleus

p0 u c J/Y e+e- Annihilation p0 Decay J/+- The coupling strength determines “order of magnitude” of the matrix element. For particles interacting/decaying via EM interaction: typical values for cross-sections/lifetimes sem ~ 10-2 mb tem ~ 10-20 s u p0 c J/Y

Scattering in QED Examples: Calculate the “spin-less” cross-sections for the two processes: (1) (2) Fermi’s Golden rule and Born Approximation (see page 48): For both processes write the SAME matrix element is the strength of the interaction. measures the probability that the photon carries 4-momentum i.e. smaller probability for higher mass. e p Electron-proton scattering Electron-positron annihilation

(1) “Spin-less” e-p Scattering q2 is the four-momentum transfer: Neglecting electron mass: i.e. and Therefore for ELASTIC scattering e e RUTHERFORD SCATTERING

Discovery of Quarks Virtual g carries 4-momentum Large q  Large , small Large , large High q wave-function oscillates rapidly in space and time  probes short distances and short time. Rutherford Scattering q2 small E = 8 GeV Excited states q2 increases Expected Rutherford scattering E q2 large l<< size of proton q2 > 1 (GeV)2 Elastic scattering from quarks in proton

(2) “Spin-less” e+e- Annihilation Same formula, but different 4-momentum transfer: Assuming we are in the centre-of-mass system Integrating gives total cross-section:

This is not quite correct, because we have neglected spin This is not quite correct, because we have neglected spin. The actual cross-section (using the Dirac equation) is Example: Cross-section at GeV (i.e. 11 GeV electrons colliding with 11 GeV positrons)

The Drell-Yan Process d u Can also annihilate as in the Drell-Yan process. Example: See example sheet (Question 25) d u

Experimental Tests of QED QED is an extremely successful theory tested to very high precision. Example: Magnetic moments of e,  : For a point-like spin ½ particle: Dirac Equation However, higher order terms introduce an anomalous magnetic moment i.e. g not quite 2. v v     O(a) O(a4) 12672 diagrams O(1)

O(a3) Agreement at the level of 1 in 108. QED provides a remarkable precise description of the electromagnetic interaction ! Experiment Theory

+     +     Higher Orders So far only considered lowest order term in the perturbation series. Higher order terms also contribute Second order suppressed by a2 relative to first order. Provided a is small, i.e. perturbation is small, lowest order dominates. Lowest Order +     Second Order Third Order +    

Running of  specifies the strength of the interaction between an electron and a photon. BUT a is NOT a constant. Consider an electric charge in a dielectric medium. Charge Q appears screened by a halo of +ve charges. Only see full value of charge Q at small distance. Consider a free electron. The same effect can happen due to quantum fluctuations that lead to a cloud of virtual e+e- pairs The vacuum acts like a dielectric medium The virtual e+e- pairs are polarised At large distances the bare electron charge is screened. At shorter distances, screening effect reduced and see a larger effective charge i.e. a. -Q + - e+ e- g

Measure a(q2) from etc a increases with increasing q2 (i.e. closer to the bare charge) At q2=0: a=1/137 At q2=(100 GeV)2: a=1/128

Summary QED is the physics of the photon + “charged particle” vertex: Every Feynman vertex has: An arrow going in (lepton or quark) An arrow going out (lepton or quark) And a photon Does not change the type of lepton or quark “passing through” Conserves charge, energy and momentum. The coupling α “runs”, and is proportional to the electric charge of the lepton or quark QED has been tested at the level of 1 part in 108. g