Standard Model

Standard Model

Standard Model, in physics: see elementary particles.

In physics, the combination of two theories of particle physics into a single framework to describe all interactions of subatomic particles except those due to gravity (see gravitation). The two theories, the electroweak theory and the theory of quantum chromodynamics, describe the interactions between particles in terms of the exchange of intermediary particles. The model has proved highly accurate in predicting certain interactions, but it does not explain all aspects of subatomic particles. For example, it cannot say how many particles there should be or what their masses are. The search goes on for a more complete theory, and in particular a unified field theory describing the strong, weak, and electromagnetic forces.

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The Standard Model of particle physics is a theory that describes three of the four known fundamental interactions together with the elementary particles that take part in these interactions. These particles make up all matter in the universe except for the dark matter. The standard model is a non-abelian gauge theory of the electroweak and strong interactions with the symmetry group SU(3)×SU(2)×U(1). To date, almost all experimental tests of the three forces described by the Standard Model have agreed with its predictions.

The Standard Model falls short of being a complete theory of fundamental interactions, primarily because of its lack of inclusion of gravity, the fourth known fundamental interaction.

The recent observation of neutrino oscillations will result in certain modifications of some of the parameters of the standard model.

Historical background

The formulation of the unification of the electromagnetic and weak interactions in the Standard Model is due to Steven Weinberg, Abdus Salam and, subsequently, Sheldon Glashow. The unification model was initially proposed by Steven Weinberg in 1967, and completed integrating it with the proposals by P. Higgs, G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, and F. Englert and R. Brout of spontaneous symmetry breaking which gives origin to the masses of all particles described in the model.

After the discovery, made at CERN of the existence of neutral weak currents, mediated by the boson, foreseen in the Standard Model, Glashow, Salam, and Weinberg received the Nobel Prize in Physics in 1979.

Overview

In physics, the dynamics of both matter and energy in nature is presently best understood in terms of the kinematics and interactions of fundamental particles. To date, science has managed to reduce the laws which seem to govern the behavior and interaction of all types of matter and energy we are aware of, to a small core of fundamental laws and theories. A major goal of physics is to find the 'common ground' that would unite all of these into one integrated model of everything, in which all the other laws we know of would be special cases, and from which the behavior of all matter and energy can be derived (at least in principle). "Details can be worked out if the situation is simple enough for us to make an approximation, which is almost never, but often we can understand more or less what is happening." (Feynman's lectures on Physics, Vol 1. 2–7)

The standard model is a grouping of two major theories — quantum electroweak and quantum chromodynamics — which provides an internally consistent theory describing interactions between all experimentally observed particles. Technically, quantum field theory provides the mathematical framework for the standard model. The standard model describes each type of particle in terms of a mathematical field. For a technical description of the fields and their interactions, see standard model (mathematical formulation).

Particle Content

The particles of the standard model are organized into three classes according to their spin: fermions (spin-½ particles of matter), gauge bosons (spin-1 force-mediating particles), and the (spin-0) Higgs boson.

Particles of matter

All fermions in the Standard Model are spin-½, and follow the Pauli Exclusion Principle in accordance with the spin-statistics theorem.

Organization of Fermions
  Generation 1 Generation 2 Generation 3
Quarks Up
Charm
Top
Down
Strange
Bottom
Leptons Electron
Neutrino
Muon
Neutrino
Tau
Neutrino
Electron Muon Tau
Apart from their antiparticle partners, a total of twelve different fermions are known and accounted for. They are classified according to how they interact (or equivalently, what charges they carry): six of them are classified as quarks (up, down, charm, strange, top, bottom), and the other six as leptons (electron, muon, tau, and their corresponding neutrinos).

Pairs from each classification are grouped together to form a generation, with corresponding particles exhibiting similar physical behavior (see table of fermions).

The defining property of the quarks is that they carry color charge, and hence, interact via the strong force. The infrared confining behavior of the strong force results in the quarks being perpetually bound to one another forming color-neutral composite particles (hadrons) of either two quarks (mesons) or three quarks (baryons). The familiar proton and the neutron are examples of the two lightest baryons. Quarks also carry electric charges and weak isospin. Hence they interact with other fermions electromagnetically and via the weak nuclear interactions.

The remaining six fermions that do not carry color charge are defined to be the leptons. The three neutrinos do not carry electric charge either, so their motion is directly influenced only by means of the weak nuclear force. For this reason neutrinos are notoriously difficult to detect in laboratories. However, the electron, muon and the tau lepton carry an electric charge so they interact electromagnetically, too.

Force mediating particles

Forces in physics are the ways that particles interact and influence each other. At a macro level, the electromagnetic force allows particles to interact with one another via electric and magnetic fields, and the force of gravitation allows two particles with mass to attract one another in accordance with Newton's Law of Gravitation. The standard model explains such forces as resulting from matter particles exchanging other particles, known as force mediating particles. When a force mediating particle is exchanged, at a macro level the effect is equivalent to a force influencing both of them, and the particle is therefore said to have mediated (i.e., been the agent of) that force. Force mediating particles are believed to be the reason why the forces and interactions between particles observed in the laboratory and in the universe exist.

The known force mediating particles described by the Standard Model also all have spin (as do matter particles), but in their case, the value of the spin is 1, meaning that all force mediating particles are bosons. As a result, they do not follow the Pauli Exclusion Principle. The different types of force mediating particles are described below.

  • Photons mediate the electromagnetic force between electrically charged particles. The photon is massless and is well-described by the theory of quantum electrodynamics.
  • The , , and gauge bosons mediate the weak interactions between particles of different flavors (all quarks and leptons). They are massive, with the being more massive than the . The weak interactions involving the act on exclusively left-handed particles and right-handed antiparticles. Furthermore, the carry an electric charge of +1 and −1 and couple to the electromagnetic interactions. The electrically neutral boson interacts with both left-handed particles and antiparticles. These three gauge bosons along with the photons are grouped together which collectively mediate the electroweak interactions.
  • The eight gluons mediate the strong interactions between color charged particles (the quarks). Gluons are massless. The eightfold multiplicity of gluons is labeled by a combination of color and an anticolor charge (e.g., red–antigreen). Because the gluon has an effective color charge, they can interact among themselves. The gluons and their interactions are described by the theory of quantum chromodynamics.

The interactions between all the particles described by the Standard Model are summarized in the illustration immediately above and to the right.

Force Mediating Particles
Electromagnetic Force Weak Nuclear Force Strong Nuclear Force
Photon , , and
Gauge Bosons
, , Gluons

The Higgs boson

The Higgs particle is a hypothetical massive scalar elementary particle predicted by the Standard Model, and the only fundamental particle predicted by that model which has not been directly observed as yet. This is because it requires an exceptionally large amount of energy and beam luminosity to create and observe at high energy colliders. It has no intrinsic spin, and thus, (like the force mediating particles, which also have integral spin) is also classified as a boson.

The Higgs boson plays a unique role in the Standard Model, and a key role in explaining the origins of the mass of other elementary particles, in particular the difference between the massless photon and the very heavy W and Z bosons. Elementary particle masses, and the differences between electromagnetism (caused by the photon) and the weak force (caused by the W and Z bosons), are critical to many aspects of the structure of microscopic (and hence macroscopic) matter. In electroweak theory it generates the masses of the massive leptons (electron, muon and tau); and also of the quarks.

As of September 2008, no experiment has directly detected the existence of the Higgs boson, but there is some indirect evidence for it. It is hoped that upon the completion of the Large Hadron Collider, experiments conducted at CERN would bring experimental evidence confirming the existence of the particle.

Science, a journal of original scientific research, has reported: "...experimenters may have already overlooked a Higgs particle, argues theorist Chien-Peng Yuan of Michigan State University in East Lansing and his colleagues. They considered the simplest possible supersymmetric theory. Ordinarily, theorists assume that the lightest of theory's five Higgses is the one that drags on the W and Z. Those interactions then feed back on Higgs and push its mass above 121 times the mass of the proton, the highest mass searched for at CERN's Large Electron–Positron (LEP) collider, which ran from 1989 to 2000. But it's possible that the lightest Higgs weighs as little as 65 times the mass of a proton and has been missed, Yuan and colleagues argue in a paper to be published in Physical Review Letters`.

Theoretical Aspects

Construction of the Standard Model Lagrangian

Parameters of the Standard Model
Symbol Description Renormalization
scheme (point)
Value
m_e Electron mass 511 keV
m_mu Muon mass 106 MeV
m_tau Tau lepton mass 1.78 GeV
m_u Up quark mass (mu_{overline{text{MS}}}=2text{ GeV}) 1.9 MeV
m_d Down quark mass (mu_{overline{text{MS}}}=2text{ GeV}) 4.4 MeV
m_s Strange quark mass (mu_{overline{text{MS}}}=2text{ GeV}) 87 MeV
m_c Charm quark mass (mu_{overline{text{MS}}}=m_c) 1.32 GeV
m_b Bottom quark mass (mu_{overline{text{MS}}}=m_b) 4.24 GeV
m_t Top quark mass (on-shell scheme) 172.7 GeV
theta_{12} CKM 12-mixing angle 0.229
theta_{23} CKM 23-mixing angle 0.042
theta_{13} CKM 13-mixing angle 0.004
delta CKM CP-Violating Phase 0.995
g_1 U(1) gauge coupling (mu_{overline{text{MS}}}=M_text{Z}) 0.357
g_2 SU(2) gauge coupling (mu_{overline{text{MS}}}=M_text{Z}) 0.652
g_3 SU(3) gauge coupling (mu_{overline{text{MS}}}=M_text{Z}) 1.221
theta_text{QCD} QCD Vacuum Angle ~0
mu Higgs quadratic coupling Unknown
lambda Higgs self-coupling strength Unknown
Technically, quantum field theory provides the mathematical framework for the standard model, in which a Lagrangian controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical field that pervades space-time. The construction of the standard model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.

The global Poincaré symmetry is postulated for all relativistic quantum field theories. It consists of the familiar translational symmetry, rotational symmetry and the inertial reference frame invariance central to the theory of special relativity. The local SU(3)timesSU(2)timesU(1) gauge symmetry is an internal symmetry that essentially defines the standard model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. The fields fall into different representations of the various symmetry groups of the Standard Model (see table). 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 in the table at right.

The QCD sector

The electroweak sector

The electroweak sector is a Yang-Mills gauge theory with the symmetry group U(1)times SU(2)_L,
mathcal{L}_{EW}=sum_psibarpsigamma^muleft( ipartial_mu-g'{1over2}Y_WB_mu -g{1over2}vectau_Lvec W_mu right)psi +mathcal{L}_{YM}(B_mu) +mathcal{L}_{YM}(vec W_mu), where B_mu is the U(1) gauge field; Y_W is the weak hypercharge — the generator of the U(1) group; vec{W}_mu is the three-component SU(2) gauge field; vec{tau}_L are the Pauli matrices — infinitesimal generators of the SU(2) group, the subscript {}_L indicates that they only act on left fermions; g' and g are coupling constants.

The Higgs sector

In the standard model the Higgs field is a complex spinor of the group SU(2)_L,

varphi={1oversqrt{2}} left( begin{array}{c} varphi^+ varphi^0 end{array} right);, where the indexes {}^+ and {}^0 indicate the Q-charges of the components; the Y_W-charge of both components is equal 1.

Before the symmetry breaking the Higgs Lagrangian is given as

mathcal{L}_H=varphi^dagger left( stackrel{leftarrow}{partial_mu}-ig'{1over2}Y_WB_mu -ig{1over2}vectauvec W_mu right) left( partial_mu+ig'{1over2}Y_WB_mu +ig{1over2}vectauvec W_mu right)varphi -{lambda^2over4}left(varphi^daggervarphi-v^2right)^2

Additional Symmetries of the Standard Model

From the theoretical point of view, the standard model exhibits additional global symmetries that were not postulated at the outset of its construction. There are four such symmetries and are collectively called accidental symmetries, all of which are continuous U(1) global symmetries. The transformations leaving the Lagrangian invariant are
psi_text{q}(x)rightarrow e^{ialpha/3}psi_text{q}
E_Lrightarrow e^{ibeta}E_Ltext{ and }(e_R)^crightarrow e^{ibeta}(e_R)^c
M_Lrightarrow e^{ibeta}M_Ltext{ and }(mu_R)^crightarrow e^{ibeta}(mu_R)^c
T_Lrightarrow e^{ibeta}T_Ltext{ and }(tau_R)^crightarrow e^{ibeta}(tau_R)^c.
The first transformation rule is shorthand to mean that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields M_L, T_L and (mu_R)^c, (tau_R)^c are the 2nd (muon) and 3rd (tau) generation analogs of E_L and (e_R)^c fields.

By Noether's theorem, each of these symmetries yields an associated conservation law. They are the conservation of baryon number, electron number, muon number, and tau number. Each quark carries 1/3 of a baryon number, while each antiquark carries -1/3 of a baryon number. The conservation law implies that the total number of quarks minus number of antiquarks stays constant throughout time. Within experimental limits, no violation of this conservation law has been found.

Similarly, each electron and its associated neutrino carries +1 electron number, while the antielectron and the associated antineutrino carry -1 electron number, the muons carry +1 muon number and the tau leptons carry +1 tau number. The standard model predicts that each of these three numbers should be conserved separately in a manner similar to the baryon number. These numbers are collectively known as lepton family numbers (LF). The difference in the symmetry structures between the quark and the lepton sectors is due to the masslessness of neutrinos in the standard model. However, it was recently found that neutrinos have small mass, and oscillate between flavors, signaling the violation of these three quantum numbers.

In addition to the accidental (but exact) symmetries described above, the standard model exhibits a set of approximate symmetries. These are the SU(2) Custodial Symmetry and the SU(2) or SU(3) quark flavor symmetry.

Symmetries of the Standard Model and Associated Conservation Laws
Symmetry Lie Group Symmetry Type Conservation Law
Poincaré TranslationsrtimesSO(3,1) Global symmetry Energy, Momentum, Angular momentum
Gauge SU(3)timesSU(2)timesU(1) Local symmetry Electric charge, Weak isospin, Color charge
Baryon phase U(1) Accidental Global symmetry Baryon number
Electron phase U(1) Accidental Global symmetry Electron number
Muon phase U(1) Accidental Global symmetry Muon number
Tau phase U(1) Accidental Global symmetry Tau-lepton number

Field content of the Standard Model
Field
(1st generation)
Spin Gauge Group
Representation
Baryon
Number
Electron
Number
Left-handed quark Q_text{L} 1/2 (mathbf{3}, mathbf{2}, +1/6) 1/3 0
Right-handed up quark (u_text{R})^c 1/2 (barmathbf{3}, mathbf{1}, -2/3) -1/3 0
Right-handed down quark (d_text{R})^c 1/2 (barmathbf{3}, mathbf{1}, +1/3) -1/3 0
Left-handed lepton E_text{L} 1/2 (mathbf{1}, mathbf{2}, -1/2) 0 1
Right-handed electron (e_text{R})^c 1/2 (mathbf{1}, mathbf{1}, +1) 0 -1
Hypercharge gauge field B_mu 1 (barmathbf{1}, mathbf{1}, 0) 0 0
Isospin gauge field W_mu 1 (mathbf{1}, mathbf{3}, 0) 0 0
Gluon field G_mu 1 (mathbf{8}, mathbf{1}, 0) 0 0
Higgs field H 0 (mathbf{1}, mathbf{2}, +1/2) 0 0

List of standard model fermions

This table is based in part on data gathered by the Particle Data Group ().

Left-handed fermions in the Standard Model
Generation 1
Fermion
(left-handed)
Symbol Electric
charge
Weak
isospin
Weak
hypercharge
Color
charge
 *
Mass **
Electron e^-, -1, -1/2, -1, bold{1}, 511 keV
Positron e^+, +1, 0, +2, bold{1}, 511 keV
Electron-neutrino nu_e, 0, +1/2, -1, bold{1}, < 2 eV ****
Up quark u, +2/3, +1/2, +1/3, bold{3}, ~ 3 MeV ***
Up antiquark bar{u}, -2/3, 0, -4/3, bold{bar{3}}, ~ 3 MeV ***
Down quark d, -1/3, -1/2, +1/3, bold{3}, ~ 6 MeV ***
Down antiquark bar{d}, +1/3, 0, +2/3, bold{bar{3}}, ~ 6 MeV ***
 
Generation 2
Fermion
(left-handed)
Symbol Electric
charge
Weak
isospin
Weak
hypercharge
Color
charge *
Mass **
Muon mu^-, -1, -1/2, -1, bold{1}, 106 MeV
Antimuon mu^+, +1, 0, +2, bold{1}, 106 MeV
Muon-neutrino nu_mu, 0, +1/2, -1, bold{1}, < 2 eV ****
Charm quark c, +2/3, +1/2, +1/3, bold{3}, ~ 1.337 GeV
Charm antiquark bar{c}, -2/3, 0, -4/3, bold{bar{3}}, ~ 1.3 GeV
Strange quark s, -1/3, -1/2, +1/3, bold{3}, ~ 100 MeV
Strange antiquark bar{s}, +1/3, 0, +2/3, bold{bar{3}}, ~ 100 MeV
 
Generation 3
Fermion
(left-handed)
Symbol Electric
charge
Weak
isospin
Weak
hypercharge
Color
charge *
Mass **
Tau lepton tau^-, -1, -1/2, -1, bold{1}, 1.78 GeV
Anti-tau lepton tau^+, +1, 0, +2, bold{1}, 1.78 GeV
Tau-neutrino nu_tau, 0, +1/2, -1, bold{1}, < 2 eV ****
Top quark t, +2/3, +1/2, +1/3, bold{3}, 171 GeV
Top antiquark bar{t}, -2/3, 0, -4/3, bold{bar{3}}, 171 GeV
Bottom quark b, -1/3, -1/2, +1/3, bold{3}, ~ 4.2 GeV
Bottom antiquark bar{b}, +1/3, 0, +2/3, bold{bar{3}}, ~ 4.2 GeV
Notes:

  • * These are not ordinary abelian charges, which can be added together, but are labels of group representations of Lie groups.
  • ** 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 basis or to suggest a left-handed electron antineutrino.
  • *** The masses of baryons and hadrons and various cross-sections are the experimentally measured quantities. Since quarks can't be isolated because of QCD confinement, the quantity here is supposed to be the mass of the quark at the renormalization scale of the QCD scale.
  • **** The Standard Model assumes that neutrinos are massless. However, several contemporary experiments prove that neutrinos oscillate between their 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 eigenstates are still open. See Neutrino#Mass.

Tests and predictions

The Standard Model predicted the existence of W and Z bosons, the gluon, the top quark and the charm quark before these particles had been observed. Their predicted properties were experimentally confirmed with good precision.

The Large Electron-Positron Collider at CERN tested various predictions about the decay of Z bosons, and found them confirmed.

To get an idea of the success of the Standard Model a comparison between the measured and the predicted values of some quantities are shown in the following table:

Quantity Measured (GeV) SM prediction (GeV)
Mass of W boson 80.398±0.025 80.3900±0.0180
Mass of Z boson 91.1876±0.0021 91.1874±0.0021

Challenges to the standard model

The Standard Model of particle physics has been empirically determined through experiments over the past fifty years. Currently the Standard Model predicts that there is one more particle to be discovered, the Higgs boson. One of the reasons for building the Large Hadron Collider is that the increase in energy is expected to make the Higgs observable. However, as of August 2008, there are only indirect experimental indications for the existence of the Higgs boson and it can not be claimed to be found.

There has been a great deal of both theoretical and experimental research exploring whether the Standard Model could be extended into a complete theory of everything. This area of research is often described by the term 'Beyond the Standard Model'. There are several motivations for this research. First, the Standard Model does not attempt to explain gravity, and it is unknown how to combine quantum field theory which is used for the Standard Model with general relativity which is the best physical model of gravity. This means that there is not a good theoretical model for phenomena such as the early universe.

Another avenue of research is related to the fact that the standard model seems very ad-hoc and inelegant. For example, the theory contains many seemingly unrelated parameters of the theory — 21 in all (18 parameters in the core theory, plus G, c and h; there are believed to be an additional 7 or 8 parameters required for the neutrino masses, although neutrino masses are outside the standard model and the details are unclear). Research also focuses on the Hierarchy problem (why the weak scale and Planck scale are so disparate), and attempts to reconcile the emerging Standard Model of Cosmology with the Standard Model of particle physics. Many questions relate to the initial conditions that led to the presently observed Universe. Examples include: Why is there a matter/antimatter asymmetry? Why is the Universe isotropic and homogeneous at large distances?

See also

Notes

References

Introductory textbooks

  • Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 0-471-60386-4.
  • D.A. Bromley (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-67672-4.
  • Gordon L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5.

Advanced textbooks

  • Cheng, Ta Pei; Li, Ling Fong Gauge theory of elementary particle physics. Oxford University Press. ISBN 0-19-851961-3.
  • :— introduction to all aspects of gauge theories and the Standard Model.
  • Donoghue, J. F.; Golowich, E.; Holstein, B. R. Dynamics of the Standard Model. Cambridge University Press. ISBN 978-0521476522.
  • :— highlights dynamical and phenomenological aspects of the Standard Model.
  • O'Raifeartaigh, L. Group structure of gauge theories. Cambridge University Press. ISBN 0-521-34785-8.
  • :— highlights group-theoretical aspects of the Standard Model.

Journal articles

  • S.F. Novaes, Standard Model: An Introduction, hep-ph/0001283
  • D.P. Roy, Basic Constituents of Matter and their Interactions — A Progress Report, hep-ph/9912523
  • Y. Hayato et al., Search for Proton Decay through p → νK+ in a Large Water Cherenkov Detector. Phys. Rev. Lett. 83, 1529 (1999).
  • Ernest S. Abers and Benjamin W. Lee, Gauge theories. Physics Reports (Elsevier) C9, 1–141 (1973).

External links

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