V() ½ 2 + ¼ 4

A translation (x) = 0 + u(x) → u(x) ≡ (x) – 0

V() V(u +) ½(u +)2 + ¼ (u +)4

selects one of the minima by moving into a new basisredefining the functional form of in the new basis

(in order to study deviations in energy from the minimum 0)

√λ

√λ

V +u2 + √ u3 + ¼u4

energy scalewe can neglect

The observable field describes particles of ordinary mass |/2.

plus new self-interaction terms

some calculus of variations UU1

UU22

2

2

1412

2

2

1

2

21 )()(

)( 2

2

2

111

2 Letting = *

1

2

11

2 )( UU2

2

2

22

2 )(

= 0

= 0

Extrema occur not only for )0,0(021

but also for

22

2

2

1

But since = * > 0 This must mean 2 > 0 2 < 0

and I guess we’ve sort of been assuming was a mass term!

02

22

21

defines a circle of radius | | /

in the 2 vs 1 plane

1

2

not x-y “space”

Furthermore, from

2UU1

2

2

1

2

2

2

1

2 2)(

UU1

)( 2

2

2

111

2

we note

2

2

2

1

2 3

which at = 0 gives just 2 < 0

making = 0 the location of a local MAXIMUM!

1

2

Lowest energy states exist in thiscircular valley/rut of radius v = /2

This clearly shows the U(1)SO(2) symmetry of the Lagrangian

But only one final state can be “chosen”

Because of the rotational symmetry all are equivalent

We can chose the one that will simplify our expressions(and make it easier to identify the meaningful terms)

vxx )()(1

)()(2

xx shift to the

selected ground state

expanding the field about the ground state: 1(x)=+(x)

vxx )()(1

)()(2

xx v = /2

Scalar (spin=0) particle Lagrangian

with these substitutions:

becomes

L=½11 + ½22 ½1

2 + 2

2 ¼12 + 2

2

L=½ + ½ ½2

+2v+v2+ 2 ¼2

+2v+v2 + 2

L=½ + ½ ½2

+ 2 v ½v2

¼2 +2 ¼22

+ 22v+v2 ¼2v+v2

L=½ + ½ ½(2v)2

v2 + 2¼2

+2 + ¼v4

Explicitly expressed in real quantities and v

this is now an ordinary mass term! “appears” as a scalar (spin=0)

particle with a mass 22 vm

½ ½(2v)2

22

½ “appears” as a massless scalar

There is NO mass term!

Of course we want even this Lagrangian to be invariant to

LOCAL GAUGE TRANSFORMATIONS D=+igGLet’s not worry about the higher order symmetries…yet…

FFigGigG4

1*

4*

2

1*

2

1 22 L

free field for the gauge particle introduced

Recall: F=GG

FFGGg

Gig

Gig

4

1*

4*

2

1*

2

*2

*2

*2

1

222

L

again we define: 1 + i2 * 12 + i2

2

FFGGg

Gig

4

1

422

**22

1

2

1

22

2

2

1

2

2

2

1

22

2

2

1

2

2211

L

FFGGg

Gig

4

1

422

**22

1

2

1

22

2

2

1

2

2

2

1

22

2

2

1

2

2211

L

212121212

iiiiGig

22211211

22211211

2

ii

iiG

ig

1221

22

Gg

FFGGg

gG

4

1

422

2

1

2

1

22

2

2

1

2

2

2

1

22

2

2

1

2

12212211

L

Exactly the same potential U U as before!

so, also as before:v

1

2

with22 v

Note:v =0

L = [ + v22 ] + [ ] + [ FF+ GG] gvG

12

12

-14

g2v2

2

+{ gG[] + [2+2v+2]GG g2

2

+ [2v3+v4+2v22

12vv2v3v

v22v4 ) ]}

L = [ + v22 ] + [ ] + [ FF+ GG] gvG12

12

-14

g2v2

2

+{ gG[] + [2+2v+2]GG g2

2

+ [ v4]2

4v( 22 ]

12

which includes a numerical constant

v4

4

and many interactionsbetween and

The constants , v give thecoupling strengths of each

which we can interpret as:

L = scalar field with

22 vm

masslessscalar

free Gaugefield withmass=gv

gvG + a whole bunch of 3-4 legged

vertex couplings

But no MASSLESS scalar particle has ever been observed

is a ~massless spin-½ particleis a massless spin-1 particle

plus gvG seems to describe

G

Is this an interaction?A confused mass term?G not independent? ( some QM oscillation between mixed states?)

Higgs suggested: have not correctly identified thePHYSICALLY OBSERVABLE fundamental particles!

spinless , have plenty of mass!

NoteRemember L is U(1) invariant rotationally invariant in , (1, 2) space –

i.e. it can be equivalently expressedunder any gauge transformation in the complex plane

)(xie /

or

/=(cos + i sin )(1 + i2)

=(1 cos 2 sin ) + i(1 sin + 2 cos)With no loss of generality we are free to pick the gaugefor examplepicking:

121tan

/2 0 and

/ becomes real!

cossin21

1

2

ring of possible ground states

equivalent torotating thesystem byangle

1

2tan

2

2

2

1

2sin

2

2

2

1

1cos

2

2

2

1

21212

2

2

1

2

2

2

1

i

(x) (x) = 0

£

With real, the field vanishes and our Lagrangian reduces to

443222

2222

44

24

2

1

2

1

vvv

vv

GGgGGg

GGg

FF

introducing a MASSIVE Higgs scalar field, , and “getting” a massive vector gauge field G

Notice, with the field gone, all those extra

, , and interaction terms have vanished

Can we employ this same technique to explain massive Z and W vector bosons?

Now apply these techniques: introducing scalar Higgs fields with a self-interaction term and then expanding fields about the ground state of the broken symmetry

to the SUL(2)×U(1)Y Lagrangian in such a way as to

endow W,Zs with mass but leave s massless.

+

0

These two separate cases will follow naturally by assuming the Higgs field is a weak iso-doublet (with a charged and uncharged state)

with Q = I3+Yw /2 and I3 = ±½

Higgs= for Q=0 Yw = 1

Q=1 Yw = 1

couple to EW UY(1) fields: B

Higgs= with Q=I3+Yw /2 and I3 = ±½

Yw = 1

+

0

Consider just the scalar Higgs-relevant terms

£Higgs

22 ) (4

1

2

1) (

2

1 02 with† † †

not a single complex function now, but a vector (an isodoublet)

Once again with each field complex we write

+ = 1 + i2 0 = 3 + i4

† 12 + 2

2 + 32 + 4

2

£Higgs

244114411

2

442211

)(4

1)(

2

1

)(2

1

† † †

† † † †

U =½2† + ¹/4 † )2

LHiggs

244114411

2

442211

)(4

1)(

2

1

)(2

1

† † †

† † † †

just like before:

12 + 2

2 + 32 + 4

2 = 22

Notice how 12, 2

2 … 42 appear interchangeably in the Lagrangian

invariance to SO(4) rotations

Just like with SO(3) where successive rotations can be performed to align a vector

with any chosen axis,we can rotate within this 1-2-3-4 space to

a Lagrangian expressed in terms of a SINGLE PHYSICAL FIELD

Were we to continue without rotating the Lagrangian to its simplest termswe’d find EXTRANEOUS unphysical fields with the kind of bizarre interactions

once again suggestion non-contributing “ghost particles” in our expressions.

So let’s pick ONE field to remain NON-ZERO.

1 or 2 3 or 4

because of the SO(4) symmetry…all are equivalent/identical

might as well make real!

+

0Higgs=

Can either choose v+H(x)

00

v+H(x)or

But we lose our freedom to choose randomly. We have no choice.Each represents a different theory with different physics!

Let’s look at the vacuum expectation values of each proposed state.

v+H(x)0

0v+H(x)

or

000

00 000

Aren’t these just orthogonal?Shouldn’t these just be ZERO?

Yes, of course…for unbroken symmetric ground states.

If non-zero would imply the “empty” vacuum state “OVERLPS with”or contains (quantum mechanically decomposes into) some of + or 0.

But that’s what happens in spontaneous symmetry breaking: the vacuum is redefined “picking up” energy from the field

which defines the minimum energy of the system.

0)(000 xHv )(000 xHv

0)(000 xHv )(000 xHv

)(000 xHv 0

1

= v a non-zerov.e.v.!

This would be disastrous for the choice + = v + H(x)since 0|+ = v implies the vacuum is not chargeless!

But 0| 0 = v is an acceptable choice.

If the Higgs mechanism is at work in our world, this must be nature’s choice.

Let’s recap:We’ve worked through 2 MATHEMATICAL MECHANISMS

for manipulating Lagrangains

Introducing SELF-INTERACTION terms (generalized “mass” terms)showed that a specific GROUND STATE of a system need NOT display the full available symmetry of the Lagrangian

Effectively changing variables by expanding the field about the GROUND STATE (from which we get the physically meaningful ENERGY values, anyway) showed

•The scalar field ends up with a mass term; a 2nd (extraneous) apparently massless field (ghost particle) can be gauged away.

•Any GAUGE FIELD coupling to this scalar (introduced by local invariance) acquires a mass as well!

We then applied these techniques by introducing the scalar Higgs fields

through a weak iso-doublet (with a charged and uncharged state)

+

0Higgs=

0v+H(x)

=

which, because of the explicit SO(4) symmetry, the proper

gauge selection can rotate us within the1, 2, 3, 4 space,

reducing this to a single observable real field which we we expand about the vacuum expectation value v.