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Derivation of Electro-Weak Unification and Final Form of St Derivation of Electro-Weak Unification and Final Form of St

Derivation of Electro-Weak Unification and Final Form of St - PowerPoint Presentation

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Derivation of Electro-Weak Unification and Final Form of St - PPT Presentation

1 W 1 2 W 2 3 W 3 Substitute B cos W A sin W Z 0 Sum over first generation particles Flavor changing interactions ID: 261950

terms gluon weak flavor gluon terms flavor weak interactions vertex quark color term interaction note qcd changing proton sum

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Slide1

Derivation of Electro-Weak Unification and Final Form of Standard Model with QCD and Gluons

1

W

1

+ 

2

W

2

+ 

3

W

3Slide2

Substitute B

=

cos

W

A

+ sin 

W Z0 

Sum over first generation particles.

Flavor changing interactions.

Left handed only

Flavor up

Flavor down

up

down

text error

text errorSlide3

Weak interaction terms

flavor changing: leptons

flavor changing: quarksSlide4

We want the coefficient for the electron-photon term to be

-e

-e

f

= 0 for neutrino and = 1 for others

A

A

Z

0

Z

0

Slide5

Consider only the A

term:

e

a

1

e

a2

gives agreement with experiment.

C

f

= 2T

3

 = -1

RSlide6

The following values for the constants

give the correct charge for all the particles. Slide7

Coefficients for the

Z0

 term

Z

0

Z0

g

1

=

g

2

sin

W

/

cos W

f

Y

fR = 2

Q f

C

f

cos

W

= 2T

3

[

1-sin

2

W

]/

cos

W

 = -1

Y

f

L

=

2 [

Q

f

– T

3

]

Slide8

A

Z

0

Final Form for Electro-Weak Interaction Terms(the QCD terms have yet to be written in terms of the color/anti-color gluons)

In the text by Gordon Kane, p.92, Eq. 7.32, there appears to be a typo in the sign of the third term. Slide9

(E & M) QED interactions

weak neutral current interactions

weak flavor changing interactions

QCD color interactions

-

+

The Standard Model Interaction

Lagrangian

for the 1

st

generationSlide10

Weak neutral current interactions

Z

0

Z

0

Z

0

Z

0

note: no flavor changesSlide11

quarks

leptons

Weak charged flavor changing interactions

-g

2

-g

2Slide12

Quantum

Chromodynamics

(QCD): color forces

Only non-zero

components of 

contribute. Slide13

To find the final form of the QCD terms, we rewrite the above sum,

collecting similar quark “color” combinations.Slide14

The QCD interaction

Lagrangian

density Slide15

The

red

, anti-green gluon

The green

, anti-blue gluon

Note that there are only 8 possibilities:

r

g

g

r

g

-

g

g

b

-Slide16

At any time the proton is color neutral. That is,

it contains one

red

, one blue

and one green quark.

The gluon forces hold the

proton together

protonSlide17

neutron

proton

beta decay

u

d

u

d

d

W doesn’t see color

u

W

-Slide18

decay of

-

-

u

d

-Slide19

p

p

d

u

u

u

u

d

-

-

-

W

production from

p

-

p

p

p

-

-

W

+Slide20

The nuclear force

n

p

u

d

d

u

d

u

u

u

d

u

p

n

d

d

u

W

-

Note that W

-

d

+

u

=

-

In older theories, one would consider rather the exchange of a 

-

between the n and p.

-Slide21
Slide22

Cross sections and Feynman diagrams

everything happens here

transition probability amplitude

must sum over all possible Feynman diagram amplitudes with the same initial and final states .Slide23
Slide24

Feynman rules applied to a 2-vertex electron positron scattering diagram

left vertex function

right vertex function

M

fi

=

spin

spin

time

propagator

metric tensor

The next steps are to do the sum over

 and  and carry out the matrix multiplications.

Note that

is a 4x4 matrix and the

spinors

are 4-component vectors. The result is a

a function of the

momenta

only, and the four spin (

helicity

) states.

coupling constant –

one for each vertex

Note that each vertex is

generated by the interaction

Lagrangian

density. Slide25

Confinement of quarks

free quark terms

free gluon terms

quark- gluon interactions

The free gluon terms have products of 2, 3 and 4 gluon field operators. These

terms lead to the interaction of gluons with other gluons.

Slide26

G



G



quark

loop

gluon

loop

N

f

N

c

N

f

= # flavors

N

c

= # colors

normal free gluon term

3-gluon vertex

Note signSlide27

momentum squared of exchanged gluon

N

f

N

c

N

c

N

f

In QED one has no terms corresponding to the number of colors (the 3-gluon) vertex.

This term has a negative sign.

-7

 M

2

quark

The terms given explicitly in M are only those loops shown in the previous diagram. Higher order terms are indicated by “ + … “Slide28

Quark confinement arises from the increasing strength of the interaction at

long range. At short range the gluon force is weak; at long range it is strong.

This confinement arises from the SU(3) symmetry – with it’s non-commuting

(non-

abelian) group elements. This non-commuting property generatesterms in the Lagrangian density which produce 3-gluon vertices – and gluonloops in the exchanged gluon “propagator”. Slide29

Conditions on the SU(2) Gauge Particle Fields which

complete the Invariance of the Lagrangian DensitySlide30

D’

’ =

D

[

ei/2 ] =

ei/2

D

=

cancel

We want to find

W

such that the following is satisfied: Slide31

0 and the

k

are linearly independent s

o [ …] = 0 and the following is the expression for

W

 Slide32

The Higgs

Lagrangian

Contribution