Mathematical formulation of the Standard Model

Standard Model of Particle Physics. The diagram shows the elementary particles of the Standard Model (the Higgs boson, the three generations of quarks and leptons, and the gauge bosons), including their names, masses, spins, charges, chiralities, and interactions with the strong, weak and electromagnetic forces. It also depicts the crucial role of the Higgs boson in electroweak symmetry breaking, and shows how the properties of the various particles differ in the (high-energy) symmetric phase (top) and the (low-energy) broken-symmetry phase (bottom).

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 Template:Math. 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 pattern of weak isospin Template:Math, weak hypercharge Template:Mvar, and color charge of all known elementary particles, rotated by the weak mixing angle to show electric charge Template:Mvar, roughly along the vertical. The neutral Higgs field (gray square) breaks the electroweak symmetry and interacts with other particles to give them mass.

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, Template:Mvar, which account for "matter particles";
• the electroweak boson fields $W_1$, $W_2$, $W_3$, and Template:Mvar;
• the gluon field, Template:Mvar; and
• the Higgs field, Template:Mvar.

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 Template:Mvar, 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 Template:Math (describing the electron and its antiparticle the positron) is then the original Template:Mvar field of quantum electrodynamics, which was later accompanied by Template:Mvar and Template:Mvar fields for the muon and tauon respectively (and their antiparticles). Electroweak theory added ${\psi }_{{\nu }_{\mathrm{e}}},{\psi }_{{\nu }_{\mu }}$, and ${\psi }_{{\nu }_{\tau }}$ 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 bispinor, for a total of 96 complex-valued components for the fermion field.

An important definition is the barred fermion field $\overline{\psi }$, which is defined to be ${\psi }^{†}{\gamma }^0$, where $†$ denotes the Hermitian adjoint of Template:Mvar, and Template:Math is the zeroth gamma matrix. If Template:Mvar is thought of as an Template:Math matrix then $\overline{\psi }$ should be thought of as a [[row vector|Template:Math matrix]].

A chiral theory

An independent decomposition of Template:Mvar is that into chirality components: Template:Unbulleted list where ${\gamma }_5$ 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 Template:Math 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 Template:Math), 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, Template:Math acts differently on ${\psi }_{\mathrm{e}}^{\mathrm{L}}$ and ${\psi }_{\mathrm{e}}^{\mathrm{R}}$ (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" (Template:SubatomicParticle, Template:SubatomicParticle, or Template:SubatomicParticle) 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: Template:Math. 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

Weinberg angle Template:Math, and relation between coupling constants g, g′, and e. Adapted from T D Lee's book Particle Physics and Introduction to Field Theory (1981).

Due to the Higgs mechanism, the electroweak boson fields $W_1$, $W_2$, $W_3$, and $B$ "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: $$Z=\cos {\theta }_{\mathrm{W}}{W}_3-\sin {\theta }_{\mathrm{W}}B$$ The massless neutral boson: $$A=\sin {\theta }_{\mathrm{W}}{W}_3+\cos {\theta }_{\mathrm{W}}B$$ The massive charged W bosons: $$W^{\pm }=\frac{1}{\sqrt{2}}(W_1\mp i{W}_2)$$ where Template:Math is the Weinberg angle.

The Template:Mvar field is the photon, which corresponds classically to the well-known electromagnetic four-potential – i.e. the electric and magnetic fields. The Template:Mvar 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 $ℒ={ℒ}_0+{ℒ}_{\mathrm{I}}$ 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. Template:See also

Free fields

Under the usual free/interaction decomposition, which is suitable for low energies, the free fields obey the following equations:

• The fermion field Template:Mvar satisfies the Dirac equation; $(i\hslash {\gamma }^{\mu }{\partial }_{\mu }-m_{\mathrm{f}}c){\psi }_{\mathrm{f}}=0$ for each type $f$ of fermion.
• The photon field Template:Mvar satisfies the wave equation ${\partial }_{\mu }{\partial }^{\mu }{A}^{\nu }=0$.
• The Higgs field Template:Mvar satisfies the Klein–Gordon equation.
• The weak interaction fields Template:Math satisfy the Proca equation.

These equations can be solved exactly. One usually does so by considering first solutions that are periodic with some period Template:Mvar along each spatial axis; later taking the limit: Template:Math will lift this periodicity restriction.

In the periodic case, the solution for a field Template:Mvar (any of the above) can be expressed as a Fourier series of the form $$F(x)=\beta \sum _{𝐩}\sum _r{E}_{𝐩}^{-\frac{1}{2}}(a_r(𝐩)u_r(𝐩)e^{-\frac{ipx}{\hslash }}+b_r^{†}(𝐩)v_r(𝐩)e^{\frac{ipx}{\hslash }})$$ where:

• Template:Mvar is a normalization factor; for the fermion field ${\psi }_{\mathrm{f}}$ it is $\sqrt{m_{\mathrm{f}}{c}^2/V}$, where $V=L^3$ is the volume of the fundamental cell considered; for the photon field Template:Mvar it is $\hslash c/\sqrt{2V}$.
• The sum over Template:Math is over all momenta consistent with the period Template:Mvar, i.e., over all vectors $\frac{2\pi \hslash }{L}(n_1,n_2,n_3)$ where $n_1,n_2,n_3$ are integers.
• The sum over Template:Mvar covers other degrees of freedom specific for the field, such as polarization or spin; it usually comes out as a sum from Template:Math to Template:Math or from Template:Math to Template:Math.
• Template:Math is the relativistic energy for a momentum Template:Math quantum of the field, $=\sqrt{m^2{c}^4+c^2{𝐩}^2}$ when the rest mass is Template:Mvar.
• Template:Math and $b_r^{†}(𝐩)$ are annihilation and creation operators respectively for "a-particles" and "b-particles" respectively of momentum Template:Math; "b-particles" are the antiparticles of "a-particles". Different fields have different "a-" and "b-particles". For some fields, Template:Mvar and Template:Mvar are the same.
• Template:Math and Template:Math are non-operators that carry the vector or spinor aspects of the field (where relevant).
• $p=(E_{𝐩}/c,𝐩)$ is the four-momentum for a quantum with momentum Template:Math. $px=p_{\mu }{x}^{\mu }$ denotes an inner product of four-vectors.

In the limit Template:Math, the sum would turn into an integral with help from the Template:Mvar hidden inside Template:Mvar. The numeric value of Template:Mvar also depends on the normalization chosen for $u_r(𝐩)$ and $v_r(𝐩)$.

Technically, $a_r^{†}(𝐩)$ is the Hermitian adjoint of the operator Template:Math in the inner product space of ket vectors. The identification of $a_r^{†}(𝐩)$ and Template:Math as creation and annihilation operators comes from comparing conserved quantities for a state before and after one of these have acted upon it. $a_r^{†}(𝐩)$ can for example be seen to add one particle, because it will add Template:Math to the eigenvalue of the a-particle number operator, and the momentum of that particle ought to be Template:Math 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 $†$ 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 Template:Mvar and Template:Mvar above from their corresponding vector or spinor factors Template:Mvar and Template:Mvar. The vertices of Feynman graphs come from the way that Template:Mvar and Template:Mvar from different factors in the interaction Lagrangian fit together, whereas the edges come from the way that the Template:Mvars and Template:Mvars 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

Interactions in the Standard Model. All Feynman diagrams in the model are built from combinations of these vertices. q is any quark, g is a gluon, X is any charged particle, γ is a photon, f is any fermion, m is any particle with mass (with the possible exception of the neutrinos), m_B is any boson with mass. In diagrams with multiple particle labels separated by / one particle label is chosen. In diagrams with particle labels separated by | the labels must be chosen in the same order. For example, in the four boson electroweak case the valid diagrams are WWWW, WWZZ, WWγγ, WWZγ. The conjugate of each listed vertex (reversing the direction of arrows) is also allowed.

The full expanded form of the Standard Model Lagrangian

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 Template:Math 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 $$i\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi$$ where the notations are carried from earlier in the article. Template:Mvar 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 $$F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$$ for a given gauge field (here we use Template:Mvar), with gauge coupling constant Template:Mvar. The quantity Template:Math is the structure constant of the particular gauge group, defined by the commutator $$[t_a,t_b]=i{f}^{abc}{t}_c,$$ where Template:Mvar are the generators of the group. In an abelian (commutative) group (such as the Template:Math we use here) the structure constants vanish, since the generators Template:Mvar all commute with each other. Of course, this is not the case in general – the standard model includes the non-Abelian Template:Math and Template:Math 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 Template:Math.

• The gluon field tensor will be denoted by $G_{\mu \nu }^a$, where the index Template:Mvar labels elements of the Template:Math representation of color SU(3). The strong coupling constant is conventionally labelled Template:Math (or simply Template:Mvar 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 $W_{\mu \nu }^a$ will be used for the gauge field tensor of Template:Math where Template:Mvar runs over the Template:Math generators of this group. The coupling can be denoted Template:Math or again simply Template:Mvar. The gauge field will be denoted by $W_{\mu }^a$.
• The gauge field tensor for the Template:Math of weak hypercharge will be denoted by Template:Math, the coupling by Template:Math, and the gauge field by Template:Mvar.

The kinetic term can now be written as $${ℒ}_{\mathrm{k}\mathrm{i}\mathrm{n}}=-\frac{1}{4}{B}_{\mu \nu }{B}^{\mu \nu }-\frac{1}{2}\mathrm{t}\mathrm{r}{W}_{\mu \nu }{W}^{\mu \nu }-\frac{1}{2}\mathrm{t}\mathrm{r}{G}_{\mu \nu }{G}^{\mu \nu }$$ where the traces are over the Template:Math and Template:Math indices hidden in Template:Mvar and Template:Mvar respectively. The two-index objects are the field strengths derived from Template:Mvar and Template:Mvar the vector fields. There are also two extra hidden parameters: the theta angles for Template:Math and Template:Math.

Coupling terms

The next step is to "couple" the gauge fields to the fermions, allowing for interactions.

Electroweak sector

Template:Main The electroweak sector interacts with the symmetry group Template:Math, where the subscript L indicates coupling only to left-handed fermions. $${ℒ}_{\mathrm{E}\mathrm{W}}=\sum _{\psi }\overline{\psi }{\gamma }^{\mu }(i{\partial }_{\mu }-g^{\prime }\frac{1}{2}{Y}_{\mathrm{W}}{B}_{\mu }-g\frac{1}{2}\mathit{\tau }{𝐖}_{\mu })\psi$$ where Template:Mvar is the Template:Math gauge field; Template:Math is the weak hypercharge (the generator of the Template:Math group); Template:Math is the three-component Template:Math gauge field; and the components of Template:Math are the Pauli matrices (infinitesimal generators of the Template:Math group) whose eigenvalues give the weak isospin. Note that we have to redefine a new Template:Math symmetry of weak hypercharge, different from QED, in order to achieve the unification with the weak force. The electric charge Template:Mvar, third component of weak isospin Template:Math (also called Template:Math or Template:Mvar) and weak hypercharge Template:Math are related by $$Q=T_3+\frac{1}{2}{Y}_{\mathrm{W}},$$ (or by the alternative convention Template:Math). 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 $${𝐣}_{\mu }=\frac{1}{2}{\overline{\psi }}_{\mathrm{L}}{\gamma }_{\mu }\mathit{\tau }{\psi }_{\mathrm{L}}$$ and for weak hypercharge as $$j_{\mu }^Y=2(j_{\mu }^{\mathrm{e}\mathrm{m}}-j_{\mu }^3),$$ where $j_{\mu }^{\mathrm{e}\mathrm{m}}$ is the electric current and $j_{\mu }^3$ 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 Template:Mvar, for example $$-\frac{g}{2}({\overline{\nu }}_e\overline{e}){\tau }^{+}{\gamma }_{\mu }(W^{+}{)}^{\mu }\left(\begin{array}{c}{\nu }_e\\ e\end{array}\right)=-\frac{g}{2}{\overline{\nu }}_e{\gamma }_{\mu }(W^{+}{)}^{\mu }e$$ where the particles are understood to be left-handed, and where $${\tau }^{+}\equiv \frac{1}{2}({\tau }^1+i{\tau }^2)=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$$

This is an interaction corresponding to a "rotation in weak isospin space" or in other words, a transformation between Template:Math and Template:Math via emission of a Template:Math boson. The Template:Math symmetry, on the other hand, is similar to electromagnetism, but acts on all "weak hypercharged" fermions (both left- and right-handed) via the neutral Template:Math, as well as the charged fermions via the photon.

Quantum chromodynamics sector

Template:Main The quantum chromodynamics (QCD) sector defines the interactions between quarks and gluons, with Template:Math symmetry, generated by Template:Mvar. 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 $${ℒ}_{\mathrm{Q}\mathrm{C}\mathrm{D}}=i\bar{U}({\partial }_{\mu }-i{g}_s{G}_{\mu }^a{T}^a){\gamma }^{\mu }U+i\bar{D}({\partial }_{\mu }-i{g}_s{G}_{\mu }^a{T}^a){\gamma }^{\mu }D.$$ where Template:Mvar and Template:Mvar 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 Template:Mvar) is $-m\overline{\psi }\psi$, which is not invariant under the electroweak symmetry. This can be seen by writing Template:Mvar in terms of left and right-handed components (skipping the actual calculation): $$-m\overline{\psi }\psi =-m({\overline{\psi }}_{\mathrm{L}}{\psi }_{\mathrm{R}}+{\overline{\psi }}_{\mathrm{R}}{\psi }_{\mathrm{L}})$$ i.e. contribution from ${\overline{\psi }}_{\mathrm{L}}{\psi }_{\mathrm{L}}$ and ${\overline{\psi }}_{\mathrm{R}}{\psi }_{\mathrm{R}}$ 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 Template:Math 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. Template:Mvar, 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.

Higgs mechanism

Template:Main 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 field of the group Template:Math: $$\varphi =\frac{1}{\sqrt{2}}\left(\begin{array}{c}{\varphi }^{+}\\ {\varphi }^0\end{array}\right),$$ where the superscripts Template:Math and Template:Math indicate the electric charge (Template:Mvar) of the components. The weak hypercharge (Template:Math) of both components is Template:Math.

The Higgs part of the Lagrangian is $${ℒ}_{\mathrm{H}}={\left[({\partial }_{\mu }-ig{W}_{\mu }^a{t}^a-i{g}^{\prime }{Y}_{\varphi }{B}_{\mu })\varphi \right]}^2+{\mu }^2{\varphi }^{†}\varphi -\lambda ({\varphi }^{†}\varphi {)}^2,$$ where Template:Math and Template:Math, 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 ${\varphi }^{+}=0$ and make ${\varphi }^0$ real. Then $⟨{\varphi }^0⟩=v$ 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 that is not dimensionless. It is also much smaller than the Planck scale and about twice the Higgs mass, setting the scale for the mass of all other particles in the Standard Model. This is the only real fine-tuning to a small nonzero value in the Standard Model. Quadratic terms in Template:Mvar and Template:Mvar arise, which give masses to the W and Z bosons: $$\begin{array}{cc}{M}_{\mathrm{W}}& =\frac{1}{2}vg\\ M_{\mathrm{Z}}& =\frac{1}{2}v\sqrt{g^2+{g^{\prime }}^2}\end{array}$$

The mass of the Higgs boson itself is given by $M_{\mathrm{H}}=\sqrt{2{\mu }^2}\equiv \sqrt{2\lambda v^2}.$

Yukawa interaction

The Yukawa interaction terms are $${ℒ}_{\text{Yukawa}}=(Y_{\text{u}}{)}_{mn}({\overline{q}}_{\text{L}}{)}_m\tilde{\phi }(u_{\text{R}}{)}_n+(Y_{\text{d}}{)}_{mn}({\overline{q}}_{\text{L}}{)}_m\phi (d_{\text{R}}{)}_n+(Y_{\text{e}}{)}_{mn}({\overline{L}}_{\text{L}}{)}_m\tilde{\phi }(e_{\text{R}}{)}_n+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}$$ where $Y_{\text{u}}$, $Y_{\text{d}}$, and $Y_{\text{e}}$ are Template:Math matrices of Yukawa couplings, with the Template:Mvar term giving the coupling of the generations Template:Mvar and Template:Mvar, and h.c. means Hermitian conjugate of preceding terms. The fields $q_{\text{L}}$ and $L_{\text{L}}$ are left-handed quark and lepton doublets. Likewise, $u_{\text{R}}$, $d_{\text{R}}$ and $e_{\text{R}}$ are right-handed up-type quark, down-type quark, and lepton singlets. Finally $\phi$ is the Higgs doublet and $\tilde{\phi }=i{\tau }_2{\phi }^{*}$

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 Template:Math, which requires the addition of a new Dirac mass term in the Yukawa sector: $${ℒ}_{\nu }^{\text{Dir}}=(Y_{\nu }{)}_{mn}({\overline{L}}_L{)}_m\phi ({\nu }_R{)}_n+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}$$

This field however must be a sterile neutrino, since being right-handed it experimentally belongs to an isospin singlet (Template:Math) and also has charge Template:Math, implying Template:Math (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 a new Majorana mass term is added to the Yukawa sector: $${ℒ}_{\nu }^{\text{Maj}}=-\frac{1}{2}m(\stackrel{C}{\stackrel{‾}{\nu }}\nu +\bar{\nu }{\nu }^C)$$ where Template:Mvar denotes a charge conjugated (i.e. anti-) particle, and the $\nu$ 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 fermionic field. By CPT symmetry, there is a set of fermions and antifermions with opposite parity and charges. If a left-handed fermion spans some representation its antiparticle (right-handed antifermion) spans the dual representation (note that $\overline{\mathrm{𝟐}}=\mathrm{𝟐}$ for SU(2), because it is pseudo-real). The column "representation" indicates under which representations of the gauge groups that each field transforms, in the order (SU(3), SU(2), U(1)) and for the U(1) group, the value of the weak hypercharge is listed. There are twice as many left-handed lepton field components as right-handed lepton field components in each generation, but an equal number of left-handed quark and right-handed quark field components.

Field content of the standard model
Spin 1 – the gauge fields
Symbol	Associated charge	Group	Coupling	Representation
$B$	Weak hypercharge	Template:Math	$g^{\prime }$ or $g_1$	$(\mathrm{𝟏},\mathrm{𝟏},0)$
$W$	Weak isospin	Template:Math	$g_w$ or $g_2$	$(\mathrm{𝟏},\mathrm{𝟑},0)$
$G$	color	Template:Math	$g_s$ or $g_3$	$(\mathrm{𝟖},\mathrm{𝟏},0)$
Spin Template:Frac – the fermions
Symbol	Name	Baryon number	Lepton number	Representation
$q_{\mathrm{L}}$	Left-handed quark	$\frac{1}{3}$	$0$	$(\mathrm{𝟑},\mathrm{𝟐},\frac{1}{3})$
$u_{\mathrm{R}}$	Right-handed quark (up)	$\frac{1}{3}$	$0$	$(\mathrm{𝟑},\mathrm{𝟏},\frac{4}{3})$
$d_{\mathrm{R}}$	Right-handed quark (down)	$\frac{1}{3}$	$0$	$(\mathrm{𝟑},\mathrm{𝟏},-\frac{2}{3})$
${\ell }_{\mathrm{L}}$	Left-handed lepton	$0$	$1$	$(\mathrm{𝟏},\mathrm{𝟐},-1)$
${\ell }_{\mathrm{R}}$	Right-handed lepton	$0$	$1$	$(\mathrm{𝟏},\mathrm{𝟏},-2)$
Spin 0 – the scalar boson
Symbol	Name	Representation
$H$	Higgs boson	$(\mathrm{𝟏},\mathrm{𝟐},1)$

Fermion content

This table is based in part on data gathered by the Particle Data Group.

Left-handed fermions in the Standard Model
Generation 1
Fermion (left-handed)	Symbol	Electric charge	Weak isospin	Weak hypercharge	Color charge	Mass
Electron	Template:Math	$-1$	$-\frac{1}{2}$	$-1$	$\mathrm{𝟏}$	511 keV
Positron	Template:Math	$+1$	$0$	$+2$	$\mathrm{𝟏}$	511 keV
Electron neutrino	Template:Math	$0$	$+\frac{1}{2}$	$-1$	$\mathrm{𝟏}$	< 0.28 eV
Electron antineutrino	Template:Math	$0$	$0$	$0$	$\mathrm{𝟏}$	< 0.28 eV
Up quark	Template:Math	$+\frac{2}{3}$	$+\frac{1}{2}$	$+\frac{1}{3}$	$\mathrm{𝟑}$	~ 3 MeV
Up antiquark	Template:Math	$-\frac{2}{3}$	$0$	$-\frac{4}{3}$	$\overline{\mathrm{𝟑}}$	~ 3 MeV
Down quark	Template:Math	$-\frac{1}{3}$	$-\frac{1}{2}$	$+\frac{1}{3}$	$\mathrm{𝟑}$	~ 6 MeV
Down antiquark	Template:Math	$+\frac{1}{3}$	$0$	$+\frac{2}{3}$	$\overline{\mathrm{𝟑}}$	~ 6 MeV
Generation 2
Fermion (left-handed)	Symbol	Electric charge	Weak isospin	Weak hypercharge	Color charge	Mass
Muon	Template:Math	$-1$	$-\frac{1}{2}$	$-1$	$\mathrm{𝟏}$	106 MeV
Antimuon	Template:Math	$+1$	$0$	$+2$	$\mathrm{𝟏}$	106 MeV
Muon neutrino	Template:Math	$0$	$+\frac{1}{2}$	$-1$	$\mathrm{𝟏}$	< 0.28 eV
Muon antineutrino	Template:Math	$0$	$0$	$0$	$\mathrm{𝟏}$	< 0.28 eV
Charm quark	Template:Math	$+\frac{2}{3}$	$+\frac{1}{2}$	$+\frac{1}{3}$	$\mathrm{𝟑}$	~ 1.3 GeV
Charm antiquark	Template:Math	$-\frac{2}{3}$	$0$	$-\frac{4}{3}$	$\overline{\mathrm{𝟑}}$	~ 1.3 GeV
Strange quark	Template:Math	$-\frac{1}{3}$	$-\frac{1}{2}$	$+\frac{1}{3}$	$\mathrm{𝟑}$	~ 100 MeV
Strange antiquark	Template:Math	$+\frac{1}{3}$	$0$	$+\frac{2}{3}$	$\overline{\mathrm{𝟑}}$	~ 100 MeV
Generation 3
Fermion (left-handed)	Symbol	Electric charge	Weak isospin	Weak hypercharge	Color charge	Mass
Tau	Template:Math	$-1$	$-\frac{1}{2}$	$-1$	$\mathrm{𝟏}$	1.78 GeV
Antitau	Template:Math	$+1$	$0$	$+2$	$\mathrm{𝟏}$	1.78 GeV
Tau neutrino	Template:Math	$0$	$+\frac{1}{2}$	$-1$	$\mathrm{𝟏}$	< 0.28 eV
Tau antineutrino	Template:Math	$0$	$0$	$0$	$\mathrm{𝟏}$	< 0.28 eV
Top quark	Template:Math	$+\frac{2}{3}$	$+\frac{1}{2}$	$+\frac{1}{3}$	$\mathrm{𝟑}$	171 GeV
Top antiquark	Template:Math	$-\frac{2}{3}$	$0$	$-\frac{4}{3}$	$\overline{\mathrm{𝟑}}$	171 GeV
Bottom quark	Template:Math	$-\frac{1}{3}$	$-\frac{1}{2}$	$+\frac{1}{3}$	$\mathrm{𝟑}$	~ 4.2 GeV
Bottom antiquark	Template:Math	$+\frac{1}{3}$	$0$	$+\frac{2}{3}$	$\overline{\mathrm{𝟑}}$	~ 4.2 GeV

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 the 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.

Parameters of the Standard Model
Symbol	Description	Renormalization scheme (point)	Value
me	electron mass		Template:Physconst/c2
mμ	muon mass		Template:Physconst/c2
mτ	tau mass		Template:Physconst/c2
mu	up quark mass	μ[[MSbar scheme|Template:Overline]] = 2 GeV	Template:Val
md	down quark mass	μTemplate:Overline = 2 GeV	Template:Val
ms	strange quark mass	μTemplate:Overline = 2 GeV	Template:Val
mc	charm quark mass	μTemplate:Overline = mc	Template:Val
mb	bottom quark mass	μTemplate:Overline = mb	Template:Val
mt	top quark mass	on-shell scheme	Template:Val
θ12	CKM 12-mixing angle		13.1°
θ23	CKM 23-mixing angle		2.4°
θ13	CKM 13-mixing angle		0.2°
δ	CKM CP-violating Phase		0.995
g1 or g′	U(1) gauge coupling	μTemplate:Overline = mZ	0.357
g2 or g	SU(2) gauge coupling	μTemplate:Overline = mZ	0.652
g3 or gs	SU(3) gauge coupling	μTemplate:Overline = mZ	1.221
θQCD	QCD vacuum angle		~ 0
v	Higgs vacuum expectation value		Template:Val
mH	Higgs mass		Template:Val

The choice of free parameters is somewhat arbitrary. In the table above, gauge couplings are listed as free parameters, therefore with this choice the Weinberg angle is not a free parameter – it is defined as $\tan {\theta }_{\mathrm{W}}=g_1/g_2$. Likewise, the fine-structure constant of QED is $\alpha =\frac{1}{4\pi }\frac{(g_1{g}_2{)}^2}{g_1^2+g_2^2}$. Instead of fermion masses, dimensionless Yukawa couplings can be chosen as free parameters. For example, the electron mass depends on the Yukawa coupling of the electron to the Higgs field, and its value is $m_{\mathrm{e}}=y_{\mathrm{e}}v/\sqrt{2}$. Instead of the Higgs mass, the Higgs self-coupling strength $\lambda =\frac{m_{\mathrm{H}}^2}{2v^2}$, which is approximately 0.129, can be chosen as a free parameter. Instead of the Higgs vacuum expectation value, the ${\mu }^2$ parameter directly from the Higgs self-interaction term ${\mu }^2{\varphi }^{†}\varphi -\lambda ({\varphi }^{†}\varphi {)}^2$ can be chosen. Its value is ${\mu }^2=\lambda v^2=m_{\mathrm{H}}^2/2$, or approximately $\mu$ = Template:Val.

The value of the vacuum energy (or more precisely, the renormalization scale used to calculate this energy) may also be treated as an additional free parameter. The renormalization scale may be identified with the Planck scale or fine-tuned to match the observed cosmological constant. However, both options are problematic.

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: $${\psi }_{\text{q}}\to e^{i\alpha /3}{\psi }_{\text{q}}$$ $$E_{\mathrm{L}}\to e^{i\beta }{E}_{\mathrm{L}}\text{ and }(e_{\mathrm{R}}{)}^{\text{c}}\to e^{i\beta }(e_{\mathrm{R}}{)}^{\text{c}}$$ $$M_{\mathrm{L}}\to e^{i\beta }{M}_{\mathrm{L}}\text{ and }({\mu }_{\mathrm{R}}{)}^{\text{c}}\to e^{i\beta }({\mu }_{\mathrm{R}}{)}^{\text{c}}$$ $$T_{\mathrm{L}}\to e^{i\beta }{T}_{\mathrm{L}}\text{ and }({\tau }_{\mathrm{R}}{)}^{\text{c}}\to e^{i\beta }({\tau }_{\mathrm{R}}{)}^{\text{c}}$$

The first transformation rule is shorthand meaning that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields Template:Math and $({\mu }_{\mathrm{R}}{)}^{\text{c}},({\tau }_{\mathrm{R}}{)}^{\text{c}}$ are the 2nd (muon) and 3rd (tau) generation analogs of Template:Math and $(e_{\mathrm{R}}{)}^{\text{c}}$ 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 $\frac{1}{3}$, while each antiquark is assigned a baryon number of $-\frac{1}{3}$. 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 imply 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".

Symmetries of the Standard Model and associated conservation laws
Symmetry	Lie group	Symmetry Type	Conservation law
Poincaré	Translations⋊SO(3,1)	Global symmetry	Energy, Momentum, Angular momentum
Gauge	SU(3)×SU(2)×U(1)	Local symmetry	Color charge, Weak isospin, Electric charge, Weak hypercharge
Baryon phase	U(1)	Accidental Global symmetry	Baryon number
Electron phase	U(1)	Accidental Global symmetry	Electron number
Muon phase	U(1)	Accidental Global symmetry	Muon number
Tau phase	U(1)	Accidental Global symmetry	Tau number

U(1) symmetry

For the leptons, the gauge group can be written Template:Math. The two Template:Math factors can be combined into Template:Math, where Template:Math is the lepton number. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group Template:Math. A similar argument in the quark sector also gives the same result for the electroweak theory.

Charged and neutral current couplings and Fermi theory

The charged currents $j^{\mp }=j^1\pm i{j}^2$ are $$j_{\mu }^{-}={\bar{U}}_{i\mathrm{L}}{\gamma }_{\mu }{D}_{i\mathrm{L}}+{\bar{\nu }}_{i\mathrm{L}}{\gamma }_{\mu }{l}_{i\mathrm{L}}.$$ These charged currents are precisely those that entered the Fermi theory of beta decay. The action contains the charge current piece $${ℒ}_{\mathrm{C}\mathrm{C}}=\frac{g}{\sqrt{2}}(j_{\mu }^{+}{W}^{-\mu }+j_{\mu }^{-}{W}^{+\mu }).$$ For energy much less than the mass of the W-boson, the effective theory becomes the current–current contact interaction of the Fermi theory, $2\sqrt{2}{G}_{\mathrm{F}}{J}_{\mu }^{+}{J}^{\mu -}$.

However, gauge invariance now requires that the component $W^3$ of the gauge field also be coupled to a current that lies in the triplet of SU(2). However, this mixes with the Template:Math, and another current in that sector is needed. These currents must be uncharged in order to conserve charge. So neutral currents are also required, $$j_{\mu }^3=\frac{1}{2}({\bar{U}}_{i\mathrm{L}}{\gamma }_{\mu }{U}_{i\mathrm{L}}-{\bar{D}}_{i\mathrm{L}}{\gamma }_{\mu }{D}_{i\mathrm{L}}+{\bar{\nu }}_{i\mathrm{L}}{\gamma }_{\mu }{\nu }_{i\mathrm{L}}-{\bar{l}}_{i\mathrm{L}}{\gamma }_{\mu }{l}_{i\mathrm{L}})$$ $$j_{\mu }^{\mathrm{e}\mathrm{m}}=\frac{2}{3}{\bar{U}}_i{\gamma }_{\mu }{U}_i-\frac{1}{3}{\bar{D}}_i{\gamma }_{\mu }{D}_i-{\bar{l}}_i{\gamma }_{\mu }{l}_i.$$ The neutral current piece in the Lagrangian is then $${ℒ}_{\mathrm{N}\mathrm{C}}=e{j}_{\mu }^{\mathrm{e}\mathrm{m}}{A}^{\mu }+\frac{g}{\cos {\theta }_{\mathrm{W}}}(J_{\mu }^3-{\sin }^2{\theta }_{\mathrm{W}}{J}_{\mu }^{\mathrm{e}\mathrm{m}})Z^{\mu }.$$

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
  • Flavor
  • Quantum chromodynamics
  • Quark model
• Weak interactions
  • Electroweak interaction
  • Fermi's interaction
• Weinberg angle
• Symmetry in quantum mechanics
• Quantum Field Theory in a Nutshell by A. Zee

References and external links

• An introduction to quantum field theory, by M.E. Peskin and D.V. Schroeder (HarperCollins, 1995) Template:ISBN.
• Gauge theory of elementary particle physics, by T.P. Cheng and L.F. Li (Oxford University Press, 1982) Template:ISBN.
• 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) Template:ISBN.
• Quantum Field Theory in a Nutshell (Second Edition), by A. Zee (Princeton University Press, 2010) Template:ISBN.
• An Introduction to Particle Physics and the Standard Model, by R. Mann (CRC Press, 2010) Template:ISBN
• Physics From Symmetry by J. Schwichtenberg (Springer, 2015) Template:ISBN. Especially page 86