Mathematical formulation of the Standard Model
Standard Model of particle physics 

Quantum field theory 

History 
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 describing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.
The Standard Model is renormalizable and mathematically selfconsistent, 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 , and B;
 the gluon field, G_{a}; and
 the Higgs field, φ.
That these are quantum rather than classical fields has the mathematical consequence that they are operatorvalued. 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 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 ψ_{e} (describing the electron and its antiparticle the positron) is then the original ψ field of quantum electrodynamics, which was later accompanied by ψ_{μ} and ψ_{τ} fields for the muon and tauon respectively (and their antiparticles). Electroweak theory added , and for the corresponding neutrinos. The quarks add still further components. In order to be fourspinors 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 complexvalued components for the fermion field.
An important definition is the barred fermion field , which is defined to be , where denotes the Hermitian adjoint of ψ, and γ^{0} is the zeroth gamma matrix. If ψ is thought of as an n × 1 matrix then should be thought of as a 1 × n matrix.
A chiral theory
An independent decomposition of ψ is that into chirality components:
 "Left" chirality:
 "Right" chirality:
where 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 lefthanded particles are weakisospin doublets, whereas the righthanded are singlets – i.e. the weak isospin of ψ_{R} is zero. Put more simply, the weak interaction could rotate e.g. a lefthanded electron into a lefthanded neutrino (with emission of a W^{−}), but could not do so with the same righthanded particles. As an aside, the righthanded 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, righthanded neutrinos must exist in reality. This does not however change the (experimentallyproven) chiral nature of the weak interaction.
Furthermore, U(1) acts differently on and (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" (^{}
_{}ν^{}
_{e}, ^{}
_{}ν^{}
_{μ}, or ^{}
_{}ν^{}
_{τ}) 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 , and "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:
The A field is the photon, which corresponds classically to the wellknown electromagnetic fourpotential – 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 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 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 timedependent. 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 twoline 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 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; for each type of fermion.
 The photon field A satisfies the wave equation .
 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
 β is a normalization factor; for the fermion field it is , where is the volume of the fundamental cell considered; for the photon field A^{μ} it is .
 The sum over p is over all momenta consistent with the period L, i.e., over all vectors where 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.
 E_{p} is the relativistic energy for a momentum p quantum of the field, when the rest mass is m.
 a_{r}(p) and are annihilation and creation operators respectively for "aparticles" and "bparticles" respectively of momentum p; "bparticles" are the antiparticles of "aparticles". Different fields have different "a" and "bparticles". For some fields, a and b are the same.
 u_{r}(p) and v_{r}(p) are nonoperators which carry the vector or spinor aspects of the field (where relevant).
 is the fourmomentum for a quantum with momentum p. denotes an inner product of fourvectors.
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 and .
Technically, is the Hermitian adjoint of the operator a_{r}(p) in the inner product space of ket vectors. The identification of and a_{r}(p) as creation and annihilation operators comes from comparing conserved quantities for a state before and after one of these have acted upon it. can for example be seen to add one particle, because it will add 1 to the eigenvalue of the aparticle number operator, and the momentum of that particle ought to be p since the eigenvalue of the vectorvalued 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 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 referenceframe 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 which relates to the "motion" of the fields.
Fermion fields
The kinetic term for a Dirac fermion is
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 spin1 fields, first define the field strength tensor
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
where t_{i} 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 t_{a} all commute with each other. Of course, this is not the case in general – the standard model includes the nonAbelian 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 , where the index a labels elements of the 8 representation of colour SU(3). The strong coupling constant is conventionally labelled g_{s} (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 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 g_{w} or again simply g. The gauge field will be denoted by .
 The gauge field tensor for the U(1) of weak hypercharge will be denoted by B_{μν}, the coupling by g′, and the gauge field by B_{μ}.
The kinetic term can now be written as
where the traces are over the SU(2) and SU(3) indices hidden in W and G respectively. The twoindex 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 lefthanded fermions.
where B_{μ} is the U(1) gauge field; Y_{W} is the weak hypercharge (the generator of the U(1) group); W_{μ} is the threecomponent 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 T_{3} (also called T_{z}, I_{3} or I_{z}) and weak hypercharge Y_{W} are related by
One may then define the conserved current for weak isospin as
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 (lefthanded) fermion doublet contained in ψ, for example
This is an interaction corresponding to a "rotation in weak isospin space" or in other words, a transformation between e_{L} and ν_{eL} 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 righthanded) via the neutral Z^{0}, 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 T_{a}. 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
where U and D are the Dirac spinors associated with up and downtype 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 which is not invariant under the electroweak symmetry. This can be seen by writing ψ in terms of left and righthanded components (skipping the actual calculation):
i.e. contribution from and terms do not appear. We see that the massgenerating interaction is achieved by constant flipping of particle chirality. The spinhalf 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 spinhalf 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. A^{μ}A_{μ}, 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 field of the group SU(2)_{L}:
where the superscripts + and 0 indicate the electric charge (Q) of the components. The weak hypercharge (Y_{W}) of both components is 1.
The Higgs part of the Lagrangian is
where λ > 0 and μ^{2} > 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 and make real. Then is the nonvanishing vacuum expectation value of the Higgs field. 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 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 finetuning to a small nonzero value in the Standard Model. Quadratic terms in W_{μ} and B_{μ} arise, which give masses to the W and Z bosons:
The mass of the Higgs boson itself is given by
The Yukawa interaction
The Yukawa interaction terms are
where , , and are 3 × 3 matrices of Yukawa couplings, with the mn term giving the coupling of the generations m and n, and h.c. means Hermitian conjugate of preceding terms. The fields and are lefthanded quark and lepton doublets. Likewise, and are righthanded uptype quark, downtype quark, and lepton singlets. Finally is the Higgs doublet and
Neutrino masses
As previously mentioned, evidence shows neutrinos must have mass. But within the standard model, the righthanded neutrino does not exist, so even with a Yukawa coupling neutrinos remain massless. An obvious solution is to simply add a righthanded neutrino ν_{R}, which requires the addition of a new Dirac mass term in the Yukawa sector:
This field however must be a sterile neutrino, since being righthanded it experimentally belongs to an isospin singlet (T_{3} = 0) and also has charge Q = 0, implying Y_{W} = 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 a new Majorana mass term is added to the Yukawa Sector:
where C denotes a charge conjugated (i.e. anti) particle, and the terms are consistently all left (or all right) chirality (note that a leftchirality projection of an antiparticle is a righthanded field; care must be taken here due to different notations sometimes used). Here we are essentially flipping between lefthanded neutrinos and righthanded antineutrinos (it is furthermore possible but not necessary that neutrinos are their own antiparticle, so these particles are the same). However, for leftchirality 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 rightchirality 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 Diracmassonly approach) can provide a “natural” explanation for the smallness of the observed neutrino masses, by linking the righthanded neutrinos to yetunknown 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 lefthanded fermion spans some representation its antiparticle (righthanded antifermion) spans the dual representation (note that for SU(2), because it is pseudoreal). 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 lefthanded lepton field components as righthanded lepton field components in each generation, but an equal number of lefthanded quark and righthanded quark field components.
Field content of the standard model  

Spin 1 – the gauge fields  
Symbol  Associated charge  Group  Coupling  Representation 
Weak hypercharge  U(1)_{Y}  or  
Weak isospin  SU(2)_{L}  or  
colour  SU(3)_{C}  or  
Spin 1⁄2 – the fermions  
Symbol  Name  Baryon number  Lepton number  Representation 
Lefthanded quark  
Righthanded quark (up)  
Righthanded quark (down)  
Lefthanded lepton  
Righthanded lepton  
Spin 0 – the scalar boson  
Symbol  Name  Representation  
Higgs boson 
Fermion content
This table is based in part on data gathered by the Particle Data Group.
Lefthanded fermions in the Standard Model  

Generation 1  
Fermion (lefthanded) 
Symbol  Electric charge 
Weak isospin 
Weak hypercharge 
Color charge 
Mass  
Electron  ^{} _{}e^{−} _{} 
511 keV  
Positron  ^{} _{}e^{+} _{} 
511 keV  
Electron neutrino  ^{} _{}ν^{} _{e} 
< 0.28 eV  
Electron antineutrino  ^{} _{}ν^{} _{e} 
< 0.28 eV  
Up quark  ^{} _{}u^{} _{} 
~ 3 MeV  
Up antiquark  ^{} _{}u^{} _{} 
~ 3 MeV  
Down quark  ^{} _{}d^{} _{} 
~ 6 MeV  
Down antiquark  ^{} _{}d^{} _{} 
~ 6 MeV  
Generation 2  
Fermion (lefthanded) 
Symbol  Electric charge 
Weak isospin 
Weak hypercharge 
Color charge 
Mass  
Muon  ^{} _{}μ^{−} _{} 
106 MeV  
Antimuon  ^{} _{}μ^{+} _{} 
106 MeV  
Muon neutrino  ^{} _{}ν^{} _{μ} 
< 0.28 eV  
Muon antineutrino  ^{} _{}ν^{} _{μ} 
< 0.28 eV  
Charm quark  ^{} _{}c^{} _{} 
~ 1.3 GeV  
Charm antiquark  ^{} _{}c^{} _{} 
~ 1.3 GeV  
Strange quark  ^{} _{}s^{} _{} 
~ 100 MeV  
Strange antiquark  ^{} _{}s^{} _{} 
~ 100 MeV  
Generation 3  
Fermion (lefthanded) 
Symbol  Electric charge 
Weak isospin 
Weak hypercharge 
Color charge 
Mass  
Tau  ^{} _{}τ^{−} _{} 
1.78 GeV  
Antitau  ^{} _{}τ^{+} _{} 
1.78 GeV  
Tau neutrino  ^{} _{}ν^{} _{τ} 
< 0.28 eV  
Tau antineutrino  ^{} _{}ν^{} _{τ} 
< 0.28 eV  
Top quark  ^{} _{}t^{} _{} 
171 GeV  
Top antiquark  ^{} _{}t^{} _{} 
171 GeV  
Bottom quark  ^{} _{}b^{} _{} 
~ 4.2 GeV  
Bottom antiquark  ^{} _{}b^{} _{} 
~ 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  Experimental uncertainty 
m_{e}  Electron mass  510.9989461 keV  ±3.1 meV  
m_{μ}  Muon mass  105.6583745 MeV  ±2.4 eV  
m_{τ}  Tau mass  1.77686 GeV  ±0.12 MeV  
m_{u}  Up quark mass  μ_{MS} = 2 GeV  2.16 MeV  +0.49 −0.26 MeV 
m_{d}  Down quark mass  μ_{MS} = 2 GeV  4.67 MeV  +0.48 −0.17 MeV 
m_{s}  Strange quark mass  μ_{MS} = 2 GeV  93.4 MeV  +8.6 −3.4 MeV 
m_{c}  Charm quark mass  μ_{MS} = m_{c}  1.27 GeV  ±0.02 GeV 
m_{b}  Bottom quark mass  μ_{MS} = m_{b}  4.18 GeV  +0.03 −0.02 GeV 
m_{t}  Top quark mass  Onshell scheme  172.69 GeV  ±0.30 GeV 
θ_{12}  CKM 12mixing angle  13.1°  
θ_{23}  CKM 23mixing angle  2.4°  
θ_{13}  CKM 13mixing angle  0.2°  
δ  CKM CPviolating Phase  0.995  
g_{1} or g'  U(1) gauge coupling  μ_{MS} = m_{Z}  0.357  
g_{2} or g  SU(2) gauge coupling  μ_{MS} = m_{Z}  0.652  
g_{3} or g_{s}  SU(3) gauge coupling  μ_{MS} = m_{Z}  1.221  
θ_{QCD}  QCD vacuum angle  ~0  
v  Higgs vacuum expectation value  246.2196 GeV  ±0.2 MeV  
m_{H}  Higgs mass  125.18 GeV  ±0.16 GeV 
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 . Likewise, the finestructure constant of QED is . 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 . Instead of the Higgs mass, the Higgs selfcoupling strength , which is approximately 0.129, can be chosen as a free parameter. Instead of the Higgs vacuum expectation value, the parameter directly from the Higgs selfinteraction term can be chosen. Its value is , or approximately GeV.
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 finetuned 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:
The first transformation rule is shorthand meaning that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields M_{L}, T_{L} and are the 2nd (muon) and 3rd (tau) generation analogs of E_{L} and 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 , while each antiquark is assigned a baryon number of . 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 antielectron and the associated antineutrino 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."
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 
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 are
However, gauge invariance now requires that the component 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,
Physics beyond the Standard Model
Beyond the Standard Model 

Standard Model 
Physics beyond the Standard Model (BSM) refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the inability to explain the fundamental parameters of the standard model, the strong CP problem, neutrino oscillations, matter–antimatter asymmetry, and the nature of dark matter and dark energy. Another problem lies within the mathematical framework of the Standard Model itself: the Standard Model is inconsistent with that of general relativity, and one or both theories break down under certain conditions, such as spacetime singularities like the Big Bang and black hole event horizons.
Theories that lie beyond the Standard Model include various extensions of the standard model through supersymmetry, such as the Minimal Supersymmetric Standard Model (MSSM) and NexttoMinimal Supersymmetric Standard Model (NMSSM), and entirely novel explanations, such as string theory, Mtheory, and extra dimensions. As these theories tend to reproduce the entirety of current phenomena, the question of which theory is the right one, or at least the "best step" towards a Theory of Everything, can only be settled via experiments, and is one of the most active areas of research in both theoretical and experimental physics.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
 Weak interactions
 Weinberg angle
 Symmetry in quantum mechanics
 Quantum Field Theory in a Nutshell by A. Zee
References and external links
 ^ In fact, there are mathematical issues regarding quantum field theories still under debate (see e.g. Landau pole), but the predictions extracted from the Standard Model by current methods are all selfconsistent. For a further discussion see e.g. R. Mann, chapter 25.
 ^ Overbye, Dennis (11 September 2023). "Don't Expect a 'Theory of Everything' to Explain It All  Not even the most advanced physics can reveal everything we want to know about the history and future of the cosmos, or about ourselves". The New York Times. Archived from the original on 11 September 2023. Retrieved 11 September 2023.
 ^ Lindon, Jack (2020). Particle Collider Probes of Dark Energy, Dark Matter and Generic Beyond Standard Model Signatures in Events With an Energetic Jet and Large Missing Transverse Momentum Using the ATLAS Detector at the LHC (PhD). CERN.
 ^ ^{a} ^{b} Raby, Stuart; Slansky, Richard. "Neutrino Masses  How to add them to the Standard Model" (PDF). FAS Project on Government Secrecy. Retrieved 3 November 2023.
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{{cite web}}
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 ^ McCabe, Gordon. (2007). The structure and interpretation of the standard model. Amsterdam: Elsevier. pp. 160–161. ISBN 9780444531124. OCLC 162131565.
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 ^ The lepton number in SM is only conserved at the classical level. There are nonperturbative effects which do not conserve lepton number: see FuentesMartín, J.; Portolés, J.; RuizFemenía, P. (January 2015). "Instantonmediated baryon number violation in nonuniversal gauge extended models". Journal of High Energy Physics. 2015 (1): 134. arXiv:1411.2471. Bibcode:2015JHEP...01..134F. doi:10.1007/JHEP01(2015)134. ISSN 10298479. or Baryon and lepton numbers in particle physics beyond the standard model
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