48240012 SP06 Neutron Decay Real Virtual Particle Interactions

8/2/2019 48240012 SP06 Neutron Decay Real Virtual Particle Interactions

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Lecture Notes, Standard Model (6), Susskind. The Weak Interaction and why the decay times arevery long.The video is available on Stanford iTunes or on the Stanford Susskind web site

SP06 Neutron Decay, Real, Virtual Particle Interactions2010-02-15

ContentsQuick Review Notes ................................................................................................................................. 1

Symmetries ............................ ........................ ................................ ....................... ............................... 2

Coupling Constants (10:00 ................................................................................................................... 2

The Decay of the Neutron (14:00 ............................................................................................................. 3

The W-boson Propagator ..................................................................................................................... 3

Discussions (40:00 ................................................................................................................................... 5

Virtual W-bosons, Uncertainty of Energy, Time (42:30 ......................................... ....................... ......... 5

Discussion on Real, Virtual Particles (55:00 .............................................................................................. 6

Vacuum Energy Fluctuation Discussion (64:20...................................................................................... 7

Photon (boson) Interactions (74:00 ......................... ................................ ...................... ....................... 7

Spontaneous Symmetry Breaking Overview (82:00 ..................... ................................ ...................... .... 8

What distinguishes Spontaneous Symmetry breaking from Normal Symmetry breaking?..................... 8

Quick Review Notes

Flavor or family groupsud cs t b

red x x x x x x Color groups blue x x x x x xgreen x x x x x x

Leptons e e

Table notes:y all fermions, spin ½ particles



y (u,c,t) quarks have +2/3 charge; (d,s,b) quarks have -1/3 charge;y electrons (muon, tau) have -1 charge;y neutrinos (nu symbol) (electron, muon, tau) have 0 chargey W boson emitted when charge difference is ±1

SymmetriesColor symmetry. QCD

y SU(3), change color between quarks.y Gluons are the gauge boson force generator.y change occurs simultaneously across all groups (u,d,c,s,t,b). Exactly the same gluon

.Charge symmetry QED

y photon emission, U(1)

Flavor symmetryy Weak interactions change quarks (u, c, t b) and (electrons

y y y SU(2) group

SU(3)×SU(2)×U(1) Form a group SU(6)

Coupling Constants (10:00

At each intersection in a Feynman diagram a coupling constant gives the strength of theinteraction (varies depending upon kinetic energy, particles involved, etc.)

QED, Electron Photon emission, the Fine Structure Constanty U(1)coupling constant is the electric charge, associated with the emission of a photony probability of finding a photon = (electric charge).squared Fine structure constanty either the virtual emission of a photon (which gets quickly reabsorbed) or the actual

emission if the electron is affected

Fine structure constant ~1/137. When an electron ³hits´ something only 1/137 probability of emitting a photon

QCD, strong Quarks Gluon emissiony SU(3) coupling constant g_s associated with the emission of a gluony also a number, close to 1/5. when a quark ³hits´ something a gluon will almost always be

emitted

Weak Interactionsy The coupling constant for weak interactions is about the same order of magnitude as the

QED coupling constant;y That is not what makes the ³weak´ interaction.



y other factors affect interaction (below)

The Decay of t he Neutron (14:00

(from lecture 5)

W- boson can become electron and antineutrino

Call ³gw´ the coupling constant for emission of a W-boson¶ ³gw´,is a number, same order of magnitude as the electron charge. Not a very small number so the ³g

w´ coupling constant does

not account for the improbable nature of the weak interactions.

³gw´ occurs at both ends of W-boson propagator branch (multiply)

The W-boson Propagator

Propagator: the factors inserted into a Feynman diagram to account for the motion of a particlefrom one location to another.

Propagator (Quantum Theory) -Wiki In quantum mechanics and quantum field theory, the propagator gives the probabilityamplitude for a particle to travel from one place to another in a given time, or to travelwith a certain energy and momentum. Propagators are used to represent the contributionof virtual particles on the internal lines of Feynman diagrams. They also can be viewed asthe inverse of the wave operator appropriate to the particle, and are therefore often calledGreen's functions.

propagator branch in Feynman diagram

CharmedStrange interaction, emits a W+ boson (+2/3- -1/3)=+1gW coupling constant

gW W+

GS

GC

Neutron decays to a proton and W- boson which in turn decays to anelectron, anti-neutrino

Energy and momentum are conserved in the initial (neutron) stageand the final (proton, electron, anti-neutrino) stage.

The W- boson is an intermediate stage (virtual)

u d d

u d u

W-

e



We determine where this drop-off occurs by doing dimensional analysis in units of

mass has same units as energy, momentum and inverse

length

drop off distance is ~ 1/(mass of the particle)

or at the Compton wave-length of the particle

at a distance of /mc the propagator (W boson) suddenly dies down in timeThe square of delta (t) is the probability it can go (distance) before emitting a (e, ) pair (that is, give up energy as (e, ) pair

Usually measured in momentum. Pn momentum of neutron; Pp momentum of proton

(Pn-Pp) momentum of W boson, which gives the momentum of the (e, ) pair

You can calculate (not measure) the distance between 2 points. The momentum and distance arerelated with a Fourier Transform

time is a function of distance between 2 pointslarge at small distances, grows by 1\d2(~square of distance)if particle had no mass curve just continues smoothly

if particle (W-) has mass decreases exponentially suddenly

vertical axis is the size of delta, proper timedistance

u d d

u d u

W-

e

time of W- between two points (d,u, W-) (e, , W-)

(distance) function of space-time distance between the 2 points(relativistic proper time)

Note: assumes momentum is small so that the ³distance´ is space-like. If the momentum is large (time-like) energy must be takeninto consideration.



The Fourier Transform of the propagator is a function of the (square of the momentum of the W boson) and, parametrically, the mass of the W boson

Fourier Transform of the propagator

Where

is a symbol for the Fourier transform of k momentum of the W bosonmW mass of the W boson

This is a factor (of the coupling constant) that you insert into the Feynman diagram

If a photon;y no mass, so factor is 1/k 2 y momentum of photon quite low (~100eV) so factor is quite large

with W bosony lowest value is 1/m2 (neutron, proton at rest)y square is probability of emitting a W bosony will always have a fairly large value

The amplitude for emitting a W boson is the (coupling constants) *

(factor)

the probability is the amplitude squared

mass of W boson is ~100 times mass of proton

Discussions (40:00

Generally the amount of energy transferred from the neutron to the proton is very smallcompared to the mass of the W boson. Because of energy conservation things cannot happenwhen an insufficient amount of energy is available except for virtual energy

originally all that was known was the factor (g\m) called capital G

_Fermi

V irt ual W-bosons, Uncertaint y of Energy, Time (42:30

rates are controlled by:y coupling constants (vary with momentum);y masses of particles (or propagators) in intermediate statesy available kinetic energy



The total energy is conserved. This means that there is not enough energy to create a W-bosonunless the Neutron is given sufficient kinetic energy (as in a collider). QM allows for virtual W- bosons to be created provided the time interval is short enough that is sufficient energy is³borrowed´ from the virtual vacuum to allow for creation of a W-boson and the subsequent

decay of the Neutron.

Uncertainty relation between energy and t ime

To move across the energy barrier, an energy of E is required however energy can be borrowedwithin the t time requirements.y see also QM wave-particle functiony the borrowed energy is returned to the vacuum or creates new particles

With Neutron decay there is a small probability that a Neutron will create a (virtual) W-boson.The borrowed energy is either expended as a (e, ) pair or returned to the vacuum.

probability per unit time. The inverse is related to the decay time.

With sufficient kinetic energy real W-bosons can be created. These bosons decay to a flavor

group pair within a certain t. Note that the greater kinetic energy gives a W-boson of greater velocity and subsequent longer special relativistic ³proper decay time´ and of course if a W- boson decays to a (e, ) pair those particles will have tremendous kinetic energy.

Discussion on Real, V irt ual Part icles (55:00

To see a virtual particle you require a photon with a frequency t comparable to the uncertainty(Et < - but thephoton then has enough energy E to create a real particle the ³catch 22´of quantum mechanics.

protons, neutrons continually fluctuate

virtual W-bosons are emitted, absorbed continually as protons neutrons

pn

p

W+ W-

n p

n

tunneling An example from Quantum Mechanics

Consider an alpha particle, at rest, inside the nucleus and preventedfrom moving outside by an energy barrier.

t

E



If you hit a particle hard enough almost any particle can and will appear no such thing as acompletely determined single reaction.

V acuum Energy Fluct uat ion Discussion (64:20

Photon (boson) Interact ions (74:00

The probability of an event is related to the time duration. An event that has a short duration isalways at a cost of probabilities.

SU(2) Group Structure ± next lecture (80:00The SU(2) group has 3 generators (W+, W- , Z) bosons. In subsequent lectures discuss how photons fit into the structure

With enough energy; 2 photons colliding can create a (electron, positron) (e,e ) pair which can then create a pair of photons.

If not enough energy in the photons to create a (e,e ) pair , avirtual (e,e ) pair can be created.

e- e+

vacuum fluctuation resulting from a (e, ) pair creating a W- boson if not enough energy in the (e, ) pair the interactionresults in avacuum fluctuatione W

If vacuum fluctuation hit with high energy photon you may create(electron, positron) pair with an additional photon

Feynman diagram for vacuum fluctuations particles from a vacuum fluctuation appear disappear quickly



Spontaneous Symmetry Breaking Overview (82:00 Consider a simple system composed of heads or tails of equal probability

P(H)=P(T).

Energy between equal pairs (HH) or (TT) is lower than energy (HT) or (TH).

There are two states of lowest energy, all heads or all tails. Consider the effect of a ³single head´far removed from the system this will have the effect of making the all heads state of slightlyless energy than the all tails state. The presence of a tail in the ³now lowest energy state of allheads´ would constitute symmetry breaking it costs energy.

What dist inguishes Spontaneous Symmetry breaking fromNormal Symmetry

breaking?

Consider a system where, on one side the coins are in a lowest state of ³all heads´; and on theother side the coins are in a lowest state of ³all tails´. At the point where the two sides meet a³boundary´ condition exists.

The presence of a boundary condition is required for

Spontaneous Symmetry breaking.

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48240012 SP06 Neutron Decay Real Virtual Particle Interactions