Editing Electroweak interaction (section)

== 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>