Editing Electroweak interaction

{{short description|Unified description of electromagnetism and the weak interaction}}
{{Standard model of particle physics|cTopic=Some models}}

In [[particle physics]], the '''electroweak interaction''' or '''electroweak force''' is the [[unified field theory|unified description]] of two of the [[fundamental interaction]]s of nature: [[electromagnetism|electromagnetism (electromagnetic interaction)]] and the [[weak interaction]]. Although these two forces appear very different at everyday low energies, the theory models them as two different aspects of the same force. Above the [[electroweak scale|unification energy]], on the order of 246&nbsp;[[GeV]],<ref group="lower-alpha">The particular number 246&nbsp;GeV is taken to be the [[vacuum expectation value]] <math>v = (G_\text{F} \sqrt{2})^{-1/2}</math> of the [[Higgs field]] (where <math>G_\text{F}</math> is the [[Fermi coupling constant]]).</ref> they would merge into a single force. Thus, if the temperature is high enough – approximately 10<sup>15</sup>&nbsp;[[Kelvin|K]] – then the electromagnetic force and weak force merge into a combined electroweak force.

During the [[quark epoch]] (shortly after the [[Big Bang]]), the electroweak force split into the electromagnetic and [[weak force]]. It is thought that the required temperature of 10<sup>15</sup>&nbsp;K has [[Orders of magnitude (temperature)|not been seen widely throughout the universe]] since before the quark epoch, and currently the highest human-made temperature in thermal equilibrium is around {{val|5.5|e=12|u=K}} (from the [[Large Hadron Collider]]).

[[Sheldon Glashow]],<ref>Glashow, S. (1959). "The renormalizability of vector meson interactions." ''Nucl. Phys.'' '''10''', 107.</ref> [[Abdus Salam]],<ref>{{cite journal |first1=A. |last1=Salam |author-link=Abdus Salam   |first2=J. C. |last2=Ward |title=Weak and electromagnetic interactions |journal=Nuovo Cimento |volume=11 |issue=4 |year=1959 |pages=568–577 |doi= 10.1007/BF02726525|bibcode=1959NCim...11..568S |s2cid=15889731 }}</ref> and [[Steven Weinberg]]<ref name=Weinberg1967>{{cite journal | last1 = Weinberg | first1 = S | year = 1967 | title = A Model of Leptons | url = http://astrophysics.fic.uni.lodz.pl/100yrs/pdf/12/066.pdf | archive-url = https://web.archive.org/web/20120112142352/http://astrophysics.fic.uni.lodz.pl/100yrs/pdf/12/066.pdf | url-status = dead | archive-date = 2012-01-12 | journal = Phys. Rev. Lett. | volume = 19 | issue = 21 | pages = 1264–66 | doi = 10.1103/PhysRevLett.19.1264 | bibcode = 1967PhRvL..19.1264W }}</ref> were awarded the 1979 [[Nobel Prize in Physics]] for their contributions to the unification of the weak and electromagnetic interaction between [[elementary particle]]s, known as the '''Weinberg–Salam theory'''.<ref>
{{cite book
 |author=S. Bais
 |year=2005
 |title=The Equations: Icons of knowledge
 |page=[https://archive.org/details/veryspecialrelat0000bais/page/84 84]
 |isbn=0-674-01967-9
 |url=https://archive.org/details/veryspecialrelat0000bais/page/84
 }}</ref><ref>
{{cite web
|url=http://nobelprize.org/nobel_prizes/physics/laureates/1979/
 |title=The Nobel Prize in Physics 1979
 |publisher=[[The Nobel Foundation]]
 |access-date=2008-12-16
}}</ref> The existence of the electroweak interactions was experimentally established in two stages, the first being the discovery of [[neutral current]]s in neutrino scattering by the [[Gargamelle]] collaboration in 1973, and the second in 1983 by the [[UA1]] and the [[UA2]] collaborations that involved the discovery of the [[W and Z bosons|W and Z]] [[gauge boson]]s in proton–antiproton collisions at the converted [[Super Proton Synchrotron]]. In 1999, [[Gerardus 't Hooft]] and [[Martinus Veltman]] were awarded the Nobel prize for showing that the electroweak theory is [[renormalizable]].

== History ==
After the [[Wu experiment]] in 1956 discovered [[parity violation]] in the [[weak interaction]], a search began for a way to relate the [[weak interaction|weak]] and [[electromagnetic interaction]]s. Extending his [[doctoral advisor]] [[Julian Schwinger]]'s work, [[Sheldon Glashow]] first experimented with introducing two different symmetries, one [[Chirality (physics)|chiral]] and one achiral, and combined them such that their overall symmetry was unbroken. This did not yield a [[renormalization|renormalizable]] [[gauge theory|theory]], and its gauge symmetry had to be broken by hand as no [[Spontaneous symmetry breaking|spontaneous mechanism]] was known, but it predicted a new particle, the [[Z boson]]. This received little notice, as it matched no experimental finding.

In 1964, [[Abdus Salam|Salam]] and [[John Clive Ward]]<ref>{{Cite journal|last1=Salam|first1=A.|last2=Ward|first2=J.C.|date=November 1964|title=Electromagnetic and weak interactions|url=https://linkinghub.elsevier.com/retrieve/pii/0031916364907115|journal=Physics Letters|language=en|volume=13|issue=2|pages=168–171|doi=10.1016/0031-9163(64)90711-5|bibcode=1964PhL....13..168S |url-access=subscription}}</ref> had the same idea, but predicted a massless [[photon]] and three massive [[gauge boson]]s with a manually broken symmetry. Later around 1967, while investigating [[spontaneous symmetry breaking]], Weinberg found a set of symmetries predicting a massless, neutral [[gauge boson]]. Initially rejecting such a particle as useless, he later realized his symmetries produced the electroweak force, and he proceeded to predict rough masses for the [[W and Z bosons]]. Significantly, he suggested this new theory was renormalizable.<ref name=Weinberg1967/> In 1971, [[Gerard 't Hooft]] proved that spontaneously broken gauge symmetries are renormalizable even with massive gauge bosons.

== Formulation ==
{{main|Mathematical formulation of the Standard Model}}

[[File:Weinberg angle (relation between coupling constants).svg|upright=1.25|thumb|Weinberg's weak mixing angle {{mvar|θ}}{{sub|W}}, and relation between coupling constants {{mvar|g, g′}}, and {{mvar|e}}. Adapted from Lee (1981).<ref>{{cite book |first=T.D. |last=Lee |year=1981 |title=Particle Physics and Introduction to Field Theory}}</ref>]]
[[File:Electroweak.svg|upright=1.25|thumb|The pattern of [[weak isospin]], {{mvar|T}}{{sub|3}}, and [[weak hypercharge]], {{mvar|Y}}{{sub|{{sc|w}}}}, of the known elementary particles, showing the electric charge, {{mvar|Q}}, along the [[weak mixing angle]]. The neutral Higgs field (circled) breaks the electroweak symmetry and interacts with other particles to give them mass. Three components of the Higgs field become part of the massive {{SubatomicParticle|W boson}} and {{SubatomicParticle|Z boson}} bosons.]]

Mathematically, electromagnetism is unified with the weak interactions as a [[Yang–Mills theory|Yang–Mills field]] with an {{nowrap|1=[[SU(2)]] × [[Unitary group|U(1)]]}} [[gauge theory|gauge group]], which describes the formal operations that can be applied to the electroweak gauge fields without changing the dynamics of the system. These fields are the weak isospin fields {{mvar|W}}{{sub|1}}, {{mvar|W}}{{sub|2}}, and {{mvar|W}}{{sub|3}}, and the weak hypercharge field {{mvar|B}}.
This invariance is known as '''electroweak symmetry'''.

The [[Generating set of a group|generators]] of [[SU(2)]] and [[Unitary group|U(1)]] are given the name [[weak isospin]] (labeled {{mvar|T}}) and [[weak hypercharge]] (labeled {{mvar|Y}}) respectively. These then give rise to the gauge bosons that mediate the electroweak interactions – the three {{math|W}} bosons of weak isospin ({{math|''W''}}{{sub|1}}, {{math|''W''}}{{sub|2}}, and {{math|''W''}}{{sub|3}}), and the {{math|''B''}} boson of weak hypercharge, respectively, all of which are "initially" massless. These are not physical fields yet, before [[spontaneous symmetry breaking]] and the associated [[Higgs mechanism]].

In the [[Standard Model]], the observed physical particles, the [[W and Z bosons|{{SubatomicParticle|W boson+-}} and {{SubatomicParticle|Z boson0}} bosons]], and the [[photon]], are produced through the [[spontaneous symmetry breaking]] of the electroweak symmetry SU(2) × U(1){{sub|{{sc|y}}}} to U(1){{sub|em}},{{efn|Note that {{math|U(1)}}{{sub|{{sc|y}}}} and {{math|U(1)}}{{sub|em}} are distinct instances of generic {{math|U(1)}}: Each of the two forces gets its own, independent copy of the unitary group.}} effected by the [[Higgs mechanism]] (see also [[Higgs boson]]), an elaborate quantum-field-theoretic phenomenon that "spontaneously" alters the realization of the symmetry and rearranges degrees of freedom.<ref>
{{cite journal
 |last1=Englert |first1=F.
 |last2=Brout |first2=R.
 |year=1964
 |title=Broken symmetry and the mass of gauge vector mesons
 |journal=[[Physical Review Letters]]
 |volume=13 |issue=9 |pages=321–323
 |doi=10.1103/PhysRevLett.13.321 |doi-access=free
 |bibcode=1964PhRvL..13..321E
}}
</ref><ref name="Peter W. Higgs 1964 508-509">
{{cite journal
 |last=Higgs |first=P.W.
 |year=1964
 |title=Broken symmetries and the masses of gauge bosons
 |journal=[[Physical Review Letters]]
 |volume=13 |issue=16 |pages=508–509
 |doi=10.1103/PhysRevLett.13.508 |doi-access=free
 |bibcode=1964PhRvL..13..508H
}}
</ref><ref>
{{cite journal
 |author1=Guralnik, G.S.
 |author2=Hagen, C.R.
 |author3=Kibble, T.W.B.
 |year=1964
 |title=Global conservation laws and massless particles
 |journal=[[Physical Review Letters]]
 |volume=13 |issue=20 |pages=585–587
 |doi=10.1103/PhysRevLett.13.585 |doi-access=free
 |bibcode=1964PhRvL..13..585G
}}
</ref><ref>
{{cite journal
 |author=Guralnik, G.S.
 |year=2009
 |title=The history of the Guralnik, Hagen, and Kibble development of the theory of spontaneous symmetry breaking and gauge particles
 |journal=[[International Journal of Modern Physics A]]
 |volume=24 |issue=14 | pages=2601–2627
 |doi=10.1142/S0217751X09045431 |arxiv=0907.3466
 |bibcode=2009IJMPA..24.2601G |s2cid=16298371
}}
</ref>

The electric charge arises as the particular linear combination (nontrivial) of {{mvar|Y}}{{sub|{{sc|w}}}} (weak hypercharge) and the {{mvar|T}}{{sub|3}} component of weak isospin (<math>Q = T_3 + \tfrac{1}{2}\,Y_\mathrm{W}</math>) that does ''not'' couple to the [[Higgs boson]]. That is to say: the Higgs and the electromagnetic field have no effect on each other, at the level of the fundamental forces ("tree level"), while any ''other'' combination of the hypercharge and the weak isospin must interact with the Higgs. This causes an apparent separation between the weak force, which interacts with the Higgs, and electromagnetism, which does not. Mathematically, the electric charge is a specific combination of the hypercharge and {{mvar|T}}{{sub|3}} outlined in the figure.

{{math|U(1)}}{{sub|em}} (the symmetry group of electromagnetism only) is defined to be the group generated by this special linear combination, and the symmetry described by the {{math|U(1)}}{{sub|em}} group is unbroken, since it does not ''directly'' interact with the Higgs.{{efn|Although electromagnetism – e.g. the photon – does not ''directly'' interact with the [[Higgs boson]], it does interact ''indirectly'', through [[quantum fluctuations]].}}

The above spontaneous symmetry breaking makes the {{mvar|W}}{{sub|3}} and {{mvar|B}} bosons coalesce into two different physical bosons with different masses – the {{SubatomicParticle|Z boson0}} boson, and the photon ({{math|{{SubatomicParticle|photon}}}}),
: <math> \begin{pmatrix}
\gamma \\
Z^0 \end{pmatrix} = \begin{pmatrix}
\cos \theta_\text{W} & \sin \theta_\text{W} \\
-\sin \theta_\text{W} & \cos \theta_\text{W} \end{pmatrix} \begin{pmatrix}
B \\
W_3 \end{pmatrix} ,</math>
where {{mvar|θ}}{{sub|{{sc|w}}}} is the ''[[weak mixing angle]]''. The axes representing the particles have essentially just been rotated, in the ({{mvar|W}}{{sub|3}}, {{mvar|B}}) plane, by the angle {{mvar|θ}}{{sub|{{sc|w}}}}. This also introduces a mismatch between the mass of the {{SubatomicParticle|Z boson0}} and the mass of the {{SubatomicParticle|W boson+-}} particles (denoted as {{mvar|m}}{{sub|{{sc|z}}}} and {{mvar|m}}{{sub|{{sc|w}}}}, respectively),
: <math>m_\text{Z} = \frac{m_\text{W}}{\,\cos\theta_\text{W}\,} ~.</math>

The {{mvar|W}}{{sub|1}} and {{mvar|W}}{{sub|2}} bosons, in turn, combine to produce the charged massive bosons {{SubatomicParticle|W boson+-}}:<ref>{{cite book
 | author=D. J. Griffiths
 | year=1987
 | title=Introduction to Elementary Particles
 | publisher=John Wiley & Sons
 | isbn=0-471-60386-4
}}</ref>
: <math>W^{\pm} = \frac{1}{\sqrt{2\,}}\,\bigl(\,W_1 \mp i W_2\,\bigr) ~.</math>

== Lagrangian ==

=== Before electroweak symmetry breaking ===
The [[Lagrangian (field theory)|Lagrangian]] for the electroweak interactions is divided into four parts before [[electroweak symmetry breaking]] manifests,
: <math>\mathcal{L}_{\mathrm{EW}} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y~.</math>

The <math>\mathcal{L}_g</math> term describes the interaction between the three {{mvar|W}}  vector bosons and the {{mvar|B}} [[vector boson]], 
: <math>\mathcal{L}_g = -\tfrac{1}{4} W_{a}^{\mu\nu}W_{\mu\nu}^a - \tfrac{1}{4} B^{\mu\nu}B_{\mu\nu},</math>
where <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 gauge fields.

<math>\mathcal{L}_f</math> is the [[kinetic term]] for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through the [[gauge covariant derivative]],
: <math>\mathcal{L}_f = \overline{Q}_j iD\!\!\!\!/\; Q_j+ \overline{u}_j iD\!\!\!\!/\; u_j+ \overline{d}_j iD\!\!\!\!/\; d_j + \overline{L}_j iD\!\!\!\!/\; L_j + \overline{e}_j iD\!\!\!\!/\; e_j,</math>
where the subscript {{mvar|j}} sums over the three generations of fermions; {{mvar|Q}}, {{mvar|u}}, and {{mvar|d}} are the left-handed doublet, right-handed singlet up, and right handed singlet down quark fields; and {{mvar|L}} and {{mvar|e}} are the left-handed doublet and right-handed singlet electron fields.
The [[Feynman slash notation|Feynman slash]] <math>D\!\!\!\!/</math> means the contraction of the 4-gradient with the [[Dirac matrices]], defined as
: <math>D\!\!\!\!/ \equiv \gamma^\mu\ D_\mu,</math>
and the covariant derivative (excluding the [[gluon]] gauge field for the [[strong interaction]]) is defined as
: <math>\ D_\mu \equiv \partial_\mu - i\ \frac{g'}{2}\ Y\ B_\mu - i\ \frac{g}{2}\ T_j\ W_\mu^j.</math>
Here <math>\ Y\ </math> is the weak hypercharge and the <math>\ T_j\ </math> are the components of the weak isospin.

The <math>\mathcal{L}_h</math> term describes the [[Higgs field]] <math>h</math> and its interactions with itself and the gauge bosons,
: <math>\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\ ,</math>
where <math>v</math> is the [[vacuum expectation value]].

The <math>\ \mathcal{L}_y\ </math> term describes the [[Yukawa interaction]] with the fermions, 
: <math>\mathcal{L}_y = - y_{u}^{ij}\epsilon^{ab}\ h_b^\dagger\ \overline{Q}_{ia} u_j^c - y_{d}^{ij}\ h\ \overline{Q}_i d^c_j - y_{e}^{ij}\ h\ \overline{L}_i e^c_j + \mathrm{h.c.} ~,</math>
and generates their masses,  manifest when the Higgs field acquires a nonzero vacuum expectation value, discussed next. The <math>\ y_k^{ij}\ ,</math> for <math>\ k \in \{ \mathrm{u, d, e} \}\ ,</math> are matrices of Yukawa couplings.

=== After electroweak symmetry breaking ===
The Lagrangian reorganizes itself as the Higgs field acquires a non-vanishing vacuum expectation value dictated by the potential of the previous section. As a result of this rewriting, the symmetry breaking becomes manifest. In the history of the universe, this is believed to have happened shortly after the hot big bang, when the universe was at a temperature {{val|159.5|1.5|ul=GeV}}<ref>
{{cite journal
  |author1   = D'Onofrio, Michela
  |author2   = Rummukainen, Kari
  |year      = 2016
  |title     = Standard model cross-over on the lattice
  |journal   = Phys. Rev. D
  |volume    = 93  |number = 2  |page = 025003
  |doi       = 10.1103/PhysRevD.93.025003  |s2cid  = 119261776
  |bibcode   = 2016PhRvD..93b5003D  |hdl = 10138/159845 |hdl-access= free
  |arxiv     = 1508.07161
}}
</ref>
(assuming the Standard Model of particle physics).

Due to its complexity, this Lagrangian is best described by breaking it up into several parts as follows.
: <math>\mathcal{L}_{\mathrm{EW}} = \mathcal{L}_\mathrm{K} + \mathcal{L}_\mathrm{N} + \mathcal{L}_\mathrm{C} + \mathcal{L}_\mathrm{H} + \mathcal{L}_{\mathrm{HV}} + \mathcal{L}_{\mathrm{WWV}} + \mathcal{L}_{\mathrm{WWVV}} + \mathcal{L}_\mathrm{Y} ~.</math>

The kinetic term <math>\mathcal{L}_K</math> contains all the quadratic terms of the Lagrangian, which include the dynamic terms (the partial derivatives) and the mass terms (conspicuously absent from the Lagrangian before symmetry breaking)
: <math> 
\begin{align}
\mathcal{L}_\mathrm{K} = \sum_f \overline{f}(i\partial\!\!\!/\!\;-m_f)\ f - \frac{1}{4}\ A_{\mu\nu}\ A^{\mu\nu} - \frac{1}{2}\ W^+_{\mu\nu}\ W^{-\mu\nu} + m_W^2\ W^+_\mu\ W^{-\mu} 
\\
\qquad -\frac{1}{4}\ Z_{\mu\nu}Z^{\mu\nu} + \frac{1}{2}\ m_Z^2\ Z_\mu\ Z^\mu + \frac{1}{2}\ (\partial^\mu\ H)(\partial_\mu\ H) - \frac{1}{2}\ m_H^2\ H^2 ~,
\end{align}
</math>
where the sum runs over all the fermions of the theory (quarks and leptons), and the fields <math>\ A_{\mu\nu}\ ,</math> <math>\ Z_{\mu\nu}\ ,</math> <math>\ W^-_{\mu\nu}\ ,</math> and <math>\ W^+_{\mu\nu} \equiv (W^-_{\mu\nu})^\dagger\ </math> are given as
: <math>X^{a}_{\mu\nu} = \partial_\mu X^{a}_\nu - \partial_\nu X^{a}_\mu + g f^{abc}X^{b}_{\mu}X^{c}_{\nu} ~,</math>
with <math>X</math> to be replaced by the relevant field (<math>A,</math> <math>Z,</math> <math>W^\pm</math>) and  {{mvar|f&thinsp;{{sup|abc}} }} by the structure constants of the appropriate gauge group.

The neutral current <math>\ \mathcal{L}_\mathrm{N}\ </math> and charged current <math>\ \mathcal{L}_\mathrm{C}\ </math> components of the Lagrangian contain the interactions between the fermions and gauge bosons,
: <math>\mathcal{L}_\mathrm{N} = e\ J_\mu^\mathrm{em}\ A^\mu + \frac{g}{\ \cos\theta_W\ }\ (\ J_\mu^3 - \sin^2\theta_W\ J_\mu^\mathrm{em}\ )\ Z^\mu ~,</math>
where <math>~e = g\ \sin \theta_\mathrm{W} = g'\ \cos \theta_\mathrm{W} ~.</math> The electromagnetic current <math>\; J_\mu^{\mathrm{em}} \;</math> is
: <math>J_\mu^\mathrm{em} = \sum_f \ q_f\ \overline{f}\ \gamma_\mu\ f ~,</math>
where <math>\ q_f\ </math> is the fermions' electric charges. 
The neutral weak current <math>\ J_\mu^3\ </math> is
: <math>J_\mu^3 = \sum_f\ T^3_f\ \overline{f}\ \gamma_\mu\ \frac{\ 1-\gamma^5\ }{2}\ f ~,</math>
where <math>T^3_f</math> is the fermions' weak isospin.{{efn|name=note_chiral_factors| Note the factors <math>~\tfrac{1}{2}\ (1-\gamma^5)~</math> in the weak coupling formulas: These factors are deliberately inserted to expunge any left-[[chirality (physics)|chiral]] components of the spinor fields. This is why electroweak theory is said to be a '<nowiki/>''[[chiral theory]]''<nowiki/>'.}}

The charged current part of the Lagrangian is given by
: <math>\mathcal{L}_\mathrm{C} = -\frac{g}{\ \sqrt{2 \;}\ }\ \left[\ \overline{u}_i\ \gamma^\mu\ \frac{\ 1 - \gamma^5\ }{2} \; M^{\mathrm{CKM}}_{ij}\ d_j + \overline{\nu}_i\ \gamma^\mu\;\frac{\ 1-\gamma^5\ }{2} \; e_i\ \right]\ W_\mu^{+} + \mathrm{h.c.} ~,</math>
where <math>\ \nu\ </math> is the right-handed singlet neutrino field, and the [[CKM matrix]] <math>M_{ij}^\mathrm{CKM}</math> determines the mixing between mass and weak eigenstates of the quarks.{{efn|name=note_chiral_factors}}

<math>\mathcal{L}_\mathrm{H}</math> contains the Higgs three-point and four-point self interaction terms,
: <math>\mathcal{L}_\mathrm{H} = -\frac{\ g\ m_\mathrm{H}^2\,}{\ 4\ m_\mathrm{W}\ }\;H^3 - \frac{\ g^2\ m_\mathrm{H}^2\ }{32\ m_\mathrm{W}^2}\;H^4 ~.</math>

<math>\mathcal{L}_{\mathrm{HV}}</math> contains the Higgs interactions with gauge vector bosons,
: <math>\mathcal{L}_\mathrm{HV}  =\left(\ g\ m_\mathrm{HV} + \frac{\ g^2\ }{4}\;H^2\ \right)\left(\ W^{+}_\mu\ W^{-\mu} + \frac{1}{\ 2\ \cos^2\ \theta_\mathrm{W}\ }\;Z_\mu\ Z^\mu\ \right) ~.</math>

<math>\mathcal{L}_{\mathrm{WWV}}</math> contains the gauge three-point self interactions,
: <math>\mathcal{L}_{\mathrm{WWV}} = -i\ g\ \left[\; \left(\ W_{\mu\nu}^{+}\ W^{-\mu} - W^{+\mu}\ W^{-}_{\mu\nu}\ \right)\left(\ A^\nu\ \sin \theta_\mathrm{W} - Z^\nu\ \cos\theta_\mathrm{W}\ \right) + W^{-}_\nu\ W^{+}_\mu\ \left(\ A^{\mu\nu}\ \sin \theta_\mathrm{W} - Z^{\mu\nu}\ \cos \theta_\mathrm{W}\ \right) \;\right] ~.</math>

<math>\mathcal{L}_{\mathrm{WWVV}}</math> contains the gauge four-point self interactions,
: <math>
\begin{align}
\mathcal{L}_{\mathrm{WWVV}} = -\frac{\ g^2\ }{4}\ \Biggl\{\ &\Bigl[\ 2\ W^{+}_\mu\ W^{-\mu} + (\ A_\mu\ \sin \theta_\mathrm{W} - Z_\mu\ \cos \theta_\mathrm{W} \ )^2\ \Bigr]^2
\\
&- \Bigl[\ W_\mu^{+}\ W_\nu^{-} + W^{+}_\nu\ W^{-}_\mu + \left(\ A_\mu\ \sin \theta_\mathrm{W} - Z_\mu\ \cos \theta_\mathrm{W}\ \right)\left(\ A_\nu\ \sin \theta_\mathrm{W} - Z_\nu\ \cos \theta_\mathrm{W}\ \right)\ \Bigr]^2\,\Biggr\} ~.
\end{align}
</math>

<math>\ \mathcal{L}_\mathrm{Y}\ </math> contains the Yukawa interactions between the fermions and the Higgs field,
: <math>\mathcal{L}_\mathrm{Y} = -\sum_f\ \frac{\ g\ m_f\ }{2\ m_\mathrm{W}} \; \overline{f}\ f\ H ~.</math>

== See also ==
* [[Electroweak star]]
* [[Fundamental force]]s
* [[History of quantum field theory]]
* [[Standard Model (mathematical formulation)]]
* [[Unitarity gauge]]
* [[Weinberg angle]]
* [[Yang–Mills theory]]

== Notes ==
{{notelist}}

== References ==
{{reflist|25em}}

== Further reading ==
=== General readers ===
* {{cite book
 |author=B. A. Schumm
 |year=2004
 |title=Deep Down Things: The Breathtaking Beauty of Particle Physics
 |publisher=Johns Hopkins University Press
 |isbn=0-8018-7971-X
 |url-access=registration
 |url=https://archive.org/details/deepdownthingsbr00schu
 }} Conveys much of the [[Standard Model]] with no formal mathematics. Very thorough on the weak interaction.

=== Texts ===
* {{cite book
 | author=D. J. Griffiths
 | year=1987
 | title=Introduction to Elementary Particles
 | publisher=John Wiley & Sons
 | isbn=0-471-60386-4
}}
* {{cite book
 |author1=W. Greiner |author2=B. Müller | year=2000
 | title=Gauge Theory of Weak Interactions
 | publisher=Springer
 | isbn=3-540-67672-4
}}
* {{cite book
 | author=E. A. Paschos
 | year=2023
 | title=Electroweak Theory
 | publisher=[[Cambridge University Press]]
 | isbn=9781009402378
 | url=https://www.cambridge.org/core/books/electroweak-theory/89127BFB5372ED4AA87704E0FA2D93F3
}}

=== Articles ===
* {{cite journal
 |author1=E. S. Abers |author2=B. W. Lee |year=1973
 |title=Gauge theories
 |journal=[[Physics Reports]]
 |volume=9 |issue=1 |pages=1–141
 |doi=10.1016/0370-1573(73)90027-6
 |bibcode = 1973PhR.....9....1A
}}
* {{cite journal
 |author=Y. Hayato
 |display-authors=etal
 |year=1999
 |title=Search for Proton Decay through p → νK<sup>+</sup> in a Large Water Cherenkov Detector
 |journal=[[Physical Review Letters]]
 |volume=83 |pages=1529–1533
 |doi=10.1103/PhysRevLett.83.1529
 |bibcode=1999PhRvL..83.1529H
 |arxiv = hep-ex/9904020
 |issue=8 |s2cid=118326409
}}
*{{cite journal
 |author=J. Hucks
 |year=1991
 |title=Global structure of the standard model, anomalies, and charge quantization
 |journal=[[Physical Review D]]
 |volume=43 |pages=2709–2717
 |doi=10.1103/PhysRevD.43.2709
 |bibcode = 1991PhRvD..43.2709H
 |issue=8 |pmid=10013661
}}
* {{cite arXiv
 |author=S. F. Novaes
 |year=2000
 |title=Standard Model: An Introduction
 |eprint=hep-ph/0001283
}}
*{{cite arXiv
 |author=D. P. Roy
 |year=1999
 |title=Basic Constituents of Matter and their Interactions – A Progress Report
 |eprint=hep-ph/9912523
}}

{{Standard model of physics}}
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