Chandrasekhar's white dwarf equation

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In astrophysics, Chandrasekhar's white dwarf equation is an initial value ordinary differential equation introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar, in his study of the gravitational potential of completely degenerate white dwarf stars. The equation reads as

${\displaystyle \frac{1}{{\eta }^2}\frac{d}{d\eta }\left({\eta }^2\frac{d\phi }{d\eta }\right)+({\phi }^2-C{)}^{3/2}=0}$

with initial conditions

${\displaystyle \phi (0)=1,{\phi }^{\prime }(0)=0}$

where ${\displaystyle \phi }$ measures the density of white dwarf, ${\displaystyle \eta }$ is the non-dimensional radial distance from the center and ${\displaystyle C}$ is a constant which is related to the density of the white dwarf at the center. The boundary ${\displaystyle \eta ={\eta }_{\mathrm{\infty }}}$ of the equation is defined by the condition

${\displaystyle \phi ({\eta }_{\mathrm{\infty }})=\sqrt{C}.}$

such that the range of ${\displaystyle \phi }$ becomes ${\displaystyle \sqrt{C}\le \phi \le 1}$. This condition is equivalent to saying that the density vanishes at ${\displaystyle \eta ={\eta }_{\mathrm{\infty }}}$.

Derivation

From the quantum statistics of a completely degenerate electron gas (all the lowest quantum states are occupied), the pressure and the density of a white dwarf are calculated in terms of the maximum electron momentum ${\displaystyle p_0}$standardized as ${\displaystyle x=p_0/mc}$,

with pressure ${\displaystyle P=Af(x)}$

and density ${\displaystyle \rho =B{x}^3}$

where

${\displaystyle \begin{array}{rl}& A=\frac{\pi m_e^4{c}^5}{3h^3}=6.02\times {10}^{21}\text{ Pa},\\ & B=\frac{8\pi }{3}{m}_p{\mu }_e{\left(\frac{m_{e}c}{h}\right)}^3=9.82\times {10}^8{\mu }_e{\text{ kg/m}}^3,\\ & f(x)=x(2x^2-3)(x^2+1{)}^{1/2}+3{\sinh }^{-1}x,\end{array}}$

${\displaystyle {\mu }_e}$ is the mean molecular weight of the gas, and ${\displaystyle h}$ is the height of a small cube of gas with only two possible states.

When this is substituted into the hydrostatic equilibrium equation

${\displaystyle \frac{1}{r^2}\frac{d}{dr}\left(\frac{r^2}{\rho }\frac{dP}{dr}\right)=-4\pi G\rho }$

where ${\displaystyle G}$ is the gravitational constant and ${\displaystyle r}$ is the radial distance, we get

${\displaystyle \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\sqrt{x^2+1}}{dr}\right)=-\frac{\pi G{B}^2}{2A}{x}^3}$

and letting ${\displaystyle y^2=x^2+1}$, we have

${\displaystyle \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dy}{dr}\right)=-\frac{\pi G{B}^2}{2A}(y^2-1{)}^{3/2}}$

If we denote the density at the origin as ${\displaystyle {\rho }_o=B{x}_o^3=B(y_o^2-1{)}^{3/2}}$, then a non-dimensional scale

${\displaystyle r={\left(\frac{2A}{\pi G{B}^2}\right)}^{1/2}\frac{\eta }{y_o},y=y_o\phi }$

gives

${\displaystyle \frac{1}{{\eta }^2}\frac{d}{d\eta }\left({\eta }^2\frac{d\phi }{d\eta }\right)+({\phi }^2-C{)}^{3/2}=0}$

where ${\displaystyle C=1/y_o^2}$. In other words, once the above equation is solved the density is given by

${\displaystyle \rho =B{y}_o^3{\left({\phi }^2-\frac{1}{y_o^2}\right)}^{3/2}.}$

The mass interior to a specified point can then be calculated

${\displaystyle M(\eta )=-\frac{4\pi }{B^2}{\left(\frac{2A}{\pi G}\right)}^{3/2}{\eta }^2\frac{d\phi }{d\eta }.}$

The radius-mass relation of the white dwarf is usually plotted in the plane ${\displaystyle {\eta }_{\mathrm{\infty }}}$-${\displaystyle M({\eta }_{\mathrm{\infty }})}$.

Solution near the origin

In the neighborhood of the origin, ${\displaystyle \eta \ll 1}$, Chandrasekhar provided an asymptotic expansion as

${\displaystyle \begin{array}{rl}\phi =& 1-\frac{q^3}{6}{\eta }^2+\frac{q^4}{40}{\eta }^4-\frac{q^5(5q^2+14)}{7!}{\eta }^6\\ & +\frac{q^6(339q^2+280)}{3\times 9!}{\eta }^8-\frac{q^7(1425q^4+11346q^2+4256)}{5\times 11!}{\eta }^{10}+\cdots \end{array}}$

where ${\displaystyle q^2=C-1}$. He also provided numerical solutions for the range ${\displaystyle C=0.01-0.8}$.

Equation for small central densities

When the central density ${\displaystyle {\rho }_o=B{x}_o^3=B(y_o^2-1{)}^{3/2}}$ is small, the equation can be reduced to a Lane-Emden equation by introducing

${\displaystyle \xi =\sqrt{2}\eta ,\theta ={\phi }^2-C={\phi }^2-1+x_o^2+O(x_o^4)}$

to obtain at leading order, the following equation

${\displaystyle \frac{1}{{\xi }^2}\frac{d}{d\xi }\left({\xi }^2\frac{d\theta }{d\xi }\right)=-{\theta }^{3/2}}$

subjected to the conditions ${\displaystyle \theta (0)=x_o^2}$ and ${\displaystyle {\theta }^{\prime }(0)=0}$. Note that although the equation reduces to the Lane-Emden equation with polytropic index ${\displaystyle 3/2}$, the initial condition is not that of the Lane-Emden equation.

Limiting mass for large central densities

When the central density becomes large, i.e., ${\displaystyle y_o\to \mathrm{\infty }}$ or equivalently ${\displaystyle C\to 0}$, the governing equation reduces to

${\displaystyle \frac{1}{{\eta }^2}\frac{d}{d\eta }\left({\eta }^2\frac{d\phi }{d\eta }\right)=-{\phi }^3}$

subjected to the conditions ${\displaystyle \phi (0)=1}$ and ${\displaystyle {\phi }^{\prime }(0)=0}$. This is exactly the Lane-Emden equation with polytropic index ${\displaystyle 3}$. Note that in this limit of large densities, the radius

${\displaystyle r={\left(\frac{2A}{\pi G{B}^2}\right)}^{1/2}\frac{\eta }{y_o}}$

tends to zero. The mass of the white dwarf however tends to a finite limit

${\displaystyle M\to -\frac{4\pi }{B^2}{\left(\frac{2A}{\pi G}\right)}^{3/2}{\left({\eta }^2\frac{d\phi }{d\eta }\right)}_{\eta ={\eta }_{\mathrm{\infty }}}=5.75{\mu }_e^{-2}{M}_{\odot }.}$

The Chandrasekhar limit follows from this limit.

Chandrasekhar limit

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Chandrasekhar_limit

The Chandrasekhar limit (/tʃʌndrəˈseɪkər/) is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about 1.4 M_☉ (2.765×10³⁰ kg).

White dwarfs resist gravitational collapse primarily through electron degeneracy pressure, compared to main sequence stars, which resist collapse through thermal pressure. The Chandrasekhar limit is the mass above which electron degeneracy pressure in the star's core is insufficient to balance the star's own gravitational self-attraction. Consequently, a white dwarf with a mass greater than the limit is subject to further gravitational collapse, evolving into a different type of stellar remnant, such as a neutron star or black hole. Those with masses up to the limit remain stable as white dwarfs. The Tolman–Oppenheimer–Volkoff limit is theoretically a next level to reach in order for a neutron star to collapse into a denser form such as a black hole.

The limit was named after Subrahmanyan Chandrasekhar. Chandrasekhar improved upon the accuracy of the calculation in 1930 by calculating the limit for a polytrope model of a star in hydrostatic equilibrium, and comparing his limit to the earlier limit found by E. C. Stoner for a uniform density star. Importantly, the existence of a limit, based on the conceptual breakthrough of combining relativity with Fermi degeneracy, was indeed first established in separate papers published by Wilhelm Anderson and E. C. Stoner in 1929. The limit was initially ignored by the community of scientists because such a limit would logically require the existence of black holes, which were considered a scientific impossibility at the time. The fact that the roles of Stoner and Anderson are often overlooked in the astronomy community has been noted.

The priority dispute has been discussed at length by Virginia Trimble: "Chandrasekhar famously, perhaps even notoriously did his critical calculation on board ship in 1930, and ... was not aware of either Stoner's or Anderson's work at the time. His work was therefore independent, but, more to the point, he adopted Eddington's polytropes for his models which could, therefore, be in hydrostatic equilibrium, which constant density stars cannot, and real ones must be."

Physics

Radius–mass relations for a model white dwarf. The green curve uses the general pressure law for an ideal Fermi gas, while the blue curve is for a non-relativistic ideal Fermi gas. The black line marks the ultrarelativistic limit.

Electron degeneracy pressure is a quantum-mechanical effect arising from the Pauli exclusion principle. Since electrons are fermions, no two electrons can be in the same state, so not all electrons can be in the minimum-energy level. Rather, electrons must occupy a band of energy levels. Compression of the electron gas increases the number of electrons in a given volume and raises the maximum energy level in the occupied band. Therefore, the energy of the electrons increases on compression, so pressure must be exerted on the electron gas to compress it, producing electron degeneracy pressure. With sufficient compression, electrons are forced into nuclei in the process of electron capture, relieving the pressure.

In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form P = K₁ρ^5/3, where P is the pressure, ρ is the mass density, and K₁ is a constant. Solving the hydrostatic equation leads to a model white dwarf that is a polytrope of index 3/2 – and therefore has radius inversely proportional to the cube root of its mass, and volume inversely proportional to its mass.

As the mass of a model white dwarf increases, the typical energies to which degeneracy pressure forces the electrons are no longer negligible relative to their rest masses. The velocities of the electrons approach the speed of light, and special relativity must be taken into account. In the strongly relativistic limit, the equation of state takes the form P = K₂ρ^4/3. This yields a polytrope of index 3, which has a total mass, M_limit, depending only on K₂.

For a fully relativistic treatment, the equation of state used interpolates between the equations P = K₁ρ^5/3 for small ρ and P = K₂ρ^4/3 for large ρ. When this is done, the model radius still decreases with mass, but becomes zero at M_limit. This is the Chandrasekhar limit. The curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored blue and green, respectively. μ_e has been set equal to 2. Radius is measured in standard solar radii or kilometers, and mass in standard solar masses.

Calculated values for the limit vary depending on the nuclear composition of the mass. Chandrasekhar^{, eq. (36),, eq. (58),, eq. (43)} gives the following expression, based on the equation of state for an ideal Fermi gas:

$${\displaystyle M_{\text{limit}}=\frac{{\omega }_3^0\sqrt{3\pi }}{2}{\left(\frac{\hslash c}{G}\right)}^{\frac{3}{2}}\frac{1}{({\mu }_{\text{e}}{m}_{\text{H}}{)}^2}}$$

where:

• ħ is the reduced Planck constant
• c is the speed of light
• G is the gravitational constant
• μ_e is the average molecular weight per electron, which depends upon the chemical composition of the star
• m_H is the mass of the hydrogen atom
• ω⁰
  ₃ ≈ 2.018236 is a constant connected with the solution to the Lane–Emden equation

As √ħc/G is the Planck mass, the limit is of the order of

$${\displaystyle \frac{M_{\text{Pl}}^3}{m_{\text{H}}^2}}$$

The limiting mass can be obtained formally from the Chandrasekhar's white dwarf equation by taking the limit of large central density.

A more accurate value of the limit than that given by this simple model requires adjusting for various factors, including electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature. Lieb and Yau have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation.

History

In 1926, the British physicist Ralph H. Fowler observed that the relationship between the density, energy, and temperature of white dwarfs could be explained by viewing them as a gas of nonrelativistic, non-interacting electrons and nuclei that obey Fermi–Dirac statistics. This Fermi gas model was then used by the British physicist Edmund Clifton Stoner in 1929 to calculate the relationship among the mass, radius, and density of white dwarfs, assuming they were homogeneous spheres. Wilhelm Anderson applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1.37×10³⁰ kg. In 1930, Stoner derived the internal energy–density equation of state for a Fermi gas, and was then able to treat the mass–radius relationship in a fully relativistic manner, giving a limiting mass of approximately 2.19×10³⁰ kg (for μ_e = 2.5). Stoner went on to derive the pressure–density equation of state, which he published in 1932. These equations of state were also previously published by the Soviet physicist Yakov Frenkel in 1928, together with some other remarks on the physics of degenerate matter. Frenkel's work, however, was ignored by the astronomical and astrophysical community.

A series of papers published between 1931 and 1935 had its beginning on a trip from India to England in 1930, where the Indian physicist Subrahmanyan Chandrasekhar worked on the calculation of the statistics of a degenerate Fermi gas. In these papers, Chandrasekhar solved the hydrostatic equation together with the nonrelativistic Fermi gas equation of state, and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above. Chandrasekhar reviews this work in his Nobel Prize lecture. This value was also computed in 1932 by the Soviet physicist Lev Landau, who, however, did not apply it to white dwarfs and concluded that quantum laws might be invalid for stars heavier than 1.5 solar mass.

Chandrasekhar–Eddington dispute

Main article: Chandrasekhar–Eddington dispute

Chandrasekhar's work on the limit aroused controversy, owing to the opposition of the British astrophysicist Arthur Eddington. Eddington was aware that the existence of black holes was theoretically possible, and also realized that the existence of the limit made their formation possible. However, he was unwilling to accept that this could happen. After a talk by Chandrasekhar on the limit in 1935, he replied:

The star has to go on radiating and radiating and contracting and contracting until, I suppose, it gets down to a few km radius, when gravity becomes strong enough to hold in the radiation, and the star can at last find peace. ... I think there should be a law of Nature to prevent a star from behaving in this absurd way!

Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law P = K₁ρ^5/3 universally applicable, even for large ρ. Although Niels Bohr, Fowler, Wolfgang Pauli, and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar. Through the rest of his life, Eddington held to his position in his writings, including his work on his fundamental theory. The drama associated with this disagreement is one of the main themes of Empire of the Stars, Arthur I. Miller's biography of Chandrasekhar. In Miller's view:

Chandra's discovery might well have transformed and accelerated developments in both physics and astrophysics in the 1930s. Instead, Eddington's heavy-handed intervention lent weighty support to the conservative community astrophysicists, who steadfastly refused even to consider the idea that stars might collapse to nothing. As a result, Chandra's work was almost forgotten.

However, in 1983 in recognition for his work, Chandrasekhar shared a Nobel prize "for his theoretical studies of the physical processes of importance to the structure and evolution of the stars" with William Alfred Fowler.

Applications

The core of a star is kept from collapsing by the heat generated by the fusion of nuclei of lighter elements into heavier ones. At various stages of stellar evolution, the nuclei required for this process are exhausted, and the core collapses, causing it to become denser and hotter. A critical situation arises when iron accumulates in the core, since iron nuclei are incapable of generating further energy through fusion. If the core becomes sufficiently dense, electron degeneracy pressure will play a significant part in stabilizing it against gravitational collapse.

If a main-sequence star is not too massive (less than approximately 8 solar masses), it eventually sheds enough mass to form a white dwarf having mass below the Chandrasekhar limit, which consists of the former core of the star. For more-massive stars, electron degeneracy pressure does not keep the iron core from collapsing to very great density, leading to formation of a neutron star, black hole, or, speculatively, a quark star. (For very massive, low-metallicity stars, it is also possible that instabilities destroy the star completely.) During the collapse, neutrons are formed by the capture of electrons by protons in the process of electron capture, leading to the emission of neutrinos. The decrease in gravitational potential energy of the collapsing core releases a large amount of energy on the order of 10⁴⁶ joules (100 foes). Most of this energy is carried away by the emitted neutrinos and the kinetic energy of the expanding shell of gas; only about 1% is emitted as optical light. This process is believed responsible for supernovae of types Ib, Ic, and II.

Type Ia supernovae derive their energy from runaway fusion of the nuclei in the interior of a white dwarf. This fate may befall carbon–oxygen white dwarfs that accrete matter from a companion giant star, leading to a steadily increasing mass. As the white dwarf's mass approaches the Chandrasekhar limit, its central density increases, and, as a result of compressional heating, its temperature also increases. This eventually ignites nuclear fusion reactions, leading to an immediate carbon detonation, which disrupts the star and causes the supernova.

A strong indication of the reliability of Chandrasekhar's formula is that the absolute magnitudes of supernovae of Type Ia are all approximately the same; at maximum luminosity, M_V is approximately −19.3, with a standard deviation of no more than 0.3. A 1-sigma interval therefore represents a factor of less than 2 in luminosity. This seems to indicate that all type Ia supernovae convert approximately the same amount of mass to energy.

Super-Chandrasekhar mass supernovas

Main article: Champagne Supernova

In April 2003, the Supernova Legacy Survey observed a type Ia supernova, designated SNLS-03D3bb, in a galaxy approximately 4 billion light years away. According to a group of astronomers at the University of Toronto and elsewhere, the observations of this supernova are best explained by assuming that it arose from a white dwarf that had grown to twice the mass of the Sun before exploding. They believe that the star, dubbed the "Champagne Supernova" may have been spinning so fast that a centrifugal tendency allowed it to exceed the limit. Alternatively, the supernova may have resulted from the merger of two white dwarfs, so that the limit was only violated momentarily. Nevertheless, they point out that this observation poses a challenge to the use of type Ia supernovae as standard candles.

Since the observation of the Champagne Supernova in 2003, several more type Ia supernovae have been observed that are very bright, and thought to have originated from white dwarfs whose masses exceeded the Chandrasekhar limit. These include SN 2006gz, SN 2007if, and SN 2009dc. The super-Chandrasekhar mass white dwarfs that gave rise to these supernovae are believed to have had masses up to 2.4–2.8 solar masses. One way to potentially explain the problem of the Champagne Supernova was considering it the result of an aspherical explosion of a white dwarf. However, spectropolarimetric observations of SN 2009dc showed it had a polarization smaller than 0.3, making the large asphericity theory unlikely.

Tolman–Oppenheimer–Volkoff limit

After a supernova explosion, a neutron star may be left behind (except Ia type supernova explosion, which never leaves any remnants behind). These objects are even more compact than white dwarfs and are also supported, in part, by degeneracy pressure. A neutron star, however, is so massive and compressed that electrons and protons have combined to form neutrons, and the star is thus supported by neutron degeneracy pressure (as well as short-range repulsive neutron-neutron interactions mediated by the strong force) instead of electron degeneracy pressure. The limiting value for neutron star mass, analogous to the Chandrasekhar limit, is known as the Tolman–Oppenheimer–Volkoff limit.

High-energy nuclear physics

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/High-energy_nuclear_physics

High-energy nuclear physics studies the behavior of nuclear matter in energy regimes typical of high-energy physics. The primary focus of this field is the study of heavy-ion collisions, as compared to lighter atoms in other particle accelerators. At sufficient collision energies, these types of collisions are theorized to produce the quark–gluon plasma. In peripheral nuclear collisions at high energies one expects to obtain information on the electromagnetic production of leptons and mesons that are not accessible in electron–positron colliders due to their much smaller luminosities.

Previous high-energy nuclear accelerator experiments have studied heavy-ion collisions using projectile energies of 1 GeV/nucleon at JINR and LBNL-Bevalac up to 158 GeV/nucleon at CERN-SPS. Experiments of this type, called "fixed-target" experiments, primarily accelerate a "bunch" of ions (typically around 10⁶ to 10⁸ ions per bunch) to speeds approaching the speed of light (0.999c) and smash them into a target of similar heavy ions. While all collision systems are interesting, great focus was applied in the late 1990s to symmetric collision systems of gold beams on gold targets at Brookhaven National Laboratory's Alternating Gradient Synchrotron (AGS) and uranium beams on uranium targets at CERN's Super Proton Synchrotron.

High-energy nuclear physics experiments are continued at the Brookhaven National Laboratory's Relativistic Heavy Ion Collider (RHIC) and at the CERN Large Hadron Collider. At RHIC the programme began with four experiments— PHENIX, STAR, PHOBOS, and BRAHMS—all dedicated to study collisions of highly relativistic nuclei. Unlike fixed-target experiments, collider experiments steer two accelerated beams of ions toward each other at (in the case of RHIC) six interaction regions. At RHIC, ions can be accelerated (depending on the ion size) from 100 GeV/nucleon to 250 GeV/nucleon. Since each colliding ion possesses this energy moving in opposite directions, the maximal energy of the collisions can achieve a center-of-mass collision energy of 200 GeV/nucleon for gold and 500 GeV/nucleon for protons.

The ALICE (A Large Ion Collider Experiment) detector at the LHC at CERN is specialized in studying Pb–Pb nuclei collisions at a center-of-mass energy of 2.76 TeV per nucleon pair. All major LHC detectors—ALICE, ATLAS, CMS and LHCb—participate in the heavy-ion programme.

History

The exploration of hot hadron matter and of multiparticle production has a long history initiated by theoretical work on multiparticle production by Enrico Fermi in the US and Lev Landau in the USSR. These efforts paved the way to the development in the early 1960s of the thermal description of multiparticle production and the statistical bootstrap model by Rolf Hagedorn. These developments led to search for and discovery of quark-gluon plasma. Onset of the production of this new form of matter remains under active investigation.

First collisions

The first heavy-ion collisions at modestly relativistic conditions were undertaken at the Lawrence Berkeley National Laboratory (LBNL, formerly LBL) at Berkeley, California, U.S.A., and at the Joint Institute for Nuclear Research (JINR) in Dubna, Moscow Oblast, USSR. At the LBL, a transport line was built to carry heavy ions from the heavy-ion accelerator HILAC to the Bevatron. The energy scale at the level of 1–2 GeV per nucleon attained initially yields compressed nuclear matter at few times normal nuclear density. The demonstration of the possibility of studying the properties of compressed and excited nuclear matter motivated research programs at much higher energies in accelerators available at BNL and CERN with relativist beams targeting laboratory fixed targets. The first collider experiments started in 1999 at RHIC, and LHC begun colliding heavy ions at one order of magnitude higher energy in 2010.

CERN operation

The LHC collider at CERN operates one month a year in the nuclear-collision mode, with Pb nuclei colliding at 2.76 TeV per nucleon pair, about 1500 times the energy equivalent of the rest mass. Overall 1250 valence quarks collide, generating a hot quark–gluon soup. Heavy atomic nuclei stripped of their electron cloud are called heavy ions, and one speaks of (ultra)relativistic heavy ions when the kinetic energy exceeds significantly the rest energy, as it is the case at LHC. The outcome of such collisions is production of very many strongly interacting particles.

In August 2012 ALICE scientists announced that their experiments produced quark–gluon plasma with temperature at around 5.5 trillion kelvins, the highest temperature achieved in any physical experiments thus far. This temperature is about 38% higher than the previous record of about 4 trillion kelvins, achieved in the 2010 experiments at the Brookhaven National Laboratory. The ALICE results were announced at the August 13 Quark Matter 2012 conference in Washington, D.C. The quark–gluon plasma produced by these experiments approximates the conditions in the universe that existed microseconds after the Big Bang, before the matter coalesced into atoms.

Objectives

There are several scientific objectives of this international research program:

• The formation and investigation of a new state of matter made of quarks and gluons, the quark–gluon plasma QGP, which prevailed in early universe in first 30 microseconds.
• The study of color confinement and the transformation of color confining = quark confining vacuum state to the excited state physicists call perturbative vacuum, in which quarks and gluons can roam free, which occurs at Hagedorn temperature;
• The study the origins of hadron (proton, neutron etc.) matter mass believed to be related to the phenomenon of quark confinement and vacuum structure.

Experimental program

This experimental program follows on a decade of research at the RHIC collider at BNL and almost two decades of studies using fixed targets at SPS at CERN and AGS at BNL. This experimental program has already confirmed that the extreme conditions of matter necessary to reach QGP phase can be reached. A typical temperature range achieved in the QGP created

${\displaystyle T=300\text{MeV}/k_{\text{B}}=3.3\times {10}^{12}\text{K}}$

is more than 100000 times greater than in the center of the Sun. This corresponds to an energy density

${\displaystyle ϵ=10{\text{GeV/fm}}^3=1.8\times {10}^{16}{\text{g/cm}}^3}$.

The corresponding relativistic-matter pressure is

${\displaystyle P\simeq \frac{1}{3}ϵ=0.52\times {10}^{31}\text{bar}.}$

Glueball

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Glueball

In particle physics, a glueball (also gluonium, gluon-ball) is a hypothetical composite particle. It consists solely of gluon particles, without valence quarks. Such a state is possible because gluons carry color charge and experience the strong interaction between themselves. Glueballs are extremely difficult to identify in particle accelerators, because they mix with ordinary meson states. In pure gauge theory, glueballs are the only states of the spectrum and some of them are stable. Theoretical calculations show that glueballs should exist at energy ranges accessible with current collider technology. However, due to the aforementioned difficulty (among others), they have so far not been observed and identified with certainty, although phenomenological calculations have suggested that an experimentally identified glueball candidate, denoted ${\displaystyle f_0(1710)}$, has properties consistent with those expected of a Standard Model glueball.

The prediction that glueballs exist is one of the most important predictions of the Standard Model of particle physics that has not yet been confirmed experimentally. Glueballs are the only particles predicted by the Standard Model with total angular momentum (J) (sometimes called "intrinsic spin") that could be either 2 or 3 in their ground states.

Experimental evidence was announced in 2021, by the TOTEM collaboration at the LHC in collaboration with the DØ collaboration at the former Tevatron collider at Fermilab, of odderon (a composite gluonic particle with odd C-parity) exchange. This exchange, associated with a quarkless three-gluon vector glueball, was identified in the comparison of proton–proton and proton–antiproton scattering.

Properties

In principle, it is theoretically possible for all properties of glueballs to be calculated exactly and derived directly from the equations and fundamental physical constants of quantum chromodynamics (QCD) without further experimental input. So, the predicted properties of these hypothetical particles can be described in exquisite detail using only Standard Model physics which have wide acceptance in the theoretical physics literature. But, there is considerable uncertainty in the measurement of some of the relevant key physical constants, and the QCD calculations are so difficult that solutions to these equations are almost always numerical approximations (calculated using several very different methods). This can lead to variation in theoretical predictions of glueball properties, like mass and branching ratios in glueball decays.

Constituent particles and color charge

Theoretical studies of glueballs have focused on glueballs consisting of either two gluons or three gluons, by analogy to mesons and baryons that have two and three quarks respectively. As in the case of mesons and baryons, glueballs would be QCD color charge neutral. The baryon number of a glueball is zero.

Total angular momentum

Double-gluon glueballs can have total angular momentum J = 0 (which are either scalar or pseudo-scalar) or J = 2 (tensor). Triple-gluon glueballs can have total angular momentum J = 1 (vector boson) or 3 (third-order tensor boson). All glueballs have integer total angular momentum which implies that they are bosons rather than fermions.

Glueballs are the only particles predicted by the Standard Model with total angular momentum ( J ) (sometimes called "intrinsic spin") that could be either 2 or 3 in their ground states, although mesons made of two quarks with J = 0 and J = 1 with similar masses have been observed and excited states of other mesons can have these values of total angular momentum.

Electric charge

All glueballs would have an electric charge of zero, as gluons themselves do not have an electric charge.

Mass and parity

Glueballs are predicted by quantum chromodynamics to be massive, despite the fact that gluons themselves have zero rest mass in the Standard Model. Glueballs with all four possible combinations of quantum numbers P (spatial parity) and C (charge parity) for every possible total angular momentum have been considered, producing at least fifteen possible glueball states including excited glueball states that share the same quantum numbers but have differing masses with the lightest states having masses as low as 1.4 GeV/c² (for a glueball with quantum numbers J = 0, P = +1, C = +1, or equivalently J ^PC = 0⁺⁺), and the heaviest states having masses as great as almost 5 GeV/c² (for a glueball with quantum numbers J = 0, P = +1, C = −1, or J ^PC = 0⁺⁻).

These masses are on the same order of magnitude as the masses of many experimentally observed mesons and baryons, as well as to the masses of the tau lepton, charm quark, bottom quark, some hydrogen isotopes, and some helium isotopes.

Stability and decay channels

Just as all Standard Model mesons and baryons, except the proton, are unstable in isolation, all glueballs are predicted by the Standard Model to be unstable in isolation, with various QCD calculations predicting the total decay width (which is functionally related to half-life) for various glueball states. QCD calculations also make predictions regarding the expected decay patterns of glueballs. For example, glueballs would not have radiative or two photon decays, but would have decays into pairs of pions, pairs of kaons, or pairs of eta mesons.

Practical impact on macroscopic low energy physics

Feynman diagram of a glueball (G) decaying to two pions (
π
). Such decays help the study of and search for glueballs.

Because Standard Model glueballs are so ephemeral (decaying almost immediately into more stable decay products) and are only generated in high energy physics, glueballs only arise synthetically in the natural conditions found on Earth that humans can easily observe. They are scientifically notable mostly because they are a testable prediction of the Standard Model, and not because of phenomenological impact on macroscopic processes, or their engineering applications.

Lattice QCD simulations

Lattice QCD provides a way to study the glueball spectrum theoretically and from first principles. Some of the first quantities calculated using lattice QCD methods (in 1980) were glueball mass estimates. Morningstar and Peardon computed in 1999 the masses of the lightest glueballs in QCD without dynamical quarks. The three lowest states are tabulated below. The presence of dynamical quarks would slightly alter these data, but also makes the computations more difficult. Since that time calculations within QCD (lattice and sum rules) find the lightest glueball to be a scalar with mass in the range of about 1000–1700 MeV/c². Lattice predictions for scalar and pseudoscalar glueballs, including their excitations, were confirmed by Dyson–Schwinger/Bethe–Salpeter equations in Yang–Mills theory.

J PC	mass
0++	1730±80 MeV/c2
2++	2400±120 MeV/c2
0−+	2590±130 MeV/c2

Experimental candidates

Particle accelerator experiments are often able to identify unstable composite particles and assign masses to those particles to a precision of approximately 10 MeV/c², without being able to immediately assign to the particle resonance that is observed all of the properties of that particle. Scores of such particles have been detected, although particles detected in some experiments but not others can be viewed as doubtful. Some of the candidate particle resonances that could be glueballs, although the evidence is not definitive, include the following:

Vector, pseudo-vector, or tensor glueball candidates

• X(3020) observed by the BaBar collaboration is a candidate for an excited state of the J ^PC = 2⁻⁺, 1⁺⁻ or 1⁻⁻ glueball states with a mass of about 3.02 GeV/c².

Scalar glueball candidates

• f₀(500) also known as σ – the properties of this particle are possibly consistent with a glueball of mass 1000 MeV/c² or 1500 MeV/c².
• f₀(980) – the structure of this composite particle is consistent with the existence of a light glueball.
• f₀(1370) – existence of this resonance is disputed but is a candidate for a glueball–meson mixing state
• f₀(1500) – existence of this resonance is undisputed but its status as a glueball–meson mixing state or pure glueball is not well established.
• f₀(1710) – existence of this resonance is undisputed but its status as a glueball–meson mixing state or pure glueball is not well established.

Other candidates

• Gluon jets at the LEP experiment show a 40% excess over theoretical expectations of electromagnetically neutral clusters which suggests that electromagnetically neutral particles expected in gluon-rich environments such as glueballs are likely to be present.

Many of these candidates have been the subject of active investigation for at least eighteen years. The GlueX experiment has been specifically designed to produce more definitive experimental evidence of glueballs.