|
|
| Line 1: |
Line 1: |
| [[Image:Standard Model of Elementary Particles.svg|thumb|right|300px|The Standard Model of Physics]]
| | == each division has its own heavy fire spirit == |
| {{Quantum field theory}}
| |
| {{for|a less mathematical description|Standard Model}}
| |
|
| |
|
| This article describes the mathematics of the '''Standard Model''' of [[particle physics]], a [[gauge theory|gauge]] [[quantum field theory]] containing the [[Field_(physics)#Symmetries_of_fields|internal symmetries]] of the [[unitary group|unitary]] [[direct product of groups|product group]] {{math|[[special unitary group|SU]](3) × SU(2) × [[U(1)]]}}. The theory is commonly viewed as containing the fundamental set of particles – the [[lepton]]s, [[quark]]s, [[gauge boson]]s and the [[Higgs boson|Higgs particle]], however due to the nature of quantum field theory, these "fundamental" particles can be viewed differently depending on the situation ([[#Alternative Presentations of the Fields|see below]]).
| | On that side of the bones, 一枚 white 'color' is satisfied that every now and then the sky, fall into their hands, flexor of a bomb, a roll of pale 'color' of the [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html 腕時計 メンズ casio] reel is present in its front, above the reel, drawing some Quintana beast birds braved flames.<br><br>'This is the year of the famous old [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ gps 時計] lady fighting skills,' five from the Fire, 'Oh, that's fighting skills are also some wrong, it should be said that the law must [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html casio 腕時計 データバンク] control the fire.' Venerable Skyfire will then reel [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ 掛け時計] thrown Xiao Yan, said: 'As long as you can help the old lady, then this thing will be yours, if you doubt the sincerity of the old lady, you can take the reel, was found to be no problem, then help me to repair the soul too late.'<br><br>cautiously took the reel, Xiao Yan hesitated a moment, just cautious gradually spread.<br><br>'five from Fire, fire control method formula, France is divided into five weight, according to the zoomorphic and identified, wolves, leopards, lions, tigers, Jiao, each division has its own heavy [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html 時計 カシオ] fire spirit, France tactic Dacheng, five animals gathered , |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://www.nnzqm.com/forum.php?mod=viewthread&tid=686610 http://www.nnzqm.com/forum.php?mod=viewthread&tid=686610]</li> |
| | |
| | <li>[http://www.pyjiazhuang.cn/forum.php?mod=viewthread&tid=44024 http://www.pyjiazhuang.cn/forum.php?mod=viewthread&tid=44024]</li> |
| | |
| | <li>[http://www.nwcommunities.org/author-diana-leafe-christian-to-present-talk-at-east-west-bookshop-tea-garden-august-21-2013/comment-page-/#comment- http://www.nwcommunities.org/author-diana-leafe-christian-to-present-talk-at-east-west-bookshop-tea-garden-august-21-2013/comment-page-/#comment-]</li> |
| | |
| | </ul> |
|
| |
|
| The Standard Model is [[renormalizable]] and mathematically self-consistent,<ref>In fact, there are mathematical issues regarding quantum field theories still under debate (see e.g. [[Landau pole]]), but the predictions extracted from the Standard Model by current methods are all self-consistent. For a further discussion see e.g. R. Mann, chapter 25.</ref> however despite having huge and continued successes in providing experimental predictions it does leave some [[beyond the standard model|unexplained phenomena]]. In particular, although the physics of [[special relativity]] is incorporated, [[general relativity]] is not, and the Standard Model will fail at energies or distances where the [[graviton]] is expected to emerge. Therefore in a modern field theory context, it is seen as an [[effective field theory]].
| | == his eyes also promised some more excitement' color. ' == |
|
| |
|
| This article requires some background in physics and mathematics, but is designed as both an introduction and a reference.
| | Thing! 'Words to the end, on the face of the emergence of a large Presbyterian touch ruddy, his eyes also [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 腕時計 ソーラー] promised some more excitement' color. '<br><br>hear this big elders has not been determined, Xiao Yan and Medusa are a heavy sigh of relief, perhaps this is [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ腕時計 メンズ] just an accident<br><br>'Whether this is true, you have to remember, perhaps later to be extra careful to wait a period of time, that is, your body is able to determine what the reason.' Qiaode It looks like Medusa, a large Presbyterian brow wrinkled, Chen Sheng said.<br><br>'this time is how long?' Xiao Yan stem asked with a smile.<br><br>'snake Terran Unlike humans, especially blood or Medusa, if indeed pregnant, time will not short, three to five years is the normal things.' great elders faint.<br><br>Wen Yan, Xiao Yan mind [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ ソーラー 腕時計] again relieved, okay, if this two-month [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 腕時計 バンド] period to [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ 腕時計 ソーラー 電波] determine the case, then he really jumped from the wall |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://www.xinzhinang.com.cn/plus/feedback.php?aid=167 http://www.xinzhinang.com.cn/plus/feedback.php?aid=167]</li> |
| | |
| | <li>[http://commonnet.no-ip.biz/mediawiki/index.php/User:Xvydprnbiw#after_a_moment http://commonnet.no-ip.biz/mediawiki/index.php/User:Xvydprnbiw#after_a_moment]</li> |
| | |
| | <li>[http://jirostyle.com/bbs/index.cgi http://jirostyle.com/bbs/index.cgi]</li> |
| | |
| | </ul> |
|
| |
|
| ==Quantum Field Theory== | | == vague hoarse voice == |
|
| |
|
| [[File:Standard Model.svg|300px|right|thumb|The pattern of [[weak isospin]] ''T''<sub>3</sub>, [[weak hypercharge]] ''Y''<sub>W</sub>, and [[color charge]] of all known elementary particles, rotated by the [[weak mixing angle]] to show electric charge ''Q'', roughly along the vertical. The neutral [[Higgs field]] (gray square) breaks the [[electroweak symmetry]] and interacts with other particles to give them mass.]] | | Under its frustration when it will exit so state, that is about to dissipate depleted grain dust pregnant spirit within my mind, but it is suddenly a burst of intense light, etc. In this [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ電波ソーラー腕時計] light, even even Xiao Yan soul are all feeling a sharp pain in the emergence.<br><br>pain gradually fade, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオソーラー時計] in some panic among Xiao Yan, at the outbreak of that light, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html casio 腕時計 スタンダード] it was a faint sound came between extremely low blurred the old 'Yin'<br><br>'very soul [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオの時計] of ... close ... Carolina Ling Ling Shou days forging Soul'<br><br>vague hoarse voice, whispering in XiaoYan minds back to 'swing', like Fine resounded like to make people kind of get a [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 phys] sense of trance.<br>Trance does not make sense<br>Xiao Yan therefore absence, and that the sound of the minds of old, although vague, but it is possible he jotted down a lot, and this old sound, in retrospect it three times After that is thoroughly dissipated away<br><br>room |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://skwlkr.s18.23dns.net/forum/forum.php?mod=viewthread&tid=594324 http://skwlkr.s18.23dns.net/forum/forum.php?mod=viewthread&tid=594324]</li> |
| | |
| | <li>[http://www.aiyingfang.cn/bbs/showtopic-904878.aspx http://www.aiyingfang.cn/bbs/showtopic-904878.aspx]</li> |
| | |
| | <li>[http://www.webngardha.blogdehi.com/shockwave/download/download.cgi http://www.webngardha.blogdehi.com/shockwave/download/download.cgi]</li> |
| | |
| | </ul> |
|
| |
|
| The standard model is a [[quantum field theory]], meaning its fundamental objects are ''quantum fields'' which are defined at all points in spacetime. These fields are
| | == wooden box open == |
| * the fermion field, <math>\psi</math>, which accounts for "matter particles";
| |
| * the electroweak boson fields <math>W_1</math>, <math>W_2</math>, <math>W_3</math>, and <math>B</math>;
| |
| * the gluon field, <math>G_a</math>; and
| |
| * the Higgs field, <math>\phi</math>.
| |
| That these are ''quantum'' rather than ''classical'' fields has the mathematical consequence that they are [[operator (physics)|operator]]-valued. In particular, values of the fields generally do not commute. As operators, they act upon the quantum state ([[ket vector]]).
| |
|
| |
|
| The dynamics of the quantum state and the fundamental fields are determined by the [[Lagrangian density]] <math>\mathcal{L}</math> (usually for short just called the Lagrangian). This plays a role similar to that of the [[Schrödinger equation]] in non-relativistic quantum mechanics, but a Lagrangian is not an equation — rather, it is a polynomial function of the fields and their derivatives. While it would be possible to derive a system of differential equations governing the fields from the Langrangian, it is more common to use other techniques to compute with quantum field theories.
| | Smile, the two boxes, open simultaneously.<br><br>wooden box open, 'Lucy,' one [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ電波ソーラー腕時計] of the two items, one is a fist-sized red 'color' round beads, the second is a gray brown 'color' of unusual bamboo.<br><br>Hsiao go far people's eyes, first brought together the egg on top of red beads, they can feel it being quite violent and contains an enormous fire is 'sexual' energy.<br><br>'This is a magic nucleus?' Xiao Yan eyes firmly fixed on the red beads While [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ レディース 電波ソーラー腕時計] some less certain way, so a high level of magic nucleus, but the first time I saw him.<br><br>'ah, is indeed a magic nuclear grade, but not low, but still kind of afraid of the higher-order form of Warcraft, or difficult to have such a' color [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html カシオ 腕時計 ソーラー 電波] 'and energy.'<br><br>Soviet [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html casio 腕時計 説明書] Union thousands stroking his beard, his eyes passing touch alarmed, said: 'This is magic nucleus, as I guess [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html 電波時計 カシオ] it should be a seven-order fire is' sexual 'magic nucleus, no |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://co.ctbu.edu.cn/forum.php?mod=viewthread&tid=892593 http://co.ctbu.edu.cn/forum.php?mod=viewthread&tid=892593]</li> |
| | |
| | <li>[http://chadayamangalam.com/cgi-bin/guestbook.cgi http://chadayamangalam.com/cgi-bin/guestbook.cgi]</li> |
| | |
| | <li>[http://www.otis.ne.jp/97tokiwa-topics/diary.cgi http://www.otis.ne.jp/97tokiwa-topics/diary.cgi]</li> |
| | |
| | </ul> |
|
| |
|
| The standard model is furthermore a [[gauge theory]], which means there are degrees of freedom in the mathematical formalism which do not correspond to changes in the physical state. The [[gauge group]] of the standard model is <math>\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)</math>, where U(1) acts on <math>B</math> and <math>\phi</math>, SU(2) acts on <math>W</math> and <math>\phi</math>, and SU(3) acts on <math>G</math>. The fermion field <math>\psi</math> also transforms under these symmetries, although all of them leave some parts of it unchanged.
| | == may seem Lin Xiu cliff this and other real powerful figures == |
|
| |
|
| ===The role of the quantum fields=== | | Here, then, is his eyes glanced Xiao Yan, laughing, 'Yin' 'Yin' and said: 'If you come up with all the means, I think, even if I and Liu Qing, and you with war, want wins, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 腕時計 バンド] then fear it is not easy. '<br><br>'This is indeed a high forest long look at me. '<br><br>Xiao Yan smiled and [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 激安] shook his head, his brow suddenly a pick, his eyes slowly turned to the field, such as [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ腕時計 メンズ] the Tower saw who like Ares, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html 電波時計 casio] sharp eyes, the two are firmly locked himself, and with the latter eyes The move, that the whole line of sight, are also stuck in a moment he and Lin Xiu Cliff body.<br><br>when the stands people found Liu Qing gaze seems to Xiao Yan parcels included, but it is somewhat suspect 'confusion' to Lin Xiu Cliff's strength, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 腕時計 バンド] which he attached so much importance possessed nothing can be words of Xiao Yan, although he was within School strongest dark horse, may seem Lin Xiu cliff this and other real powerful figures, the gap should not be small, right?<br><br>no |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://222.243.160.155/forum.php?mod=viewthread&tid=7962891 http://222.243.160.155/forum.php?mod=viewthread&tid=7962891]</li> |
| | |
| | <li>[http://www.cqxgws.com/plus/feedback.php?aid=277 http://www.cqxgws.com/plus/feedback.php?aid=277]</li> |
| | |
| | <li>[http://www.harmonicaclub.com/cgi-bin/harmonicaclub/pm.cgi http://www.harmonicaclub.com/cgi-bin/harmonicaclub/pm.cgi]</li> |
| | |
| | </ul> |
|
| |
|
| In [[classical mechanics]], the state of a system can usually be captured by a small set of variables, and the dynamics of the system is thus determined by the time evolution of these variables. In [[classical field theory]], the ''field'' is part of the state of the system, so in order to describe it completely one effectively introduces separate variables for every point in spacetime (even though there are many restrictions on how the values of the field "variables" may vary from point to point, for example in the form of field equations involving partial derivatives of the fields).
| | == but did not say anything == |
|
| |
|
| In [[quantum mechanics]], the classical variables are [[first quantization|turned into]] operators, but these do not capture the state of the system, which is instead encoded into a [[wavefunction]] <math>\psi</math> or more abstract [[ket vector]]. If <math>\psi</math> is an [[eigenstate]] with respect to an operator <math>P</math>, then <math> P \psi = \lambda \psi </math> for the corresponding eigenvalue <math>\lambda</math>, and hence letting an operator <math>P</math> act on <math>\psi</math> is analogous to multiplying <math>\psi</math> by the value of the classical variable to which <math>P</math> corresponds. By extension, a classical formula where all variables have been replaced by the corresponding operators will behave like an operator which, when it acts upon the state of the system, multiplies it by the analogue of the quantity that the classical formula would compute. The formula as such does however not contain any information about the state of the system; it would evaluate to the same operator regardless of what state the system is in.
| | It is difficult to look, but did not say anything, she knew what to say now [http://nrcil.net/sitemap.xml http://nrcil.net/sitemap.xml] is not a role.<br><br>'Since such a [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 gps] large selection elders, that the deity can only say, I hope you can not regret that time.'<br><br>evil demon king flower face 'color' is quite ugly, people today not only lost, but also to do great things smashed, the back, and certainly [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html casio 腕時計 レディース] will not be a good cross.<br><br>'It will not bother you labor,' white-haired old woman waved faint.<br><br>evil demon king mouth slightly twitching flowers, things have become a foregone conclusion that he is not good to stay here a disgrace, the moment Yifu Xiupao, is turned toward the outside of flowers were storm swept away, while turning it to look Xiao Yan's eyes, but it is full of eerie pernicious 'color' Obviously, today face the big throw, he [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 腕時計 バンド] is thoroughly Xiao Yan [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 phys] to the hate.<br><br>for his eyes, Xiao Yan just smile, |
| | | 相关的主题文章： |
| Quantum fields relate to quantum mechanics as classical fields do to classical mechanics, i.e., there is a separate operator for every point in spacetime, and these operators do not carry any information about the state of the system; they are merely used to exhibit some aspect of the state, at the point to which they belong. In particular, the quantum fields are ''not'' wavefunctions, even though the equations which govern their time evolution may be deceptively similar to those of the corresponding wavefunction in a [[Old quantum theory|semiclassical]] formulation. There is no variation in strength of the fields between different points in spacetime; the variation that happens is rather one of [[phase factor]]s.
| | <ul> |
| | | |
| ===Vectors, scalars, and spinors===
| | <li>[http://www.chinaqfd.com/plus/feedback.php?aid=177 http://www.chinaqfd.com/plus/feedback.php?aid=177]</li> |
| | | |
| Mathematically it may look as though all of the fields are vector-valued (in addition to being operator-valued), since they all have several components, can be multiplied by matrices, etc., but physicists assign a more specific meaning to the word: a '''vector''' is something which transforms like a [[four-vector]] under [[Lorentz transformations]], and a '''scalar''' is something which does not transform under Lorentz transformations. The <math>B</math>, <math>W_j</math>, and <math>G_a</math> fields are all vectors in this sense, so the corresponding particles are said to be [[vector boson]]s. The Higgs field <math>\phi</math> is a scalar.
| | <li>[http://co.ctbu.edu.cn/forum.php?mod=viewthread&tid=901371&fromuid=17329 http://co.ctbu.edu.cn/forum.php?mod=viewthread&tid=901371&fromuid=17329]</li> |
| | | |
| The fermion field <math>\psi</math> does transform under Lorentz transformations, but not like a vector should; rotations will only turn it by half the angle a proper vector should. Therefore these constitute a third kind of quantity, which is known as a [[spinor]].
| | <li>[http://www.wanboroughprimary.org/guestbook/guestbook.cgi http://www.wanboroughprimary.org/guestbook/guestbook.cgi]</li> |
| | | |
| It is common to make use of [[abstract index notation]] for the vector fields, in which case the vector fields all come with a Lorentzian index <math>\mu</math>, like so: <math>B^\mu\,</math>, <math>W_j^\mu</math>, and <math>G_a^\mu</math>. If abstract index notation is used also for spinors then these will carry a spinorial index and the [[gamma matrices|Dirac gamma]] will carry one Lorentzian and two spinorian indices, but it is more common to regard spinors as [[column vector|column matrices]] and the Dirac gamma <math>\gamma^\mu</math> as a matrix which additionally carries a Lorentzian index. The [[Feynman slash notation]] can be used to turn a vector field into a linear operator on spinors, like so: <math> {\not}B = \gamma^\mu B_\mu </math>; this may involve [[raising and lowering indices]].
| | </ul> |
| | |
| ==Alternative Presentations of the Fields==
| |
| [[Image:Elementary_particle_interactions.svg|400px|thumb|right|Connections denoting which particles interact with each other.]]
| |
| As is common in quantum theory, there is more than one way to look at things. At first the basic fields given above may not seem to correspond well with the "fundamental particles" in the chart above, but there are several alternative presentations which, in particular contexts, may be more appropriate than those that are given above.
| |
| | |
| ===Fermions===
| |
| | |
| Rather than having one fermion field <math>\psi</math>, it can be split up into separate components for each type of particle. This mirrors the historical evolution of quantum field theory, since the electron component <math>\psi_{\mathrm e}</math> (describing the [[electron]] and its antiparticle the [[positron]]) is then the original <math>\psi</math> field of [[quantum electrodynamics]], which was later accompanied by <math>\psi_\mu</math> and <math>\psi_\tau</math> fields for the [[muon]] and [[tauon]] respectively (and their antiparticles). Electroweak theory added <math>\psi_{\nu_{\mathrm e}}</math>, <math>\psi_{\nu_\mu}</math>, and <math>\psi_{\nu_\tau}</math> for the corresponding [[neutrino]]s, and the [[quark]]s add still further components. In order to be [[Dirac spinor#Four-spinor for particles|four-spinors]] like the electron and other [[lepton]] components, there must be one quark component for every combination of [[Flavor (particle physics)|flavour]] and [[color charge|colour]], bringing the total to 24 (3 for charged leptons, 3 for neutrinos, and 2·3·3 = 18 for quarks).
| |
| | |
| An important definition is the barred fermion field <math> \bar{\psi} </math> is defined to be <math> \psi^\dagger \gamma^0 </math>, where <math> \dagger </math> denotes the [[Hermitian adjoint]] and <math>\gamma^0</math> is the zeroth [[gamma matrix]]. If <math>\psi</math> is thought of as an {{math|''n'' × 1}} matrix then <math>\bar{\psi}</math> should be thought of as a [[row vector|{{math|1 × ''n''}} matrix]].
| |
| | |
| ====A chiral theory====
| |
| | |
| An independent decomposition of <math>\psi</math> is that into [[chirality (physics)|chirality]] components:
| |
| :"Left" chirality: <math>\psi^L = \frac{1}{2}(1-\gamma_5)\psi</math>
| |
| :"Right" chirality: <math>\psi^R = \frac{1}{2}(1+\gamma_5)\psi</math>
| |
| where <math>\gamma_5</math> is [[Gamma matrices#The_Fifth_Gamma_Matrix.2C_.CE.B35|the fifth gamma matrix]]. This is very important in the Standard Model because '''left and right chirality components are treated differently by the gauge interactions'''.
| |
| | |
| In particular, under [[weak isospin]] [[special unitary group|SU(2)]] transformations the left-handed particles are weak-isospin doublets, whereas the right-handed are singlets – i.e. the weak isospin of <math>\psi^R</math> is zero. Put more simply, the weak interaction could rotate e.g. a left-handed electron into a left-handed neutrino (with emission of a W<sup>−</sup>), but could not do so with the same right-handed particles. As an aside, the right-handed neutrino originally did not exist in the standard model – but the discovery of [[neutrino oscillation]] implies that [[neutrino#Mass|neutrinos must have mass]], and since chirality can change during the propagation of a massive particle, right-handed neutrinos must exist in reality. This does not however change the (experimentally-proven) chiral nature of the weak interaction.
| |
| | |
| Furthermore, U(1) acts differently on <math>\psi^L_{\mathrm e}</math> than on <math>\psi^R_{\mathrm e}</math> (because they have different [[weak hypercharge]]s).
| |
| | |
| ====Mass and interaction eigenstates====
| |
| | |
| A distinction can thus be made between, for example, the mass and interaction [[eigenstate]]s of the neutrino. The former is the state which propagates in free space, whereas the latter is the '''different''' state that participates in interactions. Which is the "fundamental" particle? For the neutrino, it is conventional to define the "flavour" ({{SubatomicParticle|Electron Neutrino|link=yes}}, {{SubatomicParticle|Muon Neutrino|link=yes}}, or {{SubatomicParticle|Tau Neutrino|link=yes}}) by the interaction eigenstate, whereas for the quarks we define the flavour (up, down, etc.) by the mass state. We can switch between these states using the [[CKM matrix]] for the quarks, or the [[PMNS matrix]] for the neutrinos (the charged leptons on the other hand are eigenstates of both mass and flavour).
| |
| | |
| As an aside, if a complex phase term exists within either of these matrices, it will give rise to direct [[CP violation]], which could explain the dominance of matter over antimatter in our current universe. This has been proven for the CKM matrix, and is expected for the PMNS matrix.
| |
| | |
| ====Positive and negative energies====
| |
| | |
| Finally, the quantum fields are sometimes decomposed into "positive" and "negative" energy parts: <math> \psi = \psi^{+} + \psi^{-} </math>. This is not so common when a quantum field theory has been set up, but often features prominently in the process of quantizing a field theory.
| |
| | |
| ===Bosons===
| |
| | |
| Due to the [[Higgs mechanism]], the electroweak boson fields <math>W_1</math>, <math>W_2</math>, <math>W_3</math>, and <math>B</math> "mix" to create the states which are physically observable. To retain gauge invariance, the underlying fields must be massless, but the observable states '''can gain masses''' in the process. These states are:
| |
| | |
| The massive neutral boson:
| |
| : <math> Z= \cos \theta_W W_3 - \sin \theta_W B</math>
| |
| The massless neutral boson:
| |
| : <math> A = \sin \theta_W W_3 + \cos \theta_W B</math>
| |
| The massive charged [[W and Z bosons|W bosons]]:
| |
| : <math>W^{\pm} = \frac1{\sqrt2}\left(W_1 \mp i W_2\right)</math>
| |
| where <math>\theta_W</math> is the [[Weinberg angle]].
| |
| | |
| The <math>A</math> field is the [[photon]], which corresponds classically to the well-known [[electromagnetic four-potential]] – i.e. the electric and magnetic fields. The <math>Z</math> field actually contributes in every process the photon does, but due to its large mass, the contribution is usually negligible.
| |
| | |
| ==Perturbative QFT and the Interaction Picture==
| |
| | |
| Much of the qualitative descriptions of the standard model in terms of "particles" and "forces" comes from the [[perturbative quantum field theory]] view of the model. In this, the Langrangian is decomposed as <math> \mathcal{L} = \mathcal{L}_0 + \mathcal{L}_\mathrm{I} </math> into separate ''free field'' and ''interaction'' Langrangians. The free fields care for particles in isolation, whereas processes involving several particles arise through interactions. The idea is that the state vector should only change when particles interact, meaning a free particle is one whose quantum state is constant. This corresponds to the [[interaction picture]] in quantum mechanics.
| |
| | |
| In the more common [[Schrödinger picture]], even the states of free particles change over time: typically the phase changes at a rate which depends on their energy. In the alternative [[Heisenberg picture]], state vectors are kept constant, at the price of having the operators (in particular the [[observable]]s) be time-dependent. The interaction picture constitutes an intermediate between the two, where some time dependence is placed in the operators (the quantum fields) and some in the state vector. In QFT, the former is called the free field part of the model, and the latter is called the interaction part. The free field model can be solved exactly, and then the solutions to the full model can be expressed as perturbations of the free field solutions, for example using the [[Dyson series]].
| |
| | |
| It should be observed that the decomposition into free fields and interactions is in principle arbitrary. For example [[Renormalization#Renormalization_in_QED|renormalization]] in [[Quantum electrodynamics|QED]] modifies the mass of the free field electron to match that of a physical electron (with an electromagnetic field), and will in doing so add a term to the free field Lagrangian which must be cancelled by a counterterm in the interaction Lagrangian, that then shows up as a two-line vertex in the [[Feynman diagrams]]. This is also how the Higgs field is thought to give particles [[Invariant mass|mass]]: the part of the interaction term which corresponds to the (nonzero) vacuum expectation value of the Higgs field is moved from the interaction to the free field Lagrangian, where it looks just like a mass term having nothing to do with Higgs.
| |
| | |
| ===Free fields===
| |
| | |
| Under the usual free/interaction decomposition, which is suitable for low energies, the free fields obey the following equations:
| |
| * The fermion field <math>\psi</math> satisfies the [[Dirac equation]]; <math> (i \hbar {\not}\partial - m_f c) \psi_f = 0 </math> for each type <math>f</math> of fermion.
| |
| * The photon field <math>A</math> satisfies the [[wave equation]] <math> \partial_\mu \partial^\mu A^\nu = 0 </math>.
| |
| * The Higgs field <math>\phi</math> satisfies the [[Klein–Gordon equation]].
| |
| * The weak interaction fields <math>Z</math>, <math>W^+</math>, and <math>W^-</math> also satisfy the [[Klein–Gordon equation]].
| |
| These equations can be solved exactly. One usually does so by considering first solutions that are periodic with some period <math>L</math> along each spatial axis; later taking the <math>L \rightarrow \infty</math> limit will lift this periodicity restriction.
| |
| | |
| In the periodic case, the solution for a field <math>F</math> (any of the above) can be expressed as a [[Fourier series]] of the form
| |
| :<math> F(x) = \beta \sum_{\mathbf{p}} \sum_r E_{\mathbf{p}}^{-1/2} \left( a_r(\mathbf{p}) u_r(\mathbf{p}) e^{-ipx/\hbar} + b^\dagger_r(\mathbf{p}) v_r(\mathbf{p}) e^{ipx/\hbar} \right)</math>
| |
| where:
| |
| * <math>\beta</math> is a normalization factor; for the fermion field <math>\psi_f</math> it is <math> \sqrt{ m_f c^2 / V} </math>, where <math> V = L^3 </math> is the volume of the fundamental cell considered; for the photon field <math> A^\mu </math> it is <math> \hbar c / \sqrt{2V} </math>.
| |
| * The sum over <math>\mathbf{p}</math> is over all momenta consistent with the period <math>L</math>, i.e., over all vectors <math> \frac{2\pi\hbar}{L}(n_1,n_2,n_3)</math> where <math>n_1,n_2,n_3</math> are integers.
| |
| * The sum over <math>r</math> covers other degrees of freedom specific for the field, such as polarization or spin; it usually comes out as a sum from <math>1</math> to <math>2</math> or from <math>1</math> to <math>3</math>.
| |
| * <math>E_{\mathbf{p}}</math> is the relativistic energy for a momentum <math>\mathbf{p}</math> quantum of the field, <math> = \sqrt{ m^2 c^4 + c^2 \mathbf{p}^2 } </math> when the rest mass is <math>m</math>.
| |
| * <math>a_r(\mathbf{p})</math> and <math>b^\dagger_r(\mathbf{p})</math> are [[Creation and annihilation operators|annihilation and creation]] respectively operators for "a-particles" and "b-particles" respectively of momentum <math>\mathbf{p}</math>; "b-particles" are the [[antiparticle]]s of "a-particles". Different fields have different "a-" and "b-particles". For some fields, <math>a</math> and <math>b</math> are the same.
| |
| * <math>u_r(\mathbf{p})</math> and <math>v_r(\mathbf{p})</math> are non-operators which carry the vector or spinor aspects of the field (where relevant).
| |
| * <math>p = (E_{\mathbf{p}}/c, \mathbf{p})</math> is the [[four-momentum]] for a quanta with momentum <math>\mathbf{p}</math>. <math>px = p_\mu x^\mu</math> denotes an inner product of [[four-vector]]s.
| |
| In the <math>L \rightarrow \infty</math> limit, the sum would turn into an integral with help from the <math>V</math> hidden inside <math>\beta</math>. The numeric value of <math>\beta</math> also depends on the normalization chosen for <math>u_r(\mathbf{p})</math> and <math>v_r(\mathbf{p})</math>.
| |
| | |
| Technically, <math>a^\dagger_r(\mathbf{p})</math> is the [[Hermitian adjoint]] of the operator <math>a_r(\mathbf{p})</math> in the [[inner product space]] of [[ket vector]]s. The identification of <math>a^\dagger_r(\mathbf{p})</math> and <math>a_r(\mathbf{p})</math> as [[creation and annihilation operators]] comes from comparing conserved quantities for a state before and after one of these have acted upon it. <math>a^\dagger_r(\mathbf{p})</math> can for example be seen to add one particle, because it will add <math>1</math> to the eigenvalue of the a-particle [[number operator]], and the momentum of that particle ought to be <math>\mathbf{p}</math> since the eigenvalue of the vector-valued [[momentum operator]] increases by that much. For these derivations, one starts out with expressions for the operators in terms of the quantum fields. That the operators with <math>\dagger</math> are creation operators and the one without annihilation operators is a convention, imposed by the sign of the commutation relations postulated for them.
| |
| | |
| An important step in preparation for calculating in perturbative quantum field theory is to separate the "operator" factors <math>a</math> and <math>b</math> above from their corresponding vector or spinor factors <math>u</math> and <math>v</math>. The vertices of [[Feynman graph]]s come from the way that <math>u</math> and <math>v</math> from different factors in the interaction Lagrangian fit together, whereas the edges come from the way that the <math>a</math>s and <math>b</math>s must be moved around in order to put terms in the Dyson series on normal form.
| |
| | |
| ===Interaction terms and the path integral approach===
| |
| | |
| The Lagrangian can also be derived without using creation and annihilation operators (the "canonical" formalism), by using a "path integral" approach, pioneered by Feynman building on the earlier work of Dirac. See e.g. [[Path_integral_formulation#Quantum_field_theory|Path integral formulation on Wikipedia]] or A. Zee's [http://press.princeton.edu/releases/m9227.html QFT in a nutshell]. This is one possible way that the [[Feynman diagram]]s, which are pictorial representations of interaction terms, can be derived relatively easily. A quick derivation is indeed presented at the article on [[Feynman diagram]]s.
| |
| | |
| ==Lagrangian Formalism==
| |
| [[Image:Standard Model Feynman Diagram Vertices.png|250px|thumb|right|The above interactions show some basic interaction vertices – Feynman diagrams in the standard model are built from these vertices. Higgs boson interactions are however not shown, and neutrino oscillations are commonly added. The charge of the W bosons are dictated by the fermions they interact with.]]
| |
| We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model [[Lagrangian#Lagrangians_and_Lagrangian_densities_in_field_theory|Lagrangian density]]. Any such term must be both gauge and reference-frame invariant, otherwise the laws of physics would depend on an arbitrary choice or the frame of an observer. Therefore the [[Global symmetry|global]] [[Poincaré group|Poincaré symmetry]], consisting of [[translational symmetry]], [[rotational symmetry]] and the inertial reference frame invariance central to the theory of [[special relativity]] must apply. The [[Local symmetry|local]] {{math|SU(3) × SU(2) × U(1)}} gauge symmetry is the [[Internal symmetries|internal symmetry]]. The three factors of the gauge symmetry together give rise to the three fundamental interactions, after some appropriate relations have been defined, as we shall see.
| |
| | |
| A complete formulation of the Standard Model Lagrangian with all the terms written together can be found e.g. [http://einstein-schrodinger.com/Standard_Model.pdf here].
| |
| | |
| ===Kinetic terms===
| |
| | |
| A free particle can be represented by a mass term, and a ''kinetic'' term which relates to the "motion" of the fields.
| |
| | |
| ====Fermion fields====
| |
| The kinetic term for a Dirac fermion is
| |
| :<math>i\bar{\psi}\gamma_{\mu}\partial^{\mu}\psi</math>
| |
| where the notations are carried from earlier in the article. <math>\psi</math> can represent any, or all, Dirac fermions in the standard model. Generally, as below, this term is included within the couplings (creating an overall "dynamical" term).
| |
| | |
| ====Gauge fields====
| |
| For the spin-1 fields, first define the field strength tensor
| |
| :<math>F^a_{ \mu\nu}=\partial_{\mu}A^{a}_{ \nu} - \partial_{\nu}A^{a}_{ \mu} + g f^{abc}A^{b}_{\mu}A^{c}_{\nu}</math>
| |
| for a given gauge field (here we use <math>A</math>), with gauge [[coupling constant]] ''g''. The quantity <math>f^{abc}</math> is the [[Algebra over a field#Structure coefficients|structure constant]] of the particular gauge group, defined by the commutator <math>[t_{a}, t_{b}] = i f^{abc} t_{c}</math>, where <math>t_{i}</math> are the <!-- don't link [[Generating set]] or so! -->[[Lie algebra#Generators and dimension|generators]] of the group. In an [[Abelian group|Abelian (commutative) group]] (such as the U(1) we use here), since the generators <math>t_{a}</math> all commute with each other, the structure constants vanish. Of course, this is not the case in general – the standard model includes the non-Abelian SU(2) and SU(3) groups (such groups lead to what is called a [[Yang–Mills theory|Yang–Mills gauge theory]]).
| |
| | |
| We need to introduce three gauge fields corresponding to each of the subgroups {{math|SU(3) × SU(2) × [[U(1)]]}}.
| |
| *The gluon field tensor will be denoted by <math>G^{a}_{\mu\nu}</math>, where the index <math>a</math> labels elements of the '''8''' representation of colour [[special unitary group|SU(3)]]. The strong coupling constant is conventionally labelled ''g''<sub>s</sub> (or simply ''g'' where there is no ambiguity). ''The observations leading to the discovery of this part of the Standard Model are discussed in the article in [[quantum chromodynamics]].''
| |
| *The notation <math>W^{a}_{\mu\nu}</math> will be used for the gauge field tensor of [[special unitary group|SU(2)]] where ''a'' runs over the ''3'' generators of this group. The coupling can be denoted ''g''<sub>w</sub> or again simply ''g''. The gauge field will be denoted by <math>W^{a}_{\mu}</math>.
| |
| *The gauge field tensor for the [[Circle group|U(1)]] of weak hypercharge will be denoted by <math>B_{\mu\nu}</math>, the coupling by <math>g'</math>, and the gauge field by ''B''<sub>μ</sub>.
| |
| | |
| The kinetic term can now be written simply as
| |
| :<math>\int - {1\over 4} B_{\mu\nu} B^{\mu\nu} - {1\over 4}\mathrm{tr} W_{\mu\nu}W^{\mu\nu} - {1\over 4} \mathrm{tr}G_{\mu\nu} G^{\mu\nu}</math>
| |
| where the traces are over the SU(2) and SU(3) indices hidden in ''W'' and ''G'' respectively. The two-index objects are the field strengths derived from ''W'' and ''G'' the vector fields. There are also two extra hidden parameters: the theta angles for SU(2) and SU(3).
| |
| | |
| ===Coupling terms===
| |
| | |
| The next step is to "couple" the gauge fields to the fermions, allowing for interactions.
| |
| | |
| ====Electroweak sector====
| |
| {{Main|Electroweak interaction}}
| |
| The electroweak sector interacts with the symmetry group {{math|U(1) × SU(2)<sub>L</sub>}}, where the subscript L indicates coupling only to left-handed fermions.
| |
| :<math>
| |
| \mathcal{L}_\mathrm{EW} =
| |
| \sum_\psi\bar\psi\gamma^\mu
| |
| \left(i\partial_\mu-g^\prime{1\over2}Y_\mathrm{W}B_\mu-g{1\over2}\boldsymbol{\tau}\mathbf{W}_\mu\right)\psi</math>
| |
| | |
| Where ''B''<sub>''μ''</sub> is the U(1) gauge field; ''Y''<sub>W</sub> is the [[weak hypercharge]] (the generator of the U(1) group); <math>\mathbf{W}_\mu</math> is the
| |
| three-component SU(2) gauge field; and the components of <math>\boldsymbol{\tau}</math> are the [[Pauli matrices]] (infinitesimal generators of the SU(2) group) whose eigenvalues give the weak isospin. Note that we have to redefine a new U(1) symmetry of ''weak hypercharge'', different from QED, in order to achieve the unification with the weak force. The [[electric charge]] ''Q'', third component of [[weak isospin]] ''T''<sub>3</sub> (also called ''T''<sub>''z''</sub>, ''I''<sub>3</sub> or ''I''<sub>''z''</sub>) and weak hypercharge ''Y''<sub>W</sub> are related by
| |
| :<math> Q = T_3 + \frac{1}{2} Y_W \, ,</math> or by the alternate convention <math> Q = T_3 + Y_W \, .</math>
| |
| The first convention (used in this article) is equivalent to the earlier [[Gell-Mann–Nishijima formula]]. We can then define the [[conserved current]] for weak isospin as
| |
| :<math>\mathbf{j}_\mu = {1\over 2}\bar{\psi}_L \gamma_\mu\boldsymbol{\tau}\psi_L</math>
| |
| and for weak hypercharge as
| |
| :<math>j_{\mu}^{Y}=2(j_{\mu}^{em}-j_{\mu}^3)</math>
| |
| where <math>j_{\mu}^{em}</math> is the electric current and <math>j_{\mu}^{3}</math> the third weak isospin current. As explained [[#Bosons|above]], ''these currents mix'' to create the physically observed bosons, which also leads to testable relations between the coupling constants.
| |
| | |
| '''To explain in a simpler way''', we can see the effect of the electroweak interaction by picking out terms from the Lagrangian. We see that the SU(2) symmetry acts on each (left-handed) fermion doublet contained in <math>\psi</math>, for example
| |
| :<math>-{g\over 2}(\bar{\nu}_e \;\bar{e})\tau^+ \gamma_{\mu}(W^-)^{\mu}
| |
| \left(
| |
| \begin{array}{c}
| |
| {\nu_e} \\ e
| |
| \end{array}
| |
| \right)
| |
| \;=\;
| |
| -{g\over 2}\bar{\nu}_e\gamma_{\mu}(W^-)^{\mu}e
| |
| </math>
| |
| where the particles are understood to be left-handed, and where
| |
| :<math>\tau^{\pm}\equiv {1 \over 2}(\tau^1{\pm}i\tau^2)=
| |
| \begin{pmatrix}
| |
| 0 & 1 \\
| |
| 0 & 0
| |
| \end{pmatrix}
| |
| \; .</math>
| |
| This is an interaction corresponding to a "rotation in weak isospin space" or in other words, a ''transformation between e<sub>L</sub> and ν<sub>eL</sub> via emission of a W<sup>−</sup> boson''. The U(1) symmetry, on the other hand, is similar to electromagnetism, but acts on all "''weak hypercharged''" fermions (both left and right handed) via the neutral Z<sup>0</sup>, as well as the ''charged'' fermions via the photon.
| |
| | |
| ====Quantum chromodynamics sector====
| |
| {{Main|Quantum chromodynamics}}
| |
| The quantum chromodynamics (QCD) sector defines the interactions between [[quark]]s and [[gluon]]s, with SU(3) symmetry, generated by ''T''<sup>''a''</sup>. Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by
| |
| ::<math>\mathcal{L}_{\mathrm{QCD}} = i\overline U (\partial_\mu-ig_sG_\mu^a T^a)\gamma^\mu U + i\overline D (\partial_\mu-i g_s G_\mu^a T^a)\gamma^\mu D.</math>
| |
| where D and U are the Dirac spinors associated with up- and down-type quarks, and other notations are continued from the previous section.
| |
| | |
| ===Mass terms and the Higgs mechanism===
| |
| | |
| ====Mass terms====
| |
| The mass term arising from the Dirac Lagrangian (for any fermion <math>\psi</math>) is <math>-m\bar{\psi}\psi</math> which is ''not'' invariant under the electroweak symmetry. This can be seen by writing <math>\psi</math> in terms of left and right handed components (skipping the actual calculation):
| |
| :<math>-m\bar{\psi}\psi\;=\;-m(\bar{\psi}_L\psi_R+\bar{\psi}_R\psi_L)</math>
| |
| i.e. contribution from <math>\bar{\psi}_L\psi_L</math> and <math>\bar{\psi}_R\psi_R</math> terms do not appear. We see that the mass-generating interaction is achieved by constant flipping of particle chirality. The spin-half particles have no right/left helicity pair with the same SU(2) and SU(3) representation and the same weak hypercharge, so assuming these gauge charges are conserved in the vacuum, none of the spin-half particles could ever swap helicity, and must remain massless. Additionally, we know experimentally that the W and Z bosons are massive, but a boson mass term contains the combination e.g. <math>A^{\mu}A_{\mu}</math>, which clearly depends on the choice of gauge. Therefore, none of the standard model fermions ''or'' bosons can "begin" with mass, but must acquire it by some other mechanism.
| |
| | |
| ====The Higgs mechanism====
| |
| {{Main|Higgs mechanism}}
| |
| The solution to both these problems comes from the [[Higgs mechanism]], which involves scalar fields (the number of which depend on the exact form of Higgs mechanism) which (to give the briefest possible description) are "absorbed" by the massive bosons as degrees of freedom, and which couple to the fermions via Yukawa coupling to create what looks like mass terms.
| |
| | |
| In the Standard Model, the [[Higgs field]] is a complex scalar of the group [[SU(2)]]<sub>L</sub>:
| |
| :<math>
| |
| \phi={1\over\sqrt{2}}
| |
| \left(
| |
| \begin{array}{c}
| |
| \phi^+ \\ \phi^0
| |
| \end{array}
| |
| \right)\;,
| |
| </math>
| |
| where the indexes + and 0 indicate the electric charge (''Q'') of the components. The weak isospin (''Y''<sub>W</sub>) of both components is 1.
| |
| | |
| The Higgs part of the Lagrangian is
| |
| ::<math>\mathcal{L}_H = [(\partial_\mu -ig W_\mu^a t^a -ig'Y_{\phi} B_\mu)\phi]^2 + \mu^2 \phi^\dagger\phi-\lambda (\phi^\dagger\phi)^2,</math>
| |
| where <math>\lambda>0</math> and <math>\mu^{2}>0</math>, so that the mechanism of [[spontaneous symmetry breaking]] can be used. There is a parameter here, at first hidden within the shape of the potential, that is very important. In a [[unitarity gauge]] one can set <math>\phi^{+}=0</math> and make <math>\phi^0</math> real. Then <math>\langle\phi^0\rangle=v</math> is the non-vanishing [[vacuum expectation value]] of the Higgs field. <math>v</math> has units of mass, and it is the only parameter in the Standard Model which is not dimensionless. It is also much smaller than the Planck scale; it is approximately equal to the Higgs mass, and sets the scale for the mass of everything else. This is the only real fine-tuning to a small nonzero value in the Standard Model, and it is called the [[Hierarchy problem]]. Quadratic terms in <math>W_{\mu}</math> and <math>B_{\mu}</math> arise, which give masses to the W and Z bosons:
| |
| :<math>M_W = \frac{v|g|}2 \qquad\qquad M_Z=\frac{v\sqrt{g^2+{g'}^2}}2.</math>
| |
| | |
| The [[Yukawa interaction]] terms are
| |
| :<math>\mathcal{L}_{YU} = \overline U_L G_u U_R \phi^0 - \overline D_L G_u U_R \phi^- + \overline U_L G_d D_R \phi^+ + \overline D_L G_d D_R \phi^0 + hc</math>
| |
| where <math> G_{u,d}</math> are {{math|3 × 3}} matrices of Yukawa couplings, with the ''ij'' term giving the coupling of the generations ''i'' and ''j''.
| |
| | |
| ====Neutrino masses====
| |
| | |
| As previously mentioned, evidence shows neutrinos must have mass. But within the standard model, the right-handed neutrino does not exist, so even with a Yukawa coupling neutrinos remain massless. An obvious solution<ref>http://www.fas.org/sgp/othergov/doe/lanl/pubs/00326607.pdf</ref> is to simply ''add a right-handed neutrino'' <math>\nu_R</math> resulting in a Dirac mass term as usual. This field however must be a [[sterile neutrino]], since being right-handed it belongs to an isospin singlet (<math>T_3=0</math>) and also has charge <math>Q=0</math>, implying <math>Y_W=0</math> (see [[#Electroweak_sector|above]]) and that it does not even participate in the weak interaction. We also know experimentally that right-handed neutrinos cannot couple to the weak interaction, but evidence for observation of sterile neutrinos is currently not convincing.<ref>http://t2k-experiment.org/neutrinos/oscillations-today/</ref>
| |
| | |
| Another possibility to consider is that the neutrino satisfies the [[Majorana equation]], which at first seems possible due to its zero electric charge. In this case the mass term is
| |
| :<math>-{m\over 2} (\bar{\nu}^C\nu + \bar{\nu}\nu^C)</math>
| |
| where C denotes a charge conjugated (i.e. anti-) particle, and each term is the left-chirality projection (note that a left-chirality projection, as defined above, of an antiparticle is a right-handed field; conflicting notations in this area can unfortunately become confusing). It is furthermore possible (but ''not'' necessary) that neutrinos ''are'' their own antiparticle. Unfortunately, in the Standard Model this term directly violates conservation of weak hypercharge, as we are essentially flipping between LH neutrinos and RH anti-neutrinos. One solution to this involves extending the Higgs field to include an extra triplet with hypercharge 2.<ref>http://www.fas.org/sgp/othergov/doe/lanl/pubs/00326607.pdf</ref>
| |
| | |
| Finally, it is also possible to introduce a RH neutrino field, but consider mass terms that combine Dirac and Majorana masses, which can provide a "natural" explanation for the smallness of the observed neutrino masses (see [[seesaw mechanism]]). Since in any case new fields must be postulated, neutrinos are an obvious place to look for physics [[beyond the Standard Model]].
| |
| | |
| ==Detailed Information==
| |
| | |
| This section provides more detail on some aspects, and some reference material.
| |
| | |
| ===Field content in detail===
| |
| | |
| The Standard Model has the following fields. These describe one ''generation'' of leptons and quarks, and there are three generations, so there are three copies of each field. By CPT symmetry, there is a set of right-handed fermions with the opposite quantum numbers. The column "'''representation'''" indicates under which [[representation theory|representations]] of the [[gauge group]]s that each field transforms, in the order (SU(3), SU(2), U(1)). Symbols used are common but not universal; superscript C denotes an antiparticle; and for the U(1) group, the value of the [[weak hypercharge]] is listed. Note that there are twice as many left-handed lepton field components as left-handed antilepton field components in each generation, but an equal number of left-handed quark and antiquark fields.
| |
| | |
| {| class="wikitable collapsible collapsed"
| |
| |-
| |
| ! colspan="5"| Field content of the standard model
| |
| |-
| |
| ! colspan="5" style="background:#ffdead"| Spin 1 – the gauge fields
| |
| |-
| |
| ! Symbol !! Associated charge !! Group !! Coupling !! Representation
| |
| |-
| |
| | <math>B</math> || [[Weak hypercharge]] || {{math|U(1)<sub>''Y''</sub>}} || <math>g'</math> || <math>(\mathbf{1},\mathbf
| |
| | |
| {1},0)</math>
| |
| |-
| |
| | <math>W</math> || [[Weak isospin]] || {{math|SU(2)<sub>''L''</sub>}} || <math>g_w</math> || <math>(\mathbf{1},\mathbf
| |
| | |
| {3},0)</math>
| |
| |-
| |
| | <math>G</math> || [[Colour charge|colour]] || {{math|SU(3)<sub>''C''</sub>}} || <math>g_s</math> || <math>(\mathbf{8},\mathbf{1},0)
| |
| | |
| </math>
| |
| |-
| |
| ! colspan="5" style="background:#ffdead"| Spin {{frac|1|2}} – the fermions
| |
| |-
| |
| ! Symbol !! Name !! Baryon number !! Lepton number !! Representation
| |
| |-
| |
| | <math>Q_L</math> || Left-handed [[quark]] || <math>\textstyle\frac{1}{3}</math> || <math>0</math> || <math>(\mathbf{3},\mathbf{2},\textstyle\frac{1}{3})</math>
| |
| |-
| |
| | <math>u_L^C</math> || Left-handed antiquark (up) || <math>-\textstyle\frac{1}{3}</math> || <math>0</math> || <math>(\bar{\mathbf{3}},\mathbf{1},-\textstyle\frac{4}{3})</math>
| |
| |-
| |
| | <math>d_L^C</math> || Left-handed antiquark (down) || <math>-\textstyle\frac{1}{3}</math> || <math>0</math> || <math>(\bar{\mathbf{3}},\mathbf{1},\textstyle\frac{2}{3})</math>
| |
| |-
| |
| | <math>L_L</math> || Left-handed [[lepton]] || <math>0</math> || <math>1</math> || <math>(\mathbf{1},\mathbf{2},-1)</math>
| |
| |-
| |
| | <math>e_L^C</math> || Left-handed antilepton || <math>0</math> || <math>-1</math> || <math>(\mathbf{1},\mathbf{1},2)</math>
| |
| |-
| |
| ! colspan="5" style="background:#ffdead"| Spin 0 – the scalar boson
| |
| |-
| |
| ! Symbol !! Name !! colspan="3"| Representation
| |
| |-
| |
| | <math>H</math> || [[Higgs boson]] || colspan="3"| <math>(\mathbf{1},\mathbf{2},1)</math>
| |
| |}
| |
| | |
| ===Fermion content===
| |
| | |
| This table is based in part on data gathered by the [[Particle Data Group]].<ref>
| |
| {{cite journal
| |
| |author=W.-M. Yao ''et al''. ([[Particle Data Group]])
| |
| |year=2006
| |
| |title=Review of Particle Physics: Quarks
| |
| |url=http://pdg.lbl.gov/2006/tables/qxxx.pdf
| |
| |journal=[[Journal of Physics G]]
| |
| |volume=33 |page=1
| |
| |doi=10.1088/0954-3899/33/1/001
| |
| |arxiv = astro-ph/0601168 |bibcode = 2006JPhG...33....1Y }}</ref>
| |
| | |
| {| class="wikitable collapsible collapsed"
| |
| ! colspan="8"| '''Left-handed fermions in the Standard Model'''
| |
| |-
| |
| !colspan="8" style="background:#ffdead"|Generation 1
| |
| |- style="background:#fdd;"
| |
| !Fermion<br />(left-handed)
| |
| !Symbol
| |
| ![[Electric charge|Electric<br />charge]]
| |
| ![[Weak isospin|Weak<br />isospin]]
| |
| ![[Weak hypercharge|Weak<br />hypercharge]]
| |
| ![[Color charge|Color<br />charge]] <ref group="lhf" name="s1">These are not ordinary [[abelian group|abelian]] [[electric charge|charges]], which can be added together, but are labels of [[group representation]]s of [[Lie group]]s.</ref>
| |
| ![[Mass]]<ref group="lhf" name="s2">Mass is really a coupling between a left-handed fermion and a right-handed fermion. For example, the mass of an electron is really a coupling between a left-handed electron and a right-handed electron, which is the [[antiparticle]] of a left-handed [[positron]]. Also neutrinos show large mixings in their mass coupling, so it's not accurate to talk about neutrino masses in the [[Flavor (particle physics)|flavor]] basis or to suggest a left-handed electron antineutrino.</ref>
| |
| |-
| |
| |style="background:#efefef"|[[Electron]]
| |
| |{{SubatomicParticle|Electron}}
| |
| |<math>-1</math>
| |
| |<math>-1/2</math>
| |
| |<math>-1</math>
| |
| |<math>\bold{1}</math>
| |
| |511 keV
| |
| |-
| |
| |style="background:#efefef"|[[Positron]]
| |
| |{{SubatomicParticle|Positron}}
| |
| |<math>+1</math>
| |
| |<math>0</math>
| |
| |<math>+2</math>
| |
| |<math>\bold{1}</math>
| |
| |511 keV
| |
| |-
| |
| |style="background:#efefef"|[[Electron neutrino]]
| |
| |{{SubatomicParticle|Electron Neutrino}}
| |
| |<math>0</math>
| |
| |<math>+1/2</math>
| |
| |<math>-1</math>
| |
| |<math>\bold{1}</math>
| |
| |< 0.28 eV<ref group="lhf" name="s4">The Standard Model assumes that neutrinos are massless. However, several contemporary experiments prove that [[neutrino oscillation|neutrinos oscillate]] between their [[flavour (particle physics)|flavour]] states, which could not happen if all were massless. It is straightforward to extend the model to fit these data but there are many possibilities, so the mass [[eigenstate]]s are still [[open question|open]]. See [[neutrino mass]].</ref><ref group="lhf" name="s4b">{{cite journal
| |
| |author=W.-M. Yao ''et al''. ([[Particle Data Group]])
| |
| |year=2006
| |
| |title=Review of Particle Physics: Neutrino mass, mixing, and flavor change
| |
| |url=http://pdg.lbl.gov/2007/reviews/numixrpp.pdf
| |
| |journal=[[Journal of Physics G]]
| |
| |volume=33 |page=1
| |
| |doi= 10.1088/0954-3899/33/1/001
| |
| |arxiv = astro-ph/0601168 |bibcode = 2006JPhG...33....1Y }}</ref>
| |
| |-
| |
| |style="background:#efefef"|[[Electron antineutrino]]
| |
| |{{SubatomicParticle|Electron Neutrino}}
| |
| |<math>0</math>
| |
| |<math>0</math>
| |
| |<math>0</math>
| |
| |<math>\bold{1}</math>
| |
| |< 0.28 eV<ref group="lhf" name="s4" /><ref group="lhf" name="s4b" />
| |
| |-
| |
| |style="background:#efefef"|[[Up quark]]
| |
| |{{SubatomicParticle|Up Quark}}
| |
| |<math>+2/3</math>
| |
| |<math>+1/2</math>
| |
| |<math>+1/3</math>
| |
| |<math>\bold{3}</math>
| |
| |~ 3 MeV<ref group="lhf" name="s3">The [[mass]]es of [[baryon]]s and [[hadron]]s and various cross-sections are the experimentally measured quantities. Since quarks can't be isolated because of [[Quantum chromodynamics|QCD]] [[colour confinement|confinement]], the quantity here is supposed to be the mass of the quark at the [[renormalization]] scale of the QCD scale.</ref>
| |
| |-
| |
| |style="background:#efefef"|[[Up antiquark]]
| |
| |{{SubatomicParticle|Up Antiquark}}
| |
| |<math>-2/3</math>
| |
| |<math>0</math>
| |
| |<math>-4/3</math>
| |
| |<math>\bold{\bar{3}}</math>
| |
| |~ 3 MeV<ref group="lhf" name="s3" />
| |
| |-
| |
| |style="background:#efefef"|[[Down quark]]
| |
| |{{SubatomicParticle|Down Quark}}
| |
| |<math>-1/3</math>
| |
| |<math>-1/2</math>
| |
| |<math>+1/3</math>
| |
| |<math>\bold{3}</math>
| |
| |~ 6 MeV<ref group="lhf" name="s3" />
| |
| |-
| |
| |style="background:#efefef"|[[Down antiquark]]
| |
| |{{SubatomicParticle|Down Antiquark}}
| |
| |<math>+1/3</math>
| |
| |<math>0</math>
| |
| |<math>+2/3</math>
| |
| |<math>\bold{\bar{3}}</math>
| |
| |~ 6 MeV<ref group="lhf" name="s3" />
| |
| |-
| |
| !colspan="8"|
| |
| |-
| |
| !colspan="8" style="background:#ffdead"|Generation 2
| |
| |- style="background:#fdd;"
| |
| !Fermion<br />(left-handed)
| |
| !Symbol
| |
| !Electric<br />charge
| |
| !Weak<br />isospin
| |
| !Weak<br />hypercharge
| |
| !Color<br />charge <ref group="lhf" name="s1" />
| |
| !Mass <ref group="lhf" name="s2" />
| |
| |-
| |
| |style="background:#efefef"|[[Muon]]
| |
| |{{SubatomicParticle|Muon}}
| |
| |<math>-1</math>
| |
| |<math>-1/2</math>
| |
| |<math>-1</math>
| |
| |<math>\bold{1}</math>
| |
| |106 MeV
| |
| |-
| |
| |style="background:#efefef"|[[Antimuon]]
| |
| |{{SubatomicParticle|Antimuon}}
| |
| |<math>+1</math>
| |
| |<math>0</math>
| |
| |<math>+2</math>
| |
| |<math>\bold{1}</math>
| |
| |106 MeV
| |
| |-
| |
| |style="background:#efefef"|[[Muon neutrino]]
| |
| |{{SubatomicParticle|Muon Neutrino}}
| |
| |<math>0</math>
| |
| |<math>+1/2</math>
| |
| |<math>-1</math>
| |
| |<math>\bold{1}</math>
| |
| |< 0.28 eV<ref group="lhf" name="s4" /><ref group="lhf" name="s4b" />
| |
| |-
| |
| |style="background:#efefef"|[[Muon antineutrino]]
| |
| |{{SubatomicParticle|Muon Antineutrino}}
| |
| |<math>0</math>
| |
| |<math>0</math>
| |
| |<math>0</math>
| |
| |<math>\bold{1}</math>
| |
| |< 0.28 eV<ref group="lhf" name="s4" /><ref group="lhf" name="s4b" />
| |
| |-
| |
| |style="background:#efefef"|[[Charm quark]]
| |
| |{{SubatomicParticle|Charm Quark}}
| |
| |<math>+2/3</math>
| |
| |<math>+1/2</math>
| |
| |<math>+1/3</math>
| |
| |<math>\bold{3}</math>
| |
| |~ 1.3 GeV
| |
| |-
| |
| |style="background:#efefef"|[[Charm antiquark]]
| |
| |{{SubatomicParticle|Charm Antiquark}}
| |
| |<math>-2/3</math>
| |
| |<math>0</math>
| |
| |<math>-4/3</math>
| |
| |<math>\bold{\bar{3}}</math>
| |
| |~ 1.3 GeV
| |
| |-
| |
| |style="background:#efefef"|[[Strange quark]]
| |
| |{{SubatomicParticle|Strange Quark}}
| |
| |<math>-1/3</math>
| |
| |<math>-1/2</math>
| |
| |<math>+1/3</math>
| |
| |<math>\bold{3}</math>
| |
| |~ 100 MeV
| |
| |-
| |
| |style="background:#efefef"|[[Strange antiquark]]
| |
| |{{SubatomicParticle|Strange Antiquark}}
| |
| |<math>+1/3</math>
| |
| |<math>0</math>
| |
| |<math>+2/3</math>
| |
| |<math>\bold{\bar{3}}</math>
| |
| |~ 100 MeV
| |
| |-
| |
| !colspan="8"|
| |
| |-
| |
| !colspan="8" style="background:#ffdead"|Generation 3
| |
| |- style="background:#fdd;"
| |
| !Fermion<br />(left-handed)
| |
| !Symbol
| |
| !Electric<br />charge
| |
| !Weak<br />isospin
| |
| !Weak<br />hypercharge
| |
| !Color<br />charge<ref group="lhf" name="s1" />
| |
| !Mass<ref group="lhf" name="s2" />
| |
| |-
| |
| |style="background:#efefef"|[[Tau (particle)|Tau]]
| |
| |{{SubatomicParticle|Tau}}
| |
| |<math>-1</math>
| |
| |<math>-1/2</math>
| |
| |<math>-1</math>
| |
| |<math>\bold{1}</math>
| |
| |1.78 GeV
| |
| |-
| |
| |style="background:#efefef"|[[Antitau]]
| |
| |{{SubatomicParticle|Antitau}}
| |
| |<math>+1</math>
| |
| |<math>0</math>
| |
| |<math>+2</math>
| |
| |<math>\bold{1}</math>
| |
| |1.78 GeV
| |
| |-
| |
| |style="background:#efefef"|[[Tau neutrino]]
| |
| |{{SubatomicParticle|Tau Neutrino}}
| |
| |<math>0</math>
| |
| |<math>+1/2</math>
| |
| |<math>-1</math>
| |
| |<math>\bold{1}</math>
| |
| |< 0.28 eV<ref group="lhf" name="s4" /><ref group="lhf" name="s4b" />
| |
| |-
| |
| |style="background:#efefef"|[[Tau antineutrino]]
| |
| |{{SubatomicParticle|Tau Antineutrino}}
| |
| |<math>0</math>
| |
| |<math>0</math>
| |
| |<math>0</math>
| |
| |<math>\bold{1}</math>
| |
| |< 0.28 eV<ref group="lhf" name="s4" /><ref group="lhf" name="s4b" />
| |
| |-
| |
| |style="background:#efefef"|[[Top quark]]
| |
| |{{SubatomicParticle|Top Quark}}
| |
| |<math>+2/3</math>
| |
| |<math>+1/2</math>
| |
| |<math>+1/3</math>
| |
| |<math>\bold{3}</math>
| |
| |171 GeV
| |
| |-
| |
| |style="background:#efefef"|[[Top antiquark]]
| |
| |{{SubatomicParticle|Top Antiquark}}
| |
| |<math>-2/3</math>
| |
| |<math>0</math>
| |
| |<math>-4/3</math>
| |
| |<math>\bold{\bar{3}}</math>
| |
| |171 GeV
| |
| |-
| |
| |style="background:#efefef"|[[Bottom quark]]
| |
| |{{SubatomicParticle|Bottom Quark}}
| |
| |<math>-1/3</math>
| |
| |<math>-1/2</math>
| |
| |<math>+1/3</math>
| |
| |<math>\bold{3}</math>
| |
| |~ 4.2 GeV
| |
| |-
| |
| |style="background:#efefef"|[[Bottom antiquark]]
| |
| |{{SubatomicParticle|Bottom Antiquark}}
| |
| |<math>+1/3</math>
| |
| |<math>0</math>
| |
| |<math>+2/3</math>
| |
| |<math>\bold{\bar{3}}</math>
| |
| |~ 4.2 GeV
| |
| |-
| |
| |colspan="8"|
| |
| |}
| |
| <references group="lhf" />
| |
| | |
| ===Free parameters===
| |
| | |
| Upon writing the most general Lagrangian, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment. The parameters are summarized here (note: with the Higgs mass is at 125 GeV, the Higgs self-coupling strength ''λ'' ~ 1/8).
| |
| | |
| {| class="wikitable collapsible collapsed"
| |
| ! colspan="4"| Parameters of the Standard Model
| |
| |-
| |
| ! Symbol
| |
| ! Description
| |
| ! Renormalization<br /> scheme (point)
| |
| ! Value
| |
| |-
| |
| |''m''<sub>e</sub>
| |
| |Electron mass
| |
| |
| |
| |511 keV
| |
| |-
| |
| |''m''<sub>μ</sub>
| |
| |Muon mass
| |
| |
| |
| |105.7 MeV
| |
| |-
| |
| |''m''<sub>τ</sub>
| |
| |Tau mass
| |
| |
| |
| |1.78 GeV
| |
| |-
| |
| |''m''<sub>u</sub>
| |
| |Up quark mass
| |
| |''μ''<sub>[[MSbar scheme|{{overline|MS}}]]</sub> = 2 GeV
| |
| |1.9 MeV
| |
| |-
| |
| |''m''<sub>d</sub>
| |
| |Down quark mass
| |
| |''μ''<sub>{{overline|MS}}</sub> = 2 GeV
| |
| |4.4 MeV
| |
| |-
| |
| |''m''<sub>s</sub>
| |
| |Strange quark mass
| |
| |''μ''<sub>{{overline|MS}}</sub> = 2 GeV
| |
| |87 MeV
| |
| |-
| |
| |''m''<sub>c</sub>
| |
| |Charm quark mass
| |
| |''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>c</sub>
| |
| |1.32 GeV
| |
| |-
| |
| |''m''<sub>b</sub>
| |
| |Bottom quark mass
| |
| |''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>b</sub>
| |
| |4.24 GeV
| |
| |-
| |
| |''m''<sub>t</sub>
| |
| |Top quark mass
| |
| |[[On-shell scheme]]
| |
| |172.7 GeV
| |
| |-
| |
| |''θ''<sub>12</sub>
| |
| |CKM 12-mixing angle
| |
| |
| |
| |13.1°
| |
| |-
| |
| |''θ''<sub>23</sub>
| |
| |CKM 23-mixing angle
| |
| |
| |
| |2.4°
| |
| |-
| |
| |''θ''<sub>13</sub>
| |
| |CKM 13-mixing angle
| |
| |
| |
| |0.2°
| |
| |-
| |
| |''δ''
| |
| |CKM [[CP violation|CP-violating]] Phase
| |
| |
| |
| |0.995
| |
| |-
| |
| |''g''<sub>1</sub> or ''g'''
| |
| |U(1) gauge coupling
| |
| |''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>Z</sub>
| |
| |0.357
| |
| |-
| |
| |''g''<sub>2</sub> or ''g''
| |
| |SU(2) gauge coupling
| |
| |''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>Z</sub>
| |
| |0.652
| |
| |-
| |
| |''g''<sub>3</sub> or ''g''<sub>s</sub>
| |
| |SU(3) gauge coupling
| |
| |''μ''<sub>{{overline|MS}}</sub> = ''m''<sub>Z</sub>
| |
| |1.221
| |
| |-
| |
| |''θ''<sub>QCD</sub>
| |
| |QCD [[vacuum angle]]
| |
| |
| |
| |~0
| |
| |-
| |
| |''v''
| |
| |Higgs vacuum expectation value
| |
| |
| |
| |246 GeV
| |
| |-
| |
| |''m''<sub>H</sub>
| |
| |Higgs mass
| |
| |
| |
| |~ 125 GeV (tentative)
| |
| |}
| |
| | |
| ===Additional symmetries of the Standard Model===
| |
| | |
| From the theoretical point of view, the Standard Model exhibits four additional global symmetries, not postulated at the outset of its construction, collectively denoted '''accidental symmetries''', which are continuous [[U(1)]] [[global symmetry|global symmetries]]. The transformations leaving the Lagrangian invariant are:
| |
| :<math>\psi_\text{q}(x)\rightarrow e^{i\alpha/3}\psi_\text{q}</math>
| |
| :<math>E_L\rightarrow e^{i\beta}E_L\text{ and }(e_R)^c\rightarrow e^{i\beta}(e_R)^c</math>
| |
| :<math>M_L\rightarrow e^{i\beta}M_L\text{ and }(\mu_R)^c\rightarrow e^{i\beta}(\mu_R)^c</math>
| |
| :<math>T_L\rightarrow e^{i\beta}T_L\text{ and }(\tau_R)^c\rightarrow e^{i\beta}(\tau_R)^c.</math>
| |
| The first transformation rule is shorthand meaning that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields <math>M_L</math>, <math>T_L</math> and <math>(\mu_R)^c</math>, <math>(\tau_R)^c</math> are the 2nd (muon) and 3rd (tau) generation analogs of <math>E_L</math> and <math>(e_R)^c</math> fields.
| |
| | |
| By [[Noether's theorem]], each symmetry above has an associated [[conservation law]]: the conservation of [[baryon number]], [[lepton number|electron number]], [[lepton number|muon number]], and [[lepton number|tau number]]. Each quark is assigned a baryon number of <math>{}_{\frac{1}{3}}</math>, while each antiquark is assigned a baryon number of <math>{}_{-\frac{1}{3}}</math>. Conservation of baryon number implies that the number of quarks minus the number of antiquarks is a constant. Within experimental limits, no violation of this conservation law has been found.
| |
| | |
| Similarly, each electron and its associated neutrino is assigned an electron number of +1, while the [[Positron|anti-electron]] and the associated anti-neutrino carry a −1 electron number. Similarly, the muons and their neutrinos are assigned a muon number of +1 and the tau leptons are assigned a tau lepton number of +1. The Standard Model predicts that each of these three numbers should be conserved separately in a manner similar to the way baryon number is conserved. These numbers are collectively known as [[lepton family number]]s (LF).
| |
| | |
| In addition to the accidental (but exact) symmetries described above, the Standard Model exhibits several '''approximate symmetries'''. These are the "SU(2) custodial symmetry" and the "SU(2) or SU(3) quark flavor symmetry."
| |
| | |
| {| class="wikitable collapsible collapsed"
| |
| ! colspan="4" |Symmetries of the Standard Model and Associated Conservation Laws
| |
| |-
| |
| ! [[Symmetry in physics|Symmetry]]
| |
| ! [[Lie Group]]
| |
| ! Symmetry Type
| |
| ! [[Conservation law|Conservation Law]]
| |
| |-
| |
| |[[Poincaré group|Poincaré]]
| |
| | style="text-align:center;"|[[Translational symmetry|Translations]][[Semidirect product|×]][[Lorentz group|SO(3,1)]]
| |
| | style="text-align:center;"|[[Global symmetry]]
| |
| |[[Energy]], [[Momentum]], [[Angular momentum]]
| |
| |-
| |
| |[[Gauge group|Gauge]]
| |
| | style="text-align:center;"|[[SU(3)]]×[[SU(2)]]×[[U(1)]]
| |
| | style="text-align:center;"|[[Local symmetry]]
| |
| |[[Color charge]], [[Weak isospin]], [[Electric charge]], [[Weak hypercharge]]
| |
| |-
| |
| |[[Baryon]] phase
| |
| | style="text-align:center;"|[[U(1)]]
| |
| | style="text-align:center;"|Accidental [[Global symmetry]]
| |
| |[[Baryon number]]
| |
| |-
| |
| |[[Electron]] phase
| |
| | style="text-align:center;"|[[U(1)]]
| |
| | style="text-align:center;"|Accidental [[Global symmetry]]
| |
| |[[Lepton number|Electron number]]
| |
| |-
| |
| |[[Muon]] phase
| |
| | style="text-align:center;"|[[U(1)]]
| |
| | style="text-align:center;"|Accidental [[Global symmetry]]
| |
| |[[Lepton number|Muon number]]
| |
| |-
| |
| |[[Tau (particle)|Tau]] phase
| |
| | style="text-align:center;"|[[U(1)]]
| |
| | style="text-align:center;"|Accidental [[Global symmetry]]
| |
| |[[Lepton number|Tau number]]
| |
| |}
| |
| | |
| ===The U(1) symmetry===
| |
| | |
| For the [[lepton]]s, the gauge group can be written {{math|SU(2)<sub>l</sub> × U(1)<sub>L</sub> × U(1)<sub>R</sub>}}. The two U(1) factors can be combined into {{math|U(1)<sub>Y</sub> × U(1)<sub>l</sub>}} where l is the [[lepton number]]. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group {{math|SU(2)<sub>L</sub> × U(1)<sub>Y</sub>}}. A similar argument in the quark sector also gives the same result for the electroweak theory.
| |
| | |
| ===The charged and neutral current couplings and Fermi theory===
| |
| The charged currents <math>j^{\pm}=j^{1}\pm i j^{2}</math> are
| |
| ::<math>j^+_\mu = \overline U_{iL}\gamma_\mu D_{iL} +\overline \nu_{iL}\gamma_\mu l_{iL}.</math>
| |
| These charged currents are precisely those that entered the [[Fermi theory of beta decay]]. The action contains the charge current piece
| |
| ::<math>\mathcal{L}_{CC} = \frac g{\sqrt2}(j_\mu^+W^{-\mu}+j_\mu^-W^{+\mu}).</math>
| |
| For energy much less than the mass of the W-boson, the effective theory becomes the current–current interaction of the Fermi theory.
| |
| | |
| However, gauge invariance now requires that the component <math>W^{3}</math> of the gauge field also be coupled to a current that lies in the triplet of SU(2). However, this mixes with the U(1), and another current in that sector is needed. These currents must be uncharged in order to conserve charge. So we require the '''neutral currents'''
| |
| :<math>j_\mu^3 = \frac12(\overline U_{iL}\gamma_\mu U_{iL} - \overline D_{iL}\gamma_\mu D_{iL} + \overline \nu_{iL}\gamma_\mu \nu_{iL} - \overline l_{iL}\gamma_\mu l_{iL})</math>
| |
| :<math>j_\mu^{em} = \frac23\overline U_i\gamma_\mu U_i -\frac13\overline D_i\gamma_\mu D_i - \overline l_i\gamma_\mu l_i.</math>
| |
| The neutral current piece in the Lagrangian is then
| |
| :<math>\mathcal{L}_{NC} = e j_\mu^{em} A^\mu + \frac g{\cos\theta_W}(J_\mu^3-\sin^2\theta_WJ_\mu^{em})Z^\mu.</math>
| |
| | |
| ==See also==
| |
| *Overview of [[Standard Model]] of [[particle physics]]
| |
| *[[Fundamental interaction]]
| |
| *[[Noncommutative standard model]]
| |
| *Open questions: [[CP violation]], [[neutrino|Neutrino mass]]es, [[QCD matter|Quark matter]]
| |
| *[[Physics beyond the Standard Model]]
| |
| *[[Strong interaction]]s: [[flavour (particle physics)|Flavour]], [[Quantum chromodynamics]], [[Quark model]]
| |
| *[[Weak interaction]]s: [[Electroweak interaction]], [[Fermi's interaction]]
| |
| *[[Weinberg angle]]
| |
| * [[Symmetry in quantum mechanics]]
| |
| | |
| ==References and external links==
| |
| {{Reflist}}
| |
| *''An introduction to quantum field theory'', by M.E. Peskin and D.V. Schroeder (HarperCollins, 1995) ISBN 0-201-50397-2.
| |
| *''Gauge theory of elementary particle physics'', by T.P. Cheng and L.F. Li (Oxford University Press, 1982) ISBN 0-19-851961-3.
| |
| *[http://nuclear.ucdavis.edu/~tgutierr/files/stmL1.html Standard Model Lagrangian with explicit Higgs terms] (T.D. Gutierrez, ca 1999) (PDF, PostScript, and LaTeX version)
| |
| *''The quantum theory of fields'' (vol 2), by S. Weinberg (Cambridge University Press, 1996) ISBN 0-521-55002-5.
| |
| *''Quantum Field Theory in a Nutshell'' (Second Edition), by A. Zee (Princeton University Press, 2010) ISBN 978-1-4008-3532-4.
| |
| *''An Introduction to Particle Physics and the Standard Model'', by R. Mann (CRC Press, 2010) ISBN 978-1420082982
| |
| | |
| {{DEFAULTSORT:Standard Model (Mathematical Formulation)}}
| |
| [[Category:Particle physics]]
| |
| [[Category:Standard Model]]
| |
| [[Category:Electroweak theory]]
| |
| | |
| [[ca:Model estàndard de la física]]
| |
| [[da:Standardmodellen]]
| |
| [[de:Standardmodell]]
| |
| [[es:Modelo estándar de física de partículas]]
| |
| [[fr:Modèle standard]]
| |
| [[ko:표준모형]]
| |
| [[it:Modello standard (fisica)]]
| |
| [[he:המודל הסטנדרטי]]
| |
| [[nl:Standaardmodel]]
| |
| [[ja:標準模型]]
| |
| [[pl:Model Standardowy]]
| |
| [[ru:Стандартная Модель]]
| |
| [[sv:Standardmodellen]]
| |
| [[vi:Mô hình chuẩn]]
| |
| [[zh:标准模型理论]]
| |