Electroweak interaction

Standard model of particle physics

Background
Particle physics Quantum field theory Gauge theory Spontaneous symmetry breaking Higgs mechanism

Constituents
Electroweak interaction Quantum chromodynamics CKM matrix

Limitations
Strong CP problem Hierarchy problem Neutrino oscillations

Theorists
Sudarshan · Marshak · Feynman · Gell-Mann · Sakata · Glashow · Zweig · Nambu · Han · Cabibbo · Weinberg · Salam · Kobayashi · Maskawa · 't Hooft · Veltman · Gross · Politzer · Wilczek

In particle physics, the electroweak interaction 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 100 GeV, they would merge into a single electroweak force. Thus if the universe is hot enough (approximately 10¹⁵ K, a temperature reached shortly after the Big Bang) then the electromagnetic force and weak force will merge into a combined electroweak force.

For contributions to the unification of the weak and electromagnetic interaction between elementary particles, Abdus Salam, Sheldon Glashow and Steven Weinberg were awarded the Nobel Prize in Physics in 1979.^[1]^[2] The existence of the electroweak interactions was experimentally established in two stages: the first being the discovery of neutral currents 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 gauge bosons in proton-antiproton collisions at the converted Super Proton Synchrotron.

1 Formulation
2 Lagrangian
- 2.1 Before Electroweak Symmetry Breaking
- 2.2 After Electroweak Symmetry Breaking
3 See also
4 References

Formulation

Mathematically, the unification is accomplished under an SU(2) × U(1) gauge group. The corresponding gauge bosons are the three W bosons of weak isospin (W⁺, W⁰, and W⁻), and the B⁰ boson of weak hypercharge, respectively, all of which are massless.

In the Standard Model, the W^± and Z⁰ bosons, and the photon, are produced by the spontaneous symmetry breaking of the electroweak symmetry from SU(2) × U(1)_Y to U(1)_em, caused by the Higgs mechanism (see also Higgs boson). U(1)_Y and U(1)_em are different copies of U(1); the generator of U(1)_em is given by Q = Y/2 + I₃, where Y is the generator of U(1)_Y (called the weak hypercharge), and I₃ is one of the SU(2) generators (a component of weak isospin).

The spontaneous symmetry breaking causes the W⁰ and B⁰ bosons to coalesce together into two different bosons - the Z⁰ boson, and the photon, as follows:

$\begin{pmatrix} Photon \\ Z^0 \end{pmatrix} = \begin{pmatrix} \cos \theta_W & \sin \theta_W \\ -\sin \theta_W & \cos \theta_W \end{pmatrix} \begin{pmatrix} B^0 \\ W^0 \end{pmatrix}$

Where θ_W is the weak mixing angle. The axes representing the particles have essentially just been rotated, in the (W⁰, B⁰) plane, by the angle θ_W. This also introduces a discrepancy between the mass of the Z⁰ and the mass of the W^± particles (denoted as M_Z and M_W, respectively);

$M_Z=\frac{M_W}{\cos\theta_W}$

The distinction between electromagnetism and the weak force arises because there is a (nontrivial) linear combination of Y and I₃ that vanishes for the Higgs boson (it is an eigenstate of both Y and I₃, so the coefficients may be taken as −I₃ and Y): U(1)_em is defined to be the group generated by this linear combination, and is unbroken because it does not interact with the Higgs.

Lagrangian

Before Electroweak Symmetry Breaking

The Lagrangian for the electroweak interactions is divided into four parts before electroweak symmetry breaking

$\mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.$

The g term describes the interaction between the three W particles and the B particle.

$\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}$

The f term gives the kinetic term for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through the covariant derivative.

$\mathcal{L}_f = \overline{Q}_i iD\!\!\!\!/\; Q_i+ \overline{u}_i^c iD\!\!\!\!/\; u^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i$

The h term describes the Higgs field F.

$\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2$

The y term gives the Yukawa interaction that generates the fermion masses after the Higgs acquires a vacuum expectation value.

$\mathcal{L}_y = - y_{u\, ij} \epsilon^{ab} \,h_b^\dagger\, \overline{Q}_{ia} u_j^c - y_{d\, ij}\, h\, \overline{Q}_i d^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.$

After Electroweak Symmetry Breaking

The Lagrangian reorganizes itself after the Higgs boson acquires a vacuum expectation value. Due to its complexity, this Lagrangian is best described by breaking it up into several parts as follows.

$\mathcal{L}_{EW} = \mathcal{L}_K + \mathcal{L}_N + \mathcal{L}_C + \mathcal{L}_H + \mathcal{L}_{HV} + \mathcal{L}_{WWV} + \mathcal{L}_{WWVV} + \mathcal{L}_Y$

The kinetic term $\mathcal{L}_K$ 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)

$\mathcal{L}_K = \sum_f \overline{f}(i\partial\!\!\!/\!\;-m_f)f-\frac14A_{\mu\nu}A^{\mu\nu}-\frac12W^+_{\mu\nu}W^{-\mu\nu}+m_W^2W^+_\mu W^{-\mu}-\frac14Z_{\mu\nu}Z^{\mu\nu}+\frac12m_Z^2Z_\mu Z^\mu+\frac12(\partial^\mu H)(\partial_\mu H)-\frac12m_H^2H^2$

where the sum runs over all the fermions of the theory (quarks and leptons), and the fields $A_{\mu\nu}^{}$ , $Z_{\mu\nu}^{}$ , $W^-_{\mu\nu}$ , and $W^+_{\mu\nu}\equiv(W^-_{\mu\nu})^\dagger$ are given as

$X_{\mu\nu}=\partial_\mu X_\nu - \partial_\nu X_\mu + g f^{abc}X^{b}_{\mu}X^{c}_{\nu}$ , (replace X by the relevant field, and f^abc with the structure constants for the gauge group).

The neutral current $\mathcal{L}_N$ and charged current $\mathcal{L}_C$ components of the Lagrangian contain the interactions between the fermions and gauge bosons.

$\mathcal{L}_{N} = e J_\mu^{em} A^\mu + \frac g{\cos\theta_W}(J_\mu^3-\sin^2\theta_WJ_\mu^{em})Z^\mu$ ,

where the electromagnetic current $J_\mu^{em}$ and the neutral weak current $J_\mu^3$ are

$J_\mu^{em} = \sum_f q_f\overline{f}\gamma_\mu f$ ,

and

$J_\mu^3 = \sum_f I^3_f\overline{f}\gamma_\mu f$

$q_f^{}$ and $I_f^3$ are the fermions' electric charges and weak isospin.

The charged current part of the Lagrangian is given by

$\mathcal{L}_C=-\frac g{\sqrt2}\left[\overline u_i\gamma^\mu\frac{1-\gamma^5}2M^{CKM}_{ij}d_j+\overline\nu_i\gamma^\mu\frac{1-\gamma^5}2e_i\right]W_\mu^++h.c.$

$\mathcal{L}_H$ contains the Higgs three-point and four-point self interaction terms.

$\mathcal{L}_H=-\frac{gm_H^2}{4m_W}H^3-\frac{g^2m_H^2}{32m_W^2}H^4$

$\mathcal{L}_{HV}$ contains the Higgs interactions with gauge vector bosons.

$\mathcal{L}_{HV}=\left(gm_WH+\frac{g^2}4H^2\right)\left(W_\mu^+W^{-\mu}+\frac1{2\cos^2\theta_W}Z_\mu Z^\mu\right)$

$\mathcal{L}_{WWV}$ contains the gauge three-point self interactions.

$\mathcal{L}_{WWV}=-ig[(W_{\mu\nu}^+W^{-\mu}-W^{+\mu}W_{\mu\nu}^-)(A^\nu\sin\theta_W-Z^\nu\cos\theta_W)+W_\nu^-W_\mu^+(A^{\mu\nu}\sin\theta_W-Z^{\mu\nu}\cos\theta_W)]$

$\mathcal{L}_{WWVV}$ contains the gauge four-point self interactions

$\mathcal{L}_{WWVV} = -\frac{g^2}4 \left\{[2W_\mu^+W^{-\mu} + (A_\mu\sin\theta_W - Z_\mu\cos\theta_W)^2]^2 - [W_\mu^+W_\nu^- + W_\nu^+W_\mu^- + (A_\mu\sin\theta_W - Z_\mu\cos\theta_W) (A_\nu\sin\theta_W - Z_\nu\cos\theta_W)]^2\right\}$

and $\mathcal{L}_Y$ contains the Yukawa interactions between the fermions and the Higgs field.

$\mathcal{L}_Y = -\sum_f \frac{gm_f}{2m_W}\overline ffH$

References

↑ S. Bais (2005). The Equations: Icons of knowledge. p. 84. ISBN 0-674-01967-9.
↑ "The Nobel Prize in Physics 1979". The Nobel Foundation. http://nobelprize.org/nobel_prizes/physics/laureates/1979/. Retrieved 2008-12-16.

General readers

B.A. Schumm (2004). Deep Down Things: The Breathtaking Beauty of Particle Physics. John Hopkins University Press. ISBN 0-8018-7971-X. Conveys much of the Standard Model with no formal mathematics. Very thorough on the weak interaction.

Texts

D.J. Griffiths (1987). Introduction to Elementary Particles. John Wiley & Sons. ISBN 0-471-60386-4.
W. Greiner, B. Müller (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-67672-4.
G.L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5.

Articles

E.S. Abers, B.W. Lee (1973). "Gauge theories". Physics Reports 9: 1–141. doi:10.1016/0370-1573(73)90027-6.
Y. Hayato et al. (1999). "Search for Proton Decay through p → νK⁺ in a Large Water Cherenkov Detector". Physical Review Letters 83: 1529. doi:10.1103/PhysRevLett.83.1529.
J. Hucks (1991). "Global structure of the standard model, anomalies, and charge quantization". Physical Review D 43: 2709–2717. doi:10.1103/PhysRevD.43.2709.
S.F. Novaes (2000). "Standard Model: An Introduction". arΧiv:hep-ph/0001283 [hep-ph].
D.P. Roy (1999). "Basic Constituents of Matter and their Interactions — A Progress Report". arΧiv:hep-ph/9912523 [hep-ph].