Lepton

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Generations of matter

Type	First	Second	Third
Quarks
up-type	up	charm	top
down-type	down	strange	bottom
Leptons
charged	electron	muon	tau
neutral	electron neutrino	muon neutrino	tau neutrino

Lepton

Leptons are involved in several processes such as beta decay.
Composition	Elementary particle
Statistics	Fermionic
Generation	1st, 2nd, 3rd
Interactions	Electromagnetism, Gravitation, Weak
Symbol	l
Antiparticle	Antilepton ( l )
Types	6 (electron, electron neutrino, muon, muon neutrino, tau, tau neutrino)
Electric charge	+1 e, 0 e, −1 e
Color charge	No
Spin	​1⁄2

In particle physics, a lepton is an elementary particle of half-integer spin (spin ​¹⁄₂) that does not undergo strong interactions.^[1] Two main classes of leptons exist: charged leptons (also known as the electron-like leptons), and neutral leptons (better known as neutrinos). Charged leptons can combine with other particles to form various composite particles such as atoms and positronium, while neutrinos rarely interact with anything, and are consequently rarely observed. The best known of all leptons is the electron.

There are six types of leptons, known as flavours, grouped in three generations.^[2] The first-generation Leptons, also called electronic leptons, comprise the electron (
e⁻) and the electron neutrino (
ν
_e); the second are the muonic leptons, comprising the muon (
μ⁻) and the muon neutrino (
ν
_μ); and the third are the tauonic leptons, comprising the tau (
τ⁻) and the tau neutrino (
ν
_τ). Electrons have the least mass of all the charged leptons. The heavier muons and taus will rapidly change into electrons and neutrinos through a process of particle decay: the transformation from a higher mass state to a lower mass state. Thus electrons are stable and the most common charged lepton in the universe, whereas muons and taus can only be produced in high energy collisions (such as those involving cosmic rays and those carried out in particle accelerators).

Leptons have various intrinsic properties, including electric charge, spin, and mass. Unlike quarks however, leptons are not subject to the strong interaction, but they are subject to the other three fundamental interactions: gravitation, the weak interaction, and to electromagnetism, that is proportional to charge, thus is zero for the electrically neutral neutrinos.

For every lepton flavor there is a corresponding type of antiparticle, known as an antilepton, that differs from the lepton only in that some of its properties have equal magnitude but opposite sign. According to certain theories, neutrinos may be their own antiparticle. It is not currently known whether this is the case.

The first charged lepton, the electron, was theorized in the mid-19th century by several scientists^[3][4][5] and was discovered in 1897 by J. J. Thomson.^[6] The next lepton to be observed was the muon, discovered by Carl D. Anderson in 1936, which was classified as a meson at the time.^[7] After investigation, it was realized that the muon did not have the expected properties of a meson, but rather behaved like an electron, only with higher mass. It took until 1947 for the concept of "leptons" as a family of particle to be proposed.^[8] The first neutrino, the electron neutrino, was proposed by Wolfgang Pauli in 1930 to explain certain characteristics of beta decay.^[8] It was first observed in the Cowan–Reines neutrino experiment conducted by Clyde Cowan and Frederick Reines in 1956.^[8][9] The muon neutrino was discovered in 1962 by Leon M. Lederman, Melvin Schwartz, and Jack Steinberger,^[10] and the tau discovered between 1974 and 1977 by Martin Lewis Perl and his colleagues from the Stanford Linear Accelerator Center and Lawrence Berkeley National Laboratory.^[11] The tau neutrino remained elusive until July 2000, when the DONUT collaboration from Fermilab announced its discovery.^[12][13]

Leptons are an important part of the Standard Model. Electrons are one of the components of atoms, alongside protons and neutrons. Exotic atoms with muons and taus instead of electrons can also be synthesized, as well as lepton–antilepton particles such as positronium.

Etymology

The name lepton comes from the Greek λεπτός leptós, "fine, small, thin" (neuter nominative/accusative singular form: λεπτόν leptón);^[14][15] the earliest attested form of the word is the Mycenaean Greek 𐀩𐀡𐀵, re-po-to, written in Linear B syllabic script.^[16] Lepton was first used by physicist Léon Rosenfeld in 1948:^[17]

Following a suggestion of Prof. C. Møller, I adopt—as a pendant to "nucleon"—the denomination "lepton" (from λεπτός, small, thin, delicate) to denote a particle of small mass.

The etymology incorrectly implies that all the leptons are of small mass. When Rosenfeld named them, the only known leptons were electrons and muons, whose masses are indeed small compared to nucleons—the mass of an electron (0.511 MeV/c²)^[18] and the mass of a muon (with a value of 105.7 MeV/c²)^[19] are fractions of the mass of the "heavy" proton (938.3 MeV/c²).^[20] However, the mass of the tau (discovered in the mid 1970s) (1777 MeV/c²)^[21] is nearly twice that of the proton, and about 3,500 times that of the electron.

History

A muon transmutes into a muon neutrino by emitting a
W⁻ boson. The
W⁻ boson subsequently decays into an electron and an electron antineutrino.

The first lepton identified was the electron, discovered by J.J. Thomson and his team of British physicists in 1897.^[22][23] Then in 1930 Wolfgang Pauli postulated the electron neutrino to preserve conservation of energy, conservation of momentum, and conservation of angular momentum in beta decay.^[24] Pauli theorized that an undetected particle was carrying away the difference between the energy, momentum, and angular momentum of the initial and observed final particles. The electron neutrino was simply called the neutrino, as it was not yet known that neutrinos came in different flavours (or different "generations").

Nearly 40 years after the discovery of the electron, the muon was discovered by Carl D. Anderson in 1936. Due to its mass, it was initially categorized as a meson rather than a lepton.^[25] It later became clear that the muon was much more similar to the electron than to mesons, as muons do not undergo the strong interaction, and thus the muon was reclassified: electrons, muons, and the (electron) neutrino were grouped into a new group of particles—the leptons. In 1962, Leon M. Lederman, Melvin Schwartz, and Jack Steinberger showed that more than one type of neutrino exists by first detecting interactions of the muon neutrino, which earned them the 1988 Nobel Prize, although by then the different flavours of neutrino had already been theorized.^[26]

The tau was first detected in a series of experiments between 1974 and 1977 by Martin Lewis Perl with his colleagues at the SLAC LBL group.^[27] Like the electron and the muon, it too was expected to have an associated neutrino. The first evidence for tau neutrinos came from the observation of "missing" energy and momentum in tau decay, analogous to the "missing" energy and momentum in beta decay leading to the discovery of the electron neutrino. The first detection of tau neutrino interactions was announced in 2000 by the DONUT collaboration at Fermilab, making it the latest particle of the Standard Model to have been directly observed,^[28] apart from the Higgs boson, which has been discovered in 2012.

Although all present data is consistent with three generations of leptons, some particle physicists are searching for a fourth generation. The current lower limit on the mass of such a fourth charged lepton is 100.8 GeV/c²,^[29] while its associated neutrino would have a mass of at least 45.0 GeV/c².^[30]

Properties

Spin and chirality

Left-handed and right-handed helicities

Leptons are spin-​¹⁄₂ particles. The spin-statistics theorem thus implies that they are fermions and thus that they are subject to the Pauli exclusion principle: No two leptons of the same species can be in exactly the same state at the same time. Furthermore, it means that a lepton can have only two possible spin states, namely up or down.

A closely related property is chirality, which in turn is closely related to a more easily visualized property called helicity. The helicity of a particle is the direction of its spin relative to its momentum; particles with spin in the same direction as their momentum are called right-handed and otherwise they are called left-handed. When a particle is massless, the direction of its momentum relative to its spin is frame independent, while for massive particles it is possible to 'overtake' the particle by a Lorentz transformation flipping the helicity. Chirality is a technical property (defined through the transformation behaviour under the Poincaré group) that agrees with helicity for (approximately) massless particles and is still well defined for massive particles.

In many quantum field theories, such as quantum electrodynamics and quantum chromodynamics, left- and right-handed fermions are identical. However, in the Standard Model, left-handed and right-handed fermions are treated asymmetrically. Only left-handed fermions participate in the weak interaction, while there are no right-handed neutrinos. This is an example of parity violation. In the literature, left-handed fields are often denoted by a capital L subscript (e.g.
e⁻_L) and right-handed fields are denoted by a capital R subscript.

Electromagnetic interaction

Lepton–photon interaction

One of the most prominent properties of leptons is their electric charge, Q. The electric charge determines the strength of their electromagnetic interactions. It determines the strength of the electric field generated by the particle (see Coulomb's law) and how strongly the particle reacts to an external electric or magnetic field (see Lorentz force). Each generation contains one lepton with Q = −e (conventionally the charge of a particle is expressed in units of the elementary charge) and one lepton with zero electric charge. The lepton with electric charge is commonly simply referred to as a 'charged lepton' while the neutral lepton is called a neutrino. For example, the first generation consists of the electron
e⁻ with a negative electric charge and the electrically neutral electron neutrino
ν
_e.

In the language of quantum field theory, the electromagnetic interaction of the charged leptons is expressed by the fact that the particles interact with the quantum of the electromagnetic field, the photon. The Feynman diagram of the electron-photon interaction is shown on the right.

Because leptons possess an intrinsic rotation in the form of their spin, charged leptons generate a magnetic field. The size of their magnetic dipole moment μ is given by

${\displaystyle \mu =g{\frac {Q\hbar }{4m}},}$

where m is the mass of the lepton and g is the so-called g-factor for the lepton. First order approximation quantum mechanics predicts that the g-factor is 2 for all leptons. However, higher order quantum effects caused by loops in Feynman diagrams introduce corrections to this value. These corrections, referred to as the anomalous magnetic dipole moment, are very sensitive to the details of a quantum field theory model and thus provide the opportunity for precision tests of the standard model. The theoretical and measured values for the electron anomalous magnetic dipole moment are within agreement within eight significant figures.^[31]

Weak interaction

The weak interactions of the first generation leptons.

In the Standard Model, the left-handed charged lepton and the left-handed neutrino are arranged in doublet (
ν
_eL,
e⁻_L) that transforms in the spinor representation (T = ​¹⁄₂) of the weak isospin SU(2) gauge symmetry. This means that these particles are eigenstates of the isospin projection T₃ with eigenvalues ​¹⁄₂ and −​¹⁄₂ respectively. In the meantime, the right-handed charged lepton transforms as a weak isospin scalar (T = 0) and thus does not participate in the weak interaction, while there is no evidence that a right-handed neutrino exists at all.

The Higgs mechanism recombines the gauge fields of the weak isospin SU(2) and the weak hypercharge U(1) symmetries to three massive vector bosons (
W⁺,
W⁻,
Z⁰) mediating the weak interaction, and one massless vector boson, the photon, responsible for the electromagnetic interaction. The electric charge Q can be calculated from the isospin projection T₃ and weak hypercharge Y_W through the Gell-Mann–Nishijima formula,

Q = T₃ + ½ Y_W

To recover the observed electric charges for all particles, the left-handed weak isospin doublet (
ν
_eL,
e⁻_L) must thus have Y_W = −1, while the right-handed isospin scalar e⁻
_R must have Y_W = −2. The interaction of the leptons with the massive weak interaction vector bosons is shown in the figure on the left.

Mass

In the Standard Model, each lepton starts out with no intrinsic mass. The charged leptons (i.e. the electron, muon, and tau) obtain an effective mass through interaction with the Higgs field, but the neutrinos remain massless. For technical reasons, the masslessness of the neutrinos implies that there is no mixing of the different generations of charged leptons as there is for quarks. This is in close agreement with current experimental observations.^[32]

However, it is known from experiments—most prominently from observed neutrino oscillations^[33]—that neutrinos do in fact have some very small mass, probably less than 2 eV/c².^[34] This implies the existence of physics beyond the Standard Model. The currently most favoured extension is the so-called seesaw mechanism, which would explain both why the left-handed neutrinos are so light compared to the corresponding charged leptons, and why we have not yet seen any right-handed neutrinos.

Leptonic numbers

The members of each generation's weak isospin doublet are assigned leptonic numbers that are conserved under the Standard Model.^[35] Electrons and electron neutrinos have an electronic number of L_e = 1, while muons and muon neutrinos have a muonic number of L_μ = 1, while tau particles and tau neutrinos have a tauonic number of L_τ = 1. The antileptons have their respective generation's leptonic numbers of −1.
Conservation of the leptonic numbers means that the number of leptons of the same type remains the same, when particles interact. This implies that leptons and antileptons must be created in pairs of a single generation. For example, the following processes are allowed under conservation of leptonic numbers:

Each generation forms a weak isospin doublet.

e⁻ +
e⁺ →
γ
+
γ
,

τ⁻ +
τ⁺ →
Z⁰ +
Z⁰,

but not these:

γ
→
e⁻ +
μ⁺,

W⁻ →
e⁻ +
ν
_τ,

Z⁰ →
μ⁻ +
τ⁺.

However, neutrino oscillations are known to violate the conservation of the individual leptonic numbers. Such a violation is considered to be smoking gun evidence for physics beyond the Standard Model. A much stronger conservation law is the conservation of the total number of leptons (L), conserved even in the case of neutrino oscillations, but even it is still violated by a tiny amount by the chiral anomaly.

Universality

The coupling of the leptons to gauge bosons are flavour-independent (i.e., the interactions between leptons and gauge bosons are the same for all leptons).^[35] This property is called lepton universality and has been tested in measurements of the tau and muon lifetimes and of Z boson partial decay widths, particularly at the Stanford Linear Collider (SLC) and Large Electron-Positron Collider (LEP) experiments.^{[36]:241–243[37]:138}

The decay rate (Γ) of muons through the process
μ⁻ →
e⁻ +
ν
_e +
ν
_μ is approximately given by an expression of the form (see muon decay for more details)^[35]

${\displaystyle \Gamma \left(\mu ^{-}\rightarrow e^{-}+{\bar {\nu _{e}}}+\nu _{\mu }\right)=K_{1}G_{F}^{2}m_{\mu }^{5},}$

where K₁ is some constant, and G_F is the Fermi coupling constant. The decay rate of tau particles through the process
τ⁻ →
e⁻ +
ν
_e +
ν
_τ is given by an expression of the same form^[35]

${\displaystyle \Gamma \left(\tau ^{-}\rightarrow e^{-}+{\bar {\nu _{e}}}+\nu _{\tau }\right)=K_{2}G_{F}^{2}m_{\tau }^{5},}$

where K₂ is some constant. Muon–Tauon universality implies that K₁ = K₂. On the other hand, electron–muon universality implies^[35]

${\displaystyle \Gamma \left(\tau ^{-}\rightarrow e^{-}+{\bar {\nu _{e}}}+\nu _{\tau }\right)=\Gamma \left(\tau ^{-}\rightarrow \mu ^{-}+{\bar {\nu _{\mu }}}+\nu _{\tau }\right).}$

This explains why the branching ratios for the electronic mode (17.85%) and muonic (17.36%) mode of tau decay are equal (within error).^[21]

Universality also accounts for the ratio of muon and tau lifetimes. The lifetime of a lepton (τ_l) is related to the decay rate by^[35]

${\displaystyle \tau _{l}={\frac {B\left(l^{-}\rightarrow e^{-}+{\bar {\nu _{e}}}+\nu _{l}\right)}{\Gamma \left(l^{-}\rightarrow e^{-}+{\bar {\nu _{e}}}+\nu _{l}\right)}},}$

where B(x → y) and Γ(x → y) denotes the branching ratios and the resonance width of the process x → y.

The ratio of tau and muon lifetime is thus given by^[35]

${\displaystyle {\frac {\tau _{\tau }}{\tau _{\mu }}}={\frac {B\left(\tau ^{-}\rightarrow e^{-}+{\bar {\nu _{e}}}+\nu _{\tau }\right)}{B\left(\mu ^{-}\rightarrow e^{-}+{\bar {\nu _{e}}}+\nu _{\mu }\right)}}\left({\frac {m_{\mu }}{m_{\tau }}}\right)^{5}.}$

Using the values of the 2008 Review of Particle Physics for the branching ratios of muons^[19] and tau^[21] yields a lifetime ratio of ~1.29×10⁻⁷, comparable to the measured lifetime ratio of ~1.32×10⁻⁷. The difference is due to K₁ and K₂ not actually being constants; they depend on the mass of leptons.

Recent tests of lepton universality in B meson decays, performed by the LHCb, BaBar and Belle experiments, have shown consistent deviations from the Standard Model predictions. However the statistical significance is not yet high enough to claim an observation of new physics.^[38]

Baryon asymmetry

From Wikipedia, the free encyclopedia

In physics, the baryon asymmetry problem, also known as the matter asymmetry problem or the matter-antimatter asymmetry problem,^[1][2] is the observed imbalance in baryonic matter (the type of matter experienced in everyday life) and antibaryonic matter in the observable universe. Neither the standard model of particle physics, nor the theory of general relativity provides a known explanation for why this should be so, and it is a natural assumption that the universe be neutral with all conserved charges.^[3] The Big Bang should have produced equal amounts of matter and antimatter. Since this does not seem to have been the case, it is likely some physical laws must have acted differently or did not exist for matter and antimatter. Several competing hypotheses exist to explain the imbalance of matter and antimatter that resulted in baryogenesis. However, there is as of yet no consensus theory to explain the phenomenon. As remarked in a 2012 research paper, "The origin of matter remains one of the great mysteries in physics."^[4]

Sakharov conditions

In 1967, Andrei Sakharov proposed^[5] a set of three necessary conditions that a baryon-generating interaction must satisfy to produce matter and antimatter at different rates. These conditions were inspired by the recent discoveries of the cosmic background radiation^[6] and CP-violation in the neutral kaon system.^[7] The three necessary "Sakharov conditions" are:

• Baryon number ${\displaystyle B}$ violation.
• C-symmetry and CP-symmetry violation.
• Interactions out of thermal equilibrium.

Baryon number violation

Baryon number violation is obviously a necessary condition to produce an excess of baryons over anti-baryons. But C-symmetry violation is also needed so that the interactions which produce more baryons than anti-baryons will not be counterbalanced by interactions which produce more anti-baryons than baryons. CP-symmetry violation is similarly required because otherwise equal numbers of left-handed baryons and right-handed anti-baryons would be produced, as well as equal numbers of left-handed anti-baryons and right-handed baryons. Finally, the interactions must be out of thermal equilibrium, since otherwise CPT symmetry would assure compensation between processes increasing and decreasing the baryon number.^[8]

Currently, there is no experimental evidence of particle interactions where the conservation of baryon number is broken perturbatively: this would appear to suggest that all observed particle reactions have equal baryon number before and after. Mathematically, the commutator of the baryon number quantum operator with the (perturbative) Standard Model hamiltonian is zero: ${\displaystyle [B,H]=BH-HB=0}$. However, the Standard Model is known to violate the conservation of baryon number only non-perturbatively: a global U(1) anomaly. To account for baryon violation in baryogenesis, such events (including proton decay) can occur in Grand Unification Theories (GUTs) and supersymmetric (SUSY) models via hypothetical massive bosons such as the X boson.

CP-Symmetry Violation

The second condition for generating baryon asymmetry – violation of charge-parity symmetry – is that a process is able to happen at a different rate to its antimatter counterpart. In the Standard Model, CP violation appears as a complex phase in the quark mixing matrix of the weak interaction. There may also be a non-zero CP-violating phase in the neutrino mixing matrix, but this is currently unmeasured. CP violation was first observed in the 1964 Fitch-Cronin experiment with neutral kaons, which resulted in the 1980 Nobel Prize in physics (direct CP-violation, that is violation of CP-symmetry in a decay process, was discovered later, in 1999). Due to CPT symmetry, violation of CP-symmetry demands violation of time inversion symmetry, or T-symmetry. Despite the allowance for CP-violation in the Standard Model, it is insufficient to account for the observed baryon asymmetry of the universe given the limits on baryon number violation, meaning that beyond-Standard Model sources are needed.

A possible new source of CP violation was found at the Large Hadron Collider (LHC) by the LHCb collaboration during the first three years of LHC operations. The experiment analyzed the decays of two particles, the bottom Lambda (Λ_b⁰) and its antiparticle, and compared the distributions of decay products. The data showed an asymmetry of up to 20% of CP-violation sensitive quantities, implying a breaking of CP-symmetry. This analysis will need to be confirmed by more data from subsequent runs of the LHC.^[9]

Interactions out of thermal equilibrium

In the out-of-equilibrium decay scenario,^[10] the last condition states that the rate of a reaction which generates baryon-asymmetry must be less than the rate of expansion of the universe. In this situation the particles and their corresponding antiparticles do not achieve thermal equilibrium due to rapid expansion decreasing the occurrence of pair-annihilation.

Other explanations

Regions of the universe where antimatter dominates

Another possible explanation of the apparent baryon asymmetry is that matter and antimatter are essentially separated into different, widely separated regions of the universe. The formation of antimatter galaxies was originally thought to explain the baryon asymmetry, as from a distance, antimatter atoms are indistinguishable from matter atoms; both produce light (photons) in the same way. Along the boundary between matter and antimatter regions, however, annihilation (and the subsequent production of gamma radiation) would be detectable, depending on its distance and the density of matter and antimatter. Such boundaries, if they exist, would likely lie in deep intergalactic space. The density of matter in intergalactic space is reasonably well established at about one atom per cubic meter.^[11][12] Assuming this is a typical density near a boundary, the gamma ray luminosity of the boundary interaction zone can be calculated. No such zones have been detected, but 30 years of research have placed bounds on how far they might be. On the basis of such analyses, it is now deemed unlikely that any region within the observable universe is dominated by antimatter.^[4]

One attempt to explain the lack of observable interfaces between matter and antimatter dominated regions is that they are separated by a Leidenfrost layer of very hot matter created by the energy released from annihilation. This is similar to the manner in which water may be separated from a hot plate by a layer of evaporated vapor, delaying the evaporation of more water.

Electric dipole moment

The presence of an electric dipole moment (EDM) in any fundamental particle would violate both parity (P) and time (T) symmetries. As such, an EDM would allow matter and antimatter to decay at different rates leading to a possible matter-antimatter asymmetry as observed today. Many experiments are currently being conducted to measure the EDM of various physical particles. All measurements are currently consistent with no dipole moment. However, the results do place rigorous constraints on the amount of symmetry violation that a physical model can permit. The most recent EDM limit, published in 2014, was that of the ACME Collaboration, which measured the EDM of the electron using a pulsed beam of thorium monoxide (ThO) molecules.^[13]

Baryon asymmetry parameter

The challenges to the physics theories are then to explain how to produce this preference of matter over antimatter, and also the magnitude of this asymmetry. An important quantifier is the asymmetry parameter,

${\displaystyle \eta ={\frac {n_{B}-n_{\bar {B}}}{n_{\gamma }}}}$.

This quantity relates the overall number density difference between baryons and antibaryons (n_B and n_B, respectively) and the number density of cosmic background radiation photons n_γ.

According to the Big Bang model, matter decoupled from the cosmic background radiation (CBR) at a temperature of roughly 3000 kelvin, corresponding to an average kinetic energy of 3000 K / (10.08×10³ K/eV) = 0.3 eV. After the decoupling, the total number of CBR photons remains constant. Therefore, due to space-time expansion, the photon density decreases. The photon density at equilibrium temperature T per cubic centimeter, is given by

${\displaystyle n_{\gamma }={\frac {1}{\pi ^{2}}}{\left({\frac {k_{B}T}{\hbar c}}\right)}^{3}\int _{0}^{\infty }{\frac {x^{2}}{e^{x}-1}}dx={\frac {2\zeta (3)}{\pi ^{2}}}{\left({\frac {k_{B}T}{\hbar c}}\right)}^{3}\approx 20.3\left({\frac {T}{1{\text{K}}}}\right)^{3}{\text{cm}}^{-3}}$ 

with k_B as the Boltzmann constant, ħ as the Planck constant divided by 2π and c as the speed of light in vacuum, and ζ(3) as Apéry's constant. At the current CBR photon temperature of 2.725 K, this corresponds to a photon density n_γ of around 411 CBR photons per cubic centimeter.

Therefore, the asymmetry parameter η, as defined above, is not the "good" parameter. Instead, the preferred asymmetry parameter uses the entropy density s,

${\displaystyle \eta _{s}={\frac {n_{B}-n_{\bar {B}}}{s}}}$

because the entropy density of the universe remained reasonably constant throughout most of its evolution. The entropy density is

${\displaystyle s\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathrm {entropy} }{\mathrm {volume} }}={\frac {p+\rho }{T}}={\frac {2\pi ^{2}}{45}}g_{*}(T)T^{3}}$

with p and ρ as the pressure and density from the energy density tensor T_μν, and g_* as the effective number of degrees of freedom for "massless" particles (inasmuch as mc² ≪ k_BT holds) at temperature T,

${\displaystyle g_{*}(T)=\sum _{\mathrm {i=bosons} }g_{i}{\left({\frac {T_{i}}{T}}\right)}^{3}+{\frac {7}{8}}\sum _{\mathrm {j=fermions} }g_{j}{\left({\frac {T_{j}}{T}}\right)}^{3}}$,

for bosons and fermions with g_i and g_j degrees of freedom at temperatures T_i and T_j respectively. At the present era, s = 7.04n_γ.