Electroweak interaction

In particle physics, the electroweak interaction is the unified description of two of the four 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 102 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. [Sander Bais (2005), "The Equations. Icons of knowledge" ISBN 0-674-01967-9 p 84] 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.

Formulation

Mathematically, the unification is accomplished under an "SU"(2) × "U"(1) gauge group. The corresponding gauge bosons are the photon of electromagnetism and the W and Z bosons of the weak force. In the Standard Model, the weak gauge bosons get their mass from 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). The subscripts are used to indicate that these are different copies of "U"(1); the generator of "U"(1)em is given by "Q" = "Y"/2 + "I"3, where "Y" is the generator of "U"(1)"Y" (called the weak hypercharge), and "I"3 is one of the "SU"(2) generators (a component of weak isospin). The distinction between electromagnetism and the weak force arises because there is a (nontrivial) linear combination of "Y" and "I"3 that vanishes for the Higgs boson (it is an eigenstate of both "Y" and "I"3, so the coefficients may betaken as −"I"3 and "Y"): "U"(1)em is defined to be the group generated by this linear combination, and is unbroken because it doesn't 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}_{E-W} = 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 u}W_{mu u}^a - frac{1}{4}B^{mu u}B_{mu u}

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} ight)^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}_{E-W} = 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 (conspicuosly absent from the Lagrangian before symmetry breaking)

:mathcal{L}_K = sum_f overline{f}(ipartial!!!/!;-m_f)f-frac14A_{mu u}A^{mu u}-frac12W^+_{mu u}W^{-mu u}+m_W^2W^+_mu W^{-mu}-frac14Z_{mu u}Z^{mu u}+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 u}^{}, Z_{mu u}^{}, W^-_{mu u}, and W^+_{mu u}equiv(W^-_{mu u})^dagger are given as

:X_{mu u}=partial_mu X_ u - partial_ u X_mu + g f^{abc}X^{b}_{mu}X^{c}_{ u}, (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 heta_W}(J_mu^3-sin^2 heta_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_foverline{f}gamma_mu f,

and

:J_mu^3 = sum_f I^3_foverline{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_igamma^mufrac{1-gamma^5}2M^{CKM}_{ij}d_j+overline u_igamma^mufrac{1-gamma^5}2e_i ight] 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 ight)left(W_mu^+W^{-mu}+frac1{2cos^2 heta_W}Z_mu Z^mu ight)

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

mathcal{L}_{WWV}=-ig [(W_{mu u}^+W^{-mu}-W^{+mu}W_{mu u}^-)(A^ usin heta_W-Z^ ucos heta_W)+W_ u^-W_mu^+(A^{mu u}sin heta_W-Z^{mu u}cos heta_W)]

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

mathcal{L}_{WWVV} = -frac{g^2}4 left{ [2W_mu^+W^{-mu} + (A_musin heta_W - Z_mucos heta_W)^2] ^2 - [W_mu^+W_ u^- + W_ u^+W_mu^- + (A_musin heta_W - Z_mucos heta_W) (A_ usin heta_W - Z_ ucos heta_W)] ^2 ight}

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

ee also

*Fundamental forces
*Formulation of the standard model

References

Textbooks

*cite book | author=Griffiths, David J. | title=Introduction to Elementary Particles | publisher=Wiley, John & Sons, Inc | year=1987 | id=ISBN 0-471-60386-4

*cite book | author=D.A. Bromley | title=Gauge Theory of Weak Interactions | publisher=Springer | year=2000 | id=ISBN 3-540-67672-4

*cite book | author=Gordon L. Kane | title=Modern Elementary Particle Physics | publisher=Perseus Books | year=1987 | id=ISBN 0-201-11749-5

Journal Articles

* S.F. Novaes, "Standard Model: An Introduction", [http://arxiv.org/abs/hep-ph/0001283 hep-ph/0001283]
* D.P. Roy, "Basic Constituents of Matter and their Interactions — A Progress Report", [http://arxiv.org/abs/hep-ph/9912523 hep-ph/9912523]
* Y. Hayato "et al.", "Search for Proton Decay through p → νK+ in a Large Water Cherenkov Detector". Phys. Rev. Lett. 83, 1529 (1999).
* Ernest S. Abers and Benjamin W. Lee, "Gauge theories". Physics Reports (Elsevier) C9, 1-141 (1973).
* J. Hucks, "Global structure of the standard model, anomalies, and charge quantization", Phys. Rev. D 43, 2709–2717 (1991). [http://prola.aps.org/abstract/PRD/v43/i8/p2709_1]


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