Yang–Mills theory

From Wikipedia, the free encyclopedia

Yang–Mills theory is a gauge theory based on the SU(N) group, or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-Abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics.

History and theoretical description

In a private correspondence, Wolfgang Pauli formulated in 1953 a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock and others to a higher-dimensional internal space.^[1] However, there is no evidence that Pauli developed the Lagrangian of a gauge field or the quantization of it. Because Pauli found that his theory "leads to some rather unphysical shadow particles”, he refrained from publishing his results formally.^[1] Although Pauli did not publish his six-dimensional theory, he gave two talks about it in Zürich.^[2] Recent research shows that an extended Kaluza–Klein theory is in general not equivalent to Yang–Mills theory, as the former contains additional terms.^[3]

In early 1954, Chen Ning Yang and Robert Mills^[4] extended the concept of gauge theory for abelian groups, e.g. quantum electrodynamics, to nonabelian groups to provide an explanation for strong interactions. The idea by Yang–Mills was criticized by Pauli,^[5] as the quanta of the Yang–Mills field must be massless in order to maintain gauge invariance. The idea was set aside until 1960, when the concept of particles acquiring mass through symmetry breaking in massless theories was put forward, initially by Jeffrey Goldstone, Yoichiro Nambu, and Giovanni Jona-Lasinio.

This prompted a significant restart of Yang–Mills theory studies that proved successful in the formulation of both electroweak unification and quantum chromodynamics (QCD). The electroweak interaction is described by SU(2) × U(1) group while QCD is an SU(3) Yang–Mills theory. The electroweak theory is obtained by combining SU(2) with U(1), where quantum electrodynamics (QED) is described by a U(1) group, and is replaced in the unified electroweak theory by a U(1) group representing a weak hypercharge rather than electric charge. The massless bosons from the SU(2) × U(1) theory mix after spontaneous symmetry breaking to produce the 3 massive weak bosons, and the photon field. The Standard Model combines the strong interaction with the unified electroweak interaction (unifying the weak and electromagnetic interaction) through the symmetry group SU(2) × U(1) × SU(3). In the current epoch the strong interaction is not unified with the electroweak interaction, but from the observed running of the coupling constants it is believed^{[citation needed]} they all converge to a single value at very high energies.

Phenomenology at lower energies in quantum chromodynamics is not completely understood due to the difficulties of managing such a theory with a strong coupling. This may be the reason why confinement has not been theoretically proven, though it is a consistent experimental observation. Proof that QCD confines at low energy is a mathematical problem of great relevance, and an award has been proposed by the Clay Mathematics Institute for whoever is also able to show that the Yang–Mills theory has a mass gap and its existence.

Mathematical overview

Yang–Mills theories are a special example of gauge theory with a non-abelian symmetry group given by the Lagrangian

${\displaystyle {\mathcal {L}}_{\mathrm {gf} }=-{\frac {1}{2}}\operatorname {Tr} (F^{2})=-{\frac {1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}}$

with the generators of the Lie algebra, indexed by a, corresponding to the F-quantities (the curvature or field-strength form) satisfying

${\displaystyle \operatorname {Tr} (T^{a}T^{b})={\frac {1}{2}}\delta ^{ab},\quad [T^{a},T^{b}]=if^{abc}T^{c},}$

where the f^abc are structure constants of the Lie algebra, and the covariant derivative defined as

${\displaystyle D_{\mu }=I\partial _{\mu }-igT^{a}A_{\mu }^{a}}$

where I is the identity matrix (matching the size of the generators), ${\displaystyle A_{\mu }^{a}}$ is the vector potential, and g is the coupling constant. In four dimensions, the coupling constant g is a pure number and for a SU(N) group one has ${\displaystyle a,b,c=1\ldots N^{2}-1.}$

The relation

${\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c}}$

can be derived by the commutator

${\displaystyle [D_{\mu },D_{\nu }]=-igT^{a}F_{\mu \nu }^{a}.}$

The field has the property of being self-interacting and equations of motion that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this theory only by perturbation theory, with small nonlinearities.

Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for a indices (e.g. ${\displaystyle f^{abc}=f_{abc}}$), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature, ${\displaystyle \eta _{\mu \nu }={\rm {diag}}(+---)}$.

From the given Lagrangian one can derive the equations of motion given by

${\displaystyle \partial ^{\mu }F_{\mu \nu }^{a}+gf^{abc}A^{\mu b}F_{\mu \nu }^{c}=0.}$

Putting ${\displaystyle F_{\mu \nu }=T^{a}F_{\mu \nu }^{a}}$, these can be rewritten as

${\displaystyle (D^{\mu }F_{\mu \nu })^{a}=0.}$

A Bianchi identity holds

${\displaystyle (D_{\mu }F_{\nu \kappa })^{a}+(D_{\kappa }F_{\mu \nu })^{a}+(D_{\nu }F_{\kappa \mu })^{a}=0}$

which is equivalent to the Jacobi identity

${\displaystyle [D_{\mu },[D_{\nu },D_{\kappa }]]+[D_{\kappa },[D_{\mu },D_{\nu }]]+[D_{\nu },[D_{\kappa },D_{\mu }]]=0}$

since ${\displaystyle [D_{\mu },F_{\nu \kappa }^{a}]=D_{\mu }F_{\nu \kappa }^{a}}$. Define the dual strength tensor ${\displaystyle {\tilde {F}}^{\mu \nu }={\frac {1}{2}}\varepsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }}$, then the Bianchi identity can be rewritten as

${\displaystyle D_{\mu }{\tilde {F}}^{\mu \nu }=0.}$

A source ${\displaystyle J_{\mu }^{a}}$ enters into the equations of motion as

${\displaystyle \partial ^{\mu }F_{\mu \nu }^{a}+gf^{abc}A^{b\mu }F_{\mu \nu }^{c}=-J_{\nu }^{a}.}$

Note that the currents must properly change under gauge group transformations.

We give here some comments about the physical dimensions of the coupling. In D dimensions, the field scales as ${\displaystyle [A]=[L^{\frac {2-D}{2}}]}$^{[citation needed]} and so the coupling must scale as ${\displaystyle [g^{2}]=[L^{D-4}]}$. This implies that Yang–Mills theory is not renormalizable for dimensions greater than four. Furthermore, for D = 4, the coupling is dimensionless and both the field and the square of the coupling have the same dimensions of the field and the coupling of a massless quartic scalar field theory. So, these theories share the scale invariance at the classical level.

Quantization

A method of quantizing the Yang–Mills theory is by functional methods, i.e. path integrals. One introduces a generating functional for n-point functions as

${\displaystyle Z[j]=\int [dA]\exp \left[-{\frac {i}{2}}\int d^{4}x\operatorname {Tr} (F^{\mu \nu }F_{\mu \nu })+i\int d^{4}x\,j_{\mu }^{a}(x)A^{a\mu }(x)\right],}$

but this integral has no meaning as it is because the potential vector can be arbitrarily chosen due to the gauge freedom. This problem was already known for quantum electrodynamics but here becomes more severe due to non-abelian properties of the gauge group. A way out has been given by Ludvig Faddeev and Victor Popov with the introduction of a ghost field that has the property of being unphysical since, although it agrees with Fermi–Dirac statistics, it is a complex scalar field, which violates the spin–statistics theorem. So, we can write the generating functional as

${\displaystyle {\begin{aligned}Z[j,{\bar {\varepsilon }},\varepsilon ]&=\int [dA][d{\bar {c}}][dc]\exp \left\{iS_{F}[\partial A,A]+iS_{gf}[\partial A]+iS_{g}[\partial c,\partial {\bar {c}},c,{\bar {c}},A]\right\}\\&\exp \left\{i\int d^{4}xj_{\mu }^{a}(x)A^{a\mu }(x)+i\int d^{4}x[{\bar {c}}^{a}(x)\varepsilon ^{a}(x)+{\bar {\varepsilon }}^{a}(x)c^{a}(x)]\right\}\end{aligned}}}$

being

${\displaystyle S_{F}=-{\frac {1}{2}}\operatorname {Tr} (F^{\mu \nu }F_{\mu \nu })}$

for the field,

${\displaystyle S_{gf}=-{\frac {1}{2\xi }}(\partial \cdot A)^{2}}$

for the gauge fixing and

${\displaystyle S_{g}=-({\bar {c}}^{a}\partial _{\mu }\partial ^{\mu }c^{a}+g{\bar {c}}^{a}f^{abc}\partial _{\mu }A^{b\mu }c^{c})}$

for the ghost. This is the expression commonly used to derive Feynman's rules. Here we have c^a for the ghost field while α fixes the gauge's choice for the quantization. Feynman's rules obtained from this functional are the following





















These rules for Feynman diagrams can be obtained when the generating functional given above is rewritten as

${\displaystyle {\begin{aligned}Z[j,{\bar {\varepsilon }},\varepsilon ]&=\exp \left(-ig\int d^{4}x\,{\frac {\delta }{i\delta {\bar {\varepsilon }}^{a}(x)}}f^{abc}\partial _{\mu }{\frac {i\delta }{\delta j_{\mu }^{b}(x)}}{\frac {i\delta }{\delta \varepsilon ^{c}(x)}}\right)\\&\qquad \times \exp \left(-ig\int d^{4}xf^{abc}\partial _{\mu }{\frac {i\delta }{\delta j_{\nu }^{a}(x)}}{\frac {i\delta }{\delta j_{\mu }^{b}(x)}}{\frac {i\delta }{\delta j^{c\nu }(x)}}\right)\\&\qquad \qquad \times \exp \left(-i{\frac {g^{2}}{4}}\int d^{4}xf^{abc}f^{ars}{\frac {i\delta }{\delta j_{\mu }^{b}(x)}}{\frac {i\delta }{\delta j_{\nu }^{c}(x)}}{\frac {i\delta }{\delta j^{r\mu }(x)}}{\frac {i\delta }{\delta j^{s\nu }(x)}}\right)\\&\qquad \qquad \qquad \times Z_{0}[j,{\bar {\varepsilon }},\varepsilon ]\end{aligned}}}$

with

${\displaystyle Z_{0}[j,{\bar {\varepsilon }},\varepsilon ]=\exp \left(-\int d^{4}xd^{4}y{\bar {\varepsilon }}^{a}(x)C^{ab}(x-y)\varepsilon ^{b}(y)\right)\exp \left({\tfrac {1}{2}}\int d^{4}xd^{4}yj_{\mu }^{a}(x)D^{ab\mu \nu }(x-y)j_{\nu }^{b}(y)\right)}$

being the generating functional of the free theory. Expanding in g and computing the functional derivatives, we are able to obtain all the n-point functions with perturbation theory. Using LSZ reduction formula we get from the n-point functions the corresponding process amplitudes, cross sections and decay rates. The theory is renormalizable and corrections are finite at any order of perturbation theory.

For quantum electrodynamics the ghost field decouples because the gauge group is abelian. This can be seen from the coupling between the gauge field and the ghost field that is ${\displaystyle {\bar {c}}^{a}f^{abc}\partial _{\mu }A^{b\mu }c^{c}}$. For the abelian case, all the structure constants ${\displaystyle f^{abc}}$ are zero and so there is no coupling. In the non-abelian case, the ghost field appears as a useful way to rewrite the quantum field theory without physical consequences on the observables of the theory such as cross sections or decay rates.

One of the most important results obtained for Yang–Mills theory is asymptotic freedom. This result can be obtained by assuming that the coupling constant g is small (so small nonlinearities), as for high energies, and applying perturbation theory. The relevance of this result is due to the fact that a Yang–Mills theory that describes strong interaction and asymptotic freedom permits proper treatment of experimental results coming from deep inelastic scattering.

To obtain the behavior of the Yang–Mills theory at high energies, and so to prove asymptotic freedom, one applies perturbation theory assuming a small coupling. This is verified a posteriori in the ultraviolet limit. In the opposite limit, the infrared limit, the situation is the opposite, as the coupling is too large for perturbation theory to be reliable. Most of the difficulties that research meets is just managing the theory at low energies. That is the interesting case, being inherent to the description of hadronic matter and, more generally, to all the observed bound states of gluons and quarks and their confinement (see hadrons). The most used method to study the theory in this limit is to try to solve it on computers (see lattice gauge theory). In this case, large computational resources are needed to be sure the correct limit of infinite volume (smaller lattice spacing) is obtained. This is the limit the results must be compared with. Smaller spacing and larger coupling are not independent of each other, and larger computational resources are needed for each. As of today, the situation appears somewhat satisfactory for the hadronic spectrum and the computation of the gluon and ghost propagators, but the glueball and hybrids spectra are yet a questioned matter in view of the experimental observation of such exotic states. Indeed, the σ resonance^[6][7] is not seen in any of such lattice computations and contrasting interpretations have been put forward. This is a hotly debated issue.

Open problems

Yang–Mills theories met with general acceptance in the physics community after Gerard 't Hooft, in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisor Martinus Veltman. (Their work^[8] was recognized by the 1999 Nobel prize in physics.) Renormalizability is obtained even if the gauge bosons described by this theory are massive, as in the electroweak theory, provided the mass is only an "acquired" one, generated by the Higgs mechanism.

Concerning the mathematics, it should be noted that the Yang–Mills theory is a very active field of research, yielding e.g. invariants of differentiable structures on four-dimensional manifolds via work of Simon Donaldson. Furthermore, the field of Yang–Mills theories was included in the Clay Mathematics Institute's list of "Millennium Prize Problems". Here the prize-problem consists, especially, in a proof of the conjecture that the lowest excitations of a pure Yang–Mills theory (i.e. without matter fields) have a finite mass-gap with regard to the vacuum state. Another open problem, connected with this conjecture, is a proof of the confinement property in the presence of additional Fermion particles.

In physics the survey of Yang–Mills theories does not usually start from perturbation analysis or analytical methods, but more recently from systematic application of numerical methods to lattice gauge theories.

Ice giant

From Wikipedia, the free encyclopedia

Uranus photographed by Voyager 2 in January 1986

 

Neptune photographed by Voyager 2 in August 1989

An ice giant is a giant planet composed mainly of elements heavier than hydrogen and helium, such as oxygen, carbon, nitrogen, and sulfur. There are two known ice giants in the Solar System, Uranus and Neptune.

In astrophysics and planetary science the term "ices" refers to volatile chemical compounds with freezing points above about 100 K, such as water, ammonia, or methane, with freezing points of 273 K, 195 K, and 91 K, respectively (see Volatiles). In the 1990s, it was realized that Uranus and Neptune are a distinct class of giant planet, separate from the other giant planets, Jupiter and Saturn. They have become known as ice giants. Their constituent compounds were solids when they were primarily incorporated into the planets during their formation^{[citation needed]}, either directly in the form of ices or trapped in water ice. Today, very little of the water in Uranus and Neptune remains in the form of ice. Instead, water primarily exists as supercritical fluid at the temperatures and pressures within them.^[1]

Ice giants consist of only about 20% hydrogen and helium in mass, as opposed to the Solar System's gas giants, Jupiter and Saturn, which are both more than 90% hydrogen and helium in mass.

Terminology

In 1952, science fiction writer James Blish coined the term gas giant^[2] and it was used to refer to the large non-terrestrial planets of the Solar System. However, in the 1990s, the compositions of Uranus and Neptune were discovered to be significantly different from those of Jupiter and Saturn. They are primarily composed of elements heavier than hydrogen and helium, constituting a separate type of giant planet altogether. Because during their formation Uranus and Neptune incorporated their material as either ices or gas trapped in water ice, the term ice giant came into use.^[1]

Formation

Modelling the formation of the terrestrial and gas giants is relatively straightforward and uncontroversial. The terrestrial planets of the Solar System are widely understood to have formed through collisional accumulation of planetesimals within the protoplanetary disc. The gas giants—Jupiter, Saturn, and their extrasolar counterpart planets—are thought to have formed after solid cores around 10 Earth masses (M_⊕) formed through the same process, while accreting gaseous envelopes from the surrounding solar nebula over the course of a few to several million years (Ma),^[3][4] although alternative models of core formation based on pebble accretion have recently been proposed.^[5] Some extrasolar giant planets may instead have formed via gravitational disk instabilities.^[4][6]

The formation of Uranus and Neptune through a similar process of core accretion is far more problematic. The escape velocity for the small protoplanets about 20 astronomical units (AU) from the centre of the Solar System would have been comparable to their relative velocities. Such bodies crossing the orbits of Saturn or Jupiter would have been liable to be sent on hyperbolic trajectories ejecting them from the system. Such bodies, being swept up by the gas giants, would also have been likely to just be accreted into the larger planets or thrown into cometary orbits.^[6]

In spite of the trouble modelling their formation, many ice giant candidates have been observed orbiting other stars since 2004. This indicates that they may be common in the Milky Way.^[1]

Migration

Considering the orbital challenges of protoplanets 20 AU or more from the centre of the Solar System would experience, a simple solution is that the ice giants formed between the orbits of Jupiter and Saturn before being gravitationally scattered outward to their now more distant orbits.^[6]

Disk instability

Gravitational instability of the protoplanetary disk could also produce several gas giant protoplanets out to distances of up to 30 AU. Regions of slightly higher density in the disk could lead to the formation of clumps that eventually collapse to planetary densities.^[6] A disk with even marginal gravitational instability could yield protoplanets between 10 and 30 AU in over one thousand years (ka). This is much shorter than the 100,000 to 1,000,000 years required to produce protoplanets through core accretion of the cloud and could make it viable in even the shortest-lived disks, which exist for only a few million years.^[6]

A problem with this model is determining what kept the disk stable prior to the instability. There are several possible mechanisms allowing gravitational instability to occur during disk evolution. A close encounter with another protostar could provide a gravitational kick to an otherwise stable disk. A disk evolving magnetically is likely to have magnetic dead zones, due to varying degrees of ionization, where mass moved by magnetic forces could pile up, eventually becoming marginally gravitationally unstable. A protoplanetary disk may simply accrete matter slowly, causing relatively short periods of marginal gravitational instability and bursts of mass collection, followed by periods where the surface density drops below what is required to sustain the instability.^[6]

Photoevaporation

Observations of photoevaporation of protoplanetary disks in the Orion Trapezium Cluster by extreme ultraviolet (EUV) radiation emitted by θ¹ Orionis C suggests another possible mechanism for the formation of ice giants. Multiple-Jupiter-mass gas-giant protoplanets could have rapidly formed due to disk instability before having the majority of their hydrogen envelopes stripped off by intense EUV radiation from a nearby massive star.^[6]

In the Carina Nebula, EUV fluxes are approximately 100 times higher than in Trapezium's Orion Nebula. Protoplanetary disks are present in both nebulae. Higher EUV fluxes make this an even more likely possibility for ice-giant formation. The stronger EUV would increase the removal of the gas envelopes from the protoplanets before they could collapse sufficiently to resist further loss.^[6]

Characteristics

These cut-aways illustrate interior models of the giant planets. The planetary cores of gas giants Jupiter and Saturn are overlaid by a deep layer of metallic hydrogen, whereas the mantles of the ice giants Uranus and Neptune are composed of heavier elements.

The ice giants represent one of two fundamentally different categories of giant planets present in the Solar System, the other group being the more-familiar gas giants, which are composed of more than 90% hydrogen and helium (by mass). Their hydrogen is thought to extend all the way down to their small rocky cores, where hydrogen molecular ion transitions to metallic hydrogen under the extreme pressures of hundreds of gigapascals (GPa).^[1]

The ice giants are primarily composed of heavier elements. Based on the abundance of elements in the universe, oxygen, carbon, nitrogen, and sulfur are most likely. Although the ice giants also have hydrogen envelopes, these are much smaller. They account for less than 20% of their mass. Their hydrogen also never reaches the depths necessary for the pressure to create metallic hydrogen.^[1] These envelopes nevertheless limit observation of the ice giants' interiors, and thereby the information on their composition and evolution.^[1]

Although Uranus and Neptune are referred to as ice giant planets, it is thought that there is a supercritical water ocean beneath their clouds, which accounts for about two-thirds of their total mass.^[7][8]

Atmosphere and weather

The gaseous outer layers of the ice giants have several similarities to those of the gas giants. These include long-lived, high-speed equatorial winds, polar vortices, large-scale circulation patterns, and complex chemical processes driven by ultraviolet radiation from above and mixing with the lower atmosphere.^[1]

Studying the ice giants' atmospheric pattern also gives insights into atmospheric physics. Their compositions promote different chemical processes and they receive far less sunlight in their distant orbits than any other planets in the Solar System (increasing the relevance of internal heating on weather patterns).^[1]

The largest visible feature on Neptune is the recurring Great Dark Spot. It forms and dissipates every few years, as opposed to the similarly sized Great Red Spot of Jupiter, which has persisted for centuries. Of all known giant planets in the Solar System, Neptune emits the most internal heat per unit of absorbed sunlight, a ratio of approximately 2.6. Saturn, the next-highest emitter, only has a ratio of about 1.8. Uranus emits the least heat, one-tenth as much as Neptune. It is suspected that this may be related to its extreme 98˚ axial tilt. This causes its seasonal patterns to be very different from those of any other planet in the Solar System.^[1]

There are still no complete models explaining the atmospheric features observed in the ice giants.^[1] Understanding these features will help elucidate how the atmospheres of giant planets in general function.^[1] Consequently, such insights could help scientists better predict the atmospheric structure and behaviour of giant exoplanets discovered to be very close to their host stars (pegasean planets) and exoplanets with masses and radii between that of the giant and terrestrial planets found in the Solar System.^[1]

Interior

Because of their large sizes and low thermal conductivities, the planetary interior pressures range up to several hundred GPa and temperatures of several thousand kelvins (K).^[9]

In March 2012, it was found that the compressibility of water used in ice-giant models could be off by one third.^[10] This value is important for modeling ice giants, and has a ripple effect on understanding them.^[10]

Magnetic fields

The magnetic fields of Uranus and Neptune are both unusually displaced and tilted.^[11] Their field strengths are intermediate between those of the gas giants and those of the terrestrial planets, being 50 and 25 times that of Earth's, respectively.^[11] Their magnetic fields are believed to originate in an ionized convecting fluid-ice mantle.^[11]