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*A. Bay Beijing October 20051 Summary Standard Model of Particles (SM) Symmetries, Gauge theories,...*

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Standard Model of Particles (SM) Symmetries, Gauge theories, Higgs, LEP, LHC

A. Bay Beijing October 2005

Symmetriessyn-: togethermetron : measure

A. Bay Beijing October 2005

What does it mean being "symmetric" 6 equivalentpositions for theobserver

A. Bay Beijing October 2005

What does it mean being "symmetric" .2the number of possibilities is

A. Bay Beijing October 2005

What does it mean being "symmetric" .3The concept of symmetry contains the idea ofnon-measurability and invariance. Of a snow flake or the liquid water, which oneis "more symmetric" ?

A. Bay Beijing October 2005

Do we always need symmetry ?GGDDGDNot too much symmetry is better for the aesthetics perception

A. Bay Beijing October 2005

Emmy Noetherhttp://www.emmynoether.comToday theories are based on the work ofE. Noether. She studies the dynamicconsequences of symmetries of a system.

In 1915-1917 she shows that every symmetryof nature yields a conservation law, andreciprocally.

The Noether theorem: SYMMETRIES CONSERVATION LAW

A. Bay Beijing October 2005

SYMMETRIES CONSERVATION LAW

Examples of continuous symmetries:

SymmetryConservation lawTranslation in time EnergyTranslation in space MomentumRotation Angular momentumGaugetransformation ChargeEx.: translation in space r r + dif the observer cannot do any measurement on a systemwhich can detect the "absolute position" then p is conserved.d is a displacement

A. Bay Beijing October 2005

Translation in space and conservation of pConsider 2 bodies initially at rest, interacting (by gravitation,for instance). Initial total momentum is p = 0.Suppose that there is some kind of non-homogeneity in thatregion of space and that the interaction strength is not identicalat the two positions. Suppose F1 > F2 , then there is atotal net force acting on the system.=> The total momentum p is notconstant with time.

A. Bay Beijing October 2005

Symmetries in particle physics

Non-observablessymmetry transformationsconservation law/ selection rulesdifference between permutationB.E. / F.D. statis. identical particlesabsolute position r r + d p conservedabsolute time t r + t E conservedabsolute spatial directionrotation r r' J conservedabsolute velocityLorentz transf. generators L. groupabsolute right (or left)r -r Paritysign of electric chargeq -qCharge conjugationrelative phase between states with different charge qy eiqq ycharge conserved different baryon nbr By eiBq yB conserved different lepton nbr Ly eiLq yL conserveddifference between coherent mixture of (p,n)isospin

A. Bay Beijing October 2005

An introduction to gauge theoriesSome history.We observe that the total electric charge of a system is conserved.

Wigner demonstrated that if one assumesconservation of Energythe "gauge" invariance of the electric potential V

=> than the electric charge must be conserved

Point 2) means that the absolute value of V is not important,any system is invariant under the "gauge" change V V+v(in other terms only differences of potential matter)

A. Bay Beijing October 2005

Wigner conservation of e.m. chargeSuppose that we can build a machine to create and destroy charges.Let's operate that machine in a region with an electric field:V1V2V1V2creation of qneeds work WV1V2move charge to V2destroy q, regain Wregaining W cannotdepend on theparticular valueof V (inv. gauge)here we gain q(V2-V1)1234E conservationis violated !

A. Bay Beijing October 2005

Maxwell assures local charge conservationDifferential equations in 1868:Taking the divergence of the last equation:if the charge density is not constant in time in the element ofvolume considered, this violates the continuity equation:To restore local charge conservation Maxwell introduces in theequation a link to the field E:The concept of global charge conservation has been transformedinto a local one. We had to introduce a link between the two fields.

A. Bay Beijing October 2005

Gauge in Maxwell theoryIntroduce scalar and potential vectors: V, and AWe have the freedom to change the "gauge":for instance we can do where c is an arbitrary function.To leave E (and B) unchanged, we need to change also A:In conclusion: E and B still satisfy Maxwell eqs, hencecharge conservation. We had to act simultaneously on V and A.

Note that one can rebuild Maxwell eqs, starting from A,V,requiring gauge invariance, and adding some relativity: A,V add gauge invariance Maxwell eqs

A. Bay Beijing October 2005

Gauge in QMIn QM a particle are described by wave function. Take y(r,t)solution of the Schreodinger eq. for a free particle.We have the freedom to change the global phase :

still satisfy to the Schroedinger equation for the free particle.

We can rewrite the phase introducing the charge q of the particleWe cannot measure the absolute global phase: this is a symmetryof the system. One can show that this brings to the conservationof the charge q: it is an instance of the Noether theorem. y add global gauge invariance charge conservationindependenton r and t

A. Bay Beijing October 2005

Gauge in QM .2If now we try a local phase change:we obtain a y which does not satisfy the free Schroedinger eq.If we insist on this local gauge, the only way out is to introduce a new field ("gauge field") to compensate the bad behaviour. This compensating field corresponds to an interaction => the Schrdinger eq. is no more free ! y add local gauge invariance interaction fieldThis is a powerful program to determine the dynamics of a systemof particles starting from some hypothesis on its symmetries.

A. Bay Beijing October 2005

QED from the gauge invarianceThe electron of charge q is represented by the wavefunction y,satisfying the free Schroedinger eq. (or Dirac, or...)

The symmetry is U(1) : multiplication of y by a phase eiqq

* Requiring global gauge symmetry we get conservation of charge:we recover a continuity equation

* Requiring local gauge symmetry we have to introduce the massless field (the photon), i.e. the potentials (A,V), and the way itcouples with the electron: the Schroedinger eq. with e.m. interaction

A. Bay Beijing October 2005

QED from the gauge invariance .2The corresponding relativistic Lagrangian is this graphexplains theinteraction termqyyAno mass termfor the photon A !A mass term for a bosonfield looks like this ina Lagrangian: M2A2

A. Bay Beijing October 2005

EW theory from gauge invarianceParticles: the set of leptons and quarks of the SM.

The symmetry is SU(2)U(1) U(1) multiplication by a phase eiqq SU(2) similar: multiplication by exp(igqT) but T are three 22 matrices and q is a vector with three components.

This is an instance of a Yang and Mills theory.

Applying gauge invariance brings to a dynamics with 4 massless fields (called "gauge" fields).Fine for the photon, but how to explain that W+ W- and Z have a mass ~ 100 GeV ?

We introduce now the Higgs mechanism.

A. Bay Beijing October 2005

Higgs mechanismAnalogy: interaction of the e.m. field with the Cooper pairsin a superconductor. For a T below some critical value Tcthe material becomes superconductor and "slow down" the penetrationof the e.m. field. This looks like if the photon has acquired a mass.

Suppose that an e.m. wave A induces a current J close to the surfaceof the material, J A. Let's write J = -M2A.In the Lorentz gauge: A = JReplacing: A = -M2A or A + M2A = 0

This is a massive wave equation:the photon, interacting with the(bosonic) Cooper pairs field f has acquired a "mass" M

A. Bay Beijing October 2005

Higgs mechanism in EWWe apply the same principle to the gauge fields of the EW theory. We have to postulated the existence of a new field, the Higgs field, which is present everywhere (or at least in the proximity of particles).

The Higgs generates the mass of the W and Z. The algebra of the theory allows to keep the photon mass-less, and we obtain the correct relations between couplings and masses:On the other hand, the model does not predict the values of themasses and couplings: only the relations between them.

A. Bay Beijing October 2005

Higgs mechanism in EW .2A new boson is created by quantum fluctuation of vacuum: the Higgs.Consider a complex field and its potentialnormalvacuumV is minimal on the circle of radiuswhile f = 0 is a local max !Any point on the circle is equivalent...

Let's choose an easy one: A fluctuation around this

point is given by:H is the bosonic fieldNature hasalso to choose

A. Bay Beijing October 2005

Spontaneous Symmetry BreakingNature has to choose the phase of f. All the choices are equivalent.Continue analogy with superconductor: superconductivity appearswhen T becomes lower than Tc. It is a phase transition.Assume that the Higgs potential V(f ) at high temperature (earlyBigBang) is more parabolic. The phase transition appears whenthe Universe has a temperature corresponding to E ~ 0.5-1 TeV High T Low TNature has to makea choice for f.Maybe different choicesin different parts of theUniverse.Are there "domains"with different phases ?

A. Bay Beijing October 2005

Spontaneous Symmetry Breakingat dinnerbefore dinneronce dinner starts

A. Bay Beijing October 2005