peak, varvell. lectures on the standard model of particle physics

The Standard Model seeks to describe the fundamental “point-like” particles of Particle Physics and the interactions between them.. As mentioned above, the Standard Model also seeks to i

THE PHYSICS OF THE STANDARD MODEL

OVERVIEW

The Standard Model seeks to describe the fundamental “point-like” particles of Particle Physics and the interactions between them It has been known for quite some time now that the normal particles such as protons and neutrons, pions kaons etc are not fundamental The fact that the neutron possesses a magnetic moment is evidence enough that the neutron must be a composite particle and this was known many decades ago In fact, we now

believe that the fundamental particles are the quarks and leptons, which seem to exhibit no evidence for internal structure down to the smallest measurable distances (101 m) Table 1 summarises the situation for both composite and fundamental particles Note that all

existing particles can be divided into two camps namely Fermions (which obey the Pauli

Exclusion Principle) and Bosons (which don’t)

As mentioned above, the Standard Model also seeks to incorporate the fundamental

interactions between these particles in terms of the exchange of an intermediate messenger called an “exchange particle” Quantum Mechanics shows how both attractive and

repulsive forces can result from the exchange of these intermediate particles (which are

bosons) Table 2 summarises the situation as far as these forces are concerned

e The Gravitational Force is too weak to play a major role in any particle physics scenario except perhaps for Black Holes and the earliest stages of the universe It comes about due to the exchange of gravitons.We will neglect it for the remainder of this course

e The Weak Force is evident in decays of particles and nuclei (beta decay etc) as well as

in the interactions of neutrinos It comes about due to the exchange of Intermediate Vector Bosons

e The Electromagnetic Force is responsible for just about all of the everyday physics

we see around us It results from the exchange of massless photons

e The residual Strong Nuclear Force was first observed in the late 40s and is due to the exchange of pions (originally called the “one pion exchange force’) From the

uncertainty principle the lower the mass of the exchanged particle the greater the range

of the force (m R =1 with #=c =1 ) As the pion is the lightest nuclear active particle, this pion exchange force has the largest range (~10°'°m) and hence was the easiest to observe It is however not fundamental as it can be explained at a lower level in terms

of exchanged quarks It is akin to the Van de Waals force which is a residual molecular force when all the constituent charges are added together vectorially

e The fundamental Strong Nuclear Force This is the basic force between quarks resulting from the exchange of Gluons

The Standard Model incorporates the Electroweak Force of Weinberg Salam with the standard model of the fundamental strong force, Quantum Chromodynamics (QCD) It therefore handles the strong, electromagnetic and weak forces in one description Although the electroweak description of Weinberg and Salam is a true unification of electromagnetic and weak phenomena, QCD does not derive naturally from the same theory and hence is an

“add-on”

The electroweak theory was the next major advance in the quest to unify all the forces of nature after Maxwell’s electromagnetic theory in 1864 Theorists are still seeking the “Holy Grail’ in a quest to unify all the forces into one complete and naturally renormalisable theory

The elecroweak theory invokes a “weak hypercharge” Y which is a scalar field just like ordinary charge The Group describing this is U(1) see later There is a messenger

conveying differences in weak hypercharge from place to place called the B’ intermediate vector boson

Also invoked is a “weak isospin’, I, rather like ordinary spin but relating to the weak species type The mathematics is identical to that of ordinary spin and is summarised by SU(2) — also see later There are two basic spin states equivalent to “up” and “down” The messengers conveying differences in weak isospin are the W', W” and W’ intermediate vector bosons Sometimes these are written equivalently W*, W and W° Weak isospin should not be confused with strong isospin which we probably won’t mention much in this course but often is encountered in textbooks and articles

The theory of strong interactions is best described by the exchange of gluons, changing the colour of the quarks in the process The group theory used here is SU(3) and there are three basic colours which we will call red, green and blue For SU(n) there are (n?-1)

independent messengers required — so there are 8 gluons in the theory and these carry

“colour charge” making this a non-Abelian group theory This makes it different in many ways to QED which is Abelian as the photons are chargeless — ie neutral

The Group Theory summary of the Standard Model is therefore

UM) weak hypercharge X SU(2) weak isospin X SU(3) colour

Although as mentioned earlier, the QCD description of colour is tacked on in a non-

fundamental manner at this stage

We are able to summarise the properties of the fundamental quarks and leptons as in the following table These are tags that are always the same for the particles, just like charge, spin, strangeness etc They are fundamental to the particle and an intrinsic part of their

very nature

PARTICLE HYPERCHARGE ISOSPIN CHARGE

(a) It is easy to see that the relationship is Q = 13+ Y/2

(b) Note that the hypercharge is the same for each isospin doublet

(c) The fundamental particles seem to occur in lefthanded doublets but right handed

singlets The W bosons do not couple to right handed particles as their weak isospin

the “Cabibbo-Kobayashi-Maskawa Mixing MatrIx”

(e) Note that all the parameters are zero for right handed neutrinos, The Standard Model does not predict any such beasts! However right handed neutrinos are needed for

neutrinos to have a non zero mass; so any observation of neutrino masses would take us

beyond the Standard Model There is now strong evidence that neutrino flavour

oscillations take place spontaneously requiring a small non zero mass for at least some

of the types

INTERACTIONS

Electroweak charged current interactions come about from the exchange of Ws and hence only involve left-handed particles Neutral currents on the other hand can come about from the exchange of a B’ and a W” The observed neutral currents are therefore a mixture of these

Weak Neutral Current Z°?=W° cos Ow —B’ sin Ow

Electromagnetic

Neutral Current A° = WY? sin Ow + B’ cos Ow

Ow is called the Weinberg Salam mixing angle and has been measured in a wide variety of experiments (see Experimental Tests of the Standard Model later) The experiments all agree in a spectacular way adding much credence to the theory The tabulated value is currently sin* @w = 0.23124(24) and is now listed in the particle physics data book along

with all the other fundamental constants such as k, h, c, fine structure constant © etc

A is the vector potential, the field appropriate for the description of photons — so we can see how the electromagnetic and weak neutral currents are intertwined

From the theory (see later) we can predict the masses of the bosons

The corresponding experimental values are 80.4 and 91.2 GeV It is not surprising that the

W and the Z have slightly different masses as they are not equivalent in the theory

SPONTANEOUS SYMMETRY BREAKING and the HIGGS FIELD

For the fields described above to be renormalisable, we need to have gauge invariance This means that the physical outcomes need to be invariant to arbitrary changes in isospin, hypercharge, colour etc anywhere in space This can only occur if the ranges of the forces are infinite and hence if all the propagator particles have zero mass We therefore need to find a mechanism to give the W and Z particles enormous mass (about 80 proton masses) whilst keeping the photon identically massless

In the Standard Model, particles are given mass by interacting with an all-pervading Higgs Field The quantum of this field is called the Higgs Boson which has not been seen

experimentally yet Much effort however is going into experimental searches to uncover the Higgs Particle The standard model needs three such particles to be absorbed to give the masses to the W*, W and Z” whilst leaving the massless photon field untouched

+

The minimal standard model proposes a scalar field of two complex components ) thus defining 4 real fields There 1s therefore at least one leftover field, and that 1s the field being searched for The Higgs mechanism gives us some understanding as to why there is such an

enormous difference between the electron (0.511 MeV) and the tauon (1777 MeV) They

are both pointlike but one couples much more strongly to this Higgs field

The W and Z were both discovered 1n historic experiments in 1983 at CERN The

experimental signatures were unambiguous One didn’t even need statistics to prove their existence!

INTRODUCTION TO FOUR VECTORS

We have a = (a,, 1a ) and b = (b,, 1b ) examples are (E, ik), (t, 1x) and (0, 1A)

2 2 2

d.a = ag —a = ag —_ aydax —_ dydy —_ azaz

a.b = agbo — axbx — ayby — a,b, etc

And we define contravariant 4 vectors a” = (a, a)

And covariant 4 vectors a, = (a, -a)

As’ = Ad,Aa" = g„Aa“Aa* is an invariant

and the scalar product which is also, of course, invariant

is defined as g,,a"b" = a,b” =a"b, = agby —a.b since a, = 84,4"

Here we use the standard convention where a repeated index implies summation over all values

THE LAGRANGIAN

The Lagrangian is defined as KE— PE = T-V

T.q) is the kinetic energy of the system and V(q) is the potential energy

We define the ACTION $ = [L(q,q)dt

t

And S is stationary for the particular path determined by the equations of motion

8§=0=ð [L(q.q)dt= 0

to OL

=1;

Where we have integrated by parts We have also used the fact that the variations

dq must be zero at both ends of the range considered

Example 1 Free particle moving in one dimension (x)

It is important to note that the Lagrangian is not unique We can add to L any function of

the form d/dt f(q,t) as the contribution f(q2, tz) —f(q1,t1) will be independent of the path and

hence the equations of motion will remain unchanged

Example 2 Energy Conservation

If the Lagrangian is not explicitly dependent upon time (as we have assumed up to now)

hence the expression becomes 2T - (-V) = T + VỀ which 1s obviously the energy!

This is a particular case of Noethers Theorem which links fundamental symmetries with conserved quantities — such as time and energy, position and momentum, angular displacement and angular momentum etc etc

Example 3 — Simple Pendulum

Now our variables are r and 9

at (mrˆ9)+ mgr sin9 =0 or 9= cÑ) sin§ for r approximately

constant and equal to the length 1

Continuous Fields

When we have continuous fields, we need to define a Lagrangian Density L

S= |Ldxdydzdt whereL = L(Ø 3,)

- (- —_V “A—(— ——_ -V — œq#Y

Op co” and Zˆ ø CN: YØ) =8 2,0

The Field Equation becomes

aL» 9L

9ò ` 9(0,9)

1 and with the Lagrangian Density of L = 5 [0,0 0" — m’*o° |

2

The field equation becomes -0,0"9- m*d=0 ie - = + Vfọ- m ¿=0

As an exercise, see if you can prove this! This is the Klein Gordon Equation!

You get the same result by saying

E=p +m h=if and p=-iV giving -29=-v"p +m"

DESCRIBING EXTENDED SYSTEMS

We need to make a transition from the understanding of an equation of motion such as the Klein Gordon Equation as the description of a particle to a more general description of extended systems using fields

This can be done quite naturally and conveniently using Lagrangians

We should acknowledge firstly that a "particle" is actually an extended thing If we

construct a wave packet, this is the sum total of constituent waves that each extend to te Even in the ultimate particle limit when the wave function is a delta function , this is easily shown using the definition

The first step is to introduce the concepts of a Lagrangian Density that needs to be

integrated over all space-time to define the action

From this one can define a momentum density —> and an energy density —> o—L and the

a

conservation laws apply to the integrals over all available space-time

So a particle (wave packet) would show up as a region where the energy and momentum density would be high Hence the particle could be thought of as a "ripple" in the field @

This is not a new concept! You are already familiar with fields in electromagnetism where

we have electric and magnetic fields as well as massless and massive particles (photons and electrons) Here the energy density is the familiar

~(@&E +—)=~(&E +uH 5 Eo i 5 (Eo Hy )

When we move on to the formal Field Theory of QED we describe interactions in terms of fundamental vertices as follows

None of these vertices conserve energy so they all involve virtual particles which can only

"exist" momentarily

For the first case of electron scattering we talk about three things happening as the electron interacts with the field at the vertex point

e An electron is annihilated at the vertex

e Another electron of different energy, momentum and phase is created at the vertex

e A photon is created at the vertex

Thus we have creation and annihilation operators and the static field can be viewed as a

"storage bank" of quantised energy (stored in harmonic oscillators) The particles couple to the field with a strength given by a coupling constant

1

If we take the Klein Gordon Lagrangian we met previously, L = 2 [2,02 ˆø— m øØ] this

1 gives rise to the energy density xe +(Vo) +m]

The field @ is a scalar field which is defined by a single number at each point in space-time There is thus no spin involved in the solutions - but such an equation would be pertinent for the description of x’ mesons (almost!) and for the scalar components of the Higgs fields mentioned earlier We will be talking about the Higgs Field in much greater detail later in the course

In order to describe electrons, and spin, we need to resort to the Dirac Equation

Dirac started with the linear form

0100

and the appropriate Lagrangian densityis y'7"liy“0,,- mly = wliy" 2„T—m]V

The solutions give rise to four component entities - and they have the form for a particle travelling in the z direction

The first two correspond to positive energy particle states (spin up and spin down)

Whilst the latter two correspond to negative energy states (E = 4E| ) also with spin up and down

So the Dirac equation gives rise in a natural way to both the concept of antiparticles and spin!

Once again, we can postulate a field - this time a spinor field with creation and annihilation operators creating and annihilating electrons and positrons

Noethers Theorem etc

Suppose we consider displacement in the x direction

2 and

Energy and Time

We can perform the same calculation with energy and time displacement

iEổ Tö0)=1+ Sðt =1- dt asE=iie dt

So E=ih Lim, ,, mee

For MACROSCOPIC transitions t = nédt

-!E¡

y(t) = Lim — = ự(0)=e 1) y(0) and we get our familar time development

GROUP THEORY

A “Group” is a set of elements such that a combination of any two gives another member

of the group The elements are associative in that a(bc) = (ab)c

There 1s an identity element, I, such thatal =Ia=a

Also, for every member there is an inverse a’ such that aa’! = 1 =I the identity element

If ab = ba, the Group is said to be Abelian such as 2D rotations and QED

If ab# ba the Group is non-Abelian such as 3D rotations and QCD

As an example, consider a rotation of axes about the origin

x= “* R’ x’ and if we consider rotations about the z (x°) axis, we get

X=XCcos 0+ Ysin 9 Y.=-Xsin 0+ Ycos 9

The scalar product is defined as xy =x! y=xy' =x'y'+x’°y+xy°

It is easy to show that this is invariant: for x’’ y’ = (Rx)'y’ =x'R'Ry=x' y

Provided R'R=1 R is called an ORTHOGONAL matrix with R’ = R'!

And a SPECIAL ORTHOGONAL matrix has |R|=1

UNITARY and HERMITIAN MATRICES

A UNITARY matrix is defined by U'U=1

and nxn unitary matrices form a Group U(n)

det (U U' ) = det (U) det (U*) = det(U) [det U]* = 1

Therefore det (U) =e and if we take Special Unitary Groups det(U) = 1

Therefore U(2) =SU (2) e*

Note also that det (U,U2) = det (U;) det(U2) = 1 so this satisfies the definition of a Group

Proposition I

A matrix is HERMITIAN if H” =H and for each Hermitian matrix there exiss a

unitary matrix U such that UHU” = UHU Ï = Hp which is a real diagonal matrix

We can see this as follows

Suppose U is unitary and we want UHU™ =H,,

we proceed to show that H must be Hermitian

UHU*=H, so (UHU')' =H) =H,

(U')'H'U' =H, UTH'U =H, UH'U' = H,

Therefore H = H' and His Hermitian

If the trace of H is zero, the determinant of U is 1

These are equivalent statements

det(U) — det (e'") — Tl, eiHnn =e 1>Hnn =e iTrH

So if TrH = 0 then det(U) =1

SU(2) In general a 2 x 2 Hermitian matrix can be written

and the o are called the GENERATORS of the Group

We can also define stepping operators

and we can have the equivalent representation oto” and o°

It is easy to see the following

So the step operators can move around between the members of

the group but cannot go beyond This is why the group members

constitute an irreducible multiplet

If we put y= cosh 8 we have pai = tanhé

We can rewrite in more familiar form as the Lorentz Transformations

WAVE FUNCTIONS for SU(2) and SU(3)

In the Standard Model SU(2) corresponds to weak isospin So we can write

¥ — oF ao ¥

g is a coupling constant

X = x (t, x ) determines a phase which varies over time and space

It is the W s which convey this phase information from one spacetime point to another

As before the o are the generators of the group

In the Standard Model, SU(3) corresponds to the Gauge Invariance of QCD and the system

of quarks and gluons The generators of this group are

mm" '— B

0 i 0 0 0 -2

analogous to the Pauli matrices Note they are all traceless as are the Pauli spin matrices

The wave functions become

yA

yw,|= e*~ | w,| and there are now 8 different phases to consider, and we need 8 gluons

to convey this phase information As before 7 is a function both of time and place

We can also construct 9 equivalent operators which move you around from one member

of the group to another and define the eigenvalues

CAPTIONS TO TABLES TABLE 1

Classes of particle both composite and fundamental as observed in nature TABLE 2

Classification of the fundamental forces All these forces can be derived from

the exchange of intermediate gauge bosons.

THE PHYSICS OF THE STANDARD MODEL

GAUGE INVARIANCE, WEINBERG SALAM

L PEAK and K VARVELL

GAUGE THEORY Particle physics attempts to produce a common unifying description of

particles and their interactions

The successful theories of particle physics:

¢Q.E.D (Electromagnetism)

2 Weinbere/Salam (Electroweak)

e Weinberg/Salam (Electroweak)

«.Q.C.D (Strong)

are examples of Gauge Theories

Gauge invariance is a quite general principle, not confined to particle physics

Gauge invariance is associated with certain transformations of the fields

representing a particle or system.

Calculations in particle physics are often done using perturbation theory, where to calculate, say, a cross section, sums of amplitudes over increasingly complex (higher order) Feynman diagrams are made

e.g Rate for ete — ete™ includes diagrams

cte

This works in principle because higher order diagrams involve powers of the

“coupling constant” and are small corrections, at least in Q.E.D

œ = €2/4m ~ 1/137

However, calculation of some diagrams involves integrals which actually give infinite answers (“diverge”) The calculations can only be done using a

technique known as renormalization to handle these infinities

For renormalization to work (and hence for sensible results to be obtained from calculations), it appears that the Quantum Field Theory being used

must be a gauge theory saus 3

QED, QCD and the Weinberg/Salam theory (to be covered soon) are all

gauge theories and renormalizable Hence the interest in gauge theory in particle physics theory

Note : Renormalization is beyond the level of this course See e.g Cottingham and

MAXWELL’S EQUATIONS Maxwell unified electricity and magnetism in 1864, expressed through the equations

(Remember / = 0, 1, 2,3, and sum over repeated indices)

Rather than work with E and B, we can define scalar and 3-vector potentials

Are these potentials any more than convenient mathematical inventions?

There is some “arbitrariness” in these definitions For example, if

Maxwell’s equations are invariant (hence so is the physics) under the above transformations

They are in fact an example of a gauge transformation

They are also local transformations, since by appropriate choice of the

arbitrary function \(t,x) we can change the potentials independently at each point in space-time

Covariant form of Maxwell’s Equations

(Notice that unlike ¢ and A, which form a 4-vector 4“, E and B are mixed

up as the components oŸ #““ which transforms as a tensor under Lorentz

transformations: see e.g Cottingham and Greenwood Ch.2 if interested)

The Lagrangian density from which Maxwell’s equations can be derived is:

1 WY TH L= qh P — JPA,

How do we interpret this?

« Thị ® (4-vector) field A

electromagnetism)

e The externally specified “current” J“ is “coupled” to the photon field in

the second term

e The first term is in some sense a “kinetic energy” term for the photon

field (it can be compared with the term (1/2)0,,¢0"¢ in the

Klein-Gordon equation)

11

The gauge transformation can now be written in “covariant” notation as

Al Al* = AP + Oly The “gauge transformed” Lagrangian density would look like

1 Li=

4

/ FV ,

Fiyk™ — JPA, The first term is invariant, since substitution and solving will show that

Fly FO = (OpA'y — OVA) (OX A” — OY A™)

= (Fav + OpOVX — OOux) (FY + OHA" x — OYA")

= Fy FH”

In the last step we have used

9“9”x—978“x=0 since the order of differentation of a scalar function is unimportant

The second term adds an extra

The solutions for y are 4-component spinors as has been seen previously

Suppose that we change the phase of y in the following way

U(a) > Y (a) = ed (2) This operation is a member of the U(1) group that we met before

14

If a is just a number, the Dirac equation is clearly unchanged This is a

global gauge transformation

(iy"O, — m)u! (iy, — m)e"

= e' (i740, —m)v

= 0

However, suppose a = a(2) is a function of the space-time point x

(iO, — mv = (Oy, — Mme

= in", (ey) — mee) a)

= int (Oe Jy + iyHe~/S)8 nụ

in the presence of an electromagnetic field

which can be written

ph — ph — qA®

Noting that p, — i0, in quantum mechanics, we might expect the Dirac

equation to become, in the presence of an electromagnetic field

[y* (iu — q4„) — ml =0

Suppose we write a(2) = q x(a) where q is the charge of the fermion (e.g

q = —e for an electron) Now consider our gauge transformations

which exactly cancels the extra term

Ht (Oper) es = +q7" (Dux) COO

that we found when transforming the free particle Dirac equation

17

Interpretation of these transformations

e The transformation w — e~“?*v is an example of a

local gauge transformation, changing the phase a = qy of the field w at

each point in space-time (since y = x(2))

Internal space Phase angle of

of e.m

Think of it as a rotation in the “internal space” of all possible phase

angles for the field, by an amount a(z)

e Symmetry of the equations describing the physics under these

transformations forces the introduction of the field A, = (ø, A) (the

photon), transforming as A, — A, + Ox

18

e The photon field A,, couples to the fermion field with coupling

strength g given by the charge of the particle

e Interaction with the photon field transmits the phase difference in the

wave function from one space-time point to another as the fermion moves through space-time

INTERNAL

SPACE (PHASEANGLE)} | | _| -_- PHASE

a

e Since the difference in phase can be arbitrarily changed at any two

space-time points, no matter how far apart (because x(a’) can be chosen

arbitrarily), the range of the photon field must be infinite and hence

photons must be massless

19

e The set of all phase transformations w — e~*?w on the field of a fermion

of charge g forms a symmetry group U(1), in this case for

electromagnetism i.e

The photon is the massless gauge boson of U(1)e.m

FULL LAGRANGIAN FOR FERMION AND PHOTON

Combine the gauge-invariant Lagrangian density describing a fermion field in the presence of an electromagnetic field with that for the e.m field itself

£ = U[ “(iu — qÂu) — m] b — Tuy — JIMA,

=_ U[“iỡ,— m]t— 11t” — (2“ + qU+“9) Au Note that the term coupling to the photon field A,, consists of two parts

e The external current density J”

e A term corresponding to the fermion field itself qyy""w This is called the electromagnetic current (think flow of the fermion charge) and when

coupled to A,, describes the interaction vertex

21

MASSIVE PHOTONS?

What would a mass term for the photon look like in the Lagrangian density?

We can use an analogy with the Klein-Gordon case

where the mass of the scalar field ¢ comes in the term m?¢? Perhaps for the

(vector) photon field A” we could introduce a term m?A,,A“

Unfortunately this is not gauge invariant, since

mA’, AM =m? (Aj, +Ø„x)(A“# + ory)

=m2(A,A* + (8„x) A* + (0x) Au„ + Øu„xГx)

~m A, AM

The term containing 0,,yO"y is harmless (it does not contribute to the

equations of motion) but the terms linear in A“ do This is not a problem for electromagnetism since the photon is massless, but it will be a problem for

the weak interaction 22

GAUGE INVARIANCE AND SU(2)

A theory having SU(2) gauge invariance would require at least a doublet of

fields with components v; and v2, transforming as, for example

where there are now 22 — 1 = 3 phase angles œ, a2, a3

a(x) = gXx(%) g some coupling constant

and the 7Ý are the Pauli matrices

By analogy with the electromagnetism case, we would need 3 massless gauge

bosons Wii, W,7, Wj to satisfy gauge invariance and transmit the phases

23

In the weak interaction, there are 3 gauge bosons, W+, W-, Z°, but these

are heavy (m ~ 80 — 90 GeV), and the force short range How to get around

this?

SPONTANEOUS SYMMETRY BREAKING

Suppose we generalise the Klein-Gordon Lagrangian density to the case

where the scalar field ® is complex, or equivalently is a pair of real scalar fields @; and ¢2 such that

The state of lowest energy of a system is known as the ground state, or in field theory terminology, the vacuum

V(I®)

In the above Lagrangian, the potential

energy density V(®'®) = m?01O is a

where the sign of the potential term has been changed from — to +, i.e

V(®'®) = —m?61®, This has no minimum, and is therefore not too useful But suppose we modify it to (do a real constant)

Nature chooses one of these as the physical vacuum

and “breaks” this symmetry This phenomenon is

known as spontaneous symmetry breaking

27

How does spontaneous symmetry breaking help? Suppose we expand the

field ® around the chosen vacuum state, by writing

2

Pick out the “free particle” pieces by writing

L= Liree + Lint

We can see that we have

e A massive, spinless scalar boson field y of mass V2m This is called a Higgs boson

e A massless, spinless scalar boson field 7 This is called a

Goldstone boson

The Higgs boson is like a fluctuation around the vacuum point in the

direction in which the potential density increases The Goldstone boson is

like a fluctuation in the direction in which the potential density is flat

At this point, we seem to have introduced new fields into our toy theory and not gained a lot However, the full theory must be locally gauge invariant, which is not yet the case

30

For local gauge invariance we require invariance under

O(a) > O (x) = e P(x)

and the introduction of a gauge field A,, transforming as

Au() — A(œ) = Au() + Øu0(z)

with the Lagrangian looking like

Again the vacuum state is when |®(2)| = ¢o, and since 0(2) is arbitrary, we

can choose it so that ®(2) is real, breaking the symmetry

— —Eu,Fttr A4 — —>|V2óoh + —h? so [V200 +5 |

and we can again write

L= Liree + Lint