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Feynmann Diagrams

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Feynman Diagrams

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Feynman Diagrams

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Vertices

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Virtual processes

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M. Cobal, PIF 2006/7

Real processes

- For a real process there must be energy conservation it has to be a combination of virtual processes.

Electron-electron scattering, single exchange

- Any real process receives contributions from all possible virtual processes.

M. Cobal, PIF 2006/7

- Points at which 3 or more particles meet are called vertices Each vertex corresponds to a term in the transition matrix elementwhich includes the structure and strength of the interaction.

-The nr. of vertices in a diagram is called order

-Each vertex has an associated probability proportional to a coupling constant, usually denoted as In the em processes this is:

-For the real processes, a diagram of order n gives a contribution of order n

-Providing that is small enough, higher order contributions to many real processes can be neglected

1371

em

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-Diagrams which differ only by time-ordering are usually implied by drawing only one of them

-This kind of process:

-Implies 3!=6 different time orderings

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Annihilation diagrams

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Exchange diagrams

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Virtual particles

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-From the order of diagrams the ratio of rates of processes can be estimated:

For example, this ratio measured appears to be: R = 0.9 x 10-3, smaller than em (but estimate is only a first order prediction)

-For nucleus, coupling is proportional to Z2, hence the rate for this process is of order Z23

)()()(

O

eeRateeeRateR

M. Cobal, PIF 2006/7

Exchange of a massive boson

-In the rest frame of particle A: where:

From this one can estimate the max distanceover which X can propagate:

This energy violation can be only for The interaction range is:

)',()','(),( pEXpEApEA x

2222 ',''

),0,0,0(,

XxA

A

MpEMpE

pME

XAX MMEEE '

Et /XMcr /

M. Cobal, PIF 2006/7

- For a massless exchanged particle, the interaction has an infinite range (e.g: em)

-If the exchanged particle is very heavy (like the W boson in the weak interaction), the interaction can be approiximated by a zero-range or point interaction

-If one consider the particle X as an electrostatic potential V(r), then the Klein-Gordon equation looks like

mxcGeV

cMcR

WW

1818

2 1024.80103.197

)/4.80(

)(1)( 222

2 rVMrVr

rrrV X

M. Cobal, PIF 2006/7

-Integration of previous equation gives:

-g is an integration constant, interpreted as the strength of the coupling of particle X to particle A and B (strong nuclear charge)

-Yukawa theory described the short-range nature of force between n and p in nucleus: interaction is due to massive quanta

-In Yukawa theory, g is analogous to the electric charge in QED, and the analogue of em is: x = g2/4

x characterizes the interaction strength at distances: r < R

Rre

rgrV /

2

4)(

Yukawa Potential (1935)

M. Cobal, PIF 2006/7

- A particle is scattered by this potential, thus receivig a momentum transfer .

-The potential has an amplitude (which is its Fourier transform):

- Using polar coordinates:- and assuming: , the amplitude is:

-For the point interaction, M2X >> q2 hence :

The point-interaction is characterized not only by X, but also by MX

q

xdexVqf xqi 3)()(

drddrxd sin23 )()( rVxV

22

22)sin()(4)(

xMqgdrr

qrqrrVgqf

24)(X

XM

Gqf

M. Cobal, PIF 2006/7

For nuclear forces with a range R ~ 10-15 m, Yukawa hypothesis predicted a spinless quantum of mass: Mc2 = hc/2R 100 MeV.

The pion observed in 1947 had M = 140 MeV, spin 0 and strong nuclear interactions.

Nowadays: pion exchange still accounted for the longer-range part of nuclear potential. However, full details of interaction are more complicated

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Electromagnetism

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Strong Interactions

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Ew Interactions

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Use of Feynmann Diagrams

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Exercises1) Knowing the dimensions of neutron and protons, you can estimate the mass of the particle, which is responsible for the interaction between the nucleus

Hint: The interaction between charged particles is carried by photons. The range of this interaction is infinite, the rest mass of photon has to be zero. The interaction between nucleons is limited to a range of about 10-15 m.

The Heisemberg uncertainty relation allows fluctuation of energy for a very short time, so that ”virtual” particles can be created which are responsible for the interaction

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Solution:

r =c.t t = r/c = 3.34.10-24 s

Et h/4 E = h/(4t) ~ 100 MeV

The uncertainty relation for a distance from 10-15 m allows a max energy deviation of about 100 MeV.

A particle which is responsible for the interaction of two nucleonsshould have a mass in the order of 100 MeV but can only exist for about 10-24 s