The Weak Interaction

1

Sun sun sunRising sun the creatorMid day blazing sun the destroyer RudraSetting sun the maintainer and continuanceGreatest of allSun sun sun(Gajanan Mishra)

1. Costituents of Matter2. Fundamental Forces3. Particle Detectors 4. Symmetries and Conservation Laws5. Relativistic Kinematics6. The Static Quark Model7. The Weak Interaction8. Introduction to the Standard Model9. CP Violation in the Standard Model (N. Neri)

The Weak Nuclear Interactions concerns all Quarks and all Leptons

The Weak Interaction takes place whenever some conservation law (isospin, strangeness, charm, beauty, top) forbids Strong or EM to take place

In the Weak Interaction leptons appear in doublets:

Q L(e) = +1 L(μ) = +1 L(τ) = +1

0

-1

e e

…and the relevant anti-leptons. For instance:

Doublets are characterized by electron, muon, tau numbers (each conserved, except in neutrino oscillations) whose sum is conserved.

(see the section on Fundamental Interactions)

Simple facts

2

Weak Nuclear Interaction violates Parity

The Parity violation is maximal

Simple facts

3

Weak Nuclear Interaction violates CP

This fact will need to be incorporated in the theory: a phase in the CKM matrix.

Discovered first in the Wu experiment.

Confirmed in all other experiments on Weak Interactions.

The fundamental weak couplings are to fully left-handed fundamental fermions (and fully right-handed fundamental antifermions).

CPT is conserved by Weak Interactions

Weak Interactions violate P, C, CP, T but not the combination CPT.

Neutron decay 103 s Long lifetime due to the small mass difference

Inverse n decay

10-43

cm2 Has only weak interactions

Lamda decayp-

10-10 s S=1: strong/e.m. interactions forbidden

Pion decay+

10-8 s Leptons are the lightest particles

Weak Interactions allow for processes otherwise impossible

At low energy: Fermi Theory

At high (and low) energies: Electroweak Theory

The first theory of Weak Interactions was developed by Enrico Fermi in close analogy with Quantum Electrodynamics. The process to be explained was the nuclear beta decay.

Nature rejected his paper “because it contained speculations too remote to be of interest to the reader.”

‘Tentativo di una teoria…’ ,

Ric. Scientifica 4, 491, 1933.

4

Fermi Theory of the Beta Decay

( , ) ( 1, 1) eA Z N A Z N e

en p e

ed u e At the fundamental (constituents) level

u d

2

1

WZM

g

gweakJ

'weakJ

FG '

2

2'

JJ

M

gJJGL

WFFermi

The rate of decay (transizions per unit time) will be:

0

222

dE

dNMGW F

2M Integration over spins and angles

0E Energy of the final state

5

F: Fermi transitions.

No nuclear spin change∆J (Nuclear Spin) = 0Leptonic state: spin singlet ↑↓│M│2 ≈ 1

GT: transizioni alla Gamow-Teller.

Nuclear Spin change∆J (Nuclear Spin) = +1,-1Letponic state: spin triplet ↑↑│M│2 ≈ 3

Several transitions are mixed transitions (F e GT).

In the assumption of no interference, one typically has :

2222222GTAFVFF McMcGMG

25.1/

1

VA

V

cc

cWith weights:

6

0dE dN

0ETPproton ,,

Epelectron ,,

Eqneutrino ,,

0

0

EEET

pqP

In the rest frame of the neutron :

The recoil kinetic energy of the nucleon Is negligible : MeVMP 32 102/

EEcq 0

7

Beta Decay Kinematics

Q-value

Energy carried away by the neutrino :

Final state

dE0 arises from the finite lifetime of the initial state

Number of neutrino and electron available states with electron and neutrino momenta in the ranges p,p+dp e q,q+dq

dpph

dV 23

dqq

h

dV 23

Choosing a normalized volume and integrating over the angles :

dpph

23

4dqq

h2

3

4

Neglecting dynamical correlations between p,q… Moreover, there is no free phase space for the proton, since given p,q its momentum is fixed:

The phase space is :

qpP

dqdpqph

Nd 226

22 16

Now, expressing q as a function of the total available energy and E :

EEcq 0 cdEdq /0

8

plotKurieEEp

N

dEdpEEpdpEEpch

Nd

02

02

022

02

36

22 )()(

16

General form

dpEE

mEEppZFdppN

dpEEppZFdppN

20

22

02

20

2

)(1)(),()(

)(),()(

Coulombian Correction F(Z,p)

Coulombian Correction and non-zero neutrino mass

Kurie plot

9

Beta Decay Spectrum in short

10

The coupling constant enters here

over the electron spectrum.This quantity features a sharp dependence on the Q-value E0

Total decay rate

dpEEpdpEEpchdE

Nd 20

220

236

2

0

2

)()(16

This can be quickly appreciated in the (somewhat crude) relativistic electron (E = pc) approximation :

30

2)(503

0422

00

20

20 EEdEEEdEEdEEEEEdEN

ESargent’s rule

11

The total decay rate depends on the coupling constant and the phase space.

For a fixed coupling constant, the rate is the integral of :

12

The Weak Fermi constant 5 231.2 10

( )FG GeVc

42

2

3 8

2

)( cM

g

c

G

W

F

35101.9 fmMeVGF

Coupling constants : Eelectromagnetic and Weak

A reminder :

cmerg

cmcmdyne

c

e 137

12

In rationalized and natural unitse is adimensional :

09.0137

1

4

2

ee

65.02

8 222 wWFw gcMGg

5.29

1

4

2

ww

g

The Weak Coupling constant is actually bigger than the fine structure constant.

But at low energies it is damped by the W mass into the small GF constant

Weak Decays and Phase Space in the low energy regime

13

5242 )( mGmmG FFn

n

According to the Sargent’s rule one has – roughly : The neutron lifetime :

And this has a general validity. In fact :

ssmm

mmn

e

pn 71035

5

10107.210)(

)()()(

The muon lifetime :

For a charmed particle :

ssmmm

mmnD

KD

pn 115

35

5

101000

3.110

)(

)()()(

Electromagnetic

14

f f

f f

e

e

22e

q

ig

f f

f f

e

e

22e

q

ig

f f

f f

W

g

g

2222

22 )/(g

cMq

cMqqgi

High Energy Matrix Element

Weak

Low Energy Matrix Element

f f

f f

g

g

2222

2222

22 )/(FGg

cM

igg

cMq

cMqqgi

Inverse Beta Decay

enpe

e

pn

e

0

222

dE

dNMGW F

2221

pMGF

p is the momentum of the neutron/positron system in their CM

This is a mixed (Fermi + Gamow-Teller) transition 42 M

22243 )/()(10 cMeVpcm A very small cross sectionThe cross section increases with E

15

16

Neutrino discovery:

Principle of the experiment

In a nuclear power reactor, antineutrinos come from decay of radioactive nuclei produced by 235U and 238U fission. And their flux is very high.

Water and Water and cadmiumcadmium

Liquid Liquid scintillatorscintillator

enpeInverse Beta Decay

1. The antineutrino reacts with a proton and forms n and e+

2. The e+ annihilates immediately in gammas

3. The n gets slowed down and captured by a Cd nucleus with the emission of gammas (after several microseconds delay)

4. Gammas are detected by the scintillator: the signature of the event is the delayed gamma signal

24310)( cmnepe

1956: Reines and Cowan at the Savannah nuclear power reactor

17

The size of the detector might be important. And this is because of the small cross neutrino section.

Not a specific detector. But… the typical configuration of a low energy, low background undergound neutrino detector :

Neutrino beamMassive, instrumented detectorDetector transparent to signal carriersBackground control!

« I went to the general store but they did not sell me anything specific»

18

Parity violation in Beta Decay

1956: Lee-Yang, studying the decay of charged K mesons hypotesized that Weak Interactions cold not conserve Parity.

1957: esperiment by Wu et al. eeNiCo 6028

6027

A sample of Co-60 nuclei at 10 mK in a magnetic field.The Co-60 spin (J=5) get statistically aligned by the magnetic field.The daughter nucleus (Ni*) has spin 4

The experimentally observed distribution for the emitted electron has the form :

cos11)(c

v

E

pI

p e

)60(CoJ

zH

19

cos11)(c

v

E

pI

p e

)60(CoJ

zH

:P

ppP

:

ppP :

This term violates Parity, by correlating the momentum of the electron to the Co-60 spin. This alignment fades away with increasing energy.

20

BBP

:

V-A structure of Weak Interactions

The helicities of neutrino and electron are :

11//

cvcv

ee

This property must be part of a consistent theory of Weak Interaction: the description of Dirac-type elementary constituents

enpepepe

Electromagnetic Weak

leptonsbaryonsJJq

eM

2

2

ppbaryonJ eeleptonJ

weakleptons

weakbaryons

W

w JJMq

gM

22

2

21

Neutrinos are considered massless !

«Electroweak analogy». What is the structure of the weak current(s) ?

weakleptons

weakbaryons

W

w JJMq

gM

22

2

Charged weak currents

OOG

M enpF

weak2

At low energy

According to the original idea by Fermi : O

22

In the earliest days of the parity violation discovery, it was natural to guess that the violation itself might be a special property of neutrinos.

The two component neutrino theory: if neutrinos were massless , then they could be polarized only parallel to the direction of motion (positive helicity) or antiparallel to it (negative helicity).

But parity violation was seen also in reactions like

And was found to be a general property of the Weak Interactions.

p

A theory of the Weak Interactions had to be based on concepts like universality and parity violation.

23

The two-component theory of the (massless) Neutrino

The spin-1/2 pointlike particle wave function obeys the Dirac Equation : 0

mi

Four components : two spin states of particle two spin states of antiparticle

2

• Massive particle: both spin states must be described by the same wavefunction because the spin direction is not Lorentz-invariant.

• Massless particle: it always travel at the speed of light, so its spin direction can be defined in a Lorentz-covariant way (parallel or antiparallel to the direction of the momentum, i.e. positive or negative helicity).

In the Weyl representation of the Gamma Matrices:

0

0

01

100

k

kk

24

Introducing the bispinors (upper and lower components) :

v

u

0 mi

Dirac Equation in the Weyl representation muvi

t

vi

mvuit

ui

For a massles fermion, the upper and lower components are decoupled :

vit

vi

uit

ui

ppvforpE

ppuforpE

For a massles particle, E= p

0

uR

vL

0

Right-handed spinor

Left-handed spinor

25

Let us now introduce the Gamma-5 matrix (in the Weyl representation) :

10

0132105 i

L

R

v

u

0

2

1

02

1

5

5One can then build right-handed or left-handed wavefunctions by using the projectors

More in general, in the case of massive particles :

2

1 5RP gives a v/c polarization along the direction of p (+1 when v=c)

2

1 5LP gives a -v/c polarization along the direction of p (-1 when v=c)

Before the Parity violation experiments, there was no reason con consider the right and left-handed spinors as particularly useful. However, detailed evidence was found that only the left-handed spinor occurs in Weak Interactions

26

Only left-handed spinor particles (and right-handed spinor antiparticles) take part in the Weak Interactions. This has several consequences :

a) The existence of a two-component massless neutrino theory

b) Maximal Parity violation

c) Maximal C violation

d) T conservation

e) CP conservation

If we carry out the P operation on the neutrino described by ψL, we obtain a neutrino described by ψR, which is unallowed in the theory.

(see before)

If we carry out the C operation on the neutrino described by ψL, we obtain an antineutrino described by ψL, which is unallowed in the theory.

This is because T reverses both spin and linear momentum.

(see the lecture on Symmetries and Conservation Laws)

There exists – however – tiny violations of CP and T invariance in the Weak Interactions (see lecture on CP violation)

27

Notable properties of the projection operators

2

1 5RP

2

1 5LP

32105 i 2

LLL PPP

555525

55

12

1)2(1

4

121

4

1

2

1

2

1

RRR PPP

555525

55

12

1)2(1

4

121

4

1

2

1

2

1

014

11

4

1

2

1

2

1 5525

5555

RLLR PPPP

RR PP

12

1

2

1

2

1 525

555

5

LL PP

12

1

2

1

2

1 525

555

5

28

LR PP

55

12

1

2

1

And this is because : 5321032105 )1( ii

(an odd number of exchanges with a different matrix)

The Universal Four-Fermion Matrix Element

A C

B D

g

g

2222

2222

22 )/(FGg

cM

igg

cMq

cMqqgi

Propagator and

coupling constant

LALCB

LD

Fweak OO

GM

2

Now, which is the form of the current ? We know that it has to be of the form :

ALRCLA

LC OPPO

RL PP

55

12

1

2

1

29

ALRC

CAC

OPPnsInteractioWeak

OnetismElectromag

Now, which is the form of the current ? We know that it has to be of the form :

(because of Lorentz invariance requirements)

arPseudoscal

VectorAxial

Tensori

Vector

Scalar

5

5

2

In the case of the Weak Interactions :

Scalar (originates F transitions)

Vector (produces F transitions)

Axial Vector(GT transitions)

Pseudoscalar

Tensor (GT transitions)

0 LRLR PPOOPP

LLLLRLR PPPPPOPP

LLLLLLLLRLR PPPPPPPPPOPP 555

05 LRLRLR PPPPOPP

0

)(

LRLR

LLLLLRLR

PPPP

PPPPPPOPP

30

The Universal Four-Fermion Matrix Element :

ABFCD GM )1()1( 55

Weak Current Weak Current

Low-E «propagator»

..can be constructed with the only non-zero matrix elements (V and A). A general form could be :

2

5AV CC

The fact that a massless neutrino is produced in a pure helicity eigenstate requires CA= - CV giving precisely the helicity projector in the current :

2

1 5

In general, this holds for any massive fermion, leading to the general form :

1V

A

C

C

31

Corrections to the V-A current structure ?

They need to be considered when the Weak Interaction involves Hadrons !

Let us first consider the electric charge of a proton

The proton is a complicate object, continually emitting and absorbing quark-antiquark pairs as well as gluons

The charge of the proton – however – is equal to the charge of the (elementary) electron !The electric current (a vector current V) is conserved by the Strong Interaction

What about the Weak interaction V-A current ? AV CC 5 The general experimental situation indicates that the V part is conserved (Conserved Vector Current, CVC hypotesis. Goldberger-Treiman). The A part of the corrent gets (most or all of) the Strong Interaction corrections :

26.1/ VAe CCepn 72.0/

VAe CCep

34.0/ VAe CCen

Pion decay and V-A structure of Weak Interactions

e

Pion has spin 0 Neutrino and muon must have antiparallel spins (J conserved) Neutrino has -1 helicity For a massless neutrino helicity is an exact quantum number Muon MUST have negative helicity (the «wrong» helicity!)

H Negative

e

From fundamental physics viewpoint, coupling constant, Feynman diagrams, they are essentially the same thing! The main difference is the phase space.

u

dW

),( el

32

If we compare the two processes :

Let us compare the decays :

From the point of view of the phase space, the decay in the electron is largely favored

But…in this decay the LEPTON is forced to have an «unnatural» helicity !

0,, mp mlp ,,

H Negative

)1960.,(

103.1 4

aletAnderson

eR

Experimentally, one has the following electron energy spectrum from stopping pions

33

34

Introducing the W and the Z0

11

5.22

0021.01876.91015.0385.80

GeVGeV

GeVMGeVM

ZW

ZW

And the relevant expression for the propagator : 2

22

222

22

)(

)/(g

Mc

gig

cMq

cMqqgi

Low energy limit Lifetimes ?

sssmGeV

fmMeV 2538

15

8103

10103

10100

1032

200

35

The Weak Charged Current and the Weak Neutral Current

States connected by a W

States connected by a Z(no flavor change whatsoever)

In fact, there is no (flavor changing) tc,tu,bs,bd,cu

36

W

e

Now let us consider – as a meaningful example – the neutrino scattering process in ordinary matter :

ee

If the neutrino is an electron neutrino :

Ze e

e

e e

e e

If the neutrino is a muon neutrino :

Z

e e

There is no annihilation diagram possible,leaving only the Z possibility (exchange of a Z between the two leptons). Only NC

Recalling the discovery of the third leptonic family: the Tau

SLAC, 1975, Martin Perl et al., studying the products of e+e- collisions

ee

ee

With hindsight :

This indicates intermediate states emitting invisibile leptons (neutrinos). This is because the Lepton Numbers (elettronic, muonic) are violated.

Is this the only possible interpretation of an eμ final state?

37

Detection of final states featuring an electron and a muon

The important point was that these events took place when the energy was greater than 3.56 GeV :

GeVmGeV 78.12)(256.3

This has to be disentangled from events with two charged particles produced by the process :

DDPsiee )3740(

0KD

eeKD 0

Featuring the same leptonic final state

With the discovery of the Tau (and the Tau Neutrino in 2002) the fundamental leptons are :

e

e

38

Energy threshold of 3740 MeV (as opposed to 3560)

Additional hadronic particles in the final state (K, pions, muons)

A note on neutrino experimental characterization : flavors and currents

Key point: different interaction in materials of a neutrino beam. Charged Currents (CC) and Neutral Currents (NC)

Muon in the final state (CC event).

Muonic neutrino arriving!

Electron in the final state (CC)

Electron neutrino arriving!

No final state lepton

Neutral current (NC) !

Neutrino flavor unknown.

Tau neutrino tau interactions

Tau lepton decaying in different ways (including muon, electron)

39

e

e

40

The Weak Charged Currents

W

),( el

The weak charged coupling to leptons is characterised by the fundamental vertex :

)1(22

5 ig

Weak Vertex Factor

The Weak Coupling Constant :

5.29

1

4

2

ww

g

Charged Currents Weak Interactions at low energy: the muon lifetime

ee

The Weak Interaction (CC) lowest order Feynman Diagram :

)( 1p

)( 3p

)( 4pe

)(qW

)( 2pe

41

)( 1p

)( 3p

)( 4pe

)(qW

)( 2pe

42

2

3 8

2

)( cM

g

c

G

W

F

)2()1(

22)4(

)(

1)1()1(

22)3( 5

25

w

W

w g

cM

gM

The muon lifetime result is : 2

34

812

cmgm

M

w

W

At low energies, MW and gw always enter in observable quantities as a ratio, which makes it possible to write :

452

73192

cmGF

The best Weak Coupling Constant determination at low energies

42

The Weak Charged Currents : Leptons and Quarks

e

eThe coupling of W to leptons takes place strictly within a given generation:

W W W

Purely leptonic Charged Current Weak Processes only involve leptons. Their general structure is : W

ee

ee

Weak decays of leptons into other leptons

Scattering between leptons (observed only if electrons are present to act as suitable targets)ee

ee ee

lJ

lJ

43

The coupling of W to Quarks :

b

t

s

c

d

uSimilar to the Quark case, there is coupling within a generation :

W W W

But cross-generational couplings are also there (6 couplings, since bu and td are not shown) :

b

t

s

c

d

u

Charged Current involving Quarks can originate :

W

hJ

lJ

Semileptonic processes Hadronic processes

W

hJ

hJ

0D K

en p e

lp l

d udu

W-ee

l n l p

They all feature a leptonic and a hadronic charged current

0lB D l

ud

44

Charged Current semileptonic processes :

The neutron decays (and beta decays)

The «inversa beta decay» kind of reaction

The decay kind of a heavy baryon

Beauty and Charm decays

u d u d There are weak processed conserving flavor ut they are dwarfed by the much stronger Strong Interaction

They are possible (and the only possibility) when the flavor is changed. Other forms of interactions are not allowed. They can connect quarks in the same generation, like in a cs decay :

c u d s 0D K c

dW

sdu

0K p 45

Charged Current purely hadronic processes :

b

dW

udu

They can connect quarks in different generations, like in a bu decay :

They of course involve Mesons and Baryons as well :

hJ

hJ

occurs more frequently than

46

Weak Charged Currents : the Cabibbo theory of Mixing (1963)

Weak Charged Interactions have been characterized with a unique coupling constant (and the phase space). However, the intergenerational processes seemed to take place less often than the decays within the same generation :

sd

u

The charm quark was not known at that time

WW

Experiments say that :

Cabibbo proposed that the quarks entered the Weak Charged Interactions as “rotated” states :

CCCC dssd

u

sincossincos

47

The Weak Interaction Eigenstates at the time of the Cabibbo Theory (no Neutral Currents, yet, no taus, no c, b and t) :

cCe sd

ue

Weak Interaction Eigenstates related to Mass Eigenstates by :

d

s

d

s

CC

CC

C

C

cossin

sincos

Mixing determined by the Cabibbo angle

97.0cos

22.0sin130

C

CC

The new interaction vertices for Weak Charged currents are :

u

s

W

51 ( sin )2 2

C

ig

u

d

W

51 (cos )2 2

C

ig

accounting for both the V-A structure and the quark mixing

48

The experimental evidence :

2

220

220

2214

sin

cos

cos)(

Fe

CFee

CFee

CFee

Ge

GeuseK

Geude

GeduOnep

CFw

WC

w Gg

cM

gM

cos)2()1(22

)4()(

1)1(cos)1(

22)3( 5

25

)( 1pu

)( 3pd

)( 4pe

)(qW

)( 2pe

Actually the rate of these processes is the motivation for introducing the mixing.

All leptonic processes are unaffected. All hadronic processes are affected.

Cabibbo-allowed

Cabibbo-allowed

Cabibbo-suppressed

Leptonic

In a semileptonic process like a beta-decay :

Experimental values of the magnitude of CKM elements are close to a unit matrix :

Same-generation transitions are favoured :

In the Standard Model, all flavors are mixed, as represented by the CKM (Cabibbo-Kobayashi-Maskawa) Matrix :

D K favored

D suppressed

favored

suppressed

d

u

s

c

b

t

49

Mass eigenstates

Weak Interaction eigenstates

The CKM 3-quark mixing is a generalization of the 2-flavor Cabibbo style mixing

Electromagnetism (the photon) couples to charged particles.

The Charge Current coupling will couple according to the weak charge

i i ii

e Q q q

( , )2

gW a f f

If we are considering leptons, one should write :

( , ) ( , ) ( , )ea e a a All the other components are zero because of the lepton numbers conservation.

( , ) ( , ) 1 0

( , ) ( , ) 0 1e ea e a

a e a

For instance, in the case of two families :

50

The flavor structure of Weak (and Electromagnetic) Interactions

In the case of quarks, we now have all four couplings that are different from zero.

( , ) ( , )

( , ) ( , )

a u d a u s

a c d a c s

This matrix is not diagonal and this is because the mass states are not eigenstates. It can be diagonalized by means of a rotation:

' cos sinC Cd d s ' cos sinC Cs s d

The rotation by the Cabibbo angle θC bring us to the Weak Eigenstates

In this base : ' '

' '

1 0( , ) ( , )

0 1( , ) ( , )

a u d a u s

a c d a c s

In considering d’,s’,b’ (eigenstates of the Weak Interaction) instead of d,s,b, we can maintain the concept that Quarks and Leptons have the same coupling to the W boson (Universality of the Weak Interactions)

51

( , )2

gW a f f

The full expression for the Weak Charged Current : ' ' '

2e

gW e d u s c b t

In case of just two families of Quarks and Leptons :

The study of the relative intensities of weak decays (comparison of different decay odes) allows to determine the Cabibbo Angle: about 130.

When just two families are considered, one can divide all Charge Current Weak Decays of Quarks into Cabibbo “allowed” and Cabibbo “suppressed” decays

52

cduscsude

g

csudeg

CCe

e

sincos2

2''

Theoretical and experimental problems showed up when considering a Weak Interaction theory with Carged Currents alone

1) Theoretical inconsistencies : divergences in the Weak Interaction theory

2) Experimental problems: the discovery of weak processes that cannot be explained by the charged currents

The problem of divergences

We require for a Quantum Field Theory to be renormalizable.

Renormalizability (e.g. the QED case) consists in the possibility of re-absorbing divergent diagrams by redefining bare charges and masses of the theory.

A theory is renormalizable if (at all orders of the perturbative expansions, and possibly at all energies) the amplitudes of the processes can be kept finite by suitably tuning a finite number of parameters (charges and masses).

53

Introducing the Weak Neutral Currents

Let us consider the weak process e ee e

with a cross section given (in the Fermi theory) by :2F

tot

G s

This cross section increases arbitrarily with energy, ultimately violating the Unitary Limit

The W propagator has the effect of mitigating the divergence by introducing a term of this kind in the scattering amplitude :

2

2

1

1W

qM

The Fermi pointlike interaction gets “spread out” in a finite range having a size proportional to 1

WM

This mitigates the divergence problems. However, divergences of the type

still remain, as in the process W W tot s

For these reasons, Glashow, Salam, Weinberg started to develop a theory that would unify Weak and Electromagnetic Interactions. These theory is renormalizable (as demonstrated later by t’Hooft) and predicts the existence of a massive neutral boson and of Weak Neutral Currents

54

The observation of weak neutral current processes

All interactions observed up to 1973 were compatible with just weak processed induced by the W

Weak neautral process are instead mediated by th Z0:

0Z

N

X (Hadrons)

Processes of this kind were observed in 1973 with the Gargamelle bubble chamber, at CERN.

55

The rate of these processes was about one-third of the rate of the related CC events

XN

lending credibility to the idea of a NC process taking place

Gargamelle was a giant particle detector at CERN, designed mostly for the detection of neutrinos. With a diameter of nearly 2 meter and 4.8 meter in length, Gargamelle was a bubble chamber that held nearly 12 cubic

meters of freon (CF3Br). It operated from 1970 to 1978 at the CERN Proton Synchrotron and

Super Proton Synchrotron. Weak neutral currents were predicted in 1973 and confirmed

shortly thereafter, in 1974, in Gargamelle.The name derives from the giantess

Gargamelle in the works of Rabelais; she was Gargantua's mother. (www.wikipedia.org)

A Neutral Current ecent in E815-NuTeV at Fermilab

A muon neutrino is coming from the left-hand side. An hadronic shower with no muons is generated (but a neutrino is present in the final state)

56

57

An event of the type :

e e

f f

Z0

ee can only proceed :

Note that at low energies, Z induced events are dwarfed by e.m. interactions (unless neutrinos are involved)

e e

58

The GIM (Glashow-Iliopolous-Maiani) mechanism

If neutral currents are admitted in the model, one should have them in forms like :

cCe sd

ue

d

s

d

s

CC

CC

C

C

cossin

sincos

Particles known in 1970 :

A charged current is of the type:

CduJ

A neutral current can be formed in the more general way as :

cossinsincos

sincossincos

22

0

sddsssdduu

sd

usdu

d

uduJ

CCCC

CC

ΔS=0 ΔS=1

59

cossinsincos 220 sddsssdduuJ

It seems that the neutral current should have both ΔS=0 and ΔS=1 components .

Experiments however say that when ΔS=1, Neutral currents are suppressed :

50

10)(

)(

CCK

NCK

The GIM proposal : a fourth quark to complete the doublet.

CCcCCC ds

c

s

c

sd

u

d

u

sincossincos

And the new neutral current built in this way, does not have any ΔS=1 terms :

CC

CC s

csc

d

uduJ 0

The charged current now has the form :

C

C

CC

CC

s

dcu

s

dcuJ

cossin

sincos

We introduce the concept of Weak isospin, to classify the states of the fundamental fermion. This in fact can be considered as “spin”. As usual, the transformation between the two states bear a formal analogy with space rotations.

Starting with the electron and its neutrino:

e

e

3 1/ 2T

3 1/ 2T T = ½ is the Weak Isospin for this doublet of fermion states

e

e

3 1/ 2T

3 1/ 2T

An equivalent SU(2) structure is considered for the quark doublets

The anti-electron and anti-neutrino doublet can be obtained from the electron/neutrino one by changing charge, lepton number T3

Let us now form composite states, using the rule of addition of the spin:

e e

e e

Isospin e Hypercharge of fundamental fermions

60

11 ee

01

1

2e e ee

T = 1, T3= +1

T = 1, T3= 0

11 ee T = 1, T3= -1

0

1

2e e ee T = 0, T3= 0

We can now see that the Weak Charged Current :

' ' '

2e

gW e d u s c b t

can be written (for the leptons of the first family) as the SU(2) current:

11

2 2e

g gW e 1

12 2

e

g gW e

The composition gives origin to the usual tripet and singlet states

Rotationally invariant in the T space

61

It is interesting to note that Isospin invariance REQUIRES the existence of01

0 01 22

e e

g gW ee

This term of course correspond to processes like :

0e W e 0e e W 0

e eW 0e eW

and similar processes for other Isospin doublets

ee W

ee W

which implies the existence of processes like :

62

11

2 2e

g gW e 1

12 2

e

g gW e

63

Z

f

f

In a Neutral Current vertex the very same fundamental fermion enters and exits (unaltered)

s

csc

d

udu

s

csc

d

uduJ

CC

CC

0

which generalizes to the case of three families (since the CKM matrix is unitary) While the coupling of quarks and leptons to the W is the universal coupling described before, the coupling of the Z has the form :

W

In the case of a NC process it does not matter if one uses the mass or the charged weak interaction eigenstates. In fact, for the case of two generations one can easily verify that :

f

f Z

5122

wig

e

ν

52

fA

fV

Z ccig

64

u

d Z

52

fA

fV

Z ccig

The coupling of the Z depends on the

specific fermion being considered

All these couplings (and the M,Z mass relaitionship) depends on the very same single parameter, which is part of the Glashow-Salam-Weinberg theory of the Electroweak Interactions.

This parameter is the Weinberg angle θW

The Weinberg angle is a characteristic of nature :

But what are gz and the c coefficients ?

2314.0sin75.28 20 WW

5(1 )2 2

Wig

W vertices

5( )2

f fZV A

igc c

Z vertices

, ,e

Vc Ac

1

2

1

2

, ,e 12sin

2 W 1

2

, ,u c t 21 4sin

2 3 W 1

2

, ,d s b 21 2sin

2 3 W 1

2

65

W

WZ

gg

cos

W

WZ

MM

cos

Electroweak Z parameters are defined by means of the Weinberg angle

66

The concept of Electroweak Unification

11 ee

01

1

2e e ee

11 ee

0

1

2e e ee

Weak Isospin states

11

2 2e

g gW e

11

2 2e

g gW e

0 01 22

e e

g gW ee

We now introduce a field corresponding to the T=0 state as well :

Weak Isospin fields

0'0 2 QgB

A weak Isospin scalarU(1) group symmetry implied here

<Q> is the average charge of the Isospin multiplet (-1/2). A different coupling constant is introduced, called g’

Note: the B0 field averages on the particles of the multiplet

The value of <Q> :

Lepton doublets :

2

1)01(

2

1Q

Quark doublets :

6

1)3/13/2(

2

1Q

We have introduced four spin-1 fields fields dictated by the Isospin Symmetry: W+,W-,W0,B0

These are not the physical fields.

67

0

1

2e e ee

0'0 2 QgB

For instance B0 does not look like any physical field, with its coupling to electrons and neutrinos :

A e ee An electromagnetic field – for instance – should have a coupling like this :

The e.m. interaction should have the form: A e ee

But let us consider the combination 0 ' 0

2 '2

g B g W

g g

0 ' 0 ' ' 0 ' 00 1 0 1

12

2 2

gg B g W g g Q g gg

And the result is 0 ' 0 '

2 '2 2 '2

g B g W ggA ee

g g g g

The Electromagnetic Interaction can be introduced as a linear combination of the T3=0 isospin states if we just assume:

'

2 '2

gge

g g

And calculate it for the first leptonic generation

68

2

1)(' eeeegg eeee

What about the first generation of Quarks?'

u

d

Let us build states with T3 = 0 (one with T = 0 and the other with T = 1)

0 ' '1

1

2uu d d ' '

0

1

2uu d d

Making the calculation as before, we obtain the electromagnetic interaction of Quarks :

0 ' 0 '' ' ' '

2 '2 2 '2

2 1 2 1

3 3 3 3

g B g W ggA uu d d e uu d d

g g g g

(recalling <Q>=1/6).

'd

u

T3= + 1/2

T3= - 1/2

So, this is the Electromagnetic Interaction

2 '201 0 '

1 12

2em

g gQ A A

gg e

69

Let us now consider the combination orthogonal to A

0 ' 0

0

2 '2

g W g BZ

g g

This is a neutral field which is independent from the one of the photon!

0 ' 0

0 0 ' '1 02 '2 2 '2 2 '2

1 12

2

g W g B gZ g g g Q

g g g g g g

2 0 '21 02 '2

1 12

2g g Q

g g

And one can show that : 0 0A Z

70

it is the physical Weak Neutral Current !

11

2

gW 0 2 0 '2

1 02 '2

1 12

2Z g g Q

g g

2 '2

01 0'

12

2em

g gA Q

g g

The physical fields (A and Z) as a function of the Weak Isospin fields

To summarize, we started from:11

2

gW 1

12

gW

0 01

2

gW 0 '

02B g Q

and we made a rotation between the neutral fields (the Weinberg angle):

0 ' 0

0 0

2 '2cos sinW W

g B g WA B W

g g

' 0 0

0 0 0

2 '2sin cosW W

g B g WZ B W

g g

2 '2cos W

g

g g

'

2 '2sin W

g

g g

71

ffQdduuee

dduueedduuee

QAgg

gg

eeee

em

''

''''

001'

2'2

3

1

3

2

)(2

1

6

12

2

1

2

12

2

1

2

1

22

1

Let us elaborate the concept a bit more, using the first generation (Quarks and Leptons)

e

e

e

e

'

u

d

'd

u

T3= + 1/2

T3= + 1/2

T3= - 1/2

T3= - 1/2

And also :

72

The electromagnetic field is given by :

ffTdduueeee 3

''01 2

2

1

2

1

2

1

2

12In addition :

0100

01 222

2

1 emem QQ

We can now write the Z current as a function of the electromagnetic and the Φ10 :

0 23 sin

cos WW

gZ T Q f f

The coupling of th Z to the members of the isospin doublets

which can also be written as :

The free constants of the theory :

', , ,sin We g g Four quantities, subjected to two conditions

'

2 '2

gge

g g

'

2 '2sin W

g

g g

Two independent quantities

73

emWW

em

em

gg

gg

ggg

gg

gggg

Qgggg

Z

sin2

1

cos)(2

1

2)(2

12

2

11

'012'2

2'01

2'01

2

2'2

01

2'01

2

2'202'0

12

2'2

0

emW

W

gZ

20

10 sin

2

1

cos

using :Wg

g tan'

La scoperta del W+ e della Z0 (1983)1979: decisione del CERN di convertire l’SPS in un collisore protoni-antiprotoni.

(e disponibilita’ di un significativo numero di antiprotoni grazie allo “stochastic cooling”)

I possibili processi di produzione:

u d W

u d W

0u u Z

0d d Z

u

d

W

u

u

Z

I possibili modi di decadimento: lW l , l lZ l l

l

l

l

l

Protoni a 270 GeV Antiprotoni a 270 GeV

74

puu

dW

e

e

u

u

d

p

( ) 1p p W e nb

puu

d

e

d

u

u

pZ

e

( ) 0.1p p Z ee nb

75

Ma la sezione d’urto totale e’ dell’ordine dei 40 mb, determinata dalla sezione d’urto di interazione forte !

Gli eventi interessanti vanno estratti dal fondo adronico sfruttandone le loro caratteristiche peculiari.

Il calorimetro di UA1 Il rivelatore UA2

Momento trasverso elevato, bilancio energetico globale

76

Un evento in cui e’ prodotto un W che decade:

eW e

• Un elettrone ad alto momento trasverso

• Uno sbilanciamento in momento trasverso di tutto l’evento consistente con il momento trasverso dell’elettrone e corrispondente al neutrino che non viene osservato.

402W

T

Mp GeV

77

Scoperta della Z0: decadimento 0Z e e

Caratteristiche dell’evento:Un elettrone ad alto pT

Un positrone ad alto pT Nessuna energia trasversa mancante

“LEGO plot” nello spazio ,

E naturalmente anche il decadimento 0Z Caratteristiche dell’evento:

• Due muoni di segno opposto ad alto pT

• Nessuna energia trasversa mancante

Using all data from 1982-3, and combining results from UA1 and UA2:

mW = 82.1 1.7 GeV

mZ0 = 93.0 1.7 GeV

Current values (Particle Data Group 2006):

M(W±) = 80.403 ± 0.029 GeVM(Z0) = 91.1876 ± 0.0021 GeV

78

Summarizing the fundamental ideas of the Electroweak Unification

Idea by GSW (Glashow, Salam Weinberg): let us treat Electromagnetic and Weak Interactions as a part of a unified theory.

Fundamental idea: SU(2) and U(1) symmetries to predict 4 bosons :

0 0, , ,W W W B

Neutral bosons do mix up to generate physical bosons : 0, , ,W W B

0 0,

W

W B

W

0 ,

W

Z

W

0 0 0cos sinW WZ W B 0 0sin cosW WW B

79

Neutral currents are a cure to divergent processes like : e e W W

e

e

W+

W-

e

e

e W+

W-

e

e W+

W-

0Z

The full set of three graphs is now convergent (which is NOT without the Z):

E in particolare le correnti deboli neutre:

• They are mediated by the neutral vector boson Z0

• They do not change flavor (no flavor changing neutral currents)

• The Z0 couplings to fermions are a mixture of electromagnetic and weak couplings, i.e. they are both vector (V) and V-A

• The relative intensity of Z0 couplings depends on a single parameter:

80

A short summary on the Weak Neutral Currents :

2314.0sin

75.282

0

W

W

Electroweak effects in e+e-

At low energy, neutral current effects are not easily visible because of the presence of electromagnetic effects

Z0

e e

f f

e e

f f

However, when we are near to the Z mass, the propagator increases dramatically :

22 2 2

/ Z

Z

i g q q M c

q M c

4

222 2 2 2

( )

( ) (2 ) ( )Z Z Z

e e Z E

e e E M c M c

As an example, if one considers the 2-muons final state :

resonating at the Z0 mass

If 2E<<MZc2

4

2Z

Z

E

M c

(negligible)

If 2E~MZc2

21

10Z Z

Z

M c

(dominating, >200)

81

82

The Electroweak Theory

Proposed in 1961 by Glashow-Salam-Weinberg (GSW)

Treat the Electromagnetic and Weak Interaction as only one interaction

At high-energy: Electroweak Interaction

At low energy: electroweak symmetry is broken into Weak and Electromagnetic

Some problems that needed to be solved :

•Disparity in strength between Weak and Electromagnetic forces•The photon is massless, while W,Z are massive•Electromagnetic interactions are V, while W couplings are V-A

The use of chiral spinors makes it easy to overcome the last difficulty :

)()(2

1 5

pupu L

A particle that has helicity -1 in the ultra-relativistic limit

)()(2

1 5

pvpv R An antiparticle that has helicity +1 in the ultra-

relativistic limit

Left-handed means helicity -1 only in the massles (ultra-relativistic) limit

83

Particles Antiparticles

uuL 2

1 5

uuR 2

1 5 vvR 2

1 5

vvL 2

1 5

2

1 5uuL

2

1 5uuR

2

1 5vvL

2

1 5vvR

( ) ( )ipx ipxu p e v p e A Dirac free particle wavefunction :

particle antiparticle

By using this notation, Weak and Electromagnetic interactions are written in a form that makes it easy to see how they can be unified.

Let us consider the W vertex coupling to a lepton (say an electron) :

e

W

e

We write the leptonic current as (for the electron and electron neutrino case :

ej2

1 5

One can easily show that :

84

Based on the properties of the Gamma matrices :

LLe

ee

ee

eej

2

1

2

1

2

1

2

1

2

1

55

555

The weak vertex is now purely a vector vertex, but left-handed spinor particles are used.

Since in general this expression holds : 5 51 1

2 2 L Ru u u u u

The Electromagnetic current as a function of the chiral spinors :

emL L R Rj l l l l l l

The Electromagnetic Current couples to both left and right fermions.

And in addition :

02

1

2

1

2

1

2

1 5555

eee eeReR

85

Weak Isospin and Hypercharge currents

The weak charged currents can be written as :

e

W

e

LL ej

eW

e

LLej

And in a more compact notation, by introducing the left-handed doublet :

eL

Le

0 1 0 0

0 0 1 0

Introducing the two matrices :

one can write :

LLj

1 21( )2

i

86

LLj

LLL

LL

L

LLL

L

LLL

ee

e

ee

eej

0

00

10

Weak Isospin and Hypercharge currents

For example :

We note that the tau matrices are linear combinations of the Pauli matrices

Introducing the third matrix would give a full Weak Isosping symmetry SU(2)

But which is the relative current ?

10

01

2

1

2

1 3

LLLLL

LLLLL ee

eej

2

1

2

1

2

33

Let us calculate :

87

LLLL eej 2

13 It actually is a neutral current, but not the right neutral current (for instance, it is pure V-A, so it just couples to LH states)

….and recall that the electromagnetic current is : RRLL

em eeeej

In analogy with the Hypercharge, we introduce the Weak Hypercharge : 23

YIQ

Its relevant current being : 32 2 2Y emR L LR L L

j j j e e e e

In terms of the couplings to the vertices, we will introduce a coupling constant g for the weak isospin triplet and a coupling constant g’ for the hypercharge singlet

The description of Weak Interactions (and Electromagnetism) with an SU(2) and U(1) symmetry makes it possible to have a gauge theory according to a symmetry group. This is important for the renormalization of the theory.

The underlying symmetry has a SU(2)L U(1)Y structure

Weak Isospin Weak Hypercharge

And the currents are:

LLe j

L LL Le e 32 j

LLe j

2 R L LR L Le e e e Yj

And this structure can form the electromagnetic current by means of a combination

3 1

2em Y

L RL Rj j j e e e e e e

88

as well as the weak neutral current

The constituents of the Electroweak Standard Model

SU(2)L U(1)Y

e

Le

L

L

'

L

u

d

'

L

c

s

'

L

t

b

3

1

2

2 2

L L

Y em

j

j j j

Three Weak Isospin currents

A Weak Hypercharge current

eR,μR,τR,uR,cR,tR,d’R,s’R,b’R

89

The Electroweak Lagrangian

We have introduced three Weak Isospin and a Hypercharge current :

The symmetry group is related to the following fields :

3

332211

22

2

1

2

1

2

1

jjj

jjj

emY

LLLLLL

90

A little bith of math :LLLLLL jjj

332211

2

1

2

1

2

1

This Weak Isospin currents are relative to Weak Isospin charges : xdjT ii 0

And these follow an SU(2)L algebra : kijk

ji TiTT ,

Summary of quantum numbers (just one generation) :

Lepton T T3 Q Y

νe 1/2 1/2 0 -1

e-L 1/2 -1/2 -1 -1

e-R 0 0 -1 -2

Quark T T3 Q Y

uL ½ ½ 2/3 1/3

dL ½ -1/2 -1/3 1/3

uR 0 0 2/3 4/3

dR 0 0 -1/3 -2/3

91

The vector part :332211

WjWjWjWj

The vector part becomes : 33

2

1

2

1

WjWjWjWj

21 jijj By using : 21

2

1 WiWW

For what concern the neutral part, we have two fields here: W3 and B0

The Electroweak Lagrangian is now being written as :

'

L2

Ygg j W j B

;;;;;;;;;;;;;;;;;;;;;;;;;;;;

which now contains the physical W+ vector boson of the Weak Interactions ♫

They are the symmetry group fields, not the physical fields.

92

The physical fields are generated via a Weinberg rotation.

Zj

gjgAj

gjgBj

gWjg Y

WWY

WWY

sin

2coscos

2sin

2

'3

'3

'33

If we want that the A coupling describes the electromagnetic interaction :

AjjgAjg Y

eem

e

2

13

which happens if : eWW ggg cossin '

WW

WW

WBZ

WWWBA

cossin

sincos3

033

WW

WW

ZAW

ZAB

cossin

sincos3

The neutral part becomes :

inverting the rotation

3 2 0L (sin ) ( sin )cos2

em emW W

W

g gj W j W j j Z g j A

Weak Charged Pure E.M.Weak Neutral and e.m.

93

'3 2 0L (sin ) ( sin )

2 cos2Y em em

W WW

g g gg j W j B j W j W j j Z g j A

;;;;;;;;;;;;;;;;;;;;;;;;;;;;

The full Electroweak Interaction Lagrangian can be written as

There is only one field, the ElectroWeak Field !

In addition, the coupling to the Z is :

Zjj

gZj

gjg em

WWW

eYWW

23'

3 sincossin

sin2

cos

Note: this Lagrangian does not include masses of W,Z and fermion masses. It is just the Electroweak Interaction part.