## Thursday, February 6, 2020

### Selected 2019 Particle Data Group Physical Constant Values

There are 30 experimentally determined fundamental physical constants in the core theory made up of the Standard Model and General Relativity. This is close to, but not quite, a minimal set of physical constants for these theories, as a small number of them can be derived in principle from some of the others (e.g. the Z boson mass could be, in principle derived from the W boson mass and two of the three coupling constants).  As the data below indicate, these are known to varying degrees of precision. Lighter particle masses and smaller quantities are known to greater absolute precision, while relative precision varies considerably.

One of the great unsolved problems of physics is how to derive more of these experimentally measured constants from a smaller number of fundamental quantities through a "deeper" or "within the Standard Model" theory, as they naively appear to have some sort of pattern and functional relationship to each other, but no widely accepted physics theories has done so successfully in the half a century since the Standard Model was devised.

There are also integer or simple fraction constants in the Standard Model that are set by theory like the quantum numbers of the various standard model fundamental particles, the masses of the photon and the gluon (zero), and the CP violation parameter of the strong force (zero).

The global average of the best available measurements of these measurements, compiled by the Particle Data Group (link in sidebar) which is the "default" source for these values by physicists, with a handful of additional notable data points, are set forth below. These are updated at least once a year, and sometimes more often (for physical constants for which there have been new experimental measurements) which is why I am posting the current values for easy reference.

In addition to the Standard Model constants, the core equations of the Standard Model are summed up in a Standard Model Lagrangian and some related renormalization beta functions, along with definitions of what the terms in it mean and conventions regarding how they are operationalized, together with some foundational concepts of quantum mechanics generally. The Lagrangian take a full t-shirt to set forth in full detail in fairly small print, and each of the beta functions (one for each of the 26 constants particular to the Standard Model) is similar in size.

The electromagnetic and weak force components of the Standard Model can be applied in a practical manner basically directly from the Lagrangian with high precision.

The QCD component is, in practice, to difficult to calculate with analytically from first principles in all but the most simplified, stylized and symmetric circumstances, and so a variety of approximations with limited ranges of applicability including Lattice QCD for most non-perturbative QCD calculations, and various perturbative QCD methods for higher energy scale applications, are used rather than the exact QCD Lagrangian terms, in most cases. There are roughly half a dozen to a dozen perturbative QCD approximations used to do actual calculations in wide use.

Core theory is known to be an incomplete description of the fundamental laws of nature because it does not explain the phenomena attributed to dark matter (although it does explain "dark energy").

Standard Model fundamental boson masses (3)

Higgs boson mass 125.10 ± 0.14 GeV

W boson mass 80.379 ± 0.012 GeV (the W boson mass/Z boson mass ratio is 0.088147 ± 0.00013)

Z boson mass 91.1876 ± 0.0021 GeV

Standard Model fundamental charged fermion masses (9)

The heavy quarks (c, b ad t) and the charged lepton masses are pole masses (i.e. evaluated at the energy scale where the rest mass of the particle and the energy scale are the same) and at a 2 GeV energy scale for the light quarks (u, d and s).

Electron mass 0.5109989461 ± 0.0000000031 MeV

Muon mass 105.6583745 ± 0.0000024 MeV

Tau lepton mass 1,776.86 ± 0.12 MeV

Up quark mass 2.16 + 0.49 - 0.26 MeV

Down quark mass 4.57 + 0.48 - 0.17 MeV

Strange quark mass 93 + 11 -5 MeV

Charm quark mass 1.27 ± 0.02 GeV

Bottom quark mass 4.18 + 0.03 - 0.02 GeV

Top quark mass 172.9 ± 0.4 GeV (direct measurements)

The charged lepton masses and top quark masses are direct measurements. The up, down, strange, charm and bottom quarks are only observed indirectly since they are always confined in hadrons, and in practice, are determined via Lattice QCD methods (which sometimes purport to measure these quantities with relative precisions greater than shown above from the Particle Data Group).

It is not feasible to measure the pole masses of the up, down and strange quarks, which are ill defined, because these quarks are always confined in hadrons (i.e. composite particles with components bound by the strong force) which characteristic mass-energies far in excess of their masses a regime in which perturbative QCD in which pole mass is well defined, is not applicable. The lightest hadrons containing only up and down valence quarks are the proton, the neutron and the pions. The lightest hadrons containing only strange quarks are kaons.

Strictly speaking, the "Yukawas" of these particles, rather than their masses, are the fundamental quantities in the Standard Model. A Yukawa is basically the "Higgs field charge" of a particle, while the mass of each of these particles is a function of the Higgs vev (discussed below) and its Yukawa.

Standard Model Coupling Constants (3)

Strong force, aka SU(3), coupling constant 0.1179 ± 0.0010 (this is a downward revision as of 2019) evaluated at the Z boson mass-energy scale.

Fermi coupling constant 1.166 378 7 ± 0.000 0006 × 10^−5 GeV^−2  (functionally related to the weak force, aka SU(2), coupling constant)

Fine structure constant 7.297 352 5693 ± 0.000 000 0011 ×10^−3 = 1/137.035 999 084 ± 0.000 000 021 (functionally related to the electromagnetic force, aka U(1), coupling constant)

Standard Model CKM Matrix Physical Constants (4)

The CKM matrix in the Standard Model sets forth the probability of each kind of quark producing a different flavor of quark via a W boson mediated interaction (a flavor changing charged current). This has nine components, the square of which is a probability, and can be summarized with four parameters that can be defined in multiple ways, all of which produce the same matrix elements.

Standard Model Neutrino Physical Constants (7)

Neutrino masses and mixings (7)

In principle, the Standard Model has seven parameters related to neutrinos: three neutrino mass eigenvalues and four parameters that describe the PMNS matrix which describes the probability of a neutrino of a particular flavor oscillating into a neutrino of a different flavor in a manner closely analogous to that of the CKM matrix (as in the case of the CKM matrix there are multiple ways to parameterize the PMNS matrix which produce identical values for its nine entires).

In practice, some of the PMNS matrix parameters (particularly the charge parity violation phase) are known only very imprecisely, and we know the magnitude of the differences between the mass eigenvalues and have certain floors and ceilings to the sum of the three neutrino eigenvalues (from very different methodologies) and to the maximum masses of each of the three neutrino mass eigenvalues, but have only very imprecise ranges of values for the absolute masses of the neutrino mass eigenstate. In particular, it is not yet certain if the neutrino masses observe the same lightest, medium and heaviest mass hierarchy with generation seen in the other fundamental fermions of the Standard Model or not, although cosmology and mixing parameter values taken together increasingly favor a normal mass hierarchy.

The Standard Model and the PMNS matrix, implicitly assume that neutrinos have what is known as "Dirac mass" rather than Majorana mass (in which case there would be additional CP violating phases), but without associating that with the Higgs mechanism use for the fundamental charged particles of the Standard Model and without truly explaining this with the neat consistency of the electroweak model explanation of other fundamental particle masses in the Standard Model at all.

The data below are in the form in which the actual values are directly measured (sine squared values of real valued mixing angles and squared values of mass differences) rather than the underlying parameter values which are easily derived from them with a scientific calculator.

in^2(theta23):

The full data for the parameters shown as ". . . " in the chart above are as follows:

sin^2(theta23) theta23 could be either side of a 45 degree angle based upon existing measurements and assuming a "normal" mass hierarchy for the neutrino masses. But existing experiments, while capable of determining that theta23 is not 45 degrees, but can't determine if it is greater or smaller than those values, which is why there is both an octant I and an octant II value.

 $\mathbf{0.512}{}_{\mathbf{-}\mathbf{0.022}}^{\mathbf{+}\mathbf{0.019}}$ OUR FIT  Normal ordering, octant I $\mathbf{0.542}{}_{\mathbf{-}\mathbf{0.022}}^{\mathbf{+}\mathbf{0.019}}$ OUR FIT  Normal ordering, octant II −0.028$\mathbf{0.536}{}_{\mathbf{-}\mathbf{0.028}}^{\mathbf{+}\mathbf{0.023}}$ OUR FIT  Inverted ordering

Delta m32^2 with normal ordering is 2.444 ± 0.034 (the number shown in chart is inverse mass hierarchy).

Sum of neutrino masses Σmν < 0.12 eV (95%, CMB + BAO); ≥ 0.06 eV (mixing).

Directly measured neutrino mass limits:

Neutrino Flavors

The number of neutrino flavors in the Standard Model is a theoretically determined, rather than experimentally measured value, but the experimental measurements are consistent at the two sigma level with the Standard Model value of 3:

Effective number of neutrino flavors Neff 2.99 ± 0.17 (cosmology measurements) (the expected value of this measured physical constant with exactly three types of neutrinos is 3.045 rather than zero for technical reasons related to the way that radiation impacts the relevant observables). This measurement includes all light neutrinos (up to the order of roughly 1-10 eV in mass) that oscillate with each other, and is independent of whether or not the interact via the weak force.

Number of light (i.e. less than 45 GeV) neutrino flavors from Z boson decays Nν = 2.984 ± 0.008. The Standard Model theoretical value is 3.

General Relativity Physical Constants (2)

Newton's constant G 6.708 83 ± 0.000 15 × 10^−39 hc (GeV/c^2)^−2

Cosmological constant Λ 1.088 ± 0.030 × 10^−56 cm^−2

General relativity can be boiled down to a single tensor valued equation with definitions of the symbols and terms used in it and the conventions about how this equation is applied. It is difficult, although not entirely intractable to compute predictions with analytically outside highly stylized and symmetric circumstances, and in other circumstances, approximations motivated by the core equation are used.

The cosmological constant is small enough that it can be ignored even in quite large scale general relativity calculations, such as those on the scale of whole galaxies. The cosmological constant is one of multiple ways of describing what is also known as "dark energy".

Note also that the Standard Model and General Relativity are inconsistent at a fundamental level. The Standard Model is a quantum field theory, while General Relativity is a deterministic classical physics theory. Efforts to formulate a theory of quantum gravity to bridge this gap have been elusive. The Standard Model is fully consistent, however, with Special Relativity. Fortunately, because there are few overlaps between the domains of applicability of the Standard Model an those of General Relativity, and there are simplifications and approximations that can be used in the places where there are overlaps (like the internal properties of neutron stars), the inconsistencies between the Standard Model and General Relativity doesn't greatly impair practical applications of these core theories. Usually, general relativity can be safely ignored in Standard Model applications, and the Standard Model's subatomic scale laws of nature are irrelevant to applications of General Relativity.

Other fundamental constants and unit conversions (2)

In the SI system (the most up to date version of the metric system) to physical constants, the speed of light and Planck's constant, have been used to define the meter per second, and the Joule*second, respectively, at values as close as experimental measurements as of 2019 allowed to the values of the more arbitrary values of the meter, second and Joule prior to being defined in terms of fundamental constants, in order to insure back compatibility.

Both of these constants are used in the Standard Model. The speed of light is a physical constant also used in General Relativity.

Speed of light in vacuum c 299 792 458 m s−1  (exact due to revised definition of units used)

Planck constant h 6.626 070 15×10−34 J s (exact due to revised definition of units used)

The reduced Planck constant a.k.a. "h bar" h (h divided by two pi) is 6.582 119 569. . . × 10−22 MeV s. This value is exact, but it a transcendental number since it includes a factor of pi (i.e. it has an infinite number of base ten digits).

The units of electromagnetic charge in the SI system, in contrast, is not defined directly in terms of fundamental electron charge. One electron charge in SI units is 1.602 176 634×10^−19 C

Footnote Regarding Derived Physical Constants

Many other physical constants, in principle, are possible to derive from the fundamental constants in the Standard Model. In practice, however, direct measurements of these quantities are actually used. Among them are:

* The decay width of every fundamental particle and hadron (this is inversely proportional to the half-life and mean lifetime of a an unstable particle). In theory, electrons, photons and gluons are stable and do not decay or oscillate in the Standard Model, as are protons and as are neutrons that are bound in stable atomic nuclei. The decay width of a free quark other than a top quark, however, isn't terribly meaningful or well defined in the Standard Model, as other flavors of quarks are always confined outside quark-gluon plasma.

* The electric and magnetic dipole moments of every particle.

* The decay width and branching fraction of every possible decay path of fundamental particles and hadrons that are not stable. The hypothetical lepton number violating processes like neutrinoless double beta decay and flavor changing neutral currents are not possible at the tree level in the Standard Model and are profoundly suppressed beyond the tree level in the Standard Model.

* The nuclear binding force between protons and neutrons in an atom.

* The half-life of every unstable atomic isotype.

* The rest mass of every possible hadron.

* The chemical properties of every atomic isotype.

* All of the Standard Model constants other than c and Planck's constant, at energy scales other than those at which they are conventionally quoted per renormalization beta functions, all coefficients of which can be derived from theoretical first principles. This may be ill defined at extremely high Planck energy scales that have never existed anywhere in the universe at any time more than tiny factions of a nanosecond after the Big Bang.

* The characteristic QCD energy scale called Λ.

* The Higgs vacuum expectation value (vev), which is also derived from the Fermi coupling constant, per a non-PDG source using PDG values, is 246,227.9579 MeV with an uncertainty on the order of 0.0010 MeV.

Some of the derived physical constants, such as the masses of the proton and neutron, are known to far greater precision than the fundamental constants from which they are derived.

This is particularly common for physical constants that require QCD to determine. This is because doing calculations with QCD is computationally extremely difficult to do to any given level of precision, and for the closely related reason that as a result of this computational difficulty, it is very difficult to measure the quark masses and the strong force coupling constant to high relative precision. The roughly 1% relative precision in the strong force coupling constant places a ceiling on the precision of all QCD calculations to the extent that they depend upon this fundamental physical constant as all QCD calculations do to some degree or another.

Footnote regarding non-fundamental, non-derived physical constants

Fundamental physical constants are those necessary to apply the "laws of nature" set forth in the Standard Model and General Relativity and any beyond the Standard Model theory that adds additional such constants. Derived physical constants are those that can, in principle, be determined from the fundamental physical constants.

There are also a variety of observationally measured physical constants which are neither fundamental, nor capable of being derived from fundamental physical constants. These are mostly descriptive constants of particular physical systems including the universe.

These include: the mass of the Earth, the Sun, the Milky Way galaxies, the universe and various other physical objections or systems in space; the age of the universe; Hubble's constant (although this is closely related to the cosmological constant); the aggregate baryon number of the universe, the aggregate lepton number of the universe; the number of bosons in existence at any one time in the universe; the mix of atomic elements in the universe or other smaller physical systems (e.g. the planet Earth or the Sun); the distances between particular places on Earth and/or in the Universe; the proportionate mix of ordinary matter other than neutrinos, neutrinos, radiation, dark matter,  and dark energy in the total mass-energy of the universe.

neo said...

i'm unclear how gravity in GR as 4D spacetime can be represnted as a spin-2 boson of QFT

2 seemingly different categories

andrew said...

This is not a short or simple question, although the leading textbooks in GR contain derivations of their analysis. Worthy of a full post, at a minimum.

andrew said...

FWIW, in my humble opinion, the two are not, in fact, exactly equivalent, although a spin-2 boson of QFT in the classical limit, is approximately equivalent to GR in 4D.

For example, generically, in a QFT where a force is transmitted via a carrier boson, the energy associated with the force can be localized subject to the limitation of the Heisenberg uncertainty principle, which is not true in classical GR. Similarly, the QFT has both local and global conservation of mass-energy which is inconsistent with the cosmological constant. GR conserves mass-energy only locally, and not globally.