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Standard Model in Physics[edit]

The Standard Model of elementary particles, with the gauge bosons in the rightmost column.
Summary of interactions between particles described by the Standard Model.


The standard model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the early and middle 20th century, the current formulation was finalized in the mid 1970s upon experimental confirmation of the existence of quarks. Since then, discoveries of the bottom quark (1977), the top quark (1995) and the tau neutrino (2000) have given credence to the standard model. Because of its success in explaining a wide variety of experimental results, the standard model is sometimes regarded as a theory of almost everything.

Still, the standard model falls short of being a complete theory of fundamental interactions because it does not incorporate the physics of general relativity, such as gravitation and dark energy. The theory does not contain any viable dark matter particle that possesses all of the required properties deduced from observational cosmology. It also does not correctly account for neutrino oscillations (and their non-zero masses). Although the standard model is theoretically self-consistent, it has several unnatural properties giving rise to puzzles like the strong CP problem and the hierarchy problem.

Nevertheless, the standard model is important to theoretical and experimental particle physicists alike. For theoreticians, the standard model is a paradigm example of a quantum field theory, which exhibits a wide range of physics including spontaneous symmetry breaking, anomalies, non-perturbative behavior, etc. It is used as a basis for building more exotic models which incorporate hypothetical particles, extra dimensions and elaborate symmetries (such as supersymmetry) in an attempt to explain experimental results at variance with the standard model such as the existence of dark matter and neutrino oscillations. In turn, the experimenters have incorporated the standard model into simulators to help search for new physics beyond the standard model from relatively uninteresting background.

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)×SU(2)×U(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.

Recently, the standard model has found applications in other fields besides particle physics such as astrophysics and cosmology, in addition to nuclear physics.

Historical background[edit]

The first step towards the Standard Model was Sheldon Glashow's discovery, in 1960, of a way to combine the electromagnetic and weak interactions.[1] In 1967, Steven Weinberg[2] and Abdus Salam[3] incorporated the Higgs mechanism[4][5][6] into Glashow's electroweak theory, giving it its modern form.

The Higgs mechanism is believed to give rise to the masses of all the elementary particles in the Standard Model. This includes the masses of the W and Z bosons, and the masses of the fermions - i.e. the quarks and leptons.

After the neutral weak currents caused by
Z
boson exchange were discovered at CERN in 1973,[7][8][9][10] the electroweak theory became widely accepted and Glashow, Salam, and Weinberg shared the 1979 Nobel Prize in Physics for discovering it. The W and Z bosons were discovered experimentally in 1981, and their masses were found to be as the Standard Model predicted.

The theory of the strong interaction, to which many contributed, acquired its modern form around 1973–74, when experiments confirmed that the hadrons were composed of fractionally charged quarks.

Overview[edit]

At present, matter and energy are best understood in terms of the kinematics and interactions of elementary particles. To date, physics has reduced the laws governing the behavior and interaction of all known forms of matter and energy to a small set of fundamental laws and theories. A major goal of physics is to find the "common ground" that would unite all of these theories into one integrated theory of everything, of which all the other known laws would be special cases, and from which the behavior of all matter and energy could be derived (at least in principle).[11]

The Standard Model groups two major extant theories—quantum electroweak and quantum chromodynamics—into an internally consistent theory that describes the interactions between all known particles in terms of quantum field theory. For a technical description of the fields and their interactions, see Standard Model (mathematical formulation).

Particle content[edit]

Fermions[edit]

Organization of Fermions
  Charge First generation Second generation Third generation
Quarks +23 Up

u
Charm

c
Top

t
13 Down

d
Strange

s
Bottom

b
Leptons −1 Electron
e
Muon
μ
Tau
τ
0 Electron neutrino
ν
e
Muon neutrino
ν
μ
Tau neutrino
ν
τ

The Standard Model includes 12 elementary particles of spin-12 known as fermions. According to the spin-statistics theorem, fermions respect the Pauli exclusion principle. Each fermion has a corresponding antiparticle.

The fermions of the Standard Model are classified according to how they interact (or equivalently, by what charges they carry). There are six quarks (up, down, charm, strange, top, bottom), and six leptons (electron, electron neutrino, muon, muon neutrino, tau, tau neutrino). Pairs from each classification are grouped together to form a generation, with corresponding particles exhibiting similar physical behavior (see table).

The defining property of the quarks is that they carry color charge, and hence, interact via the strong interaction. A phenomenon called color confinement results in quarks being perpetually (or at least since very soon after the start of the Big Bang) bound to one another, forming color-neutral composite particles (hadrons) containing either a quark and an antiquark (mesons) or three quarks (baryons). The familiar proton and the neutron are the two baryons having the smallest mass. Quarks also carry electric charge and weak isospin. Hence they interact with other fermions both electromagnetically and via the weak nuclear interaction.

The remaining six fermions do not carry color charge and are called leptons. The three neutrinos do not carry electric charge either, so their motion is directly influenced only by the weak nuclear force, which makes them notoriously difficult to detect. However, by virtue of carrying an electric charge, the electron, muon, and tau all interact electromagnetically.

Each member of a generation has greater mass than the corresponding particles of lower generations. The first generation charged particles do not decay; hence all ordinary (baryonic) matter is made of such particles. Specifically, all atoms consist of electrons orbiting atomic nuclei ultimately constituted of up and down quarks. Second and third generations charged particles, on the other hand, decay with very short half lives, and are observed only in very high-energy environments. Neutrinos of all generations also do not decay, and pervade the universe, but rarely interact with baryonic matter.

Gauge bosons[edit]

In the Standard Model, gauge bosons are force carriers that mediate the strong, weak, and electromagnetic fundamental interactions.

Interactions in physics are the ways that particles influence other particles. At a macroscopic level, electromagnetism allows particles to interact with one another via electric and magnetic fields, and gravitation allows particles with mass to attract one another in accordance with Einstein's general relativity. The standard model explains such forces as resulting from matter particles exchanging other particles, known as force mediating particles (Strictly speaking, this is only so if interpreting literally what is actually an approximation method known as perturbation theory, as opposed to the exact theory[citation needed]). When a force mediating particle is exchanged, at a macroscopic 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. The Feynman diagram calculations, which are a graphical form of the perturbation theory approximation, invoke "force mediating particles" and when applied to analyze high-energy scattering experiments are in reasonable agreement with the data. Perturbation theory (and with it the concept of "force mediating particle") in other situations fails. These include low-energy QCD, bound states, and solitons.

The gauge bosons of the Standard Model also all have spin (as do matter particles), but in their case, the value of the spin is 1, making them bosons. As a result, they do not follow the Pauli exclusion principle. The different types of gauge bosons 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
    W+
    ,
    W
    , and
    Z
    gauge bosons mediate the weak interactions between particles of different flavors (all quarks and leptons). They are massive, with the
    Z
    being more massive than the
    W±
    . The weak interactions involving the
    W±
    act on exclusively left-handed particles and right-handed antiparticles. Furthermore, the
    W±
    carry an electric charge of +1 and −1 and couple to the electromagnetic interactions. The electrically neutral
    Z
    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).[nb 1] 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 by the diagram at the top of this section.

Higgs boson[edit]

The Higgs particle is a hypothetical massive scalar elementary particle theorized by Robert Brout, François Englert, Peter Higgs, Gerald Guralnik, C. R. Hagen, and Tom Kibble in 1964 (see 1964 PRL symmetry breaking papers) and is a key building block in the Standard Model.[12][13][14][15] It has no intrinsic spin, and for that reason is classified as a boson (like the gauge bosons, which have integer spin). Because an exceptionally large amount of energy and beam luminosity are theoretically required to observe a Higgs boson in high energy colliders, it is the only fundamental particle predicted by the Standard Model that has yet to be observed.

The Higgs boson plays a unique role in the Standard Model, by explaining why the other elementary particles, the photon and gluon excepted, are massive. In particular, the Higgs boson would explain why the photon has no mass, while the W and Z bosons are very heavy. Elementary particle masses, and the differences between electromagnetism (mediated by the photon) and the weak force (mediated by the W and Z bosons), are critical to many aspects of the structure of microscopic (and hence macroscopic) matter. In electroweak theory, the Higgs boson generates the masses of the leptons (electron, muon, and tau) and quarks.

As yet, no experiment has directly detected the existence of the Higgs boson. It is hoped that the Large Hadron Collider at CERN will confirm the existence of this particle. It is also possible that the Higgs boson may already have been produced but overlooked.[16]

Field content[edit]

The standard model has the following fields:

Spin 1[edit]

  1. A U(1) gauge field Bμν with coupling g′ (weak U(1), or weak hypercharge)
  2. An SU(2) gauge field Wμν with coupling g (weak SU(2), or weak isospin)
  3. An SU(3) gauge field Gμν with coupling gs (strong SU(3), or color charge)

Spin 12[edit]

The spin 12 particles are in representations of the gauge groups. For the U(1) group, we list the value of the weak hypercharge instead. The left-handed fermionic fields are:

  1. An SU(3) triplet, SU(2) doublet, with U(1) weak hypercharge 13 (left-handed quarks)
  2. An SU(3) triplet, SU(2) singlet, with U(1) weak hypercharge 23 (left-handed down-type antiquark)
  3. An SU(3) singlet, SU(2) doublet with U(1) weak hypercharge −1 (left-handed lepton)
  4. An SU(3) triplet, SU(2) singlet, with U(1) weak hypercharge −43 (left-handed up-type antiquark)
  5. An SU(3) singlet, SU(2) singlet with U(1) weak hypercharge 2 (left-handed antilepton)

By CPT symmetry, there is a set of right-handed fermions with the opposite quantum numbers.

This describes one generation of leptons and quarks, and there are three generations, so there are three copies of each field. 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.

Spin 0[edit]

  1. An SU(2) doublet H with U(1) hyper-charge −1 (Higgs field)

Note that |H|2, summed over the two SU(2) components, is invariant under both SU(2) and under U(1), and so it can appear as a renormalizable term in the Lagrangian, as can its square.

This field acquires a vacuum expectation value, leaving a combination of the weak isospin, I3, and weak hypercharge unbroken. This is the electromagnetic gauge group, and the photon remains massless. The standard formula for the electric charge (which defines the normalization of the weak hypercharge, Y, which would otherwise be somewhat arbitrary) is:[nb 2]

Lagrangian[edit]

The Lagrangian for the spin 1 and spin 12 fields is the most general renormalizable gauge field Lagrangian with no fine tunings:

  • Spin 1:

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).

The spin-12 particles can have no mass terms because there is no right/left helicity pair with the same SU(2) and SU(3) representation and the same weak hypercharge. This means that if the gauge charges were conserved in the vacuum, none of the spin 12 particles could ever swap helicity, and they would all be massless.

For a neutral fermion, for example a hypothetical right-handed lepton N (or Nα in relativistic two-spinor notation), with no SU(3), SU(2) representation and zero charge, it is possible to add the term:

This term gives the neutral fermion a Majorana mass. Since the generic value for M will be of order 1, such a particle would generically be unacceptably heavy. The interactions are completely determined by the theory – the leptons introduce no extra parameters.

Higgs mechanism[edit]

The Lagrangian for the Higgs includes the most general renormalizable self interaction:

The parameter v2 has dimensions of mass squared, and it gives the location where the classical Lagrangian is at a minimum. In order for the Higgs mechanism to work, v2 must be a positive number. v 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.

It is traditional to choose the SU(2) gauge so that the Higgs doublet in the vacuum has expectation value (v,0).

Masses and CKM matrix[edit]

The rest of the interactions are the most general spin-0 spin-12 Yukawa interactions, and there are many of these. These constitute most of the free parameters in the model. The Yukawa couplings generate the masses and mixings once the Higgs gets its vacuum expectation value.

The terms L*HR generate a mass term for each of the three generations of leptons. There are 9 of these terms, but by relabeling L and R, the matrix can be diagonalized. Since only the upper component of H is nonzero, the upper SU(2) component of L mixes with R to make the electron, the muon, and the tau, leaving over a lower massless component, the neutrino. {Neutrino oscillation show neutrinos have mass. http://operaweb.lngs.infn.it/spip.php?rubrique14 31May2010 Press Release.}

The terms QHU generate up masses, while QHD generate down masses. But since there is more than one right-handed singlet in each generation, it is not possible to diagonalize both with a good basis for the fields, and there is an extra CKM matrix.


The QCD sector[edit]

The QCD sector defines the interactions between quarks and gluons, with SU(3) symmetry, generated by Ta. Since leptons do not interact with gluons, they are not affected by this sector.

is the gluon field strength, are the Dirac matrices, D stands for the isospin doublet section, U stands for a unitary matrix, and gs is the strong coupling constant.

The electroweak sector[edit]

The electroweak sector is a Yang–Mills gauge theory with the symmetry group U(1)×SU(2)L,

where Bμ is the U(1) gauge field; YW is the weak hypercharge—the generator of the U(1) group; is the three-component SU(2) gauge field; 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[edit]

In the Standard Model, the Higgs field is a complex spinor of the group SU(2)L:

where the indexes + and 0 indicate the electric charge (Q) of the components. The weak isospin (YW) of both components is 1.

Before symmetry breaking, the Higgs Lagrangian is:

which can also be written as:

Additional symmetries of the Standard Model[edit]

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 symmetries. The transformations leaving the Lagrangian invariant are:

The first transformation rule is shorthand meaning that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields , and , are the 2nd (muon) and 3rd (tau) generation analogs of and fields.

By Noether's theorem, each symmetry above has an associated conservation law: the conservation of baryon number, electron number, muon number, and tau number. Each quark is assigned a baryon number of 1/3, while each antiquark is assigned a baryon number of -1/3. 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 antielectron and the associated antineutrino carry −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 numbers (LF). Symmetry works differently for quarks than for leptons, mainly because the Standard Model predicts that neutrinos are massless. However, it was recently found that neutrinos have small masses and oscillate between flavors, signaling that the conservation of lepton family number is violated.

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."

Tests and predictions[edit]

The Standard Model (SM) predicted the existence of the W and Z bosons, gluon, and the top and charm quarks before these particles were observed. Their predicted properties were experimentally confirmed with good precision. To give an idea of the success of the SM, the following table compares the measured masses of the W and Z bosons with the masses predicted by the SM:

Quantity Measured (GeV) SM prediction (GeV)
Mass of W boson 80.398 ± 0.025 80.390 ± 0.018
Mass of Z boson 91.1876 ± 0.0021 91.1874 ± 0.0021

The SM also makes several predictions about the decay of Z bosons, which have been experimentally confirmed by the Large Electron-Positron Collider at CERN.

Challenges to the standard model[edit]

Unsolved Physics problems[edit]

  • What gives rise to the Standard Model of particle physics?
  • Why do particle masses and coupling constants have the values that we measure?
  • Does the Higgs boson really exist?
  • Why are there three generations of particles?


There is some experimental evidence consistent with neutrinos having mass, which the Standard Model does not allow.[17] To accommodate such findings, the Standard Model can be modified by adding a non-renormalizable interaction of lepton fields with the square of the Higgs field. This is natural in certain grand unified theories, and if new physics appears at about 1016 GeV, the neutrino masses are of the right order of magnitude.

Currently, there is one elementary particle predicted by the Standard Model that has yet to be observed: the Higgs boson. A major reason for building the Large Hadron Collider is that the high energies of which it is capable are expected to make the Higgs observable. However, as of August 2008, there is only indirect empirical evidence for the existence of the Higgs boson, so that its discovery cannot be claimed. Moreover, there are serious theoretical reasons for supposing that elementary scalar Higgs particles cannot exist (see Quantum triviality).

A fair amount of theoretical and experimental research has attempted to extend the Standard Model into a Unified Field Theory or a Theory of everything, a complete theory explaining all physical phenomena including constants. Inadequacies of the Standard Model that motivate such research include:

  • It does not attempt to explain gravitation, and unlike for the strong and electroweak interactions of the Standard Model, there is no known way of describing general relativity, the canonical theory of gravitation, consistently in terms of quantum field theory. The reason for this is among other things that quantum field theories of gravity generally break down before reaching the Planck scale. As a consequence, we have no reliable theory for the very early universe;
  • It seems rather ad-hoc and inelegant, requiring 19 numerical constants whose values are unrelated and arbitrary. Although the Standard Model, as it now stands, can explain why neutrinos have masses, the specifics of neutrino mass are still unclear. It is believed that explaining neutrino mass will require an additional 7 or 8 constants, which are also arbitrary parameters;
  • The Higgs mechanism gives rise to the hierarchy problem if any new physics (such as quantum gravity) is present at high energy scales. In order for the weak scale to be much smaller than the Planck scale, severe fine tuning of Standard Model parameters is required;
  • It should be modified so as to be consistent with the emerging "standard model of cosmology." In particular, the Standard Model cannot explain the observed amount of cold dark matter (CDM) and gives contributions to dark energy which are far too large. It is also difficult to accommodate the observed predominance of matter over antimatter (matter/antimatter asymmetry). The isotropy and homogeneity of the visible universe over large distances seems to require a mechanism like cosmic inflation, which would also constitute an extension of the Standard Model.

Currently no proposed Theory of everything has been conclusively verified.

See also[edit]

Notes and references[edit]

Notes[edit]
  1. ^ Technically, there are nine such color–anticolor combinations. However there is one color symmetric combination that can be constructed out of a linear superposition of the nine combinations, reducing the count to eight.
  2. ^ The normalization Q = I3 + Y is sometimes used instead.
References[edit]
  1. ^ S.L. Glashow (1961). "Partial-symmetries of weak interactions". Nuclear Physics. 22: 579–588. doi:10.1016/0029-5582(61)90469-2. 
  2. ^ S. Weinberg (1967). "A Model of Leptons". Physical Review Letters. 19: 1264–1266. doi:10.1103/PhysRevLett.19.1264. 
  3. ^ A. Salam (1968). N. Svartholm, ed. Elementary Particle Physics: Relativistic Groups and Analyticity. Nobel Symposium. Stockholm: Almquvist and Wiksell. p. 367.  Text "Eighth Nobel Symposium " ignored (help)
  4. ^ F. Englert, R. Brout (1964). "Broken Symmetry and the Mass of Gauge Vector Mesons". Physical Review Letters. 13: 321–323. doi:10.1103/PhysRevLett.13.321. 
  5. ^ P.W. Higgs (1964). "Broken Symmetries and the Masses of Gauge Bosons". Physical Review Letters. 13: 508–509. doi:10.1103/PhysRevLett.13.508. 
  6. ^ G.S. Guralnik, C.R. Hagen, T.W.B. Kibble (1964). "Global Conservation Laws and Massless Particles". Physical Review Letters. 13: 585–587. doi:10.1103/PhysRevLett.13.585. 
  7. ^ F.J. Hasert; et al. (1973). "Search for elastic muon-neutrino electron scattering". Physics Letters B. 46: 121. doi:10.1016/0370-2693(73)90494-2. 
  8. ^ F.J. Hasert; et al. (1973). "Observation of neutrino-like interactions without muon or electron in the gargamelle neutrino experiment". Physics Letters B. 46: 138. doi:10.1016/0370-2693(73)90499-1. 
  9. ^ F.J. Hasert; et al. (1974). "Observation of neutrino-like interactions without muon or electron in the Gargamelle neutrino experiment". Nuclear Physics B. 73: 1. doi:10.1016/0550-3213(74)90038-8. 
  10. ^ D. Haidt (4 October 2004). "The discovery of the weak neutral currents". CERN Courier. Retrieved 2008-05-08. 
  11. ^ "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." from The Feynman Lectures on Physics, Vol 1. pp. 2–7
  12. ^ F. Englert, R. Brout (1964). "Broken Symmetry and the Mass of Gauge Vector Mesons". Physical Review Letters. 13: 321–323. doi:10.1103/PhysRevLett.13.321. 
  13. ^ P.W. Higgs (1964). "Broken Symmetries and the Masses of Gauge Bosons". Physical Review Letters. 13: 508–509. doi:10.1103/PhysRevLett.13.508. 
  14. ^ G.S. Guralnik, C.R. Hagen, T.W.B. Kibble (1964). "Global Conservation Laws and Massless Particles". Physical Review Letters. 13: 585–587. doi:10.1103/PhysRevLett.13.585. 
  15. ^ G.S. Guralnik (2009). "The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles". International Journal of Modern Physics A. 24: 2601–2627. arXiv:0907.3466Freely accessible. doi:10.1142/S0217751X09045431. 
  16. ^ A. Cho (23 January 2008). "Higgs Hiding in Plain Sight?". ScienceNOW. Retrieved 2008-05-08. 
  17. ^ http://press.web.cern.ch/press/PressReleases/Releases2010/PR08.10E.html

Further reading[edit]

  • R. Oerter (2006). The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics. Plume. 
  • B.A. Schumm (2004). Deep Down Things: The Breathtaking Beauty of Particle Physics. Johns Hopkins University Press. ISBN 0-8018-7971-X. 
  • V. Stenger (2000). Timeless Reality. Prometheus Books.  See chapters 9–12 in particular.
Introductory textbooks
Advanced textbooks
Journal articles

External links[edit]