I. Lagrangians Why we love symmetries, even to the point of seemingly imagining them in all sorts of new non-geometrical spaces.

II. Introducing interactions into Lagrangians: SU(n) symmetries

III. Symmetry Breaking Where’s the ground state? What the heck are Goldstone bosons?

The Higgs Mechanism

Dan ClaesApril 8 & 15, 2005

An Outline

Weekly MassFriday, 4:00 Service

201 Brace Hall

“The Lagrangian”

an explicit function only of the dynamical variables ofthe field components and their derivatives

derived from the Lagrange function: here for a classical systems of mass points

The precise dynamical behavior of a system of particles can be inferred from the Lagrangian equations of motion

0

iiq

L

q

L

dt

d

VTL

Extended to the case of continuous (wave) function(s) ),( xtk

),()( 3 xtdxtL

L),(

kLL

x

k

0)/(

LLxx

Euler-Lagrangeequation

),( k

LL

x

k

• Does not depend explicitly on spatial coordinates (absolute positions)

- the “Ruler Postulate” - otherwise would violate relativistic invariance

Conservation of Linear Momentum!

• Does not depend explicitly on t (absolute time)-

Conservation of Energy!

translation invariance

time translation invariance

0dt

dH

00

3

t

Pg

xxd

t r

rr

L

• Similarly its invariance under spacial rotations- guarantees Conservation of Angular Momentum!

jijixRx )(

),( k

LL

x

k

• The value of , i.e., at , must depend only on value(s) of and its derivatives at .

• For linear wave equations need terms at least quadratic in and

no “non-local” terms

),( txL ),( tx

which, in general, create problems in causality and with non-real physics quantities

non-local terms never appear in any Standard Model field theorythough often considered in theories seeking to extend field theory“beyond the Standard Model”

THINK: wormholes and time travel

x

To generate differential equations not higher than 2nd order restrict terms to factors of the field components and their 1st derivative

note: renormalizability demands no higher powers in the fields than and than n = 5.nx )(

n)(

),( txk

),( tx

),( k

LL

x

k

• L should be real (field operators Hermitian)guarantees the dynamical variables (energy, momentum, currents)

are real.

• L should be relativistically invariant

so restrictive is this requirement, it guarantees the derived equations of motion are automatically Lorentz invariant

)()( xx LL

the simplest Lagrangian with (x) and dependence is

2222 )())((

mc

xx

£

the (hopefully) familiar Klein-Gordon equation!

from which /

yields

£ £

0)( 2

h

mc

Real Scaler Field

0)(2)(2 2 h

mc

kkip

From the starting point for a relativistic QM equation:

42222 cmcpE

together with the quantum mechanical prescriptionstiE /

1920E. Schrödinger

O. KleinW. Gordon

Matter fields (like the QM wave function of an electron) are known only up to a phase factor ),( xtei

a totally non-geometric attribute

If we choose, we can write this as

21

2

1 i

21

2

1* ior

*2

11

*2

12

i

2

2

2222

1

211

xxxx₤

or

** 2

xx₤

₤

** 2

xx

₤/

₤

0** 2

₤₤/ 02

Then treating and * as independent fields, we find field equations

There’s a new symmetry hidden here: the Lagragian is completelyinvariant under any arbitrary rotation in the complex plane

ie** ie

sincos211

cossin

212

or

two real fields describing particles of identical mass

** 2

xx₤

ie** ie

For an infinitesimally small rotation i ** i

₤

)()/(

xx

)( *

)/*(*

*

xx

₤ ₤ ₤ ₤

)()/( xx

*

)/*(*)(

xx

*

)/*()/(

xxx

= 0

*

*

xxxi ₤

₤ ₤ ₤ ₤

₤ ₤

a conserved 4-vector

satisfying the continuity equation!and changing sign with

a charged current density

This is easily extended to 3 (or more) related, but independent fields

for example:

3

1

2

xx

₤

allows us to consider a class of unitary transformationswider than the single-phase U(1)

the field now has 4 components

2))((

£

the (hopefully still) familiar Klein-Gordon equation!

02

Vector Field Spin-1 particlesγ, g, W, Z

Though

has all the 4-vectors, tensors contracted energy is not positive definite

unless we impose a restriction 0

i.e., only 3 linearly independent components

This is equivalent to replacing the 1st term with the invariant expression

2))((

2 FF

Dirac Field Spin- ½ particlese, , quarksthis field includes 4 independent

components in spinor space

)( 2mci LDIRAC

0 mcpDirac’s equation

with a current vector: )()()( xxxJ

0)()( * xx

We might expect a realistic Lagrangian that involves systems of particles

= LVector + LDIRAC L(r,t)describes

e+e objectsdescribesphotons

but each termdescribes free

non-interactingparticles

L+ L INT

But what should an interaction term look like?How do we introduce the interactions they experience?

These have been single free particle Lagrangians

need something like:

Again: a local Hermitian Lorentz-invariant construction of the various fields and their derivatives reflecting any additional symmetries

the interaction has been observed to respect

The simplest here would be a bilinear form like:

~

2

21 ))((

)( mi

Or consider this: Our free particle equations of motion were all homogeneous differential equations.

02 02

0 mcp

When the field is due to a source, like the electromagnetic (photon!)field you know you need to make the eq. of motion inhomogeneous:

2

charged 4-current density

and

INTL would do the trick!

)()( xxe Dirac electron current

exactly theproposedbilinear form!

Just crack open Jackson:

A charge interacts with a field through:

)(INT AJV

L);(

);(

AVA

JJ

AJ

current-field interactions

the fermion(electron)

the boson(photon) field

Ae )(INT L from the Dirac

expression for J

antiparticle(hermitian conjugate)

field

particlefield

LDirac=iħcmc2

Now let’s look back at the FREE PARTICLE Dirac Lagrangian

Which is OBVIOUSLY invariant under the transformation

ei (a simple phase change)

because ei and in all pairings this added phase cancels!

Dirac matrices Dirac spinors(Iso-vectors, hypercharge)

This one parameter unitary U(1) transformationis called a “GLOBAL GAUGE TRANSFORMATION.”

What if we GENERALIZE this? Introduce more flexibility to the transformation? Extend to:

but still enforce UNITARITY?ei(x)

LOCAL GAUGE TRANSFORMATION

Is the Lagrangian still invariant?

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

LDirac=iħcmc2

So:

L'Dirac = ħc((x))

iħcei(x)( )ei(x)mc2

L'Dirac =

ħc((x))iħc( )mc2

LDirac

For convenience (and to make subsequent steps obvious) define:

(x) (x)ħc q

L'Dirac = q

()LDirac

then this is re-written as

recognize this????

cqe /

L=[iħcmc2qA

L'Dirac = q

()LDirac

If we are going to demand the complete Lagrangian be invariant under even such a LOCAL gauge transformation,

it forces us to ADD to the “free” Dirac Lagrangian something that can ABSORB (account for) that extra term,

i.e., we must assume the full LagrangianHAS TO include a current-field interaction:

AAand that defines its transformation

under the same local gauge transformation

L=[iħcmc2qA

•We introduced the same interaction term moments ago following electrodynamic arguments (Jackson)

• the form of the current density is correctly reproduced

•the transformation rule

A' = A + is exactly (check your Jackson notes!) the rule for GAUGE TRANSFORMATIONS already introduced in e&m!

The exploration of this “new” symmetry shows that for an SU(1)- invariant Lagrangian, the free Dirac Lagrangain is “INCOMPLETE.”

If we chose to allow gauge invariance, it forces to introduce a vector field (here that means A ) that “couples” to

We can generalize our procedures into a PRESCRIPTION to be followed,noting the difference between LOCAL and GLOBAL transformations

are due to derivatives:

c

qi

for U(1) this is a1×1 unitary matrix

(just a number)

/ = [e+iq/ħc] = e+iq/ħc the extra term

that gets introduced

If we replace every derivative in the original free particle Lagrangianwith the “co-variant derivative”

D= + i A

għc

then the gauge transformation of A will cancel the term that appears through

(D )/ = e-iq/ħcD restores the invariance of Li.e.

SU(3) color symmetry of strong interactions

)/( cgieU

3

2

1

The field is assumed to exist in any of 3

possible independent color states

This same procedure, generalized to symmetries in new spaces

3-dimensional matrix formed by linear combinations

of 8 independentfundamental matrices

SU(3) “rotations” occur in an 8-dim “space”

8-dimvector

8 3x3 “generators”

Demanding invariance of the Lagrangian under SU(3)

rotations introduces the massless gluon fields we

believe are responsible for the strong force.

ddu

ud

e

e

_

u

??

u

d_

+

??

hadron decays involve the

transmutation of individual quarks

neutron decay proton

pion

e

e

_

muon decay??

in an effort to explain decays:THEN

duu e+

ud

??e

_

neutrino capture by protons

e

e

as well as the observed inverse of some of these processes:

??

d

neutrino capture by muons

in terms of the gauge model of photon-mediated charged particle interactions

e

e p

p

???

e

en

p

e e

W

d e

W

W u e

u e

e

W

d

required the existence of 3 “weakons” :W , W , Z

SU(2) electro-weak symmetry

2/

ieU1

2

“Rotational symmetry”within weakly coupled left-handed isodoublet states introduces 3 weakons:W+, W, Z and an associated weak isospin “charge”

01

10x

0

0

i

iy

10

01z

10

01

0

0

01

10

i

i

ud

e

e

L

L

L=[iħcmc2F Fg·G

14

This SU(2) theory then

2

describes doublet Dirac particle states in interaction with

3 massless vector fields G

(think of something like the -fields, A)

This followed just by insisting on local SU(2) invariance!

In the Quantum Mechanical view:•These Dirac fermions generate 3 currents, J = (g

)2

•These particles carry a “charge” g, which is the source for the 3 “gauge” fields

The Weak Force so named because unlike the PROMPT processes

e+

e

or the electromagnetic decay: which involves a 1017 sec lifetimepath length (gap) in photographic emulsions mere nm!

e+

e

qr

qg

rg_

which seem instantaneous

weak decays are “SLOW” processes…the particles

involved: , ±, are nearly “stable.”

106 sec700 m pathlengths 108 sec

+++

7 m pathlengths

887 sec

and their inverse processes: scattering or neutrino capture are rare small probability of occurrence

(small rates…small cross sections!).

Such “small cross section” seemed to suggest aSHORT RANGE force…weaker with distance

compared to the infinite range of the Coulomb force

or powerful confinement of the color force

This seems at odds with the predictions of ordinary gauge theory

in which the VECTOR PARTICLES introducedto mediate the forces

like photons and gluons are massless.

This means the symmetry cannot be exact.The symmetry is BROKEN.

Some Classical Fields

The gravitational field around a point source (e.g. the earth) is a scalar field

2

0

2

0

2

0)()()(

),,(zzyyxx

MGzyxg earth

An electric field is classical example of a vector field:

),,(ˆ),,(ˆ),,(ˆ),,( zyxEzyxEzyxEzyxzyx

kjiE

effectively 3 independent fields

Ψ(x,y,z,t;ms) =

Once spin has been introduced, we’ve grown accustomed to writing the total wave function as a two-component “vector”

ψ↑(x,y,z)

ψ↓(x,y,z)

Ψ = e-iEt/ħψ(x,y,z)g(ms) time-

dependent part

spatial part

spinspace

Ψ = e-iEt/ħR(r)S(θ)T(φ)g(ms)

Yℓm(θ,φ)

for a spherically symmetric potential

αβ

But spin is 2-dimensional only for spin-½ systems.

The 2-dim form is better recognized as just one fundamental representation of angular momentum.

Recognizing the most general solutions involve ψ/ψ* (particle/antiparticle) fields, the Dirac formalism modifies this to 4-component fields!

That 2-dim spin-½ space is operated on by

01

10x

0

0

i

iy

10

01z

10

01

0

0

01

10c

i

iba

or more generally by

The SU(2) transformation group (generalized “rotations” in 2-dim space)is based on operators: 2/

ieU

“generated by”traceless Hermitian

matrices

What’s the most general traceless HERMITIAN 22 matrices?

c aiba+ib c

aibaib

cc

and check out:

= a +b +c 0 11 0

0 -ii 0

1 00 -1

10

01

0

0

01

10c

i

iba

SU(3)FOR

What’s the form of the most general traceless HERMITIAN 3×3 matrix?

a3 a1ia2

a1ia2

Diagonal termshave to be real!

a4ia5

a4ia5

a6ia7

a6ia7

Transposed positionsmust be

conjugates!

Must be traceless!

= a1 +a2 +a3 +a4

+a3 +a6 +a7 +a8

0 1 01 0 00 0 0

0 -i 0i 0 00 0 0

0 0 10 0 01 0 0

0 0 -i0 0 0i 0 0

0 0 00 0 10 1 0

0 0 00 0 -i0 i 0

SU(3) “rotations” occur in an 8-dim “space”

ei· /2ħ

8 independentparameters

8 3x3 operators

SU(2) “rotations” occur in an 3-dim “space”

ei· /2ħ

3 independentparameters

three 2x2 matrix operators

U(1) local gauge transformation (of simple phase)electrical charge-coupled photon field mediates EM interactions

8 simultaneous gauge transformations 8 vector boson fields

3 simultaneous gauge transformations 3 vector boson fields

Spontaneous Symmetry BreakingEnglert & Brout, 1964Higgs 1964, 1966Guralnick, Hagen & Kibble 1964Kibble 1967

The Lagrangian & derived equations of motion for a system possess symmetries which simply do NOT hold for a specific ground state

of the system. (The full symmetry MAY be re-stored at higher energies.)

(1) A flexible rod under longitudinal compression.(2) A ball dropped down a flask with a convex bottom.• Lagrangian symmetric with respect to rotations about the central axis

• once force exceeds some critical value it must buckle sideways forming an arc in SOME arbitrary direction

Although one direction is chosen, the complete set of all possible final shapes DOES show the full symmetry.

What is the GROUND STATE? lowest energy state

What does GROUND STATE mean inQuantum Field Theory?

Shouldn’t that just be the vacuum state? | 0

pp 0)(†

which has an 2242 cpcmE

compared to 0.

Fields are fluctuations about the GROUND STATE.Virtual particles are created from the VACUUM.

The field configuration of MINIMUM ENERGY is

usually just the obvious 0

(e.g. out of away from a particle’s location)

Following the definition of the discrete classical L = T Vwe separate out the clearly identifiable “kinetic” part

L = T( ∂ ) – V( )

For the simple scalar field considered earlier V()=½m22

is a 2nd-order parabola:

V()The → 0 case corresponds to a stable minimum of the potential.

Notice in this simple model V is symmetric to reflections of

Quantization of the fieldcorresponds to small oscillations

about the position of equilibrium there.

Obviously as 0 there are no intereactions between the fieldsand we will have only free particle states.

we have the empty state | 0 And as (or in regions where) 0

representing the lowest possible energy stateand serving as the vacuum.

The exact numerical value of the energy content/density of | 0 is totally arbitrary…relative.

We measure a state’s or system’s energy with respect to it and usually assume it is or set it to 0.

What if the EMPTY STATE did NOT carry the lowest achievable energy?

qp 00 †

qp=

We will call 0| |0 = vev the “vacuum expectation value” of an operator state.

Now let’s consider a model with a quartic (“self-interaction”) term:

£=½½ 2¼ 4

Such models were 1st consideredfor observed interactions like

+ → 0 0

V() ½ 2 + ¼ 4 ½ 2( ½λ 2)

Now the extrema at = 0is a local maximum!

Stable minima at = ±

a doubly degenerate vacuum state

√λ

√λ

√λ

The depth of the potential at 0 is

4

4

0

V

with this sign, we’ve introduced a term that looks like an imaginary

mass (in tachyon models)

V() ½ 2 + ¼ 4

A translation (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

Complex Scalar field

£=½+½¼

Note: OBVIOUSLY globally invariant under U(1) transformations

ei

£=½11 + ½22 ½1

2 + 2

2 ¼12 + 2

2

Which is ROTATIONALLY invariant under SO(2)!! 1 2

Our Lagrangian yields the field equation:

1 + 12 + 11

2 + 2

2 = 0 or equivalently

2 + 22 + 21

2 + 2

2 = 0 some sort of

interaction betweenthe independent states

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 Lfree field for the gauge

particle introduced

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

gvG

12

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

Recall: F=GG

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

This is the technique employed to explain massive Z and W vector bosons…

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 inavariance) acquires a mass as well!

18

( )H +v† ( )H +v( 2g22W+

W+ + (g12+g2

2) ZZ )†

No AA term is introduced! The photon remains massless!

But we do get the terms

18

v22g22W+

W+† MW = vg2

18 (g1

2+g22 )Z Z MZ = v√g1

2 + g221

2

MW

MZ

g2

√g12 + g2

2

At this stage we may not know precisely the values of g1 and g2, but note:

=

12

g1g2

g12+g1

2 ψe ψeA

When repeated on a U(1) and SU(2) symmetric Lagrangianfind the terms:

shifted scalar

field, (x)

MW = MZcosθw

e e

W

u e

e

W

d

e+e + N p + e+e

~gW =e

sinθW( )2 2

g1g2

g12+g1

2 = e

and we do know THIS much about g1 and g2

to extraordinary precision!

from other weak processes:

lifetimes (decay rate cross sections) give us sin2θW

Notice = cos W according to this theory.MW

MZ

where sin2W=0.2325 +0.00159.0019

We don’t know v, but information on the coupling constants g1 and g2 follow from

• lifetime measurements of -decay: neutron lifetime=886.7±1.9 sec and • a high precision measurement of muon lifetime=2.19703±0.00004 sec and • measurements (sometimes just crude approximations perhaps) of the cross-sections for the inverse reactions:

e- + p n + eelectron capture

e + p e+ + n anti-neutrino absorption

as well as e + e- e- + e neutrino scattering

By early 1980s had the following theoretically predicted masses:

MZ = 92 0.7 GeV MW = cosWMZ = 80.2 1.1 GeV

Late spring, 1989 Mark II detector, SLAC August 1989 LEP accelerator at CERN

discovered opposite-sign lepton pairs with an invariant mass ofMZ=92 GeV

and lepton-missing energy (neutrino) invariant masses ofMW=80 GeV

Current precision measurements give: MW = 80.482 0.091 GeV MZ = 91.1885 0.0022 GeV

Electroweak Precision Tests

LEP Line shape: mZ(GeV)ΓZ(GeV)0

h(nb)Rℓ≡Γh / Γℓ

A0,ℓFB

τ polarization: Aτ

Aε

heavy flavor: Rb≡Γb / Γb

Rc≡Γc / Γb

A0,bFB

A0,cFB

qq charge asymmetry: sin2θw

91.1884 ± 0.00222.49693 ± 0.0032 41.488 ± 0.078 20.788 ± 0.032 0.0172 ± 0.012 0.1418 ± 0.0075 0.1390 ± 0.0089 0.2219 ± 0.0017 0.1540 ± 0.0074 0.0997 ± 0.0031 0.0729 ± 0.0058 0.2325 ± 0.0013

2.4985 41.462 20.760 0.0168 0.1486 0.1486 0.2157 0.1722 0.1041 0.0746 0.2325

SLC A0,ℓFB

Ab

Ac

pp mW

0.1551 ± 0.00400.841 ± 0.0530.606 ± 0.090

80.26 ± 0.016

0.14860.9350.669

80.40

Can the mass terms of the regular Dirac particles in theDirac Lagrangian also be generated from “first principles”?

Theorists noted there is an additional gauge-invariant termwe could try adding to the Lagrangian:

A Yukawa coupling which, for electrons, for example, would read

L

eRRLe e

eeeG

)()( 0

0int L

which with Higgs=0

v+H(x)becomes

Gv[eLeR + eReL] + GH[eLeR + eReL] _ _ _ _

e e_

e e_

from which we can identify: me = Gv

or eHev

meem e

e

Gv[eLeR + eReL] + GH[eLeR + eReL] _ _ _ _

Bibliography

Classical Mechanics, H. Goldstein Lagrangians, symmetries and conservation lawsAddison-Wesley (2nd edition) 1983

Classical Electrodynamics covariant form of Maxwell’s equationsJ. D. Jackson (3rd Edition) gauge transformation on the 4-potentialJohn Wiley & Sons 1998 electron-photon interaction Lagrangian

Relativistic Quantum Fields Klein Gordon Equation, Dirac EquationJ. Bjorken, S. DrellMcGraw-Hill 1965

Introduction to High Energy Physics gauge transformation & conserved charges Donald H. Perkins (4th Edition)Cambridge University Press 2000

Advanced Quantum Mechanics neutral and complex scalar fieldsJ. J. Sakurai gauge transformations & conserved chargesAddison-Wesley 1967 vector potentials in quantum mechanics

Quantum Fields real scalar fields, vector fields, Dirac fieldsN. Bogoliubov, D. ShirkovBenjamin/Cummings 1983

Weak Interactions of Leptons & Quarks Electro-weak unification, U(1), SU(2), SU(3)E. Commins, P. Bucksbaum electro-weak symmetry breakingCambridge University Press 1983 the Higgs field

Appendix

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 “OVERLAPS 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.

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.

With the choice of gauge settled: +

0Higgs=0

v+H(x)=

Let’s try to couple these scalar “Higgs” fields to W, B which means

WigBig

22 21

YDreplace:

which makes the 1st term in our Lagrangian:

WigBigWigBig

22222

12121

YY †

The “mass-generating” interaction is identified by simple constantsproviding the coefficient for a term simply quadratic in the gauge fields

so let’s just look at:

vHWigBig

vHWigBig

0

22

0

222

12121

YY †

where Y =1 for the coupling to B

vHWigBig

vHWigBig

0

22

10

22

1

2

12121

†

recall that

τ ·W→ →

= W1 + W2 + W30 11 0

0 -ii 0

1 00 -1

= W3 W1iW2

W1iW2 W3

12

= ( )

3

21

22W

gBg

2

321

22W

gBg

)(2

2 g

)(2

2 g

0

H + v0 H +v

18

= ( ) 321

WgBg 2

Zgg 2

2

2

1

Wg2

2 0

H + v0 H +v

†

†

18

= ( )H +v† ( )H +v

W1iW2

W1iW2

Wg2

2

( 2g22W+

W+ + (g12+g2

2) ZZ )†

18

= ( )H +v† ( )H +v( 2g22W+

W+ + (g12+g2

2) ZZ )†

No AA term has been introduced! The photon is massless!

But we do get the terms

18

v22g22W+

W+† MW = vg2

18 (g1

2+g22 )Z Z MZ = v√g1

2 + g221

2

MW

MZ

2g2

√g12 + g2

2

At this stage we may not know precisely the values of g1 and g2, but note:

=

12