This article describes the mathematics of
the Standard Model of particle physics, a
gauge quantum field theory containing the
internal symmetries of the unitary product
group SU(3) × SU(2) × U(1). The
theory is commonly viewed as containing the
fundamental set of particles – the leptons,
quarks, gauge bosons and the Higgs particle.
The Standard Model is renormalizable and mathematically
self-consistent, however despite having huge
and continued successes in providing experimental
predictions it does leave some unexplained
phenomena. In particular, although the physics
of special relativity is incorporated, general
relativity is not, and the Standard Model
will fail at energies or distances where the
graviton is expected to emerge. Therefore,
in a modern field theory context, it is seen
as an effective field theory.
This article requires some background in physics
and mathematics, but is designed as both an
introduction and a reference.
== Quantum field theory ==
The standard model is a quantum field theory,
meaning its fundamental objects are quantum
fields which are defined at all points in
spacetime. These fields are
the fermion fields, ψ, which account for
"matter particles";
the electroweak boson fields
W
1
,
W
2
,
W
3
{\displaystyle W_{1},W_{2},W_{3}}
, and B;
the gluon field, Ga; and
the Higgs field, φ.That these are quantum
rather than classical fields has the mathematical
consequence that they are operator-valued.
In particular, values of the fields generally
do not commute. As operators, they act upon
the quantum state (ket vector).
The dynamics of the quantum state and the
fundamental fields are determined by the Lagrangian
density
L
{\displaystyle {\mathcal {L}}}
(usually for short just called the Lagrangian).
This plays a role similar to that of the Schrödinger
equation in non-relativistic quantum mechanics,
but a Lagrangian is not an equation of motion
– rather, it is a polynomial function of
the fields and their derivatives, and used
with the principle of least action. While
it would be possible to derive a system of
differential equations governing the fields
from the Lagrangian, it is more common to
use other techniques to compute with quantum
field theories.
The standard model is furthermore a gauge
theory, which means there are degrees of freedom
in the mathematical formalism which do not
correspond to changes in the physical state.
The gauge group of the standard model is SU(3) × SU(2) × U(1),
where U(1) acts on B and φ, SU(2) acts on
W and φ, and SU(3) acts on G. The fermion
field ψ also transforms under these symmetries,
although all of them leave some parts of it
unchanged.
=== The role of the quantum fields ===
In classical mechanics, the state of a system
can usually be captured by a small set of
variables, and the dynamics of the system
is thus determined by the time evolution of
these variables. In classical field theory,
the field is part of the state of the system,
so in order to describe it completely one
effectively introduces separate variables
for every point in spacetime (even though
there are many restrictions on how the values
of the field "variables" may vary from point
to point, for example in the form of field
equations involving partial derivatives of
the fields).
In quantum mechanics, the classical variables
are turned into operators, but these do not
capture the state of the system, which is
instead encoded into a wavefunction ψ or
more abstract ket vector. If ψ is an eigenstate
with respect to an operator P, then Pψ = λψ
for the corresponding eigenvalue λ, and hence
letting an operator P act on ψ is analogous
to multiplying ψ by the value of the classical
variable to which P corresponds. By extension,
a classical formula where all variables have
been replaced by the corresponding operators
will behave like an operator which, when it
acts upon the state of the system, multiplies
it by the analogue of the quantity that the
classical formula would compute. The formula
as such does however not contain any information
about the state of the system; it would evaluate
to the same operator regardless of what state
the system is in.
Quantum fields relate to quantum mechanics
as classical fields do to classical mechanics,
i.e., there is a separate operator for every
point in spacetime, and these operators do
not carry any information about the state
of the system; they are merely used to exhibit
some aspect of the state, at the point to
which they belong. In particular, the quantum
fields are not wavefunctions, even though
the equations which govern their time evolution
may be deceptively similar to those of the
corresponding wavefunction in a semiclassical
formulation. There is no variation in strength
of the fields between different points in
spacetime; the variation that happens is rather
one of phase factors.
=== Vectors, scalars, and spinors ===
Mathematically it may look as though all of
the fields are vector-valued (in addition
to being operator-valued), since they all
have several components, can be multiplied
by matrices, etc., but physicists assign a
more specific physical meaning to the word:
a vector is something which transforms like
a four-vector under Lorentz transformations,
and a scalar is something which is invariant
under Lorentz transformations. The B, Wj,
and Ga fields are all vectors in this sense,
so the corresponding particles are said to
be vector bosons. The Higgs field φ is a
scalar.
The fermion field ψ does transform under
Lorentz transformations, but not like a vector
should; rotations will only turn it by half
the angle a proper vector should. Therefore,
these constitute a third kind of quantity,
which is known as a spinor.
It is common to make use of abstract index
notation for the vector fields, in which case
the vector fields all come with a Lorentzian
index μ, like so:
B
μ
,
W
j
μ
{\displaystyle B^{\mu },W_{j}^{\mu }}
, and
G
a
μ
{\displaystyle G_{a}^{\mu }}
. If abstract index notation is used also
for spinors then these will carry a spinorial
index and the Dirac gamma will carry one Lorentzian
and two spinorian indices, but it is more
common to regard spinors as column matrices
and the Dirac gamma γμ as a matrix which
additionally carries a Lorentzian index. The
Feynman slash notation can be used to turn
a vector field into a linear operator on spinors,
like so:
⧸
B
=
γ
μ
B
μ
{\displaystyle {\not }B=\gamma ^{\mu }B_{\mu
}}
; this may involve raising and lowering indices.
== Alternative presentations of the fields
==
As is common in quantum theory, there is more
than one way to look at things. At first the
basic fields given above may not seem to correspond
well with the "fundamental particles" in the
chart above, but there are several alternative
presentations which, in particular contexts,
may be more appropriate than those that are
given above.
=== Fermions ===
Rather than having one fermion field ψ, it
can be split up into separate components for
each type of particle. This mirrors the historical
evolution of quantum field theory, since the
electron component ψe (describing the electron
and its antiparticle the positron) is then
the original ψ field of quantum electrodynamics,
which was later accompanied by ψμ and ψτ
fields for the muon and tauon respectively
(and their antiparticles). Electroweak theory
added
ψ
ν
e
,
ψ
ν
μ
{\displaystyle \psi _{\nu _{\mathrm {e} }},\psi
_{\nu _{\mu }}}
, and
ψ
ν
τ
{\displaystyle \psi _{\nu _{\tau }}}
for the corresponding neutrinos, and the quarks
add still further components. In order to
be four-spinors like the electron and other
lepton components, there must be one quark
component for every combination of flavour
and colour, bringing the total to 24 (3 for
charged leptons, 3 for neutrinos, and 2·3·3
= 18 for quarks). Each of these is a four
component bispinor, for a total of 96 complex-valued
components for the fermion field.
An important definition is the barred fermion
field
ψ
¯
{\displaystyle {\bar {\psi }}}
, which is defined to be
ψ
†
γ
0
{\displaystyle \psi ^{\dagger }\gamma ^{0}}
, where
†
{\displaystyle \dagger }
denotes the Hermitian adjoint and γ0 is the
zeroth gamma matrix. If ψ is thought of as
an n × 1 matrix then
ψ
¯
{\displaystyle {\bar {\psi }}}
should be thought of as a 1 × n matrix.
==== A chiral theory ====
An independent decomposition of ψ is that
into chirality components:
"Left" chirality:
ψ
L
=
1
2
(
1
−
γ
5
)
ψ
{\displaystyle \psi ^{L}={\frac {1}{2}}(1-\gamma
_{5})\psi }
"Right" chirality:
ψ
R
=
1
2
(
1
+
γ
5
)
ψ
{\displaystyle \psi ^{R}={\frac {1}{2}}(1+\gamma
_{5})\psi }
where
γ
5
{\displaystyle \gamma _{5}}
is the fifth gamma matrix. This is very important
in the Standard Model because left and right
chirality components are treated differently
by the gauge interactions.
In particular, under weak isospin SU(2) transformations
the left-handed particles are weak-isospin
doublets, whereas the right-handed are singlets
– i.e. the weak isospin of ψR is zero.
Put more simply, the weak interaction could
rotate e.g. a left-handed electron into a
left-handed neutrino (with emission of a W−),
but could not do so with the same right-handed
particles. As an aside, the right-handed neutrino
originally did not exist in the standard model
– but the discovery of neutrino oscillation
implies that neutrinos must have mass, and
since chirality can change during the propagation
of a massive particle, right-handed neutrinos
must exist in reality. This does not however
change the (experimentally-proven) chiral
nature of the weak interaction.
Furthermore, U(1) acts differently on
ψ
e
L
{\displaystyle \psi _{\mathrm {e} }^{L}}
and
ψ
e
R
{\displaystyle \psi _{\mathrm {e} }^{R}}
(because they have different weak hypercharges).
==== Mass and interaction eigenstates ====
A distinction can thus be made between, for
example, the mass and interaction eigenstates
of the neutrino. The former is the state which
propagates in free space, whereas the latter
is the different state that participates in
interactions. Which is the "fundamental" particle?
For the neutrino, it is conventional to define
the "flavour" (νe, νμ, or ντ) by the
interaction eigenstate, whereas for the quarks
we define the flavour (up, down, etc.) by
the mass state. We can switch between these
states using the CKM matrix for the quarks,
or the PMNS matrix for the neutrinos (the
charged leptons on the other hand are eigenstates
of both mass and flavour).
As an aside, if a complex phase term exists
within either of these matrices, it will give
rise to direct CP violation, which could explain
the dominance of matter over antimatter in
our current universe. This has been proven
for the CKM matrix, and is expected for the
PMNS matrix.
==== Positive and negative energies ====
Finally, the quantum fields are sometimes
decomposed into "positive" and "negative"
energy parts: ψ = ψ+ + ψ−. This is not
so common when a quantum field theory has
been set up, but often features prominently
in the process of quantizing a field theory.
=== Bosons ===
Due to the Higgs mechanism, the electroweak
boson fields
W
1
,
W
2
,
W
3
{\displaystyle W_{1},W_{2},W_{3}}
, and
B
{\displaystyle B}
"mix" to create the states which are physically
observable. To retain gauge invariance, the
underlying fields must be massless, but the
observable states can gain masses in the process.
These states are:
The massive neutral (Z) boson:
Z
=
cos
⁡
θ
W
W
3
−
sin
⁡
θ
W
B
{\displaystyle Z=\cos \theta _{W}W_{3}-\sin
\theta _{W}B}
The massless neutral boson:
A
=
sin
⁡
θ
W
W
3
+
cos
⁡
θ
W
B
{\displaystyle A=\sin \theta _{W}W_{3}+\cos
\theta _{W}B}
The massive charged W bosons:
W
±
=
1
2
(
W
1
∓
i
W
2
)
{\displaystyle W^{\pm }={\frac {1}{\sqrt {2}}}\left(W_{1}\mp
iW_{2}\right)}
where θW is the Weinberg angle.
The A field is the photon, which corresponds
classically to the well-known electromagnetic
four-potential – i.e. the electric and magnetic
fields. The Z field actually contributes in
every process the photon does, but due to
its large mass, the contribution is usually
negligible.
== Perturbative QFT and the interaction picture
==
Much of the qualitative descriptions of the
standard model in terms of "particles" and
"forces" comes from the perturbative quantum
field theory view of the model. In this, the
Langrangian is decomposed as
L
=
L
0
+
L
I
{\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}+{\mathcal
{L}}_{\mathrm {I} }}
into separate free field and interaction Langrangians.
The free fields care for particles in isolation,
whereas processes involving several particles
arise through interactions. The idea is that
the state vector should only change when particles
interact, meaning a free particle is one whose
quantum state is constant. This corresponds
to the interaction picture in quantum mechanics.
In the more common Schrödinger picture, even
the states of free particles change over time:
typically the phase changes at a rate which
depends on their energy. In the alternative
Heisenberg picture, state vectors are kept
constant, at the price of having the operators
(in particular the observables) be time-dependent.
The interaction picture constitutes an intermediate
between the two, where some time dependence
is placed in the operators (the quantum fields)
and some in the state vector. In QFT, the
former is called the free field part of the
model, and the latter is called the interaction
part. The free field model can be solved exactly,
and then the solutions to the full model can
be expressed as perturbations of the free
field solutions, for example using the Dyson
series.
It should be observed that the decomposition
into free fields and interactions is in principle
arbitrary. For example, renormalization in
QED modifies the mass of the free field electron
to match that of a physical electron (with
an electromagnetic field), and will in doing
so add a term to the free field Lagrangian
which must be cancelled by a counterterm in
the interaction Lagrangian, that then shows
up as a two-line vertex in the Feynman diagrams.
This is also how the Higgs field is thought
to give particles mass: the part of the interaction
term which corresponds to the (nonzero) vacuum
expectation value of the Higgs field is moved
from the interaction to the free field Lagrangian,
where it looks just like a mass term having
nothing to do with Higgs.
=== Free fields ===
Under the usual free/interaction decomposition,
which is suitable for low energies, the free
fields obey the following equations:
The fermion field ψ satisfies the Dirac equation;
(
i
ℏ
⧸
∂
−
m
f
c
)
ψ
f
=
0
{\displaystyle (i\hbar {\not }\partial -m_{f}c)\psi
_{f}=0}
for each type
f
{\displaystyle f}
of fermion.
The photon field A satisfies the wave equation
∂
μ
∂
μ
A
ν
=
0
{\displaystyle \partial _{\mu }\partial ^{\mu
}A^{\nu }=0}
.
The Higgs field φ satisfies the Klein–Gordon
equation.
The weak interaction fields Z, W± also satisfy
the Proca equation.These equations can be
solved exactly. One usually does so by considering
first solutions that are periodic with some
period L along each spatial axis; later taking
the limit: L → ∞ will lift this periodicity
restriction.
In the periodic case, the solution for a field
F (any of the above) can be expressed as a
Fourier series of the form
F
(
x
)
=
β
∑
p
∑
r
E
p
−
1
2
(
a
r
(
p
)
u
r
(
p
)
e
−
i
p
x
ℏ
+
b
r
†
(
p
)
v
r
(
p
)
e
i
p
x
ℏ
)
{\displaystyle F(x)=\beta \sum _{\mathbf {p}
}\sum _{r}E_{\mathbf {p} }^{-{\frac {1}{2}}}\left(a_{r}(\mathbf
{p} )u_{r}(\mathbf {p} )e^{-{\frac {ipx}{\hbar
}}}+b_{r}^{\dagger }(\mathbf {p} )v_{r}(\mathbf
{p} )e^{\frac {ipx}{\hbar }}\right)}
where:
β is a normalization factor; for the fermion
field
ψ
f
{\displaystyle \psi _{f}}
it is
m
f
c
2
/
V
{\displaystyle {\sqrt {m_{f}c^{2}/V}}}
, where
V
=
L
3
{\displaystyle V=L^{3}}
is the volume of the fundamental cell considered;
for the photon field Aμ it is
ℏ
c
/
2
V
{\displaystyle \hbar c/{\sqrt {2V}}}
.
The sum over p is over all momenta consistent
with the period L, i.e., over all vectors
2
π
ℏ
L
(
n
1
,
n
2
,
n
3
)
{\displaystyle {\frac {2\pi \hbar }{L}}(n_{1},n_{2},n_{3})}
where
n
1
,
n
2
,
n
3
{\displaystyle n_{1},n_{2},n_{3}}
are integers.
The sum over r covers other degrees of freedom
specific for the field, such as polarization
or spin; it usually comes out as a sum from
1 to 2 or from 1 to 3.
Ep is the relativistic energy for a momentum
p quantum of the field,
=
m
2
c
4
+
c
2
p
2
{\displaystyle ={\sqrt {m^{2}c^{4}+c^{2}\mathbf
{p} ^{2}}}}
when the rest mass is m.
ar(p) and
b
r
†
(
p
)
{\displaystyle b_{r}^{\dagger }(\mathbf {p}
)}
are annihilation and creation respectively
operators for "a-particles" and "b-particles"
respectively of momentum p; "b-particles"
are the antiparticles of "a-particles". Different
fields have different "a-" and "b-particles".
For some fields, a and b are the same.
ur(p) and vr(p) are non-operators which carry
the vector or spinor aspects of the field
(where relevant).
p
=
(
E
p
/
c
,
p
)
{\displaystyle p=(E_{\mathbf {p} }/c,\mathbf
{p} )}
is the four-momentum for a quantum with momentum
p.
p
x
=
p
μ
x
μ
{\displaystyle px=p_{\mu }x^{\mu }}
denotes an inner product of four-vectors.In
the limit L → ∞, the sum would turn into
an integral with help from the V hidden inside
β. The numeric value of β also depends on
the normalization chosen for
u
r
(
p
)
{\displaystyle u_{r}(\mathbf {p} )}
and
v
r
(
p
)
{\displaystyle v_{r}(\mathbf {p} )}
.
Technically,
a
r
†
(
p
)
{\displaystyle a_{r}^{\dagger }(\mathbf {p}
)}
is the Hermitian adjoint of the operator ar(p)
in the inner product space of ket vectors.
The identification of
a
r
†
(
p
)
{\displaystyle a_{r}^{\dagger }(\mathbf {p}
)}
and ar(p) as creation and annihilation operators
comes from comparing conserved quantities
for a state before and after one of these
have acted upon it.
a
r
†
(
p
)
{\displaystyle a_{r}^{\dagger }(\mathbf {p}
)}
can for example be seen to add one particle,
because it will add 1 to the eigenvalue of
the a-particle number operator, and the momentum
of that particle ought to be p since the eigenvalue
of the vector-valued momentum operator increases
by that much. For these derivations, one starts
out with expressions for the operators in
terms of the quantum fields. That the operators
with
†
{\displaystyle \dagger }
are creation operators and the one without
annihilation operators is a convention, imposed
by the sign of the commutation relations postulated
for them.
An important step in preparation for calculating
in perturbative quantum field theory is to
separate the "operator" factors a and b above
from their corresponding vector or spinor
factors u and v. The vertices of Feynman graphs
come from the way that u and v from different
factors in the interaction Lagrangian fit
together, whereas the edges come from the
way that the as and bs must be moved around
in order to put terms in the Dyson series
on normal form.
=== Interaction terms and the path integral
approach ===
The Lagrangian can also be derived without
using creation and annihilation operators
(the "canonical" formalism), by using a "path
integral" approach, pioneered by Feynman building
on the earlier work of Dirac. See e.g. Path
integral formulation on Wikipedia or A. Zee's
QFT in a nutshell. This is one possible way
that the Feynman diagrams, which are pictorial
representations of interaction terms, can
be derived relatively easily. A quick derivation
is indeed presented at the article on Feynman
diagrams.
== Lagrangian formalism ==
We can now give some more detail about the
aforementioned free and interaction terms
appearing in the Standard Model Lagrangian
density. Any such term must be both gauge
and reference-frame invariant, otherwise the
laws of physics would depend on an arbitrary
choice or the frame of an observer. Therefore,
the global Poincaré symmetry, consisting
of translational symmetry, rotational symmetry
and the inertial reference frame invariance
central to the theory of special relativity
must apply. The local SU(3) × SU(2) × U(1)
gauge symmetry is the internal symmetry. The
three factors of the gauge symmetry together
give rise to the three fundamental interactions,
after some appropriate relations have been
defined, as we shall see.
A complete formulation of the Standard Model
Lagrangian with all the terms written together
can be found e.g. here.
=== Kinetic terms ===
A free particle can be represented by a mass
term, and a kinetic term which relates to
the "motion" of the fields.
==== Fermion fields ====
The kinetic term for a Dirac fermion is
i
ψ
¯
γ
μ
∂
μ
ψ
{\displaystyle i{\bar {\psi }}\gamma ^{\mu
}\partial _{\mu }\psi }
where the notations are carried from earlier
in the article. ψ can represent any, or all,
Dirac fermions in the standard model. Generally,
as below, this term is included within the
couplings (creating an overall "dynamical"
term).
==== Gauge fields ====
For the spin-1 fields, first define the field
strength tensor
F
μ
ν
a
=
∂
μ
A
ν
a
−
∂
ν
A
μ
a
+
g
f
a
b
c
A
μ
b
A
ν
c
{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu
}A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu
}^{b}A_{\nu }^{c}}
for a given gauge field (here we use A), with
gauge coupling constant g. The quantity f
abc is the structure constant of the particular
gauge group, defined by the commutator
[
t
a
,
t
b
]
=
i
f
a
b
c
t
c
,
{\displaystyle [t_{a},t_{b}]=if^{abc}t_{c},}
where ti are the generators of the group.
In an Abelian (commutative) group (such as
the U(1) we use here), since the generators
ta all commute with each other, the structure
constants vanish. Of course, this is not the
case in general – the standard model includes
the non-Abelian SU(2) and SU(3) groups (such
groups lead to what is called a Yang–Mills
gauge theory).
We need to introduce three gauge fields corresponding
to each of the subgroups SU(3) × SU(2) × U(1).
The gluon field tensor will be denoted by
G
μ
ν
a
{\displaystyle G_{\mu \nu }^{a}}
, where the index a labels elements of the
8 representation of colour SU(3). The strong
coupling constant is conventionally labelled
gs (or simply g where there is no ambiguity).
The observations leading to the discovery
of this part of the Standard Model are discussed
in the article in quantum chromodynamics.
The notation
W
μ
ν
a
{\displaystyle W_{\mu \nu }^{a}}
will be used for the gauge field tensor of
SU(2) where a runs over the 3 generators of
this group. The coupling can be denoted gw
or again simply g. The gauge field will be
denoted by
W
μ
a
{\displaystyle W_{\mu }^{a}}
.
The gauge field tensor for the U(1) of weak
hypercharge will be denoted by Bμν, the
coupling by g′, and the gauge field by Bμ.The
kinetic term can now be written simply as
L
k
i
n
=
−
1
4
B
μ
ν
B
μ
ν
−
1
2
t
r
W
μ
ν
W
μ
ν
−
1
2
t
r
G
μ
ν
G
μ
ν
{\displaystyle {\mathcal {L}}_{\rm {kin}}=-{1
\over 4}B_{\mu \nu }B^{\mu \nu }-{1 \over
2}\mathrm {tr} W_{\mu \nu }W^{\mu \nu }-{1
\over 2}\mathrm {tr} G_{\mu \nu }G^{\mu \nu
}}
where the traces are over the SU(2) and SU(3)
indices hidden in W and G respectively. The
two-index objects are the field strengths
derived from W and G the vector fields. There
are also two extra hidden parameters: the
theta angles for SU(2) and SU(3).
=== Coupling terms ===
The next step is to "couple" the gauge fields
to the fermions, allowing for interactions.
==== Electroweak sector ====
The electroweak sector interacts with the
symmetry group U(1) × SU(2)L, where the
subscript L indicates coupling only to left-handed
fermions.
L
E
W
=
∑
ψ
ψ
¯
γ
μ
(
i
∂
μ
−
g
′
1
2
Y
W
B
μ
−
g
1
2
τ
W
μ
)
ψ
{\displaystyle {\mathcal {L}}_{\mathrm {EW}
}=\sum _{\psi }{\bar {\psi }}\gamma ^{\mu
}\left(i\partial _{\mu }-g^{\prime }{1 \over
2}Y_{\mathrm {W} }B_{\mu }-g{1 \over 2}{\boldsymbol
{\tau }}\mathbf {W} _{\mu }\right)\psi }
Where Bμ is the U(1) gauge field; YW is the
weak hypercharge (the generator of the U(1)
group); Wμ is the three-component SU(2) gauge
field; and the components of τ are the Pauli
matrices (infinitesimal generators of the
SU(2) group) whose eigenvalues give the weak
isospin. Note that we have to redefine a new
U(1) symmetry of weak hypercharge, different
from QED, in order to achieve the unification
with the weak force. The electric charge Q,
third component of weak isospin T3 (also called
Tz, I3 or Iz) and weak hypercharge YW are
related by
Q
=
T
3
+
1
2
Y
W
,
{\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{W},}
(or by the alternative convention Q = T3 +
YW). The first convention, used in this article,
is equivalent to the earlier Gell-Mann–Nishijima
formula. It makes the hypercharge be twice
the average charge of a given isomultiplet.
One may then define the conserved current
for weak isospin as
j
μ
=
1
2
ψ
¯
L
γ
μ
τ
ψ
L
{\displaystyle \mathbf {j} _{\mu }={1 \over
2}{\bar {\psi }}_{L}\gamma _{\mu }{\boldsymbol
{\tau }}\psi _{L}}
and for weak hypercharge as
j
μ
Y
=
2
(
j
μ
e
m
−
j
μ
3
)
,
{\displaystyle j_{\mu }^{Y}=2(j_{\mu }^{em}-j_{\mu
}^{3})~,}
where
j
μ
e
m
{\displaystyle j_{\mu }^{em}}
is the electric current and
j
μ
3
{\displaystyle j_{\mu }^{3}}
the third weak isospin current. As explained
above, these currents mix to create the physically
observed bosons, which also leads to testable
relations between the coupling constants.
To explain this in a simpler way, we can see
the effect of the electroweak interaction
by picking out terms from the Lagrangian.
We see that the SU(2) symmetry acts on each
(left-handed) fermion doublet contained in
ψ, for example
−
g
2
(
ν
¯
e
e
¯
)
τ
+
γ
μ
(
W
+
)
μ
(
ν
e
e
)
=
−
g
2
ν
¯
e
γ
μ
(
W
+
)
μ
e
{\displaystyle -{g \over 2}({\bar {\nu }}_{e}\;{\bar
{e}})\tau ^{+}\gamma _{\mu }(W^{+})^{\mu }{\begin{pmatrix}{\nu
_{e}}\\e\end{pmatrix}}=-{g \over 2}{\bar {\nu
}}_{e}\gamma _{\mu }(W^{+})^{\mu }e}
where the particles are understood to be left-handed,
and where
τ
+
≡
1
2
(
τ
1
+
i
τ
2
)
=
(
0
1
0
0
)
{\displaystyle \tau ^{+}\equiv {1 \over 2}(\tau
^{1}{+}i\tau ^{2})={\begin{pmatrix}0&1\\0&0\end{pmatrix}}}
This is an interaction corresponding to a
"rotation in weak isospin space" or in other
words, a transformation between eL and νeL
via emission of a W− boson. The U(1) symmetry,
on the other hand, is similar to electromagnetism,
but acts on all "weak hypercharged" fermions
(both left- and right-handed) via the neutral
Z0, as well as the charged fermions via the
photon.
==== Quantum chromodynamics sector ====
The quantum chromodynamics (QCD) sector defines
the interactions between quarks and gluons,
with SU(3) symmetry, generated by Ta. Since
leptons do not interact with gluons, they
are not affected by this sector. The Dirac
Lagrangian of the quarks coupled to the gluon
fields is given by
L
Q
C
D
=
i
U
¯
(
∂
μ
−
i
g
s
G
μ
a
T
a
)
γ
μ
U
+
i
D
¯
(
∂
μ
−
i
g
s
G
μ
a
T
a
)
γ
μ
D
.
{\displaystyle {\mathcal {L}}_{\mathrm {QCD}
}=i{\overline {U}}\left(\partial _{\mu }-ig_{s}G_{\mu
}^{a}T^{a}\right)\gamma ^{\mu }U+i{\overline
{D}}\left(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a}\right)\gamma
^{\mu }D.}
where D and U are the Dirac spinors associated
with up- and down-type quarks, and other notations
are continued from the previous section.
=== Mass terms and the Higgs mechanism ===
==== Mass terms ====
The mass term arising from the Dirac Lagrangian
(for any fermion ψ) is
−
m
ψ
¯
ψ
{\displaystyle -m{\bar {\psi }}\psi }
which is not invariant under the electroweak
symmetry. This can be seen by writing ψ in
terms of left- and right-handed components
(skipping the actual calculation):
−
m
ψ
¯
ψ
=
−
m
(
ψ
¯
L
ψ
R
+
ψ
¯
R
ψ
L
)
{\displaystyle -m{\bar {\psi }}\psi =-m({\bar
{\psi }}_{L}\psi _{R}+{\bar {\psi }}_{R}\psi
_{L})}
i.e. contribution from
ψ
¯
L
ψ
L
{\displaystyle {\bar {\psi }}_{L}\psi _{L}}
and
ψ
¯
R
ψ
R
{\displaystyle {\bar {\psi }}_{R}\psi _{R}}
terms do not appear. We see that the mass-generating
interaction is achieved by constant flipping
of particle chirality. The spin-half particles
have no right/left chirality pair with the
same SU(2) representations and equal and opposite
weak hypercharges, so assuming these gauge
charges are conserved in the vacuum, none
of the spin-half particles could ever swap
chirality, and must remain massless. Additionally,
we know experimentally that the W and Z bosons
are massive, but a boson mass term contains
the combination e.g. AμAμ, which clearly
depends on the choice of gauge. Therefore,
none of the standard model fermions or bosons
can "begin" with mass, but must acquire it
by some other mechanism.
==== The Higgs mechanism ====
The solution to both these problems comes
from the Higgs mechanism, which involves scalar
fields (the number of which depend on the
exact form of Higgs mechanism) which (to give
the briefest possible description) are "absorbed"
by the massive bosons as degrees of freedom,
and which couple to the fermions via Yukawa
coupling to create what looks like mass terms.
In the Standard Model, the Higgs field is
a complex scalar of the group SU(2)L:
ϕ
=
1
2
(
ϕ
+
ϕ
0
)
,
{\displaystyle \phi ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\phi
^{+}\\\phi ^{0}\end{pmatrix}},}
where the superscripts + and 0 indicate the
electric charge (Q) of the components. The
weak hypercharge (YW) of both components is
1.
The Higgs part of the Lagrangian is
L
H
=
[
(
∂
μ
−
i
g
W
μ
a
t
a
−
i
g
′
Y
ϕ
B
μ
)
ϕ
]
2
+
μ
2
ϕ
†
ϕ
−
λ
(
ϕ
†
ϕ
)
2
,
{\displaystyle {\mathcal {L}}_{H}=\left[\left(\partial
_{\mu }-igW_{\mu }^{a}t^{a}-ig'Y_{\phi }B_{\mu
}\right)\phi \right]^{2}+\mu ^{2}\phi ^{\dagger
}\phi -\lambda (\phi ^{\dagger }\phi )^{2},}
where λ > 0 and μ2 > 0, so that the mechanism
of spontaneous symmetry breaking can be used.
There is a parameter here, at first hidden
within the shape of the potential, that is
very important. In a unitarity gauge one can
set φ+ = 0 and make φ0 real. Then
⟨
ϕ
0
⟩
=
v
{\displaystyle \langle \phi ^{0}\rangle =v}
is the non-vanishing vacuum expectation value
of the Higgs field. v has units of mass, and
it is the only parameter in the Standard Model
which is not dimensionless. It is also much
smaller than the Planck scale; it is approximately
equal to the Higgs mass, and sets the scale
for the mass of everything else. This is the
only real fine-tuning to a small nonzero value
in the Standard Model, and it is called the
Hierarchy problem. Quadratic terms in Wμ
and Bμ arise, which give masses to the W
and Z bosons:
M
W
=
1
2
v
g
M
Z
=
1
2
v
g
2
+
g
′
2
{\displaystyle {\begin{aligned}M_{W}&={\tfrac
{1}{2}}vg\\M_{Z}&={\tfrac {1}{2}}v{\sqrt {g^{2}+{g'}^{2}}}\end{aligned}}}
The mass of the Higgs boson itself is given
by
M
H
=
2
μ
2
≡
2
λ
v
2
.
{\displaystyle M_{H}={\sqrt {2\mu ^{2}}}\equiv
{\sqrt {2\lambda v^{2}}}.}
The Yukawa interaction terms are
L
Y
U
=
U
¯
L
G
u
U
R
ϕ
0
−
D
¯
L
G
u
U
R
ϕ
−
+
U
¯
L
G
d
D
R
ϕ
+
+
D
¯
L
G
d
D
R
ϕ
0
+
h
c
{\displaystyle {\mathcal {L}}_{YU}={\overline
{U}}_{L}G_{u}U_{R}\phi ^{0}-{\overline {D}}_{L}G_{u}U_{R}\phi
^{-}+{\overline {U}}_{L}G_{d}D_{R}\phi ^{+}+{\overline
{D}}_{L}G_{d}D_{R}\phi ^{0}+hc}
where Gu,d are 3 × 3 matrices of Yukawa
couplings, with the ij term giving the coupling
of the generations i and j.
==== Neutrino masses ====
As previously mentioned, evidence shows neutrinos
must have mass. But within the standard model,
the right-handed neutrino does not exist,
so even with a Yukawa coupling neutrinos remain
massless. An obvious solution is to simply
add a right-handed neutrino νR resulting
in a Dirac mass term as usual. This field
however must be a sterile neutrino, since
being right-handed it experimentally belongs
to an isospin singlet (T3 = 0) and also has
charge Q = 0, implying YW = 0 (see above)
i.e. it does not even participate in the weak
interaction. The experimental evidence for
sterile neutrinos is currently inconclusive.Another
possibility to consider is that the neutrino
satisfies the Majorana equation, which at
first seems possible due to its zero electric
charge. In this case the mass term is
−
m
2
(
ν
¯
C
ν
+
ν
¯
ν
C
)
{\displaystyle -{m \over 2}\left({\overline
{\nu }}^{C}\nu +{\overline {\nu }}\nu ^{C}\right)}
where C denotes a charge conjugated (i.e.
anti-) particle, and the terms are consistently
all left (or all right) chirality (note that
a left-chirality projection of an antiparticle
is a right-handed field; care must be taken
here due to different notations sometimes
used). Here we are essentially flipping between
left-handed neutrinos and right-handed anti-neutrinos
(it is furthermore possible but not necessary
that neutrinos are their own antiparticle,
so these particles are the same). However,
for left-chirality neutrinos, this term changes
weak hypercharge by 2 units – not possible
with the standard Higgs interaction, requiring
the Higgs field to be extended to include
an extra triplet with weak hypercharge = 2
– whereas for right-chirality neutrinos,
no Higgs extensions are necessary. For both
left and right chirality cases, Majorana terms
violate lepton number, but possibly at a level
beyond the current sensitivity of experiments
to detect such violations.
It is possible to include both Dirac and Majorana
mass terms in the same theory, which (in contrast
to the Dirac-mass-only approach) can provide
a “natural” explanation for the smallness
of the observed neutrino masses, by linking
the right-handed neutrinos to yet-unknown
physics around the GUT scale (see seesaw mechanism).
Since in any case new fields must be postulated
to explain the experimental results, neutrinos
are an obvious gateway to searching physics
beyond the Standard Model.
== Detailed information ==
This section provides more detail on some
aspects, and some reference material. Explicit
Lagrangian terms are also provided here.
=== Field content in detail ===
The Standard Model has the following fields.
These describe one generation of leptons and
quarks, and there are three generations, so
there are three copies of each field. By CPT
symmetry, there is a set of right-handed fermions
with the opposite quantum numbers. The column
"representation" indicates under which representations
of the gauge groups that each field transforms,
in the order (SU(3), SU(2), U(1)). Symbols
used are common but not universal; superscript
C denotes an antiparticle; and for the U(1)
group, the value of the weak hypercharge is
listed. Note that there are twice as many
left-handed lepton field components as left-handed
antilepton field components in each generation,
but an equal number of left-handed quark and
antiquark fields.
=== Fermion content ===
This table is based in part on data gathered
by the Particle Data Group.
=== Free parameters ===
Upon writing the most general Lagrangian with
massless neutrinos, one finds that the dynamics
depend on 19 parameters, whose numerical values
are established by experiment. Straightforward
extensions of Standard Model with massive
neutrinos need 7 more parameters, 3 masses
and 4 PMNS matrix parameters, for a total
of 26 parameters. The neutrino parameter values
are still uncertain. The 19 certain parameters
are summarized here.
The choice of free parameters is somewhat
arbitrary. In the table above, gauge couplings
are listed as free parameters, therefore with
this choice Weinberg angle is not a free parameter
- it is defined as
tan
⁡
θ
W
=
g
1
g
2
{\displaystyle \tan \theta _{W}={\frac {g_{1}}{g_{2}}}}
. Likewise, fine structure constant of QED
is
α
=
1
4
π
(
g
1
g
2
)
2
g
1
2
+
g
2
2
{\displaystyle \alpha ={\frac {1}{4\pi }}{\frac
{(g_{1}g_{2})^{2}}{g_{1}^{2}+g_{2}^{2}}}}
.
Instead of fermion masses, dimensionless Yukawa
couplings can be chosen as free parameters.
For example, electron mass depends on the
Yukawa coupling of electron to Higgs field,
and its value is
m
e
=
y
e
2
v
{\displaystyle m_{e}={\frac {y_{e}}{\sqrt
{2}}}v}
.
Instead of the Higgs mass, the Higgs self-coupling
strength
λ
=
m
H
2
2
v
2
{\displaystyle \lambda ={\frac {m_{H}^{2}}{2v^{2}}}}
, which is approximately 0.129, can be chosen
as a free parameter.
Instead of the Higgs vacuum expectation value,
μ
2
{\displaystyle \mu ^{2}}
parameter directly from Higgs self-interaction
term
μ
2
ϕ
†
ϕ
−
λ
(
ϕ
†
ϕ
)
2
{\displaystyle \mu ^{2}\phi ^{\dagger }\phi
-\lambda (\phi ^{\dagger }\phi )^{2}}
can be chosen. Its value is
μ
2
=
λ
v
2
=
m
H
2
2
{\displaystyle \mu ^{2}=\lambda v^{2}={\frac
{m_{H}^{2}}{2}}}
, or approximately
μ
=
88.45
{\displaystyle \mu =88.45}
GeV.
=== Additional symmetries of the Standard
Model ===
From the theoretical point of view, the Standard
Model exhibits four additional global symmetries,
not postulated at the outset of its construction,
collectively denoted accidental symmetries,
which are continuous U(1) global symmetries.
The transformations leaving the Lagrangian
invariant are:
ψ
q
(
x
)
→
e
i
α
/
3
ψ
q
{\displaystyle \psi _{\text{q}}(x)\to e^{i\alpha
/3}\psi _{\text{q}}}
E
L
→
e
i
β
E
L
and
(
e
R
)
c
→
e
i
β
(
e
R
)
c
{\displaystyle E_{L}\to e^{i\beta }E_{L}{\text{
and }}(e_{R})^{c}\to e^{i\beta }(e_{R})^{c}}
M
L
→
e
i
β
M
L
and
(
μ
R
)
c
→
e
i
β
(
μ
R
)
c
{\displaystyle M_{L}\to e^{i\beta }M_{L}{\text{
and }}(\mu _{R})^{c}\to e^{i\beta }(\mu _{R})^{c}}
T
L
→
e
i
β
T
L
and
(
τ
R
)
c
→
e
i
β
(
τ
R
)
c
{\displaystyle T_{L}\to e^{i\beta }T_{L}{\text{
and }}(\tau _{R})^{c}\to e^{i\beta }(\tau
_{R})^{c}}
The first transformation rule is shorthand
meaning that all quark fields for all generations
must be rotated by an identical phase simultaneously.
The fields ML, TL and
(
μ
R
)
c
,
(
τ
R
)
c
{\displaystyle (\mu _{R})^{c},(\tau _{R})^{c}}
are the 2nd (muon) and 3rd (tau) generation
analogs of EL and
(
e
R
)
c
{\displaystyle (e_{R})^{c}}
fields.
By Noether's theorem, each symmetry above
has an associated conservation law: the conservation
of baryon number, electron number, muon number,
and tau number. Each quark is assigned a baryon
number of
1
3
{\displaystyle {}_{\frac {1}{3}}}
, while each antiquark is assigned a baryon
number of
−
1
3
{\displaystyle {}_{-{\frac {1}{3}}}}
. Conservation of baryon number implies that
the number of quarks minus the number of antiquarks
is a constant. Within experimental limits,
no violation of this conservation law has
been found.
Similarly, each electron and its associated
neutrino is assigned an electron number of
+1, while the anti-electron and the associated
anti-neutrino carry a −1 electron number.
Similarly, the muons and their neutrinos are
assigned a muon number of +1 and the tau leptons
are assigned a tau lepton number of +1. The
Standard Model predicts that each of these
three numbers should be conserved separately
in a manner similar to the way baryon number
is conserved. These numbers are collectively
known as lepton family numbers (LF). (This
result depends on the assumption made in Standard
Model that neutrinos are massless. Experimentally,
neutrino oscillations demonstrate that individual
electron, muon and tau numbers are not conserved.)
In addition to the accidental (but exact)
symmetries described above, the Standard Model
exhibits several approximate symmetries. These
are the "SU(2) custodial symmetry" and the
"SU(2) or SU(3) quark flavor symmetry."
=== The U(1) symmetry ===
For the leptons, the gauge group can be written
SU(2)l × U(1)L × U(1)R. The two
U(1) factors can be combined into U(1)Y × U(1)l
where l is the lepton number. Gauging of the
lepton number is ruled out by experiment,
leaving only the possible gauge group SU(2)L × U(1)Y.
A similar argument in the quark sector also
gives the same result for the electroweak
theory.
=== The charged and neutral current couplings
and Fermi theory ===
The charged currents
j
∓
=
j
1
±
i
j
2
{\displaystyle j^{\mp }=j^{1}\pm ij^{2}}
are
j
μ
−
=
U
¯
i
L
γ
μ
D
i
L
+
ν
¯
i
L
γ
μ
l
i
L
.
{\displaystyle j_{\mu }^{-}={\overline {U}}_{iL}\gamma
_{\mu }D_{iL}+{\overline {\nu }}_{iL}\gamma
_{\mu }l_{iL}.}
These charged currents are precisely those
that entered the Fermi theory of beta decay.
The action contains the charge current piece
L
C
C
=
g
2
(
j
μ
+
W
−
μ
+
j
μ
−
W
+
μ
)
.
{\displaystyle {\mathcal {L}}_{CC}={\frac
{g}{\sqrt {2}}}(j_{\mu }^{+}W^{-\mu }+j_{\mu
}^{-}W^{+\mu }).}
For energy much less than the mass of the
W-boson, the effective theory becomes the
current–current contact interaction of the
Fermi theory,
2
2
G
F
J
μ
+
J
μ
−
{\displaystyle 2{\sqrt {2}}G_{F}~~J_{\mu }^{+}J^{\mu
~~-}}
.
However, gauge invariance now requires that
the component
W
3
{\displaystyle W^{3}}
of the gauge field also be coupled to a current
that lies in the triplet of SU(2). However,
this mixes with the U(1), and another current
in that sector is needed. These currents must
be uncharged in order to conserve charge.
So neutral currents are also required,
j
μ
3
=
1
2
(
U
¯
i
L
γ
μ
U
i
L
−
D
¯
i
L
γ
μ
D
i
L
+
ν
¯
i
L
γ
μ
ν
i
L
−
l
¯
i
L
γ
μ
l
i
L
)
{\displaystyle j_{\mu }^{3}={\frac {1}{2}}({\overline
{U}}_{iL}\gamma _{\mu }U_{iL}-{\overline {D}}_{iL}\gamma
_{\mu }D_{iL}+{\overline {\nu }}_{iL}\gamma
_{\mu }\nu _{iL}-{\overline {l}}_{iL}\gamma
_{\mu }l_{iL})}
j
μ
e
m
=
2
3
U
¯
i
γ
μ
U
i
−
1
3
D
¯
i
γ
μ
D
i
−
l
¯
i
γ
μ
l
i
.
{\displaystyle j_{\mu }^{em}={\frac {2}{3}}{\overline
{U}}_{i}\gamma _{\mu }U_{i}-{\frac {1}{3}}{\overline
{D}}_{i}\gamma _{\mu }D_{i}-{\overline {l}}_{i}\gamma
_{\mu }l_{i}.}
The neutral current piece in the Lagrangian
is then
L
N
C
=
e
j
μ
e
m
A
μ
+
g
cos
⁡
θ
W
(
J
μ
3
−
sin
2
⁡
θ
W
J
μ
e
m
)
Z
μ
.
{\displaystyle {\mathcal {L}}_{NC}=ej_{\mu
}^{em}A^{\mu }+{\frac {g}{\cos \theta _{W}}}(J_{\mu
}^{3}-\sin ^{2}\theta _{W}J_{\mu }^{em})Z^{\mu
}.}
== See also ==
Overview of Standard Model of particle physics
Fundamental interaction
Noncommutative standard model
Open questions: CP violation, Neutrino masses,
Quark matter
Physics beyond the Standard Model
Strong interactions: Flavour, Quantum chromodynamics,
Quark model
Weak interactions: Electroweak interaction,
Fermi's interaction
Weinberg angle
Symmetry in quantum mechanics
== References and external links ==
An introduction to quantum field theory, by
M.E. Peskin and D.V. Schroeder (HarperCollins,
1995) ISBN 0-201-50397-2.
Gauge theory of elementary particle physics,
by T.P. Cheng and L.F. Li (Oxford University
Press, 1982) ISBN 0-19-851961-3.
Standard Model Lagrangian with explicit Higgs
terms (T.D. Gutierrez, ca 1999) (PDF, PostScript,
and LaTeX version)
The quantum theory of fields (vol 2), by S.
Weinberg (Cambridge University Press, 1996)
ISBN 0-521-55002-5.
Quantum Field Theory in a Nutshell (Second
Edition), by A. Zee (Princeton University
Press, 2010) ISBN 978-1-4008-3532-4.
An Introduction to Particle Physics and the
Standard Model, by R. Mann (CRC Press, 2010)
ISBN 978-1420082982
Physics From Symmetry by J. Schwichtenberg
(Springer, 2015) ISBN 3319192000. Especially
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