Editing Standard Model (section)

== Theoretical aspects ==
{{Main|Mathematical formulation of the Standard Model}}

=== Construction of the Standard Model Lagrangian ===
{| class="wikitable collapsible collapsed"
!colspan="6"|Parameters of the Standard Model
|-
! #
! Symbol
! Description
! Renormalization<br /> scheme (point)
! Value
|-
|1
|''m''<sub>e</sub>
|Electron mass
|
|0.511 MeV
|-
|2
|''m''<sub>μ</sub>
|Muon mass
|
|105.7 MeV
|-
|3
|''m''<sub>τ</sub>
|Tau mass
|
|1.78 GeV
|-
|4
|''m''<sub>u</sub>
|Up quark mass
|''μ''<sub>{{overline|MS}}</sub> = 2 GeV
|1.9 MeV
|-
|5
|''m''<sub>d</sub>
|Down quark mass
|''μ''<sub>{{overline|MS}}</sub> = 2 GeV
|4.4 MeV
|-
|6
|''m''<sub>s</sub>
|Strange quark mass
|''μ''<sub>{{overline|MS}}</sub> = 2 GeV
|87 MeV
|-
|7
|''m''<sub>c</sub>
|Charm quark mass
|''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>c</sub>
|1.32 GeV
|-
|8
|''m''<sub>b</sub>
|Bottom quark mass
|''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>b</sub>
|4.24 GeV
|-
|9
|''m''<sub>t</sub>
|Top quark mass
| On shell scheme
|173.5 GeV
|-
|10
|''θ''<sub>12</sub>
|CKM 12-mixing angle
|
|13.1°
|-
|11
|''θ''<sub>23</sub>
|CKM 23-mixing angle
|
|2.4°
|-
|12
|''θ''<sub>13</sub>
|CKM 13-mixing angle
|
|0.2°
|-
|13
|''δ''
|CKM CP violation Phase
|
|0.995
|-
|14
|''g''<sub>1</sub> or ''g''{{'}}
|U(1) gauge coupling
|''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>Z</sub>
|0.357
|-
|15
|''g''<sub>2</sub> or ''g''
|SU(2) gauge coupling
|''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>Z</sub>
|0.652
|-
|16
|''g''<sub>3</sub> or ''g''<sub>s</sub>
|SU(3) gauge coupling
|''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>Z</sub>
|1.221
|-
|17
|''θ''<sub>QCD</sub>
|QCD vacuum angle
|
|~0
|-
|18
|''v''
|Higgs vacuum expectation value
|
|246 GeV
|-
|19
|''m''<sub>H</sub>
|Higgs mass
|
|{{val|125.09|0.24|u=GeV}}
|}

Technically, [[quantum field theory]] provides the mathematical framework for the Standard Model, in which a [[Lagrangian (field theory)|Lagrangian]] controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical [[field (physics)|field]] that pervades [[space-time]].<ref>{{cite journal | author=Gregg Jaeger | year=2021 | title=The Elementary Particles of Quantum Fields | journal=[[Entropy]] | volume=23 | issue=11 | pages=1416 | doi=10.3390/e23111416 | pmid=34828114 | pmc=8623095 | bibcode=2021Entrp..23.1416J | doi-access=free }}</ref>
The construction of the Standard Model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general [[renormalization|renormalizable]] Lagrangian from its particle (field) content that observes these symmetries.

The [[Global symmetry|global]] [[Poincaré group|Poincaré symmetry]] is postulated for all relativistic quantum field theories. It consists of the familiar [[translational symmetry]], [[rotational symmetry]] and the inertial reference frame invariance central to the theory of [[special relativity]]. The [[Local symmetry|local]] SU(3) × SU(2) × U(1) [[gauge symmetry]] is an [[Internal symmetries|internal symmetry]] that essentially defines the Standard Model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. The fields fall into different [[Representation of a Lie group|representations]] of the various symmetry groups of the Standard Model (see table). Upon writing the most general Lagrangian, one finds that the dynamics depends on 19 parameters, whose numerical values are established by experiment. The parameters are summarized in the table (made visible by clicking "show") above.

==== Quantum chromodynamics sector ====
{{Main|Quantum chromodynamics}}
The quantum chromodynamics (QCD) sector defines the interactions between quarks and gluons, which is a [[Yang–Mills theory|Yang–Mills gauge theory]] with SU(3) symmetry, generated by <math>T^a = \lambda^a/2</math>. 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
<math display="block">\mathcal{L}_\text{QCD} = \overline{\psi} i\gamma^\mu D_{\mu} \psi - \frac{1}{4} G^a_{\mu\nu} G^{\mu\nu}_a,</math>
where <math>\psi</math> is a three component column vector of [[Dirac spinor]]s, each element of which refers to a quark field with a specific [[color charge]] (i.e. red, blue, and green) and summation over [[Flavour (particle physics)|flavor]] (i.e. up, down, strange, etc.) is implied. 

The gauge covariant derivative of QCD is defined by <math>D_{\mu} \equiv \partial_\mu - i g_\text{s}\frac{1}{2}\lambda^a G_\mu^a</math>, where
* {{math|''γ''{{isup|''μ''}}}} are the [[Dirac matrices]],
* {{math|''G''{{su|lh=0.9|b=''μ''|p=''a''}}}} is the 8-component (<math>a = 1, 2, \dots, 8</math>) SU(3) gauge field,
* {{math|''λ''{{su|lh=0.9|p=''a''}}}} are the 3 × 3 [[Gell-Mann matrices]], generators of the SU(3) color group,
* {{math|''G''{{su|lh=0.9|b=''μν''|p=''a''}}}} represents the [[gluon field strength tensor]], and
* {{math|''g''<sub>s</sub>}} is the strong coupling constant.

The QCD Lagrangian is invariant under local SU(3) gauge transformations; i.e., transformations of the form <math>\psi \rightarrow \psi' = U\psi</math>, where <math>U = e^{-i g_\text{s}\lambda^a \phi^{a}(x)}</math> is 3 × 3 unitary matrix with determinant 1, making it a member of the group SU(3), and <math>\phi^{a}(x)</math> is an arbitrary function of spacetime.

==== Electroweak sector ====
{{Main|Electroweak interaction}}
The electroweak sector is a [[Yang–Mills theory|Yang–Mills gauge theory]] with the symmetry group {{nowrap|U(1) × SU(2)<sub>L</sub>}},
<math display="block">\mathcal{L}_\text{EW} = \overline{Q}_{\text{L}j} i\gamma^\mu D_{\mu} Q_{\text{L}j} + \overline{u}_{\text{R}j} i\gamma^\mu D_{\mu} u_{\text{R}j} + \overline{d}_{\text{R}j} i\gamma^\mu D_{\mu} d_{\text{R}j} + \overline{\ell}_{\text{L}j} i\gamma^\mu D_{\mu} \ell_{\text{L}j} + \overline{e}_{\text{R}j} i\gamma^\mu D_{\mu} e_{\text{R}j} - \tfrac{1}{4} W_a^{\mu\nu} W_{\mu\nu}^a - \tfrac{1}{4} B^{\mu\nu} B_{\mu\nu}, </math>
where the subscript <math>j</math> sums over the three generations of fermions; <math>Q_\text{L}, u_\text{R}</math>, and <math>d_\text{R}</math> are the left-handed doublet, right-handed singlet up type, and right handed singlet down type quark fields; and <math>\ell_\text{L}</math> and <math>e_\text{R}</math> are the left-handed doublet and right-handed singlet lepton fields. 

The electroweak [[gauge covariant derivative]] is defined as <math>D_\mu \equiv \partial_\mu - ig' \tfrac12 Y_\text{W} B_\mu - ig \tfrac{1}{2} \vec\tau_\text{L} \vec W_\mu</math>, where 
* {{mvar|B<sub>μ</sub>}} is the U(1) gauge field,
* {{math|''Y''<sub>W</sub>}} is the [[weak hypercharge]] – the generator of the U(1) group,
* {{math|{{vec|''W''}}<sub>''μ''</sub>}} is the 3-component SU(2) gauge field,
* {{math|{{overset|lh=0.5|→|''τ''}}<sub>L</sub>}} are the [[Pauli matrices]] – infinitesimal generators of the SU(2) group – with subscript L to indicate that they only act on ''left''-chiral fermions,
* {{mvar|g'}} and {{mvar|g}} are the U(1) and SU(2) coupling constants respectively,
* <math>W^{a\mu\nu}</math> (<math>a = 1, 2, 3</math>) and <math>B^{\mu\nu}</math> are the [[field strength tensor]]s for the weak isospin and weak hypercharge fields.

Notice that the addition of fermion mass terms into the electroweak Lagrangian is forbidden, since terms of the form <math>m\overline\psi\psi</math> do not respect {{nowrap|U(1) × SU(2)<sub>L</sub>}} gauge invariance. Neither is it possible to add explicit mass terms for the U(1) and SU(2) gauge fields. The Higgs mechanism is responsible for the generation of the gauge boson masses, and the fermion masses result from Yukawa-type interactions with the Higgs field.

==== Higgs sector ====
{{Main|Higgs mechanism}}
In the Standard Model, the [[Higgs field]] is an SU(2){{sub|L}} doublet of complex [[Scalar (physics)|scalar]] fields with four degrees of freedom:
<math display="block"> 
\varphi = \begin{pmatrix} \varphi^+ \\ \varphi^0 \end{pmatrix}
= \frac{1}{\sqrt{2}} \begin{pmatrix} \varphi_1 + i\varphi_2 \\ \varphi_3 + i\varphi_4 \end{pmatrix},
</math>
where the superscripts + and 0 indicate the electric charge <math>Q</math> of the components. The weak hypercharge <math>Y_\text{W}</math> of both components is 1. Before symmetry breaking, the Higgs Lagrangian is
<math display="block"> \mathcal{L}_\text{H} = \left(D_{\mu}\varphi\right)^{\dagger} \left(D^{\mu}\varphi \right) - V(\varphi),</math>
where <math>D_{\mu}</math> is the electroweak gauge covariant derivative defined above and <math>V(\varphi)</math> is the potential of the Higgs field. The square of the covariant derivative leads to three and four point interactions between the electroweak gauge fields <math>W^{a}_{\mu}</math> and <math>B_{\mu}</math> and the scalar field <math>\varphi</math>. The scalar potential is given by
<math display="block"> V(\varphi) = -\mu^2\varphi^{\dagger}\varphi + \lambda \left( \varphi^{\dagger}\varphi \right)^2, </math>
where <math>\mu^2>0</math>, so that <math>\varphi</math> acquires a non-zero [[Vacuum expectation value]], which generates masses for the Electroweak gauge fields (the Higgs mechanism), and <math>\lambda>0</math>, so that the potential is bounded from below. The quartic term describes self-interactions of the scalar field <math>\varphi</math>.

The minimum of the potential is degenerate with an infinite number of equivalent [[ground state]] solutions, which occurs when <math>\varphi^{\dagger}\varphi = \tfrac{\mu^2}{2\lambda}</math>. It is possible to perform a [[Unitary gauge|gauge transformation]] on <math>\varphi</math> such that the ground state is transformed to a basis where <math>\varphi_1 = \varphi_2 = \varphi_4 = 0</math> and <math>\varphi_3 = \tfrac{\mu}{\sqrt{\lambda}} \equiv v </math>. This breaks the symmetry of the ground state. The expectation value of <math>\varphi</math> now becomes 
<math display="block"> \langle \varphi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ v \end{pmatrix},</math>
where <math>v</math> has units of mass and sets the scale of electroweak physics. This is the only dimensional parameter of the Standard Model and has a measured value of ~{{val|246|u=GeV/c2}}. 

After symmetry breaking, the masses of the W and Z are given by <math>m_\text{W}=\frac{1}{2}gv</math> and <math> m_\text{Z}=\frac{1}{2}\sqrt{g^2+g'^2}v</math>, which can be viewed as predictions of the theory. The photon remains massless. The mass of the [[Higgs boson]] is <math>m_\text{H}=\sqrt{2\mu^2}=\sqrt{2\lambda}v</math>. Since <math>\mu</math> and <math>\lambda</math> are free parameters, the Higgs's mass could not be predicted beforehand and had to be determined experimentally.

==== Yukawa sector ====
The [[Yukawa interaction]] terms are:
<math display="block">\mathcal{L}_\text{Yukawa} = (Y_\text{u})_{mn}(\bar{Q}_\text{L})_m \tilde{\varphi}(u_\text{R})_n + (Y_\text{d})_{mn}(\bar{Q}_\text{L})_m \varphi(d_\text{R})_n + (Y_\text{e})_{mn}(\bar{\ell}_\text{L})_m {\varphi}(e_\text{R})_n + \mathrm{h.c.} </math>
where <math>Y_\text{u}</math>, <math>Y_\text{d}</math>, and <math>Y_\text{e}</math> are {{math|3 × 3}} matrices of Yukawa couplings, with the {{mvar|mn}} term giving the coupling of the generations {{mvar|m}} and {{mvar|n}}, and h.c. means Hermitian conjugate of preceding terms. The fields <math>Q_\text{L}</math> and <math>\ell_\text{L}</math> are left-handed quark and lepton doublets. Likewise, <math>u_\text{R}, d_\text{R}</math> and <math>e_\text{R}</math> are right-handed up-type quark, down-type quark, and lepton singlets. Finally <math>\varphi</math> is the Higgs doublet and <math>\tilde{\varphi} = i\tau_2\varphi^{*}</math> is its charge conjugate state.

The Yukawa terms are invariant under the SU(2){{sub|L}} × U(1){{sub|Y}} gauge symmetry of the Standard Model and generate masses for all fermions after spontaneous symmetry breaking.