Download Spontaneous breaking of continuous symmetries

Spontaneous breaking of continuous symmetries
based on S-32
It is also instructive to use a different parameterization:
real scalar fields
Consider the theory of a complex scalar field:
the U(1) transformation takes a simple form:
the lagrangian is invariant under the U(1) transformation:
for
negative, the minimum of the potential
parameterizes the flat direction
now we get:
is achieved for
is massless!
spectrum and scattering amplitudes are independent of field redefinitions!
The Goldstone boson remains massless even after including loop corrections!
we have a continuous family of minima (ground states):
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To see it, note that if the GB remains massless, the exact propagator:
is achieved for
has to have a pole at
we have a continuous family of minima (ground states):
for
we can write:
and we get:
b is massless!
real scalar fields
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
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.
we can calculate it by summing 1PI diagrams
with two external
lines with
.
The vertices are proportional to momenta of
(because of derivatives) and so they vanish.
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.
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Let’s now generalize our results for a nonabelian case:
Let’s work it out in detail; we can write our lagrangian as:
where
summed from 1 to N
the lagrangian is invariant under the SO(N) transformation:
Let’s shift the field by the VEV and define:
set of
for
infinitesimal parameters
set of antisymmetric generator matrices with a
single nonzero entry -i above the main diagonal
, the minimum of the potential is achieved for:
we get:
where:
with
summed from 1 to N-1
N-1 massless particles
- Goldstone bosons
the vacuum expectation value, N-component vector with arbitrary direction
we can choose the coordinate system (SO(N) rotation) so that
SO(N-1) - manifest symmetry of the lagrangian!
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Spontaneous breaking of gauge symmetries
Let’s make an infinitesimal SO(N) transformation; this changes the VEV:
based on S-84,85
Consider scalar electrodynamics:
zero for all the generators with no
non-zero entry in the last column UNBROKEN GENERATORS
where
non-zero for N-1 generators with a
non-zero entry in the last column BROKEN GENERATORS
for
these do not change the VEV
of the field, and correspond
to the unbroken symmetry
For SO(N) we have
unbroken generators
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.
, the minimum of the potential is achieved for:
we write:
and the scalar potential becomes:
SO(N-1) - obvious symmetry of the lagrangian!
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the general solution is:
The gauge symmetry allows us to shift the phase of
spacetime function; we can set:
by an arbitrary
Unitary gauge
with this choice the kinetic term becomes:
mass term for the gauge field!
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
orthonormal and complete set:
form an
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.
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Part of the lagrangian quadratic in gauge fields:
the equation of motion:
acting on both sides with
In analogy with the scalar field theory or QED the propagator for the
“massive” gauge field is:
we find:
this is not a gauge-fixing condition!
thus each component obeys the Klein-Gordon equation:
and interactions can be read off of:
the general solution is:
and
with
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There are complications with calculating loop diagrams!
taking the unitary gauge,
, corresponds to inserting a functional
delta function:
cross term between
the vector field and
the b-field
to
For abelian gauge theory in the absence of spontaneous symmetry breaking we fixed
the gauge by adding the gauge-fixing and ghost terms:
ghost-ghost-scalar vertex:
arbitrary constant
ghost propagator:
no ghost-gauge field interactions
the 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.
For spontaneously broken theory we choose:
can
ghost fields have no interactions and can be ignored.
l
and
ce
to the path integral. Changing integration variables from
and
we must include the functional determinant:
rearranging the kinetic term:
and we get:
mass term for b
integration by parts
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The
gauge doesn’t suffer from this problem:
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Let’s choose:
real scalar fields
W
IE
gauge-fixing function
with this parameterization, the potential is:
then
V
E
-1 for
and the covariant derivative is:
and so the kinetic term can be written as:
or
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R
we will get the
gauge
acting as an anticommuting object
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Collecting terms quadratic in the vector field:
W
IE
now we can easily perform the path integral over B:
in the momentum space we have:
it is equivalent to solving the classical equation of motion,
V
E
and substituting the result back to the formula:
we obtained the gauge fixing lagrangian and the ghost lagrangian
R
which can be easily inverted:
propagator for the massive vector field in the
gauge
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propagator for the massive vector field in the
gauge:
For spontaneously broken theory we choose:
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
and for the ghost term we find:
Propagator highly simplifies for
:
ghost has the same mass as the b-field!
typically used for loop calculations
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Summary of
Spontaneously broken nonabelian gauge theory
gauge:
gauge field (3 polarizations) with mass:
based on S-84,86
Generalization to a nonabelian gauge theory is straightforward:
ghost and the b-field propagators:
physical Higgs field with mass:
interactions are collected here:
the potential is minimized for:
plugging the vev to the kinetic term we get a mass term for gauge fields:
where
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It is instructive to consider
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plugging the vev to the kinetic term we get a mass term for gauge fields:
limit:
gauge field propagator:
where
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
gauge fields corresponding to unbroken
generators,
, remain massless
gauge fields corresponding to broken
generators,
, get a mass
(these are the gauge fields of
unbroken gauge symmetry)
(and they form representations of the
unbroken group)
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...
is equivalent to unitary gauge!
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The Standard Model: gauge and Higgs sector
the potential is (the usual one):
based 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).
Glashow-Weinberg-Salam model
All particles acquire mass from
interactions with the Higgs field
in the process of electroweak
symmetry breaking
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:
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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:
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we have to diagonalize the mass matrix for gauge fields, we define:
the weak mixing angle
the covariant derivative is:
then we have
can be conveniently written as
remains massless,
it is the gauge field of
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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 also have:
where:
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
and we get another part of the lagrangian:
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Putting all the terms together:
finally, let’s look at the kinetic terms for the gauge fields:
where
we can combine
where
charge of
is
,
e is positive (opposite sign that we used in QED)
The lagrangian exhibits manifest
and we get a part of the lagrangian:
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gauge invariance!
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The Standard Model: Lepton sector
Lagrangians for spinor fields
based on S-88
as far as we know there are three generations of leptons:
W
IE
based on S-36
we want to find a suitable lagrangian for left- and right-handed spinor fields.
it should be:
Lorentz invariant and hermitian
quadratic in
V
E
and
equations of motion will be linear with plane wave solutions
(suitable for describing free particles)
R
terms with no derivative:
and they are described by introducing left-handed Weyl fields:
+ h.c.
terms with derivatives:
just a name
would lead to a hamiltonian unbounded from below
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to get a bounded hamiltonian the kinetic term has to contain both
and
, a candidate is:
the covariant derivatives are:
W
IE
is hermitian up to a total divergence
and the kinetic term is the usual one for Weyl fields:
there is no gauge group singlet contained in any of the products:
R
are hermitian
it is not possible to write any mass term!
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V
E
does not contribute to the action
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Our complete lagrangian is:
W
IE
it is convenient to label the components of the lepton doublet as:
the phase of m can be absorbed into the definition of fields
V
E
and so without loss of generality we can take m to be real and positive.
Equation of motion:
R
Taking hermitian conjugate:
are hermitian
then we have:
we define a Dirac field for the electron
and we see that the electron has acquired a mass:
and the neutrino remained massless!
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consider a theory of two left-handed spinor fields:
however we can write a Yukawa term of the form:
W
IE
i = 1,2
the lagrangian is invariant under the SO(2) transformation:
V
E
it can be written in the form that is manifestly U(1) symmetric:
there are no other terms that have mass dimension four or less!
in the unitary gauge we have:
R
and the Yukawa term becomes:
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Thus the lagrangian can be written as:
Equations of motion for this theory:
W
IE
we can define a four-component Dirac field:
V
E
Dirac equation
we want to write the lagrangian in terms of the Dirac field:
R
Let’s define:
W
IE
The U(1) symmetry is obvious:
R
V
E
The Nether current associated with this symmetry is:
numerically
but different spinor index structure
later we will see that this is the electromagnetic current
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Then we find:
R
V
E
Thus we have:
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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:
W
IE
W
IE
just a name
We can define left and right projection matrices:
R
V
E
And for a Dirac field we find:
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We can describe the neutrino with a Majorana field:
since we have:
and it is also convenient to define:
the electric charge assignments are as expected:
a “Dirac field” for the neutrino
covariant derivatives in terms of the four-component fields:
because then the kinetic term:
can be written as:
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Now we have to figure out lepton-lepton-gauge boson interaction terms:
(we want to write covariant derivatives in terms of
,
and
)
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Putting pieces together,
we have
we find the interaction lagrangian:
and
where
we identify it with electric charge:
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The connection with Fermi theory:
We can calculate e.g. muon decay:
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:
the relevant part of the charged current:
and we get 4-fermion-interaction terms:
we get the effective lagrangian:
and the rest is straightforward...
where the Fermi constant is:
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The Standard Model: Quark sector
Generalization to three lepton generations:
based on S-89
as far as we know there are three generations of quarks:
the most general Yukawa term:
currents remain diagonal,
just add the generation index
a complex 3x3 matrix;
it can be diagonalized (with positive
or zero entries on the diagonal)
by a bi-unitary transformation
and they are described by introducing left-handed Weyl fields:
fermion masses are then given by:
just a name
diagonal entries
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the covariant derivatives are:
it is convenient to label the components of the quark doublet as:
then we have:
and the kinetic terms are the usual ones for Weyl fields:
we define Dirac fields for the down and up quarks
there is no gauge group singlet contained in any of the products;
it is not possible to write any mass term!
and we see that the up and down quarks have acquired masses:
...work out interactions with gauge fields, and generalization to three families...
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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:
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