Spontaneous breaking of continuous symmetriesbased on S-32

Consider the theory of a complex scalar field:

the lagrangian is invariant under the U(1) transformation:

for negative, the minimum of the potential

is achieved for

we have a continuous family of minima (ground states):

327

is achieved for

we have a continuous family of minima (ground states):

all the minima are equivalent, there is a flat direction in the

field space (we can move along without changing the energy)

the physical consequence of a flat direction is the existence of a massless particle - Goldstone boson

for we can write:

b is massless!

and we get:

real scalar fields

328

It is also instructive to use a different parameterization:real scalar fields

now we get:

the U(1) transformation takes a simple form:

parameterizes the flat direction

is massless!spectrum and scattering amplitudes are independent of field redefinitions!

The Goldstone boson remains massless even after including loop corrections!

329

To see it, note that if the GB remains massless, the exact propagator:

has to have a pole at .

The vertices are proportional to momenta of (because of derivatives) and so they vanish.

we can calculate it by summing 1PI diagrams with two external lines with .

Note: the theory expanded around the minimum of the potential has interactions that were not present in the original theory. However it turns out that divergent parts of the counterterms for the theory with (where the symmetry is unbroken) will also serve to cancel the divergencies in the theory with where the symmetry is spontaneously broken. This is a general rule for theories with spontaneous symmetry breaking.

330

Let’s now generalize our results for a nonabelian case:

the lagrangian is invariant under the SO(N) transformation:

set of antisymmetric generator matrices with a single nonzero entry -i above the main diagonal

set of infinitesimal parameters

for , the minimum of the potential is achieved for:

with

the vacuum expectation value, N-component vector with arbitrary direction

we can choose the coordinate system (SO(N) rotation) so that

331

Let’s make an infinitesimal SO(N) transformation; this changes the VEV:

zero for all the generators with no non-zero entry in the last column - UNBROKEN GENERATORS

non-zero for N-1 generators with a non-zero entry in the last column - BROKEN GENERATORS

these do not change the VEV of the field, and correspond to the unbroken symmetry

these change the VEV of the field but not the energy; each of them corresponds to a flat direction in

field space; each flat direction implies the existence of a massless

particle - Goldstone boson.For SO(N) we have

unbroken generators SO(N-1) - obvious symmetry of the lagrangian!

332

Let’s work it out in detail; we can write our lagrangian as:

where

summed from 1 to N

Let’s shift the field by the VEV and define:

we get:

where: summed from 1 to N-1

SO(N-1) - manifest symmetry of the lagrangian!

N-1 massless particles - Goldstone bosons

333

Spontaneous breaking of gauge symmetriesbased on S-84,85

Consider scalar electrodynamics:

where

for , the minimum of the potential is achieved for:

we write:

and the scalar potential becomes:

334

The gauge symmetry allows us to shift the phase of by an arbitrary spacetime function; we can set:

with this choice the kinetic term becomes:Unitary gauge

mass term for the gauge field!

Higgs mechanism: the Goldstone boson disappears and the gauge field acquires a mass. The Goldstone boson has become the longitudinal polarization of the massive gauge field.

The scalar field that is used to break the gauge symmetry is called the Higgs field.

335

Part of the lagrangian quadratic in gauge fields:

the equation of motion:

acting on both sides with we find:

thus each component obeys the Klein-Gordon equation:

this is not a gauge-fixing condition!

the general solution is:

with

336

the general solution is:

in the rest frame we can choose the polarization vector to correspond to definite spin along the z-axis:

polarization vectors together with the timelike unit vector form an orthonormal and complete set:

337

In analogy with the scalar field theory or QED the propagator for the “massive” gauge field is:

and interactions can be read off of:

and

338

taking the unitary gauge, , corresponds to inserting a functional delta function:

There are complications with calculating loop diagrams!

to the path integral. Changing integration variables from and to and we must include the functional determinant:

arbitrary constant

ghost propagator:

ghost-ghost-scalar vertex:

no ghost-gauge field interactionsthe ghost propagator doesn’t contain momentum and so loop diagrams with arbitrarily many external lines diverge more and more (also the gauge field propagator scales like for large momenta); difficult to establish renormalizability.

339

The gauge doesn’t suffer from this problem:real scalar fields

with this parameterization, the potential is:

and the covariant derivative is:

and so the kinetic term can be written as:

340

mass term for b

rearranging the kinetic term:

cross term between the vector field and

the b-field

For abelian gauge theory in the absence of spontaneous symmetry breaking we fixed the gauge by adding the gauge-fixing and ghost terms:

ghost fields have no interactions and can be ignored.

For spontaneously broken theory we choose:

and we get:

canc

el

integration by parts

341

Let’s choose:

gauge-fixing function

we will get the gaugethen

-1 for acting as an anticommuting object

or REVIEW

342

now we can easily perform the path integral over B:

it is equivalent to solving the classical equation of motion,

and substituting the result back to the formula:

we obtained the gauge fixing lagrangian and the ghost lagrangian

REVIEW

343

and for the ghost term we find:

For spontaneously broken theory we choose:

ghost has the same mass as the b-field!

344

Collecting terms quadratic in the vector field:

in the momentum space we have:

which can be easily inverted:

propagator for the massive vector field in the gauge

345

Propagator highly simplifies for :

propagator for the massive vector field in the gauge:

transverse components propagate with mass M

longitudinal component propagate with mass(the same as ghost and b)

since scattering amplitudes do not depend on they all are unphysical particles

typically used for loop calculations

346

Summary of gauge:

interactions are collected here:

ghost and the b-field propagators:

physical Higgs field with mass:

gauge field (3 polarizations) with mass:

347

It is instructive to consider limit:

gauge field propagator:

this is what we found in the unitary gauge

ghost and the b-fields become infinitely heavy

but we cannot just drop the ghost field because its interaction becomes infinite as well; instead we can define:

and consider the ghost propagator to be and the vertex

this is what we found in the unitary gauge

The in the limit is equivalent to unitary gauge!

348

Spontaneously broken nonabelian gauge theorybased on S-84,86

Generalization to a nonabelian gauge theory is straightforward:

the potential is minimized for:

plugging the vev to the kinetic term we get a mass term for gauge fields:

where

349

gauge fields corresponding to unbroken generators, , remain massless

plugging the vev to the kinetic term we get a mass term for gauge fields:

where

gauge fields corresponding to broken generators, , get a mass

(and they form representations of the unbroken group)

(these are the gauge fields of unbroken gauge symmetry)

we can repeat everything that we did for the abelian case, see S-86, instead of working out details for the general case let’s focus on a specific example...

350

The Standard Model: gauge and Higgs sectorbased on S-87

The Standard Model of elementary particles describes almost everything we know about our world. It is based on SU(3)xSU(2)xU(1) gauge symmetry spontaneously broken to SU(3)xU(1).

All particles acquire mass from interactions with the Higgs field in the process of electroweak

symmetry breaking

Glashow-Weinberg-Salam model

351

the covariant derivative is:

can be conveniently written as

Let’s start with the electroweak part which is described by SU(2)xU(1) gauge symmetry and one complex scalar field - the Higgs field in the representation:

352

the potential is (the usual one):

the scalar field get a vev, and we can use a global SU(2) transformation to bring it to the form:

plugging the vev to the kinetic term:

we get the mass term for gauge fields:

353

we have to diagonalize the mass matrix for gauge fields, we define:

then we have

the weak mixing angle

remains massless, it is the gauge field of

354

let’s work out the complete lagrangian:we will use unitary gauge

(it is simple and sufficient for tree-level calculations)

a complex scalar field (which is a doublet under SU(2)) can be written in terms of 4 real scalar fields; three of them become the longitudinal components of W and Z, thus we can write the Higgs doublet as: real scalar field

the Higgs field has the usual kinetic term and the potential is now:

we obtained the mass terms for gauge fields by inserting just the vev; to get the whole lagrangian that contains also interactions of Higgs and gauge fields we simply multiply the mass term for gauge fields by

355

we can combine

finally, let’s look at the kinetic terms for the gauge fields:

where

where

and we get a part of the lagrangian:

356

we also have:

where:

and we get another part of the lagrangian:

357

Putting all the terms together:

charge of is ,e is positive (opposite sign that we used in QED)

The lagrangian exhibits manifest gauge invariance!

358

The Standard Model: Lepton sectorbased on S-88

as far as we know there are three generations of leptons:

and they are described by introducing left-handed Weyl fields:

just a name

359

the covariant derivatives are:

and the kinetic term is the usual one for Weyl fields:

there is no gauge group singlet contained in any of the products:

it is not possible to write any mass term!

360

Lagrangians for spinor fields

we want to find a suitable lagrangian for left- and right-handed spinor fields.

based on S-36

Lorentz invariant and hermitian

it should be:

quadratic in and equations of motion will be linear with plane wave solutions

(suitable for describing free particles)

terms with no derivative:

terms with derivatives:

+ h.c.

would lead to a hamiltonian unbounded from belowREV

IEW

361

to get a bounded hamiltonian the kinetic term has to contain both and , a candidate is:

is hermitian up to a total divergence

does not contribute to the action

are hermitianREVIEW

362

Taking hermitian conjugate:

Our complete lagrangian is:

the phase of m can be absorbed into the definition of fields

and so without loss of generality we can take m to be real and positive.

Equation of motion:

are hermitianREVIEW

363

however we can write a Yukawa term of the form:

there are no other terms that have mass dimension four or less!

in the unitary gauge we have:

and the Yukawa term becomes:

364

it is convenient to label the components of the lepton doublet as:

then we have:

and we see that the electron has acquired a mass:

and the neutrino remained massless!

we define a Dirac field for the electron

365

consider a theory of two left-handed spinor fields:

i = 1,2

the lagrangian is invariant under the SO(2) transformation:

it can be written in the form that is manifestly U(1) symmetric:

REVIEW

366

Equations of motion for this theory:

we can define a four-component Dirac field:

Dirac equation

we want to write the lagrangian in terms of the Dirac field:

Let’s define:numerically

but different spinor index structureREVIEW

367

Then we find:

Thus we have:REVIEW

368

Thus the lagrangian can be written as:

The U(1) symmetry is obvious:

The Nether current associated with this symmetry is:

later we will see that this is the electromagnetic currentREVIEW

369

If we want to go back from 4-component Dirac or Majorana fields to the two-component Weyl fields, it is useful to define a projection matrix:

just a name

We can define left and right projection matrices:

And for a Dirac field we find:

REVIEW

370

We can describe the neutrino with a Majorana field:

and it is also convenient to define:

because then the kinetic term:

can be written as:

a “Dirac field” for the neutrino

371

Now we have to figure out lepton-lepton-gauge boson interaction terms:

(we want to write covariant derivatives in terms of , and )

we have

and

we identify it with electric charge:

372

since we have:

the electric charge assignments are as expected:

covariant derivatives in terms of the four-component fields:

373

Putting pieces together,

we find the interaction lagrangian:

where

374

The connection with Fermi theory:

if we want to calculate some scattering amplitude (or a decay rate) of leptons with momenta well below masses of W and Z, the propagators effectively become:

and we get 4-fermion-interaction terms:

where the Fermi constant is:

375

Generalization to three lepton generations:

the most general Yukawa term:

a complex 3x3 matrix;it can be diagonalized (with positive or zero entries on the diagonal)by a bi-unitary transformation

fermion masses are then given by:

diagonal entries

currents remain diagonal, just add the generation index

376

We can calculate e.g. muon decay:

the relevant part of the charged current:

we get the effective lagrangian:

and the rest is straightforward...

377

The Standard Model: Quark sectorbased on S-89

as far as we know there are three generations of quarks:

and they are described by introducing left-handed Weyl fields:

just a name

378

the covariant derivatives are:

and the kinetic terms are the usual ones for Weyl fields:

there is no gauge group singlet contained in any of the products;

it is not possible to write any mass term!

379

however we can write a Yukawa terms of the form:

there are no other terms that have mass dimension four or less!

in the unitary gauge we have:

and the Yukawa terms become:

380

it is convenient to label the components of the quark doublet as:

then we have:

and we see that the up and down quarks have acquired masses:

...work out interactions with gauge fields, and generalization to three families...

we define Dirac fields for the down and up quarks

381