Weak interactions and vector bosons - 2013mep/phy305/weak_interaction_phy305.pdf · 2008-02-18 ·...

Weak interactions and vector bosons

• What do we know now about weak interactions?• Theory of weak interactions

– Fermi's theory of weak interactions– V-A theory– Current - current theory, current algebra– W and Z bosons– Electroweak theory

• Neutral current reactions

Weak interactions• Weak interactions occur for all fundamental particles except

gluons and photons. Weak interactions involve the exchange orproduction of W or Z bosons.

• Weak forces are very short-ranged. In ordinary matter, theireffects are negligible except in cases where they allow an effectthat is otherwise forbidden. There is a number of conservationlaws that are valid for strong and electromagnetic interactions, butbroken by weak processes. So, despite their slow rate and shortrange, weak interactions play a crucial role in the make-up of theworld we observe.

• Any process where the number of particles minus the number ofantiparticles of a given quark or lepton type changes, is weakdecay process and involves a W-boson. Weak decays are thusresponsible for the fact that ordinary stable matter contains onlyup and down type quarks and electrons. Matter containing anymore massive quark or lepton types is unstable. If there were noweak interactions, then many more types of matter would bestable.

Quarks and their weak decays• Quark flavour never

changes except byweak interactions,like beta decay, thatinvolve W bosons.

• Any quark type canconvert to any otherquark type with adifferent electriccharge by emitting orabsorbing a W boson.Different possibletransitions are shownschematically in thediagram.

Weak interactions

• Decay processes always proceed from a moremassive quark to a less massive quark, because thereverse process would violate conservation ofenergy. Scattering processes can involve thereverse transitions, provided sufficient energy isavailable.

• Lepton number conservation rulesEach charged lepton is converted only to its ownneutrino type by emitting or absorbing a W boson.This leads to the three lepton number conservationlaws.

Leptonic

Baryon

NonleptonicMeson

Baryon

Semi-leptonicMeson

Charged/neutral

neutralchargedInteraction

!

µ " e##

# µe" µ#e

$ " l##

!

" µe#" µe

" µe#" µe

e+e$# l

+l$

!

"ee#"

ee

" ee#"

ee

e+e$#"

e" e

Fermi's theory of weak interactionsBeginning of the story:

• As an example Fermi considered β-decay of theneutron:

• Assumption: at a single point in the space-time, thewavefunction of the neutron is transformed into that ofthe proton, and the wavefunction of incoming neutrino(equivalent to outgoing antineutrino which we actuallysee) is transformed into that of electron.

• The schematic diagram for this process:

!

n" p + e#

+$ e

n

e-

p

!

"e

Such a process is called four-fermioninteraction.

Relativistic interaction currents

!

Jµ

EM = "# e$µ#e

electromagnetic interaction from QED

!

HEM

= Jµ

EMA

µ = "e# $µ#Aµ

Fermi (1934) introduced a weak hadronic current density

!

Vµ

C + =" P#µ"n (charged weak hadronic current)

in analogy to and (photon)EMJ µ

µA

weak field ( boson)CVµ

±W W

Fermi constructed further

!

lµC ="

e#µ"v

( )++ += ccccVllV

GH µ

µ

µ

µ!

!2

V-A theory• The description of this process is given by the amplitude:

where the factors Γ are responsible for particle transformation,and GF is the very well known Fermi coupling constant whichdetermines the strength of the weak interaction (the rate of thedecay).

• In 1956 Feynman and Gell-Mann proposed that the interactionfactors Γ are a mixture of vector and axial-vector quantities, toaccount for parity violation in weak interactions.– Vector - changes sign if rotated through 180o.– Axial vector - like vector under rotation, but opposite sign to vector under

parity transformation.• V-A theory was proven by many experiments at that time, in

particular by the fact that neutrino is left-handed (helicity -1/2) -its spin is antiparallel to its momentum.

!

M =GF (" p#"n )(" e#"$ )

leptonic current density, which results in left-handed neutrinos:

!

lµC ="

e#µ 1$ # 5( )"v

hadronic current

!

hµ

C + =" p#µ 1$ cA# 5( )"n (V-A structure)

Co60 Parity-violation ! decay

eKPSrr

!Pseudoscalar eveNiCo ++!

"6060

Co60 nucleus !!

KS

rKS

r

KS

r

!e

!e

!e

epr

epr

epr

Rotation invariance

P

1 2 3Wu et al(57)

Vector and axial vector current

!

jµ =" #µ"

jµ5 =" #µ# 5"

!

i"µ jµ = i"µ (# $ µ#) = i("µ# )$ µ# + i# $ µ

("µ#)

= %m# # + m# #

i("µ# )$ µ = %m#

i$ µ("µ#) = m#

&"µ jµ = 0

Vector current is conserved

!

i"µ jµ5 = i("µ# )$ µ$ 5# + i# $ µ$ 5("µ#)

= i("µ# )$ µ$ 5# % i# $ 5$µ("µ#)

= %m# $ 5# %m# $ 5#

= %2m# $ 5#

The axial vector current is not conserved for m≠0m: current mass of the quark

Current - current theory• We can also describe weak interactions in terms of the

interactions of two "currents". In β-decay one currentconverts a neutron into a proton, and the other createsan electron and its antineutrino (absorbs or destroysneutrino).

• For simplicity consider leptonic reactions whichinvolves only leptons. As a result of lepton numberconservation law, the absorption of an electronneutrino is always accompanied by the creation of anelectron and vice verse. Equivalently, the creation of apositron should be accompanied by the creation of anelectron neutrino (or the absorption/destruction ofelectron antineutrino)

!

LW ="

e#"$

e

+" µ#"$ µ+"%#"$%

!

LW

="#e

$"e

+"# µ$"µ +"%$"#%

the leptonic current is extended in an obvious wayto the muonic & tau-lepton sectors:

!

lµC ="

e#µ 1$ # 5( )"ve

+" µ#µ 1$ # 5( )"vµ+" %#µ 1$ # 5( )"v%

Leptonic currents

• The leptonic current should be multiplied by itsantiworld partner to generate all observed weakinteractions of leptons.

• The first-order amplitude of the all leptonic processescan be written as:

µ-

!

"µ

!

LW

"

µ-

!

"µe-

!

"e

!

LW

"

!

M =GFLW

LW"

e-

!

"e

!

"ee- e-

!

"e

+

+ + +e-e-

!

"e

!

"µ

!

"µ

!

"µ

µ-

µ-

µ-

!

"e

µ-

!

"µ

+ +

+

+…..

Hadronic current• There are also semileptonic reactions which involve

both leptons and hadrons, such as neutron β-decay,and hadronic weak reactions which involve onlyhadrons, such as the decays of neutral kaons into 2 or3 pions.

• Hadronic weak current can be characterised in termsof its effect on the quantum numbers of hadrons.

• The total weak current is a sum of leptonic andhadronic components:

• Then, the amplitude of weak interactions can bepresented as: !

JW

= LW

+ HW

!

GFJW

JW

=GF(L

W

LW

+ LW

HW

+HW

LW

+ HW

HW)

Hadronic current and current algebra• Similar to the leptonic current, hadronic current is a mixture of

vector and axial-vector components, to ensure the correct parity-violating effects. But hadronic current should have also a partwhich conserves the strangeness of the participating hadrons andanother part which changes the strangeness:HW = h± cosθC + s± sinθC , where θC is the Cabibbo angle whichgoverns the relative strength of the two components. This is for 4-quark scheme. For 6-quark scheme the cosθC and sinθC arereplaced with a matrix.

• So, the hadronic weak current has four separate components• The effect of different parts of the current on the quantum

numbers: h± → (ΔY,ΔI,ΔI3) = (0,1,±1) s± → (ΔY,ΔI,ΔI3) =(±1,1/2,±1/2) (Y=B+S)

• Cabibbo angle was measured in the experiments: cosθC ≈0.97, θC≈12o.

Quark mixing and CKM matrix

• Observation: ΔS=1 weak decay suppressedthan ΔS=0 weak process

!

n" p + e#

+ $ e (1)

%#" n + e

#+ $ e (2)

Process (2) is suppressed by a factor of 20 compared with (1)

Look at the diagrams at the quark level for these two processes

!

u

d

"

# $ %

& ' weak

=u

d

"

# $ %

& ' strong

?

In analogous to doublet,

Introduce quark doublet:

!

u

dcos"c

+ ssin"c

#

$ %

&

' (

Then the effective coupling for

!

n" p + e#

+ $ e

%#" n + e

#+ $ e

!

Gcos"c

Gsin"c

!

e

"e

#

$ %

&

' (

!

u

" µ

#

$ %

&

' (

!

20 = (cos"

c

sin"c

)2#"

c$12.6

0

Cabibbo angle

In 1970, Glashow, Iiiopoulos and Maiani (GIM)

!

u

d'

"

# $ %

& ' =

u

dcos(c

+ ssin(c

"

# $

%

& '

!

c

s'

"

# $ %

& ' =

c

scos(c) d sin(

c

"

# $

%

& '

Which can be expressed in matrix form (charm discovered in 1974)

!

d'

s'

"

# $ %

& ' =

cos(c

sin(c

)sin(ccos(

c

"

# $

%

& ' d

s

"

# $ %

& '

Quark state in the weak interaction

Problems with Fermi's theory• Low-energy weak interactions can be well explained in

the framework of Fermi's theory (in the framework ofthe current-current theory in the V-A form).

• Problems begin when high-energy processes need to bedescribed.– The theory predicts linear rise of the neutrino cross-sections

with energy, which violates the unitarity principle - theprobability for a particular process to occur should be less thanor equal to unity. At the energy about 300 GeV the cross-section exceeds the maximal value allowed by the unitarityprinciple.

– Very large cross-section at high energies contradict alsocosmic-ray data. Neutrinos are the important part of secondarycosmic rays produced in the atmosphere but their interactionswith matter were not frequent and put strong limit on theircross-section.

!e e

!e

!e

!e

!u

dev

bosons (82 ) range 1/1000 fermi

GeVW

pe

nev

*: Fermi’s theory as a low-energy limit of an interaction mediated by theexchange of bosons.

New ideas• Fermi's basic principle: analogy with electromagnetism.• But electromagnetic processes are described as an

exchange of photons.• There should be vector particles which transmits the

weak force.• Weak interaction is of short range, hence the vector

particle should be heavy.• Since β-decay changes nuclear charge, the vector

particles should carry the charge.• Four-fermion point-like interactions were abandoned

and replaced with a particle (W) exchange mechanism.e-

!

"e

W- W+

µ+

!

"µ

Examples of basic lepton processes with the creation or destruction of W-boson

Problems with W-boson• Neutron β-decay can now be shown as an exchange of

W-boson.

• W-boson helped to solve some problems of Fermi'stheory but produced others.

• W-boson has spin 1 and non-zero mass. As aconsequence at high-momentum transfer its propagatoris proportional to p2 / M2 c2. The diagrams arediverging and the results of the theory are infinite.

W-

n p

!

"e

e-

Electroweak theory and Higgsmechanism

• First step: 1954 - C. N. Yang and R. Mills developed atheory of interacting massless particles - gauge theory(the Lagrangian describing the interaction of particlewavefunctions remains invariant under certainsymmetry transformations). The theory couldaccommodate particles like the photon and W-boson,but it required them to be massless. The infinities in themodel could be reabsorbed (the model was"renormalisable").

•1964 - Peter Higgs found a way to transform the Lagrangian describing two massless scalar particles and one massless vector gauge particle into the Lagrangian with one massive scalar particle and one massive vector gauge particle ("Higgs mechanism" for symmetry breaking - mechanism that can give masses to W-bosons).•1967 - 1968 - Sheldon Glashow, Steven Weinberg and Abdus Salam - development of the standard model of electroweak interactions, which unifies electromagnetic and weak interactions.

Unification of the interactions

• In the last decades the belief has grown that the strong,electromagnetic, weak and gravitational interactions aredifferent aspects of a single universal interaction, whichwould be manifested at very high energies inaccessibleby existing accelerators.

• In our everyday particle physics experiments atrelatively low energies this symmetry is broken.

• The first successful attempt to unify two apparentlydifferent interactions was done by Maxwell in 1865. Heshowed that electricity and magnetism could be unifiedinto a single theory involving a vector field(electromagnetic field) interacting between electriccharges and currents.

•Maxwell equations contain one arbitrary constant - the velocity of light - which is not predicted by the theory and has to be determined from experiment.•Glashow, Weinberg and Salam - unification of the electromagnetic and weak interactions into electroweakinteraction with a single coupling described by theelementary electric charge, e.

Electroweak theory and neutral current

• Start - massless Yang-Mills particles: W+, W-, Wo

and B (not to be confused with B-meson). W-particles form a triplet of the new symmetry,"weak isospin", B is an isosinglet.

• Higgs mechanism gives masses to the W-bosons.• Wo and B mix to produce two physical particles,

the photon (represented by the field A) and the Z-boson. The photon is massless. The Z-bosonacquires a mass comparable to that of W-boson.

•Early 70s - G. ‘t Hooft et al. have shown that the theory is renormalisable. Introducing a finite number of arbitrary parameters (to be determined from the experiment), it is possible to make predicted amplitudes for physical processes finite at all energies and to all orders in the coupling constant.•Very important predictions of the new theory - there should be a neutral Z-boson which does not change the charge of particles. Fermi theory predicted only charge W-boson. Z-boson is responsible for the reaction with neutral current, such as νµ e- → νµ e-, which were not allowed in the Fermi's theory.

Early searches for neutral currents• The searches for neutral current events have been

started in the early 60s when the electroweak theory ofGlashow-Weinberg-Salam was not yet believed to berenormalisable.

• The neutral-current processes were not found in thedecays:K+→ π+e-e+ and KL

o → µ+µ -.• The searches had been limited to the neutral currents

which changed strangeness. If strangeness is notchanged and the reaction goes via a Z-exchange, then inmost cases a photon can be exchanged too. Thus theeffect of Z is masked by much larger electromagneticeffect.

•One way to avoid this was to look for neutrino scattering with the emission of Z. This process could not occur via a photon exchange because neutrino does not couple to a photon. The signature for such a process is the absence of a charged lepton in the final state.

References

1. “The experimental foundation of particle physics” byR. N. Cahn and G. Goldhaber, published by CambridgeUniversity Press.

2. “Introduction to high energy physics” by D. H. Perkins,published by Addison-Wesley.

3. “The ideas of particle physics” by G. D. Coughlan andJ. E. Dodd, published by Cambridge University Press.

4. Lecture notes by Dr. Vitaly KudryavtsevSheffield Univ.