Peak car

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Peak_car

Transport modal share in the United Kingdom from 1952 to 2014

Peak car (also peak car use or peak travel) is a hypothesis that motor vehicle distance traveled per capita, predominantly by private car, has peaked and will now fall in a sustained manner. The theory was developed as an alternative to the prevailing market saturation model, which suggested that car use would saturate and then remain reasonably constant, or to GDP-based theories which predict that traffic will increase again as the economy improves, linking recent traffic reductions to the Great Recession of 2008.

The theory was proposed following reductions, which have now been observed in Australia, Belgium, France, Germany, Iceland, Japan (early 1990s), New Zealand, Sweden, the United Kingdom (many cities from about 1994) and the United States. A study by Volpe Transportation in 2013 noted that average miles driven by individuals in the United States has been declining from 900 miles (1,400 km) per month in 2004 to 820 miles (1,320 km) in July 2012, and that the decline had continued since the recent upturn in the US economy.

A number of academics have written in support of the theory, including Phil Goodwin, formerly Director of the transport research groups at Oxford University and UCL, and David Metz, a former Chief Scientist of the UK Department of Transport. The theory is disputed by the UK Department for Transport, which predicts that road traffic in the United Kingdom will grow by 50% by 2036, and Professor Stephen Glaister, Director of the RAC Foundation, who say traffic will start increasing again as the economy improves. Unlike peak oil, a theory based on a reduction in the ability to extract oil due to resource depletion, peak car is attributed to more complex and less understood causes.

History

Saturation model

The idea that car ownership would reach a saturation level and stop growing further has been around since at least 1925 when, for example, Miller McClintock noted that "[t]he most widely accepted estimate of the saturation point is five-to-one, that is, one automobile for every five persons". McClintock also predicted induced traffic: "the density of traffic will always approach a point of complete saturation. [An] increase in street capacity ... will not reduce the density of traffic, for the places thus made available, will be taken by those drivers who may be said to be on the margin of convenience".

In Traffic in Towns, a report produced in 1963 for the UK Ministry of Transport, Professor Sir Colin Buchanan suggested that traffic would saturate early in the 21st century. It has certainly been used in official traffic forecasting since the 1970s, for example in a UK Government study by Tulpule (1973) which forecast that car ownership would reach its maximum level by about 2010, with car use showing little further growth after that point.

In a series of international comparisons starting in 1993 and continuing until his death in 2011, the American researcher Lee Schipper and his colleagues noted that car traffic growth had slowed or ceased in a number of developed economies.

'Peak car' theory

The 'peak car' hypothesis was proposed after declines in traffic during the morning peak period were observed from the mid 1990s in some places and at a national level since about 2008. Local Transport Today, a professional transport journal in the United Kingdom, reported that the number "peak car traffic" entering Britain's town and city centres during the morning peak hours had declined significantly over the previous ten years; of 21 areas studied all except Leeds had seen falls. Traffic into London during the morning peak period had fallen 28% between 1994 and 2003 when the London congestion charge was introduced and a further 12% by 2004. Inbound car trip into Birmingham during the morning peak period had fallen by 29% between 1995 and 2003.

Between June and September 2010 Professor Phil Goodwin published a series of articles in the UK professional transport press suggesting that data showed not merely a plateau in vehicle miles driven but rather a decline in overall automobile usage per capita. These articles were later compiled and updated in a journal article by Goodwin, published in 2011.

David Metz, of University College London and former Chief Scientist of the UK Department of Transport noted that "peak car use came and went [in the UK] at least 15 years ago, when none of us noticed". He then published articles in 2010 and 2012 suggesting that the Department's forecasts of growth were erroneous because in the UK a saturated peak level had already been reached.

In November 2010 by Millard-Ball and Schipper presented data confirming the trend in cities in eight nations: United States, Canada, Sweden, France, Germany, the United Kingdom, Japan and Australia. Newman and Kenworthy published an article in June 2011 suggesting that the effect was also valid for Australia.

By 2016, several papers have cast doubt on the peak car phenomenon, demonstrating that economic and sociodemographic factors account for most or all of the observed slowdowns. Recent statistics in the US show total vehicle-miles traveled (VMT) increasing after several years of decline, although per-capita VMT remains below its all-time high.

Proposed causes

There is speculation about causes of a decline in automobile usage. Analysts such as Newman as well as views expressed in the journal edited by Melia entitled World Transport Policy and Practice point to various interrelated causes. Factors include:

1. Travel time budgets, a theoretical psychological limit suggesting a long term constraint on the amount of time people allocate to travel of about one hour a day. Studies using this concept (albeit not always defined in the same way) have included those of Zahavi (1974), Mogridge (1983) and Metz (2010) who suggested that saturation would naturally follow from the observation that access to destinations increased with the square of speed, but was offset by the tendency for each additional choice of destination to offer less and less extra benefit. A version was suggested by Marchetti, sometimes called 'the Marchetti Wall'; when cities become more than "one hour wide," they stop growing or they become dysfunctional, or both.
2. The growth of public transport. For example, railway travel in the United Kingdom has been undergoing a renaissance, according to one view. In the US Amtrak has posted record ridership for every year since 2000 with the exception of 2009.
3. The reversal of urban sprawl and other population shifts from suburbs to cities.
4. The growth of a culture of urbanism.
5. The ageing of cities, especially their road infrastructure, much of which is at or near the end of its intended life.
6. Rising fuel prices.
7. Increasing costs of automobile ownership, including costs for insurance and parking.
8. Traffic-reducing policies such as the "pedestrianisation" of city centres, traffic calming, parking control, congestion charging.
9. Proliferation of different ways to own and hire vehicles, such as Streetcar, Zipcar, and Whipcar, as well as other options for car sharing.
10. Reallocation of road capacity away from cars towards bikes and pedestrian traffic resulting in disappearing traffic.
11. Cultural shifts especially among young people for whom acquisition of a driving licence is now seen less as a key rite of passage into adulthood, and is reflected in recent reductions in the propensity of young people to acquire driving licences. One report suggests there has been a shift in notions about status: the car is no longer a "big prestige item" as in previous decades. For millennials and digital natives, there is less focus on ownership of things, especially big-ticket items such as cars. Millennials see cars as "mere appliances—unnecessary, pricey ones that they'll try to avoid".
12. Legal restrictions; for example, restrictions on teenagers seeking driving licences.
13. Demographic changes; for example, baby boomers drive less as they age, according to one view.
14. Economic factors, particularly unemployment.
15. Saturation of demand in the sense that there has been a "levelling-off" of possible places to travel to by car. According to this view, when road networks were expanding, there were numerous options of new places to drive to, but as road networks have generally stopped expanding, the demand for increased car travel has become saturated.
16. Growth of e-commerce such as tele-shopping, conferences, and smartphone or computer-based social networks. According to this view, these developments have reduced the need for travel by car, such that the "love affair with the phone" has replaced the "love affair with the car" for a proportion of the population, and that widespread use of cell phones and Skype meant there was less need for in-person visits. However, a contrasting view suggested that e-commerce was not a substantial factor explaining less car travel in the United States. Still, countries with higher use of the Internet correlated with fewer 20- to 24-year-olds getting drivers' licenses.

One analyst explained about changing attitudes of young people:

Virtual contact through electronic means reduces the need for actual contact among young people ... Furthermore, some young people feel that driving interferes with texting.

— Michael Sivak, University of Michigan, 2012

Another elaborated about the saturation of demand hypothesis:

They say as we get richer, we'll want to travel more. There's no limit. Our hunch was that this might not be the case. ... The data that we have shows fairly clearly that the growth in travel demand has stopped in every industrialised country that we looked at.

— Adam Millard-Ball, McGill University

The proposition that car usage has peaked has been disputed regarding vehicle usage in the United Kingdom. In December 2010, Stephen Glaister, the Director of the RAC Foundation, suggested that total traffic has grown more or less as a straight line since the 1950s and such growth will recommence when economic conditions improve; in 2011, the UK Department for Transport predicted a 50% growth in traffic in the coming 25 years. In addition, a corroborating view by Paul Watters suggested that car usage will continue to be important in Britain, and that there will not be "shattering change" by 2020. Scholars studying transport and socio-technical transitions have elaborated possible future scenarios for car use in England and the Netherlands.

The advent of autonomous cars is likely to accelerate the decline in car ownership. A recent case study by the OECD International Transport Forum suggested Lisbon could maintain current levels of mobility with an autonomous car share fleet one tenth the size of its current vehicle fleet.

Countries

• Australia
• Belgium
• Canada
• France
• Germany
• Japan
• New Zealand
• Norway
• South Korea
• Spain
• Sweden
• United Kingdom
• United States

Note, in China there is a forecast of tremendous growth in car ownership and travel, although there is also greater awareness of environmental issues as well as issues of inequality between car-owners and non-owners.

Declines in specific countries

Germany

The city of Hamburg in its so-called Green Network Plan, is considering ways of phasing out automobile traffic in the city center over the next two decades by increasing public transportation and adding special routes for cyclists and people on foot.

United Kingdom

One report suggested driving in the United Kingdom has been declining since 1990. The number of 17- to 20-year-olds with driving licences declined from 48% during the early-1990s to 35% in 2011, according to one report. Traffic by cars and taxis has declined since 2007. One report suggested renewed growth in rail travel, such that there was a "rail renaissance" underway. The City of London has been experiencing a fall in the number of cars on the roads. In 2022 the British Society of Motor Manufacturers and Traders reported a second consecutive year of declining car ownership.

United States

A report in Time Magazine suggested Americans are "driving less and less each year." It noted that fewer Americans were "commuting solo" to work. Road congestion nationwide declined by 27% in 2011. There is some evidence of a generational shift. For example, one 24-year-old with a car moved to Washington, D.C., for work purposes but did not take her car, and she explained:

I don't need (my car). My apartment is just over a mile from my office, so I walk every day... I think I might give it to my parents...

— Leslie Norrington, quoted in Scientific American, 2013

According to transportation consultant Roy Kienitz, driving habits began to change in 2004 before the 2007-2010 recession started.

Cities

Declines of total "vehicle kilometers traveled" (vkt) in selected cities as reported in the research:

Country	City	1995–2005	peak year
Australia	All cities		2004
Austria	Vienna	–7.6%
Sweden	Stockholm	–3.7%
Switzerland	Zurich	–4.7%
UK	London	–1.2%	"Early 1990s"
USA	Atlanta	–10.1%	c.1995
USA	Houston	–15.2%	c.1995
USA	Los Angeles	–2.0%
USA	San Francisco	–4.8%

Electroweak interaction

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electroweak_interaction 

 

Standard Model of particle physics
Elementary particles of the Standard Model   In particle physics, the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: 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 unification energy, on the order of 246 GeV, they would merge into a single force. Thus, if the temperature is high enough – approximately 1015 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 10¹⁵ K has not been seen widely throughout the universe since before the quark epoch, and currently the highest human-made temperature in thermal equilibrium is around 5.5×10¹² K (from the Large Hadron Collider).

Sheldon Glashow, Abdus Salam, and Steven Weinberg were awarded the 1979 Nobel Prize in Physics for their contributions to the unification of the weak and electromagnetic interaction between elementary particles, known as the Weinberg–Salam theory. 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. 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 and electromagnetic interactions. Extending his doctoral advisor Julian Schwinger's work, Sheldon Glashow first experimented with introducing two different symmetries, one chiral and one achiral, and combined them such that their overall symmetry was unbroken. This did not yield a renormalizable theory, and its gauge symmetry had to be broken by hand as no 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, Salam and John Clive 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.

Formulation

Main article: Mathematical formulation of the Standard Model

Weinberg's weak mixing angle θ_W, and relation between coupling constants g, g′, and e. Adapted from Lee (1981).

The pattern of weak isospin, T₃, and weak hypercharge, Y_W, 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 that 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.

In the Standard Model, the observed physical particles, the W^±
and Z⁰
bosons, and the photon, are produced through the spontaneous symmetry breaking of the electroweak symmetry SU(2) × U(1)_Y to U(1)_em, 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.

The electric charge arises as the particular linear combination (nontrivial) of Y_W (weak hypercharge) and the T₃ component of weak isospin (${\displaystyle Q=T_3+\frac{1}{2}{Y}_{\mathrm{W}}}$) 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.

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),

${\displaystyle \left(\begin{array}{c}\gamma \\ Z^0\end{array}\right)=\left(\begin{array}{cc}\cos {\theta }_{\text{W}}& \sin {\theta }_{\text{W}}\\ -\sin {\theta }_{\text{W}}& \cos {\theta }_{\text{W}}\end{array}\right)\left(\begin{array}{c}B\\ W_3\end{array}\right),}$

where θ_W is the weak mixing angle. The axes representing the particles have essentially just been rotated, in the (W₃, B) plane, by the angle θ_W. This also introduces a mismatch between the mass of the Z⁰
and the mass of the W^±
particles (denoted as m_Z and m_W, respectively),

${\displaystyle m_{\text{Z}}=\frac{m_{\text{W}}}{\cos {\theta }_{\text{W}}}.}$

The W₁ and W₂ bosons, in turn, combine to produce the charged massive bosons W^±
:

${\displaystyle W^{\pm }=\frac{1}{\sqrt{2}}(W_1\mp i{W}_2).}$

Lagrangian

Before electroweak symmetry breaking

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

${\displaystyle {\mathcal{L}}_{\mathrm{E}\mathrm{W}}={\mathcal{L}}_g+{\mathcal{L}}_f+{\mathcal{L}}_h+{\mathcal{L}}_y.}$

The ${\displaystyle {\mathcal{L}}_g}$ term describes the interaction between the three W vector bosons and the B vector boson,

${\displaystyle {\mathcal{L}}_g=-\frac{1}{4}{W}_a^{\mu \nu }{W}_{\mu \nu }^a-\frac{1}{4}{B}^{\mu \nu }{B}_{\mu \nu },}$

where ${\displaystyle W_a^{\mu \nu }}$ (${\displaystyle a=1,2,3}$) and ${\displaystyle B^{\mu \nu }}$ are the field strength tensors for the weak isospin and weak hypercharge gauge fields.

${\displaystyle {\mathcal{L}}_f}$ is the kinetic term for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through the gauge covariant derivative,

${\displaystyle {\mathcal{L}}_f={\overline{Q}}_{j}iD/Q_j+{\overline{u}}_{j}iD/u_j+{\overline{d}}_{j}iD/d_j+{\overline{L}}_{j}iD/L_j+{\overline{e}}_{j}iD/e_j,}$

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 ${\displaystyle D/}$ means the contraction of the 4-gradient with the Dirac matrices, defined as

${\displaystyle D/\equiv {\gamma }^{\mu }{D}_{\mu },}$

and the covariant derivative (excluding the gluon gauge field for the strong interaction) is defined as

${\displaystyle D_{\mu }\equiv {\mathrm{\partial }}_{\mu }-i\frac{g^{\prime }}{2}Y{B}_{\mu }-i\frac{g}{2}{T}_j{W}_{\mu }^j.}$

Here ${\displaystyle Y}$ is the weak hypercharge and the ${\displaystyle T_j}$ are the components of the weak isospin.

The ${\displaystyle {\mathcal{L}}_h}$ term describes the Higgs field ${\displaystyle h}$ and its interactions with itself and the gauge bosons,

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

where ${\displaystyle v}$ is the vacuum expectation value.

The ${\displaystyle {\mathcal{L}}_y}$ term describes the Yukawa interaction with the fermions,

${\displaystyle {\mathcal{L}}_y=-y_u^{ij}{ϵ}^{ab}{h}_b^{†}{\overline{Q}}_{ia}{u}_j^c-y_d^{ij}h{\overline{Q}}_i{d}_j^c-y_e^{ij}h{\overline{L}}_i{e}_j^c+\mathrm{h}.\mathrm{c}.,}$

and generates their masses, manifest when the Higgs field acquires a nonzero vacuum expectation value, discussed next. The ${\displaystyle y_k^{ij},}$ for ${\displaystyle k\in \{\mathrm{u},\mathrm{d},\mathrm{e}\},}$ 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 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.

${\displaystyle {\mathcal{L}}_{\mathrm{E}\mathrm{W}}={\mathcal{L}}_{\mathrm{K}}+{\mathcal{L}}_{\mathrm{N}}+{\mathcal{L}}_{\mathrm{C}}+{\mathcal{L}}_{\mathrm{H}}+{\mathcal{L}}_{\mathrm{H}\mathrm{V}}+{\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}}+{\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}\mathrm{V}}+{\mathcal{L}}_{\mathrm{Y}}.}$

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

${\displaystyle \begin{array}{r}{\mathcal{L}}_{\mathrm{K}}=\sum _f\overline{f}(i\mathrm{\partial }/-m_f)f-\frac{1}{4}{A}_{\mu \nu }{A}^{\mu \nu }-\frac{1}{2}{W}_{\mu \nu }^{+}{W}^{-\mu \nu }+m_W^2{W}_{\mu }^{+}{W}^{-\mu }\\ -\frac{1}{4}{Z}_{\mu \nu }{Z}^{\mu \nu }+\frac{1}{2}{m}_Z^2{Z}_{\mu }{Z}^{\mu }+\frac{1}{2}({\mathrm{\partial }}^{\mu }H)({\mathrm{\partial }}_{\mu }H)-\frac{1}{2}{m}_H^2{H}^2,\end{array}}$

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

${\displaystyle X_{\mu \nu }^a={\mathrm{\partial }}_{\mu }{X}_{\nu }^a-{\mathrm{\partial }}_{\nu }{X}_{\mu }^a+g{f}^{abc}{X}_{\mu }^b{X}_{\nu }^c,}$

with ${\displaystyle X}$ to be replaced by the relevant field (${\displaystyle A,}$ ${\displaystyle Z,}$ ${\displaystyle W^{\pm }}$) and f ^abc by the structure constants of the appropriate gauge group.

The neutral current ${\displaystyle {\mathcal{L}}_{\mathrm{N}}}$ and charged current ${\displaystyle {\mathcal{L}}_{\mathrm{C}}}$ components of the Lagrangian contain the interactions between the fermions and gauge bosons,

${\displaystyle {\mathcal{L}}_{\mathrm{N}}=e{J}_{\mu }^{\mathrm{e}\mathrm{m}}{A}^{\mu }+\frac{g}{\cos {\theta }_W}(J_{\mu }^3-{\sin }^2{\theta }_W{J}_{\mu }^{\mathrm{e}\mathrm{m}})Z^{\mu },}$

where ${\displaystyle e=g\sin {\theta }_{\mathrm{W}}=g^{\prime }\cos {\theta }_{\mathrm{W}}.}$ The electromagnetic current ${\displaystyle J_{\mu }^{\mathrm{e}\mathrm{m}}}$ is

${\displaystyle J_{\mu }^{\mathrm{e}\mathrm{m}}=\sum _f{q}_f\overline{f}{\gamma }_{\mu }f,}$

where ${\displaystyle q_f}$ is the fermions' electric charges. The neutral weak current ${\displaystyle J_{\mu }^3}$ is

${\displaystyle J_{\mu }^3=\sum _f{T}_f^3\overline{f}{\gamma }_{\mu }\frac{1-{\gamma }^5}{2}f,}$

where ${\displaystyle T_f^3}$ is the fermions' weak isospin.^[d]

The charged current part of the Lagrangian is given by

${\displaystyle {\mathcal{L}}_{\mathrm{C}}=-\frac{g}{\sqrt{2}}\left[{\overline{u}}_i{\gamma }^{\mu }\frac{1-{\gamma }^5}{2}{M}_{ij}^{\mathrm{C}\mathrm{K}\mathrm{M}}{d}_j+{\overline{\nu }}_i{\gamma }^{\mu }\frac{1-{\gamma }^5}{2}{e}_i\right]W_{\mu }^{+}+\mathrm{h}.\mathrm{c}.,}$

where ${\displaystyle \nu }$ is the right-handed singlet neutrino field, and the CKM matrix ${\displaystyle M_{ij}^{\mathrm{C}\mathrm{K}\mathrm{M}}}$ determines the mixing between mass and weak eigenstates of the quarks.

${\displaystyle {\mathcal{L}}_{\mathrm{H}}}$ contains the Higgs three-point and four-point self interaction terms,

${\displaystyle {\mathcal{L}}_{\mathrm{H}}=-\frac{g{m}_{\mathrm{H}}^2}{4m_{\mathrm{W}}}{H}^3-\frac{g^2{m}_{\mathrm{H}}^2}{32m_{\mathrm{W}}^2}{H}^4.}$

${\displaystyle {\mathcal{L}}_{\mathrm{H}\mathrm{V}}}$ contains the Higgs interactions with gauge vector bosons,

${\displaystyle {\mathcal{L}}_{\mathrm{H}\mathrm{V}}=\left(g{m}_{\mathrm{H}\mathrm{V}}+\frac{g^2}{4}{H}^2\right)\left(W_{\mu }^{+}{W}^{-\mu }+\frac{1}{2{\cos }^2{\theta }_{\mathrm{W}}}{Z}_{\mu }{Z}^{\mu }\right).}$

${\displaystyle {\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}}}$ contains the gauge three-point self interactions,

${\displaystyle {\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}}=-ig\left[\left(W_{\mu \nu }^{+}{W}^{-\mu }-W^{+\mu }{W}_{\mu \nu }^{-}\right)\left(A^{\nu }\sin {\theta }_{\mathrm{W}}-Z^{\nu }\cos {\theta }_{\mathrm{W}}\right)+W_{\nu }^{-}{W}_{\mu }^{+}\left(A^{\mu \nu }\sin {\theta }_{\mathrm{W}}-Z^{\mu \nu }\cos {\theta }_{\mathrm{W}}\right)\right].}$

${\displaystyle {\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}\mathrm{V}}}$ contains the gauge four-point self interactions,

${\displaystyle \begin{array}{rl}{\mathcal{L}}_{\mathrm{W}\mathrm{W}\mathrm{V}\mathrm{V}}=-\frac{g^2}{4}\{& [2W_{\mu }^{+}{W}^{-\mu }+(A_{\mu }\sin {\theta }_{\mathrm{W}}-Z_{\mu }\cos {\theta }_{\mathrm{W}}{)}^2{]}^2\\ & -[W_{\mu }^{+}{W}_{\nu }^{-}+W_{\nu }^{+}{W}_{\mu }^{-}+\left(A_{\mu }\sin {\theta }_{\mathrm{W}}-Z_{\mu }\cos {\theta }_{\mathrm{W}}\right)\left(A_{\nu }\sin {\theta }_{\mathrm{W}}-Z_{\nu }\cos {\theta }_{\mathrm{W}}\right){]}^2\}.\end{array}}$

${\displaystyle {\mathcal{L}}_{\mathrm{Y}}}$ contains the Yukawa interactions between the fermions and the Higgs field,

${\displaystyle {\mathcal{L}}_{\mathrm{Y}}=-\sum _f\frac{g{m}_f}{2m_{\mathrm{W}}}\overline{f}fH.}$