The Standard Model of particle physics is 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 describing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson. 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.
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. QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles. These fields are
the fermion fields, ψ, which account for "matter particles"; the electroweak boson fields W 1 {\displaystyle W_{1}} , W 2 {\displaystyle W_{2}} , W 3 {\displaystyle 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 a quantum state (ket vector).
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 that, 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. 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 flavor and color, 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 Dirac spinor, 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 of ψ, 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:
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 demonstrated) chiral nature of the weak interaction. Furthermore, U(1) acts differently on ψ e L {\displaystyle \psi _{\mathrm {e} }^{\rm {L}}} and ψ e R {\displaystyle \psi _{\mathrm {e} }^{\rm {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 that 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 "flavor" (νe, νμ, or ντ) by the interaction eigenstate, whereas for the quarks we define the flavor (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 flavor). 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 {\displaystyle W_{1}} , W 2 {\displaystyle W_{2}} , W 3 {\displaystyle W_{3}} , and B {\displaystyle B} "mix" to create the states that 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 _{\rm {W}}W_{3}-\sin \theta _{\rm {W}}B}
The massless neutral boson:
A = sin θ W W 3 + cos θ W B {\displaystyle A=\sin \theta _{\rm {W}}W_{3}+\cos \theta _{\rm {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 Lagrangian is decomposed as L = L 0 + L I {\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}+{\mathcal {L}}_{\mathrm {I} }} into separate free field and interaction Lagrangians. 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 that 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 that 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 the Higgs field.
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 γ μ ∂ μ − m f c ) ψ f = 0 {\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-m_{\rm {f}}c)\psi _{\rm {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± 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 _{\rm {f}}} it is m f c 2 / V {\textstyle {\sqrt {m_{\rm {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 {\textstyle ={\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 operators respectively 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 that 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 † {\displaystyle \dagger } are 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 formulation, pioneered by Feynman building on the earlier work of Dirac. Feynman diagrams are pictorial representations of interaction terms. 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.
Kinetic terms A free particle can be represented by a mass term, and a kinetic term that 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 fabc 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) the structure constants vanish, since the generators ta all commute with each other. 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 color 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 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 }}\cdot \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_{\rm {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 }}_{\rm {L}}\gamma _{\mu }{\boldsymbol {\tau }}\psi _{\rm {L}}}
and for weak hypercharge as
j μ Y = 2 ( j μ e m − j μ 3 ) , {\displaystyle j_{\mu }^{Y}=2(j_{\mu }^{\rm {em}}-j_{\mu }^{3})~,}
where j μ e m {\displaystyle j_{\mu }^{\rm {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}}\