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Unified description of electromagnetism and the weak interaction
In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism 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 unification energy, on the order of 246 GeV,[a] they would merge into a single force. Thus, if the universe is hot enough (approximately 1015K, a temperature not exceeded since shortly after the Big Bang), then the electromagnetic force and weak force merge into a combined electroweak force. During the quark epoch, the electroweak force splits into the electromagnetic and weak force.
In 1964, Salam and Ward had the same idea, but predicted a massless photon and three massive gauge bosons 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. In 1971, Gerard 't Hooft proved that spontaneously broken gauge symmetries are renormalizable even with massive gauge bosons.
Weinberg's weak mixing angle ?W, and relation between coupling constants g, g?, and e. Adapted from Lee (1981).
The pattern of weak isospin, T3, and weak hypercharge, YW, of the known elementary particles, showing the electric charge, 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 W and Z bosons.
Mathematically, electromagnetism is unified with the weak interactions as a Yang-Mills field with an SU(2) × U(1)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 W1, W2, and W3, and the weak hypercharge field B.
This invariance is known as electroweak symmetry.
The generators of SU(2) and U(1) are given the name weak isospin (labeled T) and weak hypercharge (labeled Y) respectively. These then give rise to the gauge bosons which mediate the electroweak interactions - the three W bosons of weak isospin (W1, W2, and W3), and the 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.
The electric charge arises as the particular linear combination (nontrivial) of YW (weak hypercharge) and the T3 component of weak isospin 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 T3 outlined in the figure.
U(1)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 U(1)em group is unbroken, since it does not directly interact with the Higgs.[c]
The above spontaneous symmetry breaking makes the W3 and B bosons coalesce into two different physical bosons with different masses - the Z0 boson, and the photon ( γ ),
where ?W is the weak mixing angle. The axes representing the particles have essentially just been rotated, in the (W3, B) plane, by the angle ?W. This also introduces a mismatch between the mass of the Z0 and the mass of the W± particles (denoted as mZ and mW, respectively),
The W1 and W2 bosons, in turn, combine to produce the charged massive bosons W± :
is the kinetic term for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through the gauge covariant derivative,
where the subscript j sums over the three generations of fermions; Q, u, and d are the left-handed doublet, right-handed singlet up, and right handed singlet down quark fields; and L and e are the left-handed doublet and right-handed singlet electron fields.
The Feynman slash means the contraction of the 4-gradient with the Dirac matrices
and the covariant derivative is (excluding the gluon gauge field for the strong interaction)
Here is the weak hypercharge and the are the components of the weak isospin.
The term describes the Higgs field and its interactions with itself and the gauge bosons,
and generates their masses, manifest when the Higgs field acquires a nonzero vacuum expectation value, discussed next. The are matrices of Yukawa couplings.
After electroweak symmetry breaking
The Lagrangian reorganizes itself as the Higgs boson 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 159.5±1.5 GeV (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.
The kinetic term 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)
where the sum runs over all the fermions of the theory (quarks and leptons), and the fields , , , and are given as
with '' to be replaced by the relevant field (, , ), and f abc by the structure constants of the appropriate gauge group.
The neutral current and charged current components of the Lagrangian contain the interactions between the fermions and gauge bosons,
where The electromagnetic current is
where is the fermions' electric charges.
The neutral weak current is
^ ab Note the factors in the weak coupling formulas: These factors are deliberately inserted to expunge any left-chiral components of the spinor fields. This is why electroweak theory is said to be a chiral theory.
^Glashow, S. (1959). "The renormalizability of vector meson interactions." Nucl. Phys.10, 107.