stephan west-standard model cabridge

7/29/2019 Stephan West-Standard Model Cabridge

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THE STANDARD MODEL

Stephen West

Department of Physics,

Royal Holloway, University of London,

Egham, Surrey,

TW20 0EX.

NExT graduate school lectures 2010-2011.

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Contents

1 Introduction, Conventions and the fermionic content of the SM 5

1.1 Books and papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Lorentz Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 C, P and CP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 Fermionic Content of SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Abelian Gauge Symmetry 16

2.1 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Summary - QED Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Non-Abelian Gauge Theories 21

3.1 Non-Abelian gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Non-Abelian Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Summary: The Lagrangian for a General non-Abelian Gauge Theory . . . . . . . . . . 26

3.5 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Quantum Chromodynamics 30

4.1 Colour and the Spin-Statistics Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Running Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 Quark (and Gluon) Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5 Parameter and Strong CP problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Spontaneous Symmetry Breaking 40

5.1 Massive gauge bosons and Renormalisability . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.4 Goldstone Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.5 The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.6 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.7 Bit more on Renormalisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.8 Spontaneous Symmetry Breaking in a Non-Abelian Gauge Theory . . . . . . . . . . . . 55

5.9 More on renormalisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Spontaneous Breaking of SU(2)L U(1)Y U(1)EM 60

7 The Electroweak Model of Leptons 63

7.1 Left- and right- handed fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.2 Fermion masses - Yukawa couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.3 Weak Interactions of Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.4 Classifying the Free Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8 Electroweak Interactions of Hadrons 71

8.1 One Generation of Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8.2 Quark Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.3 Adding Another Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.4 The GIM Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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8.5 Adding a Third Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8.6 Mass generation - a summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.7 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

9 Neutrinos and the MNS matrix 87

9.1 Dirac Neutrino masses and mixings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

9.2 Majorana Neutrinos and the see-saw mechanism . . . . . . . . . . . . . . . . . . . . . . 87

9.3 MNS matrix and Counting phases for Dirac and Majorana Neutrinos . . . . . . . . . . 90

10 Anomalies 93

10.1 Triangle diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

10.2 Chiral Anomalies and Chiral Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . 100

10.3 Cancellation of anomalies in the Standard Model. . . . . . . . . . . . . . . . . . . . . . 106

11 Chiral Lagrangian approximation 110

11.1 Explicit Chiral Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

11.2 Electroweak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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1 Introduction, Conventions and the fermionic content of the

SM

1.1 Books and papers

F Halzen and A. D. Martin, Quarks and Leptons, Wiley, 1984.

M. E. Peskin and D V Schroeder An Introduction to Quantum Field Theory Addison Wesley1995.

I J R Aitchison and A J G Hey Gauge theories in particle physics 2nd edition Adam Hilger 1989.

J D Bjorken and S D Drell Relativistic Quantum Mechanics McGraw-Hill 1964.

F Mandl and G Shaw Quantum Field Theory Wiley 1984.

P. Ramond, Journeys beyond the standard model, Perseus Books, 1999.

D. Balin and A. Love, Introduction to Gauge Field Theory, 1986.

Several reviews on Spires, e.g. Teubner, Langacker, Signer, Novaes, Evans.

Also useful S. P. Martin, A Supersymmetry primer

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1.2 Overview

The Standard Model is a description of the strong (SU(3)c), weak (SU(2)L) and electromagnetic(U(1)em) interactions in terms of gauge theories. A gauge theory is one that possesses invariance

under a set of local transformations i.e. transformations whose parameters are space-timedependent.

Electromagnetism is a gauge theory, associated with the group U(1)em.In this case the gauge transformations are local complex phase transformations of the fields of

charged particles.

Gauge invariance necessitates the introduction of a massless vector (spin-1) particle, the photon,

whose exchange mediates the electromagnetic interactions.

In the 1950s Yang and Mills considered (as a purely mathematical exercise) extending gauge

invariance to include local non-Abelian (i.e. non-commuting) transformations such as SU(2).

In this case one needs a set of massless vector fields (three in the case of SU(2)), which were

formally called Yang-Mills fields, but are now known as gauge bosons.

In order to apply such gauge theories to the weak interaction, considers particle transforming intoeach other under the weak interactions. Doublets of weak isospin.

uL

dL and

L

eL For SU(2)L we have three gauge bosons interpreted as the W and Z bosons, that mediate

weak interactions in the same way that the photon mediates electromagnetic interactions.

However, weak interactions are known to be short ranged, mediated by very massive vector-bosons,whereas Yang-Mills fields are required to be massless in order to preserve the gauge invariance.

Solved by the Higgs mechanism.

Prescription for breaking the gauge symmetry spontaneously.

Start with a theory that possesses the required gauge invariance, but the physical quantum statesof the theory are not invariant under the gauge transformations.

The breaking of the invariance arises in the quantisation of the theory, whereas the Lagrangianonly contains terms which areinvariant.

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One of the consequences of this is that the gauge bosons acquire a mass and can thus be appliedto weak interactions.

Spontaneous symmetry breaking and the Higgs mechanism has another extremely important con-

sequence. It leads to a renormalisable theory with massive vector bosons.

Theory is renormalisable

Infinities that arise in higher order calculations can be re-absorbed into the parameters of the

Lagrangian (as in the case of QED).

If we had simply broken the gauge invariance explicitly by adding mass terms for the gauge bosons,resulting theory would not be renormalisable

Could not therefore have been used to carry out perturbative calculations.

Consequence of the Higgs mechanism is the existence of a scalar (spin-0) particle - the Higgsboson.

Remaining step is to apply the ideas of gauge theories to the strong interactions.The gauge theory of strong interactions is called Quantum ChromoDynamics (QCD), associated

with the group SU(3)c.

Quarks possess an internal property called colour and the gauge transformations are localtransformations between quarks of different colours.

The gauge bosons of QCD are called gluons and they mediate the strong interactions.

The union of QCD and the electroweak gauge theory, which describes the weak and electromag-netic interactions is known as the Standard Model.

It has eighteen fundamental parameters, most of which are associated with the masses of thegauge bosons, the quarks and leptons, and the Higgs.

Not all independent and, for example, the ratio of the W and Z boson masses are (correctly)predicted by the model.

Since the theory is renormalisable, perturbative calculations can be performed which predict cross-sections and decay rates both for strongly and weakly interacting processes.

Predictions have met with considerable success when confronted with ever stringent experimental

data.

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1.3 Conventions

Natural units: = 1, c = 1

energy, momentum, mass, inverse time, inverse length all have same dimensions.

The metric convention isg = diag(1, 1, 1, 1) (1.1)

Space time coords and derivatives

x = (x0, xi) = (t, x); x = gx = (t, x),

=

t,

; =

t,

.

Lorentz invariant inner product of two vectors:

x2 = gxx = xx

= t2 x.x

1.4 Lorentz Invariance

An arbitrary Lorentz transformation can be written as

x

x = x

Lorentz transforms defined by invariance of x2 under Lorentz Group

gxx = g

x

x, x

g = g()().

Klein-Gordon field, (x), transforms as

(x) (x) = (1x)

Transformation should leave KG equation invariant, clearly 12

m22(x) is invariant, what about

the derivatives

(x) (1x) = x

(1x) =(1)x

dx

(1)x(1x)

= (1)

(1)x(1x) = (1)()(

1x).

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Transformation of the kinetic term in KG Lagrangian

()2 g((x))((x)) = g[(1)()][(1)()](1x)

= g()()(1x) = ()2(1x)

Thus the whole Lagrangian transforms as a scalar

L L(1x)

and with the action, S, being an integral over all space of the Lagrangian it is clear that the action

is invariant.

Now Dirac Fermions

Need a representation for the Lorentz group for Dirac spinors

General 4-d Lorentz transformation

L = i(x x)

The Lorentz algebra is given by

[L, L] = i(gL gL gL + gL)

We can write down a 4-d matrix representation of the Lorentz generators (as we have Dirac Spinorsin mind) defined as

S =i

4[, ]

where the matrices are written in the Weyl or Chiral representation as

= 0

0 where = (1, ) and = (1,

)

where

1 =

0 1

1 0

2 =

0 ii 0

3 =

1 0

0 1

which satisfy

ij = ij + iijkk.

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We can then write the spinor form of the Lorentz transformation as:

12

= exp

i

2S

.

such that a Dirac fermion, transforms as

(x) 12

(1x).

In addition it can be shown that the matrices transform as

112

12

= (1.2)

Using this we can show that the Dirac equation is Lorentz invariant:

[i m] i(1) m

1

2

(1x)

= 12

112

i(1) m 12

(1x)

= 12

i11

2

12

(1) m

(1x) using 1.2

= 12

i

(1) m

(1x)

= 12

[i m] (1x)

Now we need to find how to write the Lagrangian for the Dirac theory, this will involve Diracbilinears.

One guess however this transforms as 1

2

12

but 12

= 112

and so is not Lorentz invariant.

Better guess is 0 under an infinitesimal Lorentz transformation

(1 + i2

(S))0

For = 0 and = 0, associated with rotations, (S

) = S

and commutes with 0

For = 0 or = 0, associated with boosts, (S) = S and anti-commutes with 0

this means that (1 + i

2(S

))0

112

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Consequently is a Lorentz invariant and the Dirac Lagrangian isLDirac = (i m).

Weyl and Majoran Spinors

Much of the standard model (and MSSM) is written in terms of Weyl spinors. Weyl spinners aretwo-component complex anticommuting objects. We can write a Dirac spinner in terms of two

Weyl spinners and () with two distinct indices = 1, 2 and = 1, 2:

D =

and

D = D 0 11 0 =

Undotted indices are used for the first two components of a Dirac spinor

Dotted indices are used for the last two components of a Dirac spinor

is called a left-handed Weyl spinor and is a right-handed Weyl Spinor

We can show this by using:

PL =1

2

(1

5) and PR =

1

2

(1 + 5) with 5 = 1 00 1

We have:PLD =

0

, PRD =

0

Hermitian conjugate of any left-handed Weyl spinner is a right-handed Weyl Spinner:

() = ()and vice versa

() =

Raise and lower indices using the antisymmetric symbol12 = 21 = 21 = 12 = 1, 11 = 22 = 11 = 22 = 0

with

= , = ,

=

, =

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with

= = ,

= =

The Dirac Lagrangian can then be written as:

L = D(i m)D = 0 ii 0 m 00 m = i

+ i() m( + )

= i + i m( + )where the last step involves integration by parts and the identity = etc.

A four component Majorana spinor can be obtained from the Dirac spinor by imposing thecondition = such that,

D =

, D =

The Lagrangian for the Majorana spinor

LM = i2

MM 1

2MMM

= i 12

M

+

To efficiently move between the Weyl and Dirac notation we can use the chiral projection operators,

PL,R e.g.

iPLj = ij and iPRj = i

j

and

iPLj =

i

j and iPRj = i

j = ji

1.5 C, P and CP

How do these fields transform under C, P and CP

Charge conjugation and Parity are defined byC : (t, x) C(t, x) = CT(t, x)P : (t, x) P(t, x) = P (t, x)

with

C = i20 and P = 0

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Both C and P change the chirality: Consider purely LH state

PL(t, x) =

(t, x)

0

now act the parity operator on it

(PL(t, x))P = 0

(t, x)

0

=

0

(t, x)

we can see the final object is purely RH.

Now for C:

(PL(t, x))C = i20

(t, x)

0

0T

= i200

(t, x) = i200

(t, x) =

0 i2

i2 0

0 1

1 0

0

(t, x)

=

0

i2(t, x)

the final result being RH.

Charge conjugation and parity relate the LH and RH components. We will see later that LH andRH states transform differently under the Weak force hence neither C nor P are good symmetries

when describing the weak force

Combined application of both C and P leave a LH (RH) state LH (RH). Acting parity operatoron the above we have

(PL(t, x))CP =

(PL(t, x))

CP

=

0

i2(t, x)

P= 0

0

i2(t, x)

=

i2(t, x)

0

which is still LH.

CP is a better symmetry to use, but see later for CP-violation

Majorana condition:C =

we can see this by taking the complex conjugate explicitly

CM = i20TM = i20

=

i2i2

=

= M

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1.6 Fermionic Content of SM

The fermionic content (in terms of Weyl spinors) of the SM is as follows:

Qi = ud , cs , tb ,ui = u, c, t,

di = d, s, b,

Li =

e

e

,

,

,

ei = e, , .

bars are just labels not conjugates. Left handed states are written in doublets as they transform as

doublets under the SM gauge group SU(2), while RH states transform as singlets...more later.

To form Dirac spinors: e

e

Perhaps more used to writing things in terms of eL and eR, with identification e = eL and e = eRsuch that

e

e

=

(eL)

(eR)

A more transparent notation is thus:

QLi =

uL

dL

,

cL

sL

,

tL

bL

,

uRi = uR, cR, tR,

dRi = dR, sR, bR,

LLi = Le

eL , L

L , L

L ,eRi = eR, R, R.

Example

Dirac mass term for electronsmD

2

(eR)

(eL) + (eL)(eR)

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using

e =

(eL)

(eR)

e =

(eR)

(eL)

we can write

mD2

(ePLe + ePRe) = mD2

ee

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2 Abelian Gauge Symmetry

2.1 Gauge Transformations

Consider the Lagrangian density for a free Dirac field :L = (i m)

This Lagrangian density is invariant under the phase transformation of the fermion field and itsconjugate

ei ei

set of phases belong to the group U(1) and is abelian as

ei1ei2 = ei2ei1

Infinitesimal form of this transformation

ei 1 + i + O(2)

under which the wavefunction changes by

= i

and the conjugate

= iunder this lagrangian is invariant up to order 2

Now make the parameter depend on space-time

(x) = i(x)(x) and (x) = i(x)(x)

Such local (space-time) dependent transformations are called gauge transformations.

The Lagrangian is now no longer invariant due to the derivative term giving the change in thelagrangian of

(i m) (i m) = ( i (x)) (i m) ( + i (x) )= (i m) + ((x))

L = (x)((x))(x)

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However, if we modify the Lagrangian by replacing D = + ieA

(i eA m) = (i m) eA

and demand that A transforms according to:

A A = A 1

e (x)

when

= ( + i (x) ),then we will have restored local invariance!

In other words, this is an invariance which only exists if the particles are not free! There is aninteraction term.

Interpretation is: e is the electric charge of the fermion field and A is the photon field.

D = + ieA is the covariant derivative.

It is easy to find the way the object D:

(D) = D

=

+ ieA

ei(x) = ei(x) + i ( (x)) e

i(x)+ ie

A 1

e (x)

ei(x)

D = ei(x) ( + ieA) = ei(x)D D = ei(x)D,

so D transforms like . This means that D is trivially invariant!

To have a proper QFT we need Kinetic terms for the photon fields, must be gauge invariant.Define field strength,

F = A A

Easy to see this is invariant but must be Lorentz invariant so we add to the Lagrangian

14

FF

with the numerical factor included so that the equations of motion matches Maxwells equations.

We can express this as

F = ie

[D, D] = ie

[, ] + [, A] + [A, ] + ie[A, A] = A A

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Final Lagrangian density is

LQED = 14

FF + (i eA m) = 1

4FF

+ (iD m)

with D = + ieA

NOTE: no mass term for the photon such as M2AA. If we add this and make a U(1) transfor-mation

M2AA M2AA = M2AA

2M2

eA

L = 2M2

eA.

Thus gauge invariance requires a massless gauge boson.

2.2 Gauge Fixing

In order to find the Feynman rule for the photon propagator (and to quantise the electromagneticfield) we look for the parts of the action which are quadratic in the photon field.

E.g. for some field (x) we take fourier transforms of the fields and identify the terms

S =

d4p(p)O(p)(p),

with the propagator for being

iO1(p).

In the case of QED let us derive from teh action in x-space, we have

S =

d4x

1

4(F)

2

=

d4x (A A)(A A)

=

d4x A(

2g )A

Now use Fourier transform for A as

A(x) = d4p

(2)4A(p) e

ipx

S =

d4x

d4k

(2)4d4p

(2)4A(p) e

ipx(2g ) A(k) eikx

=

d4x

d4k

(2)4d4p

(2)4A(p) e

i(p+k)x(k2g kk)A(k)

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=

d4k

(2)4d4p

(2)4A(p)

4(p + k)(k2g+ kk)A(k)

=

d4p

(2)4A(p) (p2g+ pp)A(p)

However, this is not invertible, i.e.(p2g+ pp)DF (p) = i

has no solution as (p2g+ pp) is singular. Cannot find the propagator.

Issue really comes down to the fact that we have gauge invariance. Troublesome modes are thosefor which

A =1

e(x)

that is those which are gauge equivalent to

A(x) = 0.

Result is that the functional integral is badly defined...see QFT lectures

Another interpretation is due to the fact that A has four real components, introduced to maintaingauge symmetry. However the physical photon has two polarisation states.

This difficulty can be resolved by fixing the gauge (breaking the gauge symmetry) in the La-grangian in such a way as to maintain the gauge symmetry in observables.

Can solve this using method presented by Faddeev and Popov - needs functional integrals to doproperly but can be translated to adding to teh Lagrangian density a gauge fixing term

12

(.A)2

1 leading to

S =

d4p

(2)4A(p) (p2g+

1 pp)A(p)

(2.1)

using the relation p2g

1 pp

(g pp) = p2gwe can now write the photon propagator as

i

g ppp2

1

p2

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Special choice of = 0 is known as the Feynman gauge. In this gauge the propagator is particularlysimple.

We have to fix the gauge in order to be able to perform a calculation. Once we have computed a

physical measurable quantity, the dependence on the gauge cancels.

Taking = 0 is a one choice, there are other special choices but the choice does not affect theresult of a calculation of a measurable quantity.

2.3 Summary - QED Lagrangian

Theory contains fermions with charge e, and a massless vector boson, the photon.

In Feynman Gauge

LQED = 14

FF+ (iD m) 1

2(.A)2

with D = + ieA

Invariant under the transformations

A A = A 1

e (x)

(x) (x) = ei(x)(x) (x) (x)ei(x)(x)

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3 Non-Abelian Gauge Theories

3.1 Non-Abelian gauge transformations

Extend gauge concept so that gauge bosons have self- interactions as observed for the gluonsof QCD, and the W, Z of the electroweak sector.

However, the gauge bosons will still be massless. (We will see how to give the W and Z theirobserved masses in the Higgs chapter.)

Non-Abelian means different elements of the group do not commute with each other.

Transformations act between a degenerate space of states (for example between u-d quarks states

where both have the same masses in isospin)

Consider n free fermion fields , arranged in a multiplet

=

1

2

.

.

n

The Lagrangian density for a is

L = i (i m) i, (3.1)

where the index i is summed from 1 to n. Eq.(3.1) is therefore shorthand for

L = 1 (i m) 1 + 2 (i m) 2 + .... (3.2)

The Lagrangian density is invariant under the global complex transformations in field ( i) space:

= U = U

where U is an n n unitary matrix

UU = 1, det[U] = 1

These matrices will form the group SU(n).

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In order to specify an SU(n) matrix we need n2 1 real parameters.

An arbitrary SU(n) matrix can be written as

U = ein21a=1

aTa

ei

aTa

a are real parameters, and the Ta re the generators of the SU(n) group

Generators are hermitian and traceless (can prove these facts by unitarity condition and determi-nant condition respectively

For SU(n) there are n2 1 generators Ta which are normalised using the convention

tr(TaTb) =1

2ab

This means that two such transformations do not commute.ei

a1Ta

eib2Tb

=

eib2Tb

eia1Ta

.

Specifically the Lie algebra of the SU(n) group is written in terms of commutators:

[Ta, Tb] = ifabcT

where fabc are called structure functions of the SU(n) group and are totally antisymmetric.

3.2 Non-Abelian Gauge Fields

Now consider local SU(n) transformations, as with abelian symmetries the lagrangian is no longerinvariant

L L = U(i m)U = (i m) + iU(U)

In analogy with the abelian case we introduce the covariant derivative

D = + igA

where A = TaA

Covariant derivative contains n2 1 (spin 1) gauge bosons, Aa, one for each generator.

The quantity A transforms as

A A = UAU +i

g(U)U

(3.3)

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sometimes useful to use infinitesimal form which can be found by the following, using U = eiaTa,

A = UAU +

i

g(U)U

= eibTbAaT

aeibTb +

i

g(e

iaTa)eiaTa

(1

ibTb)Aa

Ta(1 + ibTb) +

i

g(

(1

iaTa))(1 + iaTa)

AaTa ibTbAaTa + iAaTabTb +1

g(

a)Ta

AaTa + i[Ta, Tb]bAa +1

g(

a)Ta

AaTa fabcTcbAa +1

g(

a)Ta

It is easy to check how D transforms:

D (D) =

+ ig(UAU +

i

g(U)U

U (3.4)

= U + igUA (U) + (U)= U + igUA

= UD

Trivial to then check invariance of the lagrangian

The kinetic term for the gauge bosons is again constructed from the field strengths Fa which aredefined from the commutator of two covariant derivatives:

F = ig

[D, D] . (3.5)

where the matrix F is given by

F = TaFa,

F = i

g [D, D] = ( + igA)( + igA) ( + igA)( + igA)= (A) (A) + ig(AA AA)= (A

a)T

a (Aa)Ta + igAbAc(TbTc TcTb)= [(A

a) (Aa)]Ta + igAbAc[Tb, Tc]

=

[(Aa) (Aa)] gfabcAbAc

Ta

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This gives usFa = A

a Aa g fabc AbAc (3.6)

From (3.3) we can deduce how F transforms

F ig

([D, D]) = i

gU [D, D] U

Easy to show that the infinitesimal version isF (F) = FaTa + fabcFbcTa

The gauge invariant term giving the gauge boson kinetic terms is:1

2FF

= 12

FaFbTaTb = 1

4FaF

a

In contrast to the Abelian case, this term contains quadratic pieces in the derivatives of the gaugeboson fields, but also interaction terms:

14

FaFa = [(A

a Aa) g fabc AbAc][(Aa Aa) g fade AdAe] (3.7)

= (Aa Aa)(Aa Aa) g fabc AbAc(Aa Aa)

g fade AdAe(Aa Aa) + g2 fabcfade AbAcAdAe

For non-Abelian gauge theories the gauge bosons interact with each other via both three-pointand four-point interaction terms.

Once again, a mass term for the gauge bosons is forbidden, since a term proportional to AaAais not gauge invariant.

3.3 Gauge Fixing

As with the abelian symmetries, we need to add a gauge fixing term in order to be able to derivea propagator for the gauge bosons.

In the Feynman gauge this means adding the term1

2(Aa)

2

to the Lagrangian density and the propagator (in momentum space) becomes

i ab gp2

.

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One unfortunate complication only needed for the purpose of performing higher loop calculationswith non-Abelian gauge theories.

A consequence of gauge-fixing is that we need extra loop diagrams in higher order which are

mathematically equivalent to interacting scalar particles known as a Faddeev-Popov ghost foreach gauge field.

These are not physical scalar particles and cannot appear as external legs on diagrams

1. They only occur inside loops.

2. They behave like fermions even though they are scalars (spin zero). This means that we need to

count a minus sign for each loop of Faddeev-Popov ghosts in any Feynman diagram.

For example, the Feynman diagrams which contribute to the one-loop corrections to the gauge boson

propagator are

+ - -

(a) (b) (c) (d)

(a) - three-point interaction between the gauge bosons

(b)- four-point interaction between the gauge bosons

(c) involves a loop of fermions,

(d) extra diagram involving the Faddeev-Popov ghosts.

Both (c) and (d) have a minus sign in front of them because both the fermions and Faddeev-Popovghosts obey Fermi statistics.

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Representation dr nr C(r)

Fundamental N 1/2 (N2 1)/2NAdjoint N2 1 N N

Table 1: In this Table we list various properties for the SU(N) symmetry group.

The Lagrangian density for a gauge theory with this group, with a fermion multiplet i is given (in

Feynman gauge) by

L = 14

FaFa+ i (D mI) 1

2(Aa)

2 + LFP (3.14)

where

Fa = Aa Aa g fabcAbAc, (3.15)

D = I + i g Ta

Aa (3.16)

and

LFP = aa + g facbaAc(b)

Under an infinitesimal gauge transformation, the N gauge bosons, Aa change by an amount that

contains a term which is not linear in Aa:

Aa(x) = fabcAb(x)

c(x) 1g

a(x), (3.17)

whereas the field strengths Fa transform by a change

Fa(x) = fabc Fb(x)

c. (3.18)

We say that the Gauge bosons transform as the adjoint representation of the group (which hasas many components as there are generators).

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3.5 Feynman Rules

The Feynman rules for such a gauge theory are given by: Propagators:

Gauge Boson

i abg/p2pa

b

Fermion

i ij(p + m)/(p

2 m2)pi j

Faddeev-Popov ghost

i ab/p2pa b

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Vertices:

(all momenta are flowing into the vertex).

a

p1

c

p3

b

p2g fabc

g(p1 p2) + g (p2 p3) + g (p3 p1)

i g2feabfecd (gg gg)i g2feacfebd (gg gg)i g2feadfebc (gg gg)

a

b

d c

a

j i

i g (Ta)ij

a

c bq

g fabc q

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4 Quantum Chromodynamics

4.1 Colour and the Spin-Statistics Problem

Motivation for colour:++

consists of uuu and has spin 3/2 and is in the l = 0 angular momentum state

wavefunction seems to be symmetric under exchange of quarks - exclusion principle says this

cannot be so

Fermions should have a wavefunction which is antisymmetricunder the interchange of the quantumnumbers of any two fermions.

Solved by allowing quarks to also carry one of three colours

As well as the flavour index, (f) a quark field carries a colour index, i, so we write a quark fieldas qif, i = 1 3.

It was further assumed that the strong interactions are invariant under colour SU(3) transforma-tions.

The quarks transform as a triplet representation of the group SU(3) which has eight generators.

Furthermore it was assumed that all the observed hadrons are singlets of this new SU(3) group.

Spin statistics problem is now solved

Baryon consists of three quarks and a colour singlet state

Colour part of the wavefunction is

|B > = ijk

|qif1q

jf2

qkf3 >,

where f1, f2, f3 are the flavours of the three quarks that make up the baryon B and i, j, k are

the colours.

Tensor ijk is totally antisymmetric under the interchange of any two indices, so that if the part ofthe wavefunction of the baryon that does notdepend on colour is symmetric, the total wavefunction

(including the colour part) is antisymmetric, as required by the spin-statistics theorem.

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4.2 QCD

QCD is where invariance under colour SU(3) transformations is promoted to an invariance underlocal SU(3) (gauge) transformations.

Quarks transform as triplets and anti-quarks transform as anti-triplets under SU(3)C a quark transforms as

q q = e i2 . qwith eight finite real parameters ( = (1, 2, ......, 8)) and stands for eight Hermitian 3 3matrices (the generators of SU(3)), = (1,....,8),

These are the Gell-Mann matrices. These eight Hermitian3 3 matrices represent the generators

of SU(3) in the fundamental representation.

1 =

0 1 01 0 00 0 0

2 = 0 i 0i 0 0

0 0 0

3 = 1 0 00 1 0

0 0 0

4 =

0 0 1

0 0 0

1 0 0

5 =

0 0 i0 0 0

i 0 0

6 =

0 0 0

0 0 1

0 1 0

7 =

0 0 00 0 i0 i 0

8 =

13

0 0

0 13

0

0 0 23

(4.1)

The eight gauge bosons which have to be introduced in order to preserve this invariance are theeight gluons.

These are taken to be the carriers which mediate the strong interactions in exactly the same waythat photons are the carriers which mediate the electromagnetic interactions.

The Feynman rules for QCD are therefore simply the Feynman rules listed in the previous lecture,with, g, taken to be the strong coupling, gs, (more about this later)

The generators Ta = 12

a taken to be the eight generators above with algebra

[Ta, Tb] = ifabcTc

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with fabc, a, b, c, = 1 8 the structure constants of SU(3) with values

f123 = 1

f147 = f156 = f246 = f257 = f345 = f367 = 12

f458 = f678 =3

2

To maintain gauge invariance we again exchange D = igs2 Aaa

Thus we now have a Quantum Field Theory which can be used to describe the strong interactions.

4.3 Running Coupling

The coupling for the strong interactions is the QCD gauge coupling, gs. We usually work in termsof s defined as

s =g2s4

.

Since the interactions are strong, we would expect s to be too large to perform reliable calcula-tions in perturbation theory.

On the other hand the Feynman rules are only useful within the context of perturbation theory.

Difficulty resolved as coupling constants are not constant at all.

The electromagnetic fine structure constant, , only has the value 1/137 at energies which arenot large compared with the electron mass.

At higher energies it is larger than this.

For example, at LEP energies it takes a value closer to 1/128.

On the other hand, it turns out that for non-Abelian gauge theories the coupling decreasesas theenergy increases.

To see how in QCD, we note that when we perform higher order perturbative calculations thereare loop diagrams which dress the couplings.

For example, the one-loop diagrams which dress the coupling between a quark and a gluon are:

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where

= + - -

are the diagrams needed to calculate the one-loop corrections to the gluon propagator.

These diagrams contain UV divergences and need to be renormalised by subtracting at somerenormalisation scale .

This scale then appears inside a logarithm for the renormalised quantities.

This means that if the square-momenta of all the external particles coming into the vertex areof order Q2, where Q , then the above diagrams give rise to a correction which contains alogarithm of the ratio Q2/2:

2s 0 ln Q2/2 . This correction is interpreted as the correction to the effective QCD coupling, s(Q2), at momen-

tum scale Q.

A calculation shows that the effective coupling obeys the differential equation s(Q

2)

ln(Q2)=

s(Q

2)

(4.2)

where has a perturbative expansion

() = 0 2 + O(3) + (4.3)

where 0 is calculated to be

0 =1

4

11

3Cadj 2

3

i

Csi 1

6

a

Cra

, (4.4)

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where where considering contributions from fields transforming as the adjoint, scalars transforming

with representation si and fermions transforming with representation ra. The Cs are the dynkin

indices for the particular representations, r, defined by

T r[T

a

r T

b

r ] = Cr

ab

(4.5)

There are no scalars charged under QCD

For the adjoint and fermions transforming under the fundamental and anti-fundamental we have

T r[TaadjTbadj ] = Nc

ab, T r[TaFTbF] =

1

2ab, T r[TaFT

bF] =

1

2ab (4.6)

So we get with Nc = 3

0 = 14 113 3 23 nf12 + nf12 = [33 2nf]12 nf is the number of active flavours, i.e. the number of flavours whose mass threshold is below the

momentum scale, Q.

Solving this do it from the differential form we find

s(Q2) =

s(2)

1 + 0s(2) lnQ2

2

Now we need a boundary value.

This is taken to be the measured value of the coupling at the Zboson mass (= 91 GeV), whichis measured to be

s(M2Z) = 0.118 0.002 (4.7)

This is one of the free parameters of the standard model and is what we use to replace and

(2).

Note that 0 is positive, which means that the effective coupling decreasesas the momentum scaleis increased.

The running of s(Q2) is shown in the figure.

We can see that for momentum scales above about 2 GeV the coupling is less than 0.3 so thatone can hope to carry out reliable perturbative calculations for QCD processes with energy scales

larger than this.

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Gauge invariance requires that the gauge coupling for the interaction between gluons must beexactly the same as the gauge coupling for the interaction between quarks and gluons.

The function could therefore have been calculated from the higher order corrections to thethree-gluon (or four-gluon) vertex

must yield the same result, despite the fact that it is calculated from a completely different setof diagrams.

4.4 Quark (and Gluon) Confinement

This argument can be inverted to provide an answer to the question why have we never seenquarks or gluons in a laboratory ? .

Asymptotic freedom, which tells us that the effective coupling between quarks becomesweaker as we go to short distances (this is equivalent to going to high energies) implies,

conversely, that effective couplings grow as we go to large distances.

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Therefore, the complicated system of gluon exchanges, which leads to the binding of quarks (andantiquarks) inside hadrons, leads to a stronger and stronger binding as we attempt to pull the

quarks apart.

This means that we can never isolate a quark (or a gluon) at large distances since we require more

and more energy to overcome the binding as the distance between the quarks grows.

Only free particles which can be observed at macroscopic distances from each other are coloursinglets.

Mechanism is known as quark confinement.

Details are not fully understood.At the level of non-perturbative field theory, lattice calculations have confirmed that the binding

energy grows as the distance between quarks increases.

Thus we have two different pictures of the world.

Short distances, or large energies, quarks and gluons are the appropriate degrees of freedomto do calculations with.

They are what we consider interacting with each other.

e.g. We can perform calculations of the scattering cross-sections between quarks and gluons (called

the hard cross-section)

This is because running coupling is sufficiently small so that we can rely on perturbation theory.

However, on the other hand in experiments, we need to take into account the fact that thesequarks and gluons bind into colour singlet hadrons

only these colour singlet states that are observed.

The mechanism for such binding is beyond the scope of perturbation theory and is not understoodin detail.

Monte Carlo programs have been developed which simulate this binding in such a way that theresults of the short-distance perturbative calculations at the level of quarks and gluons can beconfronted with experiment in a successful way.

Thus, for example, to calculate the cross-section for electron-positron annihilation into three jets(at high energies)

First calculate, in perturbation theory, the process for electron plus positron to annihilate into a

virtual photon which then decays into a quark, and antiquark and a gluon.

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The two Feynman diagrams for this process are:

e+

e

q

q

g e+

e

q

q

g

However, before we can compare with experimental data we need to perform a convolution of thiscalculated cross-section with a Monte Carlo.

This simulates the way in which the final state partons (quarks and gluons) bind with other quarks

and gluons to produce observed hadrons.

It is only after such a convolution has been performed that one can get a reliable comparison of

the calculated cross-section with data.

Likewise, if we want to calculate cross-sections for initial state hadrons we need to account for theprobability of finding a particular quark or gluon inside an initial hadron with a given fraction of

the initial hadrons momentum (these are called parton distribution functions).

More later on the structure of hadrons.

4.5 Parameter and Strong CP problem.

There is one more gauge invariant term that can be added to the Lagrangian density for QCD.This term is

L = s8

FaFa =

s8

FaFa,

where is the totally antisymmetric tensor (in four dimensions) and F is the dual field strength.

Such a term arises when one considers instantons (which are beyond the scope of these lectures.)

This term violates CP. In QED we would have

FF = E B,

and for QCD we have a similar expression except that Ea and Ba carry a colour index - they are

known as the chromoelectric and chromomagnetic fields.

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Under charge conjugation both the electric and magnetic filed change sign, but under parity theelectric field, which is a proper vector, changes sign, whereas the magnetic field, which is a polar

vector, does not change sign.

Thus we see that the term E B is odd under CP. For this reason, the parameter in front of this term must be exceedingly small in order not to

give rise to strong interaction contributions to CP violating quantities such as the electric dipole

moment of the neutron.

The current experimental limits on this dipole moment tell us that < 109 and it is probablyzero.

Nevertheless, strictly speaking is a free parameter of QCD, and is often considered to be the

nineteenth free parameter of the Standard Model.

Of course we simply could set = 0 (or a very small number), however term is regeneratedthrough loops

Even if we could set it to zero we want to know why.

The fact that we do not know why this term is absent (or so small) is the strong CP problem.

Several possible solutions to the strong CP problem that offer explanations as to why this termis absent (or small).

One possible solution: add additional symmetry, leading to the postulation of a new, hypothetical,weakly interacting particle, called the (Peccei-Quinn) axion.

Unfortunately none of these solutions have been confirmed yet and the problem is still unresolved.

Another question: why no problem in QED?

This term can be written (in QED and QCD) as a total divergence, so it seems that it can beeliminated from the Lagrangian altogether.

However, in QCD (but not in QED) there are non-perturbative effects from the non-trivial topo-logical structure of the vacuum (somewhat related to so called instantons you probably have heard

about) which prevent us from neglecting the -term

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4.6 Summary

Quarks transform as a triplet representation of colour SU(3) (each quark can have one of threecolours.)

The requirement that the observed hadrons must be singlets of colour SU(3) solves the spin-statistics problem for baryons. The wavefunction for a colour singlet state of three quarks has an

antisymmetric colour component.

QCD is the SU(3) gauge theory in which the symmetry under colour SU(3) is taken to be local.

The eight gauge bosons of QCD are the gluons which are the carriers that mediate the stronginteractions. They are massless.

The coupling of quarks to gluons (and gluons to each other) decreases as the energy scale increases.

Therefore, at high energies one can perform reliable perturbative calculations for strong interacting

processes.

As the distance between quarks increases the binding increases, such that it is impossible to isolateindividual quarks or gluons.

The only observable particles are colour singlet hadrons.

Perturbative calculations performed at the quark and gluon level must be modified by account-ing for the recombination of final state quarks and gluons into observed hadrons as well as the

probability of finding these quarks and gluons inside the initial state hadrons.

QCD admits a gauge invariant strong CP violating term with a coefficient . This parameteris known to be very small from limits on CP violating phenomena such as the electric dipole

moment of the neutron.

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5 Spontaneous Symmetry Breaking

We have seen that in an unbroken gauge theory the gauge bosons must be massless.

The only observed massless spin-1 particles are photons. In the case of QCD, the gluons are also

massless, but they cannot be observed because of confinement.

To extend the ideas of describing interactions by a gauge theory to the weak interactions, thesymmetry must somehow be broken since the carriers of weak interactions (W- and Z-bosons) are

massive (weak interactions are very short range).

We could simply break the symmetry by hand by adding a mass term for the gauge bosons, whichwe know violates the gauge symmetry.

However, this would destroy renormalizability of our theory.

Renormalizable theories are preferred because they are more predictive.

5.1 Massive gauge bosons and Renormalisability

Show in a little more detail how explicit breaking mean non-renormalisability

Higher order (loop) corrections generate ultraviolet divergences. In a renormalisable theory, these divergences can be absorbed into the parameters of the theory

and in this way can be hidden.

As we go to higher orders we need to absorb more and more terms into these parameters, butthere are only as many divergent quantities as there are parameters.

E.g. in the QED Lagrangian we have a fermion field, the gauge boson field and interactions whosestrength is controlled by e and m

All divergences of diagrams can be absorbed into these quantities

Once measured all other predictions can be written in terms of these parameters

There are conditions on allowed terms in a renormalisable theory

Furthermore all the propagators have to decrease like 1/p2 as the momentum p .

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If conditions are not fulfilled then the theory generates more and more divergent terms as onecalculates to higher orders and it is not possible to absorb these divergences into the parameters

of the theory.

Such theories are said to be non-renormalisable.

So how does adding a an explicit mass term M2AA ruin renormalisabililty?

This term modifies the propagator,1

2Ag(p2 M2) + ppA

inverting this we havei

p2 M2 g+ pp

M2 not that this propagator has a much worse UV behaviour, it goes to a constant as p .compared to

i

g pp

p2

1

p2

With the explicit mass term the theory has more UV divergences

It is the gauge symmetry that ensures renormalisabililty

There is a far more elegant way of doing this which is called spontaneous symmetry breaking.

In this scenario, the Lagrangian maintains its symmetry under a set of local gauge transformations.

On the other hand, the lowest energy state, which we interpret as the vacuum state, is not asinglet of the gauge symmetry.

There is an infinite number of states each with the same ground-state energy and Nature choosesone of these states as the true vacuum.

5.2 Spontaneous Symmetry Breaking

Spontaneous symmetry breaking is a phenomenon that is not restricted to gauge symmetries.

In order to illustrate the idea of spontaneous symmetry breaking:Consider a pen that is completely symmetric with respect to rotations around its axis.

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If we balance this pen on its tip on a table, and start to press on it with a force precisely alongthe axis we have a perfectly symmetric situation.

This corresponds to a Lagrangian which is symmetric (under rotations around the axis of the pen

in this case).

However, if we increase the force, at some point the pen will bend (and eventually break).

The question then is in which direction will it bend. Of course we do not know, since all directionsare equal.

But the pen will pick one and by doing so it will break the rotational symmetry. This is sponta-neous symmetry breaking.

Better example can be given by looking at a point mass in a potential

V(r) = 2r.r + (r.r)2

Potential is symmetric under rotations and we assume > 0 (otherwise there would be no stableground state).

For 2 > 0 potential has a minimum at r = 0, thus the point mass will simply fall to this point.

The situation is more interesting if 2

< 0.

For two dimensions the potential is a mexican hat potential.

If the point mass sits at r = 0 the system is not in the ground state and is not stable but thesituation is completely symmetric.

In order to reach the ground state, the symmetry has to be broken, i.e. if the point mass wantsto roll down, it has to decide in which direction.

Any direction is equally good, but one has to be picked. This is exactly what spontaneous symmetry breaking means.

The Lagrangian (here the potential) is symmetric (here under rotations around the z-axis), butthe ground state (here the position of the point mass once it rolled down) is not.

Mathematically: Ground state is |0.

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A spontaneously broken gauge theory is a theory whose Lagrangian is invariant under gaugetransformations but ground state is not invariant under gauge transformations.

eiaTa|0 = |0

which means

Ta|0 = |0 for some a

Thus the theory is spontaneously broken if at least one generator does not annihilate the vacuum

5.3 Spontaneous Symmetry Breaking

We start by considering a complex scalar field theory with a mass term and a quartic self-

interaction.

The Lagrangian density for such a theory may be written

L = V(), (5.1)

where the potential V(), is given by

V() = 2 + ||2 . (5.2)

This Lagrangian is invariant under global U(1) transformations

ei.

Provided 2 is positive this potential has a minimum at = 0. Draw potential

We call the = 0 state the vacuum and expand in terms of creation and annihilation operatorsthat populate the higher energy states.

In terms of a Quantum Field Theory, where is an operator, the precise statement is that theoperator, , has zero vacuum expectation value.

Suppose now that we reverse the sign of 2, so that the potential becomes

V() = 2 + ||2 . (5.3)

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| |

0

V

Re

Im

We see that this potential no longer has a minimum at = 0, but a maximum. The minimumoccurs at

=v

2= ei

2

2, (5.4)

where takes any value from 0 to 2.

There is an infinite number of states each with the same lowest energy - i.e. we have a degeneratevacuum.

The symmetry breaking occurs in the choice made for the value of which represents the truevacuum.

For convenience we shall choose = 0 to be this vacuum.

Such a choice constitutes a spontaneous breaking of the U(1) invariance, since a U(1) transfor-mation takes us to a different lowest energy state.

In other words the vacuum breaks U(1) invariance.

In Quantum Field Theory we say that the field, , has a non-zero vacuum expectation value

= v2

.

We will see that this means that there are excitations with zero energy

The only particles which can have zero energy are massless particles (with zero momentum). Wetherefore expect a massless particle in such a theory.

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To see this, we expand around its vacuum expectation value as

=1

2

+ H+ i

. (5.5)

The fields H and have zero vacuum expectation value and it is these fields that are expandedin terms of creation and annihilation operators of the particles that populate the excited states.

Now insert (5.5) into (5.3) we find

V = 2H2 +

H3 + 2H

+

4

H4 + 4 + 2H2 2

+

4

4. (5.6)

Note that in (5.6) there is a mass term for the field H, but no mass term for the field . Thus is a field for a massless particle called a Goldstone boson.

The field H will be held in a restoring potential, which corresponds precisely to a genuine massterm.

On the other hand, in the direction at this point, the field will be moving along the valley ofthe potential which means that its a massless field when quantized.

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5.4 Goldstone Bosons

Goldstones theorem extends this to spontaneous breaking of a general symmetry.

Suppose we have a theory which is invariant under a symmetry group

Gwith N generators

Assume some operator (i.e. a function of the quantum fields - which might just be a componentof one of these fields) has a non-zero vacuum expectation value, which breaks the symmetry down

to a subgroup H ofG, with n generators

The vacuum state is still invariant under transformations generated by the n generators ofH, butnot the remaining N n generators of the original symmetry group G.

Thus we have

Ta

|0 = 0, a = 1...n , T a

|0 = 0, a = n + 1...N Goldstones theorem states that there will be N n massless particles (one for each broken

generator of the group).

The case considered in this section is special in, there is only one generator of the symmetry group(i.e. N = 1) which is broken by the vacuum.

Thus, there is no generator that leaves the vacuum invariant (i.e. n = 0) and we get N n = 1Goldstone boson.

The quantum numbers of the Goldstone particle are the same as the corresponding generatorFor example: Global transformations are usually internal transformations in field space not touch-

ing Lorentz indices

Generators are Lorentz scalars - Goldstones are scalars

Not true in SUSY; generators of SUSY transformations are fermionic - massless particles arefermions - Goldstinos.

5.5 The Higgs Mechanism

Goldstones theorem has a loophole, which arises when one considers a gauge theory, i.e. whenone allows the original symmetry transformations to be local.

In a spontaneously broken gauge theory, the choice of which vacuum is the true vacuum is equiv-alent to choosing a gauge, which is necessary in order to be able to quantise the theory.

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What this means is that the Goldstone bosons, which can, in principle, transform the vacuuminto one of the states degenerate with the vacuum, now affect transitions into states which are

not consistent with the original gauge choice.

This means that the Goldstone bosons are unphysical and are often called Goldstone ghosts.

On the other hand the quantum degrees of freedom associated with the Goldstone bosons arecertainly there before a choice of gauge is made. What happens to them?

To see how this works we return to the U(1) global theory, but now we promote the symmetry toa local symmetry (hence to a gauge theory)

We must introduce a gauge boson, A. The partial derivative of the field is replaced by acovariant derivative

D = ( + i g A) .Including the kinetic term 1

4FF

for the gauge bosons, the Lagrangian density becomes

L = 14

FF+ (D)

D V() (5.7)

Now note what happens if we insert the expansion

=1

2(v + H+ i)

into the term (D) D. This generates the following terms

(D) (D) =

( + igA)

1

2(v + H+ i)

( igA)

1

2(v + H i)

=

1

2H+

i2

+ig

2vA +

ig2

AH g2

A

1

2H i

2 ig

2vA ig

2AH g

2A

= 1

2H

H+ 12

+ 1

2g2v2AA

+1

2g2AA

H2 + 2 g A ( H H ) + g v A + g2 v AAH,

(where v = /

). The gauge boson has acquired a mass term,

MA = g v,

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There is a coupling of the gauge field to the H-field,

g2vAAH = gMAAA

H

There is also the bilinear termgvA

which after integrating by parts (for the action S) may be written as

MAA

This mixes the Goldstone boson, , with the longitudinal component of the gauge boson, withstrength MA

Explicitly,MA

A = MA(AL + A

T ) = MA

AL

as AT = 0.

Later on, we will use the gauge freedom to get rid of this mixing term.

A massless vector boson has only two degrees of freedom (the two directions of polarisation of a

photon)

A massive vector (spin-one) particle has three possible values for the helicity of the particle, 2transverse and 1 longitudinal

In a spontaneously broken gauge theory, the Goldstone boson associated with each broken gener-ator provides the third degree of freedom to the gauge bosons.

This means that the gauge bosons become massive.

The Goldstone boson is said to be eaten by the gauge boson.

This is related to the mixing term between AL and

Thus, in our abelian model, the two degrees of freedom of the complex field turn out to be theHiggs field and the longitudinal component of the (now massive) gauge boson.

There is no physical, massless particle associated with the degree of freedom present in .

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5.6 Gauge Fixing

Going back to the bilinear termMA

AL

we can think of the longitudinal component of the gauge boson oscillating between the Goldstoneboson due to this mixing term

the physical particle is described by a superposition of these fields.

We consider two special cases:

The unitary gauge:

The physical field for the longitudinal component of the gauge boson is not simply AL , but thesuperposition

Aph = A +1

MA. (5.8)

(Note that this only affects the longitudinal component).

Now all the terms quadratic in A, (including the bilinear mixing term) may be written (inmomentum space) as

Aph (p)

gp2 + pp + gM2A

Aph (p). (5.9)

Writing out the Lagrangian we notice that the Goldstone boson field, , has disappeared. Theterms involving in the original expression have been absorbed (or eaten) by the redefinition

(5.8) of the gauge field.

The gauge boson propagator is the inverse of the coefficient of (dropping the superscript ph)A(p)A(p) in (5.9) , which is

i

g ppM2A

1

(p2

M2A)

, (5.10)

which is the usual expression for the propagator of a massive spin-one particle.

The only other remaining particle is the scalar, H, with mass mH =

2 which is called the

Higgs particle

This is a physical particle, which interacts with the gauge boson and also has cubic and quarticself-interactions.

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The interaction terms involving the Higgs boson are

LI(H) = e2

2AA

H2 + eMAAAH

4H4 mH

2H3, (5.11)

which leads to the following vertices and Feynman rules.

2 i e2g

2 i eMAg

6 i

6 i mH

2

Advantage of the unitary gauge is that no unphysical particles appear, no fields appear

The disadvantage is that the propagator of the gauge field behaves as a constant for p .

As we have discussed this seems to indicate that the theory is non-renormalizable.

Fortunately this is not true. In order to see that the theory is still renormalizable it is very usefulto consider a different type of gauges, namely the R gauge.

Another Quick look at the Unitary Gauge

Now use polar field coordinates (radial and angle variables) rather than the Cartesian onesabove. We set

=1

2(v + (x)) ei(x)/v.

The radial field and the angle field here replace the Cartesian H and .

There are still, of course, two (field) degrees of freedom.

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Stick this into L and find

L = (1

2

12

22) + (1

2

) + v + other interaction terms.

The mode is the one for motion around the equilibrium circle, and it is massless. The mode (radial) is restored, and has mass .

Now make the global U(1) symmetry of this model into a local symmetry by introducing a U(1)gauge field

This is the Abelian Higgs model.

We have to change all derivatives to covariant derivatives and add in the Maxwell term for theA field. This produces

L = [( + igA)] [( + igA)] 1

4FF

+1

22 | |2 1

22 | |4 .

As in the Goldstone model, expand about a point on the equilibrium circle as before

This time we shall choose to use the polar field variables, and write

=1

2(v + (x)) ei(x)/v

where v = /.

The theory is invariant under the combined transformations

= ei(x)

and

A A = A +1

g,

where is arbitrary.

The fields and transform by

and

v.

So, we can choose the field to be /v and so vanishes!

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That is, we can choose a gauge in which is real.

Remember that the gauge transformation affects the gauge field A and the complex scalar field simultaneously.

After the gauge transformation in which is reduced to zero, A is changed to

A = A +1

gv

and is

=1

2(v + ).

L in terms of these primed fields is then

L(, A) = (1

2

1

22

2

) 1

4FF

+

1

2 g2

v2

AA

+ interactions.

has a genuine mass .

But where is the massless mode ?

It has been eaten by the A field!

It is present in A viaA = A +

1

gv.

A is a massive spin-1 particle, of mass gv!

This is the Higgs mechanism, whereby the massless gauge field A has become a massive spin-1field A by eating the scalar field .

= 0 is physically appealing and is called the unitary gauge.

R Feynman gauge:

We select the Feynman gauge by adding to the Lagrangian density the term

LR 12(1 ) ( A (1 )MA)

2

= 12(1 )A

A + MAA

1 2

M2A2

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The cross-term MA A cancels the bilinear mixing term.

Again, the special value = 0 corresponds to the Feynman gauge.

The quadratic terms in the gauge boson are

1

2A(p)

g(p2 M2A) + pp

pp1

A(p)

leading to the propagator

i(p2 M2A)

g pp

p2 (1 )M2A

in the Feynman gauge the gauge boson propagator simplifies to

i g(p2 M2A) , (5.12)

which is easy to handle.

There is, however, a price to pay. The Goldstone boson is still present.

It has acquired a mass, MA, from the gauge fixing term, and it has interactions with the gaugeboson, with the Higgs scalar and with itself.

Furthermore, for the purposes of higher order corrections in non-abelian theories we again needto introduce Faddeev-Popov ghosts which in this case interact not only with the gauge boson, butalso with the Higgs scalar and Goldstone boson.

5.7 Bit more on Renormalisability

Recalling the form of the gauge boson propagator fir the Unitary gauge

i

g pp

M2A 1

(p2

M2A)

(5.13)

we see that it does not decrease as p

This would normally lead to a violation of renormalisability, thereby rendering the Quantum FieldTheory useless

However, there is no contradiction between the apparent non-renormalizability of the theory inthe unitary gauge and the manifest renormalizability in the R gauge.

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Physical quantities are gauge invariant, any physical quantity can be calculated in a gauge whererenormalizability is manifest.

The price we pay for this is that there are more particles and many more interactions, leading to

a plethora of Feynman diagrams.

Only work in such gauges if we want to compute higher order corrections.

For the rest of these lectures we shall confine ourselves to tree-level calculations and work solelyin the unitary gauge.

Nevertheless, one cannot over-stress the fact that it is only when the gauge bosons acquire massesthrough the Higgs mechanism that we have a renormalizable theory.

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5.8 Spontaneous Symmetry Breaking in a Non-Abelian Gauge Theory

Extend to non-Abelian gauge theories.

Take an SU(2) gauge theory and consider a complex doublet of scalar fields. i, i = 1, 2.

The Lagrangian density is

L = 14

FaFa+ |D|2 V(), (5.14)

where

D = + i g Wa T

a ,

(we have changed notation for the gauge bosons from Aa to Wa ), and

V() = 2ii + ii2 . (5.15) This potential has a minimum at ii = 122/.

We choose the vacuum expectation value to be in the T3 = 12

direction and to be real, i.e.

= 12

0

v

,

(v = /). This vacuum expectation value is not invariant under any SU(2) transformation.

This means that there is no unbroken subgroup

Counting goes as follows: We start with 2 component complex scalar = 4 dofthree generators are broken = three Goldstone bosons with all three of the gauge bosons acquiring

a mass

one dof left which is the massive Higgs scalar.

We expand i about its vacuum expectation value (vev)

=1

2

1 i 2

v + H + i 0

.

The a, a = 0 2 are the three Goldstone bosons and H is the physical Higgs scalar.

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All of these fields have zero vev.

If we insert this expansion into the potential (5.15) then we find that we only get a mass term forthe Higgs field, with value mH =

2.

For simplicity, move directly into the unitary gauge by setting all the three a to zero.

In this gauge D may be written

D =1

2

0

H

+ i

g

2

W3 W

1 iW2

W1 + iW2 W3

0

v + H

1

2

0

H

+ i

g

2

W0

2W+

2W W0

0

v + H

,

where we have introduced the notation W = (W1 i W2)/

2, W0 = W

3 and used the explicit

form for the generators of SU(2) in the 2 2 representation given by the Pauli matrices.

The term |D|2 then becomes

|D|2 = 12

H H+

1

4g2v2

W+ W

+1

2W0W

0

+

1

4g2H2

W+ W

+1

2W0W

0

+

1

2g2vH

W+ W

+1

2W0W

0

. (5.16)

We see from the terms quadratic in W that all three of the gauge bosons have acquired a mass

MW =gv

2

5.9 More on renormalisability

Electroweak theory is renormalisable

Hopefully the LHC will determine whether the simplest Higgs model is correct

Higgs is motivated by theory - no experimental evidence - or for any fundamental scalar.

Allows us to introduce weak boson masses without spoiling renormalisability of electroweak theory

We have seen a rough argument with propagators but we can also see the effects in diagrams.

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Structure of lowest-order amplitudes for weak processes hints that we may have trouble withoutthe Higgs.

Consider the cross section

(ee ee) =G2s

Becomes infinite as s .

This cannot be true for large s as this violates unitarity.

Can solve this by introducing a W mass and now for large s we have a diagram

e e

e

W

e

=

G2MW

2

However the introduction of the W boson causes problems due to the diagram

W W

e

e

e

=G2s

3

which diverges at large s.

However we also have the diagram proceeding via neutral current

W W

e

Z0

e

this diagram actually cancels the divergence from the charged current diagram

Beautiful demonstration of gauge theory - gauge boson self-interactions ensuring a finite answer

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Another example...W W scattering

W+ W+

W

W

+

W+ W+

W

W

+

W W

W W

We also have diagrams via Z exchange, when we collect these all together we find that the resultdiverges when s

Only way we can cure this is by introducing a scalar particle which cancels these divergencesthrough a similar diagram

W+ W+

W

h

W

Of course h is the Higgs boson.

If we had not already introduced the Higgs to generate heavy boson masses we would have beenforced to introduce it to guarantee renormalisability - i.e. to unitarise the W W scattering crosssection

Analysing this further shows that the Higgs couplings must be proportional to the particle massesin order to cancel the divergences - more evidence in favour of the structure of the SM

5.10 Summary

For a field theory in which the potential is not a minimum when all the fields take the value zero,

at least one of the fields acquires a non-zero vacuum expectation value (vev).

The symmetry is said to be spontaneously broken because one of many degenerate ground statesis chosen to be the true vacuum.

There may, in general, be a subset of transformations (i.e. a subgroup H of the original symmetrygroup G) under which the vev is invariant.

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The vacuum is then invariant under this subgroup H and we say that the original symmetrydescribed by the group G has been spontaneously broken to the subgroup H.

When a field, in a theory that is invariant under a set of transformations, acquires a vacuum

expectation value there is a massless Goldstone boson for each generator of the symmetry, whichis broken by that expectation value.

These Goldstone bosons are the excitations which effect transitions to the other states, whichare degenerate with the vacuum.

In the case of a gauge theory these Goldstone bosons provide the longitudinal component of thegauge bosons, which therefore acquire a mass.

The mass is proportional to the magnitude of the vacuum expectation value and the gauge coupling

constant.

The Goldstone bosons themselves are unphysical.

One can work in the unitary gauge where the Goldstone boson fields are set to zero.

When gauge bosons acquire masses by this (Higgs) mechanism, renormalisability is maintained.

This can be seen explicitly if one works in a gauge other than the unitary gauge, in which the gaugeboson propagator decreases as 1/p2 as p , which is a necessary condition for renormalisability.

If one does work in such a gauge, however, one needs to work with Goldstone boson fields, eventhough the Goldstone bosons are unphysical.

The number of interactions and the number of Feynman graphs required for the calculation ofsome process is greatly increased.

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6 Spontaneous Breaking ofSU(2)L U(1)Y U(1)EM Consider the particle content just consisting of one Higgs doublet, and the gauge fields associated

with the SU(2)L U(1)Y Higgs has quantum number (2, 1

2) and has the form

=

h+

h0

The Lagrangian for this is then

L = 14

FaFa 1

4GG

+ |D|2 V() (4 d.o.f)

where

Fa = Wa Wa g fabc WbWc

G = B BD = +

ig

2B +

ig

2iWi

V() = 2ii +

ii2

where is are the Pauli matrices.

Expanding the Higgs field around its VEV In the Unitary gauge the covariant derivative takesthe form

D =1

2

+ i

g

2

W0

2 W+

2 W W0

+ i

g

2B

0

v + H

(6.1)

=1

2

+ i

g2

W0 + ig

2B i

g2

2 W+

i g2

2 W i g2W0 + i g

2B

0

v + H

=1

2

i g2

2 W+ (v + H)

(

i g2

W0 + ig

2B)(v + H)

where we have taken W =W1iW2

2and W0 = W

3 . We then get

|D |2 = 12

( H)2 +

g2v2

4W+W +

v2

8

g W0 gB

2+

g2

4W+W (2vH + H

2) +1

8

g W0 gB

2(2vH + H2) . (6.2)

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We get a mass terms for the charged gauge boson and a linear combination g W0 gB. We need to diagonalise the W0 B0 system and we can do that by introducing

Z = cos wW

0

sin wBA = sin wW

0 + cos wB

i.e.

B = cos wA sin wZW0 = sin wA + cos wZ

where

tan w =g

g

Now our non-interaction part of the Lagrangian is given by

Lnonint = 12

(.H)2 2H2

14

W

1 W1

W1 W1+ 1

8g2v2W21

14

W2 W2

W2 W2

+

1

8g2v2W22

14

(Z Z) (Z Z) + 18

(g2 + g2

)v2Z2

14

(A A) (A A)

The physical interpretation of this is the following: 3 massive vector particles

M1 = M2 =gv

2= MW

andv

2(g2 + g

2

)1/2 =vg

2

1

cos w=

MW

cos w= MZ

We also have one massless vector and a massive Higgs

The masslessness of A follows from the choice of breaking condition, i.e. the form of the vev.

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There is a combination of the original generators that still annihilates the the vacuum, i.e. there isa linear combinations of generators that is still invariant - one gauge symmetry not spontaneously

broken

Y + T32 0v/2 = 1/2 00 1/2 + 1/2 00 1/2 0v/2 =

1 0

0 0

0

v/

2

=

0 0

0 0

We can rewrite this linear combination in terms of the isospin 3 = 2tisospin3 so that the linearcombination is

Y + tisospin3

Going back to the covariant derivative for the doublet we have

D = +ig

2B +

ig

2iWi

= +ig

2(cos wA sin wZ) + ig

2(3 sin wA + cos wZ) + W

1, W2 bits

= +ig

2tan w (cos wA sin wZ) + ig

2(3 sin wA + cos wZ)

= +ig

2sin w(1 +

3)A +ig

cos w

3

2 sin2 w (1 +

3)

2

Z

D = + ig sin w

1

2

+ tisospin3 A +ig

cos w 3

2 sin2 w

(1 + 3)

2 ZWe then of course identify

g sin w

1

2+ tisospin3

= eQ()

with

g sin w = e

and

Q() =1

2+ tisospin3

We can see this makes sense asg sin w

1

2+ tisospin3

= g sin w

1

2+ tisospin3

h+

h0

= g sin w

1 0

0 0

h+

h0

= e

Q 0

0 0

h+

h0

(6.3)

if we are to identify A as the photon

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7 The Electroweak Model of Leptons

Only one or two modifications are needed to the model described at the end of the last lecture toobtain the Glashow-Weinberg-Salam (GWS) model of electroweak interactions.

This was the first model that successfully unified different forces of Nature.

In this lecture we shall consider only leptons as matter fields, deferring the introduction of hadronsto the next lecture.

7.1 Left- and right- handed fermions

The weak interactions are known to violate parity they are not symmetric under interchange of left-helicity and right-helicity fermions.

First Recall a few things about writing Lagrangians in terms of Weyl and Dirac spinors.

A Dirac field, , representing a fermion, can be expressed as the sum of a left-handed part, L,and a right-handed part, R,

= L + R, (7.1)

where

L = PL

PL =(1 5)

2(7.2)

R = PR

PR =(1 + 5)

2(7.3)

PL and PR are projection operators in the sense that

PL PL = PL, PR PR = PR, and PL PR = 0. (7.4)

They project out the left-handed (negative) and right-handed (positive) helicity states of thefermion respectively.

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The kinetic term of the Dirac Lagrangian and the interaction term of a fermion with a vector fieldcan also be written as a sum of two terms each involving only one helicity.

= L L + R

R, (7.5)

A = L AL + R

AR. (7.6)

On the other hand, a mass term mixes the two helicities

m = mL R + mR L.

Thus, if the fermions are massless, we can treat the left-handed and right-handed helicities asseparate particles.

We can understand this physically:Massive fermion moving along the positivez-axis with spin positivecomponent, so that the helicity

is positive in this frame

Can always boost into a frame with the fermion moving along the negative z-axis, but the com-ponent of spin is unchanged, helicity will be negative.

For massless particle it travels with the speed of light no such boost is possible and in that casethe helicity is a good quantum number.

Since charged weak interactions are observed to occur only between left-helicity fermions, weconsider a weak isospin, SU(2), under which the left-handed leptons transform as a doublet

LL =

L

eL

,

under this SU(2), but the right-handed electron eR is a singlet.

i.eeR eR = eR

where as

LL LL = eaT

a

LL

Under U(1)Y gauge transformations

eR eR = eY(eR)eR

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where as

LL LL = eY(LL)LLwith

Y(eR) = 1 and Y(LL) = 1

2

Since this separation of the electron into its left- and right-handed helicity only makes sense for amassless electron we also need to assume that the electron is massless in the exact SU(2) limit

Also that the mass for the electron arises as a result of spontaneous symmetry breaking in thesame way that the masses for the gauge bosons arise.

7.2 Fermion masses - Yukawa couplings

Cannot have explicit mass terms for the electrons since a mass term mixes left-handed and right-handed fermions

Is not gauge invariant.

Can have an interaction between the left-handed lepton doublet, the right-handed electron andthe scalar doublet . (Slight change of definition (this Higgs has Y = 1/2 and is related to the

previous on by i2.

Such an interaction is called a Yukawa interaction and is written

LY ukawa = e lLi ieR + h.c. (7.7)

Note that this term has zero weak hypercharge. In the unitary gauge this is

e2

L eL

0

v + H

eR + h.c..

The part proportional to the vev is simply

e v2

(eL eR + eR eL) = e v2

(ePR e + ePL e) =e v

2e e,

and we see that the electron has acquired a mass, which is proportional to the vev of the scalar

field.

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This immediately gives us a relation for the Yukawa coupling in terms of the electron mass, me,and the Wmass, MW,

e = gme2 MW

.

There is, moreover, a Yukawa coupling between the electron and the scalar Higgs field g me

2 MWe H e.

Note that there is no coupling between the neutrino and the Higgs, (and of course no neutrinomass term).

7.3 Weak Interactions of Leptons

Now look at the fermionic terms the fermionic part of the Lagrangian is

LFermi = i lL D lL + ieR D eR, (7.8)

where the covariant derivatives are:

D = ig tan w2

B +ig

2iWi (7.9)

D = ig tan wB

This gives the following interaction terms between the leptons and the gauge bosons:

g2

L eL

W0

2 W

2 W+ W0

tan WB

L

eL

i g tan W eR BeR.

Expanding this out, using the physical particles Z and A in place ofB and W0 and using theprojection operators for left- and right-handed fermions to write the terms in terms of 4-cpt Dirac

spinors we obtain the following interactions:

1. A coupling of the charged vector bosons W, which mediate transitions between neutrinos and

electrons with an interaction term

g2

2

1 5 e W + h.c..

2. The usual coupling of the electron with the photon,

g sin W e e A.

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3. The coupling of neutrinos to the neutral

stephan west-standard model cabridge

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