AIMS-pp-11.pdf

Weak  Interactions  of  

Leptons  and  Quarks

Enrico Fermi

GSW Electroweak theory (1968)

Sheldon Glashow Abdus Salam Steve Weinberg



ν

−



−

W

or

ν_

W+

Leptonic  Electroweak  Theory

+

( )

₁ 5

2

2 γ γ

µ ₋

−

= igW

ν

µ

M_W = 80.398 ± 0.0259 GeV M_Z = 91.1876 ± 0.0021 GeV

Γ_W = 2.141± 0.041 GeV

Γ_Z = 2.9452 ± 0.0023 GeV

=

−

i g

_µν

−

q

µ

q

ν

M

_W2

⎛

⎝⎜

⎞

⎠⎟

q

2

−

M

_W2

+

iM

_W

Γ

_W

The vertex rule for leptons

 ν −  − W

( )

₁ 5

2

2 γ γ

µ ₋

−

= igW

Features of the Leptonic Weak Interactions

•  Changes a lepton into its

neutrino partner and vice-versa

•  W-polarization perpendicular

to its momentum

•  Purely left-handed α_W = gW

2

4πc

does not appear in GSW theory

−

i g_µν − qµqν M_W2 ⎛

⎝⎜

⎞ ⎠⎟ q2 − M_W2 + iM_WΓ_W

ig

_µν

M

_W2

q2  M_W2

Low-energy W-propagator W-propagator

(internal line)

γ5 _=iγ0_γ1_γ2_γ 3₌ 0 I

I 0

⎛ ⎝⎜

⎞ ⎠⎟

1−γ5 = I −I

−I I

⎛ ⎝⎜

e ν − e − W − µ µ ν 1 p 2

p p'2 p'1

Muon  Decay

−iM= ( )−i 2 gW

2 2 ⎛

⎝⎜ ⎞⎠⎟

2

u(p₁′)γ µ

( )

1−γ 5 u(p₁) ⎡

⎣ ⎤⎦

g_µν − q_µq_ν /M_W2 q2 − M_W2

⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥

⎥ u(p2)γ

ν

( )

_1−γ 5

v(p₂′ ) ⎡

⎣ ⎤⎦

propagator left-handed e−νe current

left-handed µ−ν_µ current

 2gW

4M_W

⎛ ⎝⎜

⎞ ⎠⎟

2

u(p₁′)γ _µ

( )

1−γ 5 u(p₁)

⎡

⎣ ⎦⎤⎡_⎣u(p2)γ µ

( )

1−γ 5 v(p2′ )⎤_⎦

µ

−

→

e

−

+

ν

_e

+

ν

_µ

u(p₁′)γµ

( )

1−γ5 u(p₁)

⎡

⎣ ⎤⎦

†

spins∑

u(p₁′)γν

( )

1−γ 5 u(p₁)

⎡

⎣ ⎤⎦

= Tr γ µ

( )

1−γ5 ( p₁+m)γν

( )

1−γ5 p₁′+m_ν µ

(

)

⎡

⎣⎢ ⎦⎥⎤=Tr⎡_⎣γ µ

( )

1−γ 5 ( p1+m)γν

( )

1−γ 5

( )

p1′ ⎤_⎦ (neglect mνµ)

= Tr_⎣⎡γ µ

( )

1−γ5 ( p₁+m)

( )

1+γ 5 γν

( )

p₁′ ⎤_⎦=Tr⎡γ µ

( )

1−γ5 2( )p₁ γν

( )

p₁′ ⎣⎢

⎤

⎦⎥ (trace of the mass term = 0)

=2Tr_⎣⎡γµ

( )

1−γ5 ( )p₁ γν

( )

p₁′ ⎤_⎦=2Tr⎡_⎣γ µ( )p₁ γν

( )

p₁′ ⎤_{⎦ −}2Tr⎡_⎣γ µγ5( )p₁ γν

( )

p₁′ ⎤_⎦

=8⎡p₁′µp₁ν + p₁µp₁′ν −gµν

( )

p₁⋅p₁′ +iεµανβp_1αp₁′_β

⎣ ⎤⎦ e ν − e − W − µ µ ν 1 p 2

p p'2 p'1 Casimir Trick: −iM 2gW

4M_W

⎛ ⎝⎜

⎞ ⎠⎟

2

u(p₁′)γ_µ( )1−γ5 u(p₁)

⎡

⎣ ⎦⎤⎡_⎣u(p2)γµ( )1−γ5 v(p′2)⎤_⎦

The other term is similar, so:

M2 = 64

2

g_W

2 2M_W

⎛ ⎝⎜

⎞ ⎠⎟

4

p₁′µp₁ν + p₁µp₁′ν −gµν

( )

p₁⋅p₁′ +iεµανβ

(

p_1α p₁′_β

)

⎡

⎣ ⎤⎦

×_⎣⎢⎡p₂′_µp_2ν + p_2µp′_2ν − g_µν

(

p₂⋅ p₂′

)

+iε_µρνσ

(

p₂′ρpσ₂

)

⎤_⎦⎥

= 1 2 g_W M_W ⎛ ⎝⎜ ⎞ ⎠⎟ 4

2

(

p₁′ ⋅p₂′

)

(p₁⋅ p₂)+2

(

p₁⋅ p₂′

)

(

p₁′ ⋅p₂

)

⎡

⎣ +i2εµανβεµρνσ

(

p1α p1′βp′2ρp2σ

)

⎤_⎦⎥ = 1 2 g_W M_W ⎛ ⎝⎜ ⎞ ⎠⎟ 4

2

(

p₁′ ⋅p₂′

)

(p₁⋅ p₂)+2

(

p₁⋅ p₂′

)

(

p₁′ ⋅p₂

)

⎡

⎣ − −

(

2

(

δραδσβ −δρβδσα

)

)

p1α p1′βp′2ρp2σ ⎤_⎦⎥ = 2 gW

M_W ⎛ ⎝⎜ ⎞ ⎠⎟ 4

p₁′ ⋅p₂

(

)

(p₁⋅ ′p₂)

e ν

−

e

− W

− µ

µ

ν

1

p

2

p p'2 p'1

Muon rest frame

neglect electron mass

Now we need to integrate over 3-body phase space

Neutrino masses are small, so set

′

p₂

( )

2 ₌ m_ν

e

2 _{ 0 p′} 1

( )

2 ₌ m_ν

µ

e ν −

e

− W

− µ

µ ν

1

p

2

p p'2 p'1

lots of work to compute this 3-body integral

Total Decay Rate:

e ν

−

e

−

W

−

µ µ

ν

1

p

2

p p'2 p'1

Muon lifetime:

Fermi’_{s constant!}

e

ν

−

e

−

W

−

µ

µ

ν

e

ν

−

e

−

µ

µ

ν

⎯

⎯

⎯ →

⎯

q2<<MW2

Electroweak Theory

(renormalizable) _{(not renormalizable)} Fermi’s Theory

The  Strength  of  the  Weak

Large compared to QED!

•  Weak interactions are weak because the W is so heavy

Quark  &  Lepton  Vertices

we never see this we always

see this



ν

−



−

W

or

ν_

+

W +

So Leptonic charged weak vertex must be:

we always see this Conservation of Lepton Number

Left-handed SU(2) Doublets

Right-handed SU(2) Singlets E

−

W

E

Electron doublet

−

W

M

M Muon doublet

−

W

T

T

Tau doublet

Write

E

−

W

E

Electron doublet Compare to QCD

QCD

carries off colour charge carries off weak charge

u-quark in blue state emits a BR-gluon and changes into a u-quark in a red state

E-particle in electron state emits a W −

and changes into an E-particle in a neutrino state

suggests

WHY NOT?

Must allow cross-generational quark flavours to change into each other

d

−

e

e ν

d

u

u

d u

neutron proton

−

W

u

d

−

W c

s

−

W t

b

We observe that neutrons decay into protons (provided the

(

)

c W

ig

θ γ

γ µ _{1 sin}

2 2 5 − − = u s − W

(

)

c

W

ig

θ γ

γ µ _{1 cos}

2 2 5 − − = − W u d

1963: Cabbibo proposes quark-model weak vertices

Nicola Cabbibo

Kaon Decay Expts

We don’t know why this angle has the value that it does.

We regard it as another constant of nature, like the speed of light, or the mass of the electron. s − K p −  

ν _p'1

2 '

p

u

µ

ν

d

+

µ

s

−

µ

GIM  Mechanism

ΔS =1 for both Why so different?

Problems with the Cabbibo theory:

Kaon decay rates were sometimes too small

an integral you compute from the diagrams

BR

(

K+ →π+ +ν_e +ν_e

)

=

(

1.7 ±1.1

)

×10−10

BR

(

K+ →π0 +ν_e + e+

)

=

(

5.07 ± 0.04

)

×10−2

Also:

K0

ΓCabbibo

(

_K0 _{→ µ}+_µ−

)

ΓCabbibo

(

_K0 _{→ all}

)

 I pi

µ_;m

u2

(

)

×sinθ_ccosθ_c

>> Γ

expt

(

_K0 _{→ µ}+_µ−

)

Γexpt

(

_K0 _{→ all}

)

= 7.3×10

(

)

c W

ig

θ γ

γ µ _{1 cos}

2 2 5 − − =

= + igW

2 2 γ

µ

1− γ 5

(

)

sinθ_c

c s − W − W c d

1971: Glashow, Iliopolis, & Maini propose that maybe a new quark exists (call it c)

(

)

c

W ig

θ γ

γ µ _{1 sin}

2 2 5 − − = u s − W

(

)

c

W

ig

θ γ

γ µ _{1 cos}

2 2 5 − − = − W u d

u

µ

ν

d

+

µ

s

−

µ

c

µ

ν

d

+

µ

s

−

µ

+

u s − W − W u d

− igW

2 2 γ

µ

(

_{1− γ} 5

)

cosθ_c + igW

2 2 γ

µ

(

_{1− γ} 5

)

sinθ_c

c

s

−

W _{c W}−

d

+

′

s = −d sinθ_c + scosθ_c

− igW

2 2 γ

µ

(

_{1− γ} 5

)

+

′

d = d cosθ_c + ssinθ_c

− igW

2 2 γ

µ

(

_{1− γ} 5

)

cosθ_c − igW

2 2 γ

µ

(

_{1− γ} 5

)

sinθ_c

− igW

2 2 γ

µ

(

_{1− γ} 5

)

The  CKM  Matrix

Left-handed SU(2) quark Doublets

weak eigenstates mass eigenstates

Cabbibo matrix

From the viewpoint of QED and QCD, d and s are the elementary particles From the viewpoint of Electroweak, d’ and s’ are the elementary particles

1973 : Kobayashi & Maskawa propose a 6-quark model

to explain



P

violation

Note: we also must have u,c, t quarks as the partners of

d, s,b

CKM matrix

(not obvious, but can prove using matrix algebra)

− W

S

S

−

W

B B

− W

D

− W

S

S

−

W

B B

− W

D

D

Regard each down-type quark and its up-type partner as 2 states of one particle!

Similar to what happens for leptons:

E

−

W

E

Electron doublet

−

W

M

M

Muon doublet

−

W

T

T

Tau doublet Down

doublet Strange _doublet

Expt:

Jarlskog Invariant

These numbers are all

constants of nature. We do not have any explanation of their values at this point in history. This is one way to parametrize the CKM matrix

The Unitarity Triangle

Does α+β+γ =180o ? → Must check by experiment!

Electroweak  Unification

+

₌

Magnetism

Electricity Electromagnetism (QED)

Electromagnetism (QED)

Radioactivity (Weak)

=

Electroweak (QFD?)

W + W −

f

Z

f

Neutral  Currents

Bludman (1958) electrically neutral

Fermion identity preserved

Problem: This interaction will be overwhelmed by electromagnetism!

How can we observe it? Answer: Gargamelle (1973)

The effects of the W’s are easy to find: they change both the charge and the type (or flavour) of particle

π− µ−

ν_µ ν_µ

iron slab

CF₃Br freon Gargamelle

ν_µ

X

Hadron without a charged lepton

Results What

=

−

ig

Z

2 2

γ

µ

c

_Vf

−

c

_Af

γ

5

(

)

f

Z

f

Exp’_t

ν

µ

=

−

i g

_µν

−

q

µ

q

ν

M

_Z2

⎛

⎝⎜

⎞

⎠⎟

q

2

−

M

_Z2

+

iM

_Z

Γ

_Z



−

i

g

_µν

M

_Z2

q2  M_Z2

Z

Z-exchange

=

−

e

1 '

p

2

p

2

'

p

1

p

Z

−

e

1

' p

2

p

1

p

γ

+

2

'

p

f f

+

e e+

f f

f f

−

e e+

e

+

e

-­‐‑

 Neutral  Current  ScaOering

(neglecting fermion masses)

Casimir

Z

CMS − e 1 ' p 2 p 1 p γ 2 ' p f + e f − e 1 ' p 2 p 2 ' p 1 p Z f + e f Total Cross Section from Z exchange Total Cross Section from photon exchange

−

e 1

'

p

2 p

2 '

p

1 p

Z

f

+

e

f

−

e

1 '

p

2

p

1

p

γ

2

' p

f

+ e

f

2 2

Compare

E  M_Zc2

E = M_Zc2

Photons dominate